Random semilinear system of differential equations with state-dependent delay

In this paper we prove the existence of mild solutions for a first-order semilinear differential with statedependent delay. The existence results are established by means of a new version of Perov’s fixed point principles combined with a technique based on vector-valued matrix and convergent to zero matrix.


Introduction
Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) that include a stochastic process in their vector field.They seem to have had a shadow existence to stochastic differential equations (SODEs), but have been around for as long as if not longer and have many important applications.In particular, RODEs play a fundamental role in the theory of random dynamical systems, it is more realistic to consider such equations as random operator equations.Therefore, it is more realistic to consider such equations as random operator equations which are much more difficult to handle mathematically than deterministic equations.Important contributions to the study of the mathematical aspects of such random equations have been undertaken in [9,7,1,16] among others.Since sometimes we can get the random distributions of some main disturbances by historical experiences and data rather than take all random disturbances into account and assume the noise to be white noises.In a separable metric space, random fixed point theorems for contraction mappings were proved by Hans [2,3], Spacek [8], Hans and, Spacek [4] and Mukherjee [5,6].In this work we prove the existence of mild solutions of the following functional differential equation with delay and random effects (random parameters) of the form: + f 1 (t, x ρ 1 (t,xt) (•, ω), y ρ 1 (t,yt) (•, ω), ω), a.e, t ∈ J := [0, a] y (t, ω) = A 2 (w)y(t, ω) Here, x(•), y(•) takes the value in the separable Hilbert space X with inner product •, • induced by the norm • , A i : Ω × X −→ X, i = 1, 2 are random operators and (Ω, F, P) is a complete probability space, w ∈ Ω, J := [0, a] for fixed a > 0 and X is a real separable Hilbert space with inner product •, • induced by norm • , φ 1 , φ 2 are two random maps and B is a phase space to be specified later.For any function x defined on (−∞, a] × Ω and any t ∈ J we denote by x t (., w) the element of B × Ω defined by x t (θ, w) = x(t + θ, w), θ ∈ (−∞, 0].Here x t (., w) represents the history of the state from time −∞, up to the present time t.We assume that the histories x t (., w) belong to the abstract phase B. To our knowledge, the literature on the local existence of random evolution equations with delay is very limited, so the present paper can be considered as a contribution to this question.We refer the reader to [11,17] for the properties of the first order abstract Cauchy problem and the semigroup theory.
The paper is organized as follows.In Section Íš,we introduce all the background material needed such as generalized metric spaces, some random fixed point theorems .In Section Íş, by some new random versions of Perov's fixed point theorems in a vector Banach space.

Preliminaries
In this section, we introduce some notations, recall some definitions, and preliminary facts which are used throughout this paper.Actually we will borrow it from [20,10].Although we could simply refer to this paper whenever we need it, we prefer to include this summary in order to make our paper as much self-contained as possible.Let (Ω, F) be a measurable space.We equip the metric space X with a σ-algebra B(X) of Borel subsets of X so that (X, B(X)) becomes a measurable space.A mapping z : Ω → X is called a random variable if ) is measurable for all z ∈ X.We also denote a random operator A on X by (i) The map (t, w) −→ g(t, z, w) is jointly measurable for all z ∈ X, (ii) The map z −→ g(t, z, w) is continuous for all t ∈ [0, b] and w ∈ Ω.
In this paper, we will employ an axiomatic definition of the phase space B introduced by Hale and Kato in [12] and follow the terminology used in [13].Thus, (B, .B ) will be a semi norm linear space of functions mapping (−∞, 0] into X, and satisfying the following axioms : the following conditions hold: where H ≥ 0 is a constant, R + → R + , K is continuous and M is locally bounded and H, K and M are independent of x.
• A 2 For the function for all θ ≤ 0. 3. From the equivalence of in the first remark, we can see that for all φ, ψ ∈ B such that φ − ψ B = 0 .
We denote by S 0 (t) the restriction of S(t) to B 0 .
• (FMS) The space B is said to be a fading memory space if it verifies axiom (C2) and S 0 φ(0) → 0 as t → ∞ for all φ ∈ B 0 .
• (UFMS) The space B is said to be a uniformly fading memory space if it verifies (C2) and S 0 (t) B → 0 as t → ∞.
We now indicate some examples of phase spaces.For other details we refer, for instance to the book by Hinoet al. [13].
Example 2.7.Let: C b the space of bounded continuous functions defined from (−∞, 0] to X, C bu the space of bounded uniformly continuous functions defined from (−∞, 0] to X, We have that the spaces C bu , C ∞ and C 0 satisfy conditions (A 1 ) − (A 3 ).However, C b satisfies (A 1 ), (A 3 ) but (A 2 ) is not satisfied.
The norm in B is defined by To simplify some estimate, in this text we always assume that g is decreasing and g(0) = 1.
Then in the space C γ the axioms (A 1 ) − (A 3 ) are satisfied.

Vector metric space and Random variable
If, x, y ∈ R n , x = (x 1 , . . ., x n ), y = (y 1 , . . ., y n ), by x ≤ y we mean x i ≤ y i for all i = 1, . . ., n.Also Definition 3.1.Let X be a nonempty set.By a vector-valued metric on X we mean a map d : X × X → R n + with the following properties: We call the pair (X, d) a generalized metric space with d(x, y) Notice that d is a generalized metric space on X if and only if d i , i = 1, . . ., n are metrics on X.
For r = (r 1 , . . ., r n ) ∈ R n + , we will denote by the open ball centered in x 0 with radius r and the closed ball centered in x 0 with radius r.We mention that for generalized metric space, the notation of open subset, closed set, convergence, Cauchy sequence and completeness are similar to those in usual metric spaces.
Definition 3.2.A square matrix of real numbers is said to be convergent to zero if and only if its spectral radius ρ(M ) is strictly less than 1.In other words, this means that all the eigenvalues of M are in the open unit disc i.e. |λ| < 1, for every λ ∈ C with det(M − λI) = 0, where I denote the unit matrix of M n×n (R).
Theorem 3.3.[18] Let M ∈ M n×n (R + ).The following assertions are equivalent: • M is convergent towards zero; • M k → 0 as k → ∞; • The matrix (I − M ) is nonsingular and Some examples of matrices convergent to zero are the following: Definition 3.4.Let (X, d) be a generalized metric space.An operator N : X → X is said to be contractive if there exists a convergent to zero matrix M such that d(N (x), N (y)) ≤ M d(x, y) for all x, y ∈ X.
For n = 1 we recover the classical Banach's contraction fixed point result.
We shall use a random version of Perov type of random differential equations of first order for different aspects of the solutions under suitable conditions Theorem 3.5.[19] Let (Ω, F) be a measurable space, X be a real separable generalized Banach space and F : Ω × X → X be a continuous random operator, and let M (w) ∈ M n×n (R + ) be a random variable matrix such that, for every w ∈ Ω , the matrix M (w) converges to 0 and Then there exists any random variable x : Ω → X which is the unique random fixed point of F .
Lemma 3.6.[19] Let X be a separable generalized metric space and F : Ω × X → X be a mapping such that F (., x) is measurable for all x ∈ X and F (w, .) is continuous for all w ∈ Ω.Then the map (w, x) → F (w, x) is jointly measurable.
Proposition 3.7.[15] Let X be a separable Banach space, and D be a dense linear subspace of X.Let L : Ω × D → X be a closed linear random operator such that, for each w ∈ Ω, L(w) is one to one and onto.
Then the operator R : Ω × X → X defined by R(w)x = L −1 (w)x is random.

Main Results
Now we give our main existence result for problem (1.1).Before starting and proving this result, we give the definition of the mild random solution.
We will need to introduce the following hypotheses which are assumed there after: There exist random variables M 1 , M 2 : Ω → (0, +∞) such that.
It is clear that the radius spectral ρ(M trice ) < 1.By Lemma 3.3, M trix (w) converges to zero.From Theorem 3.5 there exists a unique random solution of problem (1.1).We denote by (x(t, w), y(t, w)) the mild solution of (1.1).