Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 809497, 15 pages
doi:10.1155/2010/809497
Research Article
Existence of Positive Solutions to a Nonlocal
Boundary Value Problem with p-Laplacian on
Time Scales
Ting-Ting Sun, Lin-Lin Wang, and Yong-Hong Fan
School of Mathematics and Information, Ludong University, Yantai, Shandong 264025, China
Correspondence should be addressed to Lin-Lin Wang, wangll 1994@sina.com
Received 6 October 2009; Accepted 19 January 2010
Academic Editor: Paul Eloe
Copyright q 2010 Ting-Ting Sun et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The nonlocal boundary value problem, with p-Laplacian of the form Φp u thtft, ut 0, t ∈ t1 , tm T , u t1 − nj1 θj u ηj − m−2
i1 i uξi 0, u tm 0, has been considered. Two
existence criteria of at least one and three positive solutions are presented. The first one is based
on the Four functionals fixed point theorem in the work of R. Avery et al. 2008, and the second
one is based on the Five functionals fixed point theorem. Meanwhile an example is worked out to
illustrate the main result.
1. Introduction
Due to the unification of the theory of differential and difference equations, there have
been many investigations working on the existence of positive solutions to boundary value
problems for dynamic equations on time scales. Also there is much attention paid to the study
of multipoint boundary value problem with p-Laplacian; see 1–10.
For convenience, throughout this paper we denote Φp s as the p-Laplacian operator,
that is, Φp s |s|p−2 s, p > 1. Φp −1 Φq , where 1/p 1/q 1.
In 11, the author discussed the positive solutions of a m-point boundary value
problem for a second-order dynamic equation on a time scale
u t qtfut 0,
u 0 m−2
bi u ξi ,
i1
t ∈ 0, T T ,
uT m−2
ai uξi ,
i1
1.1
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where ai , bi ≥ 0 i 1, 2, . . . , m − 2, and ξi ∈ 0, ρT T with 0 < ξ1 < ξ2 < · · · < ξm−2 < ρT .
And he got the existence of at least two positive solutions of the above problem by means of
a fixed point theorem in a cone.
Zhao and Ge 9 considered the following multi-point boundary value problem with
one-dimensional p-Laplacian:
Φp x t ft, xt 0,
x 0 −
m−1
0 < t < 1,
m−1
βi x ηi 0,
x 1 ai xξi 0,
i1
1.2
i1
m−1
where ai , βi > 0, 0 < m−1
i1 ai ξi ≤ 1, 0 <
i1 βi 1 − ηi ≤ 1, i 1, 2, . . . , m − 1, 0 ξ1 < ξ2 < · · · <
ξm−1 < η1 < η2 < · · · < ηm−1 1. By using a fixed point theorem in a cone, they obtained the
existence of at least one, two, or three positive solutions under some sufficient conditions.
Motivated by the above results, in this paper, we investigate the nonlocal boundary
value problem with p-Laplacian
Φp u u t1 −
t htft, ut 0,
t ∈ t1 , tm T ,
n
m−2
θj u ηj −
i uξi 0,
j1
i1
u tm 0,
1.3
where 0 ≤ t1 < ξ1 < ξ2 < · · · < ξm−2 < tm and t1 < η1 < η2 < · · · < ηn < tm < ∞.
For convenience, we list the following hypotheses:
n
H1 i > 0, i 1, 2, . . . , m − 2, θj ≥ 0, j 1, 2, . . . , n, m−2
j1 θj < 1;
i1 i ξi H2 ft, u ∈ Ct1 , tm T × 0, ∞, 0, ∞ and f is not identically zero on any compact
subinterval of t1 , tm T × 0, ∞;
H3 ht ∈ Crd t1 , tm T , 0, ∞ and h is not identically zero on any compact
subinterval of t1 , tm T , also it satisfies
Φq
t
m
hττ
t1
< ∞,
σtm t1
Φq
t
m
hττ s < ∞.
1.4
s
By using the Four functionals fixed point theorem and Five functionals fixed point
theorem, we obtain the existence criteria of at least one positive solution and three positive
solutions for the BVP 1.3. As an application, an example is worked out finally. The
remainder of this paper is organized as follows. Section 2 is devoted to some preliminary
discussions. We give and prove our main results in Section 3.
2. Preliminaries
The basic definitions and notations on time scales can be found in 12, 13. In the following,
we will provide some background materials on the theory of cones in Banach spaces. For
more details, please refer to 14, 15.
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3
Definition 2.1. Let E be a Banach space. A nonempty, closed set P ⊂ E is said to be a cone
provided that the following hypotheses are satisfied:
1 if x, y ∈ P , α, β ≥ 0, then αx βy ∈ P ;
2 if x ∈ P , x / θ, then x ∈ P.
Every cone P ⊂ E induces a partial ordering “≤” on E defined by x ≤ y if and only if
y − x ∈ P.
Definition 2.2. A map α is said to be a nonnegative continuous concave functional on a cone P
of a real Banach space E if α : P → 0, ∞ is continuous and αtx1−ty ≥ tαx1−tαy
for all x, y ∈ P and t ∈ 0, 1. Similarly, we say that the map β is a nonnegative continuous
convex functional on a cone P of a real Banach space E if β : P → 0, ∞ is continuous and
βtx 1 − ty ≤ tβx 1 − tβy for all x, y ∈ P and t ∈ 0, 1.
Let α and Ψ be nonnegative continuous concave functionals on P , and let β and θ be
nonnegative continuous convex functionals on P ; then for positive numbers r, ι, υ, and R,
define the sets
Q α, β, r, R x ∈ P : r ≤ αx, βx ≤ R ,
UΨ, ι x ∈ Q α, β, r, R : ι ≤ Ψx ,
V θ, υ x ∈ Q α, β, r, R : θx ≤ υ .
2.1
The following lemma can be found in 16.
Lemma 2.3 four functionals fixed point theorem. If P is a cone in a real Banach space E, α and
Ψ are nonnegative continuous concave functionals on P , β, and θ are nonnegative continuous convex
functionals on P , and there exist nonnegative positive numbers r, ι, υ, and R, such that
A : Q α, β, r, R −→ P
is a completely continuous operator, and Qα, β, r, R is a bounded set. If
i {x ∈ UΨ, ι : βx < R} ∩ {x ∈ V θ, υ : r < αx} / ∅,
ii αAx ≥ r, for all x ∈ Qα, β, r, R, with αx r and υ < θAx,
iii αAx ≥ r, for all x ∈ V θ, υ, with αx r,
iv βAx ≤ R, for all x ∈ Qα, β, r, R, with βx R and ΨAx < ι,
v βAx ≤ R, for all x ∈ UΨ, ι, with βx R,
then A has a fixed point x in Qα, β, r, R.
2.2
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We are now in a position to present the Five functionals fixed point theorem see
17. Let γ, β, θ be nonnegative continuous convex functionals on P and α, ϕ nonnegative
continuous concave functionals on P . For nonnegative numbers h, a, b, d, and c, define the
following convex sets:
P γ, c x ∈ P : γx < c ,
P γ, α, a, c x ∈ P : a ≤ αx, γx ≤ c ,
Q γ, β, d, c x ∈ P : βx ≤ d, γx ≤ c ,
P γ, θ, α, a, b, c x ∈ P : a ≤ αx, θx ≤ b, γx ≤ c ,
Q γ, β, ϕ, h, d, c x ∈ P : h ≤ ϕx, βx ≤ d, γx ≤ c .
2.3
Lemma 2.4 five functionals fixed point theorem. Let P be a cone in a real Banach space E.
Suppose that there exist nonnegative numbers c and M, nonnegative continuous concave functionals
α and ϕ on P , and nonnegative continuous convex functionals γ, β, and θ on P , with
αx ≤ βx,
∀x ∈ P γ, c .
x ≤ Mγx
2.4
Suppose that A : P γ, c → P γ, c is completely continuous and there exist nonnegative numbers
h, a, k, b, with 0 < a < b such that
i {x ∈ P γ, θ, α, b, k, c : αx > b} /
∅ and αAx > b for x ∈ P γ, θ, α, b, k, c,
ii {x ∈ Qγ, β, ϕ, h, a, c : βx < a} / ∅ and βAx < a for x ∈ Qγ, β, ϕ, h, a, c,
iii αAx > b for x ∈ P γ, α, b, c with θAx > k,
iv βAx < a for x ∈ Qγ, β, a, c with ϕAx < h,
then A has at least three fixed points x1 , x2 , x3 ∈ P γ, c such that
βx1 < a,
αx2 > b,
βx3 > a
2.5
with αx3 < b.
Consider the Banach space E Ct1 , σtm T equipped with the norm u maxt∈t1 ,σtm T |ut|. Suppose , η ∈ T with t1 < < η < σtm . For the sake of convenience, we
take the notations
m−2
i ,
h0 Φq
t
m
hττ ,
M Φq
t1
η
t1
Φq
t
m
s
Φq
η
t1
t1
i1
Mη M0 t
m
hττ s,
2.6
hττ s,
s
hττ s,
Mσtm σtm t1
Φq
t
m
s
hττ s.
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5
Define a cone
P
⎧
⎨
u ∈ E : ut ≥ 0, u t1 −
⎩
n
m−2
θj u ηj −
i uξi 0,
j1
i1
2.7
for t ∈ t1 , σtm T and u t ≤ 0, u t ≥ 0 for t ∈ t1 , tm T , u tm 0
and an operator A : P → E by
Au Φq
tm
hτfτ, uττ
t1
−
tm
n
−
j1
θj Φq
m−2 ξi tm
hτfτ, uττ s
i1 i t1 Φq
s
hτfτ, uττ
ηj
t
Φq
t
m
t1
2.8
hτfτ, uττ s.
s
Lemma 2.5. A : P → P .
Proof. For u ∈ P , t ∈ t1 , σtm T ,
Aut ≥
1−
n
j1
θj −
m−2
i1
i ξi
t
Φq
t
m
t1
Φq
t
m
hτfτ, uττ
t1
2.9
hτfτ, uττ s
s
≥ 0.
From the definition of A, it is clear that
Au t Φq
t
m
hsfs, uss
≥ 0,
t
t ∈ t1 , tm T ,
2.10
is continuous, Au t1 − nj1 θj Au ηj − m−2
i1 i Auξi 0, and Auσtm is the
maximum value of Aut on t1 , σtm T .
t
Let gt tm hsfs, uss, then g : R → R is continuous, g : T → R is delta
differentiable on t1 , tm Tk , and Φq : R → R is continuously differentiable. Moreover Φq s is
monotonically increasing for s ≥ 0 and g t −htft, ut ≤ 0. Then by the chain rule 12,
Theorem 1.87, page 31, we obtain
Au t Φq gc g t ≤ 0,
where c is in the interval t, σt. So, A : P → P . This completes the proof.
2.11
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3. Main Results and an Example
Theorem 3.1. Assume that (H1), (H2), and (H3) hold, if there exist constants r, ι, υ, R with R >
max{h0 Mσtm /M0 , σtm −t1 /−t1 }ι, υ ≥ max{h0 Mη /M ι, η−t1 /−
t1 r}, r < ι and suppose that ft, u satisfies the following conditions:
A1 ft, u ≤ Φp R/h0 Mσtm for all t, u ∈ t1 , tm T × 0, R,
A2 ft, u ≥ Φp r/M0 for all t, u ∈ , ηT × r, υ,
then the BVP 1.3 has a fixed point u ∈ P such that
min ut ≥ r,
t∈,ηT
max
t∈t1 ,σtm T
ut ≤ R.
3.1
Define maps
αu Ψu min ut,
t∈,ηT
θu max ut,
t∈,ηT
βu max
t∈t1 ,σtm T
ut,
3.2
and let Qα, β, r, R, UΨ, ι and V θ, υ be defined by 2.1.
In order to complete the proof of Theorem 3.1, we first need to prove the following
lemma.
Lemma 3.2. Qα, β, r, R is bounded and A : Qα, β, r, R → P is completely continuous.
Proof. For all u ∈ Qα, β, r, R, u maxt∈t1 ,σtm T |ut| βu ≤ R, which means that
Qα, β, r, R is a bounded set.
According to Lemma 2.5, it is clear that A : Qα, β, r, R → P .
In view of the continuity of f, there exists a constant C > 0 such that ft, u < Φp C,
for all t ∈ t1 , σtm T , u ∈ Qα, β, r, R. Consider
Au Auσtm ≤
h0
Mσtm C,
3.3
which means that AQα, β, r, R is uniformly bounded.
In addition, for all t1 ≤ t ≤ t∗ ≤ σtm , we have
t
t∗
m
∗ hτfτ, uττ s ≤ Ch0 t − t∗ .
Au t − Aut Φq
t
s
3.4
Applying the Arzelà-Ascoli theorem on time scales 18, one can show that AQα, β, r, R is
relatively compact.
Now we prove that A : Qα, β, r, R → P is continuous. Let {un }n∈N be a sequence
in Qα, β, r, R which converges to u0 ∈ Qα, β, r, R uniformly on t1 , σtm T . Because
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AQα, β, r, R is relatively compact, the sequence {Aun } admits a subsequence {Aunm }
converging to vt uniformly on t1 , σtm T . In addition,
0 ≤ Aun t ≤ C
h0
Mσtm .
ω
3.5
Observe that
Φq
Aun t tm
hτfτ, un ττ
t1
n
j1
−
θj Φq
tm
ηj
−
m−2 ξi tm
hτfτ, un ττ s
i1 i t1 Φq
s
hτfτ, un ττ
t
Φq
t
m
t1
3.6
hτfτ, un ττ s.
s
Hence, by the Lebesgue’s dominated convergence theorem on time scales 19, insert unm into
the above equality and then let m → ∞, we obtain
vt Φq
tm
hτfτ, u0 ττ
t1
n
−
j1
θj Φq
tm
−
m−2 ξi tm
hτfτ, u0 ττ s
i1 i t1 Φq
s
hτfτ, u0 ττ
ηj
t
Φq
t
t1
m
3.7
hτfτ, u0 ττ s.
s
From the definition of A, we know that vt Au0 t on t1 , σtm T . This shows that
each subsequence of {Aun t}∞
n1 uniformly converges to Au0 t. Therefore the sequence
uniformly
converges
to Au0 t. This means that A is continuous at u0 ∈
{Aun t}∞
n1
Qα, β, r, R. So, A is continuous on Qα, β, r, R since u0 is arbitrary. Thus, A : Qα, β, r, R →
P is completely continuous. This completes the proof.
Proof of Theorem 3.1. Let
⎛
u0 ι ⎜
⎝
M
Φq
⎛
−
ι ⎜
⎝
M
tm
hττ
t1
n
j1
θj Φq
−
⎞
m−2 ξi tm
hττ
s
i1 i t1 Φq
s
⎟
⎠
tm
hττ
ηj
⎞
ι
⎟
⎠
M
t
t1
Φq
t
m
s
hττ s.
3.8
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Clearly, u0 ∈ P . By direct calculation,
⎛
Ψu0 ι ⎜
⎝
M
Φq
⎛
ι ⎜
−
⎝
M
ι
≥
M
βu0 ι ⎜
⎝
M
Φq
t
m
Φq
⎛
ι ⎜ Φq
<
⎝
M
⎛
ι ⎜ Φq
⎝
M
⎛
αu0 ι ⎜
⎝
M
t1
ι
⎟
⎠
M
Φq
θj Φq
Φq
t
m
t1
hττ s
s
⎞
m−2 ξi tm
Φ
hττ
s
i
q
i1
t1
s
⎟
⎠
tm
hττ
ηj
tm
hτΔτ
t1
⎞
ι
⎟
⎠
M
tm
hττ
t1
n
j1
θj Φq
σtm Φq
tm
hττ
t1
Φq
θj Φq
tm
hττ
ηj
m
s
Φq
t
m
t1
hττ s
s
⎞
⎟
hττ s⎠ ≤ R,
3.9
⎞
ξi tm
− m−2
hττ
s
i1 i t1 Φq
s
⎟
⎠
⎞
ι
⎟
⎠
M
η
Φq
t
m
−
η
Φq
t1
t
m
hττ s
s
⎞
⎟
hττ s⎠ ≤ υ,
s
⎞
m−2 ξi tm
Φ
hττ
s
i t1 q
i1
s
⎟
⎠
tm
hττ
ηj
⎞
t
m
σtm s
t1
tm
hττ
t1
n
j1
t
t1
Φq
−
ι ⎜
−
⎝
M
⎞
hττ s ι,
⎛
ι
>
M
tm
hττ
ηj
tm
hττ
t1
n
j1
ι ⎜
−
⎝
M
ι ⎜
⎝
M
⎞
m−2 ξi tm
Φ
hττ
s
i t1 q
i1
s
⎟
⎠
⎛
≤
−
s
ι ⎜
−
⎝
M
⎛
θj Φq
⎛
θu0 n
j1
t1
⎛
tm
hττ
t1
hττ s > r.
ι
⎟
⎠
M
t1
Φq
t
m
s
hττ s
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So, u0 ∈ {u ∈ UΨ, ι : βu < R}∩{u ∈ V θ, υ : r < αu}, which means that i in Lemma 2.3
is satisfied.
For all u ∈ Qα, β, r, R, with αu r and υ < θAu, we have αAu Au ≥
− t1 /η − t1 Auη − t1 /η − t1 θAu > − t1 /η − t1 υ > r, and for all
u ∈ Qα, β, r, R, with βu R and ΨAu < ι, we obtain that βAu Auσtm ≤ σtm −
t1 / − t1 Au σtm − t1 / − t1 ΨAu < σtm − t1 / − t1 ι < R. Hence, ii
and iv in Lemma 2.3 are fulfilled.
For any u ∈ V θ, υ, with αu r,
αAu Φq
tm
hτfτ, uττ
t1
n−2
j1
−
≥
θj Φq
−
m−2 ξi tm
hτfτ, uττ s
i1 i t1 Φq
s
tm
hτfτ, uττ
ηj
Φq
η
t1
Φq
t
m
t1
hτfτ, uττ s ≥
Φq
hτfτ, uττ s
3.10
s
η
hτΦp
t1
r
τ s r,
M0
and for all u ∈ UΨ, ι, with βu R,
βAu Φq
tm
hτfτ, uττ
t1
n−2
j1
−
≤
−
m−2 ξi tm
hτfτ,
uττ
s
i1 i t1 Φq
s
tm
hτfτ, uττ
ηj
Φq
θj Φq
t1
Φq
t
Mσtm m
hτΦp
s
Φq
t
t1
tm
hτΦp R/ h0
t1
σtm η
τ
m
hτfτ, uττ s
s
3.11
R
τ s
h0 Mσtm R.
Thus iii and v in Lemma 2.3 hold true. So, by Lemma 2.3, the BVP 1.3 has a fixed point
u in Qα, β, r, R. This completes the proof.
Theorem 3.3. Assume that (H1), (H2), and (H3) hold. If there exist constants h, a, b, c, k, with b <
aM0 /h0 Mη , c > h0 Mσtm /h0 Mη a, k < h0 Mη /h0 Mσtm c,
k ≥ max{η −t1 /−t1 , h0 Mη /M }b, a ≥ max{η −t1 /−t1 , h0 Mη /M }h,
further suppose that ft, u satisfies the following conditions:
B1 ft, u < Φp a/h0 Mη for all t, u ∈ t1 , tm T × 0, c,
B2 ft, u ≥ Φp b/M0 for all t, u ∈ , ηT × b, k,
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then the BVP 1.3 has at least three positive solutions u1 , u2 and u3 such that
max u1 t < a < max u3 t,
t∈,ηT
min u3 t < b < min u2 t.
t∈,ηT
t∈,ηT
3.12
t∈,ηT
Proof. Define these maps
βu θu max ut,
αu ϕu min ut,
t∈,η
γu t∈,η
max ut,
t∈t1 ,σtm 3.13
and let P γ, c, P γ, α, b, c, Qγ, β, a, c, P γ, θ, α, b, k, c and Qγ, β, ϕ, h, a, c be defined by
2.3. It is clear that
αu ≤ βu,
3.14
∀u ∈ P γ, c.
u ≤ γu,
Using similar methods as those in Lemma 3.2, we obtain that A : P γ, c → P is
completely continuous. Thus, we only need to show that A : P γ, c → P γ, c. Let u ∈
P γ, c, then
γAu Φq
tm
hτfτ, uττ
t1
n
j1
−
≤
≤
tm
θj Φq
ηj
−
hτfτ, uττ
Φq
Φq
tm
hτfτ, uττ
t1
t1
Φq
σtm t
m
hτΦp
s
Φq
Φq
Mη
m
τ
a
h0 Mη
≤ c,
which implies that AP γ, c ⊂ P γ, c.
hτfτ, uττ s
hτfτ, uττ s
m
s
t
s
t
t1
tm
hτΦp a/ h0
t1
σtm σtm t1
m−2 ξi tm
Φ
hτfτ,
uττ
s
i
q
i1
t1
s
τ s
3.15
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Let N h0 / Mη and
⎛
b ⎜
⎝
M
u1 Φq
⎛
b ⎜
−
⎝
M
⎛
a⎜
⎝
N
u2 Φq
n
j1
θj Φq
−
⎞
m−2 ξi tm
Φ
hττ
s
i
q
i1
t1
s
⎟
⎠
tm
hττ
ηj
⎞
b
⎟
⎠
M
⎛
a⎜
− ⎝
N
tm
hττ
t1
tm
hττ
t1
n
j1
θj Φq
−
t
Φq
t
m
t1
hττ s,
s
3.16
⎞
m−2 ξi tm
Φ
hττ
s
i t1 q
i1
s
⎟
⎠
tm
hττ
ηj
⎞
⎟ a
⎠
N
t
Φq
t
m
t1
hττ s,
s
we can verify that u1 , u2 ∈ P . By calculation,
⎛
αu1 b ⎜
⎝
M
Φq
⎛
b ⎜
−
⎝
M
⎛
≥
b ⎝
M
b
>
M
−
⎞
m−2 ξi tm
Φ
hττ
s
i
q
i1
t1
s
⎟
⎠
tm
hττ
ηj
n
j1 θj −
m−2
i1
i ξi
Φq
⎞
b
⎟
⎠
M
t
m
Φq
Φq
t
m
t1
hττ s
s
t
m
hττ
Φq
t
t1
t1
m
hττ s b,
⎛
b ⎜
−
⎝
M
Φq
tm
hττ
t1
n
j1
θj Φq
−
⎞
m−2 ξi tm
hττ s
i1 i t1 Φq
s
⎟
⎠
tm
hττ
ηj
⎞
tm
hττ
t1
η
t1
Φq
b
⎟
⎠
M
t
m
s
η
Φq
t1
t
m
hττ s
s
⎞
⎟
hττ s⎠ ≤ k,
⎞
hττ s⎠
s
s
⎛
b ⎜
⎝
M
θj Φq
1−
b ⎜ Φq
θu1 ⎝
M
≤
n
j1
t1
⎛
tm
hττ
t1
12
Advances in Difference Equations
⎛ ⎞
ξi tm
tm
m−2
b ⎜ Φq t1 hττ − i1 i t1 Φq s hττ s ⎟
γu1 ⎠
⎝
M
⎛
b ⎜
−
⎝
M
⎛
≤
b ⎜
⎝
M
⎛
βu2 a⎜
⎝
N
Φq
a⎜
⎝
N
⎛
ϕu2 a⎜
⎝
N
Φq
Φq
a⎝
N
a
>
N
Φq
1−
Φq
tm
hττ
t1
θj Φq
tm
hττ
t1
n
j1
θj Φq
j1 θj −
m
Φq
m
Φq
t
m
t1
hττ s
s
⎞
⎟
hττ s⎠ ≤ c,
⎞
m−2 ξi tm
Φ
hττ
s
i t1 q
i1
s
⎟
⎠
−
tm
hττ
ηj
σtm s
⎞
⎟ a
⎠
N
η
Φq
t
m
t1
tm
hττ
t1
t
η
Φq
t
m
t1
hττ s
s
⎞
⎟
hττ s⎠ a,
s
⎞
m−2 ξi tm
Φ
hττ
s
i
q
i1
t1
s
⎟
⎠
−
tm
hττ
ηj
⎞
⎟ a
⎠
N
m−2
i1
i ξi
Φq
m
t1
t
hττ s
s
m
Φq
t
hττ
Φq
t
t1
t1
m
⎞
hττ s⎠
s
hττ s > h,
s
a⎜
− ⎝
N
a⎢
⎣
N
t
t1
n
⎛
Φq
b
⎟
⎠
M
σtm ⎛
⎞
a ⎜ Φq
γu2 ⎝
N
≤
t1
⎡
tm
hττ
t1
n
j1
a⎜
− ⎝
N
≥
tm
hττ
ηj
⎛
⎛
⎛
<
θj Φq
a⎜
− ⎝
N
⎛
n
j1
tm
hττ
t1
n
j1
θj Φq
−
⎞
m−2 ξi tm
hττ s
i1 i t1 Φq
s
⎟
⎠
tm
hττ
ηj
tm
hττ
t1
σtm t1
⎞
⎟ a
⎠
N
Φq
t
m
s
σtm Φq
t1
t
m
hττ s
s
⎤
⎥
hττ s⎦ < c.
3.17
Advances in Difference Equations
13
So, u1 ∈ P γ, θ, α, b, k, c, αu1 > b, u2 ∈ Qγ, β, ϕ, h, a, c, βu2 < a, which means that
{u ∈ P γ, θ, α, b, k, c : αu > b} and {u ∈ Qγ, β, ϕ, h, a, c : βu < a} are not empty.
For u ∈ P γ, θ, α, b, k, c,
αAu Φq
tm
hτfτ, uττ
t1
n
j1
−
≥
θj Φq
−
m−2 ξi tm
hτfτ,
uττ
s
i1 i t1 Φq
s
tm
hτfτ,
uττ
ηj
1−
n
θj −
j1
m−2
i1
Φq
t
m
t1
>
Φq
Φq
t
m
i ξi
Φq
hτfτ, uττ s
s
t
m
hτfτ, uττ
t1
3.18
hτfτ, uττ s
s
η
t1
η
Φq
t1
≥
hτfτ, uττ s
hτΦp
t1
b
τ s b,
M0
and for u ∈ Qγ, β, ϕ, h, a, c,
βAu Φq
tm
hτfτ, uττ
t1
n
−
≤
<
Φq
j1
θj Φq
−
m−2 ξi tm
hτfτ, uττ s
i1 i t1 Φq
s
tm
hτfτ, uττ
ηj
tm
hτfτ, uττ
t1
Φq
Φq
t
t1
η
Φq
t
m
t1
tm
hτΦp a/ h0 Mη τ
t1
η
m
hτfτ, uττ s
s
hτfτ, uττ s
s
η
t1
Φq
t
m
hτΦp
s
a
τ s a.
h0 Mη
3.19
Thus i and ii in Lemma 2.4 hold.
On the other hand, for u ∈ P γ, α, b, c with θAu > k, we have αAu Au ≥
− t1 /η − t1 Auη − t1 /η − t1 θAu > − t1 /η − t1 k > b. And for
u ∈ P γ, β, a, c with ϕAu < h, we can obtain βAu Auη ≤ η − t1 / − t1 Au η − t1 / − t1 ϕAu < η − t1 / − t1 h < a. Thus, iii and iv in Lemma 2.4 hold.
14
Advances in Difference Equations
So, by Lemma 2.4, we obtain that the BVP 1.3 has at least three positive solutions
u1 , u2 , u3 ∈ P γ, c such that
max u1 t < a < max u3 t,
t∈,ηT
min u3 t < b < min u2 t.
t∈,ηT
t∈,ηT
t∈,ηT
3.20
This completes the proof.
Remark 3.4. Let R c, r b, υ k, we can find that the conditions of Theorem 3.1 are
contained in Theorem 3.3.
Example 3.5. Let T {0.1, 0.18} ∪ 0.2, 1 ∪ {1.2} ∪ 1.5, 2, p 2, consider the following eightpoint BVP:
Φp u u 0.1 −
t htft, ut 0,
t ∈ 0.1, 2T ,
3
3
θj u ηj − i uξi 0,
j1
u 2 0,
3.21
i1
where ht t σt, θ1 1/12, θ2 1/7, θ3 1/42, 1 1/6, 2 1/24, 3 1/8, ξ1 0.33,
ξ2 0.45, ξ3 1.65, η1 0.88, η2 1.86, η3 1.95, for all t ∈ T, and
ft, u ⎧
⎨0.00015,
0.4, 1.8T × 0.0001, 0.055,
⎩gu,
other,
3.22
where gu is continuous, 0 ≤ gu ≤ 0.0026, and g0.0001 g0.055 0.00015.
Set 0.4, η 1.8, by calculation,
3
j1
M θj 3
1
,
4
i i1
3.539656
,
3
1
,
3
Mη h0 3.99,
M0 0.924,
3.23
15.050656
,
3
Mσtm 15.282656
,
3
and let b 0.0001, k 0.055, c 0.102 585312, a 0.045, h 0.000125. Clearly, we can verify
that the conditions in Theorem 3.3 are fulfilled. Thus, by Theorem 3.3, the BVP 3.21 has at
least three positive solutions u1 , u2 and u3 such that
max u1 t < 0.045 < max u3 t,
t∈,ηT
t∈,ηT
min u3 t < 0.0001 < min u2 t.
t∈,ηT
t∈,ηT
3.24
Remark 3.6. If we let R 0.102 585312, r 0.0001, ι r 10−5 , υ 0.055, we can also verify
that the conditions in Theorem 3.1 are satisfied.
Advances in Difference Equations
15
Acknowledgments
This work is supported by the Natural Science Foundation of Ludong University 24200301,
24070301, 24070302, Program for Innovative Research Team in Ludong University, and a
Project of Shandong Province Higher Educational Science and Technology Program.
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