An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces

1 Introduction

Let E 1 and E 2 be p -uniformly convex real Banach spaces which are also uniformly smooth. Let B 1 : E 1 → 2 E 1 * , B 2 : E 2 → 2 E 2 * be maximal monotone mappings and A : E 1 → E 2 be a bounded linear mapping. In this article, we consider the split variational inclusion problem (SVIP) in real Banach spaces, which is to find x * ∈ E 1 such that

and

using the Bregman distance technique. We denote the solution set of (1) and (2) by Γ , that is,

The SVIP can be reduced to numerous problems such as convex minimization problems, split variational inequality problems, split zero problems, split equilibrium problems, split feasibility problems for modeling the intensity-modulated radiation therapy (IMRT) treatment planning and many constrained optimization problems [1,2]. Moreover, SVIP has also been used for applications in signal processing, data compression, image reconstruction, resolution enhancement, and sensor networks; for further examples, see [3,4,22].

Now, we present the following useful notions for solving the SVIP:

1. Let E * be the dual space of E , denote the value of x * ∈ E * at x ∈ E by ⟨ x * , x ⟩ .

2. Let B : E → 2 E * be a set-valued mapping, then the domain of B is defined as

   dom ( B ) = { x ∈ E : B x ≠ ∅ } ,

   where the graph of B is given as

   G ( B ) = { ( x , x * ) ∈ E × E * : x * ∈ B x } .

3. A set-valued mapping B is said to be monotone if:

   ⟨ x * − y * , x − y ⟩ ≥ 0 whenever ( x , x * ) , ( y , y * ) ∈ G ( B )

   and B is said to be maximal monotone if its graph is not contained in the graph of any other monotone operator on E ; thus, the set:

   B − 1 ( 0 ) = { x ¯ ∈ E : 0 ∈ B ( x ¯ ) }

   is closed and convex.

4. Resolvent of B is the operator Res p λ B : E → 2 E defined by

   (3) Res p λ B = ( J p E + λ B ) − 1 ∘ J p E , λ > 0 .

   The resolvent operator Res p λ B is a Bregman firmly nonexpansive operator and 0 ∈ B ( x ) if and only if x = Res p λ B ( x ) [5].

If the set-valued mapping B is defined by the following:

then the variational inclusion problem is equivalent to solving the variational inequalities, which is to find x * ∈ C such that ⟨ f ( x * ) , x − x * ⟩ ≥ 0 ∀ x ∈ C .

Many authors have proposed various iterative methods for solving the SVIP and split variational inequality problems in real Hilbert spaces. Censor et al. [6] first introduced the following algorithm for solving the split variational inequality problem in real Hilbert spaces, for x 1 ∈ H 1 , the sequence { x n } is generated by:

where γ ∈ ( 0 , 1 L ) and L is the spectral radius of the operator A * A . The authors proved a weak convergence result for the sequence generated by (5). In 2002, Byrne [3] obtained a weak convergence theorem for solving SVIP in real Hilbert spaces, using the following introduced algorithm: for a given x 0 ∈ H 1 the sequence { x n } is generated by:

where A * is the adjoint of A , the stepsize γ ∈ ( 0 , 2 L ) with L = ‖ A * A ‖ . Recently, after Byrne’s algorithm, Chuang [7] was motivated to introduce an algorithm that is a modification of (6) for solving SVIP; the iterative steps are outlined as follows:

where β n = ⟨ x n − y n , D ( x n , y n ) ⟩ ‖ D ( x n , y n ) ‖ 2 . Moreover, inspired by the work of Byrne et al., Kazmi and Rizvi [8] proposed an algorithm for approximating a common solution of SVIP, whereby the two sequences { u n } and { x n } generated by the algorithm were both proved to converge strongly to z ∈ F ( S ) ∩ Γ , where Γ is the solution set of SVIP and F ( S ) is the fixed point of a nonexpansive mapping S . The algorithm is presented as follows: For a given x 0 ∈ H 1 , let the sequences { u n } and { x n } be generated by:

Wen and Chen [9] introduced a modified general iterative method for solving SVIP and nonexpansive semigroups that are defined as follows:

where γ , α n ∈ [ 0 , 1 ] and B is a strongly bounded linear operator on H 1 and a strong convergence theorem was proved. Recently, Alofi et al. [10] extended the study of SVIP from real Hilbert space to Banach spaces. The authors proposed the following algorithm for solving SVIP between the two spaces, namely, Hilbert and Banach spaces:

where J E is the duality mapping on a Banach space, { u n } is a sequence in Hilbert space such that u n → u and the step size λ n satisfies 0 < λ n L < 2 . Also, Suantai et al. [11] introduced a viscosity modification in Banach spaces presented as follows:

where 0 < λ n L < 2 and f is a contraction. Alvarez [12], and Alvarez and Attouch [13] were motivated by the second-order time dynamical system, the heavy ball method, to introduce an inertial term that significantly updates some previous algorithms for generating the next algorithm. Furthermore, Tang [14] introduced an algorithm with an inertial term for solving SVIP in Banach spaces:

where { θ n } is in ( 0 , θ n ¯ ) and ∑ n = 1 ∞ ε n < ∞ ,

The study of the inertial scheme has shown its importance in continuous optimization due to its good convergence properties in that domain.

Motivated by the above work, we introduce a new inertial self-adaptive projection algorithm for solving problems (1) and (2) in p -uniformly convex and uniformly smooth real Banach spaces. We highlight our contributions in this article as follows:

1. The SVIP is studied in p-uniformly convex and uniformly smooth real Banach spaces, which is more general than the real Hilbert spaces and 2-uniformly convex natural Banach spaces. This extends the results of [8,11,14] to mention a few.

2. The stepsize of our proposed algorithm is determined by a self-adaptive process that is more efficient and applicable than the methods used in [15,16].

3. We used an inertial technique to accelerate the proposed algorithm’s convergence rate and compare its performance with other methods in the literature.

The rest of the article is organized as follows: Section 2 presents some essential notions and preliminary results needed in the form. In Section 3, we offer our iterative algorithm and its convergence analysis. In Section 4, we present some numerical experiments to illustrate the performance of the proposed algorithm. Finally, in Section 5, we offer the conclusion of the article.

2 Preliminaries

In this section, we present some important and useful definitions together with lemmas used in this article. Let E be a real Banach space with the norm ‖ ⋅ ‖ and E * be the dual with the norm ‖ ⋅ ‖ * . We denote the strong and weak convergence of a sequence { x n } ⊂ E to x ∈ E by x n → x and x n ⇀ x , respectively.

The normalized duality mapping J : E → 2 E * is defined by

for all x ∈ E . Let U = { x ∈ E : ‖ x ‖ = 1 } , E is said to be smooth if the limit

exists for all x , y ∈ U . The modulus of smoothness of E is the function ρ E : [ 0 , ∞ ) → [ 0 , ∞ ) defined by

E is called uniformly smooth if lim τ → 0 + ρ E ( τ ) τ = 0 and q-uniformly smooth if there exists C q > 0 such that ρ E ( τ ) ≤ C q τ q . Every uniformly smooth Banach space is smooth and reflexive.

E is said to be strictly convex if for all x , y ∈ E , x ≠ y , ‖ x ‖ = ‖ y ‖ = 1 , we have ‖ λ x + ( 1 − λ ) y ‖ < 1 ∀ λ ∈ ( 0 , 1 ) . Let 1 < q ≤ 2 ≤ p < ∞ with 1 p + 1 q = 1 . The modulus of convexity of E is the function δ E : ( 0 , 2 ] → [ 0 , 1 ] defined by

E is said to be uniformly convex if δ E ( ε ) > 0 and p-uniformly convex if there exists a constant C p > 0 such that δ E ( ε ) ≥ C p ε , for any ε ∈ ( 0 , 2 ] . The L p space is 2-uniformly convex for 1 < p ≤ 2 and p -uniformly convex for p ≥ 2 . It is known that every uniformly convex Banach space is strictly convex and reflexive; more examples can be found in [17].

The generalized duality mapping J p E : E → 2 E * is defined by

If p = 2 , (16) becomes the normalized duality mapping (14). It is known that J p E ( x ) = ‖ x ‖ p − 2 J ( x ) for all x ∈ E , x ≠ 0 . It is well known that E is uniformly smooth if and only if J p E is norm-to-norm uniformly continuous on bounded subsets of E and E is smooth if and only if J p E is single-valued. Furthermore, E is p -uniformly convex (rep. smooth) if and only if E * is q -uniformly smooth (rep. convex). Also, if E is p -uniformly convex and uniformly smooth, then the duality mapping J p E is norm-to-norm uniformly continuous on bounded subsets of E [18,19]. Examples of generalized duality mapping can be found in, for instance, [20,21]. The following lemma was proved by Xu and Roach [19].

Let x ∈ i n t d o m f . For any y ∈ E , the right-hand derivative of f at x denoted by f 0 ( x , y ) is defined by:

If the limit as τ → 0 in (17) exists for any y , then the function f is said to be Gâteaux differentiable at x . Thus, the gradient of f at x is the function ∇ f ( x ) , which is defined by ⟨ ∇ f ( x ) , y ⟩ = f 0 ( x , y ) for any y ∈ E . Therefore, the function f is said to be Gâteaux differentiable if it is Gâteaux differentiable for any x ∈ intdomf [22].

Given a Gâteaux differentiable function f , the bifunction Δ f : E × E → [ 0 , + ∞ ) given as:

is called the Bregman distance with respect to f [23]. Moreover, let f ( x ) = 1 p ‖ x ‖ p , then, the duality mapping J p E is the derivative of f . The Bregman distance with respect to p that is Δ p : E × E → [ 0 , + ∞ ) is defined by:

Note that Δ p ( x , y ) ≥ 0 and Δ p ( x , y ) = 0 if and only if x = y . Using (19), the three-point identity equality is given by:

Furthermore,

For p -uniformly convex space, the metric and Bregman distance satisfy the following [24]:

where τ > 0 is some fixed number. Lets say, if f ( x ) = ‖ x ‖ 2 , the Bregman distance is the Lyapunov functional ϕ : E × E → [ 0 , + ∞ ) defined by:

Let E be a uniformly convex Banach space with the Gâteaux differentiable norm and A : E → 2 E * be a maximal monotone operator, for more information see [25]. The role of the resolvent Res p λ A : E → 2 E defined in (3) is of importance in the approximation theory of zero points of maximal monotone operators in Banach spaces. It is well known that the resolvent operator satisfies the following properties, see, e.g., [26–28]:

Thus, if E is a real Hilbert space, then

where J λ A = ( I − λ A ) − 1 is the general resolvent and A − 1 ( 0 ) = { z ∈ E : 0 ∈ A z } .

Let C be a nonempty, closed, and convex subset of E . The metric projection is defined as:

The metric projection is the unique minimizer of the distance ([29]) and it is also characterize by the following variational inequality:

Similarly, the Bregman projection:

is the unique minimizer of the Bregman distance (see [29]). The variational inequality can also characterize it:

from which one can derive that

The metric projection generally differs from the Bregman projection, but in Hilbert spaces, both projections coincide.

Associated with the Bregman distance f p is the functional V p : E × E * → [ 0 , + ∞ ) defined by:

We can see that V p ( x , x ¯ ) ≥ 0 and the following properties are satisfied:

and

Also, V p is convex in the second variable. Then for all z ∈ E ,

where { x i } ⊂ E and { t i } ⊂ ( 0 , 1 ) with ∑ i = 1 N t i = 1 .

Let C be a convex subset of intdomf p , where f p = ( 1 p ) ‖ x ‖ p , 2 ≤ p < ∞ , thus an asymptotic fixed point of T is a point p ∈ C such that if C contains a sequence { x n } which converges weakly to p and

F ˆ ( T ) denotes the set of asymptotic fixed points of T . Then a point p ∈ C is called a strong asymptotic fixed point of T if C contains a sequence { x n } which converges strongly to p and

F ˜ ( T ) denotes the set of strong asymptotic fixed points of T . Thus, from the above definitions, we deduce that F ( T ) ⊂ F ˜ ( T ) ⊂ F ˆ ( T ) , the introduction of these asymptotic fixed points was studied in the previous study [30].

From Definition (2.3), we have noted that the class of Bregman quasi-nonexpansive contains the class of Bregman weak relatively nonexpansive, and finally the class of Bregman weak relatively nonexpansive contains the class of Bregman relatively nonexpansive, more on Bregman relatively nonexpansive mappings can be found in [31].

The following results are necessary for establishing our main results.

3 Main results

In this section, we present our algorithm and the convergence analysis.