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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 308362, 11 pages
doi:10.1155/2011/308362
Research Article
Solvability of a Class of Generalized
Neumann Boundary Value Problems for
Second-Order Nonlinear Difference Equations
Jianye Xia1 and Yuji Liu2
1
Department of Applied Mathematics, Guangdong University of Finance,
Guangzhou 510000, China
2
Department of Mathematics, Guangdong University of Business Studies,
Guangzhou 510000, China
Correspondence should be addressed to Jianye Xia, jianye xia@sohu.com
Received 18 April 2011; Accepted 17 June 2011
Academic Editor: Hassan A. El-Morshedy
Copyright q 2011 J. Xia and Y. Liu. 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.
This paper is motivated by Rachnkovab and Tisdell 2006 and Anderson et al. 2007. New sufficient conditions for the existence of at least one solution of the generalized Neumann boundary
value problems for second order nonlinear difference equations ∇Δxk fk, xk, xk 1,
k ∈ 1, n − 1, x0 ax1, xn bxn − 1, are established.
1. Introduction
Recently, there have been many papers discussed the solvability of two-point or multipoint
boundary value problems for second-order or higher-order difference equations, we refer the
readers to the text books 1, 2 and papers 3–8 and the references therein.
In a recent paper 3, Anderson et al. studied the following problem:
∇Δyk f k, yk, Δyk ,
k 1, . . . , n − 1,
Δy0 Δyn 0,
1.1
2
Discrete Dynamics in Nature and Society
where
Δyk ∇Δyk ⎧
⎨yk 1 − yk,
for k 0, . . . , n − 1,
⎩0,
for k n,
⎧
⎨yk 1 − 2yk yk − 1, for k 1, . . . , n − 1,
⎩0,
1.2
for k 0 or n.
The following result was proved.
Theorem ART
Suppose that f is continuous and there exist constants α ≤ 0, K ≥ 0 such that
f t, p, q − p ≤ α 2pf t, p, q q2 K,
t, p, q ∈ {1, . . . , n − 1} × R2 .
∗
Then BVP1.1 has at least one solution.
The methods in 3 involved new inequalities on the right-hand side of the difference
equation and Schaefer’s Theorem in the finite-dimensional space setting.
In 7, the following discrete boundary value problem BVP involving second order
difference equations and two-point boundary conditions
Δyk
∇Δyk
,
f
t
,
y
,
k
k
h
h2
y0 0,
k 1, . . . , n − 1,
1.3
yn 0,
was studied, where n ≥ 2 an integer, f is continuous, scalar-valued function, the step size
is h N/n with N a positive constant, the grid points are tk kh for k 0, . . . , n. The
differences are given by
Δyk ∇Δyk ⎧
⎨yk1 − yk , k 0, . . . , n − 1,
⎩0,
k n,
1.4
⎧
⎨yk1 − 2yk yk−1 ,
k 1, . . . , n − 1,
⎩0,
k 0 or k n.
The following two results were proved in 7.
Discrete Dynamics in Nature and Society
3
Theorem RT
Let f be continuous on 0, N × R2 and α, β, and, K be nonnegative constants. If there exist
c, d ∈ 0, 1 such that
ft, u, v ≤ α|u|c β|v|d K,
t, u, v ∈ 0, N × R2 ,
1.5
then the discrete BVP1.3 has at least one solution.
Theorem RT
Let f be continuous on 0, N × R2 and α, β, and K nonnegative constants. If
ft, u, v ≤ α|u| β|v| K,
t, u, v ∈ 0, N × R2 ,
αN 2 βN
< 1,
8
2
1.6
1.7
then the discrete BVP1.3 has at least one solution.
In paper 8, Cabada and Otero-Espinar studied the existence of solutions of a class
of nonlinear second-order difference problem with Neumann boundary conditions by using
upper and lower solution methods. Assuming the existence of a pair of ordered lower and
upper solutions γ and β, they obtained optimal existence results for the case γ ≤ β and even
for γ ≥ β.
In this paper, we study the following boundary value problem for second-order
nonlinear difference equation
∇Δxk fk, xk, xk 1,
x0 ax1,
k ∈ 1, n − 1,
xn bxn − 1,
1.8
where a, b ∈ R, n ≥ 2 is an integer, and f is continuous, scalar-valued function. We note that
when a b 1, BVP1.8 becomes the following BVP:
∇Δxk fk, xk, xk 1,
t ∈ 1, T − 1,
Δx0 0 Δxn − 1,
1.9
which is called Neumann boundary value problem of difference equation and is a special
case of BVP1.1. When a b 0, BVP1.8 is changed to
∇Δxk fk, xk, xk 1,
x0 0 xn,
t ∈ 1, T − 1,
1.10
which is the so-called Dirichlet problem for discrete difference equations and is a special case
of BVP1.3.
4
Discrete Dynamics in Nature and Society
The purpose of this paper is to improve the assumptions ∗, 1.5, and 1.6 in the
results in paper 3, 5, 7–9, by using Mawhin’s continuation theorem of coincidence degree,
to establish sufficient conditions for the existence of at least one solution of BVP1.8. It is
interesting that we allow f to be sublinear, at most linear or superlinear.
This paper is organized as follows. In Section 2, we make the main results, and in
Section 3, we give some examples, which cannot be solved by theorems in 5, 7, 9, to illustrate
the main results presented in Section 3.
2. Main Results
To get the existence results for solutions of BVP1.8, we need the following fixed point
theorems.
Let X and Y be Banach spaces, L : DL ⊂ X → Y a Fredholm operator of index
zero, and P : X → X, Q : Y → Y projectors such that Im P Ker L, Ker Q Im L, X Ker L⊕ Ker P, Y Im L⊕ Im Q. It follows that L|DL∩Ker P : DL∩Ker P → Im L is invertible;
we denote the inverse of that map by Kp .
If Ω is an open bounded subset of X, DL ∩ Ω / ∅, the map N : X → Y will be called
L-compact on Ω if QNΩ is bounded and Kp I − QN : Ω → X is compact.
Lemma 2.1 see 9. Let L be a Fredholm operator of index zero, and let N be L-compact on Ω.
Assume that the following conditions are satisfied:
i Lx /
λNx for every x, λ ∈ DL \ Ker L ∩ ∂Ω × 0, 1;
ii Nx ∈
/ Im L for every x ∈ Ker L ∩ ∂Ω;
iii deg∧QN|Ker L , Ω ∩ Ker L, 0 / 0, where ∧ : Ker L → Y/ Im L is the isomorphism.
Then the equation Lx Nx has at least one solution in DL ∩ Ω.
Lemma 2.2 see 9. Let X and Y be Banach spaces. Suppose L : DL ⊂ X → Y is a Fredholm
operator of index zero with Ker L {0}, N : X → Y is L-compact on any open bounded subset of
X. If 0 ∈ Ω ⊂ X is an open bounded subset and Lx /
λNx for all x ∈ DL ∩ ∂Ω and λ ∈ 0, 1,
then there is at least one x ∈ Ω so that Lx Nx.
Let X Rn1 , Y Rn−1 be endowed with the norms
x max |xn|,
n∈0,n
y max yk
k∈1,n−1
2.1
for x ∈ X and y ∈ Y , respectively. It is easy to see that X and Y are Banach spaces. Choose DL {x ∈ X : x0 ax1, xn bxn − 1}. Let L : X → Y, Lxk ∇Δxk, x ∈ DL, and
N : X → Y by Nxk fk, xk, xk 1.
Consider the following problem:
∇Δxk 0,
x0 ax1,
xn bxn − 1.
2.2
Discrete Dynamics in Nature and Society
5
It is easy to see that problem 2.2 has a unique solution xk 0 if and only if
a1 − b.
1 − an − 1b − n /
2.3
If 2.3 holds, we call BVP1.8 at nonresonance case. If 1 − an − 1b − n a1 − b, then
problem 2.2 has infinite nontrivial solutions. At this case, we call BVP1.8 at resonance case.
In this paper, we establish sufficient conditions for the existence of solutions of BVP1.8 at
nonresonance case, that is, 1 − an − 1b − n /
a1 − b, and at resonance case, a b 1. It is similar to get existence results for the existence of solutions at resonance case when
1 − an − 1b − n a1 − b and a / 1, b / 1.
Lemma 2.3. Suppose a b 1. Then the following results are valid.
i Ker L {x c, . . . , c ∈ X : c ∈ R}.
ii Im L {y ∈ Y : n−1
i1 yi 0}.
iii L is a Fredholm operator of index zero.
iv There are projectors P : X → X and Q : Y → Y such that Ker L Im P , Ker Q ∅; then N is L-compact on
Im L. Furthermore, let Ω ⊂ X be an open bounded subset with Ω ∩ DL /
Ω.
v x ∈ DL is a solution of Lx Nx which implies that x is a solution of BVP1.8.
The projectors P : X → X and Q : Y → Y , the isomorphism ∧ : Ker L → Y/ Im L,
and the generalized inverse Kp : Im L → DL ∩ Im P are as follows:
P xn x1,
Qyn n−1
1 yi,
n − 1 i1
∧c c,
Kp yn 2.4
k s
yi.
s1 i1
Lemma 2.4. Suppose 1 − an − 1b − n / a1 − b. Then the following results are valid.
i x ∈ DL is a solution of Lx Nx which implies that x is a solution of BVP1.8.
ii Ker L {0}.
iii L is a Fredholm operator of index zero, N is L-compact on each open bounded subset of X.
Suppose
A there exist numbers β > 0, θ ≥ 1, nonnegative sequences pn, qn, rn, functions
gn, x, y, hn, x, y such that fn, x, y gn, x, y hn, x, y and
g n, x, y x ≥ β|x|θ1 ,
h n, x, y ≤ pn|x|θ qnyθ rn,
for all n ∈ {1, . . . , n − 1}, x, y ∈ R2 ;
2.5
6
Discrete Dynamics in Nature and Society
B there exists a constant M > 0 so that
n−1
fn, c, c > 0
c
2.6
i1
for all |c| > M or
n−1
c
fn, c, c < 0
2.7
i1
for all |c| > M.
Theorem L
Suppose a2 ≤ 1, b2 ≤ 1, and that A and B hold. Then BVP1.8 has at least one solution if
p q max |b|θ1 , 1 < β.
2.8
Proof. To apply Lemma 2.1, we consider Lx λNx for λ ∈ 0, 1.
Step 1. Let Ω1 {x ∈ X : Lx λNx, λ ∈ 0, 1}. For x ∈ Ω1 , we have
xk 1 − 2xk xk − 1 λfk, xk, xk 1,
k ∈ 1, n − 1,
x0 ax1,
2.9
xn bxn − 1.
So
xk 1 − 2xk xk − 1xk λfk, xk, xk 1xk,
k ∈ 1, n − 1.
2.10
It is easy to see that
n−1
2 xk 1 − 2xk xk − 1xk
n1
n−1 −xk 12 2xkxk 1 − xk2 − xk − 12 2xk − 1xk − xk2
n1
xk 12 − 2xk2 xk − 12
n−1 n1
−xk 1 − xk2 − xk − 1 − xk2 xk 12 − 2xk2 xk − 12
Discrete Dynamics in Nature and Society
n−1 7
−xk 1 − xk2 − xk − 1 − xk2
n1
xn2 − xn − 12 − x12 x02
n−1 −xk 1 − xk2 − xk − 1 − xk2
n1
b2 − 1 xn − 12 a2 − 1 x12 .
2.11
Since a2 ≤ 1, b2 ≤ 1, we get
n−1
xk 1 − 2xk xk − 1xk ≤ 0.
2.12
n1
So, we get
n−1
fk, xk, xk 1xk ≤ 0.
2.13
gk, xk, xk 1 hk, xk, xk 1 xk ≤ 0.
2.14
λ
n1
Then
n−1
n1
It follows that
β
n−1
n−1
|xk|θ1 ≤ − hk, xk, xk 1xk
n1
k1
≤
n−1 pk|xk|θ1 qk|xk 1|θ |xk| rk|xk|
2.15
n1
n−1
n−1
n−1
≤ p |xk|θ1 q |xk 1|θ |xk| rk|xk|.
n1
k1
k1
For xi ≥ 0, yi ≥ 0, Holder’s inequality implies
s
i1
1/p 1/q
s
s
p
q
xi yi ≤
xi
yi
,
i1
i1
1 1
1, q > 0, p > 0.
p q
2.16
8
Discrete Dynamics in Nature and Society
It follows that
β
n−1
|xk|θ1
k1
θ/θ1 1/θ1
n−1
n−1
n−1
θ1
θ1
θ1
≤ p
|xk| q
|xk 1|
|xk|
n1
r
k1
k1
n−1
|xk|
k1
rn − 1
θ/θ1
1/θ1
n−1
n−1
θ1
p |xk|θ1
|xk|
n1
k1
θ/θ1 1/θ1
n−2
n−1
θ1
θ1
θ1
θ1
q
|b| |x1| |xk 1|
|xk|
k1
≤ rn − 1
θ/θ1
2.17
k1
1/θ1
n−1
n−1
θ1
p |xk|θ1
|xk|
n1
k1
θ/θ1 1/θ1
n−1
n−1
θ1
θ1
θ1
q max |b| , 1
|xk|
|xk|
k1
rn − 1
θ/θ1
k1
1/θ1
n−1
n−1
θ1
|xk|
p |xk|θ1
n1
k1
n−1
q max |b|θ1 , 1
|xk|θ1 .
k1
It follows from 2.8 that there exists a constant M1 > 0 such that
n−1
|xk|θ1 ≤ M1 .
2.18
k1
Hence |xk| ≤ M1 /n − 11/θ1 for all k ∈ {1, . . . , n − 1}. Hence ||x|| ≤ M1 /n − 11/θ1 .
So Ω1 is bounded.
Step 2. Prove that the set Ω2 {x ∈ Ker L : Nx ∈ Im L} is bounded.
For x ∈ Ker L, we have xk c for k ∈ 0, n. Thus we have Nxk fk, c, c.
Nx, y ∈ Im L implies that
n−1
fn, c, c 0.
k1
It follows from condition B that |c| ≤ M. Thus Ω2 is bounded.
2.19
Discrete Dynamics in Nature and Society
9
Step 3. Prove the set Ω3 {x ∈ Ker L : ±λ ∧ x 1 − λQNx 0, ∃λ ∈ 0, 1} is bounded.
If the first inequality of B holds, let
Ω3 {x ∈ Ker L : λ ∧ x 1 − λQNx 0, ∃λ ∈ 0, 1}.
2.20
We will prove that Ω3 is bounded. For xk c for k ∈ 0, n such that x ∈ Ω3 , and
λ ∈ 0, 1, we have
−1 − λ
n−1
fn, c, c λcn − 1.
2.21
k1
If λ 1, then c 0. If λ / 1, then
0 > −1 − λc
n−1
fn, c, c λc2 T ≥ 0,
2.22
k1
a contradiction.
If the second inequality of B holds, let
Ω3 {x ∈ Ker L : −λ ∧ x 1 − λQNx 0, ∃λ ∈ 0, 1}.
2.23
Similarly, we can get a contradiction. So Ω3 is bounded.
Step 4. Obtain open bounded set Ω such i, ii, and iii of Lemma 2.1.
In the following, we will show that all conditions of Lemma 2.1 are satisfied. Set Ω an
open bounded subset of X such that Ω ⊃ 3i1 Ωi . We know that L is a Fredholm operator of
index zero and N is L-compact on Ω. By the definition of Ω, we have Ω ⊃ Ω1 and Ω ⊃ Ω2 , thus
Lx / Im L for x ∈ Ker L ∩ ∂Ω.
/ λNx for x ∈ DL \ Ker L ∩ ∂Ω and λ ∈ 0, 1; Nx ∈
In fact, let Hx, λ ±λ ∧ x 1 − λQNx. According the definition of Ω, we know
0 for x ∈ ∂Ω ∩ Ker L, thus by homotopy property of degree,
Ω ⊃ Ω3 , thus Hx, λ /
deg QN|Ker L , Ω ∩ Ker L, 0 degH·, 0, Ω ∩ Ker L, 0
degH·, 1, Ω ∩ Ker L, 0
2.24
deg±∧, Ω ∩ Ker L, 0 /
0.
Thus by Lemma 2.1, Lx Nx has at least one solution in DL ∩ Ω, which is a solution of
BVP1.8. The proof is completed.
Theorem L
Suppose a2 ≤ 1, b2 ≤ 1, 1 − an − 1b − n /
a1 − b, and that A holds. Then BVP1.8 has
at least one solution if 2.8 holds.
10
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Proof. To apply Lemma 2.2, we consider Lx λNx for λ ∈ 0, 1. Let Ω1 {x ∈ X : Lx λNx, λ ∈ 0, 1}. For x ∈ Ω1 , we get 2.9 and 2.10. using the methods in the proof of
Theorem LX1, we get that Ω1 is bounded. Let Ω be a nonempty open bounded subset of X
such that Ω ⊃ Ω1 centered at zero. It is easy to see that L is a Fredholm operator of index zero
and N is L-compact on Ω. One can see that Lx /
λNx for all x ∈ DL ∩ ∂Ω and λ ∈ 0, 1.
Thus, from Lemma 2.2, Lx Nx has at least one solution x ∈ DL ∩ Ω, so x is a solution of
BVP1.8. The proof is complete.
3. An Example
In this section, we present an example to illustrate the main results in Section 2.
Example 3.1. Consider the following problem:
xk 1 − 2xk xk − 1 βxk2m1 pkxk2m1 qkxk 12m1 rk,
k ∈ 1, n − 1,
x0 ax1,
xn bxn − 1,
3.1
where n ≥ 2, m ≥ 1 are integers and β > 0, pn, qn, rn are sequences. Corresponding to
the assumptions of Theorem L1, we set
f k, x, y βx2m1 pkx2m1 qky2m1 rk,
g k, x, y βx2m1 ,
h k, x, y pkx2m1 qky2m1 rk,
3.2
and θ 2m 1. It is easy to see that A holds, and
fn, c, c c2m1 β pkc2m1 qkc2m1 rk
implies that there is M > 0 such that c
n ∈ 1, n − 1 and |c| > M.
n−1
i1
3.3
c2m1 β pkc2m1 qkc2m1 rk > 0 for all
It follows from Theorem L2 that 3.1 has at least one solution if a2 ≤ 1, b2 ≤ 1, 1 −
an − 1b − n / a1 − b and ||p|| ||q|| max{|b|θ1 , 1} < β. BVP3.1 has at least one solution
if a b 1 and ||p|| ||q|| max{|b|θ1 , 1} < β.
Remark 3.2. It is easy to see that BVP3.1 when a b 0 cannot be solved by using theorems
obtained in paper 7. BVP3.1 when a b 1 cannot be solved by the results obtained in
paper 3.
Discrete Dynamics in Nature and Society
11
Acknowledgments
This paper is supported by Natural Science Foundation of Hunan province, China no.
06JJ5008 and Natural Science Foundation of Guangdong province no. 7004569.
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