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Stability of Picard iteration for contractive mappings satisfying an implicit relation. (English) Zbl 1265.54152
In this paper, the author studies the existence and uniqueness of the fixed point as well as the stability of the Picard iteration for a mapping $$T:X\rightarrow X$$ satisfying $$F(d(Tx,Ty),d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx))\leq 0$$, where $$(X,d)$$ is a complete metric space and $$F:{\mathbb{R}}_+^6\rightarrow{\mathbb{R}}_+$$ is a continuous function.

MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54E50 Complete metric spaces 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.