Open Access

Existence of solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions

Boundary Value Problems20122012:100

https://doi.org/10.1186/1687-2770-2012-100

Received: 5 May 2012

Accepted: 22 August 2012

Published: 7 September 2012

Abstract

Using the Mönch fixed point theorem, this article proves the existence of mild solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions in Banach spaces. Some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, and compactness conditions of evolution operators or compactness conditions on a nonlinear term f ( t , X r , X r , X r ) have been weakened. Our results extend and improve many known results.

MSC:34G20, 34K30.

Keywords

integro-differential functional evolution equation mild solution nonlocal conditions fixed point Banach spaces

1 Introduction

Let ( X , ) be a Banach space, C [ J , X ] = { x : J = [ 0 , a ] X , x ( t )  is continuous in  J } with the norm x C = sup t J x ( t ) . It is easy to verify that C [ J , X ] is a Banach space. The space of X-valued Bochner integrable functions on J with the norm x 1 = 0 a x ( s ) d s is denoted by L [ J , X ] . Consider the following nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions in a Banach space X ( IVP ) ,
(1.1)
(1.2)
where
( K x σ 2 ) ( t ) = 0 t k ( t , s , x ( σ 2 ( s ) ) ) d s , ( H x σ 3 ) ( t ) = 0 a h ( t , s , x ( σ 3 ( s ) ) ) d s ,
(1.3)

A is the generator of a strongly continuous semigroup in the Banach space X, and F ( t ) is a bounded operator for t J , x 0 X , f C [ J × X 3 , X ] , g : C [ J , X ] X , k C [ × X , X ] , = { ( t , s ) J × J : s t } , h C [ J × J × X , X ] , σ i C [ J , J ] and σ i ( t ) t ( i = 1 , 2 , 3 ).

For the existence of mild solutions of integro-differential functional evolution equations in abstract spaces, there are many research results, see [116], and references therein. In order to obtain the existence and controllability of mild solutions in these study papers, usually, some restricted conditions on a priori estimation and compactness conditions of an evolution operator or compactness conditions on f ( t , X r , X r , X r ) are used.

Recently, using a fixed point theorem, Haribhau Laxman Tidkey and Machindra Baburao Dhakne [1] have studied the existence of mild solutions of IVP (1.1)-(1.2) when σ i ( t ) = t ( i = 1 , 2 , 3 ), the compactness of the resolvent operator and the restricted condition
M 1 [ x 0 + G 1 + L r b + L K r b 2 + L K 1 b 2 + L H r b 2 + L H 1 b 2 + L 1 b ] r
with M 1 [ L b + L K b 2 + L H b 2 ] < 1 is used. Malar [17] and Shi [18] studied the existence of mild solutions of semilinear mixed type integrodifferential evolution equations with the equicontinuous semigroup
{ x ( t ) = A x ( t ) + f ( t , x ( t ) , 0 t a ( t , s ) k ( s , x ( s ) ) d s , x ( t ) = 0 a b ( t , s ) h ( s , x ( s ) ) d s ) , t [ 0 , a ] , x ( 0 ) = x 0 + g ( x ) .
(1.4)
Solvability of the scalar equation
m ( t ) = K 1 + K 2 0 t h ( s , m ( s ) , n ( s ) , q ( s ) ) d s , t J
and the restricted condition on measure of noncompactness estimation
0 t [ η 1 ( s ) + k 1 η 2 ( s ) + k 2 η 3 ( s ) ] d s K

are used in [17]. But estimations (3.15) and (3.21) in [18] seem to be incorrect, as they have no meaning.

In this paper, using the Mönch fixed point theorem, we investigate the existence of mild solutions of IVP (1.1)-(1.2). Some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, and compactness conditions of a resolvent operator or compactness conditions on a nonlinear term f ( t , X r , X r , X r ) have been weakened. Our results extend and improve some corresponding results in papers [14, 621].

2 Preliminaries

We will make the following assumptions:

( H 1 ) A generates a strongly continuous semigroup in the Banach space X.

( H 2 ) F ( t ) B ( X ) , 0 t a . F ( t ) : Y Y and for x ( ) continuous in Y, A F ( ) x ( ) L 1 [ J , X ] . For x X , F ( t ) x is continuous in t J , where B ( X ) is the space of all linear and bounded operators on X, and Y is the Banach space formed from D ( A ) , the domain of A, endowed with the graph norm.

Definition 2.1 [5]

R ( t ) is a resolvent operator of (1.1) with f 0 if R ( t ) B ( X ) for 0 t a and satisfies the following conditions:
  1. (1)

    R ( 0 ) = I , the identity operator on X,

     
  2. (2)

    for all x X , R ( t ) x is continuous for 0 t a ,

     
  3. (3)
    R ( t ) B ( Y ) , 0 t a ; for y Y , R ( ) y C 1 [ J , X ] C [ J , Y ] and
    d d t R ( t ) y = A [ R ( t ) y + 0 t F ( t s ) R ( s ) y d s ] = R ( t ) A y + 0 t R ( t s ) A F ( s ) y d s , 0 t a .
    (2.1)
     
The resolvent operator R ( t ) is said to be equicontinuous if { t R ( t ) x : x B } is equicontinuous for the entire bounded set B X and t > 0 . If x C [ J , X ] satisfies the following integral equation:
x ( t ) = R ( t ) ( x 0 + g ( x ) ) + 0 t R ( t s ) f ( s , x ( σ 1 ( s ) ) , ( K x σ 2 ) ( s ) , ( H x σ 3 ) ( s ) ) d s , t J ,

then x is said to be a mild solution IVP (1.1)-(1.2).

Lemma 2.2 [14]

Let the conditions ( H 1 ), ( H 2 ) be satisfied. Then (1.1) with f 0 has a unique resolvent operator.

The following lemma is obvious.

Lemma 2.3 Let the resolvent operator R ( t ) be equicontinuous. If there is ρ L [ J , R + ] such that x ( t ) ρ ( t ) for a.e. t J , then the set { 0 t R ( t s ) x ( s ) d s } is equicontinuous.

Lemma 2.4 [22]

Let V C [ J , E ] be an equicontinuous bounded subset. Then α ( V ( t ) ) C [ J , R + ] ( R + = [ 0 , ) ), α ( V ) = max t J α ( V ( t ) ) .

Lemma 2.5 [23]

Let V = { x n } L [ J , E ] and there exists σ L [ J , R + ] such that x n ( t ) σ ( t ) for any x V and a.e. t J . Then α ( V ( t ) ) L [ J , R + ] and
α ( { 0 t x n ( s ) d s : n N } ) 2 0 t α ( V ( s ) ) d s , t J .

Lemma 2.6 [24] (Mönch)

Let E be a Banach space, Ω a closed convex subset in E and y 0 Ω . Suppose that the continuous operator F : Ω Ω has the following property:
V Ω countable , V co ¯ ( { y 0 } F ( V ) ) V is relatively compact .

Then F has a fixed point in Ω.

For V C [ J , X ] , let V ( t ) = { x ( t ) : x V } , V σ i ( t ) = { x ( σ i ( t ) ) : x V } ( i = 1 , 2 , 3 ), ( K V ) ( t ) = { ( K x ) ( t ) : x V } , ( H V ) ( t ) = { ( H x ) ( t ) : x V } ( t J ), X r = { x X : x r } and S r = { x C [ J , X ] : x C r } for any r > 0 . α ( ) and denote the Kuratowski measure of noncompactness in X and C [ J , X ] respectively. For details on the properties of noncompact measure, we refer the reader to [22].

3 Existence of a mild solution

We make the following assumptions for convenience.

( H 3 ) There exist constants l g > 0 , M > 0 and 4 l g M < 1 such that
g ( x ) g ( y ) l g x y C , x , y C [ J , X ] ,

and g ( 0 ) = 0 .

( H 3 ) g : C [ J , X ] E is continuous, compact and there exists a constant N 0 such that g ( x ) N .

( H 4 ) There exists q C [ J , R + ] such that
f ( t , x , y , z ) q ( t ) ( x + y + z ) , t J , x , y , z X .
( H 5 ) There exist k 0 C [ , R + ] , h 0 C [ J × J , R + ] such that
k ( t , s , x ) k 0 ( t , s ) x , ( t , s ) , x X , h ( t , s , x ) h 0 ( t , s ) x , t , s J , x X .
( H 6 ) For any r > 0 and a bounded set V i X r , there exist constants l i > 0 ( i = 1 , 2 , 3 ) such that
α ( f ( t , V 1 , V 2 , V 3 ) ) l 1 α ( V 1 ) + l 2 α ( V 2 ) + l 3 α ( V 3 ) , t J .
( H 7 ) For any r > 0 and a bounded set V X r ,
α ( k ( t , s , V ) ) k 0 ( t , s ) α ( V ) , ( t , s ) , α ( h ( t , s , V ) ) h 0 ( t , s ) α ( V ) , t , s J .
( H 8 ) The resolvent operator R ( t ) is equicontinuous and R ( t ) M e w t for t J and some positive number
w = max { 2 M q 0 ( 1 + K 0 a + H 0 a ) , 4 M ( l 1 + 2 l 2 a K 0 + 2 l 3 a H 0 ) } ,

where K 0 = max ( t , s ) k 0 ( t , s ) , H 0 = max t , s J h 0 ( t , s ) , q 0 = max t J q ( t ) .

Without loss of generality, we always suppose that x 0 = 0 .

Theorem 3.1 Let conditions ( H 1 ), ( H 2 ), ( H 3 )-( H 8 ) be satisfied. Then IVP (1.1)-(1.2) has at least one mild solution.

Proof Let
( F x ) ( t ) = R ( t ) g ( x ) + 0 t R ( t s ) f ( s , x ( σ 1 ( s ) ) , ( K x σ 2 ) ( s ) , ( H x σ 3 ) ( s ) ) d s , t J .
(3.1)
We have by ( H 3 ), ( H 4 ) and ( H 5 ),
(3.2)
Let
B R = { x C [ J , X ] : x C R } .
Then B R is a closed convex subset in C [ J , X ] , 0 B R and F : B R B R . Similar to the proof of [6] and [9], it is easy to verify that F is a continuous operator from B R into B R . For x B R , s J , ( H 4 ) and ( H 5 ) imply
f ( s , x ( σ 1 ( s ) ) , ( K x σ 2 ) ( s ) , ( H x σ 3 ) ( s ) ) q ( s ) ( 1 + 0 s k 0 ( s , r ) d r + 0 a h 0 ( s , r ) d r ) R .
(3.3)

We can show from (3.3), ( H 8 ) and Lemma 2.3 that F ( B R ) is an equicontinuous subset in C [ J , X ] .

Let V B R be a countable set and V co ¯ ( { 0 } F ( V ) ) , then
V ( t ) co ¯ ( { 0 } ( F V ) ( t ) ) .
(3.4)
From equicontinuity of F ( B R ) and (3.4), we know that V is an equicontinuous subset in C [ J , X ] . By the properties of noncompact measure, the conditions ( H 3 ), ( H 6 ), ( H 7 ), (3.4) and Lemma 2.5, we have
(3.5)

(3.5) together with Lemma 2.4 imply that , and so . Hence V is relatively compact in C [ J , X ] . Lemma 2.6 implies that F has a fixed point in C [ J , X ] . Then IVP (1.1)-(1.2) has at least one mild solution. The proof is completed. □

Theorem 3.2 Let the conditions ( H 1 ), ( H 2 ) and ( H 3 )-( H 8 ) be satisfied. Then IVP (1.1)-(1.2) has at least one mild solution.

Proof Similar to (3.2) and (3.5), it is easy to verify
( F x ) ( t ) M N + M q 0 ( 1 + K 0 a + H 0 a ) w 1 x C = M N + η x C ,

where η = M q 0 ( 1 + K 0 a + H 0 a ) w 1 < 1 . Taking R > M N ( 1 η ) 1 , let B R = { x C [ J , X ] : x C R } . We have F : B R B R and the inequality (3.5) is transformed into , t J .

The other proof is similar to the proof of Theorem 3.1, we omit it. □

4 An example

Let X = L 2 [ 0 , π ] . Consider the following partial functional integro-differential equation with a nonlocal condition,
{ u t ( t , y ) = u y ( t , y ) + 0 t F ( t s ) u y ( s , y ) d s + γ 1 sin u ( t r , y ) u t ( t , y ) = + 0 t γ 2 u ( s r , y ) d s ( 1 + t ) + 0 a γ 3 u ( s r , y ) d s ( 1 + t ) ( 1 + s ) 2 , 0 t a , u ( 0 , y ) = u 0 ( y ) + γ 4 u ( y ) ,
(4.1)
where r , γ i R ( i = 1 , 2 , 3 , 4 ), σ 1 ( t ) = σ 2 ( t ) = σ 3 ( t ) = t r , 0 r t a , F ( t ) satisfies the condition ( H 2 ),
(4.2)
(4.3)
(4.4)
Let the operator A be defined by A w = w , w D ( A ) with the domain
D ( A ) = { w E : w E , w  is almost everywhere bounded } .
Then A generates a translation semigroup R ( t ) and R ( t ) is equicontinuous. The problem (4.1) can be regarded as a form of IVP (1.1)-(1.2). We have by (4.2), (4.3) and (4.4),
f ( t , u , v , z ) | γ | ( u + v + z ) , | γ | = max { | γ 1 | , | γ 2 | , | γ 3 | } , u , v , z X , k ( t , s , u ) u , h ( t , s , u ) u , u X ,
and
g ( u ) g ( v ) | γ 4 | u v C , g ( 0 ) = 0 .
γ 4 and M can be chosen such that 4 M | γ 4 | < 1 . In addition, for any r > 0 and a bounded set V i X r ( i = 1 , 2 , 3 ), we can show that by the diagonal method,
α ( f ( t , V 1 , V 2 , V 3 ) ) | γ | ( α ( V 1 ) + α ( V 2 ) + α ( V 3 ) ) , t J , α ( k ( t , s , V 1 ) ) α ( V 1 ) , t , s , α ( h ( t , s , V 1 ) ) α ( V 1 ) , t , s [ 0 , a ] .

Hence all the conditions of Theorem 3.1 are satisfied, the problem (4.1) has at least one mild solution in C [ J , X ] .

Declarations

Acknowledgements

The work was supported by Natural Science Foundation of Anhui Province (11040606M01) and Education Department of Anhui (KJ2011A061, KJ2011Z057), China.

Authors’ Affiliations

(1)
Department of Mathematics and Physics, Anhui University of Architecture

References

  1. Tidke HL, Dhakne MB: Existence of solutions and controllability of nonlinear mixed integrodifferential equation with nonlocal conditions. Appl. Math. E-Notes 2011, 11: 12-22.MATHMathSciNetGoogle Scholar
  2. Yan Z, Wei P: Existence of solutions for nonlinear functional integrodifferential evolution equations with nonlocal conditions. Aequ. Math. 2010, 79: 213-228. 10.1007/s00010-010-0017-2MATHMathSciNetView ArticleGoogle Scholar
  3. Tidke HL, Dhakne MB: Nonlocal Cauchy problems for nonlinear mixed integro-differential equations. Tamkang J. Math. 2010, 41: 361-373.MATHMathSciNetGoogle Scholar
  4. Balachandran K, Kumar RR: Existence of solutions of integrodifferential evolution equations with time varying delays. Appl. Math. Lett. 2007, 7: 1-8.MATHMathSciNetGoogle Scholar
  5. Kumar RR: Nonlocal Cauchy problem for analytic resolvent integrodifferential equations in Banach spaces. Appl. Math. Comput. 2008, 204: 352-362. 10.1016/j.amc.2008.06.050MATHMathSciNetView ArticleGoogle Scholar
  6. Yang YL, Wang JR: On some existence results of mild solutions for nonlocal integro-differential Cauchy problems in Banach spaces. Opusc. Math. 2011, 31: 443-455.View ArticleGoogle Scholar
  7. Liu Q, Yuan R: Existence of mild solutions for semilinear evolution equations with non-local initial conditions. Nonlinear Anal. 2009, 71: 4177-4184. 10.1016/j.na.2009.02.093MATHMathSciNetView ArticleGoogle Scholar
  8. Boucherif A, Precup R: Semilinear evolution equations with nonlocal initial conditions. Dyn. Syst. Appl. 2007, 16: 507-516.MATHMathSciNetGoogle Scholar
  9. Chalishajar DN: Controllability of mixed type Volterra-Fredholm integro-differential systems in Banach space. J. Franklin Inst. 2007, 344: 12-21. 10.1016/j.jfranklin.2006.04.002MATHMathSciNetView ArticleGoogle Scholar
  10. Xue X: Existence of solutions of a semilinear nonlocal Cauchy problem in a Banach space. Elect. J. Diff. Eqs. 2005, 2005(64):1-7.Google Scholar
  11. Liu JH, Ezzinbi K: Non-autonomous integrodifferential equations with nonlocal conditions. J. Integral Equ. Appl. 2003, 15: 79-93. 10.1216/jiea/1181074946MATHMathSciNetView ArticleGoogle Scholar
  12. Ntouyas S, Tsamotas P: Global existence for semilinear evolution equations with nonlocal conditions. J. Math. Anal. Appl. 1997, 210: 679-687. 10.1006/jmaa.1997.5425MATHMathSciNetView ArticleGoogle Scholar
  13. Byszewski L, Akca H: Existence of solutions of a semilinear functional-differential evolution nonlocal problem. Nonlinear Anal. 1998, 34: 65-72. 10.1016/S0362-546X(97)00693-7MATHMathSciNetView ArticleGoogle Scholar
  14. Lin Y, Liu J: Semilinear integrodifferential equations with nonlocal Cauchy problem. Nonlinear Anal. 1996, 26: 1023-1033. 10.1016/0362-546X(94)00141-0MATHMathSciNetView ArticleGoogle Scholar
  15. Byszewski L: Existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem. Zesz. Nauk. Politech. Rzesz., Mat. Fiz. 1993, 18: 109-112.MathSciNetGoogle Scholar
  16. Byszewski L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 1991, 162: 494-505. 10.1016/0022-247X(91)90164-UMATHMathSciNetView ArticleGoogle Scholar
  17. Malar K: Existence of mild solutions for nonlocal integro-differential equations with measure of noncompactness. Int. J. Math. Sci. Comput. 2011, 1: 86-91.MathSciNetGoogle Scholar
  18. Shi H, Li W, Sun H-R: Existence of mild solutions for abstract mixed type semilinear evolution equations. Turk. J. Math. 2011, 35: 457-472.MATHMathSciNetGoogle Scholar
  19. Zhu T, Song C, Li G: Existence of mild solutions for abstract semilinear evolution equations in Banach spaces. Nonlinear Anal. 2012, 75: 177-181. 10.1016/j.na.2011.08.019MATHMathSciNetView ArticleGoogle Scholar
  20. Fan Z, Dong Q, Li G: Semilinear differential equations with nonlocal conditions in Banach spaces. Int. J. Nonlinear Sci. 2006, 2(3):131-139.MathSciNetGoogle Scholar
  21. Dong Q, Li G: Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces. Electron. J. Qual. Theory Differ. Equ. 2009, 47: 1-13.MathSciNetView ArticleGoogle Scholar
  22. Dajun G, Lakshmikantham V, Liu X: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht; 1996.MATHGoogle Scholar
  23. Heine HP: On the behavior of measure of noncompactness with respect to differentiation and integration of vector valued functions. Nonlinear Anal. 1983, 7: 1351-1371. 10.1016/0362-546X(83)90006-8MathSciNetView ArticleGoogle Scholar
  24. Mönch H: Boundary value problems for nonlinear ordinary equations of second order in Banach spaces. Nonlinear Anal. 1980, 4: 985-999. 10.1016/0362-546X(80)90010-3MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Xie; licensee Springer 2012

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.