scholarly journals Generalized Higher Order Preinvex Functions and Equilibrium-like Problems

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1875
Author(s):  
Jiangming Ma ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequalities ◽  
Arbitrary Function ◽  
Equilibrium Problems ◽  
Higher Order ◽  
Convergence Criteria ◽  
Normed Spaces ◽  
Invex Sets ◽  
Preinvex Functions

Equilibrium problems and variational inequalities are connected to the symmetry concepts, which play important roles in many fields of sciences. Some new preinvex functions, which are called generalized preinvex functions, with the bifunction ζ(.,.) and an arbitrary function k, are introduced and studied. Under the normed spaces, new parallelograms laws are taken as an application of the generalized preinvex functions. The equilibrium-like problems are represented as the minimum values of generalized preinvex functions under the kζ-invex sets. Some new inertial methods are proposed and researched to solve the higher order directional equilibrium-like problem, Convergence criteria of the our methods is discussed, along with some unresolved issues.

scholarly journals New classes of preinvex functions and variational-like inequalities

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 2081-2097
Author(s):  
Muhammad Noor ◽  
Khalida Noor
Keyword(s):  
Variational Inequalities ◽  
Monotone Operators ◽  
Higher Order ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
New Concepts ◽  
Special Cases ◽  
Preinvex Functions ◽  
Strongly Monotone Operators ◽  
And Function

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

scholarly journals Mixed quasi invex equilibrium problems

2004 ◽  
Vol 2004 (57) ◽  
pp. 3057-3067 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Strong Monotonicity ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Iterative Schemes ◽  
New Class ◽  
Special Cases ◽  
Weaker Conditions

We introduce a new class of equilibrium problems, known asmixed quasi invex equilibrium(orequilibrium-like) problems. This class of invex equilibrium problems includes equilibrium problems, variational inequalities, and variational-like inequalities as special cases. Several iterative schemes for solving invex equilibrium problems are suggested and analyzed using the auxiliary principle technique. It is shown that the convergence of these iterative schemes requires either pseudomonotonicity or partially relaxed strong monotonicity, which are weaker conditions than the previous ones. As special cases, we also obtained the correct forms of the algorithms for solving variational-like inequalities, which have been considered in the setting of convexity. In fact, our results represent significant and important refinements of the previously known results.

scholarly journals Statistical convergence of double sequences on probabilistic normed spaces defined by $[ V, \lambda, \mu ]$-summability

2014 ◽  
Vol 33 (2) ◽  
pp. 59-67
Author(s):  
Pankaj Kumar ◽  
S. S. Bhatia ◽  
Vijay Kumar
Keyword(s):  
Statistical Convergence ◽  
Convergence Criteria ◽  
Double Sequences ◽  
Normed Spaces ◽  
Real Numbers ◽  
Positive Real ◽  
Probabilistic Normed Spaces

In this paper, we aim to generalize the notion of statistical convergence for double sequences on probabilistic normed spaces with the help of two nondecreasing sequences of positive real numbers $\lambda=(\lambda_{n})$ and $\mu = (\mu_{n})$  such that each tending to zero, also $\lambda_{n+1}\leq \lambda_{n}+1, \lambda_{1}=1,$ and $\mu_{n+1}\leq \mu_{n}+1, \mu_{1}=1.$ We also define generalized statistically Cauchy double sequences on PN space and establish the Cauchy convergence criteria in these spaces.

scholarly journals Further Results for General Financial Equilibrium Problems via Variational Inequalities

2013 ◽  
Vol 03 (01) ◽  
pp. 33-52 ◽  
Author(s):  
Annamaria Barbagallo ◽  
Patrizia Daniele ◽  
Mariagrazia Lorino ◽  
Antonino Maugeri ◽  
Cristina Mirabella
Keyword(s):  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Financial Equilibrium

scholarly journals Generalized Knaster-Kuratowski-Mazurkiewicz type theorems and applications to minimax inequalities

2017 ◽  
Vol 20 (K2) ◽  
pp. 131-140
Author(s):  
Linh Manh Ha
Keyword(s):  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Applied Mathematics ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Minimax Theorems ◽  
Minimax Inequalities ◽  
Special Cases ◽  
Kkm Mappings ◽  
Definition Of

Knaster-Kuratowski-Mazurkiewicz type theorems play an important role in nonlinear analysis, optimization, and applied mathematics. Since the first well-known result, many international efforts have been made to develop sufficient conditions for the existence of points intersection (and their applications) in increasingly general settings: Gconvex spaces [21, 23], L-convex spaces [12], and FCspaces [8, 9]. Applications of Knaster-Kuratowski-Mazurkiewicz type theorems, especially in existence studies for variational inequalities, equilibrium problems and more general settings have been obtained by many authors, see e.g. recent papers [1, 2, 3, 8, 18, 24, 26] and the references therein. In this paper we propose a definition of generalized KnasterKuratowski-Mazurkiewicz mappings to encompass R-KKM mappings [5], L-KKM mappings [11], T-KKM mappings [18, 19], and many recent existing mappings. Knaster-KuratowskiMazurkiewicz type theorems are established in general topological spaces to generalize known results. As applications, we develop in detail general types of minimax theorems. Our results are shown to improve or include as special cases several recent ones in the literature.

scholarly journals Higher order strongly general convex functions and variational inequalities

2020 ◽  
Vol 5 (4) ◽  
pp. 3646-3663 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequalities ◽  
Convex Functions ◽  
Higher Order ◽  
General Convex
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