Matkowski, Janusz Nonlinear contractions in metrically convex space. (English) Zbl 0824.47047 Publ. Math. Debr. 45, No. 1-2, 103-114 (1994). Summary: We prove among other things the following fixed point theorem. Let \(T\) be a selfmapping of a complete Menger convex metric space \((X, d)\) and \(\psi: [0, \infty)\to [0, \infty)\) a function such that \[ d(T(x), T(y))\leq \psi(d(x, y)),\qquad (x, y\in X). \] Suppose that \(\psi\) is continuous at 0 and that there exists a positive sequence \(t_ n\) \((n\in \mathbb{N})\), such that \(\lim_{n\to \infty} t_ n= 0\) and \(\psi(t_ n)< t_ n\) \((n\in \mathbb{N})\). Then \(T\) has a unique fixed point. Moreover, \(T\) is \(\gamma\)-contractive for an increasing concave function \(\gamma\) and such that \(\gamma(t)< t\) for all \(t> 0\).An application to a functional equation is also given. Cited in 9 Documents MSC: 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects) 47S50 Operator theory in probabilistic metric linear spaces 26D20 Other analytical inequalities 26A16 Lipschitz (Hölder) classes 26A51 Convexity of real functions in one variable, generalizations 39B12 Iteration theory, iterative and composite equations 47H40 Random nonlinear operators Keywords:fixed point theorem; Menger convex metric space; functional equation PDFBibTeX XMLCite \textit{J. Matkowski}, Publ. Math. Debr. 45, No. 1--2, 103--114 (1994; Zbl 0824.47047)