Existence Results for Differential Inclusions with Nonlinear Growth Conditions in Banach Spaces

In the Banach space setting, the existence of viable solutions for differential inclusions with nonlinear growth; that is, x˙(t)∈F(t,x(t)) a.e. on I, x(t) ∈ S, ∀t ∈ I, x(0) = x 0 ∈ S, (∗), where S is a closed subset in a Banach space 𝕏, I = [0, T], (T > 0), F : I × S → 𝕏, is an upper semicontinuous set-valued mapping with convex values satisfying F(t, x) ⊂ c(t)(||x|| + ||x||p)𝒦, ∀(t, x) ∈ I × S, where p ∈ ℝ, with p ≠ 1, and c ∈ C([0, T], ℝ+). The existence of solutions for nonconvex sweeping processes with perturbations with nonlinear growth is also proved in separable Hilbert spaces.


Introduction
The first motivation of the study of the concept of differential inclusions comes from the development of some studies in Control Theory and Optimization; see, for instance, [1][2][3] and the references therein. Many works investigated the existence of solutions and topological properties of solution sets for first-and second-order differential inclusions [2,[4][5][6][7][8]. For example, in [4], the authors proved an existence result for the inclusioṅ( ) ∈ ( ( )) a.e. on [0, ] , ( > 0) , by assuming that the set-valued mapping is included in the subdifferential of convex lower semicontinuous (l.s.c) function : R → R. This result has been extended in many ways by many authors (see, e.g., [9][10][11] and the references therein. We state one of them from [12], in which the author proved an existence result of viable solutions in the finite dimensional setting for the differential inclusion: where is included in the subdifferential of a regular (not necessary convex) function : R → R, is closed subset in R , and : [0, ]×R is a continuous set-valued mapping. The infinite dimensional case of (2) has been studied in [13]. A very important type of differential inclusions that will be considered in this work is the following: where ( , ) is the normal cone to at ∈ . This differential inclusion is called Sweeping Process Problem (SP) and has been introduced and studied by Moreau in 1960s in the convex case [14]. This differential inclusion (SP) models a phenomena from elastoplasticity; see the excellent books in [3] and the references therein. Since the works [14], many works extended in different ways the sweeping process problem. In [15], the author introduced some new techniques from which many results can be derived, essentially the existence of a solution of (SP) for ( ) = + V( ), where is a fixed nonconvex closed set and V is a mapping with finite variation. Another important study of the inclusion (3), with a nonconvex set ( ), has been realized by the author in [16] who proved, in the finite dimensional setting, the existence of solution for (3) whenever the set-valued mapping ( ; ) ( ( ); ) ∩ B has a closed graph. Here ( ( ); ⋅) is the Clarke normal cone and B denotes the closed unit ball 2 The Scientific World Journal of R . The main example of such sets ( ) provided in [16] is that of complements of open convex sets. In [17,18] the authors independently proved the existence of a solution of (3) for general nonconvex sets moving in a Lipschitz way in a finite dimensional space. Note that the results in [18] are given for -convex sets ( ) in Hilbert spaces and under some compactness assumptions on . In [19], the authors proved an existence result in Hilbert spaces with a regular set-valued mappings of the perturbed sweeping process problem defined as follows: ( ) ∈ − ( ( ) ; ( )) + ( ; ( )) a.e. on , ( ) ∈ ( ) , ∀ ∈ , where ( ; ⋅) is an upper semicontinuous set-valued mapping with convex compact values. The class of inclusions (4) appears in particular in mathematical economy. It corresponds for ( ) = (independent of ) to modeling planning procedures introduced by Henry [20] for convex and also considered by [7] for tangentially regular.
To the best of our knowledge no existing works studied the existence of solutions for differential inclusions with nonlinear growth. The main purpose of the paper is to prove the existence of solutions for ( * ) in Banach spaces and for (4) in separable Hilbert spaces, under the nonlinear growth condition of the set-valued mapping ; that is, when : [0, ] × H → H is an upper semi-continuous setvalued mapping with closed convex values satisfying ( , ) ⊂ ( )(‖ ‖ + ‖ ‖ )K, where ∈ R, with ̸ = 1, and ∈ ([0, ], R + ) and K is a convex compact set. The paper is organized as follows. After recalling the needed concepts in Section 2, we prove in Section 3 the existence of viable solutions for ( * ) in Banach spaces. In Section 4, we prove the existence of solution for (SP) with perturbations having nonlinear growth conditions with prox-regular values of in separable Hilbert spaces.

Preliminaries
This section is devoted to recall some notations and concepts needed in the paper. Definition 1. Let X be a Banach space, let ⊂ X be a nonempty closed subset of X, and let ∈ . The Bouligand tangent cone ( ; ) is defined by where ( ) = inf{‖ − ‖ : ∈ } is the usual distance function associated with .
Recall from [21] the original definition of the class of uniformly -prox-regular sets in Hilbert spaces as the class of all closed sets satisfying the following definition. Many equivalent definitions of this class have been used for different applications; see, for example, [5,19,22].

Definition 2.
Let H be a Hilbert space. For a given ∈ (0, +∞], a subset is uniformly -prox-regular if and only if for all ∈ { ∈ H : 0 < ( ) < }, the distance function is 1 at .
(2) The union of two disjoint convex sets is not convex but it is uniformly -prox-regular with := /2, where is the distance between the two sets. More examples, details, and characterizations of this class of sets in Hilbert spaces can be found in [5,19,22].
A set-valued mapping : X X is said to be upper semicontinuous (u.s.c) at ∈ X provided for every > 0, there exists > 0 such that We say that is u.s.c. on X whenever it is u.s.c on all ∈ X. Obviously, the upper semicontinuity coincides with the continuity for single-valued mappings. The following proposition proves the u.s.c. of set-valued mappings with closed graphs under the compactness assumption on the closure of the range. For its proof, we refer the reader to Proposition 1.2 in Deimling [2].

Proposition 4. Let Ω be a nonempty closed subset in X and let
: Ω X be a set-valued mapping with closed values. If the graph of is closed and cl( (Ω)) is compact, then is upper semicontinuous.

Nonlinear Variants of Gronwall Inequalities
Before starting this section, we refer the reader to the nice book in [23] on Gronwall inequalities and applications. We recall from [24] the following variant of Gronwall inequality that can be also found in [23].
where the functions ℎ and are continuous on [ , ] and ≥ 0 The Scientific World Journal 3 for all ∈ [ , 1 ), where 1 is chosen so that the expression is positive in the subinterval [ , 1 ).
In the following lemma we extend Lemma 5 to the case of negative exponent . To the best of our knowledge no such results on Gronwall inequalities with decreasing right hand side of the inequality (7) which is the case when the exponent is assumed to be negative.

Lemma 6.
Let V be a positive differentiable function satisfying (7) with ≤ 0 and assume that is nonnegative. Then for all ∈ [ , ].

Solutions of Differential Inclusions with Nonlinear Growth
The two following consequences of Lemmas 5 and 6 are the key tools in this section. In these lemmas we take the case ℎ( ) = ( ) > 0, and we separate the results in two cases depending on the exponent . The first case is ∈ (1, ∞) and the second case is ∈ (−∞, 1).

Lemma 8. Let V be a positive differentiable function satisfying
. Then for all ∈ [ , ].
Let be a set-valued mapping satisfying the following nonlinear growth: 4 The Scientific World Journal where = [0, ], ( > 0), is a closed nonempty set in X, ∈ ( , R + ), and ∈ R with ̸ = 1. Clearly, when = 0, this assumption coincides with the well known linear growth; that is, Our main aim in this paper is to prove the existence of absolutely continuous solutions under the nonlinear growth condition (20) oḟ( ) ∈ ( , ( )) a.e. on , for any 0 ∈ . To ensure the viability of the solution on the set , we need the following classical tangential condition: where ( ; ) is the Bouligand tangent cone to at .
The following proposition is a main tool in our next proofs.

Proposition 9. Assume that is a mapping from to X satisfying
Then is bounded by where is the normalized duality mapping (for the definition we refer to [25]). For such , we havė  In all what follows let 2 and be as in Proposition 9. We recall from Deimling [2, Theorem 9, Page 117] the following existence result for u.s.c. set-valued mappings with values contained in a compact set.
Then for every 0 ∈ , there exists an absolutely continuous mapping : → such thaṫ ( ) ∈ ( , ( )) . . , We start now by proving the following proposition needed in the proof of the main result. If satisfies the nonlinear growth on × ; that is, ( , ) ⊂ ( )(‖ ‖ + ‖ ‖ )K on × , for some ∈ ( , R + ), ∈ R with ̸ = 1, and K is a convex compact set in X, then is upper semicontinuous on × with closed convex values.

Nonconvex Sweeping Process with Perturbations Having Nonlinear Growth Conditions
Our purpose, in this section, is to use the techniques developed previously to extend some existing results, in separable Hilbert spaces, of nonconvex sweeping processes with perturbations from the case of perturbation with linear growth to the case of perturbation with nonlinear growth. For this end let H be a separable Hilbert space, let := [0, ] ( > 0), and let : H be a set-valued mapping satisfying the following Lipschitz condition for any ∈ H and any , ∈ : We start with the following existence result which is a consequence of Theorem 4.1 in [19].

6
The Scientific World Journal Using the techniques from the previous section and Theorem 13 we prove our main result in this section.
Now, let us check that is a solution of (SPP) with . Clearly, we have We use now the choice of and the assumptions on the constants , , to deduce from (43) that which ensures that (2 ( + ) + 1) + 0 < 2 , and hence ‖ ( )‖ ≤ /2 which yields that (‖ ( )‖) = 1 and so ( , ( )) = ( , ( )). This means that is a solution of (40) and hence the proof is complete.