Positive solutions for a class of non-autonomous second order difference equations via a new functional fixed point theorem

Lydia Bouchal, Karima Mebarki, and Svetlin Georgiev Georgiev b

Address:
a Laboratory of Applied Mathematics, Faculty of Exact Sciences, University of Bejaia, 06000 Bejaia, Algeria
b Department of Differential Equations, Faculty of Mathematics and Informatics, University of Sofia, Sofia, Bulgaria

E-mail:
lydia.bouchal@univ-bejaia.dz
karima.mebarki@univ-bejaia.dz
svetlingeorgiev1@gmail.com

Abstract: In this paper, by using recent results on fixed point index, we develop a new fixed point theorem of functional type for the sum of two operators $T+S$ where $I-T$ is Lipschitz invertible and $S$ a $k$-set contraction. This fixed point theorem is then used to establish a new result on the existence of positive solutions to a non-autonomous second order difference equation.

AMSclassification: primary 47H10; secondary 39A27.

Keywords: fixed point, sum of operators, non-autonomous difference equations, positive solution.

DOI: 10.5817/AM2022-4-199