Electronic Journal of Qualitative Theory of Differential Equations 2006, No. 11, 1-10; http://www.math.u-szeged.hu/ejqtde/

Boundary Value Problems for Doubly Perturbed First Order Ordinary Differential Systems

Mouffak Benchohra^{1}, Smail Djebali^{2} and Toufik Moussaoui^{2}

1 Laboratoire de Mathématiques, Université de Sidi Bel Abbès, BP 89, 22000, Sidi Bel Abbès, Algérie. e-mail: benchohra@univ-sba.dz 2 Département de Mathématiques, E.N.S. B.P. 92 Kouba, Alger, Algérie. e-mail: djebali@ens-kouba.dz,moussaoui@ens-kouba.dz

Abstract

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.

2000 Mathematics Subject Classifications: 34A34, 34B15. Key words: Perturbed BVPs, Ordinary differential systems, Nonlinear alternative.

1 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)
Mx(0) + Nx(1) = η.  (2)

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]:

1Corresponding author

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} (\frac{x_{0}}{λ} )+λB_{2}x_{0}.

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.

2 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

∥x∥_{0} = sup {∥x(t)∥ : 0 ≤ t ≤ 1} .

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

∥x∥_{L^{1}} = ∫_{0}^{1} ∥x(t)∥dt for all x ∈ L^{1}([0, 1], IR^{n}).

Recall the following. Definition 2.1 A function F: [0, 1] × IR^{n} → IR^{n} is said to be Carathéodory if

(i) t ⟼ F (t, y) is measurable for each y ∈ IR^{n}, and

(ii) y ⟼ F (t, y) is continuous for almost each t ∈ [0, 1].

Definition 2.2 Given a Banach space X, we say that a mapping T : X → X is totally bounded if it maps each bounded subset of X into a relatively compact subset. If, further it is continuous, it is called completely continuous.

3 EXISTENCE OF SOLUTIONS

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

x′(t) = A(t)x(t) + h(t), a.e. t ∈ (0, 1),  (3)
Mx(0) + Nx(1) = 0.  (4)

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

x(t) = ∫_{0}^{1} k(t, s)h(s)ds

where k(t, s) is the Green function defined by

k(t, s) = \{ \begin{matrix}Φ(t)J(s), & 0 ≤ t ≤ s, \\ Φ(t)Φ(s)^{−1} + Φ(t)J(s), & s ≤ t ≤ 1 \end{matrix}

and

J(t) = −(M + NΦ(1))^{−1}NΦ(1)Φ(t)^{−1}.

As for the inhomogeneous boundary conditions, the following Lemma is easily verified:

Lemma 3.2 Consider the following inhomogeneous linear boundary value problem

x′(t) = A(t)x(t) + h(t), a.e. t ∈ (0, 1),  (5)
Mx(0) + Nx(1) = η.  (6)

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

x(t) = x_{h}(t) + Φ(t) (M + NΦ(1))^{−1} η.

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

T x(t) = ∫_{0}^{1} k(t, s)[F (s, x(s)) + G(s, x(s))] ds.

It is clear that fixed points of T are solutions for BVP (1)-(2). Let us introduce the following hypotheses which are assumed hereafter:

• (H1) The function F : [0, 1] × IR^{n} → IR^{n} is Carathéodory and satisfies:

∃ l ∈ L^{1}([0, 1], IR_{+}), ∥F (t, y_{1}) − F (t, y_{2})∥ ≤ l(t)∥y_{1} − y_{2}∥

for almost each t ∈ [0, 1] and all y_{1}, y_{2} ∈ IR^{n}.

• (H2) The function G is continuous and there exist a function q ∈ L^{1}([0, 1], IR) with q(t) > 0 for almost each t ∈ [0, 1] and a continuous nondecreasing function ψ : IR^{+} ⟶ (0, ∞) such that

∥G(t, y)∥ ≤ q(t)ψ(∥y∥) a.e t ∈ [0, 1] and for all y ∈ IR^{n}.

• (H3) Set k^{∗}: = sup_{(t,s)∈[0,1]×[0,1]} ∥k(t, s)∥ and assume that

k^{∗}∥l∥_{L^{1}} < 1.

• (H4) Set F^{∗}: = ∫_{0}^{1} ∥F (s, 0)∥ ds and assume there exists r > 0 such that

r > \frac{k^{∗} (F^{∗} + ∥q∥_{L^{1}}Ψ(r))}{1 − k^{∗}∥l∥_{L^{1}}}·  (7)

Our main result is:

Theorem 3.1 Under hypotheses (H1)-(H4), BVP (1)-(2) has at least one solution x ∈ AC([0, 1], IR^{n}).

Proof. Define the two operators B_{1} and on B_{2} on X by

B_{1}x(t): = ∫_{0}^{1} k(t, s)F (s, x(s))ds, B_{2}x(t): = ∫_{0}^{1} k(t, s)G(s, x(s)) ds.

We are going to show that the operators B_{1} and B_{2} satisfy all conditions of Theorem 1.1.

Claim 1. B_{1} is a contraction.

Let x, y ∈ X and t ∈ [0, 1]; then

∥B_{1}x(t) − B_{1}y(t)∥ = ∥ ∫_{0}^{1} k(t, s) [F (s, x(s)) − F (s, y(s))] ds∥
≤ ∫_{0}^{1} ∥k(t, s)∥∥F (s, x(s)) − F (s, y(s))∥
≤ k^{∗}∥l∥_{L^{1}}∥x − y∥_{0} < ∥x − y∥_{0}.

Thus

∥B_{1}x − B_{1}y∥_{0} ≤ ∥x − y∥_{0}.

Claim 2. B_{2} is continuous. Let x_{n}, x ∈ X such that x_{n} ⟶ x in X, that is

∀ ε > 0, ∃ n_{0} ∈ IN^{∗}, (n ≥ n_{0} ⇒ ∥x_{n} − x∥_{0} < ε).

For each t ∈ [0, 1], we have

\begin{aligned}∥B_{2}x_{n}(t) − B_{2}x(t)∥ & ≤ & ∫_{0}^{1} ∥k(t, s)∥ · ∥G(s, x_{n}(s)) − G(s, x(s))∥ ds \\ & ≤ & k^{∗} ∫_{0}^{1} ∥G(s, x_{n}(s)) − G(s, x(s))∥ ds. \end{aligned}

Since the convergence of a sequence implies its boundedness, there is a number L > 0 such that

∥x_{n}(t)∥ ≤ L, ∥x(t)∥ ≤ L, ∀ t ∈ [0, 1].

Now, the function G is uniformly continuous on the compact set

{(t, x) ∈ IR^{+} × IR^{n}, t ∈ [0, 1], ∥x∥ ≤ L} .

It follows that

∥G(s, x_{n}(s)) − G(s, x(s))∥ ≤ \frac{ε}{k^{∗}} ·

Therefore, we infer that

∥B_{2}x_{n} − B_{2}x∥_{0} ≤ ε, ∀ n ≥ n_{0}.

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)

∥B_{2}x(t)∥ = ∥ ∫_{0}^{1} k(t, s)G(s, x(s))ds∥
≤ k^{∗} ∫_{0}^{1} ∥G(s, x(s))∥ds
≤ k^{∗} ∫_{0}^{1} q(s)ψ(∥x(s)∥)ds
≤ k^{∗}ψ(∥x∥_{0})∥q∥_{L^{1}}
≤ k^{∗}ψ(M)∥q∥_{L^{1}}.

Then B_{2}(C) is uniformly bounded. In addition, the following estimates hold true:

∥B_{2}x(t_{2}) − B_{2}x(t_{1})∥ = ∥ ∫_{0}^{1}[k(t_{2}, s) − k(t_{1}, s)]G(s, x(s))ds∥
≤ ∫_{0}^{1} ∥k(t_{2}, s) − k(t_{1}, s)∥q(s)ψ(M)ds
≤ ψ(M) ∫_{0}^{1} q(s)∥k(t_{2}, s) − k(t_{1}, s)∥ ds;

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.

Consider the sphere B(0, r), r being defined by (H4). For x ∈ B(0, r), we have

∥B_{1}0 + B_{2}x∥_{0} = sup_{t∈[0,1]} ∥ ∫_{0}^{1} k(t, s)F (s, 0) ds + ∫_{0}^{1} k(t, s)G(s, x(s)) ds∥
≤ k^{∗}F^{∗} + k^{∗}∥q∥_{L^{1}}Ψ(∥x∥_{0})
≤ k^{∗}F^{∗} + k^{∗}∥q∥_{L^{1}}Ψ(r)
< r(1 − k^{∗}∥l∥_{L^{1}}).

Now, argue by contradiction and assume that there exist λ ∈ (0, 1) and x ∈ ∂B(0, r) with x = λB_{1} (\frac{x}{λ}) + λB_{2}x. Then x verifies the estimates

∥x(t)∥ ≤ k^{∗}∥l∥_{L^{1}}∥x∥_{0} + k^{∗}F^{∗} + k^{∗}∥q∥_{L^{1}}Ψ(∥x∥_{0}).

Hence

r = ∥x∥_{0} ≤ \frac{k^{∗} (F^{∗} + ∥q∥_{L^{1}}Ψ(r))}{1 − k^{∗}∥l∥_{L^{1}}}

contradicting Assumption (7). We then conclude that Assertion (a) in Theorem 1.1 is satisfied, proving the claim of Theorem 3.1.

3.1 Example

Consider the second order boundary value Sturm-Liouville problem

−x′′ + qx′ + rx = f(t, x(t), x′(t)) + g(t, x(t), x′(t)), 0 < t < 1  (8)
a_{0}x(0) − a_{1}x′(0) = c_{0}  (9)
b_{0}x(1) + b_{1}x′(1) = c_{1}  (10)

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)^{t}

F = (0, −f)^{t} G = (0, −g)^{t}

as well as

A = ( \begin{matrix}0 & 1 \\ r & q \end{matrix} ) , M = ( \begin{matrix}a_{0} & −a_{1} \\ 0 & 0 \end{matrix} ) , N = ( \begin{matrix}0 & 0 \\ b_{0} & b_{1} \end{matrix} ) ,

and finally c = (c_{0}, c_{1})^{t}.

Problem (8) − (10) is then rewritten under the matrix form

\{ \begin{matrix}X′ = AX + F + G \\ MX(0) + NX(1) = c. \end{matrix}

Under Assumption (H4) both with det(M + NΦ(1)) ≠ 0, Problem (8) − (10) has a solution x.

Remark 3.1 In case q, r are constant, notice that condition det(M + NΦ(1)) ≠ 0 is nothing but a_{0}(a_{1}e^{r_{2}} + b_{1}r_{2}e^{r_{2}}) ≠ b_{0}(a_{1}e^{r_{2}} + b_{1}r_{r}e^{r_{2}}) where r_{1} and r_{2} are the roots of the characteristic equation −s^{2} + qs + r = 0.

4 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,

C = {x ∈ X | x(t) ≥ 0, ∀ t ∈ [0, 1]}.

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

[a, b] = {x ∈ X | a ≤ x ≤ b}.

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:

Theorem 4.1 [5] 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

x_{n+1} = B_{1}(x_{n}) + B_{2}(x_{n}), x_{0} = a and y_{n+1} = B_{1}(y_{n}) + B_{2}(y_{n}), y_{0} = b.

We need the following definitions in the sequel.

Definition 4.3 A function v ∈ AC([0, 1], IR^{n}) is called a strict lower solution of BVP (1)-(2) if v′(t) ≤ A(t)v(t) + F (t, v(t)) + G(t, v(t)) a.e. t ∈ [0, 1], Mv(0) + Nv(1) ≤ η. Similarly a strict upper solution w of BVP (1)-(2) is defined by reversing the order of the above inequalities.

Definition 4.4 A solution x_{M} of BVP (1)-(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.

(H5) The functions F (t, y) and G(t, y) are strictly monotone nondecreasing in y for almost each t ∈ [0, 1].

(H6) The BVP (1)-(2) has a lower solution v and an upper solution w with v ≤ w.

(H7) The kernel k preserves the order, that is k(t, s)v(s) ≥ 0 whenever v ≥ 0.

Remark 4.1 If we assume that there exist some constant vectors \underline{y} ≤ \overline{y} such that for each t ∈ [0, 1]

A(t)\underline{y} + F (t, \underline{y}) + G(t, \underline{y}) ≥ 0, (M + N)\underline{y} ≤ η,
A(t)\overline{y} + F (t, \overline{y}) + G(t, \overline{y}) ≤ 0, (M + N)\overline{y} ≥ η,

then \underline{y}, \overline{y} are respectively lower and upper solutions for Problem (1)-(2).

Theorem 4.3 Assume that Assumptions (H1)-(H6) hold true. Then BVP (1)-(2) has minimal and maximal solutions on [0, 1].

Proof. It can be shown, as in the proof of Theorem 3.1 that B_{1} and B_{2} are respectively a contraction and compact on [a, b]. We shall show that B_{1} and B_{2} are isotone increasing on [a, b]. Let x, y ∈ [a, b] be such that x ≤ y, x ≠ y. Then by Assumptions (H5), (H7), we have for each t ∈ [0, 1]

B_{1}(x)(t) = ∫_{0}^{1} k(t, s)F (s, x(s)) ds
≤ ∫_{0}^{1} k(t, s)F (s, y(s)) ds
= B_{1}(y)(t).

Similarly, B_{2}(x) ≤ B_{2}(y). Therefore B_{1} and B_{2} are isotone increasing on [a, b]. Finally, let x ∈ [a, b] be any element. By Assumptions (H6), we deduce that

a ≤ B_{1}(a) + B_{2}(a) ≤ B_{1}(x) + B_{2}(x) ≤ B_{1}(b) + B_{2}(b) ≤ b,

which shows that B_{1}(x) + B_{2}(x) ∈ [a, b] for all x ∈ [a, b]. Thus, the functions B_{1} and B_{2} satisfy all conditions of Theorem 4.2. It follows that BVP (1)-(2) has maximal and minimal solutions on [0, 1]. This completes the proof of Theorem 4.3.

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(Received December 11, 2005)