Abstract
We prove the existence of solutions for third-order nonconvex state-dependent sweeping process with unbounded perturbations of the form: ;, , , , , where , is a nonconvex Lipschitz set-valued mapping, is an unbounded scalarly upper semicontinuous convex set-valued mapping, and is an unbounded uniformly continuous nonconvex set-valued mapping in a separable Hilbert space .
1. Introduction
In the seventies, Moreau introduced and studied in [1] the following differential inclusion: where is a set-valued mapping defined from to a Hilbert space and takes closed convex values, denotes the normal cone to the set at the position . The differential inclusion (SP) is called the Moreau’s sweeping process problem. In [2–7], the authors studied the existence of solutions for various extensions and variants of (SP). In [6], the author studied for the first time the existence of solutions for the following type of second-order differential inclusions with convex set-valued mapping . The problem (SSP) has been extended in several ways. For instance, in [8], the authors studied a variant of (SSP) with a perturbation, when the set-valued mapping is not necessarily convex and is a linear and bounded operator on a separable Hilbert space. In [9], the author studied existence results for the following general problem: where is a nonconvex set-valued mapping, is an unbounded scalarly upper semicontinuous convex set-valued mapping, and is an unbounded continuous nonconvex set-valued mapping in a separable Hilbert space . The differential inclusion (SSPMP) is called Second-order Sweeping Process with Mixed Perturbations. In this work, we prove the existence of solutions for the following form of third-order differential inclusions (TSPMP): where is a nonconvex Lipschitz set-valued mapping, is surjective bounded linear operator, is an unbounded scalarly upper semicontinuous convex set-valued mapping, and is an unbounded uniformly continuous nonconvex set-valued mapping in a separable Hilbert space . We will call it third-order nonconvex state-dependent sweeping process with mixed perturbations (in short (TSPMP)). Problem (TSPMP) includes as a special case the following differential variational inequality (DVI): given a convex compact set in , three points with : where is inf-compact and lower- function, is a real bilinear, symmetric, bounded, and elliptic form on , and is a Lipschitz function. We use our main theorem to prove that (DVI) has at least one Lipschitz solution.
This paper is organized as follows. Section 2 contains some definitions, notations, and important results needed in the paper. In Section 3, we prove an existence result for (TSPMP) when the set-valued mapping is not necessarily convex, by using ideas and techniques from Nonsmooth Analysis. The result is proved by showing that a sequence of approximate solutions converges to a solution of (TSPMP). Then, we deduce from our main theorem an existence result for a second-order nonconvex differential inclusion , where the right-hand side is not convex. In Section 4, we study the closedness and the compactness of the solution sets of (TSPMP). In Section 5, we state an application to differential variational inequalities (DVI).
2. Preliminaries
Throughout the paper, will denote a Hilbert space. We need to recall some notation and definitions that will be used in all the paper. Let be a nonempty closed subset of . We denote by the usual distance function to the subset , that is, . We recall (see [10]) that the proximal normal cone of at is given by where Equivalently can be defined by as the set of all for which there exist such that Now, let be a function and any point in where is finite. We recall that the proximal subdifferential is the set of all for which there exist such that for all Here denotes the closed unit ball centered at the origin of . Recall that for a given , a subset is uniformly prox-regular with respect to (we will say uniformly -prox-regular) (see [10, 11]) if and only if for all and all one has for all . We make the convention for . Recall that for , the uniform -prox-regularity of is equivalent to the convexity of , which makes this class of great importance.
In [12], the authors established the following characterization of the uniform prox-regularity in terms of the subdifferential of the distance function. We recall here a consequence of their result needed in the sequel.
Proposition 2.1. Let be nonempty closed subset in , and let . Assume that is -prox-regular. Then the following holds:
The following proposition summarizes some important consequences of the uniform prox-regularity needed in the sequel of the paper (see [11, 13]).
Proposition 2.2. Let be a nonempty closed subset of and . The following assertions hold.(1). (2)If is uniformly -prox-regular, then is a closed convex set in , and for any with , one has .
Now, we recall some preliminaries concerning set-valued mappings.
Let , , and let be a set-valued mapping. We will say that is Hausdorff-continuous (resp., Lipschitz with ratio ) if for any and one has (resp., if for any and one has Here denotes the Hausdorff distance relative to the norm associated with the Hilbert space defined by Now, we give the following proposition. It proves the result of closedness of the proximal subdifferential of the distance function of prox-regular set. In [12], the authors proved the result when the set-valued mapping depends only on . We adapt the proof to the case of set-valued mapping depending on two variables and . For the completeness of the paper, we give the proof.
Proposition 2.3. Let be a Hausdorff-continuous set-valued mapping with uniformly -prox regular values. For a given , the following holds:
“for any and , and with ; then ”. Here “” denotes the weak convergence in .
Proof. Fix any , and . As , and , one gets for sufficiently large and . On the other hand, since is -prox regular, one can choose (by Proposition 2.2) a point s.t. . So for every large enough, one can write and hence the Hausdorff-continuity of yields for large enough For large enough and by Proposition 2.1, we have This inequality still holds for all where because Consequently, by the continuity of the distance function with respect to (because of (2.7)), the inequality (2.12) gives, by letting , This ensures that , and so the proof is complete.
3. Existence Results for Third-Order Nonconvex State-Dependent Sweeping Process with Perturbation
Throughout this section, will denote a separable Hilbert space. Let , , , , , be open neighborhoods of (resp.) in such that , and be a Lipschitz set-valued mapping with ratio ( taking nonempty closed uniformly -prox regular values in . Assume that is a surjective bounded linear operator. Our aim is to prove the local existence of solution of (TSPMP), that is, there exist and Lipschitz mappings , and such that We begin by recalling the following lemma proved in [14] and needed in the proof of next theorem.
Lemma 3.1. Let and be two metric spaces, and let be a uniformly continuous mapping. Then for every sequence of positive numbers, there exists a strictly decreasing sequence of positive numbers converging to 0 such that(i)for any and are integers ;(ii)for any and any , one has
We prove our main theorem in this section.
Theorem 3.2. Let be two set-valued mappings, and let such that . Assume that the following assumptions are satisfied:(1)for all and , for some convex compact set and some ;(2) is scalarly u.s.c. on with nonempty convex weakly compact values;(3) is uniformly continuous on into nonempty compact subset of , for and with ;(4) and satisfy the linear growth condition, that is, for all for some . Then for every , there exist Lipschitz mappings , and satisfying (TSPMP) with and a.e. on .
Proof. We give the proof in four steps.
Step 1. Construction of the approximants.
Let and put and . Then by the linear growth condition of and we have
for all (since ).
Let . Then by the uniform continuity of and Lemma 3.1, there is a strictly decreasing sequence of positive numbers converging to 0 such that , and are integers , and the following implication holds:
for every where . Fix so that
For all , we consider the following partition of . Without loss of generality we assume that is integer:
Put for all and . For every , we define the following approximating mappings on each interval as
where and for all . Since is surjective, we can choose such that
where
This algorithm is well defined. Indeed, as
therefore, by and (3.11), we get
so that
Then by the Lipschitz property of and the relations (3.4), (3.6), (3.8), (3.10), and (3.13), we get
Therefore, as has uniformly -prox-regular values, by Proposition 2.2 one can choose a point
Define by , and
Then (3.11) becomes
So that, all the mappings are Lipschitz with ratio , and they are also equibounded, with . Observe that for all and all , one has
because
We note that
Now we define the affine approximants
Observe that
Then by (3.4), (3.8), (3.18), (3.20), and (3.22), we can write
Now, we check that the mappings are equi-Lipschitz with ratio . Indeed, by (3.10) and (3.11), one has
From the assumption (1), (3.24) becomes
So for any
By addition on all the interval , we obtain the Lipschitz property of on all . Clearly, by the definition of and (3.25), we have
and so
Let us define , for all . Then we have
Coming back to the definition of , one observes that for a.e. ,
So, for a.e. ,
Then, by properties of proximal normal cone, we have for a.e. ,
On the other hand, by (3.25) and (3.30), we have
Put . Therefore, the relations (3.32) and (3.33) and Proposition 2.2 entail for a.e. Step 2. We will prove the uniform convergence of and .
Since for all and and is a convex set in , one gets for all so that, for every , the set is relatively compact. By using Arzela-Ascoli theorem, there exists a Lipschitz mapping with ratio such that, converges uniformly to on ; weakly star converges to in .
Now, we define the Lipschitz mappings and as follows:
By (3.28), we have
and so
This ensures that
Thus, , and uniformly converge to , and , respectively.Step 3 (relative strong compactness of ). The points defining the step function were chosen arbitrarily in our construction. Nevertheless, by using the uniform continuity of the set-valued mapping over and the techniques of [14] (see also [15, 16]), the sequence can be constructed relatively strongly compact for the uniform convergence in the space of bounded functions. Therefore, there exists a bounded mapping such that .Step 4 (existence of solutions). Since on , then , as well as . Also , and converge uniformly to , , and , respectively. Then by continuity of on , the closedness of the set , , and by (3.23), we obtain a.e. on .
Since and so by the closedness and the continuity of and the continuity of , we have a.e. on . By (3.23), the sequence is bounded in with separable; so is relatively sequentially -compact in , because is the dual of the separable Banach space . Therefore, by integrating, we have
for every measurable set in and every . By assumption, the set-valued mapping is measurable with convex weakly compact values, which ensures the last inequality
Using the same technique, we get
Indeed, for every measurable set in and every , we have
Thus, as the set-valued mapping is measurable with convex weakly compact values (see [17]), it follows that
and since a.e. on , we get for a.e. ,
Thus the proof of the theorem is complete.
We deduce from our main theorem an existence result for a second-order nonconvex differential inclusion.
Corollary 3.3. Let , be open neighborhoods of (resp.) in such that , is a Lipschitz set-valued mapping with ratio ( taking nonempty closed uniformly -prox regular values in . Assume that , for some convex compact set . Then for any , there exists a Lipschitz solution of the second-order differential inclusion
Proof. Take in Theorem 3.2. Then there is a Lipschitz solution to the cauchy problem for the third-order differential inclusion which proves that has a solution almost everywhere on .
Remark 3.4. The existence result proved in Theorem 3.2 cannot be covered by the recent existence result for third-order differential inclusions established in [18] in finite-dimensional case and extended in [19] in Hilbert spaces. Indeed, The right-hand side in Theorem 3.2 contains the normal cone, which cannot be bounded nor u.s.c. as a set-valued mapping. These two assumptions are essential in the proof of the results in [18, 19].
4. Solution Set
Throughout this section, let , and let be a set valued mapping, open subsets in , and a Lipschitz set-valued mapping with ratio ( taking nonempty closed uniformly -prox regular values in . Let with . We denote the set of all triple of Lipschitz mappings such that
Proposition 4.1. Assume that the hypothesis on and in Theorem 3.2 are satisfied and let . Then the graph of the set-valued mapping is closed in .
Proof. Let and with such that uniformly converges to some and , uniformly converges to some . We have to show that . First, observe that for sufficiently large, and , where is given as in the proof of Theorem 3.2. Now, it is not difficult to check that the continuity of , the continuity of , the uniform convergence of both sequences and , and for almost all imply that for almost all . On the other hand, we have It remains then to show that For every , one has Then for every , there exist a measurable selection such that for a.e. . By Theorem 3.2, one has for sufficiently large where . By in Theorem 3.2, we have Therefore, we get the -convergence of subsequences of both and to and , respectively, in . Using the same techniques in the proof of Theorem 3.2, we can prove that a.e. . Now taking and using Proposition 2.2 yield Once again, we use the techniques employed in the proof of Theorem 3.2 to show that for a.e. Thus we get for a.e. and so the proof is complete.
5. Applications to Differential Variational Inequalities
In this section, we are interested with an application of the main result proved in Theorem 3.2 to differential variational inequalities (DVI): given a convex compact set in , three points with : where , is inf-compact and lower- function, is a real bilinear, symmetric, bounded, and elliptic form on , and is a Lipschitz function. Let be a linear and bounded operator on associated with , that is, . We use Theorem 3.2 to prove that (DVI) has at least one Lipschitz solution.
Proposition 5.1. Assume that is a separable Hilbert space and Then (DVI) has at least one Lipschitz solution.
Proof. Let and define , for all . Since is inf-compact, the set is compact in and so has compact values. Also, the lower property of the function and the assumption ensure by Theorem 3.3 in [20] the uniform prox regularity for some , and so has uniform -prox-regular values. The Lipschitz behavior of the set-valued mapping is inherited from the function . Now, we use the definition of proximal normal cones for uniform prox-regular sets to rewrite (DVI) in the form of (TSPMP) as follows: Let, now, , for all . Clearly is uniformly continuous (since is a bounded linear operator) with compact values. Also satisfies the linear growth condition with . Indeed, . Consequently, all the assumptions of Theorem 3.2 are satisfied, and so we have the existence of a Lipschitz solution of (DVI).
Acknowledgment
This project was supported by the Research Center, College of Science, King Saud University.