1Scientific Computing Key Laboratory of Shanghai Universities and Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 2Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan
Very recently, Ahmad and Yao (2009) introduced and considered a system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. In this paper we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between general system of generalized resolvent equations and general system of variational inclusions. The iterative algorithms for finding the approximate solutions of general system of generalized resolvent equations are proposed. The convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. Our results represent the generalization, improvement, supplement, and development of Ahmad and Yao corresponding ones.
1. Introduction and Preliminaries
It is well known that the theory of variational inequalities has played an important role in the investigation of a wide class of problems arising in mechanics, physics, optimization and control, nonlinear programming, elasticity, and applied sciences and so on; see, for example, [1–7] and the references therein. In recent years variational inequalities have been extended and generalized in different directions. A useful and significant generalization of variational inequalities is called mixed variational inequalities involving the nonlinear term [8], which enables us to study free, moving, obstacle, equilibrium problems arising in pure and applied sciences in a unified and general framework. Due to the presence of the nonlinear term, the projection method and its variant forms including the technique of the Wiener-Hopf equations cannot be extended to suggest the iterative methods for solving mixed variational inequalities. To overcome these drawbacks, Hassouni and Moudafi [9] introduced variational inclusions which contain mixed variational inequalities as special cases. They studied the perturbed method for solving variational inclusions. Subsequently, M. A. Noor and K. I. Noor [10] introduced and considered the resolvent equations by virtue of the resolvent operator concept and established the equivalence between the mixed variational inequalities and the resolvent equations. The technique of resolvent equations is being used to develop powerful and efficient numerical techniques for solving mixed (quasi)variational inequalities and related optimization problems. At the same time, some iterative algorithms for approximating a solution of some system of variational inequalities are also introduced and studied in Verma [11]. Pang [12], Cohen and Chaplais [13], Binachi [14], Ansari and Yao [15] considered a system of scalar variational inequalities and Pang showed that the traffic equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled as a system of variational inequalities. As generalizations of system of variational inequalities, Agarwal et al. [16] introduced a system of generalized nonlinear mixed quasi-variational inclusions and investigated the sensitivity analysis of solutions for the system of generalized mixed quasi-variational inclusions in Hilbert spaces. In 2007, Peng and Zhu [17] considered and studied a new system of generalized mixed quasi-variational inclusions with -monotone operators and Lan et al. [18] studied a new system of nonlinear -monotone multivalued variational inclusions. Furthermore, for more details in the related research work of this field, we invoke the readers to see, for instance, [19–30]. Very recently, Ahmad and Yao [31] introduced and considered a new system of variational inclusions in uniformly smooth Banach spaces, which covers the system of variational inclusions in Hilbert spaces considered by [18]. They established an equivalence relation between this system of variational inclusions and a system of generalized resolvent equations, proposed a number of iterative algorithms for this system of variational inclusions, and also gave the convergence criteria.
Let be a real Banach space with its norm the topological dual of , and the metric induced by the norm . Let (resp., ) be the family of all nonempty closed and bounded subsets (resp., all nonempty subsets) of and the Hausdorff metric on defined by
where and . We write by the normalized duality mapping defined as
where denotes the duality pairing between and .
The uniform convexity of a Banach space means that for any , there exists , such that for any ensure the following inequality:
The function
is called the modulus of convexity of .
The uniform smoothness of a Banach space means that for any given , there exists such that
holds. The function
is called the modulus of smoothness of .
It is well known that the Banach space is uniformly convex if and only if for all , and it is uniformly smooth if and only if . All Hilbert spaces, (or ) spaces (), and the Sobolov spaces are 2-uniformly smooth, while, for (or ) and spaces are -uniformly smooth.
Proposition 1.1 (see [15]). Let be a uniformly smooth Banach space. Then the normalized duality mapping is single-valued, and for any there holds the following:(i),(ii), where .
Definition 1.2 (see [32]). A mapping is said to be (i)-strongly accretive, , if for any , there exists such that(ii)Lipschitz continuous if for any , there exists a constant , such that
Definition 1.3 (see [13]). A set-valued mapping is said to be (i)accretive, if for any , there exists such that for all and ,
(ii)-strongly accretive, , if for any , there exists , such that for all and ,
(iii)-accretive if is accretive and , for every (equivalently, for some) , where is the identity mapping (equivalently, if is accretive and ). In particular, it is clear from [9] that if is a Hilbert space, than is an -accretive mapping if and only if it is a maximal monotone mapping.
Definition 1.4 (see [31]). Let be an -accretive mapping. For any , the mapping associated with defined by
is called the resolvent operator.
Definition 1.5 (see [33]). The resolvent operator is said to be a retraction if
It is well known that is a single-valued and nonexpansive mapping.
Definition 1.6 (see [10]). A set-valued mapping is said to be -Lipschitz continuous if for any , there exists a constant such that
Let and be two real Banach spaces, and single-valued mappings, and and any four multivalued mappings. Let and be any nonlinear mappings, and nonlinear mappings with and , respectively. Then we consider the problem of finding such that
which is called a general system of variational inclusions. In particular, if and , where and are single-valued mappings, then the general system of variational inclusions (1.14) reduces to the following system of variational inclusions
which was considered by Lan et al. [18] in Hilbert spaces and studied by Ahmad and Yao [31] in Banach spaces, respectively.
Proposition 1.7 (see [31, Lemma 2.1]). is a solution of the system of variational inclusions (1.15) if and only if satisfies
Proposition 1.8 (see [31, Proposition 3.1]). The system of variational inclusions (1.15) has a solution with and if and only if the following system of generalized resolvent equations
has a solution with and , where and .
Based on the above Propositions 1.7 and 1.8, Ahmad and Yao [31] presented the following algorithm and established the following strong convergence result for the sequences generated by the algorithm.
Algorithm 1.9 (see [31, Algorithm 3.1]). For given and , compute , and by the iterative scheme:
Theorem 1.10 (see [31, Theorem 3.1]). Let and be two real uniformly smooth Banach spaces with modulus of smoothness and for , respectively. Let be -Lipschitz continuous mappings with constants and , respectively, and let be -accretive mappings such that the resolvent operators associated with and are retractions. Let be both strong accretive with constants and , respectively, and Lipschitz continuous with constants and , respectively. Let be Lipschitz continuous with constants and , respectively, and let be Lipschitz continuous in the first and second arguments with constants and , respectively. If there exist constants and , such that
where and ,,,, then there exist and satisfying the system of generalized resolvent equations (1.17) (in this case, is a solution of system of variational inclusions (1.15)), and the iterative sequences , and generated by Algorithm 1.9 converge strongly to , and , respectively.
In this paper we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. Motivated and inspired by the above Proposition 1.8, we establish an equivalence relation between general system of generalized resolvent equations and general system of variational inclusions. By using Nadler [34] we propose some new iterative algorithms for finding the approximate solutions of general system of generalized resolvent equations, which include Ahmad and Yao’s corresponding algorithms as special cases to a great extent. Furthermore, the convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. There is no doubt that our results represent the generalization, improvement, supplement, and development of Ahmad and Yao corresponding ones [31].
2. Main Results
Let and be two real Banach spaces, let and be single-valued mappings, and let and be any four multivalued mappings. Let and be any nonlinear mappings, and nonlinear mappings with and , respectively. Then we consider the problem of finding such that
where and are the resolvent operators associated with and , respectively.
The corresponding general system of variational inclusions of (2.1) is the problem (1.14), that is, find such that
Proposition 2.1. are solutions of general system of variational inclusions (1.14) if and only if satisfies
Proof. The proof of Proposition 2.1 is a direct consequence of the definition of resolvent operator, and hence, is omitted.
Next we first establish an equivalence relation between general system of generalized resolvent equations (2.1) and general system of variational inclusions (1.14) and then prove the existence of a solution of (2.1) and convergence of sequences generated by the proposed algorithms.
Proposition 2.2. The general system of variational inclusions (1.14) has a solution with and if and only if general system of generalized resolvent equations (2.1) has a solution with ,, where
and and .
Proof. Let be a solution of general system of variational inclusions (1.14). Then, by Proposition 2.1, it satisfies the following system of equations
Let and . Then we have
and hence and . Thus it follows that
that is,
Therefore, is a solution of general system of generalized resolvent equations (2.1). Conversely, let be a solution of general system of generalized resolvent equations (2.1). Then
Now observe that
which leads to
and also that
which leads to
Consequently, we have
Therefore, by Proposition 2.1, is a solution of general system of variational inclusions (1.14).
Proof (Alternative). Let
Then, utilizing (2.4), we can write
which yield that
the required general system of generalized resolvent equations.
Algorithm 2.3. For given ,,, compute
For , we take such that and . Then, by Nadler [34], there exist such that
where is the Hausdorff metric on (for the sake of convenience, we also denote by the Hausdorff metric on ). Compute
By induction, we can obtain sequences ,,, by the iterative scheme:
for . The general system of generalized resolvent equations (2.1) can also be rewritten as
Utilizing this fixed-point formulation, we suggest the following iterative algorithm.
Algorithm 2.4. For given ,,,, compute
For , we take such that and . Then, by Nadler [34], there exist such that
where is the Hausdorff metric on (for the sake of convenience, we also denote by the Hausdorff metric on ). Compute
By induction, we can obtain sequences ,,, by the iterative scheme:
for . For positive stepsize , the general system of generalized resolvent equations (2.1) can also be rewritten as
This fixed point formulation enables us to propose the following iterative algorithm.
Algorithm 2.5. For given ,,,, compute and such that
Then, by Nadler [34], there exist such that
where is the Hausdorff metric on (for the sake of convenience, we also denote by the Hausdorff metric on ). Compute and such that
By induction, we can obtain sequences by the iterative scheme:
for . Note that for , Algorithm 2.5 reduces to the following algorithm which solves the general system of variational inclusions (1.14).
Algorithm 2.6. For given , compute such that
Then, by Nadler [34], there exist such that
where is the Hausdorff metric on (for the sake of convenience, we also denote by the Hausdorff metric on ). Compute such that
By induction, we can obtain sequences by the iterative scheme:
for .
We now study the convergence analysis of Algorithm 2.3. In a similar way, one can study the convergence of other algorithms.
Theorem 2.7. Let and be two real uniformly smooth Banach spaces with modulus of smoothness and for , respectively. Let , be -Lipschitz continuous mappings with constants , and , respectively, and let be -accretive mappings such that the resolvent operators associated with and are retractions. Let be both strong accretive with constants and , respectively, and Lipschitz continuous with constants and , respectively. Let be Lipschitz continuous with constants and , respectively, and Lipschitz continuous in the first and second arguments with constants and , respectively. If there exist constants and , such that
where , and ,,,, then there exist and satisfying the general system of generalized resolvent equations (2.1) (in this case, is a solution of general system of variational inclusions (1.14)), and the iterative sequences , , and generated by Algorithm 2.3 converge strongly to , and , respectively.
Proof. From Algorithm 2.3 we have
By Proposition 1.1, we have (see, e.g., the proof of [32, Theorem 3])
Since is Lipschitz continuous in both arguments, are -Lipschitz continuous, and is Lipschitz continuous, we have
Utilizing (2.41) and Proposition 1.1, we have
which implies that
