Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces

1 Introduction

Let X be a vector space and let A be a multivalued operator on X . The zero point problem of A is defined as:

The set of all zero points of A is denoted by A − 1 ( 0 ) and defined as A − 1 ( 0 ) = { x ∈ X : 0 ∈ A x } .

Recently, the split inverse problem (SIP) has attracted the attention of several authors due to its wide areas of applications, for example, in phase retrieval, image recovery, signal processing, data compression, intensity-modulated radiation therapy, among others (see [1,2] and references therein). The SIP model is formulated as follows: find a point

such that

where X and Y are two vector spaces, T : X → Y is a linear operator, IP 1 and IP 2 denote inverse problems in X and Y , respectively. In 1994, Censor and Elfving [2] introduced the first instance of the SIP known as the split feasibility problem (SFP). The SFP is defined as: Find

where D and Q are nonempty, closed, and convex subsets of real Hilbert spaces ℋ 1 and ℋ 2 , respectively, and T : ℋ 1 → ℋ 2 is a bounded linear operator. It is known that the SFP has wide applications in many fields such as signal processing, treatment planning, phase retrieval, among others (see [3,4,5] and other references therein).

In 2011, Moudafi [6] introduced another instance of the SIP called the split monotone variational inclusion problem (SMVIP). The SMVIP is formulated as follows:

where B 1 : ℋ 1 → 2 ℋ 1 and B 2 : ℋ 2 → 2 ℋ 2 are multivalued maximal monotone mappings. For more details on SMVIP the reader can see [7] and other references therein.

If f 1 = f 2 = 0 , then the problem (1) reduces to the split variational inclusion problem (SVIP) defined as follows: Find x ∗ ∈ ℋ 1 such that

and

where 0 is the zero vector, ℋ 1 and ℋ 2 are real Hilbert spaces, B 1 : ℋ 1 → 2 ℋ 1 and B 2 : ℋ 2 → 2 H 2 are multi-valued mappings, A : H 1 → H 2 is a bounded linear operator. We denote the solution set of SVIP (2) by SOLVIP ( B 1 ) and the solution set of SVIP (3) by SOLVIP ( B 1 ) . Hence, we denote the solution set of SVIP by Ω = { x ∗ ∈ ℋ 1 : x ∗ ∈ SOLVIP ( B 1 ) and A x ∗ ∈ SOLVIP ( B 2 ) } . The SVIP is also known as the split null point problem or the split zero point problem (see [8,9,10, 11,12,13, 14,15]). It includes as special cases, the split fixed point problem (FPP), the split variational inequality problem and the SFP (see [16,17,18] and other references therein), and it has wide applications in different fields such as medical treatment of the intensity-modulated radiation therapy, data compression, medical image reconstruction, signal processing, phase retrieval, among others (e.g., see [2,3,19]). The SVIP is considered to be a central problem in optimization and nonlinear analysis and has attracted the attention of several researchers because the theory provides a simple, natural and unified framework for a general treatment of many important mathematical problems such as minimization problems, network equilibrium problems, complementary problems, systems of nonlinear equations and others (see [19,20, 21,22,23, 24,25,26, 27,28] and other references therein).

Byrne et al. [29] proposed and studied the following algorithms for solving the SVIP for two maximal monotone operators A and B in Hilbert spaces. Take x 0 ∈ ℋ 1 such that

and

for μ > 0 , T ∗ the adjoint of T , λ ∈ 0 , 2 L , L = ‖ T ∗ T ‖ and J μ A ≔ ( I + μ A ) − 1 , J μ B ≔ ( I + μ B ) − 1 are the resolvent operators of A and B , respectively. Under certain conditions, the authors obtained a weak convergence result for Algorithm 4 and a strong convergence result for Algorithm 5.

Moudafi [30] first introduced the viscosity approximation method, which is defined as : Choose x 0 ∈ ℋ such that the sequence { x n } is generated by

where { α n } ⊂ ( 0 , 1 ) and f is a contraction mapping. He proved that the sequence { x n } generated by (6) converges strongly to a fixed point of a nonexpansive mapping S under some suitable control conditions.

Very recently, Suantai et al. [31] proposed the following viscosity iterative scheme to approximate the solution of SVIP between a Banach space X and a Hilbert space ℋ :

where { μ n } , { λ n } ⊂ ( 0 , ∞ ) , { α n } , { β n } , { γ n } ⊂ ( 0 , 1 ) and α n + β n + γ n = 1 , f : ℋ → ℋ is a contraction, J X is the duality mapping on X , J λ A is the resolvent of A for λ > 0 , Q μ B is the metric resolvent of B for μ > 0 . They proved that the sequence { x n } defined by (7) converges strongly to a solution of the SVIP under the following conditions for some a , b , c , d , k ∈ R + :

1. 0 < a ≤ λ n ‖ T ‖ 2 ≤ b < 2 ;

2. 0 < k ≤ μ n ;

3. 0 < c ≤ γ n ≤ d < 1 ;

4. lim n → ∞ α n = 0 and ∑ n = 1 ∞ α n = ∞ .

The FPP is another related problem, which is defined as follows:

where C is a nonempty closed convex subset of a Hilbert space ℋ and S : ℋ → ℋ is a nonlinear mapping. We denote by F ( S ) the fixed point set of S , that is

Many problems in science and engineering can be formulated as a problem of finding the solution of an FPP of an nonlinear mapping.

Motivated by the work of Byrne et al. [32], Kazmi and Rizvi [33] proposed the following algorithm for approximating a solution of SVIP, which is also a fixed point of nonexpansive mapping.

All the above algorithms for solving SVIP have a common feature, which is also their computational weakness. It is the fact that their step size λ (or λ n ) depends on the norm of the operator ‖ T ‖ , which in most cases is unknown or very difficult to calculate or even estimate. This is a major drawback with the above algorithms and several existing algorithms in the literature.

Let ℋ be a real Hilbert space and X be a uniformly convex and smooth Banach space. Let A i : ℋ → ℋ be a finite family of α i -inversely strongly monotone operators, and let B i : ℋ → 2 ℋ and C i : X → 2 X ∗ be finite families of maximal monotone operators for each i = 1 , 2 , … , m . Let S : ℋ → ℋ be a nonexpansive mapping and let T : ℋ → X be a bounded linear operator. In this paper, we study the following common solution problem of finite family of SMVIP and FPP of nonexpansive mapping:

and

We denote the solution set of problems (11)–(12) by

To accelerate the rate of convergence of iterative methods, authors often employ the inertial technique. Polyak [34] studied the convergence of the following inertial extrapolation algorithm:

where { α n } and { β n } are two real sequences. Recently, there has been an increased interest in studying inertial-type algorithm (see [35,36, 37,38,39, 40,41] and other references therein).

Very recently, Long et al. [42] introduced a new algorithm, which combines the inertial technique and the viscosity method for solving the SVIP in Hilbert spaces. They proved that the sequence { x n } generated by the algorithm converges strongly to an element in the solution set of the SVIP.

Motivated by the aforementioned works and the ongoing research activities in this direction, we propose and study a new inertial iterative algorithm with self-adaptive step size for approximating a common solution of finite family of SMVIPs and FPP for a nonexpansive mapping between a Banach space and a Hilbert space. This method combines the inertial technique with viscosity method for solving the common solution problem without a prior knowledge of the operator norm. We prove a strong convergence result for the sequence generated by our proposed method under some mild conditions and apply our results to study split feasibility problem and split minimization problem. Finally, we provide numerical experiments for the proposed method and compare the performance of our algorithm with some existing methods in the literature.

The remaining section of the paper is organized as follows. Section 2 contains basic definitions and results needed in subsequent sections. In Section 3, we present the proposed method. We investigate the convergence of the method in Section 4. The application of our proposed method is given in Section 5, while in Section 6 we perform some numerical experiments and compare the performance of our method with other methods in the literature. We then conclude in Section 7.

2 Preliminaries

Let C be a nonempty, closed and convex subset of a real Hilbert space ℋ with inner product ⟨ ⋅ , ⋅ ⟩ and norm ‖ ⋅ ‖ . We will denote the weak and strong convergence of a sequence { x n } to a point x as x n ⇀ x and x n → x , respectively. Let X be a real Banach space with norm ‖ ⋅ ‖ and let X ∗ be the dual space of X . We denote the value of y ∗ ∈ X ∗ at x ∈ X by ⟨ x , y ∗ ⟩ . Also, we denote the set of weak limits of { x n } by w ω ( x n ) , that is

The modulus of convexity δ X : [ 0 , 2 ] → [ 0 , 1 ] is defined as

A Banach space X is called uniformly convex if δ X ( ε ) > 0 for any ε ∈ ( 0 , 2 ] , and X is said to be strictly convex if for x , y ∈ S ( X ) ≔ { x ∈ X : ‖ x ‖ = 1 } and x ≠ y , one has ‖ x + y ‖ 2 < 1 . It is well known that every uniformly convex Banach space is reflexive and strictly convex and every Hilbert space is a uniformly convex space. The modulus of smoothness ρ X ( σ ) : [ 0 , ∞ ) → [ 0 , ∞ ) is defined by

Then X is called uniformly smooth if lim σ → 0 ρ X ( σ ) σ = 0 and q-uniformly smooth if there is a constant C q > 0 such that ρ X ( σ ) ≤ C q σ q for any σ > 0 . It is known that X is p -uniformly convex if and only if its dual X ∗ is q -uniformly smooth.

The normalized duality mapping J X : X → 2 X ∗ is defined by

for every x ∈ X . If X is a real Hilbert space, then J X = I , where I is the identity mapping and if the Banach space X is smooth, then J X is a single valued mapping of X into X ∗ . Also, if J X is surjective, X is reflexive while X is strictly convex if and only if J X is one-to-one (see [26]).

The norm of X is said to be Gâteaux differentiable if for each x , y ∈ X such that ‖ x ‖ = ‖ y ‖ = 1 , the limit

exists. In this case, X is called smooth. Let X be a uniformly convex and smooth Banach space with a Gâteaux differential norm and let B : X → 2 X ∗ be a maximal monotone operator. We consider the metric resolvent of B ,

It is known that the operator Q μ B is firmly nonexpansive and the fixed points of the operator Q μ B are the null points of B (see [43,44]). The resolvent plays an important role in the approximation theory for zero points of maximal monotone operators in Banach spaces.

The following are the properties of the resolvent (see [45])

in particular, if X is a real Hilbert space, then

where J μ B = ( I + μ B ) − 1 is the general resolvent, B − 1 ( 0 ) = { z ∈ X : 0 ∈ B z } is the set of null points of B . Also, we know that B − 1 ( 0 ) is closed and convex (see [46]).

Recall that for a nonempty closed and convex subset C of ℋ , the metric projection denoted by P C , is a map defined on ℋ onto C , which assigns to each x ∈ ℋ a unique point in C , denoted by P C x such that

We have the following result on the metric projection map.

3 Proposed methods

Let ℋ be a real Hilbert space and let X be a uniformly convex and smooth Banach space. Let J X be the duality mapping on X . Let T : ℋ → X be a bounded linear operator such that T ≠ 0 and T ∗ is the adjoint operator of T . Let f : ℋ → ℋ be a contraction mapping with constant ρ ∈ [ 0 , 1 ) and 0 < γ < γ ρ , and let S : ℋ → ℋ be a nonexpansive mapping. For each i = 1 , 2 , … , m , let A i : ℋ → ℋ be a finite family of α i -inverse strongly monotone operators, B i : ℋ → 2 ℋ and C i : X → 2 X ∗ be finite families of maximal monotone operators, and let J r i B i be the resolvent of B i for r i > 0 and Q μ i C i be the metric resolvent of C i for μ i > 0 . Suppose that the solution set Γ ≠ ∅ . We establish the convergence of the algorithm under the following assumptions on the control parameters:

1. { α n } , { δ n } , { γ n } ⊂ ( 0 , 1 ) such that α n + δ n + γ n = 1 , lim n → ∞ α n = 0 , ∑ n = 0 ∞ α n = ∞ ;

2. lim inf n → ∞ δ n > 0 , lim inf n → ∞ γ n > 0 , { β n , i } ⊂ ( 0 , 1 ) and ∑ i = 0 m β n , i = 1 , lim inf n → ∞ β n , 0 β n , i > 0 ;

3. 0 < a ≤ τ n ≤ b < 2 , 0 < c ≤ μ n , i , 0 < d ≤ r n , i ≤ e < 2 α i for each i = 1 , 2 , … , m ;

4. Let θ > 0 and { ε n } be a nonnegative sequence such that 0 < d ≤ ε n and ε n = ∘ ( α n ) , i.e., lim n → ∞ ε n α n = 0 .

Next, we show that the sequence of step size { λ n , i } of the algorithm defined by (16) is well defined.

4 Convergence analysis

Let p ∈ Γ . Then, it follows that

for all n ∈ N and i = 1 , 2 , … , m . From the definition of w n in Step 2 and by applying the triangle inequality, we have

Since, by Remark 3.2, lim n → ∞ θ n α n ‖ x n − x n − 1 ‖ = 0 , it follows that there exists a constant M 1 > 0 such that θ n α n ‖ x n − x n − 1 ‖ ≤ M 1 for all n ≥ 1 . Hence, it follows from (18) that

From Step 3 and the property of the resolvent (13), we have