The periodic solutions of the second order nonlinear difference equation

Publicacions Matemátiques, Vol 32 (1988), 49-56 . Abstract THE PERIODIC SOLUTIONS OF THE SECOND ORDER NONLINEAR DIFFERENCE EQUATION RYSZARD MUSIELAK AND JERZY POPENDA Periodic and asymptotically periodic solutions of the nonlinear equation O Z X~ +- a f (x n ) = 0, nEN, are studied . In several recent papers ([2],[3]) the periodicity of solutions of linear difference equations have been investigated . In this paper we examine the periodic solutions of the nonlinear equation (E)  ~ 2 xn + a nf( x n) = 0, n E N, where N = {0, 1, 2, . . .}, R is the set of real numbers, f : R --> R and a, x N ---> R are sequences of real numbers . Throughout the paper we use the following notations . By U, - t we denote the set of integers {0,1,2, . . .,t} . For the function y : N --> R the forward difference operator ~ k is defined liUn = yn+l - yn, A k Yn =  lyn) for lC > 1 . Definition 1 .  The function y will be called t-periodic if yn+t = yn for a,ll n E N . (Furihermore we suppose that no t l exists, 0 < t l < t such that Yn+ti = yn for all n E N and that t > 1) . Definition 2 . The function y will be called asymptotically t-periodic (t > 1) if y=u+v, where u is a t-periodic function and lim ny< , vn = 0 . Definition 3 .  We soy that the equation (E) has a p t -constant if there exists a constant pE R, such that the equation ( -E l)  A 2 xn + anf(xn) = p 50  R . MUSIELAK AND J . POPENDA has a t-periodic solution . We say that the equation (E) possesses a pt ° -constant if there exists a constant p E F8 such that (E l ) has an asymptotically t-periodic solution . Definition 4 . The equation (E) is said to have a p tfunction (pt ° function) if there exists a t-periodic function p : IN -> Ft such that the equation (E2)  L,2 x n + an f (xn ) = pn has a t-periodic (asymptotically t-periodic solution . Remark 1 . Note that if (E) has a p t -constant (function) then (E) has a pt°-constant (function) and if (E) has not a p'-constant (function) then it has no p t -constant (function) . Theorem 1 . Let f . F8 -> 18 be continuous on F8 and lim n -,, . a n = 0 . Then the equation (E) has not a pf° -constant for any t > 1 . Proof .. We show the proof for simplicity in the case t = 2 . Similar reasoning can be made for t > 2 . Suppose that there exists a pt°-constant q such that the equation (E3)  Q2 xn + an f (-'In) = has one asymptotically 2-periodic solution x . Let x2n -> C l , x2n+1 --) C2 as n -> oo, Cl 5~ C2 . Hence As result of the assumption we obtain L,2 x2 n-i 2C 1 - 2C2 . Q2 x2n+1 -> 2C2 - 2C1 . 2C 1 - 2C2 = q 2C 2 -2C 1 = q . The above system has a solution if and only if q = 0, but in this case we obtain C l = C 2 , which is a contradiction . Theorem 2 . Let f :,A 0 on F8 . If the equation (E) possesses a p t -constant then a is a t-periodic function . Proof . Let x be a t-periodic solution of (E3) . Then A 2x is t-periodic . By virtue of the assumption f :,í : 0 and we get 2 x n - q __ f( x n) The case PERIODIC SOLUTIONS OFDIFFERENCE EQUATIONS  51 The left hand side of the above equality is a t-periodic function so the right hand side must also be t-periodic . Remark 2 . We can prove analogously that if f :,A 0on 18, then tperiodicity of a is the necessary condition for the existence of a p t - function q for the equation (E) . However in this case we do not require for t to be the basic period . Eventually a can be a constant function . It is easy to see that if f (C 1 ) = 0 then the equation (E) has p 1 -constant q = 0 . Then a t-periodic solution takes the form x - Cl . By iR we denote the identy function on 18 . Theorem 3 . Let a : N-~ I8, let f be a continuous function on I8, f 1 0 such that the functions (1)  aR+anf :R - >F8 are surjections for every nE N . If 00 ( 2 ) j : .7iajj < o0 j=1 then the equation (E) has a pt° function for arbitrary t Proof .. Choose t >_ 1 . By assumption there exist constants C r , r = 1, 2, - * , t, C ; :7É Ci,i :,~ j, such that f(Cr) 0 0 . f(C r ) > 0,  r = 1, .2, " " ", t will be considered .  The proof for the other cases f (Ci) > 0, f (Cj) < 0 is similar . By virtue of the continuity of the function f there exist intervals Ir = [Cr+ 1 - b, Cr+ 1 + 6], r = 0, 1, - . . , t - 1 such that (5)  f (u) >0foruEI r , r=0,1,---,t-1 . From (2) it follows that 00 ( 6 )  n lim  1 : j l aj 1 = 0 . 2 =n 52 Let us denote (7) and In the space l°° of bounded sequences with the norm lixil = supo>o ix ; l we define the set T in the following way : x={x ;} ;= o ET if So xr R . MUSIELAK AND J . POPENDA D= maxo<r<t-1(max-EI,J (u» 00 ni =min{nEN, :n=tk+t-1, DI : jjajl <8} . i=n = xt+r = x2t+r = "' = xnl-t+r+l = Cr+le xtk+r E Itk+r " _ 00  00 _ [Cr+1 -D  jl a jl~ C r+1 +D  jlaj11, j=tk+r  j=tk+r r=0,1, -,t-1 :kEN ; k> t(nl+l - t) . The set T is closed, convex and bounded . Furthermore, by diam Swé mean diam S = sup{lix - Y11 ; x ES ; y E S} . (S)  diam Itk+r ---> 0  as  k -~ oo . It is easy to find a finite c-net for every e > 0 . Therefore by Hausdorff's Theorem the set T is compact . Let us define an operator A for x E T as follows : Ax = y = {yso where yr = yt+r ="  = yn l + r+l-t = Cr+1 ; r = Do lo "' 5t - le 00 for kEN, k> ,(n1+1-t), r=0,1, . .-,t-1 . Let us observe that ytk+l=Cr+l - 1 : (j+1-tk-r)ajf(xj) j=tk+r Itk+r C I r , r = 0, 1, , . . , t - 1, k > 1 (n 1 + 1 - t~ . Hence ( 9 ) Hence we have (10) From (10) it follows that By (9) the series (13) PERIODIC SOLUTIONS OFDIFFERENCE EQUATIONS  53 I E (j+1-tk-r)ajf(xj)I < i=tk+r 00 <_  1 : jiajilf(xi)I <_ D E j1ajI . i=tk+r  j=tk+r Therefore ytk+r E Itk+r, r = 0,1,--- ,t - 1, k E N, k > (n l + 1 - t)1t and this means that A : T --> T . Let us take an arbitrary sequence {x m }°° =l of elements of T convergent to some x ° E T Le . x  - X  I I ---, 0 . Sllpn>O Ixñ - xn I -> 0 as m ) 00 . Let e l be an arbitrarily taken positive real number . By the uniform continuity of f on the sets I r we have Iul - U21< 6 implies I f (u l ) - f (u2) ( < e l . Supn>OIxn - X OI < 6 for m > M(6) . Let ym = Axm , m E N ; then ( 12 )  II Ax m - Axo II = = 00  co SuPn> n, I j :(9 + 1 - n)a j f (x~ `) - j :(j + 1 - n)aj f (x°) I . i=n  i=n j :(j + 1 - n)aj f (x~'), m E N i=n are absolutely convergent . Hence, by (11) and (12) 00 II Axn` - Axc II < el  E j I aj I i=n, so that the operator A is continuous on T . By Shauder's Theorem there exists z E T such that z =Az . By definition of A this element z = {z ;}s° 0 satifies zr = zt+r = " " " = zn,+r+l-t = cr+l 54  R . MUSIELAK AND J . POPENDA Applying the operator 0 to z we obtain and consequently where Denoting (14) we obtain the equation (16) ztk+r = Cr+1 -  1 :  ( .i + 1 - tk - r)aj f (zj ) j=tk+r k> 1 (n1+1-t), r=0,1, . . . ~t-1 . Aztk+r = ztk+r+l - ztk+r = =Cr+2 - Cr+1 - E (j+tk-r)ajf(zj)+ 1 : (j+1-tk-r)ajf(zj)= j=tk+r+1  j=tk+r 00 = Cr+2 - Cr+1 +  1 :  aj f (zj ), j=tk+r 0 2 ztk+r = Oztktr+l - Oztk+r = = Cr+3 - 2Cr+2 + Cr+ 1 +  E  aj f (zj) -  1 :  aj f (zj) _ j=tk+r+1 j=tk+r = Cr+3 - 2Cr+2 + Cr+1 - atk+rf (ztk+r), r=0,1, . . .~t-1, k> 1(n1+1-t) Ct+1 = Cl, Ct+2 = C2 . qtk+r = Cr+3 - 2Cr+2 + Cr+l, r = 0,1,  ' - , t - 1, (15)  ~2 xn + a n f (xn) T qn which has an asymptotically t-periodic solution defined for n> n i . This follows from (8) and ztk+r E ftk+r, Le . ztk+r --' Cr+1 as k ---) oo . It suffices to show that there exist a solution of (15) which coincides with (13) for n > ni . For this we observe that the equation (15) can be rewritten in equivalent form In + an f (fin) = qn - -7n+ 2 + 2xn+1 PERIODIC SOLUTIONS OF DIFFERENCE EQUATIONS  55 Taking n = nl, x,,+1 = znl+l, xn+2 = znl+2 we find xn,, which by the assumptions exists (probably more than one) . Repeating this reasoning we find xi for i = 0, 1, - - - , n l - 1 . This function x is of course a solution of (15) which coincides with z for n>n l and therefore has the desired asymptotic behaviour . Remark 3 . If the functions iR + anf are one-to-one mappings of R onto R then the solution obtained in the Theorem 3 is unique . The case t = 1, Le . the solutions having the asymptotic property lim n - oo x n = C, was considered in the paper [11 . Let us observe that by Theorem 3 if we want to have some solutions which have a given asymptotically t-periodic solution, then it suffices to add to equation (E) the periodic perturbation q which can be easily found by (14) . Example . As an example we consider the difference equation of the form It is evident by d'Alembert criterion that the series is absolutely convergent . Furthermore the functions are surjections from R onto R for all n . Therefore the assumptions of the Theorem 3 hold . We show that this equation has a p'-function and find a 2-periodic solution of the form Applying the proof of the Theorem 3 we see that the p'-function q takes the form Considering the equation we can observe that this equation has the solution which is of the desired form . 1 n+l 0 2 x n + 4(2( +) (-1)n l xn = 0, n = 1, 2, . . . ( i 4[2+ (1)i] x +  (-1)n+l  x 4[2n + (-1)n]~ xn = ( - 1) n + yn qn = 4(-1)n . 1 n+l ~n,2 xn + 4[2 ( +)(-1)n, xn = 4(-1)n _ 1 x n - ( -1 ) n+ 2n 56  R . MUSIELAK AND J . POPENDA References 1 .  A . DROZDOWICZ, J . POPENDA ., Asymptotic Bahaviour of the Solutions of the Second Order Difference Equation, Proc . Amer . Math . Soc . (in press) . 2 .  R . MUSIELAK, On the Periodical Solutions of the System Linear Difference Equations, Fase . Math 15 (1985, 141-150 . 3 .  R . MUSIELAK, The Conditions for Existence of Periodical Solutions of the Second Order Difference Equation, Fasc .Math 15 (1985, 127-139 . R . Musielak : Institute of Mechanical Engineering Technical University Pozna$ ul . Piotrowo 3, 60-965 Poznaií, POLAND . J . Popenda : Institute of Mathematics Technical University Pomaií ul . Piotrowo 3, 60-965 Poznaaí, POLAND . Rebut el 10 ¿'Octubre ¿e 1986