We present some conditions for the existence and uniqueness of almost periodic solutions of Nth-order neutral differential equations with piecewise constant arguments of the form (x(t)+px(t−1))(N)=qx([t])+f(t), here [·] is the greatest integer function, p and q are nonzero constants, N is a positive integer, and f(t) is almost periodic.

1. Introduction

In this paper we study certain functional differential equations of neutral delay type with piecewise constant arguments of the form(x(t)+px(t-1))(N)=qx([t])+f(t),
here [·] is the greatest integer function, p and q are nonzero constants, N is a positive integer, and f(t) is almost periodic. Throughout this paper, we use the following notations: ℝ is the set of reals; ℝ+ the set of positive reals; ℤ the set of integers; that is, ℤ={0,±1,±2,…}; ℤ+ the set of positive integers; ℂ denotes the set of complex numbers. A function x:ℝ→ℝ is called a solution of (1.1) if the following conditions are satisfied:

x is continuous on ℝ;

the Nth-order derivative of x(t)+p(t)x(t-1) exists on ℝ except possibly at the points t=n, n∈ℤ, where one-sided Nth-order derivatives of x(t)+p(t)x(t-1) exist;

x satisfies (1.1) on each interval (n,n+1) with integer n∈ℤ.

Differential equations with piecewise constant arguments are usually referred to as a hybrid system, and could model certain harmonic oscillators with almost periodic forcing. For some excellent works in this field we refer the reader to [1–5] and references therein, and for a survey of work on differential equations with piecewise constant arguments we refer the reader to [6].

In paper [1, 2], Yuan and Li and He, respectively, studied the existence of almost periodic solutions for second-order equations involving the argument 2[(t+1)/2] in the unknown function. In paper [3], Seifert intensively studied the special case of (1.1) for N=2 and |p|<1 by using different methods. However, to the best of our knowledge, there are no results regarding the existence of almost periodic solutions for Nth-order neutral differential equations with piecewise constant arguments as (1.1) up to now.

Motivated by the ideas of Yuan [1] and Seifert [3], in this paper we will investigate the existence of almost periodic solutions to (1.1). Both the cases when |p|<1 and |p|>1 are considered.

2. The Main Results

We begin with some definitions, which can be found (or simply deduced from the theory) in any book, say [7], on almost periodic functions.

Definition 2.1.

A set K⊂ℝ is said to be relatively dense if there exists L>0 such that [a,a+L]∩K≠∅ for all a∈ℝ.

Definition 2.2.

A bounded continuous function f:ℝ→ℝ (resp., ℂ) is said to be almost periodic if the ɛ-translation set of fT(f,ɛ)={τ∈R:|f(t+τ)-f(t)|<ɛ∀t∈R}
is relatively dense for each ɛ>0. We denote the set of all such function f by AP(ℝ,ℝ) (resp., AP(ℝ,ℂ)).

Definition 2.3.

A sequence x:ℤ→ℝk (resp., ℂk), k∈ℤ, k>0, denoted by {xn}, is called an almost periodic sequence if the ɛ-translation set of {xn}T({xn},ɛ)={τ∈Z:|xn+τ-xn|<ɛ∀n∈Z}
is relatively dense for each ɛ>0, here |·| is any convenient norm in ℝk (resp., ℂk). We denote the set of all such sequences {xn} by APS(ℤ,ℝk)(resp., APS(ℤ,ℂk)).

Proposition 2.4.

{xn}={(xn1,xn2,…,xnk)}∈APS(ℤ,ℝk) (resp., APS(ℤ,ℂk)) if and only if {xni}∈APS(ℤ,ℝ) (resp., APS(ℤ,ℂ)), i=1,2,…,k.

Proposition 2.5.

Suppose that {xn}∈APS(ℤ,ℝ), f∈AP(ℝ,ℝ). Then the sets T(f,ɛ)∩ℤ and T({xn},ɛ)∩T(f,ɛ) are relatively dense.

Now one rewrites (1.1) as the following equivalent system
(x(t)+px(t-1))′=y1(t),(2.31)y1′(t)=y2(t),(2.32)⋮⋮yN-2′(t)=yN-1(t),(2.3N-1)yN-1′(t)=qx([t])+f(t).(2.3N)
Let (x(t),y1(t),…,yN-1(t))be solutions of system (2.3) on ℝ, for n≤t<n+1, n∈ℤ, using (2.3N) we obtain
yN-1(t)=yN-1(n)+qx(n)(t-n)+∫ntf(t1)dt1,
and using this with (2.3N-1) we obtain
yN-2(t)=yN-2(n)+yN-1(n)(t-n)+12qx(n)(t-n)2+∫nt∫nt2f(t1)dt1dt2.
Continuing this way, and, at last, we get
x(t)+px(t-1)=x(n)+px(n-1)+y1(n)(t-n)+12y2(n)(t-n)2+⋯+1(N-1)!yN-1(n)(t-n)N-1+1N!qx(n)(t-n)N+∫nt∫ntN⋯∫nt2f(t1)dt1dt2⋯dtN.
Since x(t) must be continuous at n+1, using these equations we get for n∈ℤ,
x(n+1)=(1-p+qN!)x(n)+y1(n)+12!y2(n)+⋯+1(N-1)!yN-1(n)+px(n-1)+fn(1),(2.71)y1(n+1)=q(N-1)!x(n)+y1(n)+y2(n)+12!y3(n)+⋅⋅⋅+1(N-2)!yN-1(n)+fn(2),(2.72)⋮⋮yN-2(n+1)=q2x(n)+yN-2(n)+yN-1(n)+fn(N-1),(2.7N-1)yN-1(n+1)=qx(n)+yN-1(n)+fn(N),(2.7N)
where
fn(1)=∫nn+1∫ntN⋯∫nt2f(t1)dt1dt2⋯dtN,…,fn(N-1)=∫nn+1∫nt2f(t1)dt1dt2,fn(N)=∫nn+1f(t1)dt1.

Lemma 2.6.

If f∈AP(ℝ,ℝ), then sequences {fn(i)}∈APS(ℤ,ℝ),i=1,2,…,N.

Proof.

We typically consider {fn(1)}forallɛ>0 and τ∈T(f,ɛ)∩ℤ, we have
|fn+τ(1)-fn(1)|=|∫n+τn+τ+1∫n+τtN⋯∫n+τt2f(t1)dt1dt2⋯dtN-∫nn+1∫ntN⋯∫nt2f(t1)dt1dt2⋯dtN|≤∫nn+1∫ntN⋯∫nt2|f(t1+τ)-f(t1)|dt1dt2⋯dtN≤ɛN!.
From Definition 2.3, it follows that {fn(1)} is an almost periodic sequence. In a manner similar to the proof just completed, we know that {fn(2)},{fn(3)},…,{fn(N)} are also almost periodic sequences. This completes the proof of the lemma.

Lemma 2.7.

The system of difference equations
cn+1=(1-p+qN!)cn+dn(1)+12!dn(2)+⋯+1(N-1)!dn(N-1)+pcn-1+fn(1),(2.101)dn+1(1)=q(N-1)!cn+dn(1)+dn(2)+12!dn(3)+⋅⋅⋅+1(N-2)!dn(N-1)+fn(2),(2.102)⋮⋮dn+1(N-2)=q2cn+dn(N-2)+dn(N-1)+fn(N-1),(2.10N-1)dn+1(N-1)=qcn+dn(N-1)+fn(N),(2.10N)
has solutions on ℤ; these are in fact uniquely determined by c0,c-1,d0(1),…,d0(N-1).

Proof.

It is easy to check that cn,dn(i), i=1,2,…,N-1 are uniquely determined in term of c0,c-1, d0(1),d0(2),…,d0(N-1) for n∈ℤ+. For n=-1, (2.10N) uniquely determines d-1(N-1), (2.10N-1) uniquely determines d-1(N-2),…,(2.102) uniquely determines d-1(1), and thus since p≠0, (2.101) uniquely determines c-2. So c-1,c-2, d-1(1),d-1(2),…,d-1(N-1) are determined. Continuing in this way, we establish the lemma.

Lemma 2.8.

For any solution (cn,dn(1),dn(2),…,dn(N-1)), n∈ℤ, of system (2.10), there exists a solution (x(t),y1(t),y2(t),…,yN-1(t)), t∈R, of (2.3) such that x(n)=cn, y1(n)=dn(1),…,yN-1(n)=dn(N-1), n∈ℤ.

Proof .

Define
w(t)=cn+pcn-1+dn(1)(t-n)+12!dn(2)(t-n)2+⋯+1(N-1)!dn(N-1)(t-n)N-1+1N!qcn(t-n)N+∫nt∫ntN⋯∫nt2f(t1)dt1dt1⋯dtN,
for n≤t<n+1, n∈ℤ. It can easily be verified that w(t) is continuous on ℝ; we omit the details.

Define x(t)=φ(t), -1≤t≤0, where φ(t) is continuous, and φ(0)=c0, φ(-1)=c-1;
x(t)=[w(t+1)-φ(t+1)]p,-2≤t<-1,x(t)=[w(t+1)-x(t+1)]p,-3≤t<-2.
Continuing this way, we can define x(t) for t<0. Similarly, define
x(t)=-pφ(t-1)+w(t),0≤t<1,x(t)=-px(t-1)+w(t),1≤t<2,
continuing in this way x(t) is defined for t≥0, and so x(t) is defined for all t∈ℝ.

Next, define y1(t)=w′(t),y2(t)=w′′(t),…,yN-1(t)=w(N-1)(t), t≠n∈ℤ, and by the appropriate one-sided derivative of w′(t),w′′(t),…,w(N-1)(t) at n∈ℤ. It is easy to see that y1(t),y2(t),…,yN-1(t) are continuous on ℝ, and (x(n),y1(n),y2(n),…,yN-1(n))=(cn,dn(1),dn(2),…,dn(N-1)) for n∈ℤ; we omit the details.

Next we express system (2.7) in terms of an equivalent system in ℝN+1 give byvn+1=Avn+hn,
where A=(1-p+qN!112!⋅⋅⋅1(N-1)!pq(N-1)!11⋅⋅⋅1(N-2)!0⋅⋅⋅⋅⋅⋅q2!00⋯10q00⋅⋅⋅10100⋅⋅⋅00),vn=(x(n),y1(n),y2(n),…,yN-1,x(n-1))T,hn=(fn(1),fn(2),…,fn(N),0)T.

Lemma 2.9.

Suppose that all eigenvalues of A are simple (denoted by λ1,λ2,…,λN+1) and |λi|≠1, 1≤i≤N+1. Then system (2.14) has a unique almost periodic solution.

Proof.

From our hypotheses, there exists a (N+1)×(N+1) nonsingular matrix P such that PAP-1=Λ, where Λ=diag(λ1,λ2,…,λN+1) and λ1,λ2,…,λN+1 are the distinct eigenvalues of A. Define v¯n=Pvn, then (2.14) becomes
v¯n+1=Λv¯n+h¯n,
where h¯n=Phn.

For the sake of simplicity, we consider first the case |λ1|<1. Define
v¯n1=∑m≤nλ1n-mh¯(m-1)1,
where h¯n=(h¯n1,h¯n2,…,h¯n(N+1))T, n∈ℤ. Clearly {h¯n1} is almost periodic, since h¯n=Phn, and {hn} is. For τ∈T({h¯n1},ɛ), we have
|v¯(n+τ)1-v¯n1|=|∑m≤n+τλ1n+τ-mh¯(m-1)1-∑m≤nλ1n-mh¯(m-1)1|(lettingm=m′+τ,thenreplacingm′bym)=|∑m≤nλ1n-mh¯(m+τ-1)1-∑m≤nλ1n-mh¯(m-1)1|=|∑m≤nλ1n-m(h¯(m+τ-1)1-h¯(m-1)1)|≤ɛ1-|λ1|,
this shows that {v¯n1}∈APS(ℤ,ℂ).

If |λi|<1, 2≤i≤N+1, in a manner similar to the proof just completed for λ1, we know that {v¯ni}∈PAS(ℤ,ℂ), 2≤i≤N+1, and so {v¯n}∈APS(ℤ,ℂN+1). It follows easily that then {P-1v¯n}={vn}∈APS(ℤ,ℝN+1) and our lemma follows.

Assume now |λ1|>1. Now define
v¯n1=∑m≤nλ1m-nh¯(m-1)1,n∈Z.
As before, the fact that {v¯n1}∈APS(ℤ,ℂ) follows easily from the fact that {h¯n1}∈APS(ℤ,ℂ). So in every possible case, we see that each component vni, i=1,2,…,N+1, of vn is almost periodic and so {vn}∈APS(ℤ,ℝN+1).

The uniqueness of this almost periodic solution {vn} of (2.14) follows from the uniqueness of the solution v¯n of (2.16) since P-1v¯n=vn, and the uniqueness of v¯n of (2.16) follows, since if ṽn were a solution of (2.16) distinct from v¯n, un=v¯n-ṽn would also be almost periodic and solve un+1=Λun,n∈ℤ. But by our condition on Λ, it follows that each component of un must become unbounded either as n→∞ or as n→-∞, and that is impossible, since it must be almost periodic. This proves the lemma.

Lemma 2.10.

Suppose that conditions of Lemma 2.9 hold, w(t) is as defined in the proof of Lemma 2.8 with (cn,dn(1),dn(2),…,dn(N-1)) the unique first N components of the almost periodic solution of (2.14) given by Lemma 2.9, then w(t) is almost periodic.

Proof.

For τ∈T({cn},ɛ)∩T({dn(1)},ɛ)∩T({dn(2)},ɛ)∩⋯∩T({dn(N-1)},ɛ)∩T(f,ɛ),
|w(t+τ)-w(t)|=|∫n+τtN⋯∫n+τt2(cn+τ-cn)+p(cn+τ-1-cn-1)+(dn+τ(1)-dn(1))(t-n)+12!(dn+τ(2)-dn(2))(t-n)2+⋯+1(N-1)!(dn+τ(N-1)-dn(N-1))(t-n)N-1+qN!(cn+τ-cn)(t-n)N+∫n+τt+τ∫n+τtN⋯∫n+τt2f(t1)dt1dt2⋯dtN-∫nt∫ntN⋯∫nt2f(t1)dt1dt2⋯dtN|≤(1+|p|+|q|N!+∑i=0N-11i!)ɛ.
It follows from definition that w(t) is almost periodic.

Theorem 2.11.

Suppose that |p|≠1 and all eigenvalues of A in (2.14) are simple (denoted by λ1,λ2,…,λN+1) and satisfy |λi|≠1, 1≤i≤N+1. Then (1.1) has a unique almost periodic solution x¯(t), which can, in fact be determined explicitly in terms of w(t) as defined in the proof of Lemma 2.8.

Proof.

Consider the following.Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M219"><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>).

For each m∈ℤ+ define xm(t) as follows:
xm(t)=w(t)-pxm(t-1),t>-m,xm(t)=ϕ(t),t≤-m,
here w(t) is as defined in the proof of Lemma 2.8, and
ϕ(t)=cn+(cn+1-cn)(t-n),n≤t<n+1,n∈Z,
where cn is the first component of the solution vn of (2.14) given by Lemma 2.9. Let l∈ℤ+, then from (2.21) we get
(-p)lxm(t-l)=(-p)lw(t-l)+(-p)l+1xm(t-l-1),t>-m.
It follows that
xm(t)=∑j=0l-1(-p)jw(t-j)+(-p)lxm(t-l),t>-m.
If l>t+m, xm(t-l)=ϕ(t-l), and so for such l,
|xm(t)-∑j=0l-1(-p)jw(t-j)|≤|p|l|ϕ(t-l)|.

Let l→∞, we get
xm(t)={∑j=0∞(-p)jw(t-j),t>-m,ϕ(t),t≤-m.

Since w(t) and ϕ(t) are uniformly continuous on ℝ, it follows that {xm(t):m∈ℤ+} is equicontinuous on each interval [-L,L],L∈ℤ+, and by the Ascoli-Arzelá Theorem, there exists a subsequence, which we again denote by xm(t), and a function x¯(t) such that xm(t)→x¯(t) uniformly on [-L,L], and by a familiar diagonalization procedure, can find a subsequence, again denoted by xm(t) which is such that xm(t)→x¯(t) for each t∈ℝ. From (2.27) it follows that
xm(t)=∑j=0∞(-p)jw(t-j),
and so x¯(t) is almost periodic since w(t-j) is almost periodic in t for each j≥0, and |p|<1. From (2.21), letting m→∞, we get x¯(t)+px¯(t-1)=w(t),t∈ℝ, and since w(t) solves (1.1), x¯(t) does also. The uniqueness of x¯(t) as an almost periodic solution of (1.1) follows from the uniqueness of the almost periodic solution vn:ℤ→ℝN+1 of (2.14) given by Lemma 2.9, which determines the uniqueness of w(t), and therefore from (2.21) the uniqueness of x¯(t).

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M263"><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>).

Rewriting (2.24) as
(-1p)lxm(t-l)=(-1p)lw(t-l)+(-1p)l+1xm(t-l-1),t>-m,
we deduce in a similar manner that
xm(t)={∑j=0∞(-1p)jw(t-j),t>-m,ϕ(t),t≤-m.

The remainder of the proof is similar to that of Case 1, we omit the details.

If p=0, the system of difference equations (2.10) of Lemma 2.7 now becomescn+1=(1+1N!q)cn+dn(1)+12!dn(2)+⋯+1(N-1)!dn(N-1)+fn(1),dn+1(1)=1(N-1)!qcn+dn(1)+dn(2)+12!dn(3)+⋯+1(N-2)!dn(N-1)+fn(2),⋮dn+1(N-2)=q2cn+dn(N-1)+dn(N-2)+fn(N-1),dn+1(N-1)=qcn+dn(N-1)+fn(N),
and system (2.14) reduces to vn+1*=A*vn*+hn*,
where A*=(1+qN!112!⋅⋅⋅1(N-2)!1(N-1)!q(N-1)!11⋅⋅⋅1(N-3)!1(N-2)!q(N-2)!01⋅⋅⋅1(N-4)!1(N-3)!⋅⋅⋅⋅⋅⋅q2!00⋅⋅⋅11q00⋅⋅⋅01)
and vn*=(x(n),y1(n),y2(n),…,yN-1)T, hn*=(fn(1),fn(2),…,fn(N))T. Then we have the following theorem.

Theorem 2.12.

Let p=0 and q≠(-1)NN!, if all eigenvalues of A* in (2.32) are simple (denoted by λ1,λ2,…,λN) and satisfy |λi|≠1,1≤i≤N, then (1.1) has a unique almost periodic solution x¯(t).

Proof.

System (2.32) has a solution on ℤ since A* is nonsingular because q≠(-1)NN!. The rest of the proof follows in the same way as the proof of Theorem 2.11 and is omitted.

Funding

This paper was supported by NNSF of China and NSF of Guangdong Province (1015160150100003).

YuanR.The existence of almost periodic solutions to two-dimensional neutral differential equations with piecewise constant argumentLiZ.HeM.The existence of almost periodic solutions of second order neutral differential equations with piecewise constant argumentSeifertG.Second-order neutral delay-differential equations with piecewise constant time dependenceYuanR.A new almost periodic type of solutions of second order neutral delay differential equations with piecewise constant argumentYuanR.Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argumentCookeK. L.WienerJ.A survey of differential equations with piecewise continuous argumentsFinkA. M.