Abstract
In this paper, we introduce an extended S-iteration scheme for G-contractive type mappings and prove ∆-convergence as well as strong convergence in a nonempty closed and convex subset of a uniformly convex and complete b-metric space with a directed graph. We also give a numerical example in support of our result and compare the convergence rate between the studied iteration and the modified S-iteration.
Highlights
In 1922, Banach gave the proof of a fixed point result, which later on came to be known as the celebrated Banach contraction principle
We introduce an extended S-iteration scheme for G-contractive type mappings and prove ∆-convergence as well as strong convergence in a nonempty closed and convex subset of a uniformly convex and complete b-metric space with a directed graph
We prove a result on ∆-convergence and strong convergence of the iteration scheme given by (2.1) in a closed convex subset of a uniformly convex b-metric space
Summary
In 1922, Banach gave the proof of a fixed point result, which later on came to be known as the celebrated Banach contraction principle. B-metric space; extended S-iteration scheme; G-contractive type mapping; ∆-convergence; strong convergence. Where {αn} and {βn} are sequences in [0, 1] is called the Jungck-Ishikawa iteration scheme (refer to [25]). Motivated by [40], in this paper, we consider a convex b-metric space (X, d) with graph and define an extended S-iteration scheme for a triplet of three G-contractive type self mappings on a nonempty closed convex subset K of X. The convergence of this iteration scheme in comparison to the existing modified S-iteration scheme is discussed with a numerical example.
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