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General constructive fixed point theorems for Ćirić-type almost contractions in metric spaces. (English) Zbl 1199.54209
This article deals with two generalizations of the classical Banach-Caccioppoli fixed point principle for contractions in complete metric spaces. In the first of them, the author considers operators \(T: X\to X\), where \((X,d)\) is a complete metric space with a metric \(d\), satisfying the following condition \[ d(Tx,Ty)\leq kM(x,y) + L d(y,Tx), \quad x, y \in X, \] \[ (M(x,y) = \max \;\{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\}), \] where \(0 \in (0,1)\), \(L \geq 0\). It is proved that \({\text{Fix}} (T) \neq \emptyset\), the convergence of the Picard iteration \(x_{n+1} = Tx_n\) to some \(x_* \in {\text Fix} (T)\) for any \(x_0 \in X\), and the estimate \[ d(x_n,x_*) \leq \frac{k^n}{(1 - k)^2} d(x_0,Tx_0), \quad n = 0,1,2,\dots;\tag{1} \] (note that \(T\) can have more than one fixed point). In the second generalization, the author considers operators \(T:X \to X\) satisfying the following condition \[ d(Tx,Ty) \leq kM(x,y) + L d(x,Tx), \quad x, y \in X, \] where also \(0 \in (0,1)\), \(L \geq 0\) and \(M(x,y)\) is defined by the same formula. It is proved that \(T\) has a unique fixed point \(x_*\), the Picard iteration \(x_{n+1} = Tx_n\) converges to \(x_*\) for any \(x_0 \in X\) and the estimate (1) holds, and, moreover, the following estimate \[ d(x_{n+1},x_*) \leq k d(x_n,x_*), \quad n = 0,1,2,\dots, \] also holds. In the end of the article, the author gives some illustrative examples and some remarks about relations between new generalizations of the Banach-Caccioppoli fixed point principle and early versions of such generalizations.

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