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Common fixed point theorems for Mann type iterations. (English) Zbl 0921.47050

The main result of this paper is the following: Let \(C\) be a nonempty closed convex subset of a Banach space \(E\) and \(A\), \(B\), \(S\), \(T\) be mappings from \(C\) into itself satisfying the following conditions:
(i) there exist constants \(\alpha, \beta, \gamma, \delta\geq 0\), such that \[ \| Sx-Ty\|\leq \alpha\| Ax-By\|+ \beta\| Ax-Sx\|+ \gamma\max\{\| By-Ty\|,\| Ax-Ty\|\}+ \delta\| By-Sx \| \] for all \(x,y\in C\), where \(0\leq \max\{\alpha+ \gamma+ \delta,\beta+ \delta\}< 1\),
(ii) for some \(x_0\in C\) there exists a constant \(k\in [0,1)\) such that \[ \| x_{n+2}- x_{n+1}\|\leq k\| x_{n+1}- x_n\|, \quad n=0,1,2,\dots\;, \] where \(\{x_n\}\subset C\) is a sequence such that \[ Ax_{2n+1}= \tfrac 12 Ax_{2n}+ \tfrac 12 Sx_{2n}, \qquad Bx_{2n+2}= \tfrac 12 Bx_{2n+1}+ \tfrac 12 Tx_{2n+1}, \tag{M} \] (iii) the pairs \(\{A,S\}\) and \(\{B,T\}\) are compatible, i.e., \(\lim_{n\to\infty}\| ASz_n-SAz_n \|=0\), whenever \(\{z_n\}\subset C\) is a sequence such that \(\lim_{n\to \infty} Az_n= \lim_{n\to\infty} Sz_n=z\) for some \(z\in C\).
Then the sequence \(\{x_n\}\) satisfying (M) converges to a point \(\xi\in C\). Further if \(A\) and \(B\) are continuous at the point \(\xi\), then \(T\xi\) is a unique common fixed point of \(A\), \(B\), \(S\) and \(T\).
This theorem extends, among others, some results of J. Górnicki and B. E. Rhoades [Indian J. Pure Appl. Math. 27, No. 1, 13-23 (1996; Zbl 0847.47038)].

MSC:

47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
54H25 Fixed-point and coincidence theorems (topological aspects)

Citations:

Zbl 0847.47038
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