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

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