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Advances in Pure Mathematics, 2011, 1, 118-127 doi:10.4236/ apm.2011.14024 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Discrete Pseudo Almost Periodic Solutions for Some Difference Equations Elhadi Ait Dads*, Khalil Ezzinbi, Lahcen Lhachimi University Cadi Ayyad, Faculty of Sciences Semlalia, Department of Mathematics, Marrakesh, Morocco E-mail: *eaitdads@gmail.com, ezzinbi@ucam.ac.ma, lllahcen@gmail.com Received February 23, 2011; revised April 26, 2011; accepted May 10, 2011 Abstract In this work, we study the existence and uniqueness of pseudo almost periodic solutions for some difference equations. Firstly, we investigate the spectrum of the shift operator on the space of pseudo almost periodic sequences to show the main results of this work. For the illustration, some applications are provided for a second order differential equation with piecewise constant arguments. Keywords: Difference Equations, Pseudo Almost Periodic Sequences, Schift Operator 1. Introduction Difference equations have many applications in popula- tions dynamics, they are used to describe the evolution of many phenomena over the course of time. For example, if a certain population has discrete generations, the size of the th generation is a function of the th generation (1)n(1)xn n() x n. This relation expresses itself in the following difference equatio n 1= ,xnf xnn. (1) The discrete processes occur in the investigation of many phenomena, mainly in the case of use of computers. One of the most widely adopted definition of a discrete process can be formulated as follows: a discrete process is a map from the additive group of the integers , into a complete metric space (,) X d, such as or with the distance function induced by the vector norm. m m We use two different notations to designate a discrete process, namely, if : f X n fn is a discrete process, we shall write instead or , dropping nn f usually the subscript “ n”, since no confusion can occur (indeed, we are not going to consider in this work discrete processes defined o n a group, othe r t han ). Of course, one of the most common sources for the discrete processes is the theory of difference equations, such as 1=, nnn xAxbn , (2) where n x stands for the unknown process, with values in or m m A is a square matrix of order with real or complex entries and n b stands for a given discrete process, with values in the same space as m n x . In practice, we deal with solutions of (2) which are only defined on subsets of , and therefore, they might be regarded as restrictions of a “complete” process to a subset of its domain of definition. 1 m Difference equations and discrete dynamic systems represent two sides of the same coin. For instance, when mathematicians talk about difference equations, they usually refer to the analytic theory of the subject, and when they talk about discrete dynamic systems, they generally refer to its geometrical and topological aspects. More sophisticated equations (or systems) than (2) are those described by the following discrete eq uation =,n m , nn xfxn (3) where (or ) is a given map, in general nonlinear in both arguments. :fm Another example, let y n be the size of a population at time . If n is the rate of growth of the population from one generation to another, then we may consider a mathematical model in the form yn 1=, >0yn . (4) If the initial population is given by 0= 0, y y >1, then the solutions are given by . If 0 n yn y = then yn increases infinitely, and If = .lim n yn =1, then 0 = y ny for all , which means that the size of the population is constan t for the indefinite future. >0n E. A. DADS ET AL. 119 However, for <1, we have and the lim=0, nyn population eventually becomes extinct. Since our main objective is to provide a criteria to get the existence of a pseudo almost periodic solution for equations of the form (2) or (3), we shall first review the basic properties of pseudo almost periodic discrete processes. This work is motivated by the results obtained in [1,2], and the main results would be some extension for some well-established results in the literature, more details can be found in [3]. This work is organized as follows. In Section , we consider geometrical properties of the shift operator in general case and, we deal with the properties of shift operator the spaces of almost periodic and on ergodic sequences. In Section 3, we a consider the existence and uniqueness solutions of some difference equations using polynomial functions. In the last section, we deal with the application of the previous results to some second order differential equation with a piecewise constant argument. 2 2. Shift Operator Acting on the Space of Pseudo Almost Periodic Sequences In this section, we give some properties on pseudo almost periodic sequences that will be used in this work. For more details in this connexion, the reader will see [4-12]. Definition 2.1: A sequence nn x with values in is called almost periodic if for all m >0, the set ,: :,< nn Txfor all nxx is rela- tively dense. The space of almost periodic sequences is denoted by . If we use the notation Let (, ) m AP =1,m().AP (, )B denote the space of bounded complex sequen- ces provide with the supremum norm. 0 denote the space of bounded complex sequences ()PAP nn x satisfying the ergodicity condition = 1 lim= 0 2 N n NnN x N . Remark 2.2: = 1 lim= 0 2 N n NnN x N , doesn’t imply that is bounded. In fact, let us consider the sequence defined by nn x 3 if = =0otherwise. n pnp x Let be such that . Then p 3 3<1pN p 3 ==1 1 11 =0 22 p N np nN k pp xk NNp . For a function we define : m f,(,)Tf by ,=:for all ,<.Tft ftft Definition 2.3: A bounded continuous function x is said to be almost periodic if the set ,Tf is rela- tively dense for all >0 . For the next denotes the space of all almost periodic functions from to . (, ) m AP m Proposition 2.4: Let and m =nn xx be a sequence with values in . Let define the function m x : by m =forall n xn xn, and x is affine in [, 1].nn Then the following re- sults are true. 1) sup= sup, n tn x tx ,,Tx Tx and (, ) m xAP if and only if , (, ) m xAP 2) if and only if 0(, ) m xPAP(, ) m xPAP. Proof. 1) is a consequence of results taken from [1]. For the proof of 2), by taking the components, real part and imaginary part, we can consider the case where 0(, )xPAP [,tnn . Then, one has For 1] one has 11 == nn nnn 1 x txxtnx x tnxnt Two cases to be considered: a) If 10 nn xx 11 ||| d= 2 nnn n | x x xt t . b) If 1<0 nn xx 22 1 11 1 11 |||| =d 22|||| ||||3|| || . 22 n nn nn n nn nnn n n xx xx x tt xx xxx x x The result is a consequence of the fact that x if and only if 0(,)PAP 1 0 d(, ) n nn xt tPAP . Definition 2.5: We define the space of pseudo almost periodic sequences by 0 () ()()PAP AP PAP . Proposition 2.6: [2] Let be such that ()x PAP Copyright © 2011 SciRes. APM 120 E. A. DADS ET AL. =, x yz for some and Then ()yAP0()zPAP y x . Difference calculus is the discrete analogue of the familiar differential and integral calculus. In this section, we introduce some basic properties of two the following operators that are essential in difference equations =1 x nxn xn and the shift operator =1Ex nx n . Then n k= k Exn x. Let I be the identity operator. Then and . The following formula are true =E 1 kk k I =EI . i ki i =0 =0 == =1 k i k i x nEI k xn Exn i x nki i i xn =0 =k i k ni (5) Similarly we have kk Ex . We should point out here that the operator is the counterpart of the derivative operator in calculus. Both operators and share one of the helpful fea- tures of the derivative operator , namely, the property of linearity. Another interesting difference, parallel to differential calculus, is the discrete analogue of the fundamental theorem of calculus. D n a 1. E exp dsgs s .k : nn xx D Remark 2.7: Exponential in differential equa- tions corresponds to the exponential and the at integral corresponds to the sum- mation: 0exp tat 11 =0 nnk k ag Let us consider the linear map defined by nn T Let F be a subspace of that is invariant by , for example B T F could be one of the following spaces 0 Let (, ,AP ) (,),PAPPAP (,). F T be the linear map induced by on T F and take y F and where [],PX[] X is the space of polynomial functions over . Next, we study the existence of solutions in F for a given y F for the following algebraic equation = F PT xy. This equation has solutions if Im , F y PT but we have to compute Im . F PTker The uniqueness problem is equivalent to determine The following re- sult is well-established. PT . F Lemma 2.8: Let []PX be non constant. Then =1 ker=: withdeg() < Nn iiii in PTQ nQm where the i s are non zero roots of with res- pective multiplicities P P . i m Remark 2.9 : If is the unique root of , then 0 ker=0.PT Lemma 2.10: Let 1| = B TT (the restriction of to B). Then T 1 ker=,=1,=1,,. n ii n P Tvectir Proof. Let be a complex number such that =1. Then 1 ker =, n n TI cc . Let 2 1 keryTI and =. B x TIy = Then 11 and nn nn =0 n x xyyx , which implies that 10 = n nn , y yx and 10 1= nn nn yyx . Since n n y is also bounded, because =1, then and 0=0x1= n yy, n also 2 11 ker=ker.TI TI By simple recurrence on we deduce that ,m for all 1,m 11 ker= ker=: mn n TITI cc . Consequently: 11 1 ker= ker =:=1,=1,, mi i ir n ii n PTT I vecti r . Lemma 2.11: Let , x . Consider y x y the sequence defined by 1 01 1 1 if> 0 =0 if= if< 0. kn k kn n kn k nk xy n xy n xyn 0. Copyright © 2011 SciRes. APM E. A. DADS ET AL. 121 Then , x yxy, is bilinear and symmetric. More- over, if we denote by =TI then 0 = x yxy xy Remark 2.12: If then ,x x can be extended to a function x which is of stepping type on in the following manner: x tx Et where Et de- notes the greatest integer function of then one has for ,t ,,xy 0d n x yn xtyntt . Proof. Using the above remark, one can see that the following map x yxy is bilinear and symme- tric. On the other ha nd, one has 1 00 1 0 1 10 00 0 00 0 1d d 1d 1dd 1d d 1d nn nn n n n n n n x ynxtynt txtyn t t xt yntt x tynttxtyn t t xyxuyn u u xt yntt xyxtxtyn t t xyxy n p In the sequel, we denote by ,=(), suchthat anddeg . n pn n nn FbQn bF Q We define the following polynomials 011 =1and =if ! p XX XXXp CCp p . Lemma 2.13: Let and y be a complex number such that =1 , Then for , the following are true p 1) where 1 ,= p p cy y ,=pnp pn n cC . 2) such that 1 Im= : p F TI yF ,,pp cyF . Proof. 1) For we claim that >0,p ,1, =. pp cyc y . y y It follows that From lemma (2.11) one has : ,, , 0 ,1, = == pp p pp cycyc cyc for all0p 10, 0, 0, 0 = == py cy cycy ,p c .y 2) One has 1 Im p F yT if and only if there exists x F such that 1 = . p F e has: =n Txy Frommma (2.8) n le and 1) o , np n x Qn cy is in F if and only if ,p cy is in ,. p F Proposition 2.14: complex number such that Let y, λ be a =1 nomial of degree .p Thfollowin ue ,be a poly en the . In particular and Q g are tr 1 Im =, suchthat p F n TI F Qn ,p n yyF 1 , Im=, suchthat . ppn Fp n TI yFnyF and 11, Im=:suchthatfor all1,, F mn i im ii n PTy Fir nyF Proof. LetThen for all 1 Im . p F yTI ,qp 1q I Im F yT , by lemma (2.13) we have For all ,qp () nn Cy qnq is in ,q F , to which is equivalent For all (, ) qn nn qpC y is in ,q F , and as 0 () q Xqp C is a [], p X basis of then , n isin p n Qny F . Conversely, assume that is n n Qn y in , p F . One has from lemma (2.11) , andareinat p xyyF which impliesth is , in p x yF . But 1 1 = nn nn QnQ n , then 1, isin n p n Qny F , by iteration, we see that 1, for all[0,],in qnis p n qp QnyF , Copyright © 2011 SciRes. APM 122 is a basis of E. A. DADS ET AL. and since 1 qQX 0qp [], p X then for , all [],isin n p p n RXRn yF , In particular F , is in pnp np n Cy , and consequently 1 p F yImT I the fact that . The end of the proof results from i Let 1 Im= Immi FF ir PTT I . be the linear map induced by T on if (,)AP It is clear that1, then I of is invertible. On the other hand if the roots P are of modulus ditfferen from 1, then P is invertible, in this case r=0P and we have theness of the solutions. Let 1 () iir ke e uniqu be the rts of P with modulus 1. Th oo en is a polynomial whose roots are of modulus differerom Proposition 2.15: 1ir P =mi i XX QX where Q nt f1. ker=:=1, =1,,. n ii P vectir n (2.10), one has Proof. From lemma =,=1,for =1 n ii vect i ker=,,=1, =1, ,...,, n ii n n P APvectir r (since is periodic). It is well known that if ] and then , n in 12 ,[QQ X 12 =1,QQ 1 Im= ImImQQQ Q 12 2 , and 1 Im= Immi i ir P I . 3. Existence of Pseudo Almost Periodic Sequences It is known that if , n AM ,m bAP and x is a bounded solution of 1 ten =, nnn xAxbn h x is almost periodic. If ,yAP and [],PX =xy , PP to a system, we deduce the following lemma. Lemma 3.1: For and , yAP,[]PX is almost every bounded solution mma (3.1) anroposition lt. ion 3.2: Let of PT =xy periodic. Consequently from led the p (2.14), we get the following resu Proposit be a complex number such that 0 , p and Q be a polynomial with degree p. Then 1 Im = p I yA , , , suchthat n p n PQn yB . In particular 1 , Im ,, suchthat p pn p n I n yB =yA P . and 11, Im=,, suchthatforall [1,], . n i im ii n PyAPi nyB m r In the next, we are concerned with the solutions in ,AP of the following equation 1=,for p Ixyp . (6) tion 3.3: Let .y Then we Proposi have 1) 1= p p TIycy , (where =( p c) ), p nn C 2) 1 Im p yI bB if and only if there exists by transforming the scalar equation such that = p yc bQ w the solutions of ith Q[]. pX 3) 1= p I xy in ,AP are = nn x bc where d c is a c onstant an 0 =pk kp byc c 1 2 12 1 =l im () =lim k pn pk kn pn kn yc n n yc nCC n Proof. Since . 0 k Xkp C is a basis of [], p X then there exit scalars 0 k kp such that kk Qc 0 = kp in this case, one h 0 as = p kk kp c . ) yc b (7 Copyright © 2011 SciRes. APM E. A. DADS ET AL. 123 Then, by applying 1 p TI to the equation (7), we obtain 1 11 =p 1 p p ycT Ibc , then 1 =lim pn yc n n . (8) ose that Supp 12 ,,, pp k are known and let us compute 1: k e has by applying k TI to the equation (7), on 2 2 = 11 k p kkk ppk cT Icc , so k yb c 2 2 1=lim p k pkknp n kn yc nCC n . (9 We conclude that for all the equation ) ,,yAP 1= p I xy admits the solutions in if d (9) ,AP (8) anand only if, the limits given by equations exist and the sequence 0 p kk c kp yc is bounded, in this case the solutions are given by 0 = p kk kp x yc c , with 0 is any constant and the (1 ) kkp are given by (8) a). Remark 3.4: By a change of variables, the equation nd (9 1= p I xy when =1 becpre- vious form (6). Indeed, let uso omes in the Since cnsider the following operator ,APP :, n M A . nn xx =,MM then == I MM MM . I So 1 = I MIM , an 1 d 11 1 = pp p I MIM . Then eq uation 1= p I xy , (10) be comes 1 11 p p = M IMx y, 1 11 1 1 = p p I Mx M , y ing differently then by putt 1 = X Mx and 1 1 1 =, p YM y and we are comhe followi equation ing down to t =. ng 1p I XY Theorem 3.5 Fol equat =,Pxy r the generaion the solution is of the form: 01 =, x xx such that r r 1=1 =i i x x , (with is the number of different roots with modulus equal to 1, i x is the solution of the equ 1= pi ation iii I xy ) and 0 x af n element o ker .P Proof. w We writeP under the form 1 =pi X Q 1i i r P ith i s are the roots of P which are of modulus equal and to 1 , i p e may c o r Evef it means to replace by wmew n t o Indeed ttin n i do 1 y let 1 Qy, =1.Q us pug =1, = p k k =1 =1. ii AU Then kk i PX , then using the Bezout idntity, we get that there exist polynomials i U r e such that i =1 =ii i Ay with r y = ii y Uy and equation =Px y(11) becomes t a solution of Equation (12), it suffices to determine a solution = =r i Px A 1ii y (12) then to buil i x of the following equation i 1= pi ii I xy , the solution is easily determined and after we take r =1 =i i x x . To obtain all solutions , we add elements of ker =,1 n in Pvecti r . ple 3.6: For all polynomial Q with all rootExam s are of modulus different from in 1, [] X one has the following decomposition , 1=m riij =1 =1 j ij i a QX the i where s are two by two dtinct of modulus isand different from 1, then we have Copyright © 2011 SciRes. APM 124 E. A. DADS ET AL. 1 , ==1 =1 m ri j iji QaI g down to the =m QX ij e cas so we are comin with m and 1 . First case: >1: =0k I= mm I= mk m k m C so for xA has ,P one = m I xy where ase: =km x . =0 k nm k yC nk Second c<1 1 m II C , =0 == mk m k as mk km hence for one h ,xAP = m I xy with k Let be bounded. Thk for a bounded r the followingtion : n =. k nm x =0 k nm k yC en we loo qua 2 28 n nn h solution fo difference e 43 1 213 8= nn nn 30 x xx x . x 8 =2 h 1 2Qxxxxx x 3 43 2 = 2133028 32 12 1 392722721 22 = Qx xx gg where 1112141 8 := x 181 := 27 21 g x and 232 11 2141 := 3927 22 gx xx . 2 For 1<||< 2 2x, we have 1=1 81 =,:= 27 2 n n nn n a ga x 2 211 :=:= 48 22 n gb =0 171 371 ,144216 2 n nnnn n xbn n . the solution n x is given by: kn =1 =0 := knknn nn x ah bh =1 2 11 17 n n =0 81 := 27 2 1 371 48144 216 222 kkn n n kn nn n xh n h Let 0,yPAP nce and uniqueness and the existe of[].PX solutions in We study 0,PAP of the following equ ation 0=Px , y where 0 is the linear map induced by on T PAP 00 ch that . Let be su ,x PAP, 0=0.Px From lemma (2.10), we have 00 0 , =1,,, ,,=0 nir PAP AP PAP ker =n i P vect . Thenhave the uniqueness of the solution. 3.7: Unlike to the almost periodic ca we Remark se, x bounded and 0=Pxy is not enough to get that In fact, we 0,xPAP example: have the following counter 1=2 ,, n nn xx n the solutions are given by 1 0=0 2,if >0 k xn . 0 1 0= =, 2, if<0 nk n k kn xx xn unded, on the other part one has: Then all solutions are bo 0 lim =2 n nxx and 0 lim =1 . n nxx . If 0 x PAP we will have 00 2= 1=0xx e solution is not in which is absurd, con- sequently, th As a consequence of proposition (2.14), we get the following result. 0.PAP Proposition 3.8: Let be a complex number such p ,that 0 , and Qal wit a polynomih degree p. Then ,uchthat .Qny PAP 1 00 Im=, s p IyPAP 0,p n In paular n rtic 1 00 0, Im=,, suchthat . p pn p n IyPAP nyPAP Copyright © 2011 SciRes. APM E. A. DADS ET AL. 125 In ergodic case, for the calculation of the solutions, the method is similar to the one given in the almost periodic case, firstly we begin with solution of the following equation 00 101, Im=,,such that [1, ],(). mn i in mii PyPAP irn yPAP Remark 3.9: 1 00 =, = p p kk kp I xyxyc c with 1 () kkp his time th are determined by equations (8 and (9) but te ) 0 is not arbitrarily, but 0 is the mean value of 1 p kk c kp solutions needs more 1 yc, then the etence of xis p kk kp ycc to have a mean value 0 then p y c 0 0kp c kk PAP. Example 3.10: Let be an absolutely convergent , 0k k a series ]X, 0 () kk z a family of complex numbers with modulus equa1, such that [P l 0 inf( )0 k kPz and =0 n k > =. nk k y az Th has ast per en the following equation y lmoiodic solutions. In fact, if we put =Px , =0 =n k nk a kk x z Pz , one n has x is well defined and =0 forall 0,== ini k ni nk a ixx k k z Pz it results that =0 =0 == nn k kkkk n nkk k a Px Pz=azy Pz the equation admits solutions in z ,AP The hypo- thesis 0 inf>0 is necessary, as we rema k kPz rk it through the fo llowing counter example : 122 =0 1 =exp nn k in xx kk . If the solution exists, then 22 11 ,exp1= i ax 2 k kk and 2 1,1ax k k which contradicts the Parsevall’s iden tity, we d educ e that thlution. 4. Application More details and the motivation on this applications can d in [4,14-20] and the references cited therein. To apply the previous results, we consider the following system e equation does not have a so be foun 21 2122 11 =1 2 nnn q px pxx 22 2 13 2 12 = 2 nn n qpx px a xpxpqxpxb 21 21222nn nnn where ,,ab AP ,pq with 0.q nn Remark 4.1: The last system comes from the re- search of solutions of the following second order differential equation with piecewise constant argument: 2 2 d1 1= 22 d t x tpxtqxft t . . where denotes the greatest integer function. In the case where 1,p the system has a unique solution nn x in AP , =1,the heintend to study sysomes re we the situtem bec ation where p 222 22 0=12 3 22 =1 , nnnn qq 21 212 22nn nnn x xx a xx qxxb or more 22 =2 nn Px a 21 22, nn n =1 I xqIxb (13) where 2 =24 32PX qXqX . P th m We know that is invertible if and only if the roots of are wiodulus different from tion 4.2: 1) Let . The equation has roo if and only P 1. Proposi 2 21 ax ax012 ,,aaa ts of modulus 0 =0a if 1 201 =aaa or 20 10 = <2. aa aa 3,a the following equation 2) For Copyright © 2011 SciRes. APM 126 ha d only if E. A. DADS ET AL. 3 x a 2 3210 =0axax a , (14) s roots of modulus 1 if an 20 1 =aaaa 3 or 22 020313 =aaa aa Proof. It is clear that are roots of ( if 20 3 <2 .aa a a 1 14) if and only 201 =aaa If the eq of modulus if and only if their pro, which is equint to the abeomes then 2<4 is e e cond 12 0 two conjugate compleroots which are ,aaa x ductqual ovition uation admits 1 vale 1b 20 =,aa c1 a< 0 2a For the secondmits as roots if and only if eqtion , it adua1 20 1 =aaa, we can assume that if not we divide the etion by, it is a prove that 3 a 3=1,a matter toqua a3 2 020 1 20 =1aaa a aa since the equation admits always a real root ,r it will haval roots with modulus equal to 1 if and ill be factorized as follows <2 , e non re only if it w 32 2 210 1with<2,xaxacxc whch implies that =r = xa xrx is a root, in the sequel . If we obtain admits com- plex th modulen from the 0 a 0 1=a 2 0201 0aaaa 2 0=0,a roots wi 21 =0xaxa us equal to 1 th previous result, we d educe that 1=1a 2<2 .a 1 , the equ as If 02 0 =1aaa ation can be written follows 2a 3232 2 220 =1 1 020 0 2 020 =1 x axax axaxaaa x a xa xa ax We will have no real roots with modulus equal to if and only if 1 20 <2.aa 4,Corollary 4.3: 1) Ifthe roots of are modulus different from q Pof 1 (we assume that 0q). 2) If =2131.X X =4,q PX Proof. It suffices to apply the previou s proposition. Proposition 4.4: If the system (13) admits solutions if and only if 4,q 2Im nn ab I . If =4,q the system (13) admits solutions if an only if d Im n aI ab 2I m nn I Proof. First case: 4:q the system becomes 1 21 () =21 nn 1 22 =2( ) nn xPa . n I xqIPa b Th admits so lutions if and only if is system 1 21 Im nn qIPabI , or yet 21Im=Im , nn qIaPb PII since P respis invertible. Make the Euclidean dision of ectively iv P 11qX by 1 X , we s th condt toee that e previousition is equivalen 24Im nn qa qbI , identically 2Im nn ab I . Second case: =4:q The system becomes 22 21 22 23 =2 =3 , nn nnn IIxa I xIx b equivalently 22 1 22 21 3= =3 . nn nn IIxa n x IIxb Let us consider the following system (( )II 21 1 22 21 ) = =3 , nnn nn xb a n x IIxb which has solutions if and only if Im =ImIm, nn I baII II which is equivalent to Im n aI 2I m nn ab I . 5. Acknowledgements The authors would like to thank the referees for their careful reading of the paper. Copyright © 2011 SciRes. APM E. A. DADS ET AL. Copyright © 2011 SciRes. APM 127 6. References [1] C. Corduneanu, “Almost Periodic Discrete Processes,” , Vol. 2, 1982, pp. 159-169. [2] J. Hong and C. Núñez, “The Almost Periodic Type Dif- ference Equations,” Mathematical and Computer Model- ling, Vol. 28, No. 12, 1998, pp. 21-31. doi:10.1016/S0895-7177(98)00171-X Libertas Mathematica n to Difference Equations,” 3rd erlin, 2000. st Periodic Solutions for Some Equation with Piecewise Con- t,” Nonlinear Analysis: Theory, Methods [3] S. Elaydi, “An Introductio Edition, Springer-Verlag, B [4] E. A. Dads and L. Lhachimi, “New Approach for the Existence of Pseudo Almo ifferential Second Order D stant Argumen and Applications, Vol. 64, No. 6, 2006, pp. 1307-1324. doi:10.1016/j.na.2005.06.037 [5] A. I. Alonso, J. Hong and J. Rojo, “A Class of Ergodic Solutions of Differentiale Quations with Piecewise Con- stant Arguments,” Dynamic Systems and Applications, Vol. 7, 1998, pp. 561-574. [6] A. M. Fink, “Almost-Periodic Differential Equations, Lecture Notes in Mathematics,” Springer-Verlag, Berlin, Vol. 377, 1974. [7] S. Zaidman, “Solutions Presque-Périodiques des Équa- tions Différentielles Abstraites,” L’Enseignement Mathé- matique, Vol. 24, No. 1-2, 1978, pp. 87-110. [8] S. Zaidman, “A Non-Linear Abstract Differential Equa- tion with Almost-Periodic Solution,” Rivista di Mate- matica della Università di Parma, Vol. 10, No. 4, 1984, . ons, Vol. 181, No. 1, 1994, pp. 62-76. pp. 331-336. [9] C. Zhang, “Pseudo Almost Periodic Functions and Their Applications,” Ph.D Thesis, University of Western On- tario, London, Canada, 1992 [10] C. Zhang, “Pseudo Almost-Periodic Solutions of Some Differential Equations,” Journal of Mathematical Analy- sis and Applicati doi:10.1006/jmaa.1994.1005 [11] C. Zhang, “Integration of Vector-Valued Pseudo Almost Periodic Functions,” Proceeding of the American Mathe- - p. 110-114. matical Society, Vol. 121, No. 1, 1994. [12] C. Zhang, “A Characterization of Pseudo Almost Periodic Functions in Fourier Analysis,” Acta Analysis Function alis Applicata, Vol. 4, 2002, p [13] C. Zhang, “Almost Periodic Type Functions and Ergodic- ity,” Kluwer Academic Publishers, Dordrecht, 2003. [14] S. M. Shah and J. Weiner, “Advanced Differential Equa- tions with Piecewise Constant Argument Deviations,” International Journal of Mathematics and Mathematical Sciences, Vol. 6, No. 4, 1983, pp. 671-703. doi:10.1155/S0161171283000599 [15] Y. Rong and H. Jialin, “The Existence of Almost Periodic 225-K Solutions for a Class of differential Equations with Piecewise Constant Argument,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 28, No 8, 1997, pp. 1439-1450. doi:10.1016/0362-546X(95)00 Periodic Solutions of [16] Y. Rong, “Existence of Almost Second Order Neutral Delay Differential Equations with Piecewise Constant Argument,” Science in China (Series A), Vol. 41, No. 3, 1998, pp. 232-241. doi:10.1007/BF02879041 [17] R. Yuan, “Pseudo-Almost Periodic Solutions of Sec- 46X(98)00316-2 ond-Order Neutral Delay Differential Equations with Piece-Wise Constant Argument,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 41, No. 7-8, 2000, pp. 871-890. doi:10.1016/S0362-5 n Quasi-Periodic Solutions of [18] Y. Rong and T. Kupper, “O Differential Equations with Piecewise Constant Argu- ment,” Journal of Mathematical Analysis and Applica- tions, Vol. 267, No. 1, 2002, pp. 173-193. doi:10.1006/jmaa.2001.7761 [19] Y. Rong, “On a New Almost Periodic Type Solution of a Class of Singularly Perturbed Differential Equations with Piecewise Constant Argument,” Science in China (Series A), Vol. 45, No. 4, 2002, pp. 484-502. doi:10.1007/BF02872337 [20] Y. Rong, “The Existence of Almost Periodic Solutions of Retarded Differential Equations with Piecewise Constant Argument,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 48, No. 7, 2002, pp. 1013-1032. |