EXISTENCE OF SOLUTIONS FOR SECOND ORDER STOCHASTIC DIFFERENTIAL INCLUSIONS

EXISTENCE OF SOLUTIONS FOR SECOND ORDER STOCHASTIC DIFFERENTIAL INCLUSIONS

IN HILBERT SPACES

P. Balasubramaniam

Department of Mathematics

Gandhigram Rural Institute-Deemed University Gandhigram – 624 302, Tamil Nadu, India

e-mail: pbalgri@rediffmail.com and

S.K. Ntouyas Department of Mathematics

University of Ioannina, 451 10 Ioannina, Greece e-mail: sntouyas@cc.uoi.gr

Abstract

In this paper, sufficient conditions are given for the existence of solutions for a class of second order stochastic differential inclusions in Hilbert space with the help of Leray-Schauder Nonlinear Alternative.

Keywords and phrases: existence, multivalued map, stochastic dif- ferential inclusions, fixed point, Hilbert space.

2000 Mathematics Subject Classification: 34F05, 60G12.

∗

The work was supported by University Grants Commission (UGC), New Delhi under

SAP(DRS) sanctioned No. F.510/6/DRS/2004(SAP-1).

1. Introduction

The theory for differential and integral inclusions in deterministic cases may be found in several papers and monographs (see for example [2, 10, 13, 23]).

Random differential and integral inclusions play an important role in characterizing many social, physical, biological and engineering problems.

Stochastic differential inclusions are important from the viewpoint of appli- cations, since they incorporate (natural) randomness into the mathemati- cal description of the phenomena, and, therefore, provide a more accurate description of them.

In many cases it is advantageous to treat the second order abstract differ- ential equations directly rather than to convert them to first order systems.

For example, Fitzgibbon [9] used the second order abstract differential equa- tions for establishing the boundedness of solutions of the equation governing the transverse motion of an extensible beam. Second order equations have been examined in [6, 29]. The deterministic version of second order systems have been thoroughly investigated by several authors (see [7, 21, 22, 28, 29]) while the stochastic version has not yet been treated satisfactorily. In fact, abstract second order stochastic evolution equations and inclusions have only recently been studied in [5, 17, 18] and [19].

Ahmed [1] obtained the existence of solutions of nonlinear stochastic differential inclusions by using the semigroup approach and Banach fixed point theorem. Balasubramaniam [3] studied the existence of solutions of the following functional differential inclusions via integral inclusions

(1) dx(t) ∈ f (t, x

)dt + G(t, x

)dw(t), a.e. t ∈ J, x(t) = φ(t), t ∈ [−r, 0]

using Kakutani’s fixed point theorem in which G is the set valued map,

{B(t)}

is a Brownian motion or Wiener process and φ(t) is a suitable

initial random variable independent of ω(t). Kree [15] showed the existence

of solution of (1) for G(t, x(t)) using a fixed point argument. Recently, the

global existence of solutions for nonlinear stochastic evolution inclusions has

been studied in [4] by using the fixed point approach. The results presented

in the paper constitute a continuation and generalization of the existence,

uniqueness results from [1, 4, 14, 15, 25, 26, 27] to the second order semilinear

stochastic evolution inclusions in Hilbert spaces settings.

In this paper, we are interested in the existence results of the following nonlinear second order stochastic differential inclusions

(2)

dx

(t) ∈ [Ax(t) + f (t, x(t))]dt + G(t, x(t))dw(t) a.e. t ∈ J = [0, b], x(0) = x

, x

(0) = x

where A is the infinitesimal generator of a strongly continuous cosine family C(t), t ∈ J, on a separable Hilbert space H with the inner product (·, ·) and norm k · k. Let K be another separable Hilbert space with the inner product (·, ·)

and norm k · k

. Suppose {w(t)}

is a given K-valued Brownian motion or Wiener process with a finite trace nuclear covariance operator Q ≥ 0. We are also employing the same notation k · k for the norm BL(K, H), where BL(K, H) denotes the space of all bounded linear operators from K into H. Further, f : J × H → H is a measurable mapping in H-norm and G : J × H → P(L

(K, H)) (P(L

(K, H)) is the family of all nonempty subsets of L

(K, H)), a multivalued measurable mapping in L

(K, H)-norm. Here L

(K, H) denotes the space of all Q-Hilbert-Schmidt operators from K into H which is going to be defined below.

The paper is organized as follows. In Section 2, we recall some necessary preliminaries and in Section 3, we prove our main result. The second order stochastic integrodifferential inclusions are considered in Section 4.

2. Preliminaries

Let (Ω, F, P ) be a complete probability space furnished with a complete family of right continuous increasing sub σ-algebras {F

, t ∈ J} satisfying F

⊂ F. An H-valued random variable is an F-measurable function x(t) : Ω → H and a collection of random variables S = {x(t, w) : Ω → H|t ∈ J} is called a stochastic process. Usually, we suppress the dependence on w ∈ Ω and write x(t) instead of x(t, w) and x(t) : J → H in the place of S. Let β

(t) (n = 1, 2, . . .) be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over (Ω, F, P ). For more details of this section the reader may refer [8]. Set

w(t) = X

p λ

β

(t)ζ

, t ≥ 0,

where λ

≥ 0, (n = 1, 2, . . .) are nonnegative real numbers and {ζ

} (n = 1, 2, . . .) is a complete orthonormal basis in K. Let Q ∈ L(K, K) be an operator defined by Qζ

= λ

ζ

with finite T r Q = P

n=1

λ

< ∞, (Tr denotes the trace of the operator). Then the above K-valued stochastic process w(t) is called a Q-Wiener process. We assume that F

= σ(w(s) : 0 ≤ s ≤ t) is the σ-algebra generated by w and F

= F. Let ϕ ∈ L(K, H) and define

kϕk

= T r(ϕQϕ

) = X

k p

λ

ϕζ

k

.

If kϕk

< ∞, then ϕ is called a Q-Hilbert-Schmidt operator. Let L

(K, H) denote the space of all Q-Hilbert-Schmidt operators ϕ : K → H. The completion L

(F, H) of L(K, H) with respect to the topology induced by the norm k · k

where kϕk

= hhϕ, ϕii is a Hilbert space with the above norm topology.

The collection of all strongly-measurable, square-integrable H-valued random variables, denoted by L

(Ω, F, P ; H) ≡ L

(Ω; H), is a Banach space equipped with the norm kx(·)k

= Ekx(·; w)k

, where the expectation E is defined by E(h) = R

Ω

h(w)dP . Let Z = C(J, L

(Ω; H)) be the Banach space of all continuous maps from J into L

(Ω; H) satisfying the condition sup

Ekx(t)k

< ∞ and let k · k

be a norm in Z defined by

kxk

=  sup

t∈J

kx(t)k



.

It is easy to verify that Z, furnished with the norm topology as defined above, is a Banach space.

In a Hilbert space H, a multivalued map M : H → P(H) is convex (closed) valued, if M (x) is convex (closed) for all x ∈ H. M is bounded on bounded sets if M (V ) = S

x∈V

M (x) is bounded in H, for any bounded set V of H (i.e., sup



sup{kyk : y ∈ M (x)}

< ∞).

M is called upper semicontinuous (u.s.c.) on H, if for each x

∈ H, the set M (x

) is a nonempty, closed subset of H, and if for each open set V of H containing M (x

), there exists an open neighborhood N of x

such that M (N ) ⊆ V .

M is said to be completely continuous if M (V ) is relatively compact, for every bounded subset V ⊆ H.

If the multivalued map M is completely continuous with nonempty com-

pact values, then M is u.s.c. if and only if M has a closed graph (i.e.,

x

→ x

, y

→ y

, y

∈ M x

imply y

∈ M x

).

M has a fixed point if there is x ∈ H such that x ∈ M x.

In the following, P

(H) denotes the set of all nonempty bounded, closed and convex subsets of H.

A multivalued map M : J → P

(H) is said to be measurable if for each x ∈ H the mean-square distance between x and M (t) is a measurable function on J. For more details on multivalued maps see ([10, 12]).

For each x ∈ L

(L

(K, H)) define the set of selections of G by g ∈ N

= {g ∈ L

(L

(K, H)) : g(t) ∈ G(t, x(t)) for a.e. t ∈ J} . The following basic result concerning the strongly continuous cosine families is needed (cf. [28, 29]) to prove our main results.

Definition 2.1. (i) A one parameter family {C(t), t ∈ J} of bounded linear operators in the Hilbert space H is called a strongly continuous cosine family if and only if

(a) C(s + t) + C(s − t) = 2C(s)C(t) for all s, t ∈ J;

(b) C(0) = I;

(c) C(t)x is continuous in t on J for each fixed x ∈ H.

(ii) The corresponding strongly continuous sine family {S(t), t ∈ J} of bounded linear operators in the Hilbert space H is defined by

S(t)x = Z

0

C(s)xds, for all x ∈ H, for all t ∈ J.

(iii) The infinitesimal generator of a strongly continuous cosine family {C(t), t ∈ J}, is the operator A : H → H defined by

Ax = d

dt

C(t)x

  t = 0, x ∈ D(A), where

D(A) = {x ∈ H : C(t)x is twice continuously differentiable in t}.

Lemma 2.2. Let A generate a strongly continuous cosine family C(t), t ∈ J, of bounded linear operators. Then, the following hold:

(i) There exist constants M ≥ 1 and ω ≥ 0 such that kC(t)k ≤ M e

and

hence kS(t)k ≤ M e

,

(ii) A R

t

S(τ )xdτ = [C(t

) − C(t)]x, for all 0 ≤ t ≤ t

< ∞, (iii) There exists N ≥ 1 such that kS(t) − S(t

)k ≤ N    R

t

e

ds    for 0 ≤ t ≤ t

< ∞.

The Uniform Boundedness Principle, together with (i) above, imply that both {C(t), t ∈ J} and {S(t), t ∈ J} are uniformly bounded by M = M e

. In addition to the familiar Young, H¨ older and Minkowskii inequalities, the inequality of the form P

i=1

a

≤ n

P

i=1

a

, follows from the convexity of x

, m ≥ 1, and is helpful to establish various estimates, where a

are nonnegative constants (i = 1, 2, . . . , n) and n ∈ N.

The considerations in this paper are based on the following alternative ([11]).

Theorem 2.3 (Nonlinear alternative for Kakutani maps). Let Y be a Hilbert space, C a closed convex subset of Y, f an open subset of C and 0 ∈ f. Suppose that F : f → P

(C) is an upper semicontinuous compact map; here P

(C) denotes the family of nonempty, compact convex subsets of C. Then either

(i) F has a fixed point in f, or

(ii) there is a v ∈ ∂f and λ ∈ (0, 1) with v ∈ λF (v).

Definition 2.4. The multivalued map F : J × H → P(H) is said to be L

-Carath´eodory if:

(i) t 7−→ F (t, y) is measurable for each y ∈ H;

(ii) y 7−→ F (t, y) is upper semicontinuous for almost all t ∈ J;

(iii) For each q > 0, there exists h

∈ L

(J, R

) such that

kF (t, y)k

:= sup {Ekgk

: g ∈ F (t, y)} ≤ h

(t) for all kyk

≤ q and for a.e. t ∈ J.

The following lemma is crucial in the proof of our main result.

Lemma 2.5 [16]. Let I be a compact interval and Y be a separable Hilbert

space. Let G be an L

-Carath´ eodory multivalued map with N

6= ∅

and let Γ be a linear continuous mapping from L

(I, Y ) to C(I, Y ).

Then the operator

Γ ◦ N

: C(I, Y ) → P

(C(I, Y )), x → (Γ ◦ N

)(x) = Γ(N

) is a closed graph operator in C(I, Y ) × C(I, Y ).

3. Existence results: stochastic differential inclusions Before stating and proving our main result, we give the definition of the mild solution.

Definition 3.1. An F

-adapted stochastic process x(t) : J → H is a mild solution of the abstract Cauchy problem (2) if there exists a function g ∈ L

L

(K, H)

, a selection of G t, x(t)

, such that for a.e. t ∈ J, the follow- ing integral equation is satisfied

(3) x(t) = C(t)x

+S(t)x

+ Z

0

S(t−s)f (s, x(s))ds+

Z

S(t−s)g(s)dw(s).

Theorem 3.2. Assume that:

(H1) A is the infinitesimal generator of a given strongly continuous bounded cosine family {C(t) : t ∈ J}, and there exists a constant M ≥ 1 such that

kC(t)k ≤ M and kS(t)k ≤ M for all t ≥ 0;

(H2) (i) the function f : J × H → H is completely continuous;

(ii) there exist constants c

> 0, c

≥ 0 such that

Ekf (t, y)k

≤ c

Ekyk

+ c

, for every t ∈ J, and y ∈ H;

(H3) G : J × H → P(L

(K, H)) is an L

-Carath´ eodory function;

(H4) there exist a continuous nondecreasing function Ψ : R

→ (0, ∞), P ∈ L

(J, R

), and nonnegative number M

> 0 such that

EkG(t, y)k

= sup 

Ekgk

: g ∈ G(t, y)

≤ P (t)Ψ(Ekyk

)

for almost all t ∈ J and y ∈ H, and

(1 − 4c

M

b

)M

(2M )



kx

k

+ kx

k

+ c

b

+ Ψ(M

)T rQ R

0

P (s)ds > 1, with 4c

M

b

< 1;

(H5) for each bounded set B ⊆ Z, and t ∈ J the set



C(t)x

+ S(t)x

+ Z

0

S(t − s)f (s, x(s))ds

+ Z

0

S(t − s)g(s)dw(s), g ∈ N



is relatively compact in H, where x ∈ B and N

= ∪{N

: x ∈ B}.

Then there exists at least one mild solution for the system (2) on J.

P roof. Consider the multivalued map Φ : Z → P(Z) defined by Φx =



h ∈ Z : h(t) = C(t)x

+ S(t)x

+ Z

0

S(t − s)f (s, x(s))ds

+ Z

0

S(t − s)g(s)dw(s), a.e t ∈ J



where g ∈ N

. We shall show that the operator Φ has a fixed point, which then is a solution of the system (2). We divide the proof into several steps.

Step I. Φx is convex for each x ∈ Z.

In fact, if h

, h

belong to Φx, then there exist g

, g

∈ N

such that h

(t) = C(t)x

+ S(t)x

+

Z

S(t − s)f (s, x(s))ds

+ Z

0

S(t − s) g

(s) dw(s), i = 1, 2, t ∈ J.

Let 0 ≤ ρ ≤ 1. Then for each t ∈ J, we have (ρh

+ (1 − ρ)h

)(t) = C(t)x

+ S(t)x

+

Z

S(t − s)f (s, x(s))ds

+ Z

0

S(t − s)[ρg

(s) + (1 − ρ)g

(s)]dw(s).

Since N

is convex (because G has convex values), then ρh

+ (1 − ρ)h

∈ Φx

which complets the proof of Step I.

Step II. Φ maps bounded sets into bounded sets in Z.

Indeed, it is enough to show that there exists a positive constant ` such that, for each h ∈ Φx, x ∈ B

= {x ∈ Z : kxk

≤ q, q ∈ N} one has khk

≤ `.

If h ∈ Φx, then there exists g ∈ N

, such that, for each t ∈ J,

(4) h(t) = C(t)x

+S(t)x

+ Z

0

S(t−s)f (s, x(s))ds+

Z

S(t−s)g(s)dw(s).

Then for each h ∈ Φ(B

) we have

E h(t)

≤ 4



EkC(t)x

k

+ EkS(t)x

k

+ b Z

0

kS(t − s)k

Ekf (s, x(s))k

ds

+ T rQ Z

0

kS(t − s)k

Ekg(s)k

ds



≤ (2M )



Ekx

k

+ Ekx

k

+ b Z

0

[c

Ekx(s)k

+ c

]ds

+ T rQ Z

0

Ekg(s)k

ds

 .

Thus,

h(t)

≤ (2M )



kx

k

+ kx

k

+ b

[c

q + c

] + T rQkh

k

:= `.

Step III. Φ maps bounded sets into equicontinuous sets of Z.

Let 0 < t

< t

≤ b. For each x ∈ B

and h ∈ Φx, there exists g ∈ N

such that (4) holds. Thus, using Lemma 2.2, we have

E

h(t

) − h(t

)

≤ 6



k[C(t

) − C(t

)]x

k

+ k[S(t

) − S(t

)]x

k

+ b Z

0

kS(t

− s) − S(t

− s)k

Ekf (s, x(s))k

ds

+ Z

t₁

kS(t

− s)k

Ekf (s, x(s))k

ds

+ T rQ Z

0

kS(t

− s) − S(t

− s)kEkg(s)k

ds

+ T rQ Z

t₁

kS(t

− s)k

Ekg(s)k

ds

 . Hence,

h(t

) − h(t

)

Z

≤ 6



k[C(t

) − C(t

)]x

k

+ k[S(t

) − S(t

)]x

k

+ b Z

0

kS(t

− s) − S(t

− s)k

{c

q + c

}ds + M

(t

− t

)

{c

q + c

}

+ e

(t

− t

)T rQ Z

0

h

(s)ds

+ M

T rQ Z

t₁

h

(s)ds

 .

As a consequence of Steps 2, 3, (H5) and the Arzel´a-Ascoli theorem we can conclude that Φ is completely continuous.

Step IV. Φ has a closed graph.

Let x

→ x

, h

∈ Φx

and h

→ h

. We shall prove that h

∈ Φx

. Now h

∈ Φx

means that there exists g

∈ N

such that

h

(t) = C(t)x

+ S(t)x

+ Z

0

S(t − s)f (s, x

(s))ds +

Z

S(t − s)g

(s)dw(s), t ∈ J.

We must prove that there exists g

∈ N

such that

h

(t) = C(t)x

+ S(t)x

+ Z

0

S(t − s)f (s, x

(s))ds +

Z

S(t − s)g

(s)dw(s), t ∈ J.

Since f is continuous, then



h

(t) − C(t)x

− S(t)x

− Z

0

S(t − s)f (s, x

(s))ds 

− 

h

(t) − C(t)x

− S(t)x

− Z

0

S(t − s)f (s, x

(s))ds 

2

→ 0

as n → ∞. Consider the linear continuous operator Γ : L

(L

(K, H)) → Z

g → Γ(g)(t) = Z

0

S(t − s)g(s)dw(s).

Clearly, Γ is linear and continuous. Indeed, one has kΓgk

≤ bM

T rQkgk

.

From Lemma 2.5, it follows that Γ◦S

is a closed graph operator. Moreover,

 h

(t) − C(t)x

− S(t)x

− Z

0

S(t − s)f (s, x

(s))ds 

∈ Γ N

.

Since x

→ x

, for some g

∈ N

, it follows from Lemma 2.5 that

h

(t) − C(t)x

− S(t)x

− Z

0

S(t − s)f (s, x

(s))ds = Z

0

S(t − s)g

(s)dw(s).

Therefore Φ is a completely continuous multivalued map, u.s.c. with convex closed values.

In order to prove that Φ has a fixed point by Theorem 2.3, we need one

more step.

Step V. We show there exists an open set U

⊆ Z with x / ∈ λΦx for λ ∈ (0, 1) and x ∈ ∂U

.

Let x ∈ U

. Then x ∈ λΦx for some λ > 1, and there exists g ∈ N

such that

x(t) = λ

C(t)x

+ S(t)x

+ λ

Z

0

S(t − s)f (s, x(s))ds + λ

Z

0

S(t − s)g(s)dw(s), t ∈ J.

By hypotheses (H1)–(H4), we have, for each t ∈ J E

x(t)

≤ 4



EkC(t)x

k

+ EkS(t)x

k

+ b Z

0

kS(t − s)k

Ekf (s, x(s))k

ds

+ T rQ Z

0

kS(t − s)k

Ekg(s)k

ds



≤ (2M )



Ekx

k

+ Ekx

k

+ b Z

0

[c

Ekx(s)k

+ c

]ds

+ T rQ Z

0

P (s)Ψ(Ekx(s)k

)ds

 . Consider the function µ defined by

µ(t) = sup {Ekx(s)k

: 0 ≤ s ≤ t}, 0 ≤ t ≤ b.

From the previous inequality we have:

µ(t) ≤ (2M )



kx

k

+ kx

k

+ b

[c

µ(t) + c

] + T rQ Z

0

P (s)Ψ(µ(s))ds

 . Therefore

(1 − 4c

M

b

)µ(t)

≤ (2M )



kx

k

+ kx

k

+ c

b

+ T rQ Z

0

P (s)Ψ(µ(s))ds



.

Consequently,

(1 − 4c

M

b

)kxk

(2M )



kx

k

+ kx

k

+ c

b

+ Ψ(kxk

)T rQ R

0

P (s)ds ≤ 1.

Then by (H4), there exists M

such that kxk

6= M

. Set

U

= {x ∈ Z : kxk

< M

+ 1}.

From the choice of U

, there is no x ∈ ∂U

such that λx = λΦ for some λ > 1. As a consequence of the nonlinear alternative of Leray-Schauder type [11], we deduce that Φ has a fixed point in U

which is a mild solution of the problem (2).

4. Existence results: stochastic integrodifferential inclusions

In this section, we are interested in the existence of the following nonlinear abstract stochastic integrodifferential inclusions

(5)

dx

(t) ∈ [Ax(t) + f (t, x(t), ξ

(t))]dt

+ G(t, x(t), ξ

x(t)) dw(t), t ∈ J := [0, b], x(0) = x

, x

(0) = x

,

in a real separable Hilbert space H, where ξ

x(t) =

Z

k

(t, τ, x(τ ))dτ, (i = 1, 2),

and A as in problem (2). Assume that k

: J × J × H → H (i = 1, 2) are given continuous measurable mappings, f : J × H × H → H, a measurable mapping in H-norm and G : J × H × H → P(L

(K, H)), a multivalued measurable mapping such that g(t) ∈ G(t, x(t), ξ

x(t)) is locally bounded in L

(K, H)-norm.

Before stating and proving our main result, we give the definition of the

mild solution and L

-Carath´eodory.

Definition 4.1. An F

-adapted stochastic process x(t) : J → H is a mild solution of the abstract Cauchy problem (5) if there exists a function g ∈ L

L

(K, H)

, a selection of G t, x(t), ξ

x(t)

, such that for a.e. t ∈ J, the following integral equation is satisfied

(6)

x(t) = C(t)x

+ S(t)x

+ Z

0

S(t − s)f (s, x(s), ξ

x(s))ds +

Z

S(t − s)g(s)dw(s), for a.e. t ∈ J.

Definition 4.2 The multivalued map F : J × H × H → P(H) is said to be L

-Carath´eodory if:

(i) t 7−→ F (t, x, y) is measurable for each (x, y) ∈ H × H;

(ii) (x, y) 7−→ F (t, x, y) is upper semicontinuous for almost all t ∈ J;

(iii) For each q > 0, there exists h

∈ L

(J, R

) such that kF (t, x, y)k

:= sup 

Ekgk

: g ∈ F (t, x, y)

≤ h

(t) for all kxk

≤ q, kyk

≤ q and for a.e. t ∈ J.

Theorem 4.3. Assume that (H1) holds. In addition, suppose that the fol- lowing conditions are satisfied:

(H5) (i) the function f : J × H × H → H is completely continuous;

(ii) there exist constants 0 < β < 1, c

> 0, c

> 0, c

≥ 0 such that kf (t, x, y)k

≤ c

kxk

+c

kyk

+c

, for every t ∈ J, and x, y ∈ H;

(H6) there exist m

: J × J → [0, ∞), (i = 1, 2), nondecreasing, a.e., with respect to the first variable, and

Z

m

(t, s)ds

are bounded on J. Moreover there exist nondecreasing functions ψ

: R

→ (0, ∞), (i = 1, 2), continuously differentiable such that

kk

(t, s, x)k

≤ m

(t, s) ψ

(kxk

), 0 ≤ s < t ≤ b, x ∈ H;

(H7) G : J × H × H → P(L

(K, H)) is an L

-Carath´ eodory function;

(H8) there exist integrable functions P

∈ L

(J, R

) (i = 1, 2) such that kG(t, x, y)k

= sup 

kgk

: g ∈ G(t, x, y)

≤ P

(t)Ψ

(kxk

) + P

(t)Ψ

(kyk

)

for almost all t ∈ J and x, y ∈ H, where Ψ

: R

→ (0, ∞), (i = 1, 2) are continuous nondecreasing functions and there exists a constant M

with

(1 − 4M

c

b)M

N

> 1, where

N

= M + (2M b)

c

m b

ψ

(M

) + (2M )

T rQΨ

(M

) Z

0

P

(s)ds + (2M )

T rQΨ

m b

ψ

(M

)
Z

0

P

(s)ds, b

m

= Z

0

m

(b, s)ds, (i = 1, 2), M = (2M )

{kx

k

+ kx

k

+ c

b},

and 4M

c

b < 1.

(H9) for each bounded set B ⊆ Z, and t ∈ J the set n C(t)x

+ S(t)x

+

Z

S(t − s)f (s, x(s), ξ

x(s))ds +

Z

S(t − s)g(s)dw(s), g ∈ N

o

is relatively compact in H, where x ∈ B and N

= ∪{N

: x ∈ B}.

Then there exists at least one mild solution for the system (5) on J.

P roof. Consider the multivalued map Φ : Z → P(Z) defined by Φx =



h ∈ Z : h(t) = C(t)x

+ S(t)x

+ Z

0

S(t − s)f (s, x(s), ξ

x(s))ds +

Z

S(t − s)g(s)dw(s), a.e t ∈ J



where g ∈ N

. We shall show that the operator Φ has a fixed point, which then is a solution of the system (5). As in Theorem 3.2 we can prove that Φ is completely continuous.

We show there exists an open set U ⊆ Z with x / ∈ λΦx for λ ∈ (0, 1) and x ∈ ∂U. Let x ∈ U. Then x ∈ λΦx for some λ ∈ (0, 1), there exists g ∈ N

such that

x(t) = λC(t)x

+ λS(t)x

+ λ Z

0

S(t − s)f (s, x(s), ξ

x(s))ds

+ λ Z

0

T (t − s)g(s)dw(s), t ∈ J.

We have, for each t ∈ J, E x(t)

≤ (2M )



Ekx

k

+ Ekx

k

+ b Z

0

[c

Ekx(s)k

+ c

Ekξ

x(s)k

+ c

]ds

+ T rQ Z

0

Ekg(s)k

ds



≤ (2M )



Ekx

k

+ Ekx

k

+ b Z

0

[c

||x(s)||

+ c

kξ

x(s)k

+ c

]ds

+ T rQ Z

0



P

(s)Ψ

(||x(s)||

) + P

(s)Ψ

Z

m

(s, τ )ψ

(||x

||

)dτ



ds



≤ (2M )



Ekx

k

+ Ekx

k

+ b



c

kx(t)k

+ c

b Z

0

m

(t, s)ψ

(kx(s)k

)ds + c



+ T rQ Z

0



P

(s)Ψ

(||x(s)||

) + P

(s)Ψ

Z

0

m

(s, τ )ψ

(||x

||

)dτ



ds

 .

Consider the function µ defined by

µ(t) = sup {Ekx(s)k

: 0 ≤ s ≤ t}, 0 ≤ t ≤ b.

By the previous inequality we have:

µ(t) ≤ M + (2M )



c

bµ(t) + c

b

Z

0

m

(t, s)ψ

(µ(s))ds

+ T rQ Z

0

P

(s)Ψ

(µ(s))ds + T rQ

Z

P

(s)Ψ

Z

m

(s, τ )ψ

(µ(τ ))dτ

 ds

 , where M = (2M )

{kx

k

+ kx

k

+ c

b}. Therefore

(1 − 4M

c

b)µ(t) ≤ M + (2M b)

c

Z

m

(b, s)ψ

(µ(s))ds

+ (2M )

T rQ Z

0



P

(s)Ψ

(µ(s)) + P

(s)Ψ

Z

m

(s, τ )ψ

(µ(τ ))dτ

  ds.

Consequently,

(1 − 4M

c

b)kµk

≤ M + (2M b)

c

m ˆ

ψ

(kµk

) + (2M )

T rQΨ

(kµk

)

Z

P

(s)ds

+ (2M )

T rQΨ

m ˆ

ψ

(kµk

)
Z

0

P

(s)ds

or (1 − 4M

c

b)kµk

N ≤ 1,

where

N = M + (2M b)

c

m b

ψ

(kµk

) + (2M )

T rQΨ

(kµk

) Z

0

P

(s)ds

+ (2M )

T rQΨ

( b m

ψ

(kµk

)) Z

0

P

(s)ds.

Then by (H8), there exists M

such that kxk

6= M

. Set U = {x ∈ Z : kxk

< M

+ 1}.

From the choice of U, there is no x ∈ ∂U such that λx = λΦ for some λ > 1.

As a consequence of the nonlinear alternative of Leray-Schauder type [11], we deduce that Φ has a fixed point in U which is a mild solution of the problem (5).

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Received 18 September 2006