Existence and Stability of Iterative Algorithm for a System of Random Set-Valued Variational Inclusion Problems Involving ()-Generalized Monotone Operators
Jittiporn Suwannawit1,2and Narin Petrot1
Academic Editor: Yeong-Cheng Liou
Received18 Jan 2012
Accepted25 Mar 2012
Published28 May 2012
Abstract
We introduce and study a class of a system of random set-valued variational inclusion
problems. Some conditions for the existence of solutions of such problems are provided, when the operators
are contained in the classes of generalized monotone operators, so-called ()-monotone operator. Further,
the stability of the iterative algorithm for finding a solution of the considered problem is also discussed.
1. Introduction
It is well known that the ideas and techniques of the variational inequalities are being applied in a variety of diverse fields of pure and applied sciences and proven to be productive and innovative. It has been shown that this theory provides the most natural, direct, simple, unified, and efficient framework for a general treatment of a wide class of linear and nonlinear problems. The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On the one hand, it reveals the fundamental facts on the qualitative aspects of the solutions to important classes of problems. On the other hand, it also enables us to develop highly efficient and powerful new numerical methods for solving, for example, obstacle, unilateral, free, moving, and complex equilibrium problems. Of course, the concept of variational inequality has been extended and generalized in several directions, and it is worth to noticed that, an important and useful generalization of variational inequality problem is the concept of variational inclusion. Many efficient ways have been studied to find solutions for variational inclusions and a related technique, as resolvent operator technique, was of great concern.
In 2006, Jin [1] investigated the approximation solvability of a type of set-valued variational inclusions based on the convergence of ()-resolvent operator technique, while the convergence analysis for approximate solutions much depends on the existence of Cauchy sequences generated by a proposed iterative algorithm. In the same year, Lan [2] first introduced a concept of ()-monotone operators, which contains the class of ()-monotonicity, -monotonicity (see [3β5]), and other existing monotone operators as special cases. In such paper, he studied some properties of ()-monotone operators and defined resolvent operators associated with ()-monotone operators. Then, by using this new resolvent operator, he constructed some iterative algorithms to approximate the solutions of a new class of nonlinear ()-monotone operator inclusion problems with relaxed cocoercive mappings in Hilbert spaces. After that, Verma [5] explored sensitivity analysis for strongly monotone variational inclusions using ()-resolvent operator technique in a Hilbert space setting. For more examples, ones may consult [6β11].
Meanwhile, in 2001, Verma [12] introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of such those problems. Furthermore, in 2004, Fang and Huang [13] introduced and studied some new systems of variational inclusions involving -monotone operators. By Using the resolvent operator associated with -monotone operators, they proved the existence and uniqueness of solutions for the such considered problem, and also some new algorithms for approximating the solutions are provided. Consequently, in 2007, Lan et al. [14] introduced and studied another system of nonlinear -monotone multivalued variational inclusions in Hilbert spaces. Recently, based on the generalized -resolvent operator method, Argarwal and Verma [15] considered the existence and approximation of solutions for a general system of nonlinear set-valued variational inclusions involving relaxed cocoercive mappings in Hilbert spaces. Notice that, the concept of a system of variational inequality is very interesting since it is well-known that a variety of equilibrium models, for example, the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium problem, and the general equilibrium programming problem, can be uniformly modelled as a system of variational inequalities. Additional researches on the approximate solvability of a system of nonlinear variational inequalities are problems; ones may see Cho et al. [16], Cho and Petrot [17], Noor [18], Petrot [19], Suantai and Petrot [20], and others.
On the other hand, the systematic study of random equations employing the techniques of functional analysis was first introduced by Ε paΔek [21] and HanΕ‘ [22], and it has received considerable attention from numerous authors. It is well known that the theory of randomness leads to several new questions like measurability of solutions, probabilistic and statistical aspects of random solutions estimate for the difference between the mean value of the solutions of the random equations and deterministic solutions of the averaged equations. The main question concerning random operator equations is essentially the same as those of deterministic operator equations, that is, a question of existence, uniqueness, characterization, contraction, and approximation of solutions. Of course, random variational inequality theory is an important part of random function analysis. This topic has attracted many scholars and experts due to the extensive applications of the random problems. For the examples of research works in these fascinating areas, ones may see Ahmad and BazΓ‘n [23], Huang [24], Huang et al. [25], Khan et al. [26], Lan [27], and Noor and Elsanousi [28].
In this paper, inspired by the works going on these fields, we introduce a system of set-valued random variational inclusion problems and provide the sufficient conditions for the existence of solutions and the algorithm for finding a solution of proposed problems, involving a class of generalized monotone operators by using the resolvent operator technique. Furthermore, the stability of the constructed iterative algorithm is also discussed.
2. Preliminaries
Let be a real Hilbert space equipped with norm and inner product , and let and denote for the family of all the nonempty subsets of and the family of all the nonempty closed bounded subsets of , respectively. As usual, we will define , the Hausdorff metric on , by
Let be a complete -finite measure space and the class of Borel -fields in . A mapping is said to be measurable if , for all . We will denote by a set of all measurable mappings on , that is, .
Let and be two real Hilbert spaces. Let and be single-valued mappings. Let , and be set-valued mappings, for . In this paper, we will consider the following problem: find measurable mappings and such that and
The problem of type (2.2) is called the system of random set-valued variational inclusion problem. If and are solutions of problem (2.2), we will denote by .
Notice that, if and are two single-valued mappings, then the problem (2.2) reduces to the following problem: find and such that
In this case, we will denote by . Other special cases of the problem (2.2) are presented the following. (I)If and , where and are two proper convex and lower semicontinuous functions and and denoted for the subdifferential operators of and , respectively, then (2.2) reduces to the following problem: find and such that and
for all . The problem (2.4) is called a system of random set-valued mixed variational inequalities. A special of problem (2.4) was studied in by Agarwal and Verma [15]. (II)Let be two nonempty closed and convex subsets and the indicator functions of for . If and for all and . Then the problem (2.2) reduces to the following problem: find and such that and
for all .(III)If and , where is proper convex and lower semicontinuous function and is denoted for the subdifferential operators of . Let be a nonlinear mapping and . If we set , and where , then problem (2.2) reduces to the following system of variational inequalities: find and such that
for all and . A special of problem (2.6) was studied by Argarwal et al. [29]. (IV)Let be a nonlinear mapping and two fixed constants. If , and . Then (2.5) reduces to the following system of variational inequalities: find such that and
for all and . Notice that, if , then (2.5), (2.7) are studied by Kim and Kim [30].
We now recall important basic concepts and definitions, which will be used in this work.
Definition 2.1. A mapping is called a random single-valued mapping if for any , the mapping is measurable.
Definition 2.2. A set-valued mapping is said to be measurable if , for all .
Definition 2.3. A set-valued mapping is called a random set-valued mapping if for any , the set-valued mapping is measurable.
Definition 2.4. A single-valued mapping is said to be random -Lipschitz continuous if there exists a measurable function such that
for all .
Definition 2.5. A set-valued mapping is said to be random --Lipschitz continuous if there exists a measurable function such that
for all and , where is the Hausdorff metric on .
Definition 2.6. A set-valued mapping is said to be -continuous if, for any , the mapping is continuous in , where is the Hausdorff metric on .
Definition 2.7. Let and be two random single-valued mappings. Then is said to be (i)random -Lipschitz continuous if there exists a measurable function such that
for all ;(ii)random -monotone if
for all ;(iii)random strictly -monotone if, is a random -monotone and
for all ;(iv)random -strongly monotone if there exists a measurable function such that
for all .
Definition 2.8. Let be a random single-valued mapping. A single-valued mapping is said to be (i)random -relaxed cocoercive with respect to in the second argument if there exist measurable functions such that
for all and ;(ii)random -Lipschitz continuous in the second argument if there exists a measurable function such that
for all .
Notice that, in a similar way, we can define the concepts of relaxed cocoercive and Lipschitz continuous in the third argument.
Definition 2.9. Let and be two random single-valued mappings. Then a set-valued mapping is said to be (i)random -relaxed monotone if there exists a measurable function such that
for all ;(ii)random -monotone if is a random -relaxed monotone and for all measurable function and , where .
Definition 2.10. Let be a random single-valued mapping and a random -monotone mapping. For each measurable function , the corresponding random -resolvent operator is defined by
where , and .
The following lemma, which related to operator, is very useful in order to prove our results.
Lemma 2.11. Let be a random single-valued mapping, a random -strongly monotone mapping, and a random -monotone mapping. If is a measurable function with for all , then the following are true. (i)The corresponding random -resolvent operator is a random single-valued mapping. (ii)If is a random -Lipschitz continuous mapping, then the corresponding random -resolvent operator is a random -Lipschitz continuous.
Proof. The proof is similar to Proposition 3.9 in [2].
In order to prove our main results, we also need the following well known facts.
Lemma 2.12 (see [31]). Let be a separable real Hilbert space and be a -continuous random set-valued mapping. Then for any measurable mapping , the set-valued mapping is measurable.
Lemma 2.13 (see [31]). Let be a separable real Hilbert space and two measurable set-valued mappings; be a constant and a measurable selection of . Then there exists a measurable selection of such that
Lemma 2.14 (see [32]). Let be a nonnegative real sequence, and let be a real sequence in such that . If there exists a positive integer such that
where for all and as , then .
3. Existence Theorems
In this section, we will provide sufficient conditions for the existence solutions of the problem (2.2). To do this, we will begin with a useful lemma.
Lemma 3.1. Let and be two real Hilbert spaces. Let and be single-valued mappings. Let , and be a set-valued mappings for . Assume that are random -monotone mappings and random -strongly monotone mappings, for . Then we have the following statements: (i)if , then for any measurable functions we have
(ii)if there exist two measurable functions such that
for all , then .
Proof. (i) Let be any measurable functions. Since , we have
Let be fixed. By , we obtain
This means
Thus
where . Similarly, if , we can show thatwhere. Hence (i) is proved. (ii) Assume that there exist two measurable functions such that
for all . Let be fixed. Since , then by the definition of , we see that
This implies that
That is,
Similarly, if we can show that . This completes the proof.
Due to Lemma 3.1, in order to prove our main theorems, the following assumptions should be needed.
Assumption and are separable real Hilbert spaces. are random -Lipschitz continuous single-valued mappings, for . are random -strongly monotone and random -Lipschitz continuous single-valued mappings, for . are random -monotone set-valued mappings, for . is a random --Lipschitz continuous set-valued mapping and is a random --Lipschitz continuous set-valued mapping. is a random single-valued mapping, which has the following conditions: (i) is a random -relaxed cocoercive with respect to in the third argument and a random -Lipschitz continuous in the third argument, (ii) is a random -Lipschitz continuous in the second argument. is a random single-valued mapping, which has the following conditions: (i) is a random -relaxed cocoercive with respect to in the second argument and a random -Lipschitz continuous in the second argument; (ii) is a random -Lipschitz continuous in the third argument.
Now, we are in position to present our main results.
Theorem 3.2. Assume that Assumption holds and there exist two measurable functions such that , for each and
for all . Then the problem (2.2) has a solution.
Proof. Let be a null sequence of positive real numbers. Starting with measurable mappings and . By Lemma 2.12, we know that the set-valued mappings and are measurable mappings. Consequently, by Himmelberg [33], there exist measurable selections of and of . We define now the measurable mappings and by
where , for all , and . Further, by Lemma 2.12, the set-valued mappings are measurable. Again, by Himmelberg [33] and Lemma 2.13, there exist measurable selections of and of such that
for all . Define measurable mappings and as follows:
for all . Continuing this process, inductively, we obtain the sequences , and of measurable mappings satisfy the following:
where and for all . Now, since is a random -Lipschitz continuous mapping, we have
for all . On the other hand, by Assumptions and , we see that
for all . This gives
for all . Meanwhile, since is a random -Lipschitz continuous mapping in the second argument, we get
for all . From (3.16), (3.18), and (3.19), we obtain that
where
for all . Similarly, by using Assumptions and , we know that
where
for all . Next, since is a random --Lipschitz continuous mapping and is a random --Lipschitz continuous mapping, by the choices of and , we have
for all . Now, by (3.20), (3.22), and (3.24), we obtain that
for all . This implies that
where
for all . Next, let us define a norm on by
It is well known that is a Hilbert space. Moreover, for each , we have
for all . Let
We see that as . Moreover, condition (3.11) yields that for all . This allows us to choose and a natural number such that for all . Using this one together with (3.30), we get
for all and . Thus, for each , we obtain
for all . So, for any , we have
for all . Since , it follows that converges to 0, as . This means that is a Cauchy sequence, for each . Thus, there are and such that and as , for each . Next, we will show that and converge to an element of and , for all . Indeed, for , we have from (3.24) and (3.33) that
where , for each . This implies that is a Cauchy sequence in , for all . Therefore, there exist and such that and as , for each . Furthermore,
Since and as , we have from the closedness property of and (3.35) that , for all . Similarly, we can show that , for all . Finally, in view of (3.15) and applying the continuity of and , for , we see that
for all . Thus Lemma 3.1(ii) implies that is a solution to problem (2.2). This completes the proof.
In particular, we have the following result.
Theorem 3.3. Let and be two random single-valued mappings. Assume that Assumption holds and there exist measurable functions satisfing (3.11). Then problem (2.3) has a unique solution.
Proof. From Theorem 3.2, we know that the problem (2.3) has a solution. So it remains to prove that, in fact, it has the unique solution. Assume that and such that are solutions of the problem (2.3). Using the same lines as obtaining (3.20) and (3.22), by replacing with and with , we have
and, by replacing with and with , we obtain that
where and are defined as in (3.21) and (3.23), respectively. From (3.37) and (3.38), we get
where is defined as in (3.30). Since , it follows that , for all . This completes the proof.
4. Stability Analysis
In the proof of Theorem 3.3, in fact, we have constructed a sequence of measurable mappings and show that its limit point is nothing but the unique element of . In this section, we will consider the stability of such a constructed sequence.
We start with a definition for stability analysis.
Definition 4.1. Let be real Hilbert spaces. Let , and let define an iterative procedure which yields a sequence of points in , where is an iterative procedure involving the mapping . Let and that converges to a random fixed point of . Let be an arbitrary sequence in and let , for each and . For each , if implies that , then the iteration procedure defined by is said to be -stable or stable with respect to .
Let , and , for , be random mappings defined as in Theorem 3.2. Now, for each , if is any sequence in . We will consider the sequence , which is defined by
where and and . Consequently, we put
Meanwhile, let be defined by
for all . In view of Lemma 3.1, we see that if and only if .
Now, we prove the stability of the sequence with respect to mapping , defined by (4.3).
Theorem 4.2. Assume that Assumption holds and there exist satisfing (3.11). Then for each , we have if and only if , where are defined by (4.2) and .
Proof. According to Theorem 3.3, the solution set of problem (2.3) is a singleton set, that is, . For each , let be any sequence in . By (4.1) and (4.2), we have
Since is a random -Lipschitz continuous mapping, by Assumptions and Lemma 3.1(i), we get
On the other hand, by Assumptions and , we see that
This gives
Meanwhile, since is a random -Lipschitz continuous mapping in the second argument, we get
From (4.5)β(4.8), we obtain that
where
Similarly, since is a random -Lipschitz continuous mapping, by Assumption , and Lemma 3.1, we obtain that
where
Thus
where , for all . So
In view of (4.14), if , we see that Lemma 2.14 implies
On the other hand, by using (4.5) and (4.11), we see that
for all . Consequently, if for each we assume , we will have . This completes the proof.
Remark 4.3. Theorem 4.2 shows that the iterative sequence , which has constructed in Theorem 3.3, is -stable.
5. Conclusion
We have introduced a new system of set-valued random variational inclusions involving -monotone operator and random relaxed cocoercive operators in Hilbert space. By using the resolvent operator technique, we have constructed an iterative algorithm and then the approximation solvability of a aforesaid problem is examined. Moreover, we have considered the stability of such iterative algorithm. It is worth noting that for a suitable and appropriate choice of the operators, as , one can obtain a large number of various classes of variational inequalities; this means that problem (2.2) is quite general and unifying. Consequently, the results presented in this paper are very interesting and improve some known corresponding results in the literature.
Acknowledgment
J. Suwannawit is supported by the Centre of Excellence in Mathematics, Commission on Higher Education, Thailand. N. Petrot is supported by the National Research Council of Thailand (Project no. R2554B102).
References
M.-M. Jin, βIterative algorithm for a new system of nonlinear set-valued variational inclusions involving -monotone mappings,β Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 2, article 72, p. 10, 2006.
R. U. Verma, β-monotonicity and applications to nonlinear variational inclusion problems,β Journal of Applied Mathematics and Stochastic Analysis, vol. 2004, no. 2, pp. 193β195, 2004.
R. U. Verma, βApproximation-solvability of a class of A-monotone variational inclusion problems,β Journal of the Korean Society for Industrial and Applied Mathematics, vol. 8, no. 1, pp. 55β66, 2004.
R. U. Verma, βSensitivity analysis for generalized strongly monotone variational inclusions based on the -resolvent operator technique,β Applied Mathematics Letters, vol. 19, no. 12, pp. 1409β1413, 2006.
Y. J. Cho and H.-Y. Lan, βGeneralized nonlinear random -accretive equations with random relaxed cocoercive mappings in Banach spaces,β Computers & Mathematics with Applications, vol. 55, no. 9, pp. 2173β2182, 2008.
H.-Y. Lan, βApproximation solvability of nonlinear random -resolvent operator equations with random relaxed cocoercive operators,β Computers & Mathematics with Applications, vol. 57, no. 4, pp. 624β632, 2009.
H. Y. Lan, Y. S. Cui, and Y. Fu, βNew approximation solvability of general nonlinear operator inclusion couples
involving ()-resolvent operators and relaxed cocoercive type operators,β Communications in Nonlinear Science and Numerical Simulation, vol. 17, pp. 1844β1851, 2012.
H.-Y. Lan, Y. J. Cho, and W. Xie, βGeneral nonlinear random equations with random multivalued operator in Banach spaces,β Journal of Inequalities and Applications, vol. 2009, Article ID 865093, 17 pages, 2009.
H.-Y. Lan, Y. Li, and J. Tang, βExistence and iterative approximations of solutions for nonlinear implicit fuzzy resolvent operator systems of -monotone type,β Journal of Computational Analysis and Applications, vol. 13, no. 2, pp. 335β344, 2011.
R. U. Verma, βProjection methods, algorithms, and a new system of nonlinear variational inequalities,β Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 1025β1031, 2001.
Y. P. Fang and N. J. Huang, β-monotone operators and system of variational inclusions,β Communications on Applied Nonlinear Analysis, vol. 11, no. 1, pp. 93β101, 2004.
H.-Y. Lan, J. H. Kim, and Y. J. Cho, βOn a new system of nonlinear -monotone multivalued variational inclusions,β Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 481β493, 2007.
R. P. Agarwal and R. U. Verma, βGeneral system of -maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms,β Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 238β251, 2010.
Y. J. Cho, Y. P. Fang, N. J. Huang, and H. J. Hwang, βAlgorithms for systems of nonlinear variational inequalities,β Journal of the Korean Mathematical Society, vol. 41, no. 3, pp. 489β499, 2004.
Y. J. Cho and N. Petrot, βOn the system of nonlinear mixed implicit equilibrium problems in Hilbert spaces,β Journal of Inequalities and Applications, vol. 2010, Article ID 437976, 12 pages, 2010.
N. Petrot, βA resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems,β Applied Mathematics Letters, vol. 23, no. 4, pp. 440β445, 2010.
S. Suantai and N. Petrot, βExistence and stability of iterative algorithms for the system of nonlinear quasi-mixed equilibrium problems,β Applied Mathematics Letters, vol. 24, no. 3, pp. 308β313, 2011.
R. Ahmad and F. F. BazΓ‘n, βAn iterative algorithm for random generalized nonlinear mixed variational inclusions for random fuzzy mappings,β Applied Mathematics and Computation, vol. 167, no. 2, pp. 1400β1411, 2005.
N.-J. Huang, X. Long, and Y. J. Cho, βRandom completely generalized nonlinear variational inclusions with non-compact valued random mappings,β Bulletin of the Korean Mathematical Society, vol. 34, no. 4, pp. 603β615, 1997.
M. F. Khan, Salahuddin, and R. U. Verma, βGeneralized random variational-like inequalities with randomly pseudo-monotone multivalued mappings,β Panamerican Mathematical Journal, vol. 16, no. 3, pp. 33β46, 2006.
H.-Y. Lan, βProjection iterative approximations for a new class of general random implicit quasi-variational inequalities,β Journal of Inequalities and Applications, vol. 2006, Article ID 81261, 17 pages, 2006.
M. A. Noor and S. A. Elsanousi, βIterative algorithms for random variational inequalities,β Panamerican Mathematical Journal, vol. 3, no. 1, pp. 39β50, 1993.
R. P. Agarwal, Y. J. Cho, and N. Petrot, βSystems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces,β Fixed Point Theory and Applications, vol. 2011, article 31, 10 pages, 2011.
J. K. Kim and D. S. Kim, βA new system of generalized nonlinear mixed variational inequalities in Hilbert spaces,β Journal of Convex Analysis, vol. 11, no. 1, pp. 117β124, 2004.
S. S. Chang, Fixed Point Theory with Applications, Chongqing Publishing, Chongqing, China, 1984.
X. Weng, βFixed point iteration for local strictly pseudo-contractive mapping,β Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 727β731, 1991.