Some variants of the hybrid extragradient algorithm in Hilbert spaces

This paper provides convergence analysis of some variants of the hybrid extragradient algorithm (HEA) in Hilbert spaces. We employ the HEA to compute the common solution of the equilibrium problem and split ﬁxed-point problem associated with the ﬁnite families of k -demicontractive mappings. We also incorporate appropriate numerical results concerning the viability of the proposed variants with respect to various real-world applications


Introduction
The class of split inverse problems (SIP) plays a prominent role in medical image reconstruction and in signal processing.One of the important generalizations of the SIP is the split common fixed-point problem (SCFPP).The class of SCFPP associated with a variety of nonlinear mappings has been analyzed in the framework of Hilbert as well as Banach spaces.In this paper, we are interested in solving the SCFPP for finite families of k-demicontractive mappings in Hilbert spaces.
In 1994, Blum and Oettli [14] proposed the (monotone-) equilibrium problem (EP) theory in Hilbert spaces.Since then, several iterative algorithms have been employed to compute the optimal solution of the (monotone-) EP as well as EP together with the fixed-point problem (FPP).In 2006, Tada and Takahashi [32] suggested a hybrid framework for the analysis of monotone EP and FPP in Hilbert spaces.However, the iterative algorithm proposed in [32] fails for the case of pseudomonotone EP.To overcome this drawback, Anh [2] employed the hybrid extragradient method, based on the seminal work of Korpelevich [27], to compute the optimal common solution of the pseudomonotone EP and the FPP.
Inspired and motivated by the ongoing research, it is natural to study the pseudomonotone EP together with the SCFPP associated with the class of k-demicontractive mappings in Hilbert spaces.We propose some accelerated variants, based on the inertial extrapolation technique [29] (see also [1, 3-12, 15-17, 19, 21-24]), of the hybrid extragradient algorithm in Hilbert spaces.
The rest of the paper is organized as follows.We present some relevant preliminary concepts and useful results regarding the pseudomonotone EP and SCFPP in Sect. 2. Section 3 comprises strong convergence results of the proposed variants of the hybrid inertial extragradient algorithm under a suitable set of constraints.In Sect.4, we provide numerical results for the demonstration of the main results in Sect. 3 as well as the viability of the proposed variants with respect to various real-world applications.

Preliminaries
Let K be a nonempty, closed, convex subset of a real Hilbert space H 1 .The metric projector P K from H 1 onto K is defined for each μ ∈ H 1 there exists a unique nearest point Throughout the rest of the paper, we denote by CB(H 1 ), CC(H 1 ), and PB(H 1 ) the families of nonempty, closed, bounded subsets, nonempty, closed, convex subsets, and nonempty, proximal, bounded subsets of H 1 .The Hausdorff metric on CB(H 1 ) is defined as: Let T : H 1 → 2 H 1 be a multivalued mapping with a nonempty, closed, and convex fixedpoint set denoted by Fix(T ) := {ν ∈ H 1 ; ν ∈ T ν}.Recall that the multivalued mapping T is said to be a (i) contraction if there exists k ∈ (0, 1) such that [20] if Fix(T ) = ∅, and there exists k ∈ [0, 1) such that D(T μ, T ν) 2 ≤ μν 2 + kd(μ, T ν) 2 for all μ ∈ H 1 , ν ∈ Fix(T ).It is worth mentioning that every multivalued quasinonexpansive mapping T with Fix(T ) = ∅ is demicontractive, but not all multivalued demicontractive mappings are quasinonexpansive (see Example 2.2 in Ref. [25] for the proper inclusion).
The set EP(g) is weakly closed and convex provided that the bifunction g satisfies Assumption 2.1.For a finite family of bifunctions g i satisfying Assumption 2.1, we can compute the same Lipschitz coefficients (d 1 , d 2 ) for the family of bifunctions g i by employing the condition (A2) as In addition, for all j = 1, 2, . . .N , let T j : H 1 → CB(H 1 ) and S j : H 2 → CB(H 2 ) be finite families of multivalued demicontractive mappings with constants k j and kj , respectively, such that T j -Id and S j -Id are demiclosed at zero.If we assume : H 1 → H 2 to be a bounded linear operator then the solution set of the SCFPP for two finite families of multivalued mappings (T j ) N j=1 and (S j ) N j=1 , is denoted as Fix(S j ) .
In [34], it was shown that the fixed-point set of a multivalued demicontractive mapping is closed and convex provided it satisfies the endpoint condition.In a similar fashion, we can choose (k, k) = sup 1≤j≤N (k j , kj ).Suppose that := ( M i=1 EP(g i )) ∩ = ∅.Then, we are interested in the following problem:

Lemma 2.5 ([33]) Assume a convex and subdifferentiable function h
, where ∂h(•) indicates the subdifferential of h and N K (ν) is the normal cone of K at ν.

Lemma 2.6 ([28]
) Let K be a nonempty, closed, convex subset of a real Hilbert space H 1 .
For every p, q, r ∈ H 1 and γ ∈ R, the following set is closed and convex:

Algorithm and convergence analysis
Our main iterative algorithm of this section has the following architecture (Algorithm 1).
The following result is crucial for the strong convergence result of Algorithm 1.
where ν k , b k , u k , and v k are defined in Algorithm 1.
Proof of Theorem 3.
Recalling the estimate (3.1), Lemmas 2.3 and 2.4, and Lemma 3.2, we obtain , since S j is kj -demicontractive, then by Lemma 2.3, we have Utilizing (3.2) and (3.3), we obtain Since T j is k j -demicontractive and by using Lemma 2.4, we have This shows that is contained in C k , for all k ≥ 0. Recalling the definition of the set C k the above estimate infers that Algorithm 1 is well defined.
If for each j = 1, 2, . . ., N , let S j = Id, then we have the following result: Assume that the conditions (C1) and (C3) hold, then the sequence (ν k ) generated by (3.22) converges strongly to an element in .
As a direct application of Theorem 3.1, we have the following result for the variational inequality problem (i.e., find ν ∈ K for which Aν, μν ≥ 0 ∀ μ ∈ K, where A : K → H 1 is a nonlinear, monotone mapping defined on a nonempty, closed, convex subset K ⊆ H 1 ): Theorem 3.4 Let K ⊆ H 1 be a nonempty, closed, convex subset of a real Hilbert space H 1 and for each i = 1, 2, . . ., M let A i : K → H 1 be a finite family of pseudomonotone and L-Lipschitz continuous mappings.For all j = 1, 2, . . ., N , let T j : H 1 → CB(H 1 ) and S j : H 2 → CB(H 2 ) be two finite families of multivalued, demicontractive mappings with constants k j and kj , respectively, such that T j -Id and S j -Id are demiclosed at zero.Let Assume that the conditions (C1)-(C3) hold, then the sequence (ν k ) generated by (3.23) converges strongly to an element in .
Proof Let g i (ν, μ) = A i (ν), μ -ν for all ν, y ∈ K and i = 1, 2, . . ., M. Since A i is L-Lipschitz continuous, we observe that for all ν, μ, ξ ∈ C This infers that g i is Lipschitz-type continuous with Moreover, the pseudomonotonicity of A i ensures the pseudomonotonicity of g i .From Algorithm 1, we have Equivalently, we have Recalling the proof of Theorem 3.1 with the above-mentioned g i (ν, μ) for all i ∈ {1, 2, . . ., M} leads to the desired result.Then, the sequence (ν k ) generated by Algorithm 2 converges strongly to an element in .
Proof Observe that the set C k can be expressed as: Algorithm 2 Hybrid Inertial Halpern-Extragradient Algorithm (Alg.2) Initialization: Choose arbitrarily q ∈ H 1 and ν 0 , ν 1 ∈ C 0 = H 1 , set k ≥ 1 and nonincreasing sequence ( αk,j ), ( βk,j ) ⊂ (0, 1), 0 < ϑ < min( If h k = z k = y k = w k then stop and ν k is the solution of problem .Otherwise, Step 2. Compute Put k =: k + 1 and go back to Step 1. Recalling the proof of Theorem 3.1, it infers that (i) the sets andC k are closed and convex, satisfying ⊂ C k+1 for all k ≥ 0; (ii) (ν k ) is bounded such that Since ν k+1 = P H 1 C k (q) ∈ C k and by the definition of C k , we have Recalling the estimate (3.24), the conditions (C1)-(C3) and the boundedness of (ν k ), we obtain Also, observe that for each ν * ∈ .Recalling the estimate (3.24), the conditions (C1)-(C3), and the boundedness of (ν k ), we obtain Recalling h k = γk q + (1 -γk )z k and the conditions (C2) and (C3) with C * , we obtain Recalling the estimate (3.25) again, the above estimate implies that The rest of the proof of Theorem 3.5 is similar to the proof of Theorem 3.1 and is therefore omitted here.
Remark 3.6 From the numerical standpoint, the condition (C1) can easily be aligned in an algorithm as ν kν k-1 is a priorly known before selecting ξ k satisfying 0 ≤ ξ k ≤ ξ k , where where {σ k } is a positive sequence such that ∞ k=1 σ k < ∞ and ξ k ∈ [0, 1).

Numerical experiment and results
This section provides the effective viability of our algorithm supported by a suitable example.
. For each j = 1, 2, . . ., N , let T j and S j be defined as: It is not difficult to show that T j and S j are 0-demicontractive, and Id -T j and Id -S j are demiclosed at zero for all j = 1, 2, . . ., N .We also define a bounded linear operator : R → R by μ = 3μ.Thus, * μ = 3μ and = 3.It is clear that 0 ∈ , where = {μ ∈ The terminating criteria of Algorithm 1 is set as Error = E k = ν kν k-1 < 10 -6 .Table 1 summarizes the computation of Algorithm 1 and its variant.
The terminating criterion E k and (ν k ) summarized in Table 1 for Algorithm 1 are depicted in Fig. 1.We can see from Table 1 and Fig. 1 that Alg.1 with ξ k = 0 outperforms Alg.1 with ξ k = 0 with respect to the reduction in the error, time consumption, and the number of iterations required for the convergence towards the common solution.

Conclusions
In this paper, we have investigated an inertial-based, parallel, hybrid, extragradient algorithm for constructing iteratively a common solution of the pseudomonotone EP and the SCFPP associated with the finite families demicontractive mappings in Hilbert spaces.The abstract formalism of the problem has been strengthened with the computer-assisted simulation for the algorithm via an appropriate numerical example.We emphasize that our proposed abstract formalism together with the computer-assisted iterative algorithm arise naturally in various forms of real-world applications and would be an important topic of future research.

Theorem 3 . 5
Setting a terminating criterion by fixing k > k max for an appropriately chosen large number k max , we now propose a Halpern-type variant of Algorithm 1.If = ∅ such that the conditions (C1)-(C3) with C * = lim k→∞ γk = 0 hold.

1
Step 1. Algorithm 1 is well defined.It is obvious by recalling Lemma 2.6 that the set C k is closed and convex.Moreover, the set is closed and convex.Therefore, is nonempty, closed, and convex.For any ν * ∈ , it follows from Algorithm 1 that

Table 1
Summary of the Numerical Computation for Algorithm 1No. of Iter.