Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces
Introduction
In the present paper we are concerned with the existence of mild solutions for an impulsive nonlocal Cauchy problem driven by a semilinear differential inclusion in a real Banach space where: is a family of linear operators in ; is a multifunction; is a function; ; for every is an impulse function; and .
In the past much has been done for nonlocal Cauchy problems without impulses. We refer the reader to, for example, the recent papers [1], [2], [3]. For the autonomous case (i.e. for every ) we refer the reader to [4], [5].
More recently Fan [6] studied a nonlocal Cauchy problem in the presence of impulses, governed by an autonomous semilinear differential equation.
Such analysis on impulsive nonlocal Cauchy problems is important from an applied viewpoint since on one hand the nonlocal condition has a better effect in the applications than the classical initial one (for more details, see e.g. [7]), and on the other hand the impulsive model is appropriate for describing real phenomena where changes occur suddenly [8], [9], [10], [11], [12].
Under a lower semicontinuity-type assumption on the nonlinearity , we provide here some theorems on the existence of impulsive mild solutions for (INP) (see Definition 3.1) for the case where the evolution system generated by is compact and for the case where it is not. We clarify that if the evolution system is compact we can weaken the assumption on the functions which describe the impulsive conditions.
In both cases, via theory on measures of noncompactness, we achieve our goals thanks to a recent selection theorem (see [13, Theorem 3.1]) joined with the classical Schauder fixed point theorem.
We wish to note that all our results are new even in the autonomous case. Moreover, they extend to multifunctions and to the nonautonomous case the existence theorems proved in [6], where the autonomous linear operator considered therein generates a compact strongly continuous semigroup (cf. Remark 3.2).
Finally, in Section 4 we present an example to illustrate an application of the results obtained.
Section snippets
Preliminaries
Let be a Hausdorff topological space and, if necessary, also linear. In the sequel we make use of the following notation: .
If is a complete metric space, we can consider the family . We recall (see e.g. [14]) that the Hausdorff measure of noncompactness on is the function defined as and that it satisfies
Existence results
We consider the impulsive nonlocal Cauchy problem
We assume the following hypothesis on the linear part of the differential inclusion:
- (A)
is a family of linear (not necessarily bounded) operators, not depending on and a dense subset of , generating a continuous evolution operator , i.e. there exists an evolution system such that, on the
Example
We study the following impulsive partial differential system with nonlocal conditions: where is a positive integer, .
In order to rewrite (17) in the abstract form, we put and denote by the Laplace operator, i.e. on the domain
Acknowledgments
The authors wish to thank the referees for their helpful comments.
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