ALMOST PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT
ALMOST PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT
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DOI：
10.1524/anly.1996.16.2.171
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
R. Yuan;Jialin Hong
中科院分区：
文献类型：
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作者：
R. Yuan;Jialin Hong
本文研究了K. L. Cooke & J. Wiener，并在某些生物医学问题中找到了应用。1991 Mathematics Subject Classification：34 K20本文的主要动机来自K.L. Cooke & J . Wiener [4]和S. M. Shah和J. Wiener [8]，他们引入了一类新的微分方程（称为具有分段常数变元的微分方程）。这些方程在单位长度的区间内具有连续动力系统的结构。解在连接任意两个连续区间的点处的连续性意味着解在这些点处的值的递归关系。因此，它们联合收割机结合了微分方程和差分方程的性质。1991年，K. L. Cooke & J . Wiener [5]对具有分段常数变元的微分方程的研究现状作了一个综述。由此可知，对微分方程所做的工作，Au是关于稳定性、振动性和周期解的存在性。在[10-12]中，Papaschinopoulos研究了这类方程的拓扑等价性和渐近性。最近，作者研究了如下方程（1）i{ 2，i=-N其中a，aj是常数，f（t）是概周期函数（即f r（i，ε）= { t}的ε-平移集）的概周期解的存在性||f（fi）是等式的解。(2)如果满足下列条件：（i）z在R上连续，（ii）x（t）的导数x（<）处处存在，可能除了点[<]，其中存在单侧导数，（iii）X满足Eq. (2)在每个区间[n，n + 1]上，n ∈ Z = {· ··，-1，0，1，· · · }.显然，如果x（t）是方程的解。(2)在R上，我们有以下关系式（3）N .t x（t）= + 1）X（n +1利用解在一点上的连续性，我们得到如下差分方程：+ X）+ / e ' ^+ i ' ^ /C^ jd^，n ^ Z。设60 = 6 + a-i（a-i），61 = 1，i = 1，±2，...，±iV，i-n +1，则我们可以重写等式（1），(4)As（5）^ HCN+i = Hn. N c i=-N概周期解» EPCA 1 7 3方程的相应齐次方程。(5)是N（6）^6icn +i = 0。根据文献[6]，我们可以求出齐次差分方程（6）的特解eisc_n = A_n。此时，A将满足以下等式
In this paper, we s tudy the existence of almost periodic solutions to differential equat ions with piecewise constant argumenta which was considered by K. L. Cooke & J. Wiener and found applications in certain biomedical problems. 1991 Mathemat ics Subject Classification: 34K20 The miiin motivation of this paper comes from papers considered by K.L. Cooke & J . Wiener [4], and S. M. Shah & J. Wiener [8], who introduced a new class of differential equations ( called differential equations with piecewise constant argument ). These equations have the structure of continuous dynamiceJ systems within intervals of unit length. Continuity of a Solution at a point joining any two consecutive intervals implies recursion relations for the values of the Solution at such points. Therefore, they combine the properties of differential equations and difference equations. In 1991, K. L. Cooke & J . Wiener [5] gave a survey of the status conceming the differential equations with piecewise consteint argument. From this, we know that aU of the work that has been done on the differentisJ equations concems the stability, the oscillation and the existence of periodic Solution. In [10-12], Papaschinopoulos studied the topological equivalence and asymptotic behavior for these equations. Recently, authors investigated the existence of almost periodic Solution for the following equations (1) i{ 2 , i=-N where a, a j are constant, f{t) is an almost periodic function ( that is, t he £—translation set of / r ( / , £ ) = { t | | / ( fi is a Solution of Eq.(2) if the following conditions are satisfied (i) z is continuous on R, (ii) the derivative x(<) of x(t) exists everywhere, with possible exception of the point [<], where one-sided derivatives exist, (iii) X satisfies Eq.(2) on each interval [n, n + 1] , n € Z = {• • • , —1,0,1, • • • }. Clearly, if x{t) is a Solution of Eq.(2) on R, we have the following relations (3) N .t x{t) = + 1) X I »' " iCn+i + / n < t < n + 1, i=-N where x(n + t ) = c„+<, -N <i< N. By using the continuity of a Solution at a point, we obteiin the following difference equation • " rn+i (4) c„+i = e-c,. + X ) + / e ' ^+ i ' ^ /C^ jd^ , n ^ Z . i=-N Setting 60 = 6" + a -^ao le" 1), 61 = 1) 1, = a ^ a ^ e " 1), i = 1 , ±2 , • • • , ±iV, /•n+l Jn then we can rewrite Eq.(4) as (5) ^ hcn+i = hn. N c i=-N Almost Periodic Solution» Of EPCA 1 7 3 The corresponding homogeneous equation of Eq.(5) is N (6) ^ 6ic„+i = 0. i=-N Following [6], we can seek the particular solutions eis c„ = A" for homogeneous difference equation (6) . At this time, A will satisfy the following equation