Nonlocal Impulsive Cauchy Problems for Evolution Equations
© J. Liang and Z. Fan. 2011
Received: 17 October 2010
Accepted: 19 November 2010
Published: 25 November 2010
Of concern is the existence of solutions to nonlocal impulsive Cauchy problems for evolution equations. Combining the techniques of operator semigroups, approximate solutions, noncompact measures and the fixed point theory, new existence theorems are obtained, which generalize and improve some previous results since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required. An application to partial differential equations is also presented.
Impulsive equations arise from many different real processes and phenomena which appeared in physics, chemical technology, population dynamics, biotechnology, medicine, and economics. They have in recent years been an object of investigations with increasing interest. For more information on this subject, see for instance, the papers (cf., e.g., [1–6]) and references therein.
On the other hand, Cauchy problems with nonlocal conditions are appropriate models for describing a lot of natural phenomena, which cannot be described using classical Cauchy problems. That is why in recent years they have been studied by many researchers (cf., e.g., [4, 7–12] and references therein).
By going a new way, that is, by combining operator semigroups, the techniques of approximate solutions, noncompact measures, and the fixed point theory, we obtain new existence results for problem (1.1), which generalize and improve some previous theorems since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required in the present paper.
The organization of this work is as follows. In Section 2, we recall some definitions, and facts about fractional powers of operators, mild solutions and Hausdorff measure of noncompactness. In Section 3, we give the existence results for problem (1.1) when the nonlocal item and impulsive functions are only assumed to be continuous. In Section 4, we give an example to illustrate our abstract results.
Throughout this paper, we assume the following.
Theorem 2.1 (see [13, Pages 69–75]).
Lemma 2.4 (see ).
Lemma 2.5 (see Darbo-Sadovskii's fixed point theorem in ).
3. Main Results
In this section, by using the techniques of approximate solutions and fixed points, we establish a result on the existence of mild solutions for the nonlocal impulsive problem (1.1) when the nonlocal item and the impulsive functions are only assumed to be continuous in and , respectively.
So, to prove our main results, we introduce the following assumptions.
Next, for the compactness of we refer to the proof of [4, Theorem 3.1].
Thus, the set is equicontinuous due to the compactness of and the strong continuity of operator . By the Arzela-Ascoli theorem, we conclude that is precompact in . The same idea can be used to prove that is precompact for each . Therefore, is precompact in , that is, the operator is compact.
This is a contradiction with inequality (3.6). Therefore, there exists such that the mapping maps into itself. By Darbu-Sadovskii's fixed point theorem, the operator has at least one fixed point in . This completes the proof.
Proof of Theorem 3.1.
The following results are immediate consequences of Theorem 3.5.
Theorems 3.5-3.6 are new even for many special cases discussed before, since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required.
The operator is the infinitesimal generator of an analytic compact semigroup on . Moreover, has a discrete spectrum, the eigenvalues are , , with the corresponding normalized eigenvectors , and the following properties are satisfied.
Assume the following.
The authors would like to thank the referees for helpful comments and suggestions. J. Liang acknowledges support from the NSF of China (10771202) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805). Z. Fan acknowledges support from the NSF of China (11001034) and the Research Fund for Shanghai Postdoctoral Scientific Program (10R21413700).
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