Search

The objective of this paper is to explore novel unified continuous and discrete versions of the Trapezium-Jensen-Mercer (TJM) inequality, incorporating the concept of convex mapping within the framework of q-calculus, and utilizing majorized tuples as a tool. To accomplish this goal, we establish two fundamental lemmas that utilize the 1q and 2q differentiability of mappings, which are critical in obtaining new left and right side estimations of the midpoint q-TJM inequality in conjunction with convex mappings. Our findings are significant in a way that they unify and improve upon existing results. We provide evidence of the validity and comprehensibility of our outcomes by presenting various applications to means, numerical examples, and graphical illustrations. [ABSTRACT FROM AUTHOR]

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called CR - γ -convex functions based on a non-negative mapping γ and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen's inequalities using CR - γ -convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard's, Hermite-Hadamard-Fejer, and Pachpatte's type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new fractional dynamical system for variational inequalities using the Wiener Hopf equations technique. We show that the fractional Wiener-Hopf dynamical system is exponentially stable and converges to its unique equilibrium point under some suitable conditions . We also discuss some special cases, which can be obtained from our main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new class of quasi variational inequalities involving three operators, which is called the extended general quasi variational inequality. It is shown that the extended general quasi variational inequalities are equivalent to the fixed problems and the extended general implicit Wiener-Hopf equations. These alternative formulations are used to suggest and analyze some iterative methods. The convergence analysis of these new methods under some suitable conditions is investigated. Several special cases are discussed. Since the extended general quasi variational inequalities include general variational inequalities, quasi variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems. Results of this paper may stimulate further research in this fascinating area. [ABSTRACT FROM AUTHOR]

In this paper, we consider extended general variational inequalities involving nonlinear operators. Using projection operator technique, we show that extended general variational inequality is equivalent to a fixed point problem. Using this alternative equivalent formulation, we propose and analyze some new algorithms for solving extended general variational inequalities and its variant forms. The convergence of new methods are discussed under some suitable conditions. Several special cases are discussed. Results obtained in this paper continue to hold for these problems. [ABSTRACT FROM AUTHOR]

It is known that the general variational inclusions involving difference of two (or more) monotone operators are equivalent to the resolvent equations. We use this equivalent form to develop the sensitivity analysis for general variational inclusions. This technique does not require the differentiability of the given data. Some special cases are discussed. Results obtained in this paper continue to hold for these problems. [ABSTRACT FROM AUTHOR]