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

Ahmet Yantir; Ireneusz Kubiaczyk; Aneta Sikorska-Nowak

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 6-15, electronic only
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Ahmet Yantir, Ireneusz Kubiaczyk, and Aneta Sikorska-Nowak. "Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces." Open Mathematics 13.1 (2015): 6-15, electronic only. <http://eudml.org/doc/268685>.

@article{AhmetYantir2015,
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.},
author = {Ahmet Yantir, Ireneusz Kubiaczyk, Aneta Sikorska-Nowak},
journal = {Open Mathematics},
keywords = {Sturm-Liouville equation; Banach space; Measure of noncompactness; Carathéodory solutions; Time scale; measure of noncompactness; time scale},
language = {eng},
number = {1},
pages = {6-15, electronic only},
title = {Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces},
url = {http://eudml.org/doc/268685},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Ahmet Yantir
AU - Ireneusz Kubiaczyk
AU - Aneta Sikorska-Nowak
TI - Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 6
EP - 15, electronic only
AB - 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.
LA - eng
KW - Sturm-Liouville equation; Banach space; Measure of noncompactness; Carathéodory solutions; Time scale; measure of noncompactness; time scale
UR - http://eudml.org/doc/268685
ER -

References

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