Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 172939, 30 pages
doi:10.1155/2012/172939
Research Article
Positive Solutions of a Second-Order Nonlinear
Neutral Delay Difference Equation
Zeqing Liu,1 Wei Sun,1 Jeong Sheok Ume,2 and Shin Min Kang3
1
Department of Mathematics, Liaoning Normal University, Liaoning, Dalian 116029, China
Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea
3
Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
2
Correspondence should be addressed to Jeong Sheok Ume, jsume@changwon.ac.kr
Received 15 August 2012; Accepted 6 November 2012
Academic Editor: Norio Yoshida
Copyright q 2012 Zeqing Liu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The purpose of this paper is to study solvability of the second-order nonlinear neutral delay
difference equation Δan, ya1n , . . . , yarn Δyn bn yn−τ fn, yf1n , . . . , yfkn cn , ∀n ≥ n0 . By
making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem,
Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions
which ensure the existence of uncountably many bounded positive solutions for the above
equation. Five nontrivial examples are given to illustrate that the results presented in this paper
are more effective than the existing ones in the literature.
1. Introduction
It is well known that the oscillation, nonoscillation, asymptotic behavior, and existence of
solutions for second-order difference equations with delays have been widely studied in
many papers over the last 20 years, see, for example, 1–9
and the references cited therein.
Recently, Cheng 5
considered the second-order neutral delay linear difference
equation with positive and negative coefficients
Δ2 yn pyn−m pn yn−k − qn yn−l 0,
∀n ≥ n0
1.1
and investigated the existence of a nonoscillatory solution of 1.1 under the condition p /
−1
by using the Banach fixed point theorem. M. Migda and J. Migda 9
and Luo and Bainov 8
2
Abstract and Applied Analysis
discussed the asymptotic behaviors of nonoscillatory solutions for the second-order neutral
difference equation with maxima
Δ2 yn pn yn−k qn max ys : n − l ≤ s ≤ n 0,
1.2
∀n ≥ 1
and the second-order neutral difference equation
Δ2 yn pyn−k f n, yn 0,
1.3
∀n ≥ 1.
Cheng and Chu 2
got sufficient and necessary conditions of the oscillatory solutions for the
second-order difference equation
γ
Δ rn−1 Δyn−1 pn yn 0,
1.4
∀n ≥ 1.
Li and Yeh 6
established some oscillation criteria of the second-order delay difference
equation
Δ an−1 Δ yn−1 pn−1 yn−1−σ qn f yn−τ 0,
∀n ≥ 1.
1.5
Using the Leray-Schauder nonlinear alternative theorem, Agarwal et al. 1
studied the
existence of nonoscillatory solutions for the discrete equation
Δ an Δ yn pyn−τ F n 1, yn1−σ 0,
∀n ≥ 1
1.6
under the condition |p| / 1. Very recently, Liu et al. 7
utilized the Banach contraction
principle to establish the global existence and multiplicity of bounded nonoscillatory
solutions for the second-order nonlinear neutral delay difference equation
Δ an Δ yn byn−τ f n, yn−d1n , yn−d2n , . . . , yn−dkn cn ,
∀n ≥ n0 .
1.7
Motivated by the results in 1–9
, in this paper, we discuss the solvability of the secondorder nonlinear neutral delay difference equation
Δ a n, ya1n , . . . , yarn Δ yn bn yn−τ f n, yf1n , . . . , yfkn cn ,
∀n ≥ n0 ,
1.8
where τ, r, k ∈ N, n0 ∈ N0 , {bn }n∈Nn ∪{cn }n∈Nn ⊂ R, a ∈ CNn0 ×Rr , R\{0}, f ∈ CNn0 ×Rk , R,
0
∪rd1 {adn }n∈Nn ⊆ Z, ∪kj1 {fjn }n∈Nn ⊆ Z and
0
0
0
lim adn lim fjn ∞,
n→∞
n→∞
d, j ∈ Jr × Jk .
1.9
It is clear that 1.1–1.7 are special cases of 1.8. By utilizing the Rothe fixed point theorem,
Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and a few
Abstract and Applied Analysis
3
new techniques, we prove the existence of uncountably many bounded positive solutions of
1.8. Five examples are constructed to illuminate our results which extend essentially the
corresponding results in 1, 7
.
2. Preliminaries
Throughout this paper, we assume that Δ is the forward difference operator defined by Δyn yn1 − yn , R −∞, ∞, R 0, ∞, Z, N and N0 stand for the sets of all integers, positive
integers, and nonnegative integers, respectively,
β min n0 − τ, inf{adn : d ∈ Jr , n ∈ Nn0 }, inf fjn : j ∈ Jk , n ∈ Nn0 ,
Nn0 {n : n ∈ N0 with n ≥ n0 },
Zβ n : n ∈ Z with n ≥ β ,
2.1
Jl {1, . . . , l} for l ∈ {r, k}.
lβ∞ denotes the Banach space of all bounded sequences y {yn }n∈Zβ with the norm
y supyn n∈Zβ
for y yn n∈Zβ ∈ lβ∞ .
2.2
For any M > N > 0, put
V N y yn n∈Zβ ∈ lβ∞ : yn ≥ N, ∀n ∈ Zβ ,
UM y ∈ V N : y < M ,
BM, N y yn n∈Zβ ∈ lβ∞ : y − M < N ,
2.3
AN, M y yn n∈Zβ ∈ lβ∞ : N ≤ yn ≤ M, ∀n ∈ Zβ .
It is easy to see that V N is a closed convex subset of lβ∞ , UM is a bounded open subset of
V N and BM, N is a bounded open convex subset of lβ∞ and AN, M is a bounded closed
and convex subset of lβ∞ .
By a solution of 1.8, we mean a sequence {yn }n∈Zβ ∈ lβ∞ with a positive integer T ≥
n0 τ |β| such that 1.8 is satisfied for all n ≥ T .
The following Lemmas play important roles in this paper.
Lemma 2.1 Discrete Arzela-Ascoli’s Theorem 3
. A bounded, uniformly Cauchy subset Y of
lβ∞ is relatively compact.
Lemma 2.2 Rothe Fixed Point Theorem 10
. Let D be a bounded convex open subset of a Banach
space E and A : D → E be a continuous, condensing mapping, and A∂D ⊆ D. Then A has a fixed
point in D.
4
Abstract and Applied Analysis
Lemma 2.3 Leray-Schauder Nonlinear Alternative Theorem 1
. Let U be an open subset of a
closed convex set K in a Banach space E with p∗ ∈ U. Let f : U → K be a continuous, condensing
mapping with fU bounded. Then either
a f has a fixed point in U; or
b there exist an x ∈ ∂U and a λ ∈ 0, 1 such that x 1 − λp∗ λfx.
Lemma 2.4 Krasnoselskill Fixed Point Theorem 5
. Let Y be a nonempty bounded closed
convex subset of a Banach space X and f, g be mappings from Y into X such that fx gy ∈ Y for
every pair x, y ∈ Y . If f is a contraction mapping and g is completely continuous, then the equation
fx gx x has at least one solution in Y .
3. Main Results
Now we use the Rothe fixed point theorem to show the existence and multiplicity of bounded
positive solutions of 1.8.
Theorem 3.1. Assume that there exist two constants N and M with M > N > 0 and two positive
sequences {an }n∈Nn and {pn }n∈Nn satisfying
0
0
|an, u1 , u2 , . . . , ur | ≥ an ,
fn, u1 , u2 , . . . , uk ≤ pn ,
∀n, ud ∈ Nn0 × N, M
, d ∈ Jr ;
∀ n, uj ∈ Nn0 × N, M
, j ∈ Jk ;
3.1
∞
∞
1
max pi , |ci | < ∞;
sn0 as is
3.2
bn 1 eventually.
3.3
Then 1.8 has uncountably many bounded positive solutions in BM, N.
Proof. Let L ∈ M − N, M N. First of all, we show that there exists a mapping SL :
BM, N → lβ∞ with SL ∂BM, N ⊆ BM, N such that SL has a fixed point y {yn }n∈Zβ ∈
BM, N, which is also a bounded positive solution of 1.8.
It follows from 3.2 and 3.3 that there exists T ≥ max{1, n0 τ |β|} satisfying
bn 1,
∀n ≥ T ;
∞
∞
1
1
pi |ci | < min{M N − L, N − M L}.
a
2
sT s is
3.4
3.5
Abstract and Applied Analysis
5
Define a mapping SL : BM, N → lβ∞ as follows:
SL y
n
⎧
∞
⎪
⎪
⎨L −
n2lτ−1
∞
1
f i, yf1i , . . . , yfki − ci ,
sn2l−1τ a s, ya1s , . . . , yars is
⎪
⎪
⎩S y ,
L
T
l1
n≥T
β≤n<T
3.6
for each y {yn }n∈Zβ ∈ BM, N. On account of 3.1, 3.5, and 3.6, we conclude that for
every y {yn }n∈Zβ ∈ ∂BM, N ⊆ BM, N and n ≥ T
∞
∞
n2lτ−1
1
SL y − M L − M −
f i, yf1i , . . . , yfki − ci n
l1 sn2l−1τ a s, ya1s , . . . , yars is
≤ |L − M| ∞
n2lτ−1
∞ 1
f i, yf , . . . , yf |ci |
1i
ki
a s, ya , . . . , ya 1s
rs
is
l1 sn2l−1τ
≤ |L − M| ∞
∞
1
pi |ci |
a
sT τ s is
< |L − M| 1
min{M N − L, N − M L}
2
≤
N
,
2
3.7
which means that
SL y − M ≤ N < N,
2
3.8
that is, SL ∂BM, N ⊆ BM, N.
Now we assert that SL is a continuous and condensing mapping in BM, N. Let yω {ynω }n∈Zβ ∈ BM, N for each ω ∈ N and y {yn }n∈Zβ ∈ BM, N with limω → ∞ yω y. Let
ε > 0. It follows from 3.2 and the continuity of a and f that there exist T1 , T2 , T3 ∈ N with
T1 > T and T2 > T1 τ − 1 satisfying
τ−1
∞
∞
∞
T1
1
1
max
pi |ci | ,
pi |ci |
a
a
sT1 τ s is
sT τ s iT2
T
max
2 −1 1 T
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki ,
a
sT τ s is
T −1
T1
τ−1 a s, y ω , . . . , y ω
2
a1s
ars − a s, ya1s , . . . , yars
<
ε
;
16
3.9
1 τ−1
sT τ
a2s
pi |ci |
is
3.10
ε
<
,
16
∀ω ≥ T3 .
6
Abstract and Applied Analysis
In view of 3.1 and 3.6–3.10, we deduce that for any ω ≥ T3
∞ ∞ n2lτ−1
1
SL yω − SL y sup
f i, yfω1i , . . . , yfωki − ci
ω
ω
n≥T l1 sn2l−1τ a s, ya1s , . . . , yars is
∞
1
− f i, yf1i , . . . , yfki − ci a s, ya1s , . . . , yars is
∞ ∞ n2lτ−1
1
ω
ω
sup
f
i,
y
−
c
,
.
.
.
,
y
i
f
f
ω
ω
1i
ki
n≥T l1 sn2l−1τ a s, ya1s , . . . , yars is
∞
1
− f i, yf1i , . . . , yfki − ci
ω
ω
a s, ya1s , . . . , yars is
∞
1
×
f i, yf1i , . . . , yfki − ci
ω
ω
a s, ya1s , . . . , yars
is
∞
1
− f i, yf1i , . . . , yfki − ci a s, ya1s , . . . , yars is
∞ n2lτ−1
1
sup
ω
ω
n≥T l1 sn2l−1τ a s, ya1s , . . . , yars
×
∞ f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki
is
1
1
− ω
ω
a s, ya1s , . . . , yars
a s, ya1s , . . . , yars
∞
×
f i, yf1i , . . . , yfki − ci is
⎧
⎨
∞ n2lτ−1
1
≤ max sup
ω
⎩ n≥T1 l1 sn2l−1τ a s, ya1s , . . . , yaωrs ×
∞ f i, yfω1i , . . . , yfωki f i, yf1i , . . . , yfki is
sup
∞
n2lτ−1
n≥T1 l1 sn2l−1τ
×
1
1
a s, yaω , . . . , yaω a s, ya , . . . , ya 1s
rs
1s
rs
∞ f i, yf , . . . , yf |ci | ,
1i
ki
is
∞
n2lτ−1
1
ω
ω a
s,
y
,
T ≤n≤T1 −1 l1 sn2l−1τ
a1s . . . , yars
sup
×
∞ f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki is
Abstract and Applied Analysis
7
1
1
sup
−
ω
ω
a s, ya1s , . . . , yars T ≤n≤T1 −1 l1 sn2l−1τ a s, ya1s , . . . , yars
∞
n2lτ−1
∞ f i, yf , . . . , yf |ci |
×
1i
ki
is
∞
∞
∞
∞
1
1
pi 2
pi |ci | ,
a
a
sT1 τ s is
sT1 τ s is
≤ max 2
≤ max
∞
∞ 1
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki a
sT τ s is
∞ ∞
1
1
pi |ci |
− ω
ω
a s, ya1s , . . . , yars is
sT τ a s, ya1s , . . . , yars
∞
∞ ε 1
,
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki 4 sT1 τ as is
T1
τ−1
sT τ
∞ 1
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki as iT2
2 −1 1 T
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki a
sT τ s is
∞ ∞
1
1
−
pi |ci |
ω
ω
a s, ya1s , . . . , yars is
sT1 τ a s, ya1s , . . . , yars
T1
τ−1
∞
1
1
pi |ci |
− ω
ω
a s, ya1s , . . . , yars iT
sT τ a s, ya1s , . . . , yars
T1
τ−1
2
T1
τ−1
−1
T
1
1
2 pi |ci |
− ω
ω
a s, ya1s , . . . , yars is
sT τ a s, ya1s , . . . , yars
T1
τ−1
∞
∞
∞
∞
∞
1
1
1
ε
ε
,2
2
≤ max
pi 2
pi pi |ci |
4 sT1 τ as is
a
16
a
sT τ s iT2
sT1 τ s is
T1
τ−1
2
≤ max
sT τ
ε 5ε
,
4 8
∞
ε
1
pi |ci | as iT2
16
< ε,
3.11
which gives that limω → ∞ SL yω SL y, that is, SL is continuous in BM, N.
8
Abstract and Applied Analysis
In light of 3.1, 3.5, and 3.6, we get that for any y {yn }n∈Zβ ∈ BM, N
∞ n2lτ−1
∞
1
SL y supL −
f i, yf1i , . . . , yfki − ci a
s,
y
,
.
.
.
,
y
a
a
n≥T 1s
rs
is
l1 sn2l−1τ
≤ L
∞
∞
1
pi |ci |
a
sT τ s is
< L
1
min{M N − L, N − M L}
2
≤
1
M N L,
2
3.12
which implies that SL BM, N is uniformly bounded.
Given ε > 0. Clearly 3.2 ensures that there exists T ∗ > T satisfying
∞
∞
1
pi |ci | < ,
a
2
sT ∗ s is
3.13
which together with 3.1 and 3.6 implies that for all y {yn }n∈Zβ ∈ BM, N and t2 > t1 ≥
T∗
SL y t2 − SL y t1 ∞
∞ t2 2lτ−1
1
f i, yf1i , . . . , yfki − ci
l1 st2 2l−1τ a s, ya1s , . . . , yars is
∞
1
−
f i, yf1i , . . . , yfki − ci l1 st1 2l−1τ a s, ya1s , . . . , yars is
∞
≤
∞
t1 2lτ−1
∞ 1
f i, yf , . . . , yf |ci |
1i
ki
a s, ya , . . . , ya 1s
rs
is
l1 st2 2l−1τ
t2 2lτ−1
∞
3.14
t1 2lτ−1
∞ 1
f i, yf , . . . , yf |ci |
1i
ki
a s, ya , . . . , ya 1s
rs
is
l1 st1 2l−1τ
∞
∞
1
pi |ci |
a
sT ∗ τ s is
≤2
< ,
which yields that SL BM, N is uniformly Cauchy. Thus Lemma 2.1 means that
SL BM, N is relatively compact. Consequently SL is condensing in BM, N.
Abstract and Applied Analysis
9
It follows from Lemma 2.2 that SL has a fixed point y {yn }n∈Zβ ∈ BM, N, that is,
yn L −
∞
n2lτ−1
∞
1
f i, yf1i , . . . , yfki − ci ,
l1 sn2l−1τ a s, ya1s , . . . , yars is
∀n ≥ T,
3.15
which yields that
yn yn−τ 2L −
∞
∞
1
f i, yf1i , . . . , yfki − ci ,
a
s,
y
,
.
.
.
,
y
a1s
ars is
sn
∞
a n, ya1n , . . . , yarn Δ yn yn−τ f i, yf1i , . . . , yfki − ci ,
∀n ≥ T τ,
3.16
∀n ≥ T τ,
in
which together with 3.3 implies that
Δ a n, ya1n , . . . , yarn Δ yn bn yn−τ −f n, yf1n , . . . , yfkn cn ,
∀n ≥ T τ,
3.17
that is, 1.8 has a bounded positive solution y ∈ BM, N.
Next we show that 1.8 has uncountably many bounded positive solutions in
BM, N. Let L1 , L2 ∈ M − N, M N and L1 / L2 . For every θ ∈ {1, 2}, we infer similarly
that there exist a constant TLθ and a mapping SLθ satisfying 3.4–3.6, where L, T , and
SL are replaced by Lθ , TLθ , and SLθ , respectively, and the mapping SLθ has a fixed point
yθ {ynθ }n∈Zβ ∈ BM, N, which is a bounded positive solution of 1.8 in BM, N, that
is,
ynθ Lθ −
∞ 1
f i, yfθ1i , . . . , yfθki − ci ,
θ
θ
l1 sn2l−1τ a s, ya1s , . . . , yars is
∞
n2lτ−1
∀n ≥ TLθ .
3.18
Equation 3.2 ensures that there exists T∗ > max{TL1 , TL2 } satisfying
∞
∞
|L1 − L2 |
1
.
pi |ci | <
a
4
sT∗ s is
3.19
In order to show that the set of bounded positive solutions of 1.8 is uncountable, it is
y2 . It follows from 3.1, 3.18, and 3.19 that for all n ≥ T∗
sufficient to prove that y1 /
1
yn − yn2 ∞
∞ n2lτ−1
1
1
1
L1 −
f
i,
y
−
c
,
.
.
.
,
y
i
f
f
1i
ki
1
1
l1 sn2l−1τ a s, ya1s , . . . , yars is
∞ 1
2
2
f
i,
y
−
c
,
.
.
.
,
y
−L2 i f1i
fki
2
2
l1 sn2l−1τ a s, ya1s , . . . , yars is
∞
n2lτ−1
10
Abstract and Applied Analysis
≥ |L1 − L2 | −
×
∞
n2lτ−1
1
a
l1 sn2l−1τ s
∞ f i, yf11i , . . . , yf1ki |ci | f i, yf21i , . . . , yf2ki |ci |
is
∞
∞
1
pi |ci |
a
sT∗ τ s is
≥ |L1 − L2 | − 2
>
1
|L1 − L2 |,
2
3.20
that is, y1 /
y2 . This completes the proof.
Theorem 3.2. Assume that there exist two constants N and M with M > N > 0 and two positive
sequences {an }n∈Nn and {pn }n∈Nn satisfying 3.1 and
0
0
∞
∞
∞
1
max pi , |ci | < ∞;
a
l1 sn lτ s is
3.21
0
bn −1
3.22
eventually.
Then 1.8 has uncountably many bounded positive solutions.
Proof. Let L ∈ M − N, M N. Firstly, we show that there exists a mapping SL : BM, N →
lβ∞ with SL ∂BM, N ⊆ BM, N such that SL has a fixed point y {yn }n∈Zβ ∈ BM, N,
which is also a bounded positive solution of 1.8. In view of 3.21 and 3.22, we choose a
sufficiently large integer T ≥ max{1, n0 τ |β|} such that
bn −1,
∀n ≥ T ;
3.23
∞
∞ ∞
1
1
pi |ci | < min{M N − L, N − M L}.
a
2
l1 sT lτ s is
3.24
Define a mapping SL : BM, N → lβ∞ as follows:
SL y
⎧
∞
∞ ⎪
⎪
⎨L n
∞
1
f i, yf1i , . . . , yfki − ci ,
a
s,
y
,
.
.
.
,
y
a1s
ars is
l1 snlτ
⎪
⎪
⎩SL y ,
T
n≥T
β≤n<T
3.25
Abstract and Applied Analysis
11
for each y {yn }n∈Zβ ∈ BM, N. It follows from 3.1, 3.24, and 3.25 that for every
y {yn }n∈Zβ ∈ ∂BM, N ⊆ BM, N and n ≥ T
∞ ∞
∞
1
SL y − M L − M f i, yf1i , . . . , yfki − ci n
l1 snlτ a s, ya1s , . . . , yars is
≤ |L − M| ∞
∞ ∞ 1
f i, yf , . . . , yf |ci |
1i
ki
a s, ya , . . . , ya 1s
rs
is
l1 snlτ
≤ |L − M| ∞
∞ ∞
1
pi |ci |
a
l1 sT lτ s is
< |L − M| 1
min{M N − L, N − M L}
2
≤
3.26
N
,
2
which means that
SL y − M ≤ N < N,
2
3.27
that is, SL ∂BM, N ⊆ BM, N.
Now we prove that SL is a continuous and condensing mapping in BM, N. Put yω {ynω }n∈Zβ ∈ BM, N for each ω ∈ N and y {yn }n∈Zβ ∈ BM, N with limω → ∞ yω y. Let
ε > 0. Using 3.21 and the continuity of a and f, we conclude that there exist four positive
integers T1 , T2 , T3 , and T4 with T3 > T , T2 > T3 T1 τ satisfying
lτ−1
∞
∞
∞ T3
∞
∞
1
1
max
pi |ci | ,
pi |ci | ,
a
a
l1 sT lτ s is
lT sT lτ s is
3
lτ−1
T
1 −1 T3
l1 sT lτ
1
∞
1
pi |ci |
as iT2
3.28
ε
<
;
16
T −1 T lτ−1 T −1
3
1
2
1 max
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki ,
a
l1 sT lτ s is
lτ−1 a
T
1 −1 T3
l1 sT lτ
−1
2
s, yaω1s , . . . , yaωrs − a s, ya1s , . . . , yars T
a2s
pi |ci |
is
<
ε
,
16
∀ω ≥ T4 .
3.29
12
Abstract and Applied Analysis
By virtue of 3.1 and 3.25–3.29, we infer that for each ω ≥ T4
∞
∞ ∞ 1
ω
ω
SL yω − SL y sup
f
i,
y
−
c
,
.
.
.
,
y
i
f1i
fki
ω
ω
n≥T l1 snlτ a s, ya1s , . . . , yars is
∞
1
−
f i, yf1i , . . . , yfki − ci l1 snlτ a s, ya1s , . . . , yars is
∞ ∞
∞
∞ ∞ 1
ω
ω
sup
f
i,
y
−
f
i,
y
,
.
.
.
,
y
,
.
.
.
,
y
f
f
1i
ki
f
f
ω
ω
1i
ki
n≥T l1 snlτ a s, ya1s , . . . , yars is
∞
∞ 1
1
− ω
ω
a s, ya1s , . . . , yars
a s, ya1s , . . . , yars
l1 snlτ
∞
×
f i, yf1i , . . . , yfki − ci is
∞
∞ 1
ω
ω n≥T3 l1 snlτ a s, ya1s , . . . , yars
≤ max sup
×
∞ f i, yfω1i , . . . , yfωki f i, yf1i , . . . , yfki is
sup
∞
∞ n≥T3 l1 snlτ
×
1
1
a s, yaω , . . . , yaω a s, ya , . . . , ya 1s
rs
1s
rs
∞ f i, yf , . . . , yf |ci | ,
1i
ki
is
∞ ∞
1
ω
ω a
s,
y
,
T ≤n≤T3 −1 l1 snlτ
a1s . . . , yars
sup
×
∞ f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki is
∞ ∞ 1
1
sup
−
ω
ω
a s, ya1s , . . . , yars T ≤n≤T3 −1 l1 snlτ a s, ya1s , . . . , yars
∞ ×
f i, yf , . . . , yf
|ci |
1i
ki
is
∞
∞
∞ ∞ ∞
∞
1
1
≤ max 2
pi 2
pi |ci | ,
a
a
l1 sT lτ s is
l1 sT lτ s is
3
3
∞ ∞
∞ 1
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki a
l1 sT lτ s is
∞ ∞ ∞
1
1
pi |ci |
− ω
ω
a s, ya1s , . . . , yars is
l1 sT lτ a s, ya1s , . . . , yars
Abstract and Applied Analysis
∞
∞ ∞ 1
ε ,
≤ max
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki 4 l1 sT lτ as is
13
3
lτ−1
T
1 −1 T3
l1 sT lτ
lτ−1
T
1 −1 T3
l1 sT lτ
2 −1 1 T
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki as is
∞ 1
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki as iT2
lτ−1
∞ T3
∞ 1
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki a
lT sT lτ s is
1
∞ ∞ ∞
1
1
−
pi |ci |
ω
ω
a
s,
y
,
.
.
.
,
y
a
s,
y
,
.
.
.
,
y
a1s
ars
a1s
ars
is
l1 sT3 lτ
−1
T
1
1
2 pi |ci |
− ω
ω
a
s,
y
,
.
.
.
,
y
a
s,
y
,
.
.
.
,
y
a1s
ars
a1s
ars
is
l1 sT lτ
lτ−1
T
∞
1 −1 T3
1
1
−
pi |ci |
ω
ω
a s, ya1s , . . . , yars iT2
l1 sT lτ a s, ya1s , . . . , yars
lτ−1
∞ T3
∞
1
1
pi |ci |
− ω
ω
a s, ya1s , . . . , yars is
lT sT lτ a s, ya1s , . . . , yars
lτ−1
T
1 −1 T3
1
≤ max
lτ−1
lτ−1
T
∞
∞ ∞ T3
∞
∞
∞
1 −1 T3
ε 1
1
1
ε
,2
2
pi pi 2
pi
4 l1 sT lτ as is
16
a
a
l1 sT lτ s iT2
lT sT lτ s is
3
1
lτ−1
T
∞
∞ ∞
∞
1 −1T3
ε
1
1
2
2
pi |ci | pi |ci |
a
16
a
l1 sT lτ s is
l1 sT lτ s iT2
3
< max
lτ−1
∞ T3
∞
1
2
pi |ci |
a
lT sT lτ s is
1
ε 7ε
,
4 8
< ε,
3.30
which implies that limω → ∞ SL yω SL y, that is, SL is continuous in BM, N.
From 3.1, 3.24, and 3.25, we infer that for any y {yn }n∈Zβ ∈ BM, N
∞
∞ ∞
1
SL y supL f i, yf1i , . . . , yfki − ci n≥T l1 snlτ a s, ya1s , . . . , yars is
14
Abstract and Applied Analysis
≤L
∞
∞ ∞
1
pi |ci |
a
l1 snlτ s is
<L
1
min{M N − L, N − M L}
2
≤
1
M N L,
2
3.31
which implies that SL BM, N is uniformly bounded.
Let ε > 0. It follows from 3.21 that there exists T ∗ > T satisfying
∞
∞
∞
ε
1
pi |ci | < ,
a
2
l1 sT ∗ lτ s is
3.32
which together with 3.1 and 3.25 yields that for all y {yn }n∈Zβ ∈ BM, N and t2 > t1 ≥
T∗
SL y t2 − SL y t1 ∞
∞ ∞
1
f i, yf1i , . . . , yfki − ci
l1 st lτ a s, ya1s , . . . , yars is
2
∞
1
−
f i, yf1i , . . . , yfki − ci l1 st1 lτ a s, ya1s , . . . , yars is
∞
∞ ≤
∞
∞ ∞ 1
f i, yf , . . . , yf |ci |
1i
ki
a s, ya , . . . , ya 1s
rs
is
l1 st2 lτ
3.33
∞
∞ ∞ 1
f i, yf , . . . , yf |ci |
1i
ki
a s, ya , . . . , ya 1s
rs
is
l1 st1 lτ
∞
∞ ∞
1
≤2
pi |ci |
a
l1 sT ∗ lτ s is
< ε,
which gives that SL BM, N is uniformly Cauchy. Hence Lemma 2.1 implies that
SL BM, N is relatively compact, that is, SL is condensing in BM, N.
It is clear that Lemma 2.2 means that SL possesses a fixed point y {yn }n∈Zβ ∈
BM, N, that is,
yn L yn−τ
∞ ∞
∞
1
f i, yf1i , . . . , yfki − ci ,
a
s,
y
,
.
.
.
,
y
a1s
ars is
l1 snlτ
∞
∞
∞
1
L
f i, yf1i , . . . , yfki − ci ,
a
s,
y
,
.
.
.
,
y
a1s
ars is
l1 snl−1τ
∀n ≥ T,
3.34
∀n ≥ T τ
Abstract and Applied Analysis
15
which lead to
∞
1
f i, yf1i , . . . , yfki − ci ,
sn a s, ya1s , . . . , yars is
∀n ≥ T τ,
∞
1
Δ yn − yn−τ f i, yf1i , . . . , yfki − ci ,
a n, ya1n , . . . , yarn in
∀n ≥ T τ,
yn − yn−τ −
∞
3.35
which together with 3.23 yields that
Δ a n, ya1n , . . . , yarn Δ yn bn yn−τ −f n, yf1n , . . . , yfkn cn ,
∀n ≥ T τ,
3.36
that is, 1.8 has a bounded positive solution in BM, N.
Next we show that 1.8 has uncountably many bounded positive solutions in
BM, N. Let L1 , L2 ∈ M − N, M N and L1 /
L2 . Similarly we infer that for each
θ ∈ {1, 2}, there exist a constant TLθ and a mapping SLθ satisfying 3.23–3.25, where L, T
and SL are replaced by Lθ , TLθ , and SLθ , respectively, and the mapping SLθ has a fixed point
yθ {ynθ }n∈Zβ ∈ BM, N, which is a bounded positive solution of 1.8 in BM, N, that is,
ynθ Lθ ∞ 1
f i, yfθ1i , . . . , yfθki − ci ,
θ
θ
l1 snlτ a s, ya1s , . . . , yars is
∞ ∞
∀n ≥ TLθ .
3.37
It follows from 3.21 that there exists T∗ > max{TL1 , TL2 } such that
∞
∞ ∞
|L1 − L2 |
1
.
pi |ci | <
a
4
l1 sT lτ s is
∗
3.38
16
Abstract and Applied Analysis
In order to show that the set of bounded positive solutions of 1.8 is uncountable, it is
y2 . By means of 3.1, 3.37 and 3.38, we infer that for each
sufficient to prove that y1 /
n ≥ T∗
∞
∞ ∞ 1
1
2
1
1
f
i,
y
−
c
,
.
.
.
,
y
yn − yn L1 i
f1i
fki
1
1
l1 snlτ a s, ya1s , . . . , yars is
∞ 1
2
2
f i, yf1i , . . . , yfki − ci −L2 −
2
2
a
s,
y
,
.
.
.
,
y
a1s
ars is
l1 snlτ
∞
∞ ≥ |L1 − L2 | −
∞
∞ ∞ 1
f i, yf11i , . . . , yf1ki |ci | f i, yf21i , . . . , yf2ki |ci |
a
l1 snlτ s is
∞
∞ ∞
1
≥ |L1 − L2 | − 2
pi |ci |
a
l1 sT lτ s is
∗
>
1
|L1 − L2 |,
2
3.39
that is, y1 /
y2 . This completes the proof.
Next we use the Leray-Schauder nonlinear alternative theorem to show the existence
and multiplicity of bounded positive solutions of 1.8.
Theorem 3.3. Assume that there exist four constants N, M, b∗ , and b∗ and two positive sequences
{an }n∈Nn and {pn }n∈Nn satisfying 3.1, 3.2 and
0
0
0 < N < 1 − b∗ − b∗ M,
b∗ ≥ 0, b∗ ≥ 0, b∗ b∗ < 1, −b∗ ≤ bn ≤ b∗ eventually.
3.40
Then 1.8 has uncountably many bounded positive solutions in UM.
Proof. Let L ∈ b∗ M N, 1 − b∗ M. Now we prove that there exists a mapping SL : UM →
V N such that it has a fixed point y {yn }n∈Zβ ∈ UM, which is also a bounded positive
solution of 1.8. It follows from 3.2, 3.40 and that there exists a sufficiently large number
T ≥ max{1, n0 τ |β|} satisfying
−b∗ ≤ bn ≤ b∗ ,
∀n ≥ T ;
∞
∞
1
pi |ci | < min{L − b∗ M − N, 1 − b∗ M − L}.
a
sT s is
3.41
3.42
Put p∗ M − ε∗ , where ε∗ ∈ 0, min{L − b∗ M − N, 1 − b∗ M − L, M − N/2} is enough small
and
∞
∞
1
pi |ci | < min{L − b∗ M − N, 1 − b∗ M − L} − ε∗ .
a
sT s is
3.43
Abstract and Applied Analysis
17
Obviously, p∗ ∈ UM. Define a mapping SL : UM → lβ∞ by
SL y
n
S1L y n S2L y n ,
n≥β
3.44
for each y {yn }n∈Zβ ∈ UM, where the mappings S1L , S2L : UM → lβ∞ are defined by
L − bn yn−τ , n ≥ T
S1L y n β ≤ n < T,
S1L y T ,
⎧ ∞
∞
⎪
1
⎪
⎨
f i, yf1i , . . . , yfki − ci ,
−
S2L y n sn a s, ya1s , . . . , yars is
⎪
⎪
⎩S2L y ,
3.45
n≥T
T
3.46
β ≤ n < T.
It follows from 3.1, 3.41, and 3.43–3.46 that for any y {yn }n∈Zβ ∈ UM and n ≥ T
SL y
n
S1L y n S2L y n
L − bn yn−τ −
≥ L − b∗ M −
∞
∞
1
f i, yf1i , . . . , yfki − ci
sn a s, ya1s , . . . , yars is
∞
∞
1
pi |ci |
sn a s, ya1s , . . . , yars
is
∞
∞
1
≥ L−b M−
pi |ci |
a
sT s is
3.47
∗
> L − b∗ M − min{L − b∗ M − N, 1 − b∗ M − L} ε∗
≥ N ε∗
> N,
which yields that SL UM ⊆ V N.
Next we show that S2L : UM → lβ∞ is a continuous and relatively compact mapping.
Let yω {ynω }n∈Zβ ∈ UM and y {yn }n∈Zβ ∈ UM with limω → ∞ yω y. By virtue of 3.2
and the continuity of a and f, we infer that there exist T1 , T2 , T3 ∈ N with T2 > T1 − 1 > T
satisfying
18
Abstract and Applied Analysis
∞
∞
∞
1 −1
T
1
1
max
pi |ci | ,
pi |ci |
a
a
sT1 s is
sT s iT2
<
ε
;
16
3.48
T −1
1
2 −1 1 T
max
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki ,
|as | is
sT
T −1
T
1 −1 a s, y ω , . . . , y ω
2
ε
a1s
ars − a s, ya1s , . . . , yars
<
,
pi |ci |
2
16
as
is
sT
3.49
∀ω ≥ T3 .
It follows from 3.1 and 3.46–3.49 that for each ω ≥ T3
∞
∞ 1
ω
ω
S2L yω − S2L y sup
f
i,
y
−
c
,
.
.
.
,
y
i
f1i
fki
ω
ω
n≥T sn a s, ya1s , . . . , yars is
∞
1
−
f i, yf1i , . . . , yfki − ci sn a s, ya1s , . . . , yars is
∞
∞
∞ 1
ω
ω
sup
f
i,
y
−
f
i,
y
,
.
.
.
,
y
,
.
.
.
,
y
f
f
1i
ki
f
f
ω
ω
1i
ki
n≥T sn a s, ya1s , . . . , yars is
∞
1
1
− ω
ω
a s, ya1s , . . . , yars
a s, ya1s , . . . , yars
sn
∞
×
f i, yf1i , . . . , yfki − ci is
∞
1
ω
ω n≥T1 sn a s, ya1s , . . . , yars
≤ max sup
×
∞ f i, yfω1i , . . . , yfωki f i, yf1i , . . . , yfki is
sup
∞
n≥T1 sn
×
1
1
a s, yaω , . . . , yaω a s, ya , . . . , ya 1s
rs
1s
rs
∞ f i, yf , . . . , yf |ci | ,
1i
ki
is
∞
1
ω
ω T ≤n≤T1 −1 sn a s, ya1s , . . . , yars
∞ × f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki sup
is
Abstract and Applied Analysis
19
∞ 1
1
sup
−
ω
ω
a s, ya1s , . . . , yars T ≤n≤T1 −1 sn a s, ya1s , . . . , yars
∞ f i, yf , . . . , yf |ci |
×
1i
ki
is
∞
∞
∞
∞
1
1
pi 2
pi |ci | ,
a
a
sT1 s is
sT1 s is
≤ max 2
≤ max
∞
∞ 1
f i, yfω1i , yfω2i , . . . , yfωki − f i, yf1i , yf2i , . . . , yfki a
sT s is
∞ ∞
1
1
pi |ci |
− ω
ω
a s, ya1s , . . . , yars is
sT a s, ya1s , . . . , yars
1 −1
2 −1 1 T
ε T
,
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki 4 sT as is
T
1 −1
sT
∞ 1
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki as iT2
∞
∞ 1
f i, yfω1i , . . . , yfωki − f i, yf1i , . . . , yfki a
sT1 s is
T
−1
T
1 −1
1
1
2 −
pi |ci |
ω
ω
a s, ya1s , . . . , yars is
sT a s, ya1s , . . . , yars
T
∞
1 −1
1
1
−
pi |ci |
ω
ω
a s, ya1s , . . . , yars iT2
sT a s, ya1s , . . . , yars
∞ ∞
1
1
pi |ci |
− ω
ω
a s, ya1s , . . . , yars is
sT a s, ya1s , . . . , yars
≤ max
≤ max
1
T
∞
∞
∞
1 −1
ε ε
1
1
,
2
pi 2
pi
4 16
a
a
sT s iT2
sT1 s is
T −1
T
1 −1 a s, y ω , . . . , y ω
2
a1s
ars − a s, ya1s , . . . , yars
pi |ci |
2
a
s
is
sT
T
−1
∞
∞
∞
1
1 1 2
pi |ci | 2
pi |ci |
a
a
sT s iT2
sT1 s is
ε 5ε
,
4 8
< ε,
3.50
which yields that limω → ∞ S2L yω − S2L y 0, that is, S2L is continuous in UM.
20
Abstract and Applied Analysis
In light of 3.1 and 3.43–3.46, we deduce that for all y {yn }n∈Zβ ∈ UM
SL y sup S1L y S2L y n
n
n≥T
≤ sup S1L y n sup S2L y n n≥T
n≥T
∞
∞
1
≤ sup L − bn yn−τ sup
f i, yf1i , . . . , yfki − ci n≥T
n≥T sn a s, ya1s , . . . , yars is
∞
∞ 1
f i, yf , . . . , yf |ci |
≤ sup L |bn |yn−τ sup 1i
ki
n≥T
n≥T sn a s, ya1s , . . . , yars
is
≤ L b∗ b∗ M ∞
∞
1
pi |ci |
a
sT s is
≤ L M min{L − b∗ M − N, 1 − b∗ M − L} − ε∗
< 2L M,
3.51
which means that SL UM and S2L UM are bounded.
Let ε > 0. Notice that 3.2 ensures that there exists T ∗ > T satisfying
∞
∞
1
pi |ci | < ,
a
2
sT ∗ s is
3.52
which together with 3.1 and 3.46 implies that for all y {yn }n∈Zβ ∈ UM and t2 > t1 ≥ T ∗
∞
∞
1
f i, yf1i , . . . , yfki − ci
S2L y t2 − S2L y t1 st a s, ya1s , . . . , yars is
2
∞
1
−
f i, yf1i , . . . , yfki − ci st1 a s, ya1s , . . . , yars is
∞
≤
∞
∞
∞
∞
1
1
pi |ci | pi |ci |
a
a
st2 s is
st1 s is
3.53
∞
∞
1
pi |ci |
a
sT ∗ s is
≤2
< ,
which means that S2L UM is uniformly Cauchy. Thus S2L UM is relatively compact.
Abstract and Applied Analysis
21
By virtue of 3.41 and 3.45, we infer that for all x {xn }n∈Zβ , y {yn }n∈Zβ ∈ UM
and n ≥ T
S1L x − S1L y |bn |xn−τ − yn−τ ≤ b∗ b∗ x − y,
n
n
3.54
S1L x − S1L y ≤ b∗ b∗ x − y,
3.55
which yields that
that is, S1L is a contraction mapping in UM. It follows that SL is a continuous and
condensing mapping.
Put
P y yn n∈Zβ ∈ lβ∞ : N ≤ yn ≤ M, n ≥ β, y M ,
3.56
Q y yn n∈Zβ ∈ lβ∞ : N ≤ yn ≤ M, n ≥ β and there exists n∗ ≥ β satisfying yn∗ N .
3.57
It is easy to see that ∂UM P ∪ Q. Suppose that there exist y {yn }n∈Zβ ∈ ∂UM and
λ ∈ 0, 1 with
y 1 − λp∗ λSL y.
3.58
Now we consider two possible cases as follows.
Case 1. Let y ∈ P . Obviously 3.41, 3.43–3.46, 3.56, and 3.58 guarantee that
yn 1 − λp∗ λSL yn
∞
1
f i, yf1i , . . . , yfki − ci
1 − λp λ L − bn yn−τ −
sn a s, ya1s , . . . , yars is
∞
∞
1
∗
≤ 1 − λM − ε λ L b∗ M pi |ci |
a
sT s is
∗
∞
3.59
< 1 − λM − ε∗ λL b∗ M min{L − b∗ M − N, 1 − b∗ M − L} − ε∗ ≤ M − ε∗ ,
∀n ≥ T,
which implies that
M y supyn ≤ M − ε∗ < M,
n≥β
which is a contradiction.
3.60
22
Abstract and Applied Analysis
Case 2. Let y ∈ Q. It follows from 3.41, 3.43–3.46, 3.57, and 3.58 that
N yn∗ 1 − λp∗ λSL yn∗
⎡
1 − λp∗ λ⎣L − bn∗ yn∗ −τ
⎤
∞
1
−
f i, yf1i , . . . , yfki − ci ⎦
smax{n∗ ,T } a s, ya1s , . . . , yars is
∞
⎤
∞
1
≥ 1 − λM − ε∗ λ⎣L − b∗ M −
pi |ci | ⎦
a
s
∗
is
smax{n ,T }
⎡
∞
> 1 − λM − ε∗ λL − b∗ M − min{L − b∗ M − N, 1 − b∗ M − L} ε∗ ≥ 1 − λM − ε∗ λN ε∗ ≥ min{M − ε∗ , N ε∗ }
N ε∗ ,
3.61
which is absurd. Thus Lemma 2.3 ensures that there exists y {yn }n∈Zβ ∈ UM satisfying
SL y S1L y S2L y y, that is,
yn L − bn yn−τ −
∞
∞
1
f i, yf1i , . . . , xhki − ci ,
a
s,
y
,
.
.
.
,
y
a1s
ars is
sn
∀n ≥ T,
3.62
which means that
∞
f i, yf1i , . . . , yfki − ci ,
a n, ya1n , . . . , yarn Δ yn bn yn−τ ∀n ≥ T,
3.63
in
which yields that
Δ a n, ya1n , . . . , yarn Δ yn bn yn−τ −f n, yf1n , . . . , yfkn cn ,
∀n ≥ T,
3.64
that is, y {yn }n∈Zβ ∈ UM is a bounded positive solution of 1.8.
Finally we show that 1.8 has uncountably many bounded positive solutions in
UM. Let L1 , L2 ∈ b∗ MN, 1−b∗ M and L1 / L2 . Similarly we infer that for each θ ∈ {1, 2},
there exists a mapping SLθ : UM → V N satisfying 3.41–3.46, where L, β, T , S1L , S2L ,
and SL are replaced by Lθ , βθ , TLθ , S1Lθ , S2Lθ and SLθ , respectively, and the mapping SLθ has
a fixed point yθ {ynθ }n∈Zβ ∈ UM, which is a bounded positive solution of 1.8 in UM,
that is,
θ
−
ynθ Lθ − bn yn−τ
∞ 1
f i, yfθ1i , . . . , yfθki − ci ,
θ
θ
sn a s, ya1s , . . . , yars is
∞
∀n ≥ TLθ .
3.65
Abstract and Applied Analysis
23
It follows from 3.2 that there exists T∗ > max{TL1 , TL2 } satisfying
∞
∞
|L1 − L2 |
1
.
pi |ci | <
a
4
sT∗ s is
3.66
In order to prove that the set of bounded positive solutions of 1.8 is uncountable, it is
sufficient to verify that y1 / y2 . In terms of 3.1, 3.65, and 3.66, we deduce that for n ≥ T∗
∞
∞ 1
1
1
2
1
1
f
i,
y
−
c
bn yn−τ
−
,
.
.
.
,
y
yn − yn2 L1 − L2 − bn yn−τ
i
f
f
1i
ki
1
1
sn a s, ya1s , . . . , yars is
∞
∞ 1
2
2
f i, yf1i , . . . , yfki − ci 2
2
sn a s, ya1s , . . . , yars is
1
2 ≥ |L1 − L2 | − |bn |yn−τ
− yn−τ
−
∞ ∞
1
f i, yf11i , . . . , yf1ki |ci | f i, yf21i , yf22i , . . . , yf2ki |ci |
sn as is
∞
∞
1
≥ |L1 − L2 | − b∗ b∗ y1 − y2 − 2
pi |ci |
a
sT∗ s is
>
1
|L1 − L2 | − b∗ b∗ y1 − y2 ,
2
3.67
which means that
1
y − y2 >
|L1 − L2 |
> 0,
21 b∗ b∗ 3.68
that is, y1 /
y2 . This completes the proof.
Theorem 3.4. Assume that there exist four constants N, M, b∗ , and b∗ and two positive sequences
{an }n∈Nn and {pn }n∈Nn satisfying 3.1, 3.2 and
0
0
1 b∗ M < 1 b∗ N < 0,
b∗ ≤ bn ≤ b∗ < −1 eventually.
3.69
Then 1.8 has uncountably many bounded positive solutions in UM.
Proof. Let L ∈ 1 b∗ M, 1 b∗ N. Now we show that there exists a mapping SL : UM →
V N such that it has a fixed point y {yn }n∈Zβ ∈ UM, which is also a bounded positive
24
Abstract and Applied Analysis
solution of 1.8. It follows from 3.2 and 3.69 that there exists T ≥ max{1, n0 τ |β|}
satisfying
b∗ ≤ bn ≤ b∗ < −1, ∀n ≥ T ;
∞
∞
1
b∗
pi |ci | < min L − 1 b∗ M, N1 b∗ − L
.
a
b∗
sT s is
3.70
3.71
Let p∗ M − ε∗ , where ε∗ ∈ 0, min{L − 1 b∗ M, b∗ /b∗ N1 b∗ − L, b∗ M − N/b∗ − 1
}
is enough small and
∞
∞
1
b∗
∗
pi |ci | < min L − 1 b M, N1 b∗ − L
− ε∗ .
a
b∗
sT s is
3.72
Obviously, p∗ ∈ UM. Define a mapping SL : UM → lβ∞ by 3.44, where the mappings
S1L , S2L : UM → lβ∞ are defined by
⎧
ynτ
L
⎪
⎨
−
, n≥T
S1L y n bnτ bnτ
⎪
⎩
β ≤ n < T,
S1L y T ,
⎧
∞
∞ !
1
1 !
⎪
⎨−
f i, yf1i , . . . , yfki − ci ,
bnτ snτ a s, ya1s , . . . , yars is
S2L y n ⎪
⎩
S2L y T ,
3.73
n≥T
3.74
β≤n<T
for each y {yn }n∈Zβ ∈ UM. By virtue of 3.1, 3.70, and 3.72–3.74, we get that for any
y {yn }n∈Zβ ∈ UM and n ≥ T
SL y
n
S1L y n S2L y n
L
bnτ
−
∞
∞
ynτ
1
1 −
f i, yf1i , . . . , yfki − ci
bnτ bnτ snτ a s, ya1s , . . . , yars is
∞
∞
1
L N
1 −
∗
pi |ci |
b∗ b∗ b sT τ as is
L N
1
b∗
1
≥
−
∗ min L − 1 b∗ M, N1 b∗ − L
− ∗ ε∗
b∗ b∗ b
b∗
b
≥
≥N−
3.75
1 ∗
ε
b∗
> N,
which gives that SL UM ⊆ V N. The rest of the proof is similar to that of Theorem 3.3
and is omitted. This completes the proof.
Abstract and Applied Analysis
25
Now we employ the Krasnoselskii fixed point theorem to prove the existence and
multiplicity of bounded positive solutions of 1.8.
Theorem 3.5. Assume that there exist four constants N, M, b∗ and b∗ and two positive sequences
{an }n∈Nn , {pn }n∈Nn satisfying 3.1, 3.2 and
0
0
0 < Nb∗ b∗ < M b∗2 − b∗ ,
1 < b∗ ≤ bn ≤ b∗ < b∗2 eventually.
3.76
Then 1.8 has uncountably many bounded positive solutions in AN, M.
Proof. Let L ∈ b∗ /b∗ M b∗ N, b∗ M. Now we show that there exist two mappings S1L , S2L :
AN, M → lβ∞ such that the equation S1L y S2L y y has a solution y {yn }n∈Zβ ∈
AN, M, which is also a bounded positive solution of 1.8. It follows from 3.2 and 3.76
that there exists T ≥ max{1, n0 τ |β|} satisfying
1 < b∗ ≤ bn ≤ b∗ < b∗2 , ∀n ≥ T ;
∞
∞
1
b∗ L
pi |ci | < min b∗ M − L, ∗ − M − b∗ N .
a
b
sT s is
3.77
3.78
Define two mappings S1L and S2L : AN, M → lβ∞ by 3.73 and 3.74, respectively. It
follows from 3.1, 3.73, 3.74, 3.77, and 3.78 that for any x {xn }n∈Zβ , y {yn }n∈Zβ ∈
AN, M and n ≥ T
S1L x − S1L y n
n
S1L xn S2L y n ≥
1
1 xnτ − ynτ ≤ x − y,
bnτ
b∗
L
bnτ
−
∞
∞
1
xnτ
1 −
f i, yf1i , . . . , yfki − ci
bnτ bnτ snτ a s, ya1s , . . . , yars is
∞
∞ 1
L M
1 f i, yf , . . . , yf |ci |
−
−
1i
ki
b∗ b∗ b∗ snτ as is
∞
∞
L M
1
1 −
−
pi |ci |
b∗ b∗ b∗ sT τ as is
L M
1
b∗ L
> ∗−
−
min b∗ M − L, ∗ − M − b∗ N
b
b∗ b∗
b
≥
≥ N,
26
Abstract and Applied Analysis
S1L xn S2L y n ≤
L
bnτ
−
∞
∞
1
xnτ
1 −
f i, yf1i , . . . , yfki − ci
bnτ bnτ snτ a s, ya1s , . . . , yars is
∞
∞ 1
L
1 f i, yf , . . . , yf |ci |
1i
ki
b∗ b∗ snτ as is
∞
∞
1
L
1 pi |ci |
b∗ b∗ sT τ as is
L
1
b∗ L
<
min b∗ M − L, ∗ − M − b∗ N
b∗ b∗
b
≤
≤ M,
3.79
which yield that
S1L x − S1L y ≤ 1 x − y,
b∗
S1L x S2L y ∈ AN, M,
∀x, y ∈ AN, M.
3.80
As in the proof of Theorem 3.3, we infer similarly that S2L is continuous in AN, M and
S2L AN, M is relatively compact. Thus S2L is completely continuous, which together with
3.77, 3.80, and Lemma 2.4, ensures that the equation S1L y S2L y y has a solution y {yn }n∈Zβ ∈ AN, M, which is also a bounded positive solution of 1.8 in AN, M. The rest
of the proof is similar to that of Theorem 3.3 and is omitted. This completes the proof.
Remark 3.6. Theorems 3.1–3.5 extend and improve Theorem 2.1 in 1
and Theorems 2.1–2.7
in 7
, respectively. The examples in Section 4 show that our results are indeed generalizations
of the corresponding results in 1, 7
.
4. Examples
Now we construct five examples to show the applications of the results presented in Section 3.
Note that none of the known results can be applied to the five examples.
Example 4.1. Consider the second-order nonlinear neutral delay difference equation
4/5
√
4
nyn1 yn1 − 1
−3n3 yn1
n
6
5
2
Δ −1 n − n 1 yn 1 Δ yn yn−τ 3
n5 ln2 n 2 yn2 − n2 1
n 2
−1 n − 5n 1
,
√
n2 sin n3 1
n4
∀n ≥ 1,
4.1
Abstract and Applied Analysis
27
where τ ∈ N is fixed. Let n0 1, r 1, k 2, N 2, M 3, β 1 − τ and
a1n n,
f1n n 1,
f2n n2 ,
bn 1,
an, u −1n n6 − n5 1 u2 1 ,
fn, u, v −3n3 u4 n5 ln2 n
√
nu u − 14/5
2 |v −
n2 | 3
1
,
−1n n2 − 5n 1
,
√
n4 n2 sin n3 1
an 3 n6 − n5 1 ,
cn √
243n3 3 n 2
,
pn n5
∀n, u, v ∈ Nn0 × R2 .
4.2
It is easy to verify that 3.1–3.3 hold. Thus Theorem 3.1 guarantees that 4.1 has
uncountably many bounded positive solutions in BM, N. But the results in 1, 7
are not
applicable for 4.1.
Example 4.2. Consider the second-order nonlinear neutral delay difference equation
9 3
√
n3 yn4 − 2 − n3 − 2n 1 y2n−1
4
Δ
y
−
y
Δ n5 n 2 yn2 2 −60 y2n−1
n
n
n−τ
2
n5 yn4 2 −3 − 3n sin nyn4 2
√
n5 − 6n3 2n − 4 − 5
,
n11 4
4.3
∀n ≥ 5,
where τ ∈ N is fixed. Let n0 5, r 2, k 3, N 1, M 3, β min{5 − τ, −35} and
a2n 2n −1n ,
f1n n 4,
f2n 2n − 1,
√
√
n5 − 6n3 2n − 4 − 5
5
n2 − 3,
bn −1,
cn n 2 u2 v 4 ,
,
an,
u,
v
n
n11 4
√
n3 u − 29 − n3 − 2n 1 v3
5
fn, u, v, w an n n 2 1,
,
2
n5 w4 − 3n sinnu 2
a1n n2 − 60,
f3n
pn 28n3 54n 27
,
n5
∀n, u, v, w ∈ Nn0 × R3 .
4.4
It is clear that 3.1, 3.21, and 3.22 hold. Hence Theorem 3.2 ensures that 4.3 has
uncountably many bounded positive solutions in BM, N. But the results in 1, 7
are not
valid for 4.3.
28
Abstract and Applied Analysis
Example 4.3. Consider the second-order nonlinear neutral delay difference equation
−1n n7 − 1
3 2 2
n 2 y3n n yn−8 1 Δ xn xn−τ
Δ −1
3n7 1
"
2
− 2
xn−5 − 11/3 xnn1/2
nln2 n 1
n6 − 30n4 8cos5 4n8 − 1
, ∀n ≥ 3,
n9 n5 2ln2 n
nn−1/2
3
4.5
where τ ∈ N is fixed. Let n0 3, r k 2, b∗ b∗ 1/3, N 2, M 7, β min{3 − τ, −5}
and
−1n n7 − 1
bn ,
a1n 3n,
a2n n − 8,
f1n n − 5,
f2n
3n7 1
n6 − 30n4 8cos5 4n8 − 1
3
nn−1/2
3
2 2
cn n
n
2
v
1
,
,
an,
u,
v
−1
u
n9 n5 2ln2 n
#
u − 11/3 |v2 − 2|
3
2
an 8n 4n 1,
fn, u, v ,
nln2 n 1
√
61/3 47
pn , ∀n, u, v ∈ Nn0 × R2 .
nln2 n
4.6
nn 1
,
2
It is clear that 3.1, 3.2 and 3.40 are satisfied. Hence Theorem 3.3 implies that 4.5
has uncountably many bounded positive solutions in UM. But the results in 1, 7
are
unapplicable for 4.5.
Example 4.4. Consider the second-order nonlinear neutral delay difference equation
3n2 − 3n − 3 cosn/2 − 1 1
xn−τ
1 Δ xn −
Δ −1 n
n2 − n − cosn/2 − 1
√
3
3 − xn2 −9 1/3 − n − 1x2n−3
4
n4 nx3n−7
1
n3 −1n n2 ln3 1 n2
, n ≥ 2,
n6 4n5 − 3n4 sin3 4n2 − 5 1
n−1 2
yn4 3 −1
4.7
Abstract and Applied Analysis
29
where τ ∈ N is fixed. Let n0 2, r 1, k 3, b∗ −4, b∗ −3, N 1, M 3, β min{2 − τ, −5} and
a1n n3 − 1,
bn −
f1n n2 − 9,
f2n 3n − 7,
3n2 − 3n − 3 cosn/2 − 1 1
,
n2 − n − cosn/2 − 1
an, u −1n−1 n2 u4 1 ,
cn an n2 ln3 n,
√
2 27 2n − 3
pn ,
n4
f3n 2n − 3,
n3 −1n n2 ln3 1 n2
,
n6 4n5 − 3n4 sin3 4n2 − 5 1
√
3 − u1/3 − 2n − 3v3
fn, u, v, w ,
n4 nw4 1
∀n, u, v, w ∈ Nn0 × R3 .
4.8
It is easy to verify that 3.1, 3.2, and 3.69 hold. Hence Theorem 3.4 implies that 4.7 has
uncountably many bounded positive solutions in UM. But the results in 1, 7
are not valid
for 4.7.
Example 4.5. Consider the second-order nonlinear neutral delay difference equation
8
Δ −1nn−1/2 n 15 y2n−3
2n3 yn2 2 −1 ln 1 nynn−1 2
11 ln 1 n2 11 sin n 10
|xn−5 − 2|3/4 xnn−2 − 2
×Δ xn xn−τ
3
ln1 n2 sin n 1
n2 3 xn−5 − n2 xnn−2 1
4.9
n7 − 5n4 − 9
, n ≥ 0,
n11 7n8 − 6n7 5n3 1
where τ ∈ N is fixed. Let n0 0, r 3, k 2, b∗ 10, b∗ 11, N 2, M 3, β min{−τ, −5} and
a1n 2n − 3,
a2n n2 − 1,
a3n nn − 1,
f1n n − 5,
f2n nn − 2,
11 ln 1 n2 11 sin n 10
n7 − 5n4 − 9
,
c
,
bn n
ln1 n2 sin n 1
n11 7n8 − 6n7 5n3 1
4.10
an n5 ,
an, u, v, w −1nn−1/2 n 15 u8 2n3 v2 ln1 n|w| ,
fn, u, v |u − 2|3/4 v − 22
n2
3
3 |u −
n2 v|
1
,
pn n6
2
,
1
∀n, u, v, w ∈ Nn0 × R3 .
It is easy to see that 3.1, 3.2, and 3.76 hold. Hence Theorem 3.5 guarantees that 4.9
possesses uncountably many bounded positive solutions in AN, M. But the results in 1, 7
are inapplicable for 4.9.
30
Abstract and Applied Analysis
Acknowledgments
This research was supported by the Science Research Foundation of Educational Department
of Liaoning Province L2012380 and the Basic Science Research Program through the
National Research Foundation of Korea NRF funded by the Ministry of Education, Science
and Technology 2012R1A1A2002165.
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