Abstract
In this paper, the auxiliary principle technique is extended to study a system of generalized nonlinear mixed variational-like inequalities problem for set-valued mappings without compact values in Banach spaces with -uniformly convex bidual spaces. First, the existence of the solutions of the related auxiliary problem is proved. Then, a new iterative algorithm based on the system of auxiliary variational inequalities is constructed. Finally, both the existence of the solutions of the original problem and the convergence of the iterative sequences generated by the algorithm are proved. And we also present a numerical example to demonstrate the result. Our results improve and extend some known results.
1. Introduction
Throughout the paper unless otherwise stated, let be an index set for each , be a real reflexive Banach space with the dual space , be the dual pairing between and , and be the family of all nonempty bounded closed subsets of . For each , let , , be nonlinear mappings and , , be set-valued mappings. We consider the following system of generalized set-valued nonlinear mixed variational-like inequalities problem: find and such that , , , , , , andwhere for each , the bifunction , which is not necessarily differentiable, satisfies the following properties:(i) is linear in the first argument;(ii) is bounded; that is, there exists a constant such that (iii), ;(iv) is convex in the second argument.
Remark 1 (see [1]). For each and arbitrary , property implies that and property implies that . Hence we have and For each , it follows from properties and , for all , that we have and and, therefore, This implies that for each , is continuous with respect to the second argument.
There are many special cases of problem (1), which can be also found in [1–11] and the references cited therein. Therefore, problem (1) is a more general and unified one.
In the system of the generalized set-valued nonlinear mixed variational-like inequalities problem (1), is a nonlinear mapping, so the projection method cannot be applied to it. This fact motivated many authors to develop the auxiliary principle technique to study the existence of solutions of generalized mixed type variational inequalities and also to develop a large number of numerical methods for solving various variational inequalities, complementarity problems, and optimization problems. The auxiliary principle technique was first introduced by Glowinski et al. [12]. Then Ding et al. [3] extended it to study the existence and algorithm of solutions of generalized strongly nonlinear mixed variational-like inequalities in Banach spaces when is a set-valued mapping. Regretfully, is actually a single-valued mapping in Theorem 3.1 of [3]. In problem (1), we consider the set-valued type mappings. Very recently, auxiliary principle technique is used to solve regularized nonconvex mixed variational inequalities and strongly mixed variational-like inequalities; see [13] and [14], respectively.
On the other hand, Noor [6] put forward that extending the projection methods and its variant forms for generalized set-valued mixed nonlinear variational inequalities involving the nonlinear form satisfying properties (i), (ii), and (iii) is still an open problem, and this needs further research efforts.
In this paper, the auxiliary principle technique is extended to study a system of generalized nonlinear mixed variational-like inequalities problem (1) for set-valued mappings without compact values in Banach spaces with -uniformly convex bidual spaces. First, the existence of the solutions of the related auxiliary problem is proved. Then a new iterative algorithm based on the system of auxiliary variational inequalities is constructed. Finally, both the existence of the solutions of the problem (1) and the convergence of the iterative sequences generated by the algorithm are proved. And we also present a numerical example to demonstrate the result. Our results improve and extend some known results.
2. Preliminaries
We recall some concepts and definitions at first.
Let be a real Banach space; the modulus of convexity of is the function defined by is uniformly convex if and only if, for all : such that . is said to be -uniformly convex, if there exists a constant , such that . Hilbert space and and Sobolev space are -uniformly convex. Hilbert space is -uniformly convex, while are -uniformly convex, if ; they are -uniformly convex, if . For , the duality mapping with gauge function is defined by In particular, is the normalized duality mapping. It is well known that is single-valued if is smooth and , . The duality mappings of and are denoted by and , respectively. The duality mapping can be equivalently defined as the subdifferential of the gauge function ; i.e.,
We assume that is the Hausdorff metric on defined by
Definition 2. Let be a smooth Banach space, and be single-valued mappings, and be set-valued mappings. (1) is said to be -H-Lipschitz continuous, if there exists a constant such that where is the Hausdorff metric on ;(2) is said to be -Lipschitz continuous, if there exists a pair of constants such that (3) is said to be -strongly mixed accretive with set-valued mappings , and single-valued mapping , if there exists a constant such that
Remark 3. If is a Hilbert space, is an identity operator on ; then is called an -strongly mixed monotone mapping; that is, which is introduced by Zeng et al. [15].
Definition 4 (see [12]). The mapping is said to be -Lipschitz continuous, if there exists a constant such that In order to obtain our results, we need the following assumption.
Assumption 5. The mapping , , satisfies the following conditions:
, ;
is affine in the second argument;
for each fixed is continuous from the weak topology to the weak topology.
Remark 6. It follows from Assumption 5(1) that and
We also need the following Lemmas.
Lemma 7 (see [16]). Let be a nonempty closed convex subset of a Hausdorff linear topological space and let be mappings satisfying the following conditions:
(1) , ;
(2) for each , is upper semicontinuous with respect to ;
(3) for each , the set is convex;
(4) if there exists a nonempty compact set and such that then there exist such that .
Lemma 8 (see [8]). Let be a real Banach space with its uniformly convex dual space . Assume that the modulus of convexity of satisfies for some and . Then duality mapping with gauge function is Hölder continuous with exponent ; i.e., there exists a constant such that
3. Auxiliary Problem and Algorithm
In this section, we will extend the auxiliary principle technique of Glowinski et al. [12] to study problem (1). And we will prove the existence of the solution of the system of auxiliary variational inequalities problem for (1). Then, by using the existence theorem, we will construct an iterative algorithm for problem (1).
For each , let be a single-valued mapping. Given , , , , , , , we consider the following problem: find such thatfor all , where are constants. Problem (20) is called the system of auxiliary variational inequalities problem for problem (1).
Theorem 9. For each , let be -Lipschitz continuous and -strongly monotone, be -Lipschitz continuous, be a function with the properties , and and be mappings. If Assumption 5 holds, then the system of auxiliary variational inequalities problem (20) has a unique solution .
Proof. For , define the mappings by and respectively. We show that for each , the mappings satisfy the conditions of Lemma 7 in the weak topology. Indeed, since is -strongly monotone, it is clear that and satisfy condition (1) of Lemma 7. Since is -Lipschitz continuous, by Assumption 5, for any , we have By Assumption 5, the function is continuous and convex. It follows that the function is weakly lower semicontinuous and so the function is weakly upper semicontinuous. It is easy to show that, for each fixed , the set is a convex set (we note that if , the set ; then this immediately implies the conclusion of Theorem 9 and hence we discuss only the case of the set below). Thus, conditions (2) and (3) of Lemma 7 hold.
Now, for each , set Then is a weakly compact subset of . For any fixed , take ; from Assumption 5(1), the Lipschitz continuity of , the strong monotonousness of , and Remark 1(2), we have Therefore, condition of Lemma 7 holds. By Lemma 7, there exists an such that , that is,For arbitrary and , let , replace by in (26); we obtain Hence, we derive that is, Let ; we have Therefore, is a solution of the system of auxiliary variational inequalities problem (20). Now, let be any two solutions of the system of auxiliary variational inequalities problem (20). Then we haveand, taking in (31) and in (32), then adding these two inequalities, we obtain Since is -strongly monotone, we have Hence, we must have and so problem (20) has a unique solution. This completes the proof.
By virtue of Theorem 9, we now construct an iterative algorithm for solving problem (1).
For given , , , , , , , , from Theorem 9, we know that the system of auxiliary variational inequalities problem (20) has a unique solution ; that is, Since , , , , , and , , by Nadler’s Theorem, there exist , , , , , and , , such that
Again by Theorem 9, the system of auxiliary variational inequalities problem (20) has a unique solution ; that is, Since , , , , , and , , by Nadler’s Theorem, there exist , , , , , and , , such that by induction, we can get the iterative algorithm for solving problem (1).
Algorithm 10. For given , , , , , , and , , let , , and satisfy the following conditions: where constants , , , and
4. Existence and Convergence Theorem
In this section, we will prove the existence of the solution of problem (1) and the convergence of the iterative sequences generated by Algorithm 10.
Theorem 11. For each , let be a real reflexive Banach space with a uniformly convex bidual space ; the modulus of convexity of satisfies for some and . Let , , , , , , and be mappings, and let be a real valued functional. Assume that the following conditions are satisfied:
(1) is -Lipschitz continuous and -strongly mixed accretive with respect to , and ;
(2) is -H-Lipschitz continuous;
(3) is -H-Lipschitz continuous;
(4) is -H-Lipschitz continuous;
(5) is -H-Lipschitz continuous;
(6) is -H-Lipschitz continuous;
(7) is -strongly monotone and -Lipschitz continuous;
(8) has the properties (i)-(iv);
(9) is -Lipschitz continuous and satisfies the following: there exists a constant such that If Assumption 5 holds and there exist constants such thatwhere is a constant appearing in Lemma 8, then there exist , , , , , , and satisfying problem (1) and as , where the sequences are defined by Algorithm 10.
Proof. First, it follows from (39) in Algorithm 10 that for any ,andTaking in (45) and in (46), respectively, we getandAdding (47) and (48), we obtain and soBy conditions and (50), we derive Therefore,Since is -uniformly convex and Lemma 8 holds with replaced by and replaced by , where is duality mapping with gauge function , by (50) and condition (1), we haveIt follows from (52) and (53) thatSecond, it follows from (40) in Algorithm 10 that, for any ,andTaking in (55) and in (56), respectively, then adding the results, we have By conditions (1)-(9), we haveFrom (54) and (58), we haveHere Let We can see that , as . Now, it follows from condition (43) that . Hence, there are a positive number and an integer such that , for all . Therefore, it follows from (59) that is a Cauchy sequence in . Let , as . Since and are -Lipschitz continuous, by (41), we have Therefore, and are also Cauchy sequences. Let , as . Since , we have Hence, we conclude Similarly, we can obtain , , , , , .
Since , , , , , , as , and weakly in and weakly in , as , we deduce from (39) and (40) that That is, This completes the proof.
Remark 12. Problem (1) is a very general system of variational inequalities in Banach spaces.
The system of auxiliary variational inequalities problem (20) is very different from the corresponding auxiliary variational inequality problems in [1, 3].
By using the concept of strongly mixed accretiveness and Lipschitz continuity in the sense of the Hausdorff metric, the restrictions imposed on the set-valued mapping and the single-valued mapping in Theorem 11 are weaker than that in [1, 3].
Example 13. Let , for , , ,, , where , , , , . It is easy to check that these mappings satisfy the conditions of Theorem 11, and the system of variational inequalities is as follows:Let ; then Algorithm 10 can be written asFurther, we choose , , . The numerical results presented in Table 1 and Figure 1 demonstrate problem (68).
Obviously, is the solution of (68).
Data Availability
The M data and the PNG data used to support example 4.3 of this study are included within the supplementary information files.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work is supported by Shanghai Polytechnic University Foundation (No. A01GY18EX03) and the key discipline (Applied Mathematics) of Shanghai Polytechnic University (No. XXKPY1604).