Abstract
The main aim of this paper is to present the concept of general Mann and general Ishikawa type double-sequences iterations with errors to approximate fixed points. We prove that the general Mann type double-sequence iteration process with errors converges strongly to a coincidence point of two continuous pseudo-contractive mappings, each of which maps a bounded closed convex nonempty subset of a real Hilbert space into itself. Moreover, we discuss equivalence from the -stabilities point of view under certain restrictions between the general Mann type double-sequence iteration process with errors and the general Ishikawa iterations with errors. An application is also given to support our idea using compatible-type mappings. MSC:47H10, 54H25.
Highlights
In the last few decades investigations of fixed points by some iterative schemes have attracted many mathematicians
The theory of operators has occupied a central place in modern research using iterative schemes because of its promise of enormous utility in fixed point theory and its applications
There are a number of papers that have studied fixed points by some iterative schemes
Summary
In the last few decades investigations of fixed points by some iterative schemes have attracted many mathematicians. The Mann iterative scheme was invented in (see [ – ]), and it is used to obtain convergence to a fixed point for many classes of mappings (see [ – ] and others). Several authors have proved some fixed point theorems for Mann-type iterations with errors using several classes of mappings (see [ – ] and others). We define the following iteration which will be called the general Mann iteration process with errors: Suk,n+ = ( – αn)Suk,n + αnTuk,n + αnun. Using this general Mann iteration process, we give a strong convergence theorem in the double-sequence setting.
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