ON AN APPLICATION OF FIXED POINT THEOREM TO NONLINEAR INCLUSIONS

. Suﬃcient conditions for existence of solutions for a class of nonlinear inclusions in inﬁnite dimensional Banach spaces are established. The results are obtained by means of ﬁxed-point theorem for set-valued maps.


1.
Introduction. We consider an inclusion problem of the form 0 ∈ x + KF (x) (1) defined in a space X = L p (Ω, E) of functions from Ω to a Banach space E, where F : X → 2 X * is a given set-valued map and K : X * → X is a nonlinear mapping with X * = L q (Ω, E * ), 1/p + 1/q = 1. We present here sufficient conditions for the existence of solutions and an existence theorem for a class of inclusion equations (1). The approach we shall take is to reduce the problem of existence to the problem of showing a certain set-valued map has a fixed point. Applications of fixed point theorems for set-valued maps to differential inclusions and related problems had been reported by various authors. In fact, there is a vast literatures covering questions of existence of solutions for finite and infinite dimensional systems. For example, previously Lasota and Opial [1] in their paper studied the existence problem of ordinary differential equations with multi-valued right hand side. Tarnove [2] obtained sufficient conditions for controllability of the nonlinear system in the form x = f (t, x, u), u(t) ∈ Q. Hermes [3] considered equations of the form x ∈ R(t, x). Subsequently, Hou [4] studied the controllability of feedback systems in the form Cesari and Hou [5] also proved existence theorems of optimal solutions for abstract equations, and in particular to quasi-linear evolution equations, based on properties of sets depending on parameters, or set-valued maps. Ahmed [6] considered existence of optimal controls for a class of systems governed by differential inclusions on a Banach space. For further results, see Agarwal et al [7] as well as Granas and Dugundji [8], and the references therein.
There are various motivations for studying the nonlinear inclusion equation (1). Let us mention some of them here. Evidently, if the set-valued map f is singlevalued, i.e. f (ω, u) = {g(ω, u)} for some function g : Ω × E → E * , then we can construct a superposition operator of the form K g x(·) := Kg(·, x(·)).
The corresponding inclusion (1) then reduces to an equation of the familiar type On the other hand, when investigating boundary value problems in control theory which define the state x of a system by an acting input h, one is led to the equation of the form where L is a linear operator on an appropriate function space. Now, if the input is perturbed, this has to be replaced by the equation with set-valued right-hand side where N is some set-valued nonlinear operator. In many cases L is some differential operator which admits a certain Green's function on a space determined by suitably boundary conditions. As such the above inclusion may be put in the form (1) by setting K = −L −1 . The rest of the paper is organized as follows. In section 2 we introduce the basic notations and recall some preliminary results. In section 3 we prove a selection lemma and give sufficient conditions for a class of nonlinear inclusion equations to have a solution.
2. Preliminaries. Let (Ω, µ) be a finite measure space. Let E be a real Banach space and E * be the topological dual of E.
The concept of measurability of a function from an abstract measure space Ω with finite measure µ into the Banach space E, is used in the sense of Dunford and Schwartz [9]. This pertains also to the concept of Bochner integrability.
Let 1 < p, q < ∞ and 1/p + 1/q = 1. The notations L p (Ω, E) and L q (Ω, E * ) denote the spaces of p-th and q-th Bochner-integrable functions with values in E and E * , respectively.
The space L p (Ω, E) is normed by x := Ω x(ω) p E dµ 1/p . The pairing between the spaces L p (Ω, E) and L q (Ω, E * ) is given by Ω z, w dµ, where z ∈ L p (Ω, E), w ∈ L q (Ω, E * ). There are sufficient conditions on E ensuring L p (Ω, E) * ∼ = L q (Ω, E * ) and the correspondence is given as follows.
for all f ∈ L p (Ω, E) and such that The following theorem is well-known (cf. Leonard [10]): We will use the following notations associated with set-valued maps. P f (U ) denotes the class of nonempty closed subsets of a Banach space U , and P f c (U ) the class of nonempty closed convex subsets of U . A set-valued map Ψ : in Ω is said to be a measurable selection of Ψ.
We also say that a measurable set-valued map Ψ : a.e. in Ω}. A sufficient condition for the set S q Ψ of selections to be relatively weakly compact in L q (Ω, U ) is given in the following.
Lemma 2.2. Let U be a separable Banach space and (Ω, µ) a finite measure space. Let A be a closed convex set in U with the property that bounded subsets of A are relatively weakly compact. Let Ψ : Ω → P f (U ) be a L q -integrably bounded, measurable set-valued map with closed values in A. Then S q Ψ is relatively weakly compact in L q (Ω, U ) for 1 < q < ∞.
In fact, since U is separable the measurable selection theorem (cf. Kuratowski and Ryll-Nardzewski [11], Aliprandtis and Border [12]) implies that Ψ admits a measurable selection ψ. Ψ being L q -integrably bounded ensures that ψ ∈ L q (Ω, U ), and whence S q Ψ is nonempty and in particular norm bounded in L q (Ω, U ). The relatively weakly compactness of S q Ψ is now an easy consequence of the following compactness theorem (cf. Seierstad [13] and Vrabie [14]). Theorem 2.3. Let U be a separable Banach space and (Ω, µ) a finite measure space. Let A be a closed convex set in U with the property that bounded subsets of A are relatively weakly compact. Let S be a norm bounded subset of the space L q (Ω, U ) made up of functions whose values are in A. Then S is relatively weakly compact for 1 < q < ∞.
Let Y and Z be Banach spaces. A set-valued map f : Ω × Z → P f c (Y ) is said to satisfy the Cesari's condition at a point z ∈ Z if for every sequence {z k } converging to z in Z, we have The set-valued map f is said to satisfy the Cesari's condition on Z if it satisfies the Cesari's condition at every point of Z. (Here the notation cl · co A stands for the closed convex hull of the set A ⊂ Y .) Remark 1. Cesari's condition is also known as Cesari's property Q. It is noted that Cesari's condition is closely related to various semicontinuity properties of set-valued maps in locally convex spaces. For further discussions on their interrelationship, see Hou [15]. In fact, due to the closure theorem as stated below Cesari's condition plays an important role in calculus of variations and the theory of optimal controls, in particular on the controllability of controlled systems in infinite dimensional spaces. (See Hou [4], Cesari and Hou [5], and Hou [16].) Central to our discussion is the following closure theorem due essentially to Cesari as modified by Hou (cf. Cesari and Hou [5], Hou [15]).
Finally, let us recall the following well known fixed point theorem for set-valued maps of Kakutani-Fan ( cf. Aliprandtis and Border [12]). 3. Main Results. Before we state the main theorem, we first apply Cesari's closure theorem to establish the following interesting selection lemma.
Lemma 3.1. Let Y and Z be Banach spaces with Y separable, and let A be a nonempty closed convex set in Y with the property that bounded subsets of A are relatively weakly compact.
Let the set-valued map f : Ω × Z → P f c (Y ) with values in A be such that 1. for every z ∈ Z the set-valued map f (·, z) : Ω → P f c (Y ) has a measurable selection; 2. for every z ∈ Z, f (ω, z) ⊂ G(ω) a.e. in Ω with G : Ω → P f (Y ) being L q -integrably bounded for 1 < q < ∞ and having values in A ; 3. the set-valued map f : Ω × Z → P f c (Y ) satisfies the Cesari's condition on Z.
By condition (2 ) we have g n ∈ S q G . In view of Lemma 2.2 the set S q G is relatively weakly compact. Thus we may assume, by passing to subsequence if necessary, that g n converges weakly to some function g ∈ L q (Ω, Y ). It follows from condition (3 ) and Cesari's closure theorem that g(ω) ∈ f (ω, p(ω)) = Q(ω) a.e. in Ω.
Let 1 < p, q < ∞ and 1 p + 1 q = 1. The following assumptions will be used in the sequel.

Hypothesis H(A):
• E is a reflexive Banach space with separable dual E * ; • K : X * → X is completely continuous from X := L p (Ω, E) to X * = L q (Ω, E * ). Hypothesis H(B): in Ω, with G : Ω → P f c (E * ) being L q -integrably bounded. • f satisfies the Cesari's condition on E. By putting, for x ∈ X, we have a set-valued map F from X to 2 X * . This gives rise to an associated inclusion problem of the type in (1), whereby a solution is an element x ∈ X satisfying the equation 0 = x + Kg for some g ∈ F (x). Proof. First we note that the Banach space E * is separable and reflexive. Clearly by hypothesis H(B) the set S q G is nonempty. Furthermore, it is closed and convex as well as weakly compact in X * = L q (Ω, E * ) in view of Lemma 2.2 and the fact that G has closed convex values in E * .
Let g ∈ S q G and consider the set-valued map Γ defined by Γ(g) := S q f (·,(−Kg)(·)) . It follows from Lemma 3.1 that for every g ∈ S q G the set Γ(g) is nonempty and lies in S q G ⊂ X * . In addition, it follows from hypothesis H(B) that Γ(g) is convex. Thus, Γ maps S q G into P f c (S q G ) with nonempty closed convex values. It is also observed that if g is a fixed point of Γ, i.e., g ∈ Γ(g), then −Kg will be a solution of the nonlinear inclusion equation (1).
Let X * be endowed with the weak topology (denoted by X * w ). We claim that Γ has a closed graph in X * w × X * w . Since S q G is weakly compact in the separable space X * , its weak topology is metrizable (cf. theorem V.6.3. in Dunford and Schwartz [9]). Thus, it suffices to show that Gr(Γ) ⊂ S q G × S q G is sequentially weakly closed. To this end, let (g n , h n ) ∈ Gr(Γ) with (g n , h n ) w×w −→ (g, f ) in S q G × S q G . We want to show that (g, f ) ∈ Gr(Γ). But note that K being completely continuous, so by passing to a subsequence if necessary we may assume u n := −Kg n → −Kg := u in X, and u n (ω) → u(ω) almost everywhere as well. Thus we may assume h n h in X * , u n (ω) → u(ω) a.e. in Ω, h n (ω) ∈ f (ω, u n (ω)) a.e. in Ω.