Abstract and Applied Analysis
Volume 2014 (2014), Article ID 659870, 11 pages
http://dx.doi.org/10.1155/2014/659870

Research Article

Stable Perturbed Iterative Algorithms for Solving New General Systems of Nonlinear Generalized Variational Inclusion in Banach Spaces

Ting-jian Xiong¹ and Heng-you Lan1,2

¹Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China
²Key Laboratory Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things, Zigong, Sichuan 643000, China

Received 19 June 2014; Accepted 26 July 2014; Published 14 October 2014

Academic Editor: Jong Kyu Kim

Copyright © 2014 Ting-jian Xiong and Heng-you Lan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce and study a new general system of nonlinear variational inclusions involving generalized -accretive mappings in Banach space. By using the resolvent operator technique associated with generalized -accretive mappings due to Huang and Fang, we prove the existence theorem of the solution for this variational inclusion system in uniformly smooth Banach space, and discuss convergence and stability of a class of new perturbed iterative algorithms for solving the inclusion system in Banach spaces. Our results presented in this paper may be viewed as an refinement and improvement of the previously known results.

1. Introduction

Let be a given positive integer, for any , a real Banach space with dual space . , all endowed with the norm , and the dual pair between and (as matter of convenience). Let denote the family of all the nonempty subsets of , , single-valued mappings, and generalized -accretive mapping for . In this paper, we consider the following new general system for nonlinear variational inclusion involving generalized -accretive mappings. Find such that

for all . Some special cases of the problem (1) had been studied by many authors. See, for example, [1–34] and the reference therein. Here, we mention some of them as follows.

Case 1. The problem (1) with , the Hilbert spaces, was introduced and studied as general system of monotone nonlinear variational inclusions problems by Peng and Zhao [29].
If and , is proper, convex, and lower semi-continuous functional on , and denote the subdifferential operators of the for , then the problem (1) is equivalent to finding such that

When , 2-uniformly smooth Banach space with the smooth constant , is a nonempty closed convex subset of , , where and and for ; the problem (2) reduces to the following system of finding such that

Further, in the problem (3), when is the indicator function of a nonempty closed convex set , in defined by

then the system (3) reduces to finding such that

which was introduced and studied by Zhu et al. [34].

Case 2. If , then the system (3) is equivalent to finding such that

It is easy to see that the mathematical model studied by Saewan and Kumam [31] is a variant of (6).

Case 3. If , then the problem (1) reduces to find such that

Problem (7) is called a system of strongly nonlinear quasivariational inclusion involving generalized -accretive mappings, it is considered and studied by Lan [19]. There are many special cases of the problems (7) that can be found in [3, 7, 12–14, 17, 20, 28, 30] and the references cited therein.

Case 4. If and , then the problem (1) reduces to finding such that

which was introduced and studied by Fang and Huang [8]. We remark that for appropriate and suitable choices of positive integer , the mappings , , and , and the spaces for , one can know that the problem (1) includes a number of general class of variational character known problems, including minimization or maximization (whether constraint or not) of functions and minimax problems et al. as special cases. For more details, see [1–34] and the reference therein.
On the other hand, many authors discussed stability of the iterative sequence generated by the algorithm for solving the problems that they studied. Lan [19] introduced the notion of -stable or stable with respect to . Moreover, Agarwal et al. [1, 2], Jin [16], Kazmi and Bhat [18], and Lan and Kim [21] constructed some stability under suitable conditions, respectively.
Motivated and inspired by the above works, the main purpose of this paper is to introduce and study the new general system of nonlinear variational inclusions (1) involving generalized -accretive mapping in uniformly smooth Banach spaces. By using the resolvent operator technique for generalized -accretive, we prove the existence theorem of the solution for this kind of system of variational inclusions in Banach spaces and discuss the convergence and stability of a new perturbed iterative algorithm for solving this general system of nonlinear variational inclusions in Banach spaces.

2. Preliminaries

In order to get the main results of the paper, we need the following concepts and lemmas. Let be a real Banach space with dual space , the dual pair between and , and denote the family of all the nonempty subsets of . The generalized duality mapping is defined by

where is a constant. In particular, is the usual normalized duality mapping. It is known that if is strictly convex or is a uniformly smooth Banach space, then is single-valued (see [33]), and if , the Hilbert space, then becomes the identity mapping on . We will denote the single-valued duality mapping by .

In order to construct convergence and stability for researching the problem (1), we need to be using the following definition and lemma.

Definition 1. Let be Banach spaces, and let be single mappings for . Then is said to be

(i)-strongly accretive with respect to th argument if for any , , there exists , such that

 where is a constant;(ii)-Lipschitz continuous if there exists constants , , such that

 for all and .

Remark 2. When , is different or the same as Hilbert spaces, (i) and (ii) in Definition 1 reduce to strongly monotonicity with respect to th argument of and -Lipschitz continuity of , respectively (see [29]).

Definition 3. Let be single-valued mapping. Then set-valued mapping is said to be

(i)accretive if

(ii)-accretive if

(iii)-accretive if is accretive and for all , where denotes the identity operator on ;(iv)generalized -accretive if is -accretive and for all .

Remark 4. When , (i)–(iv) of Definition 3 reduce to the definitions of monotone operators, -monotone operators, classical maximal monotone operators, and maximal -monotone operators; if , then (ii) and (iv) of Definition 3 reduce to the definitions of accretive and -accretive of uniformly smooth Banach spaces (see [10, 11]).

Definition 5. The mapping is said to be

(i)-strongly monotone if there exists a constant such that

(ii)-Lipschitz continuous if there exists a constant such that

In [10], Huang and Fang show that for any , inverse mapping is single-valued, if is strict monotone and is generalized -accretive mapping, where is the identity mapping. Based on this fact, Huang and Fang [10] gave the following definition.

Definition 6. Let be strictly monotone mapping, and let be generalized -accretive mapping. Then the resolvent for is defined as follows:

where is a constant and denotes the identity mapping on for .

Lemma 7 (see [10, 11]). Let be -Lipschitz continuous and -strongly monotone, and let be generalized -accretive mapping. Then for any , the resolvent operator for is -Lipschitz continuous; that is,

The modules of smoothness is a measure, it is depicted geometric structure of the underlying Banach space. The modules of smoothness of Banach space are the function defined by

A Banach space is called uniformly smooth if . is called -uniformly smooth if there exists a constant such that , where is a real number.

Remark that is single-valued if is uniformly smooth, and Hilbert space and (or ) spaces are 2-uniformly smooth Banach spaces. In what follows, we will denote the single-valued generalized duality mapping by .

In the study of characteristic inequalities in -uniformly smooth Banach spaces, Xu [35] proved the following result.

Lemma 8. Let be a given real number and let be a real uniformly smooth Banach space. Then is -uniformly smooth if and only if there exists a constant such that for all , , there holds the following inequality:

Definition 9. Let be a self-map of , , and let define an iteration procedure which yields a sequence of points in . Suppose that and converges to a fixed point of . Let and let . If implies that , then the iteration procedure defined by is said to be -stable or stable with respect to .

Lemma 10 (see [36]). Let , , and be three nonnegative real sequences satisfying the following condition: there exists a natural number such that

where , , , and . Then converges to as .

3. Existence Theorem

In this section, we will give the existence theorem of the problem (1). The solvability of the problem (1) depends on the equivalence between (1) and the problem of finding the fixed point of the associated generalized resolvent operator. It follows from the definition of generalized resolvent operator that we can obtain the following conclusion.

Lemma 11. Let , single-valued mappings, and generalized -accretive mapping for . Then the following statements are mutually equivalent.

(i)An element is a solution to the problem (1).(ii)There is an such that

 where , and is constants for all .(iii)For any given constants , the map is defined by

 for all and , has a fixed point , where maps are defined by

 for and .

Proof. We first prove that (i) (ii). Let satisfy the relation in (ii). Then, the definition of resolvent operator implies that this equality holds if and only if

for ; that is

where . Thus is the solution of the problem (1).
Next, we show (ii) (iii). If satisfy following relation:

then, for any , it follows from

that

Hence, is a fixed point of the mapping

Conversely, if is a fixed point of the mapping , then

for . Hence, from

we have

for . Therefore satisfy the relation of (ii).

Theorem 12. Let be a real -uniformly smooth Banach space with and let be -Lipschitz continuous and -strongly monotone for any . Suppose that is generalized -accretive mapping, and is -strongly accretive in the th argument and -Lipschitz continuous for . If

where is the constants as in Lemma 8 for , then problem (1) has a unique solution .

Proof. For any given and , we first define as follows:

for all . Now define on by

It is easy to see that is a Banach space. In fact

(i), the negative being satisfied;(ii)for all real number ,

 homogeneity being satisfied;(iii)for all ,

 the triangle inequality being satisfied;(iv)let ; that is, ; this implies that ; thus ; we get is a norm on the ;(v)let is Cauchy sequence; that is, for , there exists a positive integer ; let ; we have

 Thus, for all , we have ; that is, is also Cauchy sequence; thus for ; we get and is a cluster point on the ; we claim is a Banach space.

Now, by (34), for any given , define mapping by

where for .
In the sequel, we prove that is a contractive mapping on the . In fact, for any and , it follows from (34) and Lemma 7 that

By assumptions and Lemma 8, we have

From (40)-(41), we obtain

for . Equation (42) implies that

where . By (33), we know that . It follows from (43) that

This proves that is a contraction mapping. Hence, there exists a unique such that

that is, for ; that is,

By Lemma 11, is the unique solution of problem (1). This completes the proof.

Remark 13. If , then Theorem 12 reduces to Theorem 3.2 of Lan [19].

Corollary 14. Let be real Hilbert space and be -Lipschitz continuous and -strongly monotone for any . Suppose that is maximal -monotone mapping, is -strongly monotone in the th argument, and -Lipschitz continuous for . If

then problem (1) has a unique solution .

Corollary 15. Let be real Hilbert space for any . Suppose that is proper, convex, and lower semicontinuous functional on and is -strongly monotone in the th argument and -Lipschitz continuous for . If

then problem (2) has a unique solution .

4. Perturbed Iterative Algorithms

In this section, by using Definition 9 and Lemma 10, we construct a new perturbed iterative algorithm with mixed errors for solving problem (1) and prove the convergence and stability of the iterative sequence generated by the algorithm.

Algorithm 16. Let and be single-valued mappings and let be generalized -accretive mapping for . For any given initial point , the perturbed iterative sequence for problem (1) is defined by

where , , is a sequence in , and are errors to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions:

(i);(ii);(iii), for .

Let be any sequence in and define by

for .

Algorithm 17. Let and be single-valued mappings and let be maximal -monotone mapping for . For any given initial point , the perturbed iterative sequence for problem (1) is defined by

where , , is a sequence in , and are errors to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions:

(i);(ii) for .

Let be any sequence in and define by

for .

Algorithm 18. Let be single-valued mappings and is proper, convex, and lower semi-continuous functional on for . For any given initial point , the perturbed iterative sequence for problem (2) is defined by

where , , is a sequence in , are errors to take into account a possible inexact computation of the resolvent operator point satisfying the condition for . Let be any sequence in and define by

for .

Remark 19. If , then Algorithm 16 reduces to Algorithm 4.3 of Lan [19].
Next we will show the convergence and stability of Algorithm 16.

Theorem 20. Suppose that , , , and are the same as in Theorem 12. If and condition (33) holds, then the perturbed iterative sequence defined by Algorithm 16 converges strongly to the unique solution of the problem (1). Moreover, if there exists for all , then

if and only if

where is defined by (50).

Proof. From Theorem 12, we know that problem (1) has a unique solution

It follows from (49) and the proof of (42) in Theorem 12 that, for ,

It follows from (58), we have

where is the same as in (43). Letting , , and , then it follows from and (i)–(iii) of Algorithm 16 that

Setting , then (59) can be rewritten as

It follows from Lemma 10 that ; that is,

thus

Hence, we know that the sequence converges strongly to the unique solution of the problem (1).
Now we prove the second conclusion. By (50), now we know

where . As the proof of inequality (59), we have

Since , it follows from (64) and (65) that

Suppose that . Letting and , then (66) implies that

where , are the same as previously. Since and ,

It again follows from Lemma 10, we have and so

Conversely, if , it follows from (50), then we get

Combining (65) with (70), we have

This completes the proof.

Corollary 21. Suppose that , , , and are the same as in Corollary 14. If and condition (47) holds, then the perturbed iterative sequence defined by Algorithm 17 converges strongly to the unique solution of the problem (1). Moreover, if there exists for all , then if and only if

where is defined by (50).

Corollary 22. Assume that , , and are the same as in Corollary 15. If and condition (48) holds, then the perturbed iterative sequence defined by Algorithm 18 converges strongly to the unique solution of the problem (2). Moreover, if there exists for all , then if and only if

where is defined by (54).

Remark 23. If , then Theorem 20 reduces to Theorem 4.3 of Lan [19]. Further, one can easily see that our results presented in this paper may be viewed as an refinement and improvement of the previously known results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work has been partially supported by Sichuan Province Cultivation Fund Project of academic and technical Leaders, the Open Research Fund of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (2013WZJ01), and the Cultivation Project of Sichuan University of Science and Engineering (2011PY01).

References

R. P. Agarwal, Y. J. Cho, J. Li, and N. J. Huang, “Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mapping in q-uniformly smooth Banach Spaces,” Journal of Mathematical Analysis and Applications, vol. 272, no. 2, pp. 435–447, 2002.

R. P. Agarwal, J. W. Chen, Y. J. Cho, and Z. P. Wan, “Stability analysis for parametric generalized vector quasi-variational-like inequality problems,” Journal of Inequalities and Applications, vol. 2012, article 57, 2012.

M. Alimohammady and M. Roohi, “A system of generalized variational inclusion problems involving ()-monotone mappings,” Filomat, vol. 23, no. 1, pp. 13–20, 2009.

X. P. Ding, “General algorithm of solutions for nonlinear variational inequalities in Banach space,” Computers & Mathematics with Applications, vol. 34, no. 9, pp. 131–137, 1997.

X. P. Ding, “Iterative process with errors to locally strictly pseudocontractive maps in Banach spaces,” Computers & Mathematics with Applications, vol. 32, no. 10, pp. 91–97, 1996.

X. P. Ding, “Perturbed proximal point algorithms for generalized quasivariational inclusions,” Journal of Mathematical Analysis and Applications, vol. 210, no. 1, pp. 88–101, 1997.

Y. P. Fang, N. J. Huang, and H. B. Thompson, “A new system of variational inclusions with -monotone operators in Hilbert spaces,” Computers & Mathematics with Applications, vol. 49, no. 2-3, pp. 365–374, 2005.

Y. P. Fang and N. J. Huang, “H-monotone operator and resolvent operator technique for variational inclusion,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 795–803, 2003.

F. Gürsoy, V. Karakaya, and B. E. Rhoades, “Some convergence and stability results for the Kirk multistep and Kirk-SP fixed point iterative algorithms,” Abstract and Applied Analysis, vol. 2014, Article ID 806537, 12 pages, 2014.

N. J. Huang and Y. P. Fang, “Generalized m-accretive mappings in Banach spaces,” Journal of Sichuan University, vol. 38, no. 4, pp. 591–592, 2001.

N. J. Huang, Y. P. Fang, and C. X. Deng, “Nonlinear variational inclusions involving generalized m-accretive mappings,” in Preceedings of the Bellman Continum: International Workshop on Uncertain Systems and Soft Computing, pp. 323–327, Beijing, China, 2002.

J. U. Jeong, “Sensitivity analysis for a system of extended generalized nonlinear quasi-variational inclusions in -uniformly smooth Banach spaces,” International Mathematical Forum, vol. 7, no. 3, pp. 2465–2480, 2012.

M. M. Jin, “Iterative algorithm for a new system of nonlinear set-valued variational inclusions involving -monotone mappings,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 3, article 114, 2006.

M. M. Jin, “A new system of general nonlinear variational inclusions involving (A, η)-accretive mappings in Banach Spaces,” Mathematical Inequalities and Applications, vol. 11, no. 4, pp. 783–794, 2008.

M. M. Jin, “Perturbed iterative algorithms for generalized nonlinear set-valued quasivariational inclusions involving generalized m-accretive mappings,” Journal of Inequalities and Applications, vol. 2007, Article ID 29863, 12 pages, 2007.

M. M. Jin, “Perturbed algorithm and stability for strongly nonlinear quasi-variational inclusion involving -accretive operators,” Mathematical Inequalities & Applications, vol. 9, no. 4, pp. 771–779, 2006.

K. R. Kazmi and M. I. Bhat, “Iterative algorithm for a system of nonlinear variational-like inclusions,” Computers & Mathematics with Applications, vol. 48, no. 12, pp. 1929–1935, 2004.

K. R. Kazmi and M. I. Bhat, “Convergence and stability of iterative algorithms of generalized set-valued variational-like inclusions in Banach spaces,” Applied Mathematics and Computation, vol. 166, no. 1, pp. 164–180, 2005.

H. Y. Lan, “Stability of perturbed iterative algorithm for solving a system of generalized nonlinear equations,” Nonlinear Functional Analysis and Applications, vol. 14, no. 1, pp. 1–11, 2009.

H. Y. Lan and Q. K. Liu, “Iterative approximation for a system of nonlinear variational inclusions involving generalized -accretive mappings,” Nonlinear Analysis Forum, vol. 9, no. 1, pp. 33–42, 2004.

H. Y. Lan and J. K. Kim, “Stable perturbed iteration procedures for solving new strongly nonlinear operator inclusions in Banach spaces,” Nonlinear Functional Analysis and Application, vol. 18, no. 3, pp. 433–444, 2013.

H. G. Li, “A nonlinear inclusion problem involving -NODM set-valued mappings in ordered Hilbert space,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1384–1388, 2012.

H. G. Li, D. Qiu, J. M. Zheng, and M. M. Jin, “Perturbed Ishikawa-hybrid quasi-proximal point algorithm with accretive mappings for a fuzzy system,” Fixed Point Theory and Applications, vol. 2013, article 281, 2013.

H. G. Li, D. Qiu, and M. M. Jin, “GNM ordered variational inequality system with ordered Lipschitz continuous mappings in an ordered Banach space,” Journal of Inequalities and Applications, vol. 2013, article 514, 2013.

H. G. Li, D. Qiu, and Y. Zou, “Characterizations of weak-ANODD set-valued mappings with applications to an approximate solution of GNMOQV inclusions involving ⊕ operator in ordered Banach spaces,” Fixed Point Theory and Applications, vol. 2013, article 241, 2013.

H. G. Li, A. J. Xu, and M. M. Jin, “A Hybrid proximal point three-step algorithm for nonlinear set-valued quasi-variational inclusions system involving -accretive mappings,” Fixed Point Theory and Applications, vol. 2010, Article ID 635382, 2010.

H. G. Li, A. J. Xu, and M. M. Jin, “An Ishikawa-hybrid proximal point algorithm for nonlinear set-valued inclusions problem based on -accretive framework,” Fixed Point Theory and Applications, vol. 2010, Article ID 501293, 12 pages, 2010.

Z. Q. Liu, J. S. Ume, and S. M. Kang, “On a system of nonlinear variational inclusions with Hh,n-monotone operators,” Abstract and Applied Analysis, vol. 2012, Article ID 643828, 21 pages, 2012.

J. W. Peng and L. J. Zhao, “General system of -monotone nonlinear variational inclusions problems with applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 364615, 13 pages, 2009.

P. Sunthrayuth and P. Kumam, “Iterative algorithms approach to a general system of nonlinear variational inequalities with perturbed mappings and fixed point problems for nonexpansive semigroups,” Journal of Inequalities and Applications, vol. 2012, article 133, 2012.

S. Saewan and P. Kumam, “Existence and algorithm for solving the system of mixed variational inequalities in Banach spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 413468, 15 pages, 2012.

M. J. Shang, X. F. Su, and X. L. Qin, “An iterative method for a variational inequality and a fixed point problem for nonexpansive mappings,” Acta Mathematica Scientia A, vol. 30, no. 4, pp. 1126–1137, 2010.

Y. C. Xu, X. F. He, Z. B. Hou, and Z. He, “Generalized projection methods for Noor variational inequalities in Banach spaces,” Acta Mathematica Scientia Shuxue Wuli Xuebao: Chinese Edition, vol. 30, no. 3, pp. 808–817, 2010.

J. H. Zhu, S. S. Chang, and M. Liu, “Algorithms for a system of general variational inequalities in Banach spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 580158, 18 pages, 2012.

H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.

L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114–125, 1995.