Research Article Boundary Value Problems for Systems of Second-Order Dynamic Equations on Time Scales with Δ-Carathéodory Functions

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1 Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2010, Article ID , 26 pages doi: /2010/ Research Article Boundary Value Problems for Systems of Second-Order Dynamic Equations on Time Scales with -Carathéodory Functions M. Frigon 1 and H. Gilbert 2 1 Département de Mathématiques et de Statistique, Université demontréal, CP 6128, Succursale Centre-Ville, Montréal, QC, Canada H3C 3J7 2 Département de Mathématiques, Collège Édouard-Montpetit, 945 Chemin de Chambly, Longueuil, QC, Canada J4H 3M6 Correspondence should be addressed to M. Frigon, frigon@dms.umontreal.ca Received 31 August 2010; Accepted 30 November 2010 Academic Editor: J. Mawhin Copyright q 2010 M. Frigon and H. Gilbert. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We establish the existence of solutions to systems of second-order dynamic equations on time scales with the right member f, a-carathéodory function. First, we consider the case where the nonlinearity f does not depend on the -derivative, x t. We obtain existence results for Strum-Liouville and for periodic boundary conditions. Finally, we consider more general systems in which the nonlinearity f depends on the -derivative and satisfies a linear growth condition with respect to x t. Our existence results rely on notions of solution-tube that are introduced in this paper. 1. Introduction In this paper, we establish existence results for the following systems of second-order dynamic equations on time scales: x t f t, x σ t, -a.e. t T κ2 0,x BC, 1.1 x t f t, x σ t,x t, -a.e. t T κ2 0, a 0 x a x a x 0, a 1 x b γ 1 x ρ b x

2 2 Abstract and Applied Analysis Here, T is a compact time scale where a min T, b max T, and T κ2 0 is defined in 2.4. The map f : T κ 0 Rn R n is -Carathéodory see Definition 2.9, and BC denotes one of the following boundary conditions: x a x b, x a x ρ b, a 0 x a γ 0 x a x 0, a 1 x b γ 1 x ρ b x 1, where a 0,a 1,γ 0,γ 1 0, max{a 0,γ 0 } > 0, and max{a 1,γ 1 } > 0. Problem 1.1 was mainly treated in the case where it has only one equation n 1 and f is continuous. In particular, the existence of a solution of 1.1 was established by Akın 1 for the Dirichlet boundary condition and by Stehlík 2 for the periodic boundary condition. Equation 1.1 with nonlinear boundary conditions was studied by Peterson et al. 3. In all those results, the method of lower and upper solutions was used. See also 4, 5 and the references therein for other results on the problem 1.1 when n 1. Very few existence results were obtained for the system 1.1 when n>1. Recently, Henderson et al. 6 and Amster et al. 7 established the existence of solutions of 1.1 with Sturm-Liouville and nonlinear boundary conditions, respectively, assuming that f is a continuous function satisfying the following condition: R >0 such that x, f t, x > 0 if x R. 1.5 The fact that the right member in the system 1.2 depends also on the -derivative, x, increases considerably the difficulty of this problem. So, it is not surprising that there are almost no results for this problem in the literature. Atici et al. 8 studied this problem in the particular case, where there is only one equation n 1 and f is positive, continuous and satisfies a monotonicity condition. Assuming a growth condition of Wintner type and using the method of lower and upper solutions, they obtained the existence of a solution. The system 1.2 with the Dirichlet boundary condition was studied by Henderson and Tisdell 9 in the general case where n>1. They considered f a continuous function and T a regular time scale i.e., ρ t <t<σ t or T a, b. They established the existence of a solution of 1.2 under the following assumptions: A1 there exists R>max{ x 0, x 1 } such that 2 x, f t, x, y y 2 > 0if x R, 2 x, y μ t y 2, A2 there exist c, d 0suchthatd ρ b a < 1and f t, x, y c d y if x R. In the third section of this paper, we establish an existence theorem for the system 1.1. To this aim, we introduce a notion of solution-tube of 1.1 which generalizes to systems the notions of lower and upper solutions introduced in 1, 2. This notion generalizes also condition 1.5 used by Henderson et al. 6 and Amster et al. 7. Our notion of solutiontube is in the spirit of the notion of solution-tube for systems of second-order differential equations introduced in 10. Our notion is new even in the case of systems of second-order difference equations.

3 Abstract and Applied Analysis 3 In the last section of this paper, we study the system 1.2. Again, we introduce a notion of solution-tube of 1.2 which generalizes the notion of lower and upper solutions used by Atici et al. 8. This notion generalizes also condition 1.5 and the notion of solution-tube of systems of second-order differential equations introduced in 10.Inaddition,weassumethat f satisfies a linear growth condition. It is worthwhile to mention that the time scale T does not need to be regular, and we do not require the restriction d ρ b a < 1 as in assumption A2 used in 9. Moreover, we point out that the right members of our systems are not necessarily continuous. Indeed, we assume that the weaker condition: f is a -Carathéodory function. This condition is interesting in the case where the points of T are not all right scattered. We obtain the existence of solutions to 1.1 and to 1.2 in the Sobolev space W 2,1 T, Rn. To our knowledge, it is the first paper applying the theory of Sobolev spaces with topological methods to obtain solutions to 1.1 and 1.2. Solutions of second-order Hamiltonian systems on time scales were obtained in a Sobolev space via variational methods in 11. Finally, let us mention that our results are new also in the continuous case and for systems of second-order difference equations. 2. Preliminaries and Notations For sake of completeness, we recall some notations, definitions, and results concerning functions defined on time scales. The interested reader may consult 12, 13 and the references therein to find the proofs and to get a complete introduction to this subject. Let T be a compact time scale with a min T <b max T. Theforward jump operator σ : T T resp., the backward jump operator ρ : T T is defined by inf{s T : s>t} if t<b, σ t b if t b, sup{s T : s<t} if t>a, resp., ρ t. a if t a 2.1 We say that t<bis right scattered resp., t>ais left scattered if σ t >t resp., ρ t <t otherwise, we say that t is right dense resp., left dense. The set of right-scattered points of T is at most countable, see 14.Wedenoteitby R T : {t T : t<σ t } {t i : i I}, 2.2 for some I N. Thegraininess function μ : T 0, is defined by μ t σ t t.wedenote T κ T \ ρ b,b ], T0 T \ {b}. 2.3 So, T κ T if b is left dense, otherwise T κ T0.SinceT κ is also a time scale, we denote T κ2 T κ κ, T κ2 0 Tκ2 \ {b}. 2.4

4 4 Abstract and Applied Analysis In 1990, Hilger 15 introduced the concept of dynamic equations on time scales. This concept provides a unified approach to continuous and discrete calculus with the introduction of the notion of delta-derivative x t. This notion coincides with x t resp., x t inthecasewherethetimescalet is an interval resp., the discrete set {0, 1,...,N}. Definition 2.1. Amapf : T R n is -differentiable at t T κ if there exists f t R n called the -derivative of f at t such that for all ɛ>0, there exists a neighborhood U of t such that f σ t f s f t σ t s ɛ σ t s s U. 2.5 We say that f is -differentiable if f t exists for every t T κ. If f is -differentiable and if f is -differentiable at t T κ2,wecallf t f t the second -derivative of f at t. Proposition 2.2. Let f : T R n and t T κ. i If f is -differentiable at t,thenf is continuous at t. ii If f is continuous at t R T,then f t f σ t f t. 2.6 μ t iii The map f is -differentiable at t T κ \ R T if and only if f f t f s t lim. 2.7 s t t s Proposition 2.3. If f : T R n and g : T R m are -differentiable at t T κ,then i if n m, αf g t αf t g t for every α R, ii if m 1, fg t g t f t f σ t g t f t g t g σ t f t, iii if m 1 and g t g σ t / 0,then f t g t f t f t g t, 2.8 g g t g σ t iv if W R n is open and h : W h f t h f t,f t. R is differentiable at f t W and t / R T,then We denote C T, R n the space of continuous maps on T, and we denote C 1 T, R n the space of continuous maps on T with continuous -derivative on T κ. With the norm x 0 max{ x t : t T} resp., x 1 max{ x 0, max{ x t : t T κ }}, C T, R n resp., C 1 T, R n is a Banach space.

5 Abstract and Applied Analysis 5 We study the second -derivative of the norm of a map. Lemma 2.4. Let x : T R n be -differentiable. 1 On {t T κ2 : x σ t > 0 and x t exists}, x t x σ t,x t. 2.9 x σ t 2 On {t T κ2 \ R T : x σ t > 0 and x t exists}, x t x t,x t x t 2 x t x t,x t 2 x t Proof. Denote A {t T κ2 A \ R T,wehave : x σ t > 0andx t exists}. ByProposition 2.3, ontheset x t x t,x t x t 2 x t x t x t,x t, x t x t,x t 2 x t 3 x σ t,x t. x σ t 2.11 If t A is such that t<σ t σ 2 t, thenbypropositions2.2 and 2.3,wehave x t x σ t x t μ t x σ t,x σ t μ t x σ t x σ t x t μ t 2 x σ t,x t μ t x t μ t x σ t x σ t,x t μ t x t μ t 2 x σ t x t μ t x σ t,x t x σ t x σ t,x t. x σ t x σ t,x t x t μ t 2 x σ t μ t 2

6 6 Abstract and Applied Analysis If t A is such that t<σ t <σ 2 t, then x t x σ t x t μ t x σ 2 t x σ t μ σ t μ t x σ t,x σ 2 t x σ t μ σ t μ t x σ t x σ t x t μ t 2 x σ t x t μ t x σ t,x σ t μ t x σ t and we conclude as in the previous case. x σ t x t μ t 2, Let ε>0. The exponential function e ε,t 0 is defined by e ε t, t 0 exp ξ ε μ s s, 2.14 t 0,t T where ε, if h 0, ξ ε h log 1 hε, h if h> It is the unique solution to the initial value problem x t εx t, x t Here is a result on time scales, analogous to Gronwall s inequality. The reader may find the proof of this result in 13. Theorem 2.5. Let α>0, ε>0,andy C T, R. If then y t α εy s s for every t T, 2.17 a,t T y t αe ε t, a for every t T We recall some notions and results related to the theory of -measure and -Lebesgue integration introduced by Bohner and Guseinov in 12. The reader is also referred to 14

7 Abstract and Applied Analysis 7 for expressions of the -measure and the -integral in terms of the classical Lebesgue measure and the classical Lebesgue integral, respectively. Definition 2.6. AsetA T is said to be -measurable if for every set E T, m 1 E m 1 E A m 1 E T \ A, 2.19 where m 1 E { m } m inf d k c k : E c k,d k with c k,d k T if b / E, k 1 k 1 if b E The -measure on M m 1 : {A T : A is -measurable}, denoted by μ,istherestrictionof m 1 to M m 1.So, T, M m 1,μ is a complete measurable space. Proposition 2.7 see 14. Let A T, thena is -measurable if and only if A is Lebesgue measurable. Moreover, if b / A, μ A m A t i A RT σ t i t i, 2.21 where m is the Lebesgue measure. The notions of -measurable and -integrable functions f : T R n can be defined similarly to the general theory of Lebesgue integral. Let E T be a -measurable set and f : T R n a -measurable function. We say that f L 1 E, Rn provided E f s s < The set L 1 T 0, R n is a Banach space endowed with the norm f L 1 : T 0 f s s Here is an analog of the Lebesgue dominated convergence Theorem which can be proved as in the general theory of Lebesgue integration theory. Theorem 2.8. Let {f k } k N be a sequence of functions in L 1 T 0, R n. Assume that there exists a function f : T0 R n such that f k t f t -a.e. t T0, and there exists a function g L 1 T 0 such that f k t g t -a.e. t T0 and for every k N, thenf k f in L 1 T 0, R n. In our existence results, we will consider -Carathéodory functions.

8 8 Abstract and Applied Analysis Definition 2.9. Afunctionf : T0 R m hold: R n is -Carathéodory if the following conditions i t f t, x is -measurable for every x R m, ii x f t, x is continuous for -almost every t T0, iii for every r > 0, there exists h r L 1 T 0, 0, such that f t, x h r t for -almost every t T0 and for every x R m such that x r. In this context, there is also a notion of absolute continuity introduced in 16. Definition Afunctionf : T R n is said to be absolutely continuous on T if for every ɛ>0, there exists a δ>0suchthatif{ a k,b k } m k 1 with a k,b k T is a finite pairwise disjoint family of subintervals satisfying m b k a k <δ, k 1 m then f b k f a k <ɛ. k Proposition 2.11 see 17. If f : T R n is an absolutely continuous function, then the -measure of the set {t T0 \ R T : f t 0 and f t / 0} is zero. Proposition 2.12 see 17. If g L 1 T 0, R n and f : T R n is the function defined by f t : g s s, a,t T 2.25 then f is absolutely continuous and f t g t -almost everywhere on T0. Proposition 2.13 see 16. A function f : T R is absolutely continuous on T ifandonlyiff is -differentiable -almost everywhere on T0, f L 1 T 0 and a,t T f s s f t f a, for every t T We also recall a notion of Sobolev space, see 18, { W 1,1 T, Rn x L 1 T 0, R n : g L 1 T 0, R n such that x s φ s s T 0 } g s φ σ s s for every φ C 1 0,rd T T 0, 2.27

9 Abstract and Applied Analysis 9 where C 1 0,rd T { φ : T R : φ a 0 φ b,φ is -differentiable and φ is continuous at right-dense points of T and 2.28 } its left-sided limits exist at left-dense points of T. Afunctionx W 1,1 T, Rn can be identified to an absolutely continuous map. Proposition 2.14 see 18. Suppose that x W 1,1 T, R with some g L1 T 0, R satisfying 2.27, then there exists y : T R absolutely continuous such that y x, y g -a.e. on T Moreover, if g is C rd T κ, R, then there exists y C 1 T, R such that rd y x -a.e. on T0, y g -a.e. on T κ Sobolev spaces of higher order can be defined inductively as follows: { } W 2,1 T, Rn x W 1,1 T, Rn : x W 1,1 Tκ, R n With the norm x W 1,1 : x L 1 x L 1 resp., x W 2,1 W 1,1 T, Rn resp., W 2,1 T, Rn is a Banach space. : x L 1 x L 1 x L 1, Remark By Proposition 2.7, we know that μ {t} > 0 for every t R T.Fromthisfact and the previous proposition, one realizes that there is no interest to look for solutions to 1.1 and 1.2 in W 2,1 T, Rn and to consider -Carathéodory maps f inthecasewherethe time scale is such that μ T0 \ R T 0. In particular, this is the case for difference equations. Let us point out that we consider more general time scales. Nevertheless, the results that we obtained are new in both cases. As in the classical case, some embeddings have nice properties. Proposition 2.16 see 18. The inclusion j 1 : W 2,1 T, Rn C 1 T, R n is continuous. Proposition The inclusion j 0 : W 2,1 T, Rn C T, R n is a continuous, compact, linear operator. Proof. Arguing as in the proof of the Arzelà-Ascoli Theorem, we can show that the inclusion i : C 1 T, R n C T, R n is linear, continuous, and compact. The conclusion follows from the previous proposition since j 0 i j 1. We obtain a maximum principle in this context. To this aim, we use the following result.

10 10 Abstract and Applied Analysis Lemma 2.18 see 19. Let f : T R be a function with a local maximum at t 0 a, b T. If f ρ t 0 exists, then f ρ t 0 0 provided t 0 is not at the same time left dense and right scattered. Theorem Let r W 2,1 T be a function such that r t > 0 -almost everywhere on {t : r σ t > 0}. If one of the following conditions holds: T κ2 0 i a 0 r a γ 0 r a 0 and a 1 r b γ 1 r ρ b 0 where a 0, a 1, γ 0,andγ 1 are defined as in 1.4, ii r a r b and r a r ρ b, then r t 0, for every t T. Proof. If the conclusion is false, there exists t 0 T such that r t 0 max r t > 0. t T 2.32 In the case where a ρ t 0 <t 0 <b,thenr ρ t 0 exists since μ {ρ t 0 } t 0 ρ t 0 > 0andr W 2,1 T. ByLemma 2.18, r ρ t 0 0, which is a contradiction since r σ ρ t 0 r t 0 > 0. If a < ρ t 0 t 0 σ t 0 < b,thereexistst 1 > t 0 such that r σ t > 0 for every t t 0,t 1 T. On the other hand, since r t 0 is a maximum, r t 0 0, and there exists s t 0,t 1 such that r s 0, thus, 0 r s r t 0 t 0,s T r τ τ >0, 2.33 by hypothesis and Proposition 2.13, which is a contradiction. Observe that the same argument applies if a ρ t 0 t 0 σ t 0 and r t 0 0. Notice also that if ρ t 0 t 0 σ t 0 band r t 0 0, we get a contradiction with an analogous argument for a suitable t 1 <t 0. If a ρ t 0 t 0 <σ t 0 <band r t 0 r σ t 0, weargueasinthefirstcasereplacing t 0 by σ t 0 to obtain a contradiction. Inthecasewherea<ρ t 0 t 0 <σ t 0 b and r t 0 >r σ t 0, thenr t 0 < 0. Since t r t is continuous, there exists δ>0suchthatr t < 0onaninterval t 0 δ, t 0,which contradicts the maximality of r t 0. Observe that a t 0 and r a < 0 could happen if a t 0 σ t 0 or if a t 0 <σ t 0 and r a >r σ a. In this case, we get a contradiction if r satisfies condition i. On the other hand, if r satisfies condition ii, thenr b r a > 0andr ρ b < 0. So ρ b b since r b r ρ b. This contradicts the maximality of r b. On the other hand, the case where t 0 b and r satisfies condition i can be treated similarly to the previous case. Finally, we define the -differential operator associated to the problems that we will consider L : W 2,1,B T, Rn L 1 T κ 0, Rn defined by L x t : x t x σ t, 2.34

11 Abstract and Applied Analysis 11 where W 2,1,B T, Rn : {x W 2,1 T, Rn : x BC } with BC denoting the periodic boundary condition 1.3 or the Sturm-Liouville boundary condition 1.4. Proposition The operator L is invertible and L 1 is affine and continuous. Proof. If BC denotes the Sturm-Liouville boundary condition 1.4, consider the associated homogeneous boundary condition a 0 x a γ 0 x a 0, a 1 x b γ 1 x ρ b Denote W 2,1,B 0 T, R n { } x W 2,1 T, Rn : x satisfies 1.3 { } x W 2,1 T, Rn : x satisfies 2.35 if BC denotes 1.3, if BC denotes Notice that W 2,1,B 0 T, R n is a Banach space. Define L 0 : W 2,1,B 0 T, R n L 1 T κ 0, Rn by L 0 x t : x t x σ t It is obvious that L 0 is linear and continuous. It follows directly from Theorem 2.19 that L 0 is injective. If BC denotes 1.4 resp., 1.3, letg t, s be the Green function given in 13, Theorem 4.70 resp., 13, Theorem Arguingasin 13, Theorem 4.70 resp., 13, Theorem 4.89, one can verify that for any h L 1 Tκ 0, Rn, x t a,ρ b T G t, s h s s 2.38 is a solution of L 0 x h. So,L 0 is bijective and, hence, invertible with L 1 0 continuous by the inverse mapping theorem. Finally, if BC denotes 1.3,sinceL L 0, we have the conclusion. On the other hand, if BC denotes 1.4,lety be given in 13, Theorem 4.67 such that y t y σ t 0 on T κ2, a 0 y a γ 0 y a x 0, a 1 y b γ 1 y ρ b x 1, 2.39 then L 1 y L 1 0. Remark We could have considered the operator L : W 2,1,B T, Rn L 1 T κ 0, Rn defined by L x t : x t, 2.40

12 12 Abstract and Applied Analysis when the boundary condition is 1.4 with suitable constants a 0, a 1, γ 0, γ 1 such that L is injective. For simplicity, we prefer to use only the operator L. 3. Nonlinearity Not Depending on the Delta-Derivative In this section, we establish existence results for the problem x t f t, x σ t, -a.e. t T κ2 0,x BC, 3.1 where BC denotes the periodic boundary condition x a x b, x a x ρ b, 3.2 or the Sturm-Liouville boundary condition a 0 x a γ 0 x a x 0, a 1 x b γ 1 x ρ b x 1, 3.3 where a 0,a 1,γ 0,γ 1 0, max{a 0,γ 0 } > 0, and max{a 1,γ 1 } > 0. We look for solutions in W 2,1 T, Rn. We introduce the notion of solution-tube for the problem 3.1. Definition 3.1. Let v, M W 2,1 T, Rn W 2,1 T, 0,. We say that v, M is a solution-tube of 3.1 if i x v σ t,f t, x v t M σ t M t for -almost every t T κ2 0 every x R n such that x v σ t M σ t, and for ii v t f t, v σ t and M t 0for-almost every t T κ2 0 such that M σ t 0, iii a if BC denotes 3.2,thenv a v b, M a M b,and v ρ b v a M ρ b M a, b if BC denotes 3.3, x 0 a 0 v a γ 0 v a a 0 M a γ 0 M a, x 1 a 1 v b γ 1 v ρ b a 1 M b γ 1 M ρ b. We denote T v, M { } x W 2,1 T, Rn : x t v t M t t T. 3.4 We state the main theorem of this section.

13 Abstract and Applied Analysis 13 Theorem 3.2. Let f : T κ 0 Rn R n be a -Carathéodory function. If v, M W 2,1 T, Rn W 2,1 2,1 T, 0, is a solution-tube of 3.1, then the system 3.1 has a solution x W T, Rn T v, M. In order to prove this result, we consider the following modified problem: x t x σ t f t, x σ t x σ t, x BC, -a.e. t T κ2 0, 3.5 where M s x v s v s x s x v s x if x v s >M s, otherwise. 3.6 We define the operator F : C T, R n L 1 Tκ 0, Rn by F x t : f t, x σ t x σ t. 3.7 Proposition 3.3. Under the assumptions of Theorem 3.2, the operator F defined above is continuous and bounded. Proof. First of all, we show that the set F C T, R n is bounded. Fix R> v 0 M 0.Let h R L 1 Tκ 0, 0, be given by Definition 2.9 iii. Thus,foreveryx C T, Rn, F x s h R s R : h s -a.e. s T κ To prove the continuity of F, weconsider{x k } k N asequenceofc T, R n converging to x C T, R n. We already know that for every k N, F x k s h s -a.e. s T κ One can easily check that x n t x t for all t T. It follows from Definition 2.9 ii that F x k s F x s -a.e. s T κ Theorem 2.8 implies that F x k F x in L 1 Tκ 0, Rn. Lemma 3.4. Under the assumptions of Theorem 3.2, every solution x of 3.5 is in T v, M. Proof. Since x W 2,1 T, Rn resp., v W 2,1 T, Rn, M W 2,1 T, R, x t resp., v t, M t, thereexists-almost everywhere on T κ2.denote A {t T κ2 0 : x σ t v σ t >M σ t }. 3.11

14 14 Abstract and Applied Analysis By Lemma 2.4 1, x t v t M t x σ t v σ t,x t v t x σ t v σ t M t -a.e. on A We claim that x t v t M t > 0 -a.e. on A Indeed, we deduce from the fact that v, M is a solution-tube of 3.5 and from 3.12 that -almost everywhere on {t A : M σ t > 0}, x t v t M t x σ t v σ t,f t, x σ t x σ t x σ t v t x σ t v σ t x σ t v σ t,f t, x σ t v t M σ t M t x σ t v σ t M σ t M t > M σ t M t M σ t M t 0, 3.14 and -almost everywhere on {t A : M σ t 0}, x t v t M t x σ t v σ t,f t, x σ t x σ t x σ t v t x σ t v σ t x σ t v σ t,f t, v σ t v t x σ t v σ t M t x σ t v σ t M t 3.15 > M t 0. Observe that if x a v a M a > 0, x a v a x a v a,x a v a x a v a

15 Abstract and Applied Analysis 15 Indeed, this follows from Proposition 2.3 when a σ a. Fora<σ a, x a v a x σ a v σ a x a v a μ a x a v a,x σ a v σ a x a v a μ a x a v a 3.17 x a v a,x a v a. x a v a Similarly, if x b v b M b > 0, x ρ b v ρ b x b v b,x ρ b v ρ b x b v b If BC denotes 3.2, x a v a M a x b v b M b We deduce from 3.16, 3.18, anddefinition 3.1 that x a v a M a x b v b M b 0, 3.20 or x ρ b v ρ b M ρ b x a v a M a x ρ b v ρ b,x b v b x b v b x a v a,x a v a x a v a v a v ρ b,x a v a x a v a v a v ρ b M ρ b M a M ρ b M a M ρ b M a If BC denotes 3.3,wededucefrom 3.16, 3.18, anddefinition 3.1 that x a v a M a 0, 3.22

16 16 Abstract and Applied Analysis or a 0 x a v a M a γ 0 x a v a M a x a v a,a 0 x a v a γ 0 x a v a x a v a a 0 M a γ 0 M a x 0 a 0 v a γ 0 v a a0 M a γ 0 M a , and similarly x b v b M b 0, 3.24 or a 1 x b v b M b γ 1 x ρ b v ρ b M ρ b Finally, it follows from 3.13, 3.20, 3.21, 3.22, 3.23, 3.24, 3.25, and Theorem 2.19 applied to r t x t v t M t that x t v t M t for every t T. Now, we can prove the main theorem of this section. Proof of Theorem 3.2. Asolutionof 3.5 is a fixed point of the operator T : C T, R n C T, R n defined by T : j 0 L 1 F, 3.26 where L and j 0 are defined in 2.34 and Proposition 2.17, respectively. By Propositions 2.17, 2.20, and 3.3, the operator T is compact. So, the Schauder fixed point theorem implies that T has a fixed point and, hence, Problem 3.5 has a solution x. ByLemma 3.4,thissolutionisin T v, M. Thus,x is a solution of 3.1. In the particular case where n 1, as corollary of Theorem 3.2, we obtain a generalization of results of Akın 1 and Stehlík 2 for the Dirichlet and the periodic boundary conditions, respectively. Corollary 3.5. Let f : T κ 0 R R be a -Carathéodory function. Assume that there exists α, β W 2,1 T, R such that i α t β t for -almost every t T, ii α t f t, α σ t and β t f t, β σ t for -almost every t T κ2,

17 Abstract and Applied Analysis 17 iii a if BC denotes 3.2, thenα a α b, α a α ρ b, β a β b, and β a β ρ b, b if BC denotes 3.3,thena 0 α a γ 0 α a x 0 a 0 β a γ 0 β a and a 1 α b γ 1 α ρ b x 1 a 1 β b γ 1 β ρ b, then 3.1 has a solution x W 2,1 T, R such that α t x t β t for every t T. Proof. Observe that α β /2, β α /2 is a solution-tube of 3.1. The conclusion follows from Theorem 3.2. Theorem 3.2 generalizes also a result established by Henderson et al. 6 for systems of second-order dynamic equations on time scales. Let us mention that they considered a continuous map f and they assumed a strict inequality in Corollary 3.6. Let f : T κ 0 Rn constant R>0such that R n be a -Carathéodory function. Assume that there exists a x, f t, x 0 -a.e. t κ 2 T 0, x such that x R Moreover, if BC denotes 3.3, assume that x 0 a 0 R and x 1 a 1 R, then the system 3.1 has asolutionx W 2,1 T, Rn such that x t R for every t T. Here is an example in which one cannot find a solution-tube of the form 0,R. Example 3.7. Consider the system x t k t x σ t σ t c 1 -a.e. t T κ2 0, x a 0, x b 0, 3.28 where k L 1 T 0, R n \{0} and c R n \{0} is such that c max{ a, b } 1. One can check that v, M is a solution-tube of 3.28 with v t tc, M t 1. By Theorem 3.2,thisproblem has at least one solution x such that x t tc 1. Observe that there is no R>0suchthat 3.27 is satisfied. 4. Nonlinearity Depending on x In this section, we study more general systems of second-order dynamic equations on time scales. Indeed, we allow the nonlinearity f to depend also on x. We consider the problem x t f t, x σ t,x t, -a.e. t T κ2 0, a 0 x a x a x 0, a 1 x b γ 1 x ρ b x 1, 4.1 where a 0,a 1,γ 1 0andmax{a 1,γ 1 } > 0. We also introduce a notion of solution-tube for this problem.

18 18 Abstract and Applied Analysis Definition 4.1. Let v, M W 2,1 T, Rn W 2,1 T, 0,. We say that v, M is a solution-tube of 4.1 if i for -almost every t {t T κ2 0 : t σ t }, x v t,f t, x, y 2, v t y v t 2 M t M t M t 4.2 for every x, y R 2n such that x v t M t and x v t,y v t M t M t, ii for every t {t T κ2 0 : t<σ t }, x v σ t,f t, x, y v t M σ t M t, 4.3 for every x, y R 2n such that x v σ t M σ t, iii x 0 a 0 v a v a a 0 M a M a, x 1 a 1 v b γ 1 v ρ b a 1 M b γ 1 M ρ b. If T is the real interval a, b, condition ii of the previous definition becomes useless, and we get the notion of solution-tube introduced by the first author in 10 for a system of second-order differential equations. Here is the main result of this section. Theorem 4.2. Let f : T κ 0 R2n R n be a -Carathéodory function. Assume that H1 there exists v, M W 2,1 T, Rn W 2,1 T, 0, a solution-tube of 4.1, H2 there exist constants c, d > 0 such that f t, x, y c d y for -almost every t T κ 0 and for every x, y R 2n such that x v t M t, then 4.1 has a solution x W 2,1 T, Rn T v, M. To prove this existence result, we consider the following modified problem: x t x σ t g t, x σ t,x t, -a.e. t T κ2 0, where g : T κ 0 R2n a 0 x a x a x 0, a 1 x b γ 1 x ρ b x 1, R n is defined by M σ t x v σ t f t, x σ t, ỹ t x σ t M σ t g t, x, y 1 v M t t x v σ t x v σ t x v σ t if x v σ t >M σ t, f t, x σ t, ỹ t x σ t, otherwise,

19 Abstract and Applied Analysis 19 where x σ t is defined as in 3.6, ŷ t ŷ t ỹ t ŷ t M t x v σ t, ŷ t v t x v σ t M t 1 M σ t if t σ t, K y v t x v σ t x v σ t x v σ t >M σ t, x v σ t if t σ t, x v σ t M σ t, y v t >K, if t<σ t, y otherwise, K y v t v t if y v t >K, ŷ t y v t y, otherwise, 4.6 with K>0 a constant which will be fixed later. Remark Remark that x σ t v σ t M σ t, ỹ t 2K v t M t If x v σ t >M σ t, x σ t v σ t M σ t, x σ t v σ t, x t v t M σ t M t, for t σ t If x v σ t >M σ t and t σ t, 2 x t v t, x 2 t v t x t v t 2 x t v t 2 M t. x t v t Since f is -Carathéodory, by 1, thereexistsh L 1 Tκ 0, R such that for every x, y R n, g t, x, y h t -a.e. t T κ

20 20 Abstract and Applied Analysis We associate to g the operator G : C 1 T, R n L 1 Tκ 0, Rn defined by G x t : g t, x σ t,x t Proposition 4.4. Let f : T κ 0 R2n satisfied, then G is continuous. R n be a -Carathéodory function. Assume that H1 is Proof. Let {x k } be a sequence of C 1 T, R n converging to x C 1 T, R n. It is clear that x k t x t, x k t x t On {t T : t<σ t }, wehave g t, x k σ t,x k t g t, x σ t,x t, 4.13 since f is -Carathéodory. Similarly, -almost everywhere on {t T κ 0 : t σ t, x σ t v σ t / M σ t }, g t, x k σ t,x k t g t, x σ t,x t, 4.14 since x k t x t and, for k sufficiently large, x k σ t v σ t / M σ t. Denote S : {t T κ 0 : t σ t and x σ t v σ t M σ t } and I t {k N : x k σ t v σ t M σ t }. As before, it is easy to check that -almost everywhere on {t S :cardi t }, g t, x k σ t,x k t g t, x σ t,x t as k I t goes to infinity On the other hand, Proposition 2.11 implies that x σ t v σ t,x t v t M σ t M t -a.e. t S. 4.16

21 Abstract and Applied Analysis 21 So, -almost everywhere on {t S :card N \ I t }, x k t x k t M t x k σ t v σ t, x k t v t x k σ t v σ t xk σ t v σ t x k σ t v σ t x t M t x t x t x σ t v σ t, x t v t M t 1 M σ t x σ t v σ t x σ t v σ t x σ t v σ t K x σ t v σ t x t v t if x t v t >K, if x t v t K, 4.17 x t, as k N \ I t goes to infinity. Thus, -almost everywhere on {t S :card N \ I t }, g t, x k σ t,x k t g t, x σ t,x t as k N \ I t goes to infinity By Remark 4.3 4,wehave g t, x k σ t,x k t h t -a.e. t T κ Theorem 2.8 implies that G x k G x in L 1 T κ 0, Rn Lemma 4.5. Assume H1, then x T v, M for every solution x of 4.4. Proof. Observe that x t resp., v t, M t exists -almost everywhere on T κ2,since x W 2,1 T, Rn resp., v W 2,1 T, Rn,M W 2,1 T, R. Denote A {t T κ2 0 : x σ t v σ t >M σ t }. 4.21

22 22 Abstract and Applied Analysis Observe that by H1 and Remark 4.3 2,for-almost every t A, x σ t v σ t,x t v t x σ t v σ t,g t, x σ t,x t x σ t v t x σ t v σ t,f t, x σ t, x t v t M t x σ t v σ t M σ t x σ t v σ t M σ t > x σ t v σ t,f t, x σ t, x t v t 4.22 M t x σ t v σ t M σ t M t x σ t v σ t M t x σ t v σ t M t 2 x t v t 2 if t<σ t, if t σ t. This inequality with Lemma imply that for -almost every t {t A : t<σ t }, x t v t M t > Also, 4.22, Lemma 2.4 2, andremark imply that for -almost every t {t A : t σ t }, x t v t M t > M t 2 x t v t 2 x t v t 2 x t v t x t v t,x t v t 2 x t v t 3 x t v t 2 x t v t 2 x t v t x t v t, x t v t 2 x t v t,x t v t 2 x t v t 3 0 if x t v t K, K 2 x t v t 2 1 x t v t,x t v t 2 x t v t 2 x t v t x t v t 3 if x t v t >K,

23 Abstract and Applied Analysis 23 Let us denote r t x t v t M t. Inequalities 4.23 and 4.24 imply that r t > 0for-almost every t {t T κ2 0 : r σ t > 0}. Arguing as in the proof of Lemma 3.4, we can show that r a 0 or a 0 r a r a 0, r b 0 or a 1 r b γ 1 r ρ b Theorem 2.19 implies that x T v, M. We can now prove the existence theorem of this section. Proof of Theorem 4.2. We first show that for every solution x of 4.4, thereexistsaconstant K>0suchthat x t v t K for every t T κ By H2, Proposition 2.13 and Lemma 4.5, foranyx solution of 4.4, wehavefor-almost every t T κ, x t x a x s s a,t T x 0 a 0 x a a,t T f s, x σ s, x s s c 0 c d x s s a,t T c 0 c d x s v s v s c 0 c 1 a,t T a,t T a,t T c d x s v s v s d x s s, M s s M s s 4.27 where c 0 : x 0 a 0 M a v a, c 1 : c 0 c d 2 v s M s s. a,b T 4.28 Fix K v 0 c 1 e d,a 0.ByTheorem 2.5, x t v t x t v t v t c 1 e d t, a K t T κ. 4.29

24 24 Abstract and Applied Analysis Consider the operator T j 1 L 1 G : C 1 T, R n C 1 T, R n, 4.30 where L and j 1 are defined, respectively, in 2.34 and Proposition ByPropositions2.16, 2.20,and4.4, T is continuous. Moreover, T is compact. Indeed, by Remark 4.3 4, thereexists h L 1 Tκ 0, 0, such that for every z T C1 T, R n, thereexistsx C 1 T, R n such that z T x and G x s h s -a.e. t T κ Since j 1 and L 1 are continuous and affine, they map bounded sets in bounded sets. Thus, there exists a constant k 0 > 0suchthat z 1 k Moreover, z W 2,1 T, Rn and L z s z s z σ s G x s -a.e. s T κ So, for every t<τin T κ, z t z τ z σ s G x s s k 0 h s s. t,τ T t,τ T 4.34 Thus, T C 1 T, R n is bounded and equicontinuous in C 1 T, R n.byananalogofthearzelà- Ascoli Theorem for our context, T C 1 T, R n is relatively compact in C 1 T, R n. By the Schauder fixed point theorem, T has a fixed point x which is a solution of 4.4. By Lemma 4.5, x T v, M. Also,x satisfies Hence,x is also a solution of 4.1. Here is an example in which one cannot find a solution-tube of the form 0,R. Moreover, Assumption A2 stated in the introduction and assumed in 9 is not satisfied. Example 4.6. Let T 0, 1 {2} 3, 4 and consider the system x t x σ t l t x t k t -a.e. t T0, x 0 x 0, x 4 x 1, 4.35 where x 0,x 1 R n are such that x 0 1, x 1 1, l L 1 T 0, 0,,andk L 1 T 0, R n such that l 1 1, l 2 1, k t 2, and l t r for -almost every t T0 for some r>0.

25 Abstract and Applied Analysis 25 Take v t 0and 5 t if t 0, 2 T, M t t if t 3, 4 T So, v W 2,1 T, Rn, M W 2,1 T, 0,,and 1 if t 0, 1, M t 0 if t 2, 1 if t 3, 4, 0 if t 0, 1, M t 1 if t {1, 2}, 0 if t 3, One has x 0 M 0 and x 1 M 4. Observe that -almost everywhere on 0, 1 3, 4 if x M t and x, y M t M t, one has y 1and x, f t, x, y y 2 x 2 l t y x, k t y 2 M t M t k t M t M t M t 4.38 If t {1, 2} and x M σ t 3, x, f t, x, y x 2 l t y x, k t 9l t 3 k t M σ t M t. So, v, M is a solution-tube of 4.35.Moreover, f t, x, y 2 5r 5 y -a.e. t T0, all x, y such that x M t Theorem 4.2 implies that 4.35 has at least one solution x such that x t M t. Observe that if x 0 / 0orx 1 / 0, this problem has no solution-tube of the form 0,R with R a positive constant since Definition 4.1 iii would not be satisfied. This explains why Henderson and

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