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General Class of Implicit Variational Inclusions and Graph Convergence on A-Maximal Relaxed Monotonicity

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  • Ram U. Verma

    (Texas A&M University)

Abstract

Based on the generalized graph convergence, first a general framework for an implicit algorithm involving a sequence of generalized resolvents (or generalized resolvent operators) of set-valued A-maximal monotone (also referred to as A-maximal (m)-relaxed monotone, and A-monotone) mappings, and H-maximal monotone mappings is developed, and then the convergence analysis to the context of solving a general class of nonlinear implicit variational inclusion problems in a Hilbert space setting is examined. The obtained results generalize the work of Huang, Fang and Cho (in J. Nonlinear Convex Anal. 4:301–308, 2003) involving the classical resolvents to the case of the generalized resolvents based on A-maximal monotone (and H-maximal monotone) mappings, while the work of Huang, Fang and Cho (in J. Nonlinear Convex Anal. 4:301–308, 2003) added a new dimension to the classical resolvent technique based on the graph convergence introduced by Attouch (in Variational Convergence for Functions and Operators, Applied Mathematics Series, Pitman, London 1984). In general, the notion of the graph convergence has potential applications to several other fields, including models of phenomena with rapidly oscillating states as well as to probability theory, especially to the convergence of distribution functions on ℜ. The obtained results not only generalize the existing results in literature, but also provide a certain new approach to proofs in the sense that our approach starts in a standard manner and then differs significantly to achieving a linear convergence in a smooth manner.

Suggested Citation

  • Ram U. Verma, 2012. "General Class of Implicit Variational Inclusions and Graph Convergence on A-Maximal Relaxed Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 196-214, October.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:1:d:10.1007_s10957-012-0030-9
    DOI: 10.1007/s10957-012-0030-9
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    References listed on IDEAS

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    1. R. U. Verma, 2006. "A-Monotonicity and Its Role in Nonlinear Variational Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 129(3), pages 457-467, June.
    2. R. U. Verma, 2006. "General System of A-Monotone Nonlinear Variational Inclusion Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 131(1), pages 151-157, October.
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    Cited by:

    1. Javad Balooee & Shih-Sen Chang & Lin Wang & Yu Zhang & Zhao-Li Ma, 2023. "Graph Convergence, Algorithms, and Approximation of Common Solutions of a System of Generalized Variational Inclusions and Fixed-Point Problems," Mathematics, MDPI, vol. 11(4), pages 1-29, February.
    2. Javad Balooee & Shih-Sen Chang & Lin Wang & Zhaoli Ma, 2022. "Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities," Mathematics, MDPI, vol. 10(12), pages 1-40, June.

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