Rongkun Zhuang

(Dept.of Math.,Huizhou University,Huizhou 516007,Guangdong,PR China)

Hongwu Wu

(School of Mathematical Sciences,South China University of Technology, Guangzhou 510640,Guangdong,PR China)

ON ALMOST AUTOMORPHIC SOLUTIONS OF THIRD-ORDER NEUTRAL DELAY-DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT??

Rongkun Zhuang?

(Dept.of Math.,Huizhou University,Huizhou 516007,Guangdong,PR China)

Hongwu Wu

(School of Mathematical Sciences,South China University of Technology, Guangzhou 510640,Guangdong,PR China)

We present some conditions for the existence and uniqueness of almost automorphic solutions of third order neutral delay-differential equations with piecewise constant of the form

(x(t)+px(t?1))′′=a0x([t])+a1x([t?1])+f(t),

where[·]is the greatest integer function,p,a0and a1are nonzero constants, and f(t)is almost automorphic.

almost automorphic solutions;neutral delay equation;piecewise constant argument

2000 Mathematics Subject Classification 34K14

Ann.of Appl.Math.

32:4(2016),429-438

1　Introduction

In this paper we study certain functional differential equations of neutral delay type with piecewise constant argument of the form

(x(t)+px(t?1))′′=a0x([t])+a1x([t?1])+f(t),(1)

here[·]is the greatest integer function,p,a0and a1are nonzero constants,and f(t) is almost automorphic.

By a solution x(t)of(1)on ? we mean a function continuous on ?,satisfying (1)for all t∈?,t≠n∈?,and such that the one sided third derivatives of x(t)+px(t?1)exist at n∈?.

The concept of almost automorphic functions is more general than that of almost periodic functions,which were introduced by S.Bochner[1,2],for more details about this topics we refer to[3,4,6-9]and references therein.

Differential equations with piecewise constant argument(EPCA),which were firstly considered by Cooke and Wiener[11],and Shah and Wiener[12],describe the hybrid of continuous and discrete dynamical systems,which combine the properties of both differential equations and difference equations and have applications in certain biomedical models in the works of Busenberg and Cooke in[13].Therefore, there are many papers concerning the differential equations with piecewise constant argument(see e.g.[14-20]and references therein).However,there are only a few works on the almost automorphy of solutions of EPCAs.To the best of our knowledge,only Minh et al[21]in 2006,Dimbour[22]in 2011 and Li[23]in 2013 studied in this line.They give sufficient conditions for the almost automorphy of bounded solutions of differential equation EPCAs.

Motivated by the above works,in this paper we investigate the existence of almost automorphy solutions of equation(1).The paper is organized as follows.In Section 2,some notation,preliminary definitions and lemmas are presented.The man result and its proofs is put in Sections 3.

2　Preliminary Definitions and Lemmas

Throughout this paper,?,?,? and ? denote the sets of natural numbers, integers,real and complex numbers,respectively.l∞(?)denotes the space of all bounded(two-sided)sequences x:?→? with sup-norm.We always denote by|·| the Euclidean norm in ?kor ?k,and by BC(?,?)the space of bounded continuous functions u:?→?.

Definition 2.1 A continuous function f:?→? is called almost automorphic if for every sequence of real numbersthere exists a subsequence(sn)n∈?such that

is well defined for each t∈? and

for each t∈?.The collection of such functions is denoted by AA(?).

It is clear that the function g in Definition 2.1 is bounded and measurable.

Remark 2.1 A classical example of an automorphic function given by[10]is defined as follows

but f(t)is not almost periodic as it is not uniformly continuous.

Some properties of the almost automorphic functions are listed below.

Proposition 2.1[3,4]Let f,f1,f2∈AA(?).Then the following statements hold:

(i)αf1+βf2∈AA(?)for α,β∈?.

(ii)fτ:=f(·+τ)∈AA(?)for every fixed τ∈?.

(iii)?f=f(?·)∈AA(?).

(iv)The range Rfof f is precompact,so f is bounded.

(v)If{fn}?AA(?)such that fn→f uniformly on ?,then f∈AA(?).

By(v)in Proposition 2.1,AA(?)is a Banach space equipped with the sup norm

Definition 2.2[5]A sequence x∈l∞(?)is said to be almost automorphic if for any sequence of integers{k′n},there exists a subsequence{kn}such that

for any p∈?.Denote by AAS(?)the set of all such sequences.

This limit means that

is well defined for each p∈? and

for each p∈?.

It is obvious that AAS(?)is a closed subspace of l∞(?),and the range of an almost automorphic sequence is precompact.

Proposition 2.2{x(n)}={(xn1,xn2,···,xnk)}∈AAS(?k)(resp.AAS(?k)) if and only if{xni}∈AAS(?)(resp.AAS(?)),i=1,2,···,k.

Lemma 2.1[10]Let B be a bounded linear operator in ?nwith σΓ(B)(the part of the spectrum of B on the unit circle of the complex plane)being countable,and let ?nnot contain any subspace isomorphic to c0.Assume further that x={xn}∈l∞(?) satisfies

where{yn}∈AAS(?).Then x∈AAS(?).

3　Main Results

We first rewrite equation(1)to the following equivalent system

Let(x(t),y(t),z(t))be a solution of(2)-(4)on ?,for n≤t＜n+1,n∈?. Using(4)we obtain

From this with(3)we obtain

This together with(2)we obtain

Since x(t)must be continuous at n+1,using the above equations we get for n∈?,

where

Next we express system(5)in terms of an equivalent system in ?4given by

where

vn=(x(n),y(n),z(n),x(n?1))T,

Lemma 3.1 If f∈AA(?),then the sequences

Consequently,it follows from the Lebesgue dominated convergence theorem that, for each n∈?,

Lemma 3.2 Suppose that all eigenvalues of A are simple(denoted by λ1,λ2,λ3, λ4)and|λi|≠1,1≤i≤4.Then there exists a unique almost automorphic solution vn:?→?4of(7).

Proof By Lemma 3.1 we have thatIt is clear that ?4does not contain any subspace isomorphic to c0,and the bounded linear operator A on ?4has finite spectrum.So Lemma 2.1 implies that{vn}∈AA(?4).

From our hypotheses,there exists a 4×4 nonsingular matrix P with in general complex entries such that PAP?1=Λ where Λ=diag(λ1,λ2,λ3,λ4).Define= Pvn;then(7)becomes

Lemma 3.3For any solution vn=(x(n),y(n),z(n),x(n?1))T,n∈?,of (7)there exists a solution(x(t),y(t),z(t)),t∈R,of(2)-(4)such that x(n)=cn, y(n)=dn,z(n)=en,n∈?.

Proof Define

for t∈[n,n+1),n∈?.It can easily be verified that w(t)is continuous on ?.The rest proof is similar to that of Lemma 2 in[19],we omit the details.

Lemma 3.4 Let{cn},{dn},{en}∈AAS(?),f∈AA(?)and w(t)define as in (10)for t∈[n,n+1),n∈?,then w∈AA(?).

Proof The proof is divided into the following two steps.

Step 1 For any{n′k}??,there exist a subsequence{nk}of{n′k},three sequencesand a function e f:?→? such that

Let

for t∈[n,n+1),n∈?.Noticing that f andare bounded measurable,by(11) and(12),

Step 2 We consider the general case wheremay not be an integer sequence.Letfor each k.Then by Step 1, there exist subsequencesrespectively, such thatholds and for each t∈?,

Now there are two cases to be considered:Assume thatfor sufficiently large k.Noticing the boundedness of f(t),{cn},{dn}and{en},for sufficiently large k,we obtain

This together with(13)implies that

for any m∈?.Then for sufficiently large k,

This together with(13)leads to

Theorem 1 If|p|≠1.Suppose that all eigenvalues of A are simple(denoted by λ1,λ2,λ3,λ4)and|λi|≠1,1≤i≤4.Then equation(1)has a unique almost automorphic solution x(t),which can,in fact be determined explicitly in terms of w(t)as defined in the proof of Lemma 3.3.

Proof From Lemma 3.2,we know that system(7)has a unique bounded solution {vn}n∈?∈PAAS(?4).Let(cn,dn,en)be the first three components of vn,now it follows from Lemma 3.3 that(1)has a unique bounded solution x(t)such that x(n)=cn,y(n)=dn,z(n)=dn,n∈?,where y(n)and z(n)are defined in(2)-(4), and for t∈[n,n+1),n∈?,and for t∈?,

From Lemma 3.4,we have that w∈AA(?).It is easy to get

Therefore x∈AA(?)by Proposition 2.1.

The uniqueness of x(t)as an almost automorphic solution of(1)follows from the uniqueness of the almost automorphic solution vn:?→ ?4of(7)given by Lemma 3.3,which determines the uniqueness of w(t),and therefore from(16)the uniqueness of x(t).This completes the proof.

Acknowledgments The authors would like to express the great appreciation to the referees for his/her helpful comments and suggestions.

References

[1]S.Bochner,Continuous mappings of almost automorphic and almost automorphic functions,Proc.Natl.Sci.USA,52(1964),907-910.

[2]S.Bochner,A new approach to almost automorphicity,Proc.Natl.Sci.USA,48(1962), 2039-2043.

[3]G.M.N’Guérékata,Almost Automorphic and Almost periodic Functions in Abstract Spaces,Kluwer,Amsterdam,2001.

[4]G.M.N’Guérékata,Topics in Almost Automorphy,Spring-Verlag,New York,2005.

[5]N.V.Minh,T.Naito,G.M.N’Guérékata,A spectral countability condition for almost automorphy of solutions of differential equations,Proc.Amer.Math.Soc., 134(2006),3257-3266.

[6]T.Diagana,G.M.N’Guérékata,N.V.Minh,Almost automorphic solutions of evolution equations,Proc.Amer.Math.Soc.,132(2004),3289-3298.

[7]J.Liang,J.Zhang,T.J.Xiao,Composition of pseudo almost automorphic and asymptotically almost automorphic functions,J.Math.Anal.Appl.,340(2008),1493-1499.

[8]H.S.Ding,J.Liang,T.J.Xiao,Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces,Nonlinear Anal.,73(2010),1426-1438.

[9]Z.M Zheng,H.S Ding,On completeness of the space of weighted pseudo almost automorphic functions,J.Funct.Anal.,268:10(2015),3211-3218.

[10]B.M.Levitan,V.V.Zhikov,Almost Periodic Functions and Differential Equations, Moscow Univ.Publ.House,1978.English translation by Cambridge University Press, 1982.

[11]K.L.Cooke,J.Wiener,Retarded differential equations with piecewise constant delays, J.Math.Anal.Appl.,99(1984),265-297.

[12]S.M.Shah,J.Wiener,Advanced differential equations with piecewise constant argument deviations,Int.J.Math.Math.Soc.,6(1983),671-703.

[13]S.Busenberg,K.L.Cooke,Models of vertically transmitted diseases with sequentialcontinuous dynamics,in:V.Lakshmikantham(Ed.),Nonlinear Phenomena in Mathematical Sciences,Academic Press,New York,1982.

[14]R.Yuan,On the existence of almost periodic solutions of second order neutral delay differential equations with piecewise constant argument,Sci.China,41:3(1998),232-241.

[15]G.Seifert,Second-order neutral delay-differential equations with piecewise constant time dependence,J.Math.Anal.Appl.,281(2003),1-9.

[16]H.X.Li,Almost periodic solutions of second-order neutral delay-differential equations with piecewise constant arguments,J.Math.Anal.Appl.,298(2004),693-709.

[17]H.X.Li,Almost periodic solutions of second-order neutral equations with piecewise constant arguments,Nolinear Anal.,65(2006),1512-1520.

[18]E.A.Dads,L.Lhachimi,New approach for the existence of pseudo almost periodic solutions for some second order differential equation with piecewise constant argument, Nonliear Anal.,64(2006),1307-1324.

[19]R.K.Zhuang,H.W.Wu,On almost periodic solutions of third-order neutral delaydifferential equations with piecewise constant argument,Ann.of Diff.Eqs.,29:1(2013), 114-120.

[20]R.K.Zhuang,R.Yuan,Weighted pseudo almost periodic solutions of N-th order neutral differential equations with piecewise constant arguments,Acta Math.Sin.(Engl.Ser.), 30:7(2014),1259-1272.

[21]N.V.Minh,T.Dat,On the almost automorphy of bounded solutions of differential equations with piecewise constant argument,J.Math.Anal.Appl.,236(2007),165-178.

[22]W.Dimbour,Almost automorphic solutions for differential equations with piecewise constant argument in a Banach space,Nonlinear Anal.,74(2011),2351-2357.

[23]C.H.Chen,H.X.Li,Almost automorphy for bounded solutions to second-order neutral differential equations with piecewise constant arguments,Electronic Journal of Differential Equations,140(2013),1-16.

(edited by Mengxin He)

?This project was supported by National Natural Science Foundation of China(Grant Nos.11271380,11501238),Natural Science Foundation of Guangdong Province(Grant Nos. 2014A030313641,2016A030313119,S2013010013212)and the Major Project Foundation of Guangdong Province Education Department(No.2014KZDXM070).

?Manuscript April 18,2016;Revised August 31,2016

?.E-mail:rkzhuang@163.com