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Approximating fixed points of implicit almost contractions. (English) Zbl 1279.47084
Let \(\mathcal{F}\) be the set of all continuous real functions \(F:\mathbb{R}_{+}^6 \rightarrow \mathbb{R_{+}}\), for which one considers the following conditions:
\((F_{1a})\)
\(F\) is non-increasing in the fifth variable and \(F (u, v, v, u, u + v, 0) \leq 0\) for \(u, v \geq 0\) \(\Rightarrow\) \(\exists h \in [0, 1)\) such that \(u \leq h v\);
\((F_{1b})\)
\(F\) is non-increasing in the fourth variable and \(F (u, v, 0, u+v, u, v) \leq 0\) for \(u, v \geq 0\) \(\Rightarrow\) \(\exists h \in [0, 1)\) such that \(u \leq h v\);
\((F_{1c})\)
\(F\) is non-increasing in the third variable and \(F (u, v, u+v, 0, v, u) \leq 0\) for \(u, v \geq 0\) \(\Rightarrow\) \(\exists h \in [0, 1)\) such that \(u \leq h v\);
\((F_{2})\)
\(F (u, u, 0, 0, u, u) > 0\) for all \(u > 0\).
The main result of the paper establishes a stability result for the general case of implicit almost contractions and reads as follows.
Theorem 3. Let \((X,d)\) be a complete metric space, \(T:X\rightarrow X\) a self mapping for which there exists \(F\in \mathcal{F}\) such that, for all \(x,y \in X\), \[ F\left(d(Tx,Ty),d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\right)\leq 0. \] If \(F\) satisfies \((F_{1a})\) and \((F_{2})\), then:
\((p_{1})\)
\(T\) has a unique fixed point \(\overline{x}\) in \(X\).
\((p_{2})\)
The Picard iteration \(\{x_n\}^\infty_{n=0}\) defined by \[ x_{n+1}=Tx_n\,,\quad n=0,1,2,\dots \] converges to \(\overline{x}\), for any \(x_0\in X\).
\((p_{3})\)
The following estimate holds: \[ d(x_{n+i-1},\overline{x})\leq\frac{h^{i}}{1-h}\, d(x_{n},x_{n-1})\,,\quad n=0,1,2,\dots;\;i=1,2,\dots, \] where \(h\) is the constant appearing in \((F_{1a})\).
\((p_{4})\)
If, additionally, \(F\) satisfies \((F_{1c})\), then the rate of convergence of Picard iteration is given by \[ d(x_{n+1},\overline{x})\leq h d(x_{n},\overline{x})\,,\quad n=0,1,2,\dots. \]
Several stability results are also presented as corollaries of the main theorem.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54H25 Fixed-point and coincidence theorems (topological aspects)
54E50 Complete metric spaces
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