Graph Convergence Approach of Resolvent Operators to Approximate the Solutions of General Class of Variational Inclusions

Gupta, Sanjeev; Khan, Faizan Ahmad; Ali, Montaser Saudi; Eljaneid, Nidal H. E.; Alatawi, Adel; Alamrani, Fahad M.

doi:10.3390/math14111818

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Graph Convergence Approach of Resolvent Operators to Approximate the Solutions of General Class of Variational Inclusions

by

Sanjeev Gupta

^1,*

,

Faizan Ahmad Khan

²

,

Montaser Saudi Ali

²

,

Nidal H. E. Eljaneid

²

,

Adel Alatawi

²

and

Fahad M. Alamrani

^2,*

¹

Department of Mathematics, Institute of Applied Sciences and Humanities, GLA University, Mathura 281406, India

²

Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia

^*

Authors to whom correspondence should be addressed.

Mathematics 2026, 14(11), 1818; https://doi.org/10.3390/math14111818 (registering DOI)

Submission received: 1 March 2026 / Revised: 15 May 2026 / Accepted: 20 May 2026 / Published: 24 May 2026

(This article belongs to the Special Issue Variational Inequality Problem: Theory, Analysis and Applications)

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Abstract

In this study, we introduce the concept of the graph convergence for

B^{n}

-co-monotone mappings in Banach spaces and establish an equivalence between the graph convergence and resolvent operators convergence for the

B^{n}

-co-monotone operator sequence. Further, as an application, we propose an iterative algorithm for solving a class of variational inclusions in the framework of real q-uniformly smooth Banach spaces. We prove the existence and uniqueness of the solutions for the variational inclusion and convergence of iterative sequence obtained by the iterative algorithm under appropriate convergence criteria. Our work can be viewed as incorporating some existing results as special cases. Four examples are constructed using MATLAB 2015a to illustrate some of the concepts and convergence behavior used in the article.

Keywords: Bn-co-monotone mappings; resolvent operators; graph convergence; iterative methods; variational inclusions

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MDPI and ACS Style

Gupta, S.; Khan, F.A.; Ali, M.S.; Eljaneid, N.H.E.; Alatawi, A.; Alamrani, F.M. Graph Convergence Approach of Resolvent Operators to Approximate the Solutions of General Class of Variational Inclusions. Mathematics 2026, 14, 1818. https://doi.org/10.3390/math14111818

AMA Style

Gupta S, Khan FA, Ali MS, Eljaneid NHE, Alatawi A, Alamrani FM. Graph Convergence Approach of Resolvent Operators to Approximate the Solutions of General Class of Variational Inclusions. Mathematics. 2026; 14(11):1818. https://doi.org/10.3390/math14111818

Chicago/Turabian Style

Gupta, Sanjeev, Faizan Ahmad Khan, Montaser Saudi Ali, Nidal H. E. Eljaneid, Adel Alatawi, and Fahad M. Alamrani. 2026. "Graph Convergence Approach of Resolvent Operators to Approximate the Solutions of General Class of Variational Inclusions" Mathematics 14, no. 11: 1818. https://doi.org/10.3390/math14111818

APA Style

Gupta, S., Khan, F. A., Ali, M. S., Eljaneid, N. H. E., Alatawi, A., & Alamrani, F. M. (2026). Graph Convergence Approach of Resolvent Operators to Approximate the Solutions of General Class of Variational Inclusions. Mathematics, 14(11), 1818. https://doi.org/10.3390/math14111818

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Graph Convergence Approach of Resolvent Operators to Approximate the Solutions of General Class of Variational Inclusions

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