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A generalisation of contraction principle in metric spaces. (English) Zbl 1177.54024
In this paper, the main result states that a self-mapping $T: X \to X$ defined on a complete metric space $(X,d)$ has a unique fixed point if it satisfies the following inequality $$\psi(d(Tx,Ty)) \le \psi(d(x,y))-\phi(d(x,y)),$$ where $\psi, \phi: [0,\infty) \to [0,\infty)$ are two monotone nondecreasing continuous functions with $\psi(t)=0= \phi(t)$ if and only if $t=0$. When $\psi$ is the identity function on $[0,\infty)$, this reduces to a result of {\it B. E. Rhoades} [Nonlinear Anal., Theory Methods Appl. 47, No. 4, 2683--2693 (2001; Zbl 1042.47521)]. Moreover, the main result is illustrated by an example.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
fixed point
Full Text: DOI EuDML