Approximation of Fixed Points for Suzuki's Generalized Non-Expansive Mappings

Abstract

In this paper, we study a three step iterative scheme to approximate fixed points of Suzuki's generalized non-expansive mappings. We establish some weak and strong convergence results for such mappings in uniformly convex Banach spaces. Further, we show numerically that iterative scheme (1.8) converges faster than some other known iterations for Suzuki's generalized non-expansive mappings. To support our claim, we give an illustrative example and approximate fixed points of such mappings using Matlab program. Our results are new and generalize several relevant results in the literature.

Weak and strong convergence theorems for three Suzuki’s generalized nonexpansive mappings

“…In 2008, Suzuki [8] introduced the concept of generalized nonexpansive mappings which is a condition on mappings called (C) condition. Let K be a nonempty convex subset of a Banach space X, a mapping T : K → K is satisfy condition (C) if for all x, y ∈ K, 1 2 x − T x x − y implies T x − T y x − y . Suzuki [8] showed that the mapping satisfying condition (C) is weaker than nonexpansiveness and stronger than quasi-nonexpansiveness.…”

“…The mapping satisfying condition (C) is called Suzuki generalized nonexpansive mapping. Recently, fixed point theorems for Suzuki generalized nonexpansive mappings have been studied by a number of authors see e.g., [1,4,10,12].…”

“…They proved that this scheme converges to a fixed point of contraction mapping, faster than all the known iterative schemes. Ali et al [1] studied the above iteration process defined by (1.2). They proved some weak and strong convergence theorems of the above iterative method for Suzuki's generalized nonexpansive mappings in uniformly convex Banach spaces.…”

Fixed-Point Convergence Results of a Three-Step Iterative Process in CAT(0) Spaces

“…After this, many papers appeared on the topic of Suzuki maps in linear and nonlinear spaces (cf. [8][9][10][11] and references therein).…”

Some observations on generalized non-expansive mappings with an application

“…Recently, the study of fixed points of mappings satisfying Suzuki's condition (C) has been studied by several researchers, e.g. see Thakur et al (2016), Dhomphongsa et al (2009), Ali et al (2019), Uddin and Imdad (2015a), Uddin and Imdad (2015b), Uddin and Imdad (2018).…”