General biconvex functions and bivariational inequalities

Muhammad Aslam Noor; Khalida Inayat Noor

doi:10.3934/naco.2021041

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2023, Volume 13, Issue 1: 11-27. Doi: 10.3934/naco.2021041

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General biconvex functions and bivariational inequalities

Muhammad Aslam Noor^, and
Khalida Inayat Noor

Department of Mathematics, COMSATS University Islamabad, Pakistan

^* Corresponding author: Muhammad Aslam Noor
^* Corresponding author: Muhammad Aslam Noor

Received: June 2021

Revised: August 2021

Early access: September 2021

Published: March 2023

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Abstract

In this paper, we define and introduce some new concepts of the higher order strongly general biconvex functions involving the arbitrary bifunction and a function. Some new relationships among various concepts of higher order strongly general biconvex functions have been established. It is shown that the new parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine general biconvex functions, which is itself an novel application. It is proved that the optimality conditions of the higher order strongly general biconvex functions are characterized by a class of variational inequalities, which is called the higher order strongly general bivariational inequality. Auxiliary principle technique is used to suggest an implicit method for solving strongly general bivariational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

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Mathematics Subject Classification: 49J40, 90C33.

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