Abstract

This paper is mainly devoted to the study of implicit multifunction theorems in terms of Clarke coderivative in general Banach spaces. We present new sufficient conditions for the local metric regularity, metric regularity, Lipschitz-like property, nonemptiness, and lower semicontinuity of implicit multifunctions in general Banach spaces. The basic tools of our analysis involve the Ekeland variational principle, the Clarke subdifferential, and the Clarke coderivative.

1. Introduction

Let and be topological spaces, a topological vector space, a multifunction, and a pair with . The multifunction defined by is called the implicit multifunction defined by the inclusion . The problem is to find some verifiable conditions on such that has the desirable properties. In the literature, different topological, metric, and differential properties (e.g., lower semicontinuity, metric regularity, Lipschitz-like property, upper Lipschitz continuity, and -differentiability) of implicit multifunctions are considered. The structure of and its behavior around decide local properties of in a neighborhood of the point in its graph.

The study of the stability of implicit multifunctions has a long history. The pioneering works of Robinson [1–4] gave good samples for implicit multifunction theorems and their applications. Later, Ledyaev and Zhu [5] and Ngai and Théra [6] established sufficient conditions for the metric regularity of implicit multifunctions in terms of Fréchet coderivative in Banach spaces with Fréchet-smooth Lipschitz bump functions. Recently, Lee et al. [7] showed some sufficient conditions for the nonemptiness, the lower semicontinuity, the metric regularity, and the Lipschitz-like property of implicit multifunctions in terms of Mordukhovich normal coderivative in Asplund spaces. Yen and Yao [8] obtained some point-based sufficient conditions for the metric regularity of implicit multifunctions in finite-dimensional spaces. Huy and Yao [9] established another set of sufficient conditions for the local metric regularity and the Lipschitz-like property of implicit multifunctions in terms of Mordukhovich normal coderivative in Asplund spaces. Huy and Yao [10] studied the metric regularity of implicit multifunctions in terms of Mordukhovich normal coderivative in WCG Asplund spaces. Chieu et al. [11] examined the relationship between the metric regularity and the Lipschitz-like property of implicit multifunctions in finite-dimensional spaces. Chuong [12] gave new sufficient conditions for the Lipschitz-like property of implicit multifunctions in terms of Fréchet coderivative in Asplund spaces. Nghia [13] is also devoted to the study of implicit multifunction theorems in terms of Fréchet coderivative in Asplund spaces. Yang and Huang [14] gave sufficient conditions for the local metric regularity, the metric regularity, the Lipschitz-like property, the nonemptiness, and the lower semicontinuity of random implicit multifunctions in terms of Mordukhovich normal coderivative in separable Asplund spaces.

As mentioned above, the results obtained for the (local) metric regularity, the Lipschitz-like property, the nonemptiness, and the lower semicontinuity of implicit multifunctions are almost restricted in Asplund spaces. Noting that Huy et al. [15] established new sufficient conditions for both the metric regularity and the Lipschitz-like property of implicit multifunctions in terms of Clarke coderivative in general Banach spaces, it is worth mentioning that the coderivative condition of implicit multifunction theorems in [15] can be weakened. So it is natural for us to study implicit multifunction theorems under much weaker conditions in terms of Clarke coderivative in general Banach spaces.

In this paper, we present new sufficient conditions for the local metric regularity, metric regularity, Lipschitz-like property, nonemptiness, and lower semicontinuity of implicit multifunctions in general Banach spaces. The basic tools of our analysis involve the Ekeland variational principle, the Clarke subdifferential, and the Clarke coderivative.

The paper is organized as follows. Section 2 recalls some basic concepts and results from variational analysis and generalized differentiation. Section 3 presents some implicit multifunction theorems in terms of Clarke coderivative in general Banach spaces.

2. Preliminaries

Throughout this paper, unless otherwise stated, all spaces under consideration are Banach spaces whose norms are always denoted by . For any , we consider its dual space equipped with the topology , where means the canonical pairing. As usual, and stand for the closed unit balls of the Banach space and its dual , respectively. The closed ball with center and radius is denoted by . For a subset , denote the interior of .

For a closed subset of and a point , let denote the Clarke tangent cone of at ; that is, if and only if, for each sequence in converging to and each sequence in decreasing to , there exists a sequence in converging to such that for all . We denote by the Clarke normal cone of at ; that is,

Let be a multifunction between topological spaces. Denote by the domain and the range of . Each multifunction is uniquely associated with its graph: in the product space . As usual, is said to be closed if is a closed subset of . is lower semicontinuous (in short, l.s.c.) at if, for any open set satisfying , there exists a neighborhood of such that for all . For any , the Clarke coderivative of at is defined by The history of the coderivatives can be found in [16, 17].

Let be an extended real-valued function, We say that is proper if for all and . Recall that is l.s.c. at a point with if . We say that is l.s.c. around when it is l.s.c. at any point of some neighborhood of . For and , let denote the generalized directional derivative introduced by Rockafellar (cf. [18]); that is, where the expression means that and . Let denote the Clarke-Rockafellar subdifferential of at ; that is, When is convex, the Clarke-Rockafellar subdifferential reduces to the one in the sense of convex analysis; that is, For a closed subset in , let denote the indicator function of . It is known that and

The following sum rule plays important role in variational analysis and is useful for our analysis.

Lemma 1 (see [18]). Let be a Banach space and let be proper lower semicontinuous functions. Let be a local minimizer of . Suppose that one of and is locally Lipschitz around . Then

Lemma 2 (see [16]). Let be a metric space. Assume that is complete and that is a proper l.s.c. function bounded from below. Let and be given such that . Then for any there is satisfying(a),(b),(c) for all .

3. Implicit Multifunction Theorems

Theorem 3. Let and be Banach spaces, a topological space, a multifunction, the implicit multifunction defined by (1), and a pair with . Denote . Suppose that there exist constants and such that (i)for any , the multifunction is closed;(ii)for any and any with , where and .Then is locally metrically regular around with modulus . In fact, for any , we have for all with .

Proof. Fix any and any with . If , then and hence . Therefore, both sides of (13) are equal to 0 and (13) holds. Hence, we can assume that , where . It remains to show that Since , we obtain that . For each with , by the definition of the distance function, there exists such that . Define the function by We claim that is l.s.c. on due to condition (i). Fix any . Put . We see that Clearly, Applying the Ekeland variational principle via the new norm in the product space for some allows us to find satisfying This implies that , Furthermore, That is,
We now show that . Assume to the contrary that and then . Define the function by It follows from (20) that is a minimum of the function on . Noting that , it follows from Lemma 1 that This implies that there exist and such that It follows that Let and . Then , and hence . We observe that and Since , we have that and . Furthermore, Dividing both sides of the above inequality by gives us that For any , by (20), we have that This implies that Hence,
Fix any ; it follows from (27), (29), and (32) that when are chosen sufficiently small. Taking in the above gives us that , which is contrary to condition (ii). Therefore, we have shown that ; that is, . It follows that Letting , we obtain that Letting , we obtain that

Remark 4. We obtain the same result with Huy et al. [15, Theorem 3.1] under much weaker coderivative condition. Noting that the proof of Theorem 3 is much simpler than that of [15, Theorem 3.1], similar results presented in [9, Theorem 3.5], [10, Proposition 3.6], [14, Corollary 3.3], and [8, Theorem 3.1] all require the assumption of the inner semicompactness of the metric projection mapping. However, Theorem 3 does not require this assumption. Moreover, we can see from the proof of Theorem 3 that the conclusion of the theorem is still valid, if the topological space is replaced by a metric space.

Theorem 5. Suppose that all the assumptions of Theorem 3 are satisfied. Moreover, assume that(iii) is l.s.c. at .Then is metrically regular around with modulus . In fact, there exists a constant such that for all .

Proof. By Theorem 3, for any , we have for all with . Clearly, . By condition (iii), there exists a constant such that Hence,
Choose a number . Then satisfies the conclusion of Theorem 5. Indeed, for any , we have , and it follows from (40) that . Moreover, . It follows from (38) that .

Remark 6. Similar results presented in [7, Theorem 3.2] and [14, Corollary 3.6] are established in terms of Mordukhovich normal coderivative in Asplund spaces. Moreover, [7, Theorem 3.2] and [14, Corollary 3.6] all require the assumption of the inner semicompactness of the metric projection mapping. However, Theorem 5 does not require this assumption.

Theorem 7. Suppose that all the assumptions of Theorem 3 are satisfied. Moreover, assume that is a subset of a normed space and(iii)there exists a constant such that Then is Lipschitz-like around with modulus .

Proof. Choose a number . We can assert from Theorem 3 that for all with . Choose a number with . We claim that Indeed, fix any and any . Then we have , and it follows from condition (iii) that . Hence, By (42) and (44), we obtain that Therefore, It follows that (43) holds. Therefore, is Lipschitz-like around with modulus .

Remark 8. We obtain the same result with Huy et al. [15, Theorem 3.2] under much weaker coderivative condition. Similar results presented in [9, Theorem 3.5], [10, Corollary 3.9], [7, Theorem 3.3], and [14, Corollary 3.10] all require the assumption of the inner semicompactness of the metric projection mapping. However, Theorem 7 does not require this assumption. Similar result presented in [12, Theorem 3.1] does not require the assumption of the inner semicompactness of the metric projection mapping, but it is established in terms of Fréchet coderivative in Asplund spaces.

Theorem 9. Suppose that all the assumptions of Theorem 3 are satisfied. Moreover, assume that (iii)for any , the multifunction is l.s.c. at .Then there exists a constant such that the multifunction defined by is nonempty and l.s.c. on .

Proof. Since , by condition (iii), there exists a constant such that Hence, Choose a number . We show that satisfies the conclusion of Theorem 9.(a)Fix any . We prove that is nonempty. Define the function by We claim that is l.s.c. on due to condition (i). If , then , and hence . It follows that . That is, . If , then , and hence . We may assume that , where .
For each with , by the definition of the distance function, there exists such that . Let . Fix any . We see that Clearly, Applying the Ekeland variational principle via the new norm in the product space for some allows us to find such that This implies that , Furthermore, That is,
We now show that . Assume to the contrary that and then . Define the function by It follows from (55) that is a minimum of the function on . Arguing as in Theorem 3, we can deduce a contradiction with condition (ii). Therefore, we have shown that ; that is, . It follows from (57) that .(b)Fix any . We prove that is l.s.c. at . It suffices to show that, for any and any , there exists a constant such that Since , we have that and . Choose a number such that and . Arguing as above for the pair in the place of , the constant in the place of , and the ball , , and in the place of , , and , respectively, we find a constant such that Since , from (60) we get That is,

Remark 10. Similar results presented in [7, Theorem 3.1] and [14, Theorem 3.12] are established in terms of Mordukhovich normal coderivative in Asplund spaces. Moreover, [7, Theorem 3.1] and [14, Theorem 3.12] all require the assumption of the inner semicompactness of the metric projection mapping. However, Theorem 9 does not require this assumption.

Corollary 11. Let and be Banach spaces, a multifunction, and a pair with . Suppose that is closed and that there exist constants and such that, for any and any with , where and .
Then one has the following:(a)there exists a constant such that, for any , ;(b)there exist constants and such that for all satisfying ;(c) is Lipschitz-like around with modulus ;(d)there exists a constant such that the multifunction defined by is nonempty and l.s.c. on .

Proof. Put . Define and by respectively. Obviously, . It is easy to see that all the assumptions of Theorem 3 are satisfied. Indeed, implies that . Denote . We observe that and . Since is closed, we have that is closed for all . It follows that condition (i) of Theorem 3 is satisfied. Furthermore, we can prove that , and it follows that . Hence, . Then condition (ii) of Theorem 3 is satisfied.
Fix any . By Theorem 3, we have for all with . We now prove the conclusions of the corollary.(a)Choose a number . Let . Take arbitrary . Clearly, . Since , we have that . It follows from (67) that  Hence, .(b)Take any . We can get the conclusion immediately from (67).(c)Clearly, . We can verify that condition (iii) of Theorem 7 holds for with modulus . The conclusion follows immediately from Theorem 7.(d)Clearly, for any , the multifunction is l.s.c. at . The conclusion follows immediately from Theorem 9.

Remark 12. We obtain the same result with Huy et al. [15, Corollary 3.1] under much weaker coderivative condition. Similar results presented in [10, Corollary 3.10] are established in terms of Mordukhovich normal coderivative in Asplund spaces. Moreover, [10, Corollary 3.10] requires the assumption of the inner semicompactness of the metric projection mapping. However, Corollary 11 does not require this assumption.

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11226228, 71372113, and 11301254), the Science and Technology Program Project of Henan Province of China (no. 122300410256), and the Natural Science Foundation of Henan Education Department of China (no. 2011B110025).