## 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
$\psi(d(Tx,Ty)) \leq \psi(d(x,y))-\phi(d(x,y)),$
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 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.

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.

fixed point

Zbl 1042.47521
Full Text:

### References:

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