# Noncompact Perturbation of Sweeping Process with Delay in Banach Spaces

Authors: F. Aladsani; A. G. Ibrahim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction

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 [5] proposed and studied the following differential inclusion governed by sweeping process of first order:
(1)-u′(t)∈NG(t)(u(t))a.e.onI=[0,T],u(0)=u0,
where G  is a set-valued function from the interval I=[0,T] to the family of nonempty closed convex subsets of a Hilbert space H and NG(t)(u(t)) is the normal cone of the subset G(t) at the point u(t). Problem (1) corresponds to several important mechanical problems. For more details concerning the applications of (1), we refer to [5].Since then, important improvements have been developed by weakening assumptions in order to obtain the most general result of existence for sweeping processes.LetX be a Banach space, r a positive real number, Cr=CX([-r,0])the Banach space of continuous functions from [-r,0] to X endowed with the uniform norm ∥u∥∞=Sup{∥u(t)∥:t∈[-r,0]}, C a multifunction from I×X to the family of nonempty closed convex subsets of X, and F a multifunction defined on [0,T]×CX([-r,0])×CX([-r,0]) and with nonconvex noncompact values in the dual space, X*, of X. The aim of this paper is to find the sufficient conditions that guarantee the existence of two continuous functions u:[-r,T]→X*and w:[-r,T]→X such that the following perturbed sweeping process is satisfied:
(2)u(t)=J(φ(t)),w(t)=ψ(t),fort∈[-r,0],u(t)=J(φ(0))+∫0tu′(s)ds,fort∈I,J*(u(t))∈C(t,w(t)),fort∈I,w(t)=ψ(0)+∫0tJ*u(s)ds,fort∈I,u′(t)∈-NC(t,w(t))(J*u(t))+F(t,τ(t)w,τ(t)J*u),000000000000000000000000000000000a.e.fort∈I,
where φ,ψ∈CX([-r,0]) are given with φ(0)∈C(0,ψ(0)), J is the duality mapping in X, and for each t∈[0,T],τ(t):CX([-r,t])→CX([-r,0]),(τ(t)g)(s)=g(s+t), for all s∈[-r,0].In order to explain the mathematical motivation for this work, we mention some recent results in this domain. In a very recent paper, Aitalioubrahim [6] considered (2) when X is a Hilbert spaces C is a multifunction from I to the family of nonempty closed nonconvex subsets of X, and F is a multifunction defined on [0,T]×CX([-r,0]) and with nonconvex noncompact values in X. Castaing et al. [7] considered a second order sweeping process without delay in a separable Hilbert space H in the case when C is a Lipschitz multifunction defined on I×H, taking a closed ρ-proxy regular (ρ>0) values in H, and F is a convex weakly compact valued scalary uppersemicontinuous defined on I×H×H and satisfying the growth condition
(3)F(t,x,y)⊆(1+∥x∥+∥y∥)BH,
for all (t,x,y)∈[0,T]×H×H. For other results concerning (2) with or without perturbation in Hilbert spaces, we refer to [8–13]. Bounkhel and Al-Yusof [14] initiated the extension of (1) from the Hilbert setting to the Banach spaces setting. In fact, Bounkhel [8] considered (2) when C is a multifunction from I to the family of nonempty closed convex subsets of X and satisfyes a condition similar to Condition (C1) in the statement of our result (Theorem 18) and F is an uppersemicontinuous multifunction defined on [0,T]×X and with convex compact values in X* such that F(t,x)⊆Z, for all (t,x)∈[0,T]×X, for some convex compact set Z⊂X*. More recently, Ibrahim and Aladsani [15] considered a second order sweeping process without delay in a separable p-uniformly convex and q-uniformly smooth Banach space X, and the perturbation F is an upper semicontinuous defined from [0,T]×X×Xto the family of nonempty convex weakly compact sets of the topological dual space X* of X such that
(4)F(t,x,y)⊆(1+∥x∥+∥y∥)BX,0000∀(t,x,y)∈[0,T[×X×X.In this paper, our main purpose is to prove the existence of solutions of (2) in the case when the perturbation F is nonconvex noncompact values. We will adapt the discretization technique used in [6].The paper is organized as follows. Section2 is devoted to some definitions and notations needed later. In Section 3, we prove in Theorem 18 the main result.We note that ifX is a Hilbert space, then J is equal to the identity operator on X, and X 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

LetI=[0,T], T>0, and let X be a Banach space with topological dual space X*. Let B={x∈X:∥x∥≤1} and B*={z∈X*:∥z∥≤1}.Definition 1 (see [1, Definition2.2.1]).
A Banach spaceX is said to be uniformly convex if for any ε, 0<ε≤2, the inequalities ∥x∥≤1,∥y∥≤1, and ∥x-y∥≥ε imply that there exists a δ=δ(ε) such that ∥(1/2)(x+y)∥≤1-δ.Lemma 2 (see [16, Theorems 2.2.4 and2.2.8]).
(1)
Every Hilbert space is uniformly convex.
(2)
Every uniformly convex space is strictly convex.
(3)
Every uniformly convex space is reflexive.Definition 3 (see [1, 17, 18]).
Letp>1 be a real number. A Banach space X is said to be p-uniformly convex if there exists a constant λ>0 such that
(5)δX(ε)≥λεp,∀ε∈(0,2].Definition 4 (see [1, Definition2.4.1]).
LetX* be the topological dual of a Banach space X; then the multivalued mapping
(6)J:X⟶2X*,J(x)={y∈X*:〈x,y〉=∥x∥2=∥y∥2}
is said to be the duality mapping (or duality mapping) in X, and the multivalued mapping
(7)J*:X*⟶2X,J*(y)={x∈X:〈y,x〉=∥x∥2=∥y∥2}
is called the normalized duality mapping (or duality mapping) in X*.In the following lemma, we recall some properties of the duality mapping.Lemma 5 (see [1, 19] Propositions2.4.5, 2.4.12, and2.4.15).
(1)
IfX is a Hilbert space, then J(x)={x}, for all x∈X;
(2)
for eachx∈X, J(x) is nonempty closed convex and bounded subset ofX*;
(3)
J
(
λ
x
)
=
λ
J
(
x
), for all x∈X and for all λ∈ℝ;
(4)
ifX* is strictly convex, J is single valued;
(5)
ifX is strictly convex, J is one to one; that is,x≠y⇒J(x)∩J(y)=ϕ;
(6)
ifX*is uniformly convex, then J is uniformly continuous on each bounded set inX; that is, for all ε>0 and α>0, there is a δ=δ(ε,α)>0 such that
(8)∥x∥≤α,∥y∥≤α,∥x-y∥<δ⟹∥J(x)-J(y)∥<ε
(note that ifX* is uniformly convex, then it is strictly convex, and hence J is single-valued mapping);
(7)
ifX is reflexive, then J is a mapping from X ontoX*; that is,
(9)⋃{J(x):x∈X}=X*,
(8)
ifX is reflexive strictly convex space with strictly convex conjugate spaceX*, then J and J* are one-to-one, onto, and single-valued mappings and
(10)J-1=J*,JJ*=IX*,J*J=IX,
whereIX is the identity mapping on X and IX* is the identity mapping, on X*.For more properties of the duality mapping, we refer to [16, 19].Definition 6 (see [1, Definition2.8.1]).
The Banach spaceX is said to be uniformly smooth if
(11)ρX′(0)=limt→0ρX(t)t=0,
where ρX is the modulus of smoothness of X (note that every Hilbert space is uniformly smooth).Definition 7 (see [1, 17, 18]).
Letq>1 be a real number. A Banach space X is said to be q-uniformly smooth if there exist a constant c>0 such that ρX(t)≤ctq, for all t>0.Clearly, everyq-uniformly smooth is uniformly smooth (note that every Hilbert space is 2-uniformly smooth).Lemma 8 (see [1, Theorems2.8.4,2.8.5, and2.8.6]).
LetX be a Banach space.(1)
X is uniformly smooth if and only if X*is uniformly convex.(2)
X is uniformly convex if and only if X*is uniformly smooth.(3)
IfX is uniformly smooth, then X is reflexive.Lemma 9 (see [17, 18]).
LetX be a Banach space and p>1. (1)
IfX is p-uniformly convex, then X* is p′-uniformly smooth where p′=p/(p-1).(2)
IfX is p-uniformly smooth, then X*  is p′-uniformly convex where p′=p/(p-1).Now, letX be a Banach space and X*  its topological dual. Let
(12)V:X*×X⟶ℝ,V*:X×X*⟶ℝ
be two functions defined by
(13)V(φ,x)=∥φ∥2-2〈φ,x〉+∥x∥2,V*(x,φ)=V(φ,x).Based on the functionalV, the generalized projection of a point φ in X* onto a nonempty subset E of X can be defined as follows.Definition 10 (see [16]).
LetX be a Banach space, E a nonempty subset of X, and φ∈X*. If there exists a point z∈E satisfying
(14)V(φ,z)=dEV(φ),
then z is called a generalized projection of φ onto E; where dEV(φ)=infx∈EV(φ,x).
The set of all such points is denoted byπE(φ); that is,
(15)πE(φ)={z∈E:V(φ,z)=dEV(φ)}⊆X.Now, we list in the following theorem some properties ofV and πE(φ) (see [16]).Lemma 11.
LetX be a Banach space and X* its topological dual. (1)
V(J(x),x)=0.(2)
IfX is uniformly convex or uniformly smooth, then
(16)V(φ,x)=0⟺φ=J(x),∀x∈X,φ∈X*.(3)
IfX is a Hilbert space, then
(17)V(φ,x)=∥φ-x∥2.(4)
IfX is reflexive and E is a nonempty closed and convex subset of X, then
(a)
πE(φ)≠ϕ, for all φ∈X*;(b)
X is strictly convex if and only if πE(φ) is singleton, for all φ∈X*;(c)
ifX is also smooth, then, for any given φ∈X*,
(18)z∈πE(φ)⟺〈-J(z),x-z〉≤0,∀x∈E.Definition 12 (see [4]).
LetX be a Banach space with topological dual X*, E a nonempty closed convex subset of X, and z∈X. The convex normal cone of E at z is defined by
(19)NE(z)={φ∈X*:〈φ,x-z〉≤0,∀x∈E}.For more details about the convex normal cone see [4, 20].Lemma 13 (see [4]).
LetE be a nonempty, closed, and convex subset of Banach space X, and z∈X. Then one has the following.(1)
NE(z)∩B*=∂dE(z), where ∂dE(z) is the subdifferential of the function z→dE(z) (dE(z) is the distance from z to  E).(2)
IfX is reflexive and smooth, then
(20)z∈πE(φ)⟺φ-J(z)∈NE(z),∀φ∈X*,∀z∈X.Also, we need the following four Lemmas.Lemma 14 (see [14, Proposition2.2]).
LetΩ be an open subset in a normal vector space Z, X a Banach space, and C a continuous set-valued mapping defined on Ω and with nonempty compact convex values in X. Let (φn) be a sequence of X* that converges weakly to φ in X* and (xn) a sequence in X that converges strongly to x in X such that (tn) a sequence in Ω that converges to t in I. If φn∈∂dC(tn)(xn), then φ∈∂dC(t)(x).Lemma 15 (see [14, Lemma 4.1 and Proposition4.2]).
Letp,q>1, X be a p-uniformly convex and q-uniformly smooth Banach space, and let E be a nonempty bounded subset of X; then there exist two constants α>0, β>0 such that
(21)α∥x-y∥p≤V(J(x),y)≤β∥x-y∥q,∀x,y∈E.Moreover, ifE, in addition, is closed and φ∈X*, then
(22)dEV(φ)=0⟺J*(φ)∈E.Lemma 16 (see [21, Theorem2.2]).
LetΩ be a subset of a Banach space X. Assume that F:[a,b]×Ω→2x is a multifunction such that(1)
for everyx∈Ω, the multifunction F(·,x) is measurable on [a,b];(2)
for everyt∈[a,b], the multifunction F(t,·) is continuous on Ω.
Then, for any measurable functionx(·):[a,b]→Ω, the multifunction F(·,x(·)) is measurable on [a,b].Lemma 17 (see [21, Lemma2.3]).
LetX be a separable Banach space, G:[a,b]→2X a measurable multifunction, and z:[a,b]→X a measurable function. Then for any positive measurable function r:[a,b]→ℝ+, there exists a measurable selection g of G such that for almost t∈[a,b](23)∥g(t)-z(t)∥≤d(z(t),G(t))+r(t).


## 3. Existence of Solutions for Problem (2)

Theorem 18.
Letp,q>1, X a separable p-uniformly convex and q-uniformly smooth Banach space, I=[0,T], T>0, C a multifunction defined from I×X to the family of nonempty closed convex subsets of X, and F a multifunction defined from I×Cr×Cr with nonempty closed values in X*. Assume that the following hypotheses hold: (C1)
there is an absolutely continuous functionv:I→ℝ and two positive real numbers λ and γ such that, for all ξ1,ξ2∈X*, all x,y∈X, and all t,s∈I,
(24)|(dC(t,x)V(ξ1))1/q′-(dC(s,y)V(ξ2))1/q′|≤|v(t)-v(s)|+λ∥x-y∥+γ∥ξ1-ξ2∥,whereq′=q/(1-q);(C2)
there is a compact subsetK of X* such that(25)J(C(t,x))⊂K,∀(t,x)∈I×X;(F1)
for eachψ,φ∈Cr, t→F(t,ψ,φ) is a measurable;(F2)
there is a functionm∈L1(I,ℝ*) such that, for all t∈I, and for all ψ1, φ1, ψ2, φ2∈Cr,
(26)H(F(t,ψ1,φ1),F(t,ψ2,φ2))≤m(t)(∥ψ1-ψ2∥+∥φ1-φ2∥);(F3)
there is a functionp∈L1(I,ℝ+) such that, for all ψ,φ∈Cr=C([-r,0],X),
(27)∥F(t,ψ,φ)∥≤p(t)(∥ψ(0)∥+∥φ(0)∥),a.e.t∈I.
Then, for anyφ,ψ∈Cr with φ(0)∈C(0,ψ(0)), there exist two continuous functions u:[-r,T]→X* and w:[-r,T]→X such that (2) is satisfied.Proof.
At first, we note that since the Banach spaceX is p-uniformly convex, then it is uniformly convex, and hence by Lemma 2 it is reflexive and strictly convex. Moreover, because X is q-uniformly smooth, then, by Lemma 9, its conjugate X* is q′-uniformly convex, q′=q/(q-1), and hence X*is strictly convex. Then, by Lemmas 5 and 8, the duality mapping J is single valued, continuous, one-to-one, onto, and JJ*=IX* and J*J=IX. For notational convenience, we take T=1. Let k>0 such that K⊂kB*. Let n be a fixed positive integer and z:I→X* 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 un:[-r,1]→X* and wn:[-r,1]→X, two functions θn, δn:I→I and fn∈L1(I,X*) such that un and wn  are absolutely continuous function on I=[0,1] and that
(28)(i)un(t)=J(φ(t)),wn(t)=ψ(t),dddddddddddddddddfort∈[-r,0];(ii)fn(t)∈F(t,τ(δn(t))wn,τ(δn(t))J*(un)),dddddddddggddddddddddddddddddfort∈I;(iii)J*(un(θn(t)))∈C(θn(t),wn(θn(t)))∩B*(0,k),fort∈I;(iv)wn(t)=wn(δn(t))+(t-δn(t))J*(un(δn(t)))fort∈I,ffffwn′(t)=J*(un(t)),a.e.fort∈I;(v)∥fn(t)-z(t)∥≤d(z(t),F(t,τ(δn(t))wn,τ(δn(t))J*(un)))+1n,fort∈I;(vi)∥un′(t)-fn(t)∥≤1α1/q′[|v′(t)|+γ(∥ψ(0)∥+k)p(t)],a.e.fort∈I;(vii)un′(t)-fn(t)∈-NC(θn(t),wn(θn(t))(J*(un(θn(t)))),a.e.fort∈I.
In view of Lemma15, there is α>0 such that, for all z1,z2∈ηB*, we have
(29)α∥z1-z2∥q′≤V*(J*(z1),z2),
where η=k+(k+∥ψ(0)∥)∫0Tp(s)ds.
Fort∈[-r,0], we define un(t)=J(φ(t)) and wn(t)=ψ(t).  In order to define  un  and wn  on I, we define a function e:I→ℝ+  by
(30)e(t):=1α1/q′[∫0t(|v′(s)|+γ(∥ψ(0)∥+k)p(s))ds+λk],∀t∈I.
Let us consider a partition ofI=[0,1] by the points tin=i/n, 0≤i≤n. Let I0n={t0n}={0}, Ii+1n=]tin,ti+1n], and 0≤i≤n-1. In view of Lemma 16, there exists a function f0n∈L1([0,t1n],X*)  such that f0n(t)∈F(t,τ(0)ψ,τ(0)φ) and
(31)∥f0n(t)-z(t)∥≤d(z(t),F(t,τ(0)ψ,τ(0)φ)+1n,00000000000∀t∈[0,t1n].
We putx0n=φ(0)∈C(0,ψ(0)) and for t∈I1n we define
(32)wn(t)≔ψ(0)+(t-t0n)x0n,(33)un(t)≔J(x0n)+e(t)-e(t0n)e(t1n)-e(t0n)×(J(x1n)-J(x0n)-∫t0nt1nf0n(s)ds)+∫t0ntf0n(s)ds,
where
(34)x1n:=πC(t1n,wn(t1n))(J(x0n)+∫t0nt1nf0n(s)ds).
According to Lemma11, x1n is well defined. By recalling the definition of the generalized projection, we get x1n∈C(t1n,wn(t1n)) and
(35)J(x0n)+∫t0nt1nf0n(s)ds-J(x1n)∈NC(t1n,wn(t1n))(x1n).
As a result of Condition(C2), ∥φ(0)∥=∥J(φ(0))∥=∥J(x0n)∥≤k, ∥J(x1n)∥≤k. Moreover, by Condition (F3)(36)∥∫t0nt1nf0n(s)ds∥≤∫t0nt1np(s)(∥(τ(0)ψ)(0)∥+∥(τ(0)φ)(0)∥)ds≤(∥ψ(0)∥+∥φ(0∥)∫t0nt1np(s)ds≤(∥ψ(0)∥+k)∫t0nt1np(s)ds.
Therefore,(37)∥J(x0n)+∫t0nt1nf0n(s)ds∥≤k+(∥ψ(0)∥+k)∫t0nt1np(s)ds=η.
Consequently, by (29) we obtain
(38)α∥J(x1n)-(J(x0n)+∫t0nt1nf0n(s)ds)∥q′≤V*(J*(J(x1n)),J(x0n)+∫t0nt1nf0n(s)ds)=∥x1n∥2-2〈x1n,J(x0n)+∫t0nt1nf0n(s)ds〉+∥J(x0n)+∫t0nt1nf0n(s)ds∥2=V(J(x0n)+∫t0nt1nf0n(s)ds,x1n)=dC(t1n,wn(t1n))V(J(x0n)+∫t0nt1nf0n(s)ds).
We note that, from Lemma15 and the choice x0n,dC(t0n,wn(t0n))V(J(x0n))=0. Then, by (C1) and (36), the last inequality gives us
(39)α1/q′∥J(x1n)-(J(x0n)+∫t0nt1nf0n(s)ds)∥≤[dC(t1n,wn(t1n))V(J(x0n)+∫t0nt1nf0n(s)ds)]1/q′=[dC(t1n,wn(t1n))V(J(x0n)+∫t0nt1nf0n(s)ds)]1/q′-[dC(t0n,wn(t0n))V(J(x0n))]1/q′≤|v(t1n)-v(t0n)|+γ∥∫t0nt1nf0n(s)ds∥+λ∥wn(t1n)-wn(t0n)∥≤∫t0nt1nv′(s)ds+γ(∥ψ(0)∥+k)∫t0nt1np(s)ds+λk.
This inequality with (30) leads to
(40)∥J(x1n)-(J(x0n)+∫t0nt1nf0n(s)ds)∥≤1α1/q′[∫t0nt1n(|v′(s)|+γ(∥ψ(0)∥+k)p(s))ds+λk]=|e(t1n)-e(t0n)|.
Observe that from (33), for t∈]t0n,t1n[, we get
(41)un′(t)=e′(t)e(t1n)-e(t0n)×(J(x1n)-(J(x0n)+∫t0nt1nf0n(s)ds))+f0n(t).
From this equality with (40), we get
(42)∥un′(t)-f0n(t)∥≤|e′(t)|=1α1/q′[|v′(t)|+γ(∥ψ(0)∥+k)p(t)],∀t∈]t0n,t1n[.
Note that, from (33), we have
(43)J(x0n)+∫t0nt1nf0n(s)ds-J(x1n)∈NC(t1n,wn(t1n))(x1n).
Therefore, by (41),
(44)un′(t)-f0n(t)∈-NC(t1n,wn(t1n))(J*(un(t1n))),fora.e.t∈I1n.
Moreover, by (30), (33), (36), and (40), for t,s∈[0,t1n](s<t)   we have
(45)∥un(t)-un(s)∥≤e(t)-e(s)e(t1n)-e(t0n)∥J(x1n)-J(x0n)-∫t0nt1nf0n(s)ds∥+∥∫stf0n(β)dβ∥≤|e(t)-e(s)|+(∥ψ(0)∥+k)∫stp(β)dβ=1α1/q′[∫st(|v′(β)|+γ(∥ψ(0)∥+k)p(β))dβ+λk]+(∥ψ(0)∥+k)∫stp(β)dβ=1α1/q′[+α1/q′(∥ψ(0)∥+k)p(β))∫st‍dβ+λk]∫st(α1/q′|v′(β)|+γ(∥ψ(0)∥+k)p(β)+α1/q′(∥ψ(0)∥+k)p(β))∫st‍dβ+λk]=1α1/q′[∫st(α1/q′|v′(β)|+(∥ψ(0)∥+k)ffffffggggfffff×(γ+α1/q′)p(β))dβ+λk∫st(|v′(β)|],
this shows that un is absolutely continuous on [0,t1n].
We reiterate this process for constructing two sequences(fin),(xin),i=0,1,…,n-1, and two functions un:[-r,1]→X*, wn:[-r,1]→Xsuch that, for all i=0,1,…,n-1, the following properties hold:
(46)fin(t)∈F(t,τ(tin)wn,τ(tin)J*(un)),000000000000000000000000t∈Ii+1n,(47)∥fin(t)-z(t)∥≤d(z(t),F(t,τ(tin)wn,τ(tin)J*(un)))+1n,000000000000000000000000000000000.0t∈Ii+1n,(48)wn(t)=wn(tin)+(t-tin)J*(un(tin)),0000000000000000000000000t∈Ii+1n,(49)xi+1n=πC(ti+1n,wn(ti+1n))(J(xin)+∫tinti+1nfin(s)ds),0000000000000000000000000000t∈Ii+1n,(50)un(t)=J(xin)+e(t)-e(tin)e(ti+1n)-e(tin)×(J(xi+1n)-J(xin)-∫tinti+1nfin(s)ds)+∫tintfin(s)ds,t∈Ii+1n,(51)∥J(xi+1n)-(J(xin)+∫tinti+1nfin(s)ds)∥≤|e(ti+1n)-e(tin)|,(52)∥un′(t)-fin(t)∥≤|e′(t)|=1α1/q′[|v′(t)|+γ(∥ψ(0)∥+k)p(t)],fffffffffffffffffffffffffffffffffffffffffffffddddfora.e.t∈Ii+1n,(53)un′(t)-fin(t)∈-NC(ti+1n,wn(ti+1n)(J*(un(tin))),00000000000000000000000000fora.e.t∈Ii+1n,(54)xi+1n∈C(ti+1n,wn(ti+1n))∩B*(0,k),
where J*(un)(t)=J*(un(t)).
Note that, by (48), for every i=0,1,2,…,n-1,
(55)∥wn(ti+1n)∥≤∥wn(tin)∥+|ti+1n-tin|∥J*(un(tin))∥≤∥wn(tin)∥+|ti+1n-tin|k.
Then, from this recurrence relation, we can get(56)∥wn(ti+1n)∥≤∥ψ(0)∥+k,foreveryi=0,1,2,…,n-1.
This relation with (46), (54), and Condition (F3)gives us, for t∈Ii+1n,
(57)∥fin(t)∥≤p(t)(∥τ(tin)wn(0)∥+∥τ(tin)J*(un)(0)∥)≤p(t)(∥wn(tin)∥+∥J*(un(tin))∥)≤p(t)(∥ψ(0)∥+k+k)=p(t)(∥ψ(0)∥+2k).
Therefore, by (34), (50), (51), and (57), for t,s∈Ii+1n(s<t) we have
(58)∥un(t)-un(s)∥≤e(t)-e(s)e(ti+1n)-e(tin)∥J(xi+1n)-J(xin)-∫tinti+1nfin(s)ds∥+∥∫stfin(β)dβ∥≤|e(t)-e(s)|+(∥ψ(0)∥+2k)∫stp(β)dβ=1α1/q′[∫st|v′(β)|dβ+γ(∥ψ(0)∥+k)×∫stp(β)dβ+λk]+(∥ψ(0)∥+2k)∫stp(β)dβ=1α1/q′[∫st(|v′(β)|+γ(∥ψ(0)∥+k)p(β))dβ+λk]+(∥ψ(0)∥+2k)∫stp(β)dβ.
By addition, the last inequality holds fort,s∈I, with s<t; hence, un is absolutely continuous on I.
Now, for each positive integern we define the functions θn, δn:I→I and fn∈L1(I,X*) by setting
(59)θn(0)=t1n,δn(0)=0,fn(0)=fin(0),θn(t)=ti+1n,δn(t)=tin,fn(t)=fin(t)fort∈Ii+1n.
Then obviously, the relations (46), (47), (48), (52), (53), and (54) imply (28)(ii)→(vii), respectively.
Step 2. The sequences (un) and (wn)  converge uniformly to continuous functions u:[-r,1]→X*, w:[-r,1]→X, respectively, with u(t)=J(φ(t)) and w(t)=ψ(t) for t∈[-r,0] and J*(u(t))∈C(t,w(t)) for t∈I.
Indeed, by (58) for t∈I we have limn→∞∥un(θn(t))-un(t)∥=0. Moreover, by (28)(iii), for any n≥1 and any t∈I, we have un(θn(t))∈K. Then, for any t∈I, the set {un(t):n≥1} is relatively compact subset in X*. In addition, from (28)(vi), Condition (F3), and (57) we get for a.e. t∈I(60)∥un′(t)∥≤1α1/q′[|v′(t)|+γ(∥ψ(0)∥+k)p(t)]+∥fn(t)∥≤1α1/q′[|v′(t)|+γ(∥ψ(0)∥+k)p(t)]+p(t)(∥wn(δn(t))∥+∥J*(un(δn(t)))∥)≤1α1/q′[|v′(t)|+γ(∥ψ(0)∥+k)p(t)]+p(t)(∥ψ(0)∥+2k).
This, with the absolutely continuity ofun, assures the equicontinuity of un. Thus, by Arzela-Ascoli’s Theorem, we can select a subsequence of un, again denoted by un, which converges uniformly to an absolutely continuous function u on I such that, for any t∈I,
(61)u(t)=J(x0n)+∫0tu′(s)ds.
Moreover,(un′) converges weakly to u′∈L1(I,X*). We extend the definition of u in such a way that u=J(φ) on [-r,0]. Obviously, (un) converges uniformly to u on [-r,1]. Let us define un*(t)=J*(un(t)), t∈I. Since J*  is continuous on the compact set K and un(t)∈K for t∈I, then the sequence (un*) converges uniformly to a function u*=J*(u). In addition to the fact that wn′(t)=J*(un(t)), a.e. for t∈I, we confirm that the sequence (wn) converges uniformly to a function w(t)=ψ(0)+∫0tJ*(u(s))ds,t∈I. Let us show that J*(u(t))∈C(t,w(t)) for t∈I. Note that
(62)limn→∞∥un(θn(t))-u(t)∥≤limn→∞∥un(θn(t))-un(t)∥+limn→∞∥un(t)-u(t)∥=0.
Similarly,(63)limn→∞∥wn(θn(t))-w(t)∥=0.
Moreover, asJ*(un(θn(t)))=J*(un(ti+1n))=J*(J(xi+1n))=xi+1n∈C(θn(t),wn(θn(t))), it follows that
(64)dC(θn(t),wn(θn(t)))V(un(θn(t)))=0,t∈I.
Then, from (62), (63), (64), Condition (C1) and one obtains, for t∈I,
(65)(dC(t,w(t))V(u(t)))1/q′≤|(dC(t,w(t))V(u(t)))1/q′-(dC(θn(t),wn(θn(t)))V(un(θn(t))))1/q′|≤|v(t)-v(θn(t))|+γ∥un(θn(t))-u(t)∥+λ∥wn(θn(t))-w(t)∥.
By taking the limit asn→∞, we conclude that J*(u(t))∈C(t,w(t)) for t∈I.
Step 3. For any t∈I, the sequence τ(δn(t)un*) converges to τ(t)u*, and the sequence τ(δn(t)wn) converges to τ(t)w in Cr. Let t∈I. We have
(66)∥τ(δn(t))un*-τ(t)u*∥≤∥τ(δn(t))un*-τ(t)un*∥+∥τ(t)un*-τ(t)u*∥≤sup-r≤s≤0∥un*(δn(t)+s)-un*(t+s)∥+∥τ(t)un*-τ(t)u*∥≤sup-r≤s1≤s2≤1|s1-s2|≤1/n∥un*(s1)-un*(s2)∥+∥τ(t)un*-τ(t)u*∥≤sup-r≤s1,s2≤0|s1-s2|≤1/n∥un*(s1)-un*(s2)∥+sup-r≤s1≤0≤s2≤1|s1-s2|≤1/n∥un*(s1)-un*(s2)∥+sup0≤s1≤s2≤1∥un*(s1)-un*(s2)∥+∥τ(t)un*-τ(t)u*∥≤sup-r≤s1,s2≤0|s1-s2|≤1/n∥φ(s1)-φ(s2)∥+sup-r≤s1≤0|s1|≤1/n∥un*(s1)-un*(0)∥+sup0≤s2≤1|s2|≤1/n∥un*(0)-un*(s2)∥+sup0≤s1≤s2≤1|s1-s2|≤1/n∥un*(s1)-un*(s2)∥+∥τ(t)un*-τ(t)u*∥≤2sup-r≤s1,s2≤0|s1-s2|≤1/n∥φ(s1)-φ(s2)∥+2sup0≤s1,s2≤1|s1-s2|≤1/n∥un*(s1)-un*(s2)∥+∥τ(t)un*-τ(t)u*∥.
By the continuity ofφ, the uniform convergence of un* towards u*, and the preceding estimate, we get
(67)limn→∞∥τ(δn(t))un*-τ(t)u*∥=0.
Similarly, we can show thatlimn→∞∥τ(δn(t))wn-τ(t)w∥=0.
Step 4. The sequence (fn) converges pointwise to a function f∈L1(I,X*) satisfying f(t)∈F(t,τ(t)w,τ(t)u*), t∈I.
Lett∈I be fixed. In view of (28)(v) and Condition  (F2), we obtain for n≥1(68)∥fn+1(t)-fn(t)∥≤H(F(t,τ(δn(t))wn,τ(δn(t))un*),ffffffF(t,τ(δn+1(t))wn+1,τ(δn+1(t))un+1*))+1n+1≤m(t)(∥τ(δn(t))wn-τ(δn+1(t))wn+1∥000000+∥τ(δn(t))un*-τ(δn+1(t))un+1*∥)+1n+1.
From Step 3, the right-hand side of (68) tends to zero when n→∞. Hence, (fn(t)) is a Cauchy sequence in X*; then it converges pointwise to a function f∈L1(I,X*). Moreover, by (28)(v) and Condition (F2), we get
(69)d(f(t),F(t,τ(t)w,τ(t)u*))≤∥f(t)-fn(t)∥+H(F(t,τ(t)w,τ(t)u*),F(t,τ(δn(t))wn,τ(δn(t))un*))≤∥f(t)-fn(t)∥+m(t)×(∥τ(t)w-τ(δn(t))wn∥+∥τ(t)u*-τ(δn(t))un*∥).
Again, by Step 3, the right-hand side of this inequality tends to zero whenn→∞. Hence, f(t)∈F(t,τ(t)w,τ(t)J*(u)), and t∈I.
Step 5. For almost t∈I, u′(t)-f(t)∈-NC(t,w(t))(J*(u(t))).
We note the weak convergence ofun′-fn towards u′-f in L1(I,X*), and Mazur’s Lemma gives us(70)u′(t)-f(t)∈⋂nCo¯{um′(t)-fm(t):m≥n},0000000000000000000000000000fora.e.t∈I.
Fix anyt such that the preceding relation is satisfied, and consider x∈X. The last relation above yields
(71)〈u′(t)-f(t),x〉≤infnSupm≥n〈um′(t)-fm(t),x〉.
Moreover, the relations (28)(vi) and (vii) tell us for a.e. t∈I(72)un′(t)-fn(t)∈-NC(θn(t),wn(θn(t)))J*(un(θn(t)))∩B*(0,e′(t)),
where the function e  is defined by (34). Hence, by Lemma 13, for a.e. t∈I,  we have
(73)un′(t)-fn(t)∈-e′(t)∂dC(θn(t),wn(θn(t)))(J*(un(θn(t)))).
In view of (63) and (64), for x∈X and a.e. t∈I, one obtains
(74)〈u′(t)-f(t),x〉≤limn→∞Sup〈um′(t)-fm(t),x〉≤e′(t)limn→∞Supσx,-∂dC(θn(t),wn(θn(t)))(J*(un(θn(t)))),
where σ is the support function. From the uppersemicontinuity property, Lemma 14 and the last relation above yield for x∈X and a.e. t∈I(75)〈u′(t)-f(t),x〉≤e′(t)σ(x,-∂dC(t,w(t))J*(u(t))).
As the set∂dC(t,wn(t))(J*(u(t))) is closed and convex, we get
(76)u′(t)-f(t)∈-NC(t,w(t))(J*(u(t))),fora.e.t∈I.
This proves the claim.
Finally, by Steps 4 and 5 we have for almostt∈I(77)u′(t)∈-NC(t,w(t))(J*(u(t)))+F(t,τ(t)w,τ(t)J*(u(t))),
and the proof is complete.


## 4. Conclusion

In this paper, existence problem of solution of a functional sweeping process in Banach space with noncompact perturbation has been considered. Some sufficient conditions have been obtained. The importance of this work due to the fact that the values of the perturbation are not necessarily convex or compact and the considerable space is a Banach space.



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*Source: 567094-2013-10-31.xml*