Boundary Value Problems for Doubly Perturbed First Order Ordinary Differential Systems

The aim of this paper is to present new results on existence theory for perturbed BVPs for first order ordinary differential systems. A nonlinear alternative for the sum of a contraction and a compact mapping is used.


INTRODUCTION
This paper is devoted to the question of existence of solutions for a doubly perturbed boundary value problem (BVP) associated with first order ordinary differential systems of the form: x (t) = A(t)x(t) + F (t, x(t)) + G(t, x(t)), a.e.t ∈ [0, 1]; (1) Here the functions F, G : [0, 1] × IR n −→ IR n are Carathéodory, A(.) is a continuous (n × n) matrix function, M and N are constant (n × n) matrices, and η ∈ IR n .Problem (1)-( 2) encompasses second order differential equation with periodic condition or Sturm-Liouville nonlinear problem (see the example in Section 3).We shall denote by x the norm of any element x of the euclidian space IR n and by A the norm of any matrix A. The notation : = means throughout to be equal to.In this paper, we shall prove the existence of solutions for Problem (1)- (2) under suitable conditions on the nonlinearities F and G. Our approach will be based, for the existence of solutions, on a fixed point theorem for the sum of a contraction map and a completely continuous map due to Ntouyas and Tsamatos [7] which we recall hereafter; it can be seen as a generalization of Burton and Kirk's Alternative [3]: Theorem 1.1 [7] Let (X, • ) be a Banach space, B 1 , B 2 be operators from X into X such that B 1 is a γ−contraction, and B 2 is completely continuous.Assume also that (H) There exists a sphere B(0, r) in X with center 0 and radius r such that for every y ∈ B(0, r), r(1 − γ) ≥ B 1 0 + B 2 y .Then either (a) the operator equation x = (B 1 + B 2 )x has a solution with x ≤ r, or (b) there exists a point x 0 ∈ ∂B(0, r) and λ ∈ (0, 1) such that x 0 = λB 1 Mappings which are equal to the sum of a contraction and a completely continuous function play an important role in fixed point theory (see [6]).Through Hamerstein operators, one can construct compact mapping and then apply Theorem 1.1 to BVPs associated with second order ODEs (see [2,4,6,8]).In this paper, we extend those results to the case of systems doubly perturbed with contraction and Carathéodory functions satisfying specific growth.

Preliminaries
In this section, we introduce notations, and preliminaries used throughout this paper.Recall that C([0, 1], IR n ) is the Banach space of all continuous functions from [0, 1] into IR n with the norm Let AC((0, 1), IR n ) be the space of differentiable functions x : (0, 1) → IR n , which are absolutely continuous.We denote by L 1 ([0, 1], IR n ) the Banach space of measurable functions x: [0, 1] −→ IR n which are Lebesgue integrable normed by In this section, we are concerned with the existence of solutions to Problem (1)- (2).We first state an auxiliary result from linear differential systems theory [1].Lemma 3.1 Consider the following linear mixed boundary value problem Let Φ(t) be a fundamental matrix solution of x (t) = A(t)x(t), such that Φ(0) = I, the (n × n) identity matrix.We can easily show that if det(M + NΦ(1)) = 0, then the linear inhomogeneous problem (3)-( 4) has a unique solution given by where k(t, s) is the Green function defined by As for the inhomogeneous boundary conditions, the following Lemma is easily verified: Lemma 3.2 Consider the following inhomogeneous linear boundary value problem Let x h be the solution of the homogeneous boundary value problem ( 3)-( 4).Keeping the same notations as in Lemma 3.1, the solution of Problem ( 5)-( 6) reads Next, we transform BVP (1)-( 2) into a fixed point problem.Consider the Banach space X = C([0, 1], IR n ) endowed with the sup-norm.Let the operator T : X −→ X be defined by It is clear that fixed points of T are solutions for BVP (1)-( 2).Let us introduce the following hypotheses which are assumed hereafter: EJQTDE, 2006 No. 11, p. 3 for almost each t ∈ [0, 1] and all y 1 , y 2 ∈ IR n .
k(t, s) and assume that • (H4) Set F * : = 1 0 F (s, 0) ds and assume there exists r > 0 such that Our main result is: Proof.Define the two operators B 1 and on B 2 on X by We are going to show that the operators B 1 and B 2 satisfy all conditions of Theorem 1.1.
For each t ∈ [0, 1], we have Since the convergence of a sequence implies its boundedness, there is a number L > 0 such that Now, the function G is uniformly continuous on the compact set The continuity of B 2 is proved.Claim 3. B 2 is totally bounded.
Consider the closed ball C = {x ∈ X; x 0 ≤ M}.We prove that the image B 2 (C) is relatively compact in X.We have, by (H2) Then B 2 (C) is uniformly bounded.In addition, the following estimates hold true: EJQTDE, 2006 No. 11, p. 5 the right-hand side term tends to 0 as t 2 −→ t 2 for any x ∈ C.Then, B 2 (C) is equicontinuous.By the Arzela-Ascoli Theorem, the mapping B 2 is completely continuous on X.
Claim 4. Now, we prove that, under Assumption (7), the second alternative of Theorem 1.1 is not valid.

Example
Consider the second order boundary value Sturm-Liouville problem where a 0 , a 1 and b 0 , b 1 are nonnegative real numbers satisfying a 0 + a 1 > 0, b 0 + b 1 > 0 and (c 0 , c 1 ) ∈ IR 2 .The functions f, g: [0, 1] × IR 2 → IR are assumed Carathéodory; the function f satisfies Lipschitz condition with respect to the last two arguments while g verifies a growth condition as in Assumption (H2).The functions q, r: [0, 1] → IR are continuous.v t being the transpose of the vector v, we adopt the notations x = y, X = (x, y) as well as and finally c = (c 0 , c 1 ) t .

Existence of Extremal Solutions
In this section we shall prove the existence of maximal and minimal solutions of BVP (1)-( 2) under suitable monotonicity conditions on the functions involved in it.We define the usual co-ordinate-wise order relation ≤ in IR n as follows.Let x = (x 1 , x 2 , ..., x n ) and y = (y 1 , y 2 , ..., y n ) be any two elements.Then by x ≤ y, we mean x i ≤ y i for all i = 1, ..., n.We equip the space X = C([0, 1], IR n ) with the order relation ≤ induced by the natural positive cone C in X, that is, It is known that the cone C is normal in X. Cones and their properties are detailed in [5].Let a, b ∈ X be such that a ≤ b.Then, by an order interval [a, b] we mean a set of points in X given by Definition 4.1 Let X be an ordered Banach space.A mapping T : X → X is called isotone increasing if T (x) ≤ T (y) for any x, y ∈ X with x < y.Similarly, T is called isotone decreasing if T (x) ≥ T (y) whenever x < y.
Definition 4.2 [5] We say that x ∈ X is the least fixed point of G in X if x = Gx and x ≤ y whenever y ∈ X and y = Gy.The greatest fixed point of G in X is defined similarly by reversing the inequality.If both least and greatest fixed point of G in X exist, we call them extremal fixed point of G in X.
The following fixed point theorem is due to Heikkila and Lakshmikantham: We need the following definitions in the sequel.
Similarly a strict upper solution w of BVP (1)-( 2) is defined by reversing the order of the above inequalities.2) is said to be maximal if for any other solution x of BVP ( 1)-( 2) on [0, 1], we have that x(t) ≤ x M (t) for each t ∈ [0, 1].Similarly a minimal solution of BVP (1)-( 2) is defined by reversing the order of the inequalities.Definition 4.5 A function F (t, x) is called strictly monotone increasing in x almost everywhere for t ∈ J, if F (t, x) ≤ F (t, y) a.e.t ∈ J for all x, y ∈ IR n with x < y.Similarly F (t, x) is called strictly monotone decreasing in x almost everywhere for t ∈ J, if F (t, x) ≥ F (t, y) a.e.t ∈ J for all x, y ∈ IR n with x < y.
We consider the following assumptions in the sequel.(H7) The kernel k preserves the order, that is k(t, s)v(s) ≥ 0 whenever v ≥ 0.

EJQTDE, 2006 7 Theorem 4 . 1 [ 5 ]
No. 11, p.Let [a, b] be an order interval in an order Banach space X and let Q : [a, b] → [a, b] be a nondecreasing mapping.If each sequence (Qx n ) ⊂ Q([a, b]) converges, whenever (x n ) is a monotone sequence in [a, b], then the sequence of Q−iteration of a converges to the least fixed point x * of Q and the sequence of Q−iteration of b converges to the greatest fixed point x * of Q. Moreover x * = min{y ∈ [a, b], y ≥ Qy} and x * = max{y ∈ [a, b], y ≤ Qy} As a consequence, Dhage and Henderson have proved the following Theorem 4.2 [4].Let K be a cone in the Banach space X and let [a, b] be an order interval in a Banach space and let B 1 , B 2 : [a, b] → X be two functions satisfying (a) B 1 is a contraction, (b) B 2 is completely continuous, (c) B 1 and B 2 are strictly monotone increasing, and(d) B 1 (x) + B 2 (x) ∈ [a, b], ∀ x ∈ [a, b].Further if the cone K in X is normal, then the equation x = B 1 (x) + B 2 (x)has a least fixed point x * and a greatest fixed point x * ∈ [a, b].Moreover x * = lim n→∞ x n and x * = lim n→∞ y n , where {x n } and {y n } are the sequences in [a, b] defined by