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Asymptotic behavior results for nonlinear neutral delay difference equations
Applied Mathematics and Computation 217 (2011) 7184–7190
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Asymptotic behavior results for nonlinear neutral delay difference equations Gengping Wei Department of Mathematics, Huaihua College, Huaihua, Hunan 418008, China
a r t i c l e
i n f o
a b s t r a c t This paper is concerned with the nonlinear neutral delay difference equation
Keywords: Neutral delay difference equation Boundedness Asymptotic behavior Liapunov functional
D½xðnÞ cðnÞxðn mÞ þ pðnÞf ðxðn kÞÞ ¼ 0;
n 2 Nðn0 Þ;
ðÞ
where D is the forward difference operator defined by D x(n) = x(n + 1) x(n), {c(n)} is a sequence of real numbers, {p(n)} is a positive sequence, f 2 C(R, R), m and k are positive integers, n0 is a nonnegative integer and N(n0) = {n0, n0 + 1, n0 + 2, . . .}. Sufficient conditions are obtained under which every solution of equation (⁄) is bounded and tends to a constant as n ? 1. Our results improve and extend some known results. One example is given to illustrate our results. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Qualitative theory of discrete processes has drawn considerable attention in recent years. In particular, oscillation properties of discrete analogs of delay differential equations have been studied by a number of authors, see, for example, [1–5] and the references cited therein. On the other hand, relatively little is known about the asymptotic behavior of all solutions of these discrete equations, see, for example, [6–15]. This paper considers the following nonlinear neutral delay difference equation
D½xðnÞ cðnÞxðn mÞ þ pðnÞf ðxðn kÞÞ ¼ 0;
n 2 Nðn0 Þ;
ð1Þ
where D is the forward difference operator defined by Dx(n) = x(n + 1) x(n), {c(n)} is a sequence of real numbers, {p(n)} is a positive sequence, f 2 C(R, R), m and k are positive integers, n0 is a nonnegative integer and N(n0) = {n0, n0 + 1, n0 + 2, . . .}. We note that when f(x) x, Eq. (1) reduces to the linear difference equation
D½xðnÞ cðnÞxðn mÞ þ pðnÞxðn kÞ ¼ 0;
n 2 Nðn0 Þ:
ð2Þ
The asymptotic behavior of solutions of linear difference equations has been studied by several authors, see, for example, [6–13]. In [10], it is proved that if c(n) 0, {p(n)} is a positive sequence and k is a positive integer such that
lim sup umni¼nk pðiÞ < 1 and n!1
1 X
pðnÞ ¼ 1;
n¼n0
then every solution of Eq. (2) tends to zero as n ? 1. While in [11], the authors studied the attractivity of Eq. (2) with c(n) c and obtained an improvement of the above result, which states that if c(n) c, jcj < 1, {p(n)} is a positive sequence and m, k are positive integers such that E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.02.004
G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190
( ) X nþk pðn þ m þ kÞ þ lim sup jcj 1 þ pðiÞ < 2 and pðn þ kÞ n!1 i¼nk
1 X
7185
pðnÞ ¼ 1;
n¼n0
then every solution of Eq. (2) tends to zero as n ? 1. However, relatively little is known about the asymptotic behavior of nonlinear difference equations, see, for example, [14,15]. When c(n) 0 in (1), in [15] the authors studied the nonlinear delay difference equation
DyðnÞ þ Fðn; yðn kÞÞ ¼ 0;
n 2 Nðn0 Þ
ðEÞ
and obtained the following asymptotic behavior result. Theorem. Let F: N(n0) R ? R be a real-value function, and for any n 2 N(n0), F(n, ) be a continuous function with F(n, 0) = 0. Assume that there exists a nonnegative sequence {q(n)} defined on N(n0) such that
0<
Fðn; xÞ 6 qðnÞ for n 2 Nðn0 Þ; x 2 R; x – 0; x
and
lim sup n!1
nþ2k X
qðiÞ < 2:
i¼n
Also suppose that for any constant a – 0 1 X
jFðn; aÞj ¼ 1:
i¼n
Then every solution of equation (E) tends to zero as n ? 1. The aim of this paper is to establish sufficient conditions, so that every solution of (1) is bounded and tends to a constant as n ? 1. The approach to the problem here is based on the Liapunov’s direct method. Our asymptotic behavior results include the main theorems in [10,11,15] as some special cases. So, our results improve and extend some known results. Let q = max{m, k}. By a solution of Eq. (1) we mean a sequence {x(n)} of real numbers which is defined for all n 2 N(n0 q) = {n0 q, n0 q + 1, n0 q + 2, . . .} and satisfies Eq. (1) for n 2 N(n0). It is easy to see that for any given n0 and initial conditions of the form x(n0 + j) = aj, j = q, q + 1, q + 2, . . . , 0, Eq. (1) has a unique solution {x(n)} which is defined for n 2 N(n0 q) and satisfies the above initial conditions. As is customary, a solution of (1) or (2) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, it will be called oscillatory. For the general background on difference equations, one can refer to [16,17]. 2. Main results In connection with the nonlinear function f, we assume that (H) there are constants L > 0 and M > 0 such that
xf ðxÞ > 0;
for x 2 R; x – 0 and L ¼ inf
f ðxÞ kxj > 0 ; x
M ¼ sup
f ðxÞ kxj > 0 : x
Theorem 1. Let (H) hold. Assume that
lim sup jcðnÞj ¼ l < 1; n!1 ! " # nþk X pðn þ m þ kÞ 2 þ lim sup l 1 þ 2 pði þ kÞ < : M n!1 L pðn þ kÞ i¼nk
ð3Þ ð4Þ
Then every solution of (1) is bounded. Proof. Let {x(n)} be any solution of (1). We shall prove that {x(n)} is bounded. For this purpose, we rewrite (1) in the form
D½xðnÞ cðnÞxðn mÞ
n1 X
pði þ kÞf ðxðiÞÞ þ pðn þ kÞf ðxðnÞÞ ¼ 0;
n P n0 :
i¼nk
From (3) and (4), we can select an e > 0 sufficiently small such that l + e < 1 and
ð5Þ
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G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190
" lim sup ðl þ eÞ 1 þ n!1
! pðn þ m þ kÞ
nþk X
þ
L2 pðn þ kÞ
# pði þ kÞ <
i¼nk
2 ; M
ð6Þ
also, we select n⁄ > n0 sufficiently large such that
for n P n ;
jcðnÞj 6 l þ e;
ð7Þ
and noting (H), we have
jcðnÞjx2 ðn mÞ 6
lþe L2
f 2 ðxðn mÞÞ;
n P n :
ð8Þ
Now we introduce three sequences as
"
n1 X
V 1 ðnÞ ¼ xðnÞ cðnÞxðn mÞ
#2 pði þ kÞf ðxðiÞÞ
n P n0 ;
;
ð9Þ
i¼nk
V 2 ðnÞ ¼
n1 X
pði þ 2kÞ
pðj þ kÞf 2 ðxðjÞÞ;
n P n0 ;
ð10Þ
j¼i
i¼nk
V 3 ðnÞ ¼
n1 X
n1 lþe X
L2
pði þ m þ kÞf 2 ðxðiÞÞ;
n P n0 :
ð11Þ
i¼nm
In what follows, for the sake of convenience, when we write a sequence inequality without specifying its domain of validity, we mean that it holds for all sufficiently large n. Calculating DV1(n), DV2(n) and DV3(n), we have
"
DV 1 ðnÞ ¼ D xðnÞ cðnÞxðn mÞ
#
n1 X
pði þ kÞf ðxðiÞÞ
i¼nk
"
n X
xðn þ 1Þ cðn þ 1Þxðn þ 1 mÞ
pði þ kÞf ðxðiÞÞ
i¼nkþ1
þxðnÞ cðnÞxðn mÞ
n1 X
#
pði þ kÞf ðxðiÞÞ
i¼nk
¼ pðn þ kÞ½2xðnÞf ðxðnÞÞ 2cðnÞxðn mÞf ðxðnÞÞ n1 X
2
pði þ kÞf ðxðiÞÞf ðxðnÞÞ pðn þ kÞf 2 ðxðnÞÞ
i¼nk
6 pðn þ kÞ 2xðnÞf ðxðnÞÞ jcðnÞjðx2 ðn mÞ þ f 2 ðxðnÞÞÞ # n1 X pði þ kÞðf 2 ðxðiÞÞ þ f 2 ðxðnÞÞÞ pðn þ kÞf 2 ðxðnÞÞ i¼nk
¼ pðn þ kÞ 2xðnÞf ðxðnÞÞ jcðnÞjx2 ðn mÞ jcðnÞjf 2 ðxðnÞÞ # n1 n X X 2 2 pði þ kÞf ðxðiÞÞ pði þ kÞf ðxðnÞÞ i¼nk
ð12Þ
i¼nk
DV 2 ðnÞ ¼ pðn þ kÞ
n1 X
pði þ kÞf 2 ðxðiÞÞ þ pðn þ kÞf 2 ðxðnÞÞ
i¼nk
n X
pði þ 2kÞ
ð13Þ
i¼nkþ1
and
DV 3 ðnÞ ¼
lþe L2
pðn þ m þ kÞf 2 ðxðnÞÞ
lþe L2
pðn þ kÞf 2 ðxðn mÞÞ:
Set V(n) = V1(n) + V2(n) + V3(n). By (12), (13), (14) and (6), (7), (8), we get
DVðnÞ ¼ DV 1 ðnÞ þ DV 2 ðnÞ þ DV 3 ðnÞ 6 pðn þ kÞ 2xðnÞf ðxðnÞÞ jcðnÞjx2 ðn mÞ jcðnÞjf 2 ðxðnÞÞ n1 n X X pði þ kÞf 2 ðxðiÞÞ f 2 ðxðnÞÞ pði þ kÞ i¼nk
þ
n1 X i¼nk
i¼nk
pði þ kÞf 2 ðxðiÞÞ f 2 ðxðnÞÞ
n X i¼nkþ1
pði þ 2kÞ
ð14Þ
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G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190
pðn þ m þ kÞ l þ e 2 þ f ðxðn mÞÞ pðn þ kÞ L2 L2 " nþk X 2xðnÞ jcðnÞj pði þ kÞ 6 pðn þ kÞf 2 ðxðnÞÞ f ðxðnÞÞ i¼nk l þ e pðn þ m þ kÞ 2 pðn þ kÞ L " ! 2xðnÞ pðn þ m þ kÞ ðl þ eÞ 1 þ 2 6 pðn þ kÞf 2 ðxðnÞÞ f ðxðnÞÞ L pðn þ kÞ # nþk X pði þ kÞ
lþe
f 2 ðxðnÞÞ
i¼nk
(
" ! 2 pðn þ m þ kÞ ðl þ eÞ 1 þ 2 6 pðn þ kÞf ðxðnÞÞ M L pðn þ kÞ #) nþk X þ pði þ kÞ ; 2
ð15Þ
i¼nk
which, together with (6), implies 1 X
pðn þ kÞf 2 ðxðnÞÞ < 1;
ð16Þ
n¼n0
and, hence, for any positive integer r we have
lim
n!1
n1 X
pði þ kÞf 2 ðxðiÞÞ ¼ 0:
ð17Þ
i¼nr
Noting (6), there is a sufficiently large positive integer n1 P n0 such that for n P n1 + k we have
0 6 V 2 ðnÞ ¼
n1 X i¼nk
pði þ 2kÞ
n1 X
n1 X
pðj þ kÞf 2 ðxðjÞÞ 6
j¼i
pði þ 2kÞ
i¼nk
n1 X j¼nk
pðj þ kÞf 2 ðxðjÞÞ 6
n1 2 X pðj þ kÞf 2 ðxðjÞÞ M j¼nk
ð18Þ
and
0 6 V 3 ðnÞ 6
n1 2 X pði þ kÞf 2 ðxðiÞÞ: M i¼nm
ð19Þ
(18) and (19) together with (17) imply limn?1V2(n) = 0 and limn?1V3(n) = 0. On the other hand, by (6) and (15), we see that V(n) is eventually decreasing. In view of V(n) P 0, the limit limn?1V(n) = a exists and is finite, thus, limn?1V(n) = limn?1V1(n) = a, that is,
"
n1 X
lim xðnÞ cðnÞxðn mÞ
n!1
#2 pði þ kÞf ðxðiÞÞ
¼ a:
ð20Þ
i¼nk
Let
yðnÞ ¼ xðnÞ cðnÞxðn mÞ
n1 X
pði þ kÞf ðxðiÞÞ;
i¼nk
then
DyðnÞ þ pðn þ kÞf ðxðnÞÞ ¼ 0;
n P n0
ð21Þ
and
lim y2 ðnÞ ¼ a;
n!1
that is 1
lim jyðnÞj ¼ a2 :
n!1
We claim that {y(n)} converges. In fact, this is clear if a = 0. If a > 0, it suffices to show that {y(n)} is eventually positive or 1 eventually negative. Otherwise, choose a number g such that 0 < g < a2 and let N be a positive integer such that
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G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190 1
1
a2 g < jyðnÞj < a2 þ g; n P N;
ð22Þ
and let
J ¼ fn P N : yðnÞ < 0g;
K ¼ fn P N : yðnÞ > 0g:
Since {y(n)} is neither eventually positive nor eventually negative, it follows that J and K are unbounded, thus we may choose a divergent sequence of integers {nj} such that N 6 n1 < n2 < < nj < , nj 2 K, nj + 1 2 J. Then, y(nj + 1) < 0 and y(nj) > 0. Furthermore, by (22), we easily see that
1 1 2 a2 g < yðnj þ 1Þ yðnj Þ < 2 a2 þ g ;
j P 1:
Therefore, in view of (21), we have
1 1 0 < 2 a2 g < pðnj þ kÞf ðxðnj ÞÞ < 2 a2 þ g ;
j P 1:
ð23Þ
On the other hand, by (17) and (23) we see that {f(x(nj))} converges to zero. Noting the fact that condition (4) implies {p(n)} is bounded, we get
pðnj þ kÞf ðxðnj ÞÞ ! 0;
as j ! 1;
which contradicts (23). Thus {y(n)} must converge. So,
" lim yðnÞ ¼ lim xðnÞ cðnÞxðn mÞ
n!1
where b ¼ n1 X
n!1
n1 X
# pði þ kÞf ðxðiÞÞ ¼ b;
ð24Þ
i¼nk
pffiffiffi
pffiffiffi
a or b ¼ a and is finite. In view of (21), we have
pði þ kÞf ðxðiÞÞ ¼ yðn kÞ yðnÞ:
i¼nk
So,
lim
n!1
n1 X
pði þ kÞf ðxðiÞÞ ¼ 0:
ð25Þ
i¼nk
By (24) and (25), we have
lim ½xðnÞ cðnÞxðn mÞ ¼ b:
ð26Þ
n!1
Next, we shall show {jx(n)j} is bounded. In fact, if {jx(n)j} is unbounded, then there exists a divergent sequence of integers {nj} such that jx(nj)j ? 1, as j ? 1 and
jxðnj Þj ¼
sup
jxðnÞj:
ð27Þ
n0 q6n6nj
Thus, noticing (7) and (27), we have
jxðnj Þ cðnj Þxðnj mÞj P jxðnj Þj jcðnj Þjjxðnj mÞj P jxðnj Þjð1 jcðnj ÞjÞ P jxðnj Þj½1 ðl þ eÞ ! 1;
as j ! 1
which contradicts (26). So {jx(n)j} is bounded. The proof of Theorem 1 is complete. h Now we study asymptotic behavior of solutions of (1). Theorem 2. Let (H) hold. Assume that c(n) P 0 or c(n) 6 0 for sufficiently large n and
lim jcðnÞj ¼ c < 1
ð28Þ
n!1
and
" lim sup c 1 þ n!1
pðn þ m þ kÞ 2
L pðn þ kÞ
! þ
nþk X
# pði þ kÞ <
i¼nk
2 : M
ð29Þ
Then every solution of (1) tends to a constant as n ? 1. Proof. Let {x(n)} be any solution of (1). From the proof of Theorem 1 we know that {jx(n)j} is bounded and (26) holds. Next, we shall prove that
lim xðnÞ exists and is finite:
n!1
ð30Þ
G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190
7189
If c = 0, clearly limn?1x(n) = b, which shows (30) holds. If 0 < c < 1, let
lim sup xðnÞ ¼ x1 ; n!1
lim inf xðnÞ ¼ x2 ; n!1
and let {si} and {ti} be two integer sequences such that si ? 1, ti ? 1 as i ? 1, and
lim xðsi Þ ¼ x1 ; i!1
lim xðt i Þ ¼ x2 : i!1
For n > n2, where n2 is a sufficiently large integer, we consider the following two possible cases. Case 1. 1 < c(n) 6 0 for n > n2, we have
b ¼ lim½xðsi Þ cðsi Þxðsi mÞ ¼ x1 þ c lim xðsi mÞ P x1 þ cx2 ; i!1
i!1
and
b ¼ lim½xðti Þ cðti Þxðt i mÞ ¼ x2 þ c lim xðt i mÞ 6 x2 þ cx1 : i!1
i!1
b Thus, 0 6 x1 x2 6 c(x1 x2), so that x1 ¼ x2 ¼ 1þc , which shows (30) holds. Case 2. 0 6 c(t) < 1 for n > n2, we have
x1 ¼ lim xðsi Þ ¼ lim½xðsi Þ cðsi Þxðsi mÞ þ cðsi Þxðsi mÞ ¼ b þ c lim xðsi mÞ 6 b þ cx1 ; i!1
i!1
i!1
and
x2 ¼ lim xðti Þ ¼ lim½xðti Þ cðti Þxðti mÞ þ cðti Þxðti mÞ ¼ b þ c lim xðti mÞ P b þ cx2 ; i!1
i!1
b 1c
Thus, x1 6 6 x2 , which together with x1 P x2 implies x1 ¼ x2 ¼ complete. h
i!1
b , 1c
so that (30) holds. The proof of Theorem 2 is
By Theorem 2, we have the following asymptotic behavior result immediately. Theorem 3. The conditions of Theorem 2 imply that every oscillatory solution of (1) tends to zero as n ? 1. Theorem 4. The conditions in Theorem 2 together with 1 X
pðnÞ ¼ 1
ð31Þ
n¼n0
imply that every solution of (1) tends to zero as n ? 1. Proof. Condition (H) implies that for any d > 0 there is r > 0 such that
jf ðxÞj P r;
for jxj P d:
ð32Þ
By theorem 3, we only have to prove that every nonoscillatory solution of (1) tends to zero as n ? 1. Without loss of generality, let {x(n)} be an eventually positive solution of (1), we shall prove limn?1x(n) = 0. As in the proof of Theorem 1, we can rewrite (1) in the form of (21). Summing from n0 to n on both sides of (21) produces n X
pði þ kÞf ðxðiÞÞ ¼ yðn0 Þ yðn þ 1Þ:
i¼n0
By using (24) we have 1 X
pði þ kÞf ðxðiÞÞ < 1;
i¼n0
which, together with (31) yields liminfn?1f(x(n)) = 0. We claim that
lim inf xðnÞ ¼ 0: n!1
ð33Þ
Let {ui} be an integer sequence such that ui ? 1 as i ? 1 and limi?1f(x(ui)) = 0. We must have liminfi?1x(ui) = d = 0. In fact, if d > 0, then there is a subsequence fuij g of {ui} such that xðuij Þ P d=2 for sufficiently large j. By (32) we have f ðxðuij ÞÞ P n for some n > 0 and sufficiently large j, which yields a contradiction because of limj!1 f ðxðuij ÞÞ ¼ 0: Therefore, (33) holds. On the other hand, by Theorem 2, we have limn?1x(n) exists. Therefore limn?1x(n) = 0. Thus the proof is complete. h As application of Theorems 2 and 4, taking f(x) x, for the linear equation (2) we have the following two corollaries.
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G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190
Corollary 1. Let c(n) P 0 or c(n) 6 0 for sufficiently large n. Assume that(28) holds and
" lim sup cð1 þ n!1
# nþk X pðn þ m þ kÞ Þþ pði þ kÞ < 2: pðn þ kÞ i¼nk
Then every solution of Eq. (2) tends to a constant as n ? 1. Furthermore, if (31) holds, then every solution of Eq. (2) tends to zero as n ? 1. Corollary 2. Let c(n) c and jcj < 1. Assume that (31) holds and
(
) nþk X pðn þ m þ kÞ lim sup jcjð1 þ pði þ kÞ < 2: Þþ pðn þ kÞ n!1 i¼nk Then every solution of Eq. (2) tends to zero as n ? 1. 3. Example Consider the difference equation
D½xðnÞ
n1 1 2 xðn 1Þ þ ½1 þ sin xðn 2Þxðn 2Þ ¼ 0; 2n ðn 1Þa
where a > 0 is a real constant, f ðxÞ ¼ ð1 þ sin xÞx; cðnÞ ¼ n1 ; pðnÞ ¼ 1=ðn 1Þa ; m ¼ 1; k ¼ 2. By simple estimation, we have 2n 2
1 2 2 2jxj 6 jð2 þ sin xÞxj 6 3jxj;
l ¼ n!1 lim jcðnÞj ¼ < 1;
2
x2 ð1 þ sin xÞ > 0 ðx – 0Þ
and
" lim sup
lð1 þ
n!1
pðn þ m þ kÞ 22 pðn þ kÞ
Þþ
nþk X i¼nk
# pði þ kÞ ¼
5 2 < : 8 3
If a > 0, we may conclude from Theorems 1 and 2 that every solution of this equation is bounded and tends to a constant as n ? 1; and if 0 < a 6 1, from Theorem 4, every solution of this equation tends to zero as n ? 1. Acknowledgements Supported by the NNSF of China (No. 10871062) and by Hunan Provincial Natural Science Foundation of China (No. 07JJ6010). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Asymptotic behavior results for nonlinear neutral delay difference equations Gengping Wei Department of Mathematics, Huaihua College, Huaihua, Hunan 418008, China
a r t i c l e
i n f o
a b s t r a c t This paper is concerned with the nonlinear neutral delay difference equation
Keywords: Neutral delay difference equation Boundedness Asymptotic behavior Liapunov functional
D½xðnÞ cðnÞxðn mÞ þ pðnÞf ðxðn kÞÞ ¼ 0;
n 2 Nðn0 Þ;
ðÞ
where D is the forward difference operator defined by D x(n) = x(n + 1) x(n), {c(n)} is a sequence of real numbers, {p(n)} is a positive sequence, f 2 C(R, R), m and k are positive integers, n0 is a nonnegative integer and N(n0) = {n0, n0 + 1, n0 + 2, . . .}. Sufficient conditions are obtained under which every solution of equation (⁄) is bounded and tends to a constant as n ? 1. Our results improve and extend some known results. One example is given to illustrate our results. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Qualitative theory of discrete processes has drawn considerable attention in recent years. In particular, oscillation properties of discrete analogs of delay differential equations have been studied by a number of authors, see, for example, [1–5] and the references cited therein. On the other hand, relatively little is known about the asymptotic behavior of all solutions of these discrete equations, see, for example, [6–15]. This paper considers the following nonlinear neutral delay difference equation
D½xðnÞ cðnÞxðn mÞ þ pðnÞf ðxðn kÞÞ ¼ 0;
n 2 Nðn0 Þ;
ð1Þ
where D is the forward difference operator defined by Dx(n) = x(n + 1) x(n), {c(n)} is a sequence of real numbers, {p(n)} is a positive sequence, f 2 C(R, R), m and k are positive integers, n0 is a nonnegative integer and N(n0) = {n0, n0 + 1, n0 + 2, . . .}. We note that when f(x) x, Eq. (1) reduces to the linear difference equation
D½xðnÞ cðnÞxðn mÞ þ pðnÞxðn kÞ ¼ 0;
n 2 Nðn0 Þ:
ð2Þ
The asymptotic behavior of solutions of linear difference equations has been studied by several authors, see, for example, [6–13]. In [10], it is proved that if c(n) 0, {p(n)} is a positive sequence and k is a positive integer such that
lim sup umni¼nk pðiÞ < 1 and n!1
1 X
pðnÞ ¼ 1;
n¼n0
then every solution of Eq. (2) tends to zero as n ? 1. While in [11], the authors studied the attractivity of Eq. (2) with c(n) c and obtained an improvement of the above result, which states that if c(n) c, jcj < 1, {p(n)} is a positive sequence and m, k are positive integers such that E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.02.004
G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190
( ) X nþk pðn þ m þ kÞ þ lim sup jcj 1 þ pðiÞ < 2 and pðn þ kÞ n!1 i¼nk
1 X
7185
pðnÞ ¼ 1;
n¼n0
then every solution of Eq. (2) tends to zero as n ? 1. However, relatively little is known about the asymptotic behavior of nonlinear difference equations, see, for example, [14,15]. When c(n) 0 in (1), in [15] the authors studied the nonlinear delay difference equation
DyðnÞ þ Fðn; yðn kÞÞ ¼ 0;
n 2 Nðn0 Þ
ðEÞ
and obtained the following asymptotic behavior result. Theorem. Let F: N(n0) R ? R be a real-value function, and for any n 2 N(n0), F(n, ) be a continuous function with F(n, 0) = 0. Assume that there exists a nonnegative sequence {q(n)} defined on N(n0) such that
0<
Fðn; xÞ 6 qðnÞ for n 2 Nðn0 Þ; x 2 R; x – 0; x
and
lim sup n!1
nþ2k X
qðiÞ < 2:
i¼n
Also suppose that for any constant a – 0 1 X
jFðn; aÞj ¼ 1:
i¼n
Then every solution of equation (E) tends to zero as n ? 1. The aim of this paper is to establish sufficient conditions, so that every solution of (1) is bounded and tends to a constant as n ? 1. The approach to the problem here is based on the Liapunov’s direct method. Our asymptotic behavior results include the main theorems in [10,11,15] as some special cases. So, our results improve and extend some known results. Let q = max{m, k}. By a solution of Eq. (1) we mean a sequence {x(n)} of real numbers which is defined for all n 2 N(n0 q) = {n0 q, n0 q + 1, n0 q + 2, . . .} and satisfies Eq. (1) for n 2 N(n0). It is easy to see that for any given n0 and initial conditions of the form x(n0 + j) = aj, j = q, q + 1, q + 2, . . . , 0, Eq. (1) has a unique solution {x(n)} which is defined for n 2 N(n0 q) and satisfies the above initial conditions. As is customary, a solution of (1) or (2) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, it will be called oscillatory. For the general background on difference equations, one can refer to [16,17]. 2. Main results In connection with the nonlinear function f, we assume that (H) there are constants L > 0 and M > 0 such that
xf ðxÞ > 0;
for x 2 R; x – 0 and L ¼ inf
f ðxÞ kxj > 0 ; x
M ¼ sup
f ðxÞ kxj > 0 : x
Theorem 1. Let (H) hold. Assume that
lim sup jcðnÞj ¼ l < 1; n!1 ! " # nþk X pðn þ m þ kÞ 2 þ lim sup l 1 þ 2 pði þ kÞ < : M n!1 L pðn þ kÞ i¼nk
ð3Þ ð4Þ
Then every solution of (1) is bounded. Proof. Let {x(n)} be any solution of (1). We shall prove that {x(n)} is bounded. For this purpose, we rewrite (1) in the form
D½xðnÞ cðnÞxðn mÞ
n1 X
pði þ kÞf ðxðiÞÞ þ pðn þ kÞf ðxðnÞÞ ¼ 0;
n P n0 :
i¼nk
From (3) and (4), we can select an e > 0 sufficiently small such that l + e < 1 and
ð5Þ
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G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190
" lim sup ðl þ eÞ 1 þ n!1
! pðn þ m þ kÞ
nþk X
þ
L2 pðn þ kÞ
# pði þ kÞ <
i¼nk
2 ; M
ð6Þ
also, we select n⁄ > n0 sufficiently large such that
for n P n ;
jcðnÞj 6 l þ e;
ð7Þ
and noting (H), we have
jcðnÞjx2 ðn mÞ 6
lþe L2
f 2 ðxðn mÞÞ;
n P n :
ð8Þ
Now we introduce three sequences as
"
n1 X
V 1 ðnÞ ¼ xðnÞ cðnÞxðn mÞ
#2 pði þ kÞf ðxðiÞÞ
n P n0 ;
;
ð9Þ
i¼nk
V 2 ðnÞ ¼
n1 X
pði þ 2kÞ
pðj þ kÞf 2 ðxðjÞÞ;
n P n0 ;
ð10Þ
j¼i
i¼nk
V 3 ðnÞ ¼
n1 X
n1 lþe X
L2
pði þ m þ kÞf 2 ðxðiÞÞ;
n P n0 :
ð11Þ
i¼nm
In what follows, for the sake of convenience, when we write a sequence inequality without specifying its domain of validity, we mean that it holds for all sufficiently large n. Calculating DV1(n), DV2(n) and DV3(n), we have
"
DV 1 ðnÞ ¼ D xðnÞ cðnÞxðn mÞ
#
n1 X
pði þ kÞf ðxðiÞÞ
i¼nk
"
n X
xðn þ 1Þ cðn þ 1Þxðn þ 1 mÞ
pði þ kÞf ðxðiÞÞ
i¼nkþ1
þxðnÞ cðnÞxðn mÞ
n1 X
#
pði þ kÞf ðxðiÞÞ
i¼nk
¼ pðn þ kÞ½2xðnÞf ðxðnÞÞ 2cðnÞxðn mÞf ðxðnÞÞ n1 X
2
pði þ kÞf ðxðiÞÞf ðxðnÞÞ pðn þ kÞf 2 ðxðnÞÞ
i¼nk
6 pðn þ kÞ 2xðnÞf ðxðnÞÞ jcðnÞjðx2 ðn mÞ þ f 2 ðxðnÞÞÞ # n1 X pði þ kÞðf 2 ðxðiÞÞ þ f 2 ðxðnÞÞÞ pðn þ kÞf 2 ðxðnÞÞ i¼nk
¼ pðn þ kÞ 2xðnÞf ðxðnÞÞ jcðnÞjx2 ðn mÞ jcðnÞjf 2 ðxðnÞÞ # n1 n X X 2 2 pði þ kÞf ðxðiÞÞ pði þ kÞf ðxðnÞÞ i¼nk
ð12Þ
i¼nk
DV 2 ðnÞ ¼ pðn þ kÞ
n1 X
pði þ kÞf 2 ðxðiÞÞ þ pðn þ kÞf 2 ðxðnÞÞ
i¼nk
n X
pði þ 2kÞ
ð13Þ
i¼nkþ1
and
DV 3 ðnÞ ¼
lþe L2
pðn þ m þ kÞf 2 ðxðnÞÞ
lþe L2
pðn þ kÞf 2 ðxðn mÞÞ:
Set V(n) = V1(n) + V2(n) + V3(n). By (12), (13), (14) and (6), (7), (8), we get
DVðnÞ ¼ DV 1 ðnÞ þ DV 2 ðnÞ þ DV 3 ðnÞ 6 pðn þ kÞ 2xðnÞf ðxðnÞÞ jcðnÞjx2 ðn mÞ jcðnÞjf 2 ðxðnÞÞ n1 n X X pði þ kÞf 2 ðxðiÞÞ f 2 ðxðnÞÞ pði þ kÞ i¼nk
þ
n1 X i¼nk
i¼nk
pði þ kÞf 2 ðxðiÞÞ f 2 ðxðnÞÞ
n X i¼nkþ1
pði þ 2kÞ
ð14Þ
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G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190
pðn þ m þ kÞ l þ e 2 þ f ðxðn mÞÞ pðn þ kÞ L2 L2 " nþk X 2xðnÞ jcðnÞj pði þ kÞ 6 pðn þ kÞf 2 ðxðnÞÞ f ðxðnÞÞ i¼nk l þ e pðn þ m þ kÞ 2 pðn þ kÞ L " ! 2xðnÞ pðn þ m þ kÞ ðl þ eÞ 1 þ 2 6 pðn þ kÞf 2 ðxðnÞÞ f ðxðnÞÞ L pðn þ kÞ # nþk X pði þ kÞ
lþe
f 2 ðxðnÞÞ
i¼nk
(
" ! 2 pðn þ m þ kÞ ðl þ eÞ 1 þ 2 6 pðn þ kÞf ðxðnÞÞ M L pðn þ kÞ #) nþk X þ pði þ kÞ ; 2
ð15Þ
i¼nk
which, together with (6), implies 1 X
pðn þ kÞf 2 ðxðnÞÞ < 1;
ð16Þ
n¼n0
and, hence, for any positive integer r we have
lim
n!1
n1 X
pði þ kÞf 2 ðxðiÞÞ ¼ 0:
ð17Þ
i¼nr
Noting (6), there is a sufficiently large positive integer n1 P n0 such that for n P n1 + k we have
0 6 V 2 ðnÞ ¼
n1 X i¼nk
pði þ 2kÞ
n1 X
n1 X
pðj þ kÞf 2 ðxðjÞÞ 6
j¼i
pði þ 2kÞ
i¼nk
n1 X j¼nk
pðj þ kÞf 2 ðxðjÞÞ 6
n1 2 X pðj þ kÞf 2 ðxðjÞÞ M j¼nk
ð18Þ
and
0 6 V 3 ðnÞ 6
n1 2 X pði þ kÞf 2 ðxðiÞÞ: M i¼nm
ð19Þ
(18) and (19) together with (17) imply limn?1V2(n) = 0 and limn?1V3(n) = 0. On the other hand, by (6) and (15), we see that V(n) is eventually decreasing. In view of V(n) P 0, the limit limn?1V(n) = a exists and is finite, thus, limn?1V(n) = limn?1V1(n) = a, that is,
"
n1 X
lim xðnÞ cðnÞxðn mÞ
n!1
#2 pði þ kÞf ðxðiÞÞ
¼ a:
ð20Þ
i¼nk
Let
yðnÞ ¼ xðnÞ cðnÞxðn mÞ
n1 X
pði þ kÞf ðxðiÞÞ;
i¼nk
then
DyðnÞ þ pðn þ kÞf ðxðnÞÞ ¼ 0;
n P n0
ð21Þ
and
lim y2 ðnÞ ¼ a;
n!1
that is 1
lim jyðnÞj ¼ a2 :
n!1
We claim that {y(n)} converges. In fact, this is clear if a = 0. If a > 0, it suffices to show that {y(n)} is eventually positive or 1 eventually negative. Otherwise, choose a number g such that 0 < g < a2 and let N be a positive integer such that
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G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190 1
1
a2 g < jyðnÞj < a2 þ g; n P N;
ð22Þ
and let
J ¼ fn P N : yðnÞ < 0g;
K ¼ fn P N : yðnÞ > 0g:
Since {y(n)} is neither eventually positive nor eventually negative, it follows that J and K are unbounded, thus we may choose a divergent sequence of integers {nj} such that N 6 n1 < n2 < < nj < , nj 2 K, nj + 1 2 J. Then, y(nj + 1) < 0 and y(nj) > 0. Furthermore, by (22), we easily see that
1 1 2 a2 g < yðnj þ 1Þ yðnj Þ < 2 a2 þ g ;
j P 1:
Therefore, in view of (21), we have
1 1 0 < 2 a2 g < pðnj þ kÞf ðxðnj ÞÞ < 2 a2 þ g ;
j P 1:
ð23Þ
On the other hand, by (17) and (23) we see that {f(x(nj))} converges to zero. Noting the fact that condition (4) implies {p(n)} is bounded, we get
pðnj þ kÞf ðxðnj ÞÞ ! 0;
as j ! 1;
which contradicts (23). Thus {y(n)} must converge. So,
" lim yðnÞ ¼ lim xðnÞ cðnÞxðn mÞ
n!1
where b ¼ n1 X
n!1
n1 X
# pði þ kÞf ðxðiÞÞ ¼ b;
ð24Þ
i¼nk
pffiffiffi
pffiffiffi
a or b ¼ a and is finite. In view of (21), we have
pði þ kÞf ðxðiÞÞ ¼ yðn kÞ yðnÞ:
i¼nk
So,
lim
n!1
n1 X
pði þ kÞf ðxðiÞÞ ¼ 0:
ð25Þ
i¼nk
By (24) and (25), we have
lim ½xðnÞ cðnÞxðn mÞ ¼ b:
ð26Þ
n!1
Next, we shall show {jx(n)j} is bounded. In fact, if {jx(n)j} is unbounded, then there exists a divergent sequence of integers {nj} such that jx(nj)j ? 1, as j ? 1 and
jxðnj Þj ¼
sup
jxðnÞj:
ð27Þ
n0 q6n6nj
Thus, noticing (7) and (27), we have
jxðnj Þ cðnj Þxðnj mÞj P jxðnj Þj jcðnj Þjjxðnj mÞj P jxðnj Þjð1 jcðnj ÞjÞ P jxðnj Þj½1 ðl þ eÞ ! 1;
as j ! 1
which contradicts (26). So {jx(n)j} is bounded. The proof of Theorem 1 is complete. h Now we study asymptotic behavior of solutions of (1). Theorem 2. Let (H) hold. Assume that c(n) P 0 or c(n) 6 0 for sufficiently large n and
lim jcðnÞj ¼ c < 1
ð28Þ
n!1
and
" lim sup c 1 þ n!1
pðn þ m þ kÞ 2
L pðn þ kÞ
! þ
nþk X
# pði þ kÞ <
i¼nk
2 : M
ð29Þ
Then every solution of (1) tends to a constant as n ? 1. Proof. Let {x(n)} be any solution of (1). From the proof of Theorem 1 we know that {jx(n)j} is bounded and (26) holds. Next, we shall prove that
lim xðnÞ exists and is finite:
n!1
ð30Þ
G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190
7189
If c = 0, clearly limn?1x(n) = b, which shows (30) holds. If 0 < c < 1, let
lim sup xðnÞ ¼ x1 ; n!1
lim inf xðnÞ ¼ x2 ; n!1
and let {si} and {ti} be two integer sequences such that si ? 1, ti ? 1 as i ? 1, and
lim xðsi Þ ¼ x1 ; i!1
lim xðt i Þ ¼ x2 : i!1
For n > n2, where n2 is a sufficiently large integer, we consider the following two possible cases. Case 1. 1 < c(n) 6 0 for n > n2, we have
b ¼ lim½xðsi Þ cðsi Þxðsi mÞ ¼ x1 þ c lim xðsi mÞ P x1 þ cx2 ; i!1
i!1
and
b ¼ lim½xðti Þ cðti Þxðt i mÞ ¼ x2 þ c lim xðt i mÞ 6 x2 þ cx1 : i!1
i!1
b Thus, 0 6 x1 x2 6 c(x1 x2), so that x1 ¼ x2 ¼ 1þc , which shows (30) holds. Case 2. 0 6 c(t) < 1 for n > n2, we have
x1 ¼ lim xðsi Þ ¼ lim½xðsi Þ cðsi Þxðsi mÞ þ cðsi Þxðsi mÞ ¼ b þ c lim xðsi mÞ 6 b þ cx1 ; i!1
i!1
i!1
and
x2 ¼ lim xðti Þ ¼ lim½xðti Þ cðti Þxðti mÞ þ cðti Þxðti mÞ ¼ b þ c lim xðti mÞ P b þ cx2 ; i!1
i!1
b 1c
Thus, x1 6 6 x2 , which together with x1 P x2 implies x1 ¼ x2 ¼ complete. h
i!1
b , 1c
so that (30) holds. The proof of Theorem 2 is
By Theorem 2, we have the following asymptotic behavior result immediately. Theorem 3. The conditions of Theorem 2 imply that every oscillatory solution of (1) tends to zero as n ? 1. Theorem 4. The conditions in Theorem 2 together with 1 X
pðnÞ ¼ 1
ð31Þ
n¼n0
imply that every solution of (1) tends to zero as n ? 1. Proof. Condition (H) implies that for any d > 0 there is r > 0 such that
jf ðxÞj P r;
for jxj P d:
ð32Þ
By theorem 3, we only have to prove that every nonoscillatory solution of (1) tends to zero as n ? 1. Without loss of generality, let {x(n)} be an eventually positive solution of (1), we shall prove limn?1x(n) = 0. As in the proof of Theorem 1, we can rewrite (1) in the form of (21). Summing from n0 to n on both sides of (21) produces n X
pði þ kÞf ðxðiÞÞ ¼ yðn0 Þ yðn þ 1Þ:
i¼n0
By using (24) we have 1 X
pði þ kÞf ðxðiÞÞ < 1;
i¼n0
which, together with (31) yields liminfn?1f(x(n)) = 0. We claim that
lim inf xðnÞ ¼ 0: n!1
ð33Þ
Let {ui} be an integer sequence such that ui ? 1 as i ? 1 and limi?1f(x(ui)) = 0. We must have liminfi?1x(ui) = d = 0. In fact, if d > 0, then there is a subsequence fuij g of {ui} such that xðuij Þ P d=2 for sufficiently large j. By (32) we have f ðxðuij ÞÞ P n for some n > 0 and sufficiently large j, which yields a contradiction because of limj!1 f ðxðuij ÞÞ ¼ 0: Therefore, (33) holds. On the other hand, by Theorem 2, we have limn?1x(n) exists. Therefore limn?1x(n) = 0. Thus the proof is complete. h As application of Theorems 2 and 4, taking f(x) x, for the linear equation (2) we have the following two corollaries.
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G. Wei / Applied Mathematics and Computation 217 (2011) 7184–7190
Corollary 1. Let c(n) P 0 or c(n) 6 0 for sufficiently large n. Assume that(28) holds and
" lim sup cð1 þ n!1
# nþk X pðn þ m þ kÞ Þþ pði þ kÞ < 2: pðn þ kÞ i¼nk
Then every solution of Eq. (2) tends to a constant as n ? 1. Furthermore, if (31) holds, then every solution of Eq. (2) tends to zero as n ? 1. Corollary 2. Let c(n) c and jcj < 1. Assume that (31) holds and
(
) nþk X pðn þ m þ kÞ lim sup jcjð1 þ pði þ kÞ < 2: Þþ pðn þ kÞ n!1 i¼nk Then every solution of Eq. (2) tends to zero as n ? 1. 3. Example Consider the difference equation
D½xðnÞ
n1 1 2 xðn 1Þ þ ½1 þ sin xðn 2Þxðn 2Þ ¼ 0; 2n ðn 1Þa
where a > 0 is a real constant, f ðxÞ ¼ ð1 þ sin xÞx; cðnÞ ¼ n1 ; pðnÞ ¼ 1=ðn 1Þa ; m ¼ 1; k ¼ 2. By simple estimation, we have 2n 2
1 2 2 2jxj 6 jð2 þ sin xÞxj 6 3jxj;
l ¼ n!1 lim jcðnÞj ¼ < 1;
2
x2 ð1 þ sin xÞ > 0 ðx – 0Þ
and
" lim sup
lð1 þ
n!1
pðn þ m þ kÞ 22 pðn þ kÞ
Þþ
nþk X i¼nk
# pði þ kÞ ¼
5 2 < : 8 3
If a > 0, we may conclude from Theorems 1 and 2 that every solution of this equation is bounded and tends to a constant as n ? 1; and if 0 < a 6 1, from Theorem 4, every solution of this equation tends to zero as n ? 1. Acknowledgements Supported by the NNSF of China (No. 10871062) and by Hunan Provincial Natural Science Foundation of China (No. 07JJ6010). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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