Research Article | Open Access
Noncompact Perturbation of Sweeping Process with Delay in Banach Spaces
We have proven an existence theorem concerning the existence of solutions for a functional evolution inclusion governed by sweeping process with closed convex sets depending on time and state and with a noncompact nonconvex perturbation in Banach spaces. This work extends some recent existence theorems concerning sweeping processes from Hilbert spaces setting to Banach spaces setting. Moreover, it improves some recent existence results for sweeping processes in Banach spaces.
Differential inclusions represent an important generalization of differential equations. Moreover, they have several applications in different branches of mathematical sciences such as Control Theory, Viscosity, Optimization, and Mechanical problems (see [1–4]). In his leading paper, Moreau  proposed and studied the following differential inclusion governed by sweeping process of first order: where is a set-valued function from the interval to the family of nonempty closed convex subsets of a Hilbert space and is the normal cone of the subset at the point . Problem (1) corresponds to several important mechanical problems. For more details concerning the applications of (1), we refer to .
Since then, important improvements have been developed by weakening assumptions in order to obtain the most general result of existence for sweeping processes.
Let be a Banach space, a positive real number, the Banach space of continuous functions from to endowed with the uniform norm , a multifunction from to the family of nonempty closed convex subsets of , and a multifunction defined on and with nonconvex noncompact values in the dual space, , of . The aim of this paper is to find the sufficient conditions that guarantee the existence of two continuous functions and such that the following perturbed sweeping process is satisfied: where are given with , is the duality mapping in , and for each , for all .
In order to explain the mathematical motivation for this work, we mention some recent results in this domain. In a very recent paper, Aitalioubrahim  considered (2) when is a Hilbert spaces is a multifunction from to the family of nonempty closed nonconvex subsets of , and is a multifunction defined on and with nonconvex noncompact values in . Castaing et al.  considered a second order sweeping process without delay in a separable Hilbert space in the case when is a Lipschitz multifunction defined on , taking a closed proxy regular () values in , and is a convex weakly compact valued scalary uppersemicontinuous defined on and satisfying the growth condition for all . For other results concerning (2) with or without perturbation in Hilbert spaces, we refer to [8–13]. Bounkhel and Al-Yusof  initiated the extension of (1) from the Hilbert setting to the Banach spaces setting. In fact, Bounkhel  considered (2) when is a multifunction from to the family of nonempty closed convex subsets of and satisfyes a condition similar to Condition in the statement of our result (Theorem 18) and is an uppersemicontinuous multifunction defined on and with convex compact values in such that , for all , for some convex compact set . More recently, Ibrahim and Aladsani  considered a second order sweeping process without delay in a separable uniformly convex and uniformly smooth Banach space , and the perturbation is an upper semicontinuous defined from to the family of nonempty convex weakly compact sets of the topological dual space of such that
We note that if is a Hilbert space, then is equal to the identity operator on , and is 2-uniformly convex and 2-uniformly smooth Banach space. Therefore, our work extends many problems from Hilbert spaces setting to Banach spaces setting. Moreover, our work improves many results in the literature concerning the existence of solutions for some evolution inclusions governed by sweeping process in Banach spaces, for example, [14, 15]. In addition our technique allows us to discuss some sweeping process problems with noncompact perturbation in Banach spaces.
2. Preliminaries and Notations
Let , , and let be a Banach space with topological dual space . Let and .
Definition 1 (see [1, Definition]). A Banach space is said to be uniformly convex if for any , , the inequalities , and imply that there exists a such that .
Lemma 2 (see [16, Theorems and]). (1)Every Hilbert space is uniformly convex.(2)Every uniformly convex space is strictly convex.(3)Every uniformly convex space is reflexive.
Definition 4 (see [1, Definition]). Let be the topological dual of a Banach space ; then the multivalued mapping is said to be the duality mapping (or duality mapping) in , and the multivalued mapping is called the normalized duality mapping (or duality mapping) in .
In the following lemma, we recall some properties of the duality mapping.
Lemma 5 (see [1, 19] Propositions, , and). (1)If is a Hilbert space, then , for all ;(2)for each, is nonempty closed convex and bounded subset of;(3), for all and for all ;(4)if is strictly convex, is single valued;(5)if is strictly convex, is one to one; that is,;(6)if is uniformly convex, then is uniformly continuous on each bounded set in; that is, for all and , there is a such that (note that if is uniformly convex, then it is strictly convex, and hence is single-valued mapping);(7)if is reflexive, then is a mapping from onto; that is, (8)if is reflexive strictly convex space with strictly convex conjugate space, then and are one-to-one, onto, and single-valued mappings and where is the identity mapping on and is the identity mapping, on .
Definition 6 (see [1, Definition]). The Banach space is said to be uniformly smooth if where is the modulus of smoothness of (note that every Hilbert space is uniformly smooth).
Clearly, every uniformly smooth is uniformly smooth (note that every Hilbert space is 2-uniformly smooth).
Lemma 8 (see [1, Theorems,, and]). Let be a Banach space.(1) is uniformly smooth if and only if is uniformly convex.(2) is uniformly convex if and only if is uniformly smooth.(3)If is uniformly smooth, then is reflexive.
Now, let be a Banach space and its topological dual. Let be two functions defined by
Based on the functional , the generalized projection of a point in onto a nonempty subset of can be defined as follows.
Definition 10 (see ). Let be a Banach space, a nonempty subset of , and . If there exists a point satisfying
then is called a generalized projection of onto ; where .
The set of all such points is denoted by ; that is,
Now, we list in the following theorem some properties of and (see ).
Lemma 11. Let be a Banach space and its topological dual. (1).(2)If is uniformly convex or uniformly smooth, then (3)If is a Hilbert space, then (4)If is reflexive and is a nonempty closed and convex subset of , then(a), for all ;(b) is strictly convex if and only if is singleton, for all ;(c)if is also smooth, then, for any given ,
Definition 12 (see ). Let be a Banach space with topological dual , a nonempty closed convex subset of , and . The convex normal cone of at is defined by
Lemma 13 (see ). Let be a nonempty, closed, and convex subset of Banach space , and . Then one has the following.(1), where is the subdifferential of the function ( is the distance from to ).(2)If is reflexive and smooth, then
Also, we need the following four Lemmas.
Lemma 14 (see [14, Proposition]). Let be an open subset in a normal vector space , a Banach space, and a continuous set-valued mapping defined on and with nonempty compact convex values in . Let be a sequence of that converges weakly to in and a sequence in that converges strongly to in such that a sequence in that converges to in . If , then .
Lemma 15 (see [14, Lemma 4.1 and Proposition]). Let , be a uniformly convex and uniformly smooth Banach space, and let be a nonempty bounded subset of ; then there exist two constants , such that
Moreover, if , in addition, is closed and , then
Lemma 16 (see [21, Theorem]). Let be a subset of a Banach space . Assume that is a multifunction such that(1)for every , the multifunction is measurable on ;(2)for every , the multifunction is continuous on .
Then, for any measurable function , the multifunction is measurable on .
Lemma 17 (see [21, Lemma]). Let be a separable Banach space, a measurable multifunction, and a measurable function. Then for any positive measurable function , there exists a measurable selection of such that for almost
3. Existence of Solutions for Problem (2)
Theorem 18. Let , a separable uniformly convex and uniformly smooth Banach space, , , a multifunction defined from to the family of nonempty closed convex subsets of , and a multifunction defined from with nonempty closed values in . Assume that the following hypotheses hold: there is an absolutely continuous function and two positive real numbers and such that, for all , all , and all ,
where ; there is a compact subset of such that for each , is a measurable; there is a function such that, for all , and for all , , , ,
there is a function such that, for all ,
Then, for any with , there exist two continuous functions and such that (2) is satisfied.
Proof. At first, we note that since the Banach space is uniformly convex, then it is uniformly convex, and hence by Lemma 2 it is reflexive and strictly convex. Moreover, because is uniformly smooth, then, by Lemma 9, its conjugate is uniformly convex, , and hence is strictly convex. Then, by Lemmas 5 and 8, the duality mapping is single valued, continuous, one-to-one, onto, and and . For notational convenience, we take . Let such that . Let be a fixed positive integer and a measurable function. In order to make it easier for the reader, we will divide the proof in the following steps.
Step 1. We will construct two continuous functions and , two functions , and such that and are absolutely continuous function on and that
In view of Lemma 15, there is such that, for all , we have where .
For , we define and . In order to define and on , we define a function by
Let us consider a partition of by the points , . Let , , and . In view of Lemma 16, there exists a function such that and
We put and for we define where
According to Lemma 11, is well defined. By recalling the definition of the generalized projection, we get and
As a result of Condition , , . Moreover, by Condition
Consequently, by (29) we obtain
We note that, from Lemma 15 and the choice ,. Then, by and (36), the last inequality gives us
This inequality with (30) leads to
Observe that from (33), for , we get From this equality with (40), we get
Note that, from (33), we have
Therefore, by (41),
Moreover, by (30), (33), (36), and (40), for we have this shows that is absolutely continuous on .
We reiterate this process for constructing two sequences ,,, and two functions , such that, for all , the following properties hold: where .
Note that, by (48), for every ,
Then, from this recurrence relation, we can get
This relation with (46), (54), and Condition gives us, for ,
Therefore, by (34), (50), (51), and (57), for we have
By addition, the last inequality holds for , with ; hence, is absolutely continuous on .
Now, for each positive integer we define the functions , and by setting
Then obviously, the relations (46), (47), (48), (52), (53), and (54) imply (28), respectively.
Step 2. The sequences and converge uniformly to continuous functions , , respectively, with and for and for .
Indeed, by (58) for we have . Moreover, by (28)(iii), for any and any , we have . Then, for any , the set is relatively compact subset in . In addition, from (28)(vi), Condition , and (57) we get for a.e.
This, with the absolutely continuity of , assures the equicontinuity of . Thus, by Arzela-Ascoli’s Theorem, we can select a subsequence of , again denoted by , which converges uniformly to an absolutely continuous function on such that, for any ,
Moreover, converges weakly to . We extend the definition of in such a way that on . Obviously, converges uniformly to on . Let us define , . Since is continuous on the compact set and for , then the sequence converges uniformly to a function . In addition to the fact that , a.e. for , we confirm that the sequence converges uniformly to a function . Let us show that for . Note that
Moreover, as , it follows that Then, from (62), (63), (64), Condition and one obtains, for ,
By taking the limit as , we conclude that for .
Step 3. For any , the sequence converges to , and the sequence converges to in . Let . We have
By the continuity of , the uniform convergence of towards , and the preceding estimate, we get
Similarly, we can show that .
Step 4. The sequence converges pointwise to a function satisfying , .
Let be fixed. In view of (28)(v) and Condition , we obtain for
From Step 3, the right-hand side of (68) tends to zero when . Hence, is a Cauchy sequence in ; then it converges pointwise to a function . Moreover, by (28)(v) and Condition , we get
Again, by Step 3, the right-hand side of this inequality tends to zero when . Hence, , and .
Step 5. For almost , .
We note the weak convergence of towards in , and Mazur’s Lemma gives us
Fix any such that the preceding relation is satisfied, and consider . The last relation above yields
Moreover, the relations (28)(vi) and (vii) tell us for a.e. where the function is defined by