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.
Reviewer: Peter Zabreiko (Minsk)