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

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 . In addition, we assume that f satisfies a linear growth condition. It is worthwhile to mention that the time scale 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 Δ-Carathéodory function. This condition is interesting in the case where the points of are not all right scattered. We obtain the existence of solutions to 1.1 and to 1.2 in the Sobolev space W Δ , n . 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-orderHamiltonian 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 readermay consult 12, 13 and the references therein to find the proofs and to get a complete introduction to this subject. Let be a compact time scale with a min < b max . The forward jump operator σ : → resp., the backward jump operator ρ : → is defined by σ t ⎧ ⎨ ⎩ inf{s ∈ : s > t} if t < b, b if t b, ⎛ ⎝resp., ρ t ⎧ ⎨ ⎩ sup{s ∈ : s < t} if t > a, a if t a ⎞ ⎠. 2.1 We say that t < b is right scattered resp., t > a is 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 is at most countable, see 14 . We denote it by R : {t ∈ : t < σ t } {ti : i ∈ I}, 2.2 for some I ⊂ . The graininess function μ : → 0,∞ is defined by μ t σ t − t. We denote κ \ (ρ b , b], 0 \ {b}. 2.3 So, κ if b is left dense, otherwise κ 0. Since κ is also a time scale, we denote κ2 κ , κ 2 0 κ2 \ {b}. 2.4 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 xΔ t . This notion coincides with x′ t resp., Δx t in the case where the time scale is an interval resp., the discrete set {0, 1, . . . ,N} . Definition 2.1. A map f : → n is Δ-differentiable at t ∈ κ if there exists fΔ t ∈ n called the Δ-derivative of f at t such that for all > 0, there exists a neighborhood U of t such that ∥∥ ( f σ t − f s − fΔ t σ t − s ∥∥ ≤ |σ t − s| ∀s ∈ U. 2.5 We say that f is Δ-differentiable if fΔ t exists for every t ∈ . If f isΔ-differentiable and if fΔ isΔ-differentiable at t ∈ κ2 , we call fΔΔ t fΔ Δ t the second Δ-derivative of f at t. Proposition 2.2. Let f : → n and t ∈ . i If f is Δ-differentiable at t, then f is continuous at t. ii If f is continuous at t ∈ R , then fΔ t f σ t − f t μ t . 2.6 iii The map f is Δ-differentiable at t ∈ κ \ R if and only if fΔ t lim s→ t f t − f s t − s . 2.7 Proposition 2.3. If f : → n and g : → m are Δ-differentiable at t ∈ , then i if n m, αf g Δ t αfΔ t gΔ t for every α ∈ , ii if m 1, fg Δ t g t fΔ t f σ t gΔ t f t gΔ t g σ t fΔ t , iii if m 1 and g t g σ t / 0, then ( f g )Δ t g t fΔ t − f t gΔ t g t g σ t , 2.8 iv if W ⊂ n is open and h : W → is differentiable at f t ∈ W and t /∈R , then h ◦ f Δ t 〈h′ f t , fΔ t 〉. We denote C , n the space of continuous maps on , and we denote C1 , n the space of continuous maps on with continuous Δ-derivative on . With the norm ‖x‖0 max{‖x t ‖ : t ∈ } resp., ‖x‖1 max{‖x‖0,max{‖x t ‖ : t ∈ κ}} , C , n resp., C1 , n is a Banach space. Abstract and Applied Analysis 5 We study the second Δ-derivative of the norm of a map. Lemma 2.4. Let x : → n be Δ-differentiable. 1 On {t ∈ κ2 : ‖x σ t ‖ > 0 and xΔΔ t exists}, ‖x t ‖ΔΔ ≥ 〈x σ t , x ΔΔ t 〉 ‖x σ t ‖ . 2.9and Applied Analysis 5 We study the second Δ-derivative of the norm of a map. Lemma 2.4. Let x : → n be Δ-differentiable. 1 On {t ∈ κ2 : ‖x σ t ‖ > 0 and xΔΔ t exists}, ‖x t ‖ΔΔ ≥ 〈x σ t , x ΔΔ t 〉 ‖x σ t ‖ . 2.9 2 On {t ∈ κ2 \ R : ‖x σ t ‖ > 0 and xΔΔ t exists}, ‖x t ‖ΔΔ 〈x t , x ΔΔ t 〉 ‖xΔ t ‖2 ‖x t ‖ − 〈x t , xΔ t 〉 ‖x t ‖3 . 2.10 Proof. Denote A {t ∈ κ2 : ‖x σ t ‖ > 0 and xΔΔ t exists}. By Proposition 2.3, on the set A \ R , we have ‖x t ‖Δ 〈x t , x Δ t 〉 ‖x t ‖ , ‖x t ‖ΔΔ 〈x t , x ΔΔ t 〉 ‖xΔ t ‖2 ‖x t ‖ − 〈x t , xΔ t 〉 ‖x t ‖3 ≥ 〈x σ t , xΔΔ t 〉 ‖x σ t ‖ . 2.11 If t ∈ A is such that t < σ t σ2 t , then by Propositions 2.2 and 2.3, we have ‖x t ‖ΔΔ ‖x σ t ‖ Δ − ‖x t ‖Δ μ t 〈x σ t , xΔ σ t 〉 μ t ‖x σ t ‖ − ‖x σ t ‖ − ‖x t ‖ μ t 2 〈x σ t , xΔ t μ t xΔΔ t 〉 μ t ‖x σ t ‖ − 〈x σ t , x t μ t xΔ t 〉 μ t ‖x σ t ‖ ‖x t ‖ μ t 2 〈x σ t , xΔΔ t 〉 ‖x σ t ‖ − 〈x σ t , x t 〉 μ t ‖x σ t ‖ ‖x t ‖ μ t 2 ≥ 〈x σ t , x ΔΔ t 〉 ‖x σ t ‖ . 2.12 6 Abstract and Applied Analysis If t ∈ A is such that t < σ t < σ2 t , then ‖x t ‖ΔΔ ‖x σ t ‖ Δ − ‖x t ‖Δ μ t ‖x(σ2 t )‖ − ‖x σ t ‖ μ σ t μ t − ‖x σ t ‖ − ‖x t ‖ μ t 2 ≥ 〈x σ t , x ( σ2 t ) − x σ t 〉 μ σ t μ t ‖x σ t ‖ − ‖x σ t ‖ − ‖x t ‖ μ t 2 〈x σ t , xΔ σ t 〉 μ t ‖x σ t ‖ − ‖x σ t ‖ − ‖x t ‖ μ t 2 , 2.13 and we conclude as in the previous case. Let ε > 0. The exponential function eε ·, t0 is defined by eε t, t0 exp (∫ t0,t ∩ ξε ( μ s ) Δs ) , 2.14