Nonlinear contractions in metrically convex space
Copyright policy )Nonlinear contractions in metrically convex space
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 \(\psi(t_ n) 0\). An application to a functional equation is also given.
Other analytical inequalities, Fixed-point theorems, Fixed-point and coincidence theorems (topological aspects), Lipschitz (Hölder) classes, Operator theory in probabilistic metric linear spaces, Random nonlinear operators, fixed point theorem, Iteration theory, iterative and composite equations, functional equation, Convexity of real functions in one variable, generalizations, Menger convex metric space
Other analytical inequalities, Fixed-point theorems, Fixed-point and coincidence theorems (topological aspects), Lipschitz (Hölder) classes, Operator theory in probabilistic metric linear spaces, Random nonlinear operators, fixed point theorem, Iteration theory, iterative and composite equations, functional equation, Convexity of real functions in one variable, generalizations, Menger convex metric space
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