Hindawi Publishing Corporation
Journal of Function Spaces
Volume 2014, Article ID 815170, 6 pages
http://dx.doi.org/10.1155/2014/815170
Research Article
Existence of Positive Solutions for Second-Order Neumann
Difference System
Yanqiong Lu and Ruyun Ma
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Correspondence should be addressed to Yanqiong Lu; [email protected]
Received 30 October 2013; Accepted 20 December 2013; Published 8 January 2014
Academic Editor: Gen-Qi Xu
Copyright © 2014 Y. Lu and R. Ma. 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 studies a class of Neumann difference system which is not cooperative but its linear part is and this makes it possible
to establish existence and nonexistence results for nonnegative solutions of the system in terms of the principal eigenvalue of the
corresponding linearized system.
Δ2 V (𝑘) + 𝑔 (𝑘, 𝑢 (𝑘) , V (𝑘)) = 0,
1. Introduction
There has been a long history of interest in difference equations and difference systems; for example, see [1–8] and the
references therein. In particular, a wide variety of nonlinear
difference systems have been studied because they model
numerous real-life problems in biology, physics, population
dynamics, economics, and so on.
One of the difference equations that has attracted some
attention is
𝑢 (𝑘 + 2) = 𝑓 (𝑘, 𝑢 (𝑘) , 𝑢 (𝑘 + 1)) ,
Δ𝑢 (0) = 𝐴,
𝑘 ∈ {0, 1, . . . , 𝑁 − 1} ,
(1)
where Δ𝑢(𝑘) = 𝑢(𝑘 + 1) − 𝑢(𝑘) for all 𝑘 ∈ Z, 𝐴, 𝐵 ∈ R. In
2003, Cabada and Otero-Espinar [4] studied the existence of
solution of the problem (1) and obtained optimal existence
results by lower and upper solutions methods.
Besides, we note that some systems of discrete boundary
value problems are investigated by several authors in recent
years; see [5, 7, 8] and the references therein. For example,
Sun and Li [5] studied the following boundary value problem
of discrete system:
𝑘 ∈ {0, 1, 2, . . . , 𝑁} ,
𝑁 > 3,
𝑢 (0) = 𝑢 (𝑁 + 2) = 0,
V (0) = V (𝑁 + 2) = 0.
(2)
Under some assumptions on 𝑓, 𝑔, they obtained some sufficient conditions for the existence of one or two positive solutions to the system by using nonlinear alternative of LeraySchauder and the fixed point theorem in cones.
Henderson et al. [7] considered the following system of
three-point discrete boundary value problem:
Δ2 𝑢 (𝑘 − 1) + 𝜆𝑎 (𝑘) 𝑓 (𝑢 (𝑘) , V (𝑘)) = 0,
Δ𝑢 (𝑁) = 𝐵,
Δ2 𝑢 (𝑘) + 𝑓 (𝑘, 𝑢 (𝑘) , V (𝑘)) = 0,
𝑘 ∈ {0, 1, 2, . . . , 𝑁} ,
𝑘 ∈ {1, 2, . . . , 𝑁 − 1} ,
𝑁 > 4,
Δ2 V (𝑘 − 1) + 𝜇𝑏 (𝑘) 𝑔 (𝑢 (𝑘) , V (𝑘)) = 0,
𝑘 ∈ {1, 2, . . . , 𝑁 − 1} ,
𝑢 (0) = 𝛽𝑢 (𝜂) ,
𝑢 (𝑁) = 𝛼𝑢 (𝜂) ,
V (0) = 𝛽V (𝜂) ,
V (𝑁) = 𝛼V (𝜂) ,
(3)
where 𝜂 ∈ {1, 2, . . . , 𝑁−1}, 𝛼, 𝛽 > 0, 𝑓, 𝑔, 𝑎, 𝑏 are nonnegative
functions and 𝜆, 𝜇 > 0. They deduced the existence of the
eigenvalues 𝜆 and 𝜇 yielding at least one positive solution to
the system (3) under some assumptions on 𝑓, 𝑔, 𝑎, and 𝑏 with
weakly coupling behaviors. Their main tool is the fixed point
theorem in cones.
2
Journal of Function Spaces
However, very little work has been done for the existence
of positive solutions of second-order Neumann difference
systems. Inspired by the above works, we study the existence
of positive solutions of the following second order Neumann
difference system:
−Δ2 𝑢 (𝑘 − 1) = 𝑎 (𝑘) V (𝑘) − 𝑐 (𝑘) 𝑢 (𝑘)
− 𝑒𝑢 (𝑘) [𝑢 (𝑘) + V (𝑘)] ,
𝑘 ∈ I,
−Δ2 V (𝑘 − 1) = 𝑏 (𝑘) 𝑢 (𝑘) − 𝑑 (𝑘) V (𝑘)
(4)
− 𝑓V (𝑘) [𝑢 (𝑘) + V (𝑘)] ,
Δ𝑢 (0) = Δ𝑢 (𝑛) = 0,
𝑘 ∈ I,
ΔV (0) = ΔV (𝑛) = 0,
where I = {1, 2, . . . , 𝑛}, the coefficients 𝑎, 𝑏, 𝑐, and 𝑑 are positive functions on I, and 𝑒, 𝑓 are positive constants. Through
careful analysis, we have found that (4) is not cooperative but
its linear part is and this makes it possible to establish existence and nonexistence results for nonnegative solutions of
(4) in terms of the principal eigenvalue of the corresponding
linearized system. These conditions are different from those
given in [5, 7].
Although system (4) is very simple, it contains an interesting mathematical feature. It is well known that cooperative
systems can be analysed by using lower and upper solutions
method and, in general, possesses many of the properties of
scalar equations. Although system (4) is not cooperative, its
linear part
−Δ2 𝑢 (𝑘 − 1) = 𝑎 (𝑘) V − 𝑐 (𝑘) 𝑢,
𝑘 ∈ I,
−Δ2 V (𝑘 − 1) = 𝑏 (𝑘) 𝑢 − 𝑑 (𝑘) V,
𝑘 ∈ I,
Δ𝑢 (0) = Δ𝑢 (𝑛) = 0,
(5)
𝑘 ∈ I,
−Δ2 V (𝑘 − 1) = 𝐶 (𝑘) 𝑢 (𝑘) + 𝐷 (𝑘) V (𝑘) ,
𝑘 ∈ I,
Lemma 1. Assume that 𝑞 : I → (0, ∞), ℎ : I → [0, ∞)
and 𝑅1 ≤ 0, 𝑅2 ≥ 0 are constants. Then the boundary value
problem
−Δ2 𝑢 (𝑘 − 1) + 𝑞 (𝑘) 𝑢 (𝑘) = ℎ (𝑘) ,
−Δ V (𝑘 − 1) = 𝑔 (𝑢 (𝑘) , V (𝑘)) ,
𝑘 ∈ I,
𝑘 ∈ I,
𝑘 ∈ I,
Δ𝑢 (𝑛) = 𝑅2
(8)
has the solution 𝑢 that satisfies either 𝑢 ≡ 0 or 𝑢 ≫ 0 on ̂I.
Proof. Let 𝜑(𝑘), 𝜓(𝑘) be the solution of initial value problems
−Δ2 𝜑 (𝑘 − 1) + 𝑞 (𝑘) 𝜑 (𝑘) = 0,
𝜑 (0) = 1,
Δ𝜑 (0) = 0,
−Δ2 𝜓 (𝑘 − 1) + 𝑞 (𝑘) 𝜓 (𝑘) = 0,
𝜓 (𝑛) = 1,
𝑘 ∈ I,
𝑘 ∈ I,
(9)
Δ𝜓 (𝑛) = 0,
respectively. It is easy to compute and show that
(i) 𝜑(𝑘) = 1 + ∑𝑘−1
𝑗=1 (𝑘 − 𝑗)𝑞(𝑗)𝜑(𝑗) > 0 and 𝜑 is increasing
on ̂I;
(ii) 𝜓(𝑘) = 1 + ∑𝑛𝑗=𝑘+1 (𝑗 − 𝑘)𝑞(𝑗)𝜓(𝑗) > 0 and 𝜓 is
decreasing on ̂I.
The difference system
−Δ2 𝑢 (𝑘 − 1) = 𝑓 (𝑢 (𝑘) , V (𝑘)) ,
(7)
where 𝐴, 𝐵, 𝐶, and 𝐷 are functions on I with 𝐴(𝑘), 𝐷(𝑘) < 0
and 𝐵(𝑘), 𝐶(𝑘) > 0 for 𝑘 ∈ I.
Let ̂I = {0, 1, . . . , 𝑛, 𝑛 + 1} and 𝑋 = {𝑢 | 𝑢 : ̂I → R} be a
Banach space with the norm ‖𝑢‖ = max𝑡∈̂I |𝑢(𝑡)|. In addition,
we introduce the notation 𝑢 ≫ 0 on ̂I that means that 𝑢(𝑡) > 0
for 𝑡 ∈ ̂I.
In the following, we give some important Lemmas to
show the main result.
Δ𝑢 (0) = 𝑅1 ,
2. Cooperative Systems
Δ𝑢 (0) = Δ𝑢 (𝑛) = 0,
−Δ2 𝑢 (𝑘 − 1) = 𝐴 (𝑘) 𝑢 (𝑘) + 𝐵 (𝑘) V (𝑘) ,
ΔV (0) = ΔV (𝑛) = 0
is and the cooperative nature of the linear system plays a key
role in formulating and helping to prove our existence results
for (4). For recent works in the literature on continuous
Neumann boundary value problem as well as system, we refer
to [9–15] and the references therein.
Sufficiently motivated, the paper is organized as follows:
in Section 2, we discuss the properties of cooperative system
which we will require. In Section 3, we establish necessary
and sufficient conditions for the existence of a positive
solution of (4) in terms of the principal eigenvalue of the
associated linear system. Finally, an example is presented to
illustrate the main result.
2
function for any fixed V. Because of maximum principle arguments cooperative systems possesses many of the properties
of single difference equation, in this section, we will discuss
the results that we require for linear cooperative systems.
We are primarily interested in the existence of principal
eigenvalues for such systems and on the monotone behavior
of such eigenvalues with respect to coefficients of the system.
Our proofs depend on the connection between the existence
of positive upper solutions for the system and the maximum
principle. The continuous case has been obtained by Sweers
[13], López-Gómez and Molina-Meyer [14], and Brown and
Zhang [15], but as far as we known, there is no result for the
discrete case.
Throughout the rest of this section, we will consider the
cooperative system
(6)
ΔV (0) = ΔV (𝑛) = 0
is said to be cooperative if V → 𝑓(𝑢, V) is a nondecreasing
function for any fixed 𝑢 and 𝑢 → 𝑔(𝑢, V) is a nondecreasing
It is not difficult to verify that the problem (8) has the
solution
𝑛
𝑢 (𝑘) = ∑ 𝐺 (𝑘, 𝑗) ℎ (𝑗) +
𝑗=1
𝑅1
𝑅2
𝜓 (𝑘) +
𝜑 (𝑘) ,
Δ𝜓 (0)
Δ𝜑 (𝑛)
𝑘 ∈ ̂I,
(10)
Journal of Function Spaces
3
where
𝐺 (𝑘, 𝑗) = {
𝜑 (𝑘) 𝜓 (𝑗) , 𝑘 < 𝑗,
𝜑 (𝑗) 𝜓 (𝑘) , 𝑘 ≥ 𝑗.
(11)
Clearly, it follows from the properties of 𝜑, 𝜓 together with
the above that either 𝑢 ≡ 0 or 𝑢 ≫ 0 on ̂I.
Theorem 2. Suppose that there exist functions 𝑢0 , V0 ∈ 𝑋 such
that 𝑢0 ≫ 0, V0 ≫ 0 and
−Δ2 𝑢0 (𝑘 − 1) ≥ 𝐴 (𝑘) 𝑢0 (𝑘) + 𝐵 (𝑘) V0 (𝑘)
𝑓𝑜𝑟 𝑘 ∈ I,
−Δ2 V0 (𝑘 − 1) ≥ 𝐶 (𝑘) 𝑢0 (𝑘) + 𝐷 (𝑘) V0 (𝑘)
𝑓𝑜𝑟 𝑘 ∈ I,
Δ𝑢0 (0) ≤ 0,
Δ𝑢0 (𝑛) ≥ 0,
ΔV0 (0) ≤ 0,
ΔV0 (𝑛) ≥ 0,
where equality does not hold in all of the equations in (12). Then
(7) satisfies the maximum principle; that is, if 𝑢, V ∈ 𝑋 such that
−Δ2 𝑢 (𝑘 − 1) ≥ 𝐴 (𝑘) 𝑢 (𝑘) + 𝐵 (𝑘) V (𝑘)
𝑓𝑜𝑟 𝑘 ∈ I,
−Δ2 V (𝑘 − 1) ≥ 𝐶 (𝑘) 𝑢 (𝑘) + 𝐷 (𝑘) V (𝑘)
𝑓𝑜𝑟 𝑘 ∈ I,
Δ𝑢 (𝑛) ≥ 0,
ΔV (0) ≤ 0,
ΔV (𝑛) ≥ 0,
𝑘 ∈ I,
−Δ2 V (𝑘 − 1) − 𝐶 (𝑘) 𝑢 (𝑘) − 𝐷 (𝑘) V (𝑘) = ΛV (𝑘) ,
𝑘 ∈ I,
Δ𝑢 (0) = Δ𝑢 (𝑛) = 0,
ΔV (0) = ΔV (𝑛) = 0.
(15)
Proof. Let
𝑢
𝐸 = {( ) ∈ 𝑋 × 𝑋 | Δ𝑢 (0) = Δ𝑢 (𝑛) = 0, ΔV (0) = ΔV (𝑛) = 0}
V
(16)
(13)
Proof. Suppose on the contrary that there exist 𝑢1 , V1 not both
identically zero satisfying inequalities (13) but not satisfying
(ii) in the conclusion of the theorem. For 0 ≤ 𝜏 ≤ 1, define
𝑢𝜏 = (1 − 𝜏)𝑢0 + 𝜏𝑢1 and V𝜏 = (1 − 𝜏)V0 + 𝜏V1 . Then there exists
𝜏0 , 0 < 𝜏0 ≤ 1 such that 𝑢𝜏 , V𝜏 ≫ 0 for 0 ≤ 𝜏 < 𝜏0 and either
𝑢𝜏0 or V𝜏0 has a zero in ̂I. Without loss of generality, we may
assume that there exists 𝑘1 ∈ ̂I such that 𝑢𝜏0 (𝑘1 ) = 0; then,
−Δ2 𝑢𝜏0 (𝑘 − 1) ≥ 𝐴 (𝑘) 𝑢𝜏0 (𝑘) + 𝐵 (𝑘) V𝜏0 (𝑘)
−Δ2 𝑢 (𝑘 − 1)
𝑢
)
𝐿( ) = ( 2
V
−Δ V (𝑘 − 1)
(17)
and define the matrix 𝑀(𝑘) by
then either (i) 𝑢 ≡ V ≡ 0 on I or (ii) 𝑢 ≫ 0 and V ≫ 0 on I.
−Δ2 𝑢𝜏0 (𝑘 − 1) − 𝐴 (𝑘) 𝑢𝜏0 (𝑘) ≥ 0,
−Δ2 𝑢 (𝑘 − 1) − 𝐴 (𝑘) 𝑢 (𝑘) − 𝐵 (𝑘) V (𝑘) = Λ𝑢 (𝑘) ,
and 𝑌 = 𝑋 × 𝑋. Define 𝐿 : 𝐸 → 𝑌 by
(12)
Δ𝑢 (0) ≤ 0,
Theorem 4. The cooperative system (7) has a principal eigenvalue; that is, there exists Λ ∈ R and 𝑢, V ∈ 𝑋 such that
𝑢, V ≫ 0 and
for 𝑘 ∈ I,
𝑘 ∈ I.
(14)
Moreover, Δ𝑢𝜏0 (0) ≤ 0, Δ𝑢𝜏0 (𝑛) ≥ 0 and so it follows from
Lemma 1 that 𝑢𝜏0 ≡ 0 or 𝑢𝜏0 ≫ 0 on ̂I. Obviously, 𝑢𝜏0 (𝑘1 ) = 0,
so 𝑢𝜏0 ≡ 0. Hence, by (14), it also follows that V𝜏0 ≡ 0. From
𝑢1 , V1 are not both identically zero for 𝜏0 < 1, this goether with
𝑢𝜏0 = (1 − 𝜏0 )𝑢0 + 𝜏0 𝑢1 ≡ 0 and V𝜏0 = (1 − 𝜏0 )𝑢0 + 𝜏0 V1 ≡ 0,
implies that 𝑢1 and V1 equal the same negative multiples of
𝑢0 and V0 , respectively. This is impossible as 𝑢0 , V0 satisfy (12)
and 𝑢1 , V1 satisfy (13) and so the proof is complete.
Corollary 3. Suppose that 𝐴(𝑘) + 𝐵(𝑘) < 0 and 𝐶(𝑘) + 𝐷(𝑘) <
0 for all 𝑘 ∈ I. Then the cooperative system (7) satisfies the
maximum principle.
Proof. The result follows from Theorem 2 by choosing 𝑢0 ≡
V0 ≡ 𝐾 where 𝐾 is any positive number.
𝐴 (𝑘) 𝐵 (𝑘)
).
𝑀 (𝑘) = (
𝐶 (𝑘) 𝐷 (𝑘)
(18)
It is well know that if 𝐾 > 0 is sufficiently large, then 𝐿 −
𝑀 + 𝐾 is an invertible operator such that (𝐿 − 𝑀 + 𝐾)−1 is
compact. If, moreover, 𝐾 is chosen sufficiently large to ensure
that 𝐴(𝑘) + 𝐵(𝑘) − 𝐾 < 0 and 𝐶(𝑘) + 𝐷(𝑘) − 𝐾 < 0 for all
𝑘 ∈ I, it follows from Corollary 3 that (𝐿−𝑀+𝐾)−1 is strongly
positive.
Since (𝐿 − 𝑀 + 𝐾)−1 is compact and strongly positive,
(𝐿 − 𝑀 + 𝐾)−1 has a positive principal eigenvalue 𝜇. Thus
there exists 𝜙 = ( 𝑢V00 ) ∈ 𝑌 with 𝑢0 , V0 ≫ 0 such that (𝐿 − 𝑀 +
𝐾)−1 𝜙 = 𝜇𝜙. Hence (𝐿 − 𝑀)𝜙 = (1/𝜇 − 𝐾)𝜙 and 𝐿 − 𝑀 has a
principal eigenvalue.
For convenience, we will denote the principal eigenvalue
of 𝐿 − 𝑀 by 𝜆 1 (𝑀).
Corollary 5. Suppose that 𝑀1 (𝑘) and 𝑀2 (𝑘) are cooperative
matrices (i.e., matrices with positive entries in the off-diagonal
elements) such that 𝑀1 (𝑘) ≥ 𝑀2 (𝑘) (i.e., the (𝑖, 𝑗)th element
of 𝑀1 ≥ the (𝑖, 𝑗)th element of 𝑀2 ) for all 𝑘 ∈ I but 𝑀1 (𝑘) ≢
𝑀2 (𝑘)). Then 𝜆 1 (𝑀1 ) < 𝜆 1 (𝑀2 ).
Proof. There exists 𝜙0 = ( 𝑢V00 ) ∈ 𝐸 such that 𝑢0 , V0 ≫ 0 and
[𝐿 − 𝑀1 − 𝜆 1 (𝑀1 )]𝜙0 = 0. Then
[𝐿 − 𝑀2 − 𝜆 1 (𝑀1 )] 𝜙0
= [𝐿 − 𝑀1 − 𝜆 1 (𝑀1 )] 𝜙0 + (𝑀1 − 𝑀2 ) 𝜙0 ,
𝑘 ∈ I,
= (𝑀1 − 𝑀2 ) 𝜙0 ≥ 0,
(19)
but (𝑀1 − 𝑀2 )𝜙0 ≢ 0 and so by Theorem 2, the cooperative
system [𝐿 − 𝑀2 − 𝜆 1 (𝑀1 )]𝜙 ≥ 0 with 𝜙 = ( 𝑢V ) ∈ 𝐸 satisfies
4
Journal of Function Spaces
the maximum principle. Hence, if 𝜇 denotes the principal
eigenvalue for the system 𝐿 − 𝑀2 − 𝜆 1 (𝑀1 )𝐼, it follows that
𝜇 > 0. Clearly, 𝐿 − 𝑀2 has principal eigenvalue 𝜆 1 (𝑀2 ) =
𝜆 1 (𝑀1 ) + 𝜇 > 𝜆(𝑀1 ) and so 𝜆 1 (𝑀2 ) > 𝜆 1 (𝑀1 ).
Clearly, to show that (23) has a solution 𝑢(𝑘) satisfies 𝑢(𝑘) ≤
𝑢(𝑘) ≤ 𝑢(𝑘), it is enough to prove that (25) has a solution that
𝑢∗ (𝑘) satisfies 𝑢(𝑘) ≤ 𝑢∗ (𝑘) ≤ 𝑢(𝑘).
Define the operator
𝑛
𝑇𝑢 (𝑘) = ∑ 𝐺 (𝑘, 𝑗) 𝑓∗ (𝑗, 𝑝 (𝑗, 𝑢)) ,
3. Existence and Nonexistence of
Positive Solutions
System (4) can be rewritten as
𝑢
𝑢 (𝑘)
𝑢 (𝑘)
𝐿 ( ) = 𝑀 (𝑘) (
) − 𝑁(
),
V
V (𝑘)
V (𝑘)
(20)
𝑎(𝑘)
where 𝑀(𝑘) = ( −𝑐(𝑘)
𝑏(𝑘) −𝑑(𝑘) ) and 𝑁 : 𝑌 → 𝑌 such that
𝑢 (𝑘)
𝑒𝑢 [𝑢 + V]
𝑁(
)=(
).
V (𝑘)
𝑓V [𝑢 + V]
(21)
Although 𝑀(𝑘) is a cooperative matrix, (20) is not a cooperative system. The following general theorem describes how
the method of lower and upper solutions must be modified
to deal with general, possibly non-cooperative system such
as (6). The proofs of the method of lower and upper solutions
are based upon the connectivity properties of the solution sets
of parameterized families of compact vector fields; see [16,
Theorem 1.0].
Lemma 6. Let 𝑞 : I → (0, ∞), and suppose that there exist
functions 𝑢, 𝑢 ∈ 𝑋 such that 𝑢 ≤ 𝑢 and, for all 𝑘 ∈ I,
−Δ2 𝑢 (𝑘 − 1) + 𝑞 (𝑘) 𝑢 (𝑘) ≤ 𝑓 (𝑘, 𝑢 (𝑘)) ,
Δ𝑢 (0) ≥ 0,
Δ𝑢 (𝑛) ≤ 0,
−Δ2 𝑢 (𝑘 − 1) + 𝑞 (𝑘) 𝑢 (𝑘) ≥ 𝑓 (𝑘, 𝑢 (𝑘)) ,
Δ𝑢 (0) ≤ 0,
On the other hand, we have that
−Δ2 𝑤 (𝑘1 − 1) + 𝑞 (𝑘1 ) 𝑤 (𝑘1 ) ≥ 𝑓 (𝑘1 , 𝑢 (𝑘1 ))
−𝑓∗ (𝑘1 , 𝑝 (𝑘1 , 𝑢∗ (𝑘1 ))) = 0,
(29)
which is a contradiction. So 𝑢∗ (𝑘) ≤ 𝑢(𝑘). The proof is
complete.
Theorem 7. Suppose that there exist functions 𝑢, V, 𝑢, V ∈ 𝑋
such that 𝑢 ≤ 𝑢 and V ≤ V and, for all 𝑘 ∈ I,
𝑤ℎ𝑒𝑛𝑒V𝑒𝑟 V (𝑘) ≤ V ≤ V (𝑘) ,
Δ𝑢 (0) ≥ 0,
Δ𝑢 (𝑛) = 0
(28)
𝑤ℎ𝑒𝑛𝑒V𝑒𝑟 𝑢 (𝑘) ≤ 𝑢 ≤ 𝑢 (𝑘) ,
(23)
Δ𝑢 (𝑛) ≤ 0,
ΔV (0) ≥ 0,
ΔV (𝑛) ≤ 0,
− Δ2 𝑢 (𝑘 − 1) + 𝑞 (𝑘) 𝑢 (𝑘) ≥ 𝑓 (𝑢 (𝑘) , V) ,
𝑤ℎ𝑒𝑛𝑒V𝑒𝑟 V (𝑘) ≤ V ≤ V (𝑘) ,
has a solution 𝑢(𝑘) that satisfies
− Δ2 V (𝑘 − 1) + 𝑞 (𝑘) V (𝑘) ≥ 𝑔 (𝑢, V (𝑘)) ,
𝑢 (𝑘) ≤ 𝑢 (𝑘) ≤ 𝑢 (𝑘) ,
∀ 𝑘 ∈ ̂I.
−Δ2 𝑢 (𝑘 − 1) + 𝑞 (𝑘) 𝑢 (𝑘) = 𝑓∗ (𝑘, 𝑢 (𝑘)) ,
Δ𝑢 (𝑛) = 0,
Δ𝑢 (0) ≤ 0,
(25)
where 𝑓∗ (𝑘, 𝑢) = 𝑓(𝑘, 𝑝(𝑘, 𝑢)) and 𝑝(𝑘, 𝑢) is defined by
𝑢 (𝑘) > 𝑢 (𝑘) ,
𝑢 (𝑘) ≤ 𝑢 (𝑘) ≤ 𝑢 (𝑘) ,
𝑢 (𝑘) ≤ 𝑢 (𝑘) .
𝑤ℎ𝑒𝑛𝑒V𝑒𝑟 𝑢 (𝑘) ≤ 𝑢 ≤ 𝑢 (𝑘) ,
(24)
Proof. Let us consider the auxiliary problem
𝑢 (𝑘) ,
{
{
𝑝 (𝑘, 𝑢) = {𝑢 (𝑘) ,
{
{𝑢 (𝑘) ,
−Δ2 𝑤 (𝑘1 − 1) + 𝑞 (𝑘1 ) 𝑤 (𝑘1 ) < 0.
− Δ2 V (𝑘 − 1) + 𝑞 (𝑘) V (𝑘) ≤ 𝑔 (𝑢, V (𝑘)) ,
2
Δ𝑢 (0) = 0,
By the standard of compact operator argument, it is easy to
show that 𝑇 is a completely continuous operator. Obviously,
𝑓∗ is a bounded function; this combined with Lemma 1 and
Brouwer fixed point theorem and we can get that 𝑇 has a fixed
point 𝑢∗ ; that is, 𝑢∗ is a solution of the problem (25).
Next, we will show that 𝑢(𝑘) ≤ 𝑢∗ (𝑘) ≤ 𝑢(𝑘).
We only have to deal with 𝑢∗ (𝑘) ≤ 𝑢(𝑘) and the case
𝑢(𝑘) ≤ 𝑢∗ (𝑘) can be proved by a similar argument. Suppose
on the contrary that there exists a point 𝑘0 ∈ I such that 𝑢∗ (𝑘0 )
> 𝑢(𝑘0 ). Let 𝑤(𝑘) = 𝑢(𝑘)−𝑢∗ (𝑘); then, there exists 𝑘1 ∈ I such
that 𝑤(𝑘1 ) = min𝑘∈I 𝑤(𝑘) < 0 and Δ𝑤(𝑘1 ) ≥ 0, Δ𝑤(𝑘1 −1) ≤ 0.
Subsequently,
(22)
Then the problem
Δ𝑢 (0) = 0,
(27)
−Δ2 𝑢 (𝑘 − 1) + 𝑞 (𝑘) 𝑢 (𝑘) ≤ 𝑓 (𝑢 (𝑘), V) ,
Δ𝑢 (𝑛) ≥ 0.
−Δ 𝑢 (𝑘 − 1) + 𝑞 (𝑘) 𝑢 (𝑘) = 𝑓 (𝑘, 𝑢 (𝑘)) ,
𝑘 ∈ I.
𝑗=1
Δ𝑢 (𝑛) ≥ 0,
ΔV (𝑛) ≥ 0.
(30)
Then there exists a solution ( 𝑢V ) of the system
−Δ2 𝑢 (𝑘 − 1) + 𝑞 (𝑘) 𝑢 (𝑘) = 𝑓 (𝑢 (𝑘) , V (𝑘)) ,
𝑘 ∈ I,
−Δ2 V (𝑘 − 1) + 𝑞 (𝑘) V (𝑘) = 𝑔 (𝑢 (𝑘) , V (𝑘)) ,
𝑘 ∈ I,
Δ𝑢 (0) = Δ𝑢 (𝑛) = 0,
(26)
ΔV (0) ≤ 0,
(31)
ΔV (0) = ΔV (𝑛) = 0
such that 𝑢(𝑘) ≤ 𝑢(𝑘) ≤ 𝑢(𝑘) and V(𝑘) ≤ V(𝑘) ≤ V(𝑘) for all
𝑘 ∈ I.
Journal of Function Spaces
5
Proof. From Lemma 6, it is easy to show that for any fixed
V ≤ V ≤ V, there exists the solution set
𝑊 (V) := {𝑢 ∈ 𝑋 | (𝑢, V) satisfy the first equation in (31)}
≠ 0.
(32)
Moreover, by the connectivity properties of the solution sets
of parameterized families of compact vector fields [16, Theorem 1.0], the set
S:= {(𝑢, V)𝑇 ∈ 𝐸 | (𝑢, V) satisfy the first equation in (31)}
(33)
contains a connected subset Σ which joins {V} × 𝑊(V) and
{V} × 𝑊(V) with V ≤ V.
For every 𝑢 ∈ 𝑊(V), we have that
−Δ2 V (𝑘 − 1) + 𝑞 (𝑘) V (𝑘) ≤ 𝑔 (𝑢, V (𝑘)) ,
(34)
and for any 𝑢 ∈ 𝑊(V), we get that
−Δ2 V (𝑘 − 1) + 𝑞 (𝑘) V (𝑘) ≥ 𝑔 (𝑢, V (𝑘)) .
(35)
Therefore, by the connectivity of Σ, there exist V0 ∈ [V, V] and
𝑤(V0 ) ∈ 𝑊(V0 ) such that (𝑤(V0 ), V0 ) ∈ Σ and satisfies
−Δ2 V0 (𝑘 − 1) + 𝑞 (𝑘) V0 (𝑘) = 𝑔 (𝑤 (V0 ) (𝑘) , V0 (𝑘)) ,
𝑘 ∈ I.
(36)
Thus (𝑤(V0 ), V0 ) is solution of the system (31).
We can proceed in the same way, proving that there exists
𝑢0 ∈ [𝑢, 𝑢], 𝑝(𝑢0 ) ∈ 𝑃(𝑢) such that (𝑢0 , 𝑝(𝑢0 )) is a solution of
the system (31), where
𝑃 (𝑢) := {V ∈ 𝑋 | (𝑢, V) satisfy the second equation in (31)}
≠ 0.
(37)
The proof is complete.
We can give necessary and sufficient conditions for the
existence of a positive solution of the system (4).
Theorem 8. There exists a positive solution of system (4) if and
only if 𝜆 1 (𝑀) < 0.
is chosen sufficiently small and 𝐾 is chosen sufficiently large.
Let
𝐾 = max {
max𝑘∈I 𝑎 (𝑘) max𝑘∈I 𝑏 (𝑘)
,
}.
𝑒
𝑓
(39)
Then for all 𝑘 ∈ I, 𝑎(𝑘) − 𝑒𝐾 ≤ 0 and so
𝑎 (𝑘) V − 𝑐 (𝑘) 𝑢 (𝑘) − 𝑒𝑢 (𝑘) [𝑢 (𝑘) + V]
= [𝑎 (𝑘) − 𝑒𝐾] V − 𝑐 (𝑘) 𝐾 − 𝑒𝐾2
(40)
≤ 0 = −Δ2 𝑢 (𝑘 − 1) ,
whenever V ≥ 0. Similarly,
−Δ2 V (𝑘 − 1) ≥ 𝑏𝑢 − 𝑑V (𝑘) − 𝑓V (𝑘) [𝑢 + V (𝑘)]
whenever 𝑢 ≥ 0.
Let
𝜀0 = min {
min 𝑏 (𝑘)
min 𝑎 (𝑘)
,
}.
𝑒 max 𝜙1 (𝑘) 𝑓 max 𝜙2 (𝑘)
−Δ2 𝜙2 (𝑘 − 1) = 𝑏 (𝑘) 𝜙1 (𝑘) − 𝑑 (𝑘) 𝜙2 (𝑘) + 𝜆 1 (𝑀) 𝜙2 (𝑘) ,
Δ𝜙1 (0) = Δ𝜙1 (𝑛) = 0,
Δ𝜙2 (0) = Δ𝜙2 (𝑛) = 0.
(38)
𝑢
(V)
𝜙
𝜖 ( 𝜙12
( 𝑢V )
(𝐾
𝐾 ).
𝑢
(V)
=
) and
=
We will show that
and
Let
𝑢
( V ) satisfy the hypotheses of Theorem 7 provided that 𝜀 > 0
(42)
Then if 𝜀 < 𝜀0 , we get that 𝑎(𝑘) − 𝜀𝑒𝜙1 (𝑘) ≥ 0 and 𝑏(𝑘) −
𝜀𝑓𝜙2 (𝑘) ≥ 0 for all 𝑘 ∈ I. Hence, when 𝜀 < 𝜀0 , 𝑢(𝑘) = 𝜀𝜙1 (𝑘)
and V(𝑘) ≥ 𝜀𝜙2 (𝑘), we have
− Δ2 𝑢 (𝑘 − 1) − 𝑎 (𝑘) V + 𝑐 (𝑘) 𝑢 (𝑘) + 𝑒𝑢 (𝑘) [𝑢 (𝑘) + V]
= 𝜀 [−Δ2 𝜙1 (𝑘 − 1) + 𝑐 (𝑘) 𝜙1 − 𝑎 (𝑘) 𝜙2 ]
+ 𝑎 (𝑘) [𝜀𝜙2 (𝑘) − V] + 𝑒𝜀𝜙1 (𝑘) [𝜀𝜙1 (𝑘) + V]
= 𝜀𝜆 1 (𝑀) 𝜙1 (𝑘) + 𝜀𝑎 (𝑘) 𝜙2 (𝑘)
2
− [𝑎 (𝑘) − 𝜀𝑒𝜙1 (𝑘)] V + 𝜀2 𝑒[𝜙1 (𝑘)]
≤ 𝜀𝜆 1 (𝑀) 𝜙1 (𝑘) + 𝜀𝑎 (𝑘) 𝜙2 (𝑘)
− [𝑎 (𝑘) − 𝜀𝑒𝜙1 (𝑘)] 𝜀𝜙2 (𝑘) + 𝜀2 𝑒[𝜙1 (𝑘)]
2
= 𝜀𝜆 1 (𝑀) 𝜙1 (𝑘) + 𝜀2 𝑒𝜙1 (𝑘) [𝜙1 (𝑘) + 𝜙2 (𝑘)] < 0,
(43)
when 𝜀 is sufficiently small.
Similarly, if V(𝑘) = 𝜀𝜙2 (𝑘), 𝑢 ≥ 𝜀𝜙1 (𝑘), and 𝜀 is sufficiently
small, we have
− Δ2 V (𝑘 − 1) − 𝑏 (𝑘) 𝑢 + 𝑑 (𝑘) V (𝑘) + 𝑓V (𝑘) [𝑢 + V (𝑘)] < 0
∀𝑘 ∈ I.
(44)
Proof. Suppose 𝜆 1 (𝑀) < 0. Then there exist 𝜙1 , 𝜙2 ≫ 0 such
𝜙
𝜙
that (𝐿 − 𝑀) ( 𝜙12 ) = 𝜆 1 (𝑀) ( 𝜙12 ); that is,
−Δ2 𝜙1 (𝑘 − 1) = 𝑎 (𝑘) 𝜙2 (𝑘) − 𝑐 (𝑘) 𝜙1 (𝑘) + 𝜆 1 (𝑀) 𝜙1 (𝑘) ,
(41)
Hence, by Theorem 7, there exists a positive solution of system (4).
Suppose now that system (4) has a solution ( 𝑢V00 ) with
𝑢0 , V0 ≫ 0. Then ( 𝑢V00 ) is a solution of the system
−Δ2 𝑢 (𝑘 − 1) + [𝑐 (𝑘) + 𝑒𝑞 (𝑘)] 𝑢 (𝑘) − 𝑎 (𝑘) V (𝑘) = 0,
𝑘 ∈ I,
−Δ2 V (𝑘 − 1) − 𝑏 (𝑘) 𝑢 (𝑘) + [𝑑 (𝑘) + 𝑓𝑞 (𝑘)] V (𝑘) = 0,
𝑘 ∈ I,
Δ𝑢 (0) = Δ𝑢 (𝑛) = 0,
ΔV (0) = ΔV (𝑛) = 0,
(45)
6
Journal of Function Spaces
where 𝑞(𝑘) = 𝑢0 (𝑘) + V0 (𝑘). Hence ( 𝑢V00 ) may be regarded
as the principal eigenfunction corresponding to the principal
eigenvalue 𝜆 = 0 of the system (𝐿−𝑀𝑞 (𝑘))𝑢 = 0, where 𝑀𝑞 (𝑘)
−𝑐(𝑘)−𝑒𝑞(𝑘)
𝑎(𝑘)
= ( 𝑏(𝑘) −𝑑(𝑘)−𝑓𝑞(𝑘) ). Hence 𝜆 1 (𝑀𝑞 ) = 0. As 𝑀𝑞 ≤ 𝑀
but 𝑀𝑞 ≢ 𝑀, it follows from Corollary 5 that 𝜆 1 (𝑀) <
𝜆 1 (𝑀𝑞 ) = 0 and the proof is complete.
Remark 9. From Theorem 8, it is easy to see that the system
(4) has no positive solution if 𝜆 1 (𝑀) ≥ 0.
Example 10. Let us consider the Neumann difference system
− Δ2 𝑢 (𝑘 − 1) = 3V (𝑘) − 4𝑢 (𝑘) + 𝑢 (𝑘) (𝑢 (𝑘) + V (𝑘)) ,
𝑘 ∈ {1, 2, 3, 4, 5} ,
− Δ2 V (𝑘 − 1) = 2𝑢 (𝑘) − 3V (𝑘) + 2V (𝑘) (𝑢 (𝑘) + V (𝑘)) ,
𝑘 ∈ {1, 2, 3, 4, 5} ,
Δ𝑢 (0) = Δ𝑢 (5) = 0,
ΔV (0) = ΔV (5) = 0.
(46)
5 3𝐼5
Obviously, 𝑀 = ( −4𝐼
2𝐼5 −3𝐼5 ); here 𝐼5 denote the unit matrix
of order 5. By simple computing, 𝜆 1 (𝑀) = −1 < 0. From
Theorem 8, the problem (46) has at least one positive solution
𝑢 on {0, 1, . . . , 6}.
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Authors’ Contribution
Yanqiong Lu completed the main study, carried out the results
of this article, and drafted the paper. Ruyun Ma checked the
proofs and verified the calculation. All the authors read and
approved the paper.
Acknowledgments
The authors are very grateful to the anonymous referees
for their valuable suggestions. This work was supported
by NSFC (no. 11361054 and no. 11201378), SRFDP (no.
20126203110004), and Gansu provincial National Science
Foundation of China (no. 1208RJZA258).
References
[1] A. Lasota, “A discrete boundary value problem,” Annales Polonici Mathematici, vol. 20, pp. 183–190, 1968.
[2] R. P. Agarwal, “Difference equations and inequalities,” in Theory,
Methods, and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New
York, NY, USA, 2nd edition, 2000.
[3] W. G. Kelley and A. C. Peterson, “Difference equations,” in An
Introduction With Applications, Harcourt/Academic Press, San
Diego, Calif, USA, 2nd edition, 2001.
[4] A. Cabada and V. Otero-Espinar, “Fixed sign solutions of
second-order difference equations with Neumann boundary
conditions,” Computers & Mathematics with Applications, vol.
45, no. 6-9, pp. 1125–1136, 2003.
[5] J.-P. Sun and W.-T. Li, “Existence of positive solutions of boundary value problem for a discrete difference system,” Applied
Mathematics and Computation, vol. 156, no. 3, pp. 857–870,
2004.
[6] D. R. Anderson, I. Rachunková, and C. C. Tisdell, “Solvability of
discrete Neumann boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 736–741,
2007.
[7] J. Henderson, S. K. Ntouyas, and I. K. Purnaras, “Positive solutions for systems of nonlinear discrete boundary value problems,” Journal of Difference Equations and Applications, vol. 15,
no. 10, pp. 895–912, 2009.
[8] R. Zhang, “Positive solutions of BVPs for third-order discrete
nonlinear difference systems,” Journal of Applied Mathematics
and Computing, vol. 35, no. 1-2, pp. 551–575, 2011.
[9] A. Cañada, P. Magal, and J. A. Montero, “Optimal control of harvesting in a nonlinear elliptic system arising from population
dynamics,” Journal of Mathematical Analysis and Applications,
vol. 254, no. 2, pp. 571–586, 2001.
[10] S. Li, Z. Xiu, and L. Liang, “Positive solutions for nonlinear
singular second order Neumann boundary value problems,”
Journal of Mathematical and Computational Science, vol. 2, no.
5, pp. 1353–1362, 2012.
[11] R. Chen and Y. Lu, “Existence and multiplicity of positive solutions to nonlinear semipositone Neumann boundary value
problem,” Annals of Differential Equations, vol. 28, no. 2, pp. 137–
145, 2012.
[12] L. Y. Zhang, L. Y. Wang, and X. Y. Li, “Multiplicity of positive
solutions to second-order singular Neumann boundary value
problems for differential systems,” Acta Mathematicae Applicatae Sinica, vol. 31, no. 6, pp. 1035–1045, 2008.
[13] G. Sweers, “Strong positivity in 𝐶(Ω) for elliptic systems,” Mathematische Zeitschrift, vol. 209, no. 2, pp. 251–271, 1992.
[14] J. López-Gómez and M. Molina-Meyer, “The maximum principle for cooperative weakly coupled elliptic systems and some
applications,” Differential and Integral Equations, vol. 7, no. 2, pp.
383–398, 1994.
[15] K. J. Brown and Y. Zhang, “On a system of reaction-diffusion
equations describing a population with two age groups,” Journal
of Mathematical Analysis and Applications, vol. 282, no. 2, pp.
444–452, 2003.
[16] R. Ma, “Multiplicity results for a three-point boundary value
problem at resonance,” Nonlinear Analysis. Theory, Methods &
Applications, vol. 53, no. 6, pp. 777–789, 2003.
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