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1

Remarks on some variants of minimal point theorem and Ekeland variational principle with applications

Meghea Irina, Stamin Cristina Stefania

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

354--379

EN

This paper presents some variants of minimal point theorem together with corresponding variants of Ekeland variational principle. In the second part of this article, there is a discussion on Ekeland variational principle and minimal point theorem relative to it in uniform spaces. The aim of these series of important results is to highlight relations between them, some improvements and specific applications.

2

On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces

Abass Hammed A., Oyewole Olawale K., Mebawondu Akindele A., Aremu Kazeem O., Narain Ojen K.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

658--675

EN

In this article, motivated by the works of Ali Akbar and Elahe Shahrosvand [Split equality common null point problem for Bregman quasi-nonexpansive mappings, Filomat 32 (2018), no. 11, 3917–3932], Eskandani et al. [A hybrid extragradient method for solving pseudomonotone equilibrium problem using Bregman distance, J. Fixed Point Theory Appl. 20 (2018), 132], B. Ali and M. H. Harbau [Convergence theorems for Bregman K-mappings and mixed equilibrium problems in reflexive Banach spaces, J. Funct. Spaces (2016) Article ID 5161682, 18 pages], and some other related results in the literature, we introduce a hybrid extragradient iterative algorithm that employs a Bregman distance approach for approximating a split feasibility problem for a finite family of equilibrium problems involving pseudomonotone bifunctions and fixed point problems for a finite family of Bregman quasi-asymptotically nonexpansive mappings using the concept of Bregman K-mapping in reflexive Banach spaces. Using our iterative algorithm, we state and prove a strong convergence result for approximating a common solution to the aforementioned problems. Furthermore, we give an application of our main result to variational inequalities and also report a numerical example to illustrate the convergence of our method. The result presented in this article extends and complements many related results in the literature.

3

On solving pseudomonotone equilibrium problems via two new extragradient-type methods under convex constraints

Khunpanuk Chainarong, Pakkaranang Nuttapol, Pholasa Nattawut

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

297--314

EN

The primary objective of this study is to develop two new proximal-type algorithms for solving equilibrium problems in real Hilbert space. Both new algorithms are analogous to the well-known two-step extragradient algorithm for solving the variational inequality problem in Hilbert spaces. The proposed iterative algorithms use a new step size rule based on local bifunction information instead of the line search technique. Two weak convergence theorems for both algorithms are well-established by letting mild conditions. The main results are used to solve the fixed point and variational inequality problems. Finally, we present several computational experiments to demonstrate the efficiency and effectiveness of the proposed algorithms.

4

An extragradient inertial algorithm for solving split fixed-point problems of demicontractive mappings, with equilibrium and variational inequality problems

Okeke Chibueze C., Ugwunnadi Godwin C., Jolaoso Lateef O.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

506--527

EN

The purpose of this article is to study and analyse a new extragradient-type algorithm with an inertial extrapolation step for solving split fixed-point problems for demicontractive mapping, equilibrium problem, and pseudomonotone variational inequality problem in real Hilbert spaces. One of the advantages of the proposed algorithm is that a strong convergence result is achieved without a prior estimate of the Lipschitz constant of the cost operator, which is very difficult to find. In addition, the stepsize is generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the Lipschitz constant of the cost operator. Some numerical experiments are reported to show the performance and behaviour of the sequence generated by our algorithm. The obtained results in this article extend and improve many related recent results in this direction in the literature.

5

Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

Pakkaranang Nuttapol, Rehman Habib ur, Kumam Wiyada

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

280--298

EN

The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

6

An iterative algorithm for the system of split mixed equilibrium problem

Karahan Ibrahim, Jolaoso Lateef Olakunle

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

309--324

EN

In this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.

7

Approximation of common fixed points of left Bregman strongly nonexpansive mappings and solutions of equilibrium problems

Shehu Y., Ogbuisi F. U.

Journal of Applied Analysis

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2015

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Vol. 21, nr 2

63--77

EN

In this paper we propose an iterative algorithm based on the hybrid method in mathematical programming for approximating a common fixed point of an infinite family of left Bregman strongly nonexpansive mappings which also solves a finite system of equilibrium problems in a reflexive real Banach space.We further prove that our iterative sequence converges strongly to a common fixed point of an infinite family of left Bregman strongly nonexpansive mappings which is also a common solution to a finite system of equilibrium problems. Our result extends many recent and important results in the literature.

8

Viscosity approximation methods for nonexpansive multi-valued nonself mappings and equilibrium problems

Cholamjiak P., Cholamjiak W., Suantai S.

Demonstratio Mathematica

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2014

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Vol. 47, nr 2

382--395

EN

In this paper, strong convergence theorems by the viscosity approximation method for nonexpansive multi-valued nonself mappings and equilibrium problems are established under some suitable conditions in a Hilbert space. The obtained results extend and improve the corresponding results existed in the literature.

9

Modified Mann iterative algorithms by hybrid projection methods for nonexpansive semigroups and mixed equilibrium problems

Katchang P., Kumam P.

Journal of Applied Analysis

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2012

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Vol. 18, nr 2

259-273

EN

In this paper, we propose a modified Mann iterative algorithm by two hybrid projection methods for finding a common element of the set of fixed points of nonexpansive semigroups and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we obtain interesting and new strong convergence theorems for the sequences generated by these processes by using the hybrid projection methods in the mathematical programming. The results presented in this paper extend and improve the corresponding one by Nakajo and Takahashi [J. Math. Anal. Appl. 279 (2003), 372-379].

10

Approximation for fixed points of strict pseudo-contractions and solutions of equilibrium and optimization problems

Kang J., Su Y., Zhang H.

Journal of Applied Analysis

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2010

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Vol. 16, nr 1

1-13

EN

In this paper, we introduce some iterative schemes for findin a common element of the set of fixed points of a k-strict pseudo-contractive mapping, the set of solutions of the variational inequality and the set of solutions of an equilibrium problem in a Hilbert space. The authors use the convex combination technique to show that the iterative sequences converge strongly to a common element of the three sets. The results of this paper extend and improve the results of Y. Su et al. [9], S. Plubtieng and R. Punpaeng [7], X. Qin et al. [8] and S. Takahashi and W. Takahashi [10].