Nonlocal Cauchy problem for delay integrodifferential equations of Sobolev type in Banach spaces. (English) Zbl 1028.45006
L. Byszewski established the existence and uniqueness of mild, strong, and classical solutions of the nonlocal Cauchy problem [J. Math. Anal. Appl. 162, 494-505 (1991; Zbl 0748.34040)] and investigated the same problem for different types of evolution equations in Banach spaces in a joint paper with H. Akca [J. Appl. Math. Stochastic Anal. 10, 265-271 (1997; Zbl 1043.34504)]. H. Brill investigated the existence of solutions for a semilinear Sobolev evolution equation in a Banach space [J. Differ. Equations 24, 412-425 (1977; Zbl 0346.34046)]. Recently, K. Balachandran, D. G. Park and Y. C. Kwun discussed the problem for nonlinear integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces [Commun. Korean Math. Soc. 14, 223-231 (1999; Zbl 0972.45009)].
In this paper, the authors prove the existence of mild and strong solutions of nonlinear time varying delay integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. They obtain the results by using the theory of compact semigroups and Schaefer’s fixed-point theorem.
In this paper, the authors prove the existence of mild and strong solutions of nonlinear time varying delay integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. They obtain the results by using the theory of compact semigroups and Schaefer’s fixed-point theorem.
Reviewer: Kun Soo Chang (Seoul)
MSC:
| 45N05 | Abstract integral equations, integral equations in abstract spaces |
| 45J05 | Integro-ordinary differential equations |
| 45G10 | Other nonlinear integral equations |
Keywords:
nonlocal Cauchy problem; nonlocal conditions; mild and strong solutions; time varying delay integrodifferential equations of Sobolev type; Banach spaces; compact semigroups; Schaefer’s fixed-point theoremReferences:
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