On a system of monotone variational inclusion problems with fixed-point constraint

In this paper, we study the problem of finding the solution of the system of monotone variational inclusion problems recently introduced by Chang et al. (Optimization 70(12):2511–2525, 2020) with the constraint of a fixed-point set of quasipseudocontractive mappings. We propose a new iterative method that employs an inertial technique with self-adaptive step size for approximating the solution of the problem in Hilbert spaces and prove a strong-convergence result for the proposed method under more relaxed conditions. Moreover, we apply our results to study related optimization problems. Finally, we present some numerical experiments to demonstrate the performance of our proposed method, compare it with a related method as well as experiment on the dependency of the key parameters on the performance of our method.


Introduction
In recent years, the split inverse problem (SIP) has received much research attention (see [1,11,12,20,24,50] and the references therein) because of its extensive applications, for example, in phase retrieval, signal processing, image recovery, intensity-modulated radiation therapy, data compression, among others (see [13,14,42] and the references therein). The SIP model is presented as follows: Find a point x ∈ H 1 that solves IP 1 (1.1) such that y := Ax ∈ H 2 solves IP 2 , (1.2) where H 1 and H 2 are real Hilbert spaces, IP 1 denotes an inverse problem formulated in H 1 and IP 2 denotes an inverse problem formulated in H 2 , and A : H 1 → H 2 is a bounded linear operator. Censor and Elfving [14] in 1994 introduced the split feasibility problem (SFP), which was the first instance of the SIP for modeling inverse problems that arise from medical-image reconstruction. Since then, several authors have studied and developed different iterative methods for approximating the solution of the SFP. The SFP has wide areas of applications, for instance, in signal processing, approximation theory, control theory, geophysics, communications, biomedical engineering, etc. [13,30]. The SFP is formulated as follows: find a pointx ∈ C such thatŷ = Ax ∈ Q, (1.3) where C and Q are nonempty closed convex subsets of Hilbert spaces H 1 and H 2 , respectively, and A : H 1 → H 2 is a bounded linear operator.
(i) The solution set F is nonempty; (ii) λ ∈ (0, 2α), 0 < lim inf n→∞ τ n ≤ lim sup n→∞ τ n < 1; (iii) {μ n } ∞ n=1 ⊂ 1 , i.e., ∞ n=1 |μ n | < ∞. Bauschke and Combettes [6] pointed out that in solving optimization problems, the strong convergence of iterative schemes is more desirable and useful than the weak-convergence counterparts. Therefore, when solving optimization problems the authors strive to construct algorithms that generate sequences that converge strongly to the solution of the problem under investigation.
Also, very recently, Chang et al. [16] introduced and studied the following system of monotone variational inclusion problems in Hilbert spaces: find a point x * ∈ H 1 such that . . , m; and y * = Ax * solves 0 ∈ g j (y * ) + D j (y * ), j = 1, 2, . . . , k, (1.11) where for each i = 1, 2, . . . , m and j = 1, 2, . . . , k, h i and g j are ϕ i -and ϑ j -inverse strongly monotone mappings on H 1 and H 2 , respectively, where ϕ i > 0 and ϑ j > 0, B i and D j are multivalued maximal monotone operators on H 1 and H 2 , respectively, and A : then all h i and g j are φ-inverse strongly monotone mappings. Moreover, the authors proposed the following inertial forward-backward splitting algorithm with the viscosity technique for approximating the solution of problem (1.11) in Hilbert spaces: for each i = 1, 2, . . . , m and j = 1, 2, . . . , k, h i , g j , B i , D j are as defined in (1.11), and f : H 1 → H 1 is a contraction with contraction constant ρ ∈ ( 1 2 , 1). The authors proved the strongconvergence theorem for the proposed method under the following conditions: (i) The solution set is nonempty; Remark 1.1 Observe that the problem (1.11) solved by Algorithm (1.13) is more general than the problem SMVIP (1.4) and (1.5) solved by Algorithm 1. The SMVIP (1.4) and (1.5) is a special case of the problem (1.11) when i = j = 1. We also point out that the term θ n (x nx n-1 ) in Algorithm 1 and Algorithm (1.13) above is referred to as the inertial term. It is employed in algorithm design to accelerate the rate of convergence. However, we note that condition (iii) of Algorithm 1 and condition (iv) of Algorithm (1.13) imposed to incorporate the inertial term are too restrictive. These might affect the implementation of the proposed methods. Some other drawbacks with Algorithm (1.13) are that the contraction constant ρ of the contraction f is restricted to the interval ( 1 2 , 1). Moreover, the implementation of the proposed algorithm requires knowledge of the operator norm, which is often very difficult to calculate or even estimate. On the other hand, while Algorithm 1 does not require knowledge of the operator norm for its implementation the authors were only able to obtain the weak-convergence result for the proposed algorithm.
From the above discourse, it is natural to ask the following question Some of our aims in this paper are to provide affirmative answers to the above questions. Another problem we consider in this paper is the fixed-point problem (FPP). Let C be a nonempty closed convex subset of a real Hilbert space H and let S : C → C be a nonlinear mapping. A pointx ∈ C is called a fixed point of S if Sx =x. We denote by F(S), the set of all fixed points of S, i.e., (1. 15) In recent years, the study of fixed-point theory for nonlinear mappings has flourished owing to its extensive applications in various fields like economics, compressed sensing, and other applied sciences (see [4,17,38] and the references therein).
Recently, optimization problems dealing with finding a common solution of the set of fixed points of nonlinear mappings and the set of solutions of SMVIP (see, for instance, [3,22]) were considered. One of the motivations for studying such a common solution problem is in its potential application to mathematical models whose constraints can be expressed as FPPs and SMVIP. An instance of this is found in practical problems such as signal processing, network-resource allocation, and image recovery. One scenario is in the network bandwidth-allocation problem for two services in a heterogeneous wireless access networks where the bandwidth of the services are mathematically related (see, for instance, [26,31] and the references therein).
Motivated by the above results and the current research interest in this direction, in this paper, we study the problem of finding the solution of the system of monotone variational inclusion problems (1.11) with the constraint of a fixed-point set of quasipseudocontractions. Precisely, we consider the following problem: find a point x * ∈ F(S) such that ⎧ ⎨ ⎩ 0 ∈ h i (x * ) + B i (x * ), i = 1, 2, . . . , m; and y * = Ax * solves 0 ∈ g j (y * ) + D j (y * ), j = 1, 2, . . . , k, (1.16) where S : H 1 → H 1 is a quasipseudocontractive mapping, for each i = 1, 2, . . . , m and j = 1, 2, . . . , k, h i and g j are ϕ i -and ϑ j -inverse strongly monotone mappings on H 1 and H 2 , respectively, where ϕ i > 0 and ϑ j > 0, B i and D j are multivalued maximal monotone operators on H 1 and H 2 , respectively, and A : H 1 → H 2 is a bounded linear operator.
Moreover, we introduce a new inertial iterative method that employs the viscosity technique to approximate the solution of the problem in the framework of Hilbert spaces. Furthermore, under mild conditions we prove that the sequence generated by the proposed method converges strongly to a solution of the problem. We point out that the implementation of our algorithm does not require knowledge of the operator norm and the contraction constant of the contraction mapping employed in the viscosity technique can be selected in the interval (0, 1); a larger interval than the restriction to interval ( 1 2 , 1) in Algorithm (1.13). In addition, we obtained a strong-convergence result dispensing with condition (iii) of Algorithm 1 and condition (iv) of Algorithm (1.13). We further apply our results to study other optimization problems and we provide some numerical experiments with graphical illustrations to demonstrate the implementability and efficiency of the proposed method in comparison with some methods in the current literature. Our results in this study improve and extend the recent ones announced by Yao et al. [48], Chang et al. [16], and many other results in the literature.
The paper is organized as follows: In Sect. 2, we recall basic definitions and lemmas employed in the convergence analysis. Section 3 presents the proposed algorithm and highlights some of its features, while in Sect. 4 we analyze the convergence of the proposed method. Section 5 presents applications of our results to some optimization problems. In Sect. 6, we provide some numerical examples with graphical illustrations and compare the performance of our proposed method with some of the existing methods in the literature. Finally, we give some concluding remarks in Sect. 7.

Preliminaries
In this section, we present some definitions and results, which will be needed in the following.
In what follows, we denote the weak and strong convergence of a sequence {x n } to a point x ∈ H by x n x and x n → x, respectively, and w ω (x n ) denotes the set of weak limits of {x n }, that is, where H is a real Hilbert space. For a nonempty closed and convex subset C of H, the metric projection [37] The operator P C is nonexpansive and has the following properties [34,44]: 1. it is firmly nonexpansive, that is, 2. for any x ∈ H and z ∈ C, z = P C x if and only if xz, zy ≥ 0 for all y ∈ C; (2.1) 3. for any x ∈ H and y ∈ C, Definition 2.1 Let T : H → H be a nonlinear mapping and I be the identity mapping on H. The mapping I -T is said to be demiclosed at zero, if for any sequence {x n } ⊂ H that converges weakly to x and x n -Tx n → 0, then x ∈ F(T).

Definition 2.2
Let C be a nonempty closed convex subset of a real Hilbert space H.
where α ∈ (0, 1), S : C → C is nonexpansive and I is the identity mapping on C; Remark 2.3 As pointed out by Bauschke and Combettes [6], In other words, the class of directed mappings coincides with the class of firmly quasinonexpansive mappings.
Remark 2.4 From the definitions above, we observe that the class of demicontractive mappings includes several other classes of nonlinear mappings such as the directed mappings, the quasinonexpansive mappings, and the strictly pseudocontractive mappings with fixed points as special cases. Also, it is well known that every L-ism mapping is 1 L -Lipschitz continuous and monotone, and every Lipschitz continuous operator is uniformly continuous but the converse of these statements are not always true (see, for example [41]).

Definition 2.5 A nonlinear operator
The interest of pseudocontractive mappings lies in their connection with monotone mappings, that is, T is a pseudocontraction if and only if I -T is a monotone mapping. It is well known that T is pseudocontractive if and only if It is obvious that the class of quasipseudocontractive mappings includes the class of demicontractive mappings and the class of pseudocontractive mappings with a nonempty fixed-point set.
We have the following result on L-Lipschitz quasipseudocontractive mappings.
√ 1+L 2 , then the following conclusions hold: if lim sup k→∞ b n k ≤ 0 for every subsequence {a n k } of {a n } satisfying lim inf k→∞ (a n k+1 -a n k ) ≥ 0, then lim n→∞ a n = 0. Assume ∞ n=0 |c n | < ∞. Then, the following results hold: then lim n→∞ a n = 0.

Lemma 2.12 ([7, 47]) Let H be a real Hilbert space and let A, S, T, V
The composite of finitely many averaged mappings is averaged.
are averaged and have a common fixed point, then (iv) If A is β-ism and γ ∈ (0, β], then T := Iγ A is firmly nonexpansive. (

Proposed method
In this section, we present our proposed algorithm and highlight some of its important features. We assume that: It was shown in [16] that the operators U and T defined above are averaged mappings. We establish the strong-convergence result for the proposed algorithm under the following conditions on the control parameters: (C1) {α n } ⊂ (0, 1) such that lim n→∞ α n = 0 and ∞ n=0 α n = ∞; (C2) {β n }, {δ n }, {ξ n } ⊂ (0, 1) such that 0 < a ≤ β n , δ n , ξ n ≤ b < 1;

Algorithm 2
Step 0. Let x 0 , x 1 ∈ H 1 be two arbitrary initial points and set n = 1.
Set n := n + 1 and return to Step 1.
Remark 3.1 • We point out that the step size of the proposed method defined in (3.4) does not depend on the norm of the bounded linear operator. This makes our algorithm easy to implement, unlike the methods proposed in [16,22,33,50], which require knowledge of the operator norm for their implementation.
• Step 1 of the algorithm can be implemented since the value of x nx n-1 is known prior to choosing θ n . Also, observe that in incorporating the inertial term our method does not require stringent conditions, like we have in condition (iii) of Algorithm 1 and condition (iv) of Algorithm (1.13). • We note that unlike in Algorithm (1.13), the viscosity technique employed in Step 5 of our algorithm accommodates a larger class of contraction mappings since the contraction constant ρ ∈ (0, 1). Remark 3. 3 We note that in (3.4), the choice of the step size γ n is independent of the operator norm A . Also, the value of γ has no effect on the proposed algorithm but was introduced for clarity. Now, we show that the step size of the algorithm in (3.4) is well defined.

Lemma 3.4 The step sizes {γ n } of the Algorithm 2 defined by (3.4) are well defined.
Proof Let p ∈ . Then, by Lemma 2.14(I) we have that p ∈ m i=1 ((h i + B i ) -1 (0)) and Ap ∈ k j=1 ((g j + D j ) -1 (0)). From Lemma 2.12(iii) we have p ∈ F(U) and Ap ∈ F(T). Applying the fact that T is averaged together with Lemma 2.12(vii), we have

Lemma 4.3 Suppose {x n } is a sequence generated by Algorithm 2 such that conditions
(C1)-(C4) are satisfied. Then, the following inequality holds for all p ∈ and n ∈ N: Proof Let p ∈ . By applying Lemma 2.9(iii) together with (4.5), (4.9) and (4.11) we have which is the required inequality. Now, we are in a position to state and prove the strong-convergence theorem for the proposed algorithm.

Theorem 4.4
Let H 1 and H 2 be two real Hilbert spaces and let f : H 1 → H 1 be a contraction with coefficient ρ ∈ (0, 1). Suppose {x n } is a sequence generated by Algorithm 2 such that conditions (C1)-(C4) hold. Then, the sequence {x n } converges strongly to a point x ∈ , wherex = P • f (x).
Next, we claim that the sequence { x n -x } converges to zero. In order to establish this, by Lemma 2.10, it suffices to show that lim sup k→∞ f (x) -x, x n k +1 -x ≤ 0 for every sub- Again, from Lemma 4.2 we obtain By applying (4.13) together with condition (C2) and the fact that lim k→∞ α n k = 0, we have By the definition of γ n , we obtain Consequently, we have Since A * (T -I)Aw n k is bounded, it follows that (4.14) Consequently, we obtain Following a similar argument, from Lemma 4.2 we obtain β n k w n k -Vu n k → 0, k → ∞.
By condition (C2), it follows that w n k -Vu n k → 0, k → ∞. (4.16) Next, from Lemma 4.3 we obtain By (4.13), Remark 3.2, and the fact that lim k→∞ α n k = 0, we obtain Consequently, we have By Remark 3.2, we have By applying (4.16), from Step 4 we have v n kw n k ≤ β n k w n kw n k + (1β n k ) Vu n kw n k → 0, k → ∞. (4.20) Next, by applying (4.16), (4.18), (4.19) and (4.20) we obtain w n ku n k → 0, k → ∞; x n k -Vu n k → 0, k → ∞; and x n ku n k → 0, k → ∞; Now, applying (4.21) and (4.22) together with the fact that lim k→∞ α n k = 0, we obtain To complete the proof, we need to show that w ω (x n ) ⊂ . Since {x n } is bounded, then w ω (x n ) is nonempty. Let x * ∈ w ω (x n ) be an arbitrary element. Then, there exists a subsequence {x n k } of {x n } such that x n k x * as k → ∞. From (4.22), we have that u n k x * as k → ∞. Since I -V is demiclosed at zero, then it follows from (4.22) and Lemma 2.7 that x * ∈ F(V) = F(S). That is, w ω (x n ) ⊂ F(S).
Next, we show that w ω (x n ) ⊂ . From Step 3 and by applying (4.21), we have lim k→∞ U I + γ n A * (T -I)A w n kw n k = lim k→∞ u n kw n k = 0. (4.24) Since the operators U and I + γ n A * (T -I)A are averaged, it follows from Lemma 2.12(ii) that the composition U(I + γ n A * (T -I)A) is also average and consequently nonexpansive. By the Demiclosedness Principle for nonexpansive mappings, and by applying (4.19) and (4.24) we obtain U(I + γ n A * (T -I)A)x * = x * . Since = ∅, then by Lemma 2.12(iii) we have Ux * = x * and (I + γ n A * (T -I)A)x * = x * . It then follows from Lemma 2.12(iii) and Lemma 2.14(I) that Sine T is nonexpansive, then by the Demiclosedness Principle for nonexpansive mappings, and by applying (4.14) and (4.19) we have TAx * = Ax * . It then follows from Lemma 2.12(iii) and Lemma 2.14(I) that From (4.25) and (4.26), we obtain w ω (x n ) ⊂ . Consequently, we have that w ω (x n ) ⊂ . Next, from (4.21) we have that w ω {v n k } = w ω {x n k }. By the boundedness of {x n k }, there exists a subsequence {x n k j } of {x n k } such that x n k j x † and Now, from (4.23) and (4.27), we obtain Applying Lemma 2.10 to (4.12), and using (4.28) together with the fact that lim n→∞ θ n α n x nx n-1 = 0 and lim n→∞ α n = 0, we deduce that lim n→∞ x n -x = 0 as required.
Remark 4.5 The results of this paper improve the results of Yao et al. [48] and Chang et al. [16] in the following ways: (i) Our result extends the result of Yao et al. [48] and the result of Chang et al. [16] from SMVIP (1.4) and (1.5) and a system of monotone variational inclusion problems (1.11), respectively, to the problem of finding a common solution of the system of monotone variational inclusion problems (1.11) and the fixed-point problem of quasipseudocontractions. (ii) While Yao et al. [48] were only able to prove a weak-convergence result, in this paper we established a strong-convergence result for our proposed algorithm. (iii) The proposed method of Chang et al. [16] requires knowledge of the operator norm for its implementation, while our proposed method is independent of the operator norm. (iv) Our method employs a very efficient inertial technique that does not require stringent conditions, like one has in condition (iii) of Algorithm 1 of Yao et al. [48] and condition (iv) of Algorithm (1.13) of Chang et al. [16]. (v) The viscosity technique we employed accommodates a larger class of contractions than the one employed by Chang et al. [16].
Remark 4.6 Since the class of quasipseudocontractions contains several other classes of nonlinear mappings such as the pseudocontractions, the demicontractive operators, the quasinonexpansive operators, the directed operators, and the strictly pseudocontractive mappings with fixed points as special cases, our results present a unified framework for studying these classes of operators.

Applications
In this section we consider some applications of our results to approximating solutions of related optimization problems in the framework of Hilbert spaces.

System of equilibrium problems with fixed-point constraint
Let C be a nonempty closed convex subset of a real Hilbert space H, and let F : C × C → R be a bifunction. The equilibrium problem (EP) for the bifunction F on C is to find a point x * ∈ C such that F x * , y ≥ 0, ∀y ∈ C.
We denote the solution of the EP (5.1) by EP(F). The EP serves as a unifying framework for several mathematical problems, such as variational inequality problems, minimization problems, complementarity problems, saddle-point problems, mathematical programming problems, Nash-equilibrium problems in noncooperative games, and others; see [2,23,28,29,35] and the references therein. Several problems in economics, physics, and optimization can be formulated as finding a solution of EP (5.1).
In solving the EP (5.1), we assume that the bifunction F : C × C → R satisfies the following conditions: (A1) F(x, x) = 0 for all x ∈ C; (A2) F is monotone, that is, F(x, y) + F(y, x) ≤ 0 for all x, y ∈ C; (A3) F is upper hemicontinuous, that is, for all x, y, z ∈ C, lim t↓0 F(tz + (1t)x, y) ≤ F(x, y); (A4) for each x ∈ C, y → F(x, y) is convex and lower semicontinuous. The following theorem is required in establishing our next result.
Then, the following hold: where T F r is the resolvent of A F and is given by Here, we consider the following system of equilibrium problems (SEPs) with fixed-point constraint: . . . , m; and y * = Ax * solves G j (y * , y) ≥ 0, ∀y ∈ Q, j = 1, 2, . . . , k, where C and Q are nonempty closed convex subsets of real Hilbert spaces H 1 and H 2 , respectively, S : H 1 → H 1 is a quasipseudocontractive mapping, for each i = 1, 2, . . . , m and j = 1, 2, . . . , k, F i and G j are bifunctions satisfying conditions (A1)-(A4) above, and A : H 1 → H 2 is a bounded linear operator. We denote the solution set of problem (5.2) by ). Now, taking B i = A F i , i = 1, 2, . . . , m and D j = H G j , j = 1, 2, . . . , k and setting h i = g j = 0 in Theorem 4.4, we obtain the following result for approximating solutions of problem (5.2) in Hilbert spaces. 1 and H 2 , respectively, A : H 1 → H 2 be a bounded linear operator with adjoint A * , and for each i = 1, 2, . . . , m and j = 1, 2, . . . , k let F i : C × C → R and G j : Q × Q → R be bifunctions satisfying conditions (A1)-(A4). Let S : H 1 → H 1 be a K -Lipschitz continuous quasipseudocontractive mapping, which is demiclosed at zero and with K ≥ 1, and f : H 1 → H 1 be a contraction with coefficient ρ ∈ (0, 1). Suppose that the solution set = SEP ∩ F(S) = ∅, and conditions (C1)-(C4) are satisfied. Then, the sequence {x n } generated by the following algorithm converges strongly to a pointx ∈ , wherex = P • f (x).

System of convex minimization problems with fixed-point constraint
Suppose that F : H → R is a convex and differentiable function, and M : H → (-∞, +∞] is a proper convex and lower semicontinuous function. It is known that if F is 1 μ -Lipschitz

Algorithm 3
Step 0. Let x 0 , x 1 ∈ H 1 be two arbitrary initial points and set n = 1.
Set n := n + 1 and return to Step 1.
Using MATLAB 2019(b), we compare the performance of Algorithm 2 with Algorithm (1.13). The stopping criterion used for our computation is x n+1x n < 10 -3 . We plot the graphs of errors against the number of iterations in each case. The numerical results are reported in Fig. 2 and Table 2.