oscillation for a class of right fractional differential equations on … · 2019. 7. 30. ·...
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Research ArticleOscillation for a Class of Right Fractional DifferentialEquations on the Right Half Line with Damping
Hui Liu and Run Xu
School of Mathematical Sciences Qufu Normal University Qufu 273165 Shandong China
Correspondence should be addressed to Run Xu xurun 2005163com
Received 24 January 2019 Revised 15 March 2019 Accepted 17 March 2019 Published 1 April 2019
Academic Editor Chris Goodrich
Copyright copy 2019 Hui Liu and Run XuThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In this paper we discuss a class of fractional differential equations of the form 119863120572+1minus 119910(119905) sdot 119863120572minus119910(119905) minus 119901(119905)119891(119863120572minus119910(119905)) +119902(119905)ℎ(intinfin119905
(119904 minus 119905)minus120572119910(119904)119889119904) = 0119863120572minus119910(119905) is the Liouville right-sided fractional derivative of order120572 isin (0 1)We obtain some oscillationcriteria for the equation by employing a generalized Riccati transformation technique Some examples are given to illustrate thesignificance of our results
1 Introduction
The theory of fractional derivatives was originated fromGW Leibnizrsquos conjecture To this day the theory aboutfractional calculus and fractional differential equation havebeen well developed see [1ndash7] In the beginning thetheory of fractional derivatives developed mainly as a puretheoretical filed of mathematics which can be used only formathematicians However in the past few decades fractionaldifferential equations were widely used in many fields suchas fluid flow rheology electrical networks and many otherbranches of science Great attention was paid to study theproperties of solutions of fractional differential equations
Because only few differential equations can be solvedmany researches focus on the analysis of qualitative theoryfor fractional differential equations such as the existenceuniqueness of solutions numerical solutions stability andoscillation of solutions see [8ndash34] and the references thereinAmong them there have beenmany results for the oscillationof solutions for fractional differential equations
In 2013 Chen [16] studied oscillatory behavior of thefractional differential equation in the form of
119863120572+1minus 119910 (119905) minus 119901 (119905)119863120572minus119910 (119905)+ 119902 (119905) 119891(intinfin
119905(V minus 119905)minus120572 119910 (V) 119889V) = 0 (1)
for 119905 gt 0 where 119863120572minus119910 is the Liouville right-sided fractionalderivative of order 120572 isin (0 1)
In 2013 Han [17] brought up the oscillation of fractionaldifferential equations
[119903 (119905) 119892 ((119863120572minus119910) (119905))]1015840 minus 119901 (119905) 119891(intinfin119905
(119904 minus 119905)minus120572 119910 (119904))= 0 (2)
for 119905 gt 0 where 0 lt 120572 lt 1 is a real number and 119863120572minus119910 is theLiouville right-sided fractional derivative of 119910
In 2013 Xu [18] studied the oscillation of nonlinearfractional differential equations of the form
119886 (119905) [(119903 (119905) 119863120572minus119909 (119905))1015840]1205781015840minus 119865(119905 intinfin
119905(V minus 119905)minus120572 119909 (V) 119889V) = 0 119905 ge 1199050 gt 0 (3)
where 120572 isin (0 1) is a constant and 120578 is a ratio of two oddpositive integers
In 2013 based on themodified Riemann-Liouville deriva-tive Qin and Zheng [19] discussed the oscillation of a class
HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 4902718 9 pageshttpsdoiorg10115520194902718
2 Discrete Dynamics in Nature and Society
of fractional differential equations with damping term asfollows
119863120572119905 [119886 (119905) 119863120572119905 (119903 (119905) 119863120572119905 119909 (119905))] + 119901 (119905) 119863120572119905 (119903 (119905) 119863120572119905 119909 (119905))+ 119902 (119905) 119909 (119905) = 0 (4)
for 119905 ge 1199050 gt 0 0 lt 120572 lt 1 where 119863120572119905 (sdot) denotes the modifiedRiemann-Liouville derivative regarding the variable 119905 thefunction 119886 isin 119862120572([1199050infin) 119877+) 119903 isin 1198622120572([1199050infin) 119877+) 119901 119902 isin119862120572([1199050infin) 119877+) and 119862120572 denotes continuous derivative oforder
In 2014 Jehad Alzabut andThabet Abdeljawad [20] stud-ied the oscillatory theory of fractional difference equations inthe form
nabla119902119886(119902)minus1
119909 (119905) + 1198911 (119905 119909 (119905) = 119903 (119905) + 1198912 (119905 119909 (119905)) 119905 isin N119886(119902)
nablaminus(1minus119902)119886(119902)minus1
119909 (119905)100381610038161003816100381610038161003816119905=119886(119902) = 119909 (119886 (119902)) = 119888 119888 isin R(5)
where119898 minus 1 lt 119902 lt 119898119898 isin N nabla119902119886(119902)
is the Riemann-Liouvilledifference operator of order 119902 and nablaminus119902
119886(119902)is the Riemann-
Liouville sum operator where N119886(119902) = 119886(119902) + 1 119886(119902) +2 119886(119902) = 119886 + 119898 minus 1119898 = [119902] + 1 and [119902] is the greatestinteger less than or equal to 119902
In 2017 B Abdalla K Abodayeh T Abdeljawad JAlzabut [21] studied the oscillation of solutions of nonlinearforced fractional difference equations in the form
nabla119902119886(119902)minus1
119909 (119905) + 1198911 (119905 119909 (119905) = 119903 (119905) + 1198912 (119905 119909 (119905)) 119905 isin N119886(119902)
nablaminus(119898minus119902)119886(119902)minus1
119909 (119905)100381610038161003816100381610038161003816119905=119886(119902) = 119909 (119886 (119902)) = 119888 119888 isin R(6)
where 119902 gt 0119898 = [119902] + 1119898 isin N [119902] is the greatest integerless than or equal to 119902 N119886(119902) = 119886(119902) + 1 119886(119902) + 2 119886(119902) =119886 + 119898 minus 1 119891119894 N119886(119902) times R 997888rarr R(119894 = 1 2) and nablaminus119902
119886(119902)and nabla119902
119886(119902)
are the Riemann-Liouville sum and difference operatorsIn 2018 Bai and Xu [22] discussed the oscillation problem
of a class of nonlinear fractional difference equations with thedamping term in the from
Δ(119888 (119905) [Δ (119903 (119905) Δ120572119909 (119905))]120574) + 119901 (119905) [Δ (119903 (119905) Δ120572119909 (119905))]120574+ 119902 (119905) 119891(119905minus1+120572sum
119904=1199050
(119905 minus 119904 minus 1)(minus120572) 119909 (119904)) = 0119905 isin 1198731199050
(7)
where 120574 ge 1 is a quotient of two odd positive integers 0 lt 120572 le1 is a constant Δ120572 denotes the Riemann-Liouville fractionaldifference operator of order 120572 and1198731199050 = 1199050 1199050+1 1199050+2
In 2018 Bahaaeldin Abdalla andThabet Abdeljawad [23]studied the oscillation of Hadamard fractional differentialequation of the form
119863120572119886119909 (119905) + 1198911 (119905 119909) = 119903 (119905) + 1198912 (119905 119909) 119905 gt 119886lim119905997888rarr119886+
119863120572minus119895119886 119909 (119905) = 119887119895 (119895 = 1 2 119899) (8)
where 119899 = lceil120572rceil 119863120572119886 is the left-fractional Hadamard derivativeof order 120572 isin C Re(120572) ge 0 in the Riemann-Liouville setting
In 2018 J Alzabut T Abdeljawad H Alrabaiah [24]considered the following forced and damped nabla fractionaldifference equation
(1 minus 119901 (119899)) nablanabla1205720 119910 (119899) + 119901 (119899) nabla1205720 119910 (119899) + 119902 (119899) 119891 (119910 (119899))= 119892 (119899) 119899 isin N1
nablaminus(1minus120572)0 119910 (1) = 119910 (1) = 119888(9)
where nabla1205720 119910 and nablaminus1205720 119910 are the Riemann-Liouville fractionaldifference and sum operators of 119910 of order 120572 respectively120572 is a real number 119888 is constant N1 = 1 2 and 119901 119902 arereal sequences from N1 997888rarr R 119901(119899) lt 1 119902 is a positive realsequence from N1 997888rarr R+ and 119891 R 997888rarr R such that 119891(119904)119904 gt 0 for all 119904 = 0
In 2018 B Abdalla J Alzabut T Abdeljawad [25] inves-tigated the oscillation of solutions for fractional differenceequations with mixed nonlinearities in forms
nabla120572119886(120572)minus1119909 (119905) minus 119901 (119905) 119909 (119905) + 119899sum119894=1
119902119894 (119905) |119909 (119905)|120582119894minus1 119909 (119905)= V (119905) 119905 isin N119886(120572)+1
nablaminus(119898minus119886)119886(120572minus1) 119909 (119905)10038161003816100381610038161003816119905=119886(120572) = 119909 (119886 (120572)) = 119888 119888 isin R(10)
and
119888nabla120572119886(120572)119909 (119905) minus 119901 (119905) 119909 (119905) + 119899sum119894=1
119902119894 (119905) |119909 (119905)|120582119894minus1 119909 (119905)= V (119905) 119905 isin N119886(120572)
nabla119896119909 (119886 (120572)) = 119887119896 119896 isin R 119896 = 0 1 2 119898 minus 1(11)
where 119898 = [120572] + 1 120572 gt 0 119901(119905) V(119905) and 119902119894(119905)(1 le 119894 le 119899) arefunctions defined fromN119886(120572) to119877 and 120582119894(1 le 119894 le 119899) are ratiosof odd positive integers with 1205821 gt gt 120582119897 gt 1 gt 120582119897+1 gt gt120582119899
Inspired by the above results in this paper we discuss theoscillatory behavior of the fractional differential equationalwith damping
119863120572+1minus 119910 (119905) sdot 119863120572minus119910 (119905) minus 119901 (119905) 119891 (119863120572minus119910 (119905))+ 119902 (119905) ℎ (intinfin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904) = 0 119905 gt 0 (12)
where 0 lt 120572 lt 1 is a real number 119863120572minus119910 is the Liouville right-sided fractional derivative of 119910 We always assume that thefollowing conditions are valid
Discrete Dynamics in Nature and Society 3
(1198601) 119901(119905) ge 0 and 119902(119905) ge 0 are continuous functions on119905 isin [1199050infin) 1199050 gt 0(1198602) ℎ 119891 R 997888rarr R are continuous functions with119909ℎ(119909) gt 0 119909119891(119909) gt 0 for 119909 = 0 and there exist positive
constants 1198961 1198962 such that ℎ(119909)119909 ge 1198961 119909119891(119909) ge 1198962 for all119909 = 0(1198603) 1198911015840(119906) le 119906 119891minus1(119906) isin 119862(RR) are continuousfunctions with 119891minus1(119906) gt 0 for 119906 = 0 and there exists somepositive constant 1205721 such that 119891minus1(119906V) ge 1205721119891minus1(119906)119891minus1(V) for119906V = 02 Preliminaries
For convenience some background materials from fractionalcalculus are given
From [4] we can get the definition for Liouville right-side fractional integral and Liouville right-side fractionalderivative on the whole axis R of order 120573 for a function119892 R+ 997888rarr R as follows
(119868120573minus119892) (119905) fl 1Γ (120573) intinfin
119905(V minus 119905)120573minus1 119892 (V) 119889V 119905 gt 0 (13)
(119863120573minus119892) (119905) fl (minus1)lceil120573rceil 119889lceil120573rceil119889119905lceil120573rceil (119868lceil120573rceilminus120573minus 119892) (119905)= (minus1)lceil120573rceil 1Γ (lceil120573rceil minus 120573) int
infin
119905(V minus 119905)lceil120573rceilminus120573minus1 119892 (V) 119889V
119905 gt 0(14)
provided the right hand side is pointwise defined on R+where Γ(sdot) is the gamma function defined by Γ(119905) flintinfin119905
119904119905minus1119890minus119904119889119904 and fl min119911 isin Z 119911 ge 120573 is the ceilingfunction
If 120573 isin (0 1) we have119863120572minus119910 (119905) fl minus 1Γ (1 minus 120572) 119889119889119905 int
infin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 (15)
for 119905 isin R+ fl (0infin)The following relations also existed
(119863(1+120572)minus 119910) (119905) = minus (119863120572minus119910)1015840 (119905) 120572 isin (0 1) 119905 gt 0 (16)
Set
119866 (119905) fl intinfin119905
(V minus 119905)minus120572 119910 (V) 119889V 120572 isin (0 1) (17)
and then
1198661015840 (119905) = minusΓ (1 minus 120572) (119863120572minus119910) (119905) 120572 isin (0 1) (18)
3 Main Results
First we study the oscillation of (12) under the followingcondition
intinfin1199050
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904 = infin (19)
Theorem 1 Suppose that (1198601) minus (1198603) and (19) hold fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that
lim sup119905997888rarrinfin
int1199051199050
V (119904)sdot [[1198961119903 (119904) 119902 (119904) minus
(1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = infin(20)
where 1198961 1198962 are defined as in (1198602) andV (119904) fl exp(int119904
1199050
119901 (V) 119889V) 119904 ge 1199050 (21)
Then every solution of (12) is oscillatory
Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Then there exists 1199051 isin[1199050infin] such that 119910(119905) gt 0 and 119866(119905) gt 0 for 119905 isin [1199051infin] where119866 is defined in (16) From (1198603) (12) and (16) we have
[119891 (119863120572minus119910 (119905)) V (119905)]1015840 = minus1198911015840 (119863120572minus119910 (119905)) V (119905) 119863120572+1minus 119910 (119905)+ 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)
= 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)minus 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) V (119905)
ge 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)minus 119863120572+1minus 119910 (119905) 119863120572minus119910 (119905) V (119905)
= 119902 (119905) ℎ (119866 (119905)) V (119905) gt 0119905 isin [1199050infin]
(22)
Thus 119891(119863120572minus119910(119905))V(119905) is strictly increasing on [1199050infin] SinceV(119905) gt 0 for 119905 isin [1199050infin] and from (1198603) we see that 119863120572minus119910(119905) iseventually of one sign Now we can claim
119863120572minus119910 (119905) lt 0 119905 isin [1199051infin] (23)
If not then there exists 1199052 isin [1199051infin] such that 119863120572minus119910(1199052) gt0 Since 119891(119863120572minus119910(119905))V(119905) is strictly increasing on [1199051infin] it isclear that 119891(119863120572minus119910(119905))V(119905) ge 119891(119863120572minus119910(1199052))V(1199052) fl 119888 gt 0 for119905 isin [1199052infin] Therefore from (18) we have
minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1 ( 119888V (119905))
= 119891minus1 (119888 sdot exp(minusint1199051199050
119901 (V) 119889V))ge 1205721119891minus1 (119888) 119891minus1 (exp(minusint119905
1199050
119901 (V) 119889V)) 119905 isin [1199052infin]
(24)
4 Discrete Dynamics in Nature and Society
Then we get
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V)) le minus 1198661015840 (119905)1205721119891minus1 (119888) Γ (1 minus 120572) 119905 isin [1199052infin] (25)
Integrating the above inequality from 1199052 to 119905 we haveint1199051199052
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904le minusint1199051199052
1198661015840 (119904)1205721119891minus1 (119888) Γ (1 minus 120572)119889119904= minus 119866 (119905) minus 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572)
119905 isin [1199052infin]
(26)
Letting 119905 997888rarr infin we see
intinfin1199052
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904le 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt infin (27)
This is in contradiction with (19) Hence (23) holdsDefine the function 119908 as generalized Riccati substitution
119908 (119905) = 119903 (119905) minusV (119905) 119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (28)
Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (18) (22) (28)and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)
sdot minus [119891 (119863120572minus119910 (119905)) V (119905)]1015840 119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905)le 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot [minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905) ]= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) V (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198961119903 (119905) 119902 (119905) V (119905) + 1199082 (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))119903 (119905) 119891 (119863120572minus119910 (119905)) V (119905)le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905) V (119905) minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905)
(29)
That is
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905) (30)
Taking 119909 = 119908(119905) 119886 = 1198962Γ(1 minus 120572)119903(119905)V(119905) (119886 = 0) and 119887 =1199031015840(119905)119903(119905) from 1198861199092+119887119909 le minus11988724119886 and (30)we could concludethat
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + V (119905) (1199031015840 (119905))241198962Γ (1 minus 120572) 119903 (119905) (31)
Integrating both sides of inequality (31) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)
ge int1199051199050
[[minus1198961119903 (119904) 119902 (119904) V (119904) +
V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119905)]]119889119904 (32)
Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get
lim sup119905997888rarrinfin
int1199051199050
[[minus1198961119903 (119904) 119902 (119904) V (119904)
+ V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 lt 119908 (1199050) lt infin(33)
which is in contradiction with (20)If 119910(119905) is an eventually negative solution of (12) the proof
is similar hence we omit itThe proof is complete
Theorem 2 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that
lim sup119905997888rarrinfin
int1199051199050
[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= infin(34)
where V(119904) are defined inTheorem 1Then every solution of (12)is oscillatory
Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Proceeding the same asin the proof of Theorem 1 we get (23) Define the function119908(119905) as follows
119908(119905) = 119903 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (35)
Discrete Dynamics in Nature and Society 5
Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (16) (18) (22)(35) and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905) [minus119891 (119863120572minus119910 (119905))119866 (119905) ]1015840
= 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905)sdot 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) 119866 (119905) + 119891 (119863120572minus119910 (119905)) 1198661015840 (119905)1198662 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)+ 119903 (119905) 119863120572minus119910 (119905) 119863120572+1minus 119910 (119905)119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905)sdot 119901 (119905) 119891 (119863120572minus119910 (119905)) minus 119902 (119905) ℎ (119866 (119905))119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905)sdot 1199082 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 119901 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)
(36)
That is
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) + 1199031015840 (119905) minus 119903 (119905) 119901 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905) (37)
Taking 119909 = 119908(119905) 119886 = 1198962Γ(1minus120572)119903(119905) (119886 = 0) and 119887 = (1199031015840(119905) minus119903(119905)119901(119905))119903(119905) from 1198861199092 + 119887119909 le minus11988724119886 and (37) we conclude
1199081015840 (119905) le minus1198961119903 (119904) 119902 (119904) + [1199031015840 (119905) minus 119903 (119905) 119901 (119905)]241198962Γ (1 minus 120572) 119903 (119905) (38)
Integrating both sides of inequality (38) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)ge int1199051199050
[[minus1198961119903 (119904) 119902 (119904) +
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904 (39)
Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get
lim sup119905997888rarrinfin
int1199051199050
[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
lt 119908 (1199050) lt infin(40)
which is in contradiction with (34)
If 119910(119905) is an eventually negative solution of (12) the proofis similar here we omit it
The proof is complete
We define a function class119866 setD fl (119905 119904) fl 119905 ge 119904 ge 1199050D0 fl (119905 119904) fl 119905 gt 119904 ge 1199050We say119867(119905 119904) isin 119866 if119867(119905 119904) satisfy119867(119905 119905) = 0 119905 ge 1199050119867 (119905 119904) gt 0 (119905 119904) isin D0 (41)
and 119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904) fl 120597119867(119905 119904)120597119904 on D0 with respect to the secondvariable
Theorem 3 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] and a function119867 isin 119866 satisfies
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]] 119889119904
= infin(42)
Then every solution of (12) is oscillatory
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 2 we get (37)
Multiplying (37) by 119867(119905 119904) and integrating from 1199051 to 119905we get
int1199051199051
1198961119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le minusint1199051199051
119867(119905 119904) 1199081015840 (119904) 119889119904+ minusint1199051199051
119867(119905 119904) 119908 (119904) 1199031015840 (119904)119903 (119904) 119889119904minus int1199051199051
119867(119905 119904) 1199082 (119904) 1198962Γ (1 minus 120572)119903 (119904) 119889119904119905 isin [1199051infin]
(43)
Using the formula integration by parts we obtain
minusint1199051199051
119867(119905 119904) 1199081015840 (119904) 119889119904 = minus [119867 (119905 119904) 119908 (119904)]119904=119905119904=1199051+ int1199051199051
1198671015840119904 (119905 119904) 119908 (119904) 119889119904= 119867 (119905 1199051)119908 (1199051)
+ int1199051199051
1198671015840119904 (119905 119904) 119908 (119904) 119889119904119905 isin [1199051infin)
(44)
6 Discrete Dynamics in Nature and Society
Substituting (44) with (43) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)
minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)
Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget
int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904
le int1199051199051
(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904
le int1199051199051
[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905
1199051
[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904
le int1199051199051
[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905
1199051
[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904
(46)
Substituting (46) in (45) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905
1199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)
Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get
int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)
119905 isin [1199051infin) (48)
Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have
1119867 (119905 1199050) int119905
1199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051
1199050
119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)
Letting 119905 997888rarr infin we obtain
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)
which yields a contradiction to (42) The proof is complete
Discrete Dynamics in Nature and Society 7
Second we study the oscillation of (12) under the follow-ing condition
intinfin1199050
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904 lt infin (51)
Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050
intinfin119879
119891minus1 [ 1V (119905) int
119905
119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)
where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)
If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get
[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)
(53)
Integrating both sides of the above inequality from 1199052 to 119905 wehave
119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904gt 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904119905 isin [1199052infin)
(54)
Hence from (17) we get
minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )
ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )
119905 isin [1199052infin)
(55)
Integrating both sides of the last inequality from 1199052 to 119905 weobtain
119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052
119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906
119905 isin [1199052infin) (56)
Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete
Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem
4 Examples
Example 1 Consider the fractional differential equation
(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int
infin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)
In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since
intinfin1199050
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
(exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
exp(1119905 minus 11199050)119889119905 ge intinfin1199050
exp(minus 11199050)119889119905 = infin(58)
then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin
int1199051
[[1198961119903 (119904) 119902 (119904) minus
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= lim sup119905997888rarrinfin
int1199051
[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]
]119889119904= lim sup119905997888rarrinfin
int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin
(59)
which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory
8 Discrete Dynamics in Nature and Society
Example 2 Consider the fractional differential equation
(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905
(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)
In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that
(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050
Since
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
= lim sup119905997888rarrinfin
1(119905 minus 1199050)14 int119905
1199050
119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin
sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]
= lim sup119905997888rarrinfin
(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin
(61)
which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory
5 Conclusion
In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper
Authorsrsquo Contributions
Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript
Acknowledgments
This research is supported by National Science Foundation ofChina (11671227)
References
[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962
[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974
[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999
[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006
[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983
[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996
[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993
[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997
[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002
[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011
[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010
[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012
[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012
[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013
Discrete Dynamics in Nature and Society 9
[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013
[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013
[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013
[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014
[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017
[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018
[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018
[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018
[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018
[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009
[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015
[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016
[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017
[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016
[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018
[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018
[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015
[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013
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2 Discrete Dynamics in Nature and Society
of fractional differential equations with damping term asfollows
119863120572119905 [119886 (119905) 119863120572119905 (119903 (119905) 119863120572119905 119909 (119905))] + 119901 (119905) 119863120572119905 (119903 (119905) 119863120572119905 119909 (119905))+ 119902 (119905) 119909 (119905) = 0 (4)
for 119905 ge 1199050 gt 0 0 lt 120572 lt 1 where 119863120572119905 (sdot) denotes the modifiedRiemann-Liouville derivative regarding the variable 119905 thefunction 119886 isin 119862120572([1199050infin) 119877+) 119903 isin 1198622120572([1199050infin) 119877+) 119901 119902 isin119862120572([1199050infin) 119877+) and 119862120572 denotes continuous derivative oforder
In 2014 Jehad Alzabut andThabet Abdeljawad [20] stud-ied the oscillatory theory of fractional difference equations inthe form
nabla119902119886(119902)minus1
119909 (119905) + 1198911 (119905 119909 (119905) = 119903 (119905) + 1198912 (119905 119909 (119905)) 119905 isin N119886(119902)
nablaminus(1minus119902)119886(119902)minus1
119909 (119905)100381610038161003816100381610038161003816119905=119886(119902) = 119909 (119886 (119902)) = 119888 119888 isin R(5)
where119898 minus 1 lt 119902 lt 119898119898 isin N nabla119902119886(119902)
is the Riemann-Liouvilledifference operator of order 119902 and nablaminus119902
119886(119902)is the Riemann-
Liouville sum operator where N119886(119902) = 119886(119902) + 1 119886(119902) +2 119886(119902) = 119886 + 119898 minus 1119898 = [119902] + 1 and [119902] is the greatestinteger less than or equal to 119902
In 2017 B Abdalla K Abodayeh T Abdeljawad JAlzabut [21] studied the oscillation of solutions of nonlinearforced fractional difference equations in the form
nabla119902119886(119902)minus1
119909 (119905) + 1198911 (119905 119909 (119905) = 119903 (119905) + 1198912 (119905 119909 (119905)) 119905 isin N119886(119902)
nablaminus(119898minus119902)119886(119902)minus1
119909 (119905)100381610038161003816100381610038161003816119905=119886(119902) = 119909 (119886 (119902)) = 119888 119888 isin R(6)
where 119902 gt 0119898 = [119902] + 1119898 isin N [119902] is the greatest integerless than or equal to 119902 N119886(119902) = 119886(119902) + 1 119886(119902) + 2 119886(119902) =119886 + 119898 minus 1 119891119894 N119886(119902) times R 997888rarr R(119894 = 1 2) and nablaminus119902
119886(119902)and nabla119902
119886(119902)
are the Riemann-Liouville sum and difference operatorsIn 2018 Bai and Xu [22] discussed the oscillation problem
of a class of nonlinear fractional difference equations with thedamping term in the from
Δ(119888 (119905) [Δ (119903 (119905) Δ120572119909 (119905))]120574) + 119901 (119905) [Δ (119903 (119905) Δ120572119909 (119905))]120574+ 119902 (119905) 119891(119905minus1+120572sum
119904=1199050
(119905 minus 119904 minus 1)(minus120572) 119909 (119904)) = 0119905 isin 1198731199050
(7)
where 120574 ge 1 is a quotient of two odd positive integers 0 lt 120572 le1 is a constant Δ120572 denotes the Riemann-Liouville fractionaldifference operator of order 120572 and1198731199050 = 1199050 1199050+1 1199050+2
In 2018 Bahaaeldin Abdalla andThabet Abdeljawad [23]studied the oscillation of Hadamard fractional differentialequation of the form
119863120572119886119909 (119905) + 1198911 (119905 119909) = 119903 (119905) + 1198912 (119905 119909) 119905 gt 119886lim119905997888rarr119886+
119863120572minus119895119886 119909 (119905) = 119887119895 (119895 = 1 2 119899) (8)
where 119899 = lceil120572rceil 119863120572119886 is the left-fractional Hadamard derivativeof order 120572 isin C Re(120572) ge 0 in the Riemann-Liouville setting
In 2018 J Alzabut T Abdeljawad H Alrabaiah [24]considered the following forced and damped nabla fractionaldifference equation
(1 minus 119901 (119899)) nablanabla1205720 119910 (119899) + 119901 (119899) nabla1205720 119910 (119899) + 119902 (119899) 119891 (119910 (119899))= 119892 (119899) 119899 isin N1
nablaminus(1minus120572)0 119910 (1) = 119910 (1) = 119888(9)
where nabla1205720 119910 and nablaminus1205720 119910 are the Riemann-Liouville fractionaldifference and sum operators of 119910 of order 120572 respectively120572 is a real number 119888 is constant N1 = 1 2 and 119901 119902 arereal sequences from N1 997888rarr R 119901(119899) lt 1 119902 is a positive realsequence from N1 997888rarr R+ and 119891 R 997888rarr R such that 119891(119904)119904 gt 0 for all 119904 = 0
In 2018 B Abdalla J Alzabut T Abdeljawad [25] inves-tigated the oscillation of solutions for fractional differenceequations with mixed nonlinearities in forms
nabla120572119886(120572)minus1119909 (119905) minus 119901 (119905) 119909 (119905) + 119899sum119894=1
119902119894 (119905) |119909 (119905)|120582119894minus1 119909 (119905)= V (119905) 119905 isin N119886(120572)+1
nablaminus(119898minus119886)119886(120572minus1) 119909 (119905)10038161003816100381610038161003816119905=119886(120572) = 119909 (119886 (120572)) = 119888 119888 isin R(10)
and
119888nabla120572119886(120572)119909 (119905) minus 119901 (119905) 119909 (119905) + 119899sum119894=1
119902119894 (119905) |119909 (119905)|120582119894minus1 119909 (119905)= V (119905) 119905 isin N119886(120572)
nabla119896119909 (119886 (120572)) = 119887119896 119896 isin R 119896 = 0 1 2 119898 minus 1(11)
where 119898 = [120572] + 1 120572 gt 0 119901(119905) V(119905) and 119902119894(119905)(1 le 119894 le 119899) arefunctions defined fromN119886(120572) to119877 and 120582119894(1 le 119894 le 119899) are ratiosof odd positive integers with 1205821 gt gt 120582119897 gt 1 gt 120582119897+1 gt gt120582119899
Inspired by the above results in this paper we discuss theoscillatory behavior of the fractional differential equationalwith damping
119863120572+1minus 119910 (119905) sdot 119863120572minus119910 (119905) minus 119901 (119905) 119891 (119863120572minus119910 (119905))+ 119902 (119905) ℎ (intinfin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904) = 0 119905 gt 0 (12)
where 0 lt 120572 lt 1 is a real number 119863120572minus119910 is the Liouville right-sided fractional derivative of 119910 We always assume that thefollowing conditions are valid
Discrete Dynamics in Nature and Society 3
(1198601) 119901(119905) ge 0 and 119902(119905) ge 0 are continuous functions on119905 isin [1199050infin) 1199050 gt 0(1198602) ℎ 119891 R 997888rarr R are continuous functions with119909ℎ(119909) gt 0 119909119891(119909) gt 0 for 119909 = 0 and there exist positive
constants 1198961 1198962 such that ℎ(119909)119909 ge 1198961 119909119891(119909) ge 1198962 for all119909 = 0(1198603) 1198911015840(119906) le 119906 119891minus1(119906) isin 119862(RR) are continuousfunctions with 119891minus1(119906) gt 0 for 119906 = 0 and there exists somepositive constant 1205721 such that 119891minus1(119906V) ge 1205721119891minus1(119906)119891minus1(V) for119906V = 02 Preliminaries
For convenience some background materials from fractionalcalculus are given
From [4] we can get the definition for Liouville right-side fractional integral and Liouville right-side fractionalderivative on the whole axis R of order 120573 for a function119892 R+ 997888rarr R as follows
(119868120573minus119892) (119905) fl 1Γ (120573) intinfin
119905(V minus 119905)120573minus1 119892 (V) 119889V 119905 gt 0 (13)
(119863120573minus119892) (119905) fl (minus1)lceil120573rceil 119889lceil120573rceil119889119905lceil120573rceil (119868lceil120573rceilminus120573minus 119892) (119905)= (minus1)lceil120573rceil 1Γ (lceil120573rceil minus 120573) int
infin
119905(V minus 119905)lceil120573rceilminus120573minus1 119892 (V) 119889V
119905 gt 0(14)
provided the right hand side is pointwise defined on R+where Γ(sdot) is the gamma function defined by Γ(119905) flintinfin119905
119904119905minus1119890minus119904119889119904 and fl min119911 isin Z 119911 ge 120573 is the ceilingfunction
If 120573 isin (0 1) we have119863120572minus119910 (119905) fl minus 1Γ (1 minus 120572) 119889119889119905 int
infin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 (15)
for 119905 isin R+ fl (0infin)The following relations also existed
(119863(1+120572)minus 119910) (119905) = minus (119863120572minus119910)1015840 (119905) 120572 isin (0 1) 119905 gt 0 (16)
Set
119866 (119905) fl intinfin119905
(V minus 119905)minus120572 119910 (V) 119889V 120572 isin (0 1) (17)
and then
1198661015840 (119905) = minusΓ (1 minus 120572) (119863120572minus119910) (119905) 120572 isin (0 1) (18)
3 Main Results
First we study the oscillation of (12) under the followingcondition
intinfin1199050
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904 = infin (19)
Theorem 1 Suppose that (1198601) minus (1198603) and (19) hold fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that
lim sup119905997888rarrinfin
int1199051199050
V (119904)sdot [[1198961119903 (119904) 119902 (119904) minus
(1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = infin(20)
where 1198961 1198962 are defined as in (1198602) andV (119904) fl exp(int119904
1199050
119901 (V) 119889V) 119904 ge 1199050 (21)
Then every solution of (12) is oscillatory
Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Then there exists 1199051 isin[1199050infin] such that 119910(119905) gt 0 and 119866(119905) gt 0 for 119905 isin [1199051infin] where119866 is defined in (16) From (1198603) (12) and (16) we have
[119891 (119863120572minus119910 (119905)) V (119905)]1015840 = minus1198911015840 (119863120572minus119910 (119905)) V (119905) 119863120572+1minus 119910 (119905)+ 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)
= 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)minus 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) V (119905)
ge 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)minus 119863120572+1minus 119910 (119905) 119863120572minus119910 (119905) V (119905)
= 119902 (119905) ℎ (119866 (119905)) V (119905) gt 0119905 isin [1199050infin]
(22)
Thus 119891(119863120572minus119910(119905))V(119905) is strictly increasing on [1199050infin] SinceV(119905) gt 0 for 119905 isin [1199050infin] and from (1198603) we see that 119863120572minus119910(119905) iseventually of one sign Now we can claim
119863120572minus119910 (119905) lt 0 119905 isin [1199051infin] (23)
If not then there exists 1199052 isin [1199051infin] such that 119863120572minus119910(1199052) gt0 Since 119891(119863120572minus119910(119905))V(119905) is strictly increasing on [1199051infin] it isclear that 119891(119863120572minus119910(119905))V(119905) ge 119891(119863120572minus119910(1199052))V(1199052) fl 119888 gt 0 for119905 isin [1199052infin] Therefore from (18) we have
minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1 ( 119888V (119905))
= 119891minus1 (119888 sdot exp(minusint1199051199050
119901 (V) 119889V))ge 1205721119891minus1 (119888) 119891minus1 (exp(minusint119905
1199050
119901 (V) 119889V)) 119905 isin [1199052infin]
(24)
4 Discrete Dynamics in Nature and Society
Then we get
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V)) le minus 1198661015840 (119905)1205721119891minus1 (119888) Γ (1 minus 120572) 119905 isin [1199052infin] (25)
Integrating the above inequality from 1199052 to 119905 we haveint1199051199052
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904le minusint1199051199052
1198661015840 (119904)1205721119891minus1 (119888) Γ (1 minus 120572)119889119904= minus 119866 (119905) minus 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572)
119905 isin [1199052infin]
(26)
Letting 119905 997888rarr infin we see
intinfin1199052
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904le 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt infin (27)
This is in contradiction with (19) Hence (23) holdsDefine the function 119908 as generalized Riccati substitution
119908 (119905) = 119903 (119905) minusV (119905) 119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (28)
Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (18) (22) (28)and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)
sdot minus [119891 (119863120572minus119910 (119905)) V (119905)]1015840 119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905)le 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot [minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905) ]= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) V (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198961119903 (119905) 119902 (119905) V (119905) + 1199082 (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))119903 (119905) 119891 (119863120572minus119910 (119905)) V (119905)le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905) V (119905) minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905)
(29)
That is
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905) (30)
Taking 119909 = 119908(119905) 119886 = 1198962Γ(1 minus 120572)119903(119905)V(119905) (119886 = 0) and 119887 =1199031015840(119905)119903(119905) from 1198861199092+119887119909 le minus11988724119886 and (30)we could concludethat
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + V (119905) (1199031015840 (119905))241198962Γ (1 minus 120572) 119903 (119905) (31)
Integrating both sides of inequality (31) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)
ge int1199051199050
[[minus1198961119903 (119904) 119902 (119904) V (119904) +
V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119905)]]119889119904 (32)
Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get
lim sup119905997888rarrinfin
int1199051199050
[[minus1198961119903 (119904) 119902 (119904) V (119904)
+ V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 lt 119908 (1199050) lt infin(33)
which is in contradiction with (20)If 119910(119905) is an eventually negative solution of (12) the proof
is similar hence we omit itThe proof is complete
Theorem 2 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that
lim sup119905997888rarrinfin
int1199051199050
[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= infin(34)
where V(119904) are defined inTheorem 1Then every solution of (12)is oscillatory
Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Proceeding the same asin the proof of Theorem 1 we get (23) Define the function119908(119905) as follows
119908(119905) = 119903 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (35)
Discrete Dynamics in Nature and Society 5
Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (16) (18) (22)(35) and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905) [minus119891 (119863120572minus119910 (119905))119866 (119905) ]1015840
= 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905)sdot 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) 119866 (119905) + 119891 (119863120572minus119910 (119905)) 1198661015840 (119905)1198662 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)+ 119903 (119905) 119863120572minus119910 (119905) 119863120572+1minus 119910 (119905)119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905)sdot 119901 (119905) 119891 (119863120572minus119910 (119905)) minus 119902 (119905) ℎ (119866 (119905))119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905)sdot 1199082 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 119901 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)
(36)
That is
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) + 1199031015840 (119905) minus 119903 (119905) 119901 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905) (37)
Taking 119909 = 119908(119905) 119886 = 1198962Γ(1minus120572)119903(119905) (119886 = 0) and 119887 = (1199031015840(119905) minus119903(119905)119901(119905))119903(119905) from 1198861199092 + 119887119909 le minus11988724119886 and (37) we conclude
1199081015840 (119905) le minus1198961119903 (119904) 119902 (119904) + [1199031015840 (119905) minus 119903 (119905) 119901 (119905)]241198962Γ (1 minus 120572) 119903 (119905) (38)
Integrating both sides of inequality (38) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)ge int1199051199050
[[minus1198961119903 (119904) 119902 (119904) +
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904 (39)
Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get
lim sup119905997888rarrinfin
int1199051199050
[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
lt 119908 (1199050) lt infin(40)
which is in contradiction with (34)
If 119910(119905) is an eventually negative solution of (12) the proofis similar here we omit it
The proof is complete
We define a function class119866 setD fl (119905 119904) fl 119905 ge 119904 ge 1199050D0 fl (119905 119904) fl 119905 gt 119904 ge 1199050We say119867(119905 119904) isin 119866 if119867(119905 119904) satisfy119867(119905 119905) = 0 119905 ge 1199050119867 (119905 119904) gt 0 (119905 119904) isin D0 (41)
and 119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904) fl 120597119867(119905 119904)120597119904 on D0 with respect to the secondvariable
Theorem 3 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] and a function119867 isin 119866 satisfies
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]] 119889119904
= infin(42)
Then every solution of (12) is oscillatory
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 2 we get (37)
Multiplying (37) by 119867(119905 119904) and integrating from 1199051 to 119905we get
int1199051199051
1198961119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le minusint1199051199051
119867(119905 119904) 1199081015840 (119904) 119889119904+ minusint1199051199051
119867(119905 119904) 119908 (119904) 1199031015840 (119904)119903 (119904) 119889119904minus int1199051199051
119867(119905 119904) 1199082 (119904) 1198962Γ (1 minus 120572)119903 (119904) 119889119904119905 isin [1199051infin]
(43)
Using the formula integration by parts we obtain
minusint1199051199051
119867(119905 119904) 1199081015840 (119904) 119889119904 = minus [119867 (119905 119904) 119908 (119904)]119904=119905119904=1199051+ int1199051199051
1198671015840119904 (119905 119904) 119908 (119904) 119889119904= 119867 (119905 1199051)119908 (1199051)
+ int1199051199051
1198671015840119904 (119905 119904) 119908 (119904) 119889119904119905 isin [1199051infin)
(44)
6 Discrete Dynamics in Nature and Society
Substituting (44) with (43) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)
minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)
Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget
int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904
le int1199051199051
(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904
le int1199051199051
[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905
1199051
[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904
le int1199051199051
[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905
1199051
[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904
(46)
Substituting (46) in (45) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905
1199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)
Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get
int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)
119905 isin [1199051infin) (48)
Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have
1119867 (119905 1199050) int119905
1199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051
1199050
119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)
Letting 119905 997888rarr infin we obtain
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)
which yields a contradiction to (42) The proof is complete
Discrete Dynamics in Nature and Society 7
Second we study the oscillation of (12) under the follow-ing condition
intinfin1199050
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904 lt infin (51)
Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050
intinfin119879
119891minus1 [ 1V (119905) int
119905
119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)
where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)
If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get
[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)
(53)
Integrating both sides of the above inequality from 1199052 to 119905 wehave
119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904gt 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904119905 isin [1199052infin)
(54)
Hence from (17) we get
minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )
ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )
119905 isin [1199052infin)
(55)
Integrating both sides of the last inequality from 1199052 to 119905 weobtain
119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052
119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906
119905 isin [1199052infin) (56)
Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete
Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem
4 Examples
Example 1 Consider the fractional differential equation
(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int
infin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)
In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since
intinfin1199050
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
(exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
exp(1119905 minus 11199050)119889119905 ge intinfin1199050
exp(minus 11199050)119889119905 = infin(58)
then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin
int1199051
[[1198961119903 (119904) 119902 (119904) minus
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= lim sup119905997888rarrinfin
int1199051
[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]
]119889119904= lim sup119905997888rarrinfin
int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin
(59)
which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory
8 Discrete Dynamics in Nature and Society
Example 2 Consider the fractional differential equation
(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905
(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)
In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that
(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050
Since
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
= lim sup119905997888rarrinfin
1(119905 minus 1199050)14 int119905
1199050
119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin
sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]
= lim sup119905997888rarrinfin
(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin
(61)
which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory
5 Conclusion
In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper
Authorsrsquo Contributions
Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript
Acknowledgments
This research is supported by National Science Foundation ofChina (11671227)
References
[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962
[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974
[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999
[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006
[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983
[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996
[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993
[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997
[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002
[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011
[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010
[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012
[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012
[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013
Discrete Dynamics in Nature and Society 9
[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013
[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013
[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013
[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014
[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017
[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018
[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018
[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018
[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018
[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009
[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015
[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016
[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017
[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016
[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018
[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018
[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015
[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013
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Discrete Dynamics in Nature and Society 3
(1198601) 119901(119905) ge 0 and 119902(119905) ge 0 are continuous functions on119905 isin [1199050infin) 1199050 gt 0(1198602) ℎ 119891 R 997888rarr R are continuous functions with119909ℎ(119909) gt 0 119909119891(119909) gt 0 for 119909 = 0 and there exist positive
constants 1198961 1198962 such that ℎ(119909)119909 ge 1198961 119909119891(119909) ge 1198962 for all119909 = 0(1198603) 1198911015840(119906) le 119906 119891minus1(119906) isin 119862(RR) are continuousfunctions with 119891minus1(119906) gt 0 for 119906 = 0 and there exists somepositive constant 1205721 such that 119891minus1(119906V) ge 1205721119891minus1(119906)119891minus1(V) for119906V = 02 Preliminaries
For convenience some background materials from fractionalcalculus are given
From [4] we can get the definition for Liouville right-side fractional integral and Liouville right-side fractionalderivative on the whole axis R of order 120573 for a function119892 R+ 997888rarr R as follows
(119868120573minus119892) (119905) fl 1Γ (120573) intinfin
119905(V minus 119905)120573minus1 119892 (V) 119889V 119905 gt 0 (13)
(119863120573minus119892) (119905) fl (minus1)lceil120573rceil 119889lceil120573rceil119889119905lceil120573rceil (119868lceil120573rceilminus120573minus 119892) (119905)= (minus1)lceil120573rceil 1Γ (lceil120573rceil minus 120573) int
infin
119905(V minus 119905)lceil120573rceilminus120573minus1 119892 (V) 119889V
119905 gt 0(14)
provided the right hand side is pointwise defined on R+where Γ(sdot) is the gamma function defined by Γ(119905) flintinfin119905
119904119905minus1119890minus119904119889119904 and fl min119911 isin Z 119911 ge 120573 is the ceilingfunction
If 120573 isin (0 1) we have119863120572minus119910 (119905) fl minus 1Γ (1 minus 120572) 119889119889119905 int
infin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 (15)
for 119905 isin R+ fl (0infin)The following relations also existed
(119863(1+120572)minus 119910) (119905) = minus (119863120572minus119910)1015840 (119905) 120572 isin (0 1) 119905 gt 0 (16)
Set
119866 (119905) fl intinfin119905
(V minus 119905)minus120572 119910 (V) 119889V 120572 isin (0 1) (17)
and then
1198661015840 (119905) = minusΓ (1 minus 120572) (119863120572minus119910) (119905) 120572 isin (0 1) (18)
3 Main Results
First we study the oscillation of (12) under the followingcondition
intinfin1199050
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904 = infin (19)
Theorem 1 Suppose that (1198601) minus (1198603) and (19) hold fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that
lim sup119905997888rarrinfin
int1199051199050
V (119904)sdot [[1198961119903 (119904) 119902 (119904) minus
(1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = infin(20)
where 1198961 1198962 are defined as in (1198602) andV (119904) fl exp(int119904
1199050
119901 (V) 119889V) 119904 ge 1199050 (21)
Then every solution of (12) is oscillatory
Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Then there exists 1199051 isin[1199050infin] such that 119910(119905) gt 0 and 119866(119905) gt 0 for 119905 isin [1199051infin] where119866 is defined in (16) From (1198603) (12) and (16) we have
[119891 (119863120572minus119910 (119905)) V (119905)]1015840 = minus1198911015840 (119863120572minus119910 (119905)) V (119905) 119863120572+1minus 119910 (119905)+ 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)
= 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)minus 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) V (119905)
ge 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)minus 119863120572+1minus 119910 (119905) 119863120572minus119910 (119905) V (119905)
= 119902 (119905) ℎ (119866 (119905)) V (119905) gt 0119905 isin [1199050infin]
(22)
Thus 119891(119863120572minus119910(119905))V(119905) is strictly increasing on [1199050infin] SinceV(119905) gt 0 for 119905 isin [1199050infin] and from (1198603) we see that 119863120572minus119910(119905) iseventually of one sign Now we can claim
119863120572minus119910 (119905) lt 0 119905 isin [1199051infin] (23)
If not then there exists 1199052 isin [1199051infin] such that 119863120572minus119910(1199052) gt0 Since 119891(119863120572minus119910(119905))V(119905) is strictly increasing on [1199051infin] it isclear that 119891(119863120572minus119910(119905))V(119905) ge 119891(119863120572minus119910(1199052))V(1199052) fl 119888 gt 0 for119905 isin [1199052infin] Therefore from (18) we have
minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1 ( 119888V (119905))
= 119891minus1 (119888 sdot exp(minusint1199051199050
119901 (V) 119889V))ge 1205721119891minus1 (119888) 119891minus1 (exp(minusint119905
1199050
119901 (V) 119889V)) 119905 isin [1199052infin]
(24)
4 Discrete Dynamics in Nature and Society
Then we get
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V)) le minus 1198661015840 (119905)1205721119891minus1 (119888) Γ (1 minus 120572) 119905 isin [1199052infin] (25)
Integrating the above inequality from 1199052 to 119905 we haveint1199051199052
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904le minusint1199051199052
1198661015840 (119904)1205721119891minus1 (119888) Γ (1 minus 120572)119889119904= minus 119866 (119905) minus 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572)
119905 isin [1199052infin]
(26)
Letting 119905 997888rarr infin we see
intinfin1199052
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904le 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt infin (27)
This is in contradiction with (19) Hence (23) holdsDefine the function 119908 as generalized Riccati substitution
119908 (119905) = 119903 (119905) minusV (119905) 119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (28)
Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (18) (22) (28)and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)
sdot minus [119891 (119863120572minus119910 (119905)) V (119905)]1015840 119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905)le 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot [minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905) ]= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) V (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198961119903 (119905) 119902 (119905) V (119905) + 1199082 (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))119903 (119905) 119891 (119863120572minus119910 (119905)) V (119905)le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905) V (119905) minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905)
(29)
That is
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905) (30)
Taking 119909 = 119908(119905) 119886 = 1198962Γ(1 minus 120572)119903(119905)V(119905) (119886 = 0) and 119887 =1199031015840(119905)119903(119905) from 1198861199092+119887119909 le minus11988724119886 and (30)we could concludethat
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + V (119905) (1199031015840 (119905))241198962Γ (1 minus 120572) 119903 (119905) (31)
Integrating both sides of inequality (31) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)
ge int1199051199050
[[minus1198961119903 (119904) 119902 (119904) V (119904) +
V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119905)]]119889119904 (32)
Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get
lim sup119905997888rarrinfin
int1199051199050
[[minus1198961119903 (119904) 119902 (119904) V (119904)
+ V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 lt 119908 (1199050) lt infin(33)
which is in contradiction with (20)If 119910(119905) is an eventually negative solution of (12) the proof
is similar hence we omit itThe proof is complete
Theorem 2 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that
lim sup119905997888rarrinfin
int1199051199050
[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= infin(34)
where V(119904) are defined inTheorem 1Then every solution of (12)is oscillatory
Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Proceeding the same asin the proof of Theorem 1 we get (23) Define the function119908(119905) as follows
119908(119905) = 119903 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (35)
Discrete Dynamics in Nature and Society 5
Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (16) (18) (22)(35) and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905) [minus119891 (119863120572minus119910 (119905))119866 (119905) ]1015840
= 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905)sdot 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) 119866 (119905) + 119891 (119863120572minus119910 (119905)) 1198661015840 (119905)1198662 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)+ 119903 (119905) 119863120572minus119910 (119905) 119863120572+1minus 119910 (119905)119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905)sdot 119901 (119905) 119891 (119863120572minus119910 (119905)) minus 119902 (119905) ℎ (119866 (119905))119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905)sdot 1199082 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 119901 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)
(36)
That is
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) + 1199031015840 (119905) minus 119903 (119905) 119901 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905) (37)
Taking 119909 = 119908(119905) 119886 = 1198962Γ(1minus120572)119903(119905) (119886 = 0) and 119887 = (1199031015840(119905) minus119903(119905)119901(119905))119903(119905) from 1198861199092 + 119887119909 le minus11988724119886 and (37) we conclude
1199081015840 (119905) le minus1198961119903 (119904) 119902 (119904) + [1199031015840 (119905) minus 119903 (119905) 119901 (119905)]241198962Γ (1 minus 120572) 119903 (119905) (38)
Integrating both sides of inequality (38) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)ge int1199051199050
[[minus1198961119903 (119904) 119902 (119904) +
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904 (39)
Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get
lim sup119905997888rarrinfin
int1199051199050
[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
lt 119908 (1199050) lt infin(40)
which is in contradiction with (34)
If 119910(119905) is an eventually negative solution of (12) the proofis similar here we omit it
The proof is complete
We define a function class119866 setD fl (119905 119904) fl 119905 ge 119904 ge 1199050D0 fl (119905 119904) fl 119905 gt 119904 ge 1199050We say119867(119905 119904) isin 119866 if119867(119905 119904) satisfy119867(119905 119905) = 0 119905 ge 1199050119867 (119905 119904) gt 0 (119905 119904) isin D0 (41)
and 119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904) fl 120597119867(119905 119904)120597119904 on D0 with respect to the secondvariable
Theorem 3 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] and a function119867 isin 119866 satisfies
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]] 119889119904
= infin(42)
Then every solution of (12) is oscillatory
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 2 we get (37)
Multiplying (37) by 119867(119905 119904) and integrating from 1199051 to 119905we get
int1199051199051
1198961119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le minusint1199051199051
119867(119905 119904) 1199081015840 (119904) 119889119904+ minusint1199051199051
119867(119905 119904) 119908 (119904) 1199031015840 (119904)119903 (119904) 119889119904minus int1199051199051
119867(119905 119904) 1199082 (119904) 1198962Γ (1 minus 120572)119903 (119904) 119889119904119905 isin [1199051infin]
(43)
Using the formula integration by parts we obtain
minusint1199051199051
119867(119905 119904) 1199081015840 (119904) 119889119904 = minus [119867 (119905 119904) 119908 (119904)]119904=119905119904=1199051+ int1199051199051
1198671015840119904 (119905 119904) 119908 (119904) 119889119904= 119867 (119905 1199051)119908 (1199051)
+ int1199051199051
1198671015840119904 (119905 119904) 119908 (119904) 119889119904119905 isin [1199051infin)
(44)
6 Discrete Dynamics in Nature and Society
Substituting (44) with (43) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)
minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)
Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget
int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904
le int1199051199051
(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904
le int1199051199051
[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905
1199051
[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904
le int1199051199051
[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905
1199051
[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904
(46)
Substituting (46) in (45) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905
1199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)
Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get
int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)
119905 isin [1199051infin) (48)
Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have
1119867 (119905 1199050) int119905
1199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051
1199050
119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)
Letting 119905 997888rarr infin we obtain
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)
which yields a contradiction to (42) The proof is complete
Discrete Dynamics in Nature and Society 7
Second we study the oscillation of (12) under the follow-ing condition
intinfin1199050
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904 lt infin (51)
Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050
intinfin119879
119891minus1 [ 1V (119905) int
119905
119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)
where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)
If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get
[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)
(53)
Integrating both sides of the above inequality from 1199052 to 119905 wehave
119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904gt 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904119905 isin [1199052infin)
(54)
Hence from (17) we get
minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )
ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )
119905 isin [1199052infin)
(55)
Integrating both sides of the last inequality from 1199052 to 119905 weobtain
119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052
119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906
119905 isin [1199052infin) (56)
Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete
Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem
4 Examples
Example 1 Consider the fractional differential equation
(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int
infin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)
In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since
intinfin1199050
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
(exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
exp(1119905 minus 11199050)119889119905 ge intinfin1199050
exp(minus 11199050)119889119905 = infin(58)
then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin
int1199051
[[1198961119903 (119904) 119902 (119904) minus
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= lim sup119905997888rarrinfin
int1199051
[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]
]119889119904= lim sup119905997888rarrinfin
int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin
(59)
which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory
8 Discrete Dynamics in Nature and Society
Example 2 Consider the fractional differential equation
(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905
(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)
In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that
(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050
Since
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
= lim sup119905997888rarrinfin
1(119905 minus 1199050)14 int119905
1199050
119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin
sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]
= lim sup119905997888rarrinfin
(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin
(61)
which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory
5 Conclusion
In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper
Authorsrsquo Contributions
Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript
Acknowledgments
This research is supported by National Science Foundation ofChina (11671227)
References
[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962
[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974
[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999
[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006
[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983
[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996
[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993
[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997
[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002
[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011
[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010
[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012
[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012
[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013
Discrete Dynamics in Nature and Society 9
[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013
[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013
[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013
[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014
[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017
[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018
[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018
[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018
[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018
[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009
[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015
[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016
[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017
[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016
[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018
[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018
[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015
[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013
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4 Discrete Dynamics in Nature and Society
Then we get
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V)) le minus 1198661015840 (119905)1205721119891minus1 (119888) Γ (1 minus 120572) 119905 isin [1199052infin] (25)
Integrating the above inequality from 1199052 to 119905 we haveint1199051199052
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904le minusint1199051199052
1198661015840 (119904)1205721119891minus1 (119888) Γ (1 minus 120572)119889119904= minus 119866 (119905) minus 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572)
119905 isin [1199052infin]
(26)
Letting 119905 997888rarr infin we see
intinfin1199052
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904le 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt infin (27)
This is in contradiction with (19) Hence (23) holdsDefine the function 119908 as generalized Riccati substitution
119908 (119905) = 119903 (119905) minusV (119905) 119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (28)
Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (18) (22) (28)and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)
sdot minus [119891 (119863120572minus119910 (119905)) V (119905)]1015840 119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905)le 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot [minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905) ]= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) V (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198961119903 (119905) 119902 (119905) V (119905) + 1199082 (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))119903 (119905) 119891 (119863120572minus119910 (119905)) V (119905)le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905) V (119905) minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905)
(29)
That is
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905) (30)
Taking 119909 = 119908(119905) 119886 = 1198962Γ(1 minus 120572)119903(119905)V(119905) (119886 = 0) and 119887 =1199031015840(119905)119903(119905) from 1198861199092+119887119909 le minus11988724119886 and (30)we could concludethat
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + V (119905) (1199031015840 (119905))241198962Γ (1 minus 120572) 119903 (119905) (31)
Integrating both sides of inequality (31) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)
ge int1199051199050
[[minus1198961119903 (119904) 119902 (119904) V (119904) +
V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119905)]]119889119904 (32)
Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get
lim sup119905997888rarrinfin
int1199051199050
[[minus1198961119903 (119904) 119902 (119904) V (119904)
+ V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 lt 119908 (1199050) lt infin(33)
which is in contradiction with (20)If 119910(119905) is an eventually negative solution of (12) the proof
is similar hence we omit itThe proof is complete
Theorem 2 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that
lim sup119905997888rarrinfin
int1199051199050
[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= infin(34)
where V(119904) are defined inTheorem 1Then every solution of (12)is oscillatory
Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Proceeding the same asin the proof of Theorem 1 we get (23) Define the function119908(119905) as follows
119908(119905) = 119903 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (35)
Discrete Dynamics in Nature and Society 5
Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (16) (18) (22)(35) and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905) [minus119891 (119863120572minus119910 (119905))119866 (119905) ]1015840
= 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905)sdot 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) 119866 (119905) + 119891 (119863120572minus119910 (119905)) 1198661015840 (119905)1198662 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)+ 119903 (119905) 119863120572minus119910 (119905) 119863120572+1minus 119910 (119905)119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905)sdot 119901 (119905) 119891 (119863120572minus119910 (119905)) minus 119902 (119905) ℎ (119866 (119905))119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905)sdot 1199082 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 119901 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)
(36)
That is
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) + 1199031015840 (119905) minus 119903 (119905) 119901 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905) (37)
Taking 119909 = 119908(119905) 119886 = 1198962Γ(1minus120572)119903(119905) (119886 = 0) and 119887 = (1199031015840(119905) minus119903(119905)119901(119905))119903(119905) from 1198861199092 + 119887119909 le minus11988724119886 and (37) we conclude
1199081015840 (119905) le minus1198961119903 (119904) 119902 (119904) + [1199031015840 (119905) minus 119903 (119905) 119901 (119905)]241198962Γ (1 minus 120572) 119903 (119905) (38)
Integrating both sides of inequality (38) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)ge int1199051199050
[[minus1198961119903 (119904) 119902 (119904) +
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904 (39)
Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get
lim sup119905997888rarrinfin
int1199051199050
[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
lt 119908 (1199050) lt infin(40)
which is in contradiction with (34)
If 119910(119905) is an eventually negative solution of (12) the proofis similar here we omit it
The proof is complete
We define a function class119866 setD fl (119905 119904) fl 119905 ge 119904 ge 1199050D0 fl (119905 119904) fl 119905 gt 119904 ge 1199050We say119867(119905 119904) isin 119866 if119867(119905 119904) satisfy119867(119905 119905) = 0 119905 ge 1199050119867 (119905 119904) gt 0 (119905 119904) isin D0 (41)
and 119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904) fl 120597119867(119905 119904)120597119904 on D0 with respect to the secondvariable
Theorem 3 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] and a function119867 isin 119866 satisfies
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]] 119889119904
= infin(42)
Then every solution of (12) is oscillatory
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 2 we get (37)
Multiplying (37) by 119867(119905 119904) and integrating from 1199051 to 119905we get
int1199051199051
1198961119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le minusint1199051199051
119867(119905 119904) 1199081015840 (119904) 119889119904+ minusint1199051199051
119867(119905 119904) 119908 (119904) 1199031015840 (119904)119903 (119904) 119889119904minus int1199051199051
119867(119905 119904) 1199082 (119904) 1198962Γ (1 minus 120572)119903 (119904) 119889119904119905 isin [1199051infin]
(43)
Using the formula integration by parts we obtain
minusint1199051199051
119867(119905 119904) 1199081015840 (119904) 119889119904 = minus [119867 (119905 119904) 119908 (119904)]119904=119905119904=1199051+ int1199051199051
1198671015840119904 (119905 119904) 119908 (119904) 119889119904= 119867 (119905 1199051)119908 (1199051)
+ int1199051199051
1198671015840119904 (119905 119904) 119908 (119904) 119889119904119905 isin [1199051infin)
(44)
6 Discrete Dynamics in Nature and Society
Substituting (44) with (43) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)
minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)
Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget
int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904
le int1199051199051
(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904
le int1199051199051
[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905
1199051
[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904
le int1199051199051
[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905
1199051
[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904
(46)
Substituting (46) in (45) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905
1199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)
Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get
int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)
119905 isin [1199051infin) (48)
Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have
1119867 (119905 1199050) int119905
1199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051
1199050
119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)
Letting 119905 997888rarr infin we obtain
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)
which yields a contradiction to (42) The proof is complete
Discrete Dynamics in Nature and Society 7
Second we study the oscillation of (12) under the follow-ing condition
intinfin1199050
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904 lt infin (51)
Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050
intinfin119879
119891minus1 [ 1V (119905) int
119905
119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)
where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)
If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get
[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)
(53)
Integrating both sides of the above inequality from 1199052 to 119905 wehave
119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904gt 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904119905 isin [1199052infin)
(54)
Hence from (17) we get
minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )
ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )
119905 isin [1199052infin)
(55)
Integrating both sides of the last inequality from 1199052 to 119905 weobtain
119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052
119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906
119905 isin [1199052infin) (56)
Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete
Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem
4 Examples
Example 1 Consider the fractional differential equation
(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int
infin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)
In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since
intinfin1199050
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
(exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
exp(1119905 minus 11199050)119889119905 ge intinfin1199050
exp(minus 11199050)119889119905 = infin(58)
then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin
int1199051
[[1198961119903 (119904) 119902 (119904) minus
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= lim sup119905997888rarrinfin
int1199051
[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]
]119889119904= lim sup119905997888rarrinfin
int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin
(59)
which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory
8 Discrete Dynamics in Nature and Society
Example 2 Consider the fractional differential equation
(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905
(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)
In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that
(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050
Since
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
= lim sup119905997888rarrinfin
1(119905 minus 1199050)14 int119905
1199050
119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin
sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]
= lim sup119905997888rarrinfin
(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin
(61)
which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory
5 Conclusion
In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper
Authorsrsquo Contributions
Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript
Acknowledgments
This research is supported by National Science Foundation ofChina (11671227)
References
[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962
[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974
[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999
[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006
[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983
[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996
[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993
[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997
[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002
[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011
[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010
[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012
[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012
[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013
Discrete Dynamics in Nature and Society 9
[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013
[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013
[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013
[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014
[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017
[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018
[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018
[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018
[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018
[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009
[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015
[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016
[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017
[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016
[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018
[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018
[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015
[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013
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Discrete Dynamics in Nature and Society 5
Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (16) (18) (22)(35) and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905) [minus119891 (119863120572minus119910 (119905))119866 (119905) ]1015840
= 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905)sdot 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) 119866 (119905) + 119891 (119863120572minus119910 (119905)) 1198661015840 (119905)1198662 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)+ 119903 (119905) 119863120572minus119910 (119905) 119863120572+1minus 119910 (119905)119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905)sdot 119901 (119905) 119891 (119863120572minus119910 (119905)) minus 119902 (119905) ℎ (119866 (119905))119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905)sdot 1199082 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 119901 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)
(36)
That is
1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) + 1199031015840 (119905) minus 119903 (119905) 119901 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905) (37)
Taking 119909 = 119908(119905) 119886 = 1198962Γ(1minus120572)119903(119905) (119886 = 0) and 119887 = (1199031015840(119905) minus119903(119905)119901(119905))119903(119905) from 1198861199092 + 119887119909 le minus11988724119886 and (37) we conclude
1199081015840 (119905) le minus1198961119903 (119904) 119902 (119904) + [1199031015840 (119905) minus 119903 (119905) 119901 (119905)]241198962Γ (1 minus 120572) 119903 (119905) (38)
Integrating both sides of inequality (38) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)ge int1199051199050
[[minus1198961119903 (119904) 119902 (119904) +
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904 (39)
Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get
lim sup119905997888rarrinfin
int1199051199050
[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
lt 119908 (1199050) lt infin(40)
which is in contradiction with (34)
If 119910(119905) is an eventually negative solution of (12) the proofis similar here we omit it
The proof is complete
We define a function class119866 setD fl (119905 119904) fl 119905 ge 119904 ge 1199050D0 fl (119905 119904) fl 119905 gt 119904 ge 1199050We say119867(119905 119904) isin 119866 if119867(119905 119904) satisfy119867(119905 119905) = 0 119905 ge 1199050119867 (119905 119904) gt 0 (119905 119904) isin D0 (41)
and 119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904) fl 120597119867(119905 119904)120597119904 on D0 with respect to the secondvariable
Theorem 3 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] and a function119867 isin 119866 satisfies
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]] 119889119904
= infin(42)
Then every solution of (12) is oscillatory
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 2 we get (37)
Multiplying (37) by 119867(119905 119904) and integrating from 1199051 to 119905we get
int1199051199051
1198961119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le minusint1199051199051
119867(119905 119904) 1199081015840 (119904) 119889119904+ minusint1199051199051
119867(119905 119904) 119908 (119904) 1199031015840 (119904)119903 (119904) 119889119904minus int1199051199051
119867(119905 119904) 1199082 (119904) 1198962Γ (1 minus 120572)119903 (119904) 119889119904119905 isin [1199051infin]
(43)
Using the formula integration by parts we obtain
minusint1199051199051
119867(119905 119904) 1199081015840 (119904) 119889119904 = minus [119867 (119905 119904) 119908 (119904)]119904=119905119904=1199051+ int1199051199051
1198671015840119904 (119905 119904) 119908 (119904) 119889119904= 119867 (119905 1199051)119908 (1199051)
+ int1199051199051
1198671015840119904 (119905 119904) 119908 (119904) 119889119904119905 isin [1199051infin)
(44)
6 Discrete Dynamics in Nature and Society
Substituting (44) with (43) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)
minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)
Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget
int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904
le int1199051199051
(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904
le int1199051199051
[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905
1199051
[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904
le int1199051199051
[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905
1199051
[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904
(46)
Substituting (46) in (45) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905
1199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)
Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get
int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)
119905 isin [1199051infin) (48)
Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have
1119867 (119905 1199050) int119905
1199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051
1199050
119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)
Letting 119905 997888rarr infin we obtain
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)
which yields a contradiction to (42) The proof is complete
Discrete Dynamics in Nature and Society 7
Second we study the oscillation of (12) under the follow-ing condition
intinfin1199050
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904 lt infin (51)
Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050
intinfin119879
119891minus1 [ 1V (119905) int
119905
119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)
where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)
If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get
[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)
(53)
Integrating both sides of the above inequality from 1199052 to 119905 wehave
119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904gt 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904119905 isin [1199052infin)
(54)
Hence from (17) we get
minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )
ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )
119905 isin [1199052infin)
(55)
Integrating both sides of the last inequality from 1199052 to 119905 weobtain
119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052
119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906
119905 isin [1199052infin) (56)
Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete
Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem
4 Examples
Example 1 Consider the fractional differential equation
(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int
infin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)
In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since
intinfin1199050
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
(exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
exp(1119905 minus 11199050)119889119905 ge intinfin1199050
exp(minus 11199050)119889119905 = infin(58)
then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin
int1199051
[[1198961119903 (119904) 119902 (119904) minus
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= lim sup119905997888rarrinfin
int1199051
[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]
]119889119904= lim sup119905997888rarrinfin
int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin
(59)
which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory
8 Discrete Dynamics in Nature and Society
Example 2 Consider the fractional differential equation
(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905
(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)
In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that
(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050
Since
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
= lim sup119905997888rarrinfin
1(119905 minus 1199050)14 int119905
1199050
119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin
sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]
= lim sup119905997888rarrinfin
(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin
(61)
which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory
5 Conclusion
In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper
Authorsrsquo Contributions
Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript
Acknowledgments
This research is supported by National Science Foundation ofChina (11671227)
References
[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962
[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974
[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999
[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006
[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983
[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996
[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993
[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997
[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002
[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011
[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010
[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012
[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012
[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013
Discrete Dynamics in Nature and Society 9
[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013
[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013
[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013
[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014
[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017
[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018
[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018
[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018
[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018
[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009
[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015
[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016
[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017
[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016
[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018
[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018
[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015
[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013
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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Discrete Dynamics in Nature and Society
Substituting (44) with (43) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)
minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)
Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget
int1199051199051
[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904
le int1199051199051
(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904
le int1199051199051
[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905
1199051
[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904
le int1199051199051
[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905
1199051
[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904
(46)
Substituting (46) in (45) we have
1198961 int1199051199051
119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905
1199051
119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)
Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get
int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)
119905 isin [1199051infin) (48)
Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have
1119867 (119905 1199050) int119905
1199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051
[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904
minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051
1199050
119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)
Letting 119905 997888rarr infin we obtain
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
le int11990511199050
119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)
which yields a contradiction to (42) The proof is complete
Discrete Dynamics in Nature and Society 7
Second we study the oscillation of (12) under the follow-ing condition
intinfin1199050
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904 lt infin (51)
Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050
intinfin119879
119891minus1 [ 1V (119905) int
119905
119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)
where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)
If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get
[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)
(53)
Integrating both sides of the above inequality from 1199052 to 119905 wehave
119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904gt 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904119905 isin [1199052infin)
(54)
Hence from (17) we get
minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )
ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )
119905 isin [1199052infin)
(55)
Integrating both sides of the last inequality from 1199052 to 119905 weobtain
119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052
119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906
119905 isin [1199052infin) (56)
Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete
Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem
4 Examples
Example 1 Consider the fractional differential equation
(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int
infin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)
In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since
intinfin1199050
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
(exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
exp(1119905 minus 11199050)119889119905 ge intinfin1199050
exp(minus 11199050)119889119905 = infin(58)
then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin
int1199051
[[1198961119903 (119904) 119902 (119904) minus
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= lim sup119905997888rarrinfin
int1199051
[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]
]119889119904= lim sup119905997888rarrinfin
int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin
(59)
which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory
8 Discrete Dynamics in Nature and Society
Example 2 Consider the fractional differential equation
(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905
(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)
In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that
(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050
Since
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
= lim sup119905997888rarrinfin
1(119905 minus 1199050)14 int119905
1199050
119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin
sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]
= lim sup119905997888rarrinfin
(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin
(61)
which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory
5 Conclusion
In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper
Authorsrsquo Contributions
Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript
Acknowledgments
This research is supported by National Science Foundation ofChina (11671227)
References
[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962
[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974
[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999
[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006
[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983
[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996
[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993
[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997
[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002
[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011
[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010
[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012
[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012
[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013
Discrete Dynamics in Nature and Society 9
[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013
[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013
[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013
[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014
[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017
[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018
[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018
[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018
[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018
[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009
[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015
[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016
[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017
[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016
[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018
[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018
[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015
[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013
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Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
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Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Discrete Dynamics in Nature and Society 7
Second we study the oscillation of (12) under the follow-ing condition
intinfin1199050
119891minus1 (exp(minusint1199041199050
119901 (V) 119889V))119889119904 lt infin (51)
Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050
intinfin119879
119891minus1 [ 1V (119905) int
119905
119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)
where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)
If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get
[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)
(53)
Integrating both sides of the above inequality from 1199052 to 119905 wehave
119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904gt 1198961119871int119905
1199052
119902 (119904) V (119904) 119889119904119905 isin [1199052infin)
(54)
Hence from (17) we get
minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )
ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )
119905 isin [1199052infin)
(55)
Integrating both sides of the last inequality from 1199052 to 119905 weobtain
119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052
119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906
119905 isin [1199052infin) (56)
Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete
Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin
1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0
From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem
4 Examples
Example 1 Consider the fractional differential equation
(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int
infin
119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)
In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since
intinfin1199050
119891minus1 (exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
(exp(minusint1199051199050
119901 (V) 119889V))119889119905= intinfin1199050
exp(1119905 minus 11199050)119889119905 ge intinfin1199050
exp(minus 11199050)119889119905 = infin(58)
then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin
int1199051
[[1198961119903 (119904) 119902 (119904) minus
[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904
= lim sup119905997888rarrinfin
int1199051
[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]
]119889119904= lim sup119905997888rarrinfin
int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin
(59)
which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory
8 Discrete Dynamics in Nature and Society
Example 2 Consider the fractional differential equation
(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905
(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)
In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that
(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050
Since
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
= lim sup119905997888rarrinfin
1(119905 minus 1199050)14 int119905
1199050
119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin
sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]
= lim sup119905997888rarrinfin
(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin
(61)
which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory
5 Conclusion
In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper
Authorsrsquo Contributions
Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript
Acknowledgments
This research is supported by National Science Foundation ofChina (11671227)
References
[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962
[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974
[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999
[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006
[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983
[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996
[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993
[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997
[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002
[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011
[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010
[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012
[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012
[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013
Discrete Dynamics in Nature and Society 9
[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013
[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013
[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013
[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014
[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017
[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018
[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018
[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018
[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018
[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009
[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015
[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016
[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017
[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016
[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018
[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018
[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015
[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Discrete Dynamics in Nature and Society
Example 2 Consider the fractional differential equation
(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905
(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)
In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that
(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050
Since
lim sup119905997888rarrinfin
1119867 (119905 1199050)sdot int1199051199050
119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904
= lim sup119905997888rarrinfin
1(119905 minus 1199050)14 int119905
1199050
119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin
sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]
= lim sup119905997888rarrinfin
(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin
(61)
which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory
5 Conclusion
In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper
Authorsrsquo Contributions
Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript
Acknowledgments
This research is supported by National Science Foundation ofChina (11671227)
References
[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962
[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974
[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999
[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006
[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983
[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996
[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993
[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997
[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002
[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011
[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010
[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012
[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012
[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013
Discrete Dynamics in Nature and Society 9
[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013
[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013
[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013
[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014
[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017
[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018
[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018
[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018
[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018
[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009
[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015
[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016
[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017
[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016
[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018
[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018
[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015
[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Discrete Dynamics in Nature and Society 9
[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013
[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013
[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013
[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014
[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017
[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018
[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018
[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018
[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018
[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009
[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015
[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016
[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017
[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016
[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018
[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018
[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015
[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
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