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Existence of solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions
Boundary Value Problems volume 2012, Article number: 100 (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 have been weakened. Our results extend and improve many known results.
MSC:34G20, 34K30.
1 Introduction
Let be a Banach space, with the norm . It is easy to verify that is a Banach space. The space of X-valued Bochner integrable functions on J with the norm is denoted by . Consider the following nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions in a Banach space ,
where
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.
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 (), the compactness of the resolvent operator and the restricted condition
with is used. Malar [17] and Shi [18] studied the existence of mild solutions of semilinear mixed type integrodifferential evolution equations with the equicontinuous semigroup
Solvability of the scalar equation
and the restricted condition on measure of noncompactness estimation
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 have been weakened. Our results extend and improve some corresponding results in papers [1–4, 6–21].
2 Preliminaries
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 [5]
is a resolvent operator of (1.1) with if for and satisfies the following conditions:
-
(1)
, the identity operator on X,
-
(2)
for all , is continuous for ,
-
(3)
, ; for , and
(2.1)
The resolvent operator is said to be equicontinuous if is equicontinuous for the entire bounded set and . If satisfies the following integral equation:
then x is said to be a mild solution IVP (1.1)-(1.2).
Lemma 2.2 [14]
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 [22]
Let be an equicontinuous bounded subset. Then (), .
Lemma 2.5 [23]
Let and there exists such that for any and a.e. . Then and
Lemma 2.6 [24] (Mönch)
Let E be a Banach space, Ω a closed convex subset in E and . Suppose that the continuous operator has the following property:
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 [22].
3 Existence of a mild solution
We make the following assumptions for convenience.
() There exist constants , and such that
and .
() is continuous, compact and there exists a constant such that .
() There exists such that
() There exist , such that
() For any and a bounded set , there exist constants () such that
() For any and a bounded set ,
() The resolvent operator is equicontinuous and for and some positive number
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.
Proof Let
We have by (), () and (),
Let
Then is a closed convex subset in , and . Similar to the proof of [6] and [9], it is easy to verify that F is a continuous operator from into . For , , () and () imply
We can show from (3.3), () and Lemma 2.3 that is an equicontinuous subset in .
Let be a countable set and , then
From equicontinuity of and (3.4), we know that V is an equicontinuous subset in . By the properties of noncompact measure, the conditions (), (), (), (3.4) and Lemma 2.5, we have
(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.
Proof Similar to (3.2) and (3.5), it is easy to verify
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. □
4 An example
Let . Consider the following partial functional integro-differential equation with a nonlocal condition,
where (), , , satisfies the condition (),
Let the operator A be defined by , with the domain
Then A generates a translation semigroup and 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),
and
and M can be chosen such that . In addition, for any and a bounded set (), we can show that by the diagonal method,
Hence all the conditions of Theorem 3.1 are satisfied, the problem (4.1) has at least one mild solution in .
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Acknowledgements
The work was supported by Natural Science Foundation of Anhui Province (11040606M01) and Education Department of Anhui (KJ2011A061, KJ2011Z057), China.
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Xie, S. Existence of solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions. Bound Value Probl 2012, 100 (2012). https://doi.org/10.1186/1687-2770-2012-100
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DOI: https://doi.org/10.1186/1687-2770-2012-100