A FIXED POINT THEOREM IN ABSTRACT SPACES WITH APPLICATION TO CAUCHY PROBLEM

We establish a new fixed point theorem in abstract spaces. We then derive two main con-sequences in topological spaces for mappings admitting precompact images or leading to a nonempty ω-limit set. The study is carried out by introducing a cone of special functions which enables us to extend, unify and improve fixed point results due to Bailey, Ćirić, Dass-Gupta, Edelstein, Hardy-Rogers, Jaggi, Karapınar, Liepiņš, Nemytskii, Popa, Popescu, Reich, Suzuki and Wardowski. Finally, we introduce the notion of ξ-Lipschitz property and we investigate the existence of solutions to a class of Cauchy problems.