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 Common fixed point iterations for a class of multi-valued mappings in $$\mathbf {CAT}(0)$$ spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 681-690
Author(s):  
Khairul Saleh ◽  
Hafiz Fukhar-ud-din
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces ◽  
Fixed Point Iterations

Abstract In this work, we propose an iterative scheme to approach common fixed point(s) of a finite family of generalized multi-valued nonexpansive mappings in a CAT(0) space. We establish and prove convergence theorems for the algorithm. The results are new and interesting in the theory of $$CAT\left( 0\right) $$ C A T 0 spaces and are the analogues of corresponding ones in uniformly convex Banach spaces and Hilbert spaces.

scholarly journals On the JK Iterative Process in Banach Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Junaid Ahmad ◽  
Hüseyin Işık ◽  
Faeem Ali ◽  
Kifayat Ullah ◽  
Eskandar Ameer ◽  
...  
Keyword(s):  
Weak Convergence ◽  
Banach Spaces ◽  
Iterative Process ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
The Other ◽  
Iterative Schemes ◽  
Strong And Weak Convergence ◽  
Nonexpansive Maps ◽  
Iterative Procedures

In the recent progress, different iterative procedures have been constructed in order to find the fixed point for a given self-map in an effective way. Among the other things, an effective iterative procedure called the JK iterative scheme was recently constructed and its strong and weak convergence was established for the class of Suzuki mappings in the setting of Banach spaces. The first purpose of this research is to obtain the strong and weak convergence of this scheme in the wider setting of generalized α -nonexpansive mappings. Secondly, by constructing an example of generalized α -nonexpansive maps which is not a Suzuki map, we show that the JK iterative scheme converges faster as compared the other iterative schemes. The presented results of this paper properly extend and improve the corresponding results of the literature.

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.

Approximating common fixed point of asymptotically nonexpansive mappings without convergence condition

2017 ◽  
Vol 33 (3) ◽  
pp. 327-334
Author(s):  
ABDUL RAHIM KHAN ◽  
◽  
HAFIZ FUKHAR-UD-DIN ◽  
NUSRAT YASMIN ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Convergence Condition ◽  
Finite Family ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive Mappings ◽  
Uniformly Convex Banach Spaces ◽  
Asymptotically Nonexpansive

In the context of a hyperbolic space, we introduce and study convergence of an implicit iterative scheme of a finite family of asymptotically nonexpansive mappings without convergence condition. The results presented substantially improve and extend several well-known resullts in uniformly convex Banach spaces.

scholarly journals Strong Convergence of a General Iterative Method for a Countable Family of Nonexpansive Mappings in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Chin-Tzong Pang ◽  
Eskandar Naraghirad
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Accretive Operator ◽  
Common Fixed Points ◽  
Countable Family ◽  
Convergence Theorems ◽  
General Iterative Method

We introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a real Banach space. We prove strong convergence theorems for the sequences produced by the methods and approximate a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. Furthermore, we apply our results for finding a zero of an accretive operator. It is important to state clearly that the contribution of this paper in relation with the previous works (Marino and Xu, 2006) is a technical method to prove strong convergence theorems of a general iterative algorithm for an infinite family of nonexpansive mappings in Banach spaces. Our results improve and generalize many known results in the current literature.

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 Approximation of the Solution of Delay Fractional Differential Equation Using AA-Iterative Scheme

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 273
Author(s):  
Mujahid Abbas ◽  
Muhammad Waseem Asghar ◽  
Manuel De la Sen
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Contraction Mapping ◽  
Iterative Scheme ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Numerical Examples ◽  
Strong And Weak Convergence ◽  
Convergence Results ◽  
Fractional Differential

The aim of this paper is to propose a new faster iterative scheme (called AA-iteration) to approximate the fixed point of (b,η)-enriched contraction mapping in the framework of Banach spaces. It is also proved that our iteration is stable and converges faster than many iterations existing in the literature. For validity of our proposed scheme, we presented some numerical examples. Further, we proved some strong and weak convergence results for b-enriched nonexpansive mapping in the uniformly convex Banach space. Finally, we approximate the solution of delay fractional differential equations using AA-iterative scheme.

scholarly journals Some Convergence Results for a Class of Generalized Nonexpansive Mappings in Banach Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Thabet Abdeljawad ◽  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Manuel de la Sen ◽  
Muhammad Naveed Khan
Keyword(s):  
Weak Convergence ◽  
Iterative Method ◽  
Fixed Points ◽  
Banach Spaces ◽  
Current Literature ◽  
Nonexpansive Mappings ◽  
Uniformly Convex ◽  
Strong And Weak Convergence ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces

This paper investigates fixed points of Reich-Suzuki-type nonexpansive mappings in the context of uniformly convex Banach spaces through an M ∗ iterative method. Under some appropriate situations, some strong and weak convergence theorems are established. To support our results, a new example of Reich-Suzuki-type nonexpansive mappings is presented which exceeds the class of Suzuki-type nonexpansive mappings. The presented results extend some recently announced results of current literature.

scholarly journals Iterative Algorithms for a Finite Family of Multivalued Quasi-Nonexpansive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
C. Diop ◽  
M. Sene ◽  
N. Djitté
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Algorithm ◽  
Convex Subset ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Finite Family ◽  
Uniformly Convex ◽  
Convergence Theorems

Let K be a nonempty closed and convex subset of a uniformly convex real Banach space E and let T1,…,Tm:K→2K be m multivalued quasi-nonexpansive mappings. A new iterative algorithm is constructed and the corresponding sequence xn is proved to be an approximating fixed point sequence of each Ti; that is, limdxn;Txn=0. Then, convergence theorems are proved under appropriate additional conditions. Our results extend and improve some important recent results (e.g., Abbas et al. (2011)).

scholarly journals Convergence results of a faster iterative scheme including multi-valued mappings in Banach spaces

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1359-1368
Author(s):  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Muhammad Khan ◽  
Naseer Muhammad
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Current Literature ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Numerical Efficiency ◽  
Convergence Results

In this paper, we study M-iterative scheme in the new context of multi-valued generalized ?-nonexpansive mappings. A uniformly convex Banach space is used as underlying setting for our approach. We also provide a new example of generalized ?-nonexpasive mappings. We connect M iterative scheme and other well known schemes with this example, to show the numerical efficiency of our results. Our results improve and extend many existing results in the current literature.

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