Algorithms for Approximating Solutions of Split Variational Inclusion and Fixed-Point Problems

Zhu, Li-Jun; Yao, Yonghong

doi:10.3390/math11030641

Cited by 12 publications

(6 citation statements)

References 30 publications

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“…For other related developments on this topic, we refer the readers to Maingé [19], who obtained common fixed-point convergence results by extending Mȃruşter's convergence theorem (Theorem 1) from one demicontractive mapping to a finite family of demicontractive mappings in a Hilbert space; Adamu and Adam [30]; Arfat et al [31], where the authors investigate a fixed-point problem involving an infinite family of k-demicontractive operators in conjunction with the split common null point problems (SCNPP) in Hilbert spaces by using an accelerated variant of the hybrid shrinking projection algorithm; Maingé and Mȃruşter [32]; Anh et al [33]; Charoensawan and Suparatulatorn [34]; Lin [35]; Mongkolkeha et al [36]; Wang et al [37]; Xiao et al [38]; Zhu and Yao [39], etc.…”

Section: The Fixed-point Problemmentioning

confidence: 99%

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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Section: The Fixed-point Problemmentioning

confidence: 99%

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

Berinde

2023

Symmetry

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“…In the above-mentioned work and related literature, we found that the step size, which is under the control of norm ∥A * A∥, is required for the convergence of iterative schemes. To overcome this regulation, a new form of iterative schemes have been proposed, see, e.g., [16][17][18][19]. L ópez et al [20] proposed a relaxed iterative scheme for (S p FP):…”

Section: Introductionmentioning

confidence: 99%

Halpern-Type Inertial Iteration Methods with Self-Adaptive Step Size for Split Common Null Point Problem

Alamer,

Dilshad

2024

Mathematics

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In this paper, two Halpern-type inertial iteration methods with self-adaptive step size are proposed for estimating the solution of split common null point problems (SpCNPP) in such a way that the Halpern iteration and inertial extrapolation are computed simultaneously in the beginning of each iteration. We prove the strong convergence of sequences driven by the suggested methods without estimating the norm of bounded linear operator when certain appropriate assumptions are made. We demonstrate the efficiency of our iterative methods and compare them with some related and well-known results using relevant numerical examples.

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“…Recently, Akutsah et al [28] introduced a new algorithm for finding a common fixed point of generalized nonexpansive mappings. Moreover, the fixed point problems have been studied in many aspects, see for instance [29][30][31][32][33]. Inspired by Bot et al [16], Artsawang and Ungchittrakool [25] introduced the inertial Mann-type algorithm for finding a fixed point of a nonexpansive mapping and applying to monotone inclusion and image restoration problems as follows:…”

Section: Introductionmentioning

confidence: 99%

“…Assume that Fix(S) ∩ Fix(T) = ∅ and by employing (33), then the following algorithm can be constructed for solving (35) as follows:…”

mentioning

confidence: 99%

Modified Mann-Type Algorithm for Two Countable Families of Nonexpansive Mappings and Application to Monotone Inclusion and Image Restoration Problems

Ungchittrakool,

Plubtieng,

Artsawang

et al. 2023

Mathematics

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In this paper, we introduce and study a modified Mann-type algorithm that combines inertial terms for solving common fixed point problems of two countable families of nonexpansive mappings in Hilbert spaces. Under appropriate assumptions on the sequences of parameters, we establish a strong convergence result for the sequence generated by the proposed method in finding a common fixed point of two countable families of nonexpansive mappings. This method can be applied to solve the monotone inclusion problem. Additionally, we employ a modified Mann-type iterative algorithm to address image restoration problems. Furthermore, we present numerical results across different scenarios to demonstrate the superior efficiency of our algorithm compared to existing algorithms.

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Algorithms for Approximating Solutions of Split Variational Inclusion and Fixed-Point Problems

Cited by 12 publications

References 30 publications

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

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

Halpern-Type Inertial Iteration Methods with Self-Adaptive Step Size for Split Common Null Point Problem

Modified Mann-Type Algorithm for Two Countable Families of Nonexpansive Mappings and Application to Monotone Inclusion and Image Restoration Problems

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