Open Access Research

Solvability of impulsive partial neutral second-order functional integro-differential equations with infinite delay

Shengli Xie

Author Affiliations

Department of Mathematics and Physics, Anhui University of Architecture, Zipeng Road, Hefei, Anhui, 230601, P.R. China

Boundary Value Problems 2013, 2013:203  doi:10.1186/1687-2770-2013-203


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/203


Received:5 April 2013
Accepted:5 August 2013
Published:9 September 2013

© 2013 Xie; licensee Springer

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.

Abstract

Using the Kuratowski measure of noncompactness and progressive estimation method, we obtain the existence results of mild solutions for impulsive partial neutral second-order functional integro-differential equations with infinite delay in Banach spaces. The compactness condition of the impulsive term, some restrictive conditions on a priori estimation and noncompactness measure estimation have been deleted. Our conditions are simple and our results essentially improve and extend some known results. As applications, some examples are provided to illustrate the obtained results.

MSC: 34K30, 34K40, 35R10, 47D09.

Keywords:
impulsive partial neutral functional integro-differential equations; mild solutions; fixed point; Banach spaces

1 Introduction

Consider the following impulsive partial neutral second-order functional integro-differential systems with infinite delay in a Banach space X:

{ d d t [ x ( t ) + g 1 ( t , x t , 0 t k 1 ( t , s , x s ) d s ) ] = A x ( t ) + g 2 ( t , x t , 0 t k 2 ( t , s , x s ) d s ) , t [ 0 , b ] , Δ x ( t i ) = I i 1 ( x t i ) , Δ x ( t i ) = I i 2 ( x t i ) , i = 1 , 2 , , n , x 0 = φ ß , x ( 0 ) = z X , (1)

{ d d t [ x ( t ) + g ( t , x t , x t ) ] = A x ( t ) + 0 t f ( s , x s , x s ) d s , t [ 0 , b ] , Δ x ( t i ) = I i 1 ( x t i , x t i ) , Δ x ( t i ) = I i 2 ( x t i , x t i ) , i = 1 , 2 , , n , x 0 = φ ß , x 0 = ψ ß , (2)

where A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators, ( C ( t ) ) t R , on X. In both cases, the history x t , x t : ( , 0 ] X , x t ( θ ) = x ( t + θ ) and x t ( θ ) = x ( t + θ ) belongs to some abstract phase space ß defined axiomatically; g, f, g j , k j , I i j ( j = 1 , 2 ) are appropriate functions; 0 = t 0 < t 1 < < t n < t n + 1 = b are fixed numbers and the symbol x ( t i ) represents the jump of the function x at t i , which is defined by x ( t i ) = x ( t i + ) x ( t i ) , i = 1 , 2 , , n .

The study of impulsive functional differential equations is linked to their utility in simulating processes and phenomena subject to short-time perturbations during their evolution. The perturbations are performed discretely and their duration is negligible in comparison with the total duration of the processes and phenomena. Now impulsive partial neutral functional differential equations have become an important object of investigation in recent years stimulated by their numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, etc. With regard to this matter, we refer the reader to [1-12] and references therein. However, in order to obtain the existence of solutions in these study papers, the compactness condition on the associated family of operators and the impulsive term, some similar restrictive conditions on a priori estimation,

μ = K b i = 1 n ( N c i 1 + N ¯ c i 2 ) < 1 , K b 1 μ 0 t ( N m g ( s ) + N ¯ m f ( s ) ) d s < c d s W ( s ) , (3)

K b [ N L g b + i = 1 n ( N L i 1 + N ¯ L i 2 ) + N ¯ lim inf ξ + W f ( ξ ) ξ 0 t m f ( s ) ] d s < 1 , (4)

are used. In [13-16], authors used a strict set contraction mapping fixed point theorem without the compactness assumption on the associated family of operators to obtain the existence results of system (1) when g i ( t , x t ) is not an integral operator and the following system:

{ d d t [ x ( t ) + g ( t , x t , x ( t ) ) ] = A x ( t ) + f ( t , x t , x ( t ) ) , t [ 0 , b ] , t t i , Δ x ( t i ) = I i 1 ( x t i , x ( t i ) ) , Δ x ( t i ) = I i 2 ( x t i , x ( t i ) ) , i = 1 , 2 , , n , x 0 = φ ß , x ( 0 ) = z , (5)

improved and generalized some results in [1,7]. However, the compactness condition of the impulsive terms I i j ( ) , some similar restrictive conditions on a priori estimation (3), (4) and the restrictive condition on measure of noncompactness estimation

K b [ N L g b + i = 1 n ( N L i 1 + N ˜ L i 2 ) ] + 0 b η ( s ) d s < 1 (6)

are used in [13-16]. So far we have not seen the existence results of system (2).

In this paper, using the Kuratowski measure of noncompactness and progressive estimation method, we obtain the existence results of mild solutions of impulsive partial neutral second-order functional integro-differential systems (1) and (2). The compactness condition of impulsive terms I i j ( ) , some restrictive conditions on a priori estimation and measure of noncompactness estimation (3), (4) and (6) have been deleted. Our conditions are simple and our results essentially improve and extend some corresponding results in papers [1,2,13,14]. As applications, some examples are provided to illustrate the obtained results.

2 Preliminaries

In this paper, X is a Banach space with the norm and A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators, ( C ( t ) ) t R , on X and S ( t ) is the sine function associated with ( C ( t ) ) t R , which is defined by S ( t ) x = 0 t C ( s ) x d s , x X , t R . We designate by N, N ¯ certain constants such that C ( t ) N and S ( t ) N ¯ for every t J = [ 0 , b ] . We refer the reader to [17] for the necessary concepts about cosine functions. Next, we only mention a few results and notations needed to establish our results. As usual we denote by [ D ( A ) ] the domain of A endowed with the graph norm x A = x + A x , x D ( A ) . Moreover, the notation E stands for the space formed by the vector x X , for which the function C ( ) x is of class C 1 . It was proved by Kisyński [18] that the space E endowed with the norm

x E = x + sup 0 t b AS ( t ) x , x E ,

is a Banach space. The operator-valued function G ( t ) = [ C ( t ) S ( t ) AS ( t ) C ( t ) ] is a strongly continuous group of linear operators on the space E × X generated by the operator A = [ 0 I A 0 ] defined on D ( A ) × E . It follows from this that AS ( t ) : E X is a bounded linear operator and that AS ( t ) x 0 ( t 0 ) for each x E . Furthermore, if x : [ 0 , ) X is a locally integrable function, then z ( t ) = 0 t S ( t s ) x ( s ) d s defines an E-valued continuous function. This is a consequence of the fact that

0 t G ( t s ) [ 0 x ( s ) ] d s = [ 0 t S ( t s ) x ( s ) d s 0 t C ( t s ) x ( s ) d s ]

defines an ( E × X ) -valued continuous function. Next, we denote N 1 = sup t J AS ( t ) L ( E , X ) , in which L ( E , X ) stands for the Banach space of bounded linear operators from E into X, and we abbreviate this notation to L ( X ) when E = X .

To describe appropriately our system (1), we say that the function u : [ σ , τ ] X is a normalized piecewise continuous function on [ σ , τ ] if u is piecewise continuous and left continuous on ( σ , τ ] . We denote by P C ( [ σ , τ ] , X ) the space formed by the normalized piecewise continuous functions from [ σ , τ ] into X. In particular, we introduce the space PC formed by all functions u : [ 0 , b ] X such that u is continuous at t t i , u ( t i ) = u ( t i ) and u ( t i + ) exists for all i = 1 , 2 , , n . It is clear that PC endowed with the norm x p c = sup t J x ( t ) is a Banach space.

For x P C , let

x ˜ i ( t ) = { x ( t ) , t ( t i , t i + 1 ] , x ( t i + ) , t = t i , i = 0 , 1 , , n .

Then x ˜ C ( [ t i , t i + 1 ] , X ) . Moreover, for V P C and i = 0 , 1 , , n , we use the notation V ˜ i for V ˜ i = { x ˜ i : x V } . From Lemma 1.1 in [1], we know that a set V P C is relatively compact if and only if each set V ˜ i = { x ˜ i : x V } is relatively compact in C ( [ t i , t i + 1 ] , X ) ( i = 0 , 1 , , n ).

For system (2), we give the precise meaning of the derivative in (2). We say that x P C is piecewise smooth if x is continuously differentiable at t t i , i = 1 , 2 , , n , and for t = t i , i = 1 , 2 , , n , there are the right derivative x ( t i + ) = lim s 0 + x ( t i + s ) x ( t i ) s and the left derivative x ( t i ) = lim s 0 x ( t i + s ) x ( t i ) s . Furthermore, we denote the space by P C 1 = { x P C : x ( t )  is continuous at  t t i , x ( t i )  and  x ( t i + )  exist , i = 1 , 2 , , n } . Then P C 1 endowed with the norm u 1 = u p c + u p c is a Banach space.

In this work we employ an axiomatic definition of the phase space ß introduced by Hale and Kato [19] which appropriated to treat retarded impulsive differential equations. For other abstract phase spaces, we can refer to [20,21].

Definition 2.1[19]

The phase space ß is a linear space of functions mapping ( , 0 ] into X endowed with a seminorm ß . We assume that ß satisfies the following axioms.

(A) If x : ( , σ + b ] X ( b > 0 ) is such that x σ ß and x | [ σ , σ + b ] P C ( [ σ , σ + b ] , X ) , then for every t [ σ , σ + b ) the following conditions hold:

(i) x t is in ß,

(ii) x ( t ) H x t ß ,

(iii) x t ß K ( t σ ) sup { x ( s ) : σ s t } + M ( t σ ) x σ ß , where H > 0 is a constant; K , M : [ 0 , ) [ 1 , ) , K is continuous, M is locally bounded and H, K, M are independent of x ( ) .

(B) The space ß is complete.

In this paper we denote by α ( ) the Kuratowski measure of noncompactness of X, by α c ( ) the Kuratowski measure of noncompactness of C ( [ 0 , b ] , X ) and by α p c ( ) the Kuratowski measure of noncompactness of PC.

The following lemma is easy to get.

Lemma 2.2If the cosine function family C ( t ) , t R , is equicontinuous and η L ( [ 0 , b ] , R + ) , then the set

{ 0 t S ( t s ) u ( s ) d s , u ( s ) η ( s )  for a.e.  s [ 0 , b ] }

is equicontinuous for t [ 0 , b ] .

Lemma 2.3[16,22]

(1) If W P C ( [ 0 , b ] , X ) is bounded, then α ( W ( t ) ) α p c ( W ) for any t [ 0 , b ] , where W ( t ) = { u ( t ) : u W } X .

(2) IfWis piecewise equicontinuous on [ 0 , b ] , then α ( W ( t ) ) is piecewise continuous for t [ 0 , b ] and α p c ( W ) = sup { α ( W ( t ) ) , t [ 0 , b ] } .

(3) If W P C ( [ 0 , b ] , X ) is bounded and piecewise equicontinuous, then α ( W ( t ) ) is piecewise continuous for t [ 0 , b ] and

α ( 0 t W ( s ) d s ) 0 t α ( W ( s ) ) d s , t [ 0 , b ] ,

where 0 t W ( s ) d s = { 0 t x ( s ) d s : x W } .

(4) If W P C 1 ( [ 0 , b ] , X ) is bounded and the elements of W are equicontinuous on each J i = ( t i , t i + 1 ] ( i = 0 , 1 , , n ), then

α p c 1 ( W ) = max { sup t [ 0 , b ] α ( W ( t ) ) , sup t [ 0 , b ] α ( W ( t ) ) } ,

where α p c 1 ( ) denotes the Kuratowski measure of noncompactness in the space P C 1 ( [ 0 , b ] , X ) .

Lemma 2.4[23]

Let h : [ 0 , b ] E be an integrable function such that h P C . Then the function v ( t ) = 0 t C ( t s ) h ( s ) d s belongs to P C 1 , the function s AS ( t s ) h ( s ) is integrable on [ 0 , t ] for t [ 0 , b ] and

v ( t ) = h ( t ) + A 0 t S ( t s ) h ( s ) d s = h ( t ) + 0 t AS ( t s ) h ( s ) d s , t [ 0 , b ] .

Lemma 2.5[24]

Let V = { x n } L 1 ( [ a , b ] , X ) . If there is σ L 1 ( [ a , b ] , R + ) ( R + = [ 0 , + ) ) such that x n ( t ) σ ( t ) for x V and a.e. t [ a , b ] , then α ( V ( t ) ) L 1 ( [ a , b ] , R + ) and

α ( { a t x n ( s ) d s : n N } ) 2 a t α ( V ( s ) ) d s , t [ a , b ] .

Lemma 2.6[25] (Mónch)

LetXbe a Banach space, Ω be a bounded open subset inXand 0 Ω . Assume that the operator F : Ω ¯ X is continuous and satisfies the following conditions:

(1) x λ F x , λ ( 0 , 1 ) , x Ω ,

(2) Dis relatively compact if D c o ¯ ( { 0 } F ( D ) ) for any countable set D Ω ¯ .

ThenFhas a fixed point in Ω ¯ .

3 Main results

Firstly, we discuss the existence of mild solutions for the impulsive second-order system (1).

Definition 3.1 A function x : ( , b ] X is said to be a mild solution of system (1) if x 0 = φ , x ( ) | J P C and

x ( t ) = C ( t ) φ ( 0 ) + S ( t ) ( z + g 1 ( 0 , φ , 0 ) ) 0 t C ( t s ) g 1 ( s , x s , 0 s k 1 ( s , r , x r ) d r ) d s + 0 t S ( t s ) g 2 ( s , x s , 0 s k 2 ( s , r , x r ) d r ) d s + 0 < t i < t C ( t t i ) I i 1 ( x t i ) + 0 < t i < t S ( t t i ) I i 2 ( x t i ) , t J . (7)

For system (1), we make the following hypotheses.

(H1) The functions g j : J × ß × X X ( j = 1 , 2 ) satisfy the following conditions:

(1) For every ( ϕ , x ) ß × X , g j ( , ϕ , x ) are strongly measurable and g j ( t , , ) are continuous for every t J ;

(2) There are integrable functions p j : J R + ( j = 1 , 2 ) such that

g j ( t , ϕ , x ) p j ( t ) ( ϕ ß + x ) , t J , ϕ ß , x X ;

(3) For any bounded set V P C , there are integrable functions γ j : J R + ( j = 1 , 2 ) such that

α ( g j ( t , V t , V ( t ) ) ) γ j ( t ) [ α ( V t ) + α ( V ( t ) ) ] , t J ,

where V t = { x t : x V } ß ( t J ).

(H2) k j : Δ × ß X ( j = 1 , 2 , Δ = { ( t , s ) J × J : 0 s t 1 } ) satisfies the following conditions:

(1) For every ϕ ß , k j ( , , ϕ ) are strongly measurable and k j ( t , s , ) are continuous for every ( t , s ) Δ ;

(2) There are continuous functions q j : Δ R + ( j = 1 , 2 ) such that

k j ( t , s , ϕ ) q j ( t , s ) ϕ ß , ( t , s ) Δ , ϕ ß ;

(3) For any bounded set V P C , there are continuous functions μ j : Δ R + ( j = 1 , 2 ) such that

α ( k j ( t , s , V s ) ) μ j ( t , s ) α ( V s ) , ( t , s ) Δ .

(H3) The functions I i j : ß X ( i = 1 , , n , j = 1 , 2 ) are continuous and there are constants c i j 0 , d i j > 0 such that

I i j ( ϕ ) c i j ϕ ß + d i j , ϕ ß .

Let the function y : ( , b ] X be defined by y 0 = φ and

y ( t ) = C ( t ) φ ( 0 ) + S ( t ) ( z + g 1 ( 0 , φ , 0 ) ) , t J .

Theorem 3.2Suppose that the cosine function family C ( t ) , t R , is equicontinuous, g 1 , g 2 satisfy the condition (H1), (H2) and (H3) are satisfied. Then the impulsive second-order system (1) has at least one mild solution.

Proof Let S ( b ) be the space S ( b ) = { x : ( , b ] X , x 0 = 0 , x | J P C } endowed with the supremum norm b . The map F : S ( b ) S ( b ) is defined by

( F x ) ( t ) = { 0 , t 0 , 0 t C ( t s ) g 1 ( s , x s + y s , 0 s k 1 ( s , r , x r + y r ) d r ) d s + 0 t S ( t s ) g 2 ( s , x s + y s , 0 s k 2 ( s , r , x r + y r ) d r ) d s + 0 < t i < t C ( t t i ) I i 1 ( x t i + y t i ) + 0 < t i < t S ( t t i ) I i 2 ( x t i + y t i ) , t J . (8)

Clearly, x t + y t ß x t ß + sup t J y t ß K b x t + M , where K b = sup 0 t b K ( t ) , M = sup t J y t ß , x t = sup 0 s t x ( s ) . Thus F is well defined with values in S ( b ) . In addition, from the axioms of phase space, the Lebesgue dominated convergence theorem and the conditions (H1), (H2) and (H3), we can show that F is continuous (see [5]). It is easy to see that if x is a fixed point of F, then x + y is a mild solution of system (1).

Firstly, we show that the set

Ω 0 = { x S ( b ) : x = λ F x  for some  λ ( 0 , 1 ) }

is bounded. In fact, if x Ω 0 , then there exists a λ ( 0 , 1 ) such that x = λ F x .

When t J 0 = [ 0 , t 1 ] , notice that x t ß K b x t and x t is continuous nondecreasing on J 0 . We have, by (8) and (H1),

x ( t ) N 0 t p 1 ( s ) ( x s + y s ß + 0 s q 1 ( s , r ) x r + y r ß d r ) d s + N ¯ 0 t p 2 ( s ) ( x s + y s ß + 0 s q 2 ( s , r ) x r + y r ß d r ) d s N M 0 t 1 ( p 1 ( s ) + 0 s q 1 ( s , r ) d r ) d s + N K b 0 t ( p 1 ( s ) + 0 s q 1 ( s , r ) d r ) x s d s + N ¯ M 0 t 1 ( p 2 ( s ) + 0 s q 2 ( s , r ) d r ) d s + N ¯ K b 0 t ( p 2 ( s ) + 0 s q 2 ( s , r ) d r ) x s d s ( N ¯ + N ) M 0 t 1 ( p 1 ( s ) + p 2 ( s ) + 0 s q 1 ( s , r ) d r + 0 s q 2 ( s , r ) d r ) d s + ( N ¯ + N ) K b 0 t ( p 1 ( s ) + p 2 ( s ) + 0 s q 1 ( s , r ) d r + 0 s q 2 ( s , r ) d r ) x s d s . (9)

Consequently,

x t ( N ¯ + N ) M 0 t 1 ( p 1 ( s ) + p 2 ( s ) + 0 s q 1 ( s , r ) d r + 0 s q 2 ( s , r ) d r ) d s + ( N ¯ + N ) K b 0 t ( p 1 ( s ) + p 2 ( s ) + 0 s q 1 ( s , r ) d r + 0 s q 2 ( s , r ) d r ) x s d s . (10)

By well-known Gronwall’s lemma and (10), there are constants G 0 > 0 independent of x and λ ( 0 , 1 ) such that x ( t ) G 0 and x t ß K b G 0 , t J 0 . It follows from this and the condition (H3) that

I 1 j ( x t 1 + y t 1 ) c 1 j ( K b G 0 + M ) + d 1 j = δ j ( j = 1 , 2 ) , x ( t 1 + ) = x ( t 1 ) + I 1 1 ( x t 1 + y t 1 ) G 0 + δ 1 .

Nextly, when t J 1 = ( t 1 , t 2 ] , let

u ( t ) = { x ( t ) , t ( t 1 , t 2 ] , x ( t 1 + ) , t = t 1 .

Then u C ( [ t 1 , t 2 ] , X ) . Similar to (10), we get

u ( t ) ( N ¯ + N ) M 0 t 2 ( p 1 ( s ) + p 2 ( s ) + 0 s q 1 ( s , r ) d r + 0 s q 2 ( s , r ) d r ) d s + ( N ¯ + N ) K b G 0 0 t 1 ( p 1 ( s ) + p 2 ( s ) + 0 s q 1 ( s , r ) d r + 0 s q 2 ( s , r ) d r ) d s + ( N ¯ + N ) K b t 1 t ( p 1 ( s ) + p 2 ( s ) + 0 s q 1 ( s , r ) d r + 0 s q 2 ( s , r ) d r ) u s d s + N I 1 1 ( x t 1 + y t 1 ) + N ¯ I 1 2 ( x t 1 + y t 1 ) , (11)

where x t sup 0 s t 1 x ( s ) + sup t 1 s t u ( s ) = : x t 1 + v ( t ) . Equation (11) implies that

v ( t ) N δ 1 + N ¯ δ 2 + ( N ¯ + N ) ( M + K b G 0 ) 0 t 2 a ( s ) d s + ( N ¯ + N ) K b t 1 t a ( s ) v ( s ) d s , t [ t 1 , t 2 ] , (12)

where a ( s ) = p 1 ( s ) + p 2 ( s ) + 0 s q 1 ( s , r ) d r + 0 s q 2 ( s , r ) d r . Using Gronwall’s lemma once again and (12), there is a constant G 1 > 0 independent of v and λ ( 0 , 1 ) such that v ( t ) G 1 , t [ t 1 , t 2 ] . Thence x ( t ) G 1 and x t ß K b ( G 0 + G 1 ) for t J 1 .

It is similar to the proof above, there is a constant G i > 0 independent of x and λ ( 0 , 1 ) such that x ( t ) G i , t J i ( i = 2 , 3 , , n ). Let G = max { G 0 , G 1 , , G n } , then x ( t ) G , t J , i.e., Ω 0 is bounded.

Lastly, we verify that all the conditions of Lemma 2.6 are satisfied. Let R > G and

Ω R = { x S ( b ) : x b < R } .

Then Ω R is a bounded open set and 0 Ω . Since R > G , we know that x λ F x for any x Ω R and λ ( 0 , 1 ) .

Nextly, let V Ω ¯ R be a countable set and V c o ¯ ( { 0 } F ( V ) ) . Then

V ( t ) c o ¯ ( { 0 } F ( V ) ( t ) ) , t [ 0 , b ] . (13)

It follows from (H1)-(H3) and Lemma 2.2 that F ( V ) is equicontinuous on every interval J ¯ i = [ t i , t i + 1 ] ( i = 0 , 1 , , n ), which together with (13) implies that V is equicontinuous on every J ¯ i ( i = 0 , 1 , , n ).

When t J 0 = [ 0 , t 1 ] , by the property of noncompactness measure, (H1)(3), (H2)(3) and Lemma 2.5, we have

α ( V ( t ) ) α ( F ( V ) ( t ) ) 2 N 0 t α ( g 1 ( s , V s + y s , 0 s k 1 ( s , r , V r + y r ) d r ) ) d s + 2 N ¯ 0 t α ( g 2 ( s , V s + y s , 0 s k 2 ( s , r , V r + y r ) d r ) ) d s 2 ( N ¯ + N ) 0 t [ ( γ 1 ( s ) + γ 2 ( s ) ) α ( V s ) + 2 0 s ( μ 1 ( s , r ) + μ 2 ( s , r ) ) α ( V r ) d r ] d s 2 ( N ¯ + N ) K b 0 t [ k = 1 2 γ k ( s ) + 2 0 s i = 1 2 μ k ( s , r ) d r ] sup 0 τ s α ( V ( τ ) ) d s , (14)

where α ( V t ) sup 0 s t α ( V ( s ) ) . Let m ( t ) = sup 0 s t α ( V ( s ) ) , t J 0 . Lemma 2.3 implies that m C ( J 0 , R + ) and

m ( t ) 2 ( N ¯ + N ) K b 0 t [ γ 1 ( s ) + γ 2 ( s ) + 2 0 s ( μ 1 ( s , r ) + μ 2 ( s , r ) ) d r ] m ( s ) d s , t J 0 .

From this and Gronwall’s lemma, we know that m ( t ) = 0 and α ( V ( t ) ) = 0 , t J 0 . Therefore V is a relative compact set in C ( J 0 , X ) . Since

0 α ( V t 1 ) α c ( V ) = sup 0 t t 1 α ( V ( t ) ) = 0

and I 1 j ( ) is continuous, α ( V t 1 + y t 1 ) α ( V t 1 ) = 0 , α ( I 1 j ( V t 1 + y t 1 ) ) = 0 ( j = 1 , 2 ).

When t J ¯ 1 = [ t 1 , t 2 ] , similar to (14), it is easy to get

α ( V ( t ) ) 2 ( N ¯ + N ) K b 0 t γ ( s ) sup 0 τ s α ( V ( τ ) ) d s + N α ( I 1 1 ( V t 1 + y t 1 ) ) + N ¯ α ( I 1 2 ( V t 1 + y t 1 ) ) 2 ( N ¯ + N ) K b t 1 t γ ( s ) sup t 1 τ s α ( V ( τ ) ) d s , t J ¯ 1 , (15)

where γ ( s ) = γ 1 ( s ) + γ 2 ( s ) + 2 0 s ( μ 1 ( s , r ) + μ 2 ( s , r ) ) d r . Let q ( t ) = sup t 1 s t α ( V ( s ) ) , t J ¯ 1 . Equation (15) implies that

q ( t ) 2 ( N ¯ + N ) t 1 t γ ( s ) q ( s ) d s , t J ¯ 1 .

Therefore α ( V ( t ) ) = 0 , t J ¯ 1 and V is a relative compact set in C ( J ¯ 1 , X ) .

Similarly, we can show that V is a relative compact set in C ( J ¯ i , X ) ( i = 2 , 3 , , n ), so V is a relative compact set in S ( b ) . Lemma 2.6 concludes that F has a fixed point in Ω ¯ R . Let x be a fixed point of F on S ( b ) . Then z = x + y is a mild solution of system (1). □

Nextly, we discuss the existence of mild solutions for the impulsive system (2).

Definition 3.3 A function x : ( , b ] X is said to be a mild solution of system (2) if x 0 = ϕ , x 0 = ψ , x ( ) | J P C 1 and

x ( t ) = C ( t ) ϕ ( 0 ) + S ( t ) ( ψ ( 0 ) + g ( 0 , ϕ , ψ ) ) 0 t C ( t s ) g ( s , x s , x s ) d s + 0 t S ( t s ) 0 s f ( r , x r , x r ) d r d s + 0 < t i < t C ( t t i ) I i 1 ( x t i , x t i ) + 0 < t i < t S ( t t i ) I i 2 ( x t i , x t i ) , t J . (16)

Differentiate (16) to get

x ( t ) = AS ( t ) ϕ ( 0 ) + C ( t ) ( ψ ( 0 ) + g ( 0 , ϕ , ψ ) ) g ( t , x t , x t ) 0 t A S ( t s ) g ( s , x s , x s ) d s + 0 t C ( t s ) 0 s f ( r , x r , x r ) d r d s + 0 < t i < t AS ( t t i ) I i 1 ( x t i , x t i ) + 0 < t i < t C ( t t i ) I i 2 ( x t i , x t i ) , t J . (17)

Let functions y , y : ( , b ] X be defined by y 0 = φ , y 0 = ψ and

y ( t ) = C ( t ) φ ( 0 ) + S ( t ) ψ ( 0 ) , y ( t ) = AS ( t ) φ ( 0 ) + C ( t ) ψ ( 0 ) , t J .

Clearly,

y t ß K b y b + M b φ ß = M ¯ , y t ß K b y b + M b ψ ß = M ,

where y b = sup 0 t b y ( t ) , y b = sup 0 t b y ( t ) .

Let S 1 ( b ) be the space S 1 ( b ) = { x : ( , b ] X : x 0 = 0 , x 0 = 0 , x ( ) | J P C 1 } endowed with the supremum norm 1 b .

We make the following hypotheses for convenience.

(Hf) f : J × ß × ß X satisfies the following conditions:

(1) For every x S 1 ( b ) , the function t f ( t , x t , x t ) is strongly measurable and f ( t , , ) is continuous for every t J ;

(2) There is an integrable function p : J R + such that

f ( t , u , v ) p ( t ) ( u ß + v ß ) , t J , u , v ß ;

(3) For any bounded set V P C 1 , there is an integrable function μ : J R + such that

α ( f ( t , V t , V t ) ) μ ( t ) ( α ( V t ) + α ( V t ) ) , t J ,

where V t = { x t : x V } , V t = { x t : x V } ß ( t J ), V P C .

(Hg) g : J × ß × ß E satisfies the following conditions:

(1) The function g ( ) is continuous, there are constants c > 0 , d 0 such that c K b < 1 and

g ( t , u , v ) E c ( u ß + v ß ) + d , t J , u , v ß ;

(2) For every bounded set Q S 1 ( b ) , the set of functions { ( ω x ˜ ) i ( t ) : x Q } is uniformly equicontinuous on J ¯ i = [ t i , t i + 1 ] for every i = 0 , 1 , , n , where ω x ( t ) = g ( t , x t , x t ) ;

(3) For any bounded set V P C 1 , α ( g ( t , V t , V t ) ) c ( α ( V t ) + α ( V t ) ) , t J .

(HI) The functions I i 1 : ß × ß E , I i 2 : ß × ß X ( i = 1 , , n ) are continuous and there are constants c i j 0 , d i j 0 such that

I i 1 ( u , v ) E c i 1 ( u ß + v ß ) + d i 1 , I i 2 ( u , v ) c i 2 ( u ß + v ß ) + d i 2 , u , v ß .

Theorem 3.4Let the conditions (Hf), (Hg) and (HI) be satisfied, the cosine function family C ( t ) , t R , be equicontinuous and φ ( 0 ) E . Then system (2) has at least one mild solution.

Proof Let the function z : ( , 0 ] X be defined by z 0 = x 0 , z ( t ) = x ( t ) , t J , the map Γ = ( Γ 1 , Γ 2 ) : S ( b ) × S ( b ) S ( b ) be defined by

Γ 1 ( x , z ) ( t ) = { 0 , t 0 , S ( t ) g ( 0 , φ , ψ ) 0 t C ( t s ) g ( s , x s + y s , z s + y s ) d s + 0 t S ( t s ) 0 s f ( r , x r , x r ) d r d s + 0 < t i < t C ( t t i ) I i 1 ( x t i + y t i , z t i + y t i ) + 0 < t i < t S ( t t i ) I i 2 ( x t i + y t i , z t i + y t i ) , t J , (18)

and Γ 1 ( x , z ) ( t ) = Γ 2 ( x , z ) ( t ) ,

Γ 2 ( x , z ) ( t ) = { 0 , t 0 , C ( t ) g ( 0 , φ , ψ ) g ( t , x t + y t , z t + y t ) 0 t A S ( t s ) g ( s , x s + y s , z s + y s ) d s + 0 t C ( t s ) 0 s f ( r , x r , x r ) d r d s + 0 < t i < t AS ( t t i ) I i 1 ( x t i + y t i , z t i + y t i ) + 0 < t i < t C ( t t i ) I i 2 ( x t i + y t i , z t i + y t i ) , t J . (19)

The product space S ( b ) × S ( b ) is endowed with the norm ( x , z ) b = x b + z b . Then Γ 1 , Γ 2 are well defined and with values in S ( b ) . In addition, from the axioms of phase space, the Lebesgue dominated convergence theorem and the conditions (Hf), (Hg) and (HI), we can show that Γ = ( Γ 1 , Γ 2 ) is continuous. It is easy to see that if ( x , z ) is a fixed point of Γ, then x + y is a mild solution of system (2).

Firstly, we show that the set

Ω 0 = { ( x , z ) S ( b ) × S ( b ) : ( x , z ) = λ Γ ( x , z )  for some  λ ( 0 , 1 ) }

is bounded. If x Ω 0 , there exists a λ ( 0 , 1 ) such that x = λ Γ 1 ( x , z ) and z = λ Γ 2 ( x , z ) .

When t J 0 = [ 0 , t 1 ] , it follows from (18), (19) and (Hf)(2), (Hg)(1), (HI) that

x ( t ) Γ 1 ( x , z ) ( t ) N ¯ g ( 0 , φ , ψ ) E + ( N + N ¯ ) 0 t ( g ( s , x s + y s , z s + y s ) + 0 s f ( r , x r + y r , z r + y r ) d r ) d s N ¯ g ( 0 , φ , ψ ) E + N 0 t [ c ( x s ß + z s ß + M ¯ + M ) + d ] d s + N ¯ 0 t 0 s p ( r ) ( x r ß + z r ß + M ¯ + M ) d r d s N ¯ g ( 0 , φ , ψ ) E + ( M ¯ + M ) 0 t 1 ( N c + N ¯ 0 s p ( r ) d r ) d s + N b d + ( N c + N ¯ ) K b 0 t ( 1 + 0 s p ( r ) d r ) ( x s + z s ) d s , (20)

z ( t ) Γ 2 ( x , z ) ( t ) N g ( 0 , φ , ψ ) E + c ( x t + y t ß + z t + y t ß ) + d + ( M ¯ + M ) 0 t 1 ( N 1 c + N 0 s p ( r ) d r ) d s + N 1 b d + ( N 1 c + N ) K b 0 t ( 1 + 0 s p ( r ) d r ) ( x s + z s ) d s N g ( 0 , φ , ψ ) E + ( M ¯ + M ) [ c + 0 t 1 ( N 1 c + N 0 s p ( r ) d r ) d s ] + ( N 1 b + 1 ) d + c K b ( x t + z t ) + ( N 1 c + N ) K b 0 t ( 1 + 0 s p ( r ) d r ) ( x s + z s ) d s . (21)

Equations (20) and (21) imply that

x t + z t 1 1 c K b [ ( N + N ¯ ) g ( 0 , φ , ψ ) E + ( N b + N 1 b + 1 ) d + ( M ¯ + M ) ( c + ( N + N 1 ) c t 1 + ( N + N ¯ ) 0 t 1 0 s p ( r ) d r d s ) ] + ( N c + N ¯ + N 1 c + N ) K b 1 c K b 0 t ( 1 + 0 s p ( r ) d r ) ( x s + z s ) d s . (22)

Since x t + z t C ( J 0 , X ) , by Gronwall’s lemma and (22), there is a constant G 0 > 0 such that x t + z t G 0 , t J 0 . Therefore x ( t ) + z ( t ) G 0 , t J 0 and x t ß K b G 0 , z t ß K b G 0 , t J 0 . It follows from this and the condition (HI) that

I 1 j ( x t 1 + y t 1 , z t 1 + y t 1 ) E c 1 j ( 2 K b G 0 + M ¯ + M ) + d 1 j = : η j ( j = 1 , 2 ) , x ( t 1 + ) = x ( t 1 ) + I 1 1 ( x t 1 + y t 1 , z t 1 + y t 1 ) G 0 + η 1 , z ( t 1 + ) = z ( t 1 ) + I 1 2 ( x t 1 + y t 1 , z t 1 + y t 1 ) G 0 + η 2 .

Nextly, when t J 1 = ( t 1 , t 2 ] , let

u ( t ) = { x ( t ) , t ( t 1 , t 2 ] , x ( t 1 + ) , t = t 1 , v ( t ) = { z ( t ) , t ( t 1 , t 2 ] , z ( t 1 + ) , t = t 1 .

Then u , v C ( [ t 1 , t 2 ] , X ) . Similar to (20) and (21), we get

u ( t ) N ¯ g ( 0 , φ , ψ ) E + ( N + N ¯ ) 0 t ( c + 0 s p ( r ) d r ) ( x s + z s + M + M ) d s + N b d + N I 1 1 ( x t 1 + y t 1 , z t 1 + y t 1 ) E + N ¯ I 1 2 ( x t 1 + y t 1 , z t 1 + y t 1 ) N ¯ g ( 0 , φ , ψ ) E + ( N + N ¯ ) [ ( 2 K b G 0 + M ¯ + M ) 0 t 1 ( c + 0 s p ( r ) d r ) d s + N b d + ( 2 K b G 0 + M ¯ + M ) t 1 t 2 ( c + 0 s p ( r ) d r ) d s ] + N η 1 + N ¯ η 2 + ( N + N ¯ ) K b t 1 t ( c + 0 s p ( r ) d r ) ( sup t 1 τ s u ( τ ) + sup t 1 τ s v ( τ ) ) d s , (23)

v ( t ) N g ( 0 , φ , ψ ) E + c ( x t + y t ß + z t + y t ß ) + d + N 1 η 1 + N η 2 + N 1 b d + ( N + N 1 ) 0 t ( c + 0 s p ( r ) d r ) ( K b x s + K b z s + M ¯ + M ) d s N g ( 0 , φ , ψ ) E + c ( 2 K b G 0 + M ¯ + M ) + N 1 η 1 + N η 2 + ( N 1 b + 1 ) d + ( N + N 1 ) ( 2 K b G 0 + M ¯ + M ) 0 t 2 ( c + 0 s p ( r ) d r ) d s + ( N + N 1 ) K b t 1 t ( c + 0 s p ( r ) d r ) ( sup t 1 τ s u ( τ ) + sup t 1 τ s v ( τ ) ) d s + c K b ( sup t 1 s t u ( s ) + sup t 1 s t v ( s ) ) . (24)

We have, by (23) and (24),

sup t 1 s t u ( s ) + sup t 1 s t v ( s ) e 1 + e 2 1 c K b + ( 2 N + N ¯ + N 1 ) K b 1 c K b t 1 t ( c + 0 s p ( r ) d r ) ( sup t 1 τ s u ( τ ) + sup t 1 τ s v ( τ ) ) d s , t [ t 1 , t 2 ] , (25)

where

e 1 = ( N + N ¯ ) [ g ( 0 , φ , ψ ) E + η 1 + η 2 + ( 2 K b G 0 + M ¯ + M ) 0 t 2 ( c + 0 s p ( r ) d r ) d s ] , e 2 = ( 2 K b G 0 + M ¯ + M ) [ c + ( N + N 1 ) 0 t 2 ( c + 0 s p ( r ) d r ) d s ] e 2 = + N 1 ( b d + η 1 ) + N ( b d + η 2 ) .

Using Gronwall’s lemma once again and (25), there is a constant G 1 > 0 such that u ( t ) + v ( t ) G 1 , t [ t 1 , t 2 ] , and so x ( t ) + z ( t ) G 1 , t J 1 .

It is similar to the proof above, there are constants G i > 0 such that x ( t ) + x ( t ) G i , t J i ( i = 2 , 3 , , n ). Let G = max { G 0 , G 1 , , G n } , then ( x , z ) b G and Ω 0 is bounded.

Let R > G and

Ω R = { ( x , z ) S ( b ) × S ( b ) : ( x , z ) b < R } .

Then Ω R is a bounded open set and ( 0 , 0 ) Ω . Since R > G , we know that ( x , z ) λ Γ ( x , z ) for any ( x , z ) Ω R and λ ( 0 , 1 ) .

Suppose that V Ω ¯ R is a countable set and V c o ¯ ( { ( 0 , 0 ) } Γ ( V ) ) . Let

V 1 = { x S ( b ) : z S ( b ) , ( x , z ) V } , V 2 = { z S ( b ) : x S ( b ) , ( x , z ) V } .

Then we have

V V 1 × V 2 c o ¯ ( { 0 } Γ 1 ( V ) ) × c o ¯ ( { 0 } Γ 2 ( V ) ) c o ¯ ( { 0 } Γ 1 ( V 1 × V 2 ) ) × c o ¯ ( { 0 } Γ 2 ( V 1 × V 2 ) ) . (26)

It follows from (18), (19) and (Hg)(2) that Γ j ( ( V 1 ˜ ) i × ( V 2 ˜ ) i ) ( j = 1 , 2 ) are equicontinuous on every interval J ¯ i ( i = 0 , 1 , , n ), which together with (26) implies that ( V k ˜ ) i ( k = 1 , 2 ) are equicontinuous on every interval J ¯ i .

In the following, we verify that the set V 1 , V 2 is relatively compact in PC. Without loss of generality, we do not distinguish V k | J i and V ˜ i , where V k | J i ( k = 1 , 2 ) is the restriction of V k on J i = ( t i , t i + 1 ] .

When t J 0 = [ 0 , t 1 ] , by the condition (Hf)(3), (Hg)(3) and Lemma 2.5, we have

α ( V 1 ( t ) ) α ( Γ 1 ( V 1 × V 2 ) ( t ) ) 2 N 0 t α ( g ( s , V 1 s + y s , V 2 s + y s ) ) d s + 2 N ¯ 0 t α ( 0 s f ( r , V 1 r + y r , V 2 r + y r ) d r ) d s 2 ( N + N ¯ ) 0 t ( c + 2 0 s μ ( r ) d r ) ( α ( V 1 s + y s ) + α ( V 2 s + y s ) ) d s 2 ( N ¯ + N ) K b 0 t ( c + 2 0 s μ ( r ) d r ) × ( sup 0 τ s α ( V 1 ( τ ) ) + sup 0 τ s α ( V 2 ( τ ) ) ) d s , (27)

α ( V 2 ( t ) ) α ( Γ 2 ( V 1 × V 2 ) ( t ) ) α ( g ( t , V 1 t + y t , V 2 t + y t ) ) + 2 ( N + N 1 ) K b 0 t ( c + 2 0 s μ ( r ) d r ) × ( sup 0 τ s α ( V 1 ( τ ) ) + sup 0 τ s α ( V 2 ( τ ) ) ) d s c K b ( sup 0 s t α ( V 1 ( s ) ) + sup 0 s t α ( V 2 ( s ) ) ) + 2 ( N + N 1 ) K b 0 t ( c + 2 0 s μ ( r ) d r ) × ( sup 0 τ s α ( V 1 ( τ ) ) + sup 0 τ s α ( V 2 ( τ ) ) ) d s . (28)

Since m j ( t ) = : sup 0 s t α ( V j ( s ) ) ( j = 1 , 2 ) are continuous nondecreasing on J 0 , (27) and (28) imply that

m 1 ( t ) + m 2 ( t ) 2 ( 2 N + N ¯ + N 1 ) K b 1 c K b 0 t ( c + 0 s μ ( r ) d r ) ( m 1 ( s ) + m 2 ( s ) ) d s . (29)

By Gronwall’s lemma and (29), we have α ( V k ( t ) ) = 0 ( k = 1 , 2 ), t J 0 . Lemma 2.3 implies that V k ( k = 1 , 2 ) is relatively compact in C ( J 0 , X ) . Note that α ( V j t 1 + y t 1 ) α ( V j t 1 ) K b sup 0 s t 1 α ( V j ( s ) ) = 0 and I 1 j ( , ) ( j = 1 , 2 ) is continuous, we have

α ( I 1 1 ( V 1 t 1 + y t 1 , V 2 t 1 + y t 1 ) ) = α ( I 1 2 ( V 1 t 1 + y t 1 , V 2 t 1 + y t 1 ) ) = 0 .

When t J ¯ 1 = [ t 1 , t 2 ] , it is similar to (27) and (28), we get

α ( V 1 ( t ) ) 2 ( N ¯ + N ) K b t 1 t ( c + 2 0 s μ ( r ) d r ) × ( sup t 1 τ s α ( V 1 ( τ ) ) + sup t 1 τ s α ( V 2 ( τ ) ) ) d s , (30)

α ( V 2 ( t ) ) 2 ( N + N 1 ) K b t 1 t ( c + 2 0 s μ ( r ) d r ) × ( sup t 1 τ s α ( V 1 ( τ ) ) + sup t 1 τ s α ( V 2 ( τ ) ) ) d s + c K b [ sup t 1 s t α ( V 1 ( s ) ) + sup t 1 s t α ( V 2 ( s ) ) ] . (31)

Equations (30) and (31) imply that

sup t 1 s t α ( V 1 ( s ) ) + sup t 1 s t α ( V 2 ( s ) ) 2 ( 2 N + N ¯ + N 1 ) K b 1 c K b t 1 t ( c + 2 0 s μ ( r ) d r ) ( sup t 1 τ s α ( V 1 ( τ ) ) + sup t 1 τ s α ( V 2 ( τ ) ) ) d s .

Consequently,