Common solution to a split equality monotone variational inclusion problem, a split equality generalized general variational-like inequality problem and a split equality fixed point problem

In this paper, the split variational inclusion problem (SVIP) and the system of equilibrium problems (EP) are considered in Hilbert spaces. Inspired by the works of Byrne et al., López et al., Moudafi and Thukur, Sobumt and Plubtieng, Sitthithakerngkiet et al. and Eslamian and Fakhri, a new self-adaptive step size algorithm is proposed to find a common element of the solution set of the problems SVIP and EP. Convergence theorems are established under suitable conditions for the algorithm and application to the common solution of the fixed point problem, and the split convex optimization problem is considered. Finally, the performances and computational experiments are presented and a comparison with the related algorithms is provided to illustrate the efficiency and applicability of our new algorithms.

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A new viscosity-type iteration for a finite family of split variational inclusion and fixed point problems between Hilbert and Banach spaces

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2201-9 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 1

Author(s):

C. Izuchukwu ◽

F. O. Isiogugu ◽

C. C. Okeke

Keyword(s):

Fixed Point ◽

Fixed Point Theory ◽

Iteration Process ◽

Point Theory ◽

Fixed Point Problem ◽

Finite Family ◽

Variational Inclusion ◽

Inclusion Problem ◽

Point Problem ◽

Common Solution

Abstract In this paper, we introduce a new viscosity-type iteration process for approximating a common solution of a finite family of split variational inclusion problem and fixed point problem. We prove that the proposed algorithm converges strongly to a common solution of a finite family of split variational inclusion problems and fixed point problem for a finite family of type-one demicontractive mappings between a Hilbert space and a Banach space. Furthermore, we applied our results to study a finite family of split convex minimization problems, and also considered a numerical experiment of our results to further illustrate its applicability. Our results extend and improve the results of Byrne et al. (J. Nonlinear Convex Anal. 13:759–775, 2012), Kazmi and Rizvi (Optim. Lett. 8(3):1113–1124, 2014), Moudafi (J. Optim. Theory Appl. 150:275–283, 2011), Shehu and Ogbuisi (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 110(2):503–518, 2016), Takahashi and Yao (Fixed Point Theory Appl. 2015:87, 2015), Chidume and Ezeora (Fixed Point Theory Appl. 2014:111, 2014), and a host of other important results in this direction.

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Solving Split Variational Inclusion Problem and Fixed Point Problem for Nonexpansive Semigroup without Prior Knowledge of Operator Norms

Mathematical Problems in Engineering ◽

10.1155/2015/408165 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 3

Author(s):

Haitao Che ◽

Meixia Li

Keyword(s):

Fixed Point ◽

Fixed Point Problem ◽

Variational Inclusion ◽

Common Element ◽

Inclusion Problem ◽

Point Problem ◽

Split Variational Inclusion Problem ◽

Variational Inclusion Problem ◽

Nonexpansive Semigroups ◽

Operator Norms

We introduce an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups. Moreover, numerical results demonstrate the performance and convergence of our result, which may be viewed as a refinement and improvement of the previously known results announced by many other researchers.

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Split Variational Inclusion Problem and Fixed Point Problem for a Class of Multivalued Mappings in CAT(0) Spaces

Mathematics ◽

10.3390/math7080749 ◽

2019 ◽

Vol 7 (8) ◽

pp. 749 ◽

Cited By ~ 1

Author(s):

Mujahid Abbas ◽

Yusuf Ibrahim ◽

Abdul Rahim Khan ◽

Manuel De la Sen

Keyword(s):

Fixed Point ◽

Fixed Point Problem ◽

Variational Inclusion ◽

Multivalued Mappings ◽

Strong Convergence Theorem ◽

Inclusion Problem ◽

Point Problem ◽

Strictly Pseudocontractive Mapping ◽

Split Variational Inclusion Problem ◽

Variational Inclusion Problem

The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in C A T ( 0 ) spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of C A T ( 0 ) spaces.

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Modified Halpern Iterative Method for Solving Hierarchical Problem and Split Combination of Variational Inclusion Problem in Hilbert Space

Mathematics ◽

10.3390/math7111037 ◽

2019 ◽

Vol 7 (11) ◽

pp. 1037

Author(s):

Bunyawee Chaloemyotphong ◽

Atid Kangtunyakarn

Keyword(s):

Iterative Method ◽

Zero Point ◽

Fixed Point Problem ◽

Variational Inclusion ◽

Strong Convergence Theorem ◽

Inclusion Problem ◽

Point Problem ◽

Numerical Example ◽

Common Solution ◽

Variational Inclusion Problem

The purpose of this paper is to introduce the split combination of variational inclusion problem which combines the concept of the modified variational inclusion problem introduced by Khuangsatung and Kangtunyakarn and the split variational inclusion problem introduced by Moudafi. Using a modified Halpern iterative method, we prove the strong convergence theorem for finding a common solution for the hierarchical fixed point problem and the split combination of variational inclusion problem. The result presented in this paper demonstrates the corresponding result for the split zero point problem and the split combination of variation inequality problem. Moreover, we discuss a numerical example for supporting our result and the numerical example shows that our result is not true if some conditions fail.

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The zeros of monotone operators for the variational inclusion problem in Hilbert spaces

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02663-2 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Pattanapong Tianchai

Keyword(s):

Fixed Point ◽

Hilbert Spaces ◽

Monotone Operators ◽

Fixed Point Problem ◽

Variational Inclusion ◽

Feasibility Problem ◽

Strong Convergence Theorem ◽

Inclusion Problem ◽

Point Problem ◽

Variational Inclusion Problem

AbstractIn this paper, we introduce a regularization method for solving the variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters, which allow us to obtain a short proof of another strong convergence theorem for this problem. We also apply our main result to the fixed point problem of the nonexpansive variational inequality problem, the common fixed point problem of nonexpansive strict pseudocontractions, the convex minimization problem, and the split feasibility problem. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique.

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