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A note on a fixed point theorem of Berinde on weak contractions. (English) Zbl 1199.54205
The article deals with two classes of operators \(T:X \to X\) (introduced by V. Berinde [Nonlinear Anal. Forum 9, No. 1, 43–53 (2004; Zbl 1078.47042)]) in a metric space \(X\): weak contractions \[ D(Tx,Ty) \leq \delta d(x,y) + Ld(y,Tx), \quad \text{for all} \quad x,y \in X \] and quasi-contractions \[ d(Tx,Ty) \leq h \max \;\{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\} \quad \text{for all} \quad x,y \in X. \] The authors present an example of a quasi-contraction that is not a weak contraction. Furthermore, they study operators \(T:X \to X\) satisfying the condition \[ d(Tx,Ty) \leq \delta d(x,y) + L\min \;\{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\} \quad \text{for all} \quad x,y \in X \] and it is proved that such operators have a unique fixed point. The new class of operators is smaller than the class of weak contractions. It is known that weak contractions can have more than one fixed point.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
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