Generalized split null point of sum of monotone operators in Hilbert spaces


 In this paper, we introduce a new type of a generalized split monotone variational inclusion (GSMVI) problem in the framework of real Hilbert spaces. By incorporating an inertial extrapolation method and an Halpern iterative technique, we establish a strong convergence result for approximating a solution of GSMVI and fixed point problems of certain nonlinear mappings in the framework of real Hilbert spaces. Many existing results are derived as corollaries to our main result. Furthermore, we present a numerical example to support our main result and propose an open problem for interested researchers in this area. The result obtained in this paper improves and generalizes many existing results in the literature.


Introduction
The concept of split feasibility problem (SFP) introduced by Censor and Elfving [1] in the framework of finite-dimensional Hilbert spaces is to find where C and D are nonempty, closed and convex subsets of real Hilbert spaces H 1 and H 2 , respectively, and → H H : 1 2 is a bounded linear operator. The SFP finds its applications in image recovery, signal processing, control theory, data compression, computer tomography and so on (see [2][3][4][5]). Due to this advantage, a lot of researchers have studied the SFP in various abstract spaces (see [6][7][8][9][10][11]). In 2009, Censor and Segal [12] further extended the notion of SFP by introducing the concept of split common fixed point problem (SCFPP) and found where ( ) F T , ( ) F S denote the set of fixed points of two nonlinear operators → T C C : and → S D D : , respectively, and → H H : 1 2 is a bounded linear operator. For surveys on methods for approximating the solutions of SFP, [13][14][15] and references therein.
The variational inclusion problems (VIPs) are being used as mathematical models for the study of several optimization problems arising in finance, economics, network, transportation, science and engineering. For a real Hilbert space H , the VIP consists of finding a point ∈ * x H such that where → B H : 2 H is a multivalued (point-to-set) operator. Whenever B is a maximal monotone operator, such element ∈ * x H is called the zero of the maximal monotone operator B. The VIP for monotone operators was introduced by Martinet [16]. For approximating the zeroes of (3), the relation generated by the fixed equation = * * Byrne et al. in [17] combined the concept of the VIP and SFP to introduce the split null point problem (SNPP). The SNPP is given as the problem of finding and established that { } x n converges weakly to a point * x in the solution set of (4).
The SNPP has been considered by many authors (see [18][19][20][21] and references therein). A generalization of the VIP (3) is a problem of finding an element ∈ * x H such that where is a single-valued operator and → B H : 2 H is a multivalued operator. In the case where A and B are monotone operators, the elements in the solution set of (6) are called the zeros of the sum of monotone operators. We note that the solution of (6) is the fixed points of the operator − ( ) J I λA λ B , when > λ 0 (see [22]). Several authors have employed different types of iterative methods for approximating the solution of (6) (see for example [23,24] and references therein).
Question: Can further generalize the SMVI and propose a natural modification of an iterative scheme to obtain a strong convergence result for this type of generalization?
On the other hand, the inertial extrapolation method has proven to be an effective way for accelerating the rate of convergence of iterative algorithms. The technique was introduced in 1964 based on a discrete version of a second-order dissipative dynamical system (see [26,27]). The inertial-type algorithm uses its two previous iterates to obtain its next iterate (see [28,29]). In [30], Moudafi and Oliny proposed the following inertial proximal point algorithm for finding the zero of sum of two maximal monotone operators: . Also, Lorenz and Pock [24] introduced a modified forwardbackward splitting algorithm with inertial term. They define the algorithm as follows: , n n n n n n n n n M is a linear self adjoint positive definite map and > λ 0 n is a step size parameter. They proved that the sequence { } x n generated by (10) converges weakly to the zero of + A B. Motivated by the results discussed above, we study a generalized split monotone variational inclusion (GSMVI) and then introduce an iterative method for approximating the solution of GSMVI and fixed point problem of the composition of two nonlinear mappings in the framework of real Hilbert spaces. We display a numerical example to show the behavior of our result.
The GSMVI 11 is much more general as it includes SFP, SNPP, SMVI as special cases and thus has more reallife applications. In addition, the GSMVI problem has wide applications in many fields such as machine learning, statistical regression, image processing and signal recovery (see [5,31,32] and references therein).
In this paper, we consider the problem of finding a common element in the solution set of the GSMVI and fixed point of a composition of two mappings T 1 and T 2 , where → T C C : 1 is an α-strongly quasinonexpansive mapping, → T C C : 2 is a firmly nonexpansive mapping and C is a nonempty, closed and convex subset of H 1 . That is, find ∈ * x C such that , such that 0 , such that 0 , such that 0 . As far as we are concerned, the GSMVI is new and has yet to be considered in any of the recent or previous research papers in this direction. For obtaining the solution of (12), we propose a modified Halpern iterative algorithm together with an inertial term and prove a strong convergence result for approximating the solution of Γ. The result in this paper generalizes, unifies and extends many related results in the literature.

Preliminaries
In this section, we begin by recalling some known and useful results which are needed in the sequel.
Let H be a real Hilbert space. The set of fixed point of T will be denoted by ( ) . We denote strong and weak convergence by "→" and " ⇀ ," respectively. For any ∈ x y H , and ∈ [ ] α 0, 1 , it is well known that (13) be an operator. Then the operator T is called It is well known that for any nonexpansive mapping T , the set of fixed point is closed and convex. Also, T satisfies the following inequality: where > λ 0 and I is the identity operator on H .
2 H be a set-valued maximal monotone mapping and > λ 0, then J λ B is singlevalued and a firmly nonexpansive mapping.
Then, the following hold: Let H be a real Hilbert space and C a nonempty, closed and convex subset of H . For any ∈ u H, there exists a unique point C P C is called the metric projection of H onto C. It is well known that P C is a nonexpansive mapping and that P C satisfies for all ∈ x H and ∈ y C. Lemma 3.1. Let H be a real Hilbert space and C be a nonempty closed and convex subset of H . Let → T C C : 1 be an α-strongly quasi-nonexpansive and → T C C : 2 be a firmly nonexpansive mapping, such that Proof. It is required that we show that Also, using (19), we have (20) . Using this fact, we have that is a quasi-nonexpansive mapping.
Proof. For all ≥ ∈ μ λ x y H , , , we have established the aforementioned results.
1. Using (18), we have Using the property of a monotone operator, we have that Proof. It is easy to see that . We now establish that Using Lemma 3.3( ) 4 and (22), we have . Using a similar approach to that in (4)  be arbitrary.
Iterative step: Step 1: Given the iterates − x n 1 and x n for all ∈ n , choose θ n such that ≤ ≤ θ θ 0n n , where  Step 3: Compute , then stop, otherwise, set ≔ + n n 1 and go back to Step 1.
Remark 3.6. The following are some of the highlights of our method: where Also, using Algorithm 3, we have Similarly, using Algorithm 3, we have Finally, using Algorithm 3 and (24), we have k n n k n n n n Now, using Algorithm 3, we have        be the space of function with norm and inner product defined, respectively, by ( )

Generalized split  373
In this work, we introduce and study a new type of GSMVI and established a strong convergence result for approximating a solution of GSMVI in the framework of real Hilbert spaces using an inertial extrapolation term and Halpern iterative technique. We have to only consider this problem for = N 3 in the framework of Hilbert spaces. It is therefore left open for interested researchers in this area of research to extend the concept to more general spaces and also consider the case when ≥ N 4. In addition, the authors in [38] introduced the notion of finding the zero of the sum of three monotone operators in the framework of Hilbert spaces. It is natural to ask, if the present results in this work can be extended to three monotone operators. Abbreviations SPF split feasibility problem SCFPP split common fixed point problem VIP variational inclusion problem SNPP split null point problem SMVI split monotone variational inclusion problem GSMVI generalized split monotone variational inclusion problem