On the asymptotic behavior of solutions of linear differential equations

Publicacions Matemàtiques, Vol 38 (1994), 3-9 . ON THE ASYMPTOTIC BEHAVIO R OF SOLUTIONS OF LINEA R DIFFERENCE EQUATION S JERZY POPENDA AND EWA SCHMEIDE L Abstract In the paper sufficient conditions for the difference equatio n r q x = i CLn X n+ i i = o to have a solution which tends to a constant, are given . Applyin g these conditions, an asymptotic formula far a solution of an m -t h order equation is presented . In this paper we give a sufficient condition for any linear differenc e equation which can be treated as a first order equation with advance d arguments to possess solutions which tend to arbitrary real constants . Using this theorem we shall study a linear m -th order difference equatio n and obtain a solution with a particular asymptotic behaviour . The mai n idea consists in seeing that any m -th order difference equation can b e studied as a first arder one with perturbed arguments . To start with, we consider the difference equatio n (E 1 )  q xn = E aWxn +z , n E N . z= c Here by N, R we denote the set of positive integers and reals respectively . For any function y : N --> R the difference operator q is define d as follows q ~Jn = yn +1 — yn 1 q xy n = q ~L,►i -1 yn~~ nEN , fori> 1 . Difference equations of the type (E 1 ) arise in numerical methods fo r solving differential equations with advanced arguments . In economics 4  J . POPENDA, E . SCHMEIDE L and biology there are simple discrete models describing processes i n which actual growth of considered parameter could be characterized i n terms of its future states . Such models can be used in forecasting, an d also to control actual growth ta get the desired quántities at some fixe d periods . Moreover as it is shown below the results far (E 1 ) can be use d with success in studying typical equations . To simplify formulae we use the conventional assumption that the voi d sum is equal to zero, while the void product is equal to one, that i s k  k E y3 . — 0, Hya : = 1 .j-=n  9= n for any k, n E N, k C n, and any sequence y . Instead of lim xn = C we shall write xn = C + o(l) and i f n--r o o lim (xn'yn) = C we write x n = yn(C + o(1)) . n---> 0 0 Theorem 1 . Let a (i) : N --4 R, 4 0} -1 for every n E N, an d 0 0 (1)  E 14 ) < Do , j= 1 f or i = 0,1, . . . , r . Then f or any arbitrary constant C E R, C 0 ther e exists a solution x of (E 1 ) such tha t (2) x n =C+o(1) . Proof : Let us see that if u is a solution of (E 1 ) such that un = C + o (1 ) with C > 0, then the sequence u n = —u n , far all n E N is also a solutio n of (E 1 ) possessing the same type of asymptotic behavior with the limi t — CCoinsteadofC> 0 . We prove our theorem for the case C > 0 . Since C > 0 there exists a positive constant E such that C — E > 0 . Let us take C 1 ---C+E, 1— [C—E,C+E] , r (3) = C 1 E E la ( j i) 1, n E N . i =a 3= n From (1) it follows that there exists n 1 E N such that far all n ~ n 1 w e !laZ-r rv,, <E . DIFFERENCE EQUATI4NS, ASYMPTOTIC BEHAVIOR  5 Let l~ denote the Banach space of bounded sequences x = (h i )°__ 1 with the norm  = supih 2 l . Moreover, let T C l,, be any set such tha t oa  h t -- C fort =1,2, . . .,n 1 - 1 x = { h z}i=1 E Tif h t E I t fort~ n i where I t = [C - - a t , C + a t ] . It is easy to check that T is bounded, convex, and closed in l oc . Fir - thermore by (1) and (3) it foliows that diam I t = 2a t -4 0 with t -3 oa . So, f o r arbitrary E1 > 0 we can set up a finite E1 -net for the sét T . Henc e by Hausdorff's theorem T is compact . Define now some operator A b y .1x = y = {b} 1 , wher e { b n C , C - E E a~i) h j+ i i=0 j=n for n= 1,2, . . .,rt 1 - 1 forn~ n i r 00 Let us see, using (1), that A is well defined on the space l c ,„, . Furthermor e forxETweobtain  r oo  r o 0 Ibn - Cl C E E l a ~i} j 1 1 h .7+2 I C C1 ~~ 3  ~ i=0 j=ra  i=0 j= n hecause h j + i E I j + i c 1, for all j ~ n i , i E {O, . ,r} . Hence, by (3 ) C - an ç b n ÇC + a n , which means that bn E I n for n ~ rt 1 . That is, A maps the set T into T . We now prove that A is continuous on T : Take E l > oandS 1 =~ . Let x = {h} 1 and y = {9i} i be any two elements of the set T such that Il x - -- MI I C S1 . Then the absolute convergence of the serie s r 0 0 E E aja} hj+z , i=0 j =n i r o 0 E a ( i ) E j i-=o j=n i 6  J . POPENDA, E . SCHMEIDE L yields r 00  É O D —  E a(i ) 3 hj+i  C—  a (i) gj + ~ ~j z i=0 j = n i=0 j = n IlAx — AMI = su p n>n l < r 0 0 Ç sup i~ la(i) ~ Ih + i - -- g j +i 1 n ~ n 1 a i=0 j= n T 0 0 Ç sup E E 1 a III x —yII Ç n ~ 'n1 i =o i = n T O D Ç  S1 sup E~ ~ = e l . n7n1 i=0 j= n Therefore the operator A is continuous on T, and by Schauder fixe d point theorem we obtain that there exists in the set T a solution of th e equation x = Ax . Let z = {d} 1 denote such a solution . Since z E T, it can be written as follows r 00 >>adj+j, . . . i=0 j = n r z = { 0 0 i=0 j =n 1 (1) d j + i , . . . , Applying the operator q to (5) we obtai n r 0 0 (5)  d n = C , — E  a~ i} d j + i , forn ~ n 1 . i—0 j= n r q d n = E aWd j + ia n ~ n 1 • i = o This means that the sequence {d} 1 fulfills equation (E 1 ) but fo r n ~ n i only . The equation (E 1 ) is a readily transformed t o r (fi) x n = —(1 + c4i0)} -1 [(a 1) — 1 }x n + 1 + E aWx n + i  n E N . i=2 DIFFERENCE EQUATIONS, ASYMPTOTIC BEHAVIOR  7 Substituting in (6) n =n 1 — 1, x n = d n for n ~ n 1 we obtain xnz _ 1 . Proceding in this way we find xn1 _2, . . . , x 1 one after the other . Consequently we get the sequence which fulfills (E 1 ) for all n E N . Moreove r this sequence coincides with z for n ~ n 1 ] hence it has the asymptoti c behaviour (2) because d n E I n and diam I n —> 4 as n ~ Qa . ■ A similar method and property for the difference equatio n q 2 xn + anF(xn) = = o can be found in [1] . Now we use the previous theorem to study solutions of the m-th orde r difference equatio n (E 2 )  q m xnanxn, nEN , m ~ 2 . Theorem 2 . Let a : N —> R be such that (—1) m an 1 for alln E N and 0o k— 1 E JJ Ii+' —1)m+ l Cln+j I < oo for k = 2, . . . ] m ] n=1j1 then for arbitrary constant C 0 there exists a solution x of (E 2 ) whic h possesses the asymptotic behaviou r xn = m —n H [1 + (—1)m+ll1~] } (C + 0(1)), n E N . j= 1 Proa : Similarly as in the proof of Theorem 1 we shall concentrate o n the case C> O . By formula m ' ámYk = E(—1)i m y k +m -i ] i= 0 we can transform equation (E 2 ) to the following for m m2 (_1) m_l mxn+ i +(_1) m xm_anxn = —  (i)iÇ m  n E N . i= 0 Henc e (7) rn - 2 (_1) m_l mx m + i_ [(_1) m+l +a n lx m = E (_1) 1 m )x+jy Ti E N . 2 i= 0 n-1 8  J . POPENDA, E . SCHMEIDE L Multiplying (7) by (_1) m + l m n and setting z n =m n x n we obtain from (7 ) (S) zn+l – [1 + (_1)m+la]zn = m-2 +i ( m )m_ m + z z m+m_i , = .  (1 ) i =o nE N . Now multiplication b y and the substitutio n yield H[ 1 + (—I) m+I a 3 1 j=1 n— 1 v n = z n H [1 + j = l (9) Ov n = m-2  n+m-i - 1 E (–1)' +i ( m )m -'+1  11 + (_ 1) m+l a i I vn+m_z j n E N . i =o  j =n+ l It is evident that (9) is of the form (E 1 ) and all assumptions of Theore m 1 hold . Therefore for arbitrary constant C o there exists a solution v of (9) such that 2Jn = C -{- 0( 1 ) , and as n— 1 v n = xmm n H [1 + ~- 1}7n+laj]~ I j= l we have in conclusio n n1 x n m n n [1 + (—1) m+l aJi -1 = C + o(l) . ■ j= 1 As an example consider the equatio n q 2 xn = (1—ñ 2 ) x~ , , nEN . By Theorem 2 it follows that for arbitrary C o there exists a solutio n of this equation such tha t x n, = 2 -n (n  1)j2 (C + o ( l )), n E N . DIFFERENCE EQUATI O NS, ASYMPTOTIC BEHAVIOR  9 Reference s 1 . A . DROZDOWICZ AND J . POPENDA, Asymptotic behaviour of solutions of the second order difference equation, Proc . Amer . Math . Soc . 99(1) (1987), 135-140 . Institute of Mathematic s Technical Universit y ul . Piotrowo 3 a 60-965 Pozna n POLAN D Primera versid rebuda el 17 de Desembre de 1991 , darrera versid rebuda el 15 de Desembre de 1993