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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