# General Existence Results for Third-Order Nonconvex State-Dependent Sweeping Process with Unbounded Perturbations

Authors: M. Bounkhel; B. Al-Senan
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

We prove the existence of solutions for third-order nonconvex state-dependent sweeping process with unbounded perturbations of the form:-A(x(3)(t))∈N(K(t,ẋ(t));A(ẍ(t)))+F(t,x(t),ẋ(t),ẍ(t))+G(x(t),ẋ(t),ẍ(t))a.e.[0,T], A(ẍ(t))∈K(t,ẋ(t)), a.e.t∈[0,T], x(0)=x0,ẋ(0)=u0, ẍ(0)=υ0, where T>0, K is a nonconvex Lipschitz set-valued mapping, F is an unbounded scalarly upper semicontinuous convex set-valued mapping, and G is an unbounded uniformly continuous nonconvex set-valued mapping in a separable Hilbert space ℍ.


---

## Body

## 1. Introduction

In the seventies, Moreau introduced and studied in [1] the following differential inclusion:(SP)-ẋ(t)∈NK(t)(x(t))a.e.on⁡I,x(0)=x0∈K(0),
where I=[0,T](T>0),K:I→ℍ is a set-valued mapping defined from I to a Hilbert space ℍ and takes closed convex values, NK(t)(x(t)) denotes the normal cone to the set K(t) at the position x(t). 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(SSP)ẍ(t)∈-N(K(x(t));ẋ(t)),x(0)=x0,ẋ(t)∈K(x(t)),
with convex set-valued mapping K. The problem (SSP) has been extended in several ways. For instance, in [8], the authors studied a variant of (SSP) with a perturbation,(1.1)-Ax(t)∈N(K(t);ẋ(t))+ẍ(t)+F(t,ẋ(t))a.e.on⁡[0,T],
when the set-valued mapping K is not necessarily convex and A is a linear and bounded operator on a separable Hilbert space. In [9], the author studied existence results for the following general problem:(SSPMP)-ẍ(t)∈N(K(x(t));ẋ(t))+F(t,x(t),ẋ(t))+G(t,x(t),ẋ(t)),a.e.I,x(0)=xo,ẋ(0)=u0,ẋ(t)∈K(x(t)),∀t∈I,
where K is a nonconvex set-valued mapping, F is an unbounded scalarly upper semicontinuous convex set-valued mapping, and G is an unbounded continuous nonconvex set-valued mapping in a separable Hilbert space ℍ. 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):(TSPMP)A(x(3)(t))∈-N(K(t,ẋ(t));A(ẍ(t)))+F(t,x(t),ẋ(t),A(ẍ(t)))+G(t,x(t),ẋ(t),A(ẍ(t)))a.e.on⁡I,A(ẍ(t))∈K(t,ẋ(t)),a.e.t∈I,x(0)=x0,ẋ(0)=u0,ẍ(0)=υ0,
where K is a nonconvex Lipschitz set-valued mapping, A is surjective bounded linear operator, F is an unbounded scalarly upper semicontinuous convex set-valued mapping, and G is an unbounded uniformly continuous nonconvex set-valued mapping in a separable Hilbert space ℍ. 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 D in ℍ, three points x0,u0∈ℍ,v0∈D+f(0,u0) with Λ(v0-f(0,u0))≤0:(1.2)FindT>0andaLipschitzmappingx:[0,T]⟶Hsuchthat(1)x(0)=x0,ẋ(0)=u0,ẍ(0)=v0;(2)Λ(ẍ(t)-f(t,ẋ(t)))≤0,ẍ(t)∈D+f(t,ẋ(t)),a.e.on⁡[0,T];(3)∀w∈D+f(t,ẋ(t))withΛ(w-f(t,ẋ(t)))≤0,wehave〈x(3)(t)-αx(t),w-ẍ(t)〉≤a(x(t)+ẋ(t)-ẍ(t),w-ẍ(t))+ρ‖w-ẍ(t)‖2,a.e.on⁡[0,T],
where α>0,Λ:ℍ→ℝ is inf-compact and lower-C2 function, a(·,·) is a real bilinear, symmetric, bounded, and elliptic form on ℍ×ℍ, and f:I×ℍ→ℍ 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. Section2 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 K 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 ẍ(t)∈K(t,ẋ(t))foralmostallt∈I, 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,ℍ will denote a Hilbert space. We need to recall some notation and definitions that will be used in all the paper. Let S be a nonempty closed subset of ℍ. We denote by dS(·) the usual distance function to the subset S, that is, dS(x):=infz∈S∥x-z∥. We recall (see [10]) that the proximal normal cone of S at x is given by(2.1)NP(S;x):={ξ∈H:∃α>0s.t.x∈Proj(x+αξ;S)},
where(2.2)Proj(x;S):={x′∈S:dS(x)=‖x-x′‖}.
Equivalently NP(S;x) can be defined by (2) as the set of all ξ∈ℍ for which there exist σ>0 such that(2.3)〈ξ,x′-x〉≤σ‖x′-x‖2,∀x′∈S.
Now, let f:ℍ→ℝ∪{+∞} be a function and x any point in ℍ where f is finite. We recall that the proximal subdifferential∂Pf(x) is the set of all ξ∈ℍ for which there exist δ,σ>0 such that for all x′∈x+δ𝔹(2.4)〈ξ,x′-x〉≤f(x′)-f(x)+σ‖x′-x‖.
Here 𝔹 denotes the closed unit ball centered at the origin of ℍ. Recall that for a given r∈]0,+∞], a subset S is uniformly prox-regular with respect to r (we will say uniformly r-prox-regular) (see [10, 11]) if and only if for all x¯∈S and all 0≠ξ∈NP(S;x¯) one has(2.5)〈ξ‖ξ‖,x-x¯〉≤12r‖x-x¯‖2,
for all x∈S. We make the convention 1/r=0 for r=+∞. Recall that for r=+∞, the uniform r-prox-regularity of S is equivalent to the convexity of S, 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.
LetS be nonempty closed subset in ℍ, and let r>0. Assume that S is r-prox-regular. Then the following holds:
(2.6)∀x∈H,withdS(x)<r,andallζ∈∂PdS(x)onehas〈ζ,x′-x〉≤8r-dS(x)‖x′-x‖2+dS(x′)-dS(x),∀x′∈HwithdS(x′)≤r.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.
LetS be a nonempty closed subset of ℍ and x∈S. The following assertions hold.(1)
∂PdS(x)=NSP(x)∩𝔹.(2)
IfS is uniformly r-prox-regular, then ∂PdS(x) is a closed convex set in ℍ, and for any x∈ℍ with dS(x)<r, one has projS(x)≠∅.Now, we recall some preliminaries concerning set-valued mappings.LetT>0, I=[0,T], and let K:I×H→ℍ be a set-valued mapping. We will say that K is Hausdorff-continuous (resp., Lipschitz with ratio (λ1,λ2)) if for any t∈I and x∈X one has(2.7)lim⁡x′→xH(K(t,x),K(t,x′))=0
(resp., if for any x,x′∈X and t,t′∈I one has(2.8)H(K(t,x),K(t′,x′))≤λ1‖t-t′‖+λ2‖x-x′‖.
Here ℋ denotes the Hausdorff distance relative to the norm associated with the Hilbert space ℍ defined by (2.9)H(A,B)=max⁡{supa∈AdB(a),supb∈BdA(b)}.
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 t∈I. We adapt the proof to the case of set-valued mapping depending on two variables t and x. For the completeness of the paper, we give the proof.Proposition 2.3.
Letr>0;K:I×Ω→ℍ be a Hausdorff-continuous set-valued mapping with uniformly r-prox regular values. For a given 0<δ<r, the following holds:
“for anyt¯∈I and z¯∈Ω,x¯∈K(t¯,z¯)+(r-δ)B,xn→x¯,tn→t¯,zn→z(tn∈I, xnisnotnecessaryinK(tn,zn)) and ξn⇀ξ¯ with ξn∈∂PdK(tn,zn)(xn); then ξ¯∈∂PdK(t¯,z¯)(x¯)”. Here “⇀” denotes the weak convergence in ℍ.Proof.
Fix anyz¯∈Ω,t¯∈I, and x¯∈K(t¯,z¯)+(r-δ)B. As tn→t¯,zn→z, and xn→x¯, one gets for n sufficiently large tn∈t¯+(δ/4)B and xn∈x¯+(δ/4)B. On the other hand, since K(t¯,z¯) is r-prox regular, one can choose (by Proposition 2.2) a point y¯∈K(t¯,z¯) s.t. ∥y¯-x¯∥=dK(t¯,z¯)(x¯). So for every n large enough, one can write
(2.10)|dK(tn,zn)(xn)-dK(t¯,z¯)(y¯)|≤H(K(tn,zn),K(t¯,z¯))+‖xn-y¯‖,
and hence the Hausdorff-continuity of K yields for n large enough
(2.11)dK(tn,zn)(xn)≤δ4+‖xn-x¯‖+‖x¯-y¯‖≤δ4+δ4+r-δ<r.
For n large enough and by Proposition 2.1, we have
(2.12)〈ξn,u-xn〉≤8r-dK(tn,zn)(xn)‖u-xn‖2+dK(tn,zn)(u)-dK(tn,zn)(xn)∀u∈H,dK(tn,zn)(u)≤r.
This inequality still holds for all u∈x¯+δ̀B where 0<δ̀<δ/4 because
(2.13)dK(tn,zn)(u)≤‖u-x¯‖+‖x¯-xn‖+dK(tn,zn)(xn)≤‖u-x¯‖+‖x¯-xn‖+dK(tn,zn)(xn)≤δ̀+δ4+r-δ2<δ4+δ4+r-δ2=r.
Consequently, by the continuity of the distance function with respect to (t,z,x) (because of (2.7)), the inequality (2.12) gives, by letting n→∞,
(2.14)〈ξ¯,u-x¯〉≤8r-dK(t¯,z¯)(x¯)‖u-x¯‖2+dK(t¯,z¯)(u)-dK(t¯,z¯)(x¯)∀u∈x¯+δ̀B.
This ensures that ξ¯∈∂PdK(t¯,z¯)(x¯), and so the proof is complete.


## 3. Existence Results for Third-Order Nonconvex State-Dependent Sweeping Process with Perturbation

Throughout this section,ℍ will denote a separable Hilbert space. Let x0,u0∈ℍ, A(υ0)∈K(0,u0), ς,ℓ>0, T>0, U0, V0 be open neighborhoods of u0,x0 (resp.) in ℍ such that x0+ς𝔹⊂U0,u0+ς𝔹⊂V0, and K:[0,ς/ℓ]×cl(U0)→ℍ be a Lipschitz set-valued mapping with ratio (λ1,λ2) taking nonempty closed uniformly r-prox regular values in ℍ. Assume that A is a surjective bounded linear operator. Our aim is to prove the local existence of solution of (TSPMP), that is, there exist T>0 and Lipschitz mappings x:[0,T]→cl(V0),u:[0,T]→cl(U0), and υ:[0,T]→ℍ such that(3.1)x(0)=x0,u(0)=u0,υ(0)=υ0,A(υ(t))∈K(t,u(t)),a.e.on⁡[0,T];x(t)=x0+∫0tu(s)ds,u(t)=u0+∫0tυ(s)ds,a.e.on⁡[0,T];-A(υ̇(t))∈NK(t,u(t))(A(υ(t)))+F(t,x(t),u(t),A(υ(t)))+G(t,x(t),u(t),A(υ(t)))a.e.[0,T].
We begin by recalling the following lemma proved in [14] and needed in the proof of next theorem.Lemma 3.1.
Let(X,dX) and (Y,dY) be two metric spaces, and let h:X→Y be a uniformly continuous mapping. Then for every sequence (ϵn)n≥1 of positive numbers, there exists a strictly decreasing sequence of positive numbers (en)n≥1 converging to 0 such that(i)
for anyn≥2,1/(en-1) and (en-1)/en are integers ≥2;(ii)
for anyn≥1 and any x,x′∈X, one has
(3.2)dX(x,x′)≤endY(h(x),h(x′))≤ϵn.We prove our main theorem in this section.Theorem 3.2.
LetF,G:[0,∞)×𝒞0×𝒞0×𝒞0→ℍ be two set-valued mappings, and let ς>0 such that u0+ς𝔹⊂U0. Assume that the following assumptions are satisfied:(1)
for allt∈[0,ς/ℓ] and u∈cl(U0),K(t,u)⊂A(κ1)⊂ℓb𝔹, for some convex compact set κ1⊂ℍ and some ℓ>0;(2)
F is scalarly u.s.c. on [0,ς/ℓ]×V0×U0×ImK with nonempty convex weakly compact values;(3)
G is uniformly continuous on [0,ς/ℓ]×β𝔹𝒞0×α𝔹𝒞0×ℓb𝔹𝒞0 into nonempty compact subset of ℍ, for α=∥u0∥+ς and β=∥x0∥+α(ς/ℓ) with x0+α(ς/ℓ)𝔹⊂V0;(4)
F and G satisfy the linear growth condition, that is,(3.3)F(t,x,u,υ)⊂ρ1(1+‖x‖+‖u‖+‖υ‖)B,G(t,x,u,υ)⊂ρ2(1+‖x‖+‖u‖+‖υ‖)B,
for all (t,x,u,υ)∈[0,ς/ℓ]×V0×U0×ImK for some ρ1,ρ2≥0. Then for every T∈(0,ς/ℓ], there exist Lipschitz mappings x:[0,T]→cl(V0),u:[0,T]→cl(U0), and υ:[0,T]→ℍ satisfying (TSPMP) with ∥u̇(t)∥≤ℓ and ∥υ̇(t)∥≤(1/b)(λ1+ℓλ2+2(ρ1+ρ2)(1+β+α+ℓb)) a.e. on [0,T].Proof.
We give the proof in four steps.
Step 1.
Construction of the approximants.
LetT∈]0,ς/ℓ] and put I=[0,T] and κ=I×β𝔹×α𝔹×ℓb𝔹. Then by the linear growth condition of F and G we have
(3.4)‖F(t,x,u,υ)‖≤ρ1(1+‖x‖+‖u‖+‖υ‖)≤ρ1(1+β+α+lb)=ζ1,‖G(t,x,u,υ)‖≤ρ2(1+‖x‖+‖u‖+‖υ‖)≤ρ2(1+β+α+lb)=ζ2,
for all (t,x,u,υ)∈κ∩(I×V0×U0×ImK)≠∅ (since (0,x0,u0,A(υ0))∈κ∩I×V0×U0×ImK).
Letϵn=1/2n,(n=1,2,3,…). Then by the uniform continuity of G and Lemma 3.1, there is a strictly decreasing sequence of positive numbers (en) converging to 0 such that en≤1, 1/en-1 and en-1/en are integers ≥2, and the following implication holds:
(3.5)‖(t,x,u,υ)-(t′,x′,u′,υ′)‖≤ηen⟹H(G(t,x,u,υ),G(t′,x′,u′,υ′))≤ϵn,
for every (t,x,u,υ),(t′,x′,u′,υ′)∈κ where η=(1+α+3ℓ+λ1+ℓ(1+T+λ2)+2(ζ1+ζ2)). Fix n0∈ℕ so that
(3.6)(λ1+lλ2+ζ1+ζ2)en0≤r2.
For all n≥n0, we consider the following partition of I. Without loss of generality we assume that T is integer:
(3.7)Pn:={tn,i=ien:i=0,1,…,μn=Ten}.
PutIn,i=[tn,i,tn,i+1) for all i=0,…,μn-1 and In,μn={T}. For every n≥n0, we define the following approximating mappings on each interval In,i as
(3.8)υn(t)=υn,i,un(t)=u0+∫0tυn(s)ds,xn(t)=x0+∫0tun(s)ds,fn(t)=fn,i∈F(tn,i,xn(tn,i),un(tn,i),A(υn,i)),gn(t)=gn,i∈G(tn,i,xn(tn,i),un(tn,i),A(υn,i)),
where υn,0=υ0 and for all i=0,…,μn-1. Since A is surjective, we can choose υn,i+1∈H such that
(3.9)A(υn,i+1)=ωn,i+1,
where
(3.10)ωn,i+1=Proj(A(υn,i)+en(fn,i+gn,i),K(tn,i+1,un(tn,i+1))).
This algorithm is well defined. Indeed, as
(3.11)A(υn(t))=ωn,i∈K(tn,i,un(tn,i))⊂A(κ1)⊂lbB,
therefore, by (1) and (3.11), we get
(3.12)b‖υn,i‖≤‖A(υn,i)‖≤lb,
so that
(3.13)‖υn,i‖≤l.
Then by the Lipschitz property of K and the relations (3.4), (3.6), (3.8), (3.10), and (3.13), we get
(3.14)dK(tn,i+1,un,i+1)(ωn,i+en(fn,i+gn,i))≤H(K(tn,i+1,un(tn,i+1),K(tn,i,un(tn,i))))+en‖fn,i+gn,i‖≤λ1‖tn,i+1-tn,i‖+λ2‖un(tn,i+1)-un(tn,i)‖+en(ζ1+ζ2)≤λ1en+λ2(tn,i+1-tn,i)‖υn,i‖+en(ζ1+ζ2)≤(λ1+λ2l+ζ1+ζ2)en≤(λ1+λ2l+ζ1+ζ2)en0≤r2<r.
Therefore, as K has uniformly r-prox-regular values, by Proposition 2.2 one can choose a point
(3.15)ωn,i+1=Proj(A(υn,i)+en(fn,i+gn,i),K(tn,i+1,un(tn,i+1))).
Define θn:I→I by θn(0)=0, and
(3.16)θn(t)=tn,i,∀t∈In,i.
Then (3.11) becomes
(3.17)A(υn(t))∈K(θn(t),un(θn(t))).
So that, all the mappings un are Lipschitz with ratio ℓ, and they are also equibounded, with ∥un(t)∥≤∥u0∥+ℓT. Observe that for all n≥n0 and all t∈I, one has
(3.18)un(t)∈αB∩U0,
because
(3.19)un(t)=u0+∫0tυn(s)ds∈u0+tlB⊂u0+ςB⊂αB∩U0.
We note that
(3.20)xn(θn(t))=xn(tn,i)=x0+∫0tn,iun(s)ds⊂x0+tn,iαB⊂x0+αςlB⊂βB∩V0.
Now we define the affine approximants
(3.21)zn(t):=υn,i+en-1(t-tn,i)(υn,i+1-υn,i),ift∈In,i.
Observe that
(3.22)A(zn(θn(t)))=A(υn,i)∈K(θn(t)),un(θn(t))⊂lbB.
Then by (3.4), (3.8), (3.18), (3.20), and (3.22), we can write
(3.23)fn(t)=fn,i∈F(θn(t),xn(θn(t)),un(θn(t)),A(zn(θn(t))))∩ζ1B,gn(t)=gn,i∈G(θn(t),xn(θn(t)),un(θn(t)),A(zn(θn(t))))∩ζ2B.
Now, we check that the mappings zn are equi-Lipschitz with ratio (1/b)(λ1+λ2ℓ+2(ζ1+ζ2)). Indeed, by (3.10) and (3.11), one has
(3.24)‖ωn,i+1-ωn,i‖≤‖ωn,i+1-ωn,i-en(fn,i+gn,i)‖+en‖fn,i+gn,i‖=‖ωn,i+1-(ωn,i+en(fn,i+gn,i))‖+en‖fn,i+gn,i‖≤dK(tn,i+1,un,i+1)(ωn,i+en(fn,i+gn,i))+en(ζ1+ζ2)≤(λ1+λ2l+2(ζ1+ζ2))en.
From the assumption (1), (3.24) becomes
(3.25)‖υn,i+1-υn,i‖≤1b‖A(υn,i+1-υn,i)‖=1b‖ωn,i+1-ωn,i‖≤1b(λ1+λ2l+2(ζ1+ζ2))en.
So for any t,s∈In,i(3.26)‖zn(t)-zn(s)‖=en-1|t-s|‖υn,i+1-υn,i‖≤1b(λ1+λ2l+2(ζ1+ζ2))|t-s|.
By addition on all the interval I, we obtain the Lipschitz property of zn on all I. Clearly, by the definition of zn(·) and (3.25), we have
(3.27)‖zn(t)-υn(t)‖≤en-1|t-tn,i|‖υn,i+1-υn,i‖≤1b(λ1+λ2l+2(ζ1+ζ2))en,
and so
(3.28)‖zn-υn‖∞⟶0asn⟶∞.
Let us define ρn(t)=tn,i+1, for all t∈In,i. Then we have
(3.29)A(zn(ρn(t)))∈K(ρn(t)),un(ρn(t)).
Coming back to the definition of zn, one observes that for a.e. t∈I,
(3.30)żn(t)=en-1(υn,i+1-υn,i).
So, for a.e. t∈I,
(3.31)en(A(żn(t))-(fn(t)+gn(t)))=ωn,i+1-(ωn,i+en(fn,i+gn,i))=ProjK(tn,i+1,un(tn,i+1))(ωn,i+en(fn,i+gn,i))-(ωn,i+en(fn,i+gn,i)).
Then, by properties of proximal normal cone, we have for a.e. t∈I,
(3.32)A(żn(t))-(fn(t)+gn(t))∈-N(K(tn,i+1,un(tn,i+1));A(υn,i+1))=-N(K(ρn(t),un(ρn(t)));A(υn(ρn(t)))).
On the other hand, by (3.25) and (3.30), we have
(3.33)‖A(żn(t))‖≤‖A‖1b(λ1+λ2l+2(ζ1+ζ2)).
Put δ=(∥A∥/b)(λ1+λ2ℓ)+((2∥A∥/b)+1)(ζ1+ζ2). Therefore, the relations (3.32) and (3.33) and Proposition 2.2 entail for a.e. t∈I(3.34)A(żn(t))-(fn(t)+gn(t))∈-δ∂dK(ρn(t),un(ρn(t)))(A(υn(ρn(t)))).Step 2.
We will prove the uniform convergence ofun and zn.
Sinceen-1(t-tn,i)≤1 for all t∈In,i and υn,i+1,υn,i∈κ1 and A-1κ1 is a convex set in ℍ, one gets zn(t)∈κ1 for all t∈I so that, for every t∈I, the set {zn(t):n≥n0} is relatively compact. By using Arzela-Ascoli theorem, there exists a Lipschitz mapping υ:I→ℍ with ratio (1/b)(λ1+λ2ℓ+2(ζ1+ζ2)) such that, (zn) converges uniformly to υ on I; (żn) weakly star converges to υ̇ in L∞(I,ℍ).
Now, we define the Lipschitz mappingsu:I→ℍ and x:I→ℍ as follows:
(3.35)u(t)=u0+∫0tυ(s)ds,∀t∈I,x(t)=x0+∫0tu(s)ds,∀t∈I.
By (3.28), we have
(3.36)‖vn-v‖∞≤T‖υn-zn‖∞+T‖zn-v‖∞⟶0asn⟶∞,
and so
(3.37)‖un(t)-u(t)‖=‖∫0t(υn(s)-υ(s))ds‖≤T‖υn-υ‖∞.
This ensures that
(3.38)‖xn-x‖∞≤T‖un-u‖∞⟶0asn⟶∞.
Thus, un,xn, and vn uniformly converge to u,x, and v, respectively.Step 3 (relative strong compactness of(gn)).
The points(gn,i) defining the step function gn were chosen arbitrarily in our construction. Nevertheless, by using the uniform continuity of the set-valued mapping G over κ and the techniques of [14] (see also [15, 16]), the sequence gn can be constructed relatively strongly compact for the uniform convergence in the space of bounded functions. Therefore, there exists a bounded mapping g:I→H such that ∥gn-g∥→0.Step 4 (existence of solutions).
Since|ρn(t)-t|≤en on [0,T], then ρn(t)→t, as well as θn(t)→t. Also (xn∘θn),(un∘θn), and (υn∘θn) converge uniformly to x, u, and υ, respectively. Then by continuity of G on κ, the closedness of the set G(t,x(t),u(t), A(υ(t))), and by (3.23), we obtain g(t)∈G(t,x(t),u(t),A(υ(t))) a.e. on I.
SinceA(υn(θn(t))∈K(θn(t),un(θn(t))) and so by the closedness and the continuity of K and the continuity of A, we have A(υ(t))∈K(t,u(t)) a.e. on I. By (3.23), the sequence fn is bounded in L∞(I,ℍ) with ℍ separable; so fn is relatively sequentially σ(L∞(I,ℍ),L1(I,ℍ))-compact in L∞(I,ℍ), because L∞(I,ℍ) is the dual of the separable Banach space L1(I,ℍ). Therefore, by integrating, we have
(3.39)∫Ω〈x,f(t)〉dt=lim⁡n∫Ω〈x,fn(t)〉dt≤limsup⁡n∫Ωσ(x,F(θn(t),xn(θn(t)),un(θn(t)),A(zn(θn(t)))))dt≤∫Ωlimsup⁡nσ(x,F(θn(t),xn(θn(t)),un(θn(t)),A(zn(θn(t)))))dt≤∫Ωσ(x,F(t,x(t),u(t),A(υ(t))))dt,
for every measurable set Ω in I and every x∈ℍ. By assumption, the set-valued mapping t↦F(t,x(t),u(t),A(υ(t))) is measurable with convex weakly compact values, which ensures the last inequality
(3.40)F(t,x(t),u(t),A(υ(t))),a.e.on⁡I.
Using the same technique, we get
(3.41)A(υ̇(t))-f(t)-g(t)∈-δ∂dK(t,u(t))(A(υ(t))).
Indeed, for every measurable set Ω in I and every ξ∈ℍ, we have
(3.42)∫Ω〈ξ,A(υ̇(t))-f(t)-g(t)〉dt=lim⁡n∫Ω〈ξ,A(żn(t))-fn(t)-gn(t)〉dt≤limsup⁡n∫Ωσ(ξ,-δ∂dK(ρn(t),un(ρn(t)))(A(υn(ρn(t)))))dt≤∫Ωlimsup⁡nσ(ξ,-δ∂dK(ρn(t),un(ρn(t)))(A(υn(ρn(t)))))dt≤∫Ωσ(ξ,-δ∂dK(t,u(t))(A(υ(t))))dt.
Thus, as the set-valued mapping t↦-δ∂dK(t,u(t))(A(v(t))) is measurable with convex weakly compact values (see [17]), it follows that
(3.43)A(υ̇(t))-f(t)-g(t)∈-δ∂dK(t,u(t))(A(υ(t))),
and since A(υ(t))∈K(t,u(t)) a.e. on I, we get for a.e. t∈I,
(3.44)A(υ̇(t))∈-NK(t,u(t))(A(υ(t)))+f(t)+g(t)⊂-NK(t,u(t))(A(υ(t)))+F(t,x(t),u(t),A(υ(t)))+G(t,x(t),u(t),A(υ(t))).
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.
Letx0,u0∈ℍ, υ0∈K(0,u0),ς,ℓ>0,U0,V0 be open neighborhoods of u0,x0 (resp.) in ℍ such that x0+ς𝔹⊂U0, u0+ς𝔹⊂V0,andK:[0,ς/ℓ]×cl(U0)→ℍ is a Lipschitz set-valued mapping with ratio (λ1,λ2) taking nonempty closed uniformly r-prox regular values in ℍ. Assume that K(t,u)⊂κ⊂ℓ𝔹,forall(t,x)∈[0,ς/ℓ]×cl(U0), for some convex compact set κ⊂ℍ. Then for any t∈(0,ς/ℓ], there exists a Lipschitz solution of the second-order differential inclusion
(3.45)ẍ(t)∈K(t,ẋ(t))a.e.on⁡I.Proof.
TakeF=G=0,A=Id in Theorem 3.2. Then there is a Lipschitz solution x:[0,T]→ℍ to the cauchy problem for the third-order differential inclusion
(3.46)-x(3)(t)∈NK(t,ẋ(t))(ẍ(t))a.e.[0,T];x(0)=x0,ẋ(0)=u0,ẍ(t)∈K(t,ẋ(t))a.e.on⁡[0,T],
which proves that ẍ(t)∈K(t,ẋ(t)) has a solution almost everywhere on [0,T].Remark 3.4.
The existence result proved in Theorem3.2 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, letT>0andI=[0,T], and let F:I×ℍ×ℍ×ℍ→ℍ be a set valued mapping, Ω1,Ω2 open subsets in ℍ, and K:I×cl(Ω)→ℍ a Lipschitz set-valued mapping with ratio (λ1,λ2) taking nonempty closed uniformly r-prox regular values in ℍ. Let x0∈Ω1,u0∈Ω2,A(υ0)∈K(0,u0) with u0+Tℓ∈Ω2. We denote S(x0,u0,υ0) the set of all triple (x,u,υ) of Lipschitz mappings x,u,υ:I→H such that(4.1)A(υ̇(t))∈-NK(t,u(t))(A(υ(t)))+F(t,x(t),u(t),A(υ(t))),a.e.on⁡I,x(t)=x0+∫0tu(s)ds,u(t)=u0+∫0tυ(s)ds,x(0)=x0,u(0)=u0,υ(0)=υ0,A(υ(t))∈K(t,u(t)),a.e.on⁡I.Proposition 4.1.
Assume that the hypothesis onF and K in Theorem 3.2 are satisfied and let G=0. Then the graph of the set-valued mapping S is closed in Ω1×Ω2×ImK×C(I;ℍ×ℍ×ℍ).Proof.
Let((x0n,u0n,υ0n))n∈Ω1×Ω2×ImK and ((xn,un,υn))n∈C(I;ℍ×ℍ×ℍ) with (xn,un,υn)∈S(x0n,u0n,υ0n) such that ((x0n,u0n,υ0n))n uniformly converges to some (x0,u0,υ0)∈Ω1×Ω2×ImK and ((xn,un,υn))n uniformly converges to some (x,u,υ)∈C(I;ℍ×ℍ×ℍ). We have to show that (x,u,υ)∈S(x0,u0,υ0). First, observe that for n sufficiently large, u0n∈u0+ς𝔹⊂U0 and x0n∈x0+α(ς/ℓ)𝔹⊂V0, where α is given as in the proof of Theorem 3.2. Now, it is not difficult to check that the continuity of K, the continuity of A, the uniform convergence of both sequences ((x0n,u0n,υ0n))n and ((xn,un,υn))n, and A(υn(t))∈K(t,un(t)) for almost all t∈I imply that A(υ(t))∈K(t,u(t)) for almost all t∈I. On the other hand, we have
(4.2)u(t)=lim⁡nun(t)=lim⁡n(u0n+∫0tυn(s)ds)=u0+∫0tυ(s)ds,∀t∈I,x(t)=lim⁡nxn(t)=lim⁡n(x0+∫0tun(s)ds)=x0+∫0tu(s)ds,∀t∈I.
It remains then to show that
(4.3)A(υ̇(t))∈-NK(t,u(t))(A(υ(t)))+F(t,x(t),u(t),A(υ(t))),a.e.on⁡I.
For every n, one has
(4.4)A(υ̇n(t))∈-NK(t,un(t))(A(υn(t)))+F(t,xn(t),un(t),A(υn(t))),a.e.on⁡I.
Then for every n, there exist a measurable selection fn such that
(4.5)fn(t)∈F(t,xn(t),un(t),A(υn(t))),A(υ̇n(t))-fn(t)∈-NK(t,un(t))(A(υn(t))),
for a.e. t∈I. By Theorem 3.2, one has for n sufficiently large
(4.6)‖υ̇n(t)‖≤1b(λ1+lλ2+2(ρ1+ρ2)ζ),
where ζ=1+∥x0∥+(1+T)(∥u0∥+Tℓ)+∥A∥ℓ. By (5) in Theorem 3.2, we have
(4.7)‖fn(t)‖≤ρ1(1+‖x0‖+(1+T)(‖u0‖+Tl)+‖A‖l).
Therefore, we get the σ(L∞(I;ℍ),L1(I;ℍ))-convergence of subsequences of both υ̇n and fn to υ̇ and f, respectively, in L∞(I;ℍ). Using the same techniques in the proof of Theorem 3.2, we can prove that f(t)∈F(t,x(t),u(t),A(υ(t))) a.e. t∈I. Now taking δ=(∥A∥/b)(λ1+ℓλ2+2(ρ1+ρ2)ζ)+2(ρ1+ρ2)ζ and using Proposition 2.2 yield
(4.8)A(υ̇n(t))-fn(t)∈-δ∂dK(t,un(t)))(A(υn(t))).
Once again, we use the techniques employed in the proof of Theorem 3.2 to show that for a.e. t∈I(4.9)-A(υ̇(t))+f(t)∈δ∂dK(t,u(t))(A(υ(t)))⊂NK(t,u(t))(A(υ(t))).
Thus we get for a.e. t∈I(4.10)A(υ̇(t))∈-NK(t,u(t))(A(υ(t)))+f(t)⊂-NK(t,u(t))(A(υ(t)))+F(t,x(t),u(t),A(υ(t))),
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 Theorem3.2 to differential variational inequalities (DVI): given a convex compact set D in ℍ, three points x0,u0∈ℍ,v0∈D+f(0,u0) with Λ(v0-f(0,u0))≤0:(5.1)FindT>0andaLipschitzmappingx:[0,T]→Hsuchthat(1)x(0)=x0,ẋ(0)=u0,ẍ(0)=v0;(2)Λ(ẍ(t)-f(t,ẋ(t)))≤0,ẍ(t)∈D+f(t,ẋ(t)),a.e.on⁡[0,T];(3)∀w∈D+f(t,ẋ(t))withΛ(w-f(t,ẋ(t)))≤0,wehave〈x(3)(t)-αx(t),w-ẍ(t)〉≤a(x(t)+ẋ(t)-ẍ(t),w-ẍ(t))+ρ‖w-ẍ(t)‖2,a.e.on⁡[0,T],
where α>0, Λ:ℍ→ℝ is inf-compact and lower-C2 function, a(·,·) is a real bilinear, symmetric, bounded, and elliptic form on ℍ×ℍ, and f:I×ℍ→ℍ is a Lipschitz function. Let A be a linear and bounded operator on ℍ associated with a(·,·), that is, a(u,v)=〈Au,v〉,forallu,v∈ℍ. We use Theorem 3.2 to prove that (DVI) has at least one Lipschitz solution.Proposition 5.1.
Assume thatℍ is a separable Hilbert space and
(5.2)inf⁡{‖ξ‖:ξ∈∂Λ(x)withx∈D}≠0.
Then (DVI) has at least one Lipschitz solution.Proof.
LetS:={x∈D:Λ(x)≤0} and define K(t,y)=S+f(t,y), for all (t,y)∈ℝ×ℍ. Since Λ is inf-compact, the set S is compact in ℍ and so K has compact values. Also, the lower C2 property of the function Λ and the assumption inf{∥ξ∥:ξ∈∂Λ(x)withx∈D}≠0 ensure by Theorem  3.3 in [20] the uniform prox regularity for some r>0, and so K has uniform r-prox-regular values. The Lipschitz behavior of the set-valued mapping K is inherited from the function f. Now, we use the definition of proximal normal cones for uniform prox-regular sets to rewrite (DVI) in the form of (TSPMP) as follows:
(5.3)x(3)(t)-A(x(t)+ẋ(t)-ẍ(t))-αx(t)∈NK(t,ẋ(t))(ẍ(t)),a.e.on⁡[0,T],x(0)=x0,ẋ(0)=u0,ẍ(0)=v0,ẍ(t)∈K(t,ẋ(t))a.e.on⁡[0,T].
Let, now, G(t,x,y,z)={A(x+y-z)-αx}, for all (t,x,y,z)∈ℝ×ℍ×ℍ×ℍ. Clearly G is uniformly continuous (since A is a bounded linear operator) with compact values. Also G satisfies the linear growth condition with ρ2=α+∥A∥. Indeed, ∥G(t,x,y,z)∥≤α∥x∥+∥A(x+y-z)∥≤(α+∥A∥)(1+∥x∥+∥y∥+∥z∥). Consequently, all the assumptions of Theorem 3.2 are satisfied, and so we have the existence of a Lipschitz solution of (DVI).



---

*Source: 695268-2012-05-07.xml*