ISRN Applied MathematicsVolume 2012 (2012), Article ID 731058, 29 pagesdoi:10.5402/2012/731058

Research Article

A New System of Random Generalized Variational Inclusions with Random Fuzzy Mappings and Random --Accretive Mappings in Banach Spaces

Sayyedeh Zahra Nazemi

Department of Mathematics, Yasouj University, Yasouj 75914, Iran

Received 25 November 2011; Accepted 19 December 2011

Academic Editor: Y. Song

Copyright © 2012 Sayyedeh Zahra Nazemi. 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 a new notion of random --accretive mappings and prove the Lipschitz continuity of the random resolvent operator associated with the random --accretive mappings. We introduce and study a new system of random generalized variational inclusions with random --accretive mappings and random fuzzy mappings in Banach spaces. By using the random resolvent operator, an iterative algorithm for solving such system of random generalized variational inclusions is constructed in Banach spaces. Under some suitable conditions, we prove the convergence of the iterative sequences generated by the algorithm.

1. Introduction

Variational inclusions and variational inequalities have wide applications to many fields including, for example, economics, optimization and control theory, operators research, transportation network modeling, and mathematical programming. For these reasons, various variational inclusions and variational inequalities have been intensively studied in recent years. For details, we refer the reader to [1–3] and references therein.

Recently, Wang and Ding [4], and Nazemi [5] introduced the cocepts of -accretive mappings, and --accretive mappings, respectively, which generalize the notion of -accretive mappings, -accretive mappings, -accretive mappings, -accretive mappings and other existing accretive mappings as special cases. They also defined the resolvent operators associated with this mappings, and shown their Lipschitz continuity.

On the other hand, it is well known that the study of the random equations involving the random operators in view of their need for dealing with probabilistic models in applied sciences is very important. In 1989, Chang and Zhu [6] and Chang [7] first introduced the concepts of variational inequalities for fuzzy mappings. Since then several classes of variational inequalities for fuzzy mappings were considered by Chang and Huang [8], Noor [9, 10], Noor and Al-Said [11], Ding [12], Ding and Park [13], and Park and Jeong [14, 15] in the setting of Hilbert spaces. Recently, the random variational inequalities and random variational inclusion problems have been introduced and studied by Dai [16], Cho and Lan [17], Lan [18], Ahmad and Bazán [19], Chang and Huang [20, 21], Zang and Zhu [22], Huang [23, 24], Husain et al. [25], etc.

Motivated and inspired by the recent research works in this fascinating area, in this paper, we introduce a new class of random --accretive mappings and study a new system of random generalized variational inclusions with random --accretive mappings and random fuzzy mappings in Banach spaces. An iterative algorithm is defined to compute the approximate solutions of system of random generalized variational inclusions. The convergence of iterative sequences generated by the algorithm is also shown. Our results improve and generalize many known corresponding results.

2. Preliminaries

Let be a real Banach space with dual space and the dual pair between and . Let be a collection of all fuzzy sets over . A mapping from into is called a fuzzy mapping on . If is a fuzzy mapping on , then (denote it by in the sequel) is a fuzzy set on and is the membership function of in . Let , , then the set is called a -cut set of .

Let be a measurable space, where is a set and is a -algebra of subsets of . Let , , , and be the class of Borel -fields in , the family of all nonempty subsets of , the family of all nonempty, closed and bounded subsets of , and the Hausdorff metric on , respectively. Recall that the generalized duality mapping is defined by where is a constant. In particular, is the usual normalized duality mapping. It is known that, in general, for all , and is single-valued if is strictly convex. In the sequel, unless otherwise specified, we always suppose that is a real Banach space such that is single-valued. If is a Hilbert space, then becomes the identity mapping on .

The modulus of smoothness of is the function defined by A Banach space is called uniformly smooth if is called -uniformly smooth if there exists a constant such that Note that is single-valued if is uniformly smooth. In the following, we give an important inequality due to Xu [26] in -uniformly smooth Banach spaces, which will play a crucial role in constructing the iterative algorithm to approximate the solution of system of variational inclusions.

Lemma 2.1 (see [26]). Let be a real uniformly smooth Banach space. Then is -uniformly smooth if and only if there exists a constant such that

Definition 2.2. A mapping is said to be measurable if for any , .

Definition 2.3. A mapping is called a random single-valued mapping if for any , is measurable. A random mapping is said to be continuous if for any , the mapping is continuous. Similarly, we can define a random mapping . We will write and for all and .It is well known that a measurable mapping is necessarily a random mapping.

Definition 2.4. A multivalued mapping is said to be measurable if for any , .

Definition 2.5. A mapping is called a measurable section of a multivalued mapping if is measurable and for any , .

Definition 2.6. A mapping is called a random multivalued mapping if for any , is measurable. A random multivalued mapping is said to be -continuous if for any , is continuous in , where is the Hausdorff metric on defined as follows: for any given ,

Definition 2.7. A fuzzy mapping is called measurable if for any , is a measurable multivalued mapping.

Definition 2.8. A fuzzy mapping is called a random fuzzy mapping if for any , is a measurable fuzzy mapping.

Lemma 2.9 (see [27]). Let be a -continuous random multivalued mapping. Then for any measurable mapping the multivalued mapping is measurable.

Lemma 2.10 (see [27]). Let be two measurable multivalued mappings, a constant, and a measurable selection of . Then there exists a measurable selection of T such that for all ,

Definition 2.11. A random mapping is said to be(i)accretive if(ii)-strongly accretive if there exists a measurable function such that(iii)-relaxed accretive if there exists a measurable function such that(iv)-relaxed cocoercive if there exist measurable functions , such that(v)-Lipschitz continuous if there exists a measurable function such that

Definition 2.12. A random multivalued mapping is said to be(i)accretive if(ii)-accretive if is accretive and holds for all and .

Remark 2.13. If is a Hilbert space, then we can obtain corresponding definitions of monotonicity, strongly monotonicity, relaxed monotonicity, and maximal monotonicity from Definitions 2.11 and 2.12.

Definition 2.14. A random mapping is said to be -Lipschitz continuous if there exists a measurable function such that

Definition 2.15. Let and be random single-valued mappings. Then a random multivalued mapping is said to be(i)-accretive if(ii)strictly -accretive if and equality holds if and only if ;(iii)-strongly -accretive if there exists a measurable function such that(iv)-relaxed -accretive if there exists a measurable function such that(v)--accretive if is -accretive and holds for all and ;(vi)-accretive if is accretive and holds for all and ;(vii)-accretive if is -accretive and holds for all and ;(viii)-accretive if is -relaxed -accretive and holds for all and .

Definition 2.16. Let be a single-valued mapping and and random single-valued mappings. Then(i) is said to be -strongly -accretive with respect to if there exists a measurable function such that (ii) is said to be -relaxed -accretive with respect to if there exists a measurable function such that (iii) is said to be -Lipschitz continuous with respect to in the first argument if there exists a measurable function such that

Example 2.17. Let be a Hilbert space and a -Lipschitz continuous and accretive mapping, , , and , for all , and . Then is -strongly -accretive with respect to and is -relaxed -accretive with respect to .

Definition 2.18. Let be real Banach spaces and for each , let be a random single-valued mapping. Then is said to be -Lipschitz continuous in the th argument if there exists a measurable function such that

Definition 2.19. Let be a Hilbert space, a single-valued mapping, and random single-valued mappings. is said to be coercive with respect to and if

Definition 2.20. Let be a Hilbert space, a single-valued mapping and random single-valued mappings. is said to be bounded with respect to and if is bounded for every bounded subset of and . is said to be hemicontinuous with respect to and if for any fixed , the function is continuous at .

3. Random --Accretive Mappings

In this section, we will introduce a new class of random accretive mappings---accretive mappings and discuss some properties of random --accretive mappings.

Definition 3.1. Let be a single-valued mapping and and random single-valued mappings. Then the random multivalued mapping is said to be --accretive with respect to and if is -relaxed -accretive and

Remark 3.2. (i) If and is -accretive, then the random --accretive mapping reduces to the random --accretive mapping. (ii) If , for all , , then the random --accretive mapping reduces to the random -accretive mapping.

Remark 3.3. (i) If is -accretive, , and , for all , , then the random --accretive mapping reduces to the random -accretive mapping studied by Uea and Kumam [28].(ii) If is -relaxed -accretive, , and , for all , , then the random --accretive mapping reduces to the random -accretive mapping studied by Cho and Lan [17].(iii) If is -accretive, , , for all , , then the random --accretive mapping reduces to the random -accretive mapping studied by Zhang [29].

Example 3.4. Let be a Hilbert space, a measurable function; for every , , ; a random maximal monotone mapping, a bounded, coercive, and hemicontinuous mapping, with respect to and , and -strongly -accretive with respect to in the first argument and -relaxed -accretive with respect to in second argument. Then is a random --accretive mapping with respect to and .

Proof. For every , is maximal monotone since is maximal monotone. Since is bounded, coercive, and hemicontinuous with respect to and -strongly -accretive with respect to in the first argument and -relaxed -accretive with respect to in the second argument, it follows from Corollary 32.26 of [30] that holds for every . Thus is a random --accretive mapping with respect to and .

Example 3.5. Let , , , , , , and , for all . Then This implies that is surjective. Thus is a random --accretive mapping with respect to and .

Definition 3.6. Let , be random single-valued mappings, be -strongly -accretive with respect to and -relaxed -accretive with respect to and be a random --accretive mapping with respect to and . Then the general resolvent operator is defined by

Remark 3.7. (i) If is -accretive, , and , for all , , then the resolvent operator reduces to the resolvent operator introduced by Uea and Kumam [28].(ii) If is -relaxed -accretive, and , for all , , then the resolvent operator reduces to the resolvent operator introduced by Cho and Lan [17].(iii) If is -accretive, , , for all , , then the resolvent operator reduces to the resolvent operator introduced by Zhang [29].

Theorem 3.8. Let be a random single-valued mapping, be -Lipschitz continuous and be -strongly -accretive with respect to and -relaxed -accretive with respect to and , . Let be a random --accretive mapping with respect to and . Then the resolvent operator is -Lipschitz continuous for , where , , that is,

Proof. Let be any given points; it follows from Definition 3.6 that This implies that Since is -relaxed -accretive, we get From (3.7) and Lipschitz continuity of , we have Hence This completes the proof.

4. A New System of Random Generalized Variational Inclusions

In this section, we will introduce a new system of random generalized variational inclusions with random --accretive mappings and construct an iterative algorithm for solving this system of random generalized variational inclusions.

Let for , , and be random fuzzy mappings satisfying the following condition .For , there exist mappings , , such that

By using the random fuzzy mappings , , we can define random multivalued mappings , as follows. where .

In the sequel, for , and are called the random multivalued mappings induced by the random fuzzy mappings and , respectively.

For , given mappings , , let , , be random fuzzy mappings, be a single-valued mapping and , , be random single-valued mappings and be a random --accretive mapping with respect to and in the second argument with , for each and . Now we consider the following system of random generalized variational inclusions:

For , find measurable mappings such that for all and for each , , , …, , such that

If for , is a Hilbert space and , for all , and , where denote the subdifferential of a proper, convex, and lower semicontinuous function , then system of random generalized variational inclusions (4.3) reduces to the following system of random generalized mixed variational inequalities.

For , find measurable mappings such that for all and for each , , , , …, such that

We remark that system of random generalized variational inclusions (4.3) and system of random generalized mixed variational inequalities (4.4) include as special cases, many kinds of random variational inequalities and random variational inclusions of [17, 19–21], etc.

Lemma 4.1. For , let be a single-valued mapping and , random single-valued mappings such that be -strongly -accretive with respect to and -relaxed -accretive with respect to . Then the set of measurable mappings is a random solution of problem (4.3) if and only if for all , , , , , and where .

Algorithm 4.2. Suppose that for , , be random fuzzy mappings satisfying the condition (C). Let for , , , be -continuous random multivalued mappings induced by and , respectively. Let be a single-valued mapping; , , random single-valued mappings; the random multivalued mapping be --accretive with respect to and in the second argument. For any given measurable mapping , the multivalued mappings , , are measurable by Lemma 2.9. Hence for , there exist measurable selections of , of of and of , by Himmelberg [31]. Let for , It is easy to see that for , is measurable. By Lemma 2.10, there exist measurable selections of , of , of and of such that for all and , Let for , then is measurable. Continuing the above process inductively, we can define the following random iterative sequences , , and for solving problem (4.3) as follows: for any , and .

Theorem 4.3. For , let be a -uniformly smooth Banach space and be a -Lipschitz continuous mapping satisfying and , for all . Let be -Lipschitz continuous, be a single-valued mapping, and random single-valued mappings such that be -strongly -accretive with respect to and -relaxed -accretive with respect to and be -Lipschitz continuous with respect to and -Lipschitz continuous with respect to . Let be -relaxed cocoercive and -Lipschitz continuous, and be -strongly accretive and -Lipschitz continuous. Suppose that be -Lipschitz continuous in the th argument for , be -Lipschitz continuous in the th argument for . Let be a random multivalued mapping such that is a --accretive mapping with respect to and in the second argument. Let , ,,, be random fuzzy mappings satisfying the condition (C) and , , and random multivalued mappings induced by and , respectively. Suppose that , , are -continuous with constants , , …, , and , respectively.In addition, if then there exist measurable mappings , such that (4.3) holds. Moreover for all , , , where , and , are random sequences obtained by Algorithm 4.2.

Proof. For , from (4.10), Lemma 4.1, and Algorithm 4.2, we have Since be -relaxed cocoercive and -Lipschitz continuous, we have Using -strongly accretivity and -Lipschitz continuity of , -Lipschitz continuity of we have Since is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to , we have Since is -Lipschitz continuous in the th argument, by (4.15), we have Since is -Lipschitz continuous in the th argument, for , we have It follows from the Lipschitz continuity of , the -continuity of that It follows from (4.12)–(4.18) that for each , Therefore, where Let Then , for all . Define on by It is easy to see that is a Banach space.Define . Then, we have From the condition (4.11), we know that for all , , and hence there exists an and such that , for all . Therefore, by (4.20) and (4.24), we have It follows from (4.25) that Hence, for any , it follows that Since , for all , it follows from (4.27) that as , and hence is a Cauchy sequence in . By the same argument, we also have that is a Cauchy sequence in . Thus there exist measurable mappings , such that for all , , as .Now we prove that , ,, . In fact, it follows from the Lipschitz continuity of , , and Algorithm 4.2 that for , From (4.28), we know that , , are also Cauchy sequences. Therefore, there exist , , such that , as . Further, for , Since is closed, we have . Similarly, , . This completes the proof.

Now we show the existence of solutions for problem (4.4).

Lemma 4.4. Problem (4.4) is equivalent to find measurable mappings , such that for all and , , , such that where is a measurable function and . If for , is a Hilbert space and , for all , and , then Algorithm 4.2 reduces to Algorithm 4.5.

Algorithm 4.5. For any given measurable mapping , compute the following random iterative sequences , , , …, for solving problem (4.4) as follows: for any , , .

From Theorem 4.3, we have the following Theorem.

Theorem 4.6. For , let be a Hilbert space, be -relaxed cocoercive and -Lipschitz continuous, and be -strongly accretive and -Lipschitz continuous. Let be -Lipschitz continuous in the th argument for , and be -Lipschitz continuous in the th argument for . Let , be random fuzzy mappings satisfying the condition (C); , be random multivalued mappings induced by , respectively. Suppose that , are -continuous with constants , , respectively. If there exists a constant such that Then there exist measurable mappings , such that (4.4) holds. Moreover, for all , , , , where , are random sequences obtained by Algorithm 4.5.

References

R. Ahmad, Q. H. Ansari, and S. S. Irfan, “Generalized variational inclusions and generalized resolvent equations in Banach spaces,” Computers & Mathematics with Applications, vol. 49, no. 11-12, pp. 1825–1835, 2005.

R. P. Agarwal, Y. J. Cho, and N.-J. Huang, “Generalized nonlinear variational inclusions involving maximal -monotone mappings,” in Nonlinear Analysis: Theory, Methods & Applications, pp. 59–73, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003, To V. Lakshmikantham on this 80th Brithday, Vol. 1, 2.

R. U. Verma, “New class of nonlinear A-monotone mixed variational inclusion problems and resolvent operator technique,” Journal of Computational Analysis and Applications, vol. 8, no. 3, pp. 275–285, 2006.

Z. B. Wang and X. P. Ding, “-accretive operators with an application for solving set-valued variational inclusions in Banach spaces,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1559–1567, 2010.

S. Z. Nazemi, “--accretive mappings and a new system of generalized variational inclusions with --accretive mappings in Banach spaces,” ISRN Applied Mathematics, vol. 2011, Article ID 709715, 20 pages, 2011.

S.-S. Chang and Y. G. Zhu, “On variational inequalities for fuzzy mappings,” Fuzzy Sets and Systems, vol. 32, no. 3, pp. 359–367, 1989.

S. S. Chang, Variational Inequality and Complementarity Problems Theory and Applications, Shanghai Scientific and Technological Literature Publishing House, Shanghai, China, 1991.

S.-S. Chang and N. J. Huang, “Generalized complementarity problems for fuzzy mappings,” Fuzzy Sets and Systems, vol. 55, no. 2, pp. 227–234, 1993.

M. A. Noor, “Variational inequalities for fuzzy mappings. I,” Fuzzy Sets and Systems, vol. 55, no. 3, pp. 309–312, 1993.

M. A. Noor, “Variational inequalities for fuzzy mappings. II,” Fuzzy Sets and Systems, vol. 97, no. 1, pp. 101–107, 1998.

M. A. Noor and E. A. Al-Said, “Quasi variational inequalities for fuzzy mappings,” Fuzzy Sets and Systems, vol. 3, no. 2, pp. 89–96, 1997.

X. P. Ding, “Generalized implicit quasi-variational inclusions with fuzzy set-valued mappings,” Computers & Mathematics with Applications, vol. 38, no. 1, pp. 71–79, 1999.

X. P. Ding and J. Y. Park, “A new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings,” Journal of Computational and Applied Mathematics, vol. 138, no. 2, pp. 243–257, 2002.

J. Y. Park and J. U. Jeong, “Generalized strongly quasivariational inequalities for fuzzy mappings,” Fuzzy Sets and Systems, vol. 99, no. 1, pp. 115–120, 1998.

J. Y. Park and J. U. Jeong, “A perturbed algorithm of variational inclusions for fuzzy mappings,” Fuzzy Sets and Systems, vol. 115, no. 3, pp. 419–424, 2000.

H.-X. Dai, “Generalized mixed variational-like inequality for random fuzzy mappings,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 20–28, 2009.

Y. J. Cho and H.-Y. Lan, “Generalized nonlinear random -accretive equations with random relaxed cocoercive mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 55, no. 9, pp. 2173–2182, 2008.

H.-Y. Lan, “Approximation solvability of nonlinear random -resolvent operator equations with random relaxed cocoercive operators,” Computers & Mathematics with Applications, vol. 57, no. 4, pp. 624–632, 2009.

R. Ahmad and F. F. Bazán, “An iterative algorithm for random generalized nonlinear mixed variational inclusions for random fuzzy mappings,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1400–1411, 2005.

S. S. Chang and N. J. Huang, “Generalized random multivalued quasi-complementarity problems,” Indian Journal of Mathematics, vol. 35, no. 3, pp. 305–320, 1993.

S. S. Chang and N. J. Huang, “Random generalized set-valued quasi-complementarity problems,” Acta Mathematicae Applicatae Sinica, vol. 16, pp. 396–405, 1993.

S. S. Zhang and Y. G. Zhu, “Problems concerning a class of random variational inequalities and random quasivariational inequalities,” Journal of Mathematical Research and Exposition, vol. 9, no. 3, pp. 385–393, 1989.

N. J. Huang, “Random general set-valued strongly nonlinear quasivariational inequalities,” Journal of Sichuan University, vol. 31, no. 4, pp. 420–425, 1994.

N. J. Huang, “Random generalized set-valued implicit variational inequalities,” Journal of Liaoning Normal University, vol. 18, pp. 89–93, 1995.

T. Husain, E. Tarafdar, and X. Z. Yuan, “Some results on random generalized games and random quasi-variational inequalities,” Far East Journal of Mathematical Sciences, vol. 2, no. 1, pp. 35–55, 1994.

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

S. S. Chang, Fixed Point Theory with Applications, Chongping Publishing House, Chongping, China, 1984.

N. O. Uea and P. Kumam, “A generalized nonlinear random equations with random fuzzy mappings in uniformly smooth Banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 728452, 15 pages, 2010.

W. B. Zhang, “Random nonlinear variational inclusions involving -accretive operator for random fuzzy mappings in Banach spaces,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 34, no. 2, pp. 389–402, 2011.

Zeidler, Nonlinear Functional Analysis and its Applications II: Monotone Operators, Springer, Berlin, Germany, 1985.

C. J. Himmelberg, “Measurable relations,” Fundamenta Mathematicae, vol. 87, pp. 53–72, 1975.