A 2nTH ORDER LINEAR DIFFERENCE EQUATION

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1 A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy for eve order formally self-adjoit scalar liear differece equatios. Sig coditios o the coefficiet fuctios esure (, )-discojugacy. This ad additioal results are obtaied with the use of a associated oliear operator. AMS Subject Classificatio. 39A10. 1 INTRODUCTION I this paper we will be cocered with the 2th-order liear differece equatio ( i r i (t i) i y(t i) ) = 0 (1) i=0 for t i the discrete iterval [a, ) {a, a + 1, a + 2,...}, where r i (t) is real-valued o [a + i, ) for 0 i, ad r (t) > 0 (2) o [a, ). A fuctio y defied o [a, ) is a solutio of (1), provided (1) holds for t a +. Note that Ahlbradt ad Peterso [1] studied (1) i the cotext of the discrete calculus of variatios, ad Peil ad Peterso [2] studied the asymptotic behavior of solutios of (1) with r i (t) 0 for 1 i 1. Here we will geeralize some of the results i [2]. Defiitio For ay fuctio y defied o [a, ), defie for t a + the oliear operator F by F y(t) := ( 1) ( 1) i i 1 y(t 1) ( r i (t 1) i y(t 1) ) ( 1) ( 1) i j i j y(t) ( j 1 r i (t j) i y(t j) ). (3) Begiig i the proof of Lemma 2 we will exted the domai of the coefficiet fuctios r i to the itegers Z by r i (t) r i (a + i) whe t a + i for 0 i. (4)

2 The for ay fuctio y defied o Z, F y(t) is defied for all t Z by (3) as well. Lemma 1 If y is a solutio of (1), the F y(t) = ( 1) r 0 (t)[y(t)] 2 + ( 1) +i r i (t 1) [ i y(t 1) ] 2. (5) I particular, if ( 1) +i r i (t) 0 for t a + i, 0 i 1, (6) the F is odecreasig alog solutios y of (1) for t a +. Proof: Assume y is a solutio of (1). The F y(t) = ( 1) ( 1) i i 1 y(t) ( r i (t 1) i y(t 1) ) (7) +( 1) ( 1) i i y(t 1) ( r i (t 1) i y(t 1) ) (8) ( 1) ( 1) i j i j y(t) ( j r i (t j) i y(t j) ) (9) ( 1) +i j i j+1 y(t) j 1( r i (t j+1) i y(t j+1) ), (10) where (7) ad (8) result from the product rule of the first advaced times the differece of the secod plus the secod times the differece of the first, while (9) ad (10) come from the product rule of the first times the differece of the secod plus the differece of the first times the secod advaced. Next break off the j = 2 terms to rewrite (10) as ( 1) ( 1) i j i j+1 y(t) ( j 1 r i (t j + 1) i y(t j + 1) ) (11) j=3 i=j ( 1) i=2 ( 1) i 2 i 1 y(t) ( r i (t 1) i y(t 1) ). (12) Sice (12) subtracts off all terms i (7) save whe i = 1, we have F y(t) = ( 1) +1 y(t) (r 1 (t 1) y(t 1)) (13) + ( 1) +i r i (t 1) [ i y(t 1) ] 2 (14) ( 1) ( 1) i j i j y(t) ( j r i (t j) i y(t j) ) (15) ( 1) ( 1) i j i j+1 y(t) j 1( r i (t j+1) i y(t j+1) ). (16) j=3 i=j From (15) we isolate the i = j terms, so that (15) becomes 1 ( 1) +1 ( 1) i j i j y(t) j ( r i (t j) i y(t j) ) (17) ( 1) i=2 y(t) i ( r i (t i) i y(t i) ). (18)

3 Note that (16) cacels (17) by replacig j 1 by j i (16). Cosequetly, usig lies (13), (14), ad (18), we have F y(t) = ( 1) +1 y(t) ( i r i (t i) i y(t i) ) + ( 1) +i r i (t 1) [ i y(t 1) ] 2. Sice y is a solutio of (1), F y(t) = ( 1) y(t) r 0 (t)y(t) + ( 1) +i r i (t 1) [ i y(t 1) ] 2, so that (5) follows. Moreover, if (6) holds, the F y(t) 0 o [a +, ). Therefore F is odecreasig alog solutios of (1) for t a +. Defiitios As i [2], y is a type I solutio of (1) if y solves (1) ad F y(t) 0 i a eighborhood of. If F y(t) > 0 ear, the y is a type II solutio. I view of Lemma 1, all solutios of (1) are type I or type II solutios if (6) holds. Followig [1], we say that a solutio y of (1) has a geeralized zero of order (at least) k at a provided y(a + m) = 0 for 0 m k 1, ad a geeralized zero of order (at least) k at t 0 > a provided ad y(t 0 1) 0, y(t 0 + m) = 0 for 0 m k 2, ( 1) k y(t 0 1)y(t 0 + k 1) 0. Equatio (1) is (, )-discojugate o [a, ) provided there is o otrivial solutio of (1) with two or more geeralized zeros of order (at least) o [a, ). Lemma 2 Suppose (6) holds, ad that y is a otrivial solutio of (1) with a geeralized zero of order (at least) at t 0 a. (a) If t 0 > a, the ad either F y(t 0 ) < 0, (19) (i) F y(t 0 + 1) > 0, if ( 1) y(t 0 1)y(t 0 + 1) > 0, or (ii) y has exactly + k cosecutive zeros startig at t 0, where 0 k 1, ad F y(t 0 + m) = 0, 1 m k + 1, (20) with F y(t 0 + k + 2) > 0. (21) (b) If t 0 = a, the case (a) (ii) holds, ad (20), (21) both follow.

4 Proof: Assume y is a otrivial solutio of (1). For part (a), assume that y has a geeralized zero of order (at least) at t 0 > a. The, by defiitio, ad As a result of (22), y(t 0 + m) = 0 for 0 m 2 (22) y(t 0 1) 0, while ( 1) y(t 0 1)y(t 0 + 1) 0. (23) i y(t 0 ) = 0 for 0 i 2. (24) To esure that everythig is defied o the same itervals, exted the domais of the coefficiet fuctios r i to the itegers as i (4). It suffices to show that (1) satisfies the lemma with these ew coefficiets o the itegers. Note that F y(t) is ow defied ad odecreasig alog solutios y of (1) o the itegers as well. Now, usig (3), (22), ad (24), we see that F y(t 0 ) = ( 1) ( 1) i i 1 y(t 0 1) ( r i (t 0 1) i y(t 0 1) ) +( 1) +1 ( 1) i j i j y(t 0 ) ( j 1 r i (t 0 j) i y(t 0 j) ) = ( 1) ( 1) i i 1 y(t 0 1) ( r i (t 0 1) i y(t 0 1) ) = = 1 ( 1) +i ( 1) i 1 y(t 0 1) ( r i (t 0 1) ( 1) i y(t 0 1) ) +( 1) 1 y(t 0 1)r (t 0 1) [y(t 0 + 1) + ( 1) y(t 0 1)] 1 ( 1) +i 1 r i (t 0 1) [y(t 0 1)] 2 r (t 0 1) [y(t 0 1)] 2 +r (t 0 1) ( 1) 1 y(t 0 1)y(t 0 + 1) = [y(t 0 1)] 2 ( 1) +i r i (t 0 1) r (t 0 1)( 1) y(t 0 1)y(t 0 + 1). (25) By (2), (6), ad (23), equatio (25) yields the iequality (19). To show case (a)(i), suppose we have (22) ad The ( 1) y(t 0 1)y(t 0 + 1) > 0. (26) F y(t 0 + 1) = ( 1) ( 1) i i 1 y(t 0 ) ( r i (t 0 ) i y(t 0 ) ) ( 1) +i j i j y(t 0 +1) j 1( r i (t 0 +1 j) i y(t 0 +1 j) ). (27) From (22) we kow that i 1 y(t 0 ) = 0 for 1 i 1,

5 ad i j y(t 0 + 1) = 0 except whe i = ad j = 2. As a result, (27) becomes F y(t 0 + 1) = ( 1) ( 1) 1 y(t 0 ) (r (t 0 ) y(t 0 )) ( 1) ( 1) 2 2 y(t 0 + 1) (r (t 0 1) y(t 0 1)) = y(t 0 + 1)r (t 0 ) y(t 0 ) y(t 0 + 1) (r (t 0 1) y(t 0 1)) = y(t 0 + 1)r (t 0 1) y(t 0 1) = y(t 0 + 1)r (t 0 1) [y(t 0 + 1) + ( 1) y(t 0 1)] = r (t 0 1) (y(t 0 + 1)) 2 > 0 +r (t 0 1) ( 1) y(t 0 1)y(t 0 + 1) by (2) ad (26). For case (a)(ii), we assume there is a k 0 such that y(t 0 + τ) = 0 for 0 τ + k 1; (28) sice y is a otrivial solutio, k 1. The, for m {1,..., k + 1}, (28) implies that F y(t 0 + m) = ( 1) ( 1) i i 1 y(t 0 + m 1) ( r i (t 0 + m 1) i y(t 0 + m 1) ) However, ( 1) ( 1) i j i j y(t 0 +m) j 1( r i (t 0 +m j) i y(t 0 +m j) ) = 0. F y(t 0 + k + 2) = ( 1) ( 1) i i 1 y(t 0 + k + 1) ( r i (t 0 + k + 1) i y(t 0 + k + 1) ) ( 1) { ( 1) i j i j y(t 0 + k + 2) j 1 ( r i (t 0 + k + 2 j) i y(t 0 + k + 2 j) )} = ( 1) ( 1) 1 y(t 0 + k + 1) (r (t 0 + k + 1) y(t 0 + k + 1)) ( 1) ( 1) 2 2 y(t 0 + k + 2) (r (t 0 + k) y(t 0 + k)) = y(t k)r (t 0 + k + 1) y(t 0 + k + 1) y(t k) (r (t 0 + k) y(t 0 + k)) = y(t k)r (t 0 + k) y(t 0 + k) = r (t 0 + k) (y(t k)) 2 > 0 usig (2) ad (28). Fially, i case (b), the fact that y has a geeralized zero of order (at least) at a implies that case (a)(ii) holds with t 0 = a.

6 2 MAIN RESULTS The followig result, which is importat i the discrete calculus of variatios, was first prove by Ahlbradt ad Peterso [1]. Here it follows easily from Lemma 2. Theorem 3 Assume (6) holds. The (1) is (, )-discojugate o [a, ). Proof: Suppose y is a otrivial solutio with a geeralized zero of order exactly at t 1 > a. The, by Lemma 2, either F y(t 1 +1) > 0 or F y(t 1 +2) > 0, so that F y(t) > 0 o [t 1 + 2, ). If y had aother geeralized zero of order exactly at t 2 t 1 +, the (19) would imply that F y(t 2 ) < 0, a cotradictio because t 2 [t 1 + 2, ). A similar cotradictio is reached if y has a geeralized zero of order greater tha at t 1 or t 2 or both. Hece, a otrivial solutio of (1) ca have at most oe geeralized zero of order (at least) o [a, ), so that (1) is (, )-discojugate o [a, ). Theorem 4 Assume (6) holds. The ay otrivial solutio of (1) with a geeralized zero of order (at least) is a type II solutio. I particular, (1) has 2 liearlyidepedet type II solutios. Proof: Assume y is a otrivial solutio of (1) with a geeralized zero of order (at least) at t 0 for some t 0 a. Agai, exted the domais of the coefficiet fuctios r i as i (4), ad ote that F y(t) is defied ad odecreasig alog solutios o the itegers, due to Lemma 1. The, usig Lemma 2, we see that either F y(t 0 + 1) > 0, or, if y has + k zeros startig at t 0 for some 0 k 1, the F y(t 0 + k + 2) > 0. Either way, there exists t 1 > t 0 such that F y(t 1 ) > 0, ad so F y(t) > 0 o [t 1, ). Hece, y is a type II solutio. We ow show that there are 2 liearly-idepedet type II solutios of (1). Let y k, 0 k 2 1, be solutios of (1) satisfyig y k (a + i) = { 0 if i k 1 if i = k, 0 i 2 1. Because the y k are otrivial solutios of (1) with at least cosecutive zeros o [a, a+2 1], we have by the above part of the proof that the y k are type II solutios for 0 k 2 1. It is clear that these solutios are liearly idepedet. Theorem 5 If (6) holds, the (1) has liearly-idepedet type I solutios. Proof: Followig the proof of Theorem 4 i [2], for each fixed s a+, we let v k (t, s) be a otrivial solutio of (1) for 1 k, satisfyig the 2 1 boudary coditios v k (a + i, s) = 0 for 0 i 1 but i k 1 v k (s + i, s) = 0 for 0 i 1. (29) The defie u k (t, s) := v k (t, s) v 2 k (a, s) + v2 k (a + 1, s) + + v2 k (a + 2 1, s) (30)

7 for 1 k ad s a +. The u k (t, s) is a solutio of (1) satisfyig Thus, for each k, the sequece 2 1 i=0 u 2 k(a + i, s) = 1. has a coverget subsequece Let for 0 i 2 1. The {u k (a, s), u k (a + 1, s),..., u k (a + 2 1, s)} s=a+ {u k (a, s jk ), u k (a + 1, s jk ),..., u k (a + 2 1, s jk )} j=1. v i+1,k := lim j u k (a + i, s jk ) (31) 2 1 i=0 Further, let y k be the solutios of (1) satisfyig v 2 i+1,k = 1. (32) y k (a + i) = v i+1,k (33) for 0 i 2 1, 1 k ; ote that the y k are otrivial solutios by (32) ad (33). Sice v k (t, s) has cosecutive zeros startig at s by (29), formula (30) implies the same for u k (t, s), 1 k. Hece, by case (a)(ii) of Lemma 2, we have as F u k (t, s jk ) is odecreasig, F u k (s jk + 1, s jk ) = 0; F u k (t, s jk ) 0 o [a +, s jk + 1] for 1 k. So, for each t 1 a +, there exists j t1 s jk > t 1 for all j k j t1. The F u k (t 1, s jk ) 0 for all j k j t1. Takig the limit as j, we get that such that F y k (t 1 ) 0 for 1 k. As t 1 a + was arbitrary, F y k (t) 0 for all t a +, for 1 k. It follows that the y k are type I solutios of (1) for 1 k. By (29),(30),(31), ad (33) we have that y k (a + i) = 0 for 0 i 1 if i k 1. (34) If y k (a + k 1) = 0, the y k would have cosecutive zeros ad thus a geeralized zero of order (at least) at a, so that y k would be a type II solutio by Theorem 4. Cosequetly, y k (a + k 1) 0 for 1 k, ad the y k are liearly idepedet.

8 Theorem 6 If (6) holds ad y is a type I solutio of (1), the t=a+ i ( 1) i r i (t) [ i y(t) ] 2 <, (35) for 0 i. If r 0 (t) 0 i a eighborhood of, the every otrivial type I solutio of (1) is a strict type I solutio. Proof: Let y be a type I solutio of (1); the F y(t) 0 for t a +. Let M := lim F y(t) 0. Summig both sides of (5) from a + to yields ( M F y(a + ) = ( 1) r 0 (t)[y(t)] 2 + ( 1) +i r i (t 1) [ i y(t 1) ] ) 2. t=a+ Thus (35) holds. Now assume r 0 (t) 0 i a eighborhood of, ad that v is a otrivial type I solutio of (1); the F v(t) 0 for t a +. Suppose there exists a t 0 a + such that F v(t 0 ) = 0. The F v(t) 0 o [t 0, ), by Lemma 1. But the F v(t) = 0 o [t 0, ), so that from (5) we have ( 1) r 0 (t)[v(t)] 2 + ( 1) +i r i (t 1) [ i v(t 1) ] 2 0 for t t 0. Sice (6) holds all terms are oegative; moreover, r 0 (t) 0 ear gives that v is the trivial solutio, cotrary to assumptio. Therefore, F v(t) < 0 for all t a +, ad v is a strict type I solutio of (1). Corollary 7 If (6) holds ad lim if ( 1) r 0 (t) > 0, the (1) has liearly-idepedet type I solutios v k satisfyig lim v k (t) = 0 for 1 k. Proof: By Theorem 5, equatio (1) has liearly-idepedet type I solutios v 1,..., v. For ay k {1,..., }, (35) implies that so that Sice lim if ( 1) t=a+ r 0 (t)[v k (t)] 2 <, lim r 0(t)[v k (t)] 2 = 0. ( 1) r 0 (t) > 0, we have that lim v k (t) = 0, for 1 k. Refereces [1] C. D. Ahlbradt ad A. C. Peterso, The (, )-discojugacy of a 2th-order liear differece equatio, Computers Math. Applic., 28, No 1-3(1994), 1-9. [2] T. Peil ad A. Peterso, Asymptotic behavior of solutios of a two-term differece equatio, Rocky Moutai Joural of Mathematics, 24, No 1(Witer 1994),