Solving Split Variational Inclusion Problem and Fixed Point Problem for Nonexpansive Semigroup without Prior Knowledge of Operator Norms

We introduce an iterativemethod to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms inHilbert spaces.We prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of commonfixed points of one-parameter nonexpansive semigroups. Moreover, numerical results demonstrate the performance and convergence of our result, whichmay be viewed as a refinement and improvement of the previously known results announced by many other researchers.

Moudafi [1] shows that SMVIP (1) includes, as special cases, the split variational inequality problem, the split common fixed point problem, split zero problem, and split feasibility problem [1][2][3][4][5][6][7] which have already been studied and used in practice as a model in intensity-modulated radiation therapy treatment planning (see [5,6]).This formalism is also at the core of modeling of many inverse problems arising for phase retrieval and other real-world problems, for instance, in sensor networks in computerized tomography and data compression [8,9].
In 2014, Kazmi and Rizvi [10] considered the strong convergence of the following iterative method: where  > 0,  * is the adjoint of ,  is the spectral radius of the operator ‖ * ‖, and  ∈ (0, 1/).They proved the sequence {  } generated by (5) strongly converges to the fixed point of nonexpansive mapping  and the solution set Γ of SVIP ((2)-( 3)).
In 2015, Sitthithakerngkiet et al. [11] proposed the hybrid steepest descent method: where   is a sequence of nonexpansive mappings,  is a strongly positive bounded linear operator,  > 0,  * is the adjoint of ,  is the spectral radius of the operator ‖ * ‖, and  ∈ (0, 1/) and  > 0. They revealed that the sequence {  } converges strongly to a point , where  =  Ω ( −  + )() is a unique solution of the variational inequalities: Note that, in algorithms (4), (5), and (6) mentioned above, the determination of the stepsize  depends on the operator (matrix) norms ‖‖ (or the largest eigenvalues of  * ).This means that, in order to implement algorithms (4), (5), and (6), one has first to compute (or, at least, estimate) operator norms of , which is not an easy work in practice.
To overcome this difficulty, López et al. [12] and Zhao and Yang [13] presented useful method for choosing the stepsizes which do not need prior knowledge of the operator norms for solving the split feasibility problems and multiple-set split feasibility problems, respectively.Motivated by the above results, we introduce a new choice of the stepsize sequence {  } which depends on where   ∈ [, ] ⊂ (0, 1).The advantage of our choice (8) of the stepsizes lies in the fact that no prior information about the operator norms of  is required, and still convergence is guaranteed.
Following the work of Moudafi [1], Kazmi and Rizvi [10], and Byrne et al. [4], we introduce and study an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces.We also prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups.Numerical results are proposed to show that our algorithm is more suitable for SVIP ((2)-( 3)) than the proposed algorithms (4) and (6).

Preliminaries
Throughout this paper, we denote  to be a nonempty closed convex subset of  1 .Let  :  1 →  1 be a mapping.A point  ∈  1 is said to be a fixed point of  provided  = .We use () to denote the fixed point set of .
Before proceeding further, we need to introduce a few concepts.
A monotone mapping  :  1 → 2  1 is called maximal if the graph () of  is not properly contained in the graph of any other monotone mapping.It is well known that a monotone mapping  is maximal if and only if for (, ) ∈  × , ⟨ − ,  − ⟩ ≥ 0 for every (, ) ∈ () implies  ∈ .
Let  :  1 → 2  1 be a multivalued maximal monotone mapping.Then, the resolvent mapping    :  1 →  1 , associated with , is defined by for some  > 0, where  stands for identity operator on  1 .We note that, for all  > 0, the resolvent operator    is singlevalued, nonexpansive, and firmly nonexpansive.
The following principles play an important role in our argument.
A mapping  :  1 →  1 is called demiclosed at the origin if for each sequence {  } which weakly converges to , and the sequence {  } strongly converges to 0, then  = 0.
is said to be semicompact, if, for any bounded sequence To establish our results, we need the following technical lemmas.

Main Results
In this section, we first describe our algorithm and then reveal the convergence analysis of the algorithm.Now, we propose our algorithm.
Remark 9. Notice that in (8)  Next, we will discuss the convergence analysis of algorithm (23) for approximating a common solution of SVIP ((2)-( 3)) and fixed point problem for nonexpansive semigroups.
Proof.Clearly, Theorem 12 is valid for a nonexpansive mapping.Therefore, the desired conclusion follows immediately from Theorem 11.This completes the proof.

Numerical Examples
We now pay our attention to show a numerical example to demonstrate the performance and convergence of our result.In the experiment, the stopping criterion is ‖ * (  2  − )  ‖ ≤ 10 −10 .In [4,11], the stopping criterion is ‖ +1 −   ‖ ≤ 10 −10 .ST denotes the initial point, IT denotes the iterative number, and SOL denotes a solution of the test problem.
We write   ⇀  to indicate that the sequence {  } converges weakly to , and   →  implies that {  } converges strongly to .We use   (  ) = { : ∃   ⇀ } standing for the weak -limit set of {  }.For any  ∈ , there exists a unique nearest point in , denoted by   , such that the choice of the stepsize   is independent of the norms ‖‖.Remark 10.The stepsize {  } is bounded.Indeed, it follows from the condition on {  } that − )        2       * ( * (  2  −)  ̸ =0   < ∞ and {  } is bounded.