# Relaxation Problems Involving Second-Order Differential Inclusions

Authors: Adel Mahmoud Gomaa
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We present relaxation problems in control theory for the second-order differential inclusions, with four boundary conditions,u¨(t)∈F(t,u(t),u˙(t)) a.e. on [0,1]; u(0)=0,u(η)=u(θ)=u(1) and, with m≥3 boundary conditions, u¨(t)∈F(t,u(t),u˙(t)) a.e. on [0,1];u˙(0)=0,u(1)=∑i=1m-2‍aiu(ξi), where 0<η<θ<1, 0<ξ1<ξ2<⋯<ξm-2<1 and F is a multifunction from [0,1]×ℝn×ℝn to the nonempty compact convex subsets of ℝn. We have results that improve earlier theorems.


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## Body

## 1. Introduction

Second-order differential inclusions of three boundary conditions were studied by many authors [1–6], using Hartman-type functions. Such a function was first introduced by [7] for two boundary conditions. Moreover, in [8] we consider second-order differential inclusions with four boundary conditions,
(1)u¨(t)∈F(t,u(t),u˙(t)),a.e.on[0,T],u(0)=x0,u(η)=u(θ)=u(T),
where 0<η<θ<T and F is a multifunction from [0,T]×ℝn×ℝn to the nonempty compact subsets of ℝn, while in [9] we study four-point boundary value problems for differential inclusions and differential equations with and without multivalued moving constraints.In the present paper, we study relaxation results for the second-order differential inclusions, with four boundary conditions,(P)u¨(t)∈F(t,u(t),u˙(t)),a.e.on[0,1],u(0)=0,u(η)=u(θ)=u(1)
and, with m≥3 boundary conditions,
(Q)u¨(t)∈F(t,u(t),u˙(t)),a.e.on[0,1],u˙(0)=0,u(1)=∑i=1m-2‍aiu(ξi),
where 0<η<θ<1, 0<ξ1<ξ2<⋯<ξm-2<1, and F is a multifunction from [0,1]×ℝn×ℝn to the non-empty compact subsets of ℝn.In conjunction with Problem(P) and Problem (Q) we also consider the following problems:
(Pe)u¨(t)∈extF(t,u(t),u˙(t)),a.e.on[0,1],u(0)=0,u(η)=u(θ)=u(1),(Qe)u¨(t)∈extF(t,u(t),u˙(t)),a.e.on[0,1],u˙(0)=0,u(1)=∑i=1m-2‍aiu(ξi).
By extF(t,u(t),u˙(t)), we denote the set of extreme points of F(t,u(t),u˙(t)).


## 2. Notations and Preliminaries

Throughout this paper we letI=[0,1] and 0<ξ1<ξ2<⋯<ξm-2<1. We will use the following definitions, notations, and summarize some results.(i) A multifunctionF from a metric space (X,d) to the set Pf(Y) of all closed subsets of another metric space Y is lower semicontinuous (l. s. c.) at x0∈X if for every open subset V in Y with F(x0)∩V≠∅ there exists an open subset U in X such that x0∈U and F(x)∩V≠∅ for all x∈U.  F is l. s. c. if it is l. s. c. at each x0∈X.(ii)F is upper semicontinuous (u. s. c.) at x0∈X if for every open subset V in Y and containing F(x0) there exists an open subset U in X such that x0∈U and F(x)⊆V, for all x∈U. F is u. s. c. if it is u. s. c. at each x0∈X.(iii) A multifunctionF from I into the set Pf(X) of all closed subsets of X is measurable if for all x∈X the function t→d(x,F(t))=inf{∥x-y∥:y∈F(t)} is measurable [10–13].(iv) Let(Ω,Σ) be a measurable space and X a separable Banach space. We say that F:Ω→Pf(X) is graph measurable if
(2)gr(F)={(z,x)∈Ω×X:x∈F(z)}∈Σ×ℬ(X),
where ℬ(X) is the Borel σ-field of X. For further details we refer to [14–16].(v)F is continuous if it is lower and upper semicontinuous.(vi) For eachA,B∈Pf(X), the Hausdorff metric is defined by
(3)dH(A,B)=max[supa∈Ad(a,B),supb∈Bd(b,A)].It is known that the space(Pf(X),dH) is a generalized metric space, if the sets are not bounded (see, for instance, [14, 15]).(vii) A multifunctionF is Hausdorff continuous (dH-continuous) if it is continuous from X into the metric space (Pf(Y),dH).(viii) IfF has compact values in Y, then F is dH-continuous if and only if it is continuous [14, 17].(ix) We denote byPkc(ℝn) the nonempty compact convex subsets of ℝn.(x) The Banach spacesC(I,ℝn), C1(I,ℝn), and C2(I,ℝn) endowed with the norms
(4)∥u∥C=maxt∈I|u(t)|,∥u∥C1=max{∥u∥C,∥u˙∥C},∥u∥C2=max{∥u∥C,∥u˙∥C,∥u¨∥C},
respectively.(xi)Lw1(I,ℝn) denotes the space L1(I,ℝn) equipped with weak norm ∥·∥w which is defined by
(5)∥h∥w=sup{∥∫ab‍h(t)dt∥:0≤a≤b≤1}.(xii)W2,1(I,ℝn) is the Sobolev space of functions u:I→ℝn, u and u˙ are both absolutely continuous functions so u¨(t)∈L1(I,ℝn) and it is equipped with the norm ∥u∥W2,1(I,ℝn)=∥u∥L1(I,ℝn)+∥u˙∥L1(I,ℝn)+∥u¨∥L1(I,ℝn).(xiii) LetR:I→2ℝn be a multifunction and δR1={h∈L1(I,ℝn):h(t)∈R(t)}.(xiv) By a solution of(P) (resp., of (Pe)) we mean a function u∈W2,1(I,ℝn) such that u¨(t)=h(t) a.e. on I with h∈δF(·,u(·),u˙(·))1 (resp., h∈δextF(·,u(·),u˙(·))1) and u(0)=0,u(η)=u(θ)=u(1).(xv) By a solution of(Q) (resp., of (Qe)) we mean a function u∈W2,1(I,ℝn) such that u¨(t)=h(t) a.e. on I with h∈δF(·,u(·),u˙(·))1 (resp., h∈δextF(·,u(·),u˙(·))1) and u˙(0)=0,u(1)=∑i=1m-2‍aiu(ξi).(xvi) In the sequel byΔP (resp., ΔPe) we denote the solution set of Problem (P) (resp., of Problem (Pe)). Moreover, by ΔQ (resp., ΔQe) we denote the solution set of Problem (Q) (resp., of Problem (Qe)).Definition 1.
LetE be a Banach space and let Y be a metric space. A multifunction G:I×Y→Pck(E) has the Scorza-Dragoni property (the SD-property) if for every ε>0 there exists a closed set A⊂I such that the Lebesgue measure μ(I∖A) is  less than ε and G|A×Y is continuous. The multifunction G is called integrably bounded on compacta in Y if, for any compact subset Q⊂Y, we can find an integrable function μQ:I→ℝ+ such that sup{∥y∥:y∈G(t,z)}≤μQ(t), for almost every z∈Q.Theorem 2 (see [18]).
LetY be a complete metric space, E a separable Banach space, Eσ the Banach space E endowed with the weak topology, M:I×Y→Pck(Eσ), and K a compact subset of C(I,Y). Furthermore, let R:K→2L1(I,E) be a multifunction defined by
(6)R(y)={g∈L1(I,E):g(t)∈M(t,y(t))a.e.onI}.
IfM has the SD-property and is integrably bounded on compacta in Y, then the set
(7)AK={f∈C(K,Lw1(I,E)):f(y)∈R(y)∀y∈K}
is nonempty complete subset of the space C(K,Lw1(I,E)). Moreover, AK=AextK¯ where Lw1(I,E) is the space of equivalence classes of Bochner-integrable functions v:I→E with the norm ∥v∥w=supt∈T∥∫0t‍v(s)ds∥ and
(8)AextK={f∈C(K,Lw1(I,E)):f(y)∈extR(y)∀y∈K}.Lemma 3 (see [19]).
Forp such that 1<p<∞ let {un,u}n∈ℕ⊆Lp(I,ℝn), supn∈ℕ∥un∥p<∞ and un→u with respect to the weak norm ∥·∥w. Then un→u weakly in Lp(I,ℝn).Next we state a preliminary lemma, for0<η<θ<1, which is useful in the study of four boundary problems for the differential equations and the differential inclusions, and moreover we summarize some properties of a Hartman-type function.Lemma 4 (see [8]).
LetG:I×I→ℝ be the function defined as follows:
as0≤t<η,
(9)G(t,τ)={-τif0≤τ≤t-tift<τ≤ηt(τ-θ)+(τ-η)θ-ηifη<τ≤θ1-τ1-θifθ<τ≤1,
when η≤t<θ,
(10)G(t,τ)={-τif0≤τ≤ητ(t-θ+1)+η(τ-t-1)θ-ηifη<τ≤tt(τ-θ)+(τ-η)θ-ηift<τ≤θ1-τ1-θifθ<τ≤1,
lastly if θ≤t≤1,
(11)G(t,τ)={-τif0≤τ≤ηη(τ-t-1)+τ(t-θ+1)θ-ηifη<τ≤θ1-τ1-θ+(t-τ)ifθ<τ≤t1-τ1-θift<τ≤1.
Then the following hold.  (i)
Ifu∈W2,1(I,ℝn) with u(0)=x0,u(1)=u(θ)=u(η), then
(12)u(t)=x0+∫01‍G(t,τ)u¨(τ)dτ,∀t∈I;(ii)
ifw∈L1(I,ℝn), then for all t∈I,
(13)∫01‍G(t,τ)w(τ)dτ=∫0t‍(t-τ)w(τ)dτ-∫0η‍t(τ-η)(t+1)θ-ηw(τ)dτ+∫0θ‍t(τ-θ)+(τ-η)θ-ηw(τ)dτ+∫θ1‍1-τ1-θw(τ)dτ;(iii)
supt,τ∈I|G(t,τ)|≤2, supt,τ∈I|∂G(t,τ)/∂t|≤1.Letc1,c2,a∈Lp(I,ℝ+),1<p<∞, and let L be a linear operator from C(I,ℝ)×C(I,ℝ) to C(I,ℝ)×C(I,ℝ) defined by L(f,g)=(f_,g_) such that, for all t∈I,
(14)f_(t)=∫0T‍|G(t,τ)|(c1(τ)f(τ)+c2(τ)g(τ))dτ,g_(t)=∫0T‍|∂G(t,τ)∂t|(c1(τ)f(τ)+c2(τ)g(τ))dτ.
If c1=c2=0, then clearly L=0. We note that if 𝒦={(h1,h2)∈C(I,ℝ)×C(I,ℝ):h1(t),h2(t)≥0,∀t∈I}, then L(𝒦)⊆𝒦. Moreover, the spectral radius r(L)=lim∥Ln∥1/n is an eigenvalue of L with an eigenvector in 𝒦 [20].


## 3. Relaxation Theorems

In this section, both Theorems5 and 7 improve [19, Theorem 4.1] with [21, Theorem 6]. Indeed in [19] Papageorgiou considered (P) and (Pe) with the two boundary conditions u(0)=u(1)=0 and in [21] Ibrahim and Gomaa study the same problems with three boundary conditions u(0)=x0,u(η)=u(1).Theorem 5.
LetF:I×ℝn×ℝn→Pkc(ℝn) be a multifunction such that(i)
for each(x,y)∈ℝn×ℝn, the multifunction F(·,x,y) is measurable,(ii)
dH(F(t,x,y),F(t,x′,y′))≤α1(t)∥x-x′∥+α2(t)∥y-y′∥ a.e. with α1,α2∈L1(I,ℝ+) and ∥α1+α2∥<1/2,(iii)
for each(t,x,y)∈I×ℝn×ℝn,
(15)∥F(t,x,y)∥=sup{∥v∥:v∈F(t,x,y)}≤a(t)+c1(t)∥x∥+c2(t)∥y∥
with a,c1,c2∈Lp(I,ℝ+)1<p<∞,(iv)
the spectral radius,r(L), is less than 1. Then for each solution u∈ΔPe, there is a sequence (um(·))m∈ℕ⊂ΔP converging to u(·) in (C1(I,ℝn),∥·∥C1).Proof.
From [9, Theorem 2.1], we obtain ΔPe≠∅. Moreover, we can say that ∥F(t,x,y)∥≤a1(t) a.e. on I for some a1∈Lp(I,ℝ+). Let u∈ΔP. Then
(16)u¨(t)=h(t),a.e.onI,u(0)=0,u(η)=u(θ)=u(1),
where h(t)∈F(t,u(t),u˙(t)) a.e. on I. Assume that f:L1(I,ℝn)→C1(I,ℝn) is a function such that, for each h∈L1(I,ℝn), f(h)∈W1,2(I,ℝn) is the unique solution of the second-order differential equation
(Ph)u¨(t)=h(t),a.e.onI,u(0)=0,u(η)=u(θ)=u(1).
Let 𝒮={u∈L1(I,ℝn):∥u(t)∥≤a1(t)a.e.onI}. It is easy to see that f(𝒮) is convex. Let (un)n∈ℕ be a sequence in f(𝒮). Hence, un∈W2,1(I,ℝn) with un(0)=x0,un(η)=un(θ)=un (17) and
(17)un(t)=x0+∫0t‍(t-τ)u¨n(τ)dτ-∫0η‍t(τ-η)(t+1)θ-ηu¨n(τ)dτ+∫0θ‍t(τ-θ)+(τ-η)θ-ηu¨n(τ)dτ+∫θ1‍1-τ1-θu¨n(τ)dτ.
Then,
(18)limn→∞un(t)=∫01‍G(t,τ)u¨(τ)dτ=u(t),
which means that f(𝒮) is a compact subset of C1(I,ℝn). Set
(19)𝒫ε(t)={x∈F(t,v(t),v˙(t)):∥h(t)-x∥<ε+d(h(t),F(t,v(t),v˙(t)))},
where ε>0 and v∈f(𝒮). Hence, for each t∈I, 𝒫ε(t)≠∅. Assume that ℬ(I) and ℬ(ℝn) are the Borel σ-fields of I and ℝn, respectively. From condition, (i) the function t→F(t,v(t),v˙(t)) is measurable. Hence, grF(·,v(·),v˙(·))∈ℬ(I)×ℬ(ℝn) and (t,x)→εd(h(t),F(t,v(t),v˙(t)))-∥h(t)-x∥ is measurable in t and continuous in x that is jointly measurable. Thus, by Aumann's selection theorem, there exists a measurable selection sε of 𝒫ε such that sε(t)∈𝒫ε(t) for each t∈I. Now we define a multifunction 𝒬ε:f(𝒮)→2L1(I,ℝn) by the following:
(20)𝒬ε(v)={x∈δF(·,v(·),v˙(·))1:∥h(t)-x∥<ε+d(h(t),F(t,v(t),v˙(t)))a.e.onI{x∈δF(·,v(·),v˙(·))1:∥h(t)-x∥<ε+d(h(t),F(t,v(t),v˙(t)))},
with 𝒬ε(v)(t)≠∅ for each v∈f(𝒮). From [22, Proposition 4], 𝒬ε is l.s.c. and clearly has decomposable values. Applying [22, Theorem 3], we have a continuous selection Sε of 𝒬ε¯. Therefore,
(21)∥h(t)-Sε(v)(t)∥≤ε+d(h(t),F(t,v(t),v˙(t)))≤ε+α1(t)∥u(t)-v(t)∥+α2(t)∥u˙(t)-v˙(t)∥a.e.onI.
From Theorem 2, we find a continuous function ξε:f(𝒮)→Lw1(I,ℝn) such that ξε(v)∈extδF(·,v(·),v˙(·))1 and ∥Sε(v)-ξε(v)∥<ε for each v∈f(𝒮). Define a multifunction R:f(𝒮)→2L1(I,ℝn) by
(22)R(u)={g∈L1(I,ℝn):g(t)∈F(t,u(t),u˙(t))a.e.onI}.
Assume that Y=ℝn×ℝn and set a multifunction M:I×Y→2ℝn such that M(t,(x,y))=F(t,x,y). From Theorem 3.1 in [23], M has SD-property. R has nonempty convex values. Let (gn)n∈ℕ be a sequence in R(u) for some u∈f(𝒮). So, for each t∈I,
(23)limn→∞gn(t)=g(t)∈F(t,u(t),u˙(t))
because F has closed values in ℝn. Therefore, g∈δF(·,u(·),u˙(·))1 which implies that R(·) has compact values in ℝn. We can apply Theorem 2 to find a continuous function θ:f(𝒮)→Lw1(I,ℝn) such that θ(u)∈ext(R(u)), for all u∈f(𝒮). We see that θ(u)(t)∈ext(M(t,(u(t),u˙(t)))) [24], hence θ(u)(t)∈extF(t,u(t),u˙(t)) a.e. on I. Assume that η:f(𝒮)→W1,2(I,ℝn) is the function which for each u∈f(𝒮), η(u)=g(θ(u)). For each u∈f(𝒮), we have ∥θ(u)(t)∥≤a1 and so θ(u)∈𝒮. Then, η is a function from f(𝒮) into f(𝒮) and also we see that η is continuous [19]. Now let εn→0, Sεn=Sn and ξn=ξεn. Then, for each n∈ℕ, the function foξn is a continuous function from the compact set f(𝒮) into itself. From Schauder's fixed point theorem, foξn has a fixed point un, but extδF(·,v(·),v˙(·))1=δextF(·,v(·),v˙(·))1 [24] so un∈ΔPe. By passing to a subsequence if necessary, we may assume that un→u^ in C1(I,ℝn). Then, we obtain
(24)∥un(t)-u(t)∥≤∫01‍∥[∫0t‍(t-τ)(ξn(τ)-h(τ))dτ-∫0η‍t(τ-η)(t+1)θ-η(ξn(τ)-h(τ))dτ+∫0θ‍t(τ-θ)+(τ-η)θ-η(ξn(τ)-h(τ))dτ+∫θ1‍1-τ1-θ(ξn(τ)-h(τ))dτ]∥ds≤∫01‍[∫0t‍(t-τ)∥ξn(τ)-Sn(τ)∥dτ+∫0t‍(t-τ)∥h(τ)-Sn(τ)∥dτ+∫0η‍t(τ-η)(t+1)θ-η(ξn(τ)-Sn(τ))dτ+∫0η‍t(τ-η)(t+1)θ-η∥h(τ)-Sn(τ)∥dτ+∫θ1‍1-τ1-θ∥ξn(τ)-Sn(τ)∥dτ+∫θ1‍1-τ1-θ∥h(τ)-Sn(τ)∥dτ]ds.
But ξn-Sn→0 with respect to the norm ∥·∥w from Lemma 3 we get ξn-Sn→0 weakly in L1(I,ℝn). So we have
(25)∫0t(t-τ)∥ξn(τ)-Sn(τ)∥dτ+∫0ηt(τ-η)(t+1)θ-η∥ξn(τ)-Sn(τ)∥dτ+∫θ11-τ1-θ∥ξn(τ)-Sn(τ)∥dτ⟶0.
Moreover,
(26)∫01‍[∫0t‍(t-τ)∥h(τ)-Sn(τ)∥dτ+∫0η‍t(τ-η)(t+1)θ-η(ξn(τ)-Sn(τ))dτ+∫θ1‍1-τ1-θ∥h(τ)-Sn(τ)∥dτ]ds≤∫01‍[∫0t‍(t-τ)(εn+α1(τ)∥u(τ)-un(τ)∥+α2(τ)∥u˙(τ)-u˙n(τ)∥)+∫0η‍t(τ-η)(t+1)θ-η(εn+α1(τ)∥u(τ)-un(τ)∥+α2(τ)∥u˙(τ)-u˙n(τ)∥)+∫θ1‍1-τ1-θ(εn+α1(τ)∥u(τ)-un(τ)∥+α2(τ)∥u˙(τ)-u˙n(τ)∥)]ds.
As n→∞, we have
(27)∥u^(t)-u(t)∥≤∫01‍[∫0t‍(t-τ)(∥u˙(τ)-u^˙(τ)∥α1(τ)∥u(τ)-u^(τ)∥+α2(τ)∥u˙(τ)-u^˙(τ)∥)dτ+∫0η‍t(τ-η)(t+1)θ-η(∥u˙(τ)-u^˙(τ)∥α1(τ)∥u(τ)-u^(τ)∥+α2(τ)∥u˙(τ)-u^˙(τ)∥)dτ+∫θ1‍1-τ1-θ(∥u˙(τ)-u^˙(τ)∥α1(τ)∥u(τ)-u^(τ)∥+α2(τ)∥u˙(τ)-u^˙(τ)∥)dτ]ds≤∥u-u^∥C1(I,ℝn)(∫0t‍(t-τ)(α1(τ)+α2(τ))dτ+∫0η‍t(τ-η)(t+1)θ-η(α1(τ)+α2(τ))dτ+∫θ1‍1-τ1-θ(α1(τ)+α2(τ))dτ)=∥u-u^∥C1(I,ℝn)∫01‍|G(t,τ)|(α1(τ)+α2(τ))dτ≤2∥u-u^∥C1(I,ℝn)∥α1(τ)+α2(τ)∥.
Since by assumption (ii),∥α1+α2∥<1/2 we get u=u^. So un→u in C1(I,ℝn) and u∈Δ¯Pe where the closure is taken in C1(I,ℝn) which means that ΔP⊆Δ¯Pe. Therefore, the proof is complete if we show that ΔP is closed. Indeed if vn∈ΔP and vn→v in C1(I,ℝn), then vn=f(yn) for yn∈δF(·,v(·),v˙(·))1. From assumption (iii) and the Dunford-Pettis theorem, {yn}n∈ℕ is weakly sequentially compact in L1(I,ℝn). So we can say that {yn}n∈ℕ in L1(I,ℝn). By [25, Theorem 3.1], we get
(28)y(t)∈conv¯lim¯{yn(t)}n∈ℕ⊆conv¯lim¯F(t,vn(t),v˙n(t))=F(t,v(t),v˙(t))a.e.onI.
Moreover,f(yn)→f(y) in L1(I,ℝn) for y∈L1(I,ℝn) and y(t)∈F(t,v(t),v˙(t)) a.e. on I. Hence, v∈ΔP; that is ΔP is closed in C1(I,ℝn).Now we consider the following assumptions:(A1)
β∈(0,π/2), ai>0 and ∑i=1m-2‍ai<1;(A2)
∑i=1m-2‍aicosβξi-cosβ>0 and Km=1/∑i=1m-2‍aicosβξi-cosβ;(A3)
C0=(sinβ/β)(1+Km) and C1=min{Km+1,Kmsin2β};(A4)
S={u∈C2(I,ℝn):u˙(0)=0,u(1)=∑i=1m-2‍aiu(ξi)};(A5)
𝒢:I×I→ℝ is defined by
(29)𝒢(t,s)={1βsinβ(t-s)if0≤s≤t≤10if0≤t≤s≤1+Kmβcosβt{sinβ(1-s)-∑i=1m-2‍aisinβ(ξi-s),if0≤s≤ξ1,sinβ(1-s)-∑i=2m-2‍aisinβ(ξi-s),ifξ1<s≤ξ2,sinβ(1-s)-∑i=3m-2‍aisinβ(ξi-s),ifξ2<s≤ξ3,⋮sinβ(1-s)-∑i=km-2‍aisinβ(ξi-s),ifξk-1<s≤ξk,⋮sinβ(1-s),ifξm-2<s≤1.Lemma 6 (see [26]).
If the assumptions(A1)–(A5) hold, then  (i)
0≤𝒢(t,s)≤C0 for all (t,s)∈I×I,(ii)
supt,s∈I|∂𝒢(t,s)/∂t|≤C1,(iii)
for eachx∈C1(I,ℝn) there exists a unique function ux∈S such that
(30)ux(t)=∫01‍𝒢(t,s)x(s)ds,(iv)
(∫01‍|𝒢(t,s)|kds)1/k≤C0 and (∫01‍|(∂𝒢/∂t)(t,s)|kds)1/k≤C1.Proof.
(ii) Since(31)∂𝒢(t,s)∂t={cosβ(t-s)if0≤s≤t≤10if0≤t≤s≤1-Kmsinβt{sinβ(1-s)-∑i=1m-2‍aisinβ(ξi-s),if0≤s≤ξ1,sinβ(1-s)-∑i=2m-2‍aisinβ(ξi-s),ifξ1<s≤ξ2,sinβ(1-s)-∑i=3m-2‍aisinβ(ξi-s),ifξ2<s≤ξ3,⋮sinβ(1-s)-∑i=km-2‍aisinβ(ξi-s),ifξk-1<s≤ξk,⋮sinβ(1-s),ifξm-2<s≤1,
then supt,s∈I∂𝒢(t,s)/∂t≤1+Km. Furthermore,
(32)∂𝒢(t,s)∂t≥Kmsinβt[∑i=1m-2‍aisin(ξi-s)-sinβ(1-β)]≥-Kmsin2β
and thus supt,s∈I|∂𝒢(t,s)/∂t|≤C1.Theorem 7.
Assume that the assumptions(A1) and (A2) hold. Let F be a multifunction from I×ℝn×ℝn to Pkc(ℝn) satisfying the following conditions:  (a)
for each(x,y)∈ℝ×ℝ, the multifunction F(·,x,y) is measurable;(b)
for eacht∈I, the function (x,y)→F(t,x,y) is continuous with respect to the Hausdorff metric dH;(c)
for each(t,x,y)∈I×ℝn×ℝn(33)∥F(t,x,y)∥≤sup{∥v∥:v∈F(t,x,y)}≤a(t)+c1(t)∥x∥+c2(t)∥y∥;(d)
the spectral radiusr(L) of L is less than one.
Then Problem(Qe) admits a solution in S.Proof.
We can say that∥F(t,x,y)∥≤a1(t) a.e. on I for some a1∈Lp(I,ℝ+) [9]. Let x∈C1(I,ℝn) and let u∈C2(I,ℝn) be the unique solution of the problem
(*)u¨(t)=x(t),a.e.onI,u˙(0)=0,u(1)=∑i=1m-2‍aiu(ξi).
From Lemma 6, we have u(t)=∫01‍𝒢(t,s)x(s)ds,∀t∈I. Thus, we define a function f:C1(I,ℝn)→C2(I,ℝn) such that f(x) is the unique solution of (*). Let
(34)𝒱={x∈C1(I,ℝn):∥x(t)∥≤a1(t)a.e.onI}.
From the Dunford-Pettis theorem, 𝒱 is weakly compact and then f(𝒱) is convex and compact subset of C2(I,ℝn). Let 𝒴=ℝn×ℝn. If 𝒦=f(𝒱),ℛ:𝒦→2L1(I,ℝn) and ℳ:I×𝒴→2ℝn, where ℛ(u)={g∈L1(I,ℝn):g(t)∈F(t,u(t),u˙(t))a.e.onI} and ℳ(t,(x,y))=F(t,x,y), then ℳ has SD-property [23]. It is easy to show that ℛ is nonempty and convex subset of L1(I,ℝn). If fn is a sequence in ℛ(u) for some u∈𝒦, then limn→∞fn(t)=f(t)∈F(t,u(t),u˙(t)), where the values of F are closed. Therefore, the values of ℛ are weakly compact. According to Theorem 5 there exists a continuous function r:𝒦→Lw1(I,ℝn) with r(u)∈ext(ℛ(u)), for all u∈𝒦. Thus, r(u)(t)∈ext(ℳ(t,u(t),u˙(t))) a.e. on I [24] which implies r(u)(t)∈ext(F(t,u(t),u˙(t))) a.e. on I. If u∈f(𝒱), then ∥r(u)(t)∥≤a1 and so r(u)∈𝒱. Put θ:f(𝒱)→W2,1(I,ℝn) such that θ(u)=f(r(u)), thus θ is a continuous function from f(𝒱) into f(𝒱) [19]. From Schauder's fixed point theorem, there exists x∈f(𝒱) such that x=θ(x)=f(r(x)) which means that there is x∈S⊆C2(I,ℝn) such that x¨(t)∈ext(F(t,x(t),x˙(t))).Theorem 8.
In the setting of Theorem7, if one replaces condition (b) by the following condition:
(b)′dH(F(t,x,y), F(t,x′,y′))≤k1∥x-x′∥+k2∥y-y′∥ a.e. with k1≥0, k2≥0 and |k1+k2|<1/2C0.
ThenΔQe is nonempty and ΔQe¯=ΔQ where the closure taken in C2(I,ℝn).Proof.
From Theorem7, we have ΔQe≠∅. Moreover, ∥F(t,x,y)∥≤b1(t) a.e. on I for some b1∈Lp(I,ℝ+). Let u∈ΔQ. Then
(35)u¨(t)=h(t),a.e.onI,u˙(0)=0,u(1)=∑i=1m-2‍aiu(ξi),
where h(t)∈F(t,u(t),u˙(t)) a.e. on I. Assume that f′:C1(I,ℝn)→C2(I,ℝn) is a function such that, for each h∈C1(I,ℝn), f′(h)∈C2(I,ℝn) is the unique solution of the second-order differential equation
(Qh)u¨(t)=h(t),a.e.onI,u˙(0)=0,u(1)=∑i=1m-2‍aiu(ξi).
Let S={u∈C1(I,ℝn):∥u(t)∥≤b1(t)a.e.onI}. So f′(S) is convex. Let (un)n∈ℕ be a sequence in f′(S). Hence, un∈C2(I,ℝn) with un(0)=0,u˙n(0)=0,un(1)=∑i=1m-2‍aiun(ξi). Then from Lemma 6,
(36)limn→∞un(t)=∫01‍𝒢(t,τ)u¨(τ)dτ=u(t),
hence, f′(S) is a compact subset of C2(I,ℝn). Set
(37)𝒬ε(t)={x∈F(t,v(t),v˙(t)):∥h(t)-x∥<ε+d(h(t),F(t,v(t),v˙(t)))},
where ε>0 and v∈f′(S). Hence, for each t∈I, 𝒬ε(t)≠∅. Assume that ℬ(I) and ℬ(ℝn) are the Borel σ-fields of I and ℝn, respectively. From condition (i), the function t→F(t,v(t),v˙(t)) is measurable. Hence, grF(·,v(·),v˙(·))∈ℬ(I)×ℬ(ℝn) and (t,x)→εd(h(t),F(t,v(t),v˙(t)))-∥h(t)-x∥ is measurable in t and continuous in x that is jointly measurable. Thus, by Aumann's selection theorem, there exists a measurable selection sε of 𝒬ε such that sε(t)∈𝒬ε(t) for each t∈I. Now we define a multifunction 𝒬ε:f′(S)→2C1(I,ℝn) by the following:
(38)𝒬ε(v)={x∈δF(·,v(·),v˙(·))1:∥h(t)-x∥<ε+d(h(t),F(t,v(t),v˙(t)))a.e.onIx∈δF(·,v(·),v˙(·))1:∥h(t)-x∥},
with 𝒬ε(v)(t)≠∅ for each v∈f′(S). From [22, Proposition 4], 𝒬ε is l.s.c. and clearly has decomposable values. Applying [22, Theorem 3], we have a continuous selection Sε of 𝒬ε¯. Therefore,
(39)∥h(t)-Sε(v)(t)∥≤ε+d(h(t),F(t,v(t),v˙(t)))≤ε+k1(t)∥u(t)-v(t)∥+k2(t)∥u˙(t)-v˙(t)∥a.e.onI.
From Theorem2, we find a continuous function ξε′:f′(S)→Lw1(I,ℝn) such that ξε′(v)∈extδF(·,v(·),v˙(·))1 and ∥Sε(v)-ξε′(v)∥<ε for each v∈f′(S). Define a multifunction R′:f′(S)→2C1(I,ℝn) by
(40)R′(u)={g∈C1(I,ℝn):g(t)∈F(t,u(t),u˙(t))a.e.onI}.
As in Theorem 5, let Y=ℝn×ℝn and set a multifunction M:I×Y→2ℝn such that M(t,(x,y))=F(t,x,y). From [23, Theorem 3.1], M has SD-property. R′ has nonempty convex values. Let (gn)n∈ℕ be a sequence in R′(u) for some u∈f′(S). So, for each t∈I,
(41)limn→∞gn(t)=g(t)∈F(t,u(t),u˙(t))
because F has closed values in ℝn. Therefore, g∈δF(·,u(·),u˙(·))1 which implies R′(·) has compact values in ℝn. We can apply Theorem 2 to find a continuous function θ′:f′(S)→Lw1(I,ℝn) such that θ′(u)∈ext(R′(u)), for all u∈f′(S). We see that θ′(u)(t)∈ext(M(t,(u(t),u˙(t)))) [24], hence θ′(u)(t)∈extF(t,u(t),u˙(t)) a.e. on I. Assume that η′:f′(S)→C2(I,ℝn) is the function which for each u∈f′(S), η′(u)=g(θ′(u)). For each u∈f′(S), we have ∥θ′(u)(t)∥≤b1 and so θ′(u)∈S. Then, η′ is a function from f′(S) into f′(S) and also we see that η′ is continuous [19]. Now let εn→0, Sεn=Sn and ξn′=ξεn′. Then, for each n∈ℕ, the function f′oξn′ is a continuous function from the compact set f′(S) into itself. From Schauder's fixed point theorem, foξn′ has a fixed point un, but extδF(·,v(·),v˙(·))1=δextF(·,v(·),v˙(·))1 [24] so un∈ΔPe. Assume that un→u^ in C2(I,ℝn). From Lemma 6, we obtain
(42)∥un(t)-u(t)∥≤∫01‍[∫01‍|𝒢(t,τ)|∥ξ′n(τ)-Sn(τ)∥dτ+∫01‍|𝒢(t,τ)|∥(Sn(τ)-h(τ))∥dτ]ds.
But ξn′-Sn→0 with respect to the norm ∥·∥w and from Lemma 3 we get ξn′-Sn→0 weakly in C1(I,ℝn). So we have
(43)∫01‍|𝒢(t,τ)|∥ξn′(τ)-Sn(τ)∥dτ⟶0.
Moreover, as n→∞ we have
(44)∥u^(t)-u(t)∥≤∥u-u^∥C1(I,ℝn)∫01|𝒢(t,τ)|‍(k1(τ)+k2(τ))dτ≤∥u-u^∥C1(I,ℝn)∥k1(τ)+k2(τ)∥C0.
Since by assumption (ii), ∥k1+k2∥<1/2C0, thus from Lemma 6, we get u=u^. So un→u in C2(I,ℝn) and u∈Δ¯Qe where the closure is taken in C2(I,ℝn) which means that ΔP⊆Δ¯Pe. If vn∈ΔQ and vn→v in C2(I,ℝn), then vn=f′(yn) for yn∈δF′(·,v(·),v˙(·))1. From assumption (iii) and the Dunford-Pettis theorem, {yn}n∈ℕ is weakly sequentially compact in C2(I,ℝn). By [25, Theorem 3.1], we get
(45)y(t)∈conv¯lim¯{yn(t)}n∈ℕ⊆conv¯lim¯F(t,vn(t),v˙n(t))=F(t,v(t),v˙(t))a.e.onI.
Moreover, f′(yn)→f′(y) in C2(I,ℝn) for y∈C2(I,ℝn) and y(t)∈F(t,v(t),v˙(t)) a.e. on I. Hence, v∈ΔQ; that is, ΔQ is closed in C2(I,ℝn).



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*Source: 792431-2013-04-10.xml*