Higher order strongly general convex functions and variational inequalities

Muhammad Aslam Noor; Khalida Inayat Noor; Muhammad Aslam Noor; Khalida Inayat Noor

doi:10.3934/math.2020236

AIMS Mathematics

2020, Volume 5, Issue 4: 3646-3663. doi: 10.3934/math.2020236

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Research article

Higher order strongly general convex functions and variational inequalities

Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan

Received: 23 December 2019 Accepted: 13 April 2020 Published: 16 April 2020
MSC : 26D25, 49J40, 90C33

In this paper, we define and consider some new concepts of the higher order strongly general convex functions with respect to an arbitrary function. Some properties of the higher order strongly general convex functions are investigated under suitable conditions. It is shown that the optimality conditions of the higher order strongly general convex functions are characterized by a class of variational inequalities, which is called the higher order strongly variational inequality. Auxiliary principle technique is used to suggest an implicit method for solving strongly general variational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. It is shown that the parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine convex functions. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.
- higher order convex functions,
- variational inequalities,
- parallelogram laws
Citation: Muhammad Aslam Noor, Khalida Inayat Noor. Higher order strongly general convex functions and variational inequalities[J]. AIMS Mathematics, 2020, 5(4): 3646-3663. doi: 10.3934/math.2020236

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Abstract

In this paper, we define and consider some new concepts of the higher order strongly general convex functions with respect to an arbitrary function. Some properties of the higher order strongly general convex functions are investigated under suitable conditions. It is shown that the optimality conditions of the higher order strongly general convex functions are characterized by a class of variational inequalities, which is called the higher order strongly variational inequality. Auxiliary principle technique is used to suggest an implicit method for solving strongly general variational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. It is shown that the parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine convex functions. 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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