Research ArticleSign-Changing Solutions for Nonlinear Operator Equations

Yanbin Sang

Department of Mathematics North University of China Taiyuan Shanxi 030051 China

Correspondence should be addressed to Yanbin Sang sangyanbin126com

Received 20 December 2013 Accepted 29 January 2014 Published 6 March 2014

Academic Editor Qingkai Kong

Copyright copy 2014 Yanbin Sang This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The existence of six solutions for nonlinear operator equations is obtained by using the topological degree and fixed point indextheory These six solutions are all nonzero Two of them are positive the other two are negative and the fifth and sixth ones areboth sign-changing solutions Furthermore the theoretical results are applied to elliptic partial differential equations

1 Introduction

In recent years motivated by some ecological problemsmuch attention has been attached to the existence of sign-changing solutions for nonlinear partial differential equations(see [1ndash4] and the references therein) We note that theproofs of main results in [1ndash4] depend upon critical pointtheory However some concrete nonlinear problems have novariational structures [5] To overcome this difficulty in [6]Zhang studied the existence of sign-changing solution fornonlinear operator equations by using the cone theory andcombining uniformly positive condition

Xu [7] studied multiple sign-changing solutions to thefollowing119898-point boundary value problems

11991010158401015840

(119905) + 119891 (119910 (119905)) = 0 0 le 119905 le 1

119910 (0) = 0 119910 (1) =

119898minus2

sum

119894=1

120572119894119910 (120578119894)

(1)

where 0 lt 120572119894 119894 = 1 2 119898minus2 0 lt 120578

1lt 1205782lt sdot sdot sdot lt 120578

119898minus2lt 1

We list some assumptions as follows

(A1) Suppose that the sequence of positive solutions to theequation

sinradic119909 =119898minus2

sum

119894=1

120572119894sin 120578119894radic119909 (2)

is 1205821lt 1205822lt sdot sdot sdot lt 120582

119899lt 120582119899+1

lt sdot sdot sdot

(A2) 0 lt sum

119898minus2

119894=1120572119894lt 1 119891 R rarr R is a continuous

function 119891(0) = 0 and 119909119891(119909) gt 0 for all 119909 isin R 0(A3) let 120573

0= lim

119909rarr0(119891(119909)119909) and 120573

infin= lim

|119909|rarrinfin(119891(119909)

119909) There exist positive integers 1198990and 1198991such that

12058221198990

lt 1205730lt 12058221198990+1

12058221198991

lt 120573infinlt 12058221198991+1

(3)

(A4) there exists 119862

0gt 1 such that

1003816100381610038161003816119891 (119909)

1003816100381610038161003816lt

2 (1 minus sum119898minus2

119894=1120572119894120578119894)

5 minus sum119898minus2

119894=1120572119894120578119894

1198620

(4)

for all 119909 isin R with |119909| le 1198620

Theorem 1 (see [7]) Suppose that conditions (1198601)ndash(1198604) are

satisfied Then the problem (1) has at least two sign-changingsolutions Moreover the problem (1) also has at least twopositive solutions and two negative solutions

Based on [7] many authors studied the sign-changingsolutions of differential and difference equations For exam-ple Yang [8] considered the existence of multiple sign-changing solutions for the problem (1) Compared withTheorem 1 Yang employed the following assumption whichis different from (A

4)

(A10158404) There exists 119879 gt 0 such that

1003816100381610038161003816119891 (119909)

1003816100381610038161003816lt 2 (1 minus 120572120578) |119909| |119909| le 119879 (5)

Hindawi Publishing CorporationJournal of OperatorsVolume 2014 Article ID 379056 6 pageshttpdxdoiorg1011552014379056

2 Journal of Operators

Pang et al [9] investigated multiple sign-changing solutionsof fourth-order differential equation boundary value prob-lems Moreover Wei and Pang [10] established the existencetheorem ofmultiple sign-changing solutions for fourth-orderboundary value problems Y Li and F Li [11] studied twosign-changing solutions of a class of second-order integralboundary value problems by computing the eigenvaluesand the algebraic multiplicities of the corresponding linearproblems He et al [12] discussed the existence of sign-changing solutions for a class of discrete boundary valueproblems and a concrete example was also given Veryrecently Yang [13] investigated the following discrete fourthNeumann boundary value problems

Δ4119906 (119905 minus 2) minus 120572Δ

2119906 (119905 minus 1) + 120573 (119905) = 119891 (119905 119906 (119905))

119905 isin [2 119879]119885

Δ119906 (1) = Δ119906 (119879) = Δ3119906 (0) = Δ

3

(119879 minus 1) = 0

(6)

The author employed similar conditions with (1198601)ndash(1198604) and

obtained a similar result to Theorem 1 (see Theorem 51 in[13])

The main purpose of this paper is to abstract moregeneral conditions from (119860

1)ndash(1198604) of Theorem 1 obtain the

existence theorem of sign-changing solutions for generaloperator equations and then apply the abstract resultobtained in this paper to nonlinear elliptic partial differentialequations

2 Preliminaries and Some Lemmas

For the discussion of the following sections we state herepreliminary definitions and known results on cones partialorderings and topological degree theory which can be foundin [14ndash18]

Let 119864 be a real Banach space Given a cone 119875 sub 119864 wedefine a partial ordering le with respect to 119875 by 119909 le 119910 if andonly if 119910minus119909 isin 119875 A cone119875 is said to be normal if there exists aconstant119873 gt 0 such that 120579 le 119909 le 119910 implies 119909 le 119873119910 thesmallest119873 is called the normal constant of 119875 119875 is called solidif it contains interior that is int119875 = 0 If 119909 le 119910 and 119909 = 119910 wewrite 119909 lt 119910 if cone 119875 is solid and 119910 minus 119909 isin int119875 we write119909 ≪ 119910 119875 is reproducing if 119875 minus 119875 = 119864 and total if 119875 minus 119875 = 119864Let 119861 119864 rarr 119864 be a bounded linear operator 119861 is said tobe positive if 119861(119875) sub 119875 An operator 119860 is strongly increasingthat is 119909 lt 119910 implies 119860119909 ≪ 119860119910 If 119860 is a linear operator 119860is strongly increasing which implies 119860 is strongly positive Afixed point 119906 of operator119860 is said to be a sign-changing fixedpoint if 119906 notin 119875cup(minus119875) If 119909

0isin 119864120579 satisfies 120582119860119909

0= 1199090 where

120582 is some real number then 120582 is called an eigenvalue of119860 and1199090is called an eigenfunction belonging to the eigenvalue 120582

Definition 2 (see [16]) Let 1198641 1198642be real Banach spaces and

let 119863 sub 1198641contain the outside of a ball 119909 119909 le 119903 and

119860 119863 rarr 1198642 The operator 119860 is called asymptotically linear

if there is a bounded linear operator 119861 1198641rarr 1198642such that

lim119909rarr+infin

119860119909 minus 119861119909

119909

= 0 (7)

The operator 119861 involved in the definition of an asymptoticallylinear operator 119860 is uniquely determined It is called thederivative of 119860 at infinity and is denoted by 1198601015840

infin

Definition 3 (see [16 18]) Let119883 be a retract of 119864 and let119880 sub

119883 be a relatively bounded open set of 119883 Suppose that 119860

119880 rarr 119883 is completely continuous and has no fixed point on120597119880 Let the positive integer 119894(119860 119880119883) be defined by

119894 (119860 119880119883) = deg (119868 minus 119860119903 119879119877cap 119903minus1

(119880) 120579) (8)

where 119903 119864 rarr 119883 is an arbitrary retraction and 119877 is a largeenough positive number such that 119880 sub 119879

119877= 119909 | 119909 isin

119864 119909 lt 119877 Then 119894(119860 119880119883) is called the fixed point indexof 119860 on 119880 with respect to119883

Lemma 4 (see [14 18]) Let 119875 be a reproducing cone and let 119861be a positive completely continuous linear operator with 119903(119861) gt0 where 119903(119861) denotes the spectral radius of 119861 Then 119861 has apositive eigenfunction in119875120579 corresponding to the eigenvalue(119903(119861))

minus1

Lemma 5 (see [18]) Let 119875 be a cone in 119864 and let Ω be abounded open set of119864 and 120579 isin Ω Assume that119860 119875capΩ rarr 119875

is a condensing operator If 119860119909 ge 119909 for every 119909 isin 119875 cap 120597Ω then119894(119860 119875 cap Ω 119875) = 1

Lemma6 (see [19]) Let119875 be a normal and total cone in119864 andlet 119860 119864 rarr 119864 be a completely continuous increasing operatorThen the following assertions hold

(a) 119860(120579) = 120579 119860 is Frechet differentiable at 120579 If 1198601015840120579119909 = 119909

forall119909 isin 119875 120579 and there exist 1205820lt 1 and 119909

0isin 119875 120579

such that 12058201198601015840

1205791199090= 1199090 then there exists 120588

0gt 0 such

that 119894(119860 119861120588cap 119875 119875) = 0 for all 120588 isin (0 120588

0] where 119861

120588=

119909 isin 119864 | 119909 lt 120588(b) 119860 is an asymptotically linear operator If 1198601015840

infin119909 = 119909

forall119909 isin 119875 120579 and there exist 1205820lt 1 and 119909

0isin 119875 120579

such that 12058201198601015840

infin1199090= 1199090 then there exists 120588

infingt 0 such

that 119894(119860 119861120588cap 119875 119875) = 0 for all 120588 ge 120588

infin

Lemma 7 (see [18]) Let119863 be an open set of 119864119860 119863 rarr 119864 becompletely continuous 119909

0isin 119863 and 119860119909

0= 1199090 Assume that 119860

is Frechet differentiable at 1199090and 1 is not an eigenvalue of1198601015840

1199090

then 1199090is an isolated fixed point and

ind (119868 minus 119860 1199090) = ind (119868 minus 1198601015840

1199090 120579) = (minus1)

120578 (9)

where 120578 is the sum of algebraic multiplicities of the realeigenvalues of 1198601015840

1199090in (0 1)

Lemma 8 (see [18]) Suppose that 119860 119864 rarr 119864 is a completelycontinuous and asymptotically linear operator If 1 is not aneigenvalue of the linear operator 1198601015840

infin then there exists 119877

0gt 0

such that

deg (119868 minus 119860 119861119877 120579) = (minus1)

120574 (10)

for all 119877 ge 1198770 where 120574 is the sum of the algebraic multiplicities

of the real eigenvalues of 1198601015840infin

in (0 1)

Journal of Operators 3

Lemma9 (see [9]) Let119883 be a solid cone in119864Ω be a relativelybounded open subset of 119883 and 119860 119883 rarr 119883 be a completelycontinuous operator If any fixed point of 119860 in Ω is an interiorpoint of 119883 then there exists an open subset 119874 of 119864 (119874 sub Ω)such that

deg (119868 minus 119860119874 120579) = 119894 (119860Ω119883) (11)

Lemma 10 (see [18]) Let Ω be a bounded open set of 119864 andlet 120579 isin Ω Assume that119860 Ω rarr 119864 is a condensing operator If

119860119909 = 120583119909 forall119909 isin 120597Ω 120583 ge 1 (12)

then deg(119868 minus 119860Ω 120579) = 1

3 Multiple Sign-Changing Solutions forNonlinear Operator Equations

Theorem 11 Let 119875 be a normal solid cone in 119864119860 119864 rarr 119864 bea completely continuous operator119860(119875120579) sub 119875∘119860(minus119875120579) subminus119875∘ and 119860120579 = 120579 Suppose that(H1) 1198601015840infin

exists and is an increasing operator 119903(1198601015840infin) gt 1

1 is not an eigenvalue of 1198601015840infin and the sum of algebraic

multiplicities for the real eigenvalues of 1198601015840infin

in (0 1) isan even number

(H2) 1198601015840120579exists and is an increasing operator 119903(1198601015840

120579) gt 1 1

is not an eigenvalue of 1198601015840120579 and the sum of algebraic

multiplicities for the real eigenvalues of 1198601015840120579in (0 1) is

an even number(H3) lim119909rarr0

sup(119860119909119909) lt 1Then119860 has at least two sign-changing fixed points two positivefixed points and two negative fixed points

Proof From condition (H3) we obtain that there exists 119903 gt 0

such that 119860119909 lt 119909 for all 0 lt 119909 lt 119903 By Lemma 5 thereexists 119903 lt 119903 such that

119894 (119860 119875 cap 119879119903 119875) = 1 (13)

119894 (119860 minus119875 cap 119879119903 minus119875) = 1 (14)

where 119879119903= 119909 isin 119864 | 119909 lt 119903

Since 119860(119875 120579) sub 119875∘ 119860(minus119875 120579) sub minus119875

∘ and 119860120579 = 120579which together with (H

2) imply that 1198601015840

120579(119875) sub 119875 and 119903(1198601015840

120579) gt

1 gt 0 According to Lemma 4 we know that there exists120593 isin 119875 120579 such that 1198601015840

120579120593 = 119903(119860

1015840

120579)120593 gt 120593 Since 1 is not

an eigenvalue of 1198601015840120579 1198601015840120579119909 = 119909 and forall119909 isin 119875 120579 By Lemma 6

there exists 1205880isin (0 119903) such that

119894 (119860 119875 cap 119879120588 119875) = 0 (15)

for all 120588 isin (0 1205880] Similarly we get that

119894 (119860 minus119875 cap 119879120588 minus119875) = 0 (16)

In the same sense we know that there exists 120588infingt 119903 such that

119894 (119860 119875 cap 119879120588 119875) = 0

119894 (119860 minus119875 cap 119879120588 minus119875) = 0

(17)

for all 120588 ge 120588infin Further combining Lemmas 7 and 8 with (H

1)

and (H2) we get that there exist 120588

1isin (0 120588

0) and 120588

2gt 120588infinsuch

that

deg (119868 minus 119860 1198791205881 120579) = 1 (18)

deg (119868 minus 119860 1198791205882 120579) = 1 (19)

By (15)ndash(17) we have

119894 (119860 119875 cap 1198791205881 119875) = 0 (20)

119894 (119860 minus119875 cap 1198791205881 minus119875) = 0 (21)

119894 (119860 119875 cap 1198791205882 119875) = 0 (22)

119894 (119860 minus119875 cap 1198791205882 minus119875) = 0 (23)

It follows from (13) (20) (22) and the additivity property offixed point index that

119894 (119860 119875 cap (119879119903 1198791205881) 119875) = 119894 (119860 119875 cap 119879

119903 119875)

minus 119894 (119860 119875 cap 1198791205881 119875) = 1 minus 0 = 1

119894 (119860 119875 cap (1198791205882 119879119903) 119875) = 119894 (119860 119875 cap 119879

1205882 119875)

minus 119894 (119860 119875 cap 119879119903 119875) = 0 minus 1 = minus1

(24)

Hence119860 has at least two fixed points 1199061and 1199062in119875cap(119879

1199031198791205881)

and 119875cap(1198791205882119879119903) respectively It is obvious that 119906

1and 1199062are

both positive Moreover it follows from (14) (21) (23) andthe additivity property of fixed point index that

119894 (119860 minus119875 cap (119879119903 1198791205881) minus119875) = 1

119894 (119860 minus119875 cap (1198791205882 119879119903) minus119875) = minus1

(25)

Consequently 119860 also has at least two fixed points 1199063and 119906

4

in minus119875 cap (119879119903 1198791205881) and minus119875 cap (119879

1205882 119879119903) respectively Evidently

1199063and 119906

4are both negative

Since 119860(119875 120579) sub 119875∘ 119860(minus119875 120579) sub minus119875∘ Let

Σ1= 119906 isin 119875 cap (119879

119903 1198791205881) 119860119906 = 119906

Σ2= 119906 isin 119875 cap (119879

1205882 119879119903) 119860119906 = 119906

Σ3= 119906 isin minus119875 cap (119879

119903 1198791205881) 119860119906 = 119906

Σ4= 119906 isin minus119875 cap (119879

1205882 119879119903) 119860119906 = 119906

(26)

4 Journal of Operators

By Lemma 9 and (24)ndash(25) we get that there exist open setsΩ119894(119894 = 1 2 3 4) of 119864 such that

Σ1sub Ω1sub 119875 cap (119879

119903 1198791205881)

Σ2sub Ω2sub 119875 cap (119879

1205882 119879119903)

Σ3sub Ω3sub minus119875 cap (119879

119903 1198791205881)

Σ4sub Ω4sub minus119875 cap (119879

1205882 119879119903)

(27)

deg (119868 minus 119860Ω1 120579) = 1 (28)

deg (119868 minus 119860Ω2 120579) = minus1 (29)

deg (119868 minus 119860Ω3 120579) = 1 (30)

deg (119868 minus 119860Ω4 120579) = minus1 (31)

By Lemma 10 we have

deg (119868 minus 119860 119879119903 120579) = 1 (32)

According to (32) (28) (30) (18) and the additivity propertyof Leray-Schauder degree we obtain

deg (119868 minus 119860 119879119903 (Ω1cup Ω3cup 1198791205881) 120579)

= 1 minus 1 minus 1 minus 1 = minus2

(33)

which yields that119860 has at least one fixed point 1199065in119879119903(Ω1cup

Ω3cup1198791205881) and then 119906

5is a sign-changing fixed point It follows

from (19) (29) (31) (32) and the additivity property of Leray-Schauder degree that we have

deg (119868 minus 119860 1198791205882 (Ω2cup Ω4cup 119879119903) 120579)

= 1 + 1 + 1 minus 1 = 2

(34)

which implies that 119860 has at least one fixed point 1199066in 1198791205882

(Ω2cupΩ4cup119879119903) and then 119906

6is also a sign-changing fixed point

The proof is completed

4 Example

The main purpose of this section is to apply our theorem tononlinear differential equations

We consider the following boundary value problem forelliptic partial differential equations

119871120593 (119909) = 119891 (119909 120593 (119909)) 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(35)

where Ω is a bounded open domain in R119899 120597Ω isin 1198622+120583 and

0 lt 120583 lt 1 119891(119909 120593) Ω timesR1 rarr R1 is continuous

119871120593 = minus

119899

sum

119894119895=1

119886119894119895(119909)

1205972120593

120597119909119894120597119909119895

+

119899

sum

119894=1

119887119894(119909)

120597120593

120597119909119894

+ 119888 (119909) 120593 (36)

is an uniformly elliptic operator that is 119886119894119895(119909) = 119886

119895119894(119909)

119887119894(119909) 119888(119909) isin 119862

120583(Ω) 119888(119909) gt 0 and there exists a constant

number 1205830gt 0 such that sum119899

119894119895=1119886119894119895(119909)120585119894120585119895ge 1205830|120585|2 for all

119909 isin Ω 120585 = (1205851 1205852 120585

119899) isin R119899 Consider

119861120593 = 119887 (119909) 120593 + 120575

119899

sum

119894=1

120573119894(119909)

120597120593

120597119909119894

(37)

which is a boundary operator where 120573 = (1205731 1205732 120573

119899) is a

vector field on 120597Ω of 1198621+120583 satisfying 120573 sdot n gt 0 (n denotes theouter unit normal vector on 120597Ω) and 119887(119909) isin 119862

1+120583(120597Ω) and

assume that one of the following cases holds

(i) 120575 = 0 and 119887(119909) equiv 1(ii) 120575 = 1 and 119887(119909) equiv 0(iii) 120575 = 1 and 119887(119909) gt 0

According to the theory of elliptic partial differentialequations (see [20 21]) we know that for each 119906 isin 119862(Ω)the linear boundary value problem

119871120593 (119909) = 119906 (119909) 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(38)

has a unique solution 120593119906isin 1198622(Ω) Define the operator119870 by

(119870119906) (119909) = 120593119906(119909) 119909 isin Ω (39)

Then 119870 119862(Ω) rarr 1198622(Ω) is a linear completely

continuous operator and has an unbounded sequence of eig-envalues

0 lt 1205821lt 1205822le 1205823le sdot sdot sdot 120582

119899997888rarr +infin (40)

and the spectral radius 119903(119870) = 120582minus11

Let

119864 = 119862 (Ω) 119875 = 120593 isin 119864 | 120593 (119909) ge 0 119909 isin Ω (41)

Then 119864 is an ordered Banach space with the norm 120593 =

sup119909isinΩ

|120593(119909)| and119875 is a normal solid cone in119864 and119870(119875) sub 119875For 120593 isin 119864 define Nemytskii operator by

(119865120593) (119909) = 119891 (119909 120593 (119909)) 119909 isin Ω (42)

Clearly 119865 119864 rarr 119864 is continuous Let 119860 = 119870119865 Then 119860

119864 rarr 119864 is completely continuousBy the proof of Lemma 41 in [22] we have that 1198601015840

120579=

1198910119870Let 119890 = 119890(119909) be the solution of the following boundary

value problem

119871120593 (119909) = 1 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(43)

In order to obtain multiple sign-changing solutions of(35) we give the following assumptions

Journal of Operators 5

(E1) 119891(119909 0) equiv 0 and forall119909 isin Ω 119891(119909 120593)120593 gt 0 forall119909 isin Ω 120593 = 0and 120593 isin (minusinfin +infin)

(E2) lim120593rarr0

(119891(119909 120593)120593) = 1198910uniformly for 119909 isin Ω and

120582119899lt 1198910lt 120582119899+1

where 119899 is an even number(E3) lim120593rarrinfin

(119891(119909 120593)120593) = 119891infin

uniformly for 119909 isin Ω and1205821198990lt 119891infinlt 1205821198990+1

where 1198990is an even number

(E4) there exists a constant number 119903 lt 1 such thatlim120593rarr0

(119891(119909 120593)120593) = 119903(1119890) uniformly for 119909 isin Ω

Theorem 12 Suppose that (1198641)ndash(1198644) are satisfied Then the

problem (35) has at least two sign-changing solutions More-over problem (35) has at least two positive solutions and twonegative solutions

Proof From condition (E1) we know that

119891 (119909 120593) gt 0 forall120593 gt 0

119891 (119909 120593) lt 0 forall120593 lt 0

(44)

Copy the proof proceed of Theorem 34 in [23] we have that119860((1198641198900cap 119875) 120579) sub int(119864

1198900cap 119875) and 119860(minus(119864

1198900cap 119875) 120579) sub

int(minus(1198641198900cap 119875)) where

1198641198900= 120593 isin 119864 there exists 120583 gt 0

such that minus 1205831198901(119909) le 120593 le 120583119890

1(119909)

int (1198641198900cap 119875) = 120593 isin (119864

1198900cap 119875) there exist 120572 gt 0

and 120573 gt 0 such that 1205721198901(119909)

le 120593 (119909) le 1205731198901(119909)

(45)

where 1198901is the first normalized eigenfunction of 119870 corre-

sponding to its first eigenvalue 1205821

It follows from (E2) and (E

3) that conditions (H

1) and

(H2) of Theorem 11 holdIn the following we prove that (H

3) of Theorem 11 is

satisfied It follows from (E4) that there exists 120575 gt 0 such that

119891 (119909 120593)

120593

lt

1 + 119903

2 119890

forall 0 lt10038161003816100381610038161205931003816100381610038161003816lt 120575 119909 isin Ω (46)

That is1003816100381610038161003816119891 (119909 120593)

1003816100381610038161003816lt

1 + 119903

2 119890

10038161003816100381610038161205931003816100381610038161003816 119909 isin Ω (47)

Thus1003817100381710038171003817119860120593

1003817100381710038171003817le

1 + 119903

2 119890

11989010038171003817100381710038171205931003817100381710038171003817 (48)

Therefore

lim120593rarr0

1003817100381710038171003817119860120593

1003817100381710038171003817

10038171003817100381710038171205931003817100381710038171003817

le

1 + 119903

2

lt 1 (49)

The proof is completed

Remark 13 It follows from conditions (E2) and (E

4) that

1198910119890 lt 1 We should point out that the initial ideas of

condition (E4) and the general one (H

3) aremotivatedby [24]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was supported financially by the National Nat-ural Science Foundation of China Tianyuan Foundation(11226119) the Scientific and Technological Innovation Pro-grams of Higher Education Institutions in Shanxi the YouthScience Foundation of Shanxi Province (2013021002-1) andShandong Provincial Natural Science Foundation China(ZR2012AQ024)

References

[1] ENDancer andYHDu ldquoExistence of changing sign solutionsfor some semilinear problems with jumping nonlinearities atzerordquo Proceedings of the Royal Society of Edinburgh vol 124 no6 pp 1165ndash1176 1994

[2] D Motreanu andM Tanaka ldquoSign-changing and constant-signsolutions for 119901-Laplacian problems with jumping nonlineari-tiesrdquo Journal of Differential Equations vol 249 no 12 pp 3352ndash3376 2010

[3] Z Zhang and S Li ldquoOn sign-changing and multiple solutionsof the 119901-Laplacianrdquo Journal of Functional Analysis vol 197 no2 pp 447ndash468 2003

[4] W Zou Sign-Changing Critical Point Theory Springer NewYork NY USA 2008

[5] X Cheng and C Zhong ldquoExistence of three nontrivial solutionsfor an elliptic systemrdquo Journal of Mathematical Analysis andApplications vol 327 no 2 pp 1420ndash1430 2007

[6] K Zhang The multiple solutions and sign-changing solutionsfor nonlinear operator equations and applications [PhD thesis]Shandong University Jinan China 2002 (Chinese)

[7] X Xu ldquoMultiple sign-changing solutions for some m-pointboundary-value problemsrdquo Electronic Journal of DifferentialEquations vol 89 pp 1ndash14 2004

[8] R Yang Existence of positive solutions sign-changing solutionsfor nonlinear differential equations [MS thesis] ShandongUniversity Jinan China 2006 (Chinese)

[9] C Pang W Dong and Z Wei ldquoMultiple solutions for fourth-order boundary value problemrdquo Journal of Mathematical Anal-ysis and Applications vol 314 no 2 pp 464ndash476 2006

[10] Z Wei and C Pang ldquoMultiple sign-changing solutions forfourth order 119898-point boundary value problemsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 66 no 4 pp 839ndash855 2007

[11] Y Li and F Li ldquoSign-changing solutions to second-orderintegral boundary value problemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 69 no 4 pp 1179ndash1187 2008

[12] T He W Yang and F Yang ldquoSign-changing solutions fordiscrete second-order three-point boundary value problemsrdquoDiscrete Dynamics in Nature and Society Article ID 705387 14pages 2010

[13] J Yang ldquoSign-changing solutions to discrete fourth-orderNeumann boundary value problemsrdquo Advances in DifferenceEquations vol 2013 article 10 2013

[14] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

6 Journal of Operators

[15] D Guo Partial Order Methods in Nonlinear Analysis ShandongScience and Technology Press Jinan China 2000 (Chinese)

[16] D Guo Nonlinear Functional Analysis Shandong Science andTechnology Press Jinan China 2nd edition 2001 (Chinese)

[17] F Li Solutions of nonlinear operator equations and applications[PhD thesis] Shandong University Jinan China 1996 (Chi-nese)

[18] J Sun Nonlinear Functional Analysis and Applications SciencePress Beijing China 2008 (Chinese)

[19] H Amann ldquoFixed points of asymptotically linear maps inordered Banach spacesrdquo Journal of Functional Analysis vol 14pp 162ndash171 1973

[20] H Amann ldquoOn the number of solutions of nonlinear equationsin orderedBanach spacesrdquo Journal of Functional Analysis vol 11pp 346ndash384 1972

[21] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[22] X Liu and J Sun ldquoComputation of topological degree of uni-laterally asymptotically linear operators and its applicationsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 96ndash106 2009

[23] J X Sun and X Y Liu ldquoComputation of the fixed point indexfor non-cone mappings and its applicationsrdquo ActaMathematicaSinica vol 53 no 3 pp 417ndash428 2010 (Chinese)

[24] J X Sun and K M Zhang ldquoExistence of multiple fixed pointsfor nonlinear operators and applicationsrdquo Acta MathematicaSinica vol 24 no 7 pp 1079ndash1088 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Operators

Pang et al [9] investigated multiple sign-changing solutionsof fourth-order differential equation boundary value prob-lems Moreover Wei and Pang [10] established the existencetheorem ofmultiple sign-changing solutions for fourth-orderboundary value problems Y Li and F Li [11] studied twosign-changing solutions of a class of second-order integralboundary value problems by computing the eigenvaluesand the algebraic multiplicities of the corresponding linearproblems He et al [12] discussed the existence of sign-changing solutions for a class of discrete boundary valueproblems and a concrete example was also given Veryrecently Yang [13] investigated the following discrete fourthNeumann boundary value problems

Δ4119906 (119905 minus 2) minus 120572Δ

2119906 (119905 minus 1) + 120573 (119905) = 119891 (119905 119906 (119905))

119905 isin [2 119879]119885

Δ119906 (1) = Δ119906 (119879) = Δ3119906 (0) = Δ

3

(119879 minus 1) = 0

(6)

The author employed similar conditions with (1198601)ndash(1198604) and

obtained a similar result to Theorem 1 (see Theorem 51 in[13])

The main purpose of this paper is to abstract moregeneral conditions from (119860

1)ndash(1198604) of Theorem 1 obtain the

existence theorem of sign-changing solutions for generaloperator equations and then apply the abstract resultobtained in this paper to nonlinear elliptic partial differentialequations

2 Preliminaries and Some Lemmas

For the discussion of the following sections we state herepreliminary definitions and known results on cones partialorderings and topological degree theory which can be foundin [14ndash18]

Let 119864 be a real Banach space Given a cone 119875 sub 119864 wedefine a partial ordering le with respect to 119875 by 119909 le 119910 if andonly if 119910minus119909 isin 119875 A cone119875 is said to be normal if there exists aconstant119873 gt 0 such that 120579 le 119909 le 119910 implies 119909 le 119873119910 thesmallest119873 is called the normal constant of 119875 119875 is called solidif it contains interior that is int119875 = 0 If 119909 le 119910 and 119909 = 119910 wewrite 119909 lt 119910 if cone 119875 is solid and 119910 minus 119909 isin int119875 we write119909 ≪ 119910 119875 is reproducing if 119875 minus 119875 = 119864 and total if 119875 minus 119875 = 119864Let 119861 119864 rarr 119864 be a bounded linear operator 119861 is said tobe positive if 119861(119875) sub 119875 An operator 119860 is strongly increasingthat is 119909 lt 119910 implies 119860119909 ≪ 119860119910 If 119860 is a linear operator 119860is strongly increasing which implies 119860 is strongly positive Afixed point 119906 of operator119860 is said to be a sign-changing fixedpoint if 119906 notin 119875cup(minus119875) If 119909

0isin 119864120579 satisfies 120582119860119909

0= 1199090 where

120582 is some real number then 120582 is called an eigenvalue of119860 and1199090is called an eigenfunction belonging to the eigenvalue 120582

Definition 2 (see [16]) Let 1198641 1198642be real Banach spaces and

let 119863 sub 1198641contain the outside of a ball 119909 119909 le 119903 and

119860 119863 rarr 1198642 The operator 119860 is called asymptotically linear

if there is a bounded linear operator 119861 1198641rarr 1198642such that

lim119909rarr+infin

119860119909 minus 119861119909

119909

= 0 (7)

The operator 119861 involved in the definition of an asymptoticallylinear operator 119860 is uniquely determined It is called thederivative of 119860 at infinity and is denoted by 1198601015840

infin

Definition 3 (see [16 18]) Let119883 be a retract of 119864 and let119880 sub

119883 be a relatively bounded open set of 119883 Suppose that 119860

119880 rarr 119883 is completely continuous and has no fixed point on120597119880 Let the positive integer 119894(119860 119880119883) be defined by

119894 (119860 119880119883) = deg (119868 minus 119860119903 119879119877cap 119903minus1

(119880) 120579) (8)

where 119903 119864 rarr 119883 is an arbitrary retraction and 119877 is a largeenough positive number such that 119880 sub 119879

119877= 119909 | 119909 isin

119864 119909 lt 119877 Then 119894(119860 119880119883) is called the fixed point indexof 119860 on 119880 with respect to119883

Lemma 4 (see [14 18]) Let 119875 be a reproducing cone and let 119861be a positive completely continuous linear operator with 119903(119861) gt0 where 119903(119861) denotes the spectral radius of 119861 Then 119861 has apositive eigenfunction in119875120579 corresponding to the eigenvalue(119903(119861))

minus1

Lemma 5 (see [18]) Let 119875 be a cone in 119864 and let Ω be abounded open set of119864 and 120579 isin Ω Assume that119860 119875capΩ rarr 119875

is a condensing operator If 119860119909 ge 119909 for every 119909 isin 119875 cap 120597Ω then119894(119860 119875 cap Ω 119875) = 1

Lemma6 (see [19]) Let119875 be a normal and total cone in119864 andlet 119860 119864 rarr 119864 be a completely continuous increasing operatorThen the following assertions hold

(a) 119860(120579) = 120579 119860 is Frechet differentiable at 120579 If 1198601015840120579119909 = 119909

forall119909 isin 119875 120579 and there exist 1205820lt 1 and 119909

0isin 119875 120579

such that 12058201198601015840

1205791199090= 1199090 then there exists 120588

0gt 0 such

that 119894(119860 119861120588cap 119875 119875) = 0 for all 120588 isin (0 120588

0] where 119861

120588=

119909 isin 119864 | 119909 lt 120588(b) 119860 is an asymptotically linear operator If 1198601015840

infin119909 = 119909

forall119909 isin 119875 120579 and there exist 1205820lt 1 and 119909

0isin 119875 120579

such that 12058201198601015840

infin1199090= 1199090 then there exists 120588

infingt 0 such

that 119894(119860 119861120588cap 119875 119875) = 0 for all 120588 ge 120588

infin

Lemma 7 (see [18]) Let119863 be an open set of 119864119860 119863 rarr 119864 becompletely continuous 119909

0isin 119863 and 119860119909

0= 1199090 Assume that 119860

is Frechet differentiable at 1199090and 1 is not an eigenvalue of1198601015840

1199090

then 1199090is an isolated fixed point and

ind (119868 minus 119860 1199090) = ind (119868 minus 1198601015840

1199090 120579) = (minus1)

120578 (9)

where 120578 is the sum of algebraic multiplicities of the realeigenvalues of 1198601015840

1199090in (0 1)

Lemma 8 (see [18]) Suppose that 119860 119864 rarr 119864 is a completelycontinuous and asymptotically linear operator If 1 is not aneigenvalue of the linear operator 1198601015840

infin then there exists 119877

0gt 0

such that

deg (119868 minus 119860 119861119877 120579) = (minus1)

120574 (10)

for all 119877 ge 1198770 where 120574 is the sum of the algebraic multiplicities

of the real eigenvalues of 1198601015840infin

in (0 1)

Journal of Operators 3

Lemma9 (see [9]) Let119883 be a solid cone in119864Ω be a relativelybounded open subset of 119883 and 119860 119883 rarr 119883 be a completelycontinuous operator If any fixed point of 119860 in Ω is an interiorpoint of 119883 then there exists an open subset 119874 of 119864 (119874 sub Ω)such that

deg (119868 minus 119860119874 120579) = 119894 (119860Ω119883) (11)

Lemma 10 (see [18]) Let Ω be a bounded open set of 119864 andlet 120579 isin Ω Assume that119860 Ω rarr 119864 is a condensing operator If

119860119909 = 120583119909 forall119909 isin 120597Ω 120583 ge 1 (12)

then deg(119868 minus 119860Ω 120579) = 1

3 Multiple Sign-Changing Solutions forNonlinear Operator Equations

Theorem 11 Let 119875 be a normal solid cone in 119864119860 119864 rarr 119864 bea completely continuous operator119860(119875120579) sub 119875∘119860(minus119875120579) subminus119875∘ and 119860120579 = 120579 Suppose that(H1) 1198601015840infin

exists and is an increasing operator 119903(1198601015840infin) gt 1

1 is not an eigenvalue of 1198601015840infin and the sum of algebraic

multiplicities for the real eigenvalues of 1198601015840infin

in (0 1) isan even number

(H2) 1198601015840120579exists and is an increasing operator 119903(1198601015840

120579) gt 1 1

is not an eigenvalue of 1198601015840120579 and the sum of algebraic

multiplicities for the real eigenvalues of 1198601015840120579in (0 1) is

an even number(H3) lim119909rarr0

sup(119860119909119909) lt 1Then119860 has at least two sign-changing fixed points two positivefixed points and two negative fixed points

Proof From condition (H3) we obtain that there exists 119903 gt 0

such that 119860119909 lt 119909 for all 0 lt 119909 lt 119903 By Lemma 5 thereexists 119903 lt 119903 such that

119894 (119860 119875 cap 119879119903 119875) = 1 (13)

119894 (119860 minus119875 cap 119879119903 minus119875) = 1 (14)

where 119879119903= 119909 isin 119864 | 119909 lt 119903

Since 119860(119875 120579) sub 119875∘ 119860(minus119875 120579) sub minus119875

∘ and 119860120579 = 120579which together with (H

2) imply that 1198601015840

120579(119875) sub 119875 and 119903(1198601015840

120579) gt

1 gt 0 According to Lemma 4 we know that there exists120593 isin 119875 120579 such that 1198601015840

120579120593 = 119903(119860

1015840

120579)120593 gt 120593 Since 1 is not

an eigenvalue of 1198601015840120579 1198601015840120579119909 = 119909 and forall119909 isin 119875 120579 By Lemma 6

there exists 1205880isin (0 119903) such that

119894 (119860 119875 cap 119879120588 119875) = 0 (15)

for all 120588 isin (0 1205880] Similarly we get that

119894 (119860 minus119875 cap 119879120588 minus119875) = 0 (16)

In the same sense we know that there exists 120588infingt 119903 such that

119894 (119860 119875 cap 119879120588 119875) = 0

119894 (119860 minus119875 cap 119879120588 minus119875) = 0

(17)

for all 120588 ge 120588infin Further combining Lemmas 7 and 8 with (H

1)

and (H2) we get that there exist 120588

1isin (0 120588

0) and 120588

2gt 120588infinsuch

that

deg (119868 minus 119860 1198791205881 120579) = 1 (18)

deg (119868 minus 119860 1198791205882 120579) = 1 (19)

By (15)ndash(17) we have

119894 (119860 119875 cap 1198791205881 119875) = 0 (20)

119894 (119860 minus119875 cap 1198791205881 minus119875) = 0 (21)

119894 (119860 119875 cap 1198791205882 119875) = 0 (22)

119894 (119860 minus119875 cap 1198791205882 minus119875) = 0 (23)

It follows from (13) (20) (22) and the additivity property offixed point index that

119894 (119860 119875 cap (119879119903 1198791205881) 119875) = 119894 (119860 119875 cap 119879

119903 119875)

minus 119894 (119860 119875 cap 1198791205881 119875) = 1 minus 0 = 1

119894 (119860 119875 cap (1198791205882 119879119903) 119875) = 119894 (119860 119875 cap 119879

1205882 119875)

minus 119894 (119860 119875 cap 119879119903 119875) = 0 minus 1 = minus1

(24)

Hence119860 has at least two fixed points 1199061and 1199062in119875cap(119879

1199031198791205881)

and 119875cap(1198791205882119879119903) respectively It is obvious that 119906

1and 1199062are

both positive Moreover it follows from (14) (21) (23) andthe additivity property of fixed point index that

119894 (119860 minus119875 cap (119879119903 1198791205881) minus119875) = 1

119894 (119860 minus119875 cap (1198791205882 119879119903) minus119875) = minus1

(25)

Consequently 119860 also has at least two fixed points 1199063and 119906

4

in minus119875 cap (119879119903 1198791205881) and minus119875 cap (119879

1205882 119879119903) respectively Evidently

1199063and 119906

4are both negative

Since 119860(119875 120579) sub 119875∘ 119860(minus119875 120579) sub minus119875∘ Let

Σ1= 119906 isin 119875 cap (119879

119903 1198791205881) 119860119906 = 119906

Σ2= 119906 isin 119875 cap (119879

1205882 119879119903) 119860119906 = 119906

Σ3= 119906 isin minus119875 cap (119879

119903 1198791205881) 119860119906 = 119906

Σ4= 119906 isin minus119875 cap (119879

1205882 119879119903) 119860119906 = 119906

(26)

4 Journal of Operators

By Lemma 9 and (24)ndash(25) we get that there exist open setsΩ119894(119894 = 1 2 3 4) of 119864 such that

Σ1sub Ω1sub 119875 cap (119879

119903 1198791205881)

Σ2sub Ω2sub 119875 cap (119879

1205882 119879119903)

Σ3sub Ω3sub minus119875 cap (119879

119903 1198791205881)

Σ4sub Ω4sub minus119875 cap (119879

1205882 119879119903)

(27)

deg (119868 minus 119860Ω1 120579) = 1 (28)

deg (119868 minus 119860Ω2 120579) = minus1 (29)

deg (119868 minus 119860Ω3 120579) = 1 (30)

deg (119868 minus 119860Ω4 120579) = minus1 (31)

By Lemma 10 we have

deg (119868 minus 119860 119879119903 120579) = 1 (32)

According to (32) (28) (30) (18) and the additivity propertyof Leray-Schauder degree we obtain

deg (119868 minus 119860 119879119903 (Ω1cup Ω3cup 1198791205881) 120579)

= 1 minus 1 minus 1 minus 1 = minus2

(33)

which yields that119860 has at least one fixed point 1199065in119879119903(Ω1cup

Ω3cup1198791205881) and then 119906

5is a sign-changing fixed point It follows

from (19) (29) (31) (32) and the additivity property of Leray-Schauder degree that we have

deg (119868 minus 119860 1198791205882 (Ω2cup Ω4cup 119879119903) 120579)

= 1 + 1 + 1 minus 1 = 2

(34)

which implies that 119860 has at least one fixed point 1199066in 1198791205882

(Ω2cupΩ4cup119879119903) and then 119906

6is also a sign-changing fixed point

The proof is completed

4 Example

The main purpose of this section is to apply our theorem tononlinear differential equations

We consider the following boundary value problem forelliptic partial differential equations

119871120593 (119909) = 119891 (119909 120593 (119909)) 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(35)

where Ω is a bounded open domain in R119899 120597Ω isin 1198622+120583 and

0 lt 120583 lt 1 119891(119909 120593) Ω timesR1 rarr R1 is continuous

119871120593 = minus

119899

sum

119894119895=1

119886119894119895(119909)

1205972120593

120597119909119894120597119909119895

+

119899

sum

119894=1

119887119894(119909)

120597120593

120597119909119894

+ 119888 (119909) 120593 (36)

is an uniformly elliptic operator that is 119886119894119895(119909) = 119886

119895119894(119909)

119887119894(119909) 119888(119909) isin 119862

120583(Ω) 119888(119909) gt 0 and there exists a constant

number 1205830gt 0 such that sum119899

119894119895=1119886119894119895(119909)120585119894120585119895ge 1205830|120585|2 for all

119909 isin Ω 120585 = (1205851 1205852 120585

119899) isin R119899 Consider

119861120593 = 119887 (119909) 120593 + 120575

119899

sum

119894=1

120573119894(119909)

120597120593

120597119909119894

(37)

which is a boundary operator where 120573 = (1205731 1205732 120573

119899) is a

vector field on 120597Ω of 1198621+120583 satisfying 120573 sdot n gt 0 (n denotes theouter unit normal vector on 120597Ω) and 119887(119909) isin 119862

1+120583(120597Ω) and

assume that one of the following cases holds

(i) 120575 = 0 and 119887(119909) equiv 1(ii) 120575 = 1 and 119887(119909) equiv 0(iii) 120575 = 1 and 119887(119909) gt 0

According to the theory of elliptic partial differentialequations (see [20 21]) we know that for each 119906 isin 119862(Ω)the linear boundary value problem

119871120593 (119909) = 119906 (119909) 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(38)

has a unique solution 120593119906isin 1198622(Ω) Define the operator119870 by

(119870119906) (119909) = 120593119906(119909) 119909 isin Ω (39)

Then 119870 119862(Ω) rarr 1198622(Ω) is a linear completely

continuous operator and has an unbounded sequence of eig-envalues

0 lt 1205821lt 1205822le 1205823le sdot sdot sdot 120582

119899997888rarr +infin (40)

and the spectral radius 119903(119870) = 120582minus11

Let

119864 = 119862 (Ω) 119875 = 120593 isin 119864 | 120593 (119909) ge 0 119909 isin Ω (41)

Then 119864 is an ordered Banach space with the norm 120593 =

sup119909isinΩ

|120593(119909)| and119875 is a normal solid cone in119864 and119870(119875) sub 119875For 120593 isin 119864 define Nemytskii operator by

(119865120593) (119909) = 119891 (119909 120593 (119909)) 119909 isin Ω (42)

Clearly 119865 119864 rarr 119864 is continuous Let 119860 = 119870119865 Then 119860

119864 rarr 119864 is completely continuousBy the proof of Lemma 41 in [22] we have that 1198601015840

120579=

1198910119870Let 119890 = 119890(119909) be the solution of the following boundary

value problem

119871120593 (119909) = 1 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(43)

In order to obtain multiple sign-changing solutions of(35) we give the following assumptions

Journal of Operators 5

(E1) 119891(119909 0) equiv 0 and forall119909 isin Ω 119891(119909 120593)120593 gt 0 forall119909 isin Ω 120593 = 0and 120593 isin (minusinfin +infin)

(E2) lim120593rarr0

(119891(119909 120593)120593) = 1198910uniformly for 119909 isin Ω and

120582119899lt 1198910lt 120582119899+1

where 119899 is an even number(E3) lim120593rarrinfin

(119891(119909 120593)120593) = 119891infin

uniformly for 119909 isin Ω and1205821198990lt 119891infinlt 1205821198990+1

where 1198990is an even number

(E4) there exists a constant number 119903 lt 1 such thatlim120593rarr0

(119891(119909 120593)120593) = 119903(1119890) uniformly for 119909 isin Ω

Theorem 12 Suppose that (1198641)ndash(1198644) are satisfied Then the

problem (35) has at least two sign-changing solutions More-over problem (35) has at least two positive solutions and twonegative solutions

Proof From condition (E1) we know that

119891 (119909 120593) gt 0 forall120593 gt 0

119891 (119909 120593) lt 0 forall120593 lt 0

(44)

Copy the proof proceed of Theorem 34 in [23] we have that119860((1198641198900cap 119875) 120579) sub int(119864

1198900cap 119875) and 119860(minus(119864

1198900cap 119875) 120579) sub

int(minus(1198641198900cap 119875)) where

1198641198900= 120593 isin 119864 there exists 120583 gt 0

such that minus 1205831198901(119909) le 120593 le 120583119890

1(119909)

int (1198641198900cap 119875) = 120593 isin (119864

1198900cap 119875) there exist 120572 gt 0

and 120573 gt 0 such that 1205721198901(119909)

le 120593 (119909) le 1205731198901(119909)

(45)

where 1198901is the first normalized eigenfunction of 119870 corre-

sponding to its first eigenvalue 1205821

It follows from (E2) and (E

3) that conditions (H

1) and

(H2) of Theorem 11 holdIn the following we prove that (H

3) of Theorem 11 is

satisfied It follows from (E4) that there exists 120575 gt 0 such that

119891 (119909 120593)

120593

lt

1 + 119903

2 119890

forall 0 lt10038161003816100381610038161205931003816100381610038161003816lt 120575 119909 isin Ω (46)

That is1003816100381610038161003816119891 (119909 120593)

1003816100381610038161003816lt

1 + 119903

2 119890

10038161003816100381610038161205931003816100381610038161003816 119909 isin Ω (47)

Thus1003817100381710038171003817119860120593

1003817100381710038171003817le

1 + 119903

2 119890

11989010038171003817100381710038171205931003817100381710038171003817 (48)

Therefore

lim120593rarr0

1003817100381710038171003817119860120593

1003817100381710038171003817

10038171003817100381710038171205931003817100381710038171003817

le

1 + 119903

2

lt 1 (49)

The proof is completed

Remark 13 It follows from conditions (E2) and (E

4) that

1198910119890 lt 1 We should point out that the initial ideas of

condition (E4) and the general one (H

3) aremotivatedby [24]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was supported financially by the National Nat-ural Science Foundation of China Tianyuan Foundation(11226119) the Scientific and Technological Innovation Pro-grams of Higher Education Institutions in Shanxi the YouthScience Foundation of Shanxi Province (2013021002-1) andShandong Provincial Natural Science Foundation China(ZR2012AQ024)

References

[1] ENDancer andYHDu ldquoExistence of changing sign solutionsfor some semilinear problems with jumping nonlinearities atzerordquo Proceedings of the Royal Society of Edinburgh vol 124 no6 pp 1165ndash1176 1994

[2] D Motreanu andM Tanaka ldquoSign-changing and constant-signsolutions for 119901-Laplacian problems with jumping nonlineari-tiesrdquo Journal of Differential Equations vol 249 no 12 pp 3352ndash3376 2010

[3] Z Zhang and S Li ldquoOn sign-changing and multiple solutionsof the 119901-Laplacianrdquo Journal of Functional Analysis vol 197 no2 pp 447ndash468 2003

[4] W Zou Sign-Changing Critical Point Theory Springer NewYork NY USA 2008

[5] X Cheng and C Zhong ldquoExistence of three nontrivial solutionsfor an elliptic systemrdquo Journal of Mathematical Analysis andApplications vol 327 no 2 pp 1420ndash1430 2007

[6] K Zhang The multiple solutions and sign-changing solutionsfor nonlinear operator equations and applications [PhD thesis]Shandong University Jinan China 2002 (Chinese)

[7] X Xu ldquoMultiple sign-changing solutions for some m-pointboundary-value problemsrdquo Electronic Journal of DifferentialEquations vol 89 pp 1ndash14 2004

[8] R Yang Existence of positive solutions sign-changing solutionsfor nonlinear differential equations [MS thesis] ShandongUniversity Jinan China 2006 (Chinese)

[9] C Pang W Dong and Z Wei ldquoMultiple solutions for fourth-order boundary value problemrdquo Journal of Mathematical Anal-ysis and Applications vol 314 no 2 pp 464ndash476 2006

[10] Z Wei and C Pang ldquoMultiple sign-changing solutions forfourth order 119898-point boundary value problemsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 66 no 4 pp 839ndash855 2007

[11] Y Li and F Li ldquoSign-changing solutions to second-orderintegral boundary value problemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 69 no 4 pp 1179ndash1187 2008

[12] T He W Yang and F Yang ldquoSign-changing solutions fordiscrete second-order three-point boundary value problemsrdquoDiscrete Dynamics in Nature and Society Article ID 705387 14pages 2010

[13] J Yang ldquoSign-changing solutions to discrete fourth-orderNeumann boundary value problemsrdquo Advances in DifferenceEquations vol 2013 article 10 2013

[14] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

6 Journal of Operators

[15] D Guo Partial Order Methods in Nonlinear Analysis ShandongScience and Technology Press Jinan China 2000 (Chinese)

[16] D Guo Nonlinear Functional Analysis Shandong Science andTechnology Press Jinan China 2nd edition 2001 (Chinese)

[17] F Li Solutions of nonlinear operator equations and applications[PhD thesis] Shandong University Jinan China 1996 (Chi-nese)

[18] J Sun Nonlinear Functional Analysis and Applications SciencePress Beijing China 2008 (Chinese)

[19] H Amann ldquoFixed points of asymptotically linear maps inordered Banach spacesrdquo Journal of Functional Analysis vol 14pp 162ndash171 1973

[20] H Amann ldquoOn the number of solutions of nonlinear equationsin orderedBanach spacesrdquo Journal of Functional Analysis vol 11pp 346ndash384 1972

[21] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[22] X Liu and J Sun ldquoComputation of topological degree of uni-laterally asymptotically linear operators and its applicationsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 96ndash106 2009

[23] J X Sun and X Y Liu ldquoComputation of the fixed point indexfor non-cone mappings and its applicationsrdquo ActaMathematicaSinica vol 53 no 3 pp 417ndash428 2010 (Chinese)

[24] J X Sun and K M Zhang ldquoExistence of multiple fixed pointsfor nonlinear operators and applicationsrdquo Acta MathematicaSinica vol 24 no 7 pp 1079ndash1088 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Operators 3

Lemma9 (see [9]) Let119883 be a solid cone in119864Ω be a relativelybounded open subset of 119883 and 119860 119883 rarr 119883 be a completelycontinuous operator If any fixed point of 119860 in Ω is an interiorpoint of 119883 then there exists an open subset 119874 of 119864 (119874 sub Ω)such that

deg (119868 minus 119860119874 120579) = 119894 (119860Ω119883) (11)

Lemma 10 (see [18]) Let Ω be a bounded open set of 119864 andlet 120579 isin Ω Assume that119860 Ω rarr 119864 is a condensing operator If

119860119909 = 120583119909 forall119909 isin 120597Ω 120583 ge 1 (12)

then deg(119868 minus 119860Ω 120579) = 1

3 Multiple Sign-Changing Solutions forNonlinear Operator Equations

Theorem 11 Let 119875 be a normal solid cone in 119864119860 119864 rarr 119864 bea completely continuous operator119860(119875120579) sub 119875∘119860(minus119875120579) subminus119875∘ and 119860120579 = 120579 Suppose that(H1) 1198601015840infin

exists and is an increasing operator 119903(1198601015840infin) gt 1

1 is not an eigenvalue of 1198601015840infin and the sum of algebraic

multiplicities for the real eigenvalues of 1198601015840infin

in (0 1) isan even number

(H2) 1198601015840120579exists and is an increasing operator 119903(1198601015840

120579) gt 1 1

is not an eigenvalue of 1198601015840120579 and the sum of algebraic

multiplicities for the real eigenvalues of 1198601015840120579in (0 1) is

an even number(H3) lim119909rarr0

sup(119860119909119909) lt 1Then119860 has at least two sign-changing fixed points two positivefixed points and two negative fixed points

Proof From condition (H3) we obtain that there exists 119903 gt 0

such that 119860119909 lt 119909 for all 0 lt 119909 lt 119903 By Lemma 5 thereexists 119903 lt 119903 such that

119894 (119860 119875 cap 119879119903 119875) = 1 (13)

119894 (119860 minus119875 cap 119879119903 minus119875) = 1 (14)

where 119879119903= 119909 isin 119864 | 119909 lt 119903

Since 119860(119875 120579) sub 119875∘ 119860(minus119875 120579) sub minus119875

∘ and 119860120579 = 120579which together with (H

2) imply that 1198601015840

120579(119875) sub 119875 and 119903(1198601015840

120579) gt

1 gt 0 According to Lemma 4 we know that there exists120593 isin 119875 120579 such that 1198601015840

120579120593 = 119903(119860

1015840

120579)120593 gt 120593 Since 1 is not

an eigenvalue of 1198601015840120579 1198601015840120579119909 = 119909 and forall119909 isin 119875 120579 By Lemma 6

there exists 1205880isin (0 119903) such that

119894 (119860 119875 cap 119879120588 119875) = 0 (15)

for all 120588 isin (0 1205880] Similarly we get that

119894 (119860 minus119875 cap 119879120588 minus119875) = 0 (16)

In the same sense we know that there exists 120588infingt 119903 such that

119894 (119860 119875 cap 119879120588 119875) = 0

119894 (119860 minus119875 cap 119879120588 minus119875) = 0

(17)

for all 120588 ge 120588infin Further combining Lemmas 7 and 8 with (H

1)

and (H2) we get that there exist 120588

1isin (0 120588

0) and 120588

2gt 120588infinsuch

that

deg (119868 minus 119860 1198791205881 120579) = 1 (18)

deg (119868 minus 119860 1198791205882 120579) = 1 (19)

By (15)ndash(17) we have

119894 (119860 119875 cap 1198791205881 119875) = 0 (20)

119894 (119860 minus119875 cap 1198791205881 minus119875) = 0 (21)

119894 (119860 119875 cap 1198791205882 119875) = 0 (22)

119894 (119860 minus119875 cap 1198791205882 minus119875) = 0 (23)

It follows from (13) (20) (22) and the additivity property offixed point index that

119894 (119860 119875 cap (119879119903 1198791205881) 119875) = 119894 (119860 119875 cap 119879

119903 119875)

minus 119894 (119860 119875 cap 1198791205881 119875) = 1 minus 0 = 1

119894 (119860 119875 cap (1198791205882 119879119903) 119875) = 119894 (119860 119875 cap 119879

1205882 119875)

minus 119894 (119860 119875 cap 119879119903 119875) = 0 minus 1 = minus1

(24)

Hence119860 has at least two fixed points 1199061and 1199062in119875cap(119879

1199031198791205881)

and 119875cap(1198791205882119879119903) respectively It is obvious that 119906

1and 1199062are

both positive Moreover it follows from (14) (21) (23) andthe additivity property of fixed point index that

119894 (119860 minus119875 cap (119879119903 1198791205881) minus119875) = 1

119894 (119860 minus119875 cap (1198791205882 119879119903) minus119875) = minus1

(25)

Consequently 119860 also has at least two fixed points 1199063and 119906

4

in minus119875 cap (119879119903 1198791205881) and minus119875 cap (119879

1205882 119879119903) respectively Evidently

1199063and 119906

4are both negative

Since 119860(119875 120579) sub 119875∘ 119860(minus119875 120579) sub minus119875∘ Let

Σ1= 119906 isin 119875 cap (119879

119903 1198791205881) 119860119906 = 119906

Σ2= 119906 isin 119875 cap (119879

1205882 119879119903) 119860119906 = 119906

Σ3= 119906 isin minus119875 cap (119879

119903 1198791205881) 119860119906 = 119906

Σ4= 119906 isin minus119875 cap (119879

1205882 119879119903) 119860119906 = 119906

(26)

4 Journal of Operators

By Lemma 9 and (24)ndash(25) we get that there exist open setsΩ119894(119894 = 1 2 3 4) of 119864 such that

Σ1sub Ω1sub 119875 cap (119879

119903 1198791205881)

Σ2sub Ω2sub 119875 cap (119879

1205882 119879119903)

Σ3sub Ω3sub minus119875 cap (119879

119903 1198791205881)

Σ4sub Ω4sub minus119875 cap (119879

1205882 119879119903)

(27)

deg (119868 minus 119860Ω1 120579) = 1 (28)

deg (119868 minus 119860Ω2 120579) = minus1 (29)

deg (119868 minus 119860Ω3 120579) = 1 (30)

deg (119868 minus 119860Ω4 120579) = minus1 (31)

By Lemma 10 we have

deg (119868 minus 119860 119879119903 120579) = 1 (32)

According to (32) (28) (30) (18) and the additivity propertyof Leray-Schauder degree we obtain

deg (119868 minus 119860 119879119903 (Ω1cup Ω3cup 1198791205881) 120579)

= 1 minus 1 minus 1 minus 1 = minus2

(33)

which yields that119860 has at least one fixed point 1199065in119879119903(Ω1cup

Ω3cup1198791205881) and then 119906

5is a sign-changing fixed point It follows

from (19) (29) (31) (32) and the additivity property of Leray-Schauder degree that we have

deg (119868 minus 119860 1198791205882 (Ω2cup Ω4cup 119879119903) 120579)

= 1 + 1 + 1 minus 1 = 2

(34)

which implies that 119860 has at least one fixed point 1199066in 1198791205882

(Ω2cupΩ4cup119879119903) and then 119906

6is also a sign-changing fixed point

The proof is completed

4 Example

The main purpose of this section is to apply our theorem tononlinear differential equations

We consider the following boundary value problem forelliptic partial differential equations

119871120593 (119909) = 119891 (119909 120593 (119909)) 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(35)

where Ω is a bounded open domain in R119899 120597Ω isin 1198622+120583 and

0 lt 120583 lt 1 119891(119909 120593) Ω timesR1 rarr R1 is continuous

119871120593 = minus

119899

sum

119894119895=1

119886119894119895(119909)

1205972120593

120597119909119894120597119909119895

+

119899

sum

119894=1

119887119894(119909)

120597120593

120597119909119894

+ 119888 (119909) 120593 (36)

is an uniformly elliptic operator that is 119886119894119895(119909) = 119886

119895119894(119909)

119887119894(119909) 119888(119909) isin 119862

120583(Ω) 119888(119909) gt 0 and there exists a constant

number 1205830gt 0 such that sum119899

119894119895=1119886119894119895(119909)120585119894120585119895ge 1205830|120585|2 for all

119909 isin Ω 120585 = (1205851 1205852 120585

119899) isin R119899 Consider

119861120593 = 119887 (119909) 120593 + 120575

119899

sum

119894=1

120573119894(119909)

120597120593

120597119909119894

(37)

which is a boundary operator where 120573 = (1205731 1205732 120573

119899) is a

vector field on 120597Ω of 1198621+120583 satisfying 120573 sdot n gt 0 (n denotes theouter unit normal vector on 120597Ω) and 119887(119909) isin 119862

1+120583(120597Ω) and

assume that one of the following cases holds

(i) 120575 = 0 and 119887(119909) equiv 1(ii) 120575 = 1 and 119887(119909) equiv 0(iii) 120575 = 1 and 119887(119909) gt 0

According to the theory of elliptic partial differentialequations (see [20 21]) we know that for each 119906 isin 119862(Ω)the linear boundary value problem

119871120593 (119909) = 119906 (119909) 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(38)

has a unique solution 120593119906isin 1198622(Ω) Define the operator119870 by

(119870119906) (119909) = 120593119906(119909) 119909 isin Ω (39)

Then 119870 119862(Ω) rarr 1198622(Ω) is a linear completely

continuous operator and has an unbounded sequence of eig-envalues

0 lt 1205821lt 1205822le 1205823le sdot sdot sdot 120582

119899997888rarr +infin (40)

and the spectral radius 119903(119870) = 120582minus11

Let

119864 = 119862 (Ω) 119875 = 120593 isin 119864 | 120593 (119909) ge 0 119909 isin Ω (41)

Then 119864 is an ordered Banach space with the norm 120593 =

sup119909isinΩ

|120593(119909)| and119875 is a normal solid cone in119864 and119870(119875) sub 119875For 120593 isin 119864 define Nemytskii operator by

(119865120593) (119909) = 119891 (119909 120593 (119909)) 119909 isin Ω (42)

Clearly 119865 119864 rarr 119864 is continuous Let 119860 = 119870119865 Then 119860

119864 rarr 119864 is completely continuousBy the proof of Lemma 41 in [22] we have that 1198601015840

120579=

1198910119870Let 119890 = 119890(119909) be the solution of the following boundary

value problem

119871120593 (119909) = 1 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(43)

In order to obtain multiple sign-changing solutions of(35) we give the following assumptions

Journal of Operators 5

(E1) 119891(119909 0) equiv 0 and forall119909 isin Ω 119891(119909 120593)120593 gt 0 forall119909 isin Ω 120593 = 0and 120593 isin (minusinfin +infin)

(E2) lim120593rarr0

(119891(119909 120593)120593) = 1198910uniformly for 119909 isin Ω and

120582119899lt 1198910lt 120582119899+1

where 119899 is an even number(E3) lim120593rarrinfin

(119891(119909 120593)120593) = 119891infin

uniformly for 119909 isin Ω and1205821198990lt 119891infinlt 1205821198990+1

where 1198990is an even number

(E4) there exists a constant number 119903 lt 1 such thatlim120593rarr0

(119891(119909 120593)120593) = 119903(1119890) uniformly for 119909 isin Ω

Theorem 12 Suppose that (1198641)ndash(1198644) are satisfied Then the

problem (35) has at least two sign-changing solutions More-over problem (35) has at least two positive solutions and twonegative solutions

Proof From condition (E1) we know that

119891 (119909 120593) gt 0 forall120593 gt 0

119891 (119909 120593) lt 0 forall120593 lt 0

(44)

Copy the proof proceed of Theorem 34 in [23] we have that119860((1198641198900cap 119875) 120579) sub int(119864

1198900cap 119875) and 119860(minus(119864

1198900cap 119875) 120579) sub

int(minus(1198641198900cap 119875)) where

1198641198900= 120593 isin 119864 there exists 120583 gt 0

such that minus 1205831198901(119909) le 120593 le 120583119890

1(119909)

int (1198641198900cap 119875) = 120593 isin (119864

1198900cap 119875) there exist 120572 gt 0

and 120573 gt 0 such that 1205721198901(119909)

le 120593 (119909) le 1205731198901(119909)

(45)

where 1198901is the first normalized eigenfunction of 119870 corre-

sponding to its first eigenvalue 1205821

It follows from (E2) and (E

3) that conditions (H

1) and

(H2) of Theorem 11 holdIn the following we prove that (H

3) of Theorem 11 is

satisfied It follows from (E4) that there exists 120575 gt 0 such that

119891 (119909 120593)

120593

lt

1 + 119903

2 119890

forall 0 lt10038161003816100381610038161205931003816100381610038161003816lt 120575 119909 isin Ω (46)

That is1003816100381610038161003816119891 (119909 120593)

1003816100381610038161003816lt

1 + 119903

2 119890

10038161003816100381610038161205931003816100381610038161003816 119909 isin Ω (47)

Thus1003817100381710038171003817119860120593

1003817100381710038171003817le

1 + 119903

2 119890

11989010038171003817100381710038171205931003817100381710038171003817 (48)

Therefore

lim120593rarr0

1003817100381710038171003817119860120593

1003817100381710038171003817

10038171003817100381710038171205931003817100381710038171003817

le

1 + 119903

2

lt 1 (49)

The proof is completed

Remark 13 It follows from conditions (E2) and (E

4) that

1198910119890 lt 1 We should point out that the initial ideas of

condition (E4) and the general one (H

3) aremotivatedby [24]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was supported financially by the National Nat-ural Science Foundation of China Tianyuan Foundation(11226119) the Scientific and Technological Innovation Pro-grams of Higher Education Institutions in Shanxi the YouthScience Foundation of Shanxi Province (2013021002-1) andShandong Provincial Natural Science Foundation China(ZR2012AQ024)

References

[1] ENDancer andYHDu ldquoExistence of changing sign solutionsfor some semilinear problems with jumping nonlinearities atzerordquo Proceedings of the Royal Society of Edinburgh vol 124 no6 pp 1165ndash1176 1994

[2] D Motreanu andM Tanaka ldquoSign-changing and constant-signsolutions for 119901-Laplacian problems with jumping nonlineari-tiesrdquo Journal of Differential Equations vol 249 no 12 pp 3352ndash3376 2010

[3] Z Zhang and S Li ldquoOn sign-changing and multiple solutionsof the 119901-Laplacianrdquo Journal of Functional Analysis vol 197 no2 pp 447ndash468 2003

[4] W Zou Sign-Changing Critical Point Theory Springer NewYork NY USA 2008

[5] X Cheng and C Zhong ldquoExistence of three nontrivial solutionsfor an elliptic systemrdquo Journal of Mathematical Analysis andApplications vol 327 no 2 pp 1420ndash1430 2007

[6] K Zhang The multiple solutions and sign-changing solutionsfor nonlinear operator equations and applications [PhD thesis]Shandong University Jinan China 2002 (Chinese)

[7] X Xu ldquoMultiple sign-changing solutions for some m-pointboundary-value problemsrdquo Electronic Journal of DifferentialEquations vol 89 pp 1ndash14 2004

[8] R Yang Existence of positive solutions sign-changing solutionsfor nonlinear differential equations [MS thesis] ShandongUniversity Jinan China 2006 (Chinese)

[9] C Pang W Dong and Z Wei ldquoMultiple solutions for fourth-order boundary value problemrdquo Journal of Mathematical Anal-ysis and Applications vol 314 no 2 pp 464ndash476 2006

[10] Z Wei and C Pang ldquoMultiple sign-changing solutions forfourth order 119898-point boundary value problemsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 66 no 4 pp 839ndash855 2007

[11] Y Li and F Li ldquoSign-changing solutions to second-orderintegral boundary value problemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 69 no 4 pp 1179ndash1187 2008

[12] T He W Yang and F Yang ldquoSign-changing solutions fordiscrete second-order three-point boundary value problemsrdquoDiscrete Dynamics in Nature and Society Article ID 705387 14pages 2010

[13] J Yang ldquoSign-changing solutions to discrete fourth-orderNeumann boundary value problemsrdquo Advances in DifferenceEquations vol 2013 article 10 2013

[14] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

6 Journal of Operators

[15] D Guo Partial Order Methods in Nonlinear Analysis ShandongScience and Technology Press Jinan China 2000 (Chinese)

[16] D Guo Nonlinear Functional Analysis Shandong Science andTechnology Press Jinan China 2nd edition 2001 (Chinese)

[17] F Li Solutions of nonlinear operator equations and applications[PhD thesis] Shandong University Jinan China 1996 (Chi-nese)

[18] J Sun Nonlinear Functional Analysis and Applications SciencePress Beijing China 2008 (Chinese)

[19] H Amann ldquoFixed points of asymptotically linear maps inordered Banach spacesrdquo Journal of Functional Analysis vol 14pp 162ndash171 1973

[20] H Amann ldquoOn the number of solutions of nonlinear equationsin orderedBanach spacesrdquo Journal of Functional Analysis vol 11pp 346ndash384 1972

[21] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[22] X Liu and J Sun ldquoComputation of topological degree of uni-laterally asymptotically linear operators and its applicationsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 96ndash106 2009

[23] J X Sun and X Y Liu ldquoComputation of the fixed point indexfor non-cone mappings and its applicationsrdquo ActaMathematicaSinica vol 53 no 3 pp 417ndash428 2010 (Chinese)

[24] J X Sun and K M Zhang ldquoExistence of multiple fixed pointsfor nonlinear operators and applicationsrdquo Acta MathematicaSinica vol 24 no 7 pp 1079ndash1088 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Operators

By Lemma 9 and (24)ndash(25) we get that there exist open setsΩ119894(119894 = 1 2 3 4) of 119864 such that

Σ1sub Ω1sub 119875 cap (119879

119903 1198791205881)

Σ2sub Ω2sub 119875 cap (119879

1205882 119879119903)

Σ3sub Ω3sub minus119875 cap (119879

119903 1198791205881)

Σ4sub Ω4sub minus119875 cap (119879

1205882 119879119903)

(27)

deg (119868 minus 119860Ω1 120579) = 1 (28)

deg (119868 minus 119860Ω2 120579) = minus1 (29)

deg (119868 minus 119860Ω3 120579) = 1 (30)

deg (119868 minus 119860Ω4 120579) = minus1 (31)

By Lemma 10 we have

deg (119868 minus 119860 119879119903 120579) = 1 (32)

According to (32) (28) (30) (18) and the additivity propertyof Leray-Schauder degree we obtain

deg (119868 minus 119860 119879119903 (Ω1cup Ω3cup 1198791205881) 120579)

= 1 minus 1 minus 1 minus 1 = minus2

(33)

which yields that119860 has at least one fixed point 1199065in119879119903(Ω1cup

Ω3cup1198791205881) and then 119906

5is a sign-changing fixed point It follows

from (19) (29) (31) (32) and the additivity property of Leray-Schauder degree that we have

deg (119868 minus 119860 1198791205882 (Ω2cup Ω4cup 119879119903) 120579)

= 1 + 1 + 1 minus 1 = 2

(34)

which implies that 119860 has at least one fixed point 1199066in 1198791205882

(Ω2cupΩ4cup119879119903) and then 119906

6is also a sign-changing fixed point

The proof is completed

4 Example

The main purpose of this section is to apply our theorem tononlinear differential equations

We consider the following boundary value problem forelliptic partial differential equations

119871120593 (119909) = 119891 (119909 120593 (119909)) 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(35)

where Ω is a bounded open domain in R119899 120597Ω isin 1198622+120583 and

0 lt 120583 lt 1 119891(119909 120593) Ω timesR1 rarr R1 is continuous

119871120593 = minus

119899

sum

119894119895=1

119886119894119895(119909)

1205972120593

120597119909119894120597119909119895

+

119899

sum

119894=1

119887119894(119909)

120597120593

120597119909119894

+ 119888 (119909) 120593 (36)

is an uniformly elliptic operator that is 119886119894119895(119909) = 119886

119895119894(119909)

119887119894(119909) 119888(119909) isin 119862

120583(Ω) 119888(119909) gt 0 and there exists a constant

number 1205830gt 0 such that sum119899

119894119895=1119886119894119895(119909)120585119894120585119895ge 1205830|120585|2 for all

119909 isin Ω 120585 = (1205851 1205852 120585

119899) isin R119899 Consider

119861120593 = 119887 (119909) 120593 + 120575

119899

sum

119894=1

120573119894(119909)

120597120593

120597119909119894

(37)

which is a boundary operator where 120573 = (1205731 1205732 120573

119899) is a

vector field on 120597Ω of 1198621+120583 satisfying 120573 sdot n gt 0 (n denotes theouter unit normal vector on 120597Ω) and 119887(119909) isin 119862

1+120583(120597Ω) and

assume that one of the following cases holds

(i) 120575 = 0 and 119887(119909) equiv 1(ii) 120575 = 1 and 119887(119909) equiv 0(iii) 120575 = 1 and 119887(119909) gt 0

According to the theory of elliptic partial differentialequations (see [20 21]) we know that for each 119906 isin 119862(Ω)the linear boundary value problem

119871120593 (119909) = 119906 (119909) 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(38)

has a unique solution 120593119906isin 1198622(Ω) Define the operator119870 by

(119870119906) (119909) = 120593119906(119909) 119909 isin Ω (39)

Then 119870 119862(Ω) rarr 1198622(Ω) is a linear completely

continuous operator and has an unbounded sequence of eig-envalues

0 lt 1205821lt 1205822le 1205823le sdot sdot sdot 120582

119899997888rarr +infin (40)

and the spectral radius 119903(119870) = 120582minus11

Let

119864 = 119862 (Ω) 119875 = 120593 isin 119864 | 120593 (119909) ge 0 119909 isin Ω (41)

Then 119864 is an ordered Banach space with the norm 120593 =

sup119909isinΩ

|120593(119909)| and119875 is a normal solid cone in119864 and119870(119875) sub 119875For 120593 isin 119864 define Nemytskii operator by

(119865120593) (119909) = 119891 (119909 120593 (119909)) 119909 isin Ω (42)

Clearly 119865 119864 rarr 119864 is continuous Let 119860 = 119870119865 Then 119860

119864 rarr 119864 is completely continuousBy the proof of Lemma 41 in [22] we have that 1198601015840

120579=

1198910119870Let 119890 = 119890(119909) be the solution of the following boundary

value problem

119871120593 (119909) = 1 119909 isin Ω

119861120593 = 0 119909 isin 120597Ω

(43)

In order to obtain multiple sign-changing solutions of(35) we give the following assumptions

Journal of Operators 5

(E1) 119891(119909 0) equiv 0 and forall119909 isin Ω 119891(119909 120593)120593 gt 0 forall119909 isin Ω 120593 = 0and 120593 isin (minusinfin +infin)

(E2) lim120593rarr0

(119891(119909 120593)120593) = 1198910uniformly for 119909 isin Ω and

120582119899lt 1198910lt 120582119899+1

where 119899 is an even number(E3) lim120593rarrinfin

(119891(119909 120593)120593) = 119891infin

uniformly for 119909 isin Ω and1205821198990lt 119891infinlt 1205821198990+1

where 1198990is an even number

(E4) there exists a constant number 119903 lt 1 such thatlim120593rarr0

(119891(119909 120593)120593) = 119903(1119890) uniformly for 119909 isin Ω

Theorem 12 Suppose that (1198641)ndash(1198644) are satisfied Then the

problem (35) has at least two sign-changing solutions More-over problem (35) has at least two positive solutions and twonegative solutions

Proof From condition (E1) we know that

119891 (119909 120593) gt 0 forall120593 gt 0

119891 (119909 120593) lt 0 forall120593 lt 0

(44)

Copy the proof proceed of Theorem 34 in [23] we have that119860((1198641198900cap 119875) 120579) sub int(119864

1198900cap 119875) and 119860(minus(119864

1198900cap 119875) 120579) sub

int(minus(1198641198900cap 119875)) where

1198641198900= 120593 isin 119864 there exists 120583 gt 0

such that minus 1205831198901(119909) le 120593 le 120583119890

1(119909)

int (1198641198900cap 119875) = 120593 isin (119864

1198900cap 119875) there exist 120572 gt 0

and 120573 gt 0 such that 1205721198901(119909)

le 120593 (119909) le 1205731198901(119909)

(45)

where 1198901is the first normalized eigenfunction of 119870 corre-

sponding to its first eigenvalue 1205821

It follows from (E2) and (E

3) that conditions (H

1) and

(H2) of Theorem 11 holdIn the following we prove that (H

3) of Theorem 11 is

satisfied It follows from (E4) that there exists 120575 gt 0 such that

119891 (119909 120593)

120593

lt

1 + 119903

2 119890

forall 0 lt10038161003816100381610038161205931003816100381610038161003816lt 120575 119909 isin Ω (46)

That is1003816100381610038161003816119891 (119909 120593)

1003816100381610038161003816lt

1 + 119903

2 119890

10038161003816100381610038161205931003816100381610038161003816 119909 isin Ω (47)

Thus1003817100381710038171003817119860120593

1003817100381710038171003817le

1 + 119903

2 119890

11989010038171003817100381710038171205931003817100381710038171003817 (48)

Therefore

lim120593rarr0

1003817100381710038171003817119860120593

1003817100381710038171003817

10038171003817100381710038171205931003817100381710038171003817

le

1 + 119903

2

lt 1 (49)

The proof is completed

Remark 13 It follows from conditions (E2) and (E

4) that

1198910119890 lt 1 We should point out that the initial ideas of

condition (E4) and the general one (H

3) aremotivatedby [24]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was supported financially by the National Nat-ural Science Foundation of China Tianyuan Foundation(11226119) the Scientific and Technological Innovation Pro-grams of Higher Education Institutions in Shanxi the YouthScience Foundation of Shanxi Province (2013021002-1) andShandong Provincial Natural Science Foundation China(ZR2012AQ024)

References

[1] ENDancer andYHDu ldquoExistence of changing sign solutionsfor some semilinear problems with jumping nonlinearities atzerordquo Proceedings of the Royal Society of Edinburgh vol 124 no6 pp 1165ndash1176 1994

[2] D Motreanu andM Tanaka ldquoSign-changing and constant-signsolutions for 119901-Laplacian problems with jumping nonlineari-tiesrdquo Journal of Differential Equations vol 249 no 12 pp 3352ndash3376 2010

[3] Z Zhang and S Li ldquoOn sign-changing and multiple solutionsof the 119901-Laplacianrdquo Journal of Functional Analysis vol 197 no2 pp 447ndash468 2003

[4] W Zou Sign-Changing Critical Point Theory Springer NewYork NY USA 2008

[5] X Cheng and C Zhong ldquoExistence of three nontrivial solutionsfor an elliptic systemrdquo Journal of Mathematical Analysis andApplications vol 327 no 2 pp 1420ndash1430 2007

[6] K Zhang The multiple solutions and sign-changing solutionsfor nonlinear operator equations and applications [PhD thesis]Shandong University Jinan China 2002 (Chinese)

[7] X Xu ldquoMultiple sign-changing solutions for some m-pointboundary-value problemsrdquo Electronic Journal of DifferentialEquations vol 89 pp 1ndash14 2004

[8] R Yang Existence of positive solutions sign-changing solutionsfor nonlinear differential equations [MS thesis] ShandongUniversity Jinan China 2006 (Chinese)

[9] C Pang W Dong and Z Wei ldquoMultiple solutions for fourth-order boundary value problemrdquo Journal of Mathematical Anal-ysis and Applications vol 314 no 2 pp 464ndash476 2006

[10] Z Wei and C Pang ldquoMultiple sign-changing solutions forfourth order 119898-point boundary value problemsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 66 no 4 pp 839ndash855 2007

[11] Y Li and F Li ldquoSign-changing solutions to second-orderintegral boundary value problemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 69 no 4 pp 1179ndash1187 2008

[12] T He W Yang and F Yang ldquoSign-changing solutions fordiscrete second-order three-point boundary value problemsrdquoDiscrete Dynamics in Nature and Society Article ID 705387 14pages 2010

[13] J Yang ldquoSign-changing solutions to discrete fourth-orderNeumann boundary value problemsrdquo Advances in DifferenceEquations vol 2013 article 10 2013

[14] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

6 Journal of Operators

[15] D Guo Partial Order Methods in Nonlinear Analysis ShandongScience and Technology Press Jinan China 2000 (Chinese)

[16] D Guo Nonlinear Functional Analysis Shandong Science andTechnology Press Jinan China 2nd edition 2001 (Chinese)

[17] F Li Solutions of nonlinear operator equations and applications[PhD thesis] Shandong University Jinan China 1996 (Chi-nese)

[18] J Sun Nonlinear Functional Analysis and Applications SciencePress Beijing China 2008 (Chinese)

[19] H Amann ldquoFixed points of asymptotically linear maps inordered Banach spacesrdquo Journal of Functional Analysis vol 14pp 162ndash171 1973

[20] H Amann ldquoOn the number of solutions of nonlinear equationsin orderedBanach spacesrdquo Journal of Functional Analysis vol 11pp 346ndash384 1972

[21] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[22] X Liu and J Sun ldquoComputation of topological degree of uni-laterally asymptotically linear operators and its applicationsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 96ndash106 2009

[23] J X Sun and X Y Liu ldquoComputation of the fixed point indexfor non-cone mappings and its applicationsrdquo ActaMathematicaSinica vol 53 no 3 pp 417ndash428 2010 (Chinese)

[24] J X Sun and K M Zhang ldquoExistence of multiple fixed pointsfor nonlinear operators and applicationsrdquo Acta MathematicaSinica vol 24 no 7 pp 1079ndash1088 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Operators 5

(E1) 119891(119909 0) equiv 0 and forall119909 isin Ω 119891(119909 120593)120593 gt 0 forall119909 isin Ω 120593 = 0and 120593 isin (minusinfin +infin)

(E2) lim120593rarr0

(119891(119909 120593)120593) = 1198910uniformly for 119909 isin Ω and

120582119899lt 1198910lt 120582119899+1

where 119899 is an even number(E3) lim120593rarrinfin

(119891(119909 120593)120593) = 119891infin

uniformly for 119909 isin Ω and1205821198990lt 119891infinlt 1205821198990+1

where 1198990is an even number

(E4) there exists a constant number 119903 lt 1 such thatlim120593rarr0

(119891(119909 120593)120593) = 119903(1119890) uniformly for 119909 isin Ω

Theorem 12 Suppose that (1198641)ndash(1198644) are satisfied Then the

problem (35) has at least two sign-changing solutions More-over problem (35) has at least two positive solutions and twonegative solutions

Proof From condition (E1) we know that

119891 (119909 120593) gt 0 forall120593 gt 0

119891 (119909 120593) lt 0 forall120593 lt 0

(44)

Copy the proof proceed of Theorem 34 in [23] we have that119860((1198641198900cap 119875) 120579) sub int(119864

1198900cap 119875) and 119860(minus(119864

1198900cap 119875) 120579) sub

int(minus(1198641198900cap 119875)) where

1198641198900= 120593 isin 119864 there exists 120583 gt 0

such that minus 1205831198901(119909) le 120593 le 120583119890

1(119909)

int (1198641198900cap 119875) = 120593 isin (119864

1198900cap 119875) there exist 120572 gt 0

and 120573 gt 0 such that 1205721198901(119909)

le 120593 (119909) le 1205731198901(119909)

(45)

where 1198901is the first normalized eigenfunction of 119870 corre-

sponding to its first eigenvalue 1205821

It follows from (E2) and (E

3) that conditions (H

1) and

(H2) of Theorem 11 holdIn the following we prove that (H

3) of Theorem 11 is

satisfied It follows from (E4) that there exists 120575 gt 0 such that

119891 (119909 120593)

120593

lt

1 + 119903

2 119890

forall 0 lt10038161003816100381610038161205931003816100381610038161003816lt 120575 119909 isin Ω (46)

That is1003816100381610038161003816119891 (119909 120593)

1003816100381610038161003816lt

1 + 119903

2 119890

10038161003816100381610038161205931003816100381610038161003816 119909 isin Ω (47)

Thus1003817100381710038171003817119860120593

1003817100381710038171003817le

1 + 119903

2 119890

11989010038171003817100381710038171205931003817100381710038171003817 (48)

Therefore

lim120593rarr0

1003817100381710038171003817119860120593

1003817100381710038171003817

10038171003817100381710038171205931003817100381710038171003817

le

1 + 119903

2

lt 1 (49)

The proof is completed

Remark 13 It follows from conditions (E2) and (E

4) that

1198910119890 lt 1 We should point out that the initial ideas of

condition (E4) and the general one (H

3) aremotivatedby [24]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was supported financially by the National Nat-ural Science Foundation of China Tianyuan Foundation(11226119) the Scientific and Technological Innovation Pro-grams of Higher Education Institutions in Shanxi the YouthScience Foundation of Shanxi Province (2013021002-1) andShandong Provincial Natural Science Foundation China(ZR2012AQ024)

References

[1] ENDancer andYHDu ldquoExistence of changing sign solutionsfor some semilinear problems with jumping nonlinearities atzerordquo Proceedings of the Royal Society of Edinburgh vol 124 no6 pp 1165ndash1176 1994

[2] D Motreanu andM Tanaka ldquoSign-changing and constant-signsolutions for 119901-Laplacian problems with jumping nonlineari-tiesrdquo Journal of Differential Equations vol 249 no 12 pp 3352ndash3376 2010

[3] Z Zhang and S Li ldquoOn sign-changing and multiple solutionsof the 119901-Laplacianrdquo Journal of Functional Analysis vol 197 no2 pp 447ndash468 2003

[4] W Zou Sign-Changing Critical Point Theory Springer NewYork NY USA 2008

[5] X Cheng and C Zhong ldquoExistence of three nontrivial solutionsfor an elliptic systemrdquo Journal of Mathematical Analysis andApplications vol 327 no 2 pp 1420ndash1430 2007

[6] K Zhang The multiple solutions and sign-changing solutionsfor nonlinear operator equations and applications [PhD thesis]Shandong University Jinan China 2002 (Chinese)

[7] X Xu ldquoMultiple sign-changing solutions for some m-pointboundary-value problemsrdquo Electronic Journal of DifferentialEquations vol 89 pp 1ndash14 2004

[8] R Yang Existence of positive solutions sign-changing solutionsfor nonlinear differential equations [MS thesis] ShandongUniversity Jinan China 2006 (Chinese)

[9] C Pang W Dong and Z Wei ldquoMultiple solutions for fourth-order boundary value problemrdquo Journal of Mathematical Anal-ysis and Applications vol 314 no 2 pp 464ndash476 2006

[10] Z Wei and C Pang ldquoMultiple sign-changing solutions forfourth order 119898-point boundary value problemsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 66 no 4 pp 839ndash855 2007

[11] Y Li and F Li ldquoSign-changing solutions to second-orderintegral boundary value problemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 69 no 4 pp 1179ndash1187 2008

[12] T He W Yang and F Yang ldquoSign-changing solutions fordiscrete second-order three-point boundary value problemsrdquoDiscrete Dynamics in Nature and Society Article ID 705387 14pages 2010

[13] J Yang ldquoSign-changing solutions to discrete fourth-orderNeumann boundary value problemsrdquo Advances in DifferenceEquations vol 2013 article 10 2013

[14] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

6 Journal of Operators

[15] D Guo Partial Order Methods in Nonlinear Analysis ShandongScience and Technology Press Jinan China 2000 (Chinese)

[16] D Guo Nonlinear Functional Analysis Shandong Science andTechnology Press Jinan China 2nd edition 2001 (Chinese)

[17] F Li Solutions of nonlinear operator equations and applications[PhD thesis] Shandong University Jinan China 1996 (Chi-nese)

[18] J Sun Nonlinear Functional Analysis and Applications SciencePress Beijing China 2008 (Chinese)

[19] H Amann ldquoFixed points of asymptotically linear maps inordered Banach spacesrdquo Journal of Functional Analysis vol 14pp 162ndash171 1973

[20] H Amann ldquoOn the number of solutions of nonlinear equationsin orderedBanach spacesrdquo Journal of Functional Analysis vol 11pp 346ndash384 1972

[21] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[22] X Liu and J Sun ldquoComputation of topological degree of uni-laterally asymptotically linear operators and its applicationsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 96ndash106 2009

[23] J X Sun and X Y Liu ldquoComputation of the fixed point indexfor non-cone mappings and its applicationsrdquo ActaMathematicaSinica vol 53 no 3 pp 417ndash428 2010 (Chinese)

[24] J X Sun and K M Zhang ldquoExistence of multiple fixed pointsfor nonlinear operators and applicationsrdquo Acta MathematicaSinica vol 24 no 7 pp 1079ndash1088 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Operators

[15] D Guo Partial Order Methods in Nonlinear Analysis ShandongScience and Technology Press Jinan China 2000 (Chinese)

[16] D Guo Nonlinear Functional Analysis Shandong Science andTechnology Press Jinan China 2nd edition 2001 (Chinese)

[17] F Li Solutions of nonlinear operator equations and applications[PhD thesis] Shandong University Jinan China 1996 (Chi-nese)

[18] J Sun Nonlinear Functional Analysis and Applications SciencePress Beijing China 2008 (Chinese)

[19] H Amann ldquoFixed points of asymptotically linear maps inordered Banach spacesrdquo Journal of Functional Analysis vol 14pp 162ndash171 1973

[20] H Amann ldquoOn the number of solutions of nonlinear equationsin orderedBanach spacesrdquo Journal of Functional Analysis vol 11pp 346ndash384 1972

[21] H Amann ldquoFixed point equations and nonlinear eigenvalueproblems in ordered Banach spacesrdquo SIAM Review vol 18 no4 pp 620ndash709 1976

[22] X Liu and J Sun ldquoComputation of topological degree of uni-laterally asymptotically linear operators and its applicationsrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 96ndash106 2009

[23] J X Sun and X Y Liu ldquoComputation of the fixed point indexfor non-cone mappings and its applicationsrdquo ActaMathematicaSinica vol 53 no 3 pp 417ndash428 2010 (Chinese)

[24] J X Sun and K M Zhang ldquoExistence of multiple fixed pointsfor nonlinear operators and applicationsrdquo Acta MathematicaSinica vol 24 no 7 pp 1079ndash1088 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of