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Generalized Set-valued Nonlinear Variational-like Inequalities and Fixed Point Problems: Existence and Approximation Solvability Results

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Abstract

The paper is devoted to the introduction of a new class of generalized set-valued nonlinear variational-like inequality problems in the setting of Banach spaces. By means of the notion of P-\(\eta \)-proximal mapping, we prove its equivalence with a class of generalized implicit Wiener–Hopf equations and employ the obtained equivalence relationship and Nadler’s technique to suggest a new iterative algorithm for finding an approximate solution of the considered problem. The existence of solution and the strong convergence of the sequences generated by our proposed iterative algorithm to the solution of our considered problem are verified. The problem of finding a common element of the set of solutions of a generalized nonlinear variational-like inequality problem and the set of fixed points of a total asymptotically nonexpansive mapping is also investigated. The final section deals with the investigation and analysis of the main results appeared in Kazmi and Bhat (Appl Math Comput 166:164–180, 2005) and some comments relating to them are given. The results presented in this article extend and improve some known results in the literature.

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Acknowledgements

The authors are indebted to both anonymous referees for their careful readings and thoughtful suggestions that allowed us to improve the original presentation significantly.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran

    Javad Balooee

  2. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    Shih-sen Chang & Jen-Chih Yao

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  1. Javad Balooee
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Correspondence to Shih-sen Chang.

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Communicated by Akhtar A. Khan.

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Balooee, J., Chang, Ss. & Yao, JC. Generalized Set-valued Nonlinear Variational-like Inequalities and Fixed Point Problems: Existence and Approximation Solvability Results. J Optim Theory Appl 197, 891–938 (2023). https://doi.org/10.1007/s10957-023-02182-8

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