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Existence results for impulsive neutral stochastic functional integro-differential inclusions with infinite delays

Advances in Difference Equations20142014:17

https://doi.org/10.1186/1687-1847-2014-17

Received: 11 September 2013

Accepted: 17 December 2013

Published: 15 January 2014

Abstract

In this paper, we prove the existence of mild solutions for a class of impulsive neutral stochastic functional integro-differential inclusions with infinite delays in Hilbert spaces. The results are obtained by using the fixed-point theorem for multi-valued operators due to Dhage. An example is provided to illustrate the theory.

MSC:93B05, 93E03.

Keywords

  • impulsive equation
  • stochastic functional inclusion
  • mild solution
  • infinite delay

1 Introduction

In this paper, we shall consider the existence of mild solutions for impulsive neutral stochastic functional integro-differential inclusions with infinite delay of the following form:
d [ x ( t ) g ( t , x t , 0 t a ( t , s , x s ) d s ) ] d t [ A x ( t ) + f ( t , x t ) ] d t + F ( t , x t ) d w ( t ) , t J = [ 0 , b ] , t t k ,
(1.1)
Δ x ( t k ) = x ( t k + ) x ( t k ) = I k ( x ( t k ) ) , k = 1 , 2 , , m ,
(1.2)
x ( t ) = ϕ ( t ) L 2 ( Ω , B h ) for a.e.  t J 0 = ( , 0 ] ,
(1.3)

where the state x ( ) takes values in a separable real Hilbert space H with inner product ( , ) and norm | | , A is the infinitesimal generator of a compact analytic resolvent operator S ( t ) , t 0 , in the Hilbert space H. Suppose that { w ( t ) : t 0 } is a given K-valued Brownian motion or Wiener process with a finite trace nuclear covariance operator Q 0 and L ( K , H ) denotes the space of all bounded linear operators from K into H. Further a : D × B h H , g : J × B h × H H , f : J × B h H and F : J × B h P ( L Q ( K , H ) ) are given functions, where D = { ( t , s ) J × J : s t } , P ( L Q ( K , H ) ) is the family of all nonempty subsets of L Q ( K , H ) and L Q ( K , H ) denotes the space of all Q-Hilbert-Schmidt operators from K into H, which will be defined in Section 2. Here, I k C ( H , H ) ( k = 1 , 2 , , m ) are bounded functions. Furthermore, the fixed times t k satisfies 0 = t 0 < t 1 < t 2 < < t m < b , x ( t k + ) and x ( t k ) denote the right and left limits of x ( t ) at t = t k . Δ x ( t k ) = x ( t k + ) x ( t k ) = I k ( x ( t k ) ) represents the jump in the state x at time t k , where I k determines the size of jump. The histories x t : Ω B h , t 0 , which are defined by setting x t = { x ( t + s ) : s ( , 0 ] } , belong to the abstract phase space B h , which will be defined in Section 2. The initial data ϕ = { ϕ ( t ) : < t 0 } is an F 0 -measurable, B h -valued random variables independent of { w ( t ) : t 0 } with finite second moment.

The theory of impulsive integro-differential inclusions has become an active area of investigation due to their applications in the fields such as mechanics, electrical engineering, medicine biology, ecology, and so on (see [1, 2] and references therein).

The existence of impulsive neutral stochastic functional integro-differential equations or inclusions with infinite delays have attracted great interest of researchers. For example, Lin and Hu [3] consider the existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions. Hu and Ren [4] studied the existence results for impulsive neutral stochastic functional integro-differential equations with infinite delays.

Motivated by the previous mentioned papers, we prove the existence of solutions for impulsive neutral stochastic functional integro-differential inclusions with infinite delays.

2 Preliminaries

Throughout this paper, ( H , | | ) and ( K , | | K ) denote two real separable Hilbert spaces. Let ( Ω , F , P ; F ) ( F = { F t } t 0 ) be a complete filtered probability space satisfying the requirement that F 0 contains all P-null sets of . An H-valued random variable is an -measurable function x ( t ) : Ω H and the collection of random variables S = { x ( t , w ) : Ω H | t J } is called a stochastic process. Suppose that { w ( t ) : t 0 } is a cylindrical K-valued Wiener process with a finite trace nuclear covariance operator Q 0 , denote T r Q = i = 1 λ i = λ < , which satisfies Q e i = λ i e i . So, actually, w ( t ) = i = 1 λ i w i ( t ) e i , where { w i ( t ) } i = 1 are mutually independent one-dimensional standard Wiener process. We assume that F t = σ { w ( s ) : 0 s t } is the σ-algebra generated by w and F T = F . Let ψ L ( K , H ) and define
| ψ | Q 2 = T r ( ψ Q ψ ) = n = 1 | λ n ψ e n | 2 .

If | ψ | Q < , then ψ is called a Q-Hilbert-Schmidt operator. Let L Q ( K , H ) denote the space of all Q-Hilbert-Schmidt operator ψ : K H . The completion L Q ( K , H ) of L ( K , H ) with respect to the topology induced by the norm | | Q , where | ψ | Q 2 = ψ , ψ is a Hilbert space with the above norm topology.

Let A : D ( A ) H be the infinitesimal generator of a compact, analytic resolvent operator S ( t ) , t 0 . Let 0 ρ ( A ) . Then it is possible to define the fractional power ( A ) α for 0 < α 1 as a closed linear operator with its domain D ( ( A ) α ) being dense in H. We denote by H α the Banach space D ( A α ) endowed with the norm x α = ( A ) α x , which is equivalent to the graph norm of ( A ) α .

Lemma 2.1 ([5])

The following properties hold:
  1. (i)

    If 0 < β < α 1 , the H α H β and the embedding is continuous and compact whenever the resolvent operator of A is compact.

     
  2. (ii)
    For every 0 < α < 1 , there exists a positive constant c α such that
    ( A ) α S ( t ) C α t α , t > 0 .
     
Now, we define the abstract phase space B h . Assume that h : ( , 0 ] ( 0 , ) is a continuous function with l = 0 h ( t ) d t < . For any a > 0 we define
B h = { ψ : ( , 0 ] H : ( E | ψ ( θ ) | 2 ) 1 2 is a bounded and measurable function on [ a , 0 ] and 0 h ( s ) sup s θ 0 ( E | ψ ( θ ) | 2 ) 1 2 d s < } .
If B h is endowed with the norm
ψ B h = 0 h ( s ) sup s θ 0 ( E | ψ ( θ ) | 2 ) 1 2 d s for all ψ B h ,
then ( B h , B h ) is a Banach space [6]. Now, we consider the space
B b = { x : ( , b ] H such that x k C ( J k , H ) and there exist x ( t k + ) and x ( t k ) with x ( t k ) = x ( t k ) , x 0 = ϕ L 2 ( Ω , B h ) on ( , 0 ] , k = 1 , 2 , , m } ,
where x k is the restriction of x to J k = ( t k , t k + 1 ] , k = 0 , 1 , , m . Let b be a seminorm in B b defined by
x b = x 0 B h + sup 0 s b ( E | x ( s ) | 2 ) 1 2 , x B b .

Lemma 2.2 ([7])

Assume that x B b , then for t J , x t B h . Moreover
l ( E | x ( t ) | 2 ) 1 2 x t B h x 0 B h + l sup 0 s t ( E | x ( t ) | 2 ) 1 2 ,

where l = 0 h ( s ) d s < .

We use the notation P ( H ) for the family of all subsets H and denote
P c l ( H ) = { Y P ( H ) : Y  is closed } , P b d ( H ) = { Y P ( H ) : Y  is bounded } , P c v ( H ) = { Y P ( H ) : Y  is convex } , P c p ( H ) = { Y P ( H ) : Y  is compact } .

A multi-valued mapping Γ : H P ( H ) is called upper semicontinuous (u.s.c) if for any x H , the set Γ ( x ) is a nonempty closed subset of H and if for each open set G of H containing Γ ( x ) , there exists an open neighborhood N of x such that Γ ( N ) G . Γ is said to be completely continuous if Γ ( B ) is relatively compact for every bounded subset of B H . If the multi-valued mapping Γ is completely continuous with nonempty compact values, then Γ is u.s.c. if and only if Γ has a closed graph, i.e., x n x , y n y , y n Γ ( x n ) imply y Γ ( x ) .

Definition 2.1 The multi-valued mapping F : J × B h P ( H ) is said to be L 2 -Carathéodory if
  1. (i)

    t F ( t , v ) is measurable for each v B h ,

     
  2. (ii)

    v F ( t , v ) is u.s.c. for almost all t J and v B h ,

     
  3. (iii)
    for each q > 0 , there exists h q L 1 ( J , R + ) such that
    F ( t , v ) 2 = sup f F ( t , v ) E ( | f | 2 ) h q ( t ) ,
     

for all v B h 2 q and for a.e. t J .

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

Lemma 2.3 ([8])

Let I be a compact interval and H be a Hilbert space. Let F be an L 2 -Carathéodory multi-valued mapping with N F , x ϕ and let Γ be a linear continuous mapping from L 2 ( I , H ) to C ( I , H ) . Then the operator
Γ N F : C ( I , H ) P c p , c v ( H ) , x ( Γ N F ) ( x ) = Γ ( N F , x )
is a closed graph operator in C ( I , H ) × C ( I , H ) , where N F , x is known as the selectors set from F; it is given by
σ N F , x = { σ L 2 ( L ( K , H ) ) : σ ( t ) F ( t , x ) for a.e. t J } .

Theorem 2.1 ([9])

Let X be a Banach space, Φ 1 : X P c l , c v , b d ( X ) and Φ 2 : X P c p , c v ( X ) be two multi-valued operators satisfying:
  1. (a)

    Φ 1 is a contraction,

     
  2. (b)

    Φ 2 is u.s.c. and completely continuous.

     
Then either
  1. (i)

    the operator inclusion λ x Φ 1 x + Φ 2 x has a solution for λ = 1 , or

     
  2. (ii)

    the set G = { x X : λ x Φ 1 x + Φ 2 x , λ > 1 } is unbounded.

     

Lemma 2.4 ([10])

Let v , w : [ 0 , b ] [ 0 , ) be continuous functions. If w is nondecreasing and there are constants θ > 0 , 0 < α < 1 such that
v ( t ) w ( t ) + θ 0 t v ( s ) ( t s ) 1 α d s , t J ,
then
v ( t ) e θ n Γ ( α ) n t n α Γ ( n α ) j = 0 n 1 ( θ b α α ) j w ( t )

for every t J and every n N such that n α > 1 and Γ ( ) is the Gamma function.

3 Main result

Let J 1 = ( , b ] . First, we present the definition of the mild solution of problem (1.1)-(1.3).

Definition 3.1 A stochastic process x : J 1 × Ω H is called a mild solution of problem (1.1)-(1.3) if
  1. (i)

    x ( t ) is measurable and F t -adapted for each t 0 ,

     
  2. (ii)

    Δ x ( t k ) = x ( t k + ) x ( t k ) , k = 1 , 2 , , m ,

     
  3. (iii)
    x ( t ) H has càdlàg paths on t J a.e. and there exists a function σ N F , x such that
    x ( t ) = S ( t ) [ ϕ ( 0 ) g ( 0 , ϕ , 0 ) ] + g ( t , x t , 0 t a ( t , s , x s ) d s ) + 0 t A S ( t s ) g ( s , x s , 0 s a ( s , τ , x τ ) d τ ) d s + 0 t A S ( t s ) f ( s , x s ) d s + 0 t S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( x ( t k ) ) , t J ,
     
  4. (iv)

    x 0 ( ) = ϕ L 2 ( Ω , B h ) on J 0 = ( , 0 ] satisfies ϕ B h 2 < .

     

Now, we assume the following hypotheses:

(H1) A is the infinitesimal generator of a compact analytic resolvent operator S ( t ) , t 0 , in the Hilbert space H and there exist positive constants M and M 1 such that
S ( t ) 2 M , A β M 1 , t J .
(H2) a : D × B h H , D = { ( t , s ) J × J : t s } is a continuous function and there exists a constant M a such that
E | 0 t [ a ( t , s , x ) a ( t , s , y ) ] d s | 2 M a x y B h 2 for all  t J , x , y B h .
(H3) There exist constants 0 < β < 1 and M g such that g is H β -valued, ( A ) β g is continuous and
E | ( A ) β g ( t , x 1 , y 1 ) ( A ) β g ( t , x 2 , y 2 ) | 2 M g [ x 1 x 2 B h 2 + E | y 1 y 2 | 2 ] .
(H4) The function f : J × B h H satisfies the following conditions:
  1. (i)

    t f ( t , s ) is measurable for each x B h ;

     
  2. (ii)

    x f ( t , x ) is continuous for almost all t J ;

     
  3. (iii)
    There exists a constant M f such that
    E | ( A ) β f ( t , x ) ( A ) β f ( t , y ) | 2 M f x y B h 2
    for all x , y B h , t J and
    E | f ( t , x ) | 2 p ( t ) ψ ( x B h 2 )
    for almost all t J , where p L 1 ( J , R ) , ψ : R + ( 0 , ) is continuous and increasing with
    0 b μ ( s ) ¯ d s B 0 k 1 1 ψ ( s ) d s , μ ¯ ( t ) = B 0 k 3 p ( t ) , k 1 = 4 ϕ B h 2 + l 2 F 1 96 l 2 ( A ) β 2 M g ( 1 + 2 M a ) , k 2 = 96 b l 2 M g ( 1 + 2 M a ) c 1 β 2 1 96 l 2 ( A ) β 2 M g ( 1 + 2 M a ) , k 3 = 48 M b l 2 1 96 l 2 ( A ) β 2 M g ( 1 + 2 M a ) , L 0 = 3 l 2 [ M g ( 1 + M a ) ( ( A ) β 2 + ( C 1 β b β ) 2 2 β 1 ) + M f ( C 1 β b β ) 2 2 β 1 ] < 1 , B 0 = e k 2 n Γ ( β ) n b n β / Γ ( n β ) j = 1 n 1 ( k 2 b β β ) j , c 1 = b 2 sup ( t , s ) D a 2 ( t , s , 0 ) , c 2 = ( A ) β 2 sup t J g ( t , 0 , 0 ) 2
    and
    F = 4 M | ϕ ( 0 ) | 2 + 96 ( M + ( A ) β 2 ) c 2 + 192 ( A ) β 2 M g c 1 + 192 b 2 β C 1 β 2 2 β 1 ( c 2 + 2 M g c 1 ) + 48 M μ L loc 1 ( J , R + ) b 2 Tr ( Q ) + 48 M m 2 k = 1 m d k + 96 M ( A ) β 2 M g ϕ B h 2 .
    (H5) The multi-valued mapping F : J × B h P b d , c l , c v ( L ( K , H ) ) is an L 2 -Carathéodory function that satisfies the following conditions:
    1. (i)
      For each t J , the function F ( t , ) : B h P b d , c l , c v ( L ( K , H ) ) is u.s.c. and for each fixed x B h , the function F ( , x ) is measurable. For each x B h , the set
      N F , x = { σ L 2 ( K , H ) : σ ( t ) F ( t , x )  for a.e.  t J }
       

    is nonempty.

     
  4. (ii)
    There exists a positive function μ L loc 1 ( J , R + ) such that
    F ( t , x ) 2 = sup σ F ( t , x ) E | σ | 2 μ ( t ) .
     
(H6) I k C ( H α , H α ) and there exist positive constants d k such that for each x H α ,
| I k ( x ) | 2 d k , k = 1 , 2 , , m .
We consider the mapping Φ : B h P ( B h ) defined by
Φ x ( t ) = { ϕ ( t ) , t ( , 0 ] , S ( t ) [ ϕ ( 0 ) g ( 0 , ϕ , 0 ) ] + g ( t , x t , 0 t a ( t , s , x s ) d s ) + 0 t A S ( t s ) g ( s , x s , 0 s a ( s , τ , x τ ) d τ ) d s + 0 t A S ( t s ) f ( s , x s ) d s + 0 t S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( x ( t k ) ) , t J ,
where σ N F , x . For each ϕ B h , we define
ϕ ˜ ( t ) = { ϕ ( t ) , t ( , 0 ] , S ( t ) ϕ ( 0 ) , t J ,
and then ϕ ˜ B h . Let x ( t ) = y ( t ) + ϕ ˜ ( t ) , t ( , b ] . Then it is easy to see that x satisfies (1.1)-(1.3) if and only if y satisfies y 0 = 0 and
y ( t ) = S ( t ) g ( 0 , ϕ , 0 ) + g ( t , y t + ϕ ˜ t , 0 t a ( t , s , y s + ϕ ˜ s ) d s ) + 0 t A S ( t s ) g ( s , y s + ϕ ˜ s , 0 s a ( s , τ , y τ + ϕ ˜ τ ) d τ ) d s + 0 t A S ( t s ) f ( s , y s + ϕ s ) d s + 0 t S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) , t J ,
where σ N F , y . Let B h = { y B h : y 0 = 0 B h } . For any y B h ,
y b = y 0 B h + sup 0 s b ( E | y ( s ) | 2 ) 1 2 = sup 0 s b ( E | y ( s ) | 2 ) 1 2
and thus ( B h , b ) is a Banach space. Set B q = { y B h : y b 2 q } for some q 0 . Then B q B h is uniformly bounded and for any y B q , from Lemma 2.2, we see that
y t + ϕ ˜ t B h 2 2 y t B h 2 + 2 ϕ ˜ t B h 2 4 l 2 sup 0 s t E | y ( s ) | 2 + 4 y 0 B h 2 + 4 l 2 sup 0 s t ϕ ˜ ( s ) 2 + 4 ϕ ˜ 0 B h 2 4 l 2 ( q + M | ϕ ( 0 ) | 2 ) + 4 ϕ ˜ B h 2 : = q .
Define the operator Φ ˜ : B h P ( B h ) by
Φ ˜ y ( t ) = { 0 , t ( , 0 ] , S ( t ) g ( 0 , ϕ , 0 ) + g ( t , y t + ϕ ˜ t , 0 t a ( t , s , y s + ϕ ˜ s ) d s ) + 0 t A S ( t s ) g ( s , y s + ϕ ˜ , 0 s a ( s , τ , y τ + ϕ ˜ τ ) d τ ) d s + 0 t A S ( t s ) f ( s , y s + ϕ ˜ s ) d s + 0 t S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) , t J ,
where σ N F , y . Obviously, the operator Φ has a fixed point is equivalent to proving that Φ ˜ has a fixed point. Now, we decompose Φ ˜ as Φ ˜ 1 + Φ ˜ 2 , where
Φ ˜ 1 y ( t ) = S ( t ) g ( 0 , ϕ , 0 ) + g ( t , y t + ϕ ˜ t , 0 t a ( t , s , y s + ϕ ˜ s ) d s ) + 0 t A S ( t s ) g ( s , y s + ϕ ˜ s , 0 s a ( s , τ , y τ + ϕ ˜ τ ) d τ ) d s + 0 t A S ( t s ) f ( s , y s + ϕ ˜ s ) d s
and
Φ ˜ 2 y ( t ) = 0 s S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) , t J ,

where σ N F , y . In what follows, we show that the operators Φ ˜ 1 and Φ ˜ 2 satisfy all the conditions of Theorem 2.1.

Lemma 3.1 Assume that the assumptions (H1)-(H6) hold. Then Φ ˜ 1 is a contraction and Φ ˜ 2 is u.s.c. and completely continuous.

Proof We give the proof in several steps:

Step 1. Φ ˜ 1 is a contraction.

Let u , v B h . Then we have
E | ϕ ˜ 1 u ( t ) ϕ ˜ 1 v ( t ) | 2 3 E | g ( t , u t + ϕ ˜ t , 0 t a ( t , s , u s + ϕ ˜ s ) d s ) g ( t , v t + ϕ ˜ t , 0 t a ( t , s , v s + ϕ ˜ s ) d s ) | 2 + 3 b E ( 0 t | A S ( t s ) [ g ( s , u s + ϕ ˜ s , 0 s a ( s , τ , u τ + ϕ ˜ τ ) d τ ) g ( s , v s + ϕ ˜ s , 0 s a ( s , τ , u τ + ϕ ˜ τ ) d τ ) ] | 2 d s ) + 3 b E ( 0 t | A S ( t s ) [ f ( s , u s + ϕ ˜ s ) f ( s , v s + ϕ ˜ s ) ] | 2 d s ) 3 ( A ) β 2 M g ( u t v t B h 2 + M a u t v t B h 2 ) + 3 b 0 t C 1 β 2 ( t s ) 2 ( 1 β ) M g ( u s v s B h 2 + M a u s v s B h 2 ) d s + 3 b 0 t C 1 β 2 ( t s ) 2 ( 1 β ) M f u s v s B h 2 d s 3 ( A ) β 2 M g ( 1 + M a ) u t v t B h 2 + 3 M g ( 1 + M a ) ( C 1 β b β ) 2 2 β 1 u t v t B h 2 + 3 M f ( C 1 β b β ) 2 2 β 1 u t v t B h 2 3 [ M g ( 1 + M a ) ( ( A ) β 2 + ( C 1 β b β ) 2 2 β 1 ) + M f ( C 1 β b β ) 2 2 β 1 ] × [ l 2 sup s [ 0 , b ] E | u ( s ) v ( s ) | 2 + u 0 B h 2 + v 0 B h 2 ] = 3 l 2 [ M g ( 1 + M a ) ( ( A ) β 2 + ( C 1 β b β ) 2 2 β 1 ) + M f ( C 1 β b β ) 2 2 β 1 ] sup s [ 0 , b ] E | u ( s ) v ( s ) | 2 = L 0 sup s [ 0 , b ] E | u ( s ) v ( s ) | 2 ,
where L 0 = 3 l 2 [ M g ( 1 + M a ) ( ( A ) β 2 + ( C 1 β b β ) 2 2 β 1 ) + M f ( C 1 β b β ) 2 2 β 1 ] < 1 and we have used the fact that u 0 B h 2 = 0 and v 0 B h 2 = 0 . Taking the supremum over t, we obtain
ϕ ˜ 1 u ϕ ˜ 1 v b 2 L 0 u v b 2

and so ϕ ˜ 1 is a contraction.

Now, we show that the operator Φ ˜ 2 is completely continuous.

Step 2. Φ ˜ 2 y is convex for each y B h .

In fact, if u 1 , u 2 Φ ˜ 2 ( y ) , then there exist σ 1 , σ 2 N F , y such that
u i ( t ) = 0 t S ( t s ) σ i ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) )
for i = 1 , 2 and t J . Let λ [ 0 , 1 ] . Then for each t J , we have
λ u 1 ( t ) + ( 1 λ ) u 2 ( t ) = 0 t S ( t s ) [ λ σ 1 ( s ) + ( 1 λ ) σ 2 ( s ) ] d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) .
Since N F , y is convex (because F has convex values), we obtain
λ u 1 ( t ) + ( 1 λ ) u 2 ( t ) Φ ˜ 2 ( y ) .

Step 3. Φ ˜ 2 maps bounded sets into bounded sets in B h .

It is enough to show that there exists a positive constant Λ such that for each u Φ ˜ 2 y , y B q = { y B h : y b q } one has u b Λ . If u Φ ˜ 2 ( y ) , there exists σ N F , y such that for each t J
u ( t ) = 0 t S ( s t ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) )
and so
E | u ( t ) | 2 = E | 0 t S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) | 2 2 E | 0 t S ( t s ) σ ( s ) d w ( s ) | 2 + 2 E | 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) | 2 2 Tr ( Q ) M b 0 b μ ( s ) d s + 2 M m 2 k = 1 m d k 2 Tr ( Q ) M b 2 μ L loc 1 ( J , R + ) + 2 M m 2 k = 1 m d k : = Λ .

Thus, for each y B h , we get u b 2 Λ .

Step 4. Φ ˜ 2 maps bounded sets into equicontinuous sets of B h .

Let 0 < τ 1 < τ 2 b . For each y B q = { y B h : y b q } and u Φ ˜ 2 ( y ) . Let τ 1 , τ 2 J { t 1 , t 2 , , t m } . Then there exists σ N F , y such that for each t J ,
u ( t ) = 0 t S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) .
Thus we have
E | u ( τ 2 ) u ( τ 1 ) | 2 = E | 0 τ 2 S ( τ 2 s ) σ ( s ) d w ( s ) + 0 < t k < τ 2 S ( τ 2 s ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) 0 τ 1 S ( τ 1 s ) σ ( s ) d w ( s ) 0 < t k < τ 1 S ( τ 1 t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) | 2 2 E | 0 τ 1 ε ( S ( τ 2 s ) σ ( s ) S ( τ 1 s ) σ ( s ) ) d w ( s ) + τ 1 ε τ 1 ( S ( τ 2 s ) σ ( s ) S ( τ 1 s ) σ ( s ) ) d w ( s ) + τ 1 τ 2 S ( τ 2 s ) σ ( s ) d w ( s ) | 2 + 2 E | 0 < t k < τ 1 [ S ( τ 2 t k ) S ( τ 1 t k ) ] I k ( y ( t k ) + ϕ ˜ ( t k ) ) + τ 1 < t k < τ 2 S ( τ 2 t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) | 2 6 ε Tr ( Q ) 0 τ 1 ε μ ( s ) S ( τ 2 s ) S ( τ 1 s ) 2 d s + 6 ε Tr ( Q ) τ 1 ε τ 1 μ ( s ) S ( τ 2 s ) S ( τ 1 s ) 2 d s + 6 ( τ 2 τ 1 ) Tr ( Q ) τ 1 τ 2 μ ( s ) S ( τ 2 s ) 2 d s + 4 m 2 0 < t k < τ 1 S ( τ 2 s ) S ( τ 1 s ) 2 d k + 4 m 2 M τ 1 < t k < τ 2 d k .

The right-hand side of the above inequality tends to zero as τ 1 τ 2 with ε sufficiently small, since S ( t ) is strongly continuous and the compactness of S ( t ) for t > 0 implies the continuity in the uniform operator topology. Thus, the set { Φ ˜ 2 y : y B q } is equicontinuous. Here we consider the case 0 < τ 1 < τ 2 b , since the case τ 1 < τ 2 0 or τ 1 0 τ 2 b is simple.

Step 5. Φ ˜ 2 maps B q into a precompact set in H.

Let 0 < t b and 0 < ε < t . For y B q and u Φ ˜ 2 ( y ) , there exists σ N F , y such that
u ( t ) = 0 t ε S ( t s ) σ ( s ) d w ( s ) + t ε t S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) .
Define
u ε ( t ) = S ( ε ) 0 t ε S ( t ε s ) σ ( s ) d w ( s ) + 0 < t k < t ε S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) .
Since S ( t ) is a compact operator, the set V ε ( t ) = { u ε ( t ) : u ε Φ ˜ 2 ( B q ) } is relatively compact in H for each ε, 0 < ε < t . Moreover,
E | u ( t ) u ε ( t ) | 2 = E | 0 t ε S ( t s ) σ ( s ) d w ( s ) + t ε t S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) S ( ε ) 0 t ε S ( t ε s ) σ ( s ) d w ( s ) 0 < t k < t ε S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) | 2 4 M b Tr ( Q ) ε μ L loc 1 ( J , R + ) + 4 m 2 M t ε < t k < t d k .

Therefore letting ε 0 , we can see that there are relative compact sets arbitrarily close to the set { u ( t ) : u Φ ˜ 2 ( B q ) } . Thus, the set { u ( t ) : u Φ ˜ 2 ( B q ) } is relatively compact in H. Hence, the Arzelá-Ascoli theorem shows that Φ ˜ 2 is a compact multi-valued mapping.

Step 6. Φ ˜ 2 has a closed graph.

Let y n y , u n Φ ˜ 2 ( y n ) and u n u . We prove that u Φ ˜ 2 ( y ) .

Indeed, u n Φ ˜ 2 ( y n ) means that there exists σ n N F , y n such that
u n ( t ) = 0 t S ( t s ) σ n ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y n ( t k ) + ϕ ˜ ( t k ) ) , t J .
Thus we must prove that there exists σ N F , y such that
u ( t ) = 0 t S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) , t J .
Since I k , k = 1 , 2 , , m , are continuous, we see that
0 < t k < t S ( t t k ) I k ( y n ( t k ) + ϕ ˜ ( t k ) ) 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) b 2 0
as n . Consider the linear continuous operator Γ : L 2 ( J , H ) C ( J , H ) with Γ ( σ ) ( t ) = 0 t S ( t s ) σ ( s ) d w ( s ) , where σ N F , y . From Lemma 2.3, it follows that Γ N F is a closed graph operator. Moreover, we have
u n ( t ) 0 < t k < t S ( t t k ) I k ( y n ( t k ) + ϕ ˜ ( t k ) ) Γ ( N F , y n ) .
Since y n y , from Lemma 2.3, we obtain
u ( t ) 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) Γ ( N F , y ) .
That is, there exists a σ N F , y such that
u ( t ) 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) = Γ ( σ ( t ) ) = 0 t S ( t s ) σ ( s ) d w ( s ) .

Therefore Φ ˜ 2 has a closed graph and Φ ˜ 2 is u.s.c. This completes the proof. □

Lemma 3.2 Assume that the assumptions (H1)-(H2) hold. Then there exists a constant K > 0 such that y t + ϕ ˜ t B h 2 K for all t J , where K is depends only on b and the functions ψ and μ ¯ .

Proof Let y be a possible solution of y λ Φ ˜ ( y ) for some 0 < λ < 1 . Then there exists σ N F , y such that for t J we have
y ( t ) = λ S ( t ) g ( 0 , ϕ , 0 ) + λ g ( t , y t + ϕ ˜ t , 0 t a ( t , s , y s + ϕ ˜ s ) d s ) + λ 0 t A S ( t s ) g ( s , y s + ϕ ˜ s , 0 s a ( s , τ , y τ + ϕ ˜ τ ) d τ ) d s + λ 0 t S ( t s ) f ( s , y s + ϕ ˜ s ) d s + 0 t S ( t s ) σ ( s ) d w ( s ) + λ 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) .
Then, by the assumptions, we deduce that
E | y ( t ) | 2 E | S ( t ) g ( 0 , ϕ , 0 ) + g ( t , y t + ϕ ˜ t , 0 t a ( t , s , y s + ϕ ˜ s ) d s ) + 0 t A S ( t s ) g ( s , y s + ϕ ˜ s , 0 s a ( s , τ , y τ + ϕ ˜ τ ) d τ ) d s + 0 t S ( t s ) f ( s , y s + ϕ ˜ s ) d s + 0 t S ( t s ) σ ( s ) d w ( s ) + 0 < t k < t S ( t t k ) I k ( y ( t k ) + ϕ ˜ ( t k ) ) | 2 12 { 2 M ( ( A ) β 2 M g ϕ B h 2 + c 2 ) + 2 ( A ) β 2 [ M g ( y s + ϕ ˜ s B h 2 + 2 M a y s + ϕ ˜ s B h 2 + 2 c 1 ) + c 2 ] + 2 b 0 t c 1 β 2 ( t s ) 2 ( 1 β ) [ M g ( y s + ϕ ˜ s B h 2 + 2 M a y s + ϕ ˜ s B h 2 + 2 c 1 ) + c 2 ] d s + M b 0 t p ( s ) ψ ( y s + ϕ ˜ s B h 2 ) d s + M μ L loc 1 ( J , R + ) b 2 Tr ( Q ) + M m 2 k = 1 m d k } = 24 ( M + ( A ) β 2 ) c 2 + 48 ( A ) β 2 M g c 1 + 48 b 2 β c 1 β 2 2 β 1 ( c 2 + 2 M g c 1 ) + 12 M μ L loc 1 ( J , R + ) b 2 Tr ( Q ) + 12 M m 2 k = 1 m d k + 24 M ( A ) β 2 M g ϕ B h 2 + 24 ( A ) β 2 M g ( 1 + 2 M a ) y s + ϕ ˜ s B h 2 + 24 b M g ( 1 + 2 M a ) c 1 β 2 0 t y s + ϕ ˜ s B h 2 ( t s ) 2 ( 1 β ) d s + 12 M b 0 t p ( s ) ψ ( y s + ϕ ˜ s B h 2 ) d s .
From Lemma 2.2 we see that
y t + ϕ ˜ t B h 2 4 l 2 sup 0 s t E | y ( s ) | 2 + 4 l 2 M | ϕ ˜ ( 0 ) | 2 + 4 ϕ ˜ B h 2 .
Thus, for any t J , we have
y t + ϕ ˜ t B h 2 4 l 2 M | ϕ ˜ ( 0 ) | 2 + 4 ϕ ˜ B h 2 + 96 l 2 ( M + ( A ) β 2 ) c 2 + 192 l 2 ( A ) β 2 M g c 1 + 192 l 2 b 2 β C 1 β 2 2 β 1 ( c 2 + 2 M g c 1 ) + 48 M μ L loc 1 ( J , R + ) b 2 l 2 Tr ( Q ) + 48 M l 2 m 2 k = 1 m d k + 96 M l 2 ( A ) β 2 M g ϕ B h 2 + 96 l 2 ( A ) β 2 M g ( 1 + 2 M a ) y s + ϕ ˜ s B h 2 + 96 b l 2 M g ( 1 + 2 M a ) C 1 β 2 0 t y s + ϕ ˜ s B h 2 ( t s ) 2 ( 1 β ) d s + 48 M b l 2 0 t p ( s ) ψ ( y s + ϕ ˜ s B h 2 ) d s = 4 ϕ B h 2 + l 2 F + 96 l 2 ( A ) β 2 M g ( 1 + 2 M a ) sup 0 s t y s + ϕ ˜ s B h 2 + 96 b l 2 M g ( 1 + 2 M a ) C 1 β 2 0 t y s + ϕ ˜ s B h 2 ( t s ) 2 ( 1 β ) d s + 48 M b l 2 0 t p ( s ) ψ ( y s + ϕ ˜ s B h 2 ) d s .
Let v ( t ) = sup 0 s t y s + ϕ ˜ s B h 2 . Then the function v ( t ) is nondecreasing in J. Thus, we obtain
v ( t ) 4 ϕ B h 2 + l 2 F + 96 l 2 ( A ) β 2 M g ( 1 + 2 M a ) v ( t ) + 96 b l 2 M g ( 1 + 2 M a ) C 1 β 2 0 t v ( s ) ( t s ) 2 ( 1 β ) d s + 48 M b l 2 0 t p ( s ) ψ ( v ( s ) ) d s .
From this we derive that
v ( t ) 4 ϕ B h 2 + l 2 F 1 96 l 2 ( A ) β 2 M g ( 1 + 2 M a ) + 96 b l 2 M g ( 1 + 2 M a ) C 1 β 2 1 96 l 2 ( A ) β 2 M g ( 1 + 2 M a ) 0 t v ( s ) ( t s ) 1 β d s + 48 M b l 2 1 96 l 2 ( A ) β 2 M g ( 1 + 2 M a ) 0 t p ( s ) ψ ( v ( s ) ) d s k 1 + k 2 0 t v ( s ) ( t s ) 1 β d s + k 3 0 t p ( s ) ψ ( v ( s ) ) d s .
By Lemma 2.4, we get
v ( t ) B 0 ( k 1 + k 3 0 t p ( s ) ψ ( v ( s ) ) d s ) ,
where
B 0 = e k 2 n Γ ( β ) n b n β / Γ ( n β ) j = 1 n 1 ( k 2 b β β ) j .
Let us take the right-hand side of the above inequality as μ ( t ) . Then μ ( 0 ) = B 0 k 1 , v ( t ) μ ( t ) , t J and
μ ( t ) B 0 k 3 p ( t ) ψ ( v ( t ) ) .
Since ψ is nondecreasing, we have
μ ( t ) B 0 k 3 p ( t ) ψ ( μ ( t ) ) = μ ¯ ( t ) ψ ( μ ( t ) ) .
It follows that
μ ( 0 ) μ ( t ) 1 ψ ( s ) d s 0 b μ ( s ) ¯ d s B 0 K 1 1 ψ ( s ) d s ,

which indicates that μ ( t ) < . Thus, there exists a constant K such that μ ( t ) K , t J . Furthermore, we see that y t + ϕ ˜ t B h 2 v ( t ) μ ( t ) K , t J . □

Theorem 3.1 Assume that the assumptions (H1)-(H6) hold. The problem (1.1)-(1.3) has at least one mild solution on J.

Proof Let us take the set
G ( Φ ) = { x B h : x λ Φ ( x )  for some  λ ( 0 , 1 ) } .
Then for any x G ( Φ ) , we have
x t B h 2 = y t + ϕ ˜ t B h 2 K , t J ,

where K > 0 is a constant in Lemma 3.2. This show that G is bounded on J. Hence from Theorem 2.1 there exists a fixed point x ( t ) for Φ on B h , which is a mild solution of (1.1)-(1.3) on J. □

4 An example

As an application of Theorem 3.1, we consider the impulsive neutral stochastic functional integro-differential inclusion of the following form:
t ( z ( t , x ) + g ( t , z ( t h , x ) , 0 t a ( t , s , z ( s h , x ) ) d s ) ) 2 x 2 z ( t , x ) + ( f ( t , z ( t h , x ) ) + [ Q 1 ( t , z ( t h , x ) ) , Q 2 ( t , z ( t h , x ) ) ] ) d w ( t ) ,
(4.1)
0 x π , t J , t t k , Δ z ( t k , x ) = z ( t k + , x ) z ( t k , x ) = I k ( z ( t k , x ) ) , k = 1 , 2 , , m ,
(4.2)
z ( t , 0 ) = z ( t , π ) = 0 , t J ,
(4.3)
z ( t , x ) = ρ ( t , x ) , < t 0 , 0 x π ,
(4.4)
where J = [ 0 , b ] , k = 1 , 2 , , m , z ( t k + , x ) = lim h 0 + z ( t k + h , x ) , z ( t k , x ) = lim h 0 z ( t k + h , x ) , Q 1 , Q 2 : J × R R are two given functions and w ( t ) is a one-dimensional standard Wiener process. We assume that for each t J , Q 1 ( t , ) is lower semicontinuous and Q 2 ( t , ) is upper semicontinuous. Let J 1 = ( , b ] and H = L 2 ( [ 0 , π ] ) with norm . Define A : H H by A v = v with domain D ( A ) = { v H : v , v  are absolutely continuous , v H , v ( 0 ) = v ( π ) = 0 } . Then
A v = n = 1 n 2 ( v , v n ) , v D ( A ) ,
where v n = 2 π sin ( n s ) , n = 1 , 2 ,  , is the orthogonal set of eigenvectors in A. It is well known that A is the infinitesimal generator of an analytic semigroup S ( t ) , t 0 in H given by
S ( t ) v = n = 1 e n 2 t ( v , v n ) v n , v H .
For every v H , ( A ) 1 2 v = n = 1 1 n ( v , v n ) v n and ( A ) 1 2 = 1 . The operator ( A ) 1 2 is given by
( A ) 1 2 v = n = 1 n ( v , v n ) v n
on the space D ( ( A ) 1 2 ) = { v H : n = 1 n ( v , v n ) v n H } . Since the analytic semigroup S ( t ) is compact [10], there exists a constant M > 0 such that S ( t ) M and satisfies (H1). Now, we give a special B h -space. Let h ( s ) = e 2 s , s < 0 . Then l = 0 h ( s ) d s = 1 2 and let
φ B h = 0 h ( s ) sup s θ 0 ( E | φ ( θ ) | 2 ) 1 2 d s .
It follows from [5] that ( B h , B h ) is a Banach space. Hence for ( t , ϕ ) [ 0 , b ] × B h , let
ϕ ( θ ) x = ϕ ( θ , x ) , ( θ , x ) ( , 0 ] × [ 0 , π ] , z ( t ) ( x ) = z ( t , x )
and
F ( t , ϕ ) ( x ) = [ Q 1 ( t , ϕ ( θ , x ) ) , Q 2 ( t , ϕ ( θ , x ) ) ] , < θ 0 , x [ 0 , π ] .

Then (4.1)-(4.4) can be rewritten as the abstract form as the system (1.1)-(1.3). If we assume that (H2)-(H6) are satisfied, then the system (4.1)-(4.4) has a mild solution on [ 0 , b ] .

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Busan National Universeity, Busan, South Korea
(2)
Department of Mathematics, Dongeui University, Busan, South Korea

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© Park and Jeong; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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