Fixed points of convex and generalized convex contractions

TLDR: In this article, the fixed points of convex contraction and generalized convex contractions were studied and the assumption of continuity condition in [11] was replaced by a relatively weaker condition of k-continuity under various settings.

Abstract: Istr$$\check{a}$$tescu (Lib Math 1:151–163, 1981) introduced the notion of convex contraction. He proved that each convex contraction has a unique fixed point on a complete metric space. In this paper we study fixed points of convex contraction and generalized convex contractions. We show that the assumption of continuity condition in [11] can be replaced by a relatively weaker condition of k-continuity under various settings. On this way a new and distinct solution to the open problem of Rhoades (Contemp Math 72:233–245, 1988) is found. Several examples are given to illustrate our results.