Eigenvalue characterization of a system of difference equations
We consider the following system of difference equations: ui(k) = λ∑gi(k, l)Pi(l, u1(l), u2(l), . . . , un(l)), k ∈ {0, 1, . . . }, 1 ≤ i ≤ n, where λ > 0 and T ≥ N ≥ 0. Our aim is to determine those values of λ such that the above system has a constant-sign solution. In addition, explicit...
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irk-123456789-1769902021-02-11T01:28:46Z Eigenvalue characterization of a system of difference equations Agarwal, R.P. O'Regan, D. P. J. Y. Wong We consider the following system of difference equations: ui(k) = λ∑gi(k, l)Pi(l, u1(l), u2(l), . . . , un(l)), k ∈ {0, 1, . . . }, 1 ≤ i ≤ n, where λ > 0 and T ≥ N ≥ 0. Our aim is to determine those values of λ such that the above system has a constant-sign solution. In addition, explicit intervals for λ will be presented. The generality of the results obtained is illustrated through applications to several well known boundary-value problems. We also extend the above problem to that on {0, 1, . . . }, ui(k) = λ∑gi(k, l)Pi(l, u1(l), u2(l), . . . , un(l)), k ∈ {0, 1, . . . , T}, 1 ≤ i ≤ n, Finally, both systems above are extended to the general case when λ is replaced by λi . Розглянуто систему диференцiальних рiвнянь ui(k) = λ∑gi(k, l)Pi(l, u1(l), u2(l), . . . , un(l)), k ∈ {0, 1, . . . }, 1 ≤ i ≤ n, де λ > 0 i T ≥ N ≥ 0. Метою статтi є знаходження тих значень λ, для яких наведена система має розв’язок постiйного знаку. Також знайдено в явному виглядi iнтервали для таких λ. Загальнiсть отриманих результатiв проiлюстровано застосуваннями до низки добре вiдомих граничних задач. Наведена вище задача також узагальнюється до такої ж задачi на {0, 1, . . . }, ui(k) = λ∑gi(k, l)Pi(l, u1(l), u2(l), . . . , un(l)), k ∈ {0, 1, . . .T }, 1 ≤ i ≤ n, На завершення цi двi системи поширюються на загальний випадок, коли λ замiнюється на λi . 2004 Article Eigenvalue characterization of a system of difference equations / R.P. Agarwal, D. O'Regan, P. J. Y. Wong // Нелінійні коливання. — 2004. — Т. 7, № 1. — С. 3-47. — Бібліогр.: 23 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176990 517.9 en Нелінійні коливання Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
We consider the following system of difference equations:
ui(k) = λ∑gi(k, l)Pi(l, u1(l), u2(l), . . . , un(l)), k ∈ {0, 1, . . . }, 1 ≤ i ≤ n,
where λ > 0 and T ≥ N ≥ 0. Our aim is to determine those values of λ such that the above system
has a constant-sign solution. In addition, explicit intervals for λ will be presented. The generality of the
results obtained is illustrated through applications to several well known boundary-value problems. We
also extend the above problem to that on {0, 1, . . . },
ui(k) = λ∑gi(k, l)Pi(l, u1(l), u2(l), . . . , un(l)), k ∈ {0, 1, . . . , T}, 1 ≤ i ≤ n,
Finally, both systems above are extended to the general case when λ is replaced by λi
. |
| format |
Article |
| author |
Agarwal, R.P. O'Regan, D. P. J. Y. Wong |
| spellingShingle |
Agarwal, R.P. O'Regan, D. P. J. Y. Wong Eigenvalue characterization of a system of difference equations Нелінійні коливання |
| author_facet |
Agarwal, R.P. O'Regan, D. P. J. Y. Wong |
| author_sort |
Agarwal, R.P. |
| title |
Eigenvalue characterization of a system of difference equations |
| title_short |
Eigenvalue characterization of a system of difference equations |
| title_full |
Eigenvalue characterization of a system of difference equations |
| title_fullStr |
Eigenvalue characterization of a system of difference equations |
| title_full_unstemmed |
Eigenvalue characterization of a system of difference equations |
| title_sort |
eigenvalue characterization of a system of difference equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/176990 |
| citation_txt |
Eigenvalue characterization of a system of difference equations / R.P. Agarwal, D. O'Regan, P. J. Y. Wong // Нелінійні коливання. — 2004. — Т. 7, № 1. — С. 3-47. — Бібліогр.: 23 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT agarwalrp eigenvaluecharacterizationofasystemofdifferenceequations AT oregand eigenvaluecharacterizationofasystemofdifferenceequations AT pjywong eigenvaluecharacterizationofasystemofdifferenceequations |
| first_indexed |
2025-07-15T14:56:53Z |
| last_indexed |
2025-07-15T14:56:53Z |
| _version_ |
1837725294468792320 |
| fulltext |
UDC 517.9
EIGENVALUE CHARACTERIZATION
OF A SYSTEM OF DIFFERENCE EQUATIONS
ХАРАКТЕРИЗАЦIЯ СИСТЕМИ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
ЗА ВЛАСНИМИ ЗНАЧЕННЯМИ
R. P. Agarwal
Florida Inst. Technology
Melbourne, Florida 32901-6975, USA
e-mail: agarwal@fit.edu
D. O’Regan
Nat. Univ. Ireland
Galway, Ireland
P. J. Y. Wong
School Electrical and Electronic Engineering, Nanyang Technol. Univ.
50 Nanyang Avenue, 639798, Singapore
e-mail: ejywong@ntu.edu.sg
We consider the following system of difference equations:
ui(k) = λ
N∑
`=0
gi(k, `)Pi(`, u1(`), u2(`), . . . , un(`)), k ∈ {0, 1, . . . , T}, 1 ≤ i ≤ n,
where λ > 0 and T ≥ N ≥ 0. Our aim is to determine those values of λ such that the above system
has a constant-sign solution. In addition, explicit intervals for λ will be presented. The generality of the
results obtained is illustrated through applications to several well known boundary-value problems. We
also extend the above problem to that on {0, 1, . . . },
ui(k) = λ
∞∑
`=0
gi(k, `)Pi(`, u1(`), u2(`), . . . , un(`)), k ∈ {0, 1, . . . }, 1 ≤ i ≤ n.
Finally, both systems above are extended to the general case when λ is replaced by λi.
Розглянуто систему диференцiальних рiвнянь
ui(k) = λ
N∑
`=0
gi(k, `)Pi(`, u1(`), u2(`), . . . , un(`)), k ∈ {0, 1, . . . , T}, 1 ≤ i ≤ n,
де λ > 0 i T ≥ N ≥ 0. Метою статтi є знаходження тих значень λ, для яких наведена си-
стема має розв’язок постiйного знаку. Також знайдено в явному виглядi iнтервали для таких λ.
Загальнiсть отриманих результатiв проiлюстровано застосуваннями до низки добре вiдомих
граничних задач. Наведена вище задача також узагальнюється до такої ж задачi на {0, 1, . . . },
ui(k) = λ
N∑
`=0
gi(k, `)Pi(`, u1(`), u2(`), . . . , un(`)), k ∈ {0, 1, . . . , T}, 1 ≤ i ≤ n,
На завершення цi двi системи поширюються на загальний випадок, коли λ замiнюється на λi.
c© R. P. Agarwal, D. O’Regan, and P. J. Y. Wong, 2004
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1 3
4 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
1. Introduction. We shall use the notation Z[a, b] = {a, a + 1, . . . , b} where a, b (> a) are
integers. In this paper two systems of difference equations are discussed. The first system is on
a finite set of integers,
ui(k) = λ
N∑
`=0
gi(k, `)Pi(`, u1(`), u2(`), . . . , un(`)), k ∈ I ≡ Z[0, T ], 1 ≤ i ≤ n, (1.1)
where T ≥ N > 0. The second system is on the infinite set of IN = {0, 1, . . . },
ui(k) = λ
∞∑
`=0
gi(k, `)Pi(`, u1(`), u2(`), . . . , un(`)), k ∈ IN, 1 ≤ i ≤ n. (1.2)
A solution u = (u1, u2, . . . , un) of (1.1) will be sought in (C(I))n = C(I) × . . . × C(I) (n
times), where C(I) denotes the class of functions continuous on I (discrete topology). We say
that u is a solution of constant sign of (1.1) if for each 1 ≤ i ≤ n, we have θiui(k) ≥ 0 for k ∈ I
where θi ∈ {1,−1} is fixed. On the other hand, a solution u = (u1, u2, . . . , un) of (1.2) will be
sought in a subset of (C(IN))n = C(IN) × . . . × C(IN) (n times) where limk→∞ ui(k) exists for
each 1 ≤ i ≤ n. Moreover, u is a solution of constant sign of (1.2) if for each 1 ≤ i ≤ n, we
have θiui(k) ≥ 0 for k ∈ IN where θi ∈ {1,−1} is fixed.
For each of (1.1) and (1.2), we shall characterize those values of λ for which the system
has a constant-sign solution. If, for a particular λ the system has a constant-sign solution u =
= (u1, u2, . . . , un), then λ is called an eigenvalue and u a corresponding eigenfunction of the
system. Let E be the set of eigenvalues, i.e.,
E = {λ | λ > 0 such that the system under consideration has a constant-sign solution}.
We shall establish criteria forE to be an interval (which may either be bounded or unbounded).
In addition explicit subintervals of E are derived.
Finally, both (1.1) and (1.2) are extended to the following systems:
ui(k) = λi
N∑
`=0
gi(k, `)Pi(`, u1(`), u2(`), . . . , un(`)), k ∈ I, 1 ≤ i ≤ n, (1.3)
ui(k) = λi
∞∑
`=0
gi(k, `)Pi(`, u1(`), u2(`), . . . , un(`)), k ∈ IN, 1 ≤ i ≤ n. (1.4)
For each of (1.3) and (1.4), we shall characterize those values of λi, 1 ≤ i ≤ n, for which the
system has a constant-sign solution. If, for a particular λ = (λ1, λ2, . . . , λn) the system has a
constant-sign solution u = (u1, u2, . . . , un), then λ is called an eigenvalue and u a corresponding
eigenfunction of the system. The set of eigenvalues is denoted by
E = {λ = (λ1, λ2, . . . , λn) | λi > 0, 1 ≤ i ≤ n such that the system
under consideration has a constant-sign solution}.
Results analogous to those for (1.1) and (1.2) will be developed for systems (1.3) and (1.4).
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1
EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 5
Recently, Agarwal and O’Regan [1] have investigated the existence of positive solutions of
the discrete equation
y(k) =
N∑
`=0
g(k, `)f(y(`)) + h(k), k ∈ Z[0, T ]. (1.5)
The continuous version of (1.5) is well known in the literature, see [2 – 4]. We remark that a
generalization of (1.5) to a system with existence criteria for single and multiple constant-sign
solutions has recently been presented in [5]. In the present paper, besides extending (1.5) to a
system, we have added in the parameter λ (or λi) and we consider constant-sign solutions. As
a result, it is the eigenvalue problem that is of interest in this paper. Note that the term h(k)
in (1.5) has been excluded as we intend to apply the results to homogeneous boundary-value
problems (in which case h(k) ≡ 0), which have received almost all the attention in the recent
literature. However, it is not difficult to develop parallel results with the inclusion of h(k) or
even hi(k), 1 ≤ i ≤ n. Many papers have discussed eigenvalues of boundary-value problems
(see the monographs [6, 7] and the references cited therein). Our eigenvalue problems (1.1) –
(1.4) generalize almost all the work done in the literature to date as we are considering systems
as well as more general nonlinear terms. Moreover, our present approach is not only generic,
but also improves, corrects and completes the arguments in many papers in the literature. It is
also noted that this paper provides a discrete extension to the recent work [8].
The outline of the paper is as follows. In Section 2, we shall state Krasnosel’skii’s fixed-
point theorem which is crucial in establishing subintervals of E. The system (1.1) is discussed in
Sections 3 and 4. In Section 3, we develop criteria for E to contain an interval, and for E to be
an interval, which may either be bounded or unbounded. Moreover, upper and lower bounds
are established for an eigenvalue λ. Explicit subintervals of E are derived in Section 4. To
illustrate the importance and generality of the results obtained, applications to six well known
boundary-value problems are included in Section 5. The treatment of systems (1.2), (1.3) and
(1.4) is respectively presented in Sections 6 – 9 and 10, 11.
2. Preliminaries. The following theorem will be needed. It is usually called Krasnosel’skii’s
fixed point theorem in a cone.
Theorem 2.1 [9]. Let B = (B, ‖ · ‖) be a Banach space, and let C ⊂ B be a cone in B.
Assume Ω1,Ω2 are open subsets of B with 0 ∈ Ω1, Ω1 ⊂ Ω2, and let
S : C ∩ (Ω2\Ω1) → C
be a completely continuous operator such that, either
(a) ‖Su‖ ≤ ‖u‖, u ∈ C ∩ ∂Ω1, and ‖Su‖ ≥ ‖u‖, u ∈ C ∩ ∂Ω2, or
(b) ‖Su‖ ≥ ‖u‖, u ∈ C ∩ ∂Ω1, and ‖Su‖ ≤ ‖u‖, u ∈ C ∩ ∂Ω2.
Then S has a fixed point in C ∩ (Ω2\Ω1).
3. Characterization of E for (1.1). Throughout we shall denote u = (u1, u2, . . .
. . . , un). Let the Banach space
B =
{
u
∣∣∣∣ u ∈ (C(I))n
}
(3.1)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1
6 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
be equipped with norm
‖u‖ = max
1≤i≤n
max
k∈I
|ui(k)| = max
1≤i≤n
|ui|0 (3.2)
where we let |ui|0 = maxk∈I |ui(k)|, 1 ≤ i ≤ n. Moreover, for fixed θi ∈ {1,−1}, 1 ≤ i ≤ n,
define
K̃ =
{
u ∈ B
∣∣∣∣ θiui ≥ 0, 1 ≤ i ≤ n
}
and
K =
{
u ∈ K̃
∣∣∣∣ θjuj > 0 for some j ∈ {1, 2, . . . , n}
}
= K̃\{0}.
For the purpose of clarity, we shall list the conditions that are needed later. Note that in
these conditions θi ∈ {1,−1}, 1 ≤ i ≤ n, are fixed.
(C1) For each 1 ≤ i ≤ n, assume that Pi : Z[0, N ]× IRn → IR is continuous and
gi(k, `) ≥ 0, (k, `) ∈ I × Z[0, N ].
(C2) For each 1 ≤ i ≤ n, there exists a constant Mi ∈ (0, 1), a continuous function Hi :
Z[0, N ] → [0,∞), and an interval Z[a, b] ⊆ Z[0, N ] such that
gi(k, `) ≥ MiHi(`) ≥ 0, (k, `) ∈ Z[a, b]× Z[0, N ].
(C3) For each 1 ≤ i ≤ n,
gi(k, `) ≤ Hi(`), (k, `) ∈ I × Z[0, N ].
(C4) For each 1 ≤ i ≤ n, assume that
θiPi(`, u) ≥ 0, u ∈ K̃, ` ∈ Z[0, N ] and θiPi(`, u) > 0, u ∈ K, ` ∈ Z[0, N ].
(C5) For each 1 ≤ i ≤ n, there exist continuous functions fi, ai, bi with fi : IRn → [0,∞)
and ai, bi : Z[0, N ] → [0,∞) such that
ai(`) ≤
θiPi(`, u)
fi(u)
≤ bi(`), u ∈ K̃, ` ∈ Z[0, N ].
(C6) For each 1 ≤ i ≤ n, the function ai is not identically zero on any nondegenerate
subinterval of Z[0, N ], and there exists a number 0 < ρi ≤ 1 such that
ai(`) ≥ ρibi(`), ` ∈ Z[0, N ].
(C7) For each 1 ≤ i, j ≤ n, if |uj | ≤ |vj |, then
θiPi(`, u1, . . . , uj−1, uj , uj+1, . . . , un) ≤ θiPi(`, u1, . . . , uj−1, vj , uj+1, . . . , un), ` ∈ Z[0, N ].
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1
EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 7
(C8) For each 1 ≤ i, j ≤ n, if |uj | ≤ |vj |, then
fi(u1, . . . , uj−1, uj , uj+1, . . . , un) ≤ fi(u1, . . . , uj−1, vj , uj+1, . . . , un).
To begin the discussion, let the operator S : B → B be defined by
Su(k) = (Su1(k), Su2(k), . . . , Sun(k)), k ∈ I, (3.3)
where
Sui(k) = λ
N∑
`=0
gi(k, `)Pi(`, u(`)), k ∈ I, 1 ≤ i ≤ n. (3.4)
Clearly, a fixed point of the operator S is a solution of the system (1.1).
Next, we define a cone in B as
C =
{
u ∈ B
∣∣∣∣ for each 1 ≤ i ≤ n, θiui(k) ≥ 0 for k ∈ I,
and min
k∈Z[a,b]
θiui(k) ≥ Miρi|ui|0
}
(3.5)
where Mi and ρi are defined in (C2) and (C6) respectively. Note that C ⊆ K̃. A fixed point of
S obtained in C or K̃ will be a constant-sign solution of the system (1.1). For R > 0, let
C(R) = {u ∈ C | ‖u‖ ≤ R}.
If (C1), (C4) and (C5) hold, then it is clear from (3.4) that for u ∈ K̃,
λ
N∑
`=0
gi(k, `)ai(`)fi(u(`)) ≤ θiSui(k) ≤ λ
N∑
`=0
gi(k, `)bi(`)fi(u(`)), k ∈ I, 1 ≤ i ≤ n. (3.6)
Lemma 3.1. Let (C1) hold. Then, the operator S is continuous and completely continuous.
Proof. Using Ascoli – Arzela Theorem as in [10], (C1) ensures that S is continuous and
completely continuous.
Lemma 3.2. Let (C1) – (C6) hold. Then, the operator S maps C into itself.
Proof. Let u ∈ C. From (3.6) we have for k ∈ I and 1 ≤ i ≤ n,
θiSui(k) ≥ λ
N∑
`=0
gi(k, `)ai(`)fi(u(`)) ≥ 0. (3.7)
Next, using (3.6) and (C3) gives for k ∈ I and 1 ≤ i ≤ n,
|Sui(k)| = θiSui(k) ≤ λ
N∑
`=0
gi(k, `)bi(`)fi(u(`)) ≤ λ
N∑
`=0
Hi(`)bi(`)fi(u(`)).
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1
8 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Hence, we have
|Sui|0 ≤ λ
N∑
`=0
Hi(`)bi(`)fi(u(`)), 1 ≤ i ≤ n. (3.8)
Now, employing (3.6), (C2), (C6) and (3.8) we find for k ∈ Z[a, b] and 1 ≤ i ≤ n,
θiSui(k) ≥ λ
N∑
`=0
gi(k, `)ai(`)fi(u(`)) ≥ λ
N∑
`=0
MiHi(`)ai(`)fi(u(`)) ≥
≥ λ
N∑
`=0
MiHi(`)ρibi(`)fi(u(`)) ≥ Miρi|Sui|0.
This leads to
min
k∈Z[a,b]
θiSui(k) ≥ Miρi|Sui|0, 1 ≤ i ≤ n. (3.9)
Inequalities (3.7) and (3.9) imply that Su ∈ C.
Theorem 3.1. Let (C1) – (C6) hold. Then, there exists c > 0 such that the interval (0, c] ⊆ E.
Proof. Let R > 0 be given. Define
c = R
[
max
1≤m≤n
sup
|uj |≤R
1≤j≤n
fm(u1, u2, . . . , un)
] N∑
`=0
Hi(`)bi(`)
−1
. (3.10)
Let λ ∈ (0, c]. We shall prove that S(C(R)) ⊆ C(R). To begin, let u ∈ C(R). By Lemma
3.2, we have Su ∈ C. Thus, it remains to show that ‖Su‖ ≤ R. Using (3.6), (C3) and (3.10), we
get for k ∈ I and 1 ≤ i ≤ n,
|Sui(k)| = θiSui(k) ≤ λ
N∑
`=0
Hi(`)bi(`)fi(u(`)) ≤
≤ λ
[
sup
|uj |≤R
1≤j≤n
fi(u1, u2, . . . , un)
] N∑
`=0
Hi(`)bi(`) ≤
≤ λ
[
max
1≤m≤n
sup
|uj |≤R
1≤j≤n
fm(u1, u2, . . . , un)
] N∑
`=0
Hi(`)bi(`) ≤
≤ c
[
max
1≤m≤n
sup
|uj |≤R
1≤j≤n
fm(u1, u2, . . . , un)
] N∑
`=0
Hi(`)bi(`) = R.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1
EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 9
It follows immediately that
‖Su‖ ≤ R.
Thus, we have shown that S(C(R)) ⊆ C(R). Also, from Lemma 3.1 the operator S is conti-
nuous and completely continuous. Schauder’s fixed point theorem guarantees that S has a fixed
point in C(R). Clearly, this fixed point is a constant-sign solution of (1.1) and therefore λ is an
eigenvalue of (1.1). Since λ ∈ (0, c] is arbitrary, we have proved that the interval (0, c] ⊆ E.
Theorem 3.2. Let (C1), (C4) and (C7) hold. Suppose that λ∗ ∈ E. Then, for any λ ∈ (0, λ∗),
we have λ ∈ E, i.e., (0, λ∗] ⊆ E.
Proof. Let u∗ = (u∗1, u
∗
2, . . . , u
∗
n) be the eigenfunction corresponding to the eigenvalue λ∗.
Thus, we have
u∗i (k) = λ∗
N∑
`=0
gi(k, `)Pi(`, u∗(`)), k ∈ I, 1 ≤ i ≤ n. (3.11)
Define
K∗ =
{
u ∈ K̃
∣∣∣∣ for each 1 ≤ i ≤ n, θiui(k) ≤ θiu
∗
i (k), k ∈ I
}
.
For u ∈ K∗ and λ ∈ (0, λ∗), applying (C1), (C4) and (C7) yields
θiSui(k) = θi
[
λ
N∑
`=0
gi(k, `)Pi(`, u(`))
]
≤ θi
[
λ∗
N∑
`=0
gi(k, `)Pi(`, u∗(`))
]
=
= θiu
∗
i (k), k ∈ I, 1 ≤ i ≤ n,
where the last equality follows from (3.11). This immediately implies that the operator S defi-
ned by (3.3) maps K∗ into K∗. Moreover, from Lemma 3.1 the operator S is continuous and
completely continuous. Schauder’s fixed point theorem guarantees that S has a fixed point in
K∗, which is a constant-sign solution of (1.1). Hence, λ is an eigenvalue, i.e., λ ∈ E.
Corollary 3.1. Let (C1), (C4) and (C7) hold. If E 6= ∅, then E is an interval.
Proof. SupposeE is not an interval. Then, there exist λ0, λ
′
0 ∈ E (λ0 < λ′0) and τ ∈ (λ0, λ
′
0)
with τ /∈ E. However, this is not possible as Theorem 3.2 guarantees that τ ∈ E. Hence, E is
an interval.
We shall now establish conditions under which E is a bounded or an unbounded interval.
For this, we need the following result.
Theorem 3.3. Let (C1) – (C6) and (C8) hold. Suppose that λ is an eigenvalue of (1.1) and
u ∈ C is a corresponding eigenfunction. Let qi = |ui|0, 1 ≤ i ≤ n. Then, for each 1 ≤ i ≤ n,
we have
λ ≥ qi
fi(q1, q2, . . . , qn)
[
N∑
`=0
Hi(`)bi(`)
]−1
(3.12)
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10 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
and
λ ≤ qi
fi(M1ρ1q1,M2ρ2q2, . . . ,Mnρnqn)
[
b∑
`=a
MiHi(`)ai(`)
]−1
. (3.13)
Proof. First, we shall prove (3.12). For each 1 ≤ i ≤ n, let k∗i ∈ I be such that
qi = |ui|0 = θiui(k∗i ), 1 ≤ i ≤ n.
Then, in view of (3.6), (C3) and (C8), we find
qi = θiui(k∗i ) = θiSui(k∗i ) = θiλ
N∑
`=0
gi(k∗i , `)Pi(`, u(`)) ≤
≤ λ
N∑
`=0
gi(k∗i , `)bi(`)fi(u(`)) ≤ λ
N∑
`=0
Hi(`)bi(`)fi(q1, q2, . . . , qn)
from which (3.12) is immediate.
Next, to verify (3.13), we employ (3.6), (C4), (C8) and the fact that mink∈Z[a,b] θiui(k) ≥
≥ Miρi|ui|0 = Miρiqi to get
qi = |ui|0 ≥ θiui(a) = θiλ
N∑
`=0
gi(a, `)Pi(`, u(`)) ≥
≥ λ
N∑
`=0
gi(a, `)ai(`)fi(u(`)) ≥ λ
b∑
`=a
MiHi(`)ai(`)fi(u(`)) ≥
≥ λ
b∑
`=a
MiHi(`)ai(`)fi(M1ρ1q1,M2ρ2q2, . . . ,Mnρnqn)
which reduces to (3.13).
Theorem 3.4. Let (C1) – (C8) hold. For each 1 ≤ i ≤ n, define
FBi =
{
f : IRn → [0,∞)
∣∣∣∣ |ui|
f(u1, u2, . . . , un)
is bounded for u ∈ IRn
}
,
F 0
i =
{
f : IRn → [0,∞)
∣∣∣∣ lim
min1≤j≤n |uj |→∞
|ui|
f(u1, u2, . . . , un)
= 0
}
,
F∞i =
{
f : IRn → [0,∞)
∣∣∣∣ lim
min1≤j≤n |uj |→∞
|ui|
f(u1, u2, . . . , un)
= 0
}
.
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 11
(a) If fi ∈ FBi for each 1 ≤ i ≤ n, then E = (0, c) or (0, c] for some c ∈ (0,∞).
(b) If fi ∈ F 0
i for each 1 ≤ i ≤ n, then E = (0, c] for some c ∈ (0,∞).
(c) If fi ∈ F∞i for each 1 ≤ i ≤ n, then E = (0,∞).
Proof. (a) This is immediate from (3.13) and Corollary 3.1.
(b) Since F 0
i ⊆ FBi , 1 ≤ i ≤ n, it follows from Case (a) that E = (0, c) or (0, c] for some
c ∈ (0,∞). In particular,
c = supE.
Let {λm}∞m=1 be a monotonically increasing sequence in E which converges to c, and let
{um = (um1 , u
m
2 , . . . , u
m
n )}∞m=1 ∈ K̃
be a corresponding sequence of eigenfunctions. Further, let qmi = |umi |0, 1 ≤ i ≤ n. Then,
(3.13) together with fi ∈ F 0
i implies that no subsequence of {qmi }∞m=1 can diverge to infinity.
Thus, there exists Ri > 0, 1 ≤ i ≤ n, such that qmi ≤ Ri, 1 ≤ i ≤ n, for all m. So umi
is uniformly bounded for each 1 ≤ i ≤ n. This together with Sum = um (note Lemma 3.1)
implies that for each 1 ≤ i ≤ n there is a subsequence of {umi }∞m=1, relabeled as the original
sequence, which converges uniformly to some ui ∈ K̃i, where
K̃i =
{
y ∈ C(I)
∣∣∣∣ θiy(k) ≥ 0, k ∈ I
}
.
Clearly, we have
umi (k) = λm
N∑
`=0
gi(k, `)Pi(`, um1 (`), um2 (`), . . . , umn (`)), k ∈ I, 1 ≤ i ≤ n. (3.14)
Since umi converges to ui and λm converges to c, letting m → ∞ in (3.14) yields
ui(k) = c
N∑
`=0
gi(k, `)Pi(`, u1(`), u2(`), . . . , un(`)), k ∈ I, 1 ≤ i ≤ n.
Hence, c is an eigenvalue with corresponding eigenfunction u = (u1, u2, . . . , un), i.e., c =
= supE ∈ E. This completes the proof for Case (b).
(c) Let λ > 0 be fixed. Choose ε > 0 so that
λ max
1≤i≤n
N∑
`=0
Hi(`)bi(`) ≤
1
ε
. (3.15)
By definition, if fi ∈ F∞i , 1 ≤ i ≤ n, then there exists R = R(ε) > 0 such that the following
holds for each 1 ≤ i ≤ n:
fi(u1, u2, . . . , un) < ε|ui|, |uj | ≥ R, 1 ≤ j ≤ n. (3.16)
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12 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
We shall prove that S(C(R)) ⊆ C(R). To begin, let u ∈ C(R).By Lemma 3.2, we have Su ∈ C.
Thus, it remains to show that ‖Su‖ ≤ R. Using (3.6), (C3), (C8), (3.16) and (3.15), we find for
k ∈ I and 1 ≤ i ≤ n,
|Sui(k)| = θiSui(k) ≤ λ
N∑
`=0
Hi(`)bi(`)fi(u(`)) ≤
≤ λfi(R,R, . . . , R)
N∑
`=0
Hi(`)bi(`) ≤ λ(εR)
N∑
`=0
Hi(`)bi(`) ≤ R.
It follows that ‖Su‖ ≤ R and hence S(C(R)) ⊆ C(R). From Lemma 3.1 the operator S is
continuous and completely continuous. Schauder’s fixed point theorem guarantees that S has a
fixed point in C(R). Clearly, this fixed point is a constant-sign solution of (1.1) and therefore λ
is an eigenvalue of (1.1). Since λ > 0 is arbitrary, we have proved that E = (0,∞).
4. Subintervals of E for (1.1). For each fi, 1 ≤ i ≤ n, introduced in (C5), we shall define
f0,i = lim sup
max1≤j≤n |uj |→0
fi(u1, u2, . . . , un)
|ui|
, f
0,i
= lim inf
max1≤j≤n |uj |→0
fi(u1, u2, . . . , un)
|ui|
,
f∞,i = lim sup
min1≤j≤n |uj |→∞
fi(u1, u2, . . . , un)
|ui|
and f ∞,i = lim inf
min1≤j≤n |uj |→∞
fi(u1, u2, . . . , un)
|ui|
.
Theorem 4.1. Let (C1) – (C6) hold. If λ satisfies
γ1,i < λ < γ2,i, 1 ≤ i ≤ n, (4.1)
where
γ1,i =
[
f ∞,i Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
and
γ2,i =
[
f0,i
N∑
`=0
Hi(`)bi(`)
]−1
,
then λ ∈ E.
Proof. Let λ satisfy (4.1) and let εi > 0, 1 ≤ i ≤ n, be such that[
(f ∞,i − εi)Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
≤ λ ≤
[
(f0,i + εi)
N∑
`=0
Hi(`)bi(`)
]−1
, 1 ≤ i ≤ n.
(4.2)
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 13
First, we choose w > 0 so that
fi(u) ≤ (f0,i + εi)|ui|, 0 < |ui| ≤ w, 1 ≤ i ≤ n. (4.3)
Let u ∈ C be such that ‖u‖ = w. Then, applying (3.6), (C3), (4.3) and (4.2) successively, we
find for k ∈ I and 1 ≤ i ≤ n,
|Sui(k)| = θiSui(k) ≤ λ
N∑
`=0
gi(k, `)bi(`)fi(u(`)) ≤
≤ λ
N∑
`=0
Hi(`)bi(`)fi(u(`)) ≤
≤ λ
N∑
`=0
Hi(`)bi(`)(f0,i + εi)|ui(`)| ≤
≤ λ
N∑
`=0
Hi(`)bi(`)(f0,i + εi)‖u‖ ≤ ‖u‖.
Hence,
‖Su‖ ≤ ‖u‖. (4.4)
If we set Ω1 = {u ∈ B | ‖u‖ < w}, then (4.4) holds for u ∈ C ∩ ∂Ω1.
Next, pick r > w > 0 such that
fi(u) ≥ (f ∞,i − εi)|ui|, |ui| ≥ r, 1 ≤ i ≤ n. (4.5)
Let u ∈ C be such that
‖u‖ = r′ ≡ max
1≤j≤n
r
Mjρj
(> w).
Suppose ‖u‖ = |uz|0 for some z ∈ {1, 2, . . . , n}. Then, for ` ∈ Z[a, b] we have
|uz(`)| ≥ Mzρz|uz|0 = Mzρz‖u‖ ≥ Mzρz
r
Mzρz
= r,
which, in view of (4.5), yields
fz(u(`)) ≥ (f ∞,z − εz)|uz(`)|, ` ∈ Z[a, b]. (4.6)
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14 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Using (3.6), (C2), (4.6) and (4.2), we find
|Suz(a)| = θzSuz(a) ≥ λ
N∑
`=0
gz(a, `)az(`)fz(u(`)) ≥
≥ λ
N∑
`=0
MzHz(`)az(`)fz(u(`)) ≥
≥ λ
b∑
`=a
MzHz(`)az(`)fz(u(`)) ≥
≥ λ
b∑
`=a
MzHz(`)az(`)(f ∞,z − εz)|uz(`)| ≥
≥ λ
b∑
`=a
MzHz(`)az(`)(f ∞,z − εz)Mzρz|uz|0 =
= λ
b∑
`=a
MzHz(`)az(`)(f ∞,z − εz)Mzρz‖u‖ ≥ ‖u‖.
Therefore, |Suz|0 ≥ ‖u‖ and this leads to
‖Su‖ ≥ ‖u‖. (4.7)
If we set Ω2 = {u ∈ B | ‖u‖ < r′}, then (4.7) holds for u ∈ C ∩ ∂Ω2.
Now that we have obtained (4.4) and (4.7), it follows from Theorem 2.1 that S has a fixed
point u ∈ C ∩ (Ω̄2\Ω1) such that w ≤ ‖u‖ ≤ r′. Since this u is a constant-sign solution of (1.1),
the conclusion of the theorem follows immediately.
The following corollary is immediate from Theorem 4.1.
Corollary 4.1. Let (C1) – (C6) hold. Then,
(γ1,i, γ2,i) ⊆ E, 1 ≤ i ≤ n,
where γ1,i and γ2,i are defined in Theorem 4.1.
Corollary 4.2. Let (C1) – (C7) hold. Then,(
min
1≤i≤n
γ1,i, max
1≤i≤n
γ2,i
)
⊆ E
where γ1,i and γ2,i are defined in Theorem 4.1.
Proof. This is immediate from Corollaries 4.1 and 3.1.
Theorem 4.2. Let (C1) – (C6) hold. If λ satisfies
γ3,i < λ < γ4,i, 1 ≤ i ≤ n, (4.8)
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 15
where
γ3,i =
[
f
0,i
Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
and
γ4,i =
[
f∞,i
N∑
`=0
Hi(`)bi(`)
]−1
,
then λ ∈ E.
Proof. Let λ satisfy (4.8) and let εi > 0, 1 ≤ i ≤ n, be such that
[
(f
0,i
− εi)Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
≤ λ ≤
[
(f∞,i + εi)
N∑
`=0
Hi(`)bi(`)
]−1
, 1 ≤ i ≤ n.
(4.9)
First, pick w̄ > 0 such that
fi(u) ≥ (f
0,i
− εi)|ui|, 0 < |ui| ≤ w̄, 1 ≤ i ≤ n. (4.10)
Let u ∈ C be such that ‖u‖ = w̄. Suppose ‖u‖ = |uz|0 for some z ∈ {1, 2, . . . , n}. Employing
(3.6), (C2), (4.10) and (4.9) successively, we get
|Suz(a)| = θzSuz(a) ≥ λ
N∑
`=0
gz(a, `)az(`)fz(u(`)) ≥
≥ λ
N∑
`=0
MzHz(`)az(`)fz(u(`)) ≥
≥ λ
N∑
`=0
MzHz(`)az(`)(f0,z
− εz)|uz(`)| ≥
≥ λ
b∑
`=a
MzHz(`)az(`)(f0,z
− εz)|uz(`)| ≥
≥ λ
b∑
`=a
MzHz(`)az(`)(f0,z
− εz)Mzρz|uz|0 =
= λ
b∑
`=a
MzHz(`)az(`)(f0,z
− εz)Mzρz‖u‖ ≥ ‖u‖.
Therefore, |Suz|0 ≥ ‖u‖ and inequality (4.7) follows immediately. By setting Ω1 =
= {u ∈ B | ‖u‖ < w̄}, we see that (4.7) holds for u ∈ C ∩ ∂Ω1.
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16 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Next, choose r̄ > w̄ > 0 such that
fi(u) ≤ (f∞,i + εi)|ui|, |ui| ≥ r̄, 1 ≤ i ≤ n. (4.11)
For each fi, 1 ≤ i ≤ n, we shall consider two cases, namely, fi is bounded and fi is unbounded.
Let Nb and Nu be subsets of {1, 2, . . . , n} such that
Nb ∩Nu = ∅, Nb ∪Nu = {1, 2, . . . , n},
fi is bounded for i ∈ Nb,
fi is unbounded for i ∈ Nu.
Case 1. Suppose that fi, i ∈ Nb, is bounded. Then, there exists some Ri > 0 such that
fi(u) ≤ Ri, u ∈ IRn, i ∈ Nb. (4.12)
We define
r′ = max
i∈Nb
γ4,i Ri
N∑
`=0
Hi(`)bi(`).
Let u ∈ C be such that ‖u‖ ≥ r′. Applying (3.6), (C3), (4.12) and (4.8) gives for i ∈ Nb and
k ∈ I,
|Sui(k)| = θiSui(k) ≤ λ
N∑
`=0
gi(k, `)bi(`)fi(u(`)) ≤
≤ λ
N∑
`=0
Hi(`)bi(`)Ri <
< γ4,i
N∑
`=0
Hi(`)bi(`)Ri ≤ r′ ≤ ‖u‖.
It follows that for u ∈ C with ‖u‖ ≥ r′,
max
i∈Nb
|Sui|0 ≤ ‖u‖. (4.13)
Case 2. Suppose that fi, i ∈ Nu, is unbounded. Then, there exists
r′′ > max{r̄, r′} (> w̄)
such that
fi(u) ≤ max
ηj∈{−1,1}
1≤j≤n
fi(η1r
′′, η2r
′′, . . . , ηnr
′′), |uj | ≤ r′′, 1 ≤ j ≤ n. (4.14)
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 17
Let u ∈ C be such that ‖u‖ = r′′. Then, successive use of (3.6), (4.14), (4.11), (C3) and (4.9)
provides for i ∈ Nu and k ∈ I,
|Sui(k)| = θiSui(k) ≤ λ
N∑
`=0
gi(k, `)bi(`)fi(u(`)) ≤
≤ λ
N∑
`=0
gi(k, `)bi(`) max
ηj∈{−1,1}
1≤j≤n
fi(η1r
′′, η2r
′′, . . . , ηnr
′′) ≤
≤ λ
N∑
`=0
gi(k, `)bi(`)(f∞,i + εi)r′′ ≤
≤ λ
N∑
`=0
Hi(`)bi(`)(f∞,i + εi)‖u‖ ≤ ‖u‖.
Therefore, we have for u ∈ C with ‖u‖ = r′′,
max
i∈Nu
|Sui|0 ≤ ‖u‖. (4.15)
Combining (4.13) and (4.15), we obtain for u ∈ C with ‖u‖ = r′′,
max
i∈Nb∪Nu
|Sui|0 ≤ ‖u‖,
which is actually (4.4). Hence, by setting Ω2 = {u ∈ B | ‖u‖ < r′′}, we see that (4.4) holds for
u ∈ C ∩ ∂Ω2.
Having obtained (4.7) and (4.4), an application of Theorem 2.1 leads to the existence of a
fixed point u of S in C ∩ (Ω̄2\Ω1) such that w̄ ≤ ‖y‖ ≤ r′′. This u is a constant-sign solution of
(1.1) and the conclusion of the theorem follows immediately.
Theorem 4.2 leads to the following corollary.
Corollary 4.3. Let (C1) – (C6) hold. Then,
(γ3,i, γ4,i) ⊆ E, 1 ≤ i ≤ n,
where γ3,i and γ4,i are defined in Theorem 4.2.
Corollary 4.4. Let (C1) – (C7) hold. Then,(
min
1≤i≤n
γ3,i, max
1≤i≤n
γ4,i
)
⊆ E
where γ3,i and γ4,i are defined in Theorem 4.2.
Proof. This is immediate from Corollaries 4.3 and 3.1.
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18 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Remark 4.1. For a fixed i ∈ {1, 2, . . . , n}, if fi is superlinear (i.e., f0,i = 0 and f ∞,i = ∞)
or sublinear (i.e., f
0,i
= ∞ and f∞,i = 0), then we conclude from Corollaries 4.1 and 4.3 that
E = (0,∞), i.e., (1.1) has a constant-sign solution for any λ > 0.We remark that superlinearity
and sublinearity conditions have also been discussed for various boundary-value problems in
the literature for the single equation case (n = 1), see for example [3, 6, 7, 11 – 14] and the
references cited therein.
5. Applications to boundary-value problems. In this section we shall illustrate the generali-
ty of the results obtained in Sections 3 and 4 by considering various well known boundary-
value problems in the literature. Indeed, we shall apply our results to systems of boundary-
value problems of the following types: (m, p), Lidstone, focal, conjugate, Hermite and Sturm –
Liouville.
Case 5.1. (m, p) Boundary-value problem. Consider the system of (m, p) boundary-value
problems
∆mui(k) + λPi(k, u(k)) = 0, k ∈ Z[0, N ],
(5.1)
∆jui(0) = 0, 0 ≤ j ≤ m− 2, ∆piui(N +m− pi) = 0
where i = 1, 2, . . . , n. It is assumed that m ≥ 2, N ≥ m − 1 and for each 1 ≤ i ≤ n,
1 ≤ pi ≤ m− 1 is fixed and Pi : Z[0, N ]× IRn → IR is continuous.
Let Gi(k, `) be the Green’s function of the boundary-value problem
−∆my(k) = 0, k ∈ Z[0, N ],
∆jy(0) = 0, 0 ≤ j ≤ m− 2; ∆piy(N +m− pi) = 0.
It is known that [6, p. 315]
(a) Gi(k, `) =
1
(m− 1)!
k(m−1)(N +m− pi − 1− `)(m−pi−1)
(N +m− pi)(m−pi−1)
− (k − `− 1)(m−1),
` ∈ Z[0, k −m];
k(m−1)(N +m− pi − 1− `)(m−pi−1)
(N +m− pi)(m−pi−1)
, ` ∈ Z[k −m+ 1, N ];
(b) ∆jGi(k, `) (w.r.t. k) ≥ 0, 0 ≤ j ≤ pi, (k, `) ∈ Z[0, N +m− j]× Z[0, N ];
(c) for (k, `) ∈ Z[m− 1, N +m− pi]× Z[0, N ], we have
Gi(k, `) ≥
pi
(N +m− pi)(m−pi−1)(N + 1)
(N +m− pi − 1− `)(m−pi−1);
(d) for (k, `) ∈ Z[0, N +m]× Z[0, N ], we have
Gi(k, `) ≤
(N +m)(m−1)
(m− 1)!(N +m− pi)(m−pi−1)
(N +m− pi − 1− `)(m−pi−1).
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 19
Now, with I = Z[0, N +m], u = (u1, u2, . . . , un) is a solution of the system (5.1) if and only
if u is a fixed point of the operator S : B → B defined by (3.3) where
Sui(k) = λ
N∑
`=0
Gi(k, `)Pi(`, u(`)), k ∈ I, 1 ≤ i ≤ n. (5.2)
In the context of Section 3, we have
gi(k, `) = Gi(k, `), I = Z[0, N +m], Z[a, b] = Z[m− 1, N ], Mi =
(m− 1)!pi
(N +m)(m)
,
(5.3)
Hi(`) =
(N +m)(m−1)
(m− 1)!(N +m− pi)(m−pi−1)
(N +m− pi − 1− `)(m−pi−1).
Then, noting (a) – (d), we see that the conditions (C1) – (C3) are fulfilled.
The results in Sections 3 and 4 reduce to the following theorem, which improves and extends
the earlier work of [11, 15] (for n = 1) — not only do we consider a more general Pi, our
method is also generic in nature.
Theorem 5.1. LetE = {λ | λ > 0 such that (5.1) has a constant-sign solution}.With gi, a, b,
Mi and Hi given in (5.3), we have the following:
(i) ( Theorem 3.1). Let (C4) – (C6) hold. Then, there exists c > 0 such that the interval (0, c] ⊆
⊆ E.
(ii) ( Theorem 3.2 and Corollary 3.1). Let (C4) and (C7) hold. Suppose that λ∗ ∈ E. Then,
for any λ ∈ (0, λ∗), we have λ ∈ E, i.e., (0, λ∗] ⊆ E. Indeed, if E 6= ∅, then E is an interval.
(iii) ( Theorem 3.3). Let (C4) – (C6) and (C8) hold. Suppose that λ ∈ E and
u ∈ C =
{
u ∈ (C(I))n
∣∣∣∣ for each 1 ≤ i ≤ n, θiui(k) ≥ 0 for k ∈ I,
and min
k∈Z[a,b]
θiui(k) ≥ Miρi|ui|0
}
is a corresponding eigenfunction. Let qi = |ui|0, 1 ≤ i ≤ n. Then, for each 1 ≤ i ≤ n, we have
λ ≥ qi
fi(q1, q2, . . . , qn)
[
N∑
`=0
Hi(`)bi(`)
]−1
and
λ ≤ qi
fi(M1ρ1q1,M2ρ2q2, . . . ,Mnρnqn)
[
b∑
`=a
MiHi(`)ai(`)
]−1
.
(iv) (Theorem 3.4). Let (C4) – (C8) hold. For each 1 ≤ i ≤ n, let FBi , F
0
i and F∞i be defined
as in Theorem 3.4.
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20 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
(a) If fi ∈ FBi for each 1 ≤ i ≤ n, then E = (0, c) or (0, c] for some c ∈ (0,∞).
(b) If fi ∈ F 0
i for each 1 ≤ i ≤ n, then E = (0, c] for some c ∈ (0,∞).
(c) If fi ∈ F∞i for each 1 ≤ i ≤ n, then E = (0,∞).
(v) ( Theorem 4.1, Corollaries 4.1 and 4.2). Let (C4) – (C6) hold. For each 1 ≤ i ≤ n, let f0,i
and, f ∞,i be defined as in Section 4. If λ satisfies
γ1,i < λ < γ2,i, 1 ≤ i ≤ n,
where
γ1,i =
[
f ∞,i Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
and
γ2,i =
[
f0,i
N∑
`=0
Hi(`)bi(`)
]−1
,
then λ ∈ E. Indeed,
(γ1,i, γ2,i) ⊆ E, 1 ≤ i ≤ n.
Moreover, if (C7) holds, then (
min
1≤i≤n
γ1,i, max
1≤i≤n
γ2,i
)
⊆ E.
(vi) (Theorem 4.2, Corollaries 4.3 and 4.4). Let (C4) – (C6) hold. For each 1 ≤ i ≤ n, let f
0,i
and f∞,i be defined as in Section 4. If λ satisfies
γ3,i < λ < γ4,i, 1 ≤ i ≤ n,
where
γ3,i =
[
f
0,i
Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
and
γ4,i =
[
f∞,i
N∑
`=0
Hi(`)bi(`)
]−1
,
then λ ∈ E. Indeed,
(γ3,i, γ4,i) ⊆ E, 1 ≤ i ≤ n.
Moreover, if (C7) holds, then (
min
1≤i≤n
γ3,i, max
1≤i≤n
γ4,i
)
⊆ E.
(vii) ( Remark 4.1). Let (C4) – (C6) hold. If fj is m superlinear (i.e., f0,j = 0 and f ∞,j = ∞)
or sublinear (i.e., f
0,j
= ∞ and f ∞,j = 0) for some j ∈ {1, 2, . . . , n}, then E = (0,∞).
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 21
Example 5.1. Consider the system of (m, p) boundary-value problems
∆3u1(k) + λ
[u1(k) + 1][u2(k) + 1]
[k(k − 1)(11− k) + 1][k(k − 1)(20− k) + 1]
= 0, k ∈ Z[0, 5],
∆3u2(k) + λ
[u1(k) + 20][u2(k) + 20]
[k(k − 1)(11− k) + 20][k(k − 1)(20− k) + 20]
= 0, k ∈ Z[0, 5], (5.4)
u1(0) = ∆u1(0) = 0, ∆u1(7) = 0; u2(0) = ∆u2(0) = 0, ∆2u2(6) = 0.
In this example, n = 2, m = 3, N = 5, p1 = 1, p2 = 2,
P1(k, u(k)) =
[u1(k) + 1][u2(k) + 1]
[k(k − 1)(11− k) + 1][k(k − 1)(20− k) + 1]
and
P2(k, u(k)) =
[u1(k) + 20][u2(k) + 20]
[k(k − 1)(11− k) + 20][k(k − 1)(20− k) + 20]
.
Fix θ1 = θ2 = 1. Clearly, (C4) and (C7) are satisfied. Now, choose
f1(u) = [u1(k) + 1][u2(k) + 1], f2(u) = [u1(k) + 20][u2(k) + 20],
a1(k) = b1(k) = {[k(k − 1)(11− k) + 1][k(k − 1)(20− k) + 1]}−1
and
a2(k) = b2(k) = {[k(k − 1)(11− k) + 20][k(k − 1)(20− k) + 20]}−1.
Then, (C5), (C6) (with ρ1 = ρ2 = 1) and (C8) are fulfilled. Moreover, we haveH1(`) = 4(6− `)
and H2(`) = 28.
It is easy to see that
f0,1 = f
0,1
= ∞, f∞,1 = f ∞,1 = 1, f0,2 = f
0,2
= ∞ and f∞,2 = f ∞,2 = 1.
Clearly, fi ∈ FBi , i = 1, 2. Hence, Theorem 5.1(iv) guarantees that
E = {λ | λ > 0 such that (5.4) has a constant-sign solution} = (0, c) or (0, c] (5.5)
for some c ∈ (0,∞).
By direct computation, we get
γ3,1 = γ3,2 = 0, γ4,1 = 0, 02271 and γ4,2 = 6, 3121.
It follows from Theorem 5.1(vi) that(
min
i=1,2
γ3,i, max
i=1,2
γ4,i
)
= (0, 63121) ⊆ E. (5.6)
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22 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Coupling with (5.5), we further conclude that E = (0, c) or (0, c] where c ≥ 6, 3121. Indeed,
when λ = 6 ∈ E, the system (5.4) has a positive solution given by
u(k) = (u1(k), u2(k)) = (k(k − 1)(11− k), k(k − 1)(20− k)) , k ∈ Z[0, 8].
Case 5.2. Lidstone boundary-value problem. Consider the system of Lidstone boundary-
value problems
(−1)m∆2mui(k) = λPi(k, u(k)), k ∈ Z[0, N ],
(5.7)
∆2jui(0) = ∆2jui(N + 2m− 2j) = 0, 0 ≤ j ≤ m− 1,
where i = 1, 2, . . . , n. It is assumed that m ≥ 1 and Pi : Z[0, N ] × IRn → IR, 1 ≤ i ≤ n, is
continuous.
Let Gm(k, `) be the Green’s function of the boundary-value problem
∆2my(k) = 0, k ∈ Z[0, N ],
∆2jy(0) = ∆2jy(N + 2m− 2j) = 0, 0 ≤ j ≤ m− 1.
It is given in [16] that
(a) Gm(k, `) =
∑N+2m−2
τ=0 G(k, τ)Gm−1(τ, `) where
G(k, `) = G1(k, `) = − 1
N + 2m
{
(N + 2m− k)(`+ 1), ` ∈ Z[0, k − 2];
k(N + 2m− 1− `), ` ∈ Z[k − 1, N + 2m− 2];
(b) (−1)mGm(k, `) ≥ 0, (k, `) ∈ Z[0, N + 2m]× Z[0, N ];
(c) for (k, `) ∈ Z[1, N + 2m− 1]× Z[0, N ], we have
(−1)mGm(k, `) ≥ βm min{`+ 1, N + 1− `} ≥ βm
N + 1
(`+ 1)(N + 1− `)
where
βm =
m∏
j=1
(N + 2j)
−1
m−1∏
j=1
T2j−1
and
Tj =
N+j∑
τ=1
min{τ+1, N+j+2−τ} =
1
4
{
(N + j)2 + 6(N + j) + 1, (N + j)is odd;
(N + j)(N + j + 6), (N + j) is even
, j ≥ 1;
(d) for (k, `) ∈ Z[0, N + 2m]× Z[0, N ], we have
(−1)mGm(k, `) ≤ αm(`+ 1)(N + 1− `)
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 23
where
αm =
m∏
j=1
(N + 2j)
−1
m−1∏
j=1
s2j
and
sj =
N+j∑
τ=0
(τ + 1)(N + j + 1− τ) =
1
6
(N + j + 3)(3), j ≥ 2.
Clearly, with I = Z[0, N + 2m], u = (u1, u2, . . . , un) is a solution of the system (5.7) if and
only if u is a fixed point of the operator S : B → B defined by (3.3) where
Sui(k) = λ
N∑
`=0
(−1)mGm(k, `)Pi(`, u(`)), k ∈ I, 1 ≤ i ≤ n. (5.8)
In the context of Section 3, let
gi(k, `) = (−1)mGm(k, `), I = Z[0, N + 2m], Z[a, b] = Z[1, N ],
(5.9)
Mi =
βm
αm(N + 1)
and Hi(`) = αm(`+ 1)(N + 1− `).
Then, the conditions (C1) – (C3) are satisfied in view of (a) – (d).
Applying the results in Sections 3 and 4, we obtain the following theorem which improves
and extends the earlier work of [16] (for n = 1). Note that the Pi considered in (5.7) as well as
the methodology used are both more general.
Theorem 5.2. Let E = {λ | λ > 0 such that (5.7) has a constant-sign solution}. With gi, a,
b,Mi and Hi given in (5.9), the statements (i) – (vii) of Theorem 5.1 hold.
Case 5.3. Focal boundary-value problem. Consider the system of focal boundary-value
problems
(−1)m−pi ∆mui(k) = λPi(k, u(k)), k ∈ Z[0, N ],
(5.10)
∆jui(0) = 0, 0 ≤ j ≤ pi − 1; ∆jui(N + 1) = 0, pi ≤ j ≤ m− 1,
where i = 1, 2, . . . , n. It is assumed that m ≥ 2, and for each 1 ≤ i ≤ n, 1 ≤ pi ≤ min{m −
−1, N} is fixed and Pi : Z[0, N ]× IRn → IR is continuous.
Let Gi(k, `) be the Green’s function of the boundary-value problem
∆my(k) = 0, k ∈ Z[0, N ],
∆jy(0) = 0, 0 ≤ j ≤ pi − 1; ∆jy(N + 1) = 0, pi ≤ i ≤ m− 1.
In [17] it is given that
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24 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
(a) Gi(k, `) = (−1)m−pi
∑̀
j=0
(k − j − 1)(pi−1)(`+m− pi − 1− j)(m−pi−1)
(pi − 1)!(m− pi − 1)!
,
` ∈ Z[0, k − 1];
k−1∑
j=0
(k − j − 1)(pi−1)(`+m− pi − 1− j)(m−pi−1)
(pi − 1)!(m− pi − 1)!
,
` ∈ Z[k,N ];
(b) the signs of the differences of Gi(k, `) w.r.t. k are as follows:
(−1)m−pi∆jGi(k, `) ≥ 0, (k, `) ∈ Z[0, N +m− j]× Z[0, N ], 0 ≤ j ≤ pi − 1,
(−1)m−pi+j∆j+piGi(k, `) ≥ 0,
(k, `) ∈ Z[0, N +m− j − pi]× Z[0, N ], 0 ≤ j ≤ m− pi − 1;
(c) for a given δi ∈ Z[pi, N ], and (k, `) ∈ Z[δi, N +m]× Z[0, N ], we have
(−1)m−piGi(k, `) ≥ Li(−1)m−piGi(N +m, `)
where
Li = min
`∈Z[0,N ]
Gi(δi, `)
Gi(N +m, `)
;
(d) (−1)m−piGi(k, `) ≤ (−1)m−piGi(N +m, `), (k, `) ∈ Z[0, N +m]× Z[0, N ].
Obviously, with I = Z[0, N + m], u = (u1, u2, . . . , un) is a solution of the system (5.10) if
and only if u is a fixed point of the operator S : B → B defined by (3.3) where
Sui(k) = λ
N∑
`=0
(−1)m−piGi(k, `)Pi(`, u(`)), k ∈ I, 1 ≤ i ≤ n. (5.11)
Let δi ∈ Z[pi, N ], 1 ≤ i ≤ n, be fixed and δ ≡ max1≤i≤n δi. In the context of Section 3, let
gi(k, `) = (−1)m−piGi(k, `), I = Z[0, N +m], Z[a, b] = Z[δ,N ],
(5.12)
Mi = Li and Hi(`) = (−1)m−piGi(N +m, `).
Then, from (a) – (d) we see that the conditions (C1) – (C3) are satisfied.
The results in Sections 3 and 4 reduce to the following theorem which improves and extends
the earlier work of [17] (for n = 1). We remark that the Pi considered in (5.10) as well as the
methodology used are both more general.
Theorem 5.3. Let E = {λ | λ > 0 such that (5.10) has a constant-sign solution}. With
gi, a, b, Mi and Hi given in (5.12), the statements (i) – (vii) of Theorem 5.1 hold.
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 25
Case 5.4. Conjugate boundary-value problem. Consider the system of conjugate boundary-
value problems
(−1)m−pi ∆mui(k) = λPi(k, u(k)), k ∈ Z[0, N ],
∆jui(0) = 0, 0 ≤ j ≤ pi − 1; ∆jui(N + pi + 1) = 0, 0 ≤ j ≤ m− pi − 1, (5.13)
where i = 1, 2, . . . , n. It is assumed that m ≥ 2, and for each 1 ≤ i ≤ n, 1 ≤ pi ≤ m − 1,
N ≥ min1≤i≤n pi and Pi : Z[0, N ]× IRn → IR is continuous.
Let Gi(k, `) be the Green’s function of the boundary-value problem
∆my(k) = 0, k ∈ Z[0, N ],
∆jy(0) = 0, 0 ≤ j ≤ pi − 1; ∆jy(N + pi + 1) = 0, 0 ≤ j ≤ m− pi − 1.
It is known that [18, 19]
(a)Gi(k, `) =
pi−1∑
j=0
[
pi−j−1∑
τ=0
(
m− pi + τ − 1
τ
)
k(j+τ)
(N +m− j)(m−pi+τ)
]
(−`− 1)(m−j−1)
j!(m− j − 1)!
×
×(N +m− k)(m−pi), ` ∈ Z[0, k − 1],
−
m−pi−1∑
j=0
[
m−pi−j−1∑
τ=0
(
pi + τ − 1
τ
)
(N + pi + j + τ − k)(j+τ)
(N + pi + 1 + j + τ)(pi+τ)
]
(−1)j×
×(N + pi − `)(m−j−1)
j!(m− j − 1)!
k(pi), ` ∈ Z[k,N ];
(b) (−1)m−piGi(k, `) ≥ 0, (k, `) ∈ Z[0, N +m]× Z[0, N ];
(c) for a given δi ∈ Z[pi, N + pi], and (k, `) ∈ Z[δi, N + pi]× Z[0, N ], we have
(−1)m−piGi(k, `) ≥ Ki‖Gi(·, `)‖
where
‖Gi(·, `)‖ = max
k∈Z[0,N+m]
|Gi(k, `)| = max
k∈Z[0,N+m]
(−1)m−piGi(k, `),
Ki = min
{
mink∈Z[δi,N+pi] v(pi + 1, k)
maxk∈Z[δi,N+pi] v(pi + 1, k)
,
mink∈Z[δi,N+pi] v(pi, k)
maxk∈Z[δi,N+pi] v(pi, k)
}
,
and the function v is defined as
v(x, k) = k(x−1)(N +m− k)(m−x);
(d) (−1)m−piGi(k, `) ≤ ‖Gi(·, `)‖, (k, `) ∈ Z[0, N +m]× Z[0, N ].
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26 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Now, with I = Z[0, N + m], u = (u1, u2, . . . , un) is a solution of the system (5.13) if and
only if u is a fixed point of the operator S : B → B defined by (3.3) where
Sui(k) = λ
N∑
`=0
(−1)m−piGi(k, `)Pi(`, u(`)), k ∈ I, 1 ≤ i ≤ n. (5.14)
Let δi ∈ Z[pi, N + pi], 1 ≤ i ≤ n, be fixed and δ ≡ max1≤i≤n δi. In the context of Section
3, let
gi(k, `) = (−1)m−piGi(k, `), I = Z[0, N +m], Z[a, b] = Z[δ,N ],
(5.15)
Mi = Ki and Hi(`) = ‖Gi(·, `)‖.
Then, (a) – (d) ensures that the conditions (C1) – (C3) are fulfilled.
Applying the results in Sections 3 and 4, we obtain the following theorem which improves
and extends the earlier work of [18] (for n = 1). Note that the Pi considered in (5.13) as well
as the methodology used are both more general.
Theorem 5.4. Let E = {λ | λ > 0 such that (5.13) has a constant-sign solution}. With
gi, a, b, Mi and Hi given in (5.15), the statements (i) – (vii) of Theorem 5.1 hold.
Case 5.5. Hermite boundary-value problem. Consider the system of Hermite boundary-
value problems
∆mui(k) = λFi(k, u(k)), k ∈ Z[0, N ],
(5.16)
∆jui(kν) = 0, j = 0, . . . ,mν − 1, ν = 1, . . . , J,
where i = 1, 2, . . . , n. It is assumed that J ≥ 2, mν ≥ 1 for ν = 1, . . . , J,
∑J
ν=1mν = m, and
kν ’s are integers such that kJ ≥ N and
0 = k1 < k1 +m1 < k2 < k2 +m2 < . . . < kJ ≤ kJ +mJ − 1 = N +m.
Moreover, for each 1 ≤ i ≤ n and k ∈ Z[0, N ], we assume
Fi(k, u(k)) =
{
(−1)γνPi(k, u(k)), k ∈ Z[kν , kν+1 − 1], ν = 1, . . . , J − 2;
(−1)γJ−1Pi(k, u(k)), k ∈ Z[kJ−1, kJ ],
(5.17)
where Pi : Z[0, N ]× IRn → IR, 1 ≤ i ≤ n, is continuous and
γν =
J∑
j=ν+1
mj , 1 ≤ ν ≤ J − 1.
We shall also use the notation
Iν = Z[kν +mν , kν+1 − 1], 1 ≤ ν ≤ J − 1.
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 27
Let G(k, `) be the Green’s function of the boundary-value problem
∆my(k) = 0, k ∈ Z[0, N ],
∆jy(kν) = 0, j = 0, . . . ,mν − 1, ν = 1, . . . , J.
It is known that [20, 21]
(a) the signs of G(k, `) are as follows:
(−1)γνG(k, `) ≥ 0, (k, `) ∈ Z[kν , kν+1]× Z[0, N ], ν = 1, . . . , J − 1,
G(k, `) = 0, (k, `) ∈ Z[kJ , N +m]× Z[0, N ];
(b) for (k, `) ∈ Iν × Z[0, N ], ν = 1, . . . , J − 1, we have
(−1)γνG(k, `) ≥ Lν‖G(·, `)‖
where
‖G(·, `)‖ = max
k∈Z[0,N+m]
|G(k, `)| = max
1≤ν≤J−1
max
k∈Z[kν ,kν+1]
(−1)γνG(k, `),
Lν = min
{
min {p(kν +mν), p(kν+1 − 1)}
maxk∈Z[0,N+m] p(k)
,
min {q(kν +mν), q(kν+1 − 1)}
maxk∈Z[0,N+m] q(k)
}
and the functions p and q are defined as
p(k) =
∣∣∣∣∣∣
J−1∏
j=1
(k − kj)(mj)
∣∣∣∣∣∣ (N +m− k)(mJ−1), q(k) = k(m1−1)
∣∣∣∣∣∣
J∏
j=2
(k − kj)(mj)
∣∣∣∣∣∣ ;
(c) (−1)γνG(k, `) ≤ ‖G(·, `)‖, (k, `) ∈ Z[0, N +m]× Z[0, N ], ν = 1, . . . , J − 1.
Clearly, with I = Z[0, N +m], u = (u1, u2, . . . , un) is a solution of the system (5.16) if and
only if u is a fixed point of the operator S : B → B defined by (3.3) where
Sui(k) = λ
N∑
`=0
G(k, `)Fi(`, u(`)), k ∈ I, 1 ≤ i ≤ n. (5.18)
In the context of Section 3, let
gi(k, `) = (−1)γνG(k, `), I = Z[0, N +m], Z[a, b] = Iν ∩ Z[0, N ],
(5.19)
Mi = Lν and Hi(`) = ‖G(·, `)‖.
Then, noting (a) – (c) the conditions (C1), (C3) and (C2) (for ν = 1, 2, . . . , J − 1) are fulfilled.
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28 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
The results in Sections 3 and 4 reduce to the following theorem, which improves and extends
the earlier work of [21] (for n = 1) — note that a more general Fi is considered by using a more
general method.
Theorem 5.5. Let E = {λ | λ > 0 such that (5.16) has a constant-sign solution}. With
gi, a, b, Mi and Hi given in (5.19), the statements (i), (ii), (iv) and (vii) of Theorem 5.1 hold.
Moreover, we have the following:
(iii) (Theorem 3.3). Let (C4) – (C6) and (C8) hold. Suppose that λ ∈ E and
u ∈ C =
{
u ∈ (C(I))n
∣∣∣∣ for each 1 ≤ i ≤ n, θiui(k) ≥ 0 for k ∈ I,
and min
k∈Iν∩Z[0,N ]
θiui(k) ≥ Lνρi|ui|0, ν = 1, 2, . . . , J − 1
}
is a corresponding eigenfunction. Let qi = |ui|0, 1 ≤ i ≤ n. Then, we have
λ ≥ qi
fi(q1, q2, . . . , qn)
[
N∑
`=0
Hi(`)bi(`)
]−1
, 1 ≤ i ≤ n, (5.20)
and
λ ≤ qi
fi(Lνρ1q1, Lνρ2q2, . . . , Lνρnqn)
∑
`∈Iν∩Z[0,N ]
LνHi(`)ai(`)
−1
,
(5.21)
1 ≤ i ≤ n, 1 ≤ ν ≤ J − 1.
(v) (Theorem 4.1, Corollaries 4.1 and 4.2). Let (C4) – (C6) hold. For each 1 ≤ i ≤ n, let f0,i
and f ∞,i be defined as in Section 4. If λ satisfies
γ1,i,ν < λ < γ2,i, 1 ≤ i ≤ n, 1 ≤ ν ≤ J − 1, (5.22)
where
γ1,i,ν =
f ∞,i Lνρi ∑
`∈Iν∩Z[0,N ]
LνHi(`)ai(`)
−1
and
γ2,i =
[
f0,i
N∑
`=0
Hi(`)bi(`)
]−1
,
then λ ∈ E. Indeed,
(γ1,i,ν , γ2,i) ⊆ E, 1 ≤ i ≤ n, 1 ≤ ν ≤ J − 1.
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 29
Moreover, if (C7) holds, then min
1 ≤ i ≤ n
1 ≤ ν ≤ J − 1
γ1,i,ν , max
1≤i≤n
γ2,i
⊆ E.
(vi) (Theorem 4.2, Corollaries 4.3 and 4.4). Let (C4) – (C6) hold. For each 1 ≤ i ≤ n, let f
0,i
and f∞,i be defined as in Section 4. If λ satisfies
γ3,i,ν < λ < γ4,i, 1 ≤ i ≤ n, 1 ≤ ν ≤ J − 1, (5.23)
where
γ3,i,ν =
f
0,i
Lνρi
∑
`∈Iν∩Z[0,N ]
LνHi(`)ai(`)
−1
and
γ4,i =
[
f∞,i
N∑
`=0
Hi(`)bi(`)
]−1
,
then λ ∈ E. Indeed,
(γ3,i,ν , γ4,i) ⊆ E, 1 ≤ i ≤ n, 1 ≤ ν ≤ J − 1.
Moreover, if (C7) holds, then min
1 ≤ i ≤ n
1 ≤ ν ≤ J − 1
γ3,i,ν , max
1≤i≤n
γ4,i
⊆ E.
Proof. (iii) Here, the cone C in (3.5) is modified to that in the statement of Theorem 5.5(iii).
The proof of (5.20) is similar to that in the proof of Theorem 3.3. To verify (5.21), let 1 ≤ i ≤ n
and 1 ≤ ν ≤ J − 1 be fixed. Using (3.6), (C2), (C8) and the fact that mink∈Iν∩Z[0,N ] θiui(k) ≥
≥ Lνρi|ui|0 = Lνρiqi, we get
qi = |ui|0 ≥ θiui(kν+1 − 1) =
= θiλ
N∑
`=0
Gi(kν+1 − 1, `)Fi(`, u(`)) ≥
≥ θiλ
∑
`∈Z[kν ,kν+1−1]∩Z[0,N ]
Gi(kν+1 − 1, `)(−1)γνPi(`, u(`)) ≥
≥ λ
∑
`∈Z[kν ,kν+1−1]∩Z[0,N ]
(−1)γνGi(kν+1 − 1, `)ai(`)fi(u(`)) ≥
≥ λ
∑
`∈Iν∩Z[0,N ]
LνHi(`)ai(`)fi(u(`)) ≥
≥ λ
∑
`∈Iν∩Z[0,N ]
LνHi(`)ai(`)fi(Lνρ1q1, Lνρ2q2, . . . , Lνρnqn)
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30 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
which reduces to (5.21).
(v) Let λ satisfy (5.22) and let εiν > 0, 1 ≤ i ≤ n, 1 ≤ ν ≤ J − 1, be such that
(f ∞,i − εiν)Lνρi
∑
`∈Iν∩Z[0,N ]
LνHi(`)ai(`)
−1
≤λ ≤
[
(f0,i + εiν)
N∑
`=0
Hi(`)bi(`)
]−1
,
(5.24)
1 ≤ i ≤ n, 1 ≤ ν ≤ J − 1.
First, we can choose w > 0 so that for 1 ≤ i ≤ n and 1 ≤ ν ≤ J − 1,
fi(u) ≤ (f0,i + εiν)|ui|, 0 < |ui| ≤ w. (5.25)
As in the proof of Theorem 4.1, it now follows that ‖Su‖ ≤ ‖u‖ for u ∈ C ∩ ∂Ω1 where
Ω1 = {u ∈ B | ‖u‖ < w}.
Next, pick T > w > 0 such that for 1 ≤ i ≤ n and 1 ≤ ν ≤ J − 1,
fi(u) ≥ (f ∞,i − εiν)|ui|, |ui| ≥ T. (5.26)
Let u ∈ C be such that
‖u‖ = T ′ ≡ max
1 ≤ j ≤ n
1 ≤ ν ≤ J − 1
T
Lνρj
(> w).
Suppose ‖u‖ = |uz|0 for some z ∈ {1, 2, . . . , n}. Let ν ∈ {1, 2, . . . , J − 1} be fixed. Then, for
` ∈ Iν ∩ Z[0, N ] we have
|uz(`)| ≥ Lνρz|uz|0 = Lνρz‖u‖ ≥ Lνρz
T
Lνρz
= T,
which, in view of (5.26), yields
fz(u(`)) ≥ (f ∞,z − εzν)|uz(`)|, l ∈ Iν ∩ Z[0, N ]. (5.27)
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 31
Using (3.6), (C2), (5.27) and (5.24), we find
|Suz(kν+1 − 1)| = θzSuz(kν+1 − 1) ≥
≥ θzλ
∑
`∈Z[kν ,kν+1−1]∩Z[0,N ]
Gz(kν+1 − 1, `)(−1)γνPz(`, u(`)) ≥
≥ λ
∑
`∈Z[kν ,kν+1−1]∩Z[0,N ]
(−1)γνGz(kν+1 − 1, `)az(`)fz(u(`)) ≥
≥ λ
∑
`∈Iν∩Z[0,N ]
LνHz(`)az(`)fz(u(`)) ≥
≥ λ
∑
`∈Iν∩Z[0,N ]
LνHz(`)az(`)(f ∞,z − εzν)|uz(`)| ≥
≥ λ
∑
`∈Iν∩Z[0,N ]
LνHz(`)az(`)(f ∞,z − εzν)Lνρz|uz|0 =
= λ
∑
`∈Iν∩Z[0,N ]
LνHz(`)az(`)(f ∞,z − εzν)Lνρz‖u‖ ≥ ‖u‖.
Therefore, |Suz|0 ≥ ‖u‖ and this leads to ‖Su‖ ≥ ‖u‖. Setting Ω2 = {u ∈ B | ‖u‖ < T ′}, we
have ‖Su‖ ≥ ‖u‖ for u ∈ C ∩ ∂Ω2.
The rest of the proof is similar to that of Theorem 4.1.
(vi) The proof is similar to that of Theorem 4.2 with analogous modification as in the proof
of Theorem 5.5(v).
Case 5.6. Sturm – Liouville boundary-value problem. Consider the system of Sturm – Liouville
boundary-value problems
∆mui(k) + λPi(k, u(k)) = 0, k ∈ Z[0, N ],
∆jui(0) = 0, 0 ≤ j ≤ m− 3, (5.28)
ζi∆m−2ui(0)− ηi∆m−1ui(0) = 0, γi∆m−2ui(N + 1) + δi∆m−1ui(N + 1) = 0,
where i = 1, 2, . . . , n. It is assumed that m ≥ 2, N ≥ m − 1, and for each 1 ≤ i ≤ n,
Pi : Z[0, N ]× IRn → IR is continuous,
ζi > 0, γi > 0, ηi ≥ 0, δi ≥ γi, ψi ≡ ζiγi(N + 1) + ζiδi + ηiγi > 0.
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32 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Let hi(k, `) be the Green’s function of the boundary-value problem
−∆my(k) = 0, k ∈ Z[0, N ],
∆jy(0) = 0, 0 ≤ j ≤ m− 3,
ζi∆m−2y(0)− ηi∆m−1y(0) = 0, γi∆m−2y(N + 1) + δi∆m−1y(N + 1) = 0.
It can be verified that [14]
Gi(k, `) = ∆m−2hi(k, `) (w.r.t. k) (5.29)
is the Green’s function of the boundary-value problem
−∆2w(k) = 0, k ∈ Z[0, N ],
ζiw(0)− ηi∆w(0) = 0, γiw(N + 1) + δi∆w(N + 1) = 0.
Further, it is known that [14]
(a) Gi(k, `) =
1
ψi
{
[ηi + ζi(`+ 1)][δi + γi(N + 1− k)], ` ∈ Z[0, k − 1];
(ηi + ζik)[δi + γi(N − `)], ` ∈ Z[k,N ];
(b) Gi(k, `) ≥ 0, (k, `) ∈ Z[0, N + 2]× Z[0, N ];
(c) for (k, `) ∈ Z[1, N ]× Z[0, N ], we have
Gi(k, `) ≥ Ai Gi(`, `)
where
Ai =
(ηi + ζi)(δi + γi)
(ηi + ζiN)(δi + γiN)
;
d) for (k, `) ∈ Z[0, N + 2]× Z[0, N ], we have
Gi(k, `) ≤ Bi Gi(`, `)
where
Bi =
ηi + ζi
ηi
, ηi > 0;
2, ηi = 0.
In the context of Section 3, let the Banach space
B =
{
u = (u1, u2, . . . , un) ∈ (C(Z[0, N +m]))n
∣∣∣∣ ∆jui(0) = 0, 0 ≤ j ≤ m− 3, 1 ≤ i ≤ n
}
(5.30)
be equipped with norm
‖u‖ = max
1≤i≤n
max
k∈Z[0,N+2]
|∆m−2ui(k)| = max
1≤i≤n
|ui|0 (5.31)
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 33
where we denote |ui|0 = maxk∈Z[0,N+2] |∆m−2ui(k)|, 1 ≤ i ≤ n. Further, define the cone C in
B as
C =
{
u = (u1, u2, . . . , un) ∈ B
∣∣∣∣for each 1 ≤ i ≤ n, θi∆m−2ui(k) ≥ 0 for k ∈ Z[0, N + 2],
and min
k∈Z[1,N ]
θi∆m−2ui(k) ≥ Mi|ui|0
}
(5.32)
where Mi =
Ai
Bi
∈ (0, 1), 1 ≤ i ≤ n. It can be shown that S maps C into C.
Lemma 5.1 [14].
(a) Let u ∈ B. For 0 ≤ j ≤ m− 2, we have
|∆jui(k)| ≤ k(m−2−j)
(m− 2− j)!
|ui|0, k ∈ Z[0, N +m− j], 1 ≤ i ≤ n. (5.33)
In particular,
|ui(k)| ≤ (N +m)(m−2)
(m− 2)!
‖u‖, k ∈ Z[0, N +m], 1 ≤ i ≤ n. (5.34)
(b) Let u ∈ C. For 0 ≤ j ≤ m− 2, we have
θi∆jui(k) ≥ 0, k ∈ Z[0, N +m− j], 1 ≤ i ≤ n, (5.35)
and
θi∆jui(k) ≥ (k − 1)(m−2−j)
(m− 2− j)!
Miρi|ui|0, k ∈ Z[1, N +m− 2− j], 1 ≤ i ≤ n. (5.36)
In particular,
θiui(k) ≥ Miρi|ui|0, k ∈ Z[m− 1, N +m− 2], 1 ≤ i ≤ n. (5.37)
Hence, if u = (u1, u2, . . . , un) ∈ C is a solution of (5.28), then it follows from (5.35) that u
is a constant-sign solution. Clearly, u is a solution of the system (5.28) if and only if u is a fixed
point of the operator S : B → B defined by (3.3) where
Sui(k) = λ
N∑
`=0
hi(k, `)Pi(`, u(`)), k ∈ Z[0, N +m], 1 ≤ i ≤ n, (5.38)
or equivalently
∆m−2(Sui)(k) = λ
N∑
`=0
Gi(k, `)Pi(`, u(`)), k ∈ Z[0, N + 2], 1 ≤ i ≤ n. (5.39)
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34 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Now, in the context of Section 3, let
gi(k, `) = Gi(k, `), I = Z[0, N + 2], Z[a, b] = Z[1, N ],
(5.40)
Mi =
Ai
Bi
and Hi(`) = Bi Gi(`, `).
Then, noting (a) – (d), we see that (C1) – (C3) are fulfilled.
The results in Sections 3 and 4 together with Lemma 5.1 lead to the following theorem,
which improves and extends the earlier work of [14, 21, 22] (for n = 1) — not only do we
consider a more general Pi, our method is also more general.
Theorem 5.6. Let E = {λ | λ > 0 such that (5.28) has a constant-sign solution}. With
gi, a, b, Mi and Hi given in (5.40), the statements (i), (ii), (iv)–(vii) of Theorem 5.1 hold.
Moreover, we have the following:
(iii) (Theorem 3.3). Let (C4) – (C6) and (C8) hold. Suppose that λ ∈ E and u ∈ C (see
(5.32)) is a corresponding eigenfunction. Let qi = |ui|0, 1 ≤ i ≤ n. Then, for each 1 ≤ i ≤ n,
we have
λ ≥ qi
[
fi
(
N (m−2)q1
(m− 2)!
,
N (m−2)q2
(m− 2)!
, . . . ,
N (m−2)qn
(m− 2)!
)
N∑
`=0
Hi(`)bi(`)
]−1
and
λ ≤ qi
[
fi(M1ρ1q1,M2ρ2q2, . . . ,Mnρnqn)
N∑
`=m−1
MiHi(`)ai(`)
]−1
.
Proof. (iii) For each 1 ≤ i ≤ n, let k∗i ∈ I be such that
qi = |ui|0 = θi∆m−2ui(k∗i ), 1 ≤ i ≤ n.
Then, applying (C3), (C8) and (5.33) gives
qi = θi∆m−2ui(k∗i ) = θi∆m−2(Sui)(k∗i ) =
= θiλ
N∑
`=0
Gi(k∗i , `)Pi(`, u(`)) ≤
≤ λ
N∑
`=0
Gi(k∗i , `)bi(`)fi(u(`)) ≤
≤ λ
N∑
`=0
Hi(`)bi(`)fi
(
N (m−2)q1
(m− 2)!
,
N (m−2)q2
(m− 2)!
, . . . ,
N (m−2)qn
(m− 2)!
)
from which the first inequality is immediate.
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 35
Next, we use (C2), (C8) and (5.37) to get
qi = |ui|0 ≥
≥ θi∆m−2ui(m− 1) =
= θiλ
N∑
`=0
Gi(m− 1, `)Pi(`, u(`)) ≥
≥ λ
N∑
`=0
Gi(m− 1, `)ai(`)fi(u(`)) ≥
≥ λ
N∑
`=m−1
MiHi(`)ai(`)fi(u(`)) ≥
≥ λ
N∑
`=m−1
MiHi(`)ai(`)fi(M1ρ1q1,M2ρ2q2, . . . ,Mnρnqn)
which reduces to the second inequality.
6. Characterization ofE for (1.2). This section extends the results in Section 3 to the system
of difference equations (1.2) on the infinite set of IN = {0, 1, . . . }. To begin, let the Banach
space B = (C(IN))n be equipped with norm
‖u‖ = max
1≤i≤n
max
k∈IN
|ui(k)| = max
1≤i≤n
|ui|0 (6.1)
where we let |ui|0 = max
k∈IN |ui(k)|, 1 ≤ i ≤ n.
We shall seek a solution u = (u1, u2, . . . , un) of (1.2) in (Cl(IN))n where
(Cl(IN))n =
{
u ∈ (C(IN))n
∣∣∣∣ lim
k→∞
ui(k) exists, 1 ≤ i ≤ n
}
. (6.2)
For the purpose of clarity, we shall list the conditions that are needed later. Note that in
these conditions θi ∈ {1,−1}, 1 ≤ i ≤ n are fixed.
(C1)∞ For each 1 ≤ i ≤ n, assume that
gki (`) ≡ gi(k, `) ≥ 0, (k, `) ∈ IN× IN,
∞∑
`=0
gki (`) < ∞, k ∈ IN (i.e., gki (`) ∈ l1(IN), k ∈ IN),
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36 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
there exists g̃i ∈ l1(IN) such that lim
k→∞
∞∑
`=0
|gki (`)− g̃i(`)| = 0 (i.e., gki → g̃i in l1(IN) as k → ∞),
Pi : IN× IRn → IR is continuous,
for each r > 0, there exists Mr,i such that for k ∈ IN and |uj | ≤ r, 1 ≤ j ≤ n, |Pi(k, u)| ≤
≤ Mr,i.
(C2)∞ For each 1 ≤ i ≤ n, there exists a constant Mi ∈ (0, 1), a continuous function
Hi : IN → [0,∞), and an interval Z[a, b] ⊆ IN such that
gi(k, `) ≥ MiHi(`) ≥ 0, (k, `) ∈ Z[a, b]× IN.
(C3)∞ For each 1 ≤ i ≤ n,
gi(k, `) ≤ Hi(`), (k, `) ∈ IN× IN.
(C4)∞ Let K̃ and K be as in Section 3 with B = (C(IN))n. For each 1 ≤ i ≤ n, assume that
θiPi(`, u) ≥ 0, u ∈ K̃, ` ∈ IN and θiPi(`, u) > 0, u ∈ K, ` ∈ IN.
(C5)∞ For each 1 ≤ i ≤ n, there exist continuous functions fi, ai, bi with fi : IRn → [0,∞)
and ai, bi : IN → [0,∞) such that
ai(`) ≤
θiPi(`, u)
fi(u)
≤ bi(`), u ∈ K̃, ` ∈ IN.
(C6)∞ For each 1 ≤ i ≤ n, the function ai is not identically zero on any nondegenerate
subinterval of IN, and there exists a number 0 < ρi ≤ 1 such that
ai(`) ≥ ρibi(`), ` ∈ IN.
(C7)∞ For each 1 ≤ i, j ≤ n, if |uj | ≤ |vj |, then
θiPi(`, u1, . . . , uj−1, uj , uj+1, . . . , un) ≤ θiPi(`, u1, . . . , uj−1, vj , uj+1, . . . , un), ` ∈ IN.
(C8)∞ For each 1 ≤ i, j ≤ n, if |uj | ≤ |vj |, then
fi(u1, . . . , uj−1, uj , uj+1, . . . , un) ≤ fi(u1, . . . , uj−1, vj , uj+1, . . . , un).
Assume (C1)∞ holds. Let the operator S : (Cl(IN))n → (Cl(IN))n be defined by
Su(k) = (Su1(k), Su2(k), . . . , Sun(k)) , k ∈ IN, (6.3)
where
Sui(k) = λ
∞∑
`=0
gi(k, `)Pi(`, u(`)), k ∈ IN, 1 ≤ i ≤ n. (6.4)
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 37
Clearly, a fixed point of the operator S is a solution of the system (1.2). We shall show that S
maps (Cl(IN))n into itself. Let u ∈ (Cl(IN))n and i ∈ {1, 2, . . . , n} be fixed. We need to show
that limk→∞ Sui(k) exists. Fix r > 0. Then, it follows from (C1)∞ that∣∣∣∣∣
∞∑
`=0
[gi(k, `)− g̃i(`)]Pi(`, u(`))
∣∣∣∣∣ ≤
∞∑
`=0
|gi(k, `)− g̃i(`)|Mr,i → 0
as k → ∞. Therefore, as k → ∞ we have
Sui(k) = λ
∞∑
`=0
gi(k, `)Pi(`, u(`)) → λ
∞∑
`=0
g̃i(`)Pi(`, u(`)).
Hence, S maps (Cl(IN))n into (Cl(IN))n if (C1)∞ holds.
Next, we define a cone in B as
C =
{
u ∈ (Cl(IN))n
∣∣∣∣for each 1 ≤ i ≤ n, θiui(k) ≥ 0 for k ∈ IN,
and min
k∈Z[a,b]
θiui(k) ≥ Miρi|ui|0
}
(6.5)
where Mi and ρi are defined in (C2)∞ and (C6)∞ respectively. Note that C ⊆ K̃. A fixed point
of S obtained in C will be a constant-sign solution of the system (1.2). For R > 0, let
C(R) = {u ∈ C | ‖u‖ ≤ R}.
If (C1)∞, (C4)∞ and (C5)∞ hold, then it is clear from (6.4) that for u ∈ K̃,
λ
∞∑
`=0
gi(k, `)ai(`)fi(u(`)) ≤ θiSui(k) ≤ λ
∞∑
`=0
gi(k, `)bi(`)fi(u(`)), k ∈ IN, 1 ≤ i ≤ n. (6.6)
Lemma 6.1. Let (C1)∞ hold. Then, the operator S is continuous and completely continuous.
Proof. As in [10] (Chapter 5), (C1)∞ ensures that S is continuous and completely conti-
nuous.
Lemma 6.2. Let (C1)∞ – (C6)∞ hold. Then, the operator S maps C into itself.
Proof. The proof is similar to that of Lemma 3.2, with the intervals Z[0, N ] and I replaced
by IN.
Theorem 6.1. Let (C1)∞ – (C6)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. Then, there exists
c > 0 such that the interval (0, c] ⊆ E.
Proof. Let R > 0 be given. Define
c = R
[
max
1≤m≤n
sup
|uj | ≤ R
1 ≤ j ≤ n
fm(u1, u2, . . . , un)
] ∞∑
`=0
Hi(`)bi(`)
−1
. (6.7)
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38 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Let λ ∈ (0, c]. Using an argument similar to that in the proof of Theorem 3.1 yields
S(C(R)) ⊆ C(R). Applying Lemma 6.1 and Schauder’s fixed point theorem, we see that S has
a fixed point in C(R). Clearly, this fixed point is a constant-sign solution of (1.2) and therefore λ
is an eigenvalue of (1.2). Since λ ∈ (0, c] is arbitrary, we have proved that the interval (0, c] ⊆ E.
Theorem 6.2. Let (C1)∞, (C4)∞ and (C7)∞ hold. Suppose that λ∗ ∈ E. Then, for any
λ ∈ (0, λ∗), we have λ ∈ E, i.e., (0, λ∗] ⊆ E.
Proof. Let u∗ = (u∗1, u
∗
2, . . . , u
∗
n) be the eigenfunction corresponding to the eigenvalue λ∗,
i.e.,
u∗i (k) = λ∗
∞∑
`=0
gi(k, `)Pi(`, u∗(`)), k ∈ IN, 1 ≤ i ≤ n. (6.8)
Define
K∗ =
{
u ∈ (Cl(IN))n
∣∣∣∣ for each 1 ≤ i ≤ n, 0 ≤ θiui(k) ≤ θiu
∗
i (k), k ∈ IN
}
.
For u ∈ K∗ and λ ∈ (0, λ∗), an application of (C1)∞, (C4)∞, (C7)∞ and (6.8) gives
θiSui(k) = θi
[
λ
∞∑
`=0
gi(k, `)Pi(`, u(`))
]
≤ θi
[
λ∗
∞∑
`=0
gi(k, `)Pi(`, u∗(`))
]
=
= θiu
∗
i (k), k ∈ IN, 1 ≤ i ≤ n.
This immediately implies that S maps K∗ into K∗. Coupling with Lemma 6.1, Schauder’s fixed
point theorem guarantees that S has a fixed point in K∗, which is a constant-sign solution of
(1.2). Hence, λ is an eigenvalue, i.e., λ ∈ E.
Corollary 6.1. Let (C1)∞, (C4)∞ and (C7)∞ hold. If E 6= ∅, then E is an interval.
Proof. The argument is similar to that in the proof of Corollary 3.1, where Theorem 6.2
(instead of Theorem 3.2) is used.
We shall now establish conditions under which E is a bounded or an unbounded interval.
For this, we need the following result.
Theorem 6.3. Let (C1)∞ – (C6)∞ and (C8)∞ hold and letHibi ∈ l1(IN), 1 ≤ i ≤ n. Suppose
that λ is an eigenvalue of (1.2) and u ∈ C is a corresponding eigenfunction. Let qi = |ui|0, 1 ≤
≤ i ≤ n. Then, for each 1 ≤ i ≤ n, we have
λ ≥ qi
fi(q1, q2, . . . , qn)
[ ∞∑
`=0
Hi(`)bi(`)
]−1
(6.9)
and
λ ≤ qi
fi(M1ρ1q1,M2ρ2q2, . . . ,Mnρnqn)
[
b∑
`=a
MiHi(`)ai(`)
]−1
. (6.10)
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 39
Proof. The proof is similar to that of Theorem 3.3, with the intervals Z[0, N ] and I replaced
by IN.
Theorem 6.4. Let (C1)∞ – (C8)∞ hold and letHibi ∈ l1(IN), 1 ≤ i ≤ n.For each 1 ≤ i ≤ n,
let FBi , F
0
i and F∞i be defined as in Theorem 3.4.
(a) If fi ∈ FBi for each 1 ≤ i ≤ n, then E = (0, c) or (0, c] for some c ∈ (0,∞).
(b) If fi ∈ F 0
i for each 1 ≤ i ≤ n, then E = (0, c] for some c ∈ (0,∞).
(c) If fi ∈ F∞i for each 1 ≤ i ≤ n, then E = (0,∞).
Proof. (a) This is immediate from (6.10) and Corollary 6.1.
(b) The argument is similar to that in the proof of Theorem 3.4, with
K̃i =
{
y ∈ C(IN)
∣∣∣∣ lim
k→∞
y(k) exists and θiy(k) ≥ 0, k ∈ IN
}
.
(c) Let λ > 0 be fixed. Choose ε > 0 so that
λ max
1≤i≤n
∞∑
`=0
Hi(`)bi(`) ≤
1
ε
. (6.11)
The rest of the proof is similar to that of Theorem 3.4, with the intervals Z[0, N ] and I replaced
by IN.
7. Subintervals of E for (1.2). For each fi, 1 ≤ i ≤ n, introduced in (C5)∞, we shall define
f0,i = lim sup
max1≤j≤n |uj |→0
fi(u1, u2, . . . , un)
|ui|
, f
0,i
= lim inf
max1≤j≤n |uj |→0
fi(u1, u2, . . . , un)
|ui|
,
f∞,i = lim sup
min1≤j≤n |uj |→∞
fi(u1, u2, . . . , un)
|ui|
and f ∞,i = lim inf
min1≤j≤n |uj |→∞
fi(u1, u2, . . . , un)
|ui|
.
Theorem 7.1. Let (C1)∞–(C6)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. If λ satisfies
γ̂1,i < λ < γ̂2,i, 1 ≤ i ≤ n, (7.1)
where
γ̂1,i =
[
f ∞,i Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
and
γ̂2,i =
[
f0,i
∞∑
`=0
Hi(`)bi(`)
]−1
,
then λ ∈ E.
Proof. The proof is similar to that of Theorem 4.1, with the intervals Z[0, N ] and I replaced
by IN.
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40 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
The following corollary is immediate from Theorem 7.1.
Corollary 7.1. Let (C1)∞ – (C6)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. Then,
(γ̂1,i, γ̂2,i) ⊆ E, 1 ≤ i ≤ n,
where γ̂1,i and γ̂2,i are defined in Theorem 7.1.
Corollary 7.2. Let (C1)∞ – (C7)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. Then,(
min
1≤i≤n
γ̂1,i, max
1≤i≤n
γ̂2,i
)
⊆ E
where γ̂1,i and γ̂2,i are defined in Theorem 7.1.
Proof. This is immediate from Corollaries 7.1 and 6.1.
Theorem 7.2. Let (C1)∞ – (C6)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. If λ satisfies
γ̂3,i < λ < γ̂4,i, 1 ≤ i ≤ n, (7.2)
where
γ̂3,i =
[
f
0,i
Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
and
γ̂4,i =
[
f∞,i
∞∑
`=0
Hi(`)bi(`)
]−1
,
then λ ∈ E.
Proof. The proof is similar to that of Theorem 4.2, with the intervals Z[0, N ] and I
replaced by IN.
Theorem 7.2 leads to the following corollary.
Corollary 7.3. Let (C1)∞ – (C6)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. Then,
(γ̂3,i, γ̂4,i) ⊆ E, 1 ≤ i ≤ n,
where γ̂3,i and γ̂4,i are defined in Theorem 7.2.
Corollary 7.4. Let (C1)∞ – (C7)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. Then,(
min
1≤i≤n
γ̂3,i, max
1≤i≤n
γ̂4,i
)
⊆ E
where γ̂3,i and γ̂4,i are defined in Theorem 7.2.
Proof. This is immediate from Corollaries 7.3 and 6.1.
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 41
Remark 7.1. For a fixed i ∈ {1, 2, . . . , n}, if fi is superlinear (i.e., f0,i = 0 and f ∞,i = ∞)
or sublinear (i.e., f
0,i
= ∞ and f∞,i = 0), then we conclude from Corollaries 7.1 and 7.3 that
E = (0,∞), i.e., (1.2) has a constant-sign solution for any λ > 0.
8. Characterization of E for (1.3). Let the Banach space B = (C(I))n be equipped with
norm ‖ · ‖ as given in (3.2). Define the operator S : B → B by (3.3) where
Sui(k) = λi
N∑
`=0
gi(k, `)Pi(`, u(`)), k ∈ I, 1 ≤ i ≤ n. (8.1)
Clearly, a fixed point of the operator S is a solution of the system (1.3).
Next, with the conditions (C1) – (C8) stated as in Section 3 and the cone C defined as in
(3.5), it is obvious that a fixed point of S obtained in C or K̃ will be a constant-sign solution of
the system (1.3).
If (C1), (C4) and (C5) hold, then it is clear from (8.1) that for u ∈ K̃,
λi
N∑
`=0
gi(k, `)ai(`)fi(u(`)) ≤ θiSui(k) ≤ λi
N∑
`=0
gi(k, `)bi(`)fi(u(`)), k ∈ I, 1 ≤ i ≤ n. (8.2)
Using similar arguments as in Section 3, we obtain the following results.
Lemma 8.1. Let (C1) hold. Then, the operator S is continuous and completely continuous.
Lemma 8.2. Let (C1) – (C6) hold. Then, the operator S maps C into itself.
Theorem 8.1. Let (C1) – (C6) hold. Then, there exist ci > 0, 1 ≤ i ≤ n, such that
(0, c1]× (0, c2]× . . .× (0, cn] ⊆ E.
Proof. Let R > 0 be given. For each 1 ≤ i ≤ n, define
ci = R
[
max
1≤m≤n
sup
|uj | ≤ R
1 ≤ j ≤ n
fm(u1, u2, . . . , un)
] N∑
`=0
Hi(`)bi(`)
−1
.
Let λi ∈ (0, ci], 1 ≤ i ≤ n. Using a similar technique as in the proof of Theorem
3.1, we can show that S(C(R)) ⊆ C(R). Also, from Lemma 8.1 the operator S is conti-
nuous and completely continuous. Schauder’s fixed point theorem guarantees that S has a fi-
xed point in C(R). Clearly, this fixed point is a constant-sign solution of (1.3) and therefore
λ = (λ1, λ2, . . . , λn) is an eigenvalue of (1.3). Since λi ∈ (0, ci] is arbitrary, we have proved that
(0, c1]× (0, c2]× . . .× (0, cn] ⊆ E.
Theorem 8.2. Let (C1), (C4) and (C7) hold. Suppose that (λ∗1, λ
∗
2, . . . , λ
∗
n) ∈ E. Then, for
any λi ∈ (0, λ∗i ), 1 ≤ i ≤ n, we have (λ1, λ2, . . . , λn) ∈ E, i.e.,
(0, λ∗1]× (0, λ∗2]× . . .× (0, λ∗n] ⊆ E.
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42 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Proof. Let u∗ = (u∗1, u
∗
2, . . . , u
∗
n) be the eigenfunction corresponding to the eigenvalue λ∗ =
= (λ∗1, λ
∗
2, . . . , λ
∗
n). Thus, we have
u∗i (k) = λ∗i
N∑
`=0
gi(k, `)Pi(`, u∗(`)), k ∈ I, 1 ≤ i ≤ n.
Define K∗ as in the proof of Theorem 3.2. For u ∈ K∗ and λi ∈ (0, λ∗i ), 1 ≤ i ≤ n, it follows
that
θiSui(k) = θi
[
λi
N∑
`=0
gi(k, `)Pi(`, u(`))
]
≤ θi
[
λ∗i
N∑
`=0
gi(k, `)Pi(`, u∗(`))
]
=
= θiu
∗
i (k), k ∈ I, 1 ≤ i ≤ n.
Hence, we have shown that S(K∗) ⊆ K∗. Moreover, from Lemma 8.1 the operator S is conti-
nuous and completely continuous. Schauder’s fixed point theorem guarantees that S has a fixed
point in K∗, which is a constant-sign solution of (1.3). Hence, λ = (λ1, λ2, . . . , λn) is an ei-
genvalue of (1.3).
Theorem 8.3. Let (C1) – (C6) and (C8) hold. Suppose that (λ1, λ2, . . . , λn) is an eigenvalue
of (1.3) and u ∈ C is a corresponding eigenfunction. Let qi = |ui|0, 1 ≤ i ≤ n. Then, for each
1 ≤ i ≤ n, we have
λi ≥
qi
fi(q1, q2, . . . , qn)
[
N∑
`=0
Hi(`)bi(`)
]−1
(8.3)
and
λi ≤
qi
fi(M1ρ1q1,M2ρ2q2, . . . ,Mnρnqn)
[
b∑
`=a
MiHi(`)ai(`)
]−1
. (8.4)
Theorem 8.4. Let (C1) – (C6) and (C8) hold. For each 1 ≤ i ≤ n, define F∞i as in Theorem
3.4. If fi ∈ F∞i for each 1 ≤ i ≤ n, then E = (0,∞)n.
Proof. Fix λ = (λ1, λ2, . . . , λn) ∈ (0,∞)n. Choose ε > 0 so that for each 1 ≤ i ≤ n,
λi max
1≤j≤n
N∑
`=0
Hj(`)bj(`) ≤
1
ε
. (8.5)
By definition, if fi ∈ F∞i , 1 ≤ i ≤ n, then there exists R = R(ε) > 0 such that the following
holds for each 1 ≤ i ≤ n:
fi(u1, u2, . . . , un) < ε|ui|, |uj | ≥ R, 1 ≤ j ≤ n. (8.6)
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 43
We shall prove that S(C(R)) ⊆ C(R). To begin, let u ∈ C(R).By Lemma 8.2, we have Su ∈ C.
Thus, it remains to show that ‖Su‖ ≤ R. Using (8.2), (C3), (C8), (8.6) and (8.5), we find for
k ∈ I and 1 ≤ i ≤ n,
|Sui(k)| = θiSui(k) ≤
≤ λi
N∑
`=0
Hi(`)bi(`)fi(u(`)) ≤
≤ λifi(R,R, . . . , R)
N∑
`=0
Hi(`)bi(`) ≤
≤ λi(εR)
N∑
`=0
Hi(`)bi(`) ≤ R.
It follows that ‖Su‖ ≤ R and hence S(C(R)) ⊆ C(R). From Lemma 8.1 the operator S is
continuous and completely continuous. Schauder’s fixed point theorem guarantees that S has
a fixed point in C(R). Clearly, this fixed point is a constant-sign solution of (1.3) and therefore
λ = (λ1, λ2, . . . , λn) is an eigenvalue of (1.3). Since λ ∈ (0,∞)n is arbitrary, we have proved
that E = (0,∞)n.
9. Subintervals of E for (1.3). Define f0,i, f0,i
, f∞,i and f ∞,i as in Section 4. Using similar
arguments as in Section 4, we obtain the following results.
Theorem 9.1. Let (C1) – (C6) hold. For each 1 ≤ i ≤ n, if λi satisfies
γ1,i < λi < γ2,i, (9.1)
where
γ1,i =
[
f ∞,i Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
and
γ2,i =
[
f0,i
N∑
`=0
Hi(`)bi(`)
]−1
,
then (λ1, λ2, . . . , λn) ∈ E.
Corollary 9.1. Let (C1) – (C6) hold. Then,
(γ1,1, γ2,1)× (γ1,2, γ2,2)× . . .× (γ1,n, γ2,n) ⊆ E
where γ1,i and γ2,i are defined in Theorem 9.1.
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44 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
Theorem 9.2. Let (C1) – (C6) hold. For each 1 ≤ i ≤ n, if λi satisfies
γ3,i < λi < γ4,i (9.2)
where
γ3,i =
[
f
0,i
Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
and
γ4,i =
[
f∞,i
N∑
`=0
Hi(`)bi(`)
]−1
,
then (λ1, λ2, . . . , λn) ∈ E.
Corollary 9.2. Let (C1) – (C6) hold. Then,
(γ3,1, γ4,1)× (γ3,2, γ4,2)× . . .× (γ3,n, γ4,n) ⊆ E
where γ3,i and γ4,i are defined in Theorem 9.2.
Remark 9.1. For each 1 ≤ i ≤ n, if fi is superlinear (i.e., f0,i = 0 and f ∞,i = ∞) or
sublinear (i.e., f
0,i
= ∞ and f∞,i = 0), then we conclude from Corollaries 9.1 and 9.2 that
E = (0,∞)n, i.e., (1.3) has a constant-sign solution for any λi > 0, 1 ≤ i ≤ n.
10. Characterization of E for (1.4). Let the Banach space B = (C(IN))n be equipped with
norm ‖ · ‖ as given in (6.1). With (Cl(IN))n given in (6.2), define the operator S : (Cl(IN))n →
→ (Cl(IN))n by (6.3) where
Sui(k) = λi
∞∑
`=0
gi(k, `)Pi(`, u(`)), k ∈ IN, 1 ≤ i ≤ n. (10.1)
Clearly, a fixed point of the operator S is a solution of the system (1.4).
Next, with the conditions (C1)∞ – (C8)∞ stated as in Section 6 and the cone C defined as in
(6.5), it is obvious that a fixed point of S obtained in C will be a constant-sign solution of the
system (1.4).
If (C1)∞, (C4)∞ and (C5)∞ hold, then it is clear from (10.1) that for u ∈ K̃,
λi
∞∑
`=0
gi(k, `)ai(`)fi(u(`)) ≤ θiSui(k) ≤ λi
∞∑
`=0
gi(k, `)bi(`)fi(u(`)), k ∈ IN, 1 ≤ i ≤ n.
(10.2)
Using similar arguments as in Section 6, we obtain the following results.
Lemma 10.1. Let (C1)∞ hold. Then, the operator S is continuous and completely continuous.
Lemma 10.2. Let (C1)∞ – (C6)∞ hold. Then, the operator S maps C into itself.
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EIGENVALUE CHARACTERIZATION OF A SYSTEM OF DIFFERENCE EQUATIONS 45
Theorem 10.1. Let (C1)∞ – (C6)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. Then, there exist
ci > 0, 1 ≤ i ≤ n, such that
(0, c1]× (0, c2]× . . .× (0, cn] ⊆ E.
Proof. Let R > 0 be given. For each 1 ≤ i ≤ n, define
ci = R
[
max
1≤m≤n
sup
|uj | ≤ R
1 ≤ j ≤ n
fm(u1, u2, . . . , un)
] ∞∑
`=0
Hi(`)bi(`)
−1
.
The rest of the proof is similar to that of Theorem 8.1.
Theorem 10.2. Let (C1)∞, (C4)∞ and (C7)∞ hold. Suppose that (λ∗1, λ
∗
2, . . . , λ
∗
n) ∈ E. Then,
for any λi ∈ (0, λ∗i ), 1 ≤ i ≤ n, we have (λ1, λ2, . . . , λn) ∈ E, i.e.,
(0, λ∗1]× (0, λ∗2]× . . .× (0, λ∗n] ⊆ E.
Proof. The proof is similar to that of Theorem 8.2, with K∗ defined as in Theorem 6.2.
Theorem 10.3. Let (C1)∞ – (C6)∞ and (C8)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n.
Suppose that (λ1, λ2, . . . , λn) is an eigenvalue of (1.4) and u ∈ C is a corresponding eigenfuncti-
on. Let qi = |ui|0, 1 ≤ i ≤ n. Then, for each 1 ≤ i ≤ n, we have
λi ≥
qi
fi(q1, q2, . . . , qn)
[ ∞∑
`=0
Hi(`)bi(`)
]−1
(10.3)
and
λi ≤
qi
fi(M1ρ1q1,M2ρ2q2, . . . ,Mnρnqn)
[
b∑
`=a
MiHi(`)ai(`)
]−1
. (10.4)
Theorem 10.4. Let (C1)∞ – (C6)∞ and (C8)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. For
each 1 ≤ i ≤ n, define F∞i as in Theorem 3.4. If fi ∈ F∞i for each 1 ≤ i ≤ n, thenE = (0,∞)n.
The proof is similar to that of Theorem 8.4, where the intervals Z[0, N ] and I are replaced
by IN, and Lemmas 10.1 and 10.2 are used instead of Lemmas 8.1 and 8.2.
11. Subintervals ofE for (1.4). Define f0,i, f0,i
, f∞,i and f ∞,i as in Section 7. Using similar
arguments as in Section 7, we obtain the following results.
Theorem 11.1. Let (C1)∞ – (C6)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. For each 1 ≤ i ≤
≤ n, if λi satisfies
γ̂1,i < λi < γ̂2,i (11.1)
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46 R.P. AGARWAL, D. O’REGAN, P.J.Y. WONG
where
γ̂1,i =
[
f ∞,i Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
and
γ̂2,i =
[
f0,i
∞∑
`=0
Hi(`)bi(`)
]−1
,
then (λ1, λ2, . . . , λn) ∈ E.
Corollary 11.1. Let (C1)∞ – (C6)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. Then,
(γ̂1,1, γ̂2,1)× (γ̂1,2, γ̂2,2)× . . .× (γ̂1,n, γ̂2,n) ⊆ E
where γ̂1,i and γ̂2,i are defined in Theorem 11.1.
Theorem 11.2. Let (C1)∞ – (C6)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. For each 1 ≤ i ≤
≤ n, if λi satisfies
γ̂3,i < λi < γ̂4,i (11.2)
where
γ̂3,i =
[
f
0,i
Miρi
b∑
`=a
MiHi(`)ai(`)
]−1
and
γ̂4,i =
[
f∞,i
∞∑
`=0
Hi(`)bi(`)
]−1
,
then (λ1, λ2, . . . , λn) ∈ E.
Corollary 11.1. Let (C1)∞ – (C6)∞ hold and let Hibi ∈ l1(IN), 1 ≤ i ≤ n. Then,
(γ̂3,1, γ̂4,1)× (γ̂3,2, γ̂4,2)× . . .× (γ̂3,n, γ̂4,n) ⊆ E
where γ̂3,i and γ̂4,i are defined in Theorem 11.2.
Remark 11.1. For each 1 ≤ i ≤ n, if fi is superlinear (i.e., f0,i = 0 and f ∞,i = ∞) or
sublinear (i.e., f
0,i
= ∞ and f∞,i = 0), then we conclude from Corollaries 11.1 and 11.2 that
E = (0,∞)n, i.e., (1.4) has a constant-sign solution for any λi > 0, 1 ≤ i ≤ n.
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