Computers and Mathematics with Applications. Sharp algebraic periodicity conditions for linear higher order

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1 Computer and Mathematic with Application 64 (2012) Content lit available at SciVere ScienceDirect Computer and Mathematic with Application journal homepage: wwweleviercom/locate/camwa Sharp algebraic periodicity condition for linear higher order difference equation Itván Győri, Lázló Horváth Department of Mathematic, Univerity of Pannonia, 8200 Vezprém, Egyetem u 10, Hungary a r t i c l e i n f o a b t r a c t Keyword: Periodic olution Higher order difference equation Companion matrix In thi paper we give eaily verifiable, but harp (in mot cae neceary and ufficient) algebraic condition for the olution of ytem of higher order linear difference equation to be periodic The main tool in our invetigation i a tranformation, recently introduced by the author, which formulate a given higher order recurion a a firt order difference equation in the phae pace The periodicity condition are formulated in term of the ocalled companion matrice and the coefficient of the given higher order equation, a well 2012 Elevier Ltd All right reerved 1 Introduction In thi paper we derive new neceary and ufficient, and ufficient algebraic condition on the periodicity of the olution of the d-dimenional ytem of the th order difference equation x(n) = A i (n)x(n i), n 0, where (C 1 ) 1 i a given integer, and A i (n) R d d for every 1 i and n 0 It i clear that the olution of (1) are uniquely determined by their initial value x(n) = ϕ(n), n 1, where ϕ(n) R d The unique olution of (1) and (2) i denoted by x(ϕ) = (x(ϕ)(n)) n, where the block vector ϕ := (ϕ( ),, ϕ( 1)) T V Here V mean the d-dimenional real vector pace of block vector with entrie in R d We believe that our reult about Eq (1) are intereting in their own right Further, we believe that thee reult offer prototype toward the development of the theory of the periodic behavior of the olution of nonlinear higher order difference equation Thi equation and it pecial cae are tudied in many textbook on difference equation uch a [1 6], and o on On p 25 in the book [3], Grove and Lada put the following two quetion: What i it that make every olution of a difference equation periodic with the ame period? I there any eaily verifiable tet that we can apply to determine whether or not thi i true? Motivated by the above quetion, and the paper [7,8], we worked out an eaily verifiable algebraic tet that we can apply to determine the p-periodic olution of a linear higher order difference equation In our reult we obtain precie analyi of the periodicity of the olution not only for the calar but for the vector cae Note that for thi latter cae, to the bet knowledge of the author, there are no imilar reult in the literature (1) (2) Correponding author addree: gyori@almouni-pannonhu (I Győri), lhorvath@almouni-pannonhu, lhorvath@almoveinhu (L Horváth) /$ ee front matter 2012 Elevier Ltd All right reerved doi:101016/jcamwa

2 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) A an illutration, we formulate two typical application of our main reult Conider the pecial cae of (1) x(n) = A i x(n i), n 0, (3) where A i R d d for every 1 i For the integer 1 p, V p denote the et of all initial vector ϕ = (ϕ( ),, ϕ( 1)) T V uch that ϕ(i) = ϕ(j) if i j( mod p)(i, j =,, 1) Of coure V = V If a R, then [a] denote the greatet integer that doe not exceed a Theorem 11 Suppoe 1 p i an integer, let u :=, and let v := up(0 v p 1) Then for every ϕ V p the olution of (3) and (2) i p-periodic if and only if one of the following condition hold: (a 1 ) v = 0 and u 1 A jp+i = O, 1 i < p; (a 2 ) 0 < v < p 1, moreover u A jp+i = O, 1 i v; and u 1 A jp+p = I (a 3 ) v = p 1 and u A jp+i = O, 1 i < p; p u 1 A jp+p = I (4) u 1 A jp+i = O, v + 1 i < p, u 1 A jp+p = I Remark 11 If p =, then v = 0 and (4) can be written in the form A i = O, 1 i 1, A = I Theorem 12 Suppoe p > i an integer Then for every ϕ V the olution of (3) and (2) i p-periodic if and only if the rank of the matrix with entrie B p I B p 1 B 2 B 1 B 1 B p I B 3 B 2 B p 1 B p 2 B 1 B p I B i := A i, 1 i and B i := O, + 1 i p i equal to d(p ) Theorem 11 and 12 are conequence of our Theorem For the definition of the periodic olution ee Section 2 below It i known that the ytem (1) can be reformulated to a d-dimenional ytem of firt order difference equation in an appropriate equence pace (ee eg [2]) The matrice of the firt order d-dimenional ytem are called companion matrice of Eq (1) and the ytem itelf i called a companion ytem of (1) It i clear that the companion matrice are block matrice defined by all A i (n) in Eq (1) Recently, Győri and Horváth introduced a new tranformation which i extremely ueful in analyzing the ummability of the olution of higher order difference equation (ee eg [9,10]) In thi paper we how that our tranformation i alo powerful in tudying the periodicity of the olution of the Eq (1) Our paper i eentially ubdivided into ix part Section 2 i fundamental for our work and contain baic reult on our tranformation of the Eq (1) into a firt order ytem with tractable companion matrice (5)

3 2264 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) Section 3 give neceary and ufficient, and ufficient periodicity condition via the companion matrice for both time dependent and independent cae The main reult are tated in Section 4 It i an amazing fact that the algebraic periodicity condition in the main theorem are urpriingly imple In Section 5 ome illutrative example are given to how the effectivene of our theory for higher order equation Some preliminary reult and the proof of the main reult can be found in Section 6 2 Definition and a phae pace tranformation Definition 21 (a) The zero matrix and the identity matrix in R d d are denoted by O and I, repectively (b) O and I mean the zero matrix and the identity matrix in the real vector pace of p p(p 1) block matrice with entrie in R d d, repectively Let (x(n)) n be a given equence in R d Then for any fixed n 0 we introduce an d-dimenional tate vector x n = (x n ( ),, x n ( 1)) T V, where the entrie x n (i) are defined by x n (i) := x(n + i)( i 1) Baed on the tate vector notation Győri and Horváth [9] introduced a new tranformation to rewrite a higher order ytem of difference equation into an d-dimenional ytem of firt order difference equation In the tate vector notation, we trancribe (1) a x k = C (k) x (k 1), k 1, with initial condition x 0 = ϕ, (6) (7) where x k = (x k ( ),, x k ( 1)) T = (x(k ),, x(k 1)) T V The block matrix C (k) (, ) C (k) (, 1) C (k) :=, k 1 (8) C (k) ( 1, ) C (k) ( 1, 1) i a d by d matrix, where the entrie are d by d matrice C (k) i called the companion matrix of Eq (1) and it i defined by the formula C (k) = I L (k) 1 U (k), (9) where O O O L (k) L (k) ( + 1, ) O O :=, (10) L (k) ( 1, ) L (k) ( 1, 2) O and U (k) (, ) U (k) (, + 1) U (k) (, 1) U (k) O U (k) ( + 1, + 1) U (k) ( + 1, 1) :=, (11) O O U (k) ( 1, 1) for any k 1 The element of the above block matrice are defined a follow: L (k) O, i j 1 (i, j) = A i j (k + i), j < i 1, (12) and U (k) A+i j (k + i), i j 1 (i, j) = O, j < i 1 (13) for any k 1

4 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) Remark 21 The matrix L (k) i nilpotent, ince L (k) = O Therefore I L (k) 1 = I + L (k) + L (k) 1, k 1 The next theorem give the relation between the olution of the higher order ytem (1) and the related firt order ytem (6) Theorem 21 Aume (C 1 ) Then (a) For any ϕ = (ϕ( ),, ϕ( 1)) T V, x(ϕ) = (x(ϕ)(n)) n i the olution of (1) and (2) exactly if (x k (ϕ)) k 1 = ((x k (ϕ)( ),, x k (ϕ)( 1)) T ) k 1 i the olution of (6) and (7), that i 1 x k (ϕ)(i) = C (k) (i, j)x (k 1) (ϕ)(j) k 1, i 1 (14) j= (b) The explicit form of the olution (x k (ϕ)) k 1 of (6) and (7) i x k (ϕ)(i) = x(ϕ)(k + i) = C (k) C (1) ϕ (i), k 1, i 1, or hortly written x k (ϕ) = C (k) C (1) ϕ, k 1 (c) The explicit form of the olution x(ϕ) of (1) and (2) i n x(ϕ)(n) = x ([ n ]+1)(ϕ) n + 1 = C ([ n ]+1) n C (1) ϕ n + 1, n 0 We illutrate the tranformation by two example Example 21 Conider the econd order difference equation x(n) = A 1 (n)x(n 1) + A 2 (n)x(n 2), n 0, (15) where A 1 (n), A 2 (n) R d d for every n 0 Then it i eay to ee that L (k) O O =, k 1, A 1 (2k 1) O and U (k) = A2 (2k 2) A 1 (2k 2), k 1 O A 2 (2k 1) Thu the companion matrice of (15) are given a follow: C (k) = I L (k) 1 U (k) A = 2 (2k 2) A 1 (2k 2), k 1 A 1 (2k 1)A 2 (2k 2) A 1 (2k 1)A 1 (2k 2) + A 2 (2k 1) Example 22 Conider the difference equation x(n) = x(n 1) + A(n)x(n ), n 0, (16) where 2 Thi i a pecial cae of Eq (1) with the matrice A 1 (n) = I, A 2 (n) = = A 1 (n) = O and A (n) = A(n) In thi cae Lemma 20 in Győri and Horváth [10] can be ued It follow that all the entrie of L (k) (k 1) except the entrie L (k) (i, i 1)( + 1 i 1) are O, and L (k) (i, i 1) = I, + 1 i 1 Similarly, all the entrie of U (k) (k 1) except the entrie U (k) (i, i)( i 1) and U (k) (, 1) are O, and U (k) (i, i) = A(k + i), i 1, U (k) (, 1) = I

5 2266 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) and With the help of (9) thee yield that C (k) = C (k) 1 + C (k) (k 1), where for all i, j 1 C (k) O, if i < j (i, j) = A(k + i), if i = j 1 A(k + j), if i > j C (k) 2 (i, j) = O, if j 2 I, if j = 1 We cloe thi ection ome baic definition about periodicity Definition 22 Aume (C 1 ) Let ϕ = (ϕ( ),, ϕ( 1)) T V be a given initial vector 2 (a 1 ) The olution x(ϕ) = (x(ϕ)(n)) n of (1) and (2) i called periodic if there exit a poitive integer p uch that x(ϕ)(n + p) = x(ϕ)(n) for all n In thi cae we ay that x(ϕ) i p-periodic (a 2 ) The olution (x k (ϕ)) k 0 of (6) and (7) i called periodic if there exit a poitive integer q uch that x (k+q) (ϕ) = x k (ϕ) for all k 0 In thi cae we ay that (x k (ϕ)) k 0 i q-periodic (b 1 ) ϕ i aid to be a p-periodic initial vector of (1) if the olution x(ϕ) of (1) and (2) i p-periodic (b 2 ) ϕ i aid to be a q-periodic initial vector of (6) if the olution (x k (ϕ)) k 0 of (6) and (7) i q-periodic (c 1 ) We ay that p i the prime period of the olution x(ϕ) of (1) and (2) if it i p-periodic and p i the mallet poitive integer having thi property (c 2 ) We ay that q i the prime period of the olution (x k (ϕ)) k 0 of (6) and (7) if it i q-periodic and q i the mallet poitive integer having thi property (d) A periodic olution of (1) i called nontrivial if it i different from the zero olution 3 Periodicity condition via the companion matrice The next theorem give a trong connection between the periodicity of the olution of the higher order difference equation (1) and the related olution of the firt order companion difference equation (6) The leat common multiple of the poitive integer u and v will be denoted by [u, v] Theorem 31 Aume (C 1 ) Let ϕ = (ϕ( ),, ϕ( 1)) T V be a given initial vector (a) If the olution x(ϕ) of (1) and (2) i p-periodic, then the olution (x k (ϕ)) k 0 of (6) and (7) i [p,] -periodic (b) If the olution (x k (ϕ)) k 0 of (6) and (7) i q-periodic, then the olution x(ϕ) of (1) and (2) i q-periodic (c) If the olution x(ϕ) of (1) and (2) i periodic with prime period p, then the prime period of the olution (x k (ϕ)) k 0 of (6) and (7) i q := [p,] (d) The olution (x k (ϕ)) k 0 of (6) and (7) i q-periodic if and only if C (lq+i) C ((l 1)q+i+1) I C (i) C (1) C (0) ϕ = 0 (17) for every l 1 and 0 i q 1, where C (0) := I Remark 31 (a) Theorem 31(a) and (b) imply that the olution x(ϕ) of (1) and (2) i periodic if and only if the olution (x k (ϕ)) k 0 of (6) and (7) i periodic (b) It i obviou that the olution x(ϕ) of (1) and (2) i m-periodic with ome poitive integer m if and only if the olution (x k (ϕ)) k 0 of (6) and (7) i m-periodic (c) Suppoe the olution x(ϕ) of (1) and (2) i periodic Theorem 31(c) how that the prime period of (x k (ϕ)) k 0 i uniquely determined by the prime period of x(ϕ) Converely, thi i not true in general: for example, if = 5 and q = 2 in Theorem 31(c), then p i either 2 or 10 (d) If the olution (x k (ϕ)) k 0 of (6) and (7) i periodic with prime period q, then Theorem 31(c) implie that the prime period p of the olution x(ϕ) of (1) and (2) i a divior of q Now we are in the poition to tate a theorem which give explicit condition in term of the companion matrice for a olution of Eq (1) to be periodic Theorem 32 Aume (C 1 ) Let ϕ = (ϕ( ),, ϕ( 1)) T V be a given initial vector (a) Let p := m with ome poitive integer m The olution x(ϕ) of (1) and (2) i p-periodic if and only if C (lm+i) C ((l 1)m+i+1) I C (i) C (1) C (0) ϕ = 0 for every l 1 and 0 i m 1

6 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) (b) If the olution x(ϕ) of (1) and (2) i p-periodic and q := [p,], then C (lq+i) C ((l 1)q+i+1) I C (i) C (1) C (0) ϕ = 0 for every l 1 and 0 i q 1 The following reult, which i a corollary of the previou theorem i an important periodicity theorem for higher order difference equation with time independent companion matrice Theorem 33 Aume (C 1 ) Let ϕ = (ϕ( ),, ϕ( 1)) T V be a given initial vector If then C (k) = C, k 1, (a) Let p := m with ome poitive integer m The olution x(ϕ) of (1) and (2) i p-periodic if and only if C m ϕ = ϕ (b) If the olution x(ϕ) of (1) and (2) i p-periodic and q := [p,], then C q ϕ = ϕ (18) 4 Main reult: imple neceary and ufficient algebraic condition Theorem 32(a) and (9) give a urpriingly imple way to determine the exitence of -periodic olution of Eq (1) The condition are formulated in term of the coefficient matrice of (1) Theorem 41 Aume (C 1 ) (a) Let ϕ = (ϕ( ),, ϕ( 1)) T V be a given initial vector The olution x(ϕ) of (1) and (2) i -periodic if and only if U (l) + L (l) ϕ = ϕ (19) for every l 1, where A (l ) A 1 (l ) A 2 (l ) A 1 (l ) A U (l) + L (l) 1 (l + 1) A (l + 1) A 3 (l + 1) A 2 (l + 1) = A 1 (l 1) A 2 (l 1) A 1 (l 1) A (l 1) for all l 1 (b) The Eq (1) ha a nontrivial -periodic olution if and only if 1 i an eigenvalue of all the matrice U (l) + L (l) (l 1) and they have a common eigenvector correponding to 1 The following reult how that if p, we can conider a new homogeneou linear difference equation p z(n) = B i (n)z(n i), n 0, where B i (n) R d d for every 1 i p and n 0, which allow the problem of the exitence of p-periodic olution of Eq (1) to tranlate into an equivalent problem about the Eq (20) The definition of the coefficient B i (n) in (20) depend on the order between and p (20) Definition 41 Aume (C 1 ), and let p be a poitive integer (a) Suppoe p <, and let u := If v = 0, define p u 1 B i (n) := A jp+i (n), 1 i p, while if v 0, define u A jp+i (n), 1 i v B i (n) := u 1 A jp+i (n), v + 1 i p and v := up(0 v p 1) (21) (22)

7 2268 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) (b) If p >, define Ai (n), 1 i B i (n) := O, + 1 i p (23) Our approache will ue that Theorem 41 give a imple neceary and ufficient condition for the exitence of p-periodic olution of the Eq (20) by uing the U (k) and L (k) (k 1) matrice belonging to (20) The analogue of the matrice U (k) and L (k) (k 1) for Eq (20) will be denoted by U (k) p, and L (k) p,(k 1) Then by (10) and (11), B p (lp p) B p 1 (lp p) B 2 (lp p) B 1 (lp p) U (l) p, + L(l) p, = for all l 1 We begin with the cae 1 p < B 1 (lp p + 1) B p (lp p + 1) B 3 (lp p + 1) B 2 (lp p + 1) B p 1 (lp 1) B p 2 (lp 1) B 1 (lp 1) B p (lp 1) Theorem 42 Aume (C 1 ) Suppoe 1 p < i an integer We conider Eq (20) with the coefficient either (21) or (22) (a) If ϕ = (ϕ( ),, ϕ( 1)) T V i a p-periodic initial vector of (1), and ψ := (ϕ( p),, ϕ( 1)) T V p, then U (l) + p, p, L(l) ψ = ψ for every l 1 Converely, if ψ = (ψ( p),, ψ( 1)) T V p uch that (24) i atified for every l 1, then ϕ = (ϕ( ),, ϕ( 1)) T V, where ϕ(i) := ψ(j) if i j(mod p)(i =,, 1), i a p-periodic initial vector of (1) (b) The Eq (1) ha a nontrivial p-periodic olution if and only if there exit a ψ V p \ {0} uch that (24) i atified for every l 1 (c) The Eq (1) ha a nontrivial p-periodic olution if and only if 1 i an eigenvalue of all the matrice U (l) p + L (l) p,(l 1) and they have a common eigenvector correponding to 1 We turn now to the cae p > (24) Theorem 43 Aume (C 1 ) Suppoe p > i an integer We conider Eq (20) with the coefficient (23) (a) If ϕ = (ϕ( ),, ϕ( 1)) T V i a p-periodic initial vector of (1), and ψ := (x(ϕ)(0),, x(ϕ)(p 1), ϕ( ),, ϕ( 1)) T V p, then (24) hold for every l 1 Converely, if ψ := (ψ( p),, ψ( 1)) T V p uch that (24) i atified for every l 1, then ϕ = (ψ( ),, ψ( 1)) T V i a p-periodic initial vector of (1) (b) The Eq (1) ha a nontrivial p-periodic olution if and only if there exit a ψ V p \ {0} uch that (24) i atified for every l 1 (c) The Eq (1) ha a nontrivial p-periodic olution if and only if 1 i an eigenvalue of all the matrice U (l) p, + L (l) p,(l 1) and they have a common eigenvector correponding to 1 Remark 41 (a) If we confine our attention to higher order difference equation with time independent coefficient, then the equation can be handled eaily, ince U (k) p, and L (k) p,(k 1) do not depend on k In thi cae we have to tudy only the equation Up, + L p, ψ = ψ, where B p B p 1 B 2 B 1 B 1 B p B 3 B 2 U p, + L p, = B p 1 B p 2 B 1 B p (b) It hould be treed that the matrice U (k) p, and L (k) p,(k 1) do not depend on k not only in the cae when the coefficient in Eq (1) are time independent, but alo in the cae when the coefficient are p-periodic

8 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) Dicuion and application In order to give example a tranparent a poible, we uppoe in the further part of thi chapter that the coefficient in the tudied equation are real number that i d = 1 Now we apply the reult in the introduction for ome equation Example 51 (a) Conider the equation x(n) = 5 A i x(n i), n 0, and uppoe that p = 3 Then, by Theorem 11, for every ϕ V 3 5, that i ϕ( 5) = ϕ( 2), ϕ( 4) = ϕ( 1), the olution of (25) correponding to ϕ i 3-periodic if and only if A 3 = 1, A 4 = A 1, A 5 = A 2 (b) Conider the equation x(n) = A 1 x(n 1) + A 2 x(n 2), n 0, (26) and uppoe that p = 5 Some eay calculation how that the rank of the matrix (5) i 3 if and only if A 3 1 A 2 + 2A 1 A 2 = 2 1 A A2 1 A 2 + A 2 = 2 0, and therefore Theorem 12 implie that every olution of Eq (26) i 5-periodic if and only if (27) i atified (27) ha two real olution 5 1 A 1 =, A 2 = 1 and A 1 = 1 + 5, A 2 = In the next example we apply Theorem 33 Example 52 Conider the equation where x(n) = A 1 (n)x(n 1) + A 2 (n)x(n 2) + A 3 (n)x(n 3), n 0, (28) A 1 (3l 3) = A 2 (3l 3) = A 3 (3l 3) = 0 A 3 (3l 2) = a, A 2 (3l 2) = b, l 1 A 1 (3l 1) = c, A 3 (3l 1) = d Here a, b, c and d are real number In thi cae C (l) = C = 0 a b, l 1 0 ac bc + d We tre that C (l) doe not depend on l in pite of A 1 (3l 2) and A 2 (3l 1) being permitted to depend on l(l 1) By Theorem 33, Eq (28) ha a 6-periodic olution different from zero if and only if det(c 2 I) = 0, which i equivalent with a 2 d 2 a 2 2abc b 2 c 2 2bcd d = 0 (29) If b 0, a, b R, then (29) give that c = 1 b (a + d + ad + 1) or c = 1 (a + d ad 1) (30) b Chooe a = 2, b = 1 and d = 1 Then (30) inure that c = 6 Now ome eay computation how that there are 6-periodic olution with prime period 6, and the 6-periodic initial vector belonging to thee olution are (0, α, 3α), α 0 (25) (27)

9 2270 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) Example 53 Conider the equation x(n) = A 1 (n)x(n 1) + A 2 (n)x(n 2), n 0, (31) where A 1 (n), A 2 (n) R for every n 0 Then by Example 21 and by Theorem 41, Eq (31) ha a nontrivial 2-periodic olution if and only if there exit ϕ = (ϕ( 2), ϕ( 1)) T R 2 \ {(0, 0)} uch that (A 2 (2l 2) 1) ϕ( 2) + A 1 (2l 2)ϕ( 1) = 0 A 1 (2l 1)ϕ( 2) + (A 2 (2l 1) 1) ϕ( 1) = 0 for all l 1 For example, if A 1 (2l 2) A 2 (2l 2) 1 = A 2(2l 1) 1 = α R, l 1, A 1 (2l 1) then ϕ = ( αt, t) T R 2 i a 2-periodic initial vector of (31) for every t R We give a concrete equation: Let A 1 (2l 2) = 2l 2, A 2 (2l 2) = l A 1 (2l 1) = 1, A 2 (2l 1) = 3, l 2, and aume A 1 (0) := 0, A 1 (1) := 1 and A 2 (1) := 3 In thi cae, if A 2 (0) := 1, then x(ϕ) i a 2-periodic olution of (31) with initial vector ϕ = (2t, t) T R 2 for every t R, but if A 2 (0) 1, then Eq (31) doe not have any nontrivial 2-periodic olution Example 54 We conider the Eq (31) Suppoe p > 2 i an integer By Theorem 43(a), Eq (31) ha a nontrivial p-periodic olution if and only if there exit a ψ R p \ {0} uch that U (l) + p,2 L(l) p,2 ψ = ψ, where and U (l) p,2 = L (l) p,2 = 0 0 A 2 (lp p) A 1 (lp p) A 2 (lp p + 1) A 1 (lp p + 1) A 2 (lp p + 2) A 1 (lp p + 2) A 2 (lp p + 3) A 2 (lp 1) A 1 (lp 1) 0 for all l 1 Suppoe p = 3 Then Eq (31) ha a 3-periodic olution if and only if there exit a (ψ( 3), ψ( 2), ψ( 1)) T R 3 uch that ψ( 3) + A 2 (3l 3)ψ( 2) + A 1 (3l 3)ψ( 1) = 0 A 1 (3l 2)ψ( 3) ψ( 2) + A 2 (3l 2)ψ( 1) = 0 A 2 (3l 1)ψ( 3) + A 1 (3l 1)ψ( 2) ψ( 1) = 0 for all l 1 For example, (2, 0) R 2 i a 3-periodic initial vector of Eq (31), if A 1 (3l 3) = l, A 2 (3l 3) = 1 2, l 1 A 1 (3l 2) = 2, A 2 (3l 2) = l 2 A 1 (3l 1) = l, A 2 (3l 1) = 2l

10 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) Example 55 Conider the equation x(n) = A 1 (n)x(n 1) + A 2 (n)x(n 2) + A 3 (n)x(n 3), n 0, (32) where A 1 (n), A 2 (n), A 3 (n) R for every n 0 By Theorem 42(a), Eq (32) ha a nontrivial 2-periodic olution if and only if there exit a ψ = (ψ( 2), ψ( 1)) T R 2 \ {(0, 0)} uch that (A 2 (2l 2) 1) ψ( 2) + (A 1 (2l 2) + A 3 (2l 2)) ψ( 1) = 0 (A 1 (2l 1) + A 3 (2l 1)) ψ( 2) + (A 2 (2l 1) 1) ψ( 1) = 0 for all l 1 For example, (1, 1, 1) R 3 i a 2-periodic initial vector of Eq (32), if A 1 (2l 2) = l, A 2 (2l 2) = 1 2l, A 3 (2l 2) = l A 1 (2l 1) = l 2 1, A 2 (2l 1) = 2l 2, A 3 (2l 1) = l 2, l 1 Example 56 We conider the equation x(n) = x(n 1) + Ax(n 5), n 0, where A R (33) i a pecial cae of (16) If 1 p 4, then the exitence of p-periodic olution can be examined by Theorem 42(a): (a) If A = 0, then ϕ = (α, α, α, α, α) i a 1-periodic initial vector for every α R If A 0, then ϕ = (0, 0, 0, 0, 0) the only 1-periodic initial vector (b) There exit a 2-periodic olution with prime period 2 if and only if A = 2 The 2-periodic initial vector belonging to thee olution are ϕ = (α, α, α, α, α), where α 0 (c) There are neither 3-periodic nor 4-periodic olution with prime period 3 or 4 By Theorem 41, there are no 5-periodic olution with prime period 5 By Theorem 43(a), there are 6-periodic olution with prime period 6 if and only if A = 1 The 6-periodic initial vector belonging to thee olution are ϕ = (α, β, β α, α, β), where αβ 0 Finally, by uing Theorem 42, we contruct an equation which ha olution with two different prime period Example 57 Conider the equation x(n) = 2 3 x(n 1) x(n 2) x(n 3) + 4 x(n 4), n 0 3 It i eay to check that the initial vector (α, α, α, α), α 0 generate olution with prime period 2, while the initial vector ( α β, α, β, α β), αβ 0 generate olution with prime period 3 6 Proof of the main reult Proof of Theorem 31 (a) Let q := [p,] It follow that q = pl with ome poitive integer l Then x (k+q) (ϕ)(i) = x(ϕ)(k + q + i) = x(ϕ)(k + pl + i) = x(ϕ)(k + i) = x k (ϕ)(i), i 1, and therefore (x k (ϕ)) k 0 i a q-periodic olution of (6) and (7) (b) By uing Theorem 21 (c), we have x(ϕ)(n + q) = x n + q (ϕ) n+q n + q n = x ([ n ]+q+1)(ϕ) n + q + q + 1 n = x ([ n ]+1)(ϕ) n + 1 = x(ϕ)(n), n, which how that x(ϕ) i a q-periodic olution of (1) and (2) (c) It follow from (a) that (x k (ϕ)) k 0 i a q-periodic olution of (6) and (7) (33)

11 2272 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) It remain to how that q i the prime period of (x k (ϕ)) k 0 In cae q = 1, it i obviou Suppoe q > 1, and uppoe that (x k (ϕ)) k 0 i k-periodic with ome 1 k < q By the definition of the integer q, there i a nonnegative integer l and an integer 1 r < p uch that k = pl + r Then Theorem 21 (c) implie a in (b) that x(ϕ)(n + r) = x(ϕ)(n), n, thu x(ϕ) i an r-periodic olution which i a contradiction (d) Suppoe firt, that the olution (x k (ϕ)) k 0 of (6) and (7) i q-periodic Then x (k+q) (ϕ) = x k (ϕ), k 0, and ince every nonnegative integer k can be written in the form k = (l 1)q + i with l 1 and 0 i q 1, (34) x k (ϕ) = x ((l 1)q+i) (ϕ) = x i (ϕ) With the help of Theorem 21(b) thee yield that C (k+q) C (k+1) x k (ϕ) = C (lq+i) C ((l 1)q+i+1) x i (ϕ) = x i (ϕ), k 0, and therefore C (lq+i) C ((l 1)q+i+1) I C (i) C (1) C (0) ϕ = 0 for every l 1 and 0 i q 1 Converely, uppoe that (17) hold for every l 1 and 0 i q 1 By uing the repreentation (34) of the nonnegative integer k, it i enough to prove that x ((l 1)q+i) (ϕ) = x i (ϕ) for every l 1 and 0 i q 1 The cae l = 1 and 0 i q 1 are trivial, and we complete the proof by induction on l for each fixed 0 i q 1 Let l be a poitive integer uch that (35) hold for a fixed 0 i q 1 By (17) C (lq+i) C ((l 1)q+i+1) C (i) C (1) C (0) ϕ = C (i) C (1) C (0) ϕ, and hence, by the induction hypothei and by Theorem 21(b) x (lq+i) (ϕ) = C (lq+i) C ((l 1)q+i+1) x ((l 1)q+i) (ϕ) = C (lq+i) C ((l 1)q+i+1) C (i) C (1) C (0) ϕ = x i (ϕ) (35) The proof of the theorem i now complete Proof of Theorem 32 (a) It follow from Remark 31(b) and Theorem 31(d) (b) It follow from Theorem 31(a) and Theorem 31(d) Proof of Theorem 33 (a) By Theorem 32(a), the olution x(ϕ) of (1) and (2) i p-periodic if and only if C m I C i ϕ = 0, 0 i m 1, which i equivalent with (18) (b) We can prove a before by applying Theorem 32(b) Proof of Theorem 41 (a) By Theorem 32(a), the olution x(ϕ) of (1) and (2) i -periodic if and only if C (l) I ϕ = 0 for every l 1, which i equivalent with (19) becaue of (9) (b) It come from (a) To prove Theorem 42 and 43 we prepare two lemma Lemma 61 Aume (C 1 ) Suppoe 1 p < i an integer, let u := p, and let v := up(0 v p 1) (a 1 ) If ϕ = (ϕ( ),, ϕ( 1)) T V i a p-periodic initial vector of (1), then ψ := (ϕ( p),, ϕ( 1)) T V p i a p-periodic initial vector of Eq (20) with coefficient either (21) or (22) (a 2 ) If ψ = (ψ( p),, ψ( 1)) T V p i a p-periodic initial vector of Eq (20) with coefficient either (21) or (22), then ϕ = (ϕ( ),, ϕ( 1)) T V, where ϕ(i) := ψ(j) if i j(mod p)(i =,, 1), i a p-periodic initial vector of (1)

12 I Győri, L Horváth / Computer and Mathematic with Application 64 (2012) Proof (a 1 ) Suppoe ϕ i a p-periodic initial vector of (1), and let ψ := (ϕ( p),, ϕ( 1)) T V p Obviouly, it i enough to check that z(ψ)(n) = x(ϕ)(n), n p Since ϕ i a p-periodic initial vector of (1), v u x(ϕ)(n) = A jp+i (n)x(ϕ)(n (jp + i)) + = = v u A jp+i (n)x(ϕ)(n i) + p B i (n)x(ϕ)(n i), n 0, p i=v+1 p i=v+1 u 1 u 1 A jp+i (n)x(ϕ)(n (jp + i)) A jp+i (n)x(ϕ)(n i) and hence by the definition of ψ, (36) can be proved by induction on n for n 0 (a 2 ) Now uppoe ψ = (ψ( p),, ψ( 1)) T V p i a p-periodic initial vector of Eq (20), and let ϕ = (ϕ( ),, ϕ( 1)) T V, where ϕ(i) := ψ(j) if i j(mod p)(i =,, 1) Set z(ψ)(i) := ψ(j) if i j(mod p), i =,, 1 By the definition of ϕ, it i enough to verify that x(ϕ)(n) = z(ψ)(n), n Becaue ψ i a p-periodic initial vector of Eq (20), v u z(ψ)(n) = A jp+i (n) z(ψ)(n i) + = = p i=v+1 v u A jp+i (n)z(ψ)(n (jp + i)) + A k (n)z(ψ)(n k), n 0, k=1 u 1 A jp+i (n) z(ψ)(n i) p i=v+1 u 1 and therefore we again complete the proof by induction on n for n 0 The proof i completed Lemma 62 Aume (C 1 ) Suppoe p > i an integer (a 1 ) If ϕ = (ϕ( ),, ϕ( 1)) T V i a p-periodic initial vector of (1), then ψ := (x(ϕ)(0),, x(ϕ)(p 1), ϕ( ),, ϕ( 1)) T V p A jp+i (n)z(ψ)(n (jp + i)) i a p-periodic initial vector of Eq (20) with coefficient (23) (a 2 ) If ψ := (ψ( p),, ψ( 1)) T V p i a p-periodic initial vector of Eq (20) with coefficient (23), then ϕ = (ψ( ),, ψ( 1)) T V i a p-periodic initial vector of (1) Proof Thi follow by an argument imilar to that for the previou lemma, and we omit the detail Proof of Theorem 42 (a) Theorem 41 and Lemma 61 can be applied (b) and (c) follow from (a) Proof of Theorem 43 (a) Theorem 41 and Lemma 62 can be applied (b) and (c) follow from (a) Proof of Theorem 11 It follow from Theorem 41 and 42 Proof of Theorem 12 It i eay to check that the firt d(p ) column vector of the matrix (5) are linearly independent, therefore the reult come from Theorem 43 Acknowledgment Thi work i upported by the Hungarian National Foundation for Scientific Reearch Grant No K73274 and TÁMOP- 422/B-10/ project (36)

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