scholarly journals A Krasnoselskii–Ishikawa Iterative Algorithm for Monotone Reich Contractions in Partially Ordered Banach Spaces with an Application

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 76
Author(s):  
Nawab Hussain ◽  
Saud M. Alsulami ◽  
Hind Alamri
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Sufficient Conditions ◽  
Iterative Algorithms ◽  
Approximate Solutions ◽  
Existence Of A Solution ◽  
Partially Ordered ◽  
Ordered Banach Spaces

Iterative algorithms have been utilized for the computation of approximate solutions of stationary and evolutionary problems associated with differential equations. The aim of this article is to introduce concepts of monotone Reich and Chatterjea nonexpansive mappings on partially ordered Banach spaces. We describe sufficient conditions for the existence of an approximate fixed-point sequence (AFPS) and prove certain fixed-point results using the Krasnoselskii–Ishikawa iterative algorithm. Moreover, we present some interesting examples to highlight the superiority of our results. Lastly, we provide both weak and strong convergence results for such mappings and consider an application of our results to prove the existence of a solution to an initial value problem.

scholarly journals Best Proximity Points for Monotone Relatively Nonexpansive Mappings in Ordered Banach Spaces

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 121
Author(s):  
Karim Chaira ◽  
Mustapha Kabil ◽  
Abdessamad Kamouss ◽  
Samih Lazaiz
Keyword(s):  
Banach Spaces ◽  
Best Proximity Point ◽  
Nonexpansive Mappings ◽  
Sufficient Conditions ◽  
Best Proximity Points ◽  
Partially Ordered ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive ◽  
Ordered Banach Spaces

In this paper, we give sufficient conditions to ensure the existence of the best proximity point of monotone relatively nonexpansive mappings defined on partially ordered Banach spaces. An example is given to illustrate our results.

scholarly journals Fixed point approximation of Presic nonexpansive mappings in product of CAT (0) spaces

2016 ◽  
Vol 32 (3) ◽  
pp. 315-322
Author(s):  
HAFIZ FUKHAR-UD-DIN ◽  
◽  
VASILE BERINDE ◽  
ABDUL RAHIM KHAN ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Banach Spaces ◽  
Point Theorem ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Control Parameters ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces

We obtain a fixed point theorem for Presiˇ c nonexpansive mappings on the product of ´ CAT (0) spaces and approximate this fixed points through Ishikawa type iterative algorithms under relaxed conditions on the control parameters. Our results are new in the literature and are valid in uniformly convex Banach spaces.

scholarly journals Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition

2020 ◽  
Vol 36 (1) ◽  
pp. 27-34 ◽  
Author(s):  
VASILE BERINDE
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Displacement Condition ◽  
Nonlinear Mappings

In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij-Mann typeassociated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generaliza-tions of the corresponding ones in [Berinde, V.,Approximating fixed points of enriched nonexpansive mappings byKrasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304.], from the setting of Hilbertspaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G.,Approximating fixed pointsof nonexpansive mappings, Proc. Amer. Math. Soc.,44(1974), No. 2, 375–380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.], byconsidering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related resultsin literature can be obtained as particular instances of our results.

scholarly journals Random Fixed Point Theorems of Random Comparable Operators and an Application

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Jiandong Yin ◽  
Zhongdong Liu
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Monotone Operators ◽  
Hammerstein Integral Equation ◽  
Random Fixed Point ◽  
Partially Ordered ◽  
Ordered Banach Spaces

We introduce the new concept of random comparable operators as a generalization of random monotone operators and prove several random fixed point theorems for such a class of operators in partially ordered Banach spaces. Part of the presented results generalize and extend some known results of random monotone operators. Finally, as an application, we consider the existence of the solution of a random Hammerstein integral equation.

scholarly journals Existence of local fractional integral equation via a measure of non-compactness with monotone property on Banach spaces

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hemant Kumar Nashine ◽  
Rabha W. Ibrahim ◽  
Ravi P. Agarwal ◽  
N. H. Can
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Banach Spaces ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Coupled Fixed Point ◽  
Partially Ordered ◽  
Fractional Integral Equation ◽  
Local Fractional Integral ◽  
Ordered Banach Spaces

AbstractIn this paper, we discuss fixed point theorems for a new χ-set contraction condition in partially ordered Banach spaces, whose positive cone $\mathbb{K}$ K is normal, and then proceed to prove some coupled fixed point theorems in partially ordered Banach spaces. We relax the conditions of a proper domain of an underlying operator for partially ordered Banach spaces. Furthermore, we discuss an application to the existence of a local fractional integral equation.

THE EXISTENCE OF A SOLUTION OF THE NONLINEAR INTEGRAL EQUATION VIA THE FIXED POINT APPROACH

2021 ◽  
Vol 10 (7) ◽  
pp. 2977-2998
Author(s):  
T.A. Adeyemi ◽  
F. Akusah ◽  
A.A. Mebawondu ◽  
M.O. Adewole ◽  
O.K. Narain
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Nonlinear Integral Equation ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Hyperbolic Spaces ◽  
Uniformly Convex ◽  
Existence Of A Solution ◽  
Uniformly Convex Banach Spaces ◽  
Uniformly Convex Hyperbolic Space

In this paper, we present some fixed point results for a generalized class of nonexpansive mappings in the framework of uniformly convex hyperbolic space and also propose a new iterative scheme for approximating the fixed point of this class of mappings in the framework of uniformly convex hyperbolic spaces. Furthermore, we establish some basic properties and some strong and $\triangle$-convergence theorems for these mappings in uniformly convex hyperbolic spaces. Finally, we present an application to the nonlinear integral equation and also, a numerical example to illustrate our main result and then display the efficiency of the proposed algorithm compared to different iterative algorithms in the literature with different choices of parameters and initial guesses. The results obtained in this paper extends and generalizes corresponding results in uniformly convex Banach spaces, CAT(0) spaces and other related results in literature.

scholarly journals Fixed Point Theorems of Binary Contraction Comparable Operators and an Application

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Zhan Liu ◽  
Chuanxi Zhu
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Large Class ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Nonlinear Integral Equation ◽  
Nonlinear Problems ◽  
Existence Of Solution ◽  
Partially Ordered ◽  
Ordered Banach Spaces

The aim of this paper is to present the concept of binary comparable operators in partially ordered Banach spaces and prove several fixed point theorems under some contractive conditions. The results of this paper can be used to investigate a large class of nonlinear problems. As an application, we study the existence of solution of a nonlinear integral equation.

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