Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces

Sikorska-Nowak, Aneta; Nowak, Grzegorz

doi:https://doi.org/10.1155/2007/65947

International Journal of Mathematics and Mathematical Sciences

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Volume 2007 | Article ID 065947 | https://doi.org/10.1155/2007/65947

Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces

Aneta Sikorska-Nowak¹and Grzegorz Nowak²

Academic Editor: Marco Squassina

Received27 Feb 2007

Revised28 Apr 2007

Accepted19 Jun 2007

Published02 Sept 2007

Abstract

We prove two existence theorems for the integrodifferential equation of mixed type: x'(t)=f(t,x(t),∫0tk1(t,s)g(s,x(s))ds,∫0ak2(t,s)h(s,x(s))ds), x(0)=x0, where in the first part of this paper f, g, h, x are functions with values in a Banach space E and integrals are taken in the sense of Henstock-Kurzweil (HK). In the second part f, g, h, x are weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis (HKP) integral. Additionally, the functions f, g, h, x satisfy some conditions expressed in terms of the measure of noncompactness or the measure of weak noncompactness.

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Copyright

Copyright © 2007 Aneta Sikorska-Nowak and Grzegorz Nowak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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