- Open Access
Graph convergence for the -mixed mappingwith an application for solving the system of generalized variationalinclusions
© Husain et al.; licensee Springer. 2013
Received: 30 July 2013
Accepted: 10 October 2013
Published: 19 November 2013
In this paper, we investigate a class of accretive mappings called the-mixed mappingsin Banach spaces. We prove that the proximal-point mapping associated with the-mixed mapping issingle-valued and Lipschitz continuous. Some examples are given to justify thedefinition of -mixed mapping.Further, a concept of graph convergence concerned with the-mixed mapping isintroduced in Banach spaces and some equivalence theorems betweengraph-convergence and proximal-point mapping convergence for the-mixed mappingssequence are proved. As an application, we consider a system of generalizedvariational inclusions involving -mixed mappingsin real q-uniformly smooth Banach spaces. Using the proximal-pointmapping method, we prove the existence and uniqueness of solution and suggest aniterative algorithm for the system of generalized variational inclusions.Furthermore, we discuss the convergence criteria for the iterative algorithmunder some suitable conditions.
MSC: 47J19, 49J40, 49J53.
Variational inclusions, as the generalization of variational inequalities, have beenwidely studied in recent years. Some of the most interesting and important problemsin the theory of variational inclusions include variational, quasi-variational,variational-like inequalities as special cases. For applications of variationalinclusions, we refer to . Various kinds ofiterative methods have been studied to solve the variational inclusions. Among thesemethods, the proximal-point mapping technique for the study of variationalinclusions has been widely used by many authors. For details, we refer to[2–20].
In 2001, Huang and Fang  were the first tointroduce the generalized m-accretive mapping and give the definition ofthe proximal-point mapping for the generalized m-accretive mapping inBanach spaces. Since then a number of researchers have investigated several classesof generalized m-accretive mappings such as H-accretive,-accretive,-proximal-point,-accretive,A-maximal relaxed accretive, -accretive mappings.For details, we refer to [2, 3, 6, 7, 11, 14, 16, 18].
Recently, Zou and Huang [19, 20] introduced and studied -accretive mappings;Kazmi et al.[8–10] introduced and studied generalized -accretive mappings,-η-proximal-point mappings. Veryrecently, Li and Huang  studied thegraph convergence for the -accretive mappingand showed the equivalence between graph convergence and proximal-point mappingconvergence for the -accretive mappingsequence in a Banach space, and Verma studied the graph convergence for an A-maximal relaxed monotone mapping andgave the equivalence between the graph convergence and the proximal-point mappingconvergence for the A-maximal relaxed monotone mapping sequence in aHilbert space. They extended the concept of graph convergence introduced andconsidered by Attouch .
Motivated by the research work going on in this direction, we consider a class ofaccretive mappings called -mixed mappings, anatural generalization of accretive (monotone) mappings in Banach spaces. Forrelated work, we refer to [2–4, 11, 14, 16, 18–20]. We prove that theproximal-point mapping of the -mixed mapping issingle-valued and Lipschitz continuous and extends the concept of proximal-pointmappings associated with the -accretive mappingsto the -mixed mappings.Further, we study the graph convergence for the -mixed mappings. Wepresent an equivalence theorem between graph convergence and proximal-point mappingconvergence for the -mixed mappingsequence in Banach spaces. As an application, we consider a system of generalizedvariational inclusions involving the -mixed mappings inreal q-uniformly smooth Banach spaces. Using the proximal-point mappingmethod, we prove the existence and uniqueness of solution and suggest an iterativealgorithm for the system of generalized variational inclusions. Furthermore, wediscuss the convergence criteria of the iterative algorithm under some suitableconditions. Our results can be viewed as a generalization of some known resultsgiven in [12, 17, 19–21].
Let X be a real Banach space equipped with the norm , and let be the topological dual space of X. Let be the dualpair between X and ,and let be the power set of X.
If a real Hilbert space, then becomesan identity mapping on H.
A Banach space X is called smooth if, for everywith ,there exists a unique such that .
- (i)uniformly smooth if
- (ii)q-uniformly smooth, for , if there exists a constant such that
Note that is single-valued if X is uniformly smooth. Concerned with thecharacteristic inequalities in q-uniformly smooth Banach spaces, Xu proved the following result.
From Lemma 2 of Liu , it is easy tohave the following lemma.
- (i)G is said to be accretive if
- (ii)G is said to be ξ-strongly accretive if there exists a constant such that
- (iii)G is said to be μ-cocoercive if there exists a constant such that
- (iv)G is said to be -Lipschitz continuous if there exists a constant such that
- (v)G is said to be α-expansive if there exists a constant such that
if ,then it is expansive.
- (i)is said to be μ-cocoercive with respect to A if there exists a constant such that
- (ii)is said to be γ-relaxed accretive with respect to B if there exists a constant such that
- (iii)is said to be -Lipschitz continuous with respect to A if there exists a constant such that
- (iv)is said to be -Lipschitz continuous with respect to B if there exists a constant such that
for all scalers and for all .
Suppose that isdefined by ,then is-cocoercivewith respect to A and m-relaxed accretive with respect toB, and -Lipschitzcontinuous with respect to A and -Lipschitzcontinuous with respect to B.
that is, is-Lipschitzcontinuous with respect to B.
- (i)η is said to be τ-Lipschitz continuous if there exists a constant such that
- (ii)M is said to be accretive if
- (iii)M is said to be -strongly accretive if there exists a constant such that
- (iv)M is said to be m-relaxed accretive if there exists a constant such that
- (v)M is said to be η-accretive if
M is said to be strictly η-accretive if M is η-accretive and equality holds if and only if ;
- (vii)M is said to be γ-strongly η-accretive if there exists a constant such that
- (viii)M is said to be α-relaxed η-accretive if there exists a constant such that
M is said to be m-accretive if M is accretive and for all , where I denotes the identity operator on X;
M is said to be generalized m-accretive if M is η-accretive and for all ;
M is said to be H-accretive if M is accretive and for all ;
M is said to be -accretive if M is η-accretive and for all ;
M is said to be -accretive if M is m-relaxed η-accretive and for all .
Let ,be three single-valued mappings. Let be a set-valued mapping. Then M is said to be -accretivewith respect to A and B if M is accretive and for all.
In this section, we introduce the -mixed mapping andshow some of its properties.
M is m-relaxed accretive;
for all .
Example 3.2 Let X, H, A, B be the same asin Example 2.8, and bedefined by ,.
Furthermore, M is also an -mixed mapping since for any.
It implies that since .By (3.1), we have ,a contradiction. This completes the proof. □
Theorem 3.4 Let the set-valued mappingbe an-mixed mappingwith respect to mappings A and B. If A is α-expansive andwith,thenis single-valued.
which is like (3.3). Hence it follows that .This implies that and sois single-valued. □
Now we prove that the proximal-point mapping defined by (3.4) is Lipschitzcontinuous.
Let and .
This completes the proof. □
4 Graph convergence for an -mixedmapping
In this section we shall introduce the graph convergence for the-mixed mapping.
Proof It follows from Theorem 3.6 that and are both-Lipschitzcontinuous.
and so .This completes the proof. □
5 An application of the -mixed mappingfor solving the system of generalized variational inclusions
Throughout the rest of the paper, unless otherwise stated, we assume that for each,is a -uniformlysmooth Banach space with the norm .
where ,are zero vectors of and,respectively. The problem of type (5.1) was studied by Zou and Huang .
- (i)κ-strongly accretive in the first argument with respect to A if there exists a constant such that
- (ii)-Lipschitz continuous in the first argument if there exists a constant such that
- (iii)-Lipschitz continuous in the second argument if there exists a constant such that
The following lemma, which will be used in the sequel, is an immediate consequence ofthe definitions of ,.
Then SGVI (5.1) has a unique solution.
Proof For ,it follows that for ,the proximal-point mappings and are-Lipschitzcontinuous and -Lipschitzcontinuous, respectively.
It follows from Lemma 5.2 that is a unique solutionof SGVI (5.1). This completes the proof. □
6 Convergence of an iterative algorithm for SGVI (5.1)
Based on Lemma 5.2, we suggest and analyze the following iterative algorithm forfinding an approximate solution for SGVI (5.1).
where and are constants.
Then the approximate solutiongenerated by Algorithm 6.1 converges strongly to the uniquesolutionof SGVI (5.1).
By condition (6.3), it follows that andLemma 2.5 implies that as .
Thus converges strongly to the unique solution of SGVI (5.1). Thiscompletes the proof. □
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