Yosida approximation equations technique for system of generalized set-valued variational inclusions
© Cao; licensee Springer. 2013
Received: 18 May 2013
Accepted: 26 September 2013
Published: 7 November 2013
In this paper, under the assumption with no continuousness, a new system of generalized variational inclusions in the Banach space is introduced. By using the Yosida approximation operator technique, the existence and uniqueness theorems for solving this kind of variational inclusion are established.
MSC:49H09, 49J40, 49H10.
Variational inclusions are useful and important extensions and generalizations of the variational inequalities with a wide range of applications in industry, mathematical finance, economics, decisions sciences, ecology, mathematical and engineering sciences. In general, the method based on the resolvent operator technique has been widely used to solve variational inclusions.
In this paper, under the assumption with no continuousness, we first introduce a new system of generalized variational inclusions in the Banach space. By using the Yosida approximation technique for m-accretive operator, we prove some existence and uniqueness theorems of solutions for this kind of system of generalized variational inclusions. Our results generalize and improve main results in [1–7].
This problem is called the system of generalized set-valued variational inclusions.
There are some special cases in literature.
where denotes the generalized duality paring. In the sequel, we shall denote the single-valued normalized duality map by j. It is well known that if E is smooth, then J is single-valued, and is uniformly convex, then j is uniformly continuous on bounded set.
We assume that E, , are smooth Banach spaces. For convenience, the norms of E, and are all denoted by . The norm of is defined by , i.e., if , then .
- (i)T is said to be accretive, if , , ,
- (ii)T is said to be α-strongly-accretive if there exists such that , , ,
T is said to be m-accretive if T is accretive and , .
- (i)The mapping is said to be accretive if , , , ,
- (ii)The mapping is said to be α-strongly-accretive if there exists such that , , , ,
The mapping is said to be m-α-strongly-accretive if is α-strongly-accretive and , , .
In a similar way, we can define the strong accretiveness of the mapping with respect to the second argument.
The resolvent operator of T is defined by , ,.
The Yosida approximation of T is defined by , , .
In the sequel, we use the notation → and ⇀ to denote strong and weak convergence, respectively.
is single-valued and , ;
is m-accretive on E, and , , ;
If is uniformly convex Banach space, then T is demiclosed, i.e., , , implies that .
is -Lipschitz continuous;
- (ii)By definition of and (i), we have
This completes the proof of (ii). □
is m--strongly-accretive in the ith argument ();
Proof (i) The fact directly follows from Kobayashi  (Theorem 5.3).
Therefore, . This completes the proof of (ii).
(iii) The proof is similar. We omit it. □
2 Main results
We assume that in the family of all nonempty closed and bounded subset of E.
Lemma 2.1 
then there exists such that , .
is -strongly-accretive in the ith argument and -mixed Lipschitz continuous, , .
- (i)for any λ in (2.1), there exists such that(2.2)
and and are bounded;
if , are bounded, then there exists unique , which is a solution of Problem (1.1), such that , as .
Remark 2.1 Equation (2.2) is called the system of Yosida approximation inclusions (equations).
where , . By (2.1), . Therefore, by Lemma 2.1, for λ in (2.1), there exists such that , , i.e., satisfies (2.3), and hence (2.2) hold.
It follows from (2.10) and (2.11) that and are bounded since .
Consequently, and are the Cauchy net. There exists such that , as from which and and , it follows that and as .
Equations (2.1), (2.13) and (2.14) imply that , . □
then Problem (1.1) has a unique solution.
Since () is uniformly continuous, map bounded set in to bounded set. Hence, (2.21) and (2.22) imply that and are bounded. □
for , , , , then Problem (1.1) has a unique solution.
which implies that . Similarly, . This completes the proof of Theorem 2.3. □
3 Conclusion and future perspective
Two of the most difficult and important problems in variation inclusions are the establishment of system of variational inclusions and the development of an efficient numerical methods. A new system of generalized variational inclusions in the Banach space under the assumption with no continuousness is introduced, and some existence and uniqueness theorems of solutions for this kind of system of generalized variational inclusions are proved by using the Yosida approximation technique for m-accretive operator.
More approaches [13–15], which have been applied in variational inequalities, could be manipulated in variational inclusions. We will make further research to solve this kind of system of generalized variational inclusions by using extragradient method and implicit iterative methods.
The author thanks for the guidance and support my supervisor Prof. Li-Wei Liu who taught at the department of mathematics in Nanchang university. The author thanks the anonymous referees for reading this paper carefully, providing valuable suggestions and comments. The work was supported by the National Science Foundation of China (No. 10561007) and the Youth Science Foundation of Jiangxi Provincial Department of Science and Technology (No. 20122BAB211021).
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