Electronic Journal of Differential Equations, Vol. 2010(2010), No. 146, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF NON-OSCILLATORY SOLUTIONS FOR A
HIGHER-ORDER NONLINEAR NEUTRAL DIFFERENCE
EQUATION
ZHENYU GUO, MIN LIU
Abstract. This article concerns the solvability of the higher-order nonlinear
neutral delay difference equation
s
“
`
´” X
∆ akn . . . ∆ a2n ∆(a1n ∆(xn + bn xn−d )) +
pjn fj (xn−rjn ) = qn ,
j=1
where n ≥ n0 ≥ 0, d, k, j, s are positive integers, fj : R → R and xfj (x) ≥ 0
for x 6= 0. Sufficient conditions for the existence of non-oscillatory solutions
are established by using Krasnoselskii fixed point theorem. Five theorems are
stated according to the range of the sequence {bn }.
1. Introduction and preliminaries
Interest in the solvability of difference equations has increased lately, as inferred
from the number of related publications; see for example the references in this article
and their references. Authors have examined various types difference equations, as
follows:
n ≥ 0,
in [14],
∆(an ∆xn ) = qn f (xn+1 ),
n ≥ 0,
in [11],
(1.2)
n ≥ n0 ,
in [6],
(1.3)
∆(an ∆xn ) + pn xg(n) = 0,
∆(an ∆xn ) = qn xn+1 ,
2
∆ (xn + pxn−m ) + pn xn−k − qn xn−l = 0,
2
∆ (xn + pxn−k ) + f (n, xn ) = 0,
∆2 (xn − pxn−τ ) =
m
X
qi fi (xn−σi ),
n ≥ 1,
(1.1)
in [10],
n ≥ n0 ,
in [9],
(1.4)
(1.5)
i=1
∆(an ∆(xn + bxn−τ )) + f (n, xn−d1n , xn−d2n , . . . , xn−dkn ) = cn ,
n ≥ n0 ,
in [8],
m
(1.6)
∆ (xn + cxn−k ) + pn xn−r = 0, n ≥ n0 ,
m
∆ (xn + cn xn−k ) + pn f (xn−r ) = 0,
n ≥ n0 ,
in [15],
in [3, 4, 12, 13],
2000 Mathematics Subject Classification. 34K15, 34C10.
Key words and phrases. Nonoscillatory solution; neutral difference equation;
Krasnoselskii fixed point theorem.
c
2010
Texas State University - San Marcos.
Submitted July 30, 2010. Published October 14, 2010.
1
(1.7)
(1.8)
2
Z. GUO, M. LIU,
∆m (xn + cxn−k ) +
u
X
EJDE-2010/146
psn fs (xn−rs ) = qn ,
n ≥ n0 ,
in [16],
(1.9)
s=1
∆m (xn + cxn−k ) + pn xn−r − qn xn−l = 0,
n ≥ n0 ,
in [17].
(1.10)
Motivated by the above publications, we investigate the higher-order nonlinear
neutral difference equation
s
X
∆ akn . . . ∆ a2n ∆(a1n ∆(xn + bn xn−d )) +
pjn fj (xn−rjn ) = qn ,
(1.11)
j=1
where n ≥ n0 ≥ 0, d, k, j, s are positive integers, {ain }n≥n0 (i = 1, 2, . . . , k),
{bn }n≥n0 , {pjn }n≥n0 (1 ≤ j ≤ s) and {qn }n≥n0 are sequences of real numbers,
rjn ∈ Z (1 ≤ j ≤ s, n0 ≤ n), fj : R → R and xfj (x) ≥ 0 for x 6= 0 (j = 1, 2, . . . , s).
Clearly, difference equations (1.1)–(1.10) are special cases of (1.11), for which we
use Krasnoselskii fixed point theorem to obtain non-oscillatory solutions.
Lemma 1.1 (Krasnoselskii Fixed Point Theorem). Let Ω be a bounded closed convex subset of a Banach space X and T1 , T2 : S → X satisfy T1 x + T2 y ∈ Ω for
each x, y ∈ Ω. If T1 is a contraction mapping and T2 is a completely continuous
mapping, then the equation T1 x + T2 x = x has at least one solution in Ω.
As usual, the forward difference ∆ is defined as ∆xn = xn+1 − xn , and for a
positive integer m the higher-order difference is defined as
∆m xn = ∆(∆m−1 xn ),
∆0 xn = xn .
In this article, R = (−∞, +∞), N is the set of positive integers, Z is the sets
of all integers, α = inf{n − rjn : 1 ≤ j ≤ s, n0 ≤ n}, β = min{n0 − d, α},
limn→∞ (n − rjn ) = +∞, 1 ≤ j ≤ s, lβ∞ denotes the set of real-valued bounded
sequences x = {xn }n≥β . It is well known that lβ∞ is a Banach space under the
supremum norm kxk = supn≥β |xn |.
For N > M > 0, let
A(M, N ) = x = {xn }n≥β ∈ lβ∞ : M ≤ xn ≤ N, n ≥ β .
Obviously, A(M, N ) is a bounded closed and convex subset of lβ∞ . Put
b = lim sup bn
and b = lim inf bn .
n→∞
n→∞
lβ∞
Definition 1.2 ([5]). A set Ω of sequences in
is uniformly Cauchy (or equiCauchy) if for every ε > 0, there exists an integer N0 such that
|xi − xj | < ε,
whenever i, j > N0 for any x = xk in Ω.
Lemma 1.3 (Discrete Arzela-Ascoli’s theorem [5]). A bounded, uniformly Cauchy
subset Ω of lβ∞ is relatively compact.
By a solution of (1.11), we mean a sequence {xn }n≥β with a positive integer
N0 ≥ n0 + d + |α| such that (1.11) is satisfied for all n ≥ N0 . As is customary, a
solution of (1.11) is said to be oscillatory about zero, or simply oscillatory, if the
terms xn of the sequence {xn }n≥β are neither eventually all positive nor eventually
all negative. Otherwise, the solution is called non-oscillatory.
EJDE-2010/146
EXISTENCE OF NON-OSCILLATORY SOLUTIONS
3
2. Existence of non-oscillatory solutions
In this section, we will give five sufficient conditions of the existence of nonoscillatory solutions of (1.11).
Theorem 2.1. If there exist constants M and N with N > M > 0 and such that
|bn | ≤ b <
∞
X
max
t=n0
N −M
,
2N
eventually,
(2.1)
1
, |pjt |, |qt | : 1 ≤ i ≤ k, 1 ≤ j ≤ s < +∞,
|ait |
(2.2)
then (1.11) has a non-oscillatory solution in A(M, N ).
Proof. Choose L ∈ (M + bN, N − bN ). By (2.1) and (2.2), an integer N0 >
n0 + d + |α| can be chosen such that
|bn | ≤ b <
N −M
, ∀n ≥ N0 ,
2N
(2.3)
and
∞
∞
X
X
t1 =N0 t2 =t1
···
∞
X
Ps
∞
X
F j=1 pjt + |qt |
≤ min{L − bN − M, N − bN − L},
Qk
aiti tk =tk−1 t=tk
i=1
(2.4)
where F = maxM ≤x≤N {fj (x) : 1 ≤ j ≤ s}. Define two mappings T1 , T2 :
A(M, N ) → X by
(
L − bn xn−d , n ≥ N0 ,
(T1 x)n =
(2.5)
(T1 x)N0 ,
β ≤ n < N0 ,

P∞ P∞
(−1)k t1 =n t2 =t1 . . .


P
P∞ Psj=1 pjt fj (xt−rjt )−qt
∞
Qk
(2.6)
(T2 x)n =
, n ≥ N0 ,
tk =tk−1
t=tk

i=1 aiti


(T2 x)N0 ,
β ≤ n < N0 ,
for all x ∈ A(M, N ).
(i) Note that T1 x + T2 y ∈ A(M, N ) for all x, y ∈ A(M, N ). In fact, for every
x, y ∈ A(M, N ) and n ≥ N0 , by (2.4), we have
∞ X
∞
∞
∞ Ps
X
X
X
j=1 pjt fj (yt−rjt ) − qt
(T1 x + T2 y)n ≥ L − bN −
···
Qk
t1 =n t2 =t1
tk =tk−1 t=tk
i=1 aiti
P
s
∞
∞
∞
∞
X
X
X
X
F j=1 pjt + |qt |
≥ L − bN −
···
≥M
Qk
aiti t1 =N0 t2 =t1
tk =tk−1 t=tk
i=1
and
(T1 x + T2 y)n ≤ L + bN +
∞
∞
X
X
···
t1 =N0 t2 =t1
≤ N.
That is, (T1 x + T2 y)(A(M, N )) ⊆ A(M, N ).
∞
X
Ps
∞
X
F j=1 pjt + |qt |
Qk
aiti tk =tk−1 t=tk
i=1
4
Z. GUO, M. LIU,
EJDE-2010/146
(ii) W show that T1 is a contraction mapping on A(M, N ). For any x, y ∈
A(M, N ) and n ≥ N0 , it is easy to derive that
(T1 x)n − (T1 y)n ≤ |bn kxn−d − yn−d | ≤ bkx − yk,
which implies
kT1 x − T1 yk ≤ bkx − yk.
−M
Then b < N2N
< 1 ensures that T1 is a contraction mapping on A(M, N ).
(iii) We show that T2 is completely continuous. First, we show T2 that is contin(u)
(u)
uous. Let x(u) = {xn } ∈ A(M, N ) be a sequence such that xn → xn as u → ∞.
Since A(M, N ) is closed, x = {xn } ∈ A(M, N ). Then, for n ≥ N0 ,
(u)
∞
∞
∞
∞ Ps
fj (xt−r
X
X
X
X
p
) − fj (xt−rjt )|
jt
j=1
jt
T2 x(u)
.
···
Qk
n − T2 xn ≤
tk =tk−1 t=tk
i=1 aiti
t1 =N0 t2 =t1
Since
Ps
j=1
Ps
(u)
(u)
pjt fj (xt−rjt ) − fj (xt−rjt )|
j=1 pjt |fj (xt−rjt )| + |fj (xt−rjt )|
≤
Qk
Qk
i=1 aiti
i=1 aiti
Ps
2F j=1 pjt ≤ Qk
ait i=1
i
(u)
|fj (xt−rjt ) − fj (xt−rjt )|
→ 0 as u → ∞ for j = 1, 2, . . . , s, it follows from (2.2)
and
and the Lebesgue dominated convergence theorem that limu→∞ kT2 x(u) −T2 xk = 0,
which means that T2 is continuous.
Next, we show that T2 A(M, N ) is relatively compact. By (2.2), for any ε > 0,
take N1 ≥ N0 large enough,
Ps
∞
∞
∞
∞
X
X
X
X
F j=1 pjt + |qt |
ε
···
< .
(2.7)
Qk
2
tk =tk−1 t=tk
i=1 aiti
t1 =N1 t2 =t1
Then, for any x = {xn } ∈ A(M, N ) and n1 , n2 ≥ N1 , (2.7) ensures that
∞
∞
∞ Ps
∞
X
X
X
X
j=1 pjt fj (yt−rjt ) − qt
T2 xn1 − T2 xn2 ≤
···
Qk
t1 =n1 t2 =t1
tk =tk−1 t=tk
i=1 aiti
∞
∞
∞
∞ Ps
X
X
X
X
j=1 pjt fj (yt−rjt ) − qt
+
···
Qk
t1 =n2 t2 =t1
tk =tk−1 t=tk
i=1 aiti
P
s
∞
∞
∞
∞
X
X
X
X
F j=1 pjt + |qt |
≤
···
Qk
tk =tk−1 t=tk
i=1 aiti
t1 =N1 t2 =t1
P
s
∞
∞
∞
∞
X
X
X
X
F j=1 pjt + |qt |
+
···
Qk
aiti t1 =N1 t2 =t1
tk =tk−1 t=tk
i=1
ε ε
< + = ε,
2 2
which implies T2 A(M, N ) begin uniformly Cauchy. Therefore, by Lemma 1.3, the
set T2 A(M, N ) is relatively compact. By Lemma 1.1, there exists x = {xn } ∈
A(M, N ) such that T1 x + T2 x = x, which is a bounded non-oscillatory solution to
(1.11). This completes the proof.
EJDE-2010/146
EXISTENCE OF NON-OSCILLATORY SOLUTIONS
5
Theorem 2.2. If (2.2) holds,
bn ≥ 0 eventually, 0 ≤ b ≤ b < 1,
2−b
M
1−b
and there exist constants M and N with N >
oscillatory solution in A(M, N ).
(2.8)
> 0 then (1.11) has a non-
b
Proof. Choose L ∈ (M + 1+b
2 N, N + 2 M ). By (2.2) and (2.8), an integer N0 >
n0 + d + |α| can be chosen such that
and
1+b
b
≤ bn ≤
, ∀n ≥ N0
2
2
Ps
∞
∞
∞
∞
X
X
X
X
F j=1 pjt + |qt |
···
Qk
aiti (2.9)
t1 =N0 t2 =t1
tk =tk−1 t=tk
i=1
(2.10)
n
b o
1+b
N, N − L + M ,
≤ min L − M −
2
2
where F = maxM ≤x≤N {fj (x) : 1 ≤ j ≤ s}. Then define T1 , T2 : A(M, N ) → X
as (2.5) and (2.6). The rest proof is similar to that of Theorem 2.1, and it is
omitted.
Theorem 2.3. If (2.2) holds,
bn ≤ 0 eventually,
−1 < b ≤ b ≤ 0,
and there exist constants M and N with N >
oscillatory solution in A(M, N ).
2+b
1+b M
(2.11)
> 0, then (1.11) has a non-
1+b
Proof. Choose L ∈ ( 2+b
2 M, 2 N ). By (2.2) and (2.11), an integer N0 > n0 +d+|α|
can be chosen such that
b−1
b
≤ bn ≤ , ∀n ≥ N0 ,
(2.12)
2
2
and
Ps
∞
∞
∞
∞
X
X
X
X
F j=1 pjt + |qt |
···
Qk
tk =tk−1 t=tk
i=1 aiti
t1 =N0 t2 =t1
(2.13)
n
o
2+b
1+b
≤ min L −
M,
N −L ,
2
2
where F = maxM ≤x≤N {fj (x) : 1 ≤ j ≤ s}. Then define T1 , T2 : A(M, N ) → X by
(2.5) and (2.6). The rest proof is similar to that of Theorem 2.1, and is omitted. Theorem 2.4. If (2.2) holds,
bn > 1 eventually,
1 < b, and b < b2 < +∞,
(2.14)
2
and there exist constants M and N with N >
non-oscillatory solution in A(M, N ).
b(b −b)
M
b(b2 −b)
> 0, then (1.11) has a
Proof. Take ε ∈ (0, b − 1) sufficiently small satisfying
1 < b − ε < b + ε < (b − ε)2
(2.15)
and
(b + ε)(b − ε)2 − (b + ε)2 N > (b + ε)2 (b − ε) − (b − ε)2 M.
(2.16)
6
Z. GUO, M. LIU,
EJDE-2010/146
b+ε
M . By (2.2) and (2.15), an integer
Choose L ∈ (b + ε)M + b−ε
N, (b − ε)N + b−ε
b+ε
N0 > n0 + d + |α| can be chosen such that
b − ε < bn < b + ε,
∀b ≥ N0
(2.17)
and
∞
∞
X
X
∞
X
···
t1 =N0 t2 =t1
Ps
∞
X
F j=1 pjt + |qt |
Qk
aiti tk =tk−1 t=tk
i=1
(2.18)
o
b−ε
L − (b − ε)M − N,
M + (b − ε)N − L ,
≤ min
b+ε
b+ε
nb − ε
where F = maxM ≤x≤N {fj (x) : 1 ≤ j ≤ s}. Define two mappings T1 , T2 :
A(M, N ) → X by
(
L
n+d
− xbn+d
, n ≥ N0 ,
(2.19)
(T1 x)n = bn+d
(T1 x)N0 ,
β ≤ n < N0 ,
 (−1)k P∞ P∞


t1 =n
t2 =t1 . . .
bn+d
P
P∞ Psj=1 pjt fj (xt−rjt )−qt
∞
Qk
T2 x)n =
(2.20)
, n ≥ N0 ,
tk =tk−1
t=tk

i=1 aiti


(T2 x)N0 ,
β ≤ n < N0 ,
for all x ∈ A(M, N ). The rest proof is similar to that in Theorem 2.1, and is
omitted.
Theorem 2.5. If (2.2) holds,
bn < −1 eventually,
−∞ < b, b < −1
(2.21)
1+b
M
1+b
and there exist constants M and N with N >
> 0, then (1.11) has a nonoscillatory solution in A(M, N ).
Proof. Take ∈ 0, −(1 + b) sufficiently small satisfying
b − < b + < −1
(2.22)
and
(2.23)
(1 + b + )N < (1 + b − )M.
Choose L ∈ (1 + b + )N, (1 + b − )M . By (2.2) and (2.22), an integer N0 >
n0 + d + |α| can be chosen such that
b − < bn < b + ,
and
∞
∞
X
X
t1 =N0 t2 =t1
···
∞
X
∀n ≥ N0 ,
Ps
∞
X
F j=1 pjt + |qt |
Qk
aiti (2.24)
tk =tk−1 t=tk
i=1
(2.25)
o
n
b+
b+
≤ min b + +
M−
L, L − (1 + b + )N ,
b−
b−
where F = maxM ≤x≤N {fj (x) : 1 ≤ j ≤ s}. Then define T1 , T2 : A(M, N ) → X
as (2.19) and (2.20). The rest proof is similar to that in Theorem 2.1, and is
omitted.
EJDE-2010/146
EXISTENCE OF NON-OSCILLATORY SOLUTIONS
7
Remark 2.6. Theorems 2.1–2.5 extend the results in Cheng [6, Theorem 1], Liu,
Xu and Kang [8, Theorems 2.3-2.7], Zhou and Huang [16, Theorems 1-5] and corresponding theorems in [3, 4, 9, 10, 11, 12, 13, 14, 15].
Acknowledgments. The authors are grateful to the anonymous referees for their
careful reading, editing, and valuable comments and suggestions.
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Zhenyu Guo
School of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China
E-mail address: guozy@163.com
Min Liu
School of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China
E-mail address: min liu@yeah.net