ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO BOUNDARY VALUE PROBLEMS ON TIME SCALES

This work formulates existence, uniqueness, and uniqueness-implies-existence theorems for solutions to two-point vector boundary value problems on time scales. The meth-ods used include maximum principles, a priori bounds on solutions, and the nonlinear alternative of Leray-Schauder.

Throughout this work the assumption is made that T has the topology that it inherits from the standard topology on the real numbers R. Also assume throughout that a < b are points in T with [a,b] = {t ∈ T : a ≤ t ≤ b}.
The jump operators σ and ρ allow the classification of points in a time scale in the following way: if σ(t) > t, then call the point t right-scattered; while if ρ(t) < t, then we call t left-scattered. If t < supT and σ(t) = t, then call the point t right-dense; while if t > inf T and ρ(t) = t, then we call t left-dense.
If T has a left-scattered maximum at m, then define T k = T − {m}. Otherwise T k = T. Definition 1.3. Fix t ∈ T and let y : T → R d . Define y ∆ (t) to be the vector (if it exists) with the property that given > 0 there is a neighbourhood U of t such that, for all s ∈ U and each i = 1,...,d, Call y ∆ (t) the (delta) derivative of y(t) at t. The following theorem is due to Hilger [12].
Theorem 1. 5. Assume that f : T → R d and let t ∈ T k .
(i) If f is differentiable at t, then f is continuous at t.
(ii) If f is continuous at t and t is right-scattered, then f is differentiable at t with (1.9) (iii) If f is differentiable and t is right-dense, then In order to prove the existence of solutions to the BVPs (1.1), (1.2) through (1.1), (1.5), the following theorem will be used, which is referred to as the nonlinear alternative of Leray-Schauder.
Theorem 1.8. Let Ω be an open, convex, and bounded subset of a Banach space X with 0 ∈ Ω and let T :Ω → X be a compact operator. If y = λT(y) for all y ∈ ∂Ω and all λ ∈ [0,1], then y = T(y) for some y ∈ Ω.
Proof. This is a special case of Lloyd [15,Theorem 4.4.11].
Recently the study of dynamic equations on time scales has attracted much interest (see [1,2,3,4,6,7,8,9,10,12,13,14]). This has been mainly due to its unification of the theory of differential and difference equations. The potential for applications is enormous-especially in those phenomena that manifest themselves partly in continuous time and partly in discrete time.
To the authors' knowledge, no papers have yet dealt with second-order systems of BVPs on time scales. The extension to systems is a natural one; for example, many occurrences in nature involve two or more populations coexisting in an environment, with the model being best described by a system of dynamic equations. (Beltrami [5, Section 5.6] discusses algae and copepod populations via second-order systems of BVPs.) This paper deals with two specific types of second-order equations. Sections 2, 3, 4, 5, and 6 treat the nonlinear equation (1.12) and Section 7 treats the linear equation where P and Q are d × d matrices functions and h is a d × 1 vector function. In particular, the paper is organized as follows.
In Section 2, the necessary a priori bounds on solutions to the BVPs (1.12), (1.2) through (1.12), (1.5) are formulated via some simple lemmas involving inequalities on f and on the boundary conditions. In Section 3, the a priori bounds from Section 2 are used in conjunction with the nonlinear alternative to prove the existence of solutions to the BVPs (1.12), (1.2) through (1.12), (1.5).
In Section 4, the inequalities on f from Section 3 are slightly strengthened and some extra qualitative information about solutions is obtained. Solutions are shown to be nonincreasing or nondecreasing in norm.
In Section 5, BVPs on infinite intervals are investigated and some existence theorems are presented. The proofs rely on the existence of solutions on finite intervals and so use the theorems of Section 3. A standard diagonalization argument is also employed.
The theory of time scales dates back to Hilger [12]. The monographs [6,14] also provide an excellent introduction. Of particular motivation for the research in this paper were the works [1,2,3,4,8,9,10].

A priori bounds on solutions
In order to apply Theorem 1.8, a priori bounds on solutions to the BVPs are needed. In this section conditions on f and on the boundary conditions are formulated, under which these bounds are guaranteed.
The following maximum principle will be very useful throughout the rest of the paper and can be found in [10].
Lemma 2.1. If a function r : T → R has a local maximum at a point c ∈ [a,σ 2 (b)], then r ∆∆ (ρ(c)) ≤ 0 provided that c is not simultaneously left-dense and right-scattered and that r ∆∆ (ρ(c)) exists.
Let y be a solution to (1.1). In what follows, the maximum principle of Lemma 2.1 will be applied to the "Lyapunov-type" function r(t) = y(t) 2 , and then used to show that r is bounded on [a,σ 2 (b)] (and therefore solutions y are bounded on [a,σ 2 (b)]). In order to guarantee that r ∆∆ (ρ(c)) exists, both y ∆∆ (t) and [y(σ(t))] ∆ must exist, since r(t) = y(t), y(t) is the inner product of two functions. As remarked in [6], the product of two functions is not necessarily differentiable even if each of the functions is twice differentiable. Therefore, for the rest of the paper, assume that σ(t) is such that for those solutions y ∈ S, [y(σ(t))] ∆ exists.
If y is a solution to (1.12) and y(t) does not achieve its maximum value at t = a or Proof. Assume that the conclusion of the lemma is false. Therefore r(t) := y(t) 2 − R 2 must have a nonnegative maximum in [a,σ 2 (b)]. By hypothesis, this maximum must occur in (a,σ 2 (b)). Choose c ∈ (a,σ 2 (b)) such that First, we show that the point c cannot be simultaneously left-dense and right-scattered. Assume the contrary by letting ρ(c) = c < σ(c). If r ∆ (c) ≥ 0, then r(σ(c)) ≥ r(c), and this Hence r(t) is strictly decreasing on (c − δ,c] and this contradicts the way c was chosen. Therefore the point c cannot be simultaneously left-dense and right-scattered. By Lemma 2.1 we must have So using the product rule (see [6]) we have since c is not simultaneously left-dense and right-scattered.) This concludes the proof.
The following lemma provides a priori bounds on solutions to the Dirichlet BVP (1.12), (1.2).

Lemma 2.3. If f and R satisfy the conditions of Lemma 2.2 with
Proof. This result follows immediately from Lemma 2.2.
The following lemma provides a priori bounds on solutions to the Sturm-Liouville BVP (1.12), (1.3).
Proofs. The proofs follow lines similar to those of Lemmas 2.3 and 2.4 and so are omitted.

Existence of solutions
In this section, some existence results are presented for the BVPs (1.12), (1.2) through (1.12), (1.5). The proofs rely on the a priori bounds on solutions of Section 2 and the nonlinear alternative.
The following theorem gives the existence of solutions to the Dirichlet BVP on time scales.
Proof. The BVP (1.12), (1.2) is equivalent (see [6,Corollary 4.76]) to the integral equation 100 Systems of BVPs on time scales Thus, we want to prove that there exists at least one y satisfying (3.1). Define an operator If we can prove that there exists a y such that T(y) = y, then there exists a solution to (3.1). To show that T has a fixed point, consider the equation Define an open, bounded subset of the Banach space S by Ω = {y ∈ S : y < R}, where here · is the sup norm. Note that (3.4) is equivalent to the BVP Now show that all solutions to (3.5) must satisfy y ∈ Ω, and consequently y ∈ ∂Ω for all λ ∈ [0,1]. Obviously y ∈ Ω for λ = 0. So consider (3.5) for λ ∈ (0,1]. Note that, by (2.1), Also λA , λB ≤ A , B < R. Therefore Lemma 2.3 is applicable to solutions of (3.5). Hence all solutions y to (3.5) must satisfy y(t) < R for t ∈ [a,σ 2 (b)]. Hence y ∈ ∂Ω.
Since f is continuous, T is continuous and it can be shown that T is a compact operator by the Arzela-Ascoli theorem. Therefore, Theorem 1.8 is applicable to T and T must have a fixed point. Hence the BVP has a solution. This concludes the proof.
The following theorem gives the existence of solutions to the Sturm-Liouville BVP on time scales.
where Thus, we want to prove that there exists at least one y satisfying (3.7). Define an operator If we can prove that there exists a y such that T(y) = y, then there exists a solution to (3.7). To show that T has a fixed point, consider the equation Define an open, bounded subset of the Banach space S by Note that (3.10) is equivalent to the BVP Now show that all solutions to (3.12) must satisfy y ∈ Ω, and consequently y ∈ ∂Ω for all λ ∈ [0,1]. Obviously y ∈ Ω for λ = 0. So consider (3.12) for λ ∈ (0,1]. Note that, by (2.1), (3.6) holds. Since λα,λβ,λγ,λδ > 0, we get that Lemma 2.4 is applicable to solutions of (3.12), hence for t ∈ [a,σ 2 (b)]. Hence all solutions y to (3.12) must satisfy for t ∈ [a,σ 2 (b)] and therefore y ∈ ∂Ω.
Since f is continuous, T is continuous and it can be shown that T is a compact operator by the Arzela-Ascoli theorem. Therefore, Theorem 1.8 is applicable to T and T must have a fixed point. Hence the BVP has a solution. This concludes the proof.
Since f is continuous, T is continuous and it can be shown that T is a compact operator by the Arzela-Ascoli theorem. Therefore Theorem 1.8 is applicable to T and T must have a fixed point. Hence the BVP has a solution. This concludes the proof.
Similarly, the following result holds.  Define an open, bounded subset of the Banach space S by Ω = {y ∈ S : y < ME + N + 1}, where here · is the sup norm, E is the bound on m and It is easy to see that λT(y) ≤ λ(ME + N) < ME + N + 1 for all λ ∈ [0,1] and that Theorem 1.8 is applicable. Therefore, the BVP (3.24), (1.2) has a solution y ∈ Ω. To show that this is a solution of the BVP (1.12), (1.2), see that, for y ≥ R, y σ ,m t, y σ = p, f (t, p) > 0, (3.29) for y ≥ R = p by (3.23) and p = Ry σ / y σ . Therefore, all solutions to (3.24), (1.2) satisfy y < R and are solutions to the BVP (1.12), (1.2). This concludes the proof.   Proofs. The proofs follow the modification technique of Theorems 3.6 and 3.7 and so are omitted for brevity.

On nonincreasing solutions
Some results about the qualitative nature of solutions for the BVPs are now proved. In particular, by strengthening inequality (2.1), the solutions furnished by Theorem 3.1 may be shown to be nondecreasing or nonincreasing in norm. Proof. Note that (4.4) implies that r(t) := y(t) 2 cannot have a nonnegative maximum in (a,σ 2 (b)) for any solution y, and therefore r must have a maximum at either t = a or t = σ 2 (b) with maxr(t) = max{r(a),r(σ 2 (b))} = A 2 . Proof. The proof is similar to that of Corollary 4.1.

BVPs on infinite intervals
This section formulates the existence theorems for solutions to the following BVPs on infinite intervals: In particular, Theorems 3.1 and 3.3 will be useful.
. Throughout this section assume that there exists t n ∈ T and n ∈ N such that a < t 1 < t 2 < ··· < t n < ··· with t n ↑ ∞ as n −→ ∞. where R > 0. Then for each A < R, the BVP (5.1), (5.2) has at least one solution y ∈ C([a,∞);R d ) with y(t) < R on [a,∞).
Proof. Fix n ∈ N and consider the BVP It is clear from Theorem 3.1 that (5.6) has a solution y n ∈ C([a,σ 2 (t n )];R d ) with y n (t) < R for t ∈ [a,t n ]. (Note also that y ∆∆ n ∈ C rd [a,σ 2 (t n )];R d .) This argument can be used for each n ∈ N. The theorem then follows from Ascoli's selection theorem (see [11]) applied to a sequence of intervals [a,t n ] as n → ∞. Theorem 5.2. Suppose that f is continuous on [a,∞) × R d and satisfies (5.5), where R > 0. If α,β > 0, then the Sturm-Liouville BVP (5.1), (5.3) has at least one solution y ∈ C([a,∞); R d ) satisfying Proof. Fix n ∈ N and consider the BVP y ∆∆ = f t, y σ , t ∈ a,t n , αy(a) − βy ∆ (a) = C, y σ 2 t n = 0. (5.8) It is clear from Theorem 3.3 that (5.8) has a solution y n ∈ C([a,σ 2 (t n )];R d ) with y n < M + 1 for t ∈ [a,t n ]. (Note also that y ∆∆ n ∈ C rd ([a,σ 2 (t n )];R d ).) This argument can be used for each n ∈ N. The theorem then follows from Ascoli's selection theorem applied to a sequence of intervals [a,t n ] as n → ∞.

On uniqueness of solutions
This section provides some results which guarantee the uniqueness of solutions to the Dirichlet BVP (1.12), (1.2).
Consider r(t) := y(t) − z(t) 2 , t ∈ [a,σ 2 (b)]. Now r must have a positive maximum at some point c ∈ [a,σ 2 (b)]. From the boundary conditions, c ∈ (a,σ 2 (b)). Choosing c in the same fashion as in the proof of Lemma 2.2, it can be shown via the same reasoning that c cannot be simultaneously left-dense and right-scattered. Therefore, by Lemma 2.1 we must have (2.4). So using the product rule we have which contradicts (2.4). (Notice at c that y σ (ρ(c)) − z σ (ρ(c)) = y(c) − z(c) , since c is not simultaneously left-dense and right-scattered.) Therefore r(t) = y(t) − z(t) 2 = 0 for t ∈ [a,σ 2 (b)], and solutions of the BVP (1.12), (1.2) must be unique. This concludes the proof.

Uniqueness implies existence
In this section, a uniqueness-implies-existence result is formulated for the BVP (1.13), (1.2). Since the nonlinear alternative is not required, the continuity requirements of the matrices P(t) and Q(t) may be relaxed to P,Q ∈ C rd . The following is a vector analogue of a result of Bohner and Peterson [6]. Proof. The proof is omitted as it follows lines similar to that of Bohner and Peterson [6] with only minor modifying changes.
Remark 7.7. The question on how to ensure that [y(σ(t))] ∆ exists is naturally raised. This "smoothness" requirement will be satisfied if, for example, σ is differentiable or when the points in T are isolated (left-scattered and right-scattered). However, if these cases are excluded, then the method of upper and lower solutions for systems of equations can be developed with the a priori bounds on solutions being obtained in a component-wise fashion. These arguments are essentially minor extensions of the ideas in [4].