Abstract
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.
1. Introduction
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. [1] (see, for instance, [2, 3]).
If we set , where and , GMVLIP equation (1) is reduced to MVLIP (see for details [4]).
Further, if we set and , GMVLIP equation (1) is reduced to VLIP: find such thatwhich is presented by Parida et al. [5].
Moreover, if , VLIP is reduced to VIP: find such thatwhich is introduced by Hartmann and Stampacchia [6].
If , , and , where is continuous and is differentiable and -convex, GMVLIP equation (1) is reduced to mathematical programming problem as [5]
Korpelevich [7] 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 [8] proposed a hybrid extragradient algorithm involving nonexpansive mapping on and studied the convergence analysis in 2006 as
The idea considered in [8] has been generalized in [9] 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 [18] 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 [23] 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 [29] 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.
2. Preliminaries
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 [19]:(i), is Gateaux differentiable on , and (ii), is Gateaux differentiable on , and
We have the following [19]:(i) be Legendre iff be Legendre mapping(ii)(iii), , (iv)The mappings and are strictly convex on and
Definition 1 (see [18]). 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 [40] 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 [26]). 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 [18]). Let be a Gateaux differentiable and convex function. The Bregman projection of onto is a unique vector with
Remark 1 (see [24]). (i) If is a smooth Banach space and , , then the Bregman projection reduces to , generalized projection (see [41]), 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 [20]). 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 [20] (Section 1.3, p.30), we notice that any uniformly convex function is totally convex but the converse is not true. Also, by [21] (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 [21]). 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 [42]). 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 [23]). Let be a Gateaux differentiable and totally convex function. If and are bounded, then is also bounded.
Lemma 4 (see [21]). 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 [25]). Let be Legendre and be Bregman quasi nonexpansive mapping with respect to . Then, is closed and convex.
Lemma 6 (see [23]). 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 [43]). 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 [1]). 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 [44], 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 By condition (iii) of Theorem 1, we get As is skew symmetric and is a generalized relaxed -monotone, By interchanging the position of and in equation (76), we get Adding equations (76) and (77), we have As , This implies that As is convex and Gateaux differentiable, By equations (80) and (81), we have Since is a Legendre function, . Hence, is single-valued.(ii)For , we obtain and Adding the above two inequalities, we have which yields by applying the concept of and , In equation (86), interchanging the position of and , we get Adding equations (86) and (87) and using , we get This implies that This means that is a Bregman firmly nonexpansive type mapping.(iii)Let ; then, Furthermore, Since is a Bregman firmly nonexpansive type mapping, in ([42], Lemma 1.3.1), is a closed and convex subset of . Therefore, by equation (90), we get that is closed and convex.(iv)Now, we show that is Bregman quasi-nonexpansive mapping. For , from (b), we have Moreover, we have Taking , we see that Hence,(v)Equation (94) implies that is Bregman quasi-nonexpansive mapping.
4. Main Result
We developed the strong convergence algorithm for the inertial iterative method to find the common solution of GMVLIP equation (1) and fixed-point problem of a Bregman relatively nonexpansive mapping in reflexive Banach space.
Iterative Algorithm 1. Let the sequences and be generated by the iterative algorithm:where and .
Theorem 2. Let with , where be a coercive Legendre function which is bounded, uniformly Frechet differentiable, and totally convex on bounded subsets of . Let satisfy all conditions of Theorem 1 with continuous , and satisfies Assumption 1, respectively. Let be a Bregman relatively nonexpansive mapping. Let . Let be generated by Iterative 1 and , with . Then, converges strongly to .
Proof. For convenience, we divide its proof into several steps as in the following.
Step 1. and are closed and convex, .
By Lemmas 5 and 9, is a closed and convex, and therefore, is well defined.
Obviously, is closed and convex. Furthermore, we prove that is closed and convex, . We can easily show that is closed and convex, . Thus, is closed and convex, .
Step 2. , , and is well defined.
Let ; then,andSubstituting equation (97) into equation (96), we haveThus, . Therefore, , . Furthermore, by induction, we show that , . As , . Suppose that , for some . Then, such that . From the definition of , we get . Since , we havewhich implies . Hence, implies , , and thus, is well defined, . Hence, is well defined.
Step 3. The sequences , , , , and are bounded.
Using the concept of , we get . By and Lemma 10 (iii), we obtainThis implies that is bounded, and hence, is bounded by Lemma 3.
Now,which implies that is bounded. Using , is bounded. Therefore, , , , and are bounded.
Step 4. ; , and .
Since and , we getwhich implies is nondecreasing. By boundedness of , exists and is finite. Furthermore,which yieldsUsing Lemma 1,From the definition of , , which implies by equation (105) thatSinceit follows from equations (105) and (106) thatUsing Lemma 2 because is uniformly Frechet differentiable, we getandBy the concept of , we get is bounded on the bounded subset of because is bounded on . Since is uniformly Frechet differentiable, it is uniformly continuous on bounded subsets. Hence, by equations (108), (109), and (111),As , we haveand hence, by equations (112) and (113),Thanks to Lemma 1,Taking into accountby equations (108) and (115), we getBy Lemma 2,andNext, we estimateSince , , , and are bounded and by equations (117)–(120), we getFurthermore, it follows from Lemma 9 (v) thatSince and are bounded, by equations (121) and (122),and hence,From equations (117) and (124), we getBy uniform Frechet differentiable of , Lemma 2, and equations (124) and (125), we haveNote thatBy equations (127) and (128) and , we getMoreover, we have from equation (129) that
Step 5. .
First, we prove that . As is bounded, a subsequence such that as . By equations (106), (117), (124), and (125), , , , and have the same asymptotic behaviour and thus subsequences of , of , and of such that , , and as . Using and equation (130), we getBy the concept of , .
Next, prove that . As , we haveUsing generalized relaxed -monotonicity of , we haveUsing the concept of , b, equation (126), and in equation (133), we obtainFor and , let . Since , we havewhich implies thatSince is hemicontinuous, we havewhich impliesHence, . Thus, .
Step 6. We prove that .
Proof of Step 6. Let . As is weakly convergent, and . By equation (100), we haveUsing Lemma 6, is strongly convergent to . Hence, by the uniqueness of the limit, converges strongly to .
5. Consequences
Finally, we get the following consequences of Theorem 2.
Corollary 1. Let with , where be a coercive Legendre function which is bounded, uniformly Frechet differentiable, and totally convex on bounded subsets of . Let satisfy conditions (i), (ii), and (iii) of Theorem 1 and be monotone, i.e.,Let satisfy Assumption 1, and Let be a Bregman relatively nonexpansive mapping. Let . Let be generated by Iterative 1 and and with . Then, converges strongly to .
Moreover, if GMVLIP equation (1) and by the concept of Example 1 for , we have the maximal monotone operator.
Corollary 2. Let with , where be a coercive Legendre function which is bounded, uniformly Frechet differentiable, and totally convex on bounded subsets of . Let be a maximal monotone operator with . Let generated bywhere and with . Then, converges strongly to .
Remark 4. If , , then Theorem 2 is reduced to the strong convergence theorem for finding the common solution of GMVLIP equation (1) and fixed-point problem of a relatively nonexpansive mapping in reflexive Banach space.
6. Numerical Example
Finally, to support our main theorem, we now give an example in infinitely dimensional spaces such that is -norm defined by where .
Example 3. Let and . Define mappings as follows:(i)Coercive Legendre function by , (ii)∀x,y,z∈C, Function by , with such that , (iii)Bifunction by , (iv)Bregman relatively nonexpansive mapping with respect to by , It is obvious that satisfies all conditions of Theorem 1 with continuous and satisfies Assumption 1, respectively. On the other hand, we considerFor the experiments in this section, we use the Cauchy error for the stopping criterion. We will start with the initialization and in two cases. We split considering all of the performances of our algorithm in four cases by considering all of the parameters that have an effect on the convergence of the algorithm.
Case 1. We start computation by comparison of the algorithm with different parameters wherewhere is the number of iterations that we want to stop, , and . We choose , , and . Then, the results are presented in Table 1.
Case 2. We compare the performance of the algorithm with different parameters by setting , , and . Then, the results are presented in Table 2.
Case 3. We compare the performance of the algorithm with different parameters by setting , , and for the initialization and and for the initialization . Then, the results are presented in Table 3.
Case 4. We compare the performance of the algorithm with different parameters by setting , , and for the initialization and and for the initialization . Then, the results are presented in Table 4.
From Tables 1–4 and Figures 1–12, we noticed that in all the above 4 cases, choosing , , , and yields the best results for the initialization . Choosing , , , and yields the best results for the initialization , and choosing , , , and yields the best results for the initialization .
7. Conclusion
In this paper, we established an inertial hybrid iterative algorithm involving Bregman relatively nonexpansive mapping to find a common solution of GMVLIP equation (1) and FPP 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. From the theoretical and application point of view, the inertial method via Bregman relatively nonexpansive mapping has a great importance on data analysis and some imaging problems. The inertial method has been studied by various researchers due to its importance (see for details [19, 24–28, 30, 31, 33–36, 39]).
Abbreviations.
| GMVLIP: | Generalized mixed variational-like inequality problem |
| GVLIP: | General variational-like inequality problem |
| MVLIP: | Mixed variational-like inequality problem |
| VLIP: | Variational-like inequality problem |
| VIP: | Variational inequality problem |
| FPP: | Fixed-point problem. |
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.