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Generalized Higher Order Preinvex Functions and Equilibrium-like Problems

: 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 k ζ -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.

The so-called (h, k) convex set and convex functions are defined by Micherda and Rajba [26], Hazy [27], and Crestescu et al. [28]. The definition and characterizations of k-convex functions are introduced and studied by Noor [29]. The modified kη-invex sets and kη-preinvex is studied. Generalized preinvex functions can be regarded as an important improvement of the (h, k) convex functions, which are investigated by Micherda et al. [26] and Hazy [27]. Their results help us to further consider the problems of directional equilibrium-like instances.
Some computing methods to solve variational inequalities, optimization problems, and equilibrium problems are discussed. Glowinski et al. [5] use the auxiliary principle approach of involving Bregman distance function to consider some iterative schemes that solve the higher order directional equilibrium-like problems. Some convergence properties of these methods are discussed by applying either pseudomonotonicity or partially relaxed strongly monotonicity, which is a weaker condition than monotonicity.
In Section 2, some new concepts and properties of generalized preinvex functions are set. Preinvex functions are viewed as novel extensions of the convex functions that are associated with variational-like inequalities. Naturally, all the results are closely related with symmetry concepts. In Section 3, main characterizations of the higher-order strongly generalized preinvex are investigated. In Section 4, we derive various parallelograms. Definition 6. The function G on Ω kζ is defined as higher-order k-convex function, if a constant β and an arbitrary function k are present, that is: If k(λ) = λ, ζ(ν, µ) = ν − µ, thus, Definition 6 is reduced to: Definition 7. The function G on Ω ζ is defined as higher-order convex function, if a constant β and an arbitrary function k are present, such that Mohsin et al. [32] study higher-order convex functions which represent a significant improvement to the concepts for higher-order convex functions that are studied by Alabdali et al. [33], Lin et al. [34], Mako et al. [35], and Olbrys [36].
When λ = 1, the k-convex function is reduced to: which is very important to the derivation of these main results.

Definition 8.
The function G on Ω kζ is defined as higher-order quasi k-preinvex function if a constant β and a function k are present, that is:

Definition 9.
A function G on Ω kζ is defined as higher-order logarithmic generalized preinvex function, if the bifunction η(., .) and a function k are present, that is: where G(·) > 0.
When β = 0, Definition 9 is reduced to: Definition 10. The function G on Ω kζ is defined as higher-order logarithmic generalized preinvex function, if a function k is present, that is: or, equivalently: Definition 11. A function G on Ω kζ is defined as higher-order logarithmic generalized preinvex function, if a function k is present, that is: From this idea, the higher-order logarithmic generalized preinvex function is defined as follows: Definition 12. A function G on Ω kζ is defined as higher-order logarithmic generalized preinvex function, if a function k is present, that is: where it seems to be a new one.
Through the above definitions, we obtain: From the above equation, we know that higher-order logarithmic generalized preinvex functions =⇒ higher-order generalized preinvex functions and higher-order generalized preinvex functions =⇒ higher-order generalized quasi preinvex functions. Otherwise, it does not hold.
If the function G satisfies these two conditions of higher-order generalized preinvex function and higher-order generalized preincave function, then, a new concept is defined as follows.
Definition 13. The function G on Ω kζ is is defined as higher-order generalized preinvex affine function, if an arbitrary function k and a constant β are present, that is: If the functions k(λ) and the bifunction ζ(., .) are selected as the suitable form, several categories of higher-order generalized preinvex functions and their variant forms are obtained.
The further assumption of the bifunction ζ(·, ·) and the function k(λ) is defined as follows.
For k(λ) = λ, Condition M is reduced to the condition C of Mohan and Neogy [18]. Next, the definition of k-directional derivative is set.

Definition 15.
The differentiable function G on t Ω ζ is defined as higher-order strongly pseudo k-invex function, iff, if a constant β > 0 is present, that is: Definition 16. The differentiable function G on Ω kζ is defined as higher order strongly quasi-kinvex function, iff, a constant β > 0 is present, that is: Definition 17. The G on Ω ζ is defined as pseudo-k-invex, iff, Definition 18. The differentiable function G on Ω ζ is defined as quasi k-invex function, iff, When ζ(ν, µ) = −ζ(ν, µ), ∀µ, ν ∈ Ω ζ , ζ(·, ·) is skew-symmetric. Thus, Definitions 7-14 are changed to the known ones. The concepts of our paper have a great improvement on the previously known concepts. The new concepts are very important to the optimization and mathematical programming.

Characterizations
Next, under the invex set Ω kζ , some features of higher-order strongly generalized preinvex functions are researched.

Theorem 1.
Under the condition M and the assumption of G which is a differentiable function on Ω kζ in H. G is a higher-order generalized preinvex function, if and only if, G is a higher-order k-invex function.
Proof. Because G is a higher-order generalized preinvex function which is based on Ω kζ , which is changed as When λ → 0, the following inequality is obtained: Thus, G is a higher-order generalized invex function. However, under the condition M, if G is a higher-order strongly generalized invex function which is based on Using the similar method, the following results are obtained: From multiplying (1) by k(λ) plus multiplying(2) by (1 − k(λ)), we obtain: From the above equation, we find that G is a higher-order generalized preinvex function.

Theorem 2.
Under the assumption of G which is a differentiable higher-order generalized preinvex function on Ω kζ . If G is a higher-order generalized invex function, thus, Proof. Because G is a higher-order generalized invex function which is based on Ω kζ , By exchanging the positions of u and v in Equation (4), the following inequality is obtained: From adding Equations (4) and (5), the following result is: We find that G k (.) is a higher-order generalized monotone operator.
When p = 2, the converse of Theorem 2 holds. Then, the following result is obtained: Assuming that the G k (.) is a higher order generalized monotone, that is: Proof. Because G k (.) is a higher-order strongly kζ-monotone. Through Equation (6), the following inequality is obtained: Under the Condition M, because Ω is an invex set, ∀µ, (7), we obtain: Assuming ξ(λ) = G(µ + k(λ)ζ(ν, µ)), through Equation (8), we obtain: Equation (9) is integrated from 0 and 1, and the following inequality is obtained: That is, From Condition A, the following inequality is obtained: Now, a necessary condition of higher-order strongly generalized pseudo-invex function is given.
Theorem 5. Assuming that the differential (µ) of a differentiable preinvex function G(µ) is Lipschitz continuous on Ω kζ under a constant β > 0, the inequality is obtained as follows: Proof. Through Noor and Noor [20,21], its proof is obtained easily.

Definition 19.
The function G is defined as a sharply higher order strongly generalized pseudo preinvex; if a constant β > 0 is present, then,

Theorem 6.
Assuming that G is a higher-order sharply generalized pseudo preinvex function on Ω kζ under a constant β > 0, the inequality is obtained as follows: Proof. Because G is a higher sharply pseudo generalized preinvex function which is based on Ω kζ , then, When λ → 0, we obtain: Definition 20. G is defined as a higher-order pseudo generalized preinvex function about strictly positive bifunction W(., .), if Theorem 7. Assuming that G is a higher-order generalized preinvex function and satisfies G(ν) < G(µ), G is a higher-order generalized pseudo preinvex function.

Parallelogram Laws
Next, we obtain some new parallelogram laws. Through Definition 13, we obtain: Bringing λ = 1 2 into Equation (13), we obtain: Thus, for higher order generalized preinvex functions, the above equation is called the generalized parallelogram-like laws under the Banach spaces.
Several special situations of the generalized parallelogram-like laws are discussed.

Equilibrium-like Problems
The directional equilibrium-like problems are introduced in this section. Next, the optimality of the differentiable higher-order generalized preinvex functions are discussed.
In order to properly select the spaces, operators, and kζ-invex sets, a large group of optimization programming, variational-like inequalities, and equilibrium problem is obtained. We find that the higher-order strongly directional equilibrium-like problems are relatively unified and flexible.
From an auxiliary principle technique which is based on Bergman functions, some iterative methods for equilibrium-like problems (23) are investigated.
Algorithm a. For µ 0 ∈ H, from the iterative scheme computing method of the approximate solution µ n+1 is obtained, where ρ > 0 is a constant. Algorithm a is defined as the proximal method which solves directional equilibriumlike problem Equation (23). When β = 0, then, Algorithm a is changed into: computing method of the approximate solution µ n+1 is obtained to solve the bifunction variational-like inequality.
From the proper selections of the spaces and the operators, some known and new algorithms are obtained to solve variational inequalities and related issues.
From the above equation, we obtain: Therefore, the {µ n } is a bounded sequence. Assuming thatμ is a cluster point of the subsequence {µ n i }, and {µ n i } is a subsequence converging toμ, according to the method of Zhu and Marcotte [13], the sequence {µ n } converges to the cluster pointū, which satisfies Equation (23).
From the auxiliary principle technique, another method is considered to solve the higher order directional equilibrium-like problem Equation (23).
In order to properly select the spaces and the operators, we obtain some new algorithms to solve higher order directional equilibrium-like problem Equation (23) and optimization problems. From analytical and numerical perspectives, it is an interesting problem.

Conclusions
Several new categories of higher order generalized preinvex functions are investigated. We discuss the new characterizations of the generalized preinvex functions, especially their relations with previously results. We derive some parallelograms laws of inner product spaces and Banach spaces. Optimality conditions of the differentiable k-preinvex functions are characterized by a category of directional variational-like inequalities. The result drives us to consider higher order equilibrium-like problems. Some iterative methods to solve higher order directional equilibrium-like problem are investigated from the auxiliary principle technique under the Bregman functions. Several Bregman distance functions are symmetric and are used to discuss the convergence criteria of proposed methods, These concepts highlight the role of symmetry. Some efficient computing methods to solve higher order directional equilibrium-like problem are also proposed and discussed.
In this paper, we have considered the theoretical aspects of the kζ-preinvex functions and variational problems. The numerical results for the the preinvex equilibrium problems and k-equilibrium are interesting problems for the future research.

Data Availability Statement:
The data used to support the findings of this study are included within the article.