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The 9th Asian Conference on Fixed Point Theory and Optimization 2016

“Advances in fixed point theory towards real world optimization problems”

MAY 18 - 20, 2016

King Mongkut’s University of Technology Thonburi, Bangkok,

Thailand

Faculty of Science, Fundamental Science Laboratory Bldg.

Sponsors:

Organizers:

Contract: Department of Mathematics,

Faculty of Science, King Mongkut’s University of Technology Thonburi

126 Prachauthit Road, Bangmod, Thung Khru Bangkok, Thailand 10140

Tel: +66-(0)2-470-8994

Email: [email protected]

URL : http://acfpto2016.kmutt.ac.th

FB: https://www.facebook.com/acfpto2016/

Message from the President of King Mongkut’s University of Technology Thonburi

I am very pleased and honored to welcome you to the 9th Asian Con-ference on Fixed Point Theory and Optimization 2016 (ACFPTO2016)with an interesting title “Advances in fixed point theory towards realworld optimization problems” which King Mongkut’s University ofTechnology Thonburi proudly hosts the event. This is the traditionof the fixed point theory and optimization meetings in Thailand andremarks our special occasion of celebrating the 40th anniversary of De-partment of Mathematics and the 56th anniversary of King Mongkut’sUniversity of Technology Thonburi.

Not only is this conference setting a stage to promote new devel-opments, ideas and methods in dynamic field of mathematics fromacademics and researchers around the world, but it also provides aplatform to build a network of researchers in both theoretical and ap-plied mathematics. The progress in the subject is indeed vital as it plays an important role in allareas of research. In this conference, there will be invited talks and oral presentations. I believethat the exchange of ideas during the conference will indeed make connections fit for the themeof the conference. I would like to take this opportunity to express my gratitude to the organizers,committees and all who make this possible. I wish to sincerely thank all honorable speakers whohave helped in the preparation and organizing the conference from the start and they are here withus to see it through, despite being occupied with their other obligations. I would like to thank oursponsors for their generosity and interest in this conference.

I wish the conference to be a great success and hope that you will find it fruitful with excitingnew developments, ideas and methods for research in theoretical and applied mathematics. Ideclare the o�cial opening of ACFPTO2016 and hope you have a pleasant and enjoyable time inBangkok and our hospitality from King Mongkut’s University of Technology Thonburi throughoutthis conference.

Associate Professor Dr. Sakarindr BhumiratanaPresident of King Mongkut’s University of Technology Thonburi

Message from the Dean of Faculty of Science,King Mongkut’s University of Technology Thonburi

On behalf of Faculty of Science, King Mongkut’s University of Tech-nology Thonburi (KMUTT), it give me a great pleasure to welcome eachand every one of you to "The 9th Asian Conference on Fixed Point Theoryand Optimization 2016 (ACFPTO2016)". It is a great honor for Depart-ment of Mathematics, Faculty of Science, King Mongkut’s Universityof Technology Thonburi to host this remarkable international confer-ence as an integral part of our celebrating activities to commemoratethe 40th year of Department of Mathematics and the 56th anniversaryof KMUTT. Over the past years, the Department of Mathematics hasbeen focused on committing excellent teaching and research in the areaof mathematics, statistics and computer science based on internationalstandard.

In this event, we are fortunate to bring together leading experts andresearchers in fixed point theory and optimization and also to assess new developments, ideas andmethods which are important on applications in related areas, as well as other sciences, such asthe natural sciences, health science, epidemiology, economics and engineering.

Thank everyone for being here to contribute, to discuss, and to share research and new ideas,we will have the opportunity to gain insightful knowledge about Fixed Point Theory and Opti-mization research projects. Through the interaction, we expect that ACFPTO2016 may provideopportunities for further networking and development of science and technology.

Assistant Professor Dr. Woranuch KerdsinchaiDean of Faculty of Science, King Mongkut’s University of Technology Thonburi

Table of Contents

Keynote and Invited Speakers i

Committees iv

Conference Program vi

Parallel Sessions vii

Abstracts of Keynote Speakers 1

Abstracts of Invited Speakers 9

Abstracts of Oral Presenters 24

Index 127

Keynote and Invited Speakers

Keynote Speakers

Wataru TakahashiWataru Takahashi is one of the most influential mathematician to-

day, especially for those who work in the area around fixed point theoryand related variational problems. He is now a professor at Keio Univer-sity, Japan, and at National Sun Yat-sen University, Taiwan. He obtainedhis Bachelor from Yokohama National University, and his master as wellas PhD degrees from Tokyo Institute of Technology, where he later spentyears for his dedication to mathematics. He has contributed throughhis life the useful novel ideas and concepts in fixed point theory. Hehas published more than 300 research articles and almost 20 books ininterantional journals and publishers, and he has been cited more than

4,000 times. He also serves as an editor of several reputed journals worldwide.

Sompong DhompongsaSompong Dhompongsa is a leading professor of mathematics

in Thailand. He is now working at Department of Mathematics,Chiang Mai University, Thailand. He finished both his BSc andMEd in 1975 from Srinakarinworot University, Thailand, and ob-tained his MSc and PhD in 1978 and 1982, respectively, from Uni-versity of Illinois at Urbana Champaign, USA. He is best knownfor his expertise in both probability theory and fixed point theory.He has win several awards nationally and internationally, whichproved his virtuosity in his works. He has published his researches

in several international journals. Also, he is listed in several editorial boards of leading journals inthe world. He is also regarded as father of fixed point theory in Thailand.

Phan Quoc KhanhPhan Quoc Khanh is a professor of Mathematics at the Inter-

national University, Vietnam National University Hochiminh City(HCMIU) and also the vice president of the Vietnam MathematicalSociety. He obtained his PhD as well as DSc degrees in 1978 and1988, respectively, at the Institute of Mathematics, Polish Academyof Sciences. He was the founder-president of HCMIU. He is alsothe founder-head of Department of Optimization and System Theoryat the University of Science of Hochiminh City, adjunct professor atthe Federation University, Australia, vice president of the Mathemat-ical Council of the National Foundation for Science and Technology

Development, member of the Scientific Council of the Vietnam Institute for Advanced Study inMathematics (VIASM) and the National Council for Mathematics Professorship. Now, he is anassociate editor of a number of international mathematical journals.

i

Qamrul H. AnsariQamrul Hasan Ansari is a professor in the Department of Mathe-

matics, Aligarh Muslim University, Aligarh, India. He was rewarded hisMaster in 1985 and his PhD in 1988 from The Aligarh Muslim University,India. He has published more than 140 research papers in internation-ally repute journal. He is the associate editor of Journal of OptimizationTheory and Applications and Journal of Inequalities and Applications.He has acted as a guest editor of several internationally repute journals,namely, Positivity, Journal of Global Optimization, Applicable Analysis,

Journal of Fixed Point Theory, etc. He is listed in the top cited 1200 mathematicians of the world.He is regular visitor of several universities around the world.

Yeol Je ChoYeol Je Cho is one of the most well-known mathematician in Korea.

He finishes his BS, MS, and PhD at Busan National University in 1976,1979, and 1984, respectively. He did his post-doctoral in USA during1987-1988. Now, he is a professor in mathematics at the Department ofMathematics Education and the RINS, Gyeongsang National University,South Korea. He is an expert in several areas in nonlinear analysisincluding fixed point theory, variational analysis, optimization, andfunctional equations. By his outstanding knowledge, he was electedas a member of several renowned societies, and had won many awardaround the world. He is in editorial boards of various internationaljournals, and has published a great number of research papers and

books. He is also listed as one of the most cited authors in pure mathematics.

Tamaki TanakaTamaki Tanaka is a professor at Niigata University, Japan, where

he had actually obtained his degrees from Bachelor through PhD. Hisresearch Expertise is convex analysis, nonlinear analysis, game theory,vector optimization and set-valued analysis. He has developed a frame-work of vector-valued minimax problems and proved saddle-point ex-istence theorems and several types of minimax inequality results. More-over, he has introduced a framework of multicriteria game whose payo↵takes its values in a vector space, and by using computational programshe has made such problems more visualizable. Recently, he and his

group have developed nonlinear scalarization methods for set-valued maps, and proved inher-ited properties on convexity and semi-continuity. Also, his group is concerned with OperationsResearch including vector optimization and global optimization.

ii

Invited Speakers

Ryszard Płuciennik, Poznan University of Technology, Poland.

Suthep Suantai, Chiang Mai University, Thailand.

Somyot Plubtieng, Naresuan University, Thailand.

Jong Kyu Kim, Kyungnam University, South Korea.

Yasunori Kimura, Toho University, Japan.

Michel De Lara, Université Paris-Est, France.

Dhananjay Gopal, S. V. National Institute of Technology, India.

Satit Saejung, Khon kaen University, Thailand.

Ali Farajzadeh, Razi University, Kermanshah, Iran.

Lam Quoc Anh, Can Tho University, Vietnam.

Rabian Wangkeeree, Naresuan University, Thailand.

Fumiaki Kohsaka, Tokai University, Japan.

Sang-Eon Han, Chonbuk National University, South Korea.

iii

Committees

Scientific Committees

W. Takahashi, Tokyo Institute of Technology & Keio University, Japan.S. Dhompongsa, Chiang Mai University, Thailand.P. Q. Khanh, International Vietnam National University (HCMC), Vietnam.S. Plubtieng, Naresuan University, Thailand.S. Suantai, Chiang Mai University, Thailand.A. T. Lau, University of Alberta, Canada.

International Program Committees

W. Takahashi, Tokyo Institute of Technology & Keio University, Japan (Chair).S. Dhompongsa, Chiang Mai University, Thailand (Co-Chair).S. Plubtieng, Naresuan University, Thailand (Co-Chair).S. Suantai, Chiang Mai University, Thailand (Co-Chair).P. Kumam, King Mongkut’ts University of Technology Thonburi, Thailand (Secretariat).T. Tanaka, Niigata University, Japan.D.T. Luc, University of Avignon, France.N. Petrot, Naresuan University, Thailand.R.L. Sheu, National Cheng-Kung University, Tainan, Taiwan.T.Q. Son, Saigon University, Vietnam.R. Wangkeeree, Naresuan University, Thailand.H.K. Xu, National Sun Yat-sen University, Taiwan.J.C. Yao, Kaohsiung Medical University, Taiwan.N.D. Yen, Institute of Mathematics, Vietnam.A. Kaewcharoen, Naresuan University, Thailand.N. Khamsemanan, Thammasat University, Thailand.B. Panyanak, Chiang Mai University, Thailand.J. K. Kim, Kyungnam University, Korea.A. Farajzadeh, Razi University, Kermanshah, Iran.

Organizing Committees

R. Wangkeeree, Naresuan University, Thailand.N. Petrot, Naresuan University, Thailand.A. Inchan, Uttaradit Rajabhat University, Thailand.N. Khamsemanan, SIIT, Thammasat University, Thailand.J. Tariboon, King Mongkut’s University of Technology North Bangkok, Thailand.B. Panyanak, Chiang Mai University, Thailand.

iv

A. Kaewcharoen, Naresuan University, Thailand.K. Nonlaopon, , Khon Kaen University, Thailand.S. Niyom, Nakhon Sawan Rajabhat University, Thailand.S. Phiangsungnoen, Rajamangala University of Technology Rattanakosin, Thailand.N. Wairojjana, Valaya Alongkorn Rajabhat University under Royal Patronage, Thailand.W. Sintunavarat, Thammasat University, Thailand.

Local Organizing Committees at KMUTT

T. Jiarasuksakun, King Mongkut’s University of Technology Thonburi, Thailand (Head ofMathematics Department).P. Kumam, King Mongkut’s University of Technology Thonburi, Thailand.K. Akkarajitsakul, King Mongkut’s University of Technology Thonburi, Thailand.D. Thongtha, King Mongkut’s University of Technology Thonburi, Thailand.T. Saleewong, King Mongkut’s University of Technology Thonburi, Thailand.A. Sae-tang, King Mongkut’s University of Technology Thonburi, Thailand.P. Sa-Ngiamsunthorn, King Mongkut’s University of Technology Thonburi, Thailand.P. Phunchongharn, King Mongkut’s University of Technology Thonburi, Thailand.C. Watchararuangwit, King Mongkut’s University of Technology Thonburi, Thailand.P. Wankowit, King Mongkut’s University of Technology Thonburi, Thailand.S. Yookong, King Mongkut’s University of Technology Thonburi, Thailand.W. Leawlien, King Mongkut’s University of Technology Thonburi, Thailand.A. Wisitsorrasak, King Mongkut’s University of Technology Thonburi, Thailand.C. La-o-vorakiat, King Mongkut’s University of Technology Thonburi, Thailand.T. Thanatphanit, King Mongkut’s University of Technology Thonburi, Thailand.

Local Coordinator

P. Kumam, King Mongkut’s University of Technology Thonburi, Thailand.

(The Secretariat Conference of ACFPTO 2016)

v

Time08.00-08.30

09.00-09.4509.45-10.00

10.30-12.0012.00-13.0013.00-13.4513.45-14.00

16.00-16.1516.15-18.00

Time09.00-09.45

10.15-10.3010.30-12.0012.00-13.0013.00-13.45

14.15-16.2016.20-16.3516.30-17.3517.35-18.0018.00-21.00

Time09.00-09.45

10.15-10.30

11.30-12.0012.00-13.00

13.00Lunch

Excursion

ClosingCeremony:AllScientificCommittees,Room600FundamentalScienceLaboratoryBuilding

Banquet:atTONGTARARiverviewHotel(CharoenKrung)

ProfessorSang-EonHan,Korea(S.Phiangsungnoen,Chair)

CoffeebreakPlenarylecture:

Prof.Dr.SompongDhompongsa,ChiangMaiUniversity,ThailandProf.Dr.SomyotPlubtieng,NaresuanUniversity,Thailand(Chair)Prof.Dr.SuthepSuantai,ChiangMaiUniversity,Thailand(Co-Chair)

Plenarylecture:Prof.Dr.QamrulHassanAnsari,India(Prof.Dr.Y.J.Cho,Chair)

InvitedSpeakerAssoc.Prof.Dr.SatitSaejung,Thailand

(K.Nonlaopon,Chair)Prof.Dr.M.DeLara,France(C.La-o-

vorakiat,Chair)

Coffeebreak

GoingtoTONGTARARiverviewHotel(CharoenKrung)OralPresentation(ParallelSessions)III

Friday,May20,2016

10.30-12.00

09.45–10.15

OralPresentation(ParallelSessions)ILunch

Plenarylecture:Prof.Dr.TamakiTanaka,Japan(Prof.Dr.QamrulHassanAnsari,Chair)

InvitedSpeaker

OralPresentation(ParallelSessions)II

CoffeebreakInvitedSpeaker

09.45–10.15

13.45-14.15

Plenarylecture:Prof.Dr.PhanQuocKhanh,Vietnam(Prof.Dr.TamakiTanakaChair)

InvitedSpeakerAsst.Prof.Dr.D.Gopal,India(W.Sintunavarat,Chair)

Prof.Dr.A.Farajzadeh,Iran(A.Kaewcharoen,Chair)

Assoc.Prof.Dr.FumiakiKohsaka,Japan(N.Petrot,Chair)

Assoc.Prof.Dr.RabianWangkeeree,Thailand(J.Tariboont,Chair)

Prof.Dr.L.Q.Anh,Vietnam(B.Panyanak,Chair)

Prof.Dr.Y.Kimura,Japan(S.SaejungChair)

OralPresentation(ParallelSessions)III

Thursday,May19,2016

Coffeebreak

〈 WelcomeSpeechandOpeningremark:Assoc.Prof.Dr.SakarindrBhumiratana,PresidentofKingMongkut’sUniversityofTechnologyThonburi,KMUTT

Plenarylecture:Prof.Dr.WataruTakahashi,Japan(Prof.Dr.SompongDhompongsa,Chair)

08.30-09.00

InvitedSpeaker14.00-16.00

Prof.Dr.RyszardPluciennik,Poland(S.Suantai,Chair)

OralPresentation(ParallelSessions)II

Prof.Dr.SuthepSuantai,Thailand(A.Farajzadeh,Chair)

Prof.Dr.J.K.Kim,Korea(G.Gopal,Chair)

Prof.Dr.SomyotPlubtieng,Thailand(Y.J.Cho,Chair)

OralPresentation(ParallelSessions)ILunch

Plenarylecture:Prof.Dr.Y.J.Cho,Korea(L.Q.Anh,Chair)

OpeningCeremony(Room600FundamentalScienceLaboratoryBldg)

10.00-10.30

Break

The9thAsianConferenceonFixedPointTheoryandOptimization2016on18-20MAY2016

FacultyofScience,KMUTT,Bangkok,Thailand

Wednesday,May18,2016

〈ConferenceReport:Asst.Prof.Dr.WoranuchKerdsinchai,DeanofFacultyofScience,KMUTT

Registration(AreafrontRoom600FundamentalScienceLaboratoryBldg)

Oral Presentation

Room : Basement Learning Space Room : Sci-Connect

10.30 - 10.45 am.

ID 070 Thanyarat Jitpeera ID 029 Huynh Thi Hong Diem

10.45 - 11.00 am.

ID 076 Sarawut Suwannaut ID 067 Thanatporn Bantaojai

11.00 - 11.15 am.

ID 109 Sachiko Atsushiba ID 068 Ariana Pitea

11.15 - 11.30 am.

ID 164 Renu Chugh ID 096 Nithirat Sisarat

11.30 - 11.45 am.

ID 169Kanokwan

SitthithakerngkietID 150 Tran Quoc Duy

11.45 - 12.00 am.

ID 170 Wutiphol Sintunavarat ID 090 Boonyarit Ngeonkam

An iterative approximation scheme for solving a split generalized equilibrium, variational inequalities and fixed point problems

On penalty method for lexicographic vector equilibrium problems

A new hybrid iterative algorithm for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings

Existence results for new extended vector variational-like inequality and equilibrium problems

An iterative method for triple-hierarchical problems

Variational convergence of bifunctions on nonrectangular domains and applications

Convergence theorem for solving the combination of equilibrium problems and fixed point problems in Hilbert spaces

Stability Analysis for Lexicographic Vector Equilibrium Problem

Attrative points, acute point and fixed point properties for nonlinear mappings

On vector optimization problems with geometric framework

Time : 10.30 am. - 12.00 am.18 May 2016 - I

Oral Presentation (Parallel Sessions )

Some convergence results for SKC mapping in hyperbolic spaces

Weak Pareto-optimality for multi-objective optimization involving tangentially convex functions

Section A : Nonlinear Functional Analysis Section B : Computational and Optimization

Chair :J.K. Kim Chair : R. Wangkeeree

Co-Chair : B. Panyanak Co-Chair : K. Nammanee

ACFPTO 2016 | May 18-20,2016 | KMUTT | Bangkok,Thailand

Oral Presentation


02.00 - 02.15 pm.

ID 010 Bancha Panyanak ID 084 Kan Buranakorn

02.15 - 02.30 pm.

ID 011 Javid Ali ID 093 Chalermchai Puripat

02.30 - 02.45 pm.

ID 013 Bancha Nanjaras ID 099 Thanatchaporn Sirichunwijit

02.45 - 03.00 pm.

ID 059 Nuttapol Pakkaranang ID 116 Porntip Promsinchai

03.00 - 03.15 pm.

ID 110 BurisTongnoi ID 128 Des Welyyanti

A general iterative method for solving convex optimization problems of the sum of two convex functions

Convergence theorems in CAT(0) space and an application

The Prediction of Drought Using Correlation between Temperature and Rainfall

18 May 2016 - IIOral Presentation (Parallel Sessions)

Time : 02.00 pm. - 04.00 pm.

Section A : Nonlinear Functional Analysis Section B : Computational and Optimization

Chair : Y.J. Cho Chair : A. Farajzadeh

Co-Chair : C. Klin-eam Co-Chair : K. Ungchittrakool

Endpoints of multi-valued nonexpansive mappings in geodesic spaces

Fixed point theorems for fundamentally nonexpansive mappings in CAT(k) spaces

Sequential optimality conditions for fractional convex optimization problem

Fixed point and convergence theorems for Suzuki type Z-contraction mappings in CAT(0) spaces

Barrier method for convex optimization problem without regularity of constraint functions

Browder's Convergence Theorem in CAT(0) Spaces Endowed with Graph

On Locating-Chromatic Number of a Complete n-ary Tree of Depth 1, 2 and 3


Oral Presentation


03.15 - 03.30 pm.

ID 126 Narongrit Puturong ID 151 Solikhatun

03.30 - 03.45 pm.

ID 014 Khanitin Samanmit ID 153 Taufan Mahardhika

03.45 - 04.00 pm.

ID 168 Chutiphon Pukdeboon

Section B : Computational and Optimization

Chair : Y.J. Cho Chair : A. Farajzadeh

18 May 2016 - II (Cont.)Time : 02.00 pm. - 04.00 pm.

Co-Chair : C. Klin-eam Co-Chair : K. Ungchittrakool

Anti-Disturbance Inverse Optimal Control for Spacecraft Position and Attitude Maneuvers with Input Saturation

Oral Presentation (Parallel Sessions)

A convergence theorem for a finite family of multivalued k-strictly Pseudononspreading mappings in R-trees

Design linear state feedback controller for bilinear system using hybrid genetic algorithm-particles swarm optimization

Existence and convergence of fixed points for a strict pseudo-contraction in CAT(0) spaces

Adaptive optimal control for a bilinear model in cancer chemotherapy

Section A : Nonlinear Functional Analysis


ACFPTO2016|May18-20,2016|KMUTT|Bangkok,Thailand

OralPresentation

04.15-04.30pm.

ID006DaruneeHunwisai

ID061 OrataiYamaod ID071 R.P.Ghimire

04.30-04.45pm.

ID039Konrawut

KhammahawongID062

KanchhaBhaiManandhar

ID073 ApirakSombat

04.45-05.00pm.

ID051Pathaithep

KumrodID063

LaddawanAiemsomboon

ID079JiraprapaMunkong

05.00-05.15pm.

ID054KanokwanSawangsup

ID083 WarutSaksirikun ID102 NayyarMehmood

FixedpointresultsforFR-

contractionsandsolvingthenonlinearmatrixequation

Somecommonfixedpointtheoremsforgeneralizedcyclicmulti-valuedcontractiveoperatorsincompletematricspaces

VariationalInequalitiesforL-fuzzyMappings

BestproximitypointmultivaluedcyclicF-contraction

Acommonfixedpointtheoremforcompatiblemappingsoftype(K)inintuitionisticfuzzymetricspace

Ahybridoptimizationofparticleswarmoptimizationandgeneticalgorithmwithmulti-parentcrossover(GA-MPC)

-fixedpointtheoremsforgeneralized-contractionmappingsinmetricspaceswithapplications

Brzd\c{e}k'sfixedpointtheoremapproachtogeneralizedhyperstabilityofthegenerallinearequation

Generalized(phi,psi)vectorcomplementarityproblemandgeneralized(phi,psi)vectorvariationalinequalityproblemwithfuzzymappings

Co-Chair:N.Wairojjana Co-Chair:T.Jitpeera Co-Chair:T.Mahardhika

SomefixedpointtheoremsformultivaluedF-fuzzycontractionmappingsinfuzzymetricspaces

FixedpointresultsforgeneralizedF-contractionsinb-metricspaces

PerformancemeasuresofE2|E2|1queueingsystemwithsinusoidalarrivalrate

Chair:D.Gopal Chair:A.Kaewcharoen Chair:J.Tariboon

Room:BasementLearningSpace

Room:Sci-Connect Room:SCL213-214

Time:04.15pm.-05.45pm.18May2016-III

OralPresentation(ParallelSessions)

SectionA:NonlinearFunctionalAnalysis

SectionB:ComputationalandOptimization

SectionC:OtherRelatedTopic


OralPresentation

05.15-05.30pm.

ID016ChaowalitPanthong

ID148 AreeratArunchai ID118Supak

Phiangsungnoen

05.30-05.45pm.

ID064 DilipJain ID142DeepeshKumar

PatelID120

KhusnulNovianingsih

NewfixedpointtheoremsofmultivaluedF-contractionsinmodularmetricspacesanditsapplicationtonon-linearintegralequations

Fixedpointsandperiodicpointsof-typeF-contractivemappings

FlightRe-timingModelstoImprovetheRobustnessofAircraftRoutings

Somecoincidencepointsformulti-valuedF-weakcontractionsoncompletemetricspacesendowedwithagraph

Fixedpointtheoremsforgeneralizedfuzzycontractivemappingswithalteringdistanceinfuzzymetricspaces






Chair:D.Gopal Chair:A.Kaewcharoen Chair:J.Tariboon

Co-Chair:N.Wairojjana Co-Chair:T.Jitpeera Co-Chair:T.Mahardhika

OralPresentation(ParallelSessions)18May2016-III(Cont.)

Time:04.15pm.-06.00pm.

AgeneralizabonofEkeland's

!-variabonalprinciplefor"-distance


OralPresentation

10.30-10.45am.

ID005 PlernSaipara ID027 NattawutPholasa ID036 ParinChaipunya

10.45-11.00am.

ID032Chirasak

MongkolkehaID080 PravitraOyjinda ID075

JittipornTangkhawiwetkul

11.00-11.15am.

ID031PraveenKumar

SharmaID077

WongvisarutKhuangsatung

ID095 PanatdaBoonman

StrongconvergencetheoremsforthemodifiedvariationalinclusionproblemsandvariousnonlinearmappingsinHilbertspace

Painleve'-Kuratowskivariationalinclusionproblems

OralPresentation(ParallelSessions)19May2016-I

Time:10.30am.-12.00am.

Co-Chair:K.Jha Co-Chair:P.Phuangphoo Co-Chair:L.Q.Anh

RandomfixedpointtheoremforarandomHardy-Rogersmappings

Modifiedforward-backwardsplittingmethodsforaccretiveoperatorsinBanachspaces

EquilibriumproblemsinHadamardmanifolds




Chair:S.Saejung Chair:W.Sintunavarat



Chair:Y.Kimura

Fixedpointtheoremsforsimulationfunctionsinb-metricspacesviathewt-distance

Numericalsimulationofanairpollutionmodelonindustrialareasbyconsideringtheinfluenceofmultiplepointsources

Sensitivityanalysisofthequasivariationalinequalityproblemonuniformlyproxregularsets

Acommonfixedpointtheoremforsixselfmapsinfuzzymetricspacesusingimplicitrelationandproperty(CLRg)


OralPresentation

11.15-11.30am.

ID041 AnitaTomar ID069 Young-HoKim ID115 PakkaponPreechasilp

11.30-11.45am.

ID056 WudthichaiOnsod ID082 PiyadaPhosri ID162 TranNgocTam

11.45-12.00am.

ID065 UmeshRajopadhyaya ID111 AnimeshGupta ID055 JamnianNantadilok

Acommonfixedpointtheoremforsequenceofmappingsinsemi-metricspacewithcompatiblemappingoftype(E)

TripledPBVPSofnonlinearsecondorderdifferentialequations

BestproximitypointtheoremsforSuzukitypeproximalcontractivemulti-maps

Numericalcomputationofawater-qualitymodelwithadvection-diffusion-reactionequationusinganupwindimplicitscheme

Stabilityforparametricprimalanddualequilibriumproblems


OralPresentation(ParallelSessions)19May2016-I(Cont.)

Co-Chair:K.Jha Co-Chair:P.Phuangphoo Co-Chair:L.Q.Anh

Onexistenceofcoincidenceandcommonfixedpointoffaintlycompatiblepairofmaps

Recentresultstoapproximatesolutionofstochasticdifferentialdelayequations

Anoteoncontinuityofsolutionsetforvectorequilibriumproblems

Commonfixedpointsfor-type(phi,psi)-weakcontractionmappinginintuitionisticfuzzymetricspaces

Time:10.30am.-12.00am.


Chair:S.Saejung




Chair:W.Sintunavarat Chair:Y.Kimura


OralPresentation

02.15-02.30pm.

ID072 KanhaiyaJha ID033 NimitNimana ID086 CholatisSuanoom

02.30-02.45pm.

ID045Somkiat

ChaipornjareansriID037 SomayyaKomal ID125 ChoonkilPark

02.45-03.00pm.

ID009 PakeetaSukprasert ID038 ChayutKongban ID091 NattanawanPiwma

OralPresentation(ParallelSessions)19May2016-II

Time:02.15pm.-03.45pm.

Fixedpointresultsforgeneralizedcontractivemappingsinrectangularb-metricspaces

BestproximitypointtheoremsformultivaluedF-contractivemappings

Anewtwo-stepfixedpointsiterativeschemefortwoasymptoticallynonexpansivemappings

FixedpointtheoremsforFW-

contractionsincompleteS-metricspaces

ABestProximityPointTheoremforGeneralizedContractioninCompleteMetricSpaces

Acommonfixedpointtheoremforsubcompatiblemappingsinfuzzymetricspace

Adaptivesub-gradientmethodforthesplitquasi-convexfeasibilityproblems

Oncoupled-nonexpansivemappings

Co-Chair:S.Phiangsungnoen Co-Chair:P.Preechasilp Co-Chair:C.Mongkolkeha






Chair:I.Inchan Chair:T.Q.Son Chair:N.Petrot

Fixedpointsandquadraac

!-funcaonalinequaliaesinBanachspaces


OralPresentation

03.00-03.15pm.

ID008 PhuminSumalai ID066Kasamsuk

UngchittrakoolID092 PongrusPhuangphoo

03.15-03.30pm.

ID007 ChatupholKhaofong ID135Thidaporn

SeangwattanaID133

PongsakornSunthrayuth

03.30-03.45pm.

ID113 LokeshBudhia ID155Chanoksuda

KhantreeID028 JitsupaDeepho

TheresolventoperatortechniqueswithperturbationsforfindingzerosofmaximalmonotoneoperatorandfixedpointproblemsinHilbertspaces

TheBorwein-Preissvariationalprinciplefornonconvexcountablesystemsofequilibriumproblems

OralPresentation(ParallelSessions)19May2016-II(Cont.)

Time:02.15pm.-03.45pm.






Chair:I.Inchan Chair:T.Q.Son

Extensionsofalmost-FandF-Suzukicontractionswithgraphandsomeapplicationstofractionalcalculus

SequentialOptimalityConditionsforGeneralizedEquilibriumProblemsinvolvingDCfunctions

Aniterativeapproximationschemeforsolvingasplitgeneralizedequilibrium,variationalinequalitiesandfixedpointproblems

Multidimensionalcoincidencepointtheoremsfor−weakcontractionsinpartiallyorderedfuzzymetricspaces

Co-Chair:S.Phiangsungnoen Co-Chair:P.Preechasilp Co-Chair:C.Mongkolkeha

Chair:N.Petrot

NewcoupledfixedpointresultsforF-contractivemappingsinametricspaceendowedwithagraphandapplications

Abestproximitypointtheoremforgeneralizednon-selfKannanandChetterjeatypemappingsandLipschitzianmappingsincompletemetricspaces

AHalperniterationforsystemofequilibriumandvariationalinequalityandfixedpointproblemsoffamiliesofquasi-phi-asymptoticallynonexpansiveinBanachspaces


OralPresentation

04.00-04.15pm.

ID074 TadchaiYuying ID089ProndanaiKaskasem

ID149 JessadaTariboon

04.15-04.30pm.

ID046AnantachaiPadcharoen

ID108 YumnamRohen ID174 LeMinhHuy

04.30-04.45pm.

ID129 PreeyanuchChuasuk ID114 PhikulSridarat ID109 SachikoAtsushiba

Co-Chair:J.Deepho Co-Chair:U.Witthayarat Co-Chair:P.Sunthrayuth

Somecommonminimum-normfixedpointsofafinitefamilyof-asymptoticallyquasi-non-expansivenon-selfmappingswithapplications

Somequadrupledbestproximitypointtheoremspartiallyorderedmetricspaces

OnthequalitativepropertiesforsolutionsequilibriumprobleminvolvingLorentzcone

Strongconvergencetheoremsbyhybridandshrinkingprojectionmethodsforsumsoftwomonotoneoperators

CoupledfixedpointtheoremsinC*-algebra-valuedmetricspaces

Impulsivequantumdifferenceequations

Time:04.00pm.-05.30pm.






Chair:P.Cholamjiak Chair:C.Labuschagne

AniterativeprocessforahybridpairofgeneralizedI-asymptoticallynon-expansivesingle-valuedmappingandgeneralizednon-expansivemulti-valuedmappingsinBanachspaces

Commonfixedpointtheoremformulti-valuedweakcontractivemappingsinmetricspaceswithgraphs

Attrativepoints,acutepointandfixedpointpropertiesfornonlinearmappings

Chair:K.Nonlaopon

OralPresentation(ParallelSessions)19May2016-III


OralPresentation

04.45-05.00pm.

ID078 Qiao-LiDong ID117JukrapongTiammee

ID166 RifaldyFajar

05.00-05.15pm.

ID058KhanitinMuangchoo-

inID121 PorpimonBoriwan ID173 AzadehHosseinpour

05.15-05.30pm.

ID131 KhanitthaPromluang ID167SeyedMasoud

AghayanID088

NatthaphonArtsawang

OralPresentation(ParallelSessions)19May2016-III(Cont.)

Fixedpointandapproximationtheoremsformonotonenon-spreadingmappingsinorderedBanachspaces

FixedpointtheoremsforPre\v{s}i\'{c}almostcontractionmappingsinorbitallycompletemetricspacesendowedwithdirectedgraphs

Anexplicitmethodforsolvingfuzzyheatequationwithintegralboundaryconditions

ViscosityapproximationmethodforsplitcommonnullpointproblemsbetweenBanachspacesandHilbertspaces

Existenceanduniquenessofcoupledbestproximityincomplexvaluedmetricspaces

Existencetheoremsforcoincidencepointsofgeneralizedcontractivemappingsinconeb-metricspaces

FastMannandCQalgorithmsforanon-expansivemapping

Fixedpointtheoremsofanewset-valuedMT-contractioninb-metricspacesendowedwithgraphsandapplications

Stabilityanalysisonmathematicalmodelofspreadingtheparasitoftoxoplasmagondiifromcattocongenitalinfectionofpregnantmotherhaveanimpactonthefetusthroughtheplacentawithherbaltherapy

Time:04.00pm.-05.30pm.






Chair:P.Cholamjiak Chair:C.Labuschagne Chair:K.Nonlaopon

Co-Chair:J.Deepho Co-Chair:U.Witthayarat Co-Chair:P.Sunthrayuth

ABSTRACTSKeynote Speakers

1

ACFPTO 2016 |May 18-20, 2016 | KMUTT | Bangkok, Thailand

Split Common Null Point Problems and Split Common FixedPoint Problems

Wataru Takahashi⇤,11 Kaohsiung Medical University, Taiwan,

Tokyo Institute of Technology and Keio University,JapanEmail: [email protected]; [email protected]

⇤Presenting author.Email: [email protected]; [email protected]

Abstract

Let H1 and H2 be two real Hilbert spaces. Let D and Q be nonempty, closed and convex subsetsof H1 and H2, respectively. Let A : H1 ! H2 be a bounded linear operator. Then the split feasibilityproblem is to find z 2 H1 such that z 2 D \ A�1Q. Given two set-valued mappings G : H1 ! 2H1 ,B : H2 ! 2H2 and a bounded linear operator A : H1 ! H2, the split common null point problem is tofind a point z 2 H1 such that z 2 G�10\A�1(B�10),where G�10 and B�10 are null point sets of G andB, respectively. Given two mappings T : H1 ! H1, U : H2 ! H2 and a bounded linear operatorA : H1 ! H2, the split common fixed point problem is to find a point z 2 H1 such that z 2 F(T)\A�1F(U),where F(T) and F(U) are fixed point sets of T and U, respectively. Defining T = PD and U = PQin the split feasibility problem, we have that z 2 D \ A�1Q is equivalent to z 2 F(T) \ A�1F(U).Furthermore, defining T = Jr and U = Qs in the split common null point problem, where Jr andQs are resolvents of G for r > 0 and B for s > 0, we get that z 2 G�10 \ A�1(B�10) is equivalentto z 2 F(T) \ A�1F(U). Thus the split common fixed point problem generalizes the split feasibilityproblem and the split common null point problem. Putting U = A⇤(I � PQ)A in the split feasibilityproblem, where A⇤ is the adjoint operator of A, we have that U : H1 ! H1 is an inverse stronglymonotone operator. Furthermore, if D \ A�1Q is nonempty, then z 2 D \ A�1Q is equivalent toz = PD(I � �A⇤(I � PQ)A)z, where � > 0. By using such results regarding nonlinear operatorsand fixed points, many authors have studied split feasibility problems, split common null pointproblems and split common fixed point problems in Hilbert spaces. However, we have not foundsuch results outside Hilbert spaces.

In this talk, motivated by split feasibility problems, split common null point problems and splitcommon fixed point problems in Hilbert spaces, we first solve split common null point problems formetric resolvents and generalized resolvents of maximal monotone operators in two Banach spaces.Furthermore, we introduce new nonlinear operators in Banach spaces which simultaneously extendwell-known mappings in Hilbert spaces and Banach spaces. Using hybrid methods, Mann’stype iterations and Halpern’s type iterations, we prove weak convergence theorems and strongconvergence theorems for such operators in Banach spaces which are connected with split feasibilityproblems, split common null point problems and split common fixed point problems in Hilbertspaces and Banach spaces.

Keywords: Maximal monotone operator; iteration procedure; split feasibility problem; split com-mon null point problem; split common fixed point problem; duality mapping

2


On my two recent papers

Sompong Dhompongsa⇤

Department of Mathematics, Faculty of Science,Chiang Mai University, Chiang Mai 50200, Thailand

⇤Presenting author.Email: [email protected]

Abstract

The talk will give more details on two papers [1,2]. For [1], a new approach will be presented.Its proof is much more simpler than the one given in [1]. As for [2], a sketch of proof as well as itsapplications in economics will be given.

References:[1] S. Dhompongsa, J. Nantadilok, A simple proof of the Brouwer fixed point theorem, Thai J. of Mathematics, V. 13

(2015), No. 3, 519-525.[2] W. Anakkamatee, S. Dhompongsa, S. Tasena, A constructive proof of the Sklar’s theorem on copulas, JNCA V. 15

(2014), No. 6, 1137-1145.

3


Weak and Strong Convergence Theorems of Accelerated Mannand CQ-Algorithms for Nonexpansive Mappings in Hilbert

Spaces

Yeol Je ChoDepartment of Mathematics Education and the RINS

, Gyeongsang National University, Jinju 660-701, Korea⇤Presenting author.


Abstract

In this talk, first, we introduce the inertial accelerated Mann and the inertial CQ-algorithms bycombining the accelerated Mann algorithm and the CQ-algorithm with the inertial extrapolation,respectively. Second, we intend to speed up the convergence of the given algorithms. Finally, wegive the numerical experiments to illustrate that the inertial accelerated Mann algorithm may hasmore advantage than other methods in computing for some cases and the inertial CQ-algorithmis more e↵ective than the CQ-algorithm. Our results improve the corresponding results given bysome authors.

Keywords: Nonexpansive mapping; the inertial algorithm; the CQ-algorithm; the inertial Mannalgorithm; the Mann algorithm; the accelerated Mann algorithm.

4


Variational Convergence and Applications in Optimization

Phan Quoc KhanhInternational Vietnam National University (HCMC), Vietnam.


Abstract

Convergence is the first basic notion in continuous mathematics which plays important rolesin most areas of this âAIJhalfâAI of mathematics. There have been many notions of convergencein analysis, probability theory, numerical methods, etc. For optimization and related fields, nat-urally kinds of convergence which preserve variational properties like being minimum points,minsup points, saddle points, extremal values, etc, are crucial from various aspects. Variationalconvergence is the general terminology for such kinds of convergence.

Epi convergence for unifunctions, epi/hypo convergence and lopside convergence for bifunc-tions are main notions of variational convergence. Epi convergence was introduced more than halfa century ago, and the other two have been developed for three decades now. In 2009, lopsideconvergence for finite-valued bifunctions was proposed and lead to a new e↵ective approach forvariational convergence and applications.

In this talk, we focus on lopside and epi/hypo convergence of finite valued bifunctions andapplications in optimization. We also aim to a systematic exposition of the theory of variationalconvergence, from epi convergence of unifunctions to lopside and epi/hypo convergence and finallyto applications in approximating optimization problems and estimating solutions of stochasticoptimization problems. We present the theory for the general setting of topological spaces, but tryto illustrate it by simple applications.

Keywords: Epi/hypo convergence; lopsided convergence; finite-valued bifunctions defined onnonrectangular domains; tightness; variational properties.

5


Generalized alternative theorems based on set-relations

GUE MYUNG LEE1, JAE HYOUNG LEE1, YUTO OGATA2, YUTAKA SAITO2, ANDTAMAKI TANAKA⇤,2

1Department of Applied Mathematics, Pukyong National University, Korea.Email: [email protected](G.M.Lee), [email protected] (J.H.Lee)

2Graduate School of Science and Technology, Niigata University, Japan.Email: [email protected],[email protected],[email protected]


Abstract

Alternative theorems such as Farkas’ lemma and Gordan’s theorem usually play importantroles in considering optimization problems and because of that, many kinds of valuable extensionshave been established; Jeyakumar [1] produces a generalized Gordan’s theorem for a vector-valuedfunction in 1986. Li [3] in 1999 and Yang et al.[5] in 2000, extend it to the case of set-valued maps.However, these theorems rely on some assumptions related to convexity to make systems in abilinear form.

In this talk, I would like to introduce alternative theorems from a set-valued analytic pointof view, using the set-relations proposed by Kuroiwa, Tanaka, and Ha [2] in 1997. They canbe considered in a topological vector space with the set-relations induced by a convex orderingcone. A similar approach with scalarizing functions for vectors had been done by Nishizawa,Onodsuka, and Tanaka [4] in 2005. They prove some alternative theorems with no convex as-sumption by nonlinear scalarizations, not bilinear forms. We reviewed these results and found away of generalizations. I show 12 types of alternative theorems given by scalarizing functions forsets. Comparing with previous studies, our results achieve the subdivision of the case and thesimplification of the forms of them simultaneously. Also, important properties are still conserved.

Reducing some conditions, we recognize that some of 12 types imply several Gordantypetheorems. This fact may allow our theorems to have a suitability of extensions of previous results.Moreover, we show some application to semidefinite optimization problems.

Keywords: Generalized alternative; set-relation

References:[1] V. Jeyakumar, A generalization of a minimax theorem of Fan via theorem of the alternative, J. Optim. Theory Appl.

48 (1986) 525-533.[2] D. Kuroiwa, T. Tanaka, and T. X. D. Ha, On convexity of set-valued maps, Nonlinear Anal. 30 (1997), 1487-1496.[3] Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps, J. Optim. Theory Appl.

100 (1990) 365-375.[4] S. Nishizawa, M. Onodsuka, and T. Tanaka, Alternative theorems for set-valued maps based on a nonlinear scalar-

ization, Paci�c J. Optim. 1 (2005) 147-159.[5] X. M. Yang, X. Q. Yang, and G. Y. Chen, Theorems of the alternative and optimization with set-valued maps, J. Optim.

Theory Appl. 3 (2000) 627-640.

6


Split type Problems in Nonlinear Analysis

Qamrul Hasan AnsariDepartment of Mathematics,Aligarh Muslim University,

Aligarh 202 002, India


Abstract

In this talk, we shall discuss some spilt type problems from nonlinear analysis, namely, convexfeasibility problems, split feasibility problems, split common fixed point problem, split variationalinequality problems, hierarchical variational inequality problems, split hierarchical variationalinequality problems, split hierarchical variational inclusion problem, etc. We shall discuss severalapplications of these problems. We shall mention several iterative methods for finding the solutionsof above mentioned problems.

7

ABSTRACTSInvited Speakers

9


�-points in Orlicz spaces

Ryszard Płuciennik⇤


Abstract

Let (X, k·kX) be a real Banach space and B(X) be the closed unit ball of X.Denote by extB (X) theset set of all extreme points of B (X) . A function � : B(X) ! [0, 1] defined for any x 2 B(X) by theformula

� (x) = sup�� 2 [0, 1] : x = �e + (1 � �)y, e 2 extB (X) , y 2 B (X)

is called �-function. In the case when extB (X) = ?,we assume that �(x) = 0 for any x 2 B(X).A point x 2 B(X) is said to a �-point of B(X) if �(x) > 0. If every point of B(X) is a �-point, then

X is said to have the �-property. Moreover, if

�X = inf {� (x) : x 2 S(X)} > 0,

then X is said to have the uniform �-property.The �-property was introduced by Aron and Lohman [1]. The �-property is important because

for Banach spaces with the �-property we have that co (extB (X)) = B (X) .Moreover, Aron, Lohmanand Granero proved in [2] that a Banach space X has the �-property if and only if it has the convexseries representation property, i.e. for each x 2 B (X) , there is a sequence (ek) of extreme points ofB (X) and a sequence of non-negative real numbers (�k) such that

P1

k=1 �k = 1 and x =P1

k=1 �kek. Ithas been also shown in [2] that the uniform �-property for a Banach space X is equivalent to theuniform convex series representation property for X, i.e. convex series representation property inwhich the sequence (�k) does not depend on x.

Among others, a criterion for �-points of the unit ball in Orlicz spaces generated by arbitraryOrlicz functions (that is Orlicz functions which vanish outside zero and which attain infinite valuesto the right of some point u > 0 are not excluded) and equipped with the Orlicz norm is given.Moreover, Orlicz spaces with �-property are characterized. In contrast to results in [5], Orliczspaces considered by us need not have always the �-property.

References:[1] R.M. Aron and R.H. Lohman, A geometric function determined by extreme points of the unit ball of a normed space, Pacific

J. Math. 127 (1987), 209-231.[2] R.M. Aron, R.H. Lohman and A. Suárez Granero, Rotundity, the C.S.R.P., and the �-property in Banach spaces, Proc.

Amer. Math. Soc. 111, (1991), 151-155.[3] A. Bohonos, R. Płuciennik, �-points in Orlicz spaces, Journal of Convex Analysis 21 (2014), 147-166.[4] A. Bohonos, R. Płuciennik, Uniform �-property in L1

\ L1, Comment Math. (accepted for publication).[5] H.Y. Sun and C.X. Wu, �-property of Orlicz space LM, Comment. Math. Univ. Carolin. 31 (1990), 731-741.

10


Fixed point and best proximity point theory with graphs and rateof convergence of some iterative methods

Suthep SuantaiDepartment of Mathematics, Faculty of Science,

Chiang Mai University (CMU),239 Huay Kaew Road, Muang District, Chiang Mai 50200, Thailand



Abstract

In this talk, we first discuss the development of fixed point theory of various classes of nonlinearmappings in both metric spaces and Banach spaces with directed graphs after that best proximitypoint theorems for some nonlinear mappings are discussed in Banach spaces with directed graphs.We also discuss rate of convergence of various iterative methods for finding fixed points of somenonlinear mappings. Finally, some open questions related to our talk are posed.

Keywords: fixed point theory; best proximity point theorems ; directed graph; weak contractionmappings; generalized nonexpansive mappings

References:[1] S.B. Nadler, “Multivalued contraction mappings,” Pacific Journal of Mathematics, vol. 30, no. 2, pp. 475–488, 1969.[2] N. Mizoguchi and W. Takahashi, “Fixed point theorems for multivalued mappings on complete metric spaces,”

Journal of Mathematical Analysis and Applications, vol. 141, pp. 177 – 188, 1989.[3] D. Klim and D. Wardowski, “Fixed point theorems for multi-valued contractions in complete metric space,” Journal

of Mathematical Analysis and Applications, vol. 334, pp. 132 – 139, 2007.[4] M. Berinde and V. Berinde, “On a general class of multi-valued weakly Picard mappings,” Journal of Mathematical

Analysis and Applications, vol. 326, pp. 772 – 782, 2007.[5] J. Tiammee and S. Suantai, “Coincidence point theorems for graph-preserving multi-valued mappings,” Fixed Point

Theory and Applications, vol. 70, 2014.[6] A. Hanjing and S. Suantai, “Coincidence point and fixed point theorems for a new type of G-contraction multivalued

mappings on a metric space endowed with a graph,” Fixed Point Theory and Applications, vol. 171, 2015.[7] I. Beg and A.R. Butt, “Fixed point of set-valued graph contractive mappings,” Journal of Inequalities and Applications,

vol. 252, 2013.[8] J. Jachymski, “The contraction principle for mappings on a metric space with a graph,” Proceedings of the American

Mathematical Society, vol. 136, no. 4, pp. 1359–1373, 2008.[9] N.A. Assad and W.A. Kirk, “Fixed point theorems for setvalued mappings of contractive type,” Pacific Journal of

Mathematics, vol. 43, pp. 533–562, 1972.

The authors were supported by the Thailand Research Fund under the project RTA 5780007 and Chiang Mai University.

11


Borwein–Preiss Variational Principle Revisited

Alexander Kruger1, Somyot Plubtieng⇤,2, Thidaporn Seangwattana 3

1Centre for Informatics and Applied Optimization,Faculty of Science and Technology,

Federation University Australia,Ballarat, Victoria 3353, Australia

Email: [email protected] of Mathematics,

Faculty of Science, Naresuan University,Phitsanulok 65000, Thailand

Email: [email protected] of Mathematics,

Faculty of Science, Naresuan University,Phitsanulok 65000, Thailand



Abstract

In this article, we refine and slightly strengthen the metric space version of the Borwein–Preiss variational principle due to Li and Shi [12], clarify the assumptions and conclusions of theirTheorem 1 as well as Theorem 2.5.2 in Borwein and Zhu [4], and streamline the proofs. Our mainresult is formulated in the metric space setting. When reduced to Banach spaces, it extends andstrengthens the smooth variational principle established in Borwein and Preiss [3], along severaldirections. Moreover, we introduce and characterize two seemingly new natural concepts of ✏-minimality, one of them dependant on the chosen element in the ordering cone and the fixed“gauge-type” function, and extend our main result to the vector setting.

Keywords: Borwein-Preiss variational principle, smooth variational principle, gauge-type func-tion, perturbation.

References:[1] D. Aze, “A unified theory for metric regularity of multifunctions,” J. Convex Anal., vol. 13, no. 2, pp. 225–252, 2006.[2] E.M. Bednarczuk and D. Zagrodny, “A smooth vector variational principle,” SIAM J. Control Optim., vol. 48, no. 6,

pp. 3735 – 3745, 2010.[3] J.M. Borwein and D. Preiss, “A smooth variational principle with applications to subdi↵erentiability and to di↵eren-

tiability of convex functions,” Trans. Amer. Math. Soc. vol. 303, pp. 517-527, 1987.[4] J.M. Borwein and Q.J. Zhu, “Techniques of Variational Analysis,” Springer Berlin Heidelberg, NewYork, 2005.

The research was supported by the Australian Research Council, projects DP110102011 and DP160100854; NaresuanUniversity, and Thailand Research Fund, the Royal Golden Jubilee Ph.D. Program, scholarship 3.M.NU/51/A.1.N.XX.

12


An iterative algorithms for generalized mixed equilibriumproblems and fixed point of nonexpansive semigroups

Jong Kyu KimDepartment of mathematics Education

Kyungnam University,Changwon, Gyeongnam, 51767, KoreaE-mail: [email protected]


Abstract

In this talk, by using the modified viscosity approximation method associated with Meir-Keeler contractions, we proved the convergence theorem for solving fixed point problem of anonexpansive semigroup and generalized mixed equilibrium problems in Hilbert spaces.

Keywords: Meir-Keeler contraction mappings, left regular, generalized mixed equilibrium prob-lems, variational inequalities, nonexpansive semigroups.

References:[1] X.S. Li, N.J. Huang, and J.K. Kim, General viscosity approximation methods for common fixed points of non-

expansive semigroups in Hilbert spaces, Fixed Point Theory and Appl., Article ID 783502, (2011), 1-12 doi:10,1155/2011/783502.

[2] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbertspaces, J. Math. Anal. Appl., 331(2007), 506-515

[3] J.K. Kim and Nguyen Buong, An iterative method for common solution of a system of equilibrium problems inHilbert spaces, Fixed Point Theory and Appl., Article ID 780764, (2011), 1-15 doi: 10,1155/2011/780764.

[4] J.K. Kim, S.Y. Cho, X.L .Qin, Some results on generalized equilibrium problems involving strictly pseudocontractivemappings, Acta Math. Sci., Series B, 31(5)(2011), 2041-2057.

[5] J.K. Kim and T.M. Tuyen Viscosity approximation method with Meir-Keeler contractions for common zero of accretiveoperators in Banach spaces, Fixed Point Theory and Appl., 2015:9 (2015), 17 pages,

[6] J.K.Kim and S.S. Chang, Generalized Mixed Equilibrium Problems for an infinite Family of Quasi-phi-nonexpansiveMappings in Banach Spaces,Nonlinear Anal. and Convex Anal., RIMS Kokyuroku, Kyoto Univ., 1923 (2014), 28-41.

[7] Li Yang, F. Zhao and J.K.Kim, Hybrid projection method for generalized mixed equilibrium problem and fixedpoint problem of infinite family of asymptotically quasi-Îe-nonexpansive mappings in Banach spaces, AppliedMathematics and Computation, 218(2012), 6072-6082

[8] U. Witthayarat, J.K. Kim and P. Kumam, A viscosity hybrid steepest-descent methods for a system of equilibriumproblems and fixed point for an infinite family of strictly pseudo-contractive mappings, Jour. of Inequalities andAppl., 2012, 2012:224, doi: 10,1186/1029-242X-2012-224.

This work was supported by the Basic Science Research Program through the National Research Foundation(NRF)Grant funded by Ministry of Education of the republic of Korea(2015R1D1A1A09058177)

13


Resolvents of convex functions in a complete geodesic space withcurvature bounded above

Yasunori Kimura*Department of Information Science,

Toho University,Miyama, Funabashi, Chiba 274-8510, Japan


Abstract

In this talk, we introduce the notion of resolvent for a proper lower semicontinuous functiondefined on a complete geodesic space with curvature bounded above by one, and show severalimportant properties of this operator. The resolvents of convex functions on Banach and Hilbertspaces have been investigated by many researchers and that defined on a Hadamard space hasbeen studied by Jost [2], Mayer [4], and others. We show that our new notion is a generalization ofclassical resonvents and it can be applied to convex optimization in a complete geodesic space.

This is the joint work with Professor Fumiaki Kohsaka in Tokai University.

Keywords: Convex function; geodesic space; minimizer; resolvent.

References:[1] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wis-

senschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999.[2] J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv. 70,

(1995), 659–673.[3] Y. Kimura and F. Kohsaka, Spherically nonspreading mappings in geodesic spaces with curvature bounded above by one,

Journal of Fixed Point Theory and Appl. 18 (2016), 93–115.[4] U. F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom. 6 (1998),

199–253.

14


Stochastic and decentralized optimizationfor smart grid energy management

Michel De Lara⇤,11CERMICS, École des Ponts ParisTech

6 et 8 avenue Blaise Pascal, Cité Descartes77455 Marne la Vallée Cedex 2, France


⇤Presenting author.

Abstract

The transformation of energy systems is accelerating. Local initiatives are blossoming, trig-gered by the drop in renewable energy costs and by the impulse of decentralized actors (individ-uals, collectivities). With myriads of decentralized intermittent sources (wind, sun) and of actors,managing an energy system is becoming more and more challenging. We present how stochas-tic and decentralized optimization can contribute to formulate and to solve problems of energymanagement with micro grids, smart grids and renewable energies.

Keywords: optimization; stochastic; decentralized; energy; smart grid

References:[1] P. Carpentier, J.-P. Chancelier, G. Cohen, M. De Lara. “Stochastic Multi-Stage Optimization. At the Crossroads

between Discrete Time Stochastic Control and Stochastic Programming”, Springer-Verlag, Berlin, 2015[2] M. De Lara, P. Carpentier, J.-P. Chancelier, V. LeclÃlre, “Optimization Methods for the Smart Grid”, report commis-

sioned by Conseil FranÃgais de l’Energie, october 2014[3] Jean-Christophe Alais, Pierre Carpentier, Michel De Lara, “Multi-usage hydropower single dam management:

chance-constrained optimization and stochastic viability”, Energy Systems, to appear

15


Recent development in fuzzy metric fixed point theory

D. Gopal⇤,1

1Department of Applied Mathematics and Humanities,SV National Institute of Technology, Surat, Gujarat (India)

Email: gopal.dhananjay@redi↵mail.com⇤Presenting author.

Email: gopal.dhananjay@redi↵mail.com

Abstract

In this talk, we discuss on the recent development in the area of fuzzy metric fixed point theory.In particular, we present several new fixed point results along with some new open problems inthis topic.

Keywords: fuzzy metric space; Fuzzy contractive mappings; fixed point

References:[1] A. George and P. Veeramani, “On some results in fuzzy metric spaces” On some results in fuzzy metric spaces, Fuzzy

Sets and Systems, vol. 64, pp. 395 – 399, 1994.[2] A. Spena, “A contribution to the study of fuzzy metric spaces,” Applied General Topology, vol. 2, no. 1, pp. 63 – 75,

2001, .[3] S. Shukla, D. Gopal, and A. Roldán, “Some fixed point theorems in 1-complete fuzzy metric-like spaces,” International

Journal of General Systems, 2016.

16


Recent results for single-valued and multi-valued mappings insome geodesic spaces

Satit SaejungDepartment of Mathematics, Faculty of Science,

Khon Kaen University,Khon Kaen, 40002, Thailand

Abstract

In this talk, we discuss our recent results in complete CAT(0) and CAT(1) spaces. We areinterested in both single-valued and multi-valued mappings. Many results are shown that theynot only extend but also significantly improve previous known ones.

Keywords: fuzzy metric space; Fuzzy contractive mappings; fixed point

References:[1] Saejung, Satit. Halpern’s iteration in CAT(0) spaces. Fixed Point Theory Appl. 2010, Art. ID 471781, 13 pp.[2] Kimura, Yasunori; Saejung, Satit; Yotkaew, Pongsakorn. The Mann algorithm in a complete geodesic space with

curvature bounded above. Fixed Point Theory Appl. 2013, 2013:336, 13 pp.[3] Saejung, Satit. Two remarks on the modified Halpern iterations in CAT(0) spaces. Fixed Point Theory 15 (2014), no.

2, 595–602.[4] Yotkaew, Pongsakorn; Saejung, Satit. Strong convergence theorems for multivalued mappings in a geodesic space

with curvature bounded above. Fixed Point Theory Appl. 2015, 2015:235, 11 pp.[5] Saejung, Satit. Remarks on endpoints of multivalued mappings in geodesic spaces. Fixed Point Theory Appl. 2016,

2016:52.

17


An equivalence relation between generalized vector equilibriumproblems and scalar minimization problems for multivalued

mappings

Ali FarajzadehDepartment of Mathematics, Razi University,

Kermanshah, 67149, IranEmail: [email protected]

Abstract

In this paper, existence of a nonempty pointed convex cone with empty topological interior andnonempty algebraic interior for an arbitrary infinite dimensional linear topological space is proved.A multivalued version of Farakas’s lemma in the setting of ordered linear spaces is established. Byusing it, an equivalence relation between the solution set of some generalized vector equilibriumproblems and the corresponding minimization problems are provided. The techniques are used inthis note di↵erent from the KKM theory and fixed point theory. Some examples in order to supportthe main results are given.

Keywords: Multivalued map, generalized vector equilibrium problems, pointed convex cone,Farkas’s Lemma, weakly e�cient solution, globally e�cient solution, supere�cient solution.

References:[1] Q.H. Ansari and F. Flores-Bazan, “Recession methods for generalized vector equilibrium problems,” J. Math. Anal.

Appl., vol. 321, pp. 132–146, 2006.[2] Q.H. Ansari, A.H. Siddiqi and S.Y. Wu, “Existence and duality of generalized vector equilibrium problems,” J.Math.

Anal. Appl., vol. 259, pp. 115–126, 2001.[3] Q.H. Ansari, I.V. Konnov and J.C.Yao, “Characterizations of solutions for vector equilibrium problems,“ J. Optim.

Theory Appl., vol. 113, pp. 435–447, 2002.[4] ] J.P. Aubin and A. Cellina, “Di↵erential Inclusions,“ Springer-Verlag, Berlin, Heidelberg, Germany, 1994.[5] M. Fakhar and J. Zafarani, “Generalized vector equilibrium problems for pseudomonotone multivalued bifunctions,“

J. Optim. Theory Appl., vol. 126, pp. 109–124, 2005.

18


On the stability conditions for equilibrium problems and relatedproblems

Lam Quoc Anh⇤ and Pham Thi VuiDepartment of Mathematics, Cantho University,

Cantho City, Vietnam⇤Presenting author.


Abstract

In this report, we consider the class of equilibrium problems. Su�cient conditions for thestability and sensitivity analysis of the solution sets of such problems are proposed. These topicsare also studied for some problems related to optimization.

Keywords: Equilibrium problem, variational inequality, optimization problem, variational inclu-sion, stability, sensitivity analysis, well-posedness

References:[1] A. George and P. Veeramani, “On some results in fuzzy metric spaces” On some results in fuzzy metric spaces, Fuzzy

Sets and Systems, vol. 64, pp. 395 – 399, 1994.[2] A. Spena, “A contribution to the study of fuzzy metric spaces,” Applied General Topology, vol. 2, no. 1, pp. 63 – 75,

2001, .[3] S. Shukla, D. Gopal, and A. Roldán, “Some fixed point theorems in 1-complete fuzzy metric-like spaces,” International

Journal of General Systems, 2016.

19


Sequential Optimality Conditions for Infinite FractionalProgramming Problem with DC Functions

Rabian Wangkeeree⇤, Chanoksuda Khantree


Abstract

In this paper, the absence of any constraint qualifications, a sequential Lagrange multiplierrule condition characterizing optimality for an infinite fractional programming problem with DCfunctions is obtained in terms of the subdi↵erentials of the functions involved at the minimizer.The significance of this result is that it yields the standard Lagrange multiplier rule conditionfor the infinite fractional programming problem under a simple closedness condition that ismuch weaker than the well-known constraint qualifications. A sequential condition characterizingoptimality involving only subdi↵erentials at nearby points to the minimizer is also investigated.As applications, the proposed approach is applied to investigate sequential optimality conditionsfor fractional with DC function, fractional and DC optimization problem.

20


Existence and approximation of minimizers of convex functionsin geodesic metric spaces

Fumiaki Kohsaka⇤

Department of Mathematical Sciences,Tokai University,

Kitakaname, Hiratsuka, Kanagawa 259-1292, JapanEmail: [email protected]


Abstract

Using the resolvents of convex functions in complete geodesic metric spaces with nonpositivecurvature and with curvature bounded above, we study the problem of approximating minimizersof convex functions in such spaces. We then obtain existence and convergence theorems for findingsolutions to this problem. Among other things, we obtain some counterparts of Rockafellar’s resultson the proximal point algorithm for convex functions in the geodesic metric space setting. This isjoint work with Professor Yasunori Kimura in Toho University, Japan.

Keywords: CAT(0) spaces, CAT(1) spaces, convex function, proximal point algorithm

References:[1] K. Aoyama, F. Kohsaka, and W. Takahashi, “Proximal point methods for monotone operators in Banach spaces”,

Taiwanese J. Math. 15 (2011), 259–281.[2] M. Bacák, “The proximal point algorithm in metric spaces”, Israel J. Math. 194 (2013), 689–701.[3] M. Bacák, “Convex analysis and optimization in Hadamard spaces”, De Gruyter, Berlin, 2014.[4] H. Brézis and P.-L. Lions, “Produits infinis de résolvantes”, Israel J. Math. 29 (1978), 329–345.[5] J. Jost, “Convex functionals and generalized harmonic maps into spaces of nonpositive curvature”, Comment. Math.

Helv. 70 (1995), 659–673.[6] S. Kamimura and W. Takahashi, “Approximating solutions of maximal monotone operators in Hilbert spaces”, J.

Approx. Theory 106 (2000), 226–240.[7] Y. Kimura and F. Kohsaka, “Spherical nonspreadingness of resolvents of convex functions in geodesic spaces”, J.

Fixed Point Theory Appl. 18 (2016), 93–115.[8] Y. Kimura and F. Kohsaka, “Two modified proximal point algorithms for convex functions in Hadamard spaces”,

Linear Nonlinear Anal., to appear.[9] U. F. Mayer, “Gradient flows on nonpositively curved metric spaces and harmonic maps”, Comm. Anal. Geom. 6

(1998), 199–253.[10] R. T. Rockafellar, “Monotone operators and the proximal point algorithm”, SIAM J. Control Optim. 14 (1976), 877–898.

21


Digital topological based fixed point theory and its applications

SANG-EON HAN⇤,11Chonbuk National University,

Jeonju-City Jeonbuk, 561-756, Republic of KoreaEmail: [email protected]⇤Presenting author.


Abstract

In this talk, we studies the fixed point theory from the viewpoint of digital topology. Motivatedby the ordinary Banach contraction principle [1], we can consider their digital versions. Moreprecisely, in digital topology, we say that a digital image (X, k) has the fixed point property if everyk-continuous map f : (X, k)! (X, k) has a fixed point x 2 X, i.e. f (x) = x. Unlike the formal researchinto the fixed point property, we have some intrinsic features in digital digital versions of fixedpoint theorems [1,2,3,4,6]. This approach can be used in certain areas in applied sciences.

Keywords: digital topology; Banach contraction principle; digital homotopy

References:[1] L.E.J. Brouwer, Uber Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1912), 97-115.[2] S.-E. Han, Non-product property of the digital fundamental group, Information Sciences 171(1-3) (2005) 73-91.[3] S.-E. Han, The k-homotopic thinning and a torus-like digital image in Zn, Journal of Mathematical Imaging and Vision

31(1) (2008) 1-16.[4] S.-E. Han, Banach fixed point theorem from the viewpoint of digital topology, Journal of Nonlinear Sciences and

Applications 9(3) (2016) 895-905.[5] S. Lefschetz, Intersections and transformations of complexes and manifolds. Trans. Amer. Math. Soc. 28 (1) (1926)

1-49.[6] A. Rosenfeld, A. Rosenfeld, Digital topology, Amer. Math. Monthly, 86 (1979) 76-87.

22

ABSTRACTSOral Presenters

24


Random fixed point theorem for a random Hardy-Rogersmappings

Plern Saipara⇤,1, Poom Kumam1,2 and Yeol Je Cho3

1Department of Mathematics, Faculty of Science,King Mongkut’s University of Technology Thonburi (KMUTT),

126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, ThailandEmail: [email protected]

2Theoretical and Computational Science (TaCS) Center,Science Laboratory Building, Faculty of Science,

King Mongkut’s University of Technology Thonburi (KMUTT),126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand

Email: [email protected] of Mathematics Education, Gyeongsang

Natoinal University , Jinju 660-701, Korea.Email: [email protected]


Abstract

The main objective of this paper is to prove theorem of a random fixed point for a randomHardy-Rogers mappings. The main result in this paper is the identification of some theorems of arandom fixed point and the interrelated application.

Keywords: a random fixed point, a random Hardy-Rogers mappings

References:[1] MC.Joshi and RK. Bose: “Some Topics in Nonlinear Functional Analysis,” Wiley, New York (1984).[2] MC.Kumari and D.Panthi: “Connecting various types of cyclic contractions and contractive self-mappings with Hardy-Rogers

self-mappings,” Fixed Point Theory Appl. 2016:15.[3] M.Saha and A.Ganguly: “Random fixed point theorem on a Ciric-type contractive mapping and its consequence,” Fixed Point Theory

Appl. 2012, Article ID 209 (2012).[4] P.Kumam: “Random common fixed points of single-valued and multivalued random operators in a uniformly convex Banach space,”

J. Comput. Anal. Appl. 13(2), 368-375 (2011).[5] W.Kumam and P.Kumam: “Random fixed point theorems for multivalued subsequentially limit-contractive maps satisfying inward-

ness conditions,” J. Comput. Anal. Appl. 14(2), 239-251 (2012)[6] W.Sintunavarat, P.Kumam and P.Patthanangkoor: “Common random fixed points for multivalued random operators without S and

T-weakly commuting random operators,” Random Oper. Stoch. Equ. 17(4), 381-388 (2009).[7] P.Kumam and S.Plubtieng: “Random fixed point theorems for multivalued nonexpansive non-self random operators,” J.Appl. Math.

Stoch. Anal. 2006, Article ID 43796 (2006).[8] P.Kumam and S.Plubtieng: “Some random fixed point theorems for non-self nonexpansive random operators,” Turk. J.Math. 30,

359-372 (2006).

The authors were supported by the Higher Education Research Promotion and National Research University Projectof Thailand, O�ce of the Higher Education Commission (NRU-CSEC No.55000613).

25


Some fixed point theorems for multivalued F-fuzzy contractionmappings in fuzzy metric spaces

Darunee Hunwisai⇤,1, Poom Kumam1,2







Abstract

In this paper, we introduce a new concept of fuzzy fixed points for multivalued F-fuzzycontraction mappings in fuzzy metric spaces. We prove the existence of fuzzy fixed poings formultivalued F-fuzzy contraction mappings on fuzzy metric spaces.

Keywords: F-fuzzy contraction mapping ; fixed point; Fuzzy mappings; Fuzzy metric spaces.

References:[1] SB. Nadler, “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, no. 2, pp. 475–488, 1969.[2] D. Wardowski, “Fixed point theory of a new type of contractive mapping in complete metric spaces,” Fixed Point

Theory Applications,94, 2012.[3] A. Ishak, M. Gulhan, and D. Hacer„ “Multivalued F-contractions on complete metric spaces. Online Journal,” Journal

of Nonlinear and Convex Anlysis, vol. 16, pp. 659 – 666, 2015.[4] S. Heilpern, “Fuzzy mappings and fixed point theorem,” Journal of Mathematical Analysis and Application, vol. 83, pp.

566–569, 1981.


26


Multidimensional coincidence point theorems for ( )�weakcontractions in partially ordered fuzzy metric spaces

Chatuphol Khaofong⇤,1, Poom Kumam1,2







Abstract

In this paper, using the notion of ( )-weak contraction is extend to partially ordered fuzzymetric spaces in the sense of George and Veeramani. The existence of coincidence points fornonlinear mappings in any number of variables, we will generalize the concept of ( )-weakcontraction in partially ordered fuzzy metric spaces. Then, coincidence point results for two mapsare obtained.

Keywords: partially ordered fuzzy metric spaces; mixed monotone property; coincidence point;( )-weak contraction.

References:[1] Ya. I. Alber and S. Guerre-Delabriere, Principles of weakly contractive maps in Hilbert spaces, in: I. Gohberg, Yu.

Lyubich (Eds.), New Results in Operator Theory: Advances and Applications, vol. 98, Birkhauser, Basel, 1997,7–22.

[2] D. Doric, Common fixed point for generalized ( ,')-weak contractions, Appl. Math. Lett., 22 (2009), 1896–1900.[3] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994), 395–399.[4] E. KarapÄsnar, A. Roldán, A note on 0n-tuplet fixed point theorems for contractive type mappings in partially ordered

metric spaces0. Journal of Inequalities and Applications 2013, 567 (2013).[5] A. Roldán, J. Martínez-Moreno, C. Roldán, Y.J. Cho, Multidimensional coincidence point results for compatible

mappings in partially ordered fuzzy metric spaces, Fuzzy Sets and Systems 251, 71-82 (2014).


27


New coupled fixed point results for F-contractive mappings in ametric space endowed with a graph and applications

Phumin Sumalai⇤,1, Poom Kumam1,2







Abstract

The purpose of this paper is to present some existence results of coupled fixed points for F-typecontraction type operators in metric spaces endowed with a directed graph. Our results generalizethe results obtained by Gnana Bhaskar and Lakshmikantham in (Nonlinear Anal. 65:1379-1393,2006). We also have applied to some integral systems.

Keywords: fixed point; coupled fixed point; F-contration; metric space; connectedgraph;

References:[1] D. Gopal, C. Vetro, M. Abbas and D.K. Patel, “Some coincidence and periodic points results in a metric space endowed

with a graph and applications,” Banach J. Math. Anal., vol. 9, no. 3, pp. 128–139, 2015.[2] C. Chifu and G. Petrusel, “New results on coupled fixed point theory in metric spaces endowed with a directed

graph” Applied Mathematics Letters, vol. 151, 2014.[3] S. Suantai, P. Charoensawan and T.A. Lampert, “Common coupled fixed point theorems for✓- -contraction mappings

endowed with a directed graph,” Fixed Point Theory and Applications, vol. 224, 2015. of Nonlinear Analysts.[4] W. Sintunavarat, P. Kumam, “Coupled coincidence and coupled common fixed point theorems in partially ordered

metric spaces,” Thai J. Math., vol. 10, pp. 551–563, 2012.[5] D. Wardowski, “Fixed points of new type of contractive mappings in complete metric spaces,” Fixed Point Theory and

Applications, vol. 94, 2012.

The authors were supported by Theoretical and Computational Science (Tacs) Center (Project Grant No.TaCS2559-2).

28


Fixed point results for generalized ( ,�)s-contractive mappingsin rectangular b-metric spaces

Pakeeta Sukprasert⇤,1, Poom Kumam1,2







Abstract

The aim of this paper is to present the definition of a weak altering distance function and anew generalized contractive mapping in rectangular b-metric spaces. We discuss the fixed pointresults of such a mapping in rectangular b-metric spaces.

Keywords: Fixed point; rectangular metric space; rectangular b-metric space; partially orderedset; weak altering distance function.

References:[1] S. Banach “Sur les operations dans les ensembles abstrait et leur application aux equations,” Fundamenta Mathematicae,

vol. 3, pp. 133–181, 1922.[2] M.-S. Khan, M. Swaleh and S. Sessa, “Fixed point theorems by altering distances between the points,” Bulletin of the

Australian Mathematical Society, vol. 30, pp. 1 – 9, 1984.[3] M.-M. Frèchet, “Sur quelques points du calcul fonctionnel,” Rendiconti del Circolo Matematico di Palermo, vol. 22, pp.

1 – 72, 1906.[4] F. Yan, Y. Su,Q. Feng, “A new contraction mapping principle in partially ordered metric spaces and applications to

ordinary di↵erential equations,” Fixed Point Theory and Applications, vol. 2012, Article ID: 152, 2012.[5] Y. Su, “Contraction mapping principle with generalized altering distance function in ordered metric spaces and

applications to ordinary di↵erential equations,” Fixed Point Theory and Applications, vol. 2014, Article ID: 227,2014.

The authors were supported by Theoretical and Computational Science Center (TaCS), Science Laboratory Building,Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT).

29


Endpoints of multivalued nonexpansive mappings in geodesicspaces

Bancha Panyanak⇤

Department of Mathematics, Faculty of Science,Chiang Mai University, Chiang Mai 50200, Thailand


Abstract

Let E be a nonempty subset of a Banach space X and T : E! K (E) be a multivalued mapping.A point x 2 E is called an endpoint of T if T(x) = {x}. It is shown that a multivalued nonexpansivemapping on a bounded closed convex subset of a uniformly convex Banach space has an endpoint ifand only if it has the approximate endpoint property. This is the first result regarding the existenceof endpoints for such kind of mappings even in Hilbert spaces. The related result in a completeCAT(0) space is also given.

Keywords: endpoint; fixed point; multivalued nonexpansive mapping; Banach space; CAT(0)space

References:[1] Amini-Harandi, A: Endpoints of set-valued contractions in metric spaces, Nonlinear Anal. 72, 132-134 (2010)[2] Amini-Harandi, A, Petrusel, A: An endpoint theorem in generalized L-spaces with applications, J. Nonlinear Convex

Anal. 16, 265-271 (2015)[3] Aubin, JP, Siegel, J: Fixed points and stationary points of dissipative multivalued maps, Proc. Amer. Math. Soc. 78,

391-398 (1980)[4] Bridson, M, Haefliger, A: Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, Heidelberg, (1999)[5] Garcia-Falset, J, Llorens-Fuster, E, Moreno-Galvez, E: Fixed point theory for multivalued generalized nonexpansive

mappings, Appl. Anal. Discrete Math. 6, 265-286 (2012)[6] Wlodarczyk, K, Plebaniak, R, Obczynski, C: Endpoints of set-valued dynamical systems of asymptotic contractions

of Meir-Keeler type and strict contractions in uniform spaces, Nonlinear Anal. 67, 1668-1679 (2007)

30


Convergence theorems in CAT(0) space and an application

Javid Ali⇤,1, Izhar Uddin1,2

1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaEmail: [email protected]

2Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India


Abstract

The aim of present paper is to introduce a new iterative process involving a finite family ofmultivalued nonexpansive mappings in CAT(0) spaces. We prove some �-convergence and strongconvergence theorems for the proposed scheme with and without end point conditions. The newlydefined iteration scheme is also utilized to an application in image recovery problem. In process,our results generalize and extend the corresponding results of Uddin et al., Abbas et al., Eslamianand Abkar, Bunyawat and Suantai, Khan, Khan and Fukhar-ud-din and Fukhar-ud-din and Khanand references cited therein.

Keywords: CAT(0) space, Fixed point, �-convergence, Opial’s property and image recovery.

References:

31


Fixed point theorems for fundamentally nonexpansive mappingsin CAT() spaces

Bancha Nanjaras⇤,1

1Department of Mathematics, Faculty of Science,Chiang Mai University, Chiang Mai, 50200, Thailand



Abstract

In this paper, we obtain fixed point theorems and �-convergence theorems for fundamentallynonexpansive mappings on CAT() spaces with > 0. Our results extend and improve someresults of Salahifard et al. [1], and many others.

Keywords: CAT() space; fixed point; �-convergence; generalized nonexpansive mapping.

References:[1] H. Salahifard, S. M. Vaezpour, and S. Dhompongsa, " Fixed point theorems for some generalized nonexpansive

mappings in CAT(0) spaces," Journal of Nonlinear Analysis and Optimization: Theory & Applications, vol. 4, no. 24,pp. 241–248, 2013.

The author was supported by the Ministry of Science and Technology, Thailand and the Graduate School, Chiang MaiUniversity, Chiang Mai, Thailand.

32


A Convergence Theorem for a Finite Family of Multivaluedk-Strictly Pseudononspreading Mappings in R-Trees

Khanitin Samanmit⇤,11Department of Mathematics, Faculty of Science,

Chiang Mai University, Chiang Mai, 50200, ThailandEmail: [email protected]


Abstract

In this work, we introduce a new m-step iterative process for finite family of kâLŠstrictlypseudononspreading multivalued mappings in R-trees. We obtain a strong convergence theoremof m-step iterative method to a common fixed point of a finite family of those multivalued mappingsinR-trees. Our results extend many known recent results in the literature. We close this work withthe first examples of kâLŠstrictly pseudononspreading multivalued mappings in R-trees

Keywords: fixed point; multivalued mapping; R-tree; kâLŠstrictly pseudononspreading; conver-gence theorems.

References:[1] W. Phuengrattana, " Approximation of common fixed points of two strictly pseudononspreading multivalued map-

pings in RâLŠtrees," Kyungpook Math. J., vol. 55, pp. 373–382, 2015.

33


Some coincidence points for multi-valued F-weak contractions oncomplete metric spaces endowed with a graph

Chaowalit Panthong⇤,1, Phumin Sumalai2, Poom Kumam2,3

1Department of Mathematics, Faculty of Science and Technology,Muban Chom Bueng Rajabhat University,

46 Chom Bueng, Ratchaburi 70150, ThailandEmail: [email protected]







Abstract

In this paper, we introduce the concepts of weak g-graph-preserving for multi-valued map-pings and weak F-G-contractions in a metric space endowed with a directed graph. We establishsome coincidence point theorems for this type of mappings in a complete metric space endowedwith a directed graph. Examples illustrating our main results are also presented. Our resultsextend and generalize various known results in the literature.

Keywords: coincidence point; F-contration; metric space; connected graph;

References:[1] D. Wardowski, “Some coincidence and periodic points results in a metric space endowed with a graph and applica-

tions,” Banach J. Math. Anal., vol. 9, no. 3, pp. 128–139, 2015.[2] SB. Nadler, “Multivalued contraction mappings” Pac. J. Math, 30 (1969) 475- 488.[3] H. Piri and P. Kumam, “Some fixed point theorems concerning F-contraction in complete metric spaces,” Fixed Point

Theory and Applications, vol. 210, 2014.[4] M. Berinde, V. Berinde, “On a general class multi-valued weakly Picard mappings,” J. Math. Anal. Appl., vol. 326,

pp. 772–782, 2007.


34


Modified Forward-Backward Splitting Methods for AccretiveOperators in Banach Spaces

N. Pholasa1,⇤, P. Cholamjiak1 and Y. J. Cho2,3

1Department of Mathematics, School of Science,University of Phayao, Phayao 56000, Thailand

Email: [email protected] of Mathematics Education and the RINSGyeongsang National University, Jinju 660-701, Korea

3Department of Mathematics, King Abdulaziz UniversityJeddah 21589, Saudi Arabia



Abstract

In this research, we propose the modified splitting method for accretive operators in Banachspaces and prove some strong convergence theorems of the proposed method under suitableconditions. Finally, we give some applications to the minimization problems.

Keywords: Accretive operator; Banach space; splitting method; forward-backward splittingmethod

References:[1] G.H.G. Chen and R.T. Rockafellar, “Convergence rates in forward-backward splitting,” SIAM J. Optim, 7, 421-444,

1997.[2] P.L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM J. Numer. Anal., 16,

964-979, 1979.[3] G. López, V. Martín-Márquez, F. Wang, and H.K. Xu, “Forward-Backward splitting methods for accretive operators

in Banach spaces,” Abstr. Appl. Anal. 2012., Art ID 109236 2012.[4] S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,” J. Math. Anal. Appl.,

75, 287-292 1980.[5] W. Takahashi, N.C. Wong, and J.C. Yao, “Two generalized strong convergence theorems of Halperns type in Hilbert

spaces and applications,” Taiwan. J. Math., 16, 1151-1172 2012.

This research was supported by the Thailand Research Fund and University of Phayao under Grant TRG5780075.

35


An iterative approximation scheme for solving a split generalizedequilibrium, variational inequalities and fixed point problems

Jitsupa Deepho⇤,1,2,3, Juan Martínez-Moreno2, Kanokwan Sitthithakerngkiet4

and Poom Kumam1,5



2Department of Mathematics,Faculty of Science, University of Jaén

Campus Las Lagunillas, s/n, 23071 Jaén, SpainEmail: [email protected]

3 Department of Mathematics, Faculty of Education,Uttaradit Rajabhat University, 27 Injaimee Road, Mueang, Uttarsdit, 53000 Thailand

4Nonlinear Dynamic Analysis Research Center, Department of Mathematics,Faculty of Applied Science,

King Mongkut’s University of North Bangkok (KMUTNB)Wongsawang, Bangsue, Bangkok, 10800, Thailand





Abstract

In this paper, we introduce a new iterative method for finding a common element of the setof solutions of the split generalized equilibrium problem, the set of the variational inequality for�-inverse strongly monotone mappings, and the set of fixed point of nonexpansive mappings inHilbert spaces. We show that the sequence converges strongly to a common element of the abovethree sets under some controlling conditions.

Keywords: Fixed point; Variational inequality; Viscosity approximation method; Nonexpansivemapping; Hilbert space; Split generalized equilibrium problem; Strong convergence

References:[1] J.-P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer-Verlag.[2] A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., Vol. 150, 275–283 (2011).[3] K. R. Kazmi and S. H. Rizvi, Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed

point problem for nonexpansive semigroup, Mathematical Sciences, Vol. 7:1 (2013), doi:10.1186/2251–7456–7–1[4] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000), 46–55.


36


Variational convergence of bifunctions on nonrectangulardomains and applications

Huynh Thi Hong Diem and Phan Quoc KhanhInternational Vietnam National University (HCMC), Vietnam.


Abstract

Variational convergence of extended-real-valued functions has been developed for half a cen-tury with many important applications. In 2009 Jofre and Wets considered variational convergenceof finite-valued bifunctions defined on rectangles and considered its variational properties. Sincethen, there have been a number of contributions in this direction including application in opti-mization. However, quasivariational problems cannot be expressed in terms of such bifunctionson rectangles, because their constraint sets depend on the variables of the problems. The aim of thispaper is to extend epi/hypo and lopsided convergence, the main kinds of variational convergenceof bifunctions, to the case of finite-valued bifunctions defined on nonrectangular domains andapply them to quasivatiational models. Their basic characterizations are established. Variationalproperties such as saddle points, minsup points, sup-projections, etc, of bifunctions are shownto be preserved for the limit bifunctions when the bifunctions epi/hypo converge to these limits(possibly under some additional assumptions) and applied to approximations of quasiequilibriumproblems. The obtained results are new and, in the special case of bifunctions defined on rectangles,they also improve some known results.

Keywords: Epi/hypo convergence; lopsided convergence; finite-valued bifunctions defined onnonrectangular domains; tightness; variational properties; saddle points; quasiequilibrium prob-lems, multiobjective quasioptimization, generalized Nash equilibria.

References:

37


A common fixed point theorem for six self maps in fuzzy metricspaces using implicit relation and property (CLRg)

Praveen Kumar SharmaDepartment of Applied Mathematics, Amity School of Engineering and Technology,

Amity University Madhyapradesh, Gwalior - 474005 (M.P.), IndiaEmail: praveen_jan1980@redi↵mail.com

⇤Presenting author.Email: praveen_jan1980@redi↵mail.com

Abstract

In this note we generalize the results of S.Kumar and S.Chouhan[S.Kumar and S.Chouhan ,common fixed point theorems using implicit relation and property (E.A.) in fuzzy metric spaces ,Annals of fuzzy mathematics and informatics, 5(1)(2013), 107-114] by using (CLRg) property andimplicit relation. The purpose of this note is to prove a common fixed point theorem for six selfmaps in fuzzy metric spaces using the property (CLRg) and contractive type implicit relation.

Keywords: fuzzy metric space, common fixed point; weakly compatible maps; implicit relationand property CLRg.

References:

38


Fixed point theorems for simulation functions in b-metric spacesvia the wt-distance†

Chirasak Mongkolkeha⇤,1, Yeol Je Cho2 and Poom Kumam1,2

1Department of Mathematics Statistics and Computer SciencesFaculty of Liberal Arts and Science, Kasetsart University

Kamphaeng-Saen Campus, Nakhonpathom 73140, ThailandEmail: [email protected]

2Department of Mathematics Education and the RINSGyeongsang National University

Chinju 660-701, KoreaCenter for General Education, China Medical University

Taichung, 40402, Taiwan3 Department of Mathematics, Faculty of Science

King Mongkut’s University of Technology Thonburi (KMUTT)Bangmod, Thrungkru, Bangkok 10140, Thailand



Abstract

The purpose of this article is to prove some fixed point theorems for simulation functionsin complete b�metric spaces with partially ordered by using wt-distance which introduced byHussain et al. (2014). Also, we give some examples to illustrate our main results.

Keywords: Fixed point; simulation function; b-metric space; wt-distance; w-distance; generalizeddistance.

References:[1] N. Hussain, R. Saadati and R. P Agrawal, “ On the topology and wt-distance onmetric type spaces,” Fixed Point

Theory Appl., 2014, 2014:88.

The first author was supported by Thailand Research Fund (Grant No. TRG5880221) and Kasetsart University.

39


Adaptive subgradient method for the split quasi-convexfeasibility problems

Nimit Nimana⇤,1, Ali P. Farajzadeh2 and Narin Petrot1

1Department of Mathematics, Faculty of Science,Naresuan University, Phitsanulok, 65000, Thailand

Email: [email protected]; [email protected] of Mathematics, Razi University, Kermanshah, 67149, Iran



Abstract

In this talk, we consider a type of the celebrated convex feasibility problem, named a splitquasi-convex feasibility problem. This problem is to find a point in a sublevel set of a quasi-convexfunction in one space and its image under a bounded linear operator is contained in a sublevelset of another quasi-convex function in the image space. We propose a new adaptive subgradientalgorithm for solving this considered problem. We then discuss the convergence analyses for twocases: the case where the functions are upper semicontinuous in the finite dimensional settings,and the second one where the functions are demicontinuous in the infinite dimensional settings.We also give a numerical example to support the convergence results.

Keywords: Split feasibility problem; Quasi-convex feasibility problem; Adaptive Subgradientmethod; Convergence

References:[1] C.L. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem," Inverse Probl., vol. 18, pp.

441–453, 2002.[2] Y. Censor and A. Segal, “Algorithms for the quasiconvex feasibility problem," J. Comput. Appl. Math., vol. 185, pp.

34–50, 2006.[3] Y.H. Hu, X.Q. Yang and C.-K. Sim, “Inexact subgradient methods for quasi-convex optimization problems," European

J. Oper. Res., vol. 240, pp. 315–327, 2015.[4] I.V. Konnov, “On Convergence Properties of a Subgradient Method," Optim. Meth. Software, vol. 18, no. 1, pp. 53–62,

2003.

N. Nimana was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (GrantNo. PHD/0079/2554) and Naresuan University.

40


Equilibrium problems in Hadamard manifolds

Parin Chaipunya⇤,1, Poom Kumam1,2







Abstract

Equilibrium problems are well-widely considered by nonlinear analysts and optimizationtheorists as a central theory of unifying nonlinear variational models. Classical model focus on anobjective bifunction defined on a squared product of a convex set. In our talk, we consider insteada product between two convex sets that are proximal to one another. Moreover, the underlyingspace in our results is assumed to be a Hadamard manifold, i.e., a complete and simply connectedRiemannian manifold with nonpositive sectional curvatures. The reason behind a Hadamardmanifold domain is that it allows us to transform various complicate constrained and nonconvexproblems into a non-constrained and convex one.

Keywords: Equilibrium problem, Best proximity point, Hadamard manifold

References:[1] E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics

Student, vol. 63, no. 1-4, pp. 123–145, 1994.[2] Q. Ansari and J.-C. Yao, “An existence result for the generalized vector equilibrium problem,” Applied Mathematics

Letters, vol. 12, no. 8, pp. 53 – 56, 1999.[3] Q. Ansari, I. Konnov, and J. Yao, “On generalized vector equilibrium problems,” Nonlinear Analysis: Theory, Methods &

Applications, vol. 47, no. 1, pp. 543 – 554, 2001, proceedings of the Third World Congress of Nonlinear Analysts.[4] M. Fakhar and J. Zafarani, “Generalized vector equilibrium problems for pseudomonotone multivalued bifunctions

1,” Journal of Optimization Theory and Applications, vol. 126, pp. 109–124, 2005.[5] H. Ben-El-Mechaiekh, S. Chebbi, and M. Florenzano, “A generalized kkmf principle,” Journal of Mathematical Analysis

and Applications, vol. 309, no. 2, pp. 583 – 590, 2005.


41


Optimal Approximate Solution: Best proximity Point theoremsfor generalized nonlinear contraction mappings

Somayya Komal1 and Poom Kumam1,2


126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand2Theoretical and Computational Science (TaCS) Center,

Science Laboratory Building, Faculty of Science,King Mongkut’s University of Technology Thonburi (KMUTT),

126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand⇤Presenting author.


Abstract

In this paper, we obtained the best proximity point theorem for ↵-Geraghty contractions in thesetting of complete metric spaces by using weak P-property. Also we presented some examplesto prove the validity of our results. Our results extended and unify many existing results in theliterature.

Keywords: Best proximity point, weak P-property, triangular ↵-admissible.


Student, vol. 63, no. 1-4, pp. 123–145, 1994.[2] Q. Ansari and J.-C. Yao, “An existence result for the generalized vector equilibrium problem,” Applied Mathematics

Letters, vol. 12, no. 8, pp. 53 – 56, 1999.[3] Q. Ansari, I. Konnov, and J. Yao, “On generalized vector equilibrium problems,” Nonlinear Analysis: Theory, Methods &

Applications, vol. 47, no. 1, pp. 543 – 554, 2001, proceedings of the Third World Congress of Nonlinear Analysts.[4] M. Fakhar and J. Zafarani, “Generalized vector equilibrium problems for pseudomonotone multivalued bifunctions

1,” Journal of Optimization Theory and Applications, vol. 126, pp. 109–124, 2005.[5] H. Ben-El-Mechaiekh, S. Chebbi, and M. Florenzano, “A generalized kkmf principle,” Journal of Mathematical Analysis

and Applications, vol. 309, no. 2, pp. 583 – 590, 2005.

42


Best proximity point theorems for multivalued F -contractivemappings

Chayut Kongban⇤,1, Poom Kumam1,2







Abstract

In this article, we introduce the notion of multivalued F -contraction mapping and we alsoprove the existence best proximity point theorems in complete metric spaces.

Keywords: Best proximity point; Fixed point; P-property; F -contraction; multivalued mapping

References:[1] Nadler, S.B. Jr., “Multivalued contraction mappinsg,” Pac. J. Math., vol. 30, pp. 475–488, 1969.[2] Sadiq Basha, “Best proximity point theorems,” J. Approx. Theory, vol. 163, pp. 1772–1781, 2011.[3] Abkar, A., Gabeleh, M., “The existence of best proximity points for multivalued non-self-mappings,” RACSAM,

vol. 107, pp. 319–325, 2012.[4] Wardowski, D., “Fixed points of a new type of contractive mappings in complete metric spaces,” Fixed Point Theory

Appl., vol. 2012, no. 94, 2012.[5] Omidvari, M., Vaezpour, S.M. and Saadati. R., “Best proximity points for F -contractive non-self-mappings,” Miskolc

Mathematical Notes., vol. 15, no. 2 , pp.615–623, 2014.

The authors were supported by the Theoretical and Computational Science (TaCS) Center (Project Grant No. TaCS2559-2).

43


Best proximity point multivalued cyclic F -contraction

Konrawut Khammahawong⇤,1, Poom Kumam1,2







Abstract

In this paper we prove the existence of best proximity point for multivalued cyclic F - contrac-tion and state some result in the complete metric space.

Keywords: best proximity point; cyclic F -contraction; multi-valued contraction; metric space

References:[1] D. Wardowski,“Fixed point of a new type of contractive mappings in complete metric space,” Fixed Point Theory and

Applications 2012., 2012:94.[2] M. Omidvari, S.M. Vaezour, and R. Saadti,“Best proximity point theorem for F -contractive non-self mappings,”

Miskolc Mathematical Notes, vol. 15, no. 2, pp. 615–623, 2014.[3] P. Kumam, H. Aydi, E. Karapinar and W. Sintunavarat,“Best proximity points and extension of Mizoguchi- Taka-

hashi’s fixed point theorems,” Fixed Point Theory and Applications 2013, 2013:242.[4] S.B. Nadler Jr.,“Multivalued contraction mappings,” Pacific J. Math., vol. 30, pp. 475–488, 1969.

The authors were supported by the Theoretical and Computational Science (TaCS) Center (Project Grant No.Tacs2559-2)

44


On Existence of Coincidence and Common Fixed Point of FaintlyCompatible Pair of Maps

Anita TomarGovernment P. G. College Dakpathar(Dehradun) India



Abstract

Motivated by the fact that a wide variety of problems appearing in distinctive areas of pureand applied mathematics can be modeled as fixed point equations of the form fx=x, the aim ofthis talk is to discuss the existence of coincidence and common fixed point of a faintly compatiblediscontinuous pair of maps without using the containment requirement of involved maps. Resultsto be discussed improve, generalize and extend many results existing in the literature and aresupported with an illustrative example.

Keywords: Coincidence point; common fixed point; conditional reciprocal continuity and faintcompatiblity.

References:[1] R. K. Bisht, Lipschitz, Type Analogue of Strict Contractive Conditions and Common Fixed Points, Thai Journal of

Mathematics, 12(03) (2014): 631âAS637.[2] R.K. Pant and R.P. Bisht, Occasionally weakly compatible mappings and fixed points, Bull. Belg. Math. Soc. Simon

Stevin, 19 (2012), 655-661.[3] R.K. Bisht and R.P. Pant, Common fixed point theorems under a new continuity condition, Ann. University, Ferrara,

58 (2012) 127-141.[4] R. K. Bisht and N. Shahzad, Faintly compatible mappings and common fixed points, Fixed point theory and appli-

cations, 2013, 2013:156.[5] G. Jungck and B. Rhoades, Fixed points for set valued functions without continuity. Indian J. Pure Appl. Math. 29(3)

(1998), 227âAS238.

45


Fixed Point Theorems for Fw-Contractions in Complete S-MetricSpaces

Somkiat Chaipornjareansri⇤⇤Department of Mathematics, Faculty of Science,

Lampang Rajabhat University, Lampang 52100, Thailand


Abstract

In this paper, we define a w-distance on a complete S-metric space, which is a generalization ofthe concept of the w-distance due to Kada, Suzuki and Takahashi. Also, we introduce the conceptof the Fw-contraction in a complete S-metric space and extend the fixed point theorem due toMalhotra and Bansal. We also discuss an example.

Keywords: w-distance; F-contraction; Fw-contraction; Complete S-metric spaces

References:[1] R. Batra and S. Vashistha, “Fixed point theorem for Fw-contractions in complete metric spaces”, J. Nonlinear Anal.

Appl., (2013).[2] N. Malhotra, B. Bansal, “Fw-contractions in a complete G-metric space”, J. Math. Anal. Appl., 28 (1969) 326C329.[3] J. Mojaradi, “Fixed point type theorem in S-metric spaces”, Middle-East Journal of Scientific Research 22(6):(2014) pp.

864–869.[4] J. Mojaradi, “Fixed point type theorem for weak contractions in S-metric spaces”, IJRRAS 22 (1) (2015) pp. 11–14.[5] O. Kada, T. Suzuki and W. Takahashi, “Nonconvex minimization theorems and fixed point theorems in complete

metric spaces”, Math. Japonica 44(1996) 381C391.[6] S. Sedghi, N. Shobe and A. Aliouche, “A generalization of fixed point theorem in S-metric spaces”, Mat. Vesnik, 64

(2012), pp. 258–266.[7] D. Wardowski, “Fixed points of a new type of contractive mappings in complete metric spaces”, Fixed Point Theory

Appl., 2012 (2012), 94.

This work was funded by Faculty of Science, Lampang Rajabhat University.

46


Some common minimum-norm fixed points of a finite family of�-asymptotically quasi-nonexpansive nonself-mappings with

applications

Anantachai Padcharoen⇤,1, Poom Kumam1,2, Yeol Je Cho3




King Mongkut’s University of Technology Thonburi (KMUTT),126 Pracha-Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand

Email: [email protected] of Mathematics Education and the RINS,

Gyeongsang National University, Chinju 660-701, KoreaEmail: [email protected]


Abstract

In this paper, we consider the two-step iteration for a finite family of �-asymptotically quasi-nonexpansive nonself-mappings and prove some strong convergence theorems of the proposedsequence {xn} for this family in real uniformly convex and uniformly smoothBanach spaces. Further,we give one an application of the main result.

Keywords: �-asymptotically quasi-nonexpansive nonself-mapping; strong convergence; fixedpoint; uniformly convex and uniformly smooth Banach space.

References:[1] Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” Theory and

Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50, Dekker, New York 1996.[2] S. S. Chang, C. K. Chan, H. W. J. Lee, “Modified block iterative algorithm for quasi-�-asymptotically nonexpansive

mappings and equilibrium problem in Banach spaces,” Appl. Math. Comput, vol. 217, pp. 7520–7530, 2011.[3] X. L. Qin, Y. J. Cho, S. M. Kang, and H. Y. Zhou, “Convergence of a modified Halpern-type iterative algorithm for

quasi-�-nonexpansive mappings,” Appl. Math. Lett., vol. 22, pp. 1051–1055, 2009.[4] H. K. Pathak, V. K. Sahu, and Y. J. Cho,“Approximation of a common minimum-norm fixed point of a finite family

of �-asymptotically quasi-nonexpansive mappings with applications,”

The authors were supported by the Petchra Pra Jom Klao Doctoral Scholarship.

47


'-fixed point theorems for generalized (F,')-contractionmappings in metric spaces with applications

Pathaithep Kumrod⇤,1, Wutiphol Sintunavarat1

Department of Mathematics and Statistic, Faculty of Science and Technology,Thammasat University Rangsit Center,

99 Moo 18, Phahon Yothin Rd., Khlong Nueng, Khlong Luang, Pathum Thani 12120 ThailandEmail 1: [email protected] (Pathaithep Kumrod)

Email 2: [email protected] (Wutiphol Sintunavarat)


Abstract

In this work, we introduce a new generalization of (F,')-contraction mappings due to Jleli etal. [1] and establish some existence results of '-fixed point for such mappings. We also state someillustrative example to support our results. Furthermore, we prove some '-fixed point results forgeneralized contraction in partial metric spaces by using the main results.

Keywords: partial metric spaces; (F,')-contraction mappings; '-fixed points

References:[1] M. Jleli, B. Samet and C. Vetro, “Fixed point theory in partial metric spaces via '-fixed point’s concept in metric

spaces,” Journal of Inequalities and Applications , vol. 426, 2014.

48


Fixed point results for F<

-contractions and solving the nonlinearmatrix equation

Kanokwan Sawangsup⇤,1, Wutiphol Sintunavarat1

1Department of Mathematics and Statistics, Faculty of Science and Technology,Thammasat University Rangsit Center (TU),

99 Moo 18, Phahon Yothin Rd., Khlong Nueng, Khlong Luang, Pathumthani 12121, Thailand.Email: [email protected], [email protected] (Kanokwan Sawangsup)

Email: [email protected], [email protected] (Wutiphol Sintunavarat)


Abstract

In this work we introduce the notion of a F<

-contraction, which by improve the idea of War-dowski [1] under a nonempty binary relation. We give some fixed point results for F

<

-contractionsin complete metric spaces and also give an illustrative example. Furthermore, multidimensionalfixed point theorems are derived from our main results. As an application, we apply our mainresult to study a nonlinear matrix equation. Additionally, we give numerical data to support ourapplication by using Matlab.

Keywords: Complete metric space; binary relation; F<

-contraction

References:[1] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory

Applications, 2012:94, 6 pages.

49


Best proximity point theorems for Suzuki type proximalcontractive multimaps

J. Nantadilok⇤,1, N. Thamboonruang 2

1Department of Mathematics, Faculty of Science,Lampang Rajabhat University(LPRU),

119 Lampang-Maetha Rd., Lampang 52100, ThailandEmail: [email protected]

2Department of Mathematics, Faculty of Science,Lampang Rajabhat University(LPRU),

119 Lampang-Maetha Rd., Lampang 52100, ThailandEmail: [email protected]


Abstract

The aim of this paper is to introduce new Suzuki type proximal contractive multimaps andprove new best proximity results for these multimaps in the setting of a metric space. Our resultsextend the recent results by Hussain et al. (Fixed Point Theory Appl.(2016) 2016:14 as well as otherresults in the literature. Some illustrative examples are provided to highlight our findings.

Keywords: multivalued mapping; best proximity point; proximal contractive multimaps; Suzukitype proximal contractive multimaps;

References:[1] N. Hussain, M. Hezarjaribi, MA. Kutbi, P. Salami, "Best proximity results for Suzuki and convex type contractions".

Fixed Point Theory Appl. 2016, 14(2016).[2] M. Jleli, B. Samet, "Best proximity points for ↵- -proximal contractive type mappings and applications". Belletin des

Sciences Mathematiques, vol. 137, no. 8, 977-995.[3] M.U. Ali, T. Kamran, N. Shazad, "Best proximity points for ↵- -proximal contractive multimaps". Abstr. Appl. Anal.

2014, 181598.[4] M.U. Ali, T. Kamran, "On (↵⇤- )- contractive multivalued mappings". Fixed Point Theory Appl. 2013, 137(2013).[5] A. Abkar, M. Gabeleh, "The existence of best proximity points for multivalued nonself mappings". Revista de la Real

Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Mathematicas, vol. 107, no. 2, 319-325.

The authors were supported by the Faculty of Science Research Fund, Lampang Rajabhat University, Lampang,Thailand.

50


Common fixed points for ↵-type (�, )-weak contraction mappingin intuitionistic fuzzy metric spaces

Wudthichai Onsod⇤,1, Poom Kumam1,2







Abstract

In this paper, we extend the notation of↵-type (�, )-weak contraction mapping in intuitionisticfuzzy metric spaces, and also prove some common fixed point results for this type mappingin intuitionistic fuzzy metric space under some suitable conditions. This result generalize andimprove the corresponding results given in the literature.

Keywords: Common fixed points, intuitionistic fuzzy metric space, ↵-admissible, (�, )-weakcontractions

References:[1] I. Beg, C. Vetro, D. Gopal and M. Imdad, “(�, )-weak contractions in intuitionistic fuzzy metric spaces,” Journal of

Intelligent & Fuzzy Systems, vol. 26, pp. 2497 – 2504, 2014.[2] D. Gopal and C. Vetro, “Some new fixed point theorems in fuzzy metric spaces,” Iranian Journal of Fuzzy Systems,

vol. 11, no. 3, pp. 95 – 107, 2014.[3] C. Vetro, D. Gopal, and M. Imdad, “Common fixed point theorems for (�, )-weak contractions fuzzy metric spaces,”

Indian Journal of Mathematics, vol. 52, pp. 573 – 590, 2010.[4] A. George and P. Veeramani, “On some result of analysis for fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 90,

pp. 365 – 368,1997.[5] A. Branciari, “A fixed point theorem for mapping satisfying a general contractive condition of integral type,”

International Journal of Mathematics and Mathematical Sciences, vol. 29, pp. 531 – 536, 2002.

The authors were supported by Theoretical and Computational Science (TaCS) Center (Project Grant No.TaCS2559-2).

51


Some common fixed points for generalized cyclic contractionmappings with implicit relation and its applications

Nantaporn Chuensupantharat⇤,1, Poom Kumam1,2







Abstract

By the concept of cyclic relation, we introduced a new generalized cyclic contraction withrespect to multi - valued mappings under implicit relation and we also consider some of furtherresults of fixed point theorems on multi - valued mappings in a complete metric space. Moreoverwe obtained some common fixed point theorems for such mappings. In addition, some examplesand applications are presented to demonstrate our results.

Keywords: cyclic contraction; implicit relation; common fixed point

References:[1] B.-K. Robati, M.-B. Pour, and C. Ionescu, “Common fixed point results for cyclic operators on complete metric spaces,”

U.P.B. Sci. Bull., vol. 77, no. 8, 2015.[2] H. K. Nashine, “Fixed points and cyclic contraction mappings under implicit relations and applications to integral

equations,” SARAJEVO JOURNAL OF MATHEMATICS, vol. 10(23), pp. 257–270, 2014.[3] I. Altun and A. Erduran, “A Suzuki Type Fixed - Point Theorem,” International Journal of Mathematics and Mathematical

Sciences, vol. 2011, 2011.[4] V. Popa. A general fixed point theorem for implicit cyclic multi-valued contraction mappings. Annales Mathematicae

Silesianae. 29, pp. 119-129, 2015.


52


Fixed point and approximation theorems for monotonenonspreading mappings in ordered Banach spaces

Khanitin Muangchoo-in⇤,1, Poom Kumam1,2







Abstract

In this work, we will prove some existence theorems of fixed points for monotone nonspreadingmappings T in a Banach space E with the partial order �. In order to finding a fixed point of sucha mapping T, moreover we also prove the convergence theorem of Ishikawa iterative schemes.

Keywords: Ordered Banach space; fixed point; monotone nonspreading mapping; Ishikawaiteration schemes.

The authors were supported by Theoretical and Computational Science (TaCS) Center (Project Grant No.TaCS2559-2).

53


Fixed point and convergence theorems for Suzuki typeZ-contraction mappings in CAT(0) spaces

Nuttapol Pakkaranang⇤,1, Poom Kumam1,2







Abstract

In this paper, we study fixed point theorems and convergence theorems for Suzuki type Z-contraction mappings in CAT(0) spaces. Our result extend and improve many results in theiterature.

Keywords: fixed point; suzuki type Z-contraction mappings; convergence theorems; CAT(0)spaces.

References:[1] B. Nanjaras, B. Panyanak and W. Phuengrattana, “Fixed point theorems and convergence theorems for Suzuki-

generalized nonexpansive mappings in CAT(0) spaces,” Nonlinear Analysis: Hybrid Systems, vol. 4, pp. 25–31,2010.

[2] T. Suzuki, “Fixed point theorems and convergence theorems for some generalized nonexpansive mapping,” Journalof mathematical analysis and applications, vol. 340, pp. 1088 – 1095, 2008.

[3] S. Dhompongsa, and B. Panyanak, “On 4-convergence theorems in CAT(0) spaces,” Computers and Mathematics withApplications, vol. 56, pp. 2572 – 2579, 2008.

[4] W.A. Kirk, and B. Panyanak, “A concept of convergence in geodesic spaces,” Nonlinear Analysis, vol. 68, pp. 3689–3696, 2008.

The first author were supported by science achievement scholarship of thailand.The authers were supported by Theoretical and Computational Science (TaCS) Center (ProjectGrant No.TaCS2559-2).

54


Fixed point results for generalized F-contractions in b-metricspaces

Oratai Yamaod⇤,1, Wutiphol Sintunavarat1

1Department of Mathematics and Statistics, Faculty of Science and Technology,Thammasat University Rangsit Center, Pathumthani 12121, Thailand

Email: [email protected] (Oratai Yamaod)Email: [email protected], [email protected] (Wutiphol Sintunavarat)


Abstract

In this paper, we introduce the concept of generalized F-contraction in b-metric spaces. Fixedpoint results for these contraction mappings in b-metric spaces are obtained. Also, we give someexamples to illustrate the main results. Our results generalize the result of Wardowski [1].

Keywords: F-contraction; b-metric spaces

References:[1] D. Wardowski, “Fixed point of a new type of contractive mapping in complete metric spaces,” Fixed Point Theory and

Applications, 2012:94

The authors were supported by Research Professional Development Project under the Science Achievement Scholar-ship of Thailand (SAST).

55


A common fixed point theorem for compatible mappings of type(K) in intuitionistic fuzzy metric space

Kanchha Bhai Manandhar⇤,1, Kanhaiya Jha2

1Department of Mathematics,Kavre Multiple Campus, Tribhuvan University(TU)

Banepa, Kavre, NepalEmail: [email protected]

2Department of Natural Sciences (Mathematics), School of Science,Kathmandu University(KUSOSc), Dhulikhel, Nepal.



Abstract

The study of common fixed point of mappings satisfying contractive type conditions hasbeen a very active field of research during the last three decades.In 2014, K. Jha, V. Popa and K.B. Manandhar introduced the concept of compatible mappings of type (K) in metric space andManandhar et al. further extended the compatible mappings of type (K) in fuzzy metric space.The purpose of this paper is to obtain a common fixed point theorem for two pairs of self-mappingsof compatible of type (K) in a complete intuitionistic fuzzy metric space with example. Our resultgeneralized and improves similar other results in literature. .

Keywords: Fuzzy metric space, Compatible mappings, Compatible mappings of type (K) andcommon fixed point.

References:[1] Alaca, C., Turkoglu.D, Yildiz. C, Fixed points in intuitionistic fuzzy metric Spaces, Chaos, Solitons and Fractals, 29,

1073-1078, 2006.[2] Jha, K., Popa, V. and Manandhar, K.B., A common fixed point theorem for compatible mapping of type (K) in metric

space, Internat. J. of Math. Sci. & Engg. Appl. (IJMSEA), 8 (I), 383-391, 2014.[3] Manandhar, K. B., Jha, K. and Porru, G., Common Fixed Point Theorem of Compatible Mappings of Type (K) in

Fuzzy Metric Space, Electronic J. Math. Analysis and Appl, 2(2), 248-253, 2014.[4] Manandhar, K. B., Jha, K. and Cho,Y.J., Common Fixed Point Theorem in Intuitionistic fuzzy metric spaces using

mpatible Mappings of Type (K),Bulletin of Society for Mathematical Services & Standards 3, 81-87, 2014.[5] Zadeh, L.A, Fuzzy Sets, Inform. and Control,8, 338-353, 1965.

56


Brzdek’s fixed point theorem approach to generalizedhyperstability of the general linear equation

Laddawan Aiemsomboon⇤,1, Wutiphol Sintunavarat1

1Department of Mathematics and Statistics, Faculty of Science and Technology,Thammasat University (TU),

99/2 M. 18 Phahonyothin Rd., Khlong Nueng, Khlong Luang, Pathumthani 12121, ThailandEmail: [email protected] (Laddawan Aiemsomboon)

2Email: [email protected], [email protected] (Wutiphol Sintunavarat)


Abstract

Let F,K be two fields of real or complex numbers and X,Y be two normed spaces over F,K.The aim of this work is to study generalized hyperstability results for general linear equation ofthe form

g(ax + by) = Ag(x) + Bg(y),where g : X ! Y is a mapping and a, b 2 F\{0}, A,B 2 K. Our results are improvement andgeneralization of main results of Piszczek [1].

Keywords: generalized hyperstability; general linear equation

References:[1] M. Piszczek, “Hyperstability of the general linear functional equation,” Bull. Korean. Math. Soc., vol. 52, pp.

1827–1838, 2015.

The authors were supported by Research Professional Development Project under the Science Achievement Scholar-ship of Thailand (SAST).

57


New fixed point theorems of multivalued F-contractions inmodular metric spaces and its application to non-linear integral

equations

Dilip Jain1, Anantachai Padcharoen2 , Poom Kumam2,§, and Dhananjay Gopal1

1Department of Applied Mathematics & Humanities, S.V. National Institute of Technology,Surat-395007, Gujarat, India

2Department of Mathematics, Faculty of Science, King Mongkut’s University of TechnologyThonburi (KMUTT),

126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.⇤Presenting author.


Abstract

In this paper, we discuss the existence of fixed point for multivalued F-contraction in the settingof modular metric spaces. In this connection, we introduce the notion of multivalued F-contractionand prove corresponding fixed point theorems in complete modular metric space. Then we applyour result to establish the existence of solutions for a certain type of non-linear integral equations.

Keywords: Fixed point, multivalued F-contractive, modular metric space

58


A common fixed point theorem for sequence of mappings insemi-metric space with compatible mapping of type (E)

Umesh Rajopadhyaya⇤,1, Kanhaiya Jha1,2

1School of Management, Kathmandu University (KUSOM),Balkumari, Lalitpur, Nepal

Email: [email protected] of Natural Sciences(Mathematics),

School of Science,Kathmandu University (KUSOSc),

Dhulikhel, NepalEmail: [email protected]


Abstract

In 1922, Polish mathematician Stephan Banach established the famous Banach’s ContractionPrincipal. Since then, it has become milestone to the researchers in analysis to establish newtheorems by generalizing this theorem. K. Menger in 1928 introduced the notion of semi-metricspace as generalization of metric space. In 2002, M. Aamri and D. El. Moutawakil established thecommon fixed point theorem for two pairs of self mappings in semi-metric space. Also,in 2007M. R. Singh and M. Y. Singh introduced the notion of Compatible mapping of type(E) in metricspace. In 2014, Rajopadhyaya et. al. established the common fixed point theorem for three pairsof self mappings in semi-metric space using various contractions. The purpose of this paper is toestablish a common fixed point theorem for sequence of self mappings in semi-metric space withcompatible mappings of type (E).

Keywords: Semi-metric space; Compatible mapping of type (E); Common fixed point

References:[1] M. Aamri and D. El. Moutawakil, “Common fixed points under contractive conditions in semi-metric space,” Appl.

Math. E-Notes, vol. 3, pp. 156 – 162, 2003.[2] S. H. Cho and D. J. Kim, “Fixed point theorems for generalized contractive type mappings in symmetric space,”

Korean J. Math. Appl., vol. 16, pp. 439 – 450, 2008.[3] U. Rajopadhyaya, K. Jha, and M. Imdad, “Common fixed point theorem in semi-metric space with compatible

mapping of type (E),” Bull. Math. Sci. and Appl., vol. 33, pp. 63 – 67, 2014.[4] M. R. Sing and M. Y. Singh, “Compatible mappings of type (E) and common fixed point theorems of Meir-keeler

type,” Internat. J. Maths. Sci. Engg. Appl., vol.2, pp. 299–315, 2007.[5] W. A. Wilson,“On semi-metric space,” Amer. J. Math., vol. 53, pp. 361 – 373, 1931.

59


A best proximity point theorem for generalized non-self Kannanand Chetterjea type mappings and Lipschitzian mappings in

complete metric spaces

Kasamsuk Ungchittrakool⇤,1,2,3

1Department of Mathematics, Faculty of Science,Naresuan University, Phitsanulok 65000, Thailand

2Research Center for Academic Excellence in Mathematics,Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

3Research Center for Academic Excellence in Nonlinear Analysis and Optimizations,Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand


Abstract

The purpose of this paper is to provide and study a best proximity point theorem for generalizednon-self Kannan and Chetterjea type mappings and Lipschitzian mappings in complete metricspaces. The significant mapping in a unified form which related to contractive mappings, Kannantype mappings and Chetterjea type mappings is established. We also provide an example toillustrate the situation corresponding to the main theorem. The main result of this paper can beviewed as a general and unified form of several previously existing results.

Keywords: Optimal approximate solution; Best proximity point; Lipschitzian mapping; General-ized Kannan and Chatterjea type mapping; Cyclic contraction

References:[1] Chatterjea1972Fix S.K. Chatterjea, “Fixed point theorems,” Comptes Rendus De L Academie Bulgare Des Sciences, vol.

25, pp. 727 – 730, 1972.[2] Kannan1968Som R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp.

71 – 76, 1968.[3] P S. Sadiq Basha, “Best proximity points: global optimal approximate solutions,” Journal of Global Optimization, vol.

49, pp. 15 – 21, 2011.[4] SadiqBashaShahzadJeyaraj2013 S. Sadiq Basha, N. Shahzad, R. Jeyaraj, “Best proximity points: approximation and

optimization,” Optimization Letters, vol. 7, no. 1, pp. 145 – 155, 2013.[5] AE C. Vetro, “Best proximity points: convergence and existence theorems for p-cyclic mappings,” Nonlinear Analysis:

Theory, Methods & Applications, vol. 73, pp. 2283 – 2291, 2010.

The author was supported by Naresuan University.

60


Stability for Lexicographic Vector Equilibrium Problems

T. Bantaojai1, L.Q. Anh2, R. Wangkeeree1 , T.Q. Duy2, and P.T. Vui2

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000,Thailand

Email: [email protected] of Mathematics, Teacher College, Cantho University, Cantho, Vietnam


Abstract

In this work, we have studied on stability for Lexicographic Vector Equilibrium Problems(LEP),that is, we study Painlevé-Kuratowski convergence of the solution sets with a sequence of mappingsconverging continuously and sequence of set converging in the sense of Painlevé-Kuratowski andwe also study PK-wellposedness for (LEP). Our main results are new and di↵erent from the existingones in the literature.

Keywords: Lexicographic Vector Equilibrium Problems, PK-wellposedness, Painlevé-Kuratowskiconvergence, Continuous convergence.

References:[1] L.Q. Anh, P.Q. Khanh : Semicontinuity of the solution set of parametric multivalued vector quasiequalibrium

problems, Journal of Mathematical Analysis and Applications, 294: 699- 711(2004).[2] L.Q. Anh, P.Q. Khanh : Continuity of solution maps of parametric quasiequilibrium problems, Journal of Global

Optimization, 46, 247-259 (2010).[3] Z.M. Fang, S.J. Li, K.L. Teo, Painlevé-Kuratowski convergences for the solution sets of set-valued weak vector

variational inequalities, J. Inequal. Appl. ID 43519 , 1-14 (2008).[4] R.T. Rockafella, R.J.-B. Wets, Variational analysis, Springer, Berlin (1998).[5] R. Wangkeeree, P. Boonman, P. Prechasilp : Lower semicontinuity of approximate solution mappings for parametric

generalized vector equilibrium problems, Journal of Inequalities and Applications, 2014-421 (2014).[6] R. Wangkeeree, T. Bantaojai, P. Yimmuang : Well-posedness for lexicographic vector quasiequilibrium problems with

lexicographic equilibrium constraints, Journal of Inequal- ities and Applications, 2015:163 (2015).

The authors were supported by the Thailand Research Fund and Naresuan University

61


On vector optimization problems with geometric framework

Ariana Pitea⇤,11Department of Mathematics & Informatics,

University “Politehnica" of Bucharest,313 Splaiul Independentei, RO-060042, Bucharest,

Email: [email protected]⇤Presenting author.


Abstract

We will survey some classes of multitime multiobjective variational problems of minimizinga vector of functionals or vector of quotients of curvilinear integral functionals (mechanical work)subject to certain partial di↵erential equations and inequations (limited resources). To state theresults on e�ciency and optimality, various types of generalized convexities are used. For theconsidered multitime multiobjective variational problems, several duality results are establishedunder these types of convexity.

Keywords: multitime multiobjective problem, e�cient solution, quasiinvexity, duality

References:[1] T. Antczak, and A. Pitea, "Parametric approach to multitime multiobjective fractional variational problems under

(F, rho)-convexity," Optimal Control: Applications and Methods, DOI: 10.1002/oca.2192.[2] A. Pitea, and T. Antczak, "Proper e�ciency and duality for a new class of nonconvex multitime multiobjective

variational problems," Journal of Inequalities and Applications, Vol. 2014, Art. No. 333.[3] A. Pitea, and M. Postolache, "Duality theorems for a new class of multitime multiobjective variational problems,"

Journal of Global Optimization, vol. 54, no. 1, pp. 47-58, 2012.[4] A. Pitea, and M. Postolache, "Minimization of vectors of curvilinear functionals on the second order jet bundle.

Su�cient e�ciency conditions," Optimization Letter, vol. 6, no. 8, pp. 1657-1669, 2012.[5] G. J. Zalmai, "Generalized (F , b,�,⇢,✓)-univex n-set functions and semiparametric duality models in multiobjective

fractional subset programming," International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 7,pp. 1109-1133, 2005.

62


Recent results to approximate solution of stochastic di↵erentialdelay equations

Young-Ho Kim⇤,1

1Department of Mathematics, Faculty of Science,Changwon National University,

20 Changwondaehak-ro, Changwon, Gyeongsangnam-do 641-773, Republic of KoreaEmail: [email protected]


Abstract

In this talk, we deal with some recent results to approximate solution of a stochastic systemsand discuss some di↵erence between an approximate solution and an accurate solution to thespecial but important class of stochastic delay systems. To make the theory more understandable,we use a non-uniform Lipschitz condition and special linear growth condition.

Keywords: approximate solutions; stochastic di↵erential delay equation; Lipshitz condition;linear growth condition

References:[1] Y. -H. Kim, “A note on the solutions of Neutral SFDEs with infinite delay,” J. Inequal. Appl., vol. 2013, no.181, pp.

1–9, 2013.[2] Y. -H. Kim, “On the pth moment estimates for the solution of stochastic di↵erential equations,” J. Inequal. Appl.,

vol.2014, no.395, pp. 1–12, 2014.[3] Y.-H. Kim, “An exponential estimates of the solution for Stochastic functional Di↵erential Equations,” Journal of

Nonlinear and Convex Analysis, vol.16, no.9, pp. 1861-1868, 2015.[4] X. Mao, “Stochastic Di↵erential Equations and Applications,” Horwood Publication Chich- ester, UK 2007.[5] G. Pavlovic and S. Jancovic, “The Razumikhin approach on general decay stability for neutral stochastic functional

di↵erential equations,” Journal of the Franklin Institute, vol.350 pp. 2124–2145, 2013.[6] F. Wei and Y. Cai, “Existence, uniqueness and stability of the solution to neutral stochastic functional di↵erential

equations with infinite delay under non-Lipschitz conditions,” Advances in Di↵erence Equations vol.2013, no.1512013.

63


An iterative method for triple-hierarchical problems

Thanyarat Jitpeera⇤,1, Kanoktip Anorat1 and Poom Kumam2,3

1Department of Mathematics, Faculty of Science and Agriculture Technology,Rajamangala University of Technology Lanna (RMUTL),

99 Prahonyothin Rd., Sai Khao, Phan, Chiang Rai 57120, ThailandEmail: [email protected], [email protected]







Abstract

In this paper, we introduce the solution of the triple-hierarchical fixed point problems in thereal Hilbert spaces. We establish the strong convergence of the proposed method under some mildconditions. The results presented in this paper extend and improve some well-known results inthe literature.

Keywords: fixed point problem, hierarchical problem, variational inequality problem

References:[1] P. L. Combettes, “A block-itrative surrogate constraint splitting method for quadratic signal recovery,” IEEE Trans.

Signal Process., vol. 51, no. 7, pp. 1771–1782, 2003.[2] L.-U. Ceng, Q.H. Ansari and J.-C Yao, “Iterative methods for triple hierarchical variational inequalities in Hilbert

spaces,” J. Optim. Theory Appl., DOI 10.1007/s10957-011-9882-7.[3] S. A. Hirstoaga, “Iterative selection method for common fixed point problems,” J. Math. Anal. Appl., vol. 324, pp.

1020 – 1035, 2006[4] P. Hartman and G. Stampacchia, “On some nonlinear elliptic di↵erential functional equations,” Acta Math., vol. 115,

pp. 271–310, 1966.[5] H. Iiduka, “Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem,”

Nonlinear Anal., vol. 71, pp. e1292 – e1297, 2009.

64


Performance measures ofE2|E2|1queueing system with sinusoidalarrival rate

R.P.Ghimire,⇤,1, Amir Adhikari 1,2

1Department of Mathematics, School of Science,Kathmandu University (KU),

Dhulikhel., Kavre, NepalEmail: [email protected]

2Department of Mathematics,Sanothimi Campus,Tribhuvan University (TU) ,Nepal



Abstract

Tthis paper deals with the study of E2|E2|1 queueing model with the provision that the cus-tomers arrive in the system follows Erlang distribution.The customers arrival rate function is takento be sinusoidal .The main objective of the paper is to find some performance measures- number ofcustomers in the system ,number of customers in the queue,expected time to failure of server.Thenumerical results have also been shown so as to show that the model under study is realistic.

Keywords: Erlang distribution; Sinusoidal; Queue

References:[1] K.-W. Wang and K.-Y.. Tai, “A queueing system with queue-dependent servers and finite capacity,,” The Applied

Mathematical Modelingt, vol. 24, pp. 807–814, 2000.[2] A.V.S. Suhasini,K.S. Rao and JP.R.S. Reddy, “Transient analysis of tandem queueing model with non homogeneous

poisson bulk arrivals having state dependent service rates,” International Journal of Advanced Computer andMathematical Sciences, vol. 3, no. 3, pp. 272 – 289, 2012.

[3] S.S Sah, and R.P Ghimire, “Transient analysis ofM|Ek |1queueing model ,” Journal of the Institute of Engineering, vol. 11,no. 1, pp. 165 –171, 2015.


65


A common fixed point theorem for subcompatible mappings infuzzy metric space

Kanhaiya JhaDepartment of Mathematical Sciences,

School of Science, Kathmandu University,Dhulikhel, Kavre, NepalEmail: [email protected]


Abstract

The classical Banach fixed point theorem in metric space is one of the fundamental results inmathematics with wide applications. Also, the study of common fixed points of self mappingsin fuzzy metric space satisfying certain contractive conditions as an extension of this Banachcontraction principle has been at the center of vigorous research activities. The purpose of thispaper is to introduce the notion of subcompatible pair of mappings and to establish a commonfixed point theorem for subcompatible pairs of reciprocally continuous self mapping in fuzzymetric space which generalizes and improve similar results of fixed points.

Keywords: fixed point; subcompatible maps; reciprocal continuity; fuzzy metric spaceoint

References:[1] H. Bouhadjera and C. Godet-Thobie, “Common fixed point theorems for pair of subcompatible mappings, ”

arxiv:[math.FA], pp. 1–16, 2011.[2] K. Jha, “A fixed point theorem for semi-compatible maps in fuzzy metric space, ” Kathmandu Univ. J. Sci. Engg. Tech.,

vol. 9, no. 1, pp. 83 – 89, 2013.[3] O. Kramosil and J. Michalek, “Fuzzy mathematics and statistical metric space ,” Kybernetika, vol. 11,pp. 326 – 334,

1975.[4] R.P. Pant, “Common fixed points of four mappings,” Bull. Calcutta Math. Soc., vol. 90, pp. 281–286, 1998.[5] L.A. Zadeh, “Fuzzy sets,” Inform and Control, vol. 89, pp. 338 – 353, 1965.

66


A hybrid optimization of particle swarm optimization andgenetic algorithm with multi-parent crossover (GA-MPC)

Apirak Sombat⇤,1, Poom Kumam1,2







Abstract

This paper proposed a hybrid optimization of particle swarm and genetic algorithm withmulti-parent crossover is proposed to solve the optimization model problem.

Keywords: equipment maintenance; preventive maintenance; maintenance period optimization;particle swarm optimization; genetic algorithm optimization

References:[1] M.-B. Biggs, B. Christianson, and M. J. Zuo, “Optimization preventive maintenance models,” Computational Opti-

mization an Applications, vol. 35, no. 2, pp. 261-279, 2006.[2] J. Kennedy and R. Eberhart, “Particle swarm optimization ,” Proceeding of the IEEE International Conference on Neural

Networks, IEEE, pp. 1942-1948, 1995.[3] R. Poli, J. Kennedy, and T. Blackwell, “Particle swarm optimization an overview,” Swarm Intelligence, vol. 1, no. 1, pp.

33-57, 2007.[4] Elsayed, S. M., Sarker, R. A., and Essam, D. L., “A new genetic algorithm for solving optimization problem,”

Engineering Applications of Artificial Intelligence, vol. 27, pp. 57-69, 2014.


67


Strong convergence theorems by hybrid and shrinking projectionmethods for sums of two monotone operators

Somyot Plubtieng1, Tadchai Yuying ⇤,2

1,2Department of Mathematics,Faculty of Science,

Naresuan University,Phitsanulok 65000, ThailandEmail: [email protected]


Abstract

In this paper, we establish the new iterative algorithm and prove strong convergence theoremsfor finding the common solution of the sum of two monotone operators and fixed point problemsby using hybrid methods and shrinking projection methods.

Keywords: Hybrid methods, Shrinking projection methods, Monotone operators and Resolvent.

References:[1] H.M.1996 H. Attouch and M. Thera, A general duality principle for the sum of two operators, J. Convex Anal. 3

(1996) , 1-24.[2] H.H.2009 H.H. Bauschke, A note on the paper by Eckstein and Svaiter on general projective splitting methods for

sums of maximal monotone operators, SIAM J. Control Optim. 48 (2009), 2513-2515.[3] Bruck.1974 R. E. Bruck, A strongly convergent iterative solution of 0 2 U(x) for a maximal monotone operator U in

Hilbert space, J. Math. Anal. Appl. 48 (1974), 114-126.[4] Brezis.1978 H. Brezis and P. L. Lions, Produits infinis de resolvants, Israel J. Math. 29 (1978), 329-345.[5] Bro.1967 F.E. Browder and W.V. Petryshyn, Construction of fied points of nonlinear mappings in Hilbert space, J.

Math. Anal. Appl. 20 (1967), 197ÂU228.[6] RS.1980 R. E. Bruck and S. Reich, A general convergence principle in nonlinear functional analysis, Nonlinear Anal.

5 (1980), 939-950.[7] Cegielski2012 A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Springer, 2012.[8] CR.T.1997 G.H.G. Chen and R.T. Rockafellar, Convergence rates in forward-backward splitting, SIAM J. Optim. 7

(1997), 421-444[9] S.2013 S. Y. Cho, Strong convergence of an iterative algorithm for sum of two monotone operators, J. Fixed Point

Theory, 6 (2013).[10] CQW. S.Y. Cho, X. Qin and L. Wang, A strong convergence theorem for solutions of zero point problems and fixed

point problems, Bull. Iriann Math. Soc. in press.[11] He.2010 S. He, C. Yang, P. Duan, Realization of the hybrid method for Mann iterations, Appl. Math. Comput. 217

(2010), 4239-4247.

The authors would like to thanks Naresuan University and Thailand Research Fund (TRF) for supporting by permitmoney of investment under of The Royal Golden Jubillee Ph.D. Program (RGJ-Ph.D.), Thailand.

68


Sensitivity analysis of the quasi variational inequality problemon uniformly prox regular sets

Jittiporn Tangkhawiwetkul⇤,1, Narin Petrot1,2

1 Faculty of Science and Technology,Pibulsongkram Rajabhat University,

156 Phlai-Chumphon, Meuang, Phitsanulok, 65000, ThailandEmail: [email protected]

2Deparment of Mathematics, Faculty of Science,Nareasuan University, Meuang, Phitsanulok, 65000, Thailand



Abstract

In this paper, we consider the sensitivity analysis of the quasi variational inequality problemover a class of nonconvex sets, as uniformly prox-regular sets. The Wiener-Hopf equation, whichequivalent to the quasi variational inequality problem on uniformly prox regular sets is considered.The sensitivity analysis of this problem is studied. The results in this paper improve and extendthe variational inequality problems which have been appeared in literature.

Keywords: Sensitivity analysis, quasi variational inequality, uniformlly prox-regular set, locallyLipschitz continuous mapping, locally strongly monotone mapping

References:[1] M. A. Noor, “Sensitivity Analysis for Quasi-Variational Inequalities,” Journal of Optimization Theory and Applications,

vol. 95, no. 2„ no. 399-407, pp. 123–145, 1997.[2] P. Shi, “Equivalence of Variational Inequalities with Wiener-Hopf Equations,” Proceedings of the American Mathematical

Society, vol. 111, no. 2, pp. 339-346, 1991.[3] J. Tangkhawiwetkul and N. Petrot, “Existence theorems for quasi-variational inequalities problems on uniformly

prox-regular sets ,” Abstract and Applied Analysis, vol. 2013, Article ID 612819, 7 pages, 2013.

The authors were supported by the National Research Council of Thailand by Pibulsongkram Rajabhat University in2016 (RID-1-59-6-18).

69


Convergence theorem for solving the combination of equilibriumproblems and fixed point problems in Hilbert spaces

Sarawut Suwannaut⇤,1, Atid Kangtunyakarn1

1Department of Mathematics, Faculty of Science,King Mongkut’s Institute of Technology Ladkrabang (KMITL),

Chalongkrung Rd., Ladkrabang, Bangkok 10520, ThailandEmail: [email protected]



Abstract

In this article, we propose a iterative algorithm for approximating a common element of a finitefamily of solution sets of equilibrium problems, the set of common fixed points of a finite family ofnonspreading mappings and the set of common fixed points of a finite family of i-strictly pseudocontractive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative schemeconverges strongly to a common element of those three sets. Finally, to support our main results,the numerical examples are given.

Keywords: strictly-pseudo contractive mapping; nonspreading mapping; equilibrium problem;fixed point; Hilbert space

References:[1] Z. Opial, “Weak convergence of the sequence of successive approximation of nonexpansive mappings”, Bull. Amer.

Math. Soc., vol. 73, pp. 591–597, 1967.[2] E. Blum and W. Oettli, “From optimization and variational inequilities to equilibrium problems” Math. Student,

vol. 63, no. 14, pp. 123–145, 1994.[3] W. Takahashi, “Nonlinear Functional Analysis” Yokohama Publishers, Yokohama, 2000.[4] H.K. Xu, “An iterative approach to quadric optimization” J. Optim. Theory Appl. vol. 116, pp. 659–678, 2003.[5] P.L. Combettes and S.A. Hirstoaga, “Equilibrium programming in Hilbert spaces” J. Nonlinear Convex Anal. vol. 6,

no. 1, pp. 117–136, 2005.[6] A. Kangtunyakarn, “Convergence theorem of -strictly pseudo-contractive mapping and a modification of general-

ized equilibrium problems” Fixed point Theory and Applications vol. 89, 2012.[7] S. Suwannaut and A. Kangtunyakarn, “The combination of the set of solutions of equilibrium problem for convergence

theorem of the set of fixed points of strictly pseudo-contractive mappings and variational inequalities problem”Fixed point Theory and Applications vol. 291, 2013.

[8] S. Suwannaut and A. Kangtunyakarn “Convergence analysis for the equilibrium problems with numerical results”Fixed point Theory and Applications vol. 167, 2014).

70


Strong convergence theorems for the modified variationalinclusion problems and various nonlinear mappings in Hilbert

space

Wongvisarut Khuangsatung⇤,1, Atid Kangtunyakarn1


Chalongkrung Rd., Ladkrabang, Bangkok 10520, ThailandEmail: [email protected]: [email protected]


Abstract

In this paper, we prove a strong convergence theorem for finding a common element of theset of fixed points of a finite family of -strictly pseudononspreading mappings and the set ofsolutions of a finite family of variational inclusion problems and the set of solutions of generalizedequilibrium problem in Hilbert space. By using our main result, we give the numerical exampleto support some of our results.

Keywords: variational inclusion problems; -strictly pseudononspreading mapping; generalizedequilibrium problem; resolvent operator; fixed point problem

References:[1] W. Takahashi, “Nonlinear Functional Analysis” Yokohama Publishers, Yokohama, 2000.[2] Z. Opial, “Weak convergence of the sequence of successive approximation of nonexpansive mappings” Bull. Amer.

Math. Soc., vol. 73, pp. 591–597, 1967.[3] E. Blum and W. Oettli, “From optimization and variational inequilities to equilibrium problems” Math. Student,

vol. 63, no. 14, pp. 123–145, 1994.[4] H.K. Xu, “An iterative approach to quadric optimization” J. Optim. Theory Appl. vol. 116, pp. 659–678, 2003.[5] P.L. Combettes and S.A. Hirstoaga, “Equilibrium programming in Hilbert spaces” J. Nonlinear Convex Anal. vol. 6,

no. 1, pp. 117–136, 2005.[6] S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonex-

pansive mapping in a Hilbert space” Nonlinear Analysis: Theory, Methods & Applications,vol. 69, no. 3, pp.1025ÂU1033, 2008.

[7] S.S. Zhang, J.H.W. Lee and C.K. Chan “Algorithms of common solutions for quasi variational inclusion and fixedpoint problems” Applied Mathematics and Mechanics, vol. 29, pp. 571ÂU581, 2008.

[8] A. Kangtunyakarn, “The methods for variational inequality problems and fixed point of i-strictly pseudononspread-ing mapping” Fixed point Theory and Applications vol. 171, 2013.

[9] W. Kangtunyakarn and A. Kangtunyakarn, “Algorithm of a new variational inclusion problem and strictly pseudonon-spreading mapping with application” Fixed point Theory and Applications vol. 209, 2014.

71


Fast Mann and CQ algorithms for a nonexpansive mapping

Qiao-Li Dong1, Han-Bo Yuan, Yeol Je Cho1College of Science, Civil Aviation University of China


Abstract

In this talk, we introduce two fast Mann and CQ algorithms and analyze their convergence andbehavior. Firstly, by pointing that Picard algorithm is the steepest descent method for solving theminimization problem, provide the accelerated Picard algorithm by using the ideas of conjugategradient methods that accelerate the steepest descent method. Then, based on the acceleratedPicard algorithm, we present accelerations of the Mann and CQ algorithms. Secondly, we introduceinertial accelerated Mann and inertial CQ algorithms by combining accelerated Mann algorithmand CQ algorithm with inertial extrapolation respectively. This strategy is intended to speed upthe convergence of algorithms. The convergence theorems established in this new setting improveknown ones.

Keywords: two fast Mann; CQ algorithms

References:[1] Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas-Rachford splitting for monotone inclusion problems. Appl.

Math. Comput. 256 (2015) 472-487.[2] Sakurai, K, Liduka, H: Acceleration of the Halpern algorithm to search for a fixed point of a nonexpansive mapping.

Fixed Point Theory Appl. 2014, 202 (2014).

72


Generalized (�, ) vector complementarity problem andgeneralized (�, ) vector variational inequality problem with

fuzzy mappings

Jiraprapa Munkong⇤,1, Kasamsuk Ungchittrakool1,2,3

1Department of Mathematics, Faculty of Science,Naresuan University, Phitsanulok 65000, Thailand,

Email: [email protected] Center for Academic Excellence in Mathematics,

Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand3Research Center for Academic Excellence in Nonlinear Analysis and Optimizations,

Faculty of Science, Naresuan University, Phitsanulok 65000, ThailandEmail: [email protected]


Abstract

In this paper, we introduce and study a generalized (�, ) vector complementarity problemwith fuzzy mappings. Under suitable conditions, we have shown that generalized (�, ) vectorcomplementarity problem with fuzzy mappings is equivalent to generalized (�, ) vector varia-tional inequality problem with fuzzy mappings. We derive some existence results for our problem.Results of this paper represent a significant improvement and refinement of the previously knownresults.

Keywords: Vector; Complementarity problem; Variational inequality; Positive homogeneous;Fuzzy mapping

References:[1] R. Ahmad, Q.H. Ansari, “An iterative algorithm for generalized nonlinear variational inclusions,” Appl. Math. Lett.

, vol.13 pp. 23-26, 2000.[2] J.-P. Aubin, I. Ekeland, “Applied Nonlinear Analysis”, Pure Appl. Math., John Wiley and Sons, New York, 1984.[3] C. Baiocchi, A. Capelo, “Variational and Quasi variational Inequalities,” Wiley, New York,, 1984.[4] C. Bardaro, R. Ceppitelli, “Some further generalizations of Knaster -Kuratowski-Mazurkiewicz-theorem and minimax

inequalities,” J. Math. Anal. Appl, vol.132 pp.484-490,1988.[5] S.S. Chang, “Coincidence theorem and fuzzy variational inequalities for fuzzy mappings,” Fuzzy Sets Syst, vol.61 pp.

359-368, 1994.

The authors were supported by The Thailand Research Fund (TRF).

73


Numerical Simulation of an Air Pollution Model on IndustrialAreas by Considering the Influence of Multiple Point Sources

Pravitra Oyjinda⇤,1, Nopparat Pochai1,2

1Department of Mathematics, Faculty of Science,King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Email: [email protected] of Excellence in Mathematics

CHE, Si Ayutthaya Rd., Bangkok 10400, ThailandEmail: [email protected]


Abstract

A numerical simulation on a two-dimensional atmospheric di↵usion equation of an air pollu-tion measurement model is proposed. The considered area is separated into two parts such as anindustrial zone and an urban zone. In this research, the air pollution measurement by releasingthe pollutant from multiple point sources above an industrial zone to the other area is simulated.The governing partial di↵erential equation of air pollutant concentration is approximated by usinga finite di↵erence technique. The approximated solutions of the air pollutant concentration onboth areas are compared. The air pollutant concentration levels that influenced by multiple pointsources are also analyzed.

Keywords: multiple point sources; finite di↵erence technique; air pollutant concentration; indus-trial zone; urban zone

References:[1] S.A. Konglok and S. Tangmanee, ”A K-model for simulating the dispersion of sulfur dioxide in a tropical area,”

Journal of Interdisciplinary Mathematics, vol. 10, no. 6, pp. 789–799, 2007.[2] S.A. Konglok and N. Pochai, ”A Numerical Treatment of Smoke Dispersion Model from Three Sources Using

Fractional Step Method,” Advanced Studies in Theoretical Physics, vol. 6, no. 5, pp. 217–223, 2012.[3] K. Lakshminarayanachari, K.L. Sudheer Pai, M. Siddalinga Prasad, and C. Pandurangappa, “Advection- Di↵usion

numerical model of air pollutants emitted from an urban area source with removal mechanisms by consideringpoint source on the boundary,” International Journal of Application or Innovation in Engineering & Management, vol.2, pp. 251–268, 2013.

[4] S.A. Konglok and N. Pochai, ”Numerical Computations of Three-dimensional Air-Quality Model with Variations onAtmospheric Stability Classes and Wind Velocities using Fractional Step Method,” IAENG International Journalof Applied Mathematics, vol. 46, no. 1, pp. 112–120, 2016.

74


Numerical computation of a water-quality model withadvection-di↵usion-reaction equation using an upwind implicit

scheme

Piyada Phosri⇤,1, Nopparat Pochai1,2


Bangkok 10520, ThailandEmail: [email protected]

2Centre of Excellence in Mathematics, Commission on higher Education (CHE),Si Ayutthaya Road, Bangkok 10400, Thailand



Abstract

In this research, numerical computations of water-quality model in a uniform flow stream areproposed. The model is governed by a one-dimensional advection-di↵usion-reaction equation thatprovided the pollutant concentrations along the stream. The pollutant concentration is approxi-mated by using a finite di↵erence technique. The upwind implicit scheme is used to approximatethe pollutant concentration in each points at each times along a uniform flow stream. The accurateof the proposed technique is compared with the analytical solutions that are shown in a numericalexperiment.

Keywords: advection-di↵usion-reaction equation; water-quality model; uniform flow stream;upwind implicit scheme

References:[1] M. Dehghan, “Weighed finite di↵erence techniques for the one-dimensional advection-di↵usion equation,” Applied

Mathematics and Computation, vol. 147, pp. 307–319, 2004.[2] H. Karahan, “Implicit finite di↵erence techniques for the advection-di↵usion equation using spreadsheets,” Advances

in Engineering Software,vol. 37, no. 9, pp. 601 – 608, 2006.[3] H. Karahan, “A third-oder upwind scheme for the advection-di↵usion equation using spreadsheets,” Advances in

Engineering Software , vol. 38, pp. 688–697, 2007.[4] N. Pochai, “Numerical treatment of a modified MacCormack scheme in a nondimensional form of the water quality

models in a nonuniform flow stream,” Journal of Applied Mathematics, 2014.

75


Some common fixed point theorems for generalized cyclicmulti-valued contractive operators in complete matric spaces

Narin Petrot1, Warut Saksirikun⇤,11Department of Mathematics, Faculty of Science,

Naresuan University, Phitsanulok, 65000, ThailandEmail: [email protected], [email protected]


Abstract

In this work, we introduce a new generalized contraction which is corresponding to the newclass of control functions under a general contractive type condition based on the Hausdor↵metricbetween subsets of a metric space and give some common fixed point results for this introducedcontraction in complete metric spaces. This results presented in this work improve and generalizesome known corresponding results in the literature.

Keywords: common fixed point; cyclic operator; Hausdor↵metric; multi-valued operator; matricspaces

References:[1] P.N. Dutta and B.S. Choudhury, "A generalisation of contraction principle in metric spaces," Fixed Point Theory and

Applications, Volume 2008, Article ID 406368, 8 pages, 2008.[2] M.S. Khan, M. Swaleh and S. Sessa, "Fixed point theorems by altering distances between the points," Bulletin of the

Australian Mathematical Society,vol. 30,no. 1,pp. 1–9, 1984.[3] W.A. Kirk, P.S. Srinivasan, and P. Veeramani, "Fixed points for mappings satisfying cyclical contractive conditions,"

Fixed Point Theory,vol. 4,no. 1,pp. 79–89, 2003.[4] S.B. Nadler, "Multi-valued contraction mappings," Pacific Journal of Mathematics,vol. 30,pp. 475–488, 1969.[5] I.A. Rus, "Cyclic representations and fixed points," Annals of the Tiberiu Popoviciu Seminar of Functional Equations,

Approximation and Convexity,vol 3,pp. 171–178, 2005.[6] N. Shahzad, E. Karap?nar and A.F. Roldan-Lopez-de-Hierro, "On some fixed point theorems under (↵, ,�)-

contractivity conditions in metric spaces endowed with transitive binary relations,"Fixed Point Theory andApplications,vol. 2015:124, 2015.

[7] W. Sintunavarat, P. Kumam, "Common fixed point theorem for cyclic generalized multi-valued contraction map-pings," Applied Mathematics Letters,vol. 25,no 11,pp. 1849–1855, 2012.

The authors were supported by the Thailand Research Fund through the Royal Golden Jubilee PhD Program (GrantNo. PHD/0248/2553) and Naresuan University.

76


A general iterative method for solving convex optimizationproblems of the sum of two convex functions

Kan Buranakorn⇤,1, Somyot Plubtieng1,2


Email: [email protected] center for Academic Excellence in Mathematics, Naresuan University



Abstract

It is well known that the proximal gradient algorithm plays an important role in solving convexoptimization problems. In this paper, we use the idea of proximal gradient algorithm, viscosityiterative method and regularization to establish a sequence generated by a general iterative methodconverges strongly to a convex optimization problems of the sum of two convex functions, whichsolves a variational inequality under suitable conditions.

Keywords: convex optimization problems; variational inequality; proximal gradient algorithm;fixed point

References:[1] P.L. Combettes and V.-R. Wajs, "Signal Recovery by Proximal Forward-Backward Splitting," Multiscale Model. Simul.,

vol. 4, pp. 1168–1200, 2006.[2] H. Iiduka and W. Takahashi, "Strong Convergence Theorems for Nonexpansive Nonself-Mappings and Inverse-

Strongly-Monotone Mappings," J. Convex. Anal., vol. 11, pp. 69–79, 2004.[3] J.J. Moreau, "Fonctions convexes duales et points proximaux dans un espace hilbertien," C. R. Acad. Sci. Paris Ser. A

Math., vol. 255, pp. 2897–2899, 1962.[4] B.T. Polyak, "Introduction to optimization," In Optimization Software, New York, 1987.[5] M. Su and H.K. Xu, "Remarks on the gradient-projection algorithm," J. Nonlinear Anal. Optim., vol. 1, pp. 35–43, 2010.[6] M. Tian, "A general iterative algorithm for nonexpansive mappings in Hilbert spaces," Nonlinear Anal., vol. 73, pp.

689–694, 2010.[7] M. Tian and M.M. Li, "A General Iterative Method for Solving Constrained Convex Minimization Problems," J. Optim.

Theory. Appl., vol. 162, pp. 202–207, 2014.[8] H.K. Xu, "Viscosity approximation methods for nonexpansive mappings," J. Math Anal. Appl., vol. 298, pp. 279–291,

2004.

The first author would like to thanks the Thailand Research Fund through the Royal Golden Jubilee PH.D. Programfor supporting by grant fund under Grant No. PHD/0249/2553, Thailand

77


On Coupled-Nonexpansive Mappings

Cholatis Suanoom⇤,1, Chakkrid Klin-eam†,11Department of Mathematics, Faculty of Science,

Naresuan University,99 Village No.9, Tha Pho Sub-district, Muang District, Phitsanulok, 65000, Thailand

†Adviser.Email: [email protected]


Abstract

In this work, we obtained the properties of the coupled fixed point set for coupled-nonexpansivemappings in Banach spaces. Moreover, we prove such properties of the coupled fixed point set forcoupled-nonexpansive mappings and prove some coupled fixed point theorems in Banach spaces.

Keywords: coupled fixed point set; coupled-nonexpansive mappings; coupled fixed point theo-rems; Banach spaces

References:[1] K. Goebel and W.A. Kirk, “Iteration processes for nonexpansive mappings,” Con- temp. Math., vol. 21, pp. 115–123,

1983.[2] S. Ishikawa, “ Fixed points and iteration of a nonexpansive mapping in a Banach space,” Proc. Amer. Math. Soc., vol.

59, 65–71, 1976.[3] Z. Opial, “ Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bull. Amer.

Math. Soc., vol. 73, 591–597, 1976.[4] T. Suzuki, “ Strongly convergence theorems for infnite families of nonexpansive mappings in general Banach spaces,”

Fixed Point Theory Appl., 103–123, 2005.[5] R.P. Agarwal, D. O’Regan and D.R. Sahu, “ Fixed Point Theory for Lipschitzian- Type Mappings with Applications,”

Springer, Heidelberg 2003.[6] W. Takahashi, “ Nonlinear Function Analysis,” Fixed point theory and its Applications, 93–106, 2000.[7] L. Leustean, “ Nonexpansive iteration in uniformly convex W-hyperbolic spaces,” In: A. Leizarowitz , BS. Mor-

dukhovich, I. Shafrir, A. Zaslavski (eds.) Contemp Math Am Math Soc AMS, vol. 513, 193–209, 2010.[8] S. Chang and et al., “ �-convergence theorems for multi-valued nonexpansive mappings in hyperbolic spaces,”

Applied Mathematics and Computation, vol. 249, 535–540, 2014.[9] T. Grana Bhaskar and V. Lakshmikantham, “ Fixed point theorems in partially ordered metric spaces and applica-

tions,” Nonlinear Anal., vol. 65, 1379–1393, 2006.[10] H.S. Ding and L. Li, “ Coupled fixed point theorems in partially ordered cone metric spaces,” Filomat, vol. 25, no. 2,

137–149, 2011.

The authors were supported by the Science Achievement Scholarship of Thailand.

78


Existence theorems for coincidence points of generalizedcontractive mappings in cone b-metric spaces

Natthaphon Artsawang⇤,1, Kasamsuk Ungchittrakool1,2,3

1Department of Mathematics, Faculty of Science,Naresuan University, Phitsanulok 65000, Thailand,





Abstract

In this paper, we establish some su�cient conditions and then prove some existence theoremsof coincidence points of generalized contractive mappings without the assumption of normality incone b-metric spaces. The results not only directly improve and generalize some fixed point resultsin metric spaces and b-metric spaces, but also expand and complement some previous results incone metric spaces.

Keywords: Cone b-metric spaces; Generalized contractive mapping; Coincidence point; Fixedpoint

References:[1] S. Banach “Sur les operations dans les ensembles abstrait et leur application aux equations, integrals,” Fundamenta

Mathematicae , vol.3 pp. 133-181, 1922.[2] L-G. Huang, X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” J. Math. Anal. Appl,

vol.332(2) pp. 1468-1476, 2007.[3] H. Huang , S. Xu , “Fixed point theorems of contractive mappings in cone b-metric spaces and applications,” Fixed

Point Theory and Applications,,vol.2013:112, 2013.[4] S. Jankovic, Z. Kadelburg, S. Radenovic, “On cone metric spaces: a survey,” Nonlinear Analysis, vol.4(7) pp.2591-

2601,2011.[5] S. Rezapour, R. Hamlbarani, “Some notes on the paper ’Cone metric spaces and fixed point theorems of contractive

mapping,” J. Math. Anal. Appl, vol.345 pp. 512-516, 2008.

The authors were supported by The Thailand Research Fund (TRF).

79


Coupled fixed point theorems in C⇤-algebra-valued metric spaces

Prondanai Kaskasem⇤,1, Chakkrid Klin-eam1,2

1Department of Mathematics, Faculty of Science, Naresuan UniversityPhitsanulok, 65000, Thailand




Abstract

Let L(H) be the set of bounded linear operators on Hilbert space H. We consider a class ofoperator equations of type

X = Q +mX

i=1

A⇤i XAi �

mX

i=1

B⇤i XBi

where Q is a positive operator and Ai,Bi are arbitary operators in L(H). In this paper, we provethe existence and uniqueness of coupled fixed point for a mixed monotone mapping in C⇤-algebra-valued metric space. As an application, we prove the existence of solution to such equations.

Keywords: C⇤-algebra; C⇤-algebra-valued metric space; coupled fixed point theorem

References:[1] M. S. Asgari and B. Mousavi, Solving a class of nonlinear matrix equations via the coupled fixed point theorem,

Applied Mathematics and Computation 259 364-373 (2015)[2] S. Batul and T. Kamran, C⇤-valued contractive type mappimgs, Fixed Point Theory and Applications, 2015.[3] V. Berinde, Generalized coupled fixed point theorems for mixed monotone mapping in partailly ordered metric

spaces. Nonlinear Anal. 74 7347-7355 (2011)[4] M. Berzig, Solving a class of matrix equations via the Bhaskar-Lakshmikantham coupled fixed point theorem, Applied

Mathematics Letters 25 1638-1643 (2012)[5] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications,

Nonlinear Analysis, 65 1379-1393 (2006)[6] M. Bota, A. Petrusel, G. Petrusel and B. Samet, Coupled fixed point theorems for single-valued operators in b-metric

spaces, Fixed Point Theory and Applications (2015)[7] R.G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, (1972)[8] K. R. Davidson, C⇤-Algebras by Example. Fields Institute Monographs, vol.6, (1996)[9] D. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear anal 11

623-632 (1987)


80


Existence results for new extended vector variational-likeinequality and equilibrium problems

Boonyarit Ngeonkam⇤,1, Kasamsuk Ungchittrakool1,2,3

1Department of Mathematics, Faculty of Science,Naresuan University, Phitsanulok 65000, Thailand





Abstract

In this paper, we establish and study some new existence theorems for a new extended vectorvariational-like inequality and equilibrium problem in Banach space. The results are proved byusing the new definition of g � f � ⌘ � � � µ�quasimonotone of Stampacchia and of Minty typemappings. The obtained results in this article can be viewed as some new and generalized formswhich can be applied to several problems.

Keywords: new extended vector variational-like inequality and equilibrium problems; Existenceresults; KKM-mapping; g � f � ⌘ � � � µ� quasimonotone of Stampacchia type; g � f � ⌘ � � �µ�quasimonotone of Minty type mapping; g � f � ⌘ � � � µ� pseudomonotone

References:[1] S. S. Irfan and R. Ahmadi, “Existence results for extended vector variational-like inequality,” Journal of the Egyptian

Mathematical Society, vol. 23, pp. 144-148, 2015.[2] Y. Zhao and Z. Xia, “Existence results for system of variational-like inequalities,” Nonlinear Anal. Real World Appl,

vol. 8, pp. 1370-1378, 2007.[3] A.P. Farajzadeh, and B.S. Lee, “Vector variational-like inequality problem and vector optimization problem,” Appl.

Math, vol. 23, pp. 48-52, 2010.[4] R. Ahmad, “Existence results for vector variational-like inequalities,” Thai J. Math, vol. 9(3), pp. 553-561, 2011.

The authors would like to thank Naresuan University

81


A new two-step fixed points iterative scheme for twoasymptotically nonexpansive mappings

N. Piwma⇤,1, T. Thianwan1

1Department of Mathematics, School of Science,University of Phayao, Phayao 56000, Thailand



Abstract

In this paper, a new two-step iteration scheme for approximating common fixed points oftwo asymptotically nonexpansive mappings is defined and we have proved weak and strongconvergence theorems in a uniformly convex Banach space. The result presented in this paperimprove and extend the recent ones announced by many others

Keywords: Asymptotically nonexpansive; Strong convergence; Weak convergence

References:[1] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc.

35 (1972), 171-174.[2] S.H. Khan, Y.J. Cho and M. Abbas, Convergence to common fixed points by a modified iteration process, J. Appl.

Math. and Comput.doi:10.1007/s12190-010-0381.[3] W. Nisrakoo and S. Saejung, A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings,

Comput. Math. Appl.18 (2006), 1026-1034.[4] Y.J. Cho, H.Y. Zhou and G. Guo, Weak and strong convergence theorems for three-step iterations with errors for

asymptotically nonexpansive mapping, Comput. Math. Appl., 47 (2004), 707-717.[5] Z. Opial, Weak and convergence of successive approximations for nonexpansive mapping, Bull. Amer. Math. Soc.

73 (1967), 591-597.

82


A Halpern Iteration for System of Equilibrium and VariationalInequality and Fixed Point Problems of Families of Quasi -�-

Asymptotically Nonexpansive in Banach Spaces

Pongrus Phuangphoo⇤,1, Poom Kumam2

1Department of Mathematics, Faculty of Education,Bansomdejchaopraya Rajabhat University (BSRU),

1061 Issaraphap Rd., Hiran Ruchi, Thon Buri, Bangkok 10600, ThailandEmail: [email protected]





Abstract

In this paper, we introduce a new iterative sequence by using a halpern algorithm for findingthe common solution of a system of equilibrium problems for a finite family of bifunctions satisfyingthe conditions and the fixed point problems for families of quasi -�- asymptotically nonexpansivemapping and the variational inequality problems for a finite family of monotone mapping. Finally,we prove some strong convergence theorem of an iteration generated by the some mild conditionsin Banach spaces.

Keywords: Halpern Iteration; Equilibrium Problem; Variational Inequality Problem; Fixed PointProblem

References:[1] A. Moudafi, “A partial complement method for approximating solutions for a primal dual fixed point problem,"

Optimization Letters, Vol. 4, Issue 3, pp. 449–456, 2010.[2] P. M. Pardalos, T. M. Rassias and A. A. Khan, “Nonlinear analysis and variational problems," Springer Optimization

and Its Applications, Vol. 35, 2010.[3] H. Zegeye and N. Shahzad, “A hybrid scheme for finite families of equilibrium, variational inequality and fixed point

problems," Nonlinear Analysis, Vol. 74, Issue 1, pp. 263–272, 2011.[4] Y. Shehu, “A new iterative scheme for a countable family of relatively nonexpansive mappings and an equilibrium

problem in Banach spaces," Journal of Global Optimization, Vol. 54, Issue 3, pp. 519–535, 2012.[5] H. Zegeye and N. Shahzad, “Strong convergence theorems for a solution of finite families of equilibrium and

variational inequality problems," Optimization, Vol. 63, Issue 2, pp. 1–17, 2011.

The first author was supported by the Bansomdejchaopraya Rajabhat University.

83


The Prediction of Drought Using Correlation betweenTemperature and Rainfall Case study: The Prediction Based on

Fuzzy Clustering and Grey Theory in The North East of Thailand.

Chalermchai Puripat1, Adisak Pongpullponsak2, Sukuman Sarikavanij3


Email: [email protected] of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000,

ThailandEmail: [email protected]

3Research center for Academic Excellence in Mathematics, Naresuan UniversityEmail: [email protected]


Abstract

In Thailand, after two severe consecutive winters, drought usually occurs as a result of watershortage caused by less-than-usual amount of rainfall during the rainy season. Unusually hightemperature can also cause the following yearâAZs drought, which seriously a↵ects agriculturalproductivity negatively [1]. Both high temperature and limited rainfall in the North East ofThailand can be used to predict the severity of ThailandâAZs drought nation-wide. In this study,the grey relational decision-making method was applied to calculate the degree between theweighted temperature and rainfall, while the amount of rainfall represented the main factor of thedecision targets and the comparison of fuzzy clustering [2]and grey system model (GM)[3] wasapplied to predict the final drought analysis.

Keywords: grey relational; fuzzy clustering; prediction drought; GM(grey system model)

References:[1] Yaoxian,L & Yufu,Z., 1987. The application of Grey System theory to long-range prediction. Acta Meteorologica

Sinica, 4,489-94.[2] Julong,D.,1985, Relational space of Grey Systems. fuzzy Mathematics, (Special Issue of Grey Systems)2, 1-10 (in

Chinese).[3] Liu Si-Feng, Yi Lin .,2010, Grey Systems Theory and Applications .Springer :complexity (in Chinese).

84


Painlevé-Kuratowski variational inclusion problems

Panatda Boonman⇤,1, Rabian Wangkeeree1,2

1Department of Mathematics, Faculty of Science,Naresuan University, 99 Moo. 9,

Tha Pho, Muang, Phisanulok 65000, ThailandEmail: [email protected]

2 Research center for Academic Excellence in Mathematics,Naresuan University



Abstract

In this paper, we aim to establish some results for the solution set of a variational inclusionproblems with set-valued mapping and we study Painlevé-Kuratowski convergence of the solutionsets with a sequence of mappings converging continuously and sequence of set converging in thesense of Painlevé-Kuratowski.

Keywords: Painlevé-Kuratowski convergence; variational inclusion problems; set-valued map-ping

References:[1] K. Kuratowski, Topology, vols. 1, 2. Academic Press, New York, NY (1968).[2] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student

63. 123-145 (1994).[3] X.B. Li, Z. Lin, Q.L. Wang, Stability of approximate solution mappings for generalized Ky Fan inequality, Springer.

DOI 10.1007/s11750-015-0385-9 (2015).


85


Weak Pareto-optimality for multiobjective optimizationinvolving tangentially convex functions

Rabian Wangkeeree and Nithirat Sisarat⇤

1Department of Mathematics, Faculty of Science,Naresuan University, 99 Moo. 9,

Tha Pho, Muang, Phisanulok 65000, Thailand⇤Presenting author.


Abstract

We consider the multiobjective optimization problem (MOP) over a feasible set which is de-scribed by inequality constraints that are tangentially convex. In this paper we present necessaryand su�cient conditions for the weakly Pareto optimal solutions of (MOP) in terms of tangentialsubdi↵erentials.

Keywords:

References:[1] K. Kuratowski, Topology, vols. 1, 2. Academic Press, New York, NY (1968).[2] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student

63. 123-145 (1994).[3] X.B. Li, Z. Lin, Q.L. Wang, Stability of approximate solution mappings for generalized Ky Fan inequality, Springer.

DOI 10.1007/s11750-015-0385-9 (2015).

86


Sequential optimality conditions for fractional convexoptimization problems

Thanatchaporn Sirichunwijit⇤,1, Rabian Wangkeeree 2


Email: [email protected] of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000,

ThailandEmail: [email protected]

⇤ Presenting author.Email: [email protected]

Abstract

In this paper, using a general approach which provides sequential optimality conditions fora general convex optimization problem and is given in terms of the ✏-subdi↵erentials, we derivenecessary and su�cient optimality conditions for fractional convex optimization problems is ob-tained in terms of the ✏-subdi↵erentials of the functions involved at the minimizer and sequentialcharacterization of optimal solution involving the convex subdi↵erential.

Keywords: Convex programming; Conjugate function; Perturbation theory; Sequential optimalityconditions

References:[1] R.I. Bot, E.R. Csetnek and G. Wanka “Sequential optimality conditions for composed convex optimization problems,”

J. Math. Anal. Appl. , vol. 342, pp. 1015 – 1025, 2008.[2] X.K. Sun, X.J. Long and Y. Chai, “Sequential Optimality Conditions for Fractional Optimization with Applications to

Vector Optimization,” J Optim Theory Appl, vol. 164, pp:479 – 499, 2015.[3] R.I. Bot,E.R. Csetnek, and G. Wanka, “Sequential optimality conditions in convex programming via perturbation

approach,” J. Math. Anal. Appl., vol. 15, no. 1, pp. 149 – 164, 2008,


87


Variational Inequalities for L-fuzzy Mappings.

Nayyar Mehmood1, Akbar Akbar2

11. Department of Mathematics and Statistics, International Islamic University, Islamabad,Pakistan.

22. Department of Mathematics, COMSATS Institute of Information Technology, Islamabad,Pakistan

⇤Presenting author.Email:

Abstract

In this article we introduce the notion of variation inequalities for L-fuzzy mappings. Wepropose some iterative algorithms by virtue of projection methods, and study the convergencecriteria for these algorithms. Our results extend/generalize some eminent results already presentin the literature.

Keywords: Variational Inequalities; L-fuzzy Mappings

References:[1] Yaoxian,L & Yufu,Z., 1987. The application of Grey System theory to long-range prediction. Acta Meteorologica

Sinica, 4,489-94.[2] Julong,D.,1985, Relational space of Grey Systems. fuzzy Mathematics, (Special Issue of Grey Systems)2, 1-10 (in

Chinese).[3] Liu Si-Feng, Yi Lin .,2010, Grey Systems Theory and Applications .Springer :complexity (in Chinese).

88


SOME QUADRUPLED BEST PROXIMITY POINT THEOREMSIN PARTIALLY ORDERED METRIC SPACES

YUMNAM ROHENDepartment of Basic Sciences and Humanities (Mathematics) National Institute of Technology

Manipur), Langol, pin-795004, Manipur, India.Email: [email protected]


Abstract

In this paper, we prove some quadrupled best proximity point theorems in partially orderedmetric space by using ( ,�) contraction. Our result generalises the result of Kumam et.al. (Cou-pled best proximity points in ordered metric spaces, Fixed point Theory and Applications 2014,2014:107.). An example is also given to verify the results obtained.

Keywords: partial ordered set, best proximity point, quadrupled fixed point, quadrupled bestproximity point.

References:[1] Fan K., Extensions of two fixed point theorems of F. E. Browder, Math. Z. 122 (1969), 234-240.[2] H. Aydi, E. Karapinar, P. Salimi, I. M. Erhan, Best proximity points of generalized almost -Geraghty contractive

non-self mappings, Fixed Point Theory Appl. 2014 (2014), Article ID 34.[3] S. S. Basha, P. Veeramani, Best approximations and best proximity pairs, Acta Sci. Math.(Szeged), 63 (1997), 289-300.[4] G. Beer, D. V. Pai, Proximal maps, prox maps and coincidence points, Number. Funct. Anal. Optim. 11(1990),

429-448.[5] G. Beer, D. V. Pai, The prox map, J. Math. Anal. Appl. 156 (1991), 428-443.[6] A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006),

1001-1006.[7] M. Jleli, B. Samet, Remarks on the paper: Best proximity point theorems: an exploration of a common solution to

approximation and optimization problems, Appl. Math. Comput. 228 (2014), 366-370.[8] P. Kumam, H. Aydi, E. Karapinar, W. Sintunavarat, Best proximity points and extension of Mizoguchi- Takahashis

fixed point theorems, Fixed Point Theory Appl. 2013 (2013), Article ID 242.[9] W. A. Kirk, S. Reich, P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal.

Optim. 24 (2003), 851-862.[10] W. K. Kim, K. H. Lee, Existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl. 316 (2006),

433-446.[11] Z. Kadelburg, S. RadenoviÃCËGtc, A note on some recent best proximity point results for non-self mappings, Gulf

J. Math. 1 (2013), 36-41.[12] E. Karapinar, W. Sintunavarat, The existence of an optimal approximate solution theorems for generalized ↵-proximal

contraction nonself mappings and applications, Fixed Point Theory Appl. 2013 (2013), Article ID 323.[13] S. Reich, Approximate selections, best approximations, fixed points and invariant sets, J. Math. Anal. 62 (1978),

104-113.

89


Attractive points, acute point and fixed point properties fornonlinear mappings

Sachiko Atsushiba⇤

Department of Science Education,Graduate School of Education Science of Teaching and Learning,

University of Yamanashi4-4-37, Takeda Kofu, Yamanashi 400-8510, Japan


Abstract

In this talk, we study the concepts of acute points of a nonlinear mapping. Further, weintroduce the concepts of common acute points of a family of nonlinear mappings. We study fixedpoint properties for nonlinear mappings. We also study some properties of acute points, attractivepoints and fixed points. Further, we prove some convergence theorems for nonlinear mappings.

Keywords: Fixed point, attractive point, acute point, iteration, weak convergence, strong conver-gence

The author is supported by Grant-in-Aid for Scienti?c Research No. 26400196 from Japan Society for the Promotionof Science.

90


Browder’s Convergence Theorem in CAT(0) Spaces Endowedwith Graph

Buris Tongnoi⇤,1, Watcharapong Anakkamatee1

1Department of Mathematics, Faculty of Science,Naresuan University,

99 Moo 9, Thapho, Muang, Phitsanulok 65000, Thailand⇤Presenting author.

Email: [email protected] (B. Tongnoi), [email protected] (W. Anakkamatee)

Abstract

In this talk, we discuss about CAT(0) spaces endowed with graph. The Browder’s convergencetheorem for G-nonexpansive mappings will be presented. This results extend and generalize theresult of Tiammee, Kaewkhao and Suantai (2015).

Keywords: CAT(0) space; directed graph; nonexpansive mapping; Browder’s convergence theo-rem

References:[1] J. Tiammee, A. Kaewkhao and S. Suantai, “On Browder’s convergence theorem and Halpern iteration process for

G-nonexpansive mappings in Hilbert spaces endowed with graphs,” Fixed Point Theory and Applications (2015),2015:187.

[2] Jachymski, J: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136(4),1359-1373 (2008)

[3] Browder, FE: Convergence of approximants to fixed points of non-expansive maps in Banach spaces. Arch. Ration.Mech. Anal. 24, 82-90 (1967)

91


TRIPLED PBVPS OF NONLINEAR SECOND ORDERDIFFERENTIAL EQUATIONS

Animesh Gupta1, Saurabh Manro2

1H.No. 93/654, Ward No.-2 Gandhi Chowk Pachmarhi,Dist. Hoshangabad- 461881, Madhya Pradesh - India,

2B.No. 33, H.No. 223, Peer Khana Road, Near Tiwari Di Kothi,Khanna, Dist. Ludhiana- 141401, Punjab - India


Abstract

The present paper proposes a new monotone iteration principle for the existence as well asapproximations of the tripled solutions for a tripled periodic boundary value problem of secondorder ordinary nonlinear di↵erential equations. An algorithm for the tripled solutions is developedand it is shown that the sequences of successive approximations defined in a certain way convergemonotonically to the tripled solutions of the related di↵erential equations under some suitablehybrid conditions. A numerical example is also indicated to illustrate the abstract theory developedin the paper. We claim that the method as well as the results of this paper are new to literature onnonlinear analysis and applications.

Keywords:

92


Extensions of almost-F and F-Suzuki contractions with graph andsome applications to fractional calculus

Lokesh Budhia⇤,1, Poom Kumam2,3, D. Gopal1

1Department of Applied Mathematics and Humanities,SV National Institute of Technology, Surat, Gujarat (India)

Email: gopal.dhananjay@redi↵mail.com1, [email protected]

2 Theoretical and Computational Science Center (TaCS) &Department of Mathematics, Faculty of Science,

King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand3 China Medical University, No. 91, Hsueh-Shih Road, Taichung, Taiwan


Abstract

In this paper, we introduce the two new concepts of an ↵-type almost-F-contraction and an↵-type F Suzuki contraction and prove some fixed point theorems for such mappings in a completemetric space. Some examples and an application to a nonlinear fractional di↵erential equation aregiven to illustrate the usability of the new theory.

Keywords: fixed point; almost-F-contraction; F-Suzuki contraction; nonlinear fractional di↵eren-tial equation

References:[1] Samet, B, Vetro, C, Vetro, P: Fixed point theorems for ↵� -contractive type mappings. Nonlinear Anal. 75, 2154-2165

(2012)[2] Wardowski, D: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory

Appl. 2012, 94 (2012). doi:10.1186/1687-1812-2012-94[3] Sgroi, M, Vetro, C: Multi-valued F-contractions and the solution of certain functional and integral equations. Filomat

27, 1259-1268 (2013)[4] Karapinar, E, Samet, B: Generalized (↵ � ) contractive type mappings and related fixed point theorems with

applications. Abstr. Appl. Anal. 2012, Article ID 793486 (2012)[5] Mohammadi, B, Rezapour, S, Shahzad, N: Some results on fixed points of ↵ � -Ciric generalized multifunctions.

Fixed Point Theory Appl. 2013, 24 (2013). doi:10.1186/1687-1812-2013-24[6] Salimi, P, Latif, A, Hussain, N: Modified (↵ � )-contractive mappings with applications. Fixed Point Theory Appl.

2013, 151 (2013). doi:10.1186/1687-1812-2013-151[7] Batra, R, Vashistha, S: Fixed points of an F-contraction on metric spaces with a graph. Int. J. Comput. Math. 91(12),

2483 (2014). doi:10.1080/00207160.2014.887700

93


Common fixed point theorem for multi-valued weak contractivemappings in metric spaces with graphs

Phikul Sridarat⇤,1, Suthep Suantai1,2

1Department of Mathematics, Faculty of Science,Chiang Mai University (CMU),

239 Huay Kaew Road, Muang District, Chiang Mai 50200, ThailandEmail: [email protected]

2 Email: [email protected]


Abstract

In this research, a new type of weak contractive multi-valued mappings in a complete metricspace with a directed graph is introduced. A common fixed point theorem of those two multi-valued mappings is established under some appropriate conditions. Moreover, some exampleillustrating our main result is also given. The obtained result extend and generalize several fixedpoint results of multi-valued mappings in the literature.

Keywords: fixed point; directed graph; multi-valued mappings; graph weak contractive map-pings

References:[1] S.B. Nadler, “Multivalued contraction mappings,” Pacific Journal of Mathematics, vol. 30, no. 2, pp. 475–488, 1969.[2] N. Mizoguchi and W. Takahashi, “Fixed point theorems for multivalued mappings on complete metric spaces,”

Journal of Mathematical Analysis and Applications, vol. 141, pp. 177 – 188, 1989.[3] D. Klim and D. Wardowski, “Fixed point theorems for multi-valued contractions in complete metric space,” Journal

of Mathematical Analysis and Applications, vol. 334, pp. 132 – 139, 2007.[4] M. Berinde and V. Berinde, “On a general class of multi-valued weakly Picard mappings,” Journal of Mathematical

Analysis and Applications, vol. 326, pp. 772 – 782, 2007.[5] J. Tiammee and S. Suantai, “Coincidence point theorems for graph-preserving multi-valued mappings,” Fixed Point

Theory and Applications, vol. 70, 2014.[6] A. Hanjing and S. Suantai, “Coincidence point and fixed point theorems for a new type of G-contraction multivalued

mappings on a metric space endowed with a graph,” Fixed Point Theory and Applications, vol. 171, 2015.[7] I. Beg and A.R. Butt, “Fixed point of set-valued graph contractive mappings,” Journal of Inequalities and Applications,

vol. 252, 2013.[8] J. Jachymski, “The contraction principle for mappings on a metric space with a graph,” Proceedings of the American

Mathematical Society, vol. 136, no. 4, pp. 1359–1373, 2008.[9] N.A. Assad and W.A. Kirk, “Fixed point theorems for setvalued mappings of contractive type,” Pacific Journal of

Mathematics, vol. 43, pp. 533–562, 1972.

The authors were supported by the Thailand Research Fund under the project RTA 5780007 and Chiang Mai University.

94


A note on continuity of solution set for vector equilibriumproblems

Pakkapon Preechasilp⇤,1, Rabian Wangkeeree2

1Program in Mathematics, Faculty of Education,Pibulsongkram Rajabhat University, Phitsanulok 65000, Thailand

Email: [email protected] of Mathematics, Faculty of Science,

Naresuan University, Phitsanulok 65000, ThailandEmail: [email protected]


Abstract

In this note, we consider the parametric weak vector equilibrium problem. By using a weakerassumption in Peng and Chang (2014), the su�cient conditions for continuity of the solutionmappings to a parametric weak vector equilibrium problem are established. Examples are providedto illustrate the essentialness of imposed assumptions. As an advantages of the results, we derivethe continuity of solution mappings for vector optimization problems.

Keywords: equilibrium problem; solution mapping; continuity; linear scalarization


Student, vol. 63, no. 1-4, pp. 123–145, 1994.[2] Z. Y. Peng, S. S. Chang, ”On the lower semicontinuity of the set of e�cient solutions to parametric generalized

systems,” Optimization Letters, vol. 8, pp. 159–169, 2014.

95


Barrier method for convex optimization problem withoutregularity of constraint functions

Porntip Promsinchai⇤,1, Narin Petrot1


Phitsanulok, 65000, ThailandEmail: [email protected]


Abstract

We consider the convex optimization problem when the objective function may not smooth andthe constraint set is represented by constraint functions that are locally Lipschitz and directionallydi↵erentiable, but neither necessarily concave nor continuously di↵erentiable. The obtained resultsimprove and extend those results that have been presented in [Dutta, J., Lalitha, C.S.: Optimalityconditions in convex optimization revisited. Optim. Lett. 7(2),221-229 (2013)], and [Dutta, J.:Barrier method in nonsmooth convex optimization without convex representation. Optim. Lett.9(6), 1177-1185 (2015)], by removing the regularity and continuously di↵erentiable assumptionson the constraint functions from the considering.

Keywords: Convex optimization; log-barrier function; locally Lipschitz function; directionallydi↵erentiable; Clarke derivative; regular function.

References:[1] Bagirov, A., Karmitsa, N., Makela, M. M., Introduction to Nonsmooth Optimization: Theory, Practice and Software,

Springer Verlag (2014).[2] Bertsekas, D., Nedic, A., Ozdaglar, E.: Convex Analysis and Optimization. Athena Scientific, Belmont, Massachusetts,

2003[3] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York, 1983[4] Dutta, J., Lalitha, C.S.: Optimality conditions in convex optimization revisited. Optim. Lett. 7(2),221-229 (2013)[5] Dutta, J.: Barrier method in nonsmooth convex optimization without convex representation. Optim. Lett. 9(6),1177-

1185 (2015)[6] Hiriart-Urruty, J.-B.: Miscellanies on nonsmooth analysis and optimization, in nondi↵erentiable optimization: moti-

vation and applications. workshop in Sopron, 1984. In: Demyanov, V.F., Pallaschke, D. (eds.) Lecture Notes inEconomics and Mathematical Systems, vol.255, pp. 8ÂU24. Springer, Berlin (1985)

[7] Lasserre, J.B.: On representations of the feasible set in convex optimization. Optim. Lett. 4(1), 1-5 (2010)[8] Lasserre, J.B.: On convex optimization without convex representation. Optim. Lett. 5(4), 549-556 (2011)[9] Rockafellar, R.T.: Convex Analysis. Princeton, New Jersey (1970)

96


Fixed point theorems of a new set-valued MT-contraction inb-metric spaces endowed with graphs

Jukrapong Tiammee⇤,1, Suthep Suantai1 and Yeol Je Cho2

1Department of Mathematics, Faculty of Science,Chiang Mai University, Chiang Mai, Thailand

Email: [email protected] (J. Tiammee)[email protected] (S. Suantai)

2Department of Education and the RINS,Gyeongsang National University, Jinju 660-701, Korea,



Abstract

In this work, we introduce a new concept of Set-Valued Mizoguchi-Takahashi G-contractionsand prove some fixed point theorems for such mappings in b- metric spaces endowed with directedgraphs. Our results improve and extend those of Mizoguchi, Takahashi [7] and Sultana, Vetrivel[19]. We also give some examples supporting our main results. As an applications, we prove theexistence of fixed points for set-valued mappings satisfying generalized MT-contractive conditionin ✏-chainable b-metric spaces.

Keywords: fixed point; Mizoguchi-Takahashi function; b-metric spaces; directed graph; set-valuedmap; ✏-chainable metric space.

References:[5] S. Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math. 3

(1922), 133–181.[6] M. Edelstein, An extension of Banach’s contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7–10.[7] S. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math. 20(1969), 475–488.[8] M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math Anal. Appl. 326(2007),

772–782.[9] V. Berinde, M. Pacurar, The role of Pompeiu-Hausdor↵metric in fixed point theory, Creat. Math. Inform. 22, 143–150.

[10] S. Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital. 5(1972), 26–42.[11] N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math.

Anal. Appl. 141(1989), 177–188.[12] M. Javahernia, A. Razani, F. Khojasteh, Common fixed point of generalized Mizoguchi-Takahashi’s type contractions,

Fixed Point Theory Appl. 2014, 2014:195.[13] I. A. Bakhtin, The contraction mapping principle in quasi-metric spaces, Functional Anal. 30(1989), 26–37 (Russian).

The authors were supported by the Thailand Research Fund under the project RTA5780007 and Chiang Mai University,Chiang Mai, Thailand.

97


Fixed point theorems for generalized fuzzy contractive mappingswith altering distance in fuzzy metric spaces

Supak Phiangsungnoen⇤

Department of Mathematics, Faculty of Liberal Arts,Rajamangala University of Technology Rattanakosin (RMUTR),

264 Chakkrawat Rd., Chakkrawat, Samphanthawong, Bangkok 10100, Thailand.Email: [email protected], [email protected]


Abstract

The aim of this work is to introduce and prove the existence and uniqueness of fixed point forgeneralized fuzzy (↵, �,')-contractive mappings in complete fuzzy metric spaces. The research isillustrated by example.

Keywords: ↵-admissible; fixed point; fuzzy metric spaces, generalized fuzzy contractive mapping.

References:[1] A. George, P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 64, pp. 395–399, 1994.[2] B. Samet, C. Vetro, P. Vetro, “Fixed point theorems for ↵- -contractive type mappings,” Nonlinear Analysisvol. 75, pp.

2154–2165, 2012.[3] T. Došenovic, D. Rakic, M. Brdara, “Fixed Point Theorem in Fuzzy Metric Spaces Using Altering Distance,” Filomat

vol. 28, no. 7, pp. 1517âAS-1524, 2014.[4] P. Salimi, C. Vetrob, P. Vetro, “Some new fixed point results in non-Archimedean fuzzy metric spaces,” Nonlinear

Analysis: Modelling and Control, vol. 18 , no. 3, pp. 344–358, 2013.

The author was supported by Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin Research.

98


Flight re-timing models to improve the robustness of aircraftroutings

Khusnul Novianingsih⇤,1,2, Rieske Hadianti1

1Department of Mathematics,Institut Teknologi Bandung,Jalan Ganesha No. 10 Bandung, Jawa Barat 40132, Indonesia

2Department of Mathematics,Universitas Pendidikan Indonesia,Jalan Dr. Setiabudhi No. 229 Bandung, Jawa Barat 40154, Indonesia


Abstract

We study a problem on improving the robustness of aircraft routings. An aircraft routingis robust if the routings can minimize the e↵ect of flight delays in day-to-day operations. Weimprove the robustness of old aircraft routings by re-timing departure time of flights. We deriveoptimization models to change departure time of flights while the feasibility of both aircraft andcrew connections are maintained. We define several alternative objective functions to obtain thebest distribution of optimal slacks that should be allocated to the connections. We construct asimulation for evaluating the robustness of the new routings. The computational results show thatthe re-timing flights can improve the robustness of aircraft routings significantly.

Keywords: robust aircraft routing; slack; delay; re-timing flight

References:[1] M. A. Aloulou, M. Haouari, and F. Z. Mansour, “Robust aircraft Routing and Flight Retiming,” Electronic Note in

Discrete Mathematics, vol. 36, pp. 367–374, 2010.[2] M. A. Aloulou, M. Haouari, and F. Z. Mansour, “A model for enhanching robustness of aircraft and passenger,”

Transportation Research Part C, vol. 32, pp. 48–60, 2013.[3] M. Dumbar, Froylandand, and C-. Wu, “Robust airline schedule planning: Minimizing propagated delay in an

integrated routing and crewing framework,” Transportation Science, vol. 46, no. 2, pp. 204–216, 2012.[4] S. Lan, J.-P. Carke, and C. Bernhart, “Planning for robust airline operation: Optimizing aircraft routings and flight

departure time to minimize passenger disruptions,” Transportation Science, vol. 40, pp. 15–28, 2006.

99


Fixed Point Theorems for Prešic almost contraction mappings inOrbitally Complete Metric Spaces Endowed with Directed

Graphs

Porpimon Boriwan⇤,1, Narin Petrot2, Suthep Suantai3


Phitsanulok 65000, ThailandEmail: [email protected]


Phitsanulok 65000, ThailandEmail: [email protected]

3Department of Mathematics, Faculty of Science,Chiang Mai University

Chiang Mai, 50200, ThailandEmail: [email protected]


Abstract

The main aim of this work is to introduce a class of generalized contractions in product spacesin the sense of Prešic. Some examples and fixed point theorems for such introduced mappings inthe setting of orbitally complete metric spaces are discussed and provided.

Keywords: Prešic operator; almost contraction mapping; orbitally complete metric space; directedgraph; fixed point

References:[1] V. Berinde, “ Approximation fixed points of weak contractions using the Picard iteration,” Nonlinear Anal Forum, 9

(2004), 43-53[2] V. Berinde, “ Some remarks on a fixed point theorem for Ciric-type almost contractions,” Carpath. J. Math. 25 (2009),

62-157[3] J. Jachymski, I. Jozwik, “Nonlinear contractive conditions: comparison and related problems,” Banach Cent. Publ. 77

(2007) 123-146.[4] M. Pacurar “ Fixed points of almost Prešicc operators by a k-step iterative method,” An. Stiint. Univ. Al. l. Cuza lasi.

Mat. (N.S.) 57 (2011), suppl. 1, 199-210[5] S. Shukla, N. Shahzad, “ G-Prešic operators on metric spaces endowed with a graph and fixed point theorems,” Fixed

Point Theory and Appl. 2014, 127 (2014)

The authors has been supported by Graduate School, Naresuan University and the Thailand Research Fund underthe project RTA5780007.

100


Existence and convergence of fixed points for a strictpseudo-contraction in CAT(0) spaces

Narongrit Puturong⇤,1, Kasamsuk Ungchittrakool2,3

1Mathematics Program, Faculty of Science,Udon Thani Rajabhat University, Udon Thani 41000, Thailand



Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Email:[email protected]


Abstract

The purposes of this paper are to study some existence and convergence theorems of fixedpoints for a strict pseudo-contraction by using an iterative projection technique with some suitableconditions. The method permits us to obtain a strong convergence iteration for finding some fixedpoints of a strict pseudo-contraction in the framework of complete CAT(0) spaces.

Keywords: Strict pseudo-contraction; Iterative projection technique; CAT(0) spaces.

References:[1] I.D. Berg and I.G. Nikolaev,“Quasilinearization and curvature of Alexandrov spaces,” Geom.Dedicata, vol. 133, pp.

195âAS218, 2008.[2] F.E. Browder and W.V. Petryshyn,“Construction of fixed points of nonlinear mappings in Hilbert spaces,” J.Math.

Anal. Appl, vol. 20, pp. 197 âAS 228, 1967.[3] H. Dehghan and J. Rooin,“A characterization of metric projection in CAT(0) spaces,” International Conference on Func-

tional Equation, Geometric Functions and Applications (ICFGA 2012) 10-12th. May 2012, Payame Noor University,Tabriz, Iran, pp. 41âAS43.

[4] H. Lu, D. Lan, Q. Hu and G. Yuan, “Fixed point theorems in CAT(0) spaces with applications,” Journal of Inequalitiesand Applications, vol. 320, pp. 1âAS26, 2014.

[5] W.A. Kirk and B. Panyanak, “A concept of convergence in geodesic spaces,” Nonlinear Anal., vol. 68, pp. 3689 âAS3696, 2008

[6] K. Ungchittrakool, “Existence and convergence of fixed points for a strict pseudo-contraction via an iterative shrinkingprojection technique,” J.Nonlinear Convex Anal., vol. 15, no. 4, pp. 693 âAS 710, 2014.

101


On Locating-Chromatic Number of a Complete n-ary Tree ofDepth 1, 2, and 3

Des Welyyanti, Edy Tri Baskoro, Rinovia Simanjuntak, Saladin UttunggadewaCombinatorial Mathematics Research DivisionFaculty of Mathematics and Natural Sciences

Institut Teknologi BandungJl. Ganesa 10 Bandung 40132, IndonesiaEmail: [email protected]

{ebaskoro, rino, s_uttunggadewa}@math.itb.ac.id⇤Presenting author.


Abstract

The concept of locating-chromatic number of a graph was introduced by Chartrand, Erwin,Henning, Slater and Zhang in 2002. The locating-chromatic number of a graph is a special case ofthe graph partition dimension. The partition dimension of graph was introduced by Chartrandal. in 1998. The locating-chromatic number of trees was firstly studied by Chartrand et al. in2002. Chartrand et al. determined the locating-chromatic of paths and double stars. Furthermore,Chartrand et al. also studied that for any integer, there exist a tree on vertices with the locating-chromatic number. Asmiati et al. determined the locating-chromatic number of firecrackers andamalgamation of stars.

Let c be a k-coloring of graph G(V,E) andQ= {C1,C2, · · · ,Ck} be the partition of V(G) induced

by c, where Ci is the set of all vertices receiving color i. The color code cQ(v) of a vertex v 2 V(G) is theordered k-tuple (d(v,C1), d(v,C2), · · · , d(v,Ck)), where d(v,Ci) = min{d(v, x)|x 2 Ci} and d(v,Ci) < 1for all i 2 [1, k]. If all vertices of H have distinct color codes, then c is called a locating-coloringof G. The locating-chromatic number of G, denoted by �L(G), is the smallest k such that G admitsa locating-coloring with k colors. Let T(n, k) be a complete n-ary tree, namely a rooted tree withdepth k, which each vertex has n children, except for its the leaves. In this paper, we determine thelocating-chromatic number of complete n-ary tree T(n, k) with k = 1, 2, and 3.

Keywords: color code, locating-chromatic number, complete n-ary tree

References:[1] Asmiati, E.T. Baskoro, H. Assiyatun, D. Suprijanto, R. Simanjuntak, and S. Uttunggadewa, “Locating-chromatic

number of firecracker graphs, Far East J. Math. Sci., vol. 63, no. 1, pp. 11-13, 2012.[2] Asmiati, H. Assiyatun and E.T Baskoro, “Locating-chromatic of amalgamation of stars, ITB J.Sci., vol. 43A, no. 1, pp.

1-8, 2011.[3] G. Chartrand, P. Zhang, and E. Salehi, “On the partition dimension of graph, Congr. Numer., vol. 130, pp. 157-168,

1998.

102


An iterative process for a hybrid pair of generalizedI-asymptotically nonexpansive single-valued mapping and

generalized nonexpansive multi-valued mappings in Banachspaces

P. Chuasuk ⇤,1, A. Kaewcharoen1

1Department of Mathematics, Faculty of Science,Naresuan University,Phitsanulok 65000, Thailand,

Email:[email protected],⇤Presenting author.


Abstract

In this paper, we establish an iterative process for approximating a common fixed point for ahybrid pair of generalized I-asymptotically nonexpansive single-valued mappings and generalizednonexpansive multi-valued mappings. The weak convergence theorems and strong convergencetheorems of the proposed iterative process in Banach spaces are proven. Our results improve andextend several results in the existing literature.

Keywords: generalized I-asymptotically nonexpansive mappings; generalized nonexpansivemulti-valued mappings; Banach spaces; common fixed points

103


Viscosity approximation method for split common null pointproblems between Banach spaces and Hilbert spaces

Khanittha Promluang⇤,1, Poom Kumam1,2







Abstract

We study an iterative scheme to approximation the split common null point problems forset-valued maximal monotone operators which combine viscosity method and some fixed pointtechnically proving method between Banach spaces and Hilbert space, without using the metricprojection. We prove that strong convergence theorem. Also, we show that our result can be solvesthe split minimization problems.

Keywords: iterative method, viscosity approximation method, fixed point problems, split com-mon null point problems, a zero point, nonexpansive operator, (metric) resolvent operator.

References:[1] C. Byrne, Y. Censor, A. Gibali, and S. Reich, “The split common null point problem,” J. Nonlinear Convex Anal., vol. 13,

pp. 759-775, (2012).[2] W. Takahashi, “The split feasibility problem in Banach spaces.,” Applied Mathematics Letters, vol. 15, pp. 1349-1355,

(2014).[3] W. Takahashi, “The split common null point problem in Banach spaces.,” Arch. Math. (Basel), vol. 104, no. 4, pp.

357-365, (2015).


104


The Comparison of Spread Poin for Initial Value Population onOptimization of Function of Two Variables Using Di↵erential

Evolution Algorithm. Case Study: Inverse Problems in a MarkovModel

Dony Permana⇤,1, Suprayogi2, Sapto Wahyu Indratno3, Udjianna S. Pasaribu3

1Study Programme Statistics,Faculty of Mathematics and Natural Sciencies,

Padang State University, IndonesiaEmail: [email protected]

2Industial System and Techno-Economy Research Division,Faculty of Industrial Technology,

Institut Teknologi Bandung, IndonesiaEmail: [email protected] Research Division,

Faculty of Mathematics and Natural Sciencies,Institut Teknologi Bandung, IndonesiaEmail: {sapto,udjianna}@math.itb.ac.id


Abstract

Di↵erential Evolution Algorithm (DEA) is one heuristic numerical methods used to find thelocation an extreme point on multivariable function both linear or non linear. Generally, DEA needrandom point population that are repeated several times. But these are only taken from randompoint uniformly distributed. In this paper, the distribution point not only used random but alsonon random or systematic. There are point fulfill area, formed lines, or follow a specific function.Each population are compared in accuracy and the number of iterations required. The accuracyis calculated based on the value of objective function. Spread di↵erent point aimed to examinethe e↵ect of the initial value of the DEA convergence towards a solution. The results obtainedshowed that the DEA is quite powerful method to find a locate of maximum or minimum valuefrom a function multivariable. Here, the method is used on a case study of inverse problem of aMarkov model. Problem on that topic is to estimate a transition matrix in a Markov model. Inverseproblem will transform matrix transition problem into the optimization problem.of function of twovariables. So, the DEA is used to solve that problem.

Keywords: Di↵erential Evolution Algorithm; random point distribution; non-random point dis-tribution; obyective function value; inverse problem

105


The resolvent operator techniques with perturbations for findingzeros of maximal monotone operator and fixed point problems in

Hilbert spaces

Pongsakorn Sunthrayuth⇤,1, Poom Kumam2

1Department of Mathematics and Computer Science,Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT),

39 Rangsit-Nakhonnayok Rd., Klong 6,Thanyaburi, Pathumthani, 12110, Thailand

2Department of Mathematics, Faculty of Science,King MongkutâAZs University of Technology Thonburi (KMUTT),

126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand


Abstract

In this paper, we introduce iterative schemes with perturbations for finding zeros point ofthe sum of two monotone operators and a fixed point problem of a nonexpansive mapping inHilbert spaces. We prove a strong convergence theorem of the proposed iterative schemes underappropriate conditions. Furthermore, we also apply our results to solving the variational inequalityand equilibrium problems.

Keywords: strong convergence; iterative method; monotone operators; fixed point; variationalinequalityReferences:

[1] S. Takahashi, W. Takahashi and M. Toyoda, âAIJStrong convergence theorems for maximal monotone operators withnonlinear mappings in Hilbert spacesâAI, Journal of Optimization Theory and Applications, vol. 147, no. 1, pp.27âAS41, 2010.

[2] H.Manaka and W.Takahashi, âAIJWeak convergence theorems for maximal monotone operators with nonspreadingmappings in a Hilbert spaceâAI, CUBO, vol. 13, pp. 11-24, 2011.

[3] S. S. Zhang, J. H. W. Lee, and C. K. Chan, âAIJAlgorithms of common solutions to quasi variational inclusion andfixed point problemsâAI, Applied Mathematics and Mechanics, vol. 29, no. 5, pp. 571âAS581, 2008.

106


The Borwein-Preiss variational principle for nonconvexcountable systems of equilibrium problems

Somyot Plubtieng1, Thidaporn Seangwattana ⇤,2

1,2Department of Mathematics,Faculty of Science,

Naresuan University,Phitsanulok 65000, ThailandEmail: [email protected]


Abstract

The aim of the present paper, by using the Borwein-Preiss variational principle, we proveexistence results for countable systems of equilibrium problems. We establish some su�cientconditions which can guarantee two existence theorems for countable systems of equilibriumproblems on closed subsets of complete metric spaces and on weakly compact subsets of realBanach spaces, respectively.

Keywords: Borwein-Preiss variational principle, bifunction, complete metric space, equilibriumproblems, gauge-type function, nonconvex.

References:[1] H.M.1996 H. Attouch and M. Thera, A general duality principle for the sum of two operators, J. Convex Anal. 3

(1996) , 1-24.[2] All B. Alleche and V. D. Radulescu, The Ekeland variational principle for equilibrium problems revisited and

applications, Nonlinear Analysis: Real World Applications, 3 (2015), 17-25.[3] Ansari Q.H. Ansari, S. Schaible, J.C. Yao, System of vector equilibrium problems and its applications. J. Optim.

Theory Appl. 107 (2000), 547-557.[4] AnsariLin Q.H. Ansari, L.-J. Lin, Ekeland-type variational principles and equilibrium problems, in: S.K. Mishra (Ed.),

Topics in Nonconvex Optimization: Optimization and its Applications, Springer, 50 (2011), 147-174.[5] Bi M. Bianchi, G. Kassay, R. Pini, Existence of equilibria via EkelandÂŠs principle. J. Math. Anal. Appl. 305 (2005),

502-512.[6] Bi1 M. Bianchi, G. Kassay, R. Pini, EkelandÂŠs principle for vector equilibrium problems, Nonlinear Analysis. 66

(2007), 1454-1464.[7] Blum E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63

(1994), 123-145.[8] BO J.M. Borwein, D. Preiss, A smooth variational principle with applications to subdi↵erentiability and to di↵eren-

tiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517-527.

The authors would like to thanks Naresuan University and Thailand Research Fund (TRF) for supporting by permitmoney of investment under of The Royal Golden Jubillee Ph.D. Program (RGJ-Ph.D.), Thailand.

107


FIXED POINTS AND PERIODIC POINTS OF ↵-TYPEF-CONTRACTIVE MAPPINGS

D.K. Patela, D. Gopalb, M. Abbasc, C. Vetrod

aDepartment of Mathematics, Visvesvaraya National Institute of Technology, Nagpur-440010,India

Email:[email protected] [email protected] of Applied Mathematics & Humanities, S.V. National Institute of Technology,

Surat-395007, IndiaEmail: [email protected]

cDepartment of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood Road,Pretoria 0002, South Africa

Email: [email protected]‘a degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi, 34 -

90123 Palermo, ItalyEmail: [email protected]


Abstract

In this paper, we introduce new concepts of ↵-type F-contractive mappings which are essen-tially weaker than the class of F-contractive mappings given in [D. Wardowski, Fixed Point Theoryand Applications 2012 2012:94 and D. Wardowski, N. Van Dung, Demonstratio Math, 2014, 47:146âAS155] and di↵erent from ↵-GF-contractions given in [N. Hussain, P. Salimi, Taiwanese J Math,2014, 18: 1879-1895]. Then, su�cient conditions for the existence and uniqueness of fixed point areestablished for these new types of contractive mappings, in the setting of complete metric space.Consequently, the obtained results encompass various generalizations of the Banach contractionprinciple.

Keywords: ↵-type F-contractive mappings.

References:

108


A Generalization of Ekeland’s "-Variational Principle for⌧-distance

Areerat Arunchai⇤,1, Somyot Plubtieng1,2

1Department of Mathematics, Faculty of Science and Teachnology,Nakhon Sawan Rajabhat University,

398 M.9, Nakhon Sawan Tok, Mueang Nakhon Sawan, Nakhon Sawan 60000, ThailandEmail: naomi [email protected]

2Department of Mathematics, Faculty of Science, Naresuan University99 M.9, Thapho, Mueang, Phitsanulok 65000, Thailand


⇤Presenting author.Email: naomi [email protected]

Abstract

In this paper, we present the Cantor Intersection Theorem for ⌧-distance and the generalizationof Ekeland’s "-Variational Principle. Our results in this paper extend and improve some knownresults in the literature.

Keywords: The Ekeland’s "-Variational Principle; Borwein-Preiss smooth variant; ⌧-distance

References:[1] I. Ekeland, “On the variational principle,” J. Math. Anal. Appl. 47, 324–353, 1974.[2] J. M. Borwein and D. Preiss, “A smooth variational principle with applications to subdi↵erentiability and to di↵er-

entiability of convex functions,” Trans. Amer. Math. Soc. 303, 517–527, 1987.[3] L. Yongxin and S. Shuzhong, “A Generalization of Ekeland’s ✏-Variational Principle and Its Borwein-Preiss Smooth

Variant,” Journal of Mathematical Analysis and Applications, 246, pp. 308 – 319, 2000.


109


Impulsive Quantum Di↵erence Equations

Jessada Tariboon⇤,11 Nonlinear Dynamic Analysis Research Center,

Department of Mathematics, Faculty of Applied Science,King Mongkut’s University of Technology North Bangkok,

Bangkok 10800, ThailandEmail: [email protected]


Abstract

Quantum di↵erence equations has been introduced in 1910 by Jackson [H.F. Jackson, q-Di↵erence equations, Am. J. Math. 32, (1910) 305-314.] In this presentation, we give somenew concepts of ordinary and fractional of quantum calculus which can be used for constructingimpulsive quantum di↵erence equations. The initial and boundary value problems of impulsivequantum di↵erence equations are shown.

Keywords: quantum derivatives; quantum integrals; impulsive quantum di↵erence equations

References:[1] J. Tariboon, S.K. Ntouyas, “Quantum calculus on finite intervals and applications to impulsive di↵erence equations,”

Advances in Di↵erence Equations, 2013, 2013:282.[2] J. Tariboon, S.K. Ntouyas, P. Agarwal, “New concepts of fractional quantum calculus and applications to impulsive

fractional q-di↵erence equations,” Advances in Di↵erence Equations, (2015) 2015:18.

110


On penalty method for lexicographic vector equilibriumproblems

Lam Quoc Anh1, Tran Quoc Duy⇤,2

1Department of Mathematics, Teacher College, Cantho University, Cantho, VietnamEmail: [email protected]

2 Department of Mathematics, Cantho Technical and Economic College, Cantho, VietnamEmail: [email protected]


Abstract

We consider vector equilibrium problems involving lexicographic cone. A penalty functionmethod for solving such problems is proposed. We prove that every solution of the originallexicographic equilibrium problem is a cluster point of the penalty trajectory of the penalizedproblem. Using the regularized gap function to obtain an error bound result for such penalizedproblems is given.

Keywords: Lexicographic cone; equilibrium problem; penalty method; gap function; error bound

111


Adaptive optimal control for a bilinear model in cancerchemotherapy

Solikhatun⇤,1,3, Roberd Saragih1, Endra Joelianto2, Janson Naiborhu1

1Industrial and Financial Mathematics Research Group, Faculty of Mathematics and NaturalSciences, Institut Teknologi Bandung, Bandung, Indonesia.

Email: [email protected] and Control Research Group, Faculty of Industrial Technology, Institut

Teknologi Bandung, Bandung, Indonesia.3Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah

Mada, Yogyakarta, Indonesia.


Abstract

The cell cycle in cancer chemotherapy is modelled in a bilinear system which are linear in thestates and inputs but not linear in both. The policy iteration (PI) is used to solve the optimal controlproblem of the bilinear system. The PI is an online control method which does not require theinternal system dynamic and avoids the direct solution of the HJB equation. The simulation resultsfor adaptive optimal control design enables the researchers to know the e↵ects of the chemotherapyand the number of cancer cells during treatment

Keywords: bilinear system; cycle-cells; cancer chemotherapy; adaptive optimal control; policyiteration

References:[1] Y. Biran and B. McInnis, "Optimal Control of Bilinear Systems: Time-Varying E↵ects of Cancer Drugs," Automatica,

vol. 15, pp. 325–329, 1979.[2] U. Ledzewicz and H. Schattiler, "Optimal Bang-bang Controls for a 2-compartement Model in Cancer Chemotherapy,"

Journal of Optimization Theory and Applications-JOTA, vol. 114, pp. 609–637, 2002.[3] U. Ledzewicz and H. Schattiler, "Analysis of A Cell Cycle Specific Model for Cancer Chemotherapy," Journal of

Biological System, vol.10, pp. 183–206, 2005.[4] D.L. Elliot, "Bilinear Control Systems: Matrices in Action," Springer, New York, 2009.[5] H. Ramenzanpour, S. Setayeshi, H. Arabalibeik and A. Jajrami, "An iterative procedure for optimal control of bilinear

systems," International Journal of Instrumentation and Control Systems, vol. 2, no. 1 pp. 1–10, 2012.[6] B. Luo, and H.N. Wu, "Online adaptive optimal control for bilinear systems" proceeding of 2012 American Control

Conference, Fairmont Queen Elizabeth, Montreal, Canada, pp. 5507–5512, 2012.

The authors were supported by Ministry of Education, Research and Technology of Indonesia via BOPTN.

112


Positive solutions for hybrid fractional q-di↵erence equations

Aphirak Aphithana⇤,1, Jessada Tariboon1

1Nonlinear Dynamic Analysis Research Center,Department of Mathematics, Faculty of Applied Science,

King Mongkut’s University of Technology North Bangkok,Bangkok 10800, Thailand

Email: [email protected],[email protected]


Abstract

In this talk, we present some new existence results for positive solutions for hybrid fractionalq-di↵erence equations. By using the hybrid fixed point of two operators, the main theorem isproved. An example will be illustrated in the last sections.

Keywords: positive solutions; fractional q-di↵erence equations; fixed point theorem

References:[1] J. Tariboon, S.K. Ntouyas, P. Agarwal, “New concepts of fractional quantum calculus and applications to impulsive

fractional q-di↵erence equations,” Advances in Di↵erence Equations, (2015) 2015:18.[2] B. Ahmad, S.K. Ntouyas, “Fractional q-di↵erence hybrid equations and inclusions with Dirichlet boundary condi-

tions,” Advances in Di↵erence Equations, (2014) 2014:199.[3] B.C. Dhage “Nonlinear functional boundary value problems involving Caratheodory,” Kyungpook Math. J., (2006)

46, pp. 427 – 441.

113


Design Linear State Feedback Controller for Bilinear Systemusing Hybrid Genetic Algorithm - Particles Swarm Optimization

Taufan Mahardhika⇤,1,3, Roberd Saragih1, Bambang Riyanto Trilaksono2

1Department of Mathematics, Faculty of Science and Mathematics,Institut Teknologi Bandung (ITB),

Jl. Ganesha no 10, Bandung 40, IndonesiaEmail: [email protected]

2Department of Electrical Engineering, School of Electrical Engineering and Informatics ,Institut Teknologi Bandung (ITB),

Jl. Ganesha no 10, Bandung 40, IndonesiaEmail: [email protected]

3Department of Chemistry,Sekolah Tinggi Analis Bakti Asih (STABA),

Jl. Padasuka Atas no 233, Bandung 40, Indonesia


Abstract

In this paper the bilinear system, x(t) = Ax(t) + Bu(t) + N(x(t),u(t)) is a system with linear onstate variables (0x(t)0 variable) and linear on input variables (0u(t)0 variable) but not both. WithN(x(t),u(t)) as a function state and input. We would like to create a controller K such that the linearstate feedback u(t) = Kx(t) make the bilinear system more stabilize than before. We join GeneticAlgorithm and Particles Swarm Optimization as a hybrid algorithm. In here we design the matrixK using a hybrid algorithm.

Keywords: Linear State Feedback Controller; Bilinear System; Hybrid Algorithm; Genetic Algo-rithm; Particle Swarm Optimization

References:[1] Mohler, Ronald R., Nonlinear Systems Volume II : Aplications to Bilinear Control, Prentice-Hall, New Jersey, 1991.[2] Goldberg, D.E., Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, 1989[3] Li, Zhijie, Xiangdong Liu, Xiaodong Duan, dan Feixue Huang, "Comparative Research on Particle Swarm Optimiza-

tion and Genetic Algorithm", Computer and Information Science Volume 3 No. 1, February, 2010[4] Amato, F., C. Cosentino, A. Merola, "Stabilization of bilinear systems via linear state feedback control", Control and

Automation, 2007

The authors were supported by Negara Kesatuan Republik Indonesia

114


Sequential Optimality Conditions for Generalized EquilibriumProblems involving DC functions

Chanoksuda Khantree⇤,1, Rabian Wangkeeree1,2





Abstract

We consider a generalized equilibrium problem involving DC functions which is called (GEP).For this problem in the absence of any constraint qualifications we establish two new sequentialLagrange multiplier rule conditions characterizing optimality for (GEP). The first one is conditionin terms of the epigraphs of conjugate functions. The second sequential condition characterizingin terms of the subdi↵erentials of the functions involved at the minimizer. The significance of theresults yield the standard Lagrange multiplier rule condition for (GEP) under simple closednesscondition and new proposed approach.

Keywords: equilibrium; DC functions; constraint qualifications; Lagrange multiplier

115


Supremacy of fixed point theory and its applications

Deepak Singh⇤

Department of Applied Sciences,NITTTR, Bhopal, Ministry of HRD, Govt. of India.



Abstract

Fixed point theory handles many areas of mathematics, such as general topology, algebraictopology, nonlinear functional analysis and ordinary and partial di↵erential equations and alsoserves as a useful tool in applied mathematics. Fixed point theory is a powerful device to determineuniqueness of solutions to dynamical systems and is widely used in theoretical and appliedanalysis. In my talk, I will emphasis on the consistency and the balance of theory and applications offixed point theorems in various abstract spaces which highlights that now a days, fixed point theoryis not only a branch of pure mathematics but of applied mathematics also. In this concern, I bring tolight on some recent papers [Applied Mathematics and Computation, 273 (2016), pp. 155-164, AppliedMathematics and Computation, 268 (2015), pp. 839-843, Journal of Inequalities and Applications (2015),2015:32]. Furthermore, recognizing aforementioned paper, my talk is focused on applications offixed point theorems relating some Boundary value problems, Dynamic programming Problemsand Integral equations arising in Science and Engineering like solution of beam equation, solutionof BesselâAZs type equations etc., which highlight the superiority of fixed point theorems and itsapplications.

116


Stability for parametric primal and dual equilibrium problems

Lam Quoc Anh1 and Tran Ngoc Tam⇤,2

1Department of Mathematics, Teacher College,Can Tho University, Can Tho, Viet Nam

2Department of Mathematics, Nam Can Tho University,Cantho, Viet Nam⇤Presenting author.


Abstract

We study the parametric primal and dual equilibrium problems in locally convex Hausdor↵topological vector spaces. Su↵cient conditions for the approximate solution maps to be Hausdor↵continuous are established. Based on scalarization method, we also discuss the Hausdor↵ con-tinuity of the approximate solution maps of parametric weak vector equilibrium problems. Asapplications, we derive the Hausdor↵ continuity of the approximate solutions maps for optimiza-tion problems and variational inequalities.

117


SOME CONVERGENCE RESULTS FOR SKC MAPPINGS INHYPERBOLIC SPACES

Renu Chugh ⇤,1

1Department of Mathematics,Maharshi Dayanand University, Rohtak-124001(INDIA)

Abstract

All normed spaces and their subsets are the examples of convex metric spaces. One suchconvex structure is hyperbolic space which was introduced by Kohlenbach [U. Kohlenbach: Somelogical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc., vol.357(2005), 89-128]. The aim of this paper is to prove some results on strong and 4-convergence ofS-iterative scheme for SKC mappings in hyperbolic spaces. The results presented here extend andimprove the results of Nanjaras et. al. [B. Nanjaras, B. Panyanak, W. Phuengrattana, Fixed pointtheorems and convergence theorems for Suzuki generalized nonexpansive mappings in CAT(0)spaces, Nonlinear Analysis:Hybrid Systems, Vol. 4(2010), 25-31], Karapinar and Tas [E. Karapinarand Kenan Tas, Generalized (C)-conditions and related fixed point theorems, Computers andMathematics with Applications, Vol. 61(2011), 3370-3380].

118



Nana Indri Kurniastuti⇤,1, Rifaldy Fajar, Intan Lisnawati1,Vella Liani2,Nur Khotimah2,Dwi Lestari, M.Sc1

1 Departement of Mathematics, Faculty of Mathematics and Natural Sciences,Yogyakarta State University

Jalan Colombo 1 Yogyakarta, IndonesiaEmail:: [email protected], [email protected], [email protected],

[email protected] of Biology, Faculty of Mathematics and Natural Sciences,

Yogyakarta State UniversityJalan Colombo 1 Yogyakarta, Indonesia

Email:[email protected], [email protected]⇤Presenting author.


Abstract

In Indonesia, the case of toxoplasmosis in humans ranges between 43-88% where as in animalsranges from 6 to 70%. Toxoplasmosis is a disease caused by Toxoplasma gondii is a parasite diseaseand on animals that can be transmitted to humans. The purpose of this research is to know themathematical model and the analysis of the stability of models on propagation of the parasiteof Toxoplasma gondii from cat to congenital infection of pregnant mother have an impact on thefetus through the placenta with herbal therapy as the treatment. This type of research using thistype of research is the development by performing studies on the literature of books, reference,national and international scientific journals. This paper is developed from a mathematical Journalof ELSEVIER with authors from Venezuela, Colombia and Spain University.

Keywords: cat, prgenant mother; stability analysis; toxoplasma gondii

References:[1] Allen, L. J. S. 2000. An Introduction to Stochastic Epidemic, Department of Mathemarics and Statistics, Texas.[2] Allen, L.J.S. dan Burgin, A.M. 2000.Comparation of deterministic and stochastic SIS and SIR models in discrete time,

Mathematical Biosciences 163:1-33. Ernawati.2008.Toxoplasmosis, Terapi dan Pencegahannya. Surabaya: FakultasKedokteran Universitas Wijaya Kusuma elib.fk.uwks.ac.id. Diakses pada 4 Oktober 2015.

[3] Hiswani. 2005. Toxoplasmosis penyakit zoonosis yang perlu diwaspadai oleh ibu hamil: Fakultas Kedokteran UniversitasSumatera Utara repository.usu.ac.id.Diakses pada 3 Oktober 2015.

[4] Rohmawati, Ika dan Wibowo, Arief.2013. Hubungan Kejadian Abortus dengan Toxoplasmosis di Puskesmas Men-taras Kabupaten Gresik.Jurnal Biometrika dan Kependudukan. Volume [2]: 173-181

119


Existence and uniqueness of coupled best proximity in complexvalued metric spaces

Seyed Masoud Aghayan⇤,1, Ahmad Zireh1,2

1Department of Mathematics ,Payame Noor University,P.O.Box 19395-3697, Tehran, Iran

Email: [email protected] of Mathematics ,Shahrood University of Technology,

P.O.Box 316-36155, Shahrood, IranEmail: [email protected]


Abstract

In this paper, we prove the existence and uniqueness of a coupled best proximity point formappings by using P-property in a complete complex valued metric spaces which is a recentlyintroduced extension of metric spaces obtained by allowing the metric function to assume valuesfrom field of complex numbers. Further, our results are illustrated with examples.

Keywords: coupled best proximity; complex valued metric spaces; coupled fixed point

References:[1] B. S. Choudhury, N. Metiya and P. Maity, “Best Proximity Point Results in Complex Valued Metric Spaces,” Int. J.

Anal., vol. 2014, ID 827862, 6pages, 2014.[2] M. A. Kutbi, A. Azam, J. Ahmad and C. D. Bari, “Some Common Coupled Fixed Point Results for Generalized

Contraction in Complex-Valued Metric Spaces,” Journal of Applied Mathematics, vol. 2013, Article ID 352927, 10pages, 2013.

[3] P. Kumam, V. Pragadeeswarar, M. Marudai, and K. Sitthithakerngkiet, “Coupled best proximity points in orderedmetric spaces,” Fixed Point Theory and Applications, vol. 2014, no. 107, 2014.

[4] W. Sintunavarat and P. Kumam*, “Coupled best proximity point theorem in metric Spaces,” Fixed Point Theory andApplications, vol. 2012, no. 93, 2012.

120


Anti-Disturbance Inverse Optimal Control for SpacecraftPosition and Attitude Maneuvers with Input Saturation

Chutiphon Pukdeboon⇤

Nonlinear Dynamic Analysis Research Center,Department of Mathematics, Faculty of Applied Science,

King Mongkut’s University of Technology North Bangkok (KMUTNB),1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800, Thailand



Abstract

In this paper, a new anti-disturbance inverse optimal translation and rotation control schemefor a rigid spacecraft with external disturbances and actuator constraint is presented. An inverseoptimal controller with input saturations is designed to achieve asymptotic convergence to thedesired translation and attitude and avoid the unwinding phenomenon. The derived optimalcontrol law can minimize a given cost functional and guarantee the stability of the closed-loopsystem. Later, a new sliding mode disturbance observer is also proposed to compensate for the totaldisturbances. A rigorous Lyapunov analysis is employed to ensure the finite-time convergence ofobserver error dynamics. A numerical simulation of position and attitude maneuvers is given toverify the performance of the developed controller.

Keywords: Translation and rotation control; input saturation; disturbance observer; inverseoptimal control

References:[1] R. Freeman and P. Kokotovic, “Inverse optimality in robust stabilization,” SIAM Journal of Control and Optimization,

vol. 34, no. 4, pp. 1365 – 1319, 1996.[2] M.J. Sidi, Spacecraft Dynamics and Control, New York. NY:Cambridge University Press. 1997.[3] J.D. Boskovic, S.-M Li and R.K. Mehra, “Robust tracking control design for spacecraft under control input saturation,”

Journal of Guidance Control and Dynamics, vol. 27, no. 4, pp. 627–633, 2004.[4] Y. Lv, Q. Hu, G. Ma, and J. Zhou, “6-DOF synchronized control for spacecraft formation flying with input constraint

and parameter uncertainties,” ISA Transactions, vol. 50, no. 4, pp. 573–580, 2011.[5] C. Pukdeboon, and P. Kumam, “Robust optimal sliding mode control for spacecraft position and attitude maneuvers,”

Aerospace Science and Technology, vol. 43, pp. 329–342, 2015.

The research was supported by King Mongkut’s University of Technology North Bangkok (KMUTNB) and ThailandResearch Fund (TRF) [Contract number TRG5780030].

121



Kanokwan Sitthithakerngkiet⇤,1, Jitsupa Deepho2,Juan Martínez-Moreno3 and Poom Kumam2,4

1 Nonlinear Dynamic Analysis Research Center,Department of Mathematics, Faculty of Applied Science,

King Mongkut’s University of Technology North Bangkok (KMUTNB),Wongsawang, Bangsue, Bangkok, 10800, Thailand

Email: [email protected] of Mathematics, Faculty of Science,


Email:[email protected] of Mathematics, Faculty of Science,

University of Jaen Campus Las Lagunillas, s/n, 23071 Jaen, SpainEmail: [email protected]





Abstract

In this paper, we introduce a new iterative method for finding a common element of the setof solutions of the split generalized equilibrium problem, the set of the variational inequality for�-inverse strongly monotone mapping, and the set of fixed point of nonexpansive mapping inHilbert spaces. We show that the sequence converges strongly to a common element of the abovethree sets under some controlling conditions. Moreover, the numerical examples are presented.

Keywords: Fixed point; Variational inequality; Viscosity approximation method; Nonexpansivemapping; Split generalized equilibrium problem

122


A new hybrid iterative algorithm for numerical reckoning fixedpoints of Suzuki’s generalized nonexpansive mappings

Wutiphol Sintunavarat⇤,1

1Department of Mathematics and Statistics, Faculty of Science and Technology,Thammasat University Rangsit Center,

Khlong Nueng, Khlong Luang, Pathumthani 12121, ThailandEmail: [email protected]


Abstract

The purpose of this talk is to propose a new hybrid iterative algorithm to approximate fixedpoint of Suzuki’s generalized nonexpansive mappings. We prove some weak and strong conver-gence theorems in uniformly convex Banach spaces. A numerical example is also given to examinethe fastness of the proposed iteration process under di↵erent control conditions and initial pointswith the well-known iterations such as Picard’s iteration, Mann’s iteration [4], Ishikawa’s iteration[3], Noor’s iteration [5] and the recent iterations of Agarwal et al. [2], Abbas et al. [1] and Thakuret al. [6].

Keywords: Fixed points; Suzuki’s generalized nonexpansive mappings; Uniformly convex Ba-nach spaces.

References:[1] M. Abbas, T. Nazir, “A new faster iteration process applied to constrained minimization and feasibility problems,"

Mat. Vesn., vol. 66, pp. 223–234, 2014.[2] R.P. Agarwal, D. O’Regan, D.R. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive

mappings," J. Nonlinear Convex Anal., vol. 8, pp. 61–79, 2007.[3] S. Ishikawa, “Fixed points by a new iteration method," Proc. Am. Math. Soc., vol. 44, pp. 147–150, 1974.[4] W.R. Mann, “Mean value methods in iteration," Proc. Am. Math. Soc., vol. 4, pp. 506–510, 1953.[5] M.A. Noor, “New approximation schemes for general variational inequalities," J. Math. Anal. Appl., vol. 251, pp.

217–229, 2000.[6] B.S. Thakur, B. Thakur, M. Postolache, “A new iterative scheme for numerical reckoning fixed points of Suzuki’s

generalized nonexpansive mappings," Appl. Math. Comput., vol. 275, pp. 147–155, 2016.

The authors were supported by the Thailand Research Fund and Thammasat University under Grant No. TRG5780013.

123


An Explicit Method to Solve Fuzzy Heat Equation with IntegralBoundary Conditions

A. Hosseinpour⇤,1

1 Young Researchers and Elite Club,Kermanshah Branch,Islamic Azad University, Kermanshah, Iran



Abstract

In this paper, we provide an explicit method to solve fuzzy heat equation with nonlocalboundary value conditions. We first express the necessary materials and definitions, then weconsider a di↵erence scheme for one dimensional heat equation. Here, boundary conditionsinclude integral equations which are approximated by the composite trapezoid rule. In the lastpart, we give an example to check numerical results. In this example, we obtain the Hausdor↵distance between exact solution and approximate solution.

Keywords: Explicit method; Fuzzy numbers; Fuzzy heat equation; Finite di↵erence scheme

References:[1] D. Dubois and H. Prade, Towards fuzzy di↵erential calculus: part 3, di↵erentiation, âAsFuzzy Sets and SystemsâAs,

vol. 8, pp. 225-233, 1982.[2] M. Friedman and M. Ming, A. Kandel, Fuzzy linear systems, âAsFuzzy Sets and SystemsâAs, vol. 96, pp. 201-209, 1998.[3] O. Kaleva, Fuzzy di↵erential equations, âAsFuzzy Sets and SystemsâAs, vol. 24, pp. 301-307, 1987.[4] O. Kaleva, The cauchy problem for fuzzy di↵erential equations, âAsFuzzy sets and systemsâAs, vol. 5, pp.389-396,

1990.[5] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets. âAsWorld ScientificâAs, Singapore, 1994

124


On the qualitative properties for solutions equilibrium probleminvolving Lorentz cone

L.M. Huy1, L.Q. Anh2, N.H. Danh3, T.T.K. Linh1

1Department of Mathematics, Cantho University, Cantho, Vietnam2Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam3Department of Mathematics, Teacher College, TayDo University, Cantho, Vietnam


Abstract

In this work, inspired by the great importance of equilibrium problems and the Lorentz cone,we consider the equilibrium problem involving Lorentz cone in Rn. Su�cient conditions forthe solution maps to such problems to be upper semicontinuous, lower semicontinuous, closedand well-posed are established. We provide numerous examples to explain that all the imposedassumptions are essential. Applications the achieved results to the variational inequality are alsodiscussed.

Keywords: Lorentz cone, well-posedness, upper semicontinuity, lower semicontinuity, existenceof solution maps, equilibrium problem, variational inequalities

References:[1] L.Q. Anh, T.Q. Duy, A.Y. Kruger, and N.H. Thao : Well-Posedness for Lexicographic Vector Equilibrium Problems,

Optimization and Its Applications, 87, 159-174 (2013).[2] L.Q. Anh, P.Q. Khanh : Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium

problems, Journal of Mathematics Analysis and Applications, 294, 699-711 (2004).[3] L.Q. Anh, P.Q. Khanh : On the stability of the solution sets of general multivalued vector quasiequilibrium problems,

Journal of Optimization Theory Applications, 135, 271-284 (2007).[4] E. Blum, W. Oettli : From optimization and variational inequalities to equilibrium problems, Mathematics Student,

63, 123-145 (1994).[5] Y. Han, N. Huang : Existence and stability of solutions for a class of generalized vector equilibrium problems,

positivity, 1-18 (2015).[6] R. Wangkeeree, P. Boonman, P. Prechasilp : Lower semicontinuity of approximate solution mappings for parametric

generalized vector equilibrium problems, Journal of Inequalities and Applications, 2014-421 (2014).

125


On Vector Optimization Problems and Vector VariationalInequalities via Convexificators

S. K. Mishra⇤,11 Professor, Department of Mathematics

Institute of ScienceBanaras Hindu University Varanasi, India



Abstract

We discuss some results which exhibit an application of convexificators in vector optimizationproblems and vector variational inequaities involving locally Lipschitz functions. We presentvector variational inequalities of Stampacchia and Minty type in terms of convexificators and usethese vector variational inequalities as a tool to find out necessary and su�cient conditions fora point to be a vector minimal point of the vector optimization problem. We also discuss thecorresponding weak versions of the vector variational inequalities and several results to find outweak vector minimal points.

126

Index

Abbas, M., 107Adhikari, A., 64Aghayan, S.M., 120Aiemsomboon, L., 56Akbar, A., 87Ali, J., 30Anakkamatee, W., 90Anh, L.Q., 19, 60, 110, 116, 125Anorat, K., 63Ansari, Q.H., 7Aphithana, A., 112Artsawang, N., 78Arunchai, A., 108Atsushiba, S., 89

Bantaojai, 60Baskoro, E.T., 101Boonman, P., 84Boriwan, P., 99Budhia, L., 92Buranakorn, K., 76

Chaipornjareansri, S., 45Chaipunya, P., 40Cho, Y.J., 4, 24, 34, 38, 46, 71, 96Cholamjiak, P., 34Chuasuk, P., 102Chuensupantharat, N., 51Chugh, R., 118

Danh, N.H., 125de Lara, M., 15Deepho, J., 35, 122Dhompongsa, S., 3Dien, H.T.H., 36Dong, Q.-L., 71Duy, T.Q., 60, 110

Fajar, R., 119Farajzadeh, A., 18Farajzadeh, A.P., 39

Ghimire, R.P., 64Gopal, D., 16, 57, 92, 107Gupta, A., 91

Hadianti, R., 98Han, S.-E., 117Hosseinpour, A., 124Hunwisai, D., 25Huy, L.M., 125

Indratno, S.W., 104

Jain, D., 57Jha, K., 55, 58, 65Jitpeera, T., 63Joelianto, E., 111

Kaewcharoen, A., 102Kangtunyakarn, A., 69, 70Kaskasem, P., 79Khammahawong, K., 43Khanh, P.Q., 5, 36Khantree, C., 20, 114Khaofong, C., 26Khotimah, N., 119Khuangsatung, W., 70Kim, J.K., 13Kim, Y.-H., 62Kimura, Y., 14Klin-eam, C., 77, 79Kohsaka, F., 21Komal, S., 41Kongban, C., 42Kruger, A., 12Kumam, P., 24–28, 33, 35, 38, 40–43, 46, 50–53, 57, 63, 66,

82, 92, 103, 105, 122Kumrod, P., 47Kurniastuti, N.I., 119

Lee, G.M., 6Lee, J.H., 6Lestari, D., 119

127


Liani, V., 119Linh, T.T.K., 125Lisnawati, I., 119

Mahardhika, T., 113Manandhar, K.B., 55Manro, S., 91Martínez-Moreno, J., 35, 122Mehmood, N., 87Mishra, S. K., 126Mongkolkeha, C., 38Muangchoo-in, K., 52Munkong, J., 72

Naiborhu, J., 111Nanjaras, B., 31Nantadilok, J., 49Ngeonkam, B., 80Nimana, N., 39Novianingsih, K., 98

Ogata, Y., 6Onsod, W., 50Oyjinda, P., 73

Płuciennik, R., 10Padcharoen, A., 46, 57Pakkaranang, N., 53Panthong, C., 33Panyanak, B., 29Pasaribu, U.S., 104Patel, D.K., 107Permana, D., 104Petrot, N., 39, 68, 75, 95, 99Phiangsungnoen, S., 97Pholasa, N., 34Phosri, P., 74Phuangphoo, P., 82Pitea, A., 61Piwma, N., 81Plubtieng, S., 67, 76, 106, 108Plubtieng. S., 12Pochai, N., 73, 74Pongpullponsak, A., 83Preechasilp, P., 94Promluang, K., 103Promsinchai, P., 95Pukdeboon, C., 121Puripat, C., 83Puturong, N., 100

Rajopadhyaya, U., 58Riyanto, T., 113Rohen, Y., 88

Saejung, S., 17Saipara, P., 24

Saito, Y., 6Saksirikum, W., 75Samanmit, K., 32Saragih, R., 113Sarigih, R., 111Sarikavanij, S., 83Sawangsup, K., 48Seangwattana, T., 12, 106Sharma, P.K., 37Simanjuntak, R., 101Singh, D., 115Sintunavarat, W., 47, 48, 54, 56, 123Sirichunwijit, T., 86Sisarat, N., 85Sitthithakerngkiet, K., 35, 122Solikhatun, 111Sombat, A., 66Sridarat, P., 93Suanoom, C., 77Suantai, S., 11, 93, 99Suantai. S., 96Sukprasert, P., 28Sumalai, P., 27, 33Sunthrayuth, P., 105Suprayogi, 104Suwannaut, S., 69

Takahashi, W., 2Tam, T.N., 116Tanaka, T., 6Tangkhawiwetkul, J., 68Tariboon, J., 109, 112Thainwan, T., 81Thamboonruang, N., 49Tiammee, J., 96Tomar, A., 44Tongnoi, B., 90

Uddin, I., 30Ungchittrakool, K., 59, 72, 78, 80, 100Uttunggadewa, S., 101

Vetro, C., 107Vui, P.T., 19, 60

Wangkeeree, R., 20, 60, 84–86, 94, 114Welyyanti, D., 101

Yamaod, O., 54Yuan, H.-B., 71Yuying, T., 67

Zireh, A., 120

128

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