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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.

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