Convergence Analysis of a Three-Step Iterative Algorithm for Generalized Set-Valued Mixed-Ordered Variational Inclusion Problem

: This manuscript aims to study a generalized, set-valued, mixed-ordered, variational inclusion problem involving H ( · , · ) -compression XOR- α M -non-ordinary difference mapping and relaxed cocoercive mapping in real-ordered Hilbert spaces. The resolvent operator associated with H ( · , · ) compression XOR- α M -non-ordinary difference mapping is deﬁned, and some of its characteristics are discussed. We prove existence and uniqueness results for the considered generalized, set-valued, mixed-ordered, variational inclusion problem. Further, we put forward a three-step iterative algorithm using a ⊕ operator, and analyze the convergence of the suggested iterative algorithm under some mild assumptions. Finally, we reconﬁrm the existence and convergence results by an illustrative numerical example.


Introduction
The theory of variational inequalities was studied in the early 1960s to solve a problem which appeared in a mechanical system. In 1964, Stampacchia [1] solved the variational inequality problem, where it was found thatx ∈ K such that important generalizations of variational inequality due to Verma [15] is the system of variational inequalities, because a number of equilibrium problems, such as the traffic equilibrium, the spatial equilibrium, the Nash equilibrium, and the general equilibrium programming problems can be designed as a system of variational inequalities. The birth of this theory can be observed as the simultaneous acquirement of two different lines of research-namely, it affirms the qualitative aspects of the solution to important classes of problems, and also empowers us to establish influential and fruitful new techniques for the solving of problems.
In 1994, Hassouni and Moudafi [16] evolved a class of mixed-type variational inequalities with single-valued mappings using the technique of a resolvent operator for monotone mapping, namely-variational inclusion problem. They developed a perturbed algorithm to estimate the solution of mixed variational inequalities. Recently, the fixed-point theory is widely applied to study problems appearing in real ordered Banach spaces, see [17][18][19][20][21]. Li [22] studied the nonlinear ordered variational inequalities and developed an iterative algorithm to estimate its solution in real ordered Banach spaces. In 2012, Li [23] planted a new class of variational inclusions for (α, λ)-NODM set-valued mappings in an ordered Hilbert space. Using the technique of resolvent operator, the existence result was proven and the convergence of sequence obtained from iterative algorithm was discussed. In [24], Li et al. investigated the solution of a general nonlinear ordered variational inclusion with (γ G , λ)-weak-GRD mappings. Recently, Li et al. [25] presented the convergence of an Ishikawa-type iterative method for the general nonlinear ordered variational inclusion with (γ G , λ)-weak-GRD set-valued mappings, and exhibited the stability of the algorithm. Very recently, ordered variational inclusions with XOR operator are considered in diverse direction-see, for example, [26,27].
Following the facts mentioned above and encouraged by recent investigations in this order, we introduce H(·, ·)-compression XOR-α M -non-ordinary difference mapping. A resolvent operator associated to this mapping is defined, and we discuss some of its characteristics. We examine a generalized, set-valued, mixed-ordered, variational inclusion with H(·, ·)-compression XOR-α M -non-ordinary difference mapping and relaxed cocoercive mapping in real ordered Hilbert spaces. We validate the existence and uniqueness of the solution for the considered ordered variational inclusion. In addition, we present a three-step iterative algorithm using a ⊕ operator and analyze the convergence of the proposed iterative algorithm under suitable assumptions. At last, a numerical example is given to show that the considered three-step iterative algorithm converges to the unique solution of a generalized, set-valued, mixed-ordered, variational inclusion.

Preliminaries and Auxiliary Results
We remind some necessary definitions, notions, and auxiliary results which are constructive tools and will be used throughout this paper.
Let H p be a real Hilbert space equipped with the norm · , and inner product ·, · where d is the metric induced by the norm · . Let C be a normal cone in H p with normal constant λ C , and " ≤ denotes the partial ordering defined by C. The Hilbert space H p equipped with partial ordering ≤ defined by C is called an ordered Hilbert space. Let CB(H p ) (respectively, 2 H p ) be the family of all non-empty closed and bounded subsets (respectively, all non-empty subsets) of H p . For any arbitrary µ, ν ∈ H p , glb{µ, ν} and lub{µ, ν} represent the greatest lower bound and least upper bound, respectively, for the set {µ, ν} with partial ordering ≤. The operators ∧, ∨, ⊕, and are called AND, OR, XOR, and XNOR operators, respectively, and defined as follows: Throughout this paper, unless otherwise stated, we denote positive, real ordered Hilbert space by H p .
Proposition 1 ([29]). Let C be a normal cone with normal constant λ C in H p . Then, for each µ, ν ∈ H p , the following relations hold: Lemma 1 ([30]). Let {ω n } be a nonnegative real sequence satisfying the following inequality for all x, y ∈ H p .
Definition 8 ([18,23]). Let F , G : H p → H p be the single-valued mappings and M : H p × H p → 2 H p be a set-valued mapping. Then, (i) M is called a comparison mapping with respect to F and G if, for any p µ ∈ M(F (µ), G(µ)), µ ∝ p µ and if µ ∝ ν, then for any p µ ∈ M(F (µ), G(µ)) and p ν ∈ M(F (ν), G(ν)), p µ ∝ p ν , for all µ, ν ∈ H p ; (ii) A comparison mapping M is called an XOR-α M -non-ordinary difference mapping with respect to A and B if there exists a constant α M > 0, for each µ, ν ∈ H p there exist p µ ∈ M(F (µ), G(µ)) and p ν ∈ M(F (ν), G(ν)) such that for all µ, ν ∈ H p ; (iv) A comparison mapping M is called -XOR-ordered strongly monotone mapping with respect to F and G if for any µ,

Proof. For any given
By utilizing (i) and (ii) of Proposition 2, we have Since M is θ M -ordered rectangular mapping with respect to F and G, H(A, ·) is β 1 and H(·, B) is β 2 -ordered compression mapping with respect to F and G. Therefore, which implies that That is, F (µ) = F (ν) and G(µ) = G(ν). Since the mappings F and G are one-one, Lemma 3. Let A, B, F , G : H p → H p and H : H p × H p → H p be the single-valued mappings such that H(·, ·) is a mixed strong comparison mapping with respect to A and B. Let M : H p × H p → 2 H p be a set-valued ρ-XOR-ordered strongly monotone mapping with respect to F and G and assume that (F (µ), G(µ)) ⊕ (F (ν), G(ν)) ∝ µ ⊕ ν. Then, the resolvent operator R ρ,M(F ,G) (µ))), ρ,M(F ,G) (ν))).
Proof. Let p µ and p ν assume the same values as in (5) and (6), respectively. Then, Since M is XOR-α M -non-ordinary difference mapping with respect to A and B, we have

Formulation of the Problem and Existence Result
This section begins with the designing of a generalized ordered variational inclusion problem involving H(·, ·)-compression XOR-α M -non-ordinary difference mapping and relaxed cocoercive mapping. We discuss the existence of a unique solution for the considered problem.
Let H p be a real ordered Hilbert space and C be a normal cone with normal constant λ C . Let A, B, F , G, P, S : H p → H p and H, ϕ : H p × H p → H p be the single-valued mappings. Let R, T : H p → CB(H p ) be the set-valued mappings and M : H p × H p → 2 H p be a set-valued H(·, ·)-compression XOR-α M -non-ordinary difference mapping. We deal with the following generalized ordered variational inclusion problem: For some a ∈ H p and any b ∈ R, find x ∈ H p , µ ∈ R(x), ν ∈ T (x) such that where τ > 0 is a constant. Problem (10) is termed as generalized, set-valued, mixed-ordered, variational inclusion problem with XOR-operator (in short, GSMOVIP). Note that GSMOVIP (10) is more general, and for appropriate selection of the mappings comprised in the designing, it includes many problems existing in the literature. Some particular cases of GSMOVIP (10) are reported below: (i) If F = I, the identity mapping, G, S = 0, M(F (x), G(x)) = M(x) and R, T are the single-valued mappings, and then GSMOVIP (10) becomes an equivalent problem of finding x ∈ H p , for some a ∈ H p and any b ∈ R such that Problem (11) was constructed and examined by Ahmad et al. [27], using (γ R , λ)-weak-RRD mappings. (ii) If τ = 1 and ϕ(·, ·) = 0, then problem (11) coincides with the problem of finding x ∈ H p such that a ∈ P (x) ⊕ M(x). (12) Problem (12) was investigated by Li et al. [18] in the framework of weak-ANODD set-valued mappings. (iii) If a = 0 and P = 0, then problem (11) coincides to the problem of finding x ∈ H p such that 0 ∈ τM(x) − bϕ(x, g(x)).
In the following Lemma, we set up an equivalence between ordered fixed-point problem and GSMOVIP (10). Proof. The proof of the lemma follows immediately by using the definition of a resolvent operator, so we omit the proof. Theorem 1. Let C be a normal cone in H p with normal constant λ C . Let A, B, F , G, P, S : H p → H p and H, ϕ : H p × H p → H p be the single-valued mappings such that P is a δ P -Lipschitzcontinuous mapping; S is δ S -Lipschitz-continuous and relaxed (α S , k S )-cocoercive; H(·, ·) is π 1ordered compression mapping with respect to A in the first argument and π 2 -ordered compression mapping with respect to B in the second argument, and ϕ(·, ·) is δ ϕ 1 and δ ϕ 2 -Lipschitz-continuous mapping with respect to the first and second argument, respectively. Let R, T : H p → CB(H p ) be D-Lipschitz-continuous mappings with constant ζ R , ζ T , respectively, and M : H p × H p → 2 H p be a set-valued H(·, ·)-compression XOR-α M -non-ordinary difference mapping. If x 1 ∝ x 2 and the following condition is satisfied: Then, GSMOVIP (10) admits a unique solution.
Proof. For the sake of convenience, we assume Q(

. It follows from Lemma 4 and Proposition 2 that
Since H(·, ·) is π 1 -ordered compression mapping with respect to A in the first argument and π 2 -ordered compression mapping with respect to B in the second argument, Since P is δ P -Lipschitz-continuous mapping and R is D-Lipschitz-continuous mapping with constant ζ R , then we have Thus, from the definition of a normal cone, utilizing the Lipschitz continuity of ϕ in the first and second argument and the D-Lipschitz continuity of T , we can write Since S is δ S -Lipschitz-continuous and relaxed (α S , k S )-cocoercive mapping, therefore Utilizing (17)- (20), (16) becomes where From condition (15), we deduce that Θ < 1. Thus, from (21), we can see that the mapping R Thus, from Lemma 5, one can conclude that (x, µ, ν), x ∈ H p , µ ∈ R(x), ν ∈ T (x) is the unique solution of GSMOVIP (10).

Convergence Analysis
This section begins with the construction of a three-step iterative algorithm. Finally, the convergence analysis of the proposed iterative algorithm to the unique solution of GSMOVIP (10) is discussed. Theorem 2. Let C be a normal cone in H p with normal constant λ C . Let the mappings A, B, F , G, P, S, H, ϕ, R, T be identical as in Theorem 1, such that all the suppositions of Theorem 1 are satisfied. Let M : H p × H p → 2 H p be a set-valued H(·, ·)-compression XOR-α M -non-ordinary difference mapping. If x n+1 ∝ x and the following condition holds: Then, the approximate sequences {x n }, {µ n } and {ν n } generated by Algorithm 1 converge strongly to the unique solution x, µ, and ν, respectively, of GSMOVIP (10).
Algorithm 1 Let C be a normal cone with normal constant λ C in H p . Let A, B, F , G, P, S : H p → H p and H, ϕ : H p × H p → H p be the single-valued mappings.
It is also clear that the condition (23) is satisfied. Hence, all the suppositions of Theorem 2 are verified. Therefore, the sequence {(x n , µ n , ν n )} converges strongly to the unique solution 0, which is the solution of GSMOVIP (10).

Conclusions
We introduced a new class of mapping, namely, H(·, ·)-compression XOR-α M -nonordinary difference mapping. A resolvent operator associated to this mapping was defined, and we discussed some of its characteristics. We examined a generalized, set-valued, mixed-ordered, variational inclusion problem involving H(·, ·)-compression XOR-α M -nonordinary difference mapping and relaxed cocoercive mapping in the setting of real ordered Hilbert spaces. An existence result was discussed for our considered ordered inclusion problem. Further, a three-step iterative algorithm using a ⊕ operator was suggested, and a convergence analysis of the proposed iterative algorithm was presented. Finally, a numerical example was given to illustrate the existence and convergence results. Further, the results presented in this paper can be generalized in ordered Banach spaces and ordered uniformly smooth Banach spaces.