Random Solutions of a System of Functional-Differential Equations in a Banach Space

In the present paper we shall prove a random fixed point theorem for the system of condensing transformations which generalizes the result of Shigeru Itoh ([5],Theorem 2. 1).Moreover, we give an application of the obtained theorem to the existence of a random solutions of a system of functional-differential equations in a Banach space.Our result generalizes theorems of A. Ambrosetti [1], R. Nussbaum [8] (Theorem 2, p. 102) and S. Itoh [5] (Theorem 4.3).2.Preliminaries.Let (T, ovalboxttsmall REJECT) be a measurable space.Let $(X, d)$ be a metric space and 2 X be the family of all subsets of X .A mapping F:Trightarrow 2^overlineX is mathrmcmathrma¹mathrmi1mathrmemathrmd measurable (weakly meas- urable) if for any closed (open) subset B of X , F⁻¹(B)=tin T:F(t)cap Bneq 0 is meas- urable.If $F(t)$ is compact for all $tin T$ then F is measurable if and only if F is weakly measurable (cf.[4]).K. Kuratowski ([7], p. 413, G. Darbo [3]) defined the measure of noncompact- ness, $alpha(A)$ , of a bounded subset A of a metric space X : $alpha(A)=inf$ { $epsilon>0$ : A can be covered by a finite number of sets of diameter less than or equal to $epsilon$ }.We shall need the following results.Lemma 1 ([8], p. 100).Let $(X, d)$ be a compact metric space and (7, d₁) a metric space.Let C(X, mathrmY)=U denote the continuous functions from X to Y with the metric rho:rho(f, g)=supd₁[f(x), g(x)]:x in X.Let $Hsubset U$ be an equicontinuous family of functions and let H(x)=h(x):hin H , x $in X$ .Then we $f'/ave$ $alpha(H)=sup$ { alpha₁[H(x)] : x in X , where $alpha$ , alpha₁ is the measure of noncompactness in U and Y respectively.Lemma 2 ([8], p. 10).Let (Xᵢ, dᵢ) , $i=1,2$ , be metric spaces and f:X₁rightarrow X₂ fulfils:d₂[f(x₁),f(x₂)]leq kd₁(x₁, x₂) , x₁ , x₂in X₁ , k -constant,