Well-Posedness of Triequilibrium-Like Problems

This work emphasizes in presenting new class of equilibrium-like problems, termed as equilibriumlike problems with trifunction. We establish some metric characterizations for the well-posed triequilibriumlike problems. We give some conditions under which the equilibrium-like problems are strongly wellposed. Our results, which give essential and adequate conditions to the well-posedness of triequilibriumlike problems, are acquired by utilizing the assumption of pseudomonotonicity. Technique and ideas of this paper inspire further research in this dynamic field.


Introduction
The theory of equilibrium problems is an engrossing and significant offshoot of variational inequalities in practice with a broad variety of industrial, physical, geographical and social applications. Across various areas of pure, applied and engineering sciences, equilibrium problem theory has shown incredible potential and great impact. In nearly all disciplines, of mathematics and engineering, this theory has registered its exceptional ever-expanding mark. A new and incisive treatment of a broad list of problems, that occur in ecology, finance, economics, elasticity, network, image reconstruction, optimization and transport, are exhaustively approached by the equilibrium problem theory. In 1994, Blum and Oettli [3] and Noor and Oettli [23] rendered equilibrium problems their existing form. The classical equilibrium problems theory revolves around the assumption of convexity of the set and objective function. Equilibrium problems cover a diverse set of applications including hemivariational inequalities, variational inequalities, game theory, Nash equilibrium, variational-like inequalities and fixed point point theory, see [1-7, 12-19, 21, 22, 24-29].
Recently, the notion of convexity has started expanding to numerous fields showing the capacity for various useful applications. Hanson [8] derived invex functions as a special extension of convex functions. Different results were presented to make this fact noticeable that what holds for convex functions in mathematical programming also holds for a generalized class of functions known as invex functions. Ben-Israel and Mond [1] and Weir and Jeyakumar [31] works led to preinvex functions as another generalized class of convex functions. Weir and Mond [30] has shown in their work the interchangeability of preinvex functions with convex functions in optimization problems. In [15], it is made evident that the minimum of preinvex on the invex set can be disciplined into variational inequalities, widely known as variational-like inequalities. Variational-like inequalities and equilibrium-like problems, owing to their specialized nature, cannot allow traditional resolvent method, projection method and their prevalent variant forms to propound any iterative methods. To bridge this gap, we resort to a technique named as auxiliary principle, proposed by Glowinski et al. [4]. To solve numerous variational inequalities and equilibrium problems, Noor [15,19,20] and Noor et al. [22,24,25,27] employed the technique of auxiliary principle to propose various iterative methods.
Research of our paper is devoted to present a new class of equilibrium problems, termed as equilibrium-like problems with trifunction. The notion of well-posedness of variational ineaqualities and equilibrium problems was introduced by Lucchetti and Patrone [9,10]. We expand the notion of well-posedness to contemplate and establish the well-posedness of triequilibrium-like problems. Several interesting and important cases are discussed as applications of the obtained results.

Basic concepts and formulations
LetH be a real Hilbert space. The inner product and norm onH are denoted by ., . and .
respectively. Let K be a nonempty invex set inH. Let : K → and (., .) :H ×H →H are continuous functions.
First we recall the following well-known results and concepts, before discussing our main results.
If the set K is invex at each ξ ∈ K , then K is also called −connected set.

Definition 2.2. ( [1]). A function
: K → is said to be a preinvex function, if there exists a bifunction (., .), such that where (ξ) is the differential of at ξ. From above definitions, it is clear that the differentiable preinvex functions are the invex functions but the converse is not true, see [21]. We note that, if (ζ, ξ) = ζ − ξ, the invex set K reduces to the convex set K and preinvex functions reduce to convex functions. There are some functions which are preinvex but not convex.
Definition 2.4. . The bifunction (., .) :H ×H → satisfies the following condition For τ 1 = 0, 1 and τ 2 = τ , we get Condition C of Mohan and Neogy [11], Mohan and Neogy [11] used definition 2.4 to show that an invex function on an invex set K , is also a preinvex function and the converse also hold.
Given an operator Υ :H → and a continuous trifunction Ψ(., ., .) :H ×H ×H → , consider the problem of finding ξ ∈ K , such that The problem (2.1) is called an equilibrium-like problem with trifunction.
If (ζ, ξ) = ζ − ξ, then an invex set reduces to a convex set and problem (2.1) is equivalent to finding which is called an equilibrium problem with trifunction and gives off an impression of being new.
Variational-like inequality with trifunction (2.2) reduces to variational inequality with trifunction of finding ξ ∈ K, such that Also variational-like inequality (2.3) is equivalent to finding ξ ∈ K, such that which is known as variational inequality, proposed and cosidered by Stampacchia [20]. In brief, for suitable and appropriate choice of the functions Ψ(., ., .), (., .) and the spaces, one can obtain a number of new and known problems as special cases of the problem (2.1). This shows that problem (2.1) is quite general and unifying.

Lemma (2.1) can be considered as the generalized form of Minty's Lemma for triequilibrium-like
problems. Result obtained in above Lemma is termed as the dual triequilibrium -like problem.

Well-Posedness
We generalize the notion of well-posedness to triequilibrium-like problems. By using the assumption of pseudomonotonicity we obtain some results for well-posed equilibrium-like problems with trifunction.
Hence triequilibrium-like problem (2.1) has at least one solution.
(iii) Theorem (3.3) concludes that, the unique solution of problem (2.1) could be calculated by utilizing ε− equilibrium-like problem with trifunction, that is, find ξ ε ∈H, such that

Perspective
In this section, we discuss some research perspective of the equilibrium-like problem with trifunction.
We show that the results derived in this paper can also be extended for a class of nonconvex equilibrium problem with trifunction.

Conclusion
In our work we focused on introducing another class of equilibrium-like problems terming it as triequilibrium-like problems. We studied and established the well-posedness of triequilibrium-like problems by using the assumption of pseudomonotonicity. The demonstrated results in this endeavour can rightly be considered as an enhancement and sophistication of already existing work.