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Modified Inertial Forward–Backward Algorithm in Banach Spaces and Its Application

Author

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  • Yanlai Song

    (College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China
    The authors contributed equally to this work.)

  • Mihai Postolache

    (Department of General Education, China Medical University, Taichung 40402, Taiwan
    Department of Interior Design, Asia University, Taichung 41354, 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)

Abstract

In this paper, we present a new modified inertial forward–backward algorithm for finding a common solution of the quasi-variational inclusion problem and the variational inequality problem in a q -uniformly smooth Banach space. The proposed algorithm is based on descent, splitting and inertial ideas. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the abovementioned problems. Numerical examples are also given to demonstrate our results.

Suggested Citation

  • Yanlai Song & Mihai Postolache, 2021. "Modified Inertial Forward–Backward Algorithm in Banach Spaces and Its Application," Mathematics, MDPI, vol. 9(12), pages 1-17, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:12:p:1365-:d:574073
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    References listed on IDEAS

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    1. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    2. Yao, Yonghong & Cho, Yeol Je & Liou, Yeong-Cheng, 2011. "Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems," European Journal of Operational Research, Elsevier, vol. 212(2), pages 242-250, July.
    3. Yanlai Song & Luchuan Ceng, 2013. "A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces," Journal of Global Optimization, Springer, vol. 57(4), pages 1327-1348, December.
    4. Genaro López & Victoria Martín-Márquez & Fenghui Wang & Hong-Kun Xu, 2012. "Forward-Backward Splitting Methods for Accretive Operators in Banach Spaces," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-25, July.
    5. Zhangsong Yao & Yan-Kuen Wu & Ching-Feng Wen & Jen-Chih Yao, 2021. "Strong Convergence Analysis of Iterative Algorithms for Solving Variational Inclusions and Fixed-Point Problems of Pseudocontractive Operators," Journal of Mathematics, Hindawi, vol. 2021, pages 1-7, April.
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