A new algorithm for two finite families of demicontractive mappings and equilibrium problems

Abkar, Ali; Tavakkoli, M.

doi:10.1016/j.amc.2015.05.078

Cited by 4 publications

(7 citation statements)

References 23 publications

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“…In a similar fashion, combining (43), (53), and (58) and the fact that for each i ≥ 1, r i < 𝛼 i , we have lim…”

Section: Assume That Conditions (I)-(v) Of Lemma 3 and The Following ...mentioning

confidence: 90%

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Inertial hybrid algorithm for generalized mixed equilibrium problems, zero problems, and fixed points of some nonlinear mappings in the intermediate sense

Ahmad,

Kumam,

Harbau

et al. 2024

Math Methods in App Sciences

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In this work, we establish the closedness and convexity of the set of fixed points of equally continuous and asymptotically demicontractive mapping in the intermediate sense. We proposed an inertial hybrid projection technique for determining an approximate common solution to three significant problems. The first is the system of generalized mixed equilibrium problems with relaxed monotone mappings, the second is the problem of fixed points of a countable family of equally continuous and asymptotically demicontractive mappings in the intermediate sense, and the third is of determining a point in a null space of a countable family of inverse strongly monotone mappings in Hilbert space. Based on these problems, we formulate a theorem and establish its strong convergence to their common solution. Additionally, we studied the applications of our algorithm to variational inequality problems and convex optimization problems. Finally, we numerically demonstrate the efficiency and robustness of our scheme. Several results available in the literature can be obtained as special cases of our result.

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“…In a similar fashion, combining (43), (53), and (58) and the fact that for each i ≥ 1, r i < 𝛼 i , we have lim…”

Section: Assume That Conditions (I)-(v) Of Lemma 3 and The Following ...mentioning

confidence: 90%

“…and a weak convergence theorem for (21) with some imposed conditions on {𝜆 n }, {𝛿 n } has been established. Lately, the following new algorithm has been proposed by Abkar and Tavakkoli [43], for equilibrium problem and fixed points of two families of demicontractives in Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

Inertial hybrid algorithm for generalized mixed equilibrium problems, zero problems, and fixed points of some nonlinear mappings in the intermediate sense

Ahmad,

Kumam,

Harbau

et al. 2024

Math Methods in App Sciences

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“…Observe that every k-strictly pseudononspreading map with a nonempty fixed-point set F(T) is k-demicontractive. Recently Abkar et al 11 proposed a new algorithm for finding a common element set of solutions found in an equilibrium problem and the set of common fixed points found in two finite families of demicontractive mappings. The sequences {x n } and {u n } are generated by…”

Section: Introductionmentioning

confidence: 99%

“…Inspired and motivated by above works and the work of other studies 10,11,17,19,[29][30][31] in this paper, we modify CQ methods for finding a common solution set of system found in equilibrium problem and the fixed-point set of a finite family of demicontractive mappings. Further, we prove a strong convergence theorem for the sequences generated by iterative method and derive some consequences which are new and interesting.…”

Section: Introductionmentioning

confidence: 99%

“…Recently Abkar et al proposed a new algorithm for finding a common element set of solutions found in an equilibrium problem and the set of common fixed points found in two finite families of demicontractive mappings. The sequences

false{x_{n} false}

and

false{u_{n} false}

are generated by

({, \begin{matrix} left \\ array F (u_{n}, y) + \frac{1}{r_{n}} ⟨ y - u_{n}, u_{n} - x_{n} ⟩ \geq 0, \forall y \in C, \\ array y_{n, i} = α_{n} u_{n} + β x_{n} + \sum_{i = 1}^{N} γ_{n, i} T_{i} u_{n}, \\ array w_{n, i} = δ_{n} y_{n} + \sum_{i = 1}^{N} δ_{n, i} S_{i} y_{n}, \\ array x_{n + 1} = a_{n} γ f (w_{n}) + (1 - a_{n} A) w_{n}, \forall n \geq 0, \end{matrix})

where

α_{n} + β + true \sum_{i = 1}^{N} γ_{n, i} = 1

and

δ_{n} + true \sum_{i = 1}^{N} δ_{}

…”

Section: Introductionmentioning

confidence: 99%

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A hybrid projective method for solving system of equilibrium problems with demicontractive mappings applicable in image restoration problems

Khan

Kazmi

Yambangwai

et al. 2020

Math Methods in App Sciences

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In this paper, we propose a general iterative scheme based on CQ projection method for finding a common solution of system of equilibrium problems and the fixed point set of a finite family of demicontractive mappings. We also prove strong convergence of the scheme to a common element of the two above‐described sets. We then give a numerical example to justify our main result. An example is given in an infinite dimensional space for supporting our main result. Moreover, we apply our main result to solve the unconstrained image restoration problems with a finite family of blurring operators. Our results extend and improve some existing results in the literature.

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Single-Valued Demicontractive Mappings: Half a Century of Developments and Future Prospects

Berinde

2023

Symmetry

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Demicontractive operators form an important class of nonexpansive type mappings whose study led researchers to the creation of some beautiful results in the framework of metric fixed-point theory. This article aims to provide an overview of the most relevant results on the approximation of fixed points of single-valued demicontractive mappings in Hilbert spaces. Subsequently, we exhibit the role of additional properties of demicontractive operators, as well as the main features of the employed iterative algorithms to ensure weak convergence or strong convergence. We also include commentaries on the use of demicontractive mappings to solve some important nonlinear problems with the aim of providing a comprehensive starting point to readers who are attempting to apply demicontractive mappings to concrete applications. We conclude with some brief statements on our view on relevant and promising directions of research on demicontractive mappings in nonlinear settings (metric spaces) and some application challenges.

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A new algorithm for two finite families of demicontractive mappings and equilibrium problems

Cited by 4 publications

References 23 publications

Inertial hybrid algorithm for generalized mixed equilibrium problems, zero problems, and fixed points of some nonlinear mappings in the intermediate sense

Inertial hybrid algorithm for generalized mixed equilibrium problems, zero problems, and fixed points of some nonlinear mappings in the intermediate sense

A hybrid projective method for solving system of equilibrium problems with demicontractive mappings applicable in image restoration problems

Single-Valued Demicontractive Mappings: Half a Century of Developments and Future Prospects

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