Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 8, 1-8; http://www.math.u-szeged.hu/ejqtde/

On the existence of solutions for a higher order differential inclusion

without convexity

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

             acernea@math.math.unibuc.ro

Abstract We prove a Filippov type existence theorem for solutions of a higher order differential inclusion in Banach spaces with nonconvex valued right hand side by applying the contraction principle in the space of the derivatives of solutions instead of the space of solutions.

Key words and phrases. Differential inclusion, contractive set-valued map, fixed point.

2000 AMS Subject Classifications. 34A60.

1 Introduction

In this note we study the n-th order differential inclusion

x^{(n)} − λx ∈ F (t, x), a.e. (I)  (1.1)

with boundary conditions of the form

x^{(i)}(0) − x^{(i)}(T) = μ_{i}, i = 0, 1, …, n − 1,  (1.2)

where λ ∈ R, E is real separable Banach space, I = [0, T], F (·, ·) : I × E → P(E) and μ_{i} ∈ E, i = 0, 1, …, n−1. When λ ≠ 0 the linear part in equation (1.1) is invertible and, in this case, the problem (1.1)-(1.2) is well known as a nonresonance problem. Moreover, if μ_{i} = 0, i = 0, 1, …, n − 1 then the conditions (1.2) are periodic boundary conditions.

The present note is motivated by a recent paper of Benchohra, Graef, Henderson and Ntouyas ([1]) in which several existence results concerning nonresonance impulsive higher order differential inclusions are obtained via fixed point techniques. The aim of our paper is to provide a Filippov type result concerning the existence of solutions to problem (1.1)-(1.2). Recall that for a differential inclusion defined by a lipschitzian set-valued map with nonconvex values, Filippov’s theorem consists in proving the existence of a solution starting from a given ”quasi” solution.

Our approach is different from the one in [1] and consists in applying the contraction principle in the space of derivatives of solutions instead of the space of solutions. In addition, as usual at a Filippov existence type theorem, our result provides an estimate between the starting ”quasi” solution and the solution of the differential inclusion. The idea of applying the set-valued contraction principle due to Covitz and Nadler ([6]) in the space of derivatives of the solutions belongs to Kannai and Tallos ([7]) and it was already used for other results concerning differential inclusions ([3,4,5] etc.).

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 result.

2 Preliminaries

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

Let (X, d) be a metric space and consider a set valued map T on X with nonempty closed values in X. T is said to be a λ-contraction if there exists 0 < λ < 1 such that:

d_{H}(T (x), T (y)) ≤ λd(x, y) ∀x, y ∈ X,

where d_{H}(·, ·) denotes the Pompeiu-Hausdorff distance. 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).

If X is complete, then every set valued contraction has a fixed point, i.e. a point z ∈ X such that z ∈ T (z) ([6]).

We denote by F ix(T) the set of all fixed points of the set-valued map T. Obviously, F ix(T) is closed.

Proposition 2.1. ([8]) Let X be a complete metric space and suppose that T_{1}, T_{2} are λ-contractions with closed values in X. Then

d_{H}(F ix(T_{1}), F ix(T_{2})) ≤ \frac{1}{1 − λ} sup_{z∈X} d(T_{1}(z), T_{2}(z)).

In what follows E is a real separable Banach space with norm |·|, C(I, E) is the Banach space of all continuous functions from I to E with the norm ∥x(·)∥_{C} = sup_{t∈I} |x(t)|, AC^{i}(I, E) is the space of i-times differentiable functions x : I → E whose i-th derivative x^{i}(·) is absolutely continuous and L^{1}(I, E) is the Banach space of (Bochner) integrable functions u(·) : I → E endowed with the norm ∥u(·)∥_{1} = ∫_{0}^{1} |u(t)|dt.

A function x(·) ∈ AC^{n−1}(I, E) is called a solution of problem (1.1)-(1.2) if there exists a function v(·) ∈ L^{1}(I, E) with v(t) ∈ F (t, x(t)), a.e. (I) such that x^{(n)}(t) − λx(t) = v(t), a.e. (I) and x(·) satisfies conditions (1.2).

In what follows we consider the Green function G(·, ·) : I × I → E associated to the periodic boundary problem

x^{(n)} − λx = 0, x^{(i)}(0) − x^{(i)}(T) = 0, i = 0, 1, …, n − 1.

For the properties of G(·, ·) we refer to [1].

The next result is well known (e.g. [1]).

Lemma 2.2. ([1]) If v(·) : [0, T] → E is an integrable function then the problem

\begin{matrix}x^{(n)}(t) − λx(t) = v(t) a.e. (I) \\ x^{(i)}(0) − x^{(i)}(T) = μ_{i}, i = 0, 1, …, n − 1. \end{matrix}

has a unique solution x(·) ∈ AC^{n−1}(I, E) given by

x(t) = P_{μ}(t) + ∫_{0}^{T} G(t, s)v(s)ds,

where

P_{μ}(t) = ∑_{i=0}^{n−1} \frac{∂^{i}}{∂t^{i}} G(t, 0)μ_{n−1−i}.  (2.1)

In the sequel we assume the following conditions on F.

Hypothesis 2.3. (i) F (·, ·) : I × E → P(E) has nonempty closed values and for every x ∈ E F (·, x) is measurable.

(ii) There exists L(·) ∈ L^{1}(I, E) such that for almost all t ∈ I, F (t, ·) is L(t)-Lipschitz in the sense that

d_{H}(F (t, x), F (t, y)) ≤ L(t)|x − y| ∀ x, y ∈ E

and d(0, F (t, 0)) ≤ L(t) a.e. (I).

Denote L_{0} := ∫_{0}^{T} L(s)ds and M_{0} := sup_{(t,s)∈I×I} |G(t, s)|.

3 The main result

We are able now to prove our main result.

Theorem 3.1. Assume that Hypothesis 2.3 is satisfied and M_{0}L_{0} < 1. Let y(·) ∈ AC^{n−1}(I, E) be such that there exists q(.) ∈ L^{1}(I, E) with d(y^{(n)}(t) − λy(t), F (t, y(t))) ≤ q(t), a.e. (I). Denote μ̃_{i} = y^{(i)}(0) − y^{(i)}(T), i = 0, 1, …, n − 1.

Then for every ε > 0 there exists x(·) a solution of (1.1)-(1.2) satisfying for all t ∈ I

|x(t) − y(t)| ≤ \frac{1}{1 − M_{0}L_{0}} sup_{t∈I} |P_{μ}(t) − Pμ̃(t)| + \frac{M_{0}}{1 − M_{0}L_{0}} ∫_{0}^{T} q(t)dt + ε,

where P_{μ}(t) is defined in (2.1).

Proof. For u(·) ∈ L^{1}(I, E) define the following set valued maps:

M_{u}(t) = F (t, P_{μ}(t) + ∫_{0}^{T} G(t, s)u(s)ds) , t ∈ I,
T (u) = {ϕ(.) ∈ L^{1}(I, E); ϕ(t) ∈ M_{u}(t) a.e. (I)}.

It follows from the definition and Lemma 2.2 that x(·) is a solution of (1.1)-(1.2) if and only if x^{(n)}(·) − λx(·) is a fixed point of T (·).

We shall prove first that T (u) is nonempty and closed for every u ∈ L^{1}(I, E). The fact that the set valued map M_{u}(·) is measurable is well known. For example the map t → P_{μ}(t) + ∫_{0}^{T} G(t, s)u(s)ds can be approximated by step functions and we can apply Theorem III. 40 in [2]. Since the values of F are closed with the measurable selection theorem (Theorem III.6 in [2]) we infer that M_{u}(·) admits a measurable selection ϕ. One has

|ϕ(t)| ≤ d(0, F (t, 0)) + d_{H} (F (t, 0), F (t, P_{μ}(t) + ∫_{0}^{T} G(t, s)u(s)ds))
≤ L(t) (1 + |P_{μ}(t)| + sup_{t,s∈I} |G(t, s)| ∫_{0}^{T} |u(s)|ds) ,

which shows that ϕ ∈ L^{1}(I, E) and T (u) is nonempty.

On the other hand, the set T (u) is also closed. Indeed, if ϕ_{n} ∈ T (u) and ∥ϕ_{n}−ϕ∥_{1} → 0 then we can pass to a subsequence ϕ_{n_{k}} such that ϕ_{n_{k}}(t) → ϕ(t) for a.e. t ∈ I, and we find that ϕ ∈ T (u).

We show next that T (·) is a contraction on L^{1}(I, E).

Let u, v ∈ L^{1}(I, E) be given, ϕ ∈ T (u) and let δ > 0. Consider the following set-valued map:

H(t) = M_{v}(t) ∩ {x ∈ E; |ϕ(t) − x| ≤ L(t) |∫_{0}^{T} G(t, s)(u(s) − v(s))ds| + δ} .

From Proposition III.4 in [2], H(·) is measurable and from Hypothesis 2.3 ii) H(·) has nonempty closed values. Therefore, there exists ψ(·) a measurable selection of H(·). It follows that ψ ∈ T (v) and according with the definition of the norm we have

∥ϕ − ψ∥_{1} = ∫_{0}^{T} |ϕ(t) − ψ(t)|dt
≤ ∫_{0}^{T} L(t) (∫_{0}^{T} |G(t, s)||u(s) − v(s)|ds) dt + ∫_{0}^{T} δdt
= ∫_{0}^{T} (∫_{0}^{T} L(t)|G(t, s)|dt) |u(s) − v(s)|ds + T δ
≤ M_{0}L_{0}∥u − v∥_{1} + T δ.

Since δ > 0 was chosen arbitrary, we deduce that

d(ϕ, T (v)) ≤ M_{0}L_{0}∥u − v∥_{1}.

Replacing u by v we obtain

d_{H}(T (u), T (v)) ≤ M_{0}L_{0}∥u − v∥_{1},

thus T (·) is a contraction on L^{1}(I, E).

We consider next the following set-valued maps

F_{1}(t, x) = F (t, x) + q(t)B, (t, x) ∈ I × E,

M_{u}^{1}(t) = F_{1} (t, Pμ̃(t) + ∫_{0}^{T} G(t, s)u(s)ds) , t ∈ I, u(·) ∈ L^{1}(I, E),

T_{1}(u) = {ψ(·) ∈ L^{1}(I, E); ψ(t) ∈ M_{u}^{1}(t) a.e. (I)},

where B denotes the closed unit ball in E. Obviously, F_{1}(·, ·) satisfies Hypothesis 2.3.

Repeating the previous step of the proof we obtain that T_{1} is also a M_{0}L_{0}-contraction on L^{1}(I, E) with closed nonempty values.

We prove next the following estimate

d_{H}(T (u), T_{1}(u)) ≤ sup_{t∈I} |P_{μ}(t) − Pμ̃(t)|L_{0} + ∫_{0}^{T} q(t)dt.  (3.1)

Let ϕ ∈ T (u), δ > 0 and define

H_{1}(t) = M_{u}^{1}(t) ∩ \{z ∈ E; |ϕ(t) − z| ≤ L(t)|P_{μ}(t) − Pμ̃(t)| + q(t) + \frac{δ}{T} \} .

With the same arguments used for the set valued map H(·), we deduce that H_{1}(·) is measurable with nonempty closed values. Hence let ψ(·) be a measurable selection of H_{1}(·). It follows that ψ ∈ T_{1}(u) and one has

∥ϕ − ψ∥_{1} = ∫_{0}^{T} |ϕ(t) − ψ(t)|dt
≤ ∫_{0}^{T} [L(t)|P_{μ}(t) − Pμ̃(t)| + q(t) + \frac{δ}{T} ] dt
≤ ∫_{0}^{T} L(t)|P_{μ}(t) − Pμ̃(t)|dt + ∫_{0}^{T} q(t) + δ
≤ L_{0} sup_{t∈I} |P_{μ}(t) − Pμ̃(t)| + ∫_{0}^{T} q(t)dt + δ.

Since δ is arbitrary, as above we obtain (3.1). We apply Proposition 2.1 and we infer that

d_{H}(F ix(T), F ix(T_{1})) ≤ \frac{L_{0}}{1 − M_{0}L_{0}} sup_{t∈I} |P_{μ}(t) − Pμ̃(t)| + \frac{1}{1 − M_{0}L_{0}} ∫_{0}^{T} q(t)dt.

Since v(·) = y^{(n)}(·) − λy(·) ∈ F ix(T_{1}) it follows that there exists u(·) ∈ F ix(T) such that for any ε > 0

∥v − u∥_{1} ≤ \frac{L_{0}}{1 − M_{0}L_{0}} sup_{t∈I} |P_{μ}(t) − Pμ̃(t)| + \frac{1}{1 − M_{0}L_{0}} ∫_{0}^{T} q(t)dt + \frac{ε}{M_{0}} .

We define x(t) = P_{μ}(t) + ∫_{0}^{T} G(t, s)u(s)ds, t ∈ I and we have

|x(t) − y(t)| ≤ |P_{μ}(t) − Pμ̃(t)| + ∫_{0}^{T} |G(t, s)||u(s) − v(s)|ds
≤ sup_{t∈I} |P_{μ}(t) − Pμ̃(t)| + sup_{t∈I} |P_{μ}(t) − Pμ̃(t)|\frac{M_{0}L_{0}}{1 − M_{0}L_{0}}

+\frac{1}{1 − M_{0}L_{0}} M_{0} ∫_{0}^{T} q(t)dt + ε

≤ \frac{1}{1 − M_{0}L_{0}} sup_{t∈I} |P_{μ}(t) − Pμ̃(t)| + \frac{M_{0}}{1 − M_{0}L_{0}} ∫_{0}^{T} q(t)dt + ε,

which completes the proof.

References

[1] M. Benchohra, J. R. Graef, J. Henderson and S. K. Ntouyas, Nonresonance impulsive higher order functional nonconvex-valued differential inclusions, Electron. J. Qual. Theory Differ. Equ. (2002), No 13, 13pp (electronic).

[2] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580, Springer, Berlin, 1977.

[3] A. Cernea, A Filippov type existence theorem for an infinite operational differential inclusion, Stud. Cerc. Mat. 50 (1998), 15-22.

[4] A. Cernea, An existence theorem for some nonconvex hyperbolic differential inclusions, Mathematica (Cluj) 45(68) (2003), 121-126.

[5] A. Cernea, Existence for nonconvex integral inclusions via fixed points, Arch. Math. (Brno) 39 (2003), 293-298.

[6] H. Covitz and S. B. Nadler jr., Multivalued contraction mapping in generalized metric spaces, Israel J. Math. 8 (1970), 5-11.

[7] Z. Kannai and P. Tallos, Stability of solution sets of differential inclusions, Acta Sci. Math. (Szeged) 61(1995), 197-207.

[8] T.C. Lim, On fixed point stability for set valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985), 436-441.

(Received December 15, 2006)