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Fixed point problems and a system of generalized nonlinear mixed variational inequalities
Fixed Point Theory and Applications volume 2013, Article number: 186 (2013)
Abstract
In this paper, we introduce and consider a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators and discuss the existence and uniqueness of solution of the aforesaid system. We use three nearly uniformly Lipschitzian mappings () to suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping , which is the unique solution of the system of generalized nonlinear mixed variational inequalities. The convergence analysis of the suggested iterative algorithms under suitable conditions is studied. In the final section, an important remark on a class of some relaxed cocoercive mappings is discussed.
MSC:47H05, 47J20, 49J40, 90C33.
1 Introduction
Variational inequality theory, which was initially introduced by Stampacchia [1] in 1964, is a branch of applicable mathematics with a wide range of applications in industry, physical, regional, social, pure, and applied sciences. This field is dynamic and is experiencing an explosive growth in both theory and applications; as a consequence, research techniques and problems are drawn from various fields. Variational inequalities have been generalized and extended in different directions using the novel and innovative techniques. An important and useful generalization is called the mixed variational inequality, or the variational inequality of the second kind, involving the nonlinear term. For applications, numerical methods, and other aspects of variational inequalities, see, for example, [1–22] and the references therein. In recent years, much attention has been given to develop efficient and implementable numerical methods including projection method and its variant forms, Wiener-Hopf (normal) equations, linear approximation, auxiliary principle, and descent framework for solving variational inequalities and related optimization problems. It is well known that the projection method and its variant forms and Wiener-Hopf equations technique cannot be used to suggest and analyze iterative methods for solving mixed variational inequalities due to the presence of the nonlinear term. These facts motivated us to use the technique of resolvent operators, the origin of which can be traced back to Martinet [11] and Brezis [4]. In this technique, the given operator is decomposed into the sum of two (or more) maximal monotone operators, whose resolvents are easier to evaluate than the resolvent of the original operator. Such a method is known as the operator splitting method. This can lead to the development of very efficient methods, since one can treat each part of the original operator independently. The operator splitting methods and related techniques have been analyzed and studied by many authors including Peaceman and Rachford [15], Lions and Mercier [9], Glowinski and Tallec [7], and Tseng [18]. For an excellent account of the alternating direction implicit (splitting) methods, see [2]. A useful feature of the forward-backward splitting method for solving the mixed variational inequalities is that the resolvent step involves the subdifferential of the proper, convex and lower semicontinuous part only and the other part facilitates the problem decomposition.
Equally important is the area of mathematical sciences known as the resolvent equations, which was introduced by Noor [12]. Noor [12] established the equivalence between the mixed variational inequalities and the resolvent equations using essentially the resolvent operator technique. The resolvent equations are being used to develop powerful and efficient numerical methods for solving the mixed variational inequalities and related optimization problems. It is worth mentioning that if the nonlinear term involving the mixed variational inequalities is the indicator function of a closed convex set in a Hilbert space, then the resolvent operator is equal to the projection operator.
On the other hand, related to the variational inequalities, we have the problem of finding the fixed points of nonexpansive mappings, which is the subject of current interest in functional analysis. It is natural to consider a unified approach to these two different problems. Motivated and inspired by the research going in this direction, Noor and Huang [14] considered the problem of finding a common element of the set of solutions of variational inequalities and the set of fixed points of nonexpansive mappings. It is well known that every nonexpansive mapping is a Lipschitzian mapping. Lipschitzian mappings have been generalized by various authors. Sahu [23] introduced and investigated nearly uniformly Lipschitzian mappings as generalization of Lipschitzian mappings.
In the present paper, we introduce and consider a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators (SGNMVID). We first verify the equivalence between the SGNMVID and the fixed point problems, and then by this equivalent formulation, we discuss the existence and uniqueness of the solution of the SGNMVID. Applying nearly uniformly Lipschitzian mappings () and the aforesaid equivalent alternative formulation, we suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding the element of the set of fixed points of the nearly uniformly Lipschitzian mapping , which is the unique solution of the SGNMVID. Also, the convergence analysis of the suggested iterative algorithms under suitable conditions is studied. In the final section, some comments on the results related to a class of strongly monotone mappings are discussed. The results presented in this paper extend and improve some known results in the literature.
2 Preliminaries and basic results
Throughout this article, we let ℋ be a real Hilbert space which is equipped with an inner product and the corresponding norm . Let and () be six nonlinear single-valued operators such that for each , is an onto operator, and let denote the subdifferential of the function (), where for each , is a proper convex lower semicontinuous function on ℋ. For any given constants , we consider the problem of finding such that
which is called the system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators (SGNMVID).
If for each , , the identity operator, and , for all , where is the indicator function of a nonempty closed convex set K in ℋ defined by
then problem (2.1) reduces to the following system:
which was introduced and studied by Cho and Qin [6].
For different choices of operators and constants, we obtain different systems and problems considered and studied in [1, 5, 8, 13, 17, 19–21] and the references therein.
Definition 2.1 A set-valued operator is said to be monotone if, for any ,
A monotone set-valued operator T is called maximal if its graph, , is not properly contained in the graph of any other monotone operator. It is well known that T is a maximal monotone operator if and only if for all , where I denotes the identity operator on ℋ.
Definition 2.2 [4]
For any maximal monotone operator T, the resolvent operator associated with T of parameter λ is defined as
It is single-valued and nonexpansive, that is,
If φ is a proper, convex and lower-semicontinuous function, then its subdifferential ∂φ is a maximal monotone operator, see Theorem 4 in [24]. In this case, we can define the resolvent operator associated with the subdifferential ∂φ of parameter λ as follows:
The resolvent operator has the following useful characterization.
Lemma 2.1 [13]
For a given , satisfies the inequality
if and only if , where is the resolvent operator associated with ∂φ of parameter .
It is well known that is nonexpansive, that is,
Let us recall that a mapping is nonexpansive if for all . In recent years, nonexpansive mappings have been generalized and investigated by various authors. In the next definitions, several generalizations of nonexpansive mappings are stated.
Definition 2.3 A nonlinear mapping is called
-
(a)
L-Lipschitzian if there exists a constant such that
-
(b)
generalized Lipschitzian [25] if there exists a constant such that
-
(c)
generalized -Lipschitzian [23] if there exist two constants such that
-
(d)
asymptotically nonexpansive [26] if there exists a sequence with such that for each ,
-
(e)
pointwise asymptotically nonexpansive [27] if, for each integer ,
where pointwise on X;
-
(f)
uniformly L-Lipschitzian if there exists a constant such that for each ,
Definition 2.4 [23]
A nonlinear mapping is said to be
-
(a)
nearly Lipschitzian with respect to the sequence if for each , there exists a constant such that
(2.3)
where is a fix sequence in with , as .
For an arbitrary, but fixed , the infimum of constants in (2.3) is called nearly Lipschitz constant and is denoted by . Notice that
Definition 2.5 [23]
A nearly Lipschitzian mapping T with the sequence is said to be
-
(a)
nearly nonexpansive if for all , that is,
-
(b)
nearly asymptotically nonexpansive if for all and , in other words, for all with ;
-
(c)
nearly uniformly L-Lipschitzian if for all , in other words, for all .
Remark 2.2 It should be pointed out that:
-
(a)
Every nonexpansive mapping is an asymptotically nonexpansive mapping, and every asymptotically nonexpansive mapping is a pointwise asymptotically nonexpansive mapping. Also, the class of Lipschitzian mappings properly includes the class of pointwise asymptotically nonexpansive mappings.
-
(b)
It is obvious that every Lipschitzian mapping is a generalized Lipschitzian mapping. Furthermore, every mapping with a bounded range is a generalized Lipschitzian mapping. It is easy to see that the class of generalized -Lipschitzian mappings is more general than the class of generalized Lipschitzian mappings.
-
(c)
Clearly, the class of nearly uniformly L-Lipschitzian mappings properly includes the class of generalized -Lipschitzian mappings and that of uniformly L-Lipschitzian mappings. Note that every nearly asymptotically nonexpansive mapping is nearly uniformly L-Lipschitzian.
Some interesting examples to investigate relations between the mappings given in Definitions 2.3, 2.4 and 2.5 can be found in [3].
3 Existence of solution and uniqueness
In this section, we prove the existence and uniqueness theorem for a solution of the system of generalized nonlinear mixed variational inequalities (2.1). For this end, we need the following lemma, in which, by using the resolvent operator technique and Lemma 2.1, the equivalence between the system of generalized nonlinear mixed variational inequalities (2.1) and fixed point problems is stated.
Lemma 3.1 Let , , (), ρ, η and γ be the same as in SGNMVID (2.1). Then is a solution of SGNMVID (2.1) if and only if
where is the resolvent operator associated with of parameter ρ, is the resolvent operator associated with of parameter η and is the resolvent operator associated with of parameter γ.
Proof is a solution of SGNMVID (2.1) if and only if
Since for each , is an onto operator, Lemma 2.1 implies that is a solution of (3.2) if and only if
This completes the proof. □
Definition 3.1 Let and be two single-valued operators. Then the operator
-
(a)
T is called monotone in the first variable if
-
(b)
T is called r-strongly monotone in the first variable if there exists a constant such that
-
(c)
T is called -relaxed cocoercive in the first variable if there exist two constants such that
-
(d)
T is said to be μ-Lipschitz continuous in the first variable if there exists a constant such that
-
(e)
g is called γ-Lipschitz continuous if there exists a constant such that
-
(f)
g is said to be ν-strongly monotone if there exists a constant such that
Theorem 3.2 Let , , (), ρ, η and γ be the same as in SGNMVID (2.1) such that for each , is -strongly monotone and -Lipschitz continuous in the first variable and is -strongly monotone and -Lipschitz continuous. If the constants ρ, η and γ satisfy the following conditions:
then SGNMVID (2.1) admits a unique solution.
Proof Define the mappings by
for all . Define on by
It is obvious that is a Banach space. Moreover, define as follows:
Now, we prove that F is a contraction mapping. For this end, let be given. By using the nonexpansivity property of the resolvent operator , we get
Because is -strongly monotone and -Lipschitz continuous, we have
Since is -strongly monotone and -Lipschitz continuous in the first variable, we conclude that
Substituting (3.7) and (3.8) in (3.6), we deduce that
Like in the proof of (3.9), we can establish that
and
From (3.9)-(3.11), it follows that
where
Applying (3.5) and (3.12), we conclude that
where . Condition (3.3) implies that and so (3.14) guarantees that the mapping F is contraction. According to the Banach fixed point theorem, there exists a unique point such that . It follows from (3.4) and (3.5) that , and . Now, it follows from Lemma 3.1 that is a unique solution of SGNMVID (2.1). This completes the proof. □
4 Some new three-step resolvent iterative algorithms
In this section, applying nearly uniformly Lipschitzian mappings () and by using the equivalent alternative formulation (3.1), we suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding an element of the set of fixed points of , which is the unique solution of SGNMVID (2.1).
Let be a nearly uniformly -Lipschitzian mapping with the sequence , let be a nearly uniformly -Lipschitzian mapping with the sequence and let be a nearly uniformly -Lipschitzian mapping with the sequence . We define the self-mapping of as follows:
Then is a nearly uniformly -Lipschitzian mapping with the sequence with respect to the norm in . To see this fact, let be arbitrary. Then, for any , we have
We denote the sets of all the fixed points of () and by and , respectively, and the set of all the solutions of system (2.1) by . It is clear that for any , if and only if , and , that is, . We now characterize the problem. Let , , (), ρ, η and γ be the same as in SGNMVID (2.1). If , then , , and . Therefore, it follows from Lemma 3.1 that for each ,
The fixed point formulation (4.2) is used to suggest the following three-step resolvent iterative algorithm with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping , which is a unique solution of SGNMVID (2.1).
Algorithm 4.1 Let , , (), ρ, η and γ be the same as in SGNMVID (2.1). For an arbitrary chosen initial point , compute the iterative sequence by the iterative processes
where () are three nearly uniformly Lipschitzian mappings, , , , , and are sequences in the interval such that , , , , , , , , and , , , , , , , , are nine sequences in ℋ to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions:
If for each , , then Algorithm 4.1 reduces to the following algorithm.
Algorithm 4.2 Let , , (), ρ, η and γ be the same as in SGNMVID (2.1). For an arbitrary chosen initial point , compute the iterative sequence in the following way:
where the sequences , , , , , , , , , , , , , and are the same as in Algorithm 4.1.
Remark 4.3 Equality (4.2) can be written as follows:
The fixed point formulation (4.5) enables us to suggest the following iterative algorithms.
Algorithm 4.4 Let , , (), ρ, η and γ be the same as in SGNMVID (2.1). For an arbitrary chosen initial point , compute the iterative sequence in the following way:
where () are three nearly uniformly Lipschitzian mappings, , are sequences in such that , , and the sequences , , , , , , , , are the same as in Algorithm 4.1 satisfying (4.4).
If , for all , then Algorithm 4.4 reduces to the following algorithm.
Algorithm 4.5 Let , , (), ρ, η and γ be the same as in SGNMVID (2.1). For an arbitrary chosen initial point , compute the iterative sequence in the following way:
where (), , , , , , and are the same as in Algorithm 4.1.
If (), then Algorithm 4.4 collapses to the following algorithm.
Algorithm 4.6 Let , , (), ρ, η and γ be the same as in SGNMVID (2.1). For an arbitrary chosen initial point , compute the iterative sequence in the following way:
where the sequences , , , , , , , , , and are the same as in Algorithm 4.1.
5 Main results
In this section, we discuss the convergence analysis of the suggested three-step resolvent iterative algorithms under suitable conditions. For this end, we need the following lemma.
Lemma 5.1 Let , and be three nonnegative real sequences satisfying the following condition: There exists a natural number such that
where , , , . Then .
Proof The proof directly follows from Lemma 2 in Liu [10]. □
Theorem 5.2 Let , , (), ρ, η and γ be the same as in Theorem 3.2 and let all the conditions of Theorem 3.2 hold. Suppose that is a nearly uniformly -Lipschitzian mapping with the sequence , that is a nearly uniformly -Lipschitzian mapping with the sequence , that is a nearly uniformly -Lipschitzian mapping with the sequence , and that the self-mapping of is defined by (4.1) such that . Further, let , where λ is the same as in (3.14). Then the iterative sequence generated by Algorithm 4.1, converges strongly to the only element of .
Proof According to Theorem 3.2, SGNMVID (2.1) has a unique solution . Accordingly, in view of Lemma 3.1, satisfies (3.1). Since is a singleton set, it follows from that and so , and . Hence, for each , we can write
where the sequences , , , , and are the same as in Algorithm 4.1. Let . It follows from (4.3), (5.1) and the assumptions that
Since is -strongly monotone and -Lipschitz continuous, and is -strongly monotone and -Lipschitz continuous in the first variable, similar to the proofs of (3.7) and (3.8), one can prove that
and
Substituting (5.3) and (5.4) in (5.2), we get
where θ is the same as in (3.13). It follows from (4.3) and (5.1) that
Since is -strongly monotone and -Lipschitz continuous, and is -strongly monotone and -Lipschitz continuous in the first variable, we can get
and
Combining (5.6)-(5.8), we conclude that
where ϱ is the same as in (3.13). From (4.3) and (5.1), it follows that
Because is -strongly monotone and -Lipschitz continuous, and is -strongly monotone and -Lipschitz continuous in the first variable, we can obtain
and
Substituting (5.11) and (5.12) in (5.10), deduce that
where ϑ is the same as in (3.13).
By using (5.13) and the fact that , we have
It follows from (5.9), (5.14) and the fact that that
Applying (5.5) and (5.15), it follows that
From , and , we infer that . Since , and , in view of (4.4), it is evident that the conditions of Lemma 5.1 are satisfied and so Lemma 5.1 and (5.16) guarantee that , as . Because , , and , we have , , and , as . Now, it follows from (4.4), (5.14) and (5.15) that and , as . Therefore, the sequence generated by Algorithm 4.1 converges strongly to the unique solution of SGNMVID (2.1), that is, the only element of . This completes the proof. □
Theorem 5.3 Suppose that , , (), ρ, η and γ are the same as in Theorem 3.2 and let all the conditions of Theorem 3.2 hold. Then the iterative sequence generated by Algorithm 4.2 converges strongly to the unique solution of SGNMVID (2.1).
Theorem 5.4 Let , , , (), , ρ, η and γ be the same as in Theorem 5.2 and let all the conditions of Theorem 5.2 hold. Then the iterative sequence generated by Algorithm 4.4 converges strongly to the only element of .
Proof Theorem 3.2 guarantees the existence of a unique solution for SGNMVID (2.1). Hence, Lemma 3.1 implies that , , . Since is a singleton set, by using , we conclude that and so , and . Hence, in view of Remark 4.3, for each , we can write
where the sequences and are the same as in Algorithm 4.4. Let . By using (4.6), (5.17) and the assumptions, we have
Since is -strongly monotone and -Lipschitz continuous, and is -strongly monotone and -Lipschitz continuous in the first variable, similar to the proofs of (3.7) and (3.8), one can prove that
and
Combining (5.18)-(5.20), we get
where θ is the same as in (3.13). It follows from (4.6) and (5.17) that
Substituting (5.22) in (5.21), conclude that
Like in the proofs of (5.18)-(5.23), we can verify that
and
where ϱ and ϑ are the same as in (3.13).
Let . Then, applying (5.23)-(5.25), we obtain
where λ is the same as in (3.14). Since , , and , in view of (4.4), we note that all the conditions of Lemma 5.1 are satisfied. Hence, Lemma 5.1 and (5.26) guarantee that , as . By using (4.6) and (5.17), we have
and
Since , , and , from inequalities (5.22), (5.27) and (5.28) it follows that , and , as . Hence, the sequence generated by Algorithm 4.4 converges strongly to the unique solution of SGNMVID (2.1), that is, the only element of . This completes the proof. □
Like in the proof of Theorem 5.4, one can prove the convergence of the iterative sequences generated by Algorithms 4.5 and 4.6, and we omit their proofs.
Theorem 5.5 Suppose that , , , (), , ρ, η and γ are the same as in Theorem 5.2 and let all the conditions of Theorem 5.2 hold. Then the iterative sequence generated by Algorithm 4.5 converges strongly to the only element of .
Theorem 5.6 Assume that , , (), ρ, η and γ are the same as in Theorem 3.2 and let all the conditions of Theorem 3.2 hold. Then the iterative sequence generated by Algorithm 4.6 converges strongly to the unique solution of SGNMVID (2.1).
6 An important remark on a relaxed cocoercive mapping
In view of Definition 3.1, we note that the relaxed cocoercivity condition is weaker than the strong monotonicity condition. In other words, the class of relaxed cocoercive mappings is more general than the class of strongly monotone mappings. However, it is worth to point out that if the considered mapping T is -relaxed cocoercive and γ-Lipschitz mapping such that , then it must be a -strongly monotone mapping. Hence, the results that appeared in this paper can be also applied to a class of relaxed cocoercive mappings. In fact, one may rewrite the results considered under relaxed cocoercivity and Lipschitzian conditions of mappings and apply a known result on the strongly monotone condition to a new form. Below, we present an example of the mentioned situation.
For given three different nonlinear operators and a continuous function , Noor [13] introduced and considered the problem of finding such that
which is called a system of general mixed variational inequalities involving three different nonlinear operators (SGMVID). He also considered some spacial cases of SGMVID (6.1).
He proposed the following two-step iterative algorithm and its special forms for solving SGMVID (6.1) and studied the convergence analysis of the proposed iterative algorithms under certain conditions.
Algorithm 6.1 ([13], Algorithm 3.1)
For arbitrary chosen initial points , compute the sequences and by
where for all .
Theorem 6.2 ([13], Theorem 3.1)
Let , be the solution of SGMVID (6.1). Suppose that is relaxed -cocoercive and -Lipschitzian in the first variable, and is relaxed -cocoercive and -Lipschitzian in the first variable. Let g be a relaxed -cocoercive and -Lipschitzian. If
where
and , , then for arbitrarily chosen initial points , and obtained from Algorithm 6.1 converge strongly to and , respectively.
Remark 6.3 In view of conditions (6.2) and (6.3) (conditions (4.1) and (4.2) in [13]), we note that . Now, condition (6.4) (condition (4.3) in [13]) and imply that . Accordingly, the condition should be added to conditions (6.2)-(6.4). On the other hand, since , from condition (6.4) it follows that .
Remark 6.4 The conditions (), and in (6.2) and (6.3) imply that for each . Since for each , is -relaxed cocoercive and -Lipschitz continuous, the condition () guarantees that for each , the operator is -strongly monotone. Similarly, since g is -relaxed cocoercive and -Lipschitz continuous, the condition implies that the operator g is -strongly monotone.
In view of the above remarks, one can rewrite Theorem 6.2 as follows.
Theorem 6.5 Let , be the solution of SGMVID (6.1). Let be -strongly monotone and -Lipschitz continuous in the first variable, and let be -strongly monotone and -Lipschitz continuous in the first variable. Further, let g be -strongly monotone and -Lipschitz continuous. If the constants ρ and η satisfy the following conditions:
and , then the iterative sequences and generated by Algorithm 6.1 converge strongly to and , respectively.
7 Conclusion
In this paper, we have introduced and considered a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators (SGNMVID). We have proved the equivalence between the SGNMVID and the fixed point problem, and then by this equivalent formulation, discussed the existence and uniqueness of solution of the SGNMVID. This equivalence and three nearly uniformly Lipschitzian mappings () are used to suggest and analyze some new three-step resolvent iterative schemes with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping , which is the unique solution of the SGNMVID. Several special cases are also considered. In Section 6, an important remark on a subclass of relaxed cocoercive mappings is discussed. It is expected that the results proved in this paper may stimulate further research regarding the numerical methods and their applications in various fields of pure and applied sciences.
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The first author is supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand.
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Petrot, N., Balooee, J. Fixed point problems and a system of generalized nonlinear mixed variational inequalities. Fixed Point Theory Appl 2013, 186 (2013). https://doi.org/10.1186/1687-1812-2013-186
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DOI: https://doi.org/10.1186/1687-1812-2013-186