Research Article Existence of Nontrivial Solutions and Sign-Changing Solutions for Nonlinear Dynamic Equations on Time Scales

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1 Discrete Dynmics in Nture nd Society Volume 2011, Article ID , 22 pges doi: /2011/ Reserch Article Existence of Nontrivil Solutions nd Sign-Chnging Solutions for Nonliner Dynmic Equtions on Time Scles Huiye Xu College of Economics nd Mngement, North University of Chin, Tiyun, Shnxi , Chin Correspondence should be ddressed to Huiye Xu, Received 26 August 2010; Revised 16 November 2010; Accepted 8 Jnury 2011 Acdemic Editor: Jinde Co Copyright q 2011 Huiye Xu. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. By using the topologicl degree theory nd the fixed point index theory, the existence of nontrivil solutions nd sign-chnging solutions for clss of boundry vlue problem on time scles is obtined. We should point out tht the integrl opertor corresponding to bove boundry vlue problem is not ssumed to be cone mpping. 1. Introduction Let T be time scle which hs the subspce topology inherited from the stndrd topology on R. ForechintervlI of R, wedefineit I T. In this pper, we study the existence of nontrivil solutions nd sign-chnging solutions of the following problem on time scles: [ r t u Δ t ] Δ σ t, t, b T, αu βu Δ 0, γu σ b δu Δ σ b 0, 1.1 where α 0, β 0, γ 0, δ 0, α 2 β 2 γ 2 δ 2 / 0, r t > 0, t, σ b T,ndr Δ exists. Some bsic definitions on time scles cn be found in 1 3. In n ordered Bnch spce, much interest hs developed regrding the computtion of the fixed-point index bout cone mppings. Bsed on it, the existence of positive solutions of vrious dynmic equtions hs been studied extensively by numerous reserchers. The reder is referred to 4 13 for some recent works on second-order boundry vlue problem on time scles.

2 2 Discrete Dynmics in Nture nd Society Dynmic equtions on time scles hve ttrcted considerble interest becuse of their bility to model economic phenomen, see nd references therein. The existence of mthemtics frmework to describe plys n importnt role in the dvncement in ll the socil sciences. Time scles clculus is likely to be pplied in mny fields of economics, which cn combine the stndrd discrete nd continuous models. Historiclly, two seprte pproches hve dominted mthemtics modeling: differentileqution, termed continuous dynmic modeling, nd difference eqution, termed discrete dynmic modeling. Certin economiclly importnt phenomen do not solely possess elements of the continuous or modeling elements of the discrete. For exmple, in discrete model, n enterprise obtins some net income in period of time nd decides how mny dividends to extend nd how mny to leve s retined ernings during the sme period. Thus, ll decisions re ssumed to be mde t evenly spced intervls. The time scles to this optimiztion problem re much more flexible nd relistic. For exmple, n enterprise obtins net income t one point in time, dividend-extending decisions re mde t different point in time, nd keeping retined ernings tkes plce t yet nother point in time. Moreover, dividendsextending nd retined erning decisions cn be modeled to occur with rbitrry, timevrying frequency. It is hrd to overestimte the dvntges of such n pproch over the discrete or continuous models used in economics. In recent yers, there is much ttention pid to the existence of positive solution for second-order boundry vlue problems on time scles; for detils, see 4 11 nd references therein. In 5, Erbe nd Peterson considered the existence of positive solutions to the following problem: x ΔΔ t f t, x σ t, t, b T, αx βx Δ 0, γx σ b δx Δ σ b 0, 1.2 where f :, σ b T R R is continuous. The following conditions re imposed on f: f 0 : lim 0, u 0 u f : lim, uniformly on t, σ b u u T 1.3 or f 0 : lim, u 0 u f : lim 0, uniformly on t, σ b u u T. 1.4 The uthors obtined the existence of positive solutions for problem 1.2 by mens of the cone expnsion nd compression fixed-point theorem. In 6, Chyn nd Henderson were concerned with determining vlues of λ for which there exist positive solutions of the following nonliner dynmic eqution: x ΔΔ t λ t f x σ t 0, t 0, 1 T, 1.5 stisfying either the conjugte boundry conditions x 0 x σ

3 Discrete Dynmics in Nture nd Society 3 or the right focl boundry conditions x 0 x Δ σ Their nlysis still relied on the cone expnsion nd compression fixed-point theorem. Furthermore, Hong nd Yeh 9 generlized the min theorems in 6 to the following problem x ΔΔ t λf t, x σ t 0, t 0, 1 T, αu 0 βx Δ 0 0, γx σ 1 δx Δ σ They lso pplied the bove-mentioned fixed-point theorem in cone to yield positive solutions of 1.8 for λ belonging to suitble open intervl. On the other hnd, there is much ttention pid to the study of spectrl theory of liner problems on time scles, see 12, 13 nd references therein. Agrwl et l. 12 obtined fundmentl result on the existence of eigenvlues nd the number of the generlized zeros of ssocited eigenfunctions. This result provides foundtion for the further study of the behvior of eigenvlues of the Sturm-Liouville problems on time scles. In 13, Kong first extended the existence result in 12 to the Sturm-Liouville problems in more generl form nd with normlized seprted boundry conditions nd then explored the dependence of the eigenvlues on the boundry condition. The uthor showed tht the nth eigenvlue λ n depends continuously on the boundry condition except t the generlized Dirichlet boundry conditions, where certin jump discontinuities my occur. Furthermore, λ n s function of the boundry condition ngles is continuously differentible wherever it is continuous. Formuls for such derivtives were obtined which revel the monotone properties of λ n in terms of the boundry condition ngles. We note tht Li et l. 10 were concerned with the existence of positive solutions for problem 1.1 under some conditions concerning the first eigenvlue corresponding to the relevnt liner opertor. Their min results improved nd generlized ones in 5 9, 11. We should point out tht the nonliner item tht ppered in the bove dynmic equtions is required to be nonnegtive; moreover, the positive solutions cn be obtined only by using these tools. However, some existing nonliner problems cnnot be ttributed to the cone mppings; thus, the cone mpping theory fils to solve these problems. At the sme time, the existence of the sign-chnging solutions of the bove dynmic equtions is being rised by n ever-incresing number of reserchers; it is lso difficult to del with sign-chnging solution problems by the cone mppings theory. Stimulted by these works, in 17, Sunnd Liu used the lttice structure to present some methods of computtion of the topologicl degree for the opertor tht is qusidditive on lttice. The opertor is not ssumed to be cone mpping. Furthermore, in 18, Liu nd Sun estblished some methods of computtion of the fixed-point index nd the topologicl degree for the unilterlly symptoticlly liner opertors tht re not ssumed to be cone mppings. Very recently, in 19, SunndLiu obtined some theorems for fixed-point index bout clss of nonliner opertors which re not cone mppings by mens of the theory of cone.

4 4 Discrete Dynmics in Nture nd Society The min purpose of this pper is to estblish some existence theorems of nontrivil solutions nd sign-chnging solutions of 1.1. Our results generlize nd complement ones in This pper is orgnized s follows. Some preliminries re given in Section 2. In Section 3, we presented some results obtined in 19 ; then, we pply these bstrct results to problem 1.1, the existence of nontrivil solutions nd sign-chnging solutions for subliner problem 1.1 is obtined. In Section 4, we re concerned with the existence of sign-chnging solutions nd nontrivil solutions for superliner problem 1.1. Our min tool relies on the computtion of the topologicl degree for superliner opertors, which re obtined in 17. In Section 5, by the use of the bstrct results obtined in 18, some existence theorems of sign-chnging solutions for unilterlly symptoticlly liner problem 1.1 re estblished. 2. Preliminries nd Some Lemms Let E be n ordered Bnch spce in which the prtil ordering is induced by cone P E. P is sid to be norml if there exists positive constnt N such tht x y implies x N y. P is clled solid if it contins interior points, tht is, int P /. Ifx y nd x / y, we write x<y;ifconep is solid nd y x int P, wewritex y. OpertorA is sid to be strongly incresing if y<ximplies Ay Ax. P is clled totl if E P P. IfP is solid, then P is totl. A fixed-point u of opertor A is sid to be sign-chnging fixed-point if u/ P P.LetB : E E be bounded liner opertor. B is sid to be positive if B P P. For the concepts nd the properties bout the cones, we refer to For w E, letp w P w {x E x w} nd P w w P {x E x w}. We cll E lttice under the prtil ordering if sup{x, y} nd inf{x, y} exist for rbitrry x, y E. Forx E, let x sup{x, }, x sup{ x, }. 2.1 x nd x re clled positive prt nd negtive prt of x, respectively. Tking x x x, then x P, nd x is clled the module of x. Onecnsee 24 for the definition nd the properties of the lttice. For convenience, we use the following nottions: x x, x x, 2.2 nd then x P, x P, x x x. 2.3 In the following, we lwys ssume tht E is Bnch spce, P is totl cone in E,nd the prtil ordering in E is induced by P. We lso suppose tht E is lttice in the prtil ordering. Let B : E E be positive completely continuous liner opertor, let r B be spectrl rdius of B, letb be the conjugted opertor of B, ndletp be the conjugted cone

5 Discrete Dynmics in Nture nd Society 5 of P. SinceP E is totl cone, ccording to the fmous Krein-Rutmn theorem see 25, we infer tht if r B / 0, then there exist ϕ P \{} nd g P \{}, suchtht Bϕ r B ϕ, B g r B g. 2.4 Fixed ϕ P \{}, g P \{} such tht 2.4 holds. For δ>0, let P ( g,δ ) { x P, g x δ x }. 2.5 Then, P g,δ is lso cone in E. Definition 2.1 see 23. LetB be positive liner opertor. The opertor B is sid to stisfy H condition if there exist ϕ P \{}, g P \{}, ndδ>0suchtht 2.4 holds, nd B mps P into P g,δ. Definition 2.2 see 23. LetD E nd A : D E be nonliner opertor. A is sid to be qusidditive on lttice if there exists v E such tht Ax Ax Ax v, x D, 2.6 where x nd x re defined by 2.2. Definition 2.3 see 18. LetE be Bnch spce with cone P, ndleta : E E be nonliner opertor. We cll A unilterlly symptoticlly liner opertor long P w if there exists bounded liner opertor B such tht Ax Bx lim 0. x P w, x x 2.7 B is sid to be the derived opertor of A long P w ndwillbedenotedbya P w. Similrly, we cn lso define unilterlly symptoticlly liner opertor long P w. For convenience, we list the following definitions on time scles which cn be found in 1 3. Definition 2.4. A time scle T is nonempty closed subset of rel numbers R.Fort<sup T nd r>inf T, define the forwrd jump opertor σ nd bckwrd jump opertor ρ, respectively, by σ t inf{τ T τ>t} T, ρ r sup{τ T τ<r} T. 2.8 For ll t, r T. Ifσ t >t, t is sid to be right scttered, nd if ρ r <r, r is sid to be left scttered; if σ t t, t is sid to be right dense, nd if ρ r r, r is sid to be left dense. If T hs right-scttered minimum m, definet k T {m}; otherwise,sett k T. IfT hs left-scttered mximum M, definet k T {M}; otherwise, set T k T.

6 6 Discrete Dynmics in Nture nd Society Definition 2.5. For f : T R nd t T k, the delt derivtive of f t the point t is defined to be the number f Δ t provided it exists, with the property tht for ech ɛ>0, there is neighborhood U of t such tht f σ t f s f Δ t σ t s ɛ σ t s, 2.9 for ll s U. Definition 2.6. If G Δ t f t, then we define the delt integrl by b f t Δt G b G Lemm 2.7 see 17. Suppose tht B : E E is positive bounded liner opertor. If the spectrl rdius r B < 1,then I B 1 exists nd is positive bounded liner opertor. Lemm 2.8 see 5. Set d γβ r αδ σ b αγ r σ b 1 r τ Δτ / Let where 1 x t y σ s, t s, d G t, s 1 x σ s y t, σ s t, d 2.12 t x t α 1 β Δτ r τ r σ b, y t γ t 1 δ Δτ r τ r σ b Then G t, s is the Green function of the following liner boundry vlue problem: [ Δ r t u t ] Δ 0, t, b T, αu βu Δ 0, γu σ b δu Δ σ b According to Lemm 2.8,problem 1.1 cn be converted into the equivlent nonliner integrl eqution u t G t, s f s, u σ s Δs, t [ ], σ 2 b T. 2.15

7 Discrete Dynmics in Nture nd Society 7 Define the opertors Au t Ku t G t, s f s, u σ s Δs, G t, s u σ s Δs, t t [ ], σ 2 b T, [ ], σ 2 b T By in 13,Theorem2.1, we cn know tht K hs the sequence of eigenvlues: 0 <λ 1 <λ 2 <λ 3 < <λ n <, n N k 0, 2.18 where N k 0 : {1, 2,...}, k, {1, 2,...,k}, k <, 2.19 the lgebric multiplicities of every eigenvlue is simple, nd the spectrl rdius r K λ 1 1. Let E { u : [, σ 2 b ] T R,uΔ t is continuous on [, σ 2 b ] T, [ r t u Δ t ] Δ is right dense continuous on, b T } Then E is n ordered Bnch spce with the supremum norm u sup t,σ 2 b T u t.define conep E by P { [ ] } u E u t 0, t, σ 2 b T It is cler tht P is norml solid cone nd E becomes lttice under the nturl ordering. Lemm 2.9 see 10. Assume tht there exists h t P \{}, suchtht G t, s h t G τ, s, t, τ, σ b T,s, b T Suppose tht there exists ψ P \{}, suchthtψ r 1 K K ψ, ψ h t / 0. Then opertor K stisfies H condition, where K is defined in Topologicl Degree for Subliner Problem 1.1 For R>0, define T R {x E x <R}. In 17, Sun nd Liu obtined the following bstrct results.

8 8 Discrete Dynmics in Nture nd Society Lemm 3.1 see 17. Let E be Bnch spce, nd let P be norml solid cone in E, w 0 E, nd let A : P w0 P w0 be completely continuous opertor. Suppose tht there exists positive bounded liner opertor B : E E with r B < 1 nd u 0 P such tht Ax Bx u 0, x P w Then there exists R 0 > 0, such tht the fixed-point index i A, T R P w0,p w0 1for ll R>R 0. Lemm 3.2 see 17. Let E be Bnch spce, nd let P be norml solid cone in E, w 0 E, nd let A : P w0 P w0 be completely continuous opertor. Suppose tht there exists positive bounded liner opertor B nd positive completely continuous opertor B 1, u 0 P, r R, r>0,suchtht Ax Bx u 0, x P, Ax B 1 x, x T r P. 3.2 If r B < 1 nd r B 1 1,thenA hs t lest one nonzero positive fixed-point in P. Lemm 3.3 see 17. Let E be Bnch spce, let P be norml solid cone in E, w 0 int P, nd let A : P w0 P w0 be completely continuous opertor. Suppose tht i there exists positive bounded liner opertor B with r B < 1 nd u 0 P such tht Ax Bx u 0, x P w0, 3.3 ii A, thefréchet derivtive A of A t exists, nd 1 is not n eigenvlue of A. Then 1 if the sum of the lgebric multiplicities for ll eigenvlues of A in 0, 1 is n odd number, A hs t lest one nonzero fixed-point; 2 if r A > 1 nd A P P, A P P, A hs t lest two nonzero fixed-points, one of which is positive, the other is negtive; 3 if r A > 1, the sum of the lgebric multiplicities for ll eigenvlues of A in 0, 1 is n even number, nd A P \ {} int P, A P \ {} int P, 3.4 then A hs t lest three nonzero fixed-points, one of which is positive nother is negtive, nd the third fixed-point is sign-chnging. By mens of the bove bstrct results, we hve the following theorems.

9 Discrete Dynmics in Nture nd Society 9 Theorem 3.4. Suppose tht there exist M>0 nd η>0 such tht M, t, σ b T,u 0, 3.5 lim sup λ 1 η, uniformly on t, σ b u u T. 3.6 Then problem 1.1 hs t lest one solution. Proof. It is esy to check tht A : E E is completely continuous opertor. Choose 0 <ε< η.denotec λ 1 η ε<λ 1. According to 3.6, we cn know tht there exists R>0suchtht u c, t, σ b T, u R. 3.7 This implies tht there exists constnt number M 0 > 0suchtht cu M 0, t, σ b T,u 0, 3.8 cu M 0, t, σ b T,u Denote B ck, wherek is defined s It is obvious tht B : E E is positive bounded liner opertor, nd r B cr K < λ 1 r K 1. By Lemm 2.7, wehvetht I B 1 exists nd is positive bounded liner opertor. Set b>0, nd let b t b, l t M 0 M, [ ] w 0 t I B 1 Kl b t, t, σ b T It is obvious tht w 0 P, nd w 0 t Bw 0 t Kl t b t, t, σ b T For ny u 0 0, it follows from 3.5 nd 3.8 tht cu 0 M M 0, u u 0, 3.12 which together with w 0 0 implies tht t cw 0 t l t, u t w 0 t. 3.13

10 10 Discrete Dynmics in Nture nd Society Note tht K P P. Combining 3.11 with 3.13 nd 2.16, wehve Au t G t, s f s, u σ s Δs w 0 t, G t, s cw 0 σ s l σ s Δs G t, s cw 0 σ s l σ s Δs b t u t w 0 t, t [ ], σ 2 b T This implies tht A P w0 P w0. Let ϑ sup{ w 0 t : t, σ b T }, κ sup { : t, σ b T, ϑ u 0 } It is esy to know from 3.9 tht cu M 0 κ cϑ, t, σ b T,u ϑ Thus, t cu t M 0 κ cϑ, u t w 0 t, t, σ b T Let e 0 t M 0 κ cϑ, u 0 t Ke 0 t, t, σ b T. It is cler tht u 0 P. It follows from 3.17 tht Au t Bu t u 0 t, u P w Thus, ll conditions in Lemm 3.1 re stisfied; the conclusion then follows from Lemm 3.1. Theorem 3.5. Suppose tht 0, t, σ b T,u 0, 3.19 nd there exists η>0 such tht lim sup λ 1 η, u u lim inf λ 1 η, uniformly on t, σ b u 0 u T Then problem 1.1 hs t lest one positive solution.

11 Discrete Dynmics in Nture nd Society 11 Proof. It is cler tht A : P P is completely continuous opertor. Choose 0 <ε<η. Denote c λ 1 η ε, c 1 λ 1 η ε then c<λ 1,c 1 >λ 1.By 3.20, we obtin tht there exists D>d>0suchtht cu, t, σ b T,u D, 3.22 c 1 u, t, σ b T, 0 u d According to 3.22, wehve cu M, t, σ b T,u 0, 3.24 where M sup. t,σ b T, u D 3.25 Denote B ck, B 1 c 1 K,whereK is still defined s 2.17 ; then, B, B 1 : E E re both completely continuous positive liner opertors, nd r B cr K <λ 1 r K 1, r B 1 c 1 r K >λ 1 r K Let v 0 K M. Then, v 0 P. It follows from 2.16, 3.19, 3.23, nd 3.24 tht Au t [ ] G t, s cu σ s M Δs Au t Bu t v 0 t, u P, t, σ b T, G t, s m 1 u σ s Δs 3.27 B 1 u t, u P, t, σ b T, u d. By Lemm 3.2, problem 1.1 hs t lest one positive solution. Remrk 3.6. Condition 3.20 implies tht lim sup <λ 1, lim inf >λ u u u 0 1, uniformly on t, σ b u T Hence, Theorem 3.5 becomes Theorem 2.1 in 10. Therefore, our min result generlizes Theorem 2.1 in 10.

12 12 Discrete Dynmics in Nture nd Society Theorem 3.7. Assume tht 3.5 nd 3.6 hold. In ddition, suppose tht f t, 0 0, t, σ b T, 3.29 lim λ, uniformly on t, σ b u 0 u T Then i if λ n <λ<λ n 1,wheren is n odd number, then problem 1.1 hs t lest one nontrivil solution; ii if λ>λ 1 nd u 0, t, σ b T,u R, 3.31 then problem 1.1 hs t lest two nontrivil solutions, one of which is positive nd the otherisnegtive. Proof. It suffices to verify tht ll the conditions of Lemm 3.3 re stisfied. Let w 0 be defined s 3.10.SinceKl b int P,wehvew 0 int P. It follows from 3.5, 3.6, nd the proof of Theorem 3.4 tht A : P w0 P w0 is completely continuous nd condition i of Lemm 3.3 holds. Furthermore, 3.29 implies tht A ; 3.30 implies tht A u t G t, s f u s, 0 u σ s Δs λ G t, s u σ s Δs, t [ ], σ 2 b T This shows tht A λk. Letι denote the sum of the lgebric multiplicities for ll eigenvlues of A in 0, 1. Sinceλ n <λ<λ n 1, we cn obtin tht 1 is not n eigenvlue of A nd ι n is n odd number. Conclusion 1 of Lemm 3.3 holds. Equqtion 3.31 implies tht A P P, A P P. Byλ>λ 1, we cn know tht 1 is not n eigenvlue of A,ndr A λr K >λ 1r K 1. This implies tht the conclusion 2 of Lemm 3.3 holds. Theorem 3.8. Assume tht 3.5, 3.6, 3.29, nd 3.30 hold. In ddition, suppose tht λ n <λ<λ n 1, where n is n even number, u >0, t, σ b T,u/ 0, u R Then the problem 1.1 hs t lest three nontrivil solutions, one of which is positive, nother is negtive, nd the third solution is sign chnging.

13 Discrete Dynmics in Nture nd Society 13 Proof. For convenience, let us introduce nother ordered Bnch spce. Letting, e 1 be the first normlized eigenfunction of K corresponding to its first eigenvlue λ 1,then e 1 1. It follows from 10, Lemm 1.4 tht e 1 > 0. Let E {x E there exists ν>0, νe 1 t x t νe 1 t } According to 20, we cn obtin tht E is n ordering Bnch spce, P E P is norml solid cone, nd K : E E is liner completely continuous opertor stisfying K P \{} int P,where int P { } x P there exist ς 1 > 0, ς 2 > 0, ς 1 e 1 t x t ς 2 e 1 t Letting l t nd B be s in Theorem 3.4,thenl P \{}.Letw 0 I B 1 Kl.Since K P P, wehvekl int P. Furthermore,weobtinw 0 int P. NotingthtK E E, we hve A E E. Similrly to the proof of Theorem 3.4,wehve A P w0 P w Furthermore, we cn know tht ) A (P w0 P w Since K : E E is completely continuous, we hve tht A : P w0 P w0 is completly continuous. According to Theorem 3.7, it is esy to show tht conditions i nd ii of Lemm 3.3 re stisfied, nd A λk. Since λ n <λ<λ n 1, we hve tht 1 is not n eigenvlue of A nd λ>λ 1. It follows tht r ( A ) λr K >λ1 r K Moreover, we cn know tht the sum of the lgebric multiplicities for ll eigenvlues of A in 0, 1 is n; this is n even number. In the following, we check the condition 3.4. By 3.33,wehve > 0, u >0, < 0, u <0, t, σ b T Notice tht K P \ {} int P. 3.40

14 14 Discrete Dynmics in Nture nd Society It follows tht ( ) A P \ {} int P, (( ) ) ( ) A P \ {} int P Therefore, condition 3.4 holds. The conclusion follows from Lemm Topologicl Degree for Superliner Problem 1.1 In 17, Sun nd Liu discussed computtion for the topologicl degree bout superliner opertors which re not cone mppings. The min tools re the prtil ordering reltion nd the lttice structure. Their min results re the following theorems. Lemm 4.1 see 17. Let E be Bnch spce, nd let A : E E be completely continuous opertor stisfying A BF, wheref is qusidditive on lttice nd B is positive bounded liner opertor stisfying H condition. Moreover, suppose tht P is solid cone in E nd i there exist 1 >r 1 B nd y 1 P, suchtht Fx 1 x y 1, x P, 4.1 ii there exist 0 < 2 <r 1 B nd y 2 P, suchtht Fx 2 x y 2, x P. 4.2 In ddition, there exist bounded open set Ω, which contins, nd positive bounded liner opertor B 0 with r B 0 1,suchtht Ax B 0 x, x Ω. 4.3 Then A hs t lest one nonzero fixed-point. Lemm 4.2 see 17. Let E be Bnch spce, nd let A : E E be completely continuous opertor stisfying A BF, wheref is qusidditive on lttice nd B is positive bounded liner opertor stisfying H condition. Moreover, suppose tht P is solid cone in E nd conditions (i) nd (ii) of Theorem 4.1 re stisfied. In ddition, ssume tht A, thefréchet derivtive A of A t exists, nd 1 is not n eigenvlue of A.ThenA hs t lest one nonzero fixed-point. In [23], Sun obtined the following further result concerning sign-chnging fixed-point. Lemm 4.3 see 23. Let the conditions of Theorem 4.2 be stisfied. In ddition, suppose tht the sum of the lgebric multiplicities for ll eigenvlues of A in the intervl 0, 1 is n even number, nd A P \ {} int P, A P \ {} int P. 4.4 Then A hs t lest three nonzero fixed-points, one of which is positive, nother is negtive, the third one is sign-chnging fixed-point.

15 Discrete Dynmics in Nture nd Society 15 In the following, we will pply the bove three bstrct theorems to problem 1.1 ;our min results re s follows. Theorem 4.4. Assume tht f :, σ b T R R is continuous. Suppose tht there exists η>0 such tht lim sup λ 1 η, uniformly on t, σ b u u T, 4.5 lim inf λ 1 η, u u uniformly on t, σ b T, 4.6 lim u 0 u λ 1 η, uniformly on t, σ b T. 4.7 Then problem 1.1 hs t lest one nontrivil solution. Proof. It is cler tht A : E E is completely continuous. According to 4.6 nd 4.5, we hve tht there exists R 1 > 0, such tht (λ 1 12 η ) u, t, σ b T,u R 1, (λ 1 12 η ) u, t, σ b T,u R Hence, there exists M 0 > 0suchtht (λ 1 12 η ) u M 0, t, σ b T, u 0, (λ 1 12 η ) u M 0, t, σ b T, u According to the proof of Theorem 3.1 in 10, wehve G t, s x t y t [ ] mx t,σ2 b T x t mx t,σ2 b T y t G τ, s, t, τ, σ 2 b T, s, σ b T By Lemm 2.9, we cn know tht liner opertor K stisfies the H condition. Let Fu t t for u E. It is esy to check tht F stisfies 2.6 nd A KF. It follows from 4.9 tht 4.1 nd 4.2 hold, where 1 λ 1 1/2 η >λ 1 r 1 K nd 2 λ 1 1/2 η <λ 1 r 1 K. By 4.7, we obtin tht there exists r 0 > 0suchtht ( λ η ) u, t, σ b T, u r

16 16 Discrete Dynmics in Nture nd Society Therefore, Au (λ 1 12 η ) K u, u E, u r Thus, 4.3 holds. An ppliction of Lemm 4.1 shows tht problem 1.1 hs t lest one nontrivil solution. Remrk 4.5. If we ssume tht f :, σ b T R R, condition 4.5, 4.6 will be removed. Therefore, Theorem 4.4 generlizes nd extends Theorem 3.4 in 10. Theorem 4.6. Assume tht f :, σ b T R suppose tht R is continuous nd 4.5 holds. In ddition, f t, 0 0, t, σ b T, lim λ, uniformly on t, σ b u 0 u T, 4.13 nd λ/ {λ 1,λ 2,...,λ n,...}.thenproblem 1.1 hs t lest one nontrivil solution. Proof. By Theorem 4.4, we only need to verify the condition moreover of Lemm 4.2. According to 4.13, we cn know tht A nd A λk. SinceK hs the sequence of eigenvlues {λ n } n 1, it is esy to see tht {λ n/λ} n 1 is the sequence of ll eigenvlues of A. Therefore, 1 is not n eigenvlue of A. The conclusion of Theorem 4.6 follows from Lemm 4.2. Theorem 4.7. Assume tht f :, σ b T R R nd 4.5, 4.6, nd 4.13 hold. If λ n <λ<λ n 1, where n is n even number, u >0, t, σ b T,u/ 0, u R Then the problem 1.1 hs t lest three nontrivil solutions, one of which is positive, nother is negtive, nd the third solution is sign chnging. Proof. From the proof of Theorem 4.6, it is esy to show tht ll conditions of Lemm 4.2 re stisfied nd A hs the sequence of eigenvlues {λ n/λ} n 1. Since λ n <λ<λ n 1 nd n is n even number, we cn know tht the sum of the lgebric multiplicities for ll eigenvlues of A in 0, 1 is n even number. Condition 4.14 implies tht A P \ {} int P, A P \ {} int P An ppliction of Lemm 4.3 shows tht problem 1.1 hs t lest three nontrivil solutions, one of which is positive, nother is negtive, nd the third solution is sign chnging. Now let us end this section by the following two exmples.

17 Discrete Dynmics in Nture nd Society 17 Exmple 4.8. Let T Z. Considering the following BVP: Δ 2 u t u 4u 2 u 4 0, u 0 u 5 0, t 1, 4, 4.16 where 1, 4 is the discrete intervl {1, 2, 3, 4}, Δu t u t 1 u t, Δ 2 u t Δ Δu t. By 26, Remrk 2.3 nd Lemm 2.9, we cn know tht λ k 4sin 2 kπ/10, k 1, 2, 3, 4. In ddition, the lgebric multiplicity of ech eigenvlue λ k is equl to 1. Setting δ , it is esy to check tht u 4u 2 u 4 lim sup u u λ 1 δ, u 4u 2 u 4 lim sup lim sup ( u u 3) u u u 4.17, ( lim sup u u 3). u Therefore, ll conditions in Theorem 4.4 re stisfied. By Theorem 4.4,BVP 4.16 hs t lest one nontrivil solution. Exmple 4.9. Let T Z. Consider the following BVP: Δ 2 u t 1 u rctnu 0, u 0 u 5 0. t 1, 4, 4.18 In virtue of Exmple 4.8, we cn know tht λ , λ , λ , nd λ Setting δ , by direct clcultion, we hve u rctnu lim sup π u u λ 1 δ,

18 18 Discrete Dynmics in Nture nd Society u rctnu lim sup π u u λ 1 δ, lim rctnu 0.39 / { , , , }. u Therefore, ll conditions in Theorem 4.6 re stisfied. By Theorem 4.6,BVP 4.18 hs t lest one nontrivil solution. 5. Topologicl Degree for Unilterlly Asymptoticlly Liner Problem 1.1 In 18, Liu nd Sun presented some methods of computing the topologicl degree for unilterlly symptoticlly liner opertors by using the lttice structure. Their min results re s follows. Lemm 5.1 see 18. Let P be norml cone in E, ndleta : E E be completely continuous nd qusidditive on lttice. Suppose tht there exist u,u 1 P nd positive bounded liner opertor L 0 : E E with r L 0 < 1,suchtht Ax u, x P, Ax L 0 x u 1, x P. 5.1 In ddition, suppose tht A P exists, A, thefréchet derivtive A of A t exists,nd1isnot n eigenvlue of A.ThenA hs t lest one nontrivil fixed-point, provide tht one of the following is stisfied: i r A P > 1 nd 1 is not n eigenvlue of A P corresponding to positive eigenvector; ii r A P < 1 nd the sum of the lgebric multiplicities for ll eigenvlues of A,lyinginthe intervl 0, 1, is n odd number. Lemm 5.2 see 18. Suppose tht i A is strongly incresing on P nd P; ii both A P nd A P exist with r A P > 1 nd r A P > 1; 1 is not n eigenvlue of A P or A corresponding to positive eigenvector; P iii A ; thefréchet derivtive A of A t is strongly positive, nd r A < 1; iv the Fréchet derivtive A of A t exists; 1 is not n eigenvlue of A ; the sum of the lgebric multiplicities for ll eigenvlues of A, lying in the intervl 0, 1 is n even number. Then A hs t lest three nontrivil fixed-points, contining one sign-chnging fixed-point.

19 Discrete Dynmics in Nture nd Society 19 Through this section, the following hypotheses re needed: E 1 lim u /u p uniformly on t, σ b T ; E 2 lim u /u q uniformly on t, σ b T ; E 3 f t, 0 0, lim u 0 /u λ uniformly on t, σ b T. Our min results re s follows. Theorem 5.3. Suppose tht f stisfies E 1 E 3.Thenproblem 1.1 hs t lest one nontrivil solution provided one of the following conditions is stisfied: i p>λ 1, 0 <q<λ 1,p, λ / λ k, k 1, 2,...; ii 0 <p<λ 1, q>λ 1, q, λ / λ k, k 1, 2,...; iii 0 <p<λ 1, 0 <q<λ 1, λ n <λ<λ n 1, n is n odd number. Proof. It follows from Definition 2.3 tht ] A P [, u t pku t, t σ 2 b T. 5.2 A P [, u t qku t, t σ 2 b ]T. Furthermore, we hve A u t G t, s f u s, 0 u σ s Δs λ λku t, G t, s u σ s Δs t [ ], σ 2 b T. 5.3 Assume tht condition i is stisfied. By p>0, we cn obtin tht there exists M 1 > 0such tht M 1, t, σ b T,u So, we hve Au t KM 1, u Since q<λ 1, we cn obtin tht q η<λ 1 for some η>0. E 2 implies tht there exists μ 0 > 0 such tht ( q η ) u μ 0, t, σ b T,u

20 20 Discrete Dynmics in Nture nd Society Consequently, Au t G t, s σ s Δs G t, s [( q η ) u σ s μ 0 ] Δs ( q η ) Ku Kμ 0, u P. 5.7 Let D 0 q η K,then r D 0 ( q η ) r K <λ 1 r K Eqution 5.5 nd 5.7 imply tht 5.1 holds. Since the sequences of ll eigenvlues of A, A P,ndA P re {λ n/λ} n 1, {λ n/p} n 1, nd {λ n /q} n 1, respectively. It is esy to see from p, λ / λ k, k 1, 2,... tht 1 is not n eigenvlue of A or A P. Lemm 5.1 i ssures tht A hs t lest one nontrivil fixed-point, nd hence problem 1.1 hs t lest one nontrivil solution. Similrly, we cn prove tht the conclusion holds in the cse tht condition ii or iii is stisfied. Theorem 5.4. Letting f stisfy E 1 E 3 nd p q : χ. In ddition, suppose tht i is strictly incresing on u for fixed t, σ b T ; ii λ n <χ<λ n 1 nd n is n even number; iii 0 <λ<λ 1. Then problem 1.1 hs t lest three nontrivil solutions, contining sign-chnging solution. Proof. Similrly to the proof of Theorem 3.8, we still need to use nother ordered Bnch spce E. From the proof of Theorem 5.3 nd Theorem 3.8, we know tht A λk nd A P A P χk. P P gives tht A P A χk,wherep is s in Theorem 3.8. Itisesytosee P tht A χk. Evidently,e 1 int P. It follows from the condition i nd K P \{} int P tht A is strongly incresing. Condition ii implies tht 1 is not n eigenvlue of A P or A P,ndr A P ra P χr K > λ 1 r K 1. By the sme method, we cn show tht A is strongly incresing. Moreover, by iii, wehver A λr K < λ 1r K 1. At lst, from the proof of Theorem 5.3, we obtin tht the sequence of ll eigenvlues of A is {λ n /χ} n 1. According to ii, we cn know tht the sum of the lgebric multiplicities for ll eigenvlues of A in 0, 1 is n even number. Hence, ll the conditions of Lemm 5.2 re stisfied. The conclusion follows from Lemm Conclusion In this pper, some existence results of sign-chnging solutions for dynmic equtions on time scles re estblished. As fr s we know, there were few ppers tht studied this topic.

21 Discrete Dynmics in Nture nd Society 21 Therefore, our results of this pper re new. It is nturl for us to pply the method of this pper to more generl form of dynmic equtions. However, we note tht the study of signchnging solutions for time scles boundry vlue problems relies on the spectrum structure of liner dynmic equtions on time scles. The reder is referred to 27, 28 for further works. In most ppers on time scles boundry vlue problem, in order to obtin the existence of positive solutions by using fixed-point theorems on cone, the nonliner term, which ppers in the right-hnd side of the eqution, is required to be nonnegtive. For our min results, the nonliner term my be sign-chnging function, thus, the integrl opertor is not necessry to be cone mpping. We should point out tht Sun nd Zhng 29 hve studied nontrivil solutions of the singulr superliner Sturm-Liouville problems of ordinry differentil equtions; the ides of this pper come from 17 19, 29. In 30, Luo focused on the ppliction of spectrl theory of liner dynmic problems to the existence of positive solutions for nonliner weighted eigenvlue problem on time scles. The uthor lso discussed the existence of nodl solutions nd the globl structure of the solution for nonliner eigenvlue problem on time scles by using the eigenvlues of the corresponding liner problem nd globl bifurction theory. Up to now, no results on the spectrum structure of liner dynmic equtions hve been estblished for multipoint boundry vlue problems. This is n open problem. Acknowledgment The uthor ws supported finncilly by the Science Foundtion of North University of Chin. References 1 M. Bohner nd A. Peterson, Dynmic Equtions on Time Scles: An Introduction with Applictions, Birkhäuser, Boston, Mss, USA, M. Bohner nd A. Peterson, Eds., Advnces in Dynmic Equtions on Time Scles, Birkhäuser, Boston, Mss, USA, V. Lkshmiknthm, S. Sivsundrm, nd B. Kymkcln, Dynmic Systems on Mesure Chins, vol. 370 of Mthemtics nd Its Applictions, Kluwer Acdemic Publishers, Dordrecht, The Netherlnds, L. Erbe nd A. Peterson, Green s functions nd comprison theorems for differentil equtions on mesure chins, Dynmics of Continuous, Discrete nd Impulsive Systems, vol. 6, no. 1, pp , L. Erbe nd A. Peterson, Positive solutions for nonliner differentil eqution on mesure chin, Mthemticl nd Computer Modelling, vol. 32, no. 5-6, pp , C. J. Chyn nd J. Henderson, Eigenvlue problems for nonliner differentil equtions on mesure chin, Journl of Mthemticl Anlysis nd Applictions, vol. 245, no. 2, pp , R. P. Agrwl nd D. O Regn, Nonliner boundry vlue problems on time scles, Nonliner Anlysis: Theory, Methods & Applictions, vol. 44, no. 4, pp , D. R. Anderson, Eigenvlue intervls for two-point boundry vlue problem on mesure chin, Journl of Computtionl nd Applied Mthemtics, vol. 141, no. 1-2, pp , C.-H. Hong nd C.-C. Yeh, Positive solutions for eigenvlue problems on mesure chin, Nonliner Anlysis: Theory, Methods & Applictions, vol. 51, no. 3, pp , H. Y. Li, J. X. Sun, nd Y. J. Cui, Positive solutions of nonliner differentil equtions on mesure chin, Chinese Annls of Mthemtics. Series A., vol. 30, no. 1, pp , 2009 Chinese. 11 Z.-C. Ho, J. Ling, nd T.-J. Xio, Existence results for time scle boundry vlue problem, Journl of Computtionl nd Applied Mthemtics, vol. 197, no. 1, pp , R. P. Agrwl, M. Bohner, nd P. J. Y. Wong, Sturm-Liouville eigenvlue problems on time scles, Applied Mthemtics nd Computtion, vol. 99, no. 2-3, pp , 1999.

22 22 Discrete Dynmics in Nture nd Society 13 Q. Kong, Sturm-Liouville problems on time scles with seprted boundry conditions, Results in Mthemtics, vol. 52, no. 1-2, pp , F. M. Atici, D. C. Biles, nd A. Lebedinsky, An ppliction of time scles to economics, Mthemticl nd Computer Modelling, vol. 43, no. 7-8, pp , S. Hilger, Anlysis on mesure chins unified pproch to continuous nd discrete clculus, Results in Mthemtics, vol. 18, no. 1-2, pp , C. C. Tisdell nd A. Zidi, Bsic qulittive nd quntittive results for solutions to nonliner, dynmic equtions on time scles with n ppliction to economic modelling, Nonliner Anlysis: Theory, Methods & Applictions, vol. 68, no. 11, pp , J. Sun nd X. Liu, Computtion of topologicl degree in ordered Bnch spces with lttice structure nd its ppliction to superliner differentil equtions, Journl of Mthemticl Anlysis nd Applictions, vol. 348, no. 2, pp , X. Liu nd J. Sun, Computtion of topologicl degree of unilterlly symptoticlly liner opertors nd its pplictions, Nonliner Anlysis: Theory, Methods & Applictions, vol. 71, no. 1-2, pp , J. X. Sun nd X. Y. Liu, Computtion of the fixed point index for non-cone mppings nd its pplictions, Act Mthemtic Sinic. Chinese Series, vol. 53, no. 3, pp , 2010 Chinese. 20 H. Amnn, Fixed point equtions nd nonliner eigenvlue problems in ordered Bnch spces, SIAM Review, vol. 18, no. 4, pp , K. Deimling, Nonliner Functionl Anlysis, Springer, Berlin, Germny, D. J. Guo nd V. Lkshmiknthm, Nonliner Problems in Abstrct Cones, vol. 5 of Notes nd Reports in Mthemtics in Science nd Engineering, Acdemic Press, Boston, Mss, USA, J. X. Sun, Nonliner Functionl Anlysis nd Applictions, Science Press, Beijing, Chin, W. A. J. Luxemburg nd A. C. Znen, Riesz Spces. Vol. I, North-Hollnd Mthemticl Librry, North-Hollnd, Amsterdm, The Netherlnds, M. G. Kreĭn nd M. A. Rutmn, Liner opertors leving invrint cone in Bnch spce, Americn Mthemticl Society Trnsltions, vol. 1950, no. 26, p. 128, T. He, W. Yng, nd F. Yng, Sign-chnging solutions for discrete second-order three-point boundry vlue problems, Discrete Dynmics in Nture nd Society, vol. 2010, Article ID , 14 pges, F. A. Dvidson nd B. P. Rynne, Globl bifurction on time scles, Journl of Mthemticl Anlysis nd Applictions, vol. 267, no. 1, pp , F. A. Dvidson nd B. P. Rynne, Curves of positive solutions of boundry vlue problems on timescles, Journl of Mthemticl Anlysis nd Applictions, vol. 300, no. 2, pp , J. Sun nd G. Zhng, Nontrivil solutions of singulr superliner Sturm-Liouville problems, Journl of Mthemticl Anlysis nd Applictions, vol. 313, no. 2, pp , H. Luo, Boundry vlue problem of nonliner dynmic equtions on time scles, Doctorl thesis, Northwest Norml University, Lnzhou, Chin, 2007.

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