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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 13, Issue 1


Volume 13 (2015)

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

Ahmet Yantir / Ireneusz Kubiaczyk / Aneta Sikorska-Nowak
Published Online: 2014-10-09 | DOI: https://doi.org/10.1515/math-2015-0002


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.

Keywords : Sturm-Liouville equation; Banach space; Measure of noncompactness; Carathéodory solutions; Time scale


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About the article

Received: 2013-06-17

Accepted: 2014-06-12

Published Online: 2014-10-09

Published in Print: 2015-01-01

Citation Information: Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0002.

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© 2015 Ahmet Yantir et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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