scholarly journals Modified Inertial-Type Multi-Choice CQ-Algorithm for Countable Weakly Relatively Non-Expansive Mappings in a Banach Space, Applications and Numerical Experiments

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 613
Author(s):  
Li Wei ◽  
Yibin Xin ◽  
Ruilan Zhang ◽  
Ravi P. Agarwal
Keyword(s):  
Banach Space ◽  
Numerical Experiments ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Space Applications ◽  
Uniformly Smooth ◽  
Special Cases ◽  
Cq Algorithm ◽  
Proof Techniques

In this paper, some new modified inertial-type multi-choice CQ-algorithms for approximating common fixed point of countable weakly relatively non-expansive mappings are presented in a real uniformly convex and uniformly smooth Banach space. New proof techniques are used to prove strong convergence theorems, which extend some previous work. The connection and application to maximal monotone operators are demonstrated. Numerical experiments are conducted to illustrate that the rate of convergence is accelerated compared to some previous ones for some special cases.

scholarly journals Convergence theorems for generalized projections and maximal monotone operators in Banach spaces

2003 ◽  
Vol 2003 (10) ◽  
pp. 621-629 ◽  
Author(s):  
Takanori Ibaraki ◽  
Yasunori Kimura ◽  
Wataru Takahashi
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Banach Space ◽  
Limit Set ◽  
Strong And Weak Convergence ◽  
Strictly Convex Banach Space ◽  
Generalized Projections

We study a sequence of generalized projections in a reflexive, smooth, and strictly convex Banach space. Our result shows that Mosco convergence of their ranges implies their pointwise convergence to the generalized projection onto the limit set. Moreover, using this result, we obtain strong and weak convergence of resolvents for a sequence of maximal monotone operators.

scholarly journals Strong and Weak Convergence of Modified Mann Iteration for New Resolvents of Maximal Monotone Operators in Banach Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-20 ◽  
Author(s):  
Somyot Plubtieng ◽  
Wanna Sriprad
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Convergence Rate ◽  
Minimization Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Minimization Problem ◽  
Mann Iteration ◽  
Strong And Weak Convergence

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).

scholarly journals Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 773 ◽  
Author(s):  
Mihai Postolache ◽  
Ashish Nandal ◽  
Renu Chugh
Keyword(s):  
Fixed Point ◽  
Splitting Method ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Special Cases ◽  
Implicit Rule

In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward–backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem.

scholarly journals Strong convergence of an iterative sequence for maximal monotone operators in a Banach space

2004 ◽  
Vol 2004 (3) ◽  
pp. 239-249 ◽  
Author(s):  
Fumiaki Kohsaka ◽  
Wataru Takahashi
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Variational Inequality Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Iterative Sequence ◽  
Strong Convergence Theorem ◽  
Convex Minimization Problem ◽  
Proximal Point

We first introduce a modified proximal point algorithm for maximal monotone operators in a Banach space. Next, we obtain a strong convergence theorem for resolvents of maximal monotone operators in a Banach space which generalizes the previous result by Kamimura and Takahashi in a Hilbert space. Using this result, we deal with the convex minimization problem and the variational inequality problem in a Banach space.

scholarly journals A Unified Iterative Treatment for Solutions of Problems of Split Feasibility and Equilibrium in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Young-Ye Huang ◽  
Chung-Chien Hong
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Zero Point ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Special Cases ◽  
Split Feasibility

We at first raise the so called split feasibility fixed point problem which covers the problems of split feasibility, convex feasibility, and equilibrium as special cases and then give two types of algorithms for finding solutions of this problem and establish the corresponding strong convergence theorems for the sequences generated by our algorithms. As a consequence, we apply them to study the split feasibility problem, the zero point problem of maximal monotone operators, and the equilibrium problem and to show that the unique minimum norm solutions of these problems can be obtained through our algorithms. Since the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, each type of our algorithms can be considered as a generalized methodology for solving the aforementioned problems.

scholarly journals Split variational inclusions for Bregman multivalued maximal monotone operators

2020 ◽  
Author(s):  
Mujahid Abbas ◽  
Faik Gürsoy ◽  
Yusuf Ibrahim ◽  
Abdul Rahim Khan
Keyword(s):  
Strong Convergence ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Strong Convergence Theorem ◽  
Variational Inclusions ◽  
Uniformly Convex ◽  
Inclusion Problems ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces

We introduce a new algorithm to approximate a solution of split variational inclusion problems of multivalued maximal monotone operators in uniformly convex and uniformly smooth Banach spaces under the Bregman distance. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. As application, we solve a split minimization problem and provide a numerical example to support better findings of our result.

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