Abstract THE PERIODIC SOLUTIONS OF THE SECOND ORDER NONLINEAR DIFFERENCE EQUATION

THE PERIODIC SOLUTIONS OF THE SECOND ORDER NONLINEAR DIFFERENCE EQUATION RYSZARD MUSIELAK AND JERZY POPENDA Periodic and asymptotically periodic solutions of the nonlinear equation OZX~ +a f(xn) = 0, n E N, 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) ~2xn + anf(xn) = 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, AkYn = 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 tl 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 limny< , vn = 0 . Definition 3. We soy that the equation (E) has a pt -constant if there exists a constant p E R, such that the equation ( -El) A2xn + 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 EF8 such that (El ) has an asymptotically t-periodic solution. Definition 4. The equation (E) is said to have a ptfunction (pt function) if there exists a t-periodic function p : IN -> Ft such that the equation (E2) L,2 xn + an f (xn ) = pn has a t-periodic (asymptotically t-periodic solution. Remark 1. Note that if (E) has a pt-constant (function) then (E) has a pt°-constant (function) and if (E) has not a p'-constant (function) then it has no pt-constant (function) . Theorem 1. Let f. F8 -> 18 be continuous on F8 and limn-,,. an = 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 -> Cl , x2n+1 --) C2 as n -> oo, Cl 5~ C2 . Hence As result of the assumption we obtain L,2 x2 n -i 2C1 2C2 . Q2 x2n+1 -> 2C2 2C1 . 2C1 2C2 = q 2C2 2C 1 = q. The above system has a solution if and only if q = 0, but in this case we obtain Cl = C2 , which is a contradiction . Theorem 2 . Let f :,A 0 on F8 . If the equation (E) possesses a pt -constant then a is a t-periodic function . Proof. Let x be a t-periodic solution of (E3) . Then A2x is t-periodic . By virtue of the assumption f :,í: 0 and we get 2 xn q __ f(xn) The case PERIODIC SOLUTIONS OF DIFFERENCE 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 0 on 18, then tperiodicity of a is the necessary condition for the existence of a ptfunction 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 n E 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 Cr , r = 1, 2, * , t, C; :7É Ci,i :,~ j, such that f(Cr) 0 0. f(Cr ) > 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) >0foruEIr , r=0,1,---,t-1 . From (2) it follows that 00 (6 ) nlim 1: j laj 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+l e xtk+r E Itk+r "_ 00 00 _ [Cr+1 -D jl ajl~ Cr+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 S wé mean diam S = sup{lix Y11 ; x E S; 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 Ir , r = 0, 1, , . . , t 1, k > 1 (n1 + 1 t~ .

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 x n + anf(xn) = 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 .
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 tl 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 where u is a t-periodic function and lim ny< , vn = 0 .Definition 3. We soy that the equation (E) has a pt -constant if there exists a constant p E R, such that the equation ( -El) A2xn + anf(xn) = p 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 (El ) has an asymptotically t-periodic solution.Remark 1.Note that if (E) has a pt-constant (function) then (E) has a pt°-constant (function) and if (E) has not a p'-constant (function) then it has no pt-constant (function) .Theorem 1.Let f.F8 -> 18 be continuous on F8 and lim n-,,.an = 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 As result of the assumption we obtain L, The above system has a solution if and only if q = 0, but in this case we obtain Cl = C2 , which is a contradiction .
Theorem 2 .Let f :,A 0 on F8 .If the equation (E) possesses a pt -constant then a is a t-periodic function.
Proof.Let x be a t-periodic solution of (E3) .Then A2 x is t-periodic.By virtue of the assumption f :,í: 0 and we get The case 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 0 on 18, then t-periodicity of a is the necessary condition for the existence of a pt-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 p1 -constant q = 0. Then a t-periodic solution takes the form x -Cl .
By iR we denote the identy function on 18 .
From (2) it follows that  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 {xm }°°= l of elements of T convergent to some x°E T Le.
Sllpn>O Ixñ -xn I -> 0 as m ) 00 .Let el be an arbitrarily taken positive real number.By the uniform continuity of f on the sets Ir we have Iul -U21 < 6 implies I f (u l ) -f (u2) ( < el .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 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, ---, nl -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 xn = 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