Common fixed point theorems for Mann type iterations (Q2757170)

The main result of the present paper is the following Theorem 1: Let \(C\) be a nonempty closed convex subset of a Banach space \((X,\|\cdot \|)\) and \(A, B, S, T\) and \(P\) be mappings from \(C\) into itself satisfying the following conditions:NEWLINENEWLINENEWLINE(1.1) there exist constants \(\alpha,\beta, \gamma, \delta\geq 0\) such that NEWLINE\[NEWLINE\begin{multlined} \|Px-Py \|\leq\alpha \cdot\|ABx-STy \|+\beta \cdot\|ABx-Px \|+\\ +\gamma\cdot \max\bigl\{\|STy-Py \|, \|ABx-Py \|\bigr\} +\delta\cdot \|STy-Px\|\end{multlined}NEWLINE\]NEWLINE for all \(x,y\in C\), where \(0\leq\alpha+ \gamma+\delta <1\) and \(0\leq\gamma <1\), NEWLINENEWLINENEWLINE(1.2) for some \(x_0\in C\), there exists a constant \(k\in[0,1)\) such that \(\|x_{n+2}- x_{n+1}\|\leq k\|x_{n+1}-x_n \|\) for \(n=0,1,2, \dots\) where \(\{x_n\}\) is a sequence in \(C\) defined byNEWLINENEWLINENEWLINE(1.3) \(ABx_{2n+1}= {1\over 2}Px_{2n}+ {1\over 2}ABx_{2n}\), \(STx_{2n+2}= {1\over 2}Px_{2n+1}+ {1\over 2} STx_{2n+1}\), NEWLINENEWLINENEWLINE(1.4) the pairs \(\{P,AB\}\) and \(\{P,ST\}\) are compatible,NEWLINENEWLINENEWLINE(1.5) \(PB=BP\), \(PT=TP\), \(AB=BA\), \(ST=TS\),NEWLINENEWLINENEWLINE(1.6) \(A,B,S,T\) are continuous at \(x\in C\).NEWLINENEWLINENEWLINEThen the sequence \(\{x_n\}\) defined by (1.3) converges to \(z\in C\) and \(Pz\) is a unique common fixed point of \(A,B,S,T\) and \(P\). Next, the authors extend Theorem 1 for six mappings. These results improve some previous theorems.