General hybrid -proximal point algorithm frameworks for finding common solutions of nonlinear operator equations and fixed point problems

2013 ◽  
Vol 18 (4) ◽  
pp. 895-904 ◽  
Author(s):  
Heng-You Lan ◽  
Lecai Cai ◽  
Shulin Wu
Keyword(s):  
Fixed Point ◽  
Proximal Point Algorithm ◽  
Nonlinear Operator ◽  
Fixed Point Problems ◽  
Proximal Point ◽  
Operator Equations ◽  
Nonlinear Operator Equations

scholarly journals ON THE CONVERGENCE RATE OF THE KRASNOSEL’SKIĬ–MANN ITERATION

2017 ◽  
Vol 96 (1) ◽  
pp. 162-170 ◽  
Author(s):  
SHIN-YA MATSUSHITA
Keyword(s):  
Fixed Point ◽  
Convergence Rate ◽  
Proximal Point Algorithm ◽  
Convergence Rates ◽  
Fixed Point Problems ◽  
Proximal Point ◽  
Mann Iteration

The Krasnosel’skiĭ–Mann (KM) iteration is a widely used method to solve fixed point problems. This paper investigates the convergence rate for the KM iteration. We first establish a new convergence rate for the KM iteration which improves the known big-$O$ rate to little-$o$ without any other restrictions. The proof relies on the connection between the KM iteration and a useful technique on the convergence rate of summable sequences. Then we apply the result to give new results on convergence rates for the proximal point algorithm and the Douglas–Rachford method.

scholarly journals On the proximal point algorithm and demimetric mappings in CAT(0) spaces

2018 ◽  
Vol 51 (1) ◽  
pp. 277-294 ◽  
Author(s):  
Kazeem O. Aremu ◽  
Chinedu Izuchukwu ◽  
Godwin C. Ugwunnadi ◽  
Oluwatosin T. Mewomo
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Proximal Point Algorithm ◽  
Finite Family ◽  
Fixed Point Problems ◽  
Proximal Point ◽  
Numerical Example ◽  
Common Solution ◽  
Minimization Problems

Abstract In this paper, we introduce and study the class of demimetric mappings in CAT(0) spaces.We then propose a modified proximal point algorithm for approximating a common solution of a finite family of minimization problems and fixed point problems in CAT(0) spaces. Furthermore,we establish strong convergence of the proposed algorithm to a common solution of a finite family of minimization problems and fixed point problems for a finite family of demimetric mappings in complete CAT(0) spaces. A numerical example which illustrates the applicability of our proposed algorithm is also given. Our results improve and extend some recent results in the literature.

scholarly journals Fixed Point Theorems on Nonlinear Binary Operator Equations with Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Baomin Qiao
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlinear Operator ◽  
Order Theory ◽  
Iteration Theory ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Monotone Iteration ◽  
Binary Operator

The existence and uniqueness for solution of systems of some binary nonlinear operator equations are discussed by using cone and partial order theory and monotone iteration theory. Furthermore, error estimates for iterative sequences and some corresponding results are obtained. Finally, the applications of our results are given.

scholarly journals Solutions of Nonlinear Operator Equations by Viscosity Iterative Methods

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Mathew Aibinu ◽  
Surendra Thakur ◽  
Sibusiso Moyo
Keyword(s):  
Fixed Point ◽  
Nonlinear Operator ◽  
Monotone Mappings ◽  
Operator Equations ◽  
Fredholm Type ◽  
Nonlinear Operator Equations ◽  
Inverse Strongly Monotone ◽  
Strongly Monotone Mappings ◽  
Implicit Iterative Algorithm ◽  
Strictly Pseudocontractive Mappings

Finding the solutions of nonlinear operator equations has been a subject of research for decades but has recently attracted much attention. This paper studies the convergence of a newly introduced viscosity implicit iterative algorithm to a fixed point of a nonexpansive mapping in Banach spaces. Our technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The scheme is effective for obtaining the solutions of various nonlinear operator equations as it involves the generalized contraction. The results are applied to obtain a fixed point of λ-strictly pseudocontractive mappings, solution of α-inverse-strongly monotone mappings, and solution of integral equations of Fredholm type.

scholarly journals Iterations converging to distinct solutions of some nonlinear operator equations in Banach space

1986 ◽  
Vol 9 (3) ◽  
pp. 583-587
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Nonlinear Operator ◽  
Operator Equations ◽  
Nonlinear Operator Equations

We examine the solvability of multilinear equations of the formMk(x,x,…,x)−k   times−=y,   k=2,3,…whereMkis ak-linear operator on a Banach spaceXandy∈Xis fixed.

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