- Open Access
Existence of solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions
© Xie; licensee Springer 2012
- Received: 5 May 2012
- Accepted: 22 August 2012
- Published: 7 September 2012
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 have been weakened. Our results extend and improve many known results.
- integro-differential functional evolution equation
- mild solution
- nonlocal conditions
- fixed point
- Banach spaces
A is the generator of a strongly continuous semigroup in the Banach space X, and is a bounded operator for , , , , , , , and ().
For the existence of mild solutions of integro-differential functional evolution equations in abstract spaces, there are many research results, see [1–16], 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 are used.
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 have been weakened. Our results extend and improve some corresponding results in papers [1–4, 6–21].
We will make the following assumptions:
() A generates a strongly continuous semigroup in the Banach space X.
() , . and for continuous in Y, . For , is continuous in , where is the space of all linear and bounded operators on X, and Y is the Banach space formed from , the domain of A, endowed with the graph norm.
Definition 2.1 
, the identity operator on X,
for all , is continuous for ,
- (3), ; for , and(2.1)
then x is said to be a mild solution IVP (1.1)-(1.2).
Lemma 2.2 
Let the conditions (), () be satisfied. Then (1.1) with has a unique resolvent operator.
The following lemma is obvious.
Lemma 2.3 Let the resolvent operator be equicontinuous. If there is such that for a.e. , then the set is equicontinuous.
Lemma 2.4 
Let be an equicontinuous bounded subset. Then (), .
Lemma 2.5 
Lemma 2.6  (Mönch)
Then F has a fixed point in Ω.
For , let , (), , (), and for any . and denote the Kuratowski measure of noncompactness in X and respectively. For details on the properties of noncompact measure, we refer the reader to .
We make the following assumptions for convenience.
() is continuous, compact and there exists a constant such that .
where , , .
Without loss of generality, we always suppose that .
Theorem 3.1 Let conditions (), (), ()-() be satisfied. Then IVP (1.1)-(1.2) has at least one mild solution.
We can show from (3.3), () and Lemma 2.3 that is an equicontinuous subset in .
(3.5) together with Lemma 2.4 imply that , and so . Hence V is relatively compact in . Lemma 2.6 implies that F has a fixed point in . Then IVP (1.1)-(1.2) has at least one mild solution. The proof is completed. □
Theorem 3.2 Let the conditions (), () and ()-() be satisfied. Then IVP (1.1)-(1.2) has at least one mild solution.
where . Taking , let . We have and the inequality (3.5) is transformed into , .
The other proof is similar to the proof of Theorem 3.1, we omit it. □
Hence all the conditions of Theorem 3.1 are satisfied, the problem (4.1) has at least one mild solution in .
The work was supported by Natural Science Foundation of Anhui Province (11040606M01) and Education Department of Anhui (KJ2011A061, KJ2011Z057), China.
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