ON THIRD-ORDER LINEAR DIFFERENCE EQUATIONS INVOLVING QUASI-DIFFERENCES ZUZANA DO ˇ SL ´ A AND ALE ˇ SKOBZA Received 30 June 2004; Revised 20 September 2004; Accepted 12 O ctober 2004 We study the third-order linear difference equation with quasi-differences and its adjoint equation. The main results of the paper describe relationships between the oscillatory and nonoscillatory solutions of both equations. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction Consider the third-order linear difference equation Δ p n Δ r n Δx n + q n x n+1 = 0(E) and its adjoint equation Δ r n+1 Δ p n Δu n − q n+1 u n+2 = 0, (E A ) where Δ is the forward difference operator defined by Δx n = x n+1 − x n ,(p n ), (r n ), and (q n ) are sequences of positive real numbers for n ∈ N. This paper has been motivated by the paper [9], where third-order difference equa- tions Δ 3 v n − p n+1 Δv n+1 + q n+1 v n+1 = 0, Δ Δ 2 u n − p n+1 u n+1 − q n+2 u n+2 = 0 (1.1) had been investigated. As it is noted here, these equations are not adjoint equations and are referred to as quasi-adjoint equations. Equation (E) is a special case of linear nth-order difference equations with quasi-differ- ences. Such equations have been widely studied in the literature, see, for example, [6, 11] and the references therein. The natural question which arises is to find the adjoint equa- tion to (E) and to examine the connection between solutions of (E) and its adjoint one. Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article I D 65652, Pages 1–13 DOI 10.1155/ADE/2006/65652 2 Third-order linear difference equations In the continuous case, it holds (see, e.g., [5, Theorem 8.33]) that 1 p(t) 1 r(t) x (t) + q(t)x(t) = 0 (1.2) is oscillatory if and only if the adjoint equation 1 r(t) 1 p(t) x (t) − q(t)x(t) = 0 (1.3) has the same property. In addition, nonoscillatory solutions of these equations satisfy some interesting relationships, see, for example, [2, 5]. The aim of this paper is to investigate oscillatory and asymptotic properties of solu- tions of (E)and(E A ). We will prove that (E A ) is the adjoint e quation to (E)andwewill give discrete analogues of the above-quoted results for third-order differential equations. Moreover, the oscillation of (E)and(E A ) is characterized by means of second-order linear difference equations and the problem of the number of oscillatory solutions in a given ba- sis for the solution space of (E)and(E A ) is investigated. Our results extend and complete results of [7–10] stated for the various forms of third-order difference equations. Asolutionx of ( E)isarealsequence(x n )definedforalln ∈ N and satisfying (E)for all n ∈ N. A solution of (E)iscallednontrivial if for any n 0 ≥ 1, there exists n>n 0 such that x n = 0. Otherwise, the solution is called trivial. A nontrivial solution x of (E)issaid to be oscillatory if for any n 0 ≥ 1, there exists n>n 0 such that x n+1 x n ≤ 0. Otherwise, the nontrivial solution is said to b e nonoscillatory. Equation (E)isoscillatory if it has an oscillatory solution. The same terminology is used for (E A ). Denote quasi-differences x [i] , i = 0,1,2, of a solution x of (E)asfollows: x [0] n = x n , x [1] n = r n Δx n , x [2] n = p n Δx [1] n , x [3] n = Δx [2] n . (1.4) Similarly, denote quasi-differences u [i] , i = 0,1,2, of a solution u of (E A )asfollows: u [0] n = u n , u [1] n = p n Δu n , u [2] n = r n+1 Δu [1] n , u [3] n = Δu [2] n . (1.5) All nonoscillatory solutions x of (E) can be a priori classified to the follow ing classes: N 0 = x : ∃n x s.t. x n x [1] n < 0, x n x [2] n > 0 ∀n ≥ n x , N 1 = x : ∃n x s.t. x n x [1] n > 0, x n x [2] n < 0 ∀n ≥ n x , N 2 = x : ∃n x s.t. x n x [1] n > 0, x n x [2] n > 0 ∀n ≥ n x , N 3 = x : ∃n x s.t. x n x [1] n < 0, x n x [2] n < 0 ∀n ≥ n x , (1.6) and similarly solutions u of (E A ) can be classified to the same classes, whereby quasi- differences u [i] , i = 1,2, are defined by (1.5), see [4, 3]. Solutions of (E)fromtheclassN 0 are called Kneser solutions and solutions of (E A ) which belong to the class N 2 are called strongly monotone solutions. Z. Do ˇ sl ´ a and A. Kobza 3 2. Relationship between (E)and(E A ) Solutions of (E)and(E A ) are related by the following properties. Theorem 2.1. (a) Let x, y be solutions of (E). Then the sequence C = (C n )(n ≥ 2) such that C n−1 = C x n−1 , y n−1 ≡ x n−1 y n−1 x [1] n −1 y [1] n −1 (2.1) is a solution of (E A ). (b) Let u,v be solutions of (E A ). Then the sequence D = (D n )(n ≥ 2) such that D n = D u n−1 ,v n−1 ≡ u n−1 v n−1 u [1] n −1 v [1] n −1 (2.2) is a solution of (E). Proof. Claim (a). For any two solutions x, y of (E), we have x n Δy [2] n −1 − y n Δx [2] n −1 =−x n q n−1 y n + y n q n−1 x n = 0. (2.3) Therefore, ΔC n−1 = x n Δy [1] n −1 + y [1] n −1 Δx n−1 − y n Δx [1] n −1 − x [1] n −1 Δy n−1 = x n Δy [1] n −1 + r n−1 Δy n−1 Δx n−1 − y n Δx [1] n −1 − r n−1 Δx n−1 Δy n−1 = x n Δy [1] n −1 − y n Δx [1] n −1 . (2.4) Using the fact x [2] n −1 = x [2] n − Δx [2] n −1 and (2.3), we obtain C [1] n −1 = p n−1 ΔC n−1 = x n y [2] n −1 − y n x [2] n −1 = x n y [2] n − Δy [2] n −1 − y n x [2] n − Δx [2] n −1 = x n y [2] n − y n x [2] n . (2.5) By a direct computation in view of (2.3), we get ΔC [1] n −1 = x n+1 Δy [2] n + y [2] n Δx n − y n+1 Δx [2] n − x [2] n Δy n = y [2] n Δx n − x [2] n Δy n , (2.6) hence C [2] n −1 = r n ΔC [1] n −1 = x [1] n y [2] n − y [1] n x [2] n . (2.7) 4 Third-order linear difference equations Finally ΔC [2] n −1 = x [1] n+1 Δy [2] n + y [2] n Δx [1] n − y [1] n+1 Δx [2] n − x [2] n Δy [1] n =−x [1] n+1 q n y n+1 + p n Δy [1] n Δx [1] n + y [1] n+1 q n x n+1 − p n Δx [1] n Δy [1] n = q n x n+1 y [1] n+1 − y n+1 x [1] n+1 = q n C n+1 , (2.8) that is, C n−1 is a solution of (E A ). Claim (b). By the similar argument as in (a), we get ΔD n = u n Δv [1] n −1 − v n Δu [1] n −1 . (2.9) Using the fact u [2] n −1 = u [2] n −2 + Δu [2] n −2 ,weobtain D [1] n = r n ΔD n = u n v [2] n −1 − v n u [2] n −1 = u n v [2] n −2 + Δv [2] n −2 − v n u [2] n −2 + Δu [2] n −2 = u n v [2] n −2 + q n−1 v n − v n u [2] n −2 + q n−1 u n = u n v [2] n −2 − v n u [2] n −2 . (2.10) Using the same argument as before, we get ΔD [1] n = v [2] n −1 Δu n + u n Δv [2] n −2 − u [2] n −1 Δv n − v n Δu [2] n −2 = v [2] n −1 Δu n − u [2] n −1 Δv n . (2.11) Hence D [2] n = p n ΔD [1] n = u [1] n v [2] n −1 − v [1] n u [2] n −1 . (2.12) Finally ΔD [2] n = u [1] n Δv [2] n −1 + v [2] n Δu [1] n − v [1] n Δu [2] n −1 − u [2] n Δv [1] n = u [1] n q n v n+1 + r n+1 Δv [1] n Δu [1] n − v [1] n q n u n+1 − r n+1 Δu [1] n Δv [1] n =−q n v [1] n Δu n + u n − u [1] n Δv n + v n =− q n p n Δv n Δu n + u n v [1] n − p n Δu n Δv n − v n u [1] n =− q n D n+1 , (2.13) that is, D n is a solution of (E). Relationship between solutions of (E)and(E A ) described in Theorem 2.1 is a discrete analogue of the relationship valid for the differential (1.2) and its adjoint (1.3). For this reason, we call (E A )theadjoint equation to (E). This is in accordance with the definition of the adjoint system to the difference system as the following remark shows. Z. Do ˇ sl ´ a and A. Kobza 5 Remark 2.2. According to [1, page 60], if X ={X n } is a nontrivial solution of the system X n+1 = A n X n , (2.14) then U ={U n },whereU n = (X T n ) −1 is a solution of the system U n = A T n U n+1 . (2.15) System (2.15)iscalledtheadjoint system of (2.14). Equation (E)canbewrittenasafirst-orderdifference system Δx [0] n = 1 r n x [1] n , Δx [1] n = 1 p n x [2] n , Δx [2] n =−q n x [0] n+1 , (2.16) for the vector X n = (x [0] n ,x [1] n ,x [2] n ). Since x [0] n+1 = x [0] n + Δx [0] n ,wehave Δx [2] n =−q n x [0] n + 1 r n x [1] n . (2.17) Using the usual convention that no index actually means the index n, otherwise the index is explicitly specified, we obtain ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ lx [0] n+1 x [1] n+1 x [2] n+1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 1 r 0 01 1 p −q − q r 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ lx [0] x [1] x [2] ⎞ ⎟ ⎟ ⎠ . (2.18) Hence (E) can be interpreted as the system of the form (2.14). Its adjoint system is ⎛ ⎜ ⎜ ⎝ lu [0] u [1] u [2] ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 10−q 1 r 1 − q r 0 1 p 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ lu [0] n+1 u [1] n+1 u [2] n+1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . (2.19) From here we get Δu [0] n = q n u [2] n+1 , Δu [1] n =− 1 r n u [0] n+1 + q n r n u [2] n+1 , Δu [2] n =− 1 p n u [1] n+1 , (2.20) 6 Third-order linear difference equations and the last equation gives Δu [1] n+1 =−Δ(p n Δu [2] n ). Replacing the shift n by n + 1 and sub- stituting into the second equation, we have −Δ p n Δu [2] n =− 1 r n+1 u [0] n+2 + q n+1 r n+1 u [2] n+2 . (2.21) Multiplying this equation by −r n+1 and differentiating it, we obtain Δ r n+1 Δ p n Δu [2] n + Δ q n+1 u [2] n+2 = Δu [0] n+2 . (2.22) Substituting from the first equation in (2.20), we get Δ r n+1 Δ p n Δu [2] n + q n+2 u [2] n+3 − q n+1 u [2] n+2 = q n+2 u [2] n+3 , (2.23) which means that the sequence v n = u [2] n satisfies (E A ). Notation 2.3. Let S denote the solution space of (E)andletS denote the solution space of (E A ). For (x,u) ∈ S × S ,defineᏸ = (ᏸ n ), where ᏸ n = ᏸ x n ,u n = x n+1 u [2] n − x [1] n+1 u [1] n + x [2] n+1 u n+1 . (2.24) The functional ᏸ has the following properties. Lemma 2.4. The s equence ᏸ : S × S → R is a constant which depends only on the choice of solutions x and u, and not on n. Proof. By a direct computation we get Δᏸ n = Δ x n+1 u [2] n − x [1] n+1 u [1] n + x [2] n+1 u n+1 = x n+2 Δu [2] n + u [2] n Δx n+1 − x [1] n+1 Δu [1] n − u [1] n+1 Δx [1] n+1 + u n+2 Δx [2] n+1 + x [2] n+1 Δu n+1 = x n+2 q n+1 u n+2 + r n+1 Δu [1] n Δx n+1 − r n+1 Δx n+1 Δu [1] n − p n+1 Δu n+1 Δx [1] n+1 − u n+2 q n+1 x n+2 + p n+1 Δx [1] n+1 Δu n+1 = 0, (2.25) which completes the proof. Lemma 2.5. Let x, y, z be solutions of (E). Let C and ᏸ be defined by (2.1)and(2.24), respectively. Then the sequence R = (R n ),where R n = x n y n z n x [1] n y [1] n z [1] n x [2] n y [2] n z [2] n (2.26) satisfies R n = ᏸ z n−1 ,C n−1 . (2.27) Z. Do ˇ sl ´ a and A. Kobza 7 Proof. Expanding R n along its third column, we obtain R n = z n x [1] n y [1] n x [2] n y [2] n − z [1] n x n y n x [2] n y [2] n + z [2] n x n y n x [1] n y [1] n . (2.28) Using (2.24), we have ᏸ z n−1 ,C n−1 = z n C [2] n −1 − z [1] n C [1] n −1 + z [2] n C n . (2.29) From here, (2.1), (2.5), and (2.7) show that (2.27)holds. 3. Nonoscillatory solutions of adjoint equations In this section, we study nonoscillatory solutions. We start with the following auxiliary results. Lemma 3.1. There always exists nonoscillatory solution u of (E A )withtheproperty u n > 0, u [1] n > 0, u [2] n > 0 for n ∈ N, (3.1) that is, (E A ) has a strong ly monotone solution. For the proof, see [4, Theorem 3.2]. Lemma 3.2. If a solution y of ( E) satisfies for some intege r m>1 that y m ≥ 0, y [1] m ≤ 0, y [2] m > 0, (3.2) then y k > 0, y [1] k < 0, y [2] k > 0 (3.3) for each k ∈ N such that 1 ≤ k<m. The proof follows from the proof of [3, Proposition 2]. The existence of Kneser solutions of (E) is ensured by the following result. Theorem 3.3. There always exists nonoscillatory solut ion x of (E)withtheproperty x n > 0, x [1] n < 0, x [2] n > 0 for n ∈ N, (3.4) that is, (E)hasaKnesersolution. Proof. Let x = (x(n)), y = (y(n)), z = (z(n)) be a basis of the solution space S of (E). For k ∈ N,define ω k (n) = a k x(n)+b k y(n)+c k z(n), (3.5) where a k , b k , c k are chosen such that ω k (k) = 0, ω k (k +1)= 0, a 2 k + b 2 k + c 2 k = 1. (3.6) 8 Third-order linear difference equations Then ω [1] k (k) = 0. By [3, Lemma 1], ω k (k +2)= 0. Without loss of generality, assume that ω k (k +2)> 0. Then ω [1] k (k +1)= r k+1 Δω k (k +1)= r k+1 ω k (k +2)− ω k (k +1) > 0, (3.7) hence ω [2] k (k) = p k Δω [1] k (k) = p k ω [1] k (k +1)− ω [1] k (k) > 0. (3.8) Since ω k (k) = 0, ω [1] k (k) = 0, ω [2] k (k) > 0, (3.9) by Lemma 3.2 ω k (n) > 0, ω [1] k (n) < 0, ω [2] k (n) > 0, for 1 ≤ n<k. (3.10) Put A k = (a k ,b k ,c k ). Then A k =1foreachk. The unit ball is compact in R 3 ,so(A k ) has a convergent subsequence (A k i ). Denote A = lim i→∞ A k i = (a, b,c). (3.11) Then a 2 + b 2 + c 2 = 1and ω(n) = lim i→∞ ω k i = lim a k i x(n)+b k i y(n)+c k i z(n) (3.12) is a nontrivial solution of (E). Then in view of (3.10) and the fact that k is arbitrary integer, we get ω(n) ≥ 0, ω [1] (n) ≤ 0, ω [2] (n) ≥ 0forn ≥ 1. (3.13) If ω(n 0 ) = 0forsomen = n 0 ,thenω(n) = 0foralln ≥ n 0 whichisacontradictionwith the fact that ω is a nontrivial solution. Thus ω(n) > 0foreveryn ≥ 1, and so Δ ω [2] (n) =− q(n)ω(n +1)< 0forn ≥ 1. (3.14) Hence, ω [2] is decreasing and so ω [2] (n) > 0forn ∈ N.FromhereΔ ω [1] (n) > 0forn ∈ N , which implies that ω [1] is increasing and ω [1] (n) < 0forn ∈ N. Theorem 3.4. Ever y nonoscillatory solut ion of (E A ) is strongly monotone if and only if every nonoscillatory solution of (E)isaKnesersolution. Proof. Let every nonoscillatory solution of (E A ) be strongly monotone. Assume by con- tradiction that there exists solution y of (E) which belongs to the class N i ,wherei ∈ { 1,2,3}.Letx be a Kneser solution of (E). Without loss of generality, we may suppose that x n > 0andy n > 0forlargen. Then the sequence C defined by (2.1)isaccordingto Z. Do ˇ sl ´ a and A. Kobza 9 Theorem 2.1 solution of (E A )andinviewof(2.5)and(2.7) it satisfies, for large n, C n−1 > 0, C [1] n −1 < 0 (if i = 1) C n−1 > 0, C [2] n −1 < 0 (if i = 2) C [1] n −1 < 0, C [2] n −1 > 0 (if i = 3). (3.15) This is a contradiction with the fact that C is strongly monotone solution. Now suppose that every solution of (E) is a Kneser solution. Assume by contradiction that there exists solution v of (E A ) which belongs to the class N i ,wherei ∈{0,1,3}.Let u be a strongly monotone solution of (E A ). Without loss of generality, we may suppose that u n > 0andv n > 0forlargen. Then the sequence D defined by (2.2)isaccordingto Theorem 2.1 solution of (E) and it satisfies, for large n, D n < 0, D [2] n > 0 (if i = 0) D [1] n < 0, D [2] n < 0 (if i = 1) D n < 0, D [1] n < 0 (if i = 3). (3.16) This is a contradiction with the fact that D is a Kneser solution. 4. Oscillatory properties of adjoint equations Lemma 4.1. Let u be a strongly monotone solution and v an oscillatory solution of (E A ). Then their Casoratian D defined by (2.2) is an oscillatory solution of (E). Proof. By Theorem 2.1, D isasolutionof(E). We will show that D is an oscillatory so- lution. Without loss of generality, we may suppose that u satisfies (3.1). Since v is an os- cillatory solution, there exist increasing sequences of positive integers (i n )and(j n ), with properties v i n ≤ 0, v [1] i n > 0forn ∈ N, v j n ≥ 0, v [1] j n < 0forn ∈ N. (4.1) From the above inequalities, (2.2), and (3.1), we have D i n +1 = u i n v [1] i n − v i n u [1] i n > 0forn ∈ N, (4.2) and similarly D j n +1 < 0forn ∈ N. Hence the sequence D is an oscillatory solution of (E). Lemma 4.2. Let x be a Kneser solution and y an oscillatory solution of (E). Then their Casoratian C defined by (2.1) is an oscillatory solution of (E A ). Proof. By Theorem 2.1, C is a solution of (E A ). We will show that C is an oscillatory solution. Without loss of generality, we may suppose that x satisfies (3.4). Because y is an oscillatory solution, there exist increasing sequences of positive integers (i n ) ∞ 1 and (j n ) ∞ 1 10 Third-order linear difference equations with properties i 1 >M, y i n ≤ 0, y [1] i n > 0forn ∈ N, j 1 >M, y j n ≥ 0, y [1] j n < 0forn ∈ N, (4.3) where M = min{n ∈ N : y n y n+1 ≤ 0}. Assume that y [2] j n > 0forsomen ∈ N.ThenbyLemma 3.2,weget y k > 0, y [1] k < 0for1≤ k<j n , (4.4) which is a contradiction with j 1 >M.Hencey [2] j n ≤ 0forn ∈ N. From here and using (2.7) follows C [2] j n −1 = x [1] j n y [2] j n − y [1] j n x [2] j n > 0forn ∈ N. (4.5) By similar argument as before, we obtain y [2] i n ≥ 0forn ∈ N, which implies that C [2] i n −1 = x [1] i n y [2] i n − y [1] i n x [2] i n < 0forn ∈ N. (4.6) By [3, Lemma 2], it follows from inequalities (4.5)and(4.6)thatC is an oscillatory solu- tion of (E A ). The proof is now complete. Our next result characterizes the existence of oscillatory solutions of the adjoint equa- tions. Theorem 4.3. Equation (E A ) is oscillatory if and only if (E)isoscillatory. The proof follows from Theorem 3.3 and Lemmas 3.1, 4.1,and4.2. In the sequel, we study the existence of an oscillatory solution in terms of second-order equations. Theorem 4.4. (a) If u is a nonoscillatory solution of (E A ), then two linearly independent solutions of (E) satisfy the second-order difference equation p n+1 Δ r n+1 Δx n+1 u n+1 + u [2] n u n+1 u n+2 x n+2 = 0. (4.7) (b) If x is a nonoscillatory solution of (E), then two linearly independent solutions of (E A ) satisfy the second-order difference equation r n+1 Δ p n Δu n x n+1 + x [2] n+1 x n+1 x n+2 u n+1 = 0. (4.8) Proof. Claim (a). Let u be a fixed nonoscillatory solution of (E A )suchthatu n > 0for n ≥ N.LetL : S → R be the functional on S defined by L(x) = ᏸ (x n ,u n ). 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(E) and strongly monotone solution of (EA ) Assume that (E) and (EA ) are oscillatory In view of Theorem 4.4 and its proof, (E) has two independent solutions which must be oscillatory Similarly, there exist two independent oscillatory solutions of (EA ) Because dim S = dimS = 3, the proof is complete Theorem 4.6 (a) If (4.7), where u is a nonoscillatory solution of (EA ), is oscillatory, then (E) and. .. dimS − 1 = 2, we get the conclusion Claim (b) Let x be a fixed nonoscillatory solution of (E) such that xn > 0 for n ≥ N Let L : S → R be the functional on S defined by L (u) = ᏸ(xn ,un ) The set K = u ∈ S : L (u) = 0 (4.15) is the kernel of linear functional defined on S Then u ∈ K satisfies [1] [2] xn+1 u[2] − xn+1 u[1] + xn+1 un+1 = 0 n n (4.16) Multiplying the last equation by (xn+1 xn+2 )−1 , we...Z Doˇl´ and A Kobza sa 11 is the kernel of linear functional L defined on S Then x ∈ K satisfies [2] [1] un+1 xn+1 − u[1] xn+1 + u[2] xn+1 = 0 n n (4.10) Multiplying the last equation by (un+1 un+2 )−1 , we get [2] [1] [1] [1] un+1 xn+1 − u[1] xn+1 u[1] xn+1 − u[1] xn+1 u[2] xn+1 n n n+1 + n+1 + = 0 un+1 un+2 un+1 un+2 un+1 un+2 (4.11) From here and using pn+1 Δ [1] xn+1 un+1 . solutions and solutions of (E A ) which belong to the class N 2 are called strongly monotone solutions. Z. Do ˇ sl ´ a and A. Kobza 3 2. Relationship between (E )and( E A ) Solutions of (E )and( E A ). ON THIRD-ORDER LINEAR DIFFERENCE EQUATIONS INVOLVING QUASI-DIFFERENCES ZUZANA DO ˇ SL ´ A AND ALE ˇ SKOBZA Received 30 June 2004; Revised 20 September. are not adjoint equations and are referred to as quasi-adjoint equations. Equation (E) is a special case of linear nth-order difference equations with quasi-differ- ences. Such equations have been