Some common fixed point theorems for biased mappings. (English) Zbl 1048.47044

Let \((X,d)\) be a metric space and \(A,B: X \rightarrow X\). Following G. Jungck and H. K. Pathak [Proc. Am. Math. Soc. 123, 2049-2060 (1995; Zbl 0824.47045)], we say that the pair \(\{A,B\}\) is \(B\)-biased if \[ \begin{aligned} \liminf _{n\rightarrow \infty } d(BAx_n,Bx_n) &\leq \liminf _{n\rightarrow \infty } d(ABx_n, Ax_n), \\ \limsup _{n\rightarrow \infty } d(BAx_n, Bx_n) &\leq \limsup _{n\rightarrow \infty }d(ABx_n, Ax_n) \end{aligned} \] whenever \(\{x_n\}\subset X\) and \(\lim _{n\rightarrow \infty }Ax_n = \lim _{n\rightarrow \infty }Bx_n\in X\). In this paper, the authors give conditions on operators \(A\) and \(B\) mapping a normed space \(X\) into itself so that \(\{A,B\}\) is \(B\)-biased, which guarantee the existence of a unique common fixed point of \(A\) and \(B\). Using concepts of “biased mappings”, existence results for a unique common fixed point of four operators are proved as well.


47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)


Zbl 0824.47045
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