����������� ������ 53 (2001), 51–58

UDK 517.927

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MULTI POINT BOUNDARY VALUE PROBLEMS FOR SECOND ORDER DIFFERENTIAL INCLUSIONS

M. Benchora and S. K. Ntouyas

Abstract. In this paper we investigate the existence of solutions on a compact interval to a multi-point boundary value problem for a class of second order differential inclusions. We shall rely on a fixed point theorem for condensing maps due to Martelli.

1. Introduction

Let a_{i}, b_{j} ∈ R, with all of the a_{i}^{,}s, (respectively, b_{j}^{,}s), having the same sign, ξ_{i}, ζ_{j} ∈ (0, 1), i = 1, 2,…, m−2, j = 1, 2,…, n−2, 0 < ξ_{1} < ξ_{2} < ⋯ < ξ_{m−2} < 1, 0 < ζ_{1} < ζ_{2} < ⋯ < ζ_{n−2} < 1. The main purpose of this paper is to get results on the solvability of the following boundary value problems (BVPs for short) for second order differential inclusions of the forms

\{ \begin{matrix}y′′(t) ∈ F (t, y(t)), t ∈ (0, 1) \\ y(0) = ∑_{i=1}^{m−2} a_{i}y′(ξ_{i}), y(1) = ∑_{j=1}^{n−2} b_{j}y(ζ_{j}) \end{matrix}  (A)

\{ \begin{matrix}y′′(t) ∈ F (t, y(t)), t ∈ (0, 1) \\ y(0) = ∑_{i=1}^{m−2} a_{i}y′(ξ_{i}), y′(1) = ∑_{j=1}^{n−2} b_{j}y′(ζ_{j}) \end{matrix}  (B)

\{ \begin{matrix}y′′(t) ∈ F (t, y(t)), t ∈ (0, 1) \\ y(0) = ∑_{i=1}^{m−2} a_{i}y(ξ_{i}), y(1) = ∑_{j=1}^{n−2} b_{j}y(ζ_{j}) \end{matrix}  (C)

and

\{ \begin{matrix}y′′(t) ∈ F (t, y(t)), t ∈ (0, 1) \\ y(0) = ∑_{i=1}^{m−2} a_{i}y(ξ_{i}), y′(1) = ∑_{j=1}^{n−2} b_{i}y′(ζ_{j}) \end{matrix}  (D)

where F: J × R ⟶ 2^{R} is a multivalued map with compact convex values.

AMS Subject Classification: 34 A 60, 34 B 10, 34 B 15

Keywords and phrases: Multi point boundary value problems, differential inclusion, existence, fixed point

The study of multi-point boundary value problems for second order ordinary differential equations was initiated by Il’in and Moiseev in [12, 13] motivated by the work of Bitsadze and Samarskii on nonlocal elliptic boundary value problems, [2, 3, 4].

Existence of solutions on compact intervals for multi-point boundary value problems for second order differential equations was given by Gupta in [6], Gupta et al in [7–10]. However, to our knowledge, this type of problems has not been studied for the multivalued case.

It is well known (c.f. [12]) that if a function y ∈ C^{1} satisfies one of the boundary conditions stated above and a_{i}, b_{j}, i = 1, 2,…, m − 2, j = 1, 2,…, n − 2 are as above, then there exist η ∈ [ξ_{1}, ξ_{m−2}], τ ∈ [ζ_{1}, ζ_{n−2}] such that

y(0) = αy′(η), y(1) = βy(τ)
y(0) = αy′(η), y′(1) = βy′(τ)
y(0) = αy(η), y(1) = βy(τ)
y(0) = αy(η), y′(1) = βy′(τ)

respectively with α = ∑_{i=1}^{m−2} a_{i}, β = ∑_{j=1}^{n−2} b_{j}. Hence the multi-point BVPs (A)– (D) can be reduced to a corresponding four-point BVP. The method of proof for the existence of a solution for a four-point BVP and for a multi-point BVP (A)–(D) is the same.

In order not to hide the main ideas behind general and technicaly complicated statements, we restrict our discussion to the following four-point BVP

y′′ ∈ F (t, y), t ∈ J = [0, 1]  (1.1)
y(0) = y′(η), y(1) = y(τ)  (1.2)

where F: J × R ⟶ 2^{R} is a multivalued map with compact convex values and η, τ ∈ (0, 1). This is a special case of the BVP (A) when α = β = 1. All the other four-point BVP and the general multi-point BVP are examined in a similar way, with obvious modifications.

The method we are going to use is to reduce the existence of solutions to problem (1.1)–(1.2) to the search for fixed points of a suitable multivalued map on the Banach space C(J, R). In order to prove the existence of fixed points, we shall rely on a fixed point theorem for condensing maps due to Martelli [15].

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which are used throughout this paper.

Let (X, ∥ · ∥) be a Banach space. A multivalued map G: X ⟶ 2^{X} is convex (closed) valued if G(x) is convex (closed) for all x ∈ X. G is bounded on bounded sets if G(B) = ⋃_{x∈B} G(x) is bounded in X for any bounded set B of X (i.e. sup_{x∈B}{sup{∥y∥ : y ∈ G(x)}} < ∞).

G is called upper semicontinuous (u.s.c.) on X if for each x_{∗} ∈ X the set G(x_{∗}) is a nonempty, closed subset of X, and if for each open set B of X containing G(x_{∗}), there exists an open neighbourhood V of x_{∗} such that G(V) ⊆ B.

G is said to be completely continuous if G(B) is relatively compact for every bounded subset B ⊆ X.

If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e. x_{n} ⟶ x_{∗}, y_{n} ⟶ y_{∗}, y_{n} ∈ G(x_{n}) imply y_{∗} ∈ G(x_{∗})).

G has a fixed point if there is x ∈ X such that x ∈ G(x).

In the following CC(X) denotes the set of all nonempty compact and convex subsets of X.

A multivalued map G: J ⟶ CC(E) is said to be measurable if for each x ∈ E the function Y : J ⟶ R defined by

Y (t) = d(x, G(t)) = inf{|x − z| : z ∈ G(t)}

is measurable.

Definition 2.1. A multivalued map F: J × R ⟶ 2^{R} is said to be an L^{1}- Carathéodory map if

(i) t ⟼ F (t, y) is measurable for each y ∈ R; (ii) y ⟼ F (t, y) is upper semicontinuous for almost all t ∈ J; (iii) for each k > 0, there exists h_{k} ∈ L^{1}(J, R_{+}) such that

∥F (t, y)∥ = sup{|v| : v ∈ F (t, y)} ≤ h_{k}(t) for all |y| ≤ k and for almost all t ∈ J.

An upper semi-continuous map G: X ⟶ 2^{X} is said to be condensing if for any subset B ⊆ X with α(B) ≠ 0, we have α(G(B)) < α(B), where α denotes the Kuratowski measure of noncompacteness. For properties of the Kuratowski measure, we refer to Banas and Goebel [1].

We remark that a completely continuous multivalued map is the easiest example of a condensing map. For more details on multivalued maps see the books of Deimling [5] and Hu and Papageorgiou [11].

We will need the following hypotheses:

(H1) F: J × R ⟶ CC(R) is an L^{1}-Carathéodory multivalued map. (H2) There exists a function H ∈ L^{1}(J, R_{+}) such that

∥F (t, y)∥ := sup{|v| : v ∈ F (t, y)} ≤ H(t) for almost all t ∈ J and all y ∈ R.

Definition 2.2. A function y: J ⟶ R is called a solution for the BVP (1.1)–(1.2) if y and its first derivative are absolutely continuous and y′′ (which exists almost everywhere) satisfies the differential inclusion (1.1) a.e. on J and the condition (1.2).

Our considerations are based on the following lemmas.

Lemma 2.3. [14] Let I be a compact real interval and X be a Banach space. If F is a multivalued map satisfying (H1) and Γ is a linear continuous mapping from L^{1}(I, X) to C(I, X), then the operator

Γ ∘ S_{F} : C(I, X) ⟶ CC(C(I, X)), y ⟼ (Γ ∘ S_{F})(y) := Γ(S_{F,y})

is a closed graph operator in C(I, X) × C(I, X).

Lemma 2.4. [15] Let X be a Banach space and N: X ⟶ CC(X) be a u.s.c. condensing map. If the set

Ω := {y ∈ X : λy ∈ Ny for some λ > 1}

is bounded, then N has a fixed point.

3. Main Result

Now, we are able to state and prove our main theorem.

Theorem 3.1. Assume that Hypotheses (H1)–(H2) hold. Then the BVP (1.1)–(1.2) has at least one solution on J.

Proof. Let C(J, R) be the Banach space provided with the norm

∥y∥_{∞} := sup{ |y(t)| : t ∈ J }, for y ∈ C(J, R).

Transform the problem (1.1)–(1.2) into a fixed point problem. Consider the multivalued map, N: C(J, R) ⟶ 2^{C(J,R)} defined by:

Ny = { h ∈ C(J, R): h(t) = ∫_{0}^{t}(t − s)g(s) ds + ∫_{0}^{η} g(s) ds

+ \frac{1 + t}{1 − τ} [∫_{0}^{τ} (τ − s)g(s) ds − ∫_{0}^{1}(1 − s)g(s) ds] \}

where

g ∈ S_{F,y} = { g ∈ L^{1}(J, R) : g(t) ∈ F (t, y(t)) for a.e. in J }.

Remark 3.2. (i) It is clear that the fixed points of N are solutions to (1.1)– (1.2).

(ii) For each y ∈ C(J, R) the set S_{F,y} is nonempty (see Lasota and Opial [14]). We shall show that N satisfies the assumptions of Lemma 2.4. The proof will

be given in several steps.

Step 1. Ny is convex for each y ∈ C(J, R).

Indeed, if h_{1}, h_{2} belong to Ny, then there exist g_{1}, g_{2} ∈ S_{F,y} such that for each t ∈ J we have

h(t) = ∫_{0}^{t}(t − s)g_{i}(s) ds + ∫_{0}^{η} g_{i}(s) ds

+ \frac{1 + t}{1 − τ} [∫_{0}^{τ} (τ − s)g_{i}(s) ds − ∫_{0}^{1}(1 − s)g_{i}(s) ds], i = 1, 2.

Let 0 ≤ α ≤ 1. Then for each t ∈ J we have

(αh_{1} + (1 − α)h_{2})(t) = ∫_{0}^{t}(t − s){αg_{1}(s) + (1 − α)g_{2}(s)} ds

+ ∫_{0}^{η}{αg_{1}(s) + (1 − α)g_{2}(s)} ds

+ \frac{1 + t}{1 − τ} [∫_{0}^{τ} (τ − s)\{αg_{1}(s) + (1 − α)g_{2}(s)\} ds

− ∫_{0}^{1}(1 − s){αg_{1}(s) + (1 − α)g_{2}(s)}ds].

Since S_{F,y} is convex (because F has convex values) then

αh_{1} + (1 − α)h_{2} ∈ Ny.

Step 2. N is bounded on bounded sets of C(J, R).

Indeed, it is enough to show that there exists a positive constant c such that for each h ∈ Ny, y ∈ B_{r} = {y ∈ C(J, R) : |y∥_{∞} ≤ r} one has ∥h∥_{∞} ≤ c.

If h ∈ Ny, then there exists g ∈ S_{F,y} such that for each t ∈ J we have

h(t) = ∫_{0}^{t}(t − s)g(s) ds + ∫_{0}^{η} g(s) ds

+ \frac{1 + t}{1 − τ} [∫_{0}^{τ} (τ − s)g(s) ds − ∫_{0}^{1}(1 − s)g(s) ds], t ∈ J.

By (H1) we have for each t ∈ J that

|h(t)| ≤ ∫_{0}^{t} h_{r}(s) ds + ∫_{0}^{η} h_{r}(s) ds + \frac{2}{1 − τ} [∫_{0}^{τ} (τ − s)h_{r}(s) ds + ∫_{0}^{1}(1 − s)h_{r}(s) ds].

Then

∥h∥_{∞} ≤ ∫_{0}^{1} h_{r}(s) ds+∫_{0}^{η} h_{r}(s) ds+\frac{2}{1 − τ} [∫_{0}^{τ} (τ−s)h_{r}(s) ds+∫_{0}^{1}(1−s)h_{r}(s) ds] = c.

Step 3. N sends bounded sets of C(J, R) into equicontinuous sets.

Let t_{1}, t_{2} ∈ J, t_{1} < t_{2} and B_{r} be a bounded set of C(J, R). For each y ∈ B_{r} and h ∈ Ny, there exists g ∈ S_{F,y} such that

h(t) = ∫_{0}^{t}(t − s)g(s) ds + ∫_{0}^{η} g(s) ds

+ \frac{1 + t}{1 − τ} [∫_{0}^{τ} (τ − s)g(s) ds − ∫_{0}^{1}(1 − s)g(s) ds], t ∈ J.

Thus we obtain

|h(t_{2}) − h(t_{1})| ≤ ∫_{0}^{t_{2}}(t_{2} − s)∥g(s)∥ ds + ∫_{t_{1}}^{t_{2}}(t_{1} − s)∥g(s)∥ ds

+ \frac{t_{2} − t_{1}}{1 − τ} [∫_{0}^{τ} (τ − s)∥g(s)∥ ds + ∫_{0}^{1}(1 − s)∥g(s)∥ ds]

≤ ∫_{0}^{t_{2}}(t_{2} − s)h_{r}(s) ds + ∫_{t_{1}}^{t_{2}}(t_{1} − s)h_{r}(s) ds

+ \frac{t_{2} − t_{1}}{1 − τ} [∫_{0}^{τ} (τ − s)h_{r}(s) ds + ∫_{0}^{1}(1 − s)h_{r}(s) ds].

As t_{2} ⟶ t_{1} the right-hand side of the above inequality tends to zero.

As a consequence of Step 2, Step 3 together with the Arzela-Ascoli theorem we can conclude that N is completely continuous.

Step 4. N has a closed graph.

Let y_{n} ⟶ y_{∗}, h_{n} ∈ Ny_{n}, and h_{n} ⟶ h_{∗}. We shall prove that h_{∗} ∈ Ny_{∗}.

h_{n} ∈ Ny_{n} means that there exists g_{n} ∈ S_{F,y_{n}} such that

h_{n}(t) = ∫_{0}^{t}(t − s)g_{n}(s) ds + ∫_{0}^{η} g_{n}(s)ds

+ \frac{1 + t}{1 − τ} [∫_{0}^{τ} (τ − s)g_{n}(s) ds − ∫_{0}^{1}(1 − s)g_{n}(s) ds], t ∈ J.

We must prove that there exists g_{∗} ∈ S_{F,y_{∗}} such that

h_{∗}(t) = ∫_{0}^{t}(t − s)g_{∗}(s) ds + ∫_{0}^{η} g_{∗}(s) ds

+ \frac{1 + t}{1 − τ} [∫_{0}^{τ} (τ − s)g_{∗}(s) ds − ∫_{0}^{1}(1 − s)g_{∗}(s) ds], t ∈ J.

Now, we consider the linear continuous operator

Γ: L^{1}(J, R) ⟶ C(J, R)
g ⟼ Γ(g)(t) = ∫_{0}^{t}(t − s)g(s) ds + ∫_{0}^{η} g(s) ds

+ \frac{1 + t}{1 − τ} [∫_{0}^{τ} (τ − s)g(s) ds − ∫_{0}^{1}(1 − s)g(s) ds], t ∈ J.

From Lemma 2.3, it follows that Γ ∘ S_{F} is a closed graph operator.

Moreover from the definition of Γ we have

h_{n}(t) ∈ Γ(S_{F,y_{n}}).

Since y_{n} ⟶ y_{∗}, it follows from Lemma 2.3 that

h_{∗}(t) = ∫_{0}^{t}(t − s)g_{∗}(s) ds + ∫_{0}^{η} g_{∗}(s) ds

+ \frac{1 + t}{1 − τ} [∫_{0}^{τ} (τ − s)g_{∗}(s) ds − ∫_{0}^{1}(1 − s)g_{∗}(s) ds], t ∈ J

for some g_{∗} ∈ S_{F,y_{∗}}.

Step 5. The set

Ω := { y ∈ C(J, R) : λy ∈ Ny for some λ > 1 }

is bounded.

Let y ∈ Ω. Then λy ∈ Ny for some λ > 1. Thus there exists g ∈ S_{F,y} such that

y(t) = λ^{−1} ∫_{0}^{t}(t − s)g(s) ds + λ^{−1} ∫_{0}^{η} g(s) ds

+ λ^{−1} \frac{1 + t}{1 − τ} [∫_{0}^{τ} (τ − s)g(s) ds − ∫_{0}^{1}(1 − s)g(s) ds], t ∈ J.

This implies by (H2) that for each t ∈ J we have

|y(t)| ≤ ∫_{0}^{t}(t−s)H(s) ds+∫_{0}^{η} H(s)ds+\frac{2}{1 − τ} [∫_{0}^{τ} (τ−s)H(s) ds+∫_{0}^{1}(1−s)H(s) ds].

Thus

∥y∥_{∞} ≤ ∫_{0}^{1}(1 − s)H(s) ds + ∫_{0}^{η} H(s) ds

+ \frac{2}{1 − τ} [∫_{0}^{τ} (τ − s)H(s) ds + ∫_{0}^{1}(1 − s)H(s) ds] = K.

This shows that Ω is bounded.

Set X := C(J, R). As a consequence of Lemma 2.4 we deduce that N has a fixed point which is a solution of (1.1)–(1.2) on J.

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(received 14.02.2001, in revised form 30.11.2001)

Département de Mathématiques, Université de Sidi Bel Abbès, BP 89, 22000 Sidi Bel Abbès, Algérie E-mail: benchohra@yahoo.com Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece E-mail: sntouyas@cc.uoi.gr