On fixed point index theory for the sum of operators and applications to a class of ODEs and PDEs

Svetlin Georgiev Georgiev

Bulgaria

University of Sofia

Department of Differential Equations, Faculty of Mathematics and Informatics

Karima Mebarki

https://orcid.org/0000-0002-6679-5059

Algeria

University of Bejaia

Laboratory of Applied Mathematics, Faculty of Exact Sciences.

Submitted: 2020-03-10

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Accepted: 2021-04-09

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Published: 2021-10-01

DOI: https://doi.org/10.4995/agt.2021.13248
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Keywords:

positive solution, fixed point index, cone, sum of operators, ODEs, PDEs

Supporting agencies:

Direction Générale de la Recherche Scientifique et du Développement Technologique DGRSDT. MESRS Algeria. Projet PRFU

C00L03UN060120180009

Abstract:

The aim of this work is two fold: first  we  extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction  obtained in \cite{DjebaMeb, Svet-Meb}, to  the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction.  Secondly, as  illustration of some our theoretical results,  we study  the existence of positive solutions  for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as  a class of  partial differential equations (PDEs for short).
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