Tatra Mt. Math. Publ. 43 2009, 5 6 DOI: 0.2478/v027-009-0024-7 t Matheatical Publications EXISTENCE OF ASYMPTOTICALLY PERIODIC SOLUTIONS OF SCALAR VOLTERRA DIFFERENCE EQUATIONS Josef Diblík Miroslava Růžičková Ewa Scheidel ABSTRACT. There is used a version of Schauder s fixed point theore to prove the existence of asyptotically periodic solutions of a scalar Volterra difference equation. Along with the existence of asyptotically periodic solutions, sufficient conditions for the nonexistence of such solutions are derived. Results are illustrated on exaples.. Introduction We consider a Volterra difference equation xn =anbnxn Kn, ixi, where n N := {0,, 2,...}, a,b,x: N R, K: N N R R denotes the set of all real nubers. By a solution of equation we ean a sequence x: N R whose ters satisfy for every n N. Throughout this paper we will assue that sequences a K are not identically equal to zero. We will also adopt the custoary notations k i=ks Oi =0, k i=ks Oi =, 2000 M a t h e a t i c s Subject Classification: 39A, 39A0. K e y w o r d s: asyptotically periodic solution, asyptotic behavior of solutions, Volterra difference equation. The first the second authors have been supported by the project APVV-0700-07 of Slovak Research Developent Agency by Grant No. /0090/09 of the Grant Agency of Slovak Republic VEGA. 5
JOSEF DIBLÍK MIROSLAVA RŮŽIČKOVÁ EWASCHMEIDEL where k is an integer, s is a positive integer O denotes the function considered independently of whether it is defined for the arguents indicated or not. Definition. Let ω be a positive integer. The sequence y : N R is called ω- -periodic if ynω = yn for all n N. The sequence y is called asyptotically ω-periodic if there exist two sequences u, v : N R such that u is ω-periodic, li vn =0yn =unvn for all n N. The background for discrete Volterra equations can be found in the well known onograph [] by A g a r w a l, as well as in E l a y d i [3] K o c i ć Ladas [7]. Unifor asyptotic stability in linear Volterra difference equations was studied by E l a y d i M u r a k a i in [4]. Periodic asyptotically periodic solutions of linear difference equations were investigated, e.g., by A g a r w a l P o p e n d a in [2], by P o p e n d a S c h e i d e l in [9, 0]. In [5] [6], F u r u o c h i considered the behavior of solutions of the following classes of Volterra difference equations xn =an xn =pn D n, i, xi P n, i, xi, i= their inter-relations. Boundedness, attractivity, convergence of solutions were investigated. 2. Asyptotically periodic solutions In this section, sufficient conditions for the existence of an asyptotically ω-periodic solution of equation are given. The following version of Schauder s fixed point theore, which can be found in [8], will be used to prove the ain result of this paper. Lea. Let Ω be a Banach space S its nonepty, closed convex subset, let T be a continuous apping such that T S is contained in S, the closure T S is copact. Then T has a fixed point in S. 52
EXISTENCE OF ASYMPTOTICALLY PERIODIC SOLUTIONS Let ω be a positive integer b: N R \{0} be ω-periodic. Then we define an ω-periodic function β : N R as n if n, βn = bj j=0 2 βω if n =0. Further we define := in { β, β2,..., βω } M := ax { β, β2,..., βω }. Theore Main result. Let ω be a positive integer b: N R \{0} be ω-periodic. Assue that ω bi =, 3 ai < 4 j=0 j Kj, i < M. 5 Then, for any nonzero constant c, there exists an asyptotically ω-periodic solution x of such that xn =unvn, n N 6 with un :=c n li vn =0 7 where n is the reainder of dividing n by ω. Proof. Wenotethatn = n ω [ ] n ω where the function [ ]isthegreatest integer function. Fro the ω-periodicity of sequence b, the definition of β by 2, the property 3 we have βn { β, β2,...,βω } for any n N. Thus, βn M 8 53
JOSEF DIBLÍK MIROSLAVA RŮŽIČKOVÁ EWASCHMEIDEL for any n N. Letc>0. We set α0 := αn :=M M ai cm M j=0 j=0 j Kj, i j Kj, i c α0 M ai i=n for n. It is easy to see that j=n j Kj, i, We show, oreover, that li αn =0. 9 αn α0 0 for any n N. Let us first reark that c α0 M α0 = M ai j Kj, i. j=0 Then, due to the convergence of series 4, 5, the inequality c α0 M α0 = M ai c α0 M M ai i=n j Kj, i j=0 j Kj, i = αn j=n obviously holds for every n N 0 is proved. Let B be the Banach space of all real bounded sequences z : N R equipped with the usual supreu nor. We define a subset S B as S := { zn B : c α0 zn c α0, n N }. It is not difficult to prove that S is a nonepty, bounded, convex, closed subset of B. Let us define a apping T : S B as follows 54 Tzn =c aiβi i=n j j=n βj Kj, izi βi
EXISTENCE OF ASYMPTOTICALLY PERIODIC SOLUTIONS for any n N. We will prove that the apping T has a fixed point in B. We first show that T S S. Indeed, if z S, then zn c α0 for n N,, by 8, we have c α0 M j Tzn c M ai Kj, i i=n j=n = αn α0. 2 Next we prove that T is continuous. Let z p be a sequence in S such that z p z as p. Because S is closed, z S. Now, by 5, 8, we get j Tz p n Tzn = βj Kj, i z p n zn βi j=n M M sup z p i zi, n N. i 0 Therefore Tz p n Tzn sup z p i zi, n N i 0 li Tz p Tz =0. p This eans that T is continuous. To prove that T S is copact, we take ε>0. Then,fro9,weconclude that [ there exists n] ε N such that αn <ε, for n n ε.wecoverthesegent c α0,c α0 with a finite nuber kε of intervals each having a length of ε. Let the points c kε be the centres of the ε-length intervals. We conclude that, for an arbitrarily sall ε>0, we can collect a finite set of intervals with centres at c kε with radii ε/2 whichcoverst S. Hence T S iscopact. By Schauder s fixed point theore see Lea, there exists a z S such that zn =Tzn for n N. Thus zn =c aiβi i=n j j=n βj Kj, izi for any n N. 3 βi 55
JOSEF DIBLÍK MIROSLAVA RŮŽIČKOVÁ EWASCHMEIDEL Due to 9 2, for fixed point z S of T,wehave li zn c = li Tzn c li αn =0 or, equivalently, li zn =c. 4 Finally, we will show that there exists a connection of the fixed point z S with the existence of asyptotically ω-periodic solution of. Considering 3 for zn zn, we get βn Δzn =anβn Kn, izi, n N. βi Hence, by 2 taking into account that β0 = βω = in view of 3, we have βn zn zn =anβn Kn, 0z0 β0 Putting βn Kn, izi βi i= n n = an zn = i= k=i Kn, 0z0 5 n Kn, izi, n N. n in 5, we get equation since xn n yields 56 xn n = an i= n xn, n N 6 n k=i n Kn, 0 Kn, i xn =anbnxnkn, 0x0 i xi, x0 n N Kn, ixi, n N. i=
EXISTENCE OF ASYMPTOTICALLY PERIODIC SOLUTIONS Consequently, x defined by 6 is a solution of. Fro 4 6, we obtain n xn =zn =c o for n where o is the Lau order sybol. Hence n n xn =c o, n. Fro 3 we get n = n. The proof is coplete since the sequence { n } is ω-periodic due to properties of Lau order sybols we have n o = o, n. If c<0, the proof, which we oit, can be carried out in a anner siilar to the one used above if x is changed to x. Reark. Tracing the proof of Theore we see that it reains valid even in the case of c = 0. Then there exists an asyptotically ω -periodic solution x of as well. The forula 6 reduces to xn =vn =o, n N. Fro the point of view of Definition, we can consider this case as follows. We set as a singular case u 0 with an arbitrary possibly other than ω period with v = o for n. In the following exaple, a sequence b is -periodic. Then it is 2-periodic, too. By virtue of Theore, there exists a 2-periodic solution of the equation in question. Exaple. Put an = n 53 3 2n3 48 2 2n 2, 3n4 bn Kn, i = 2i 4 n2 in. We consider the sequence b as a 2-periodic sequence put ω =2. Obviously, = M = j j j=0 Kj, i = j=0 2 i 4 j2 = j=0 2 j 4 j2 = 6 <. 57
JOSEF DIBLÍK MIROSLAVA RŮŽIČKOVÁ EWASCHMEIDEL Then all the assuptions of Theore are satisfied by 6, 7 there exists an asyptotically 2-periodic solution xn =unvn, n N of the equation where un =c n n =c = c n, Indeed, a sequence xn =c n 4 n with c = is such a solution. li vn =0. 3. Nonexistence of asyptotically periodic solutions Finally, we present sufficient conditions for the nonexistence of asyptotically periodic solution of satisfying soe auxiliary conditions. Let xn = un vn be an asyptotically periodic solution of such that the sequence u is ω-periodic li vn =0. Theore 2. If sequences a: N R b: N R are bounded there exists a positive integer ω such that Kn, i =Kn ω, i ω 7 for all n, i N, then the equation does not have any asyptotically ω-periodic solution xn =un vn such that ω Kω,iui 0 8 vi <. 9 P r o o f. Suppose, on the contrary, that assuptions of Theore 2 are satisfied there exists an asyptotically ω-periodic solution x of equation which satisfies conditions 8 9. Without loss of generality we ay assue that 58 ω Kω,iui > 0. 20
for all n, i N. Then K n, i vi K vi EXISTENCE OF ASYMPTOTICALLY PERIODIC SOLUTIONS Fro 7 we have [ ] [ ] n i Kn, i =K n ω,i ω = K n, i. ω ω Because xn =unvn, fro 7 the ω-periodicity of the sequence u, we have Kn, ixi = K n, i u i By 20 li sup We reark that the su K n, i vi [ ] ω n = Kω,iui ω [ n ω n K n,i ui K n, i vi. ] ω n Kω,iui =. K n,i ui is bounded for n there exists a positive constant K such that K n, i K, by 9, the series K n, i vi is absolutely convergent. Thus li sup Kn, ixi =. 59
Rewriting, we get JOSEF DIBLÍK MIROSLAVA RŮŽIČKOVÁ EWASCHMEIDEL xn an bnxn = Kn, ixi, where the left-h side of the above equation is bounded while the right-h side is unbounded. This contradiction copletes the proof. Reark 2. We will ephasize the necessity of 8 in Theore 2. If ω Kω,iui =0 then can have an asyptotically ω-periodic solution. Let, e.g., Kj, i = i /2. Then, taking sequences a b in in a proper anner, the equation will have an asyptotically 4-periodic solution xn = un vn with 4-periodic function un :=0,, 0, 2,... In this case ω ω i Kω,iui = ui = 00 00 2=0 2 8 does not hold. Then [ ] ω n li sup Kω,iui =0 ω we do not get the final contradiction in the proof of Theore 2. REFERENCES [] AGARWAL, R. P.: Difference Equations Inequalities. Theory, Methods, Applications 2nd ed., in: Pure Appl. Math., Vol. 228, Marcel Dekker, Inc., New York, 2000. [2] AGARWAL, R. P. POPENDA, J.: Periodic solutions of first order linear difference equations, Math. Coput. Modelling 22 995, 9. [3] ELAYDI, S. N.: An Introduction to Difference Equations 3rd ed., Undergrad. Texts Math., Springer-Verlag, New York, 2005. [4] ELAYDI, S. N. MURAKAMI, S.: Unifor asyptotic stability in linear Volterra difference equations, J. Difference Equ. Appl. 3 998, 203 28. [5] FURUMOCHI, T.: Periodic solutions of Volterra difference equations attractivity, Nonlinear Anal. 47 200, 403 4024. [6] FURUMOCHI, T.: Asyptotically periodic solutions of Volterra difference equations, Vietna J. Math. 30 2002, 537 550. [7] KOCIĆ, V. L. LADAS, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, in: Math. Appl., Vol. 256., Kluwer Acad. Publ., Dordrecht, 993. [8] MUSIELAK, J.: Wstep do Analizy Funkcjonalnej, PWN, Warszawa, 976. In Polish 60
EXISTENCE OF ASYMPTOTICALLY PERIODIC SOLUTIONS [9] POPENDA, J. SCHMEIDEL, E.: On the asyptotically periodic solution of soe linear difference equations, Arch. Math. Brno 35 999, 3 9. [0] POPENDA, J. SCHMEIDEL, E.: Asyptotically periodic solution of soe linear difference equations, Facta Univ. Ser. Math. Infor. 4 999, 3 40. Received Deceber 8, 2008 Josef Diblík Miroslava Růžičková Departent of Matheatics Faculty of Science University of Žilina Univerzitná 825/ SK 00-26 Žilina SLOVAKIA E-ail: josef.diblik@fpv.uniza.sk iroslava.ruzickova@fpv.uniza.sk Ewa Scheidel Institut of Matheatics Faculty of Electrical Engineering Poznań University of Technology Piotrowo 3a PL 60-965 Poznań POLAND E-ail: ewa.scheidel@put.poznan.pl 6