Asymptotic Equivalence of Second-Order Difference Equations

Journal of Mathematical Analysis and Applications 238, 91᎐100 Ž1999. Article ID jmaa.1999.6508, available online at http:rrwww.idealibrary.com on

Asymptotic Equivalence of Second-Order Difference Equations Jaroslaw Morchalo Institute of Mathematics, Poznan Uni¨ ersity of Technology, Piotrowo 3A, 60-965 Poznan, Poland Submitted by William F. Ames Received April 22, 1999

The asymptotic behavior of solutions of second-order difference equations is discussed. 䊚 1999 Academic Press

1. INTRODUCTION The asymptotic equivalence for differential and differences equations was studied by many authors, e.g., w1᎐8x. We are mainly interested in establishing asymptotic relationships between the solutions of equations ⌬ Ž pny 1 ⌬ x ny1 . q qn x n s 0,

Ž 1.

⌬ Ž pny 1 ⌬ yny1 . q qn x n s f Ž n, yn . .

Ž 2.

and

The purpose of this paper is to extend some of the results from w2x on differences equations. We suppose that n g N Ž n 0 q 1. s  n 0 q 1, n 0 q 2, . . . 4 , Ž n 0 is a fixed nonnegative integer., ⌬ is the forward difference operator; i.e., ⌬ x n s x nq 1 y x n for any function u: N Ž n 0 . ª R Ž R is the real line., p: N Ž n 0 . ª Ž0, ⬁., q: N Ž n 0 . ª R, f : N Ž n 0 q 1. = R ª R is for any n g N Ž n 0 q 1. continuous as a function of y g R. Hereafter, the term ‘‘solution’’ of Ž1. or Ž2. is always used as such a real sequence  u n4 satisfying Ž1. or Ž2. for each n g N Ž n 0 .. Such a solution we denote by u n . 91 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

92

JAROSLAW MORCHALO

Notation 1. Let M1 be the set of all solutions of the equation Ž1. and M2 the set of all solutions of the equation Ž2. that exist for all n g N Ž n 0 q 1.. Let ␮ : N Ž n 0 . ª R. The symbols O and o have the usual meaning: z n s O Ž ␮ n . denotes that there exists c1 ) 0 such that < z n < F < c1 ␮ n < for large n, and z n s oŽ ␮ n . denotes that there exists h n such that z n s ␮ n h n and lim nª⬁ h n s 0. DEFINITION 1. We say that the equations Ž1. and Ž2. are ␮0 asymptotically equivalent if for each x g M1 there exists y g M2 such that x n y yn s o Ž ␮0n . ,

Ž 3.

and conversely.

2. NONHOMOGENEOUS LINEAR DIFFERENCE EQUATIONS Let in Ž2. f Ž n, x . ' a n , where a: N Ž n 0 q 1. ª R. Then Ž2. has the form ⌬ w pny 1 ⌬ yny1 x q qn yn s a n .

Ž 4.

The method of variation of constants formula gives for each solution y of the equation Ž4. the relation yn s c1 u n q c 2¨ n y cy1 u n

n

Ý

ssn 0q1

¨ s a s q cy1 ¨ n

n

Ý

ssn 0q1

u s as ,

Ž 5.

where c1 , c 2 are arbitrary constants, u n , ¨ n are linearly independent solutions of the equation Ž1., c s pnw u n¨ nq1 y ¨ n u nq1 x. Notation 2. If u n , ¨ n are linearly independent solutions of Ž1., then yn0 sycu n Ý nssn 0q1¨ s a s yc¨ n Ý⬁ssnq1 u s a s where cy1 spn ww u, ¨ n x, ww(, (x ᎏ the Casorati matrix is a particular solution of Ž4.. THEOREM 1. The equations Ž1. and Ž4. are ␮ 0 asymptotically equi¨ alent if there exists a solution yn0 of the equation Ž4. such that yn0 s oŽ ␮0n .. Proof. Each solution of the equation Ž4. can be expressed in the form yn s x n q yn0 , where x n is an arbitrary solution of the equation Ž1.. This implies the assertion of the theorem.

SECOND-ORDER DIFFERENCE EQUATIONS

THEOREM 2.

93

Assume that ⬁

n

un

Ý

ssn 0q1

¨ s as q ¨ n

u s a s s o Ž ␮0n . ,

Ý

Ž 6.

ssnq1

then the equation Ž4. has a solution y 0 such that yn0 s oŽ ␮ 0n .. Proof. The conclusion of the theorem is an immediate consequence of the relation yn s c1 u n q c 2¨ n y cy1 u n

n

Ý

ssn 0q1

⬁

¨ s a s y cy1 ¨ n

Ý

u s as .

Ž 5⬘ .

ssnq1

COROLLARY 1. If the hypotheses of Theorem 2 hold, then the equations Ž1. and Ž4. are ␮0 asymptotically equi¨ alent.

3. EQUIVALENCE OF NONLINEAR DIFFERENCE EQUATIONS We suppose that the following hypotheses hold: f : N Ž n0 q 1. = R ª R

Ž i. Ž ii .

there exists a nonnegative function F : N Ž n 0 q 1 . = Rqª Rq which is continuous and nondecreasing with respect the last argument for each fixed n g N Ž n 0 q 1 . such that < f Ž n, z . < F F Ž n, < z < . .

Ž 7.

Here Rq is the set of all nonnegative real numbers. Notation 3. Let r 0 : N Ž n 0 . ª Ž0, ⬁. be a positive function such that u n s O Ž rn0 . ,

¨ n s O Ž rn0 . .

Ž 8.

For example, we can take rn0 s wn s < u n < q < ¨ n <. THEOREM 3. 1. 2.

Assume that

assumption Ž7. holds, for any ␣ G 0, ⬁

Ý

ssn 0q1

< u s < F Ž s, ␣ r s0 . - ⬁,

< un <

n

Ý

ssn 0q1

< ¨ s < F Ž s, ␣ r s0 . s o Ž rn0 . , Ž 9 .

94

JAROSLAW MORCHALO

3.

for each solution y g M2 , yn s O Ž rn0 . ,

4.

Ž 10 .

there exist finite limits for  u n4 and  ¨ n4 .

Then the equations Ž1. and Ž2. are ␮0 asymptotically equi¨ alent for each function ␮0 , such that for any ␣ G 0: < un <

n

Ý

ssn 0q1

< ¨ s < F Ž s, ␣ r s0 . q < ¨ n <

⬁

< u s < F Ž s, ␣ r s0 . s o Ž ␮0n . . Ž 11 .

Ý ssnq1

Proof. i. Let y g M2 . Consider a nonhomogeneous linear difference equation ⌬ Ž pny 1 ⌬ z ny1 . q qn z n s f Ž n, yn . , that possesses the solution yn . From assumption of the theorem for appropriate ␣ G 0 we have ⬁

n

un

Ý

ssn 0q1

¨ s f Ž s, ys . q ¨ n

F < un <

Ý

u s f Ž s, ys .

ssnq1 n

Ý

ssn 0q1

< ¨ s < F Ž s, ␣ r s0 . q < ¨ n <

⬁

Ý

< u s < F Ž s, ␣ r s0 . s o Ž ␮0n . .

ssnq1

Theorem 2 guarantees the existence of a solution z such that z ns oŽ ␮ 0n .. Then the solution x of the equation Ž1. takes the form x n s yn y z n . ii. Take x g M1 and consider the equation n

yn s x n y cu n

Ý

ssn 0q1

¨ s f Ž s, ys . y c¨ n

⬁

Ý

u s f Ž s, ys . .

Ž 12 .

ssnq1

It is easy to verify that a solution of Ž12. is also a solution of Ž2.. We denote by ␾ Ž N, R . the space of all functions from N Ž n 0 . into R. The topology of ␾ is the topology of uniform convergence on every set Nm Ž n 0 . s  n 0 , n 0 q 1, . . . , n 0 q m4 ,

m s 0, 1, . . . ,

that is u i ª u as i ª ⬁ in ␾ if and only if lim iª⬁ < u i Ž n. y uŽ u.< s 0 uniformly on every set NmŽ n 0 ., m s 0, 1, . . . . Note that ␾ if a locally convex space with the topology defined by the following family of seminorms < x n < m s sup  < x n < : n g Nm Ž n 0 . , m s 0, 1, . . . 4 . We defined subset B␵ of ␾ as B␵ s  ␸ g ␾ : < ␸n < F ␵ rn0 , n g N Ž n 0 .4 .

SECOND-ORDER DIFFERENCE EQUATIONS

There exists ␤˙ ) 0 such that n 1 g N Ž n 0 . so large that < u n <Ý nssn < ¨ s < F Ž s, ␵ r s0 . F 12 ␤ rn0 < c
x, u, ¨ g B␤ . Let ␵ G 2 ␤ and choose Ý ⬁ss n 1 < u s < F Ž s, ␵ r s0 . F < c
n

Ž T␸ . Ž n . s x n y cu n

¨ s f Ž s, ␸s . y c¨ n

Ý

95

ssn 1

u s f Ž s, ␸s . .

Ý ssnq1

The convergence in ⌽ is uniform convergence on each compact subset NmŽ n 0 . ⭈ T maps B␵ into B␵ . If ␸ g B␵ then < Ž T␸ . Ž n . < F ␤ rn0 q < c < < u n <

n

< ¨ s < F Ž s, ␵ r s0 . q < c < < ¨ n <

Ý ssn 1

F

␤ rn0

⬁

Ý

< u s < F Ž s, ␵ r s0 .

ssnq1

< c
Therefore TB␵ ; B␵ for n g N Ž n1 .. Denote M G maxŽ< u n < q < ¨ n <. on every NmŽ n 0 .. To establish that the mapping T is continuous, fix ␧ ) 0 and select n 2 G n1 such that ⬁

Ý

< u s < F Ž s, ␵ r s0 . -

ssn 2

␧ . 4< c< M

Ž 13 .

Let  ␸ni 4⬁is1 , ␸ i g B␵ , and ␸ni i ␸n g B␵ . Then for every n g NmŽ n 0 . we have < T␸ni y T␸ F < c < < u n < n

n2

Ý

< ¨ s < < f Ž s, ␸si . y f Ž s, ␸s . <

ssn 0

q < c< <¨n <

⬁

Ý

ssn 2q1

< u s < < f Ž s, ␸si . y f Ž s, ␸s . < .

Put ⌰s

␧ 2 < < 2 M < c <Ý nss n0 ¨ s

.

By the uniform convergence on NmŽ n 0 ., m s 0, 1, . . . , n 2 , of  ␸ni 4 it follows that < f Ž s, ␸ni . y f Ž s, ␸n . < - ⌰.

96

JAROSLAW MORCHALO

Thus < T␸ni y T␸n < F < c < M⌰

n2

Ý

<¨s < q 2 < c< M

ssn 0

⬁

Ý

ssn 2q1

␧

u s F Ž s, ␵ r s0 . -

2

q

␧ 2

s␧

for every n g NmŽ n 0 .. This estimate implies that T is continuous. TB␵ is precompact. It suffices to prove that elements of TB␵ satisfy Cauchy’s condition uniformly on TB␵ . In fact let ␸ g B␵ and n ) m, n, m g N Ž n1 .. Then we have n

T␸n y T␸m s x n y cu n

Ý

¨ s f Ž s, ␸s . q c¨ n

ssn 1

⬁

u s f Ž s, ␸s .

Ý ssnq1

⬁

m

y x m q cu m

¨ s f Ž s, ␸s . q c¨ m

Ý ssn 1

u s f Ž s, ␸s . ,

Ý ssmq1

which implies < T␸ni y T␸n < F < x n y x m < q < c < < u n <

n

Ý ssm

q< u m <

m

Ý

n

< ¨ s < F Ž s, ␵ r s0 . q < u n <

< ¨ s < F Ž s, ␵ r s0 .

ssn 1 ⬁

< ¨ s < F Ž s,␵ r s0 . q 2 c < ¨ n <

ssn 1

Ý

Ý

< u s < F Ž s, ␵ r s0 . .

ssmq1

By assumption of 2 and 4 for given n 2 ) n1 such that < T␸n y T␸m < - ␧ for all n, m ) n 2 . By Schauder’s fixed point theorem we conclude that there exists ␸ g B␵ such that ␸ s T␸ that is ␸ is solution of Ž12. The relations Ž11. and Ž12. imply Ž3.. THEOREM 4.

Assume that Ž7. holds and let for any ␣ G 0: ⬁

Ý Ž < u n < q < ¨ n <. F Ž n, ␣ rn0 . - ⬁. nsn 0

For each y g M2 let Ž10. hold. Then the equations Ž1. and Ž2. are ␮0 asymptotically equivalent for each function ␮ 0 , such that for any ␣ G 0: ⬁

Ý Ž < u n¨ s < q < u s¨ n < . F Ž s, ␣ rn0 . s o Ž ␮0n . . ssn

SECOND-ORDER DIFFERENCE EQUATIONS

97

Proof. In an aim to prove this theorem one should consider the equation yn s x n q cy1 u n

⬁

¨ s f Ž s, ys . y cy1 ¨ n

Ý ssn

⬁

Ý u s f Ž s, ys . , ssn

and follow an analogous way as in the case of Theorem 3.

4. SPECIAL CASES Suppose that < f Ž n, x . < F h n < x < ,

Ž 14 .

where h: N Ž n 0 . ª ²0, ⬁.. LEMMA 1. Let Ž8., Ž14. and sup n l 0 Ž rn0 . 2 h n F ␥ - 1 hold, where l 0 is a positi¨ e constant, then each solution of the equation Ž2. exists on N Ž n 0 . and ny1

ž

yn s 0 rn0 exp

l0

Žr0. ly␥ s

Ý

syn 0q1

2

/

.

Proof. As yn s c1 u n q c2¨ n y cy1 u n

n

¨ s f Ž s, ys . q cy1 ¨ n

Ý

ssn 0q1

n

Ý

ssn 0q1

u s f Ž s, ys . ,

Ž 15 . with appropriate constants c1 , c 2 , we have for n g N Ž n 0 q 1., < yn < F k 0 rn0 q l 0 rn0

n

Ý

ssn 0q1

h s r s0 < ys < ,

where k 0 , l 0 are positive constants and l 0 does not depend on yn . Hence < yn < F

k 0 rn0 1y␥

q

l 0 rn0 1y␥

ny1

Ý

ssn 0q1

h s r s0 < ys < .

Using a generalized Gronwall’s inequality we obtain < yn < F rn0

k0 1y␥

ny1

exp

Ý

ssn 0q1

l0 1y␥

2

h s Ž r s0 . .

98

JAROSLAW MORCHALO

THEOREM 5.

Assume that

1. Ž14. holds 2. all solutions of Ž1. are bounded, 3. sup Ž l 0 Ž ␻ n2 . h n . F ␥ - 1, 4. Ý⬁ns n 0 ␻ n2 h n - ⬁, then the equations Ž1. and Ž2. are ␮ 0 asymptotically equi¨ alent. The proof is a consequence of Theorem 4 and Lemma 1. LEMMA 2.

Assume that

1. Ž7. holds, 2. for any ␭ G 0, Ý⬁ns n 0q1 rn0 F Ž n, ␭ rn0 . - ⬁. 3. there exist ␭0 ) 0 such that sup ␭g ² ␭ 0 , ⬁.

1

⬁

␭

Ý

nsn 1q1

rn0 F Ž n, ␭ rn0 . s S - < c <

Ž 16 .

for an appropriate n1 G n 0 . Then each solution y of the equation Ž2. exists for n G n1 q 1 and yn s O Ž rn0 .. Proof. From Ž15. with appropriate constants c1 , c 2 , we have for n g ² n1 q 1, N 0 ., N 0 - ⬁, n1 G n 0 . Thus < yn < F Krn0 q < c
½

n

Ý

ssn 1q1

< ¨ s < < f Ž s, ys . < q

n

Ý

ssn 1q1

5

< u s < < f Ž s, ys . < ,

or < yn < F Krn0 q < c
n

Ý

ssn 1q1

r s0 F Ž s, < ys < . ,

where K is positive constant. Denote zm s K < c < q

m

r s0 F Ž s, < ys < .

Ž 17 .

n g ² n 0 q 1, m: .

Ž 18 .

Ý

ssn 1q1

for m g ² n1 q 1, N 0 .. Then < yn < F < c
SECOND-ORDER DIFFERENCE EQUATIONS

99

If z m - < c < ␭0 for each m g ² n1 q 1, N 0 . then < yn < F ␭ 0 rn0 ,

n g² n1 q 1, N 0 . .

If there exists m 0 g ² n1 q 1, N for m g ² n1 q 1, N 0 .. From relation Ž16. we obtain sup ␭; ² ␭ 0 , ⬁. y1

Put ␭ s < c <

1

N0

␭

Ý

ssn 1q1

0.

Ž 19 .

such that z m 0 G < c < ␭0 , then z m G < c < ␭0

r s0 F Ž s, ␭ r s0 . s S1 F S - < c < .

z m for m g ² m 0 , N 0 .. Then N0

Ý

ssn 1q1

r s0 F Ž s, < c
Now from Ž17. and Ž18. we have z m F K < c < q < c
m g² m 0 , N 0 . .

Hence zm F

K < c<

,

1 y < c
since < c
K 1 y < c
rn0

Ž 20 .

for n g ² n1 q 1, m:, m g ² m 0 , N 0 .. This estimate does not depend on m, thus Ž20. holds for each n g ² n1 q 1, N 0 .. From estimate Ž19. or Ž20. we get that y is bounded on ² n1 q 1, N 0 .. This is a contradiction and hence N 0 s ⬁. THEOREM 6. Let the assumptions of Lemma 2 hold. Then the equations Ž1. and Ž2. are ␮0 asymptotically equi¨ alent. The proof is a consequence of Theorem 4 and Lemma 2.

REFERENCES 1. F. Brauer and J. S. Wang, On the asymptotic relationship between solutions of two systems of ordinary differential equations, J. Differential Equations 6 Ž1969., 527᎐543. 2. T. G. Hallam, Asymptotic relationships between the solutions of two second order differential equations, Ann. Polon. Math. 24 Ž1971., 295᎐300.

100

JAROSLAW MORCHALO

3. J. Morchalo, Asymptotic equivalence of Volterra difference systems, Publ. Math. 39 Ž1995., 301᎐312. 4. J. Morchalo, Asymptotic equivalence of difference equations, Math. Slo¨ aca, 48 Ž1998., 57᎐68. 5. N. Omuchic, Relationship among the solutions of two systems of ordinary differential equations, Michigan Math. J. 10 Ž1963., 129᎐139. 6. M. Rab, Asymptotic relationships between the solutions of two systems of differential equations, Ann. Polon. Math. 30 Ž1974., 119᎐124. 7. M. Svec, Asymptotic relationship between solutions of two systems of differential equations, Czechaslo¨ ak. Math. J. 24 Nos. 99, 1 Ž1974., 44᎐58. 8. P. Talpalaru, Asymptotic behavior of perturbed difference equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Ser. VIIILXIV Ž1979., 563᎐571.