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Mann-type approximation scheme for solving a new class of split inverse problems in Hilbert spaces

APPLICABLE ANALYSIS 2024年第6期103卷 1118-1148页

作者： Wickramasinghe, Madushi U. Mewomo, Oluwatosin T. Alakoya, Timilehin O. Iyiola, Olaniyi S. Univ KwaZulu Natal Sch Math Stat & Comp Sci Durban South Africa Clarkson Univ Dept Math Potsdam NY 13699 USA

In this paper, we introduce and study a new class of split inverse problems, named split hierarchical monotone variational inclusion problem with multiple output sets in real Hilbert spaces. By using the inertial technique and self-adaptive step size strategy, we propose and analyze a new Mann-type iterative method for solving the problem. The convergence analysis of the proposed iterative method under some suitable conditions is studied. Also, we show that the sequence of iterates generated by this method converges strongly to a minimum-norm solution of the problem. As theoretical applications, we apply our results to approximate the solutions of other classes of split inverse problems. Finally, we present some numerical experiments to illustrate the practical potential and advantages of our proposed method.

关键词： split inverse problems mann-type algorithm inertial technique self-adaptive step size

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AN EFFICIENT ALGORITHM BASED ON VISCOSITY TSENG-TYPE METHOD FOR SOLVING split HIERARCHICAL MONOTONE VARIATIONAL INCLUSION problems WITH MULTIPLE OUTPUT SETS

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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2024年第4期21卷 2558-2595页

作者： Wickramasinghe, Madushi u. Ugwu, Odirachukwunma Alakoya, Timilehin o. Mewomo, Oluwatosin t. Iyiola, Olaniyi s. Morgan State Univ Dept Math Baltimore MD 21251 USA Univ Kwazulu Natal Sch Math Stat & Comp Sci Durban South Africa

Very recently, [Wickramasinghe et al., Mann type approximation scheme for solving a new class of split inverse problems in Hilbert spaces Applicable Anal. (2023). DOI: 10.1080/00036811. 2023.2233977] introduced a new class of split inverse problems, known as split hierarchical monotone variational inclusion problems with multiple output sets. The authors proposed a Manntype iterative method to approximate the solution of the problem in Hilbert spaces. However, to establish their convergence result, the authors required the associated single-valued operators to be co-coercive. This condition is too stringent and limits the scope of application of their results. In this study, we aim to improve their result of [Wickramasinghe et al. ] by relaxing the co-coercive condition to monotone. We also introduce a new viscosity Tsengtype iterative method for solving the problem in Hilbert spaces and obtain strong convergence results under more relaxed conditions. Furthermore, our study extends to addressing the split hierarchical variational inequality problem involving multiple output sets, the split hierarchical convex minimization problem with multiple output sets, and the generalized split hierarchical monotone variational inclusion problem. Lastly, we perform numerical experiments in both finite- and infinite-dimensional spaces and provide comparison results with existing methods discussed in the literature.

关键词： split inverse problems monotone inclusion problems viscosity type algorithm hierarchical multiple output sets

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Relaxed Inertial Method for Solving split Monotone Variational Inclusion Problem with Multiple Output Sets Without Co-coerciveness and Lipschitz Continuity

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Acta Mathematica Sinica,English Series 2024年第7期40卷 1697-1726页

作者： Timilehin Opeyemi Alakoya Oluwatosin Temitope Mewomo School of Mathematics Statistics and Computer ScienceUniversity of KwaZulu-NatalDurbanSouth Africa

In this paper,we study the concept of split monotone variational inclusion problem with multiple output *** propose a new relaxed inertial iterative method with self-adaptive step sizes for approximating the solution of the problem in the framework of Hilbert *** proposed algorithm does not require the co-coerciveness nor the Lipschitz continuity of the associated single-valued ***,some parameters are relaxed to accommodate a larger range of values for the step *** some mild conditions on the control parameters and without prior knowledge of the operator norms,we obtain strong convergence result for the proposed ***,we apply our result to study certain classes of optimization problems and we present several numerical experiments to demonstrate the implementability of the proposed *** of the existing results in the literature could be viewed as special cases of our result in this paper.

关键词： split inverse problems non-Lipschitz operators inertial technique self-adaptive step sizes

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Iterative solutions of the split common fixed point problem for strictly pseudo-contractive mappings

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JOURNAL OF FIXED POINT THEORY AND APPLICATIONS 2018年第2期20卷 1-12页

作者： Cui, Huanhuan Ceng, Luchuan Shanghai Normal Univ Dept Math Shanghai 200234 Peoples R China

In this paper, we study the the split common fixed point problem in Hilbert spaces. We establish a weak convergence theorem for the method recently introduced by Wang, which extends a existing result from firmly nonexpansive mappings to strictly pseudo-contractive mappings. Moreover, our condition that guarantees the weak convergence is much weaker than that of Wang's. A strong convergence theorem is also obtained under some additional conditions. As an application, we obtain several new methods for solving various split inverse problems and split equality problems. Numerical examples are included to illustrate the applications in signal processing of the proposed algorithm.

关键词： split common fixed-point problem strictly pseudo-contractive mappings split inverse problems split equality problem

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Mann-Type Inertial Projection and Contraction Method for Solving split Pseudomonotone Variational Inequality Problem with Multiple Output Sets

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MEDITERRANEAN JOURNAL OF MATHEMATICS 2023年第6期20卷 336-336页

作者： Alakoya, T. O. Mewomo, O. T. Univ KwaZulu Natal Sch Math Stat & Comp Sci Durban South Africa

In this paper, we study the concept of split variational inequality problem with multiple output sets when the cost operators are pseudomonotone and non-Lipschitz. We introduce a new Mann-type inertial projection and contraction method with self-adaptive step sizes for approximating the solution of the problem in the framework of Hilbert spaces. Under some mild conditions on the control parameters and without prior knowledge of the operator norms, we prove a strong convergence theorem for the proposed algorithm. We point out that while the cost operators are non-Lipschitz, our proposed method does not require any linesearch method but uses a more efficient self-adaptive step size technique that generates a non-monotonic sequence of step sizes. Finally, we apply our result to study certain classes of optimization problems and we present several numerical experiments to illustrate the applicability of the proposed method. Several of the existing results in the literature could be viewed as special cases of our result in this study.

关键词： split inverse problems variationl inequalities non-Lipschitz operators projection & contraction method inertial technique

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