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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 9, 2014

Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

  • Ahmet Yantir EMAIL logo , Ireneusz Kubiaczyk and Aneta Sikorska-Nowak
From the journal Open Mathematics

Abstract

In this paper, we present the existence result for Carathéodory type solutions for the nonlinear Sturm- Liouville boundary value problem (SLBVP) in Banach spaces on an arbitrary time scale. For this purpose, we introduce an equivalent integral operator to the SLBVP by means of Green’s function on an appropriate set. By imposing the regularity conditions expressed in terms of Kuratowski measure of noncompactness, we prove the existence of the fixed points of the equivalent integral operator. Mönch’s fixed point theorem is used to prove the main result. Finally, we also remark that it is straightforward to guarantee the existence of Carathéodory solutions for the SLBVP if Kuratowski measure of noncompactness is replaced by any axiomatic measure of noncompactness.

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Received: 2013-6-17
Accepted: 2014-6-12
Published Online: 2014-10-9
Published in Print: 2015-1-1

© 2015 Ahmet Yantir et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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