Global optimization and applications to a variational inequality problem

: In the present paper, we study the existence and convergence of the best proximity point for cyclic Θ - contractions. As consequences, we extract several ﬁ xed point results for such cyclic mappings. As an application, we present some solvability theorems in the topic of variational inequalities.


Introduction and preliminaries
Let C and D be two non-empty closed subsets of a complete metric space ϖ , In the case that ϕ ∩ ≠ C D , the mapping Ψ satisfying the contractive condition ϖ x y γϖ x y Ψ , Ψ , , where γ 0 1 < < , has a fixed point in ∩ C D(note that Ψ need not to be continuous [1]). On the other hand, an element x ∈ ∪ * C D is a best proximity point of Ψ if ϖ x x ϖ ζ ζ ζ ζ , Ψ dist , inf , : , The existence and approximation of best proximity points is an important theme in optimization theory. Elderd and Veeramani [2] extended the result of Kirk et al. [1] and proved the existence and uniqueness of best proximity points in complete metric spaces.
Thagafi and Shahzad [3] provided a constructive answer to the question proposed by Elderd and Veeramani [2] on the existence of a best proximity point for a cyclic contraction map on a reflexive Banach space. Abkar and Gabeleh [4] have shown best proximity point theorems for cyclic contractions in ordered metric spaces. For more work in this direction, one can follow [5][6][7][8][9][10][11].
In this article, we attempt to generalize the results of [1] and [2] for cyclic Θ-contractions. We also obtain certain fixed point results for such type of contractions. Some few examples are presented to prove the significance of our findings. Moreover, as an application, we prove solvability theorems for variational inequality problems.

Best proximity point results
In this section, we present the notion of cyclic Θ-contractions and discuss the existence and convergence of best proximity points for this kind of mappings in complete metric spaces.
Remark 2.1. Every cyclic contraction in a classical sense is a cyclic Θ-contraction for t e Θ . Indeed, let Ψ be a cyclic contraction on ∪ C D, so for all x ∈ C and y ∈ D, we have This implies that First, we give an approximation result.
Lemma 2.1. Let C and D be non-empty subsets of a metric space ϖ , Proof. If there is an integer n 0 so that ϖ x x , d i s t , [( ( ( )))] Taking the limit as n → ∞, we have This gives that Next, the existence of a best proximity point for a cyclic Θ-contraction mapping is as follows. { } + have convergent subsequences in C and D, respectively, then there exists x y , for some x ∈ C. One writes Taking the limit as k → ∞ in (9) and using Lemma 2.1 with (8), we get That is, so, by letting k → ∞ in (13) and using (8) and (10), Hence, Ψ is a cyclic Θ-contraction. Similarly, the inequality (4) holds for the remaining cases. Also, the sequences x Ψ 0 Letting n → ∞ in (14) and using Lemma 2.1, we get which leads to a contradiction. We can also prove that x n Proof. It follows from Lemmas 2.2 and 2.3. □

Particular case: fixed point results
In this section, we deduce some of the fixed point results for cyclic Θ-contractions. Proof. Since dist , 0 ( ) = C D , equation (15) implies that Ψ is a cyclic Θ-contraction. Hence, from Lemma 2.2, we have x ∈ C and y ∈ D with x y = such that ϖ x x , Ψ 0 ( )= . Consequently, ∩ ≠ ∅ C D and Ψ has a fixed point in ∩ C D. For uniqueness, suppose that u ∈ ∪ C D such that u u Ψ = with x u ≠ , then by (15), we get It is a contradiction. Thus, x u = . □ We claim that for ε 0 > , there exists N ∈ such that with n m N , ≥ . If not, then there exist two divergent sequences n m 2 , 2 1 for all k ∈ . We assume that m 2 1 k + is a minimal index for which (18) holds. Then for all k ∈ , From (18) and (19), we have Letting k → ∞ in (20) and using (16), we get Now, by a triangular inequality, we have and Letting k → ∞ in (22) and (23) and using (16) and (21), we get In view of (21) and (24), there is an integer k 0 so that ϖ x x , 0 Letting k tend to ∞ and using again (21) and (24) lead to a contradiction. Hence, x n 2 { } is a Cauchy sequence in C. □ Theorem 3.3. Let C and D be non-empty closed subsets of a complete metric space ϖ , Proof. Consider x 0 ∈ C, then by Lemma 3.1, there exists a Cauchy sequence x n Example 3.1. Let = X be endowed with the usual metric ϖ. Consider This implies x y x y Hence, Ψ is a cyclic Θ-contraction for all k 0, 1 ( ) ∈ . Thus, all the conditions of Theorem 3.3 are satisfied and so there is a unique 0 ∈ ∪ C D such that x x Ψ = .

Application to a variational inequality problem
Let H be a real Hilbert space with an inner product ., . ⟨ ⟩ and an induced norm .
such that for all μ ∈ H, P μ ν μ ν μ Ω : , is the set of all best approximations from μ to Ω. This metric projection plays an important role for solving the variational inequality problem.
For a non-empty, closed and convex subset Ω ⊆ H, f : Ω → H is a given operator. The Hartman-Stampacchia variational inequality (HSVI) problem is defined as − ⟩≥ ∈ * * * and the Minty variational inequality (MVI) problem is defined by The HSVI has various applications in the field of engineering, physics and industry, while the MVI facilitates in the study of solvability of HSVI (f , Ω), see [20]. The following theorem establishes a relation between the problems: P-I and P-II. It is known that, for each u ∈ H, there exists a unique nearest point P u

Conclusion
In this paper, we presented the existence and convergence of the best proximity point for cyclic Θ-contractions. We derived several fixed point results for such mappings. As an application of the obtained results, we presented some solvability theorems in the topic of variational inequalities.
Funding information: This paper received no external funding.