An Inertial Iterative Algorithm with Strong Convergence for Solving Modified Split Feasibility Problem in Banach Spaces
In this paper, we propose an iterative scheme for a special split feasibility problem with the maximal monotone operator and fixed-point problem in Banach spaces. The algorithm implements Halpern’s iteration with an inertial technique for the problem. Under some mild assumption of the monotonicity of the related mapping, we establish the strong convergence of the sequence generated by the algorithm which does not require the spectral radius of A. Finally, the numerical example is presented to demonstrate the efficiency of the algorithm.
The split feasibility problem (shortly SFP) introduced by Censor and Elfving  in 1994 can be defined as follows: find a point satisfyingwhere and are nonempty closed-convex subsets of real Hilbert spaces and , respectively, and is a bounded linear operator. The SFP has broad application in modelling real-world problems such as the inverse problem in signal processing, radiotherapy, and data compression ( for example, see [2–4]). Various algorithms have been invented by several authors for solving the SFP and related optimization problems (for example, see [5–10]). The CQ algorithm is one of the most popular solvers for SFP which was first proposed by Byrne , taking an initial point arbitrarily and defining the iterative step aswhere , is the spectral radius of , and and denote the metric projections of onto C and onto Q, respectively, that is, over all . It was proved that the sequence generated by (2) converges weakly to a solution of the SFP provided the step size . As an extension of this CQ algorithm, several iterative algorithms have been invented for solving SFP in Hilbert spaces and Banach spaces ( for example, see [12–17]).
Let be a real Hilbert space, F be a strictly convex, reflexive smooth Banach space, denotes the duality mapping on F, and C and Q be nonempty closed-convex subsets of H and F, respectively. The following Halpern’s type iteration algorithm was proposed by Alsulami and Takahashi  in 2015. Let be a sequence in H such that and :where and . It was proved that the sequence defined by (3) converges strongly to a point ; for some if , , and .
However, the previous algorithms only use the current point to get the next iteration which can lead slow convergence. Inertial technique, as an accelerated method, was first proposed by Polyak  to speed up the convergence rate of smooth convex minimization. Subsequently, F. Alvarez in  combined with a proximal method to solve the problem of finding the zero of a maximal monotone operator. The main idea of this method is to make use of two previous iterates in order to update the next iterate. Due to the fact that the presence of the inertial term in an algorithm speeds up the convergence rate, inertial type algorithms have been widely studied by authors [5, 20, 21].
In this paper, we study the following modified SFP in the real Banach space:
Find such thatwhere is a nonempty, closed-convex subset of , is a Bregman weak relatively nonexpansive mapping, and is a maximal monotone operator. and are p-uniformly convex and uniformly smooth real Banach spaces, and be the duals of and , respectively, be a bounded linear operator, and be the adjoint of . We shall denote the value of the functional at by . Obviously, the modified SFP (4) is more general than (1).
Motivated by the above results, in this paper, we present an inertial algorithm for solving (4) in p-uniformly convex and uniformly smooth Banach spaces which have strong convergence. Our algorithm is designed to employ previous iterations and to obtain the next iterative point; all the implementation process does not compute the spectral radius of , which improves the feasibility of the algorithm.
The paper is organized as follows. Section 2 reviews some preliminaries. Section 3 gives the inertial iterative algorithm and its convergence analysis. Section 4 gives a numerical experiment. Some conclusions are drawn in Section 5.
In this section, we recall some basic definitions and preliminaries’ results which will be useful for our convergence analysis in this paper. We denote the strong and weak convergence of the sequence to a point x by and , respectively.
Let be a real Banach space and and . Define the modulus of smoothness of E aswhere E is uniformly smooth if and only if and E is said to be q-uniformly smooth if there exists a constant such that .
Define the modulus of convexity of E aswhere E is uniformly convex if and only if for all and E is p-uniformly convex if there is a constant such that for all . Every uniformly convex Banach space is strictly convex and reﬂexive. It is known that if E is p-uniformly convex and uniformly smooth, then its dual is q-uniformly smooth and uniformly convex.
Definition 1 (see ). Let . Define the generalized duality mapping asIt is known that when E is uniformly smooth, then is norm-to-norm uniformly continuous on bounded subsets of E, and E is smooth if and only if is single valued. is said to be weak-to-weak continuous if
Lemma 1 (see ). Let . If E is a q-uniformly smooth Banach space, then there exists a such that
Definition 2 (see ). A function is said to be(i)proper if its effective domain is nonempty(ii)convex if (iii)lower semicontinuous at
Definition 3. Let be a differentiable and convex function. The Bregman distance denoted as is defined aswhere is the value of the gradient at .
It is worthy to note that the duality mapping is actually the gradient of the function . Hence, if f = fp in (12), the Bregman distance with respect to fp now becomesIt is generally known that the Bregman distance is not a metric as a result of absence of symmetry, but it possesses some distance-like properties which are stated as follows:The relationship between the metric and Bregman distance in the p-uniformly convex space is as follows:where τ >0 is a fixed number.
Let C be a nonempty closed-convex subset of E. The Bregman projection is defined asAnd, the metric projection can be defined similarly asThe Bregman projection is the unique minimizer of the Bregman distance and can be characterized by a variational inequality :from which we haveThe metric projection which is also the unique minimizer of the norm distance can be characterized by the following variational inequality:We define the functional bywhere . It then follows thatChuasuk et al.  proved the following inequality:Furthermore, Vp is convex in the second variable, and thus, for all , and , we have (see )Let C be a nonempty, closed, and convex subset of a smooth Banach space E, and let . A point is called an asymptotic fixed point of T if a sequence exists in C and converges weakly to such that . We denote the set of all asymptotic fixed points of T by . Moreover, a point is said to be a strong asymptotic fixed point of T if there exists a sequence in C which converges strongly to such that . We denote the set of all strong asymptotic fixed points of T by . It follows from the definitions that .
Definition 4 (see ). Let T be a mapping such that . T is said to be(i)nonexpansive if for each (ii)quasi-nonexpansive if such that , and
Definition 5 (see ). Let be a mapping. T is said to be(1)Bregman nonexpansive if(2)Bregman quasi-nonexpansive if and(3)Bregman weak relatively nonexpansive if , and(4)Bregman relatively nonexpansive if , andFrom the definitions, it is evident that the class of Bregman quasi-nonexpansive maps contains the class of Bregman weak relatively nonexpansive maps. The class of Bregman weak relatively nonexpansive maps contains the class of Bregman relatively nonexpansive maps.
Let E be a smooth, strictly convex, and reﬂexive Banach space and be a maximal monotone operator. We define a mapping by (see )This mapping is known as metric resolvent of A. Obviously, for all , we haveand . Furthermore, for all and by the monotonicity of A, we can show thatFrom (29), we have for all Since A is monotone, we can obtain (30) from (31) and (32). This implies that for all , and whenever , we have
Lemma 2 (see ). Let C be a nonempty, closed, and convex subset of a reﬂexive, strictly convex, and smooth Banach space E, and . Then, the following assertions are equivalent:(1)(2)Furthermore, for , we have
Lemma 3 (see ). Let E be a smooth and uniformly convex real Banach space. Let and be two bounded sequences in E. Then, if and only if
Lemma 4 (see ). Let and be two fixed real numbers, then a Banach space E is uniformly convex if and only if there exists a continuous, strictly, increasing, and convex function such that for all and,where .
3. Inertial Iteration Algorithm and Its Strong Convergence
In this section, we present our inertial iterative algorithm for solving the modified SFP (4) in Banach spaces. We also prove its strong convergence under some suitable conditions.
3.1. Inertial Iteration Algorithm
Now, we give our inertial iterative algorithm.
Algorithm 3.1. Suppose . Let , , , and be a real sequence, and . Assuming have been constructed, we calculate the next iterate via the following formulas:whereWe can see that during the iteration, it does not require to compute the spectral radius of ATA.
Suppose . Now, we prove the following lemmas which will be used to establish the strong convergence.
Proof. Obviously, is a nonempty closed and convex set for . Now, we show that . Let . By (22) and Bregman weak relatively nonexpansive of T, we getFrom (36b), Lemma 1, and the definition of Bregman projection, we havewhere the second inequality is from Lemma 2.
Furthermore, from (32), we haveBy the definitions of , we haveSince , we haveThen,which implies So , for all . Since , then . Ofcourse, for , we also have . It implies that for all . So, we obtain that for all . Therefore, is nonempty, and thus, is well-defined.
Lemma 6. Let be a sequence generated by Algorithm 3.1. Then,(1)(2)(3)(4)(5)
Proof. (1)Let . Since , , and , we have Thus, is bounded. We observe that , and by (17), we have Also, by (18), we have that is, . Thus, ; therefore, is a bounded monotone nondecreasing sequence. Hence, exists. From (46), we have . Thus, using Lemma 2.8, we get(2)By the uniform continuity of on bounded subsets of , from (47) we have From (36a), we obtain Then, which gives Therefore, Since is also uniformly continuous on bounded subsets of , we have(3)From (47) and (53), we obtain Note that, from the construction of , we have that Thus, ; also from Lemma 3, we have Similarly, ; it then follows that It follows from (53) and (57) that Using Lemma 4, from (36c) we have Since T is Bregman weak relatively nonexpansive, we have Hence, from (12) and (13), we get From (58), we have which implies By the property of mapping , we obtain Since is uniformly continuous on bounded subsets of , we have(4)From Algorithm 3.1, we have that Since and from (65), we have Hence, It is easy to obtain(5)From (41), we haveFrom (13) and (16), we getIt follows from (69) thatThen,Now, we present the following strong convergence theorem for Algorithm 3.1.
Theorem 1. Suppose . Then, the sequence generated by Algorithm 3.1 converges strongly to , where .
Proof. We have known in Lemma 6 (1) that exists. Now, we show that . Let , then from Lemma 2, we haveTherefore, by Lemma 3, we get that as . Thus, is a Cauchy sequence in . Since is closed and convex, it implies that there exists such that . Since , , and T is a Bregman weak relatively nonexpansive mapping, then . More so, since , then , and by the linearity of A, we have . Also from (73), . Since is a resolvent metric of B for , then for all , we haveSo for all , we haveIt follows from (73) that for all , we haveSince B is maximal monotone, then it implies that