## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Nonlinear Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

#### Abstract

This paper is devoted to prove the existence of solutions of the nonlinear Sturm-Liouville boundary value problem on time scales in Banach spaces. We obtain the sufficient conditions for the existence of solutions in terms of Kuratowski measure of noncompactness. Mönch's fixed point theorem is used to prove the main result. By the unification property of time scales, our result is valid for Sturm-Liouville differential equations and difference equations, but more interestingly by the extension property, it is also valid for Sturm-Liouville $q$-difference equation.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 4 (2013), 587-601.

Dates
First available in Project Euclid: 22 October 2013

https://projecteuclid.org/euclid.bbms/1382448182

Digital Object Identifier
doi:10.36045/bbms/1382448182

Mathematical Reviews number (MathSciNet)
MR3129061

Zentralblatt MATH identifier
1297.34103

#### Citation

Yantir, Ahmet; Kubiaczyk, Ireneusz; Sikorska-Nowak, Aneta. Nonlinear Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 4, 587--601. doi:10.36045/bbms/1382448182. https://projecteuclid.org/euclid.bbms/1382448182