Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 6, 1-9; http://www.math.u-szeged.hu/ejqtde/

Existence of solutions for a certain differential inclusion of third order

Aurelian Cernea Faculty of Mathematics and Informatics, University of Bucharest, Academiei 14, 010014 Bucharest, Romania,

            e-mail: acernea@fmi.unibuc.ro

Abstract

The existence of solutions of a boundary value problem for a third order differential inclusion is investigated. New results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.

Key words and phrases. boundary value problem, differential inclusion, fixed point.

2000 Mathematics Subject Classifications. 34A60, 26A24.

1 Introduction

This paper is concerned with the following boundary value problem

x′′′ + k^{2}x′ ∈ F (t, x), a.e. ([−1, 1]), x(−1) = x(1) = x′(1) = 0, (1.1)

where F (., .) : [−1, 1] × R → P(R) is a set-valued map and k ∈ [−π, π].

Problem (1.1) occurs in hydrodynamic and viscoelastic plates theory. For the motivation of the study of this class of problem we refer to [1] and references therein.

The present paper is motivated by a recent paper of Bartuzel and Fryszkowski ([1]), where it is considered problem (1.1) and a version of the Filippov lemma for this problem is provided. The aim of our paper is to present two other existence results for problem (1.1). Our results are essentially based on a nonlinear alternative of Leray-Schauder type and on Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with decomposable values. The methods used are standard, however their exposition in the framework of problem (1.1) is new.

We note that two other existence results for problem (1.1) obtained by the application of the set-valued contraction principle due to Covitz and Nadler jr. may be found in our previous paper [3].

The paper is organized as follows: in Section 2 we recall some preliminary facts that we need in the sequel and in Section 3 we prove our main results.

2 Preliminaries

In this section we sum up some basic facts that we are going to use later.

Let (X, d) be a metric space with the corresponding norm |.| and let I ⊂ R be a compact interval. Denote by L(I) the σ-algebra of all Lebesgue measurable subsets of I, by P(X) the family of all nonempty subsets of X and by B(X) the family of all Borel subsets of X. If A ⊂ I then χ_{A}(.) : I → {0, 1} denotes the characteristic function of A. For any subset A ⊂ X we denote by \overline{A} the closure of A.

Recall that the Pompeiu-Hausdorff distance of the closed subsets A, B ⊂ X is defined by

d_{H}(A, B) = max{d^{∗}(A, B), d^{∗}(B, A)}, d^{∗}(A, B) = sup{d(a, B); a ∈ A},

where d(x, B) = inf_{y∈B} d(x, y).

As usual, we denote by C(I, X) the Banach space of all continuous functions x(.) : I → X endowed with the norm |x(.)|_{C} = sup_{t∈I}|x(t)| and by L^{1}(I, X) the Banach space of all (Bochner) integrable functions x(.) : I → X endowed with the norm |x(.)|_{1} = ∫_{I} |x(t)|dt.

A subset D ⊂ L^{1}(I, X) is said to be decomposable if for any u(·), v(·) ∈ D and any subset A ∈ L(I) one has uχ_{A} + vχ_{B} ∈ D, where B = I\A.

Consider T : X → P(X) a set-valued map. A point x ∈ X is called a fixed point for T (.) if x ∈ T (x). T (.) is said to be bounded on bounded sets if T (B) := ∪_{x∈B}T (x) is a bounded subset of X for all bounded sets B in X. T (.) is said to be compact if T (B) is relatively compact for any bounded sets B in X. T (.) is said to be totally compact if \overline{T (X)} is a compact subset of X. T (.) is said to be upper semicontinuous if for any open set D ⊂ X, the set {x ∈ X; T (x) ⊂ D} is open in X. T (.) is called completely continuous if it is upper semicontinuous and totally bounded on X.

It is well known that a compact set-valued map T (.) with nonempty compact values is upper semicontinuous if and only if T (.) has a closed graph.

We recall the following nonlinear alternative of Leray-Schauder type and its consequences.

Theorem 2.1. ([6]) Let D and \overline{D} be the open and closed subsets in a normed linear space X such that 0 ∈ D and let T : \overline{D} → P(X) be a completely continuous set-valued map with compact convex values. Then either

i) the inclusion x ∈ T (x) has a solution, or

ii) there exists x ∈ ∂D (the boundary of D) such that λx ∈ T (x) for some λ > 1.

Corollary 2.2. Let B_{r}(0) and \overline{B_{r}(0)} be the open and closed balls in a normed linear space X centered at the origin and of radius r and let T : \overline{B_{r}(0)} → P(X) be a completely continuous set-valued map with compact convex values. Then either

i) the inclusion x ∈ T (x) has a solution, or

ii) there exists x ∈ X with |x| = r and λx ∈ T (x) for some λ > 1.

Corollary 2.3. Let B_{r}(0) and \overline{B_{r}(0)} be the open and closed balls in a normed linear space X centered at the origin and of radius r and let T : \overline{B_{r}(0)} → X be a completely continuous single valued map with compact convex values. Then either

i) the equation x = T (x) has a solution, or

ii) there exists x ∈ X with |x| = r and x = λT (x) for some λ < 1.

We recall that a multifunction T (.) : X → P(X) is said to be lower semicontinuous if for any closed subset C ⊂ X, the subset {s ∈ X; G(s) ⊂ C} is closed.

If F (., .) : I × R → P(R) is a set-valued map with compact values and x(.) ∈ C(I, R) we define

S_{F} (x) := {f ∈ L^{1}(I, R); f(t) ∈ F (t, x(t)) a.e. (I)}.

We say that F (., .) is of lower semicontinuous type if S_{F} (.) is lower semicontinuous with closed and decomposable values.

Theorem 2.4. ([2]) Let S be a separable metric space and G(.) : S → P(L^{1}(I, R)) be a lower semicontinuous set-valued map with closed decomposable values.

Then G(.) has a continuous selection (i.e., there exists a continuous mapping g(.) : S → L^{1}(I, R) such that g(s) ∈ G(s) ∀s ∈ S).

A set-valued map G : I → P(R) with nonempty compact convex values is said to be measurable if for any x ∈ R the function t → d(x, G(t)) is measurable.

A set-valued map F (., .) : I ×R → P(R) is said to be Carathéodory if t → F (t, x) is measurable for any x ∈ R and x → F (t, x) is upper semicontinuous for almost all t ∈ I.

F (., .) is said to be L^{1}-Carathéodory if for any l > 0 there exists h_{l}(.) ∈ L^{1}(I, R) such that sup\{|v|; v ∈ F (t, x)\} ≤ h_{l}(t) a.e. (I), ∀x ∈ \overline{B_{l}(0)}.

Theorem 2.5. ([5]) Let X be a Banach space, let F (., .) : I ×X → P(X) be a L^{1}-Carathéodory set-valued map with S_{F} ≠ ∅ and let Γ : L^{1}(I, X) → C(I, X) be a linear continuous mapping.

Then the set-valued map Γ ∘ S_{F} : C(I, X) → P(C(I, X)) defined by

(Γ ∘ S_{F})(x) = Γ(S_{F} (x))

has compact convex values and has a closed graph in C(I, X) × C(I, X).

Note that if dimX < ∞, and F (., .) is as in Theorem 2.5, then S_{F} (x) ≠ ∅ for any x(.) ∈ C(I, X) (e.g., [5]).

In what follows I = [−1, 1] and let AC^{2}(I, R) be the space of two times differentiable functions x(.) : I → R whose second derivative exists and is absolutely continuous on I. On AC^{2}(I, R) we consider the norm |.|_{C}.

By a solution of problem (1.1) we mean a function x(.) ∈ AC^{2}(I, R) if there exists a function f(.) ∈ L^{1}(I, R) with f(t) ∈ F (t, x(t)), a.e. (I) such that x′′′(t) + k^{2}x′(t) = f(t) a.e. (I) and x(−1) = x(1) = x′(1) = 0.

Lemma 2.6. ([1]) If f(.) : [−1, 1] → R is an integrable function and k ∈ [−π, π] then the equation

x′′′ + k^{2}x′ = f(t) a.e. (I),

with the boundary conditions x(−1) = x(1) = x′(1) = 0 has a unique solution given by

x(t) = ∫_{−1}^{1} G(t, s)f(s)ds,

where G(., .) is the associated Green function. Namely,

G(t, x) = \{ \begin{matrix}\frac{(1−cosk(1+x))(1−cosk(1−t))}{k^{2}(1−cos2k)} if − 1 ≤ x ≤ t ≤ 1, \\ \frac{(1−cosk(1+x))(1−cosk(1−t))−(1−cosk(x−t))(1−cos2k)}{k^{2}(1−cos2k)} if − 1 ≤ t ≤ x ≤ 1. \end{matrix}

Moreover, if k ≠ 0

0 ≤ G(t, x) ≤ G_{0} := \frac{k^{2}(5\sqrt{5} − 11)}{sin^{2}k} ∀ (t, x) ∈ I × R.

3 The main results

We are able now to present the existence results for problem (1.1). We consider first the case when F (., .) is convex valued.

Hypothesis 3.1. i) F (., .) : I × R → P(R) has nonempty compact convex values and is Carathéodory.

ii) There exist φ(.) ∈ L^{1}(I, R) with φ(t) > 0 a.e. (I) and there exists a nondecreasing function ψ : [0, ∞) → (0, ∞) such that

sup{|v|; v ∈ F (t, x)} ≤ φ(t)ψ(|x|) a.e. (I), ∀x ∈ R.

Theorem 3.2. Assume that Hypothesis 3.1 is satisfied and there exists r > 0 such that

r > G_{0}|φ|_{1}ψ(r).  (3.1)

Then problem (1.1) has at least one solution x(.) such that |x(.)|_{C} < r.

Proof. Let X = AC^{2}(I, R) and consider r > 0 as in (3.1). It is obvious that the existence of solutions to problem (1.1) reduces to the existence of the solutions of the integral inclusion

x(t) ∈ ∫_{−1}^{1} G(t, s)F (s, x(s))ds, t ∈ I.  (3.2)

Consider the set-valued map T : \overline{B_{r}(0)} → P(AC^{2}(I, R)) defined by

T (x) := \{v(.) ∈ AC^{2}(I, R); v(t) := ∫_{−1}^{1} G(t, s)f(s)ds, f ∈ \overline{S_{F} (x)}\}. (3.3)

We show that T (.) satisfies the hypotheses of Corollary 2.2. First, we show that T (x) ⊂ AC^{2}(I, R) is convex for any x ∈ AC^{2}(I, R). If v_{1}, v_{2} ∈ T (x) then there exist f_{1}, f_{2} ∈ S_{F} (x) such that for any t ∈ I

one has

v_{i}(t) = ∫_{−1}^{1} G(t, s)f_{i}(s)ds, i = 1, 2.

Let 0 ≤ α ≤ 1. Then for any t ∈ I we have

(αv_{1} + (1 − α)v_{2})(t) = ∫_{−1}^{1} G(t, s)[αf_{1}(s) + (1 − α)f_{2}(s)]ds.

The values of F (., .) are convex, thus S_{F} (x) is a convex set and hence αh_{1} + (1 − α)h_{2} ∈ T (x).

Secondly, we show that T (.) is bounded on bounded sets of AC^{2}(I, R). Let B ⊂ AC^{2}(I, R) be a bounded set. Then there exist m > 0 such that

|x|_{C} ≤ m ∀x ∈ B.

If v ∈ T (x) there exists f ∈ S_{F} (x) such that v(t) = ∫_{−1}^{1} G(t, s)f(s)ds. One may write for any t ∈ I

|v(t)| ≤ ∫_{−1}^{1} |G(t, s)|.|f(s)|ds ≤ ∫_{−1}^{1} |G(t, s)|φ(s)ψ(|x(t)|)ds

and therefore

|v|_{C} ≤ G_{0}|φ|_{1}ψ(m) ∀v ∈ T (x),

i.e., T (B) is bounded.

We show next that T (.) maps bounded sets into equi-continuous sets. Let B ⊂ AC^{2}(I, R) be a bounded set as before and v ∈ T (x) for some

x ∈ B. There exists f ∈ S_{F} (x) such that v(t) = ∫_{−1}^{1} G(t, s)f(s)ds. Then for any t, τ ∈ I we have

|v(t) − v(τ)| ≤ | ∫_{−1}^{1} G(t, s)f(s)ds − ∫_{−1}^{1} G(τ, s)f(s)ds| ≤

∫_{−1}^{1} |G(t, s) − G(τ, s)|.|f(s)|ds ≤ ∫_{−1}^{1} |G(t, s) − G(τ, s)|φ(s)ψ(m)ds.

It follows that |v(t) − v(τ)| → 0 as t → τ. Therefore, T (B) is an equicontinuous set in AC^{2}(I, R).

We apply now Arzela-Ascoli’s theorem we deduce that T (.) is completely continuous on AC^{2}(I, R).

In the next step of the proof we prove that T (.) has a closed graph.

Let x_{n} ∈ AC^{2}(I, R) be a sequence such that x_{n} → x^{∗} and v_{n} ∈ T (x_{n}) ∀n ∈ N such that v_{n} → v^{∗}. We prove that v^{∗} ∈ T (x^{∗}).

Since v_{n} ∈ T (x_{n}), there exists f_{n} ∈ S_{F} (x_{n}) such that v_{n}(t) = ∫_{−1}^{1} G(t, s)f_{n}(s)ds.

Define Γ : L^{1}(I, R) → AC^{2}(I, R) by (Γ(f))(t) := ∫_{−1}^{1} G(t, s)f(s)ds. One has max_{t∈I} |v_{n}(t) − v^{∗}(t)| = |v_{n}(.) − v^{∗}(.)|_{C} → 0 as n → ∞

We apply Theorem 2.5 to find that Γ ∘ S_{F} has closed graph and from the definition of Γ we get v_{n} ∈ Γ ∘ S_{F} (x_{n}). Since x_{n} → x^{∗}, v_{n} → v^{∗} it follows the existence of f^{∗} ∈ S_{F} (x^{∗}) such that v^{∗}(t) = ∫_{−1}^{1} G(t, s)f^{∗}(s)ds.

Therefore, T (.) is upper semicontinuous and compact on \overline{B_{r}(0)}. We apply Corollary 2.2 to deduce that either i) the inclusion x ∈ T (x) has a solution in \overline{B_{r}(0)}, or ii) there exists x ∈ X with |x|_{C} = r and λx ∈ T (x) for some λ > 1.

Assume that ii) is true. With the same arguments as in the second step of our proof we get r = |x(.)|_{C} ≤ G_{0}|φ|_{1}ψ(r) which contradicts (3.1). Hence only i) is valid and theorem is proved.

We consider now the case when F (., .) is not necessarily convex valued. Our existence result in this case is based on the Leray-Schauder alternative for single valued maps and on Bressan Colombo selection theorem.

Hypothesis 3.3. i) F (., .) : I × R → P(R) has compact values, F (., .) is L(I) ⊗ B(R) measurable and x → F (t, x) is lower semicontinuous for almost all t ∈ I.

ii) There exist φ(.) ∈ L^{1}(I, R) with φ(t) > 0 a.e. (I) and there exists a nondecreasing function ψ : [0, ∞) → (0, ∞) such that

sup{|v|; v ∈ F (t, x)} ≤ φ(t)ψ(|x|) a.e. (I), ∀x ∈ R.

Theorem 3.4. Assume that Hypothesis 3.3 is satisfied and there exists r > 0 such that Condition (3.1) is satisfied.

Then problem (1.1) has at least one solution on I.

Proof. We note first that if Hypothesis 3.3 is satisfied then F (., .) is of lower semicontinuous type (e.g., [4]). Therefore, we apply Theorem 2.4 to deduce that there exists f(.) : AC^{2}(I, R) → L^{1}(I, R) such that f(x) ∈ S_{F} (x) ∀x ∈ AC^{2}(I, R).

We consider the corresponding problem

x(t) = ∫_{−1}^{1} G(t, s)f(x(s))ds, t ∈ I  (3.4)

in the space X = AC^{2}(I, R). It is clear that if x(.) ∈ AC^{2}(I, R) is a solution of the problem (3.4) then x(.) is a solution to problem (1.1).

Let r > 0 that satisfies condition (3.1) and define the set-valued map T : \overline{B_{r}(0)} → P(AC^{2}(I, R)) by

(T (x))(t) := ∫_{−1}^{1} G(t, s)f(x(s))ds.

Obviously, the integral equation (3.4) is equivalent with the operator equation

x(t) = (T (x))(t), t ∈ I.  (3.5)

It remains to show that T (.) satisfies the hypotheses of Corollary 2.3. We show that T (.) is continuous on \overline{B_{r}(0)}. From Hypotheses 3.3. ii) we

have

|f(x(t))| ≤ φ(t)ψ(|x(t)|) a.e. (I)

for all x(.) ∈ AC^{2}(I, R). Let x_{n}, x ∈ \overline{B_{r}(0)} such that x_{n} → x. Then

|f(x_{n}(t))| ≤ φ(t)ψ(r) a.e. (I).

From Lebesgue’s dominated convergence theorem and the continuity of f(.) we obtain, for all t ∈ I

lim_{n→∞}(T (x_{n}))(t) = ∫_{−1}^{1} G(t, s)f(x_{n}(s))ds = ∫_{−1}^{1} G(t, s)f(x(s))ds = (T (x))(t),

i.e., T (.) is continuous on \overline{B_{r}(0)}.

Repeating the arguments in the proof of Theorem 3.2 with corresponding modifications it follows that T (.) is compact on \overline{B_{r}(0)}. We apply Corollary 2.3 and we find that either i) the equation x = T (x) has a solution in \overline{B_{r}(0)}, or ii) there exists x ∈ X with |x|_{C} = r and x = λT (x) for some λ < 1.

As in the proof of Theorem 3.2 if the statement ii) holds true, then we obtain a contradiction to (3.1). Thus only the statement i) is true and problem (1.1) has a solution x(.) ∈ AC^{2}(I, R) with |x(.)|_{C} < r.

References

[1] G. Bartuzel and A. Fryszkowski, Filippov lemma for certain differential inclusion of third order, Demonstratio Math. 41 (2008), 337-352.

[2] A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 69-86.

[3] A. Cernea, On a boundary value problem for a third order differential inclusion, submitted.

[4] M. Frignon and A. Granas, Théorèmes d’existence pour les inclusions différentielles sans convexité, C. R. Acad. Sci. Paris, Ser. I 310 (1990), 819-822.

[5] A. Lasota and Z. Opial, An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Math., Astronom. Physiques 13 (1965), 781-786.

[6] D. O’ Regan, Fixed point theory for closed multifunctions, Arch. Math. (Brno), 34 (1998), 191-197.

(Received October 7, 2008)