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Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

Author

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  • Mihai Postolache

    (Department of General Education, China Medical University, Taichung 40402, Taiwan
    Gh. Mihoc-C. Iacob Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, 050711 Bucharest, Romania
    Department of Mathematics and Informatics, University “Politehnica” of Bucharest, 060042 Bucharest, Romania
    Current address: Department of Mathematics and Informatics, University “Politehnica” of Bucharest, 060042 Bucharest, Romania.)

  • Ashish Nandal

    (Department of Mathematics, Pt NRS Government College, Rohtak 124001, India
    These authors contributed equally to this work.)

  • Renu Chugh

    (Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, India
    These authors contributed equally to this work.)

Abstract

In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019 , 11 , 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward-backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem.

Suggested Citation

  • Mihai Postolache & Ashish Nandal & Renu Chugh, 2019. "Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space," Mathematics, MDPI, vol. 7(9), pages 1-24, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:773-:d:260133
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    References listed on IDEAS

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    1. Lu-Chuan Ceng & Abdul Latif & Abdullah E. Al-Mazrooei, 2014. "Iterative Algorithms for Systems of Generalized Equilibrium Problems with the Constraints of Variational Inclusion and Fixed Point Problems," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-24, March.
    2. Lu-Chuan Ceng & Chang-yu Wang & Jen-Chih Yao, 2008. "Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(3), pages 375-390, June.
    3. Lu-Chuan Ceng & Qamrul Ansari & Siegfried Schaible, 2012. "Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems," Journal of Global Optimization, Springer, vol. 53(1), pages 69-96, May.
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    Cited by:

    1. Kumari, Sudesh & Gdawiec, Krzysztof & Nandal, Ashish & Postolache, Mihai & Chugh, Renu, 2022. "A novel approach to generate Mandelbrot sets, Julia sets and biomorphs via viscosity approximation method," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).

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