An iterative approximation of common solutions of split generalized vector mixed equilibrium problem and some certain optimization problems


 In this paper, we study the problem of finding a common solution of split generalized vector mixed equlibrium problem (SGVMEP), fixed point problem (FPP) and variational inequality problem (VIP). We propose an inertial-type iterative algorithm, which uses a projection onto a feasible set and a linesearch, which can be easily calculated. We prove a strong convergence of the sequence generated by the proposed algorithm to a common solution of SGVMEP, fixed point of a quasi-
 
 
 
 ϕ
 
 \phi 
 
 -nonexpansive mapping and VIP for a general class of monotone mapping in 2-uniformly convex and uniformly smooth Banach space 
 
 
 
 
 
 E
 
 
 1
 
 
 
 {E}_{1}
 
 and a smooth, strictly convex and reflexive Banach space 
 
 
 
 
 
 E
 
 
 2
 
 
 
 {E}_{2}
 
 . Some numerical examples are presented to illustrate the performance of our method. Our result improves some existing results in the literature.


Introduction
Let C denote a nonempty, closed and convex subset of a real Banach space E with norm ‖⋅‖ and → * J E : 2 E be the normalized duality mapping defined by where * E is the dual space of E and ⟨⋅ ⋅⟩ , is the duality pairing between the elements of E and * E . Let C and X be nonempty, closed and convex subsets of real Banach spaces E 1 and E 2 , respectively, then the split feasibility problem (SFP) (see [1,2]), consists of finding a point where → L E E : 1 2 is a bounded linear operator. The SFP is a variant of the inverse problem and finds real life applications in image processing, radiation therapy and remote sensing [3][4][5][6]. For approximating the solutions of (2) and related optimization problems in Hilbert, Banach, Hadamard and p-uniformly convex metric spaces, researchers have developed several iterative methods that adopt a fixed point approach (see [3,[7][8][9] and references therein). It has been shown that a point ∈ x C is a solution of (2) if and only if x is a fixed point of the operator ( − ( − ) ) * P I γL I P L C X (see [10]), where P C , P X are metric projections onto C and X, respectively, L is a bounded linear operator with adjoint * L and γ is a positive parameter. Let → T C C : be a mapping. A point ∈ x C is called a fixed point of T , if = x Tx. We shall denote the set of fixed points of T by ( ) T Fix , that is ( ) ≔ { ∈ = } T x C x Tx Fix : . Let X be a nonempty, convex subset of a real Banach space E. Assume that P is a proper, pointed, closed, convex cone of a real Hausdorff Y and ∈ e P int . Let × → f X X Y : be a bifunction, → ψ X E : be a nonlinear mapping and → ϕ X Y : be a function. The generalized vector mixed equilibrium problem (GVMEP) is the problem of finding a point ∈ x X such that ( ) + ⟨ − ( )⟩ + ( ) − ( ) ∈ ∀ ∈ f x y y x ψ x ϕ y ϕ x P y X , , , . ( Problem (3) so defined is called strong GVMEP. The problem is however said to be weak if the notation ∈ P in (3) is replaced by the notation ∉ P int . Now, suppose in (3) we let = ψ 0, then problem reduces to the generalized vector equilibrium problem (GVEP) studied by Kazmi and Farid (see [11]). Also, if we set = = ψ ϕ 0, then (3) reduces to a vector equilibrium problem (VEP), where VEP consists of finding ∈ x X such that Moreover, if we set = Y and = e 1, then Problem (3) reduces to the generalized mixed equilibrium problem considered by Peng and Yao [12]. Furthermore, problem (4) reduces to the classical equilibrium due to Blum and Oetlli [13]. Vector equilibrium represents a unified framework for studying several problems, including vector optimization, vector variational inequality, vector complementarity problems and so on [14,15]. In recent years, iterative algorithms for obtaining the equilibrium problems, zero points problems and related optimization problems have been studied in the literature (see [14][15][16][17][18][19][20] and references therein).
Let C be a nonempty, closed and convex subset of a real Banach space E with dual * E and → * F C E : be a mapping. The variational inequality problem (VIP) is to find a point ∈ x C such that ⟨ − ( )⟩ ≥ ∀ ∈ y x F x y C , 0 , .
We shall denote the solution set of (5) by ( ) C F VIP , . Closely related to (5) (see [21]) is the problem of finding ∈ y C such that ⟨ − ( )⟩ ≥ ∀ ∈ y x F y y C , 0 , .
Following [21], we shall refer to (6) as the dual variational inequality problem (DVIP) of (5). The VIP is one of the central problems in nonlinear analysis (see [22][23][24]) with monotonicity playing a major role in its study. For instance, monotone operators are important tools in the study of several problems in the domain of optimization, nonlinear analysis, differential equation and other related fields. However, there have also been studies of variational inequalities with weaker monotonicity conditions such as pseudomonotone, quasimonotone, strictly quasimonotone, etc. In 2019, Chang et al. [25] studied an iterative approximation of solution of VIP for a semistrictly quasimonotone operator in the framework of infinite-dimensional Hilbert spaces (see [26] and references therein). It is known that extragradient methods for solving VIP require projections onto a set which are difficult to evaluate especially when the structure of the set is not simple. He et al. [27] introduced a totally relaxed self subgradient extragradient method (TRSSEM) involving feasible sets which are easily defined for solving the VIP.
for all = … j m 1, 2, , are convex and differentiable functions. For the TRSSEM, the feasible set is defined as Furthermore, obtaining a common element in the solution set of a fixed point problem (FPP), VIP and EP has recently been considered by authors in the literature due to its various applications, see [28][29][30]. In 2012, Shan and Huang [31] introduced the concept of generalized mixed vector equilibrium problem (GMVEP). They obtain the existence result in the framework of Hilbert space for this problem. They further proposed an iterative algorithm for obtaining a common element in the solution set of GMVEP, VIP and fixed point of a nonexpansive mapping. Very recently, Farid and Kazmi [32] introduced and studied a general iterative algorithm for approximating a common solution of split generalized equilibrium problem (SGEP), VIP and FPP. They proved a strong convergence theorem for the sequences generated by the proposed algorithm. For more references on this see [26].
In this paper, motivated by Shan and Huang [31], Chang et al. [25] and He et al. [27], we study an iterative approximation of a common solution of split generalized vector mixed equlibrium problem (SGVMEP), VIP and fixed point of quasi-ϕ-nonexpansive mapping. We proposed an inertial-type iterative algorithm which uses projection onto a feasible set and a linesearch with Halpern method. We prove a strong convergence theorem for the sequence generated by this algorithm to a common solution of these problems in the frame work of 2-uniformly convex and uniformly smooth Banach space E 1 and a smooth, strictly convex and reflexive Banach space E 2 . Finally, some numerical examples are presented to illustrate the performance of our method.
The rest of the paper is organized as follows: We first recall some basic definitions, required assumptions and results in Section 2. We give an explicit statement of the problem and show that its solution set is well defined, and we also propose an iterative process and prove a strong convergence of the method to a solution of the problem in Section 3. Some numerical experiments of our results are given in Section 4. We give concluding remarks in Section 5.

Preliminaries
In this section, we give some important definitions, results and restrictions which are useful in establishing our main results. Throughout this paper, we denote the weak and strong convergence of a sequence , and ∈ ( ) λ 0, 1 , then h is said to be a convex function. h is said to be differentiable if the set Each element ∂ ( ) h x is called a subgradient of h at x or the subdifferential of h and inequality (7) is said to be the subdifferential inequality of h at x. We say that the function h is subdifferentiable at E if it is subdifferentiable at every point of E. It is known that if h is Gâteaux differentiable at x, then h is subdifferentiable at x and { ′( )} = ∂ ( ) h x h x , which implies ∂ ( ) h x is singleton (see [33]). For more details on these, see [4,34,35] and references therein.
Following [36], Albert introduced a generalized projection operator is the Lyapunov functional defined by Vector mixed equilibrium problem  337 The functional ϕ is known to satisfy the following properties: x Jx Jy y x y , .
In Hilbert space, = P Π C C , the metric projection and ( ) = ‖ − ‖ ϕ x y x y , 2 , see [37] for details on P C . We also require the functional 2 , , f o r e a c h a n d . 2 2 It is easy to see that ( , 1 for all ∈ u E and ∈ * v E . It is well known (see [40]) that if E is a reflexive, strictly convex and smooth Banach space, then for all ∈ u E and ∈ * v w E , . Let C be a nonempty, closed and convex subset of a real Banach space and → T C C : be a mapping.
itself is said to be relatively nonexpansive [41,42], if , for all ∈ x C and ∈ ( ) p T Fix . The class of quasiϕ-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [41,43] as the latter requires the strong restriction and Definition 2.8. [20,50] Let X and Y be two Hausdorff topological spaces and let D be a nonempty, convex subset of X and P be a pointed, proper, closed and convex cone of Y with ≠ ∅ P int . Let 0 be the zero point of Y , ( ) 0 be the neighbourhood set of 0, ( ) x 0 be the neighbourhood set of x 0 and → f D Y : be a mapping.
[51] Let X and Y be two real Hausdorff topological spaces, D is a nonempty, compact, convex subset of X and P is a pointed, proper, closed and convex cone of Y with ≠ ∅ P int . Assume are two vector mappings. Suppose f and Φ satisfy Vector mixed equilibrium problem  339 Then, there exists a point ∈ x D that satisfies ( For solving the GMVEP, we give the following assumptions: Let ⊂ X E 2 be a nonempty, compact, convex subset of real Banach space E 2 and Y a real Hausdorff topological space, ⊂ P Y is a proper, closed and convex cone.
be two mappings. For any ∈ x E 2 , define a mapping × → X X Y Ψ : x as follows: where r is a positive number in and ∈ e P int . Let f Ψ , x and ψ satisfy the following conditions: f x, is weakly continuous and P-convex, that is, The following result was proved in [31] in the framework of Hilbert space, but can easily be adapted for this study. as follows: Then, , define the open segment , are defined analogously.
is said to be (a) weakly hemicontinuous if F is upper semicontinuous from line segments to the weak topology of E; It is easy to check that (b) implies (a).

Lemma 2.12. [21]
A solution of DVIP is always a solution of VIP, if the operator F is weakly hemicontinuous.
Remark 2.13. It is well known that ∈ p C is a solution of (5) if and only if p is a fixed point of the operator ( − ) P I λF C for all > λ 0.
Definition 2.14. Let C be a nonempty closed and subset of a real Banach space E with dual * E . The mapping is said to be: (vii) (see [52]) semistrictly quasi-monotone on C if F is quasi-monotone on C and for all distinct of points In this section, we prove our main result. First, we explicitly state the problem considered in this paper, then we introduce a linesearch algorithm for obtaining the solution of this problem and finally discuss its convergence analysis. Let C be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E 1 , X be a nonempty, compact and convex subset of a smooth, strictly convex and reflexive Banach space E 2 and → L E E : 1 2 be a bounded linear operator with → * * * L E E : 2 1 its adjoint. Let P be a pointed, proper, closed and convex cone of a real Hausdorff topological space 2 be an m-inverse strongly monotone mapping and → T C C : be a quasi-ϕ-nonexpansive mapping. Also, let be a semistrictly quasi-monotone, sequentially weakly continuous mapping. We consider the problem of finding a point ∈ p C such that Assume ≠ ∅ Γ , where Γ denotes the solution set of problem (13). We note that Γ is closed and convex. Indeed, following [41], the fixed point of quasi-ϕ-nonexpansive mapping T is closed and convex. Also by Lemma 2.10(v), we have that ( ) f A ψ X GVMEP , , , is closed and convex and finally by Lemma 2.15, Hence, the solution of (13) is well defined. To obtain the solution of (13), we consider the following iterative algorithm: j be a family of convex, weakly lower semicontinuous and We also assume the following conditions are satisfied: For each > k 0, having the k-iterate { } x k , compute the following steps: Step I: For = … j m 1, 2, , and given the current iterate, construct the family of half spaces where = λ ρ k l k and l k is the smallest nonnegative integer such that Step II: , then = w y k k and go to step III. Otherwise, compute the next iterate by Step III: Compute Step IV: Set ≔ + k k 1 and go to step I.
Proof. Suppose = w z k k , then by the characteristics of Π Ck and (15), we have Since Hence, . We conclude therefore from this, □ In what follows, we shall show that the Armijo Linesearch rule (15) is well defined.
Hence, we consider the case where ∉ ( ) w C F VIP , k and assume the contrary, that is for > l 0 Next, we consider the following possibilities, Using the continuity of F and Π Ck , then We have from (20) and (21) that Using the continuity of J 1 on bounded subsets of E 1 , we get Vector mixed equilibrium problem  343 By (9), we obtain Letting → ∞ k and using (22), we get On the other hand, suppose ∈ w C k k , then and By using (20), (25) and (26), we obtain a contradiction. Therefore, the linesearch (15) is well defined. □ For our convergence analysis, we will assume that In the following result, we prove the boundedness of the sequence generated by our proposed method.
x k be the sequence given by Algorithm 3.1 and Proof. Fix ∈ p Γ, then from Lemma 2.3, we have that We obtained the last inequality by using the fact that ∈ p Γ and the definition of F. Indeed, for any ∈ p Γ,

By the definition of Q k and Cauchy-Schwarz inequality, we obtain
Therefore, from (27) and (28), we obtain that Now, from (17) and Lemma 2.7, we have , .
Again by using (17) and Lemma 2.7, we have Again from (P3) and (14), we have In what follows, we obtain a result, which is a consequence of the boundedness of { } x k .
Proof. First, we show that ∈ q C. Indeed, it follows from By using the Cauchy-Schwartz inequality, we have Hence, by the weakly continuity of h j , we have Thus, ∈ q C. By the definition of z k and characterization of Π Ck , we have This implies that Fix ∈ w C ki and let → ∞ i in (37), by hypothesis, > λ 0 ki and uniform continuity of J 1 on bounded subsets of E 1 , we have Thus, we have from (38), the fact that ∈ w C ki and ⊂ C C ki , that Since F is semistrictly quasi-monotone, we have that from which we get , where Π Γ is the generalized projection of E 1 onto Γ.
Proof. As in Lemma 3.4, let ∈ p Γ. Then from (8) and (17), we have for each ∈ k N. Now, consider the following two possible cases: , k k n 0 is either non-increasing or non-decreasing. Then, by the boundedness of { ( )} ϕ p x , k , it follows that { ( )} ϕ p x , k is convergent and , 0 a s .

k k1
From (30), (33) and (35), we have ϕ p x β ϕ p u β ϕ p u β ϕ p u β ϕ p y α α g J y J Ty β ϕ p u β ϕ p w α α g J y J Ty Using the property of g and the uniform continuity of − J 1 1 on bounded subsets of * E 1 , we have Also, from (34), we have Observe from (17) that   (14) and condition (B3) that that is, by Lemma 2.4, we have It is also easy to see that Now,  (55) which implies that .
Using (49), the properties of A and f , we have , .
Also, define the mappings × → f X X Y : by . Then, F is semistrictly quasi-monotone and weakly sequentially continuous on C, see [25].
Also, define the mappings × → f X X Y : by ( ) = − + ∀ ∈ f x x y x y , , 2 2 , ( ) = ∀ ∈ A x x , We plot the graphs of errors against the number of iterations in each case. The numerical results can be found in Figure 3. Vector mixed equilibrium problem  355 In this paper, we introduced an iterative algorithm of inertial form for approximating an element in the solution set of SGVMEP, which is also a fixed point of a quasi-ϕ-nonexpansive mapping and solves a VIP for a weakly sequentially continuous and quasi-monotone mapping in Banach spaces. The result obtained in this sequel extends and unifies the works of Chang et al. [25], Kazmi and Farid [11], Shan and Huang [31] and others in the literature. Using numerical example, we showed the efficacy of our method for arriving at an element in the solution set.
(i) If we take = F 0, then Theorem 3.6 reduces to the theorems for finding a common element in the solution of SGVMEP and fixed point of quasi-ϕ-nonexpansive mapping.
(ii) Let = Y and = [ +∞) P 0, , then the result presented in this paper reduces to finding a common element in the solution set of SGMEP considered in [12,55] and fixed point of quasi-ϕ-nonexpansive mapping which is also a solution of a VIP. a real Hilbert space, the result presented in this sequel is a unification of the result presented in [25] and [31].