Approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces
The article deals with common fixed points of a pair of mappings \(S\) and \(T\) in a generalized convex metric space \(X\) under the assumption \[ d(S^nx,T^ny) \leq h \, \max \;\{d(x,y),d(x,S^nx),d(y,T^ny),d(x,T^ny),d(y,S^nx)\}, \quad h < 1 \] (a generalized convex metric space is defined as usual metric space in which there is given a mapping \(w: \;X^3 \times [0,1]^3 \to X\) such that the inequality \[ d(w(x,y,z,a_n,b_n,c_n),u) \leq a_n d(x,u) + b_n d(y,u) + c_n d(z,u) \] \[ \bigg(a_n + b_n + c_n = a_n' + b_n' + c_n' = 1, \quad \sum_{n=1}^\infty (b_n + c_n) < \infty\bigg) \] holds for all elements \(x, y,z,u \in X\) and sequences \(a_n, b_n, c_n \in [0,1]\) with \(a_n + b_n + c_n = 1\). It is assumed that \(X\) is complete, \(S\) and \(T\) are defined on a nonempty closed convex subset \(K \subset X\), and \(F = F(S) \cap F(T)\) is nonempty. It is proved (a) that the iteration \[ x_{n+1} = w(x_nS^ny,u_n,a_n,b_n,c_n), \quad y_n = w(x_n,T^nx_n,v_n,a_n',b_n',c_n') \] (\(u_n, v_n \in K\) and for any positive integers \(n, m\), \(1 \leq n < m\), and the inequality \[ \delta_{nm} = \text{diam} \, \{x_i,y_i,Sy_i,Tx_i,u_i,v_i: \;n \leq i \leq m\} > 0, \] implies the inequality \[ \max \;\{d(x,y): \;x \in \{u_i,v_i\}, \;y \in \{x_j,y_j,Sy_j,Tx_j,u_j,v_j\}\} < \delta_{nm}) \] converges to the unique common fixed point of \(S\) and \(T\) if and only if \(\liminf_{n\to \infty} d(x_n,F) \to 0\); (b) under some additional conditions this iteration converges to the unique common fixed point of \(S\) and \(T\). The authors state that these results, in the partial case when \(X\) is a Banach space, extend and generalize many known results.
Ishikawa type iteration process with errors
scientific article; zbMATH DE number 5632599
scientific article