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Fixed points of convex and generalized convex contractions. (English) Zbl 1462.54046

Summary: V. I. Istratescu [Libertas Math. 1, 151–163 (1981; Zbl 0477.54032)] 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 [loc. cit.] 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 B. E. Rhoades [Contemp. Math. 72, 233–245 (1988; Zbl 0649.54024)] is found. Several examples are given to illustrate our results.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
54E50 Complete metric spaces
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