Strong Convergence of an Inertial Iterative Algorithm for Generalized Mixed Variational-like Inequality Problem and Bregman Relatively Nonexpansive Mapping in Reflexive Banach Space
In this paper, we consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems and also prove some properties for the solution set of generalized mixed variational-like inequality problem. Furthermore, we introduce and study an inertial hybrid iterative method for solving the generalized mixed variational-like inequality problem involving Bregman relatively nonexpansive mapping in Banach space. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational examples to emphasize the efficiency and relevancy of the main result.
Throughout the paper, unless otherwise stated, let be a reflexive Banach space with as its dual and be the closed convex subset of . In this paper, we consider the generalized mixed variational-like inequality problem (in brief, GMVLIP): find such thatwhere and , be bifunction and trifunction, respectively, and be the set of real numbers. Sol (GMVLIP equation (1)) stands for the solution of equation (1). If , GMVLIP equation (1) is reduced to GVLIP: find such thatwhich is introduced by Preda et al.  (see, for instance, [2, 3]).
If we set , where and , GMVLIP equation (1) is reduced to MVLIP (see for details ).
Further, if we set and , GMVLIP equation (1) is reduced to VLIP: find such thatwhich is presented by Parida et al. .
Moreover, if , VLIP is reduced to VIP: find such thatwhich is introduced by Hartmann and Stampacchia .
If , , and , where is continuous and is differentiable and -convex, GMVLIP equation (1) is reduced to mathematical programming problem as 
Korpelevich  proposed the iterative method for VIP in 1976 on Hilbert space aswhere , denotes projection of onto , and is monotone and Lipschitz continuous mapping. This method is called the extragradient iterative method.
Nadezkhina and Takahashi  proposed a hybrid extragradient algorithm involving nonexpansive mapping on and studied the convergence analysis in 2006 as
The idea considered in  has been generalized in  from Hilbert to Banach space aswhere denotes generalized projection of onto , is the Lyapunov function such that , and is the normalized duality mapping with being its inverse. For further work, see [10–17].
In 1967, an important technique was discovered by Bregman  in the light of Bregman distance function. This technique is very useful not only in design and interpretation of the iterative method but also to solve optimization and feasibility problems and to approximate equilibria, fixed point, variational inequalities, etc. (for details [19–22]).
In 2010, Reich and Sabach  introduced iterative algorithm on Banach space involving maximal monotone operators. In the light of Bregman projection, there were various iterative algorithms studied by researchers in this field (see, for instance, [19, 24–28]).
In 2008, Maingé  developed and studied an inertial Krasnosel’skiǐ–Mann algorithm as
For further work, see [30–39].
Inspired by the work in [2, 27, 29], we establish an inertial hybrid iterative algorithm involving Bregman relatively nonexpansive mapping to find a common solution of GMVLIP equation (1) and a fixed-point problem in Banach space. Moreover, we study the convergence analysis for the main result. At last, we list some consequences and computational example to emphasize the efficiency and relevancy of the main result.
Assume is a proper, convex, and lower semicontinuous mapping and is a Fenchel conjugate of , defined asAnd, for any , interior of the domain of and , the right-hand derivative of at in the direction is
A mapping is called Gateaux differentiable at if the above limit exists. So, agrees with , the value of the gradient of at . It is called Frechet differentiable at , if the limit is attained uniformly in . It is called uniformly Frechet differentiable on , if the above limit is attained uniformly for and .
The mapping is called Legendre if the following holds :(i), is Gateaux differentiable on , and (ii), is Gateaux differentiable on , and
We have the following :(i) be Legendre iff be Legendre mapping(ii)(iii), , (iv)The mappings and are strictly convex on and
Definition 1 (see ). Let be Gateaux differentiable and convex and such thatis known as Bregman distance with respect to .
We notice that the Bregman distance is not a distance in the usual sense of term. Obviously, , but may not imply . It holds if is the Legendre function. However, is neither symmetric nor satisfy the triangle inequality. We have the following important properties of  for and .(i)Two-point identity:(ii)Three-point identity:(iii)Four-point identity:
Definition 2 (see [23, 25]). Let be a mapping and , where is the set of fixed points of . Then, we have the following:(i)A point is called an asymptotic fixed point if contains a sequence with such that . We represent as the set of asymptotic fixed points of .(ii) is called Bregman quasi-nonexpansive if(iii) is called Bregman relatively nonexpansive if(iv) is called Bregman firmly nonexpansive if , or, correspondingly,
Example 1 (see ). Let be a maximal monotone mapping. If and the Legendre function is bounded on bounded subsets of and uniformly Frechet differentiable, then the resolvent with respect to ,is a single-valued, closed, and Bregman relatively nonexpansive mapping from onto and .
Definition 3 (see ). Let be a Gateaux differentiable and convex function. The Bregman projection of onto is a unique vector with
Remark 1 (see ). (i) If is a smooth Banach space and , , then the Bregman projection reduces to , generalized projection (see ), and it is defined aswhere is a Lyapunov function. (ii) If is a Hilbert space and , , then reduces to the metric projection of onto .
For all , assume . Then, a map is said to be uniformly convex on bounded subsets of , if , , where is defined as. The function is known as the gauge of uniform convexity of . The function is also said to be uniformly smooth on bounded subsets of if , for all , where is defined by. The function is said to be uniformly convex if the function , defined bysatisfies that .
Remark 2. Let be a Banach space, be a constant, and be a convex function which is uniformly convex on bounded subsets. Then,for all and , where is the gauge of uniform convexity of .
Definition 4 (see ). Let be a Gateaux differentiable and convex function. Then, is called the following:(i)Totally convex at if its modulus of total convexity at , i.e., the mapping such that is positive, for (ii)Totally convex if it is totally convex at each point of (iii)Totally convex on bounded sets if such thatBy  (Section 1.3, p.30), we notice that any uniformly convex function is totally convex but the converse is not true. Also, by  (Theorem 2.10, p.9), is totally convex on bounded sets if and only if is uniformly convex on bounded sets.
Definition 5 (see [20, 23]). A mapping is called the following:(i)Coercive if (ii)Sequentially consistent if for any with bounded,
Lemma 1 (see ). Let be a convex function with domain at least two points. Then, is sequentially consistent iff it is totally convex on bounded sets.
Lemma 2 (see ). Let be uniformly Frechet differentiable and bounded on , a bounded set. Then, is uniformly continuous on and is uniformly continuous on from the strong topology of to the strong topology of .
Lemma 3 (see ). Let be a Gateaux differentiable and totally convex function. If and are bounded, then is also bounded.
Lemma 4 (see ). Let be a Gateaux differentiable and totally convex function on . Let and , a nonempty closed convex set. If , then the following statements are equivalent:(i) is the Bregman projection of onto with respect to , i.e., (ii)The vector is the unique solution of the variational inequality:(iii)The vector is the unique solution of the inequality:
Lemma 5 (see ). Let be Legendre and be Bregman quasi nonexpansive mapping with respect to . Then, is closed and convex.
Lemma 6 (see ). Let be Gateaux differentiable and totally convex function, , and , a nonempty closed convex set. Suppose that is bounded and any weak subsequential limit of belongs to . If , then strongly converges to .
Lemma 7 (see ). Let be a nonempty subset of a Hausdroff topological vector space and let be a KKM mapping. If is closed in for all and compact for some , then .
Definition 6 (see ). A function is said to be generalized relaxed -monotone if for any , we havewhere
Remark 3. (i)If , where , we say that the mapping is a generalized - monotone(ii)In Definition 6, let and , where with , for and , then we say that is called a relaxed - monotone mapping(iii)In case (ii), if for all , then Definition 6 reduces to for all and is called a relaxed -monotone mapping(iv)In case (iii), if is a constant, then Definition 6 reduces to for all and is called a -monotone mapping(v)If , then (iii) reduces to for all and is called a monotone mappingWe construct an example for generalized relaxed -monotone mapping as follows.
Example 2. Consider , , andwhere is a constant. Thus, is generalized relaxed -monotone with
Assumption 1. Let satisfy the following:(i) is skew-symmetric, i.e., (ii) is convex in the second argument(iii) is continuous
3. Existence of Solutions and Resolvent Operator
For , assume the auxiliary problems (in short, AP) related to GMVLIP equation (1): find such thatand find such that
We have the Minty-type lemma as follows.
Lemma 8. Let be Gateaux differentiable and coercive function, and let satisfy Assumption 1 (ii). Assume with the following cases:(i) is hemicontinuous(ii) is convex(iii)(iv) is a generalized relaxed -monotoneThen, AP equation (36) and AP equation (37) are equivalent.
Proof. Let be a solution of AP equation (36) and by the concept of , we obtainwhich shows that is a solution of AP equation (37).
Conversely, let be a solution of AP equation (37). Then,For any , let , and we get . By equation (39), we haveUsing conditions (ii) and (iii), we obtainBy Assumption 1 (ii), we haveandUsing equations (40)–(43), we haveHence,Let , and by condition (i), we obtainThus, be a solution of AP equation (95).
Theorem 1. Let be a Gateaux differentiable and coercive function, satisfy Assumption 1 (ii)-(iii), and be a bifunction. Consider and for any , assume the following:(i) is hemicontinuous(ii) is convex and lower semicontinuous(iii)(iv) is a generalized relaxed -monotone(v) is lower semicontinuousThen, AP equation (36) has solution.
Proof. Let , for any , be two set-valued mappings withandObviously, solves AP equation (36) if and only if . Hence, . Next, we prove that is a KKM mapping. On the contrary, let be not a KKM mapping; then, such that ; this means there exists a , but . Then,By Theorem 1 (ii)-(iii), we getwhich is a contradiction. Thus, is a KKM mapping.
Next, we prove that . Let , for any ; then,Using the concept of , we obtainThus, , , which yields that is a KKM mapping.
Let be any sequence in with as . Then,Since is Gateaux differentiable function, is norm-to-weak continuous. By (ii) and (iii) and lower semicontinuity of , we havewhich yields that . Thus, and are the closed subset of , . As is closed convex and bounded subset in , it is weakly compact. Thus, is also compact. By Lemmas 7 and 10, we have . Therefore, AP equation (36) has a solution. □
The resolvent of with respect to is the operator , defined as follows:We obtain some properties of the resolvent operator . First, we show that for and dom under some suitable conditions.
Lemma 9. Let be a coercive and Gateaux differentiable function. If satisfies all conditions of Theorem 1 and satisfies Assumption 1, then dom .
Proof. First, we prove that for any such thatfor any . As is coercive, the function defined bysatisfiesfor each fixed . By Theorem 1 in , equation (56) holds. Now, we show that equation (56) yieldsfor any . Assume and ; we get . By equation (59) and the concept of , we getSincewe get from equation (61), Theorem 1 (ii), and Assumption 1 (ii) thatFrom Lemma 10 (iii), we haveandTherefore,As is a Gateaux differentiable function, is norm-to-weak continuous. Taking , we haveThus, for any , let ; we have such thati.e.,that is, . Hence, dom . □
Lemma 10. Let satisfy all conditions of Theorem 1, and let satisfy Assumption 1. Let be a coercive Legendre function and the resolvent operator be defined by equation (55). Then, the following holds:(i) is single-valued(ii) is Bregman firmly nonexpansive type mapping, that is,(iii) is closed and convex(iv)(v) is Bregman quasi-nonexpansive
Proof. (i)For , let . Then, and hence and Adding the above two inequalities, we get