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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.

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