Iterative Scheme for Split Variational Inclusion and a Fixed-Point Problem of a Finite Collection of Nonexpansive Mappings

This article is aimed at introducing an iterative scheme to approximate the common solution of split variational inclusion and a fixed-point problem of a finite collection of nonexpansive mappings. It is proven that under some suitable assumptions, the sequences achieved by the proposed iterative scheme converge strongly to a common element of the solution sets of these problems. Some consequences of the main theorem are also given. Finally, the convergence analysis of the sequences achieved from the iterative scheme is illustrated with the help of a numerical example.


Introduction
Let H 1 and H 2 be two real Hilbert spaces endowed with inner product h⋅ , ⋅i and induced norm k⋅k. A mapping T : H 1 ⟶ H 1 is called contraction, if ∃κ ∈ ð0, 1Þ such that kTðφÞ − TðψÞk ≤ κkφ − ψk, ∀φ, ψ ∈ H 1 . If κ = 1, then T becomes nonexpansive. A mapping T is said to have a fixed point, if ∃φ ∈ ðH 1 Þ such that TðφÞ = ðφÞ. Further, if T n : H 1 ⟶ H 1 , ðn = 1,⋯,MÞ is a finite collection of nonexpansive mappings. Then, the fixed-point problem (FPP) is defined as find φ ∈ H 1 such that It is easy to show that if T M n=1 FixðT n Þ ≠ 0, then FixðT n Þ is closed and convex. Many iterative methods have been adopted to examine the solution of a fixedpoint problem for nonexpansive mappings and its variant forms, see [1][2][3][4][5] and references therein.
We know that most of the techniques for solving the fixed-point problems can be acquired from Mann's iterative technique [3], namely, for arbitrary x 0 ∈ C, compute where T is a nonexpansive mapping from a nonempty closed convex subset C of Hilbert space H 1 to itself and α n is a control sequence, which force fx k g to converge (weak) to a fixed point of T. To obtain the strong convergence result, Moudafi [4] proposed the viscosity approximation method by combining the nonexpansive mapping T with a contraction of given mapping f over C. For an arbitrary x 0 ∈ C, compute the sequence fx k g generated by where α n ∈ ð0, 1Þ goes slowly to zero. The sequence fx k+1 g achieved from this iterative method converges strongly to a fixed point of T.
On the other hand, let us recall some work about split variational inequality/inclusion problems. A multivalued mapping G : H 1 ⟶ 2 H 1 is called maximal monotone, if its graph gphðGÞ = fðφ, ψÞ ∈ H 1 × H 1 : ψ ∈ GðφÞg is not properly contained by the graph of any other monotone mapping. A monotone mapping G is maximal monotone if and only if for ðφ, ζÞ ∈ H 1 × H 1 , hφ − ψ, ζ − ϑi ≥ 0 for every ðψ, ϑÞ ∈ gphðGÞ implies that ζ ∈ GðφÞ. If G is maximal monotone, then operator J G λ = ðI + λGÞ −1 is well defined, nonexpansive, and known as the resolvent of G with parameter λ > 0, which is defined at every point of the domain.
The idea of split variational inequality problem (SVIP) given by Censor et al. [6], which amounts to saying find a solution of variational inequality whose image, under a given bounded linear operator, solves another variational inequality. Find φ * ∈ C such that and such that where C and D are closed, convex subsets of Hilbert spaces H 1 and H 2 , respectively; A : H 1 ⟶ H 2 is a bounded linear operator, and h : H 1 ⟶ H 1 and g : H 2 ⟶ H 2 are two operators. They studied the weak convergent result to solve SVIP. Moudafi [7] generalized SVIP and introduced split monotone variational inclusion problem (S p MVIP): find φ * ∈ H 1 such that and such that where G 1 : H 1 ⟶ 2 H 1 and G 2 : H 2 ⟶ 2 H 2 are multivalued monotone mappings, A : H 1 ⟶ H 2 is a bounded linear operator, h : H 1 ⟶ H 1 and g : H 2 ⟶ H 2 are two single-valued operators. The author also composed an iterative algorithm to solve (S p MVIP) and showed that the sequence achieved by the proposed algorithm converges weakly to the solution of (S p MVIP). Numerous iterative methods have been investigated for split variational inequality/inclusion problems, split common fixed-point problems, split feasibility problems, and split zero problems and their generalizations, see [6,[8][9][10][11][12][13][14][15][16] and references therein. If h = g = 0 in S p MVIP, then we obtain the split variational inclusion problem (S p VIP) considered in [8], stated as find φ * ∈ H 1 such that such that Byrne et al. [8] proposed the following iterative scheme for S p VIP and studied the strong and weak convergence. For arbitrary x o ∈ H 1 , compute the iterative sequence achieved by the following scheme: for λ > 0: Recently, Kazmi and Rizvi [17] suggested and examined an iterative algorithm to estimate the common solution for S p VIP and a fixed-point problem of a nonexpansive mapping in Hilbert spaces. Puangpee and Sauntai [18] studied the split variational inclusion problem and fixed-point problem in Banach spaces. Haitao and Li [19] investigated the split variational inclusion problem and fixed-point problem of nonexpansive semigroup without prior calculation of operator norm. Later, many authors studied the common solution of split variational inequality/inclusion problem and fixedpoint problem of nonexpansive mappings in the framework of Hilbert/Banach spaces, see for example [18][19][20][21][22][23][24] and references therein.
Following the works in [4,7,8,17] and by the current research in this flow, we propose an iterative scheme to approximate a common solution of FPP and S p VIP. We prove that the sequences achieved by the proposed iterative scheme strongly converge to the common solution of FPP and S p VIP. The iterative scheme and results discussed in this article are new and can be viewed as generalization and refinement of the previously published work in this area.

Prelude and Auxiliary Results
In this section, we assembled some underlying definitions and supporting results. Definition 1. Let CðC ⊂ H 1 Þ, the metric projection P C onto the set C is defined as P C ðϑÞ ∈ C and kϑ − P C ðϑÞk = inf ϑ∈C kϑ − ωk, ∀ω ∈ H 1 . P C is also characterised by the facts that P C ðϑÞ ∈ C, and Remark 2 (see [25,26]). For a nonexpansive mapping T and projection P C ðϑÞ onto C, the following results hold in Hilbert spaces: 2 Journal of Function Spaces (ii) For all ðϑ, ωÞ ∈ H 1 × H 1 , Thus, for all ðϑ, ωÞ ∈ H 1 × FixðTÞ, we get Definition 3. A mapping T : (ii) τ-strongly monotone, if there exists a constant τ > 0 such that (iii) γ-inverse strongly monotone, if there exists a constant γ > 0 such that Some important characteristics of an averaged operator are mentioned below; for more details, we refer to [7,27,28].
Definition 4. A mapping T : H 1 ⟶ H 1 is called an averaged if and only if T is the average of identity mapping and a nonexpansive mapping, that is, Thus, firmly nonexpansive mappings are averaged. It can also be seen that averaged mappings are nonexpansive.

Proposition 5.
(i) Let S : H 1 ⟶ H 1 be an averaged and V : (ii) If the composite fT n g M n=1 is averaged and have a nonempty common fixed point, then Lemma 6 (see [29]). Assume that T is nonexpansive selfmapping of a closed convex subset D of a Hilbert space H 1 . If T has a fixed point, then I − T is demiclosed, i.e., whenever fω n g is a sequence in D converging weakly to some ω ∈ D and the sequence fðI − TÞω n g converges strongly to some ϖ, then ðI − TÞω = ϖ, where I is the identity mapping on H 1 .
Lemma 7 (see [5]). If fv k g is a sequence of nonnegative real numbers such that where fξ k g is a sequence in (0,1) and fω k g is a sequence in ℝ such that We denote the solution set of S p VIP by Ξ = fφ * ∈ H 1 : 0 ∈ G 1 ðφ * Þand 0 ∈ G 2 ðAφ * Þg and of FPP by ∩ M n=1 FixðT n Þ.

Iterative Scheme and Its Convergence
In this section, we present the iterative scheme and show that the sequences obtained from the proposed iterative scheme converge strongly to the common solution of FPP and S p VIP. For integer K ≥ 1, we define the mapping T ½K = T K mod M with the mod function, which is taking values from the set f1, 2,⋯,Mg, that is, if K = aM + b for some integer a ≥ 0 and

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Step 1.
update k = k + 1 and go to Step 1.
Condition C. We assume that T n : Proof. The proof of the lemma follows immediately from the definitions of resolvent operators.

Remark 10. If J G
λ is the resolvent of maximal monotone mapping G, A * is the adjoint operator of A and R is the spectral radius of AA * . Then, using the properties of averaged mapping, one can easily show that the operator ½I + μA * ðJ G λ − IÞA is averaged with λ > 0, μ ∈ ð0, 1/RÞ. Now, we prove the following lemma which guarantees the contractivity of L, which is needed to prove our main result. Lemma 11. Let H 1 and H 2 be two real Hilbert spaces and A : H 1 ⟶ H 2 be a bounded linear operator. Suppose that G 1 : H 1 ⟶ H 1 and G 2 : H 2 ⟶ H 2 are maximal monotone operators and T : where μ ∈ ð0, 1/RÞ, R is the spectral radius of the operator AA * , and A * is the adjoint operator of A. Then, the mapping L is a contraction with constant 0 < 1 − θð1 − κÞ < 1; hence, L has a unique fixed point.
Since 0 < 1 − θð1 − κÞ < 1 implies that L is a contraction, hence L has a unique fixed point.

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Now, we show that ku k − v k k ⟶ 0 as k ⟶ ∞. From (29), it follows that Therefore, Since ð1 − RμÞ > 0, α k ⟶ 0, and ku k+1 − u k k ⟶ 0, we get Since μ ∈ ð0ð1/RÞÞ and using (27) and (29), we obtain Thus, we get By (37) and (41), we have Since α k ⟶ 0, ku k+1 − u k k ⟶ 0, kðJ We recognized that the following relation holds: Journal of Function Spaces By Iterative Scheme 8, we can easily see that ku k+1 − T ½k+1 v k k ⟶ 0 as k ⟶ ∞. From (44) and nonexpansiveness of T n ðn = 1, 2,⋯,MÞ, it follows that By using (36) and (45), we conclude that Now, using (47) and (44), we write as k ⟶ ∞, that is, Boundedness of fv k g implies that there exists a subsequence fv k i g of fv k g, converging weakly to w. Because the collection of mappings fT n : 1 ≤ n ≤ Mg is finite, we can say for some integer K ∈ f1, 2,⋯,Ng Thus, from (49), we have Therefore, using Lemma 6, we conclude that Thus, by the assumptions of Condition C, we have w ∈ ∩ M n=1 FixðT n Þ. On the other hand, We know that the graph of a maximal monotone operator is weakly strongly closed; hence, by taking i ⟶ ∞ and using (37) and (44), we get Since fu k g, fv k g have the same asymptotical behaviour, fAu k i g converges weakly to Aw. Therefore, by (39), the nonexpansive property of J G 2 λ and Lemma 6, we get 0 ∈ G 2 ðAwÞ. Thus, w ∈ ∩ M n=1 FixðT n Þ ∩ Ξ. Now, we need to show that lim sup since 7 Journal of Function Spaces which implies that From Lemma 7 and (55), we conclude that u k ⟶ v and from kv k − u k k ⟶ 0, v k ⇀ w ∈ ∩ M n=1 FixT n ∩ Ξ, and u k ⟶ v as k ⟶ ∞, we achieve that v = w. This completes the proof.

Consequences
Suppose C and D are closed convex subsets of Hilbert spaces H 1 and H 2 , respectively. Then, find u ∈ H 1 such that is called the split feasibility problem (SFP), where A : H 1 ⟶ H 2 is a bounded linear operator. Byrne [9] introduced the CD algorithm to approximate the solution of (58): where P C and P D are orthogonal projections onto C and D, respectively. The split common fixed-point problem (SCFPP) is an extension of Problem (58), which has been widely investigated in the present scenario. The SCFPP is the inverse problem design to search a vector in a fixed-point set so that its image under a bounded linear operator corresponds to other fixed-point set, that is, find u ∈ H 1 such that where W : H 1 ⟶ H 1 and W : H 2 ⟶ H 2 are nonexpansive mappings. By putting W = P C and W = P D , in (59), we can have an iterative scheme, which converges to the solution of SCFPP. We denote the solution set of SFP (58) and SCFPP (60) by Ψ, and Ω, respectively. The following corollaries are given as consequences of Theorem 12.
Corollary 13. Let C and D be two closed convex subsets of Hilbert spaces H 1 and H 2 , respectively. Let A : H 1 ⟶ H 2 be a bounded linear operator and f : H 1 ⟶ H 1 be a contraction mapping with constant κ ∈ ð0, 1Þ. Let T n : H 1 ⟶ H 1 , ðn = 1, 2,⋯,MÞ, be a finite collection of nonexpansive mappings satisfying the condition C such that ∩ M n=1 FixðT n Þ ∩ Ψ ≠ 0. Let R be the spectral radius of A ⋆ A, where A ⋆ is the adjoint of A such that μ ∈ ð0, 1/RÞ and fα k g be a sequence in (0,1) with lim k→∞ α k = 0, ∑ ∞ k=1 α k = ∞, and ∑ ∞ k=1 jα k − α k−1 j < ∞. Then, the iterative sequences fv k g and fu k g generated by Iterative Scheme 8 with J G 1 λ = P C and J G 2 FixðT n Þ ∩ Ω ≠ 0. Let R be spectral radius of A ⋆ A, where A ⋆ is the adjoint of A such that μ ∈ ð0, 1/RÞ and fα k g be a sequence in (0,1) with lim n→∞ α k = 0, ∑ ∞ k=1 α k = ∞, and ∑ ∞ k=1 jα k − α k−1 j < ∞. Then, the iterative sequences fv k g and fu k g obtained from Iterative Scheme 8 with J G 1 λ = W and J G 2 FixðT n Þ∩Ω f ð vÞ.
Remark 15. If we take T 1 = T 2 = ⋯T M = T, a nonexpansive mapping, then we can obtain the iterative scheme and its convergence theorem for the common solution of S p VIP and a nonexpansive mapping T, studied in [17]. At last, we illustrate the convergence analysis of the proposed iterative scheme with the help of the following numerical example.