# Variational Methods for Almost Periodic Solutions of a Class of Neutral Delay Equations

Authors: M. Ayachi; J. Blot
Journal: Abstract and Applied Analysis
 (2008)

---

## Abstract

We provide new variational settings to study the a.p. (almost periodic)
solutions of a class of nonlinear neutral delay equations. We extend
Shu and  Xu  (2006) variational setting for
periodic solutions of nonlinear neutral delay equation to the almost periodic settings. We obtain
results on the structure of the set of the a.p. solutions, results of existence of a.p. solutions,
results of existence of a.p. solutions, and also a density result for the forced
equations.


---

## Body

## 1. Introduction

The  aim of this
paper is the study of a.p. (almost periodic) solutions of neutral delay
equations of the following form:(1.1)D1L(x(t−r),x(t−2r),x′(t−r),x′(t−2r),t−r)+D2L(x(t),x(t−r),x′(t),x′(t−r),t)=ddt[D3L(x(t−r),x(t−2r),x′(t−r),x′(t−2r),t−r)+D4L(x(t),x(t−r),x′(t),x′(t−r),t)],  where L:(ℝn)4×ℝ→ℝ is a
differentiable function, Dj denotes the
partial differential with respect to the jth vector
variable, and r∈(0,∞) is fixed. We
will  consider the
almost periodicity in the sense of Corduneau [1], and in the
sense of Besicovitch [2].A special case of (1.1) is the following forced
neutral delay equation:(1.2)D1K(x(t−r),x(t−2r),x′(t−r),x′(t−2r))+D2K(x(t),x(t−r),x′(t),x′(t−r))−ddt[D3K(x(t−r),x(t−2r),x′(t−r),x′(t−2r))+D4K(x(t),x(t−r),x′(t),x′(t−r))]=b(t),  where K:(ℝn)4→ℝ is a
differentiable function, and b:ℝ→ℝn is an a.p.
forcing term. To see (1.2) as a special case of
(1.1), it suffices to take(1.3)L(x1,x2,x3,x4,t):=K(x1,x2,x3,x4)−x1⋅b(t+r), where the point denotes the usual inner product in ℝn.Another special case of (1.1) is the following forced
second-order neutral delay equation:(1.4)x′′(t−r)+D1F(x(t−r),x(t−2r))+D2F(x(t),x(t−r))=b(t), where b:ℝn→ℝ and F:(ℝn)2→ℝ. To see that this last equation is a special case of
(1.1), it suffices to take L(x1,x2,x3,x4,t):=(1/2)∥x3∥2−F(x1,x2)+x1⋅b(t+r), where the norm is the usual Euclidian norm of ℝn. In their work [3], Shu and Xu study the periodic
solutions of this last equation by using a variational method. We want to
extend such a view point to the study of the a.p. solutions.And so our approach to the study of the a.p. solutions
of (1.1) consists to search critical points of a functional Φ defined on
suitable Banach spaces of a.p. functions by(1.5)Φ(x):=lim⁡T→+∞12T∫−TTL(x(t),x(t−r),x′(t),x′(t−r),t)dt.  At this time,
we give some historical elements. Recall that the work [4] of Elsgolc treats
the calculus of variations with a retarded argument on a bounded real interval.
This work was followed by these ones of Hughes [5] and Sabbagh [6]. Since the
variational problems can be seen as optimal control problems, recall also the
existence of the theory of the periodic optimal control with retarded argument
as developed by Colonius  [7]. For instance, we consider a periodic optimal
control problem with a criterion of the form (1/T)∫0Tg(x(t),u(t),t)dt and with an
equation of motion of the form x′(t)=f(x(t),x(t−r),u(t),t), where x(t) is the state
variable and u(t) the control
variable. In the special case, where f(x(t),x(t−r),u(t),t)=f1(x(t),x(t−r),t)+u(t), the previous optimal control problem can be
transformed into a calculus of variations problem with the criterion (1/T)∫0Tg(x(t),f1(x(t),x(t−r),t)−x′(t),t)dt, which is a special case of
(1.5). Note that the
Euler-Lagrange equation of such a variational problem is a special case of
(1.1).On another hand, calculus of variations in mean time
was developed to study the a.p. solutions of some (nonretarded) differential
equations [8–13]. Here, we extend this approach to treat equation like
(1.1).Now we describe the contents of this paper. In Section2,
we precise the notations about the function spaces used later. In
Section 3,
we establish a variational formalism suitable to the Bohr a.p. solutions; we
give a variational principle and a result on the structure of the set of the
a.p. solutions of (1.1) in the convex case. In
Section 4, we establish a
variational formalism suitable to the Besicovitch a.p. solutions, we give a
variational principle, results of existence, and a result of density for the
a.p. forced equations.


## 2. Notations

AP0(ℝn) is the space
of
the Bohr almost periodic (Bohr a.p.) functions from ℝ in ℝn; endowed with the supremum ∥⋅∥∞, it is a Banach space [1].AP1(ℝn):={x∈𝒞1(ℝ,ℝn)∩AP0(ℝn):x′∈AP0(ℝn)}; endowed with the norm ∥x∥𝒞1:=∥x∥∞+∥x′∥∞, it is a Banach space.Whenk∈ℕ∗∪{∞}, APk(ℝn):={x∈𝒞k(ℝ,ℝn):∀j≤k,djx/dtj∈AP0(ℝn)}.Whenx∈AP0(ℝn), its mean value is(2.1)ℳ{x(t)}t:=lim⁡T→+∞12T∫−TTx(t)dtexistsinℝn.The Fourier-Bohr coefficients ofx∈AP0(ℝn) are the complex
vectors(2.2)a(x;λ):=lim⁡T→+∞12T∫−TTe−iλtx(t)dt and Λ(x):={λ∈ℝ:a(x,λ)≠0}.Whenp∈[1,∞),Bp(ℝn) is the
completion of AP0(ℝn) (in Llocp(ℝ,ℝn)) with respect
to the norm ∥u∥p:=ℳ{|u|p}1/p. When p=2, B2(ℝn) is a Hilbert
spaces and its norm ∥⋅∥2 is associated
to the inner product (u∣v):=ℳ{u⋅v} [2]. The
elements of these spaces Bp(ℝn) are called
Besicovitch almost periodic (Besicovitch a.p.) functions.Recall the useful following fact: if(um)m is a sequence
in AP0(ℝn) and if u∈Llocp(ℝ,ℝn) (Lebesgue
space), which satisfy(2.3)ℳ¯{∥um−u∥p}1/p=(limsupT→∞12T∫−TT∥um−u∥pdt)1/p⟶0(m⟶0) then u∈Bp(ℝn) and we have ∥um−u∥p→0(m→0).We use the generalized derivative∇u∈B2(ℝn) of u∈B2(ℝn) (when it
exists) defined by ∥∇u−(1/s)(u(⋅+s)−u)∥2→0(s→0), and we define B1,2(ℝn):={u∈B2(ℝn):∇u∈B2(ℝn)}; endowed with the inner product 〈u∣v〉:=(u∣v)+(∇u∣∇v), B1,2(ℝn) is a Hilbert
space [11, 13].IfE and F are two
finite-dimensional-normed spaces, APU(E×ℝ,F) stands for the
space of the functions f:E×ℝ→F,(x,t)↦f(x,t), which are almost periodic in t uniformly with
respect to x in the
classical sense given in [14].To make the writing less heavy, we sometimes use the
notations(2.4)u¯(t):=(u(t),u(t−r),∇u(t),∇u(t−r)), when u∈B1,2(ℝn), and(2.5)x¯(t):=(x(t),x(t−r),x′(t),x′(t−r)), when x∈AP1(ℝn).


## 3. A Variational Setting for the Bohr a.p. Functions

We consider the
following condition:(3.1)L∈APU((ℝn)4×ℝ,ℝ),forall(X,t)∈(ℝn)4×ℝ,thepartialdifferentialDXL(X,t)exists,andDXL∈APU((ℝn)4×ℝ,ℒ((ℝn)4,ℝ)).Lemma 3.1.
Under [9] the functionalΦ:AP1(ℝn)→ℝdefined by
(1.5) which is of class𝒞1, and for allx,h∈AP1(ℝn),then(3.2)DΦ(x)⋅h=ℳ{D1L(x¯(t),t)⋅h(t)+D2L(x¯(t),t)⋅h(t−r)+D3L(x¯(t),t)⋅h′(t)+D4L(x¯(t),t)⋅h′(t−r)}t.Proof.
We
introduce the linear operator𝒯:AP1(ℝn)→(AP0(ℝn))4 by setting 𝒯(x)(t):=x¯(t). The four
components of 𝒯 are continuous
linear operators that imply the continuity of 𝒯, and therefore 𝒯 is of class 𝒞1, and for all x,h∈AP1(ℝn) we have D𝒯(x)⋅h=𝒯(h).
Under (3.1), the Nemytski operator 𝒩L:(AP0(ℝn))4→AP0(ℝn), defined by  𝒩L(X)(t):=L(X(t),t), is of class 𝒞1 (cf.
[15, Lemma 7]) and we have, for all X,H∈AP0(ℝn)4, (D𝒩L(X)⋅H)(t)=DXL(X(t),t)⋅H(t).
The linear functionalℳ:AP0(ℝn)→ℝ is continuous,
therefore it is of class 𝒞1 and we have,
for all ϕ,ψ∈AP0(ℝn), Dℳ{ϕ}⋅ψ=ℳ{ψ}.
And soΦ=ℳ◦𝒩L◦𝒯 is of class 𝒞1. Furthermore, we have(3.3)DΦ(x)⋅h=Dℳ(𝒩L◦𝒯(x))◦D𝒩L(𝒯(x))◦D𝒯(x)⋅h=ℳ{D𝒩L(𝒯(x))⋅𝒯(h)}=ℳ{DXL(x¯(t),t)⋅h¯(t)}t and expressing DXL in terms of DjL, we obtain the announced formula.Note that in
the case without delay, whenL is autonomous,
that is, L(X,t)=L(X), in [9], it is established that the functional x↦ℳ{L(x,x′)} is of class 𝒞1 when L is of class 𝒞1. In [16], we can find a proof of the
differentiability of the Nemytski operator on AP0(ℝn) which is
different to this one of [9].Theorem 3.2 (variational principle).
Under (3.1), forx∈AP1(ℝn), the following two assertions are equivalent.
(i)
DΦ(x)=0, that is, x is a critical
point of Φ in AP1(ℝn).(ii)
x is a Bohr a.p.
solution of  (1.1).Proof.
First
we assume (i). Since the mean value is translation invariant, we have(3.4)ℳ{D2L(x¯(t),t)⋅h(t−r)}t=ℳ{D2L(x¯(t+r),t+r)⋅h(t)}t,ℳ{D4L(x¯(t),t)⋅h′(t−r)}t=ℳ{D4L(x¯(t+r),t+r)⋅h′(t)}t,   and so by using
Lemma 3.1 we obtain, for all h∈AP1(ℝn),(3.5)0=ℳ{(D1L(x¯(t),t)+D2L(x¯(t+r),t+r))⋅h(t)}t+ℳ{(D3L(x¯(t),t)+D4L(x¯(t+r),t+r))⋅h′(t)}t.
Settingq(t):=D1L(x¯(t),t)+D2L(x¯(t+r),t+r), denoting by qk(t) its coordinates
for k=1,…,n, setting p(t):=D3L(x¯(t),t)+D4L(x¯(t+r),t+r), and denoting by qk(t) its coordinates
for k=1,…,n, we deduce from the previous equality that, for all ϕ∈AP∞(ℝ), we have ℳ{qk(t)⋅ϕ(t)}t=−ℳ{pk(t)⋅ϕ′(t)}t. Then by reasoning like
in the proof of Theorem 1 in
[8], we obtain that Dpk=qk in the sense of
the a.p. distributions of Schwartz [17], and by using the proposition of the
Fourier-Bohr series, we obtain that pk is ℂ1 and that pk'=qk in the ordinary
sense. From this, we obtain that p(⋅−r) is ℂ1 and that p′(t−r)=q(t−r) which is
exactly (ii).
Conversely by using the formulaℳ{l⋅y′}=−ℳ{l′⋅y}, for all l∈AP1(ℒ(ℝn,ℝ)) and y∈AP1(ℝn), and by translating
the time, we obtain from (ii),
for all h∈AP1(ℝn), the following
relation:(3.6)0=ℳ{(D1L(x¯(t),t)+D2L(x¯(t+r),t+r))⋅h(t)+(D3L(x¯(t),t)+D4L(x¯(t+r),t+r))⋅h′(t)}t=ℳ{D1L(x¯(t),t)h(t)+D2L(x¯(t),t))⋅h(t−r)+D3L(x¯(t),t)h′(t)+D4L(x¯(t),t)⋅h′(t−r)}t=DΦ(x)⋅h, and so we have
(i).Theorem3.2 is
an extension to the nonautonomous case in presence of a delay of
[8, Theorem 1]. Now we use
Theorem 3.2
to provide some results on the structure of the set
of the Bohr a.p. solutions of (1.1) in the
case, where L is autonomous
and convex.Theorem 3.3 (structure result).
Assume thatL∈𝒞1((ℝn)4,ℝ), and thatLis convex.
Then the following assertions
hold.
(i)
The set of the
Bohr a.p. solutions of (1.1) is a closed convex subset of AP1(ℝn).(ii)
Ifx1 is a T1-periodic
nonconstant solution of (1.1), if x2 is a T2-periodic
nonconstant solution of (1.1), and if T1/T2 is no rational,
then (1−θ)x1+θx2 is a Bohr a.p.
nonperiodic solution of (1.1), for all θ∈(0,1).(iii)
Ifx is a Bohr a.p.
solution of (1.1), then ℳ{x} is a constant
solution of (1.1).(iv)
Ifx is a Bohr a.p.
solution of (1.1), if T∈(0,∞) is such that a(x,2π/T)≠0, then there exists a nonconstant T-periodic
solution of (1.1).Proof.
SinceL is convex, the
functional Φ of (1.5) is
also convex on AP1(ℝn). Since L is autonomous
and of class 𝒞1, L satisfies
(3.1), and so Φ is of class 𝒞1. Therefore, we have {x:Φ(x)=infΦ}={x:DΦ(x)=0} which is closed
and convex, and (i) becomes a consequence of Theorem 3.2. The assertion (ii) is
a straightforward consequence of (i).
We introduceCT,ν(x)(t):=(1/ν)∑k=0ν−1x(t+kT), when x is a Bohr
a.p. solution of (1.1), for all ν∈ℕ∗. By using a theorem of Besicovitch
(see [2, page 144]),
there exists a T-periodic
continuous function, denoted by xT, such that lim⁡ν→∞∥CT,ν(x)−xT∥∞=0.
We easily verify thatlim⁡ν→∞∥CT,ν(x)−xT∥ℂ1=0.
SinceL is autonomous, t↦x(t+kT) is a Bohr a.p.
solution of (1.1). Since CT,ν(x) is a convex
combination of Bohr a.p. solutions of (1.1), CT,ν(x) is a Bohr a.p.
solution of (1.1), and xT also by using
the closeness of the set of Bohr a.p. solutions. And so xT is a T-periodic
solution of (1.1). By using a straightforward calculation, we see that a(CT,ν(x),2π/T)=a(x,2π/T) and
consequently a(xT,2π/T)=a(x,2π/T). When  a(x,2π/T)≠0, then xT is not constant
that proves (iv).
To prove (iii) it suffices to chooseT1∈(0,∞) such that (2π/T1)(ℤ−{0})∩Λ(x)=0, and then all the Fourier-Bohr coefficients of xT1 are zero except
(perhaps) the mean value of xT1 which is equal
to ℳ{x}.The assertions
(i) and (ii) are extensions of  [8, Theorems 3 and 4]; the assertions
(iii) and (iv) are extension to neutral delay equations of
[10, Theorem 2].The space(AP1(ℝn),∥⋅∥ℂ1) does not
possess good topological properties like to be a reflexive space. It is why in
the following section we extend our variational formalism to the Hilbert space B1,2(ℝn).


## 4. A Variational Setting for the Besicovitch a.p. Functions

E and F are Euclidean
finite-dimensional spaces.Lemma 4.1.
Letg∈APU(E×ℝ,F)be a function which satisfies the following Hölder condition:(4.1)∃α∈(0,∞),⁢∃a∈[0,∞),∀t∈ℝ,∀z,w∈E,|g(z,t)−g(w,t)|≤a⋅|z−w|α. Let p,q∈[1,∞) be such that p=αq.
Then the following two assertions hold.
(i)
Ifu∈Bp(E), then t↦g(u(t),t)∈Bq(F).(ii)
The
Nemytski operator ong, 𝒩g:Bp(E)→Bq(F) defined by
𝒩gu(t):=g(u(t),t) satisfies ∥𝒩gu−𝒩gv∥q≤a⋅∥u−v∥pα,
for all u,v∈Bp(E).Proof.
We
setb(t):=g(0,t), and so we have b∈AP0(ℝ) and the Hölder
assumption implies |g(x,t)|≤a⋅|z|α+b(t), for all z∈E,t∈ℝ. If u∈Bp(E), then we have |g(u(t),t)|≤a⋅|u(t)|α+b(t), for all t∈ℝ, and since b is continuous,
we have b∈Llocq(ℝ,ℝ) (the Lebesgue
space), and since (|u(t)|α)q=|u(t)|p, we have |u|α∈Llocq(ℝ,ℝ). Since u∈Bp(E), there exists a sequence (uj)j in AP0(E) such that lim⁡j→∞∥u−uj∥p=0. By using
[14, Theorem 2.7 page 16], setting φj(t):=g(uj(t),t), we have φj∈AP0(F), and a straightforward calculation gives us the
following inequality:(4.2)ℳ¯{|g(u(t),t)−φj(t)|q}1/q≤a⋅ℳ{|u−uj|p}1/q=a∥u−uj∥pα, and consequently we obtain(4.3)lim⁡j→∞ℳ¯{|g(u(t),t)−φj(t)|p}1/q=0 that implies t↦g(u(t),t)∈Bq(F), and so (i) is proven; moreover the last inequality
becomes the one of (ii) when we
replace φj(t) by g(v(t),t).This lemma is
an extension to the nonautonomous case of
[13, Theorem 1].Lemma 4.2.
Letf∈APU(E×ℝ,F)be a function
such that the partial differentialD1f(z,t)exists, for all(z,t)∈E×ℝ, such thatD1f∈APU(E×ℝ,ℒ(E,F)). We assume the following condition fulfilled.
(C)
There exista1∈[0,∞), such that, for all z,w∈E, and for all t∈ℝ, |D1f(z,t)−D1f(w,t)|≤a1⋅|z−w|.
Then the Nemytski
operator𝒩f:B2(E)→B1(F), defined by 𝒩f(u)(t):=f(u(t),t), is of class 𝒞1 and, for all u,h∈B2(E),(4.4)(D𝒩f(u)⋅h)(t)=D1f(u(t),t)⋅h(t).Proof.
First
step: we show that there exist a0∈[0,∞), b∈B1(E), such that, for all (z,t)∈E×ℝ, |f(z,t)|≤a0|z|2+b(t). The following hold:
|D1f(z,t)−D1f(0,t)|≤a1⋅|z|(4.5)⟹|D1f(z,t)|≤|D1f(z,t)−D1f(0,t)|+|D1f(0,t)|≤a1⋅|z|+|D1f(0,t)|. By using the
mean value theorem (see [18, page 144]), we have, for all (z,t)∈E×ℝ,(4.6)|f(z,t)|≤|f(z,t)−f(0,t)|+|f(0,t)|≤sup⁡ξ∈]0,z[|D1f(ξ,t)|⋅|z−0|+|f(0,t)|≤sup⁡ξ∈]0,z[(a1⋅|ξ|+|D1f(0,t)|)⋅|z|+|f(0,t)|=(a1⋅|z|+|D1f(0,t)|)⋅|z|+|f(0,t)|=a1⋅|z|2+|D1f(0,t)|⋅|z|+|f(0,t)|≤a1⋅|z|2+12|D1f(0,t)|2+12|z|2+|f(0,t)|=(a1+12)⋅|z|2+12|D1f(0,t)|2+|f(0,t)|. Setting b(t):=(1/2)|D1f(0,t)|2+|f(0,t)|, then a0:=a1+1/2. Since f∈APU(E×ℝ,F), and D1f∈APU(E×ℝ,ℒ(E,F)), we have b∈AP0(E)⊂B1(E).
Second step: we show that t↦f(u(t),t)∈B1(F) when u∈B2(E).
Letu∈B2(E). Then the inequality |f(u(t),t)|≤a0|u(t)|2+b(t) implies that(4.7)t⟼f(u(t),t)∈Lloc1(ℝ,F). By using Lemma 4.1 with p=2,q=2,α=1, and g=D1f, we have t↦D1f(u(t),t)∈B2(ℒ(E,F)). Let (um)m be a sequence
in AP0(E) such that ∥u−um∥2→0(m→∞). By using the mean value theorem (see [18, page 144]), we
have, for all t∈ℝ,(4.8)|f(um(t),t)−f(u(t),t)−D1f(u(t),t)⋅(um(t)−u(t))|≤(sup⁡ξ∈]u(t),um(t)[|D1f(ξ,t)−D1f(u(t),t)|)⋅|(um(t)−u(t))|≤a1⋅sup⁡ξ∈]u(t),um(t)[|ξ−u(t)|⋅|um(t)−u(t)|≤a1⋅|um(t)−u(t)|2,     and consequently we obtain(4.9)ℳ¯{|f(um(t),t)−f(u(t),t)−D1f(u(t),t)⋅(um(t)−u(t))|}t≤a1⋅∥um−u∥22. Since t↦D1f(u(t),t)∈B2(ℒ(E,F)) and since um−u∈B2(E), we have(4.10)t⟼D1f(u(t),t)⋅(um(t)−u(t))∈B1(F). By using
(see [14, Theorem 2.7 page 16] ), we have(4.11)t⟼f(um(t),t)∈AP0(F)⊂B1(F), and so, by setting(4.12)ψm(t):=f(um(t),t)−D1f(um(t),t)⋅(um(t)−u(t)), we have ψm∈B1(F). The last inequality implies(4.13)lim⁡n→∞ℳ¯{|f(u(t),t)−ψm(t)|}t=0, and therefore we have t↦f(u(t),t)∈B1(F).
Third step:
we show that, for all u∈B2(E), the operator ℒ(u):B2→B1(ℝ), defined by (ℒ(u)⋅h)(t):=D1f(u(t),t)⋅h(t), is linear continuous. We have yet seen that t↦D1f(u(t),t)⋅h(t)∈B1(F). The linearity of ℒ(u) is easy to
verify. By using a Cauchy-Schwartz-Bunyakovsky
inequality, we have(4.14)ℳ{|D1f(u(t),t)⋅h(t)|}t≤ℳ{|D1f(u(t),t)|⋅|h(t)|}t≤ℳ{|D1f(u(t),t)|2}1/2⋅ℳ{|h|2}1/2 that proves the
continuity of ℒ(u).
Fourth step: we show the differentiability of 𝒩f.
Letu∈B2(E) and h∈B2(E). By using the mean value inequality (see [18, page 144]),
we have, for all t∈ℝ,(4.15)|f(u(t)+h(t),t)−f(u(t),t)−D1f(u(t),t)⋅h(t)|≤sup⁡ξ∈]u(t),u(t)+h(t)[|D1f(ξ,t)−D1f(u(t),t)|⋅|h(t)|≤a1∥h∥2,   and by using the monotonicity of ℳ, we obtain(4.16)ℳ{|f(u(t)+h(t),t)−f(u(t),t)−D1f(u(t),t)⋅h(t)|}t≤a1∥h∥22, that is,(4.17)∥𝒩f(u+h)−𝒩f(u)−ℒ(u)⋅h∥1≤a1∥h∥22 that implies that 𝒩f is differentiable
at u and that D𝒩f(u)=ℒ(u).
Fifth step:
we show that 𝒩f is of class 𝒞1.
Letu,v∈B2(E). By using (C), for all h∈B2(E), such that ∥h∥2≤1, for all t∈ℝ, we have(4.18)|(D1f(u(t),t)−D1f(v(t),t))⋅h(t)|≤|D1f(u(t),t)−D1f(v(t),t)|⋅|h(t)|≤a1⋅|u(t)−v(t)|⋅|h(t)|.   That implies, by using the Cauchy-Schwartz-Bunyakovsky
inequality, the following majorization holds:(4.19)ℳ{|(D1f(u(t),t)−D1f(v(t),t))⋅h(t)|}t≤a1ℳ{|u(t)−v(t)|⋅|h(t)|}t≤a1∥u−v∥2⋅∥h∥2≤a1∥u−v∥2.   Therefore we have ∥D𝒩f(u)−D𝒩f(v)∥ℒ≤a1∥u−v∥2 that implies
the continuity of D𝒩f.Note that Lemma4.2 is an extension to the
nonautonomous case of  [13, Theorem 2].Theorem 4.3 (variational principle).
LetL:(ℝn)4×ℝ→ℝ,(4.20)(X,t)=(x1,x1,x1,x1,t)⟼L(X,t)=L(x1,x1,x1,x1,t), be a function and let r∈(0,∞). Assume the following conditions fulfilled:(4.21)L∈APU((ℝn)4×ℝ,ℝ),⁢thepartialdifferentialsDkL(x1,x1,x1,x1,t)exist,forall(x1,x1,x1,x1,t)∈(ℝn)4×ℝandfork=1,…,4,⁢DkL∈APU((ℝn)4×ℝ,ℒ(ℝn,ℝ));thereexistsa1∈[0,∞)suchthat|LX(X,t)−LX(Y,t)|≤a1|X−Y|,forallX,Y∈(ℝn)4,forallt∈ℝ,whereLXisthepartialdifferentialwithrespecttoX∈(ℝn)4.     Then the
functional J:B1,2(ℝn)→ℝ, defined by(4.22)J(u)=ℳ{L(u(t),u(t−r),∇u(t),∇u(t−r),t)}t, is of class 𝒞1, and the two following assertions are
equivalent.
(i)
DJ(u)=0, that is, u is a critical
point of J.(ii)
D1L(u(t−r),u(t−2r),∇u(t−r),∇u(t−2r),t−r)+D2L(u(t),u(t−r),∇u(t),∇u(t−r),t)=∇[D3L(u(t−r),u(t−2r),∇u(t−r),∇u(t−2r),t−r)+D4L(u(t),u(t−r),∇u(t),∇u(t−r),t)](equalityinB2(ℒ(ℝn,ℝ))).Definition 4.4.
Whenu∈B1,2(ℝn)
satisfies the equation of
(ii) in Theorem 4.3, we say that  u
is a weak
Besicovitch a.p. solution of (1.1).Proof.
We
consider the operatorℒ:B1,2(ℝn)→B2(ℝn)4≡B2((ℝn)4), defined by (ℒ(u))(t):=(u(t),u(t−r),∇u(t),∇u(t−r)). ℒ is clearly
linear continuous, therefore ℒ is of class 𝒞1 and we have Dℒ(u)⋅h=ℒ(h).
We consider the Nemytski operator(4.23)𝒩L:B2((ℝn)4)⟶B1(R),(𝒩L(u))(t):=L(u(t),t). By using
Lemma 4.2, 𝒩L is of  class 𝒞1 and, for all U,H∈B1,2((ℝn)4), we have(4.24)(D𝒩L(U)⋅H)(t)=LX(U(t),t)⋅H(t)=∑k=14DkL(u1(t),u2(t),u3(t),u4(t),t)⋅hk(t). The mean value ℳ:B1(ℝ)→ℝ is linear
continuous, therefore it is of class 𝒞1, and Dℳ{ϕ}⋅ψ=ℳ{ψ}, for all ϕ,ψ∈B1(ℝ).
ConsequentlyJ=ℳ◦𝒩L◦ℒ is of class 𝒞1 as a
composition of three mappings of class 𝒞1.
Letu∈B1,2(ℝn). If (i) is true then, for all h∈B1,2(ℝn), we have(4.25)0=DJ(u)⋅h=Dℳ{𝒩L◦ℒ(u)}◦D𝒩L(ℒ(u))◦Dℒ(u)⋅h=ℳ{D𝒩L(ℒ(u))⋅ℒ(h)}=ℳ{D1L(u¯(t),t)⋅h(t)+D2L(u¯(t),t)⋅h(t−r)+D3L(u¯(t),t)⋅∇h(t)+D4L(u¯(t),t)⋅∇h(t−r)}t=ℳ{(D1L(u¯(t),t)+D2L(u¯(t+r),t+r))⋅h(t)}t+ℳ{(D3L(u¯(t),t)+D4L(u¯(t+r),t+r))⋅∇h(t)}t and then we
obtain (ii) by using  [13, Proposition 10].
Conversely, if (ii) is true, thent↦D3𝒩L(u¯(t),t)+D4𝒩L(u¯(t+r),t+r)∈B1,2(ℝn), and for all h∈AP1(ℝn), we have(4.26)ℳ{(D1L(u¯(t),t)+D2L(u¯(t+r),t+r))⋅h(t)}t,−ℳ{∇(D3L(u¯(t),t)+D4L(u¯(t+r),t+r))⋅h(t)}t=0,   therefore by using
[13, Proposition 9], we obtain (4.27)0=ℳ{(D1L(u¯(t),t)+D2L(u¯(t+r),t+r))⋅h(t)+(D3L(u¯(t),t)+D4L(u¯(t+r),t+r))⋅h′(t)}t=ℳ{(D1L(u¯(t),t)⋅h(t)+D2L(u¯(t),t))⋅h(t−r)+(D3L(u¯(t),t)h′(t)+D4L(u¯(t),t))⋅h′(t−r)}t=DJ(u)⋅h. Since AP1(ℝn) is dense in B1,2(ℝn), we have DJ(u)⋅h=0, for all h∈B1,2(ℝn), therefore DJ(u)=0.Note that the
Theorem4.3
is an extension to the nonautonomous case of
[13, Theorem 4].Theorem 4.5 (existence, uniqueness).
LetL:(ℝn)4×ℝ→ℝbe a function
which satisfies (4.21)  and also satisfies the following two conditions:(4.28)L(⋅,t):(ℝn)4⟶ℝisconvex,forallt∈ℝ;(4.29)thereexistj∈{1,2},k∈{3,4},c∈(0,∞)suchthat,forall(x1,x2,x3,x4,t)∈(ℝn)4×ℝ,wehaveL(x1,x2,x3,x4,t)≥c(|xj|2+|xk|2).  Then there
exists a function u∈B1,2(ℝn) which is a weak
Besicovitch a.p. solution of  (1.1).
Moreover, if in addition the following
condition fulfilled:(4.30)thereexisti∈{1,2},l∈{3,4},c1∈(0,∞)suchthatthefunctionM:(ℝn)4×ℝ⟶ℝ,definedbyM(x1,x2,x3,x4,t):=L(x1,x2,x3,x4,t)−c12|xi|2−c12|xl|2,isconvexwithrespectto(x1,x2,x3,x4,t),forallt∈ℝ,  then the weak
Besicovitch a.p. solution of (1.1) is unique.Proof.
By
using Theorem4.3, the functional J is of class 𝒞1 and, by using
(4.28), J is a convex functional.
Assumption (4.29) ensures that, for all u∈B1,2(ℝn), we have(4.31)J(u)≥c(ℳ{|u|2}+ℳ{|∇u|2})=c⋅∥u∥1,22. Since the mean value is translation invariant, consequently J is coercive on B1,2(ℝn), and so (see [19, page 46]) there exists u∈B1,2(ℝn) such that J(u)=infJ. Therefore we have DJ(u)=0 and by using
Theorem 4.3, u is a weak
Besicovitch a.p. solution of (1.1). The existence is proven.
To treat the uniqueness, we note that, under (4.30),
the functional I:B1,2(ℝn)→ℝ, defined by I(u):=J(u)−(c1/2)ℳ{|u|2}−(c1/2)ℳ{|∇u|2}, is convex and since J is of class 𝒞1, I is also of
class 𝒞1. Note that we have DI(u)=DJ(u)−c1〈u∣⋅〉. By using the
Mint monotonicity    of the differential of a
convex functional, for all u,v∈B1,2(ℝn), we have(4.32)0≤〈DI(u)−DI(v),u−v〉=〈DJ(u)−DJ(v),u−v〉−c1〈u−v∣u−v〉⟹〈DJ(u)−DJ(v),u−v〉≥c1∥u−v∥1,22.   Now if u and v are two weak
Besicovitch a.p. solutions of (1.1),
by using Theorem 4.3 we have DJ(u)=DJ(v)=0, and consequently c1∥u−v∥1,22=0, therefore u=v.Theorem 4.6 (existence and density).
LetK∈𝒞2((ℝn)4,ℝ)be a function
which satisfies the following conditions:(4.33)thereexistsa0∈[0,∞)suchthat|K(x)|≤a0|X|2,forallX∈(ℝn)4,thefollowingholds:thereexistj∈{1,2},k∈{3,4},c∈(0,∞)suchthatthefunctionG:(ℝn)4⟶ℝ,definedbyG(x1,x2,x3,x4):=K(x1,x2,x3,x4)−c2|xj|2−c2|xk|2,isconvexandnonnegative on(ℝn)4.ThedifferentialDKisLipschitzianon(ℝn)4.        Then the
following conclusions hold.
(i)
For allb∈B2(ℝn), there exists a unique u∈B1,2(ℝn) which is a weak
Besicovitch a.p. solution of (1.2).(ii)
The set of theb∈AP0(ℝn) for which there
exists a Bohr a.p. solution of (1.2) is dense in AP0(ℝn) with respect to
the norm(4.34)∥b∥∗:=sup⁡{ℳ{b⋅h}:h∈B1,2(ℝn),⁢∥h∥1,2≤1}.Proof.
We
introduce the functionalsE and E1 from B1,2(ℝn) in ℝ setting E(u):=ℳ{K(u¯(t))}t and E1(u):=ℳ{G(u¯(t))}t. They are special cases of the functional J of the Theorem 4.3, and consequently they are of class 𝒞1. Note that E1(u)=E(u)−(c/2)∥u∥1,22. By using the F. Riesz isomorphism j:B1,2(ℝn)→B1,2(ℝn)∗, 〈j(u),v〉=〈u,v〉, for all u,v∈B1,2(ℝn), we can define the gradients gradE(u):=j−1(DE(u)) and  gradE1(u):=j−1(DE1(u)). By using the Minty-monotonicity of gradE1 (due to the
convexity of E1) we have, for
all u,v∈B1,2(ℝn),(4.35)0≤〈gradE1(u)−gradE1(v)∣u−v〉=〈gradE(u)−gradE(v)∣u−v〉−c⋅∥u−v∥1,22, that implies
that gradE is strongly
monotone and consequently (see [20, page 100]) the following property holds:(4.36)gradE⁢isahomeomorphismfromB1,2(ℝn)onB1,2(ℝn).
From each  b∈B2(ℝn) we define the linear functional b#∈B1,2(ℝn)∗  by    setting 〈b#,h〉:=ℳ{b(t+r)⋅h(t)}t.
Therefore we havej−1(b#)∈B1,2(ℝn) and by using
(4.36), there exists u∈B1,2(ℝn) such that gradE(u)=j−1(b#), that is, DE(u)=b# which means
that, for all h∈B1,2(ℝn), (4.37)ℳ{DK(u¯(t))⋅h¯(t)}t=ℳ{b(t+r)⋅h(t)}t,thatis,ℳ{[D1K(u¯(t))+D2K(u¯(t+r))−b(t+r)]⋅h(t)+[D3K(u¯(t))+D4K(u¯(t+r))−b(t+r)]⋅∇h(t)}t=0   and by using
[13, Proposition 10], we obtain that u is a weak
Besicovitch a.p. solution of (1.2).
About the uniqueness, note that ifv is a weak
Besicovitch a.p. solution of (1.2), then we verify that ℳ{DK(v¯(t))⋅h¯(t)}t=ℳ{b(t+r)⋅h(t)}t, for all h∈B1,2(ℝn), and consequently DE(v)=b#, that is, gradE(v)=j−1(b#)=gradE(u), and by using
(4.36), we have u=v. And so (i) is proven.
Now we introduce the nonlinear unbounded operator(4.38)𝒦:Dom(𝒦)⊂B1,2(ℝn)⟶B2(ℝn) defined by(4.39)(𝒦(u))(t):=D1K(u¯(t−r))+D2K(u¯(t))−∇[D3K(u¯(t−r))+D4K(u¯(t))]. And so 𝒦(u)=b means that u is a weak
Besicovitch a.p. solution of (1.2). By using the assertion (i), 𝒦 is bijective.
We verify that(4.40)∥𝒦(u)−𝒦(v)∥∗=∥gradE(u)−gradE(v)∥1,2, for all u,v∈Dom(𝒦), and by using (4.36) we see that 𝒦 is a
homeomorphism from Dom(𝒦) on B2(ℝn). Since AP2(ℝn) is dense in B1,2(ℝn), 𝒦(AP2(ℝn)) is dense in B2(ℝn) with respect to
the norm ∥⋅∥∗, and since 𝒦(AP2(ℝn))⊂AP0(ℝn)⊂B2(ℝn), we have proven (ii).This result is
an extension to the neutral delay equations of  [13, Theorem 5].



---

*Source: 153285-2008-02-21.xml*