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Global existence results for functional evolution equations with delay and random effects.

1. INTRODUCTION

Functional evolution equations play a very important role in describing many phenomena of physics, mechanics, biology, etc. For more details on this theory and on its applications we refer to the monographs of Hale and Verduyn Lunel [18], Kolmanovskii and Myshkis [21], and Wu [31], and the references therein. Recently, many authors have studied the existence of various models of semilinear evolution equations with finite and infinite delay in Frechet space; for instance, we refer to the book by Abbas and Benchohra [1] and to the papers by Baghli and Benchohra [4, 5, 6]. On the other hand, different fields of engineering problems which are of current interest in unbounded domains have also received the attention of researchers; see [2, 26, 27].

The nature of a dynamic system in engineering or natural sciences depends on the accuracy of the information obtained concerning the parameters that describe that system. If the knowledge about a dynamic system is precise, then a deterministic dynamical system arises. Yet, in most cases, the available data for the description and evaluation of parameters of a dynamic system are inaccurate, imprecise or confusing. In other words, evaluation of parameters of a dynamical system is not without uncertainties. When knowledge about the parameters of a dynamic system are of a statistical nature, that is, the information is probabilistic, the common approach in mathematical modeling of such systems is the use of random differential equations or stochastic differential equations. Random differential equations, as natural extensions of deterministic ones, arise in many applications and have been investigated by many mathematicians. We refer the reader to the monographs [7, 30], the papers [10, 9, 11, 29] and the references therein. We also refer the reader to recent results in [23, 24, 25]. There are real world phenomena with anomalous dynamics such as signals transmissions through strong magnetic fields, atmospheric diffusion of pollution, network traffic, the effect of speculations on the profitability of stocks in financial markets, and so on, where the classical models are not sufficiently good to describe these features.

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:

(1.1) y'(t ,w) = A(t)y(t, w) + f(t, [y.sub.t] (*, w), w), a.e. t [member of] J := [0, [infinity]),

(1.2) y(t, w) = [phi](t,w), t [member of] (-[infinity], 0], w [member of] [OMEGA],

where ([OMEGA], F, P) is a complete probability space, f : J x B x [OMEGA] [right arrow] E, [phi] [member of] B x [OMEGA] are given random functions which represent random nonlinearity of the system, [{A(t)}.sub.0[less than or equal to]t<+[infinity]] is a family of linear closed (not necessarily bounded) operators from E into E that generates an evolution system of operators [{U(t, s)}.sub.(t,s)[member of]JxJ] for 0 [less than or equal to] s [less than or equal to] t < +[infinity], B is the phase space to be specified later, and (E, [absolute value of (*)]) is a real Banach space. For any function y defined on (-[infinity], +[infinity]) x [OMEGA] and any t [member of] J we denote by [y.sup.t] (*,w) the element of B x [OMEGA] defined by [y.sub.t]([theta], w) = y(t + [theta], w), [theta] [member of] (-[infinity], 0]. Here yt(*, w) represents the history of the state from time -[infinity], up to the present time t. We assume that the histories yt(*, w) belong to some abstract phases B, to be specified later.

To our knowledge, the literature on the global existence of random evolution equations with delay is very limited, so the present paper can be considered as a contribution to such a class of equations.

2. PRELIMINARIES

In this section we present briefly some notations, definitions, and theorems which are used throughout this work.

In this paper, we will employ an axiomatic definition of the phase space B introduced by Hale and Kato in [17] and follow the terminology used in [19]. Thus, (B, [[parallel]*[parallel].sub.B]) will be a seminormed linear space of functions mapping (-[infinity], 0] into E, and satisfying the following axioms :

([A.sub.1]) If y : (-[infinity], T) [right arrow] E, T > 0, is continuous on J and [y.sub.0] [member of] B, then for every t [member of] J the following conditions hold:

(i) [y.sub.t] [member of] B;

(ii) There exists a positive constant H such that [absolute value of (y(t))] [less than or equal to] H [[parallel][y.sub.t][parallel].sup.B];

(iii) There exist two functions L(*), M(*) : [R.sub.+] [right arrow] [R.sub.+] independent of y with L continuous and bounded, and M locally bounded such that:

[[parallel][y.sub.t][parallel].sub.B] [less than or equal to] L (t) sup {[absolute value of (y(s))] : 0 [less than or equal to] s [less than or equal to] t} + M (t) [[parallel][y.sub.0][parallel].sub.B].

([A.sub.2]) For the function y in ([A.sub.1]), [y.sub.t] is a B--valued continuous function on J.

([A.sub.3]) The space B is complete.

Denote

[K.sub.T] = sup{L(t) : t [member of] J},

and

[M.sub.T] = sup{M(t) : t [member of] J}.

Remark 2.1. 1. (ii) is equivalent to [absolute value of ([phi](0))] [less than or equal to] H[[parallel][phi][parallel].sub.B] for every [phi] [member of] B.

2. Since [[parallel]*[parallel].sub.B] is a seminorm, two elements [phi], [psi] [member of] B can satisfy [[parallel][phi] - [psi][parallel].sub.B] = 0 without necessarily [phi]([theta]) = [psi]([theta]) for all [theta] [less than or equal to] 0.

3. From the equivalence in part 1 of this remark, we can see that for all [phi], [psi] [member of] B such that [[parallel][phi] - [psi][parallel].sub.B] = 0, we necessarily have that [phi](0) = [psi](0).

By BUC we denote the space of bounded uniformly continuous functions defined from (-[infinity], 0] into E. Finally, by BC := BC([0, +[infinity]) we denote the Banach space of bounded and continuous functions from [0, [infinity]) into E, equipped with the standard norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 2.2. A map f : J x B x [OMEGA] [right arrow] E is said to be Caratheodory if

(i) t [right arrow] f (t, y, w) is measurable for all y [member of] B and for all w [member of] [OMEGA].

(ii) y [right arrow] f (t, y, w) is continuous for almost each t [member of] J and for all w [member of] [OMEGA].

(iii) w [right arrow] f (t, y, w) is measurable for all y [member of] B, and almost each t [member of] J.

In what follows, we assume that {A(t), t [greater than or equal to] 0} is a family of closed densely defined linear unbounded operators on the Banach space E and with domain D(A(t)) independent of t.

Definition 2.3. A family of bounded linear operators

[{U(t, s)}.sub.(t,s)[member of][DELTA]] : U(t, s) : E [right arrow] E, (t, s) [member of] [DELTA] := {(t, s) [member of] J x J : 0 [less than or equal to] s [less than or equal to] t < +[infinity]}

is called an evolution system if the following properties are satisfied:

1. U(t, t) = I where I is the identity operator in E,

2. U(t, s) U(s, [tau]) = U(t, [tau]) for 0 [less than or equal to] [tau] [less than or equal to] s [less than or equal to] t < +[infinity],

3. U(t, s) [member of] B(E) the space of bounded linear operators on E, where for every (s, t) [member of] [DELTA] and for each y [member of] E, the mapping (t, s) [right arrow] U(t, s) y is continuous.

More details on evolution systems and their properties could be found on the books of Ahmed [3], Engel and Nagel [13] and Pazy [28].

Lemma 2.4 (Corduneanu [8]). Let C [subset] BC(J, E) be a set satisfying the following conditions:

(i): C is bounded in BC(J, E);

(ii): the functions belonging to C are equicontinuous on any compact interval of J;

(iii): the set C(t) := {y(t) : y [member of] C} is relatively compact on any compact interval of J;

(iv): the functions from C are equiconvergent, i.e., (given [epsilon] > 0, there corresponds T([epsilon]) > 0 such that [absolute value of (y(t) - y(+[infinity]))] < [epsilon] for any t [greater than or equal to] T([epsilon]) and y [member of] C.

Then C is relatively compact in BC (J, E).

Theorem 2.5 (Schauder fixed point [16]). Let B be a closed, convex and nonempty subset of a Banach space E. Let N : B [right arrow] B be a continuous mapping such that N(B) is a relatively compact subset of E. Then N has at least one fixed point in B.

Let Y be a separable Banach space with the Borel [sigma]-algebra [B.sub.Y]. A mapping y : [OMEGA] [right arrow] Y is said to be a random variable with values in Y if for each B [member of] [B.sub.Y], [y.sup.-1] (B) [member of] F. A mapping T : [OMEGA] x Y [right arrow] Y is called a random operator if T(*, y) is measurable for each y [member of] Y and is generally expressed as T(w, y) = T(w)y; we will use these two expressions alternatively.

Let y be a mapping of J x [OMEGA] into X. y is said to be a stochastic process if for each t [member of] J the function y(t, *) is measurable.

Next, we will give a very useful random fixed point theorem with stochastic domain.

Definition 2.6 ([12]). Let C be a mapping from [OMEGA] into [2.sup.Y]. A mapping T : {(w, y) : w [member of] [OMEGA] [conjunction] y [member of] C(w)} [right arrow] Y is called a 'random operator with stochastic domain C' if and only if C is measurable (i.e., for all closed A [subset or equal to] Y, {w [member of] [omega] : C(w) [intersection] A [not equal to] 0} [member of] F) and for all open D [subset or equal to] Y and all y [member of] Y, {w [member of] [OMEGA] : y [member of] C(w) [conjunction] T(w,y) [member of] D} [member of] F. The mapping T will be called 'continuous' if every T(w) is continuous. For a random operator T, a mapping y : [OMEGA] [right arrow] Y is called a 'random (stochastic) fixed point of T' if and only if for p-almost all w [member of] [OMEGA], y(w) [member of] C(w) and T(w)y(w) = y(w) and for all open D [subset or equal to] Y, {w [member of] [OMEGA] : y(w) [member of] D} [member of] F ('y is measurable').

Remark 2.7. If C(w) [equivalent to] Y, then the definition of random operator with stochastic domain coincides with the definition of random operator.

Lemma 2.8 ([12]). Let C : [OMEGA] [right arrow] [2.sup.Y] be measurable with C(w) closed, convex and solid (i.e., int C(w) [not equal to] 0) for all w [member of] [OMEGA]. We assume that there exists a measurable function [y.sub.0] : [OMEGA] [right arrow] Y with [y.sub.0] [member of] int C(w) for all w [member of] [OMEGA]. Let T be a continuous random operator with stochastic domain C such that for every w [member of] [OMEGA], {y [member of] C(w) : T(w)y = y} [not equal to] 0. Then T has a stochastic fixed point.

3. EXISTENCE OF MILD SOLUTIONS

Now we give our main existence result for problem (1.1)-(1.2). Before stating and proving this result, we give the definition of a mild random solution.

Definition 3.1. A stochastic process y : J x [OMEGA] [right arrow] E is said to be a random mild solution of problem (1.1)-(1.2) if y(t,w) = [phi](t, w), t [member of] (- [infinity], 0] and the restriction of y(*, w) to the interval [0, [infinity]) is continuous and satisfies the following integral equation:

(3.1) y(t, w) = U(t, 0) [phi](0, w) + [[integral].sup.t.sub.0] U(t, s)f (s, ys(*, w), w)ds, t [member of] J.

We will need to introduce the following hypotheses which are be assumed hereafter

([H.sub.1]) There exist a constant M [greater than or equal to] 1 and [alpha] > 0 such that

[[parallel]U(t, s)[parallel].sub.B(E)] [less than or equal to] [Me.sup.- [alpha](t-s)] for every (s,t) [member of] [DELTA].

([H.sub.2]) The function f : [R.sup.+] x B x [OMEGA] [right arrow] E is Caratheodory.

([H.sub.3]) There exist functions [psi] : J x [OMEGA] [right arrow] [R.sup.+] and p : J x [OMEGA] [right arrow] [R.sup.+] such that for each w [member of] [OMEGA], [psi] (*, w) is a continuous nondecreasing function and p(*, w) is integrable with:

[absolute value of (f (t, u, w))] [less than or equal to] p(t, w) [psi] ([[parallel]u[parallel].sub.B], w) for a.e. t [member of] J and each u [member of] B.

([H.sub.4]) For each w [member of] [OMEGA], [phi](*, w) is continuous and for each t, [phi](t, *) is measurable.

([H.sub.5]) For each (t, s) [member of] [DELTA] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.2. Suppose that hypotheses ([H.sub.1])-([H.sub.5]) are valid, then the problem (1.1) (1.2) has at least one mild random solution on (-[infinity], [infinity]).

Proof. Let Y be the space defined by

Y = {y : R [right arrow] E such that y[|.sub.J] [member of] BC(J, E) and [y.sub.0] [member of] B},

(where we denote by y[|.sub.J] the restriction of y to J), endowed with the uniform convergence topology, and N : [OMEGA] x Y [right arrow] Y be the random operator defined by

(3.2) (N(w)y)(t) = U(t, 0) [phi](0, w) + [[integral].sup.t.sub.0] U(t, s) f (s, [y.sub.s], w) ds, t [member of] J.

Then we show that the mapping defined by (4) is a random operator. To do this, we need to prove that for any y [member of] Y, N(*)(y) : [OMEGA] [right arrow] Y is a random variable. First, we prove that N(*)(y) : [OMEGA] [right arrow] Y is measurable since the mapping f (t, y, *), t [member of] J, y [member of] Y, is measurable by assumption ([H.sub.2]) and ([H.sub.4]).

Let R(w) be any measurable positive function and consider the set-valued map D : [OMEGA] [right arrow] [2.sup.y] defined by

D(w) = {y [member of] Y : [parallel]y[parallel] [less than or equal to] R(w)}.

D(w) is bounded, closed, convex and solid for all w [member of] [OMEGA]. Then D is measurable by Lemma 17 (see [20]).

Next, let w [member of] [OMEGA] be fixed. Then for any y [member of] D(w), and by assumption (A1), we get

[parallel][y.sub.s][parallel] [less than or equal to] L(s) [absolute value of (y(s))] + M (s) [[parallel][y.sub.0][parallel].sub.B] [less than or equal to] [K.sub.T] [absolute value of (y(s))] + [M.sub.T] [[parallel][phi][parallel].sub.b],

and by ([H.sub.3]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and define the set-valued map

G(w) = {r [greater than or equal to] 0 : [C.sub.1] + [C.sub.2] [psi]([C.sub.3] + [C.sub.4]r, w) [less than or equal to] r}.

Under a suitable choice of the constantes [C.sub.2] and [C.sub.4] we can easily show that the inequality

[C.sub.1] + [C.sub.2] [psi]([C.sub.3] + [C.sub.4]r,w) [less than or equal to] r,

has at least one solution, and hence the set-valued map G is nonempty valued. The continuity of [psi] implies that G has closed values. Notice that

G([omega]) = D(w) [omicron] h(r, w),

where h is the function defined by

h(r, w) = [C.sub.1] + [C.sub.2] [psi]([C.sub.3] + [C.sub.4]r, w).

Since D and h are measurable, the set-valued map G is measurable. The celebrated Kuratowski-Ryll-Nardzewski selection theorem ([15], Theorem 19.7) implies that the set-valued map G has a measurable selection. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This implies that N is a random operator with stochastic domain D and F(w) : D(w) [right arrow] D(w) for each w [member of] [OMEGA].

Step 1: N is continuous.

Let [y.sup.n] be a sequence such that [y.sup.n] [right arrow] y in Y. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since f (s, *, w) is continuous, we have by the Lebesgue dominated convergence theorem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus N is continuous.

Step 2: We shall prove that for every w [member of] [OMEGA], {y [member of] D(w) : N(w)y = y} [not equal to] 0. For this, we use Schauder's theorem. First, we will show that N(D(w)) is relatively compact using Corduneanu's lemma.

(a) First, it is clear that the assumption (i) holds. Then we will demonstrate that N(D(w)) is an equicontinuous set for each closed bounded interval [0,T] in J. Let [[tau].sub.1], [[tau].sub.2] [member of] [0,T] with [[tau].sub.2] > [[tau].sub.1], D(w) be a bounded set as in Step 2, and y [member of] D(w). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The right-hand of the above inequality tends to zero as [[tau].sub.2] - [[tau].sub.1] [right arrow] 0, as N is bounded and equicontinuous.

(b) Now we will prove that Z(t,w) = {(N(w)y)(t) : y [member of] D(w)} is precompact in E. Let t [member of] [0, T] be fixed and let e be a real number satisfying 0 < [epsilon] < t. For y [member of] D(w) we define

([N.sub.[epsilon]] (w)y) (t) = U(t, 0)[phi](0, w) + U(t, t - [epsilon]) [[integral].sup.t-[epsilon].sub.0] U(t - [epsilon], s)f (s, [y.sub.s], w) ds.

Since U(t, s) is a compact operator and the set [Z.sub.[epsilon]] (t,w) = {([N.sub.[epsilon]](w)y)(t) : y [member of] D(w)} is the image of a bounded subset of E, then [Z.sub.[epsilon]] (t,w) is precompact in E for every [epsilon], 0 < [epsilon] < t. Moreover

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore the set Z(t,w) = {(N(w)y)(t) : y [member of] D(w)} is precompact in E.

(c) Finally, it remains to show that N is equiconvergent.

Let y [member of] D(w). Then from ([H.sub.1]) and ([H.sub.3]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows immediately by ([H.sub.5]) that [absolute value of (N(w)y)(t)] [right arrow] 0 as t [right arrow] +[infinity]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which implies that N is equiconvergent.

As a consequence of Steps 1-2 and (a), (b), (c), we can conclude that N(w) : D(w) [right arrow] D(w) is continuous and compact. From Schauder's theorem, we deduce that N(w) has a fixed point y(w) in D(w). Since [[intersection].sub.w[member of][OMEGA]] D(w) [not equal to] 0, the hypothesis that a measurable selection of int D exists holds. By Lemma 2.8, the random operator N has a stochastic fixed point [y.sup.*](w), which is a random mild solution of the random problem (1.1)-(1.2).

4. AN EXAMPLE

Consider the following functional partial differential equation:

(4.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(4.2) z(t, 0, w) = z(t, [pi], w) = 0, t [member of] [0, +[infinity]), w [member of] [OMEGA],

(4.3) z(s, x, w) = [z.sub.0](s, x, w), s [member of] (-[infinity], 0], x [member of] [0, [pi]], w [member of] [OMEGA],

where a(t, [xi]) is a continuous function which is uniformly Holder continuous in t, K and [C.sub.0] are a real-valued random variable.

Let E = [L.sup.2] [0, [pi]] and ([OMEGA], F, P) be a complete probability space, and define A(t) by

A(t)v = a(t, [xi])v"

with domain

D(A) = {v [member of] E, v, v' are absolutely continuous, v" [member of] E, v(0) = v([pi]) = 0}.

Then A(t) generates an evolution system U(t, s) satisfying assumption ([H.sub.1]) (see [14, 22]).

Let B = BUC([R.sup.-]; E) be the space of bounded uniformly continuous functions endowed with the norm,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we put [phi] [member of] BCU([R.sup.-]; E), x [member of] [0,[pi]] and w [member of] [OMEGA],

y(t, x, w) = z(t, x, w), t [member of] [0,T], [phi](s, x, w) = [z.sub.0](s, x, w), s [member of] (-[infinity], 0], x [member of] [0,[pi]], w [member of] [OMEGA].

Set

f (t, [phi](x),w) = [[integral].sup.0.sub.-[infinity]] [e.sup.-t] [phi](s, x, w)ds,

with

[phi](s, x, w) = exp(z(t + s, x, w)).

The function f (t, [phi](x),w) is Caratheodory, and satisfies ([H.sub.2]) with

p(t,w) = K(w) [pi]/2 [e.sup.-t] and [psi](x, w) = [absolute value of ([C.sub.0](w))] [e.sup.x].

Then the problem (1.1)-(1.2) is an abstract formulation of the problem (4.1)- (4.3), and conditions ([H.sub.1])-([H.sub.5]) are satisfied. Theorem 3.2 implies that the random problem (4.1)-(4.3) has at least one random mild solution.

Received January 19, 2016

ACKNOWLEDGEMENT. The authors are grateful to the Editor and the Referee for their helpful remarks.

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EMAN ALAIDAROUS (a), AMEL BENAISSA (b), MOUFFAK BENCHOHRA (b,a), AND JOHNNY HENDERSON (c)

(a) Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia ealaidarous@kau.edu.sa

(b) Laboratory of Mathematics, University of Sidi Bel-Abbes PO Box 89, Sidi Bel-Abbes 22000, Algeria aissani_k@yahoo.fr; benchohra@univ-sba.dz

(c) Department of Mathematics, Baylor University Waco, Texas 76798-7328 USA Johnny_Henderson@baylor.edu
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