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Nonlinear contractions in metrically convex space. (English) Zbl 0824.47047

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.

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
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