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Forward–Backward–Half Forward Dynamical Systems for Monotone Inclusion Problems with Application to v-GNE

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Abstract

In this paper, the first-order forward–backward–half forward dynamical systems associated with the inclusion problem consisting of three monotone operators are analyzed. The framework modifies the forward–backward–forward dynamical system by adding a cocoercive operator to the inclusion. The existence, uniqueness, and weak asymptotic convergence of the generated trajectories are discussed. A variable metric forward–backward–half forward dynamical system with the essence of non-self-adjoint linear operators is introduced. The proposed notion, in turn, extends the forward–backward–forward dynamical system and forward–backward dynamical system in the framework of variable metric by relaxing some conditions on the metrics. The distributed dynamical system is further explored to compute a generalized Nash equilibrium in a monotone game as an application. A numerical example is provided to illustrate the convergence of trajectories.

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Acknowledgements

The first author acknowledges the financial support from Ministry of Human Resource and Development (MHRD), New Delhi, India, under Junior Research Fellow (JRF) scheme. Also the third author acknowledges the financial support from Indian Institute of Technology (BHU), Varanasi, India, in terms of teaching assistantship. The authors are grateful to an anonymous reviewer for pointing out the flaws in the initial version of the paper.

Funding

Funding was provided by University Grants Commission (IN) (Grant No. 411751).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005, India

    Pankaj Gautam, Avinash Dixit & Tanmoy Som

  2. Department of Mathematics, Banaras Hindu University, Varanasi, 221005, India

    Daya Ram Sahu

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  1. Pankaj Gautam
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  2. Daya Ram Sahu
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  3. Avinash Dixit
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  4. Tanmoy Som
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Correspondence to Avinash Dixit.

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Communicated by Radu Ioan Bot.

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Gautam, P., Sahu, D.R., Dixit, A. et al. Forward–Backward–Half Forward Dynamical Systems for Monotone Inclusion Problems with Application to v-GNE. J Optim Theory Appl 190, 491–523 (2021). https://doi.org/10.1007/s10957-021-01891-2

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