Iterative algorithms for systems of extended regularized nonconvex variational inequalities and fixed point problems

This paper deals with designing a new iteration scheme associated with a given scheme for contraction mappings. This new scheme has a similar structure to that of the given scheme, in which those two iterative schemes converge to the same fixed point of the given contraction mapping. The positive influence of feedback parameters on the convergence rate of this new scheme is investigated. Moreover, the derived convergence and comparison results can be extended to nonexpansive mappings. As an application, the derived results are utilized to study the synchronization of logistic maps. Two illustrated examples are used to reveal the effectiveness of our results.

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“…. , , then lim →∞ ‖ +1 − +1 ‖ = 0, which also implies that scheme (3) and scheme (2) converge to the same fixed point of .…”

Section: Resultsmentioning

confidence: 84%

An Iteration Scheme for Contraction Mappings with an Application to Synchronization of Discrete Logistic Maps

Ding

Kim

et al. 2017

Discrete Dynamics in Nature and Society

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“…The purpose of this paper is not to criticize the authors of the articles, but to examine what is wrong with their publications to help researchers who are interested to avoid these mistakes and pay attention when using references on system of nonlinear variational inequalities. Also, I show that there is no favor in setting up this definition and all the results obtained in [1]- [25] have no pregress in H × H. We can redirect the previous studies [1]- [25] with the logical definitions as follow: Definition 3. A mapping T : H × H → H is said to be λ-Lipschitz in the first variable if there exists constant λ > 0 such that, for all u ∈ H, for all pairs x 1 , x 2 in H,…”

Section: Discussionmentioning

confidence: 98%

“…There are a lot of papers written on System of variational inequalities, in all these publications the authors used an unclear Lipschitz continuous in the first variable and/or second variable definition. The aim of this paper is to illustrate that there is no sense in setting up this definition and all the results obtained in [1]- [25] have no benefit in H × H.…”

Section: Introductionmentioning

confidence: 99%

A Valuable Remark on Lipschitz in the First Variable Definition and System of Nonlinear Variational Inequalities

Benhadid¹

2022

Croat. oper. res. rev. (Online)

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The goal of this paper is to present a critique on the incorrect use of the Lipschitz definition concerning the first variable and/or the second variable in the literature on the system of variational inequalities by many authors. The possible impact of this paper is rather important, it questions the results of different authors, particularly when taking into account that some of these papers are published in quite good mathematical journals. As a result, not only that the proofs are wrong, but also the credibility of the theorems themselves is compromised. In addition, this paper illustrate, using a counterexample, that there is an error in setting up first variable definition and the results obtained in listed references do not hold up in H × H.

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“…where for each n ∈ N, (10), we know that max{k 1 , k 2 } ∈ (0, 1), and so ϑ ∈ (0, 1). Therefore, there exist θ ∈ (0, 1) (take θ = ϑ+1 2 ∈ (ϑ, 1)) and n 0 ∈ N such that ϑ(n) ≤ θ, for all n ≥ n 0 .…”

Section: Iterative Algorithms and Convergence Analysismentioning

confidence: 99%

Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities

Balooee

Chang²,

Wang³

et al. 2022

Mathematics

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The main aim of this paper is twofold. Our first objective is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of P-η-proximal mapping. The obtained alternative equivalent formulation is used and a new iterative algorithm for finding its approximate solution is suggested. Under some appropriate assumptions imposed on the mappings and parameters involved in the system of generalized multivalued variational-like inequalities, the existence of solution for the system mentioned above is proved and the convergence analysis of the sequences generated by our proposed iterative algorithm is discussed. The second objective of this work is to investigate and analyze the notion M-η-proximal mapping defined in the literature. Taking into account of the assumptions considered for such a mapping, we prove that every M-η-proximal mapping is actually P-η-proximal and is not a new one. At the same time, some comments relating to some existing results are pointed out.

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Iterative algorithms for systems of extended regularized nonconvex variational inequalities and fixed point problems

Cited by 19 publications

References 16 publications

An Iteration Scheme for Contraction Mappings with an Application to Synchronization of Discrete Logistic Maps

An Iteration Scheme for Contraction Mappings with an Application to Synchronization of Discrete Logistic Maps

A Valuable Remark on Lipschitz in the First Variable Definition and System of Nonlinear Variational Inequalities

Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities

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