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Journal of Function Spaces
Volume 2014 (2014), Article ID 918796, 8 pages
A System of Differential Set-Valued Variational Inequalities in Finite Dimensional Spaces
1Department of Mathematics, Sichuan University, Chengdu 610064, China
2School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China
Received 29 May 2013; Revised 5 November 2013; Accepted 19 November 2013; Published 4 February 2014
Academic Editor: John R. Akeroyd
Copyright © 2014 Wei Li et al. 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.
A system of differential set-valued variational inequalities is introduced and studied in finite dimensional Euclidean spaces. An existence theorem of weak solutions for the system of differential set-valued variational inequalities in the sense of Carathéodory is proved under some suitable conditions. Furthermore, a convergence result on Euler time-dependent procedure for solving the system of differential set-valued variational inequalities is also given.
For a set-valued mapping and a nonempty closed convex set in , the VI, is to find and such that for all . Let denote the solution set of this problem. We write for the time-derivative of a function . In this paper, we consider the following system of differential set-valued variational inequalities: where , , , , and () are given mappings.
In , Pang and Stewart introduced a class of differential variational inequalities in finite dimensional Euclidean spaces. For some related results, we refer to [2–17]. Recently, the differential variational inequalities have been used in cellular biology (see ). In , the authors needed two or more variational inequalities to formulate the switching between the metabolic models. Sometimes it is convenient to apply the differential vector variational inequalities in  to show the fermentation dynamics. However, when we study the fermentation model (20) in , we find that the system (1) in this paper can help us a lot.
In this paper, we establish an existence theorem of weak solutions for the system (1) in the sense of Carathéodory under some suitable conditions. Furthermore, we give a convergence result on Euler time-dependent procedure for solving the system (1).
In this section, we will introduce some basic notations and preliminary results.
In the rest of this paper, we will use the following assumptions (A) and (B).(A), , , , and are Lipschitz continuous functions on with Lipschitz constants , , , , and , respectively.(B) is bounded on with ; is bounded on with .
Definition 1. A set-valued map is said to be(i)monotone on a convex set if for each pair of points , and for all and , ;(ii)pseudo monotone on a convex set if for each pair of points , and for all and , implies that .
Definition 2. A function is said to be Lipschitz continuous if there exists a constant such that, for any ,
Definition 3. Let , be topological spaces and let be a set-valued mapping with nonempty values. One says that is upper semicontinuous at if and only if, for any neighborhood of , there exists a neighborhood of such that
Lemma 4 (see ). Let be an upper semicontinuous set-valued map with nonempty closed convex values. Suppose that there exists a scalar satisfying For every , the has a weak solution in the sense of Carathéodory.
Lemma 5 (see ). Let be a continuous function and let be a closed set-valued map such that, for some constant , Let be a measurable function and let be a continuous function satisfying for almost all . There exists a measurable function such that and for almost all .
Lemma 6 (see ). Let denote the Lebesgue measure on and let be a measurable function. Let be a measurable set in with . Then, for any , there exists a compact set with such that the restriction of to is continuous.
Definition 7 (see ). An acyclic set is a set whose homology is the same as the homology of the space consisting of just one point. An acyclic map is an upper semicontinuous set-valued map which has compact acyclic values.
In , we can find that every homeomorphic image of a compact convex set is an acyclic set.
Lemma 8 (see ). Every acyclic set-valued map on a compact convex set has a fixed point: for some .
3. Main Results
In this section, we obtain existence theorem for weak solutions of the differential set-valued variational inequality in the sense of Carathéodory. Furthermore, we establish a convergence result for solving differential set-valued variational inequality.
Theorem 9. Assume that satisfy conditions (A) and (B) and are upper semicontinuous with nonempty and compact values such that are pseudo monotone on for each . If is a bounded, closed, and convex subset of , then initial-value system (1) has a weak solution.
Proof. From the proofs of Lemmas 3.2, 3.3, and 3.4 and Theorem 3.1 , it is easy to see that the assumption “ is pseudo monotone on ” in there should be replaced by the assumption “ is pseudo monotone on for each .” Since is a bounded, closed, and convex subset of , it follows from Lemma 3.3 in  that are nonempty and bounded. Let , where , . Then it follows that is bounded on . Moreover, Lemma 3.4 in  shows that are closed and convex for all . Therefore, is closed and convex. Let We can prove in a similar way as Lemma 6.3 in  that has linear growth and it is upper semicontinuous on . Now it follows from Lemmas 4 and 5 that system (1) has a weak solution. This completes the proof.
Remark 10. If are monotone, then it is easy to see that are pseudo monotone on for each .
Lemma 11. Let be a continuous function, a set-valued function, and with . Suppose there exists such that, for any continuous function , one has Then, for almost all , .
Proof. We assume that the contrary holds. Then there exists a set with (where denotes the Lebesgue measure of such that, for all , . By Lemma 6, we know that there exists a closed subset of with such that and are continuous on , where . Then there exists a closed subset of with and such that and so Let We know that is an integrable function on . Since the space of continuous functions is dense in , we can approximate by continuous functions and obtain that there exists a continuous function such that which contradicts (7). This completes the proof.
Remark 12. If is an integrable function satisfying, for almost all , then the integral inequality (7) must hold for any continuous .
Now we begin to design a computational method for solving DVI (1). With , we compute by the recursion, for , where , that is,
Lemma 13. Let satisfy conditions (A) and (B). Then there exists an such that, for any , with and , in , there exists a unique vector satisfying Moreover, for any , one has
Proof. It suffices to choose satisfying The right-hand side is taken to be if . Under this choice, consider any tuple as specified. Let Then with . This shows that the map is contractive and so there exists a unique vector such that It implies that, for any , there exist and such that By (22), we have and so Now the Lipschitz continuity of implies that there exists satisfying It follows that and so This completes the proof.
Lemma 14. Let satisfy conditions (A) and (B). Suppose that and satisfy the linear growth properties Then there exist positive scalars , , , , , , and such that, for any and ,
Proof. Throughout the proof below, the scalar is taken to be sufficiently small. Let It follows from Lemma 13 that By the linear growth of solutions to VI, we have Let Then, we have Letting , one has When is sufficiently small, there exists such that In a similar way, we can prove that there exists such that It follows from (31) that Let Then It follows from Lemma 7.2 in  that there exist positive scalars , , , , , , and such that, for any and , This completes the proof.
Lemma 15. Let be a nonempty, closed, and convex set and let satisfy conditions (A) and (B). Suppose that the set-valued maps are upper semicontinuous with nonempty compact values such that are pseudo monotone on for each . For some constant , and satisfy the linear growth properties Then there exists a scalar such that, for any with and , there exists satisfying (15) for every .
Proof. Assume that is defined by (39). For any scalar sufficiently small, we define the scalars by
where is arbitrary. By the proof of Lemma 7.2 in , we can show that
Let denote the quantity on the right-hand side, which depends on but is independent of . Let satisfy
where and are as described in Lemmas 13 and 14, respectively.
Next we show that, for any fixed , there exists a triple satisfying (15) with for all . Let denote the Euclidean ball in with center at the origin and radius . For any , let denote the nonempty set . Since is Lipschitz continuous on , we know that have linear growth on in ; that is, for some positive constants and for all , By the linear growth assumption, for any , we have Define mappings from to subset of as follows: for any , Since and are upper semicontinuous with nonempty compact values such that are pseudo monotone on for each , it follows from Lemmas 3.3 and 3.4 in  that and are nonempty, closed, and convex sets. By (48), we obtain that and are compact and convex. Consider the map It is easy to see that this map is continuous. Therefore, by the Tychonoff theorem, we know that is compact and so is compact. Since the mapping is a homeomorphism for all sufficiently small, it follows that is a compact acyclic set. We need to show that is a subset of . Let be an arbitrary element in and let be such that From Lemma 13, we have By induction hypothesis and , one has Now we need to prove that the solution mapping is upper semicontinuous. To prove the upper semicontinuity of , it suffices to show that is closed. Suppose that is a sequence converging to and . Then the linear growth condition implies that is bounded and so it has a convergent subsequence with a limit . Since , there exists such that Since is upper semicontinuous on with compact values, it follows that there exists a subsequence of , denoted again by , such that . Letting , we have and so . It follows that is closed and so upper semicontinuous. In a similar way, we can prove that is upper semicontinuous. Thus, we know that is a closed set-valued mapping with compact acyclic values. By Lemma 8, has a fixed point and so there exists a triple satisfying (15). Now we show that . In fact, by (40) and Lemma 7.2 in , one has The definition of implies that . This completes the proof.
Let be the continuous piecewise linear interpolant of the family , the constant piecewise interpolant of the family , and the constant piecewise interpolant of the family ; that is, for .
Theorem 16. Let satisfy conditions (A) and (B) and let be a nonempty, closed, and convex set. Suppose that and satisfy the linear growth properties. Then there exists a sequence such that uniformly on and weakly in with weakly in . Furthermore, assume that , , , and are upper semicontinuous set-valued mappings with nonempty compact values and there exist constants such that, for any sufficiently small, Then the limit is a weak solution of the system (1).
Proof. By (31) and Lemma 14, we deduce that, for sufficiently small, there exists an , which is independent of , such that
It follows from (58) that is also Lipschitz continuous on and the Lipschitz constant is independent of . Thus, there exists an such that the family of functions is an equicontinuous family of functions. Let
From (58) and Lemma 14, we deduce that is uniformly bounded. By using the Arzelá-Ascoli theorem, there exists a sequence with such that converges in the supremum norm to a Lipschitz function on . Since and satisfy the linear growth properties, it follows from Lemma 14 that is uniformly bounded in the norm on . From (58), we know that is uniformly bounded in the norm on , which means that there exists a scalar such that
Since is a reflective Banach space, every bounded sequence has a weakly convergent subsequence and so there is a sequence such that weakly in . In a similar way, we obtain that weakly in .
Next, we show that is a weak solution of the system (1). By Lemma 11, it is sufficient to prove the following three assertions:(i)for any (ii)there exist and such that, for all continuous functions: , (iii)the initial condition .Since where and are the same as described in Theorem 7.1 in ; it follows that, for any , By a similar proof to that in Theorem 7.1 of , we can obtain that Noting the proof of Theorem 7.1 in , we have and as . Let . Since is upper semicontinuous with nonempty compact values, there exists a subsequence of , denoted again by , such that with . This implies that, for any continuous functions: , Then, in a similar way of Theorem 7.1 in , we can prove that The proof in the case is similar and so we omit it here. This completes the proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the editor and the referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11171237) and the Key Program of NSFC (Grant no. 70831005).
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