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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 $$d(Tx,Ty)\le kM(x,y) + L d(y,Tx), \quad x, y \in X,$$ $$(M(x,y) = \max \ \{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\}),$$ where $0 \in (0,1)$, $L \ge 0$. It is proved that ${\text{Fix}} (T) \ne \emptyset$, 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 $$d(x_n,x_*) \le \frac{k^n}{(1 - k)^2} d(x_0,Tx_0), \quad n = 0,1,2,\dots;\tag{1}$$ (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 $$d(Tx,Ty) \le kM(x,y) + L d(x,Tx), \quad x, y \in X,$$ where also $0 \in (0,1)$, $L \ge 0$ 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 $$d(x_{n+1},x_*) \le k d(x_n,x_*), \quad n = 0,1,2,\dots,$$ 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.

54H25Fixed-point and coincidence theorems in topological spaces