Almost \(T\)-stability of the iteration procedures with errors for strongly pseudocontractions in \(Q\)-uniformly smooth Banach spaces without continuity assumption (Q2703055)

This article deals with the almost \(T\)-stability property of Ishikawa and Man iterative procedures of approximation of fixed points for strongly pseudo-contractive mappings in \(q\)-uniformly smooth Banach spaces \((1< q<\infty)\). (An iterative process \(x_{n+1}= f_n(T,x_n)\) for the operator \(T\) is called almost \(T\)-stable if the relations \(x_n\to x_*\in \text{Fix }T\) and \(\sum^\infty_{n=1}\|y_n- f_n(T, y_n)\|< \infty\) imply \(y_n\to x_*\).) The main result is the theorem about the strong convergence of Ishikawa and Mann iterates with errors to the unique fixed point of \(T\) and almost \(T\)-stability of the corresponding processes. Some applications of this theorem to equations \(Tx= f\) and \(x+ Tx=f\) with strongly accretive and, correspondingly, accretive operators are given; these results generalize some recent theorems obtained by M. O. Osilike.