scholarly journals Approximating Fixed Points Using a Faster Iterative Method and Application to Split Feasibility Problems

Computation ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 90
Author(s):  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Muhammad Arshad ◽  
Zhenhua Ma
Keyword(s):  
Iterative Scheme ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Iterative Schemes ◽  
Split Feasibility Problems ◽  
Mild Conditions ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces ◽  
The Many ◽  
Split Feasibility

In this article, the recently introduced iterative scheme of Hassan et al. (Math. Probl. Eng. 2020) is re-analyzed with the connection of Reich–Suzuki type nonexpansive (RSTN) maps. Under mild conditions, some important weak and strong convergence results in the context of uniformly convex Banach spaces are provided. To support the main outcome of the paper, we provide a numerical example and show that this example properly exceeds the class of Suzuki type nonexpansive (STN) maps. It has been shown that the Hassan et al. iterative scheme of this example is more useful than the many other iterative schemes. We provide an application of our main results to solve split feasibility problems in the setting of RSTN maps. The presented outcome is new and compliments the corresponding results of the current literature.

scholarly journals Approximation of Fixed Points for Suzuki’s Generalized Non-Expansive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 522 ◽  
Author(s):  
Javid Ali ◽  
Faeem Ali ◽  
Puneet Kumar
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Numerical Example ◽  
Matlab Program ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces

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 the considered iterative scheme converges faster than some other known iterations for Suzuki’s generalized non-expansive mappings. To support our claim, we give an illustrative numerical example and approximate fixed points of such mappings using Matlab program. Our results are new and generalize several relevant results in the literature.

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

2019 ◽  
Author(s):  
Javid Ali ◽  
Faeem Ali ◽  
Puneet Kumar
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Matlab Program ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces

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.

scholarly journals Convergence of Infinite Family of Multivalued Quasi-Nonexpansive Mappings Using Multistep Iterative Processes

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Renu Chugh ◽  
Sanjay Kumar
Keyword(s):  
Real Banach Space ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Finite Family ◽  
Countable Family ◽  
Uniformly Convex ◽  
Iterative Schemes ◽  
Strong And Weak Convergence ◽  
Convergence Results

We prove strong and weak convergence results using multistep iterative sequences for countable family of multivalued quasi-nonexpansive mappings by using some conditions in uniformly convex real Banach space. The results presented extended and improved the corresponding result of Zhang et al. (2013), Bunyawat and Suantai (2012), and some others from finite family, one countable family, and two countable families tok-number of countable families of multivalued quasi-nonexpansive mappings. Also we used a numerical example in C++ computational programs to prove that the iterative scheme we used has better rate of convergence than other existing iterative schemes.

scholarly journals Approximation of the Fixed Point for Unified Three-Step Iterative Algorithm with Convergence Analysis in Busemann Spaces

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 26
Author(s):  
Hassan Almusawa ◽  
Hasanen A. Hammad ◽  
Nisha Sharma
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Convergence Analysis ◽  
Iterative Algorithm ◽  
Iterative Scheme ◽  
Uniformly Convex ◽  
Iterative Schemes ◽  
Convergence Results

In this manuscript, a new three-step iterative scheme to approximate fixed points in the setting of Busemann spaces is introduced. The proposed algorithms unify and extend most of the existing iterative schemes. Thereafter, by making consequent use of this method, strong and Δ-convergence results of mappings that satisfy the condition (Eμ) in the framework of uniformly convex Busemann space are obtained. Our results generalize several existing results in the same direction.

scholarly journals Common Fixed Points for Weak and Strong Convergence Results

2016 ◽  
Vol 4 (2) ◽  
pp. 340
Author(s):  
S. C. Shrivastava
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Iterative Scheme ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Convergence Results

<div><p> <em>In this paper, we study the approximation of common fixed points for more general classes of mappings through weak and strong convergence results of an iterative scheme in a uniformly convex Banach space. Our results extend and improve some known recent results.</em></p></div>

scholarly journals A New Faster Iterative Scheme for Numerical Fixed Points Estimation of Suzuki’s Generalized Nonexpansive Mappings

2020 ◽  
Vol 2020 ◽  
pp. 1-9 ◽  
Author(s):  
Shanza Hassan ◽  
Manuel De la Sen ◽  
Praveen Agarwal ◽  
Qasim Ali ◽  
Azhar Hussain
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Iteration Scheme ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Uniformly Convex Banach Spaces ◽  
The Stability

The purpose of this paper is to introduce a new four-step iteration scheme for approximation of fixed point of the nonexpansive mappings named as S∗-iteration scheme which is faster than Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, and Ullah iteration schemes. We show the stability of our proposed scheme. We present a numerical example to show that our iteration scheme is faster than the aforementioned schemes. Moreover, we present some weak and strong convergence theorems for Suzuki’s generalized nonexpansive mappings in the framework of uniformly convex Banach spaces. Our results extend, improve, and unify many existing results in the literature.

scholarly journals A New Iterative Scheme for Approximation of Fixed Points of Suzuki's Generalized Nonexpansive Mappings

2021 ◽  
Author(s):  
Shivam Rawat ◽  
Ramesh Chandra Dimri ◽  
Ayush Bartwal
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Iteration Scheme ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Uniformly Convex Banach Spaces ◽  
The Stability

In this paper, we introduce a new iteration scheme, named as the S**-iteration scheme, for approximation of fixed point of the nonexpansive mappings. This scheme is faster than Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, and Ullah iteration schemes. We show the stability of our instigated scheme and give a numerical example to vindicate our claim. We also put forward some weak and strong convergence theorems for Suzuki's generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. Our results comprehend, improve, and consolidate many results in the existing literature.

Inertial accelerated algorithms for solving split feasibility with multiple output sets in Hilbert spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Chibueze C. Okeke ◽  
Lateef O. Jolaoso ◽  
Yekini Shehu
Keyword(s):  
Strong Convergence ◽  
Prior Knowledge ◽  
Hilbert Spaces ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Weak And Strong Convergence ◽  
Numerical Examples ◽  
Mild Conditions ◽  
Convergence Results ◽  
Split Feasibility

Abstract In this paper, we propose two inertial accelerated algorithms which do not require prior knowledge of operator norm for solving split feasibility problem with multiple output sets in real Hilbert spaces. We prove weak and strong convergence results for approximating the solution of the considered problem under certain mild conditions. We also give some numerical examples to demonstrate the performance and efficiency of our proposed algorithms over some existing related algorithms in the literature.

scholarly journals Iterative methods for generalized split feasibility problems in Banach spaces

2017 ◽  
Vol 33 (1) ◽  
pp. 09-26
Author(s):  
QAMRUL HASAN ANSARI ◽  
◽  
AISHA REHAN ◽  
◽  
Keyword(s):  
Weak Convergence ◽  
Banach Spaces ◽  
Iterative Methods ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Algorithms ◽  
Approximate Solutions ◽  
Split Feasibility Problems ◽  
Convergence Results ◽  
Split Feasibility

Inspired by the recent work of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 23 (2015), 205–221], in this paper, we study generalized split feasibility problems (GSFPs) in the setting of Banach spaces. We propose iterative algorithms to compute the approximate solutions of such problems. The weak convergence of the sequence generated by the proposed algorithms is studied. As applications, we derive some algorithms and convergence results for some problems from nonlinear analysis, namely, split feasibility problems, equilibrium problems, etc. Our results generalize several known results in the literature including the results of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, SetValued Var. Anal., 23 (2015), 205–221].

scholarly journals On Best Proximity Results for a Generalized Modified Ishikawa’s Iterative Scheme Driven by Perturbed 2-Cyclic Like-Contractive Self-Maps in Uniformly Convex Banach Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
M. De la Sen ◽  
Mujahid Abbas
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Iterative Process ◽  
Iterative Scheme ◽  
Convex Banach Space ◽  
The Self ◽  
Uniformly Convex ◽  
Cyclic Structure ◽  
Best Proximity Points ◽  
Uniformly Convex Banach Spaces

This paper proposes a generalized modified iterative scheme where the composed self-mapping driving can have distinct step-dependent composition order in both the auxiliary iterative equation and the main one integrated in Ishikawa’s scheme. The self-mapping which drives the iterative scheme is a perturbed 2-cyclic one on the union of two sequences of nonempty closed subsets Ann=0∞ and Bnn=0∞ of a uniformly convex Banach space. As a consequence of the perturbation, such a driving self-mapping can lose its cyclic contractive nature along the transients of the iterative process. These sequences can be, in general, distinct of the initial subsets due to either computational or unmodeled perturbations associated with the self-mapping calculations through the iterative process. It is assumed that the set-theoretic limits below of the sequences of sets Ann=0∞ and Bnn=0∞ exist. The existence of fixed best proximity points in the set-theoretic limits of the sequences to which the iterated sequences converge is investigated in the case that the cyclic disposal exists under the asymptotic removal of the perturbations or under its convergence of the driving self-mapping to a limit contractive cyclic structure.

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