A LINEARIZATION RESULT FOR A DIFFERENTIAL EQUATION WITH PIECEWISE CONSTANT ARGUMENT
In this paper we give a linearization result for a differential equation with piecewise constant argument. 1991 Mathematics Subject Classification Numbers: 34K05. INTRODUCTION. In [2], [6-10] and in the references cited therein there has been a lot of work concerning dififerential equations with piecewise constant argvunent. These equations describe hybrid dynamical systems (a combination of continuoiis and discrete) and so combine properties of both differential and difference eqiiations. Moreover differential equations with piecewise constant argument can be applied in certain biomedical models [1], In this paper we study the differential equation with piecewise constant argument of the form (1) y'{t) = A{t)y{t) + S(t)y([t] ) + f{t, y{t), s,([t])), te]R= ( -oo , oo), where [•] denotes the greatest integer function, A(t), B(t) are k x k continuous matrices for t e jR and f : ]Rx iR* x Jl* jR* is a continuous function on iR x JZ* x ü * and Lipschitzian in X 6 iR*" and y 6 iR*. A function y : Mi-y iR* is a Solution of (1) if the following conditions are satis&ed: (i) y is continuous on JR, (ii) the first derivative y' of y exists except possibly at the points t = n, n e Z = {,... — 1,0,1,...} where one-sided derivatives exist, 162 Papaschinopoulos (iii) y satisfies (1) on every intervaJ [n, n + 1), n G Z. Consider the linear differential equation of the form (2) = « € JR, where A{t) is a fc x fc continuous matrix for t e M. In what follows we denote by | • | the Eudidean norm and by X{t) the fundamental matrix Solution of (2) such that X(0) = h , is the fc x fc identity matrix. We say that (2) has an exponential dichotomy if there exist constants a > 0, K > l and a projection P (P^ = P) such that foi t, s e R , |A-(t)PA-i(s)| < t > s (3) |A:(t)(/fc < Ke-<'-'-\ s > t. Consider the equation (4) z'(t) = A{t)z{t) + B{t)z{,[t]), where A{t), B{t) are k x. k continuous matrix for t € M. We say that (1) and (4) are topologically equivalent if there exists a function H : M ^ M'' —> M'' having the following properties: (i) H is continuous on M, (ii) for each t £ M the function Ät(x) = H{t, x) is a homeomorphism of iR*, (iii) the function L{t,x) = is continuous on JJ x JR*, (iv) if y{t) is a Solution of (1) then H(t, y{t)) is a Solution of (4). In this paper if we assume that (2) has an exponential dichotomy (3) then under some restrictions on the matrices A{t), B{t) and the function / ( t ,x ,y ) we prove that (1) and (2) are topologically equivalent. We note that in the case where B(t) = 0 for all t G iR and / ( t ,x ,y) = f{t,x) for all t e x G iR*, y G JR*̂ the above result has been proved by Palmer [4]. MAIN RESULT. We prove now our main result. We need three lemmas. In the first lemma we study existence, uniqueness and continuous dependence on a parameter and initial values of the solutions for equations of the form (1). LEMMA 1. Let A(t), B(t) be k x k continuous matrices for t G H such that (5) \A{t)\ 1 , 02R* i sa continuous function on JR x JZ* x x J?"̂ such that for all t e R , xi, X2, yi, yt € H*, C e \h(t,xuyi,0-Ht,X2,y2,0\ (6) , , „ o M{M Se^) Then for every s e R , ^ e JJ*, C & RT the problem (7) x'W = ^WxW + 5Wx(W) + M<.x(t)-x(W),C), = $ has a vinique Solution xW = â id xĈ i®. ̂ >0 is continuous in (t, s,^, C). PROOF. We define the matrix A„(() as foUows A„(t) = X{t)X-\n) + f X{t)X-'iu)B{u)du (8) . = X{t)X-'(n)ih + Enit)), E„{t) = J X{n)X''{u)B(u)du, n  JZ* with |x"| = sup {|x"(t)|, t € [n, n + 1]}. We define the operator T^^ on C" as follows (12) + jT' X(,t)X-\u)h{u, x"(u), x"(n), Odu. 1 6 4 Papaschinopoulos I t is obvioiis that is continuous with respect to t G [n.,n + 1] for every (s, C)So T^^^^j is in C " . L e t x " , X? G C"T h e n using (6), (9), ( 1 1 ) and (12) we have (13) < X?|, where 9, T are deflned in (6). Relations (6) and (13) imply that ^ is a uniform contraction o n C". Moreover it is obvioiis that 's continuovis in (s, for every x " 6 C". T h e n from Theorem 3.2 [3,p.7] the opereator T ^ ^ q has a unique fixed point x " ( * ) = x ( * i C) which is continuous in (s, C) uniformly with respect to t 6 [ n , n + 1]. This means that x ( * i ^ 1C) is the imique Solution of the problem (7) for t e [n, n + 1] and it is continuous in { t , s , i , C ) , t, s,g]R, Working inductively backward we can define the Solution x ( * ) = x ( * i t̂ C) of the problem ( 7 ) on every interval [ m , m + 1], m e { , . . . , n — 2, n — 1} as follows: L e t CTM be the space o f all continuous functions x"* • ["!•> m + 1] i-> i R * with Ix*"! = s u p { | x ' " ( i ) | , t e [ T n , T n + 1 ] } . T h e n we define the operator T ^ ^ ^ on C " in the following way: = + 1 ) + 1 ) j T ^ ' X(m + l ) X \ u ) h { u , x"'(u), x ' " ( m ) , C)duj + f X{t)X-\u)h(u, x ' " ( u ) , x ' " ( m ) , C)du Jm where t e [ m , m + l ] , m e { , . . . n 2, n 1} , is the Solution o f (7) for t e [m + 1 , m + 2], m e { , . . . n — 2, n — 1 } . Arguing as above we can prove that is a uniform contraction on CTM and so the unique fixed point x " ' ( * ) = x(*>S!^iC)) i £ [m, m + 1 ] , m g { , . . . , n — 2 , n — 1 } is a continuous function in Hence is the unique Solution of (7) for t e [m, 771 + 1], m G { , . . . , n 2, n 1 } . Finally working inductively forward we can define the Solution x ' " ( * ) = «i C) of the problem (7) on every interval [m, m + 1], m G {n + 1, n + 2 , . . . , } as follows: L e t be the operator defined on CTM, m G { n + 1 , n + 2 , . . . , } given by Jm t e [ m , m + l ] , m G { n + l , n + 2 , . . . , } , x ^ ' H O is the Solution of (7) for t G [ m l , m ] , m G { n + 1 , n + 2 , . . . , } . Arguing as above we can prove that is a uniform contraction o n C"*, m e { n + 1 , n + 2 , . . . , } and so the unique fixed point x " ( < ) = t C) of ^ . , { , 0 is a continuous function in Therefore x ( * , C ) is the unique Solution of (7) for t G [ T n . , m + 1], m G { n + 1 , n + 2 , . . . , } which is continuous in Since now m. is an arbitrary integer mmiber the proof of the lemma is completed. Piecewise constant ajgument 165 For the following lemma we need the following definition Consider the difference equation (14) t)(n + l) = 5(n)i;(/i), n, e Z, where S{n) isa.kxk invertible matrix for n e. Z. We say that (14) has an exponential dichotomy if there exist a projection = P) and constants K >1, o > 0 such that |y(n)PV-'(m.)| < n > m (15) \V{n){h < m>n, where n,m e Z and V(7i) is the fundamental matrix Solution of (14) such that V(0) = IkLEMMA 2. Suppose that A{t), B{t) are fc x fc continuous matrices on M such that equation (2) has exponential dichotomy (3) and for all t 6 iR (16) |A( i ) | 1 , 0 < 6 < M e ^ p , p = ^ m i n { ^ , where K,a are defined in (3), e is a constant, 0 < « < a, ß = a — e and 0 : iR i-> iR is a function, = fi^i. t e M. Consider also that g : iR x JR'= x Jl* x 2R* is a continuous function on iR X iR* X IR* X iR'' such that for all t 6 iR, x, V, xi, X2, yi, y j € Ä* (17) |g( i ,x ,y,OI<2/i , ß > 0 , |9 (* ,x i ,y i ,0 -9( t ,x2 ,y2 ,0 l (18) M^ < 7 ( | x i X 2 | + |y i -y2 | ) , 0 < 7 < A = {Le'^(M + 6){ß) + M) where L = 2K{€). Then the differential equation (19) V'(0 = ^WV'(t) + BWV'(M) + 9(t-V'(t),V'([«]),C), t e R has a unique bounded Solution which is continuous in (t, C), t e M, € JfC and for all teR, c e Jff ^{t, C) satisfies (20) | ^ ( t ,C) l<« , K = 2 / i e ^ M ' ( 1 + Le^(J l /+ 5)M-V(/3)) • PROOF. Since (2) has an exponential dichotomy (3), it is obvioiis that eqiiation (14) where S{n) = X(n + l)A'~>(n), n e has an exponential dichotomy (15). Then using (9), (16), the 1 6 6 Papaschinopoulos roughness of exponential dichotomies for difference equations [5,p.276] and the same argument. as in [7,p.497] we can prove that the difference equation (21) u;(n + 1) = R{n)w{n), R{n) = X{n +1) ^ ^ ' ( n ) + J"^^ X'\u)B(u)duj , n e Z has an exponential dichotomy and so there exists a projection = Q) such that (22) Gin,rn) = , n,meZ where ß, L are defined in (16) and (18) respectively and W(n) is the fimdamental matrix Solution of (21) such that Wo = Ii,. Let S be the set of the continuous and bounded functions : M x JFT i-> J?* such that IV-I = sup{|V'(t,C)l, teR, C e II ' ' } < oo. Let V e e . Then if n = [t], t € R we define the operator H on Q as follows: WV(«, C) = X{t) (x-\n) + £ X-\u)B{u)duj b(n, 0 + X{t) f X~\u)q{u, V(u, C), C), Qdu, Jn (23) (24) f>inX)= X ] G(n,m + l)gv,(m,0, m = o o /•m+1 g^(m,0 = A ( m + l ) / X-'{u)g{u,i>{uX),^{m,0,(:)