Electronic Journal of Differential Equations, Vol. 2008(2008), No. 92, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
EXISTENCE OF POSITIVE SOLUTIONS FOR p-LAPLACIAN
THREE-POINT BOUNDARY-VALUE PROBLEMS
ON TIME SCALES
HONG-RUI SUN, YING-HAI WANG
Abstract. This article shows the existence of positive solutions for a class of
p-Laplacian three-point boundary-value problem on time scales. By using several fixed point theorems in cones, we establish conditions for the existence of
at least one, two or three positive solutions for the boundary-value problems.
Our results are new even for the corresponding differential (T = R) and difference equation (T = Z), and for the general time scales setting. An example is
also given to illustrate our results.
1. Introduction
Dynamic equations on time scales not only unify differential and difference equations [13], but also exhibit much more complicated dynamics [1, 8, 9]. The study
of dynamic equations on time scales has led to important applications in the study
of insect population models, biology, heat transfer, stock market, wound healing,
and epidemic models [14, 20, 21].
Before introducing the problems of interest for this paper, we present some basic
definitions which can be found in [5, 8, 9, 13]. Another source on dynamic equations
on time scales is [17].
A time scale T is a nonempty closed subset of R with the topology inherited
from R. For notation, we shall use the convention that, for each interval J of R,
JT = J ∩ T.
The jump operators σ, ρ : T → T defined by
σ(t) = inf{τ ∈ T : τ > t} and ρ(t) = sup{τ ∈ T : τ < t}
(supplemented by inf ∅ := sup T and sup ∅ := inf T) are well defined. The point
t ∈ T is left-dense, left-scattered, right-dense, right-scattered if ρ(t) = t, ρ(t) < t,
σ(t) = t, σ(t) > t, respectively. If T has a right-scattered minimum m, define
Tκ = T − {m}; otherwise, set Tκ = T. If T has a left-scattered maximum M , define
2000 Mathematics Subject Classification. 34B15, 39A10.
Key words and phrases. Time scales; p-Laplacian; positive solution; cone; fixed point.
c
2008
Texas State University - San Marcos.
Submitted January 26, 2007. Published July 2, 2008.
Supported by grants 10571078 from the NNSF of China, 10726049 from Tianyuan Youth
Grant of China, Lzu05003 from Fundamental Research Fund for Physics and Mathematics
of Lanzhou University, and 2005038486 from China Postdoctoral Science Foundation.
1
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H.-R. SUN, Y.-H. WANG
EJDE-2008/92
Tκ = T − {M }; otherwise, set Tκ = T. The forward graininess is µ(t) := σ(t) − t.
Similarly, the backward graininess is υ(t) := t − ρ(t).
For f : T → R and t ∈ Tκ , the ∆-derivative of f at t, denoted by f ∆ (t), is
the number (provided it exists) with the property that given any > 0, there is a
neighborhood U ⊂ T of t such that
|f (σ(t)) − f (s) − f ∆ (t)[σ(t) − s]| ≤ |σ(t) − s|,
for all s ∈ U .
For f : T → R and t ∈ Tκ , the ∇-derivative [5] of f at t, denoted by f ∇ (t), is
the number (provided it exists) with the property that given any > 0, there is a
neighborhood U of t such that
|f (ρ(t)) − f (s) − f ∇ (t)[ρ(t) − s]| ≤ |ρ(t) − s|,
for all s ∈ U .
A function f : T → R is ld-continuous provided it is continuous at left dense
points in T and its right sided limit exists (finite) at right dense points in T. If
T = R, then f is ld-continuous if and only if f is continuous. If T = Z, then any
function is ld-continuous. It is known [5] that if f is ld-continuous, then there is a
function F (t) such that F ∇ (t) = f (t). In this case, we define
Z b
f (τ )∇τ = F (b) − F (a).
a
For recent results on positive solutions for second order three point boundary
value problems on time scales the reader is referred to [2, 3, 6, 10, 15, 18, 22]. Our
results have been motivated by those of Anderson, Avery and Henderson [4], and
Sun, Tang and Wang [25].
For convenience, throughout this paper we denote ϕp (u) = |u|p−2 u for p > 1
with (ϕp )−1 = ϕq , where 1/p + 1/q = 1.
Anderson, Avery and Henderson [4] considered the problem
(ϕp (u∆ (t)))∇ + c(t)f (u(t)) = 0,
u(a) − B0 (u∆ (ν)) = 0,
t ∈ (a, b)T ,
u∆ (b) = 0,
where ν ∈ (a, b)T , f ∈ Cld ([0, ∞), [0, ∞)), c ∈ Cld ((a, b)T , [0, ∞)) and Km x ≤
B0 (x) ≤ KM x for some positive constants Km , KM . They established the existence
result of at least one positive solution by a fixed point theorem of cone expansion
and compression of functional type.
In [25], the authors considered the eigenvalue problem for the p-Laplacian threepoint boundary value problem
(ϕp (u∆ (t)))∇ + λh(t)f (u(t)) = 0,
∆
∆
t ∈ (0, T )T ,
∆
u(0) − βu (0) = γu (η), u (T ) = 0.
The main tool used in [24] is Krasnoselskii’s fixed point theorem.
In this paper we study the existence of solutions for the one-dimensional pLaplacian three-point boundary value problem on time scales
(ϕp (u∆ (t)))∇ + h(t)f (t, u(t)) = 0,
∆
∆
u(0) − βu (0) = γu (η),
∆
t ∈ (0, T )T ,
u (T ) = 0.
(1.1)
(1.2)
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EXISTENCE OF POSITIVE SOLUTIONS
3
We establish sufficient conditions for the existence of at least one, two or three
positive solutions for the boundary value problem. An example is also given to illustrate the main results. The results are new even for the special cases of difference
equations and differential equations.
The rest of the paper is organized as follows. In Section 2, we first give four
lemmas which are needed throughout this paper and then state several fixed point
results: Krasnosel’skii’s fixed point theorem in a cone, a new fixed point theorem
due to Avery and Henderson and the Leggett-Williams fixed point theorem. In
Section 3 we use Krasnosel’skii’s fixed point theorem to obtain the existence of
at least one positive solutions of problem (1.1)-(1.2). Section 4 will discuss the
existence of twin positive solutions of problem (1.1)-(1.2). Two new results and
some corollaries will be presented by using a new fixed point theorem due to Avery
and Henderson. In Section 5 we develop criteria for the existence of (at least)
three positive and arbitrary odd positive solutions of problem (1.1) and (1.2). In
particular, our results in this section are new when T = R (the continuous case)
and T = Z (the discrete case). Finally, in section 6, we give an example to illustrate
our main results.
For the sake of convenience, we have the following hypotheses:
(i) T is a time scale, with 0, T ∈ T, β, γ are nonnegative constants, η ∈
(0, ρ(T ))T .
RT
(ii) h ∈ Cld ((0, T )T , [0, ∞)) such that 0 < 0 h(s)∇s < ∞, and f is in the
space C([0, ∞), (0, ∞)).
2. Preliminaries
Let the Banach space B = Cld ([0, T ]T ) (see [2]) be endowed with the norm
kuk = supt∈[0,T ]T |u(t)|, and choose the cone P ⊂ B defined by
P = {u ∈ B : u(t) ≥ 0 for t ∈ [0, T ]T and
u∆∇ (t) ≤ 0 for t ∈ (0, T )T , u∆ (T ) = 0}.
Clearly, kuk = u(T ) for u ∈ P . Define the operator A : P → B by
Z t Z T
ϕq
h(τ )f (τ, u(τ ))∇τ ∆s
Au(t) =
0
Z
s
T
+ βϕq
0
Z
h(s)f (s, u(s))∇s + γϕq
T
(2.1)
h(s)f (s, u(s))∇s
η
for t ∈ [0, T ]T .
Lemma 2.1 ([24, Lemma 2.6]). Assume g : R → R is continuous, g : T → R is
delta differentiable on Tκ , and f : R → R is continuous differentiable. Then there
exists c in the interval [ρ(t), t] with
(f ◦ g)∇ (t) = f 0 (g(c))g ∇ (t).
From the definition of A, the monotonicity of ϕq (x) and Lemma 2.1, it is
easy to see that for each u ∈ P , Au ∈ P and satisfies (1.2). In addition, since
(ϕp (u∆ (t)))∇ = −h(t)f (u(t)) < 0, and u∆ (T ) = 0, then Au(T ) is the maximum
value of Au(t).
Lemma 2.2 ([25, Lemma 2.2]). A : P → P is completely continuous.
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H.-R. SUN, Y.-H. WANG
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Lemma 2.3 ([25, Lemma 2.3]). If u ∈ P , then u(t) ≥
t
T
kuk for t ∈ [0, T ].
From the two lemmas above, we see that each fixed point of the operator A in
P is a positive solution of (1.1), (1.2).
Lemma 2.4 ([11, 16]). Let P be a cone in a Banach space B. Assume Ω1 , Ω2 are
open subsets of X with 0 ∈ Ω1 , Ω1 ⊂ Ω2 . If A : P ∩ (Ω2 \Ω1 ) → P is a completely
continuous operator such that either
(i) kAxk ≤ kxk for all x ∈ P ∩ ∂Ω1 and kAxk ≥ kxk for all x ∈ P ∩ ∂Ω2 , or
(ii) kAxk ≥ kxk for all x ∈ P ∩ ∂Ω1 and kAxk ≤ kxk for all x ∈ P ∩ ∂Ω2 .
Then A has a fixed point in P ∩ (Ω2 \Ω1 ).
In the rest of this section, we provide some background material from the theory
of cones in Banach spaces, and we then state several fixed point theorems which
we needed later.
Let B be a Banach space and P be a cone in B. A map ψ : P → [0, +∞)
is said to be a nonnegative, continuous and increasing functional provided ψ is
nonnegative, continuous and satisfies ψ(x) ≤ ψ(y) for all x, y ∈ P and x ≤ y.
Given a nonnegative continuous functional ψ on a cone P of a real Banach space
B, we define, for each d > 0, the set
P (ψ, d) = {x ∈ P : ψ(x) < d}.
Lemma 2.5 ([7]). Let P be a cone in a real Banach space E. Let α and ψ be
increasing, nonnegative continuous functional on P , and let θ be a nonnegative
continuous functional on P with θ(0) = 0 such that, for some c > 0 and H > 0,
ψ(x) ≤ θ(x) ≤ α(x)
and
kxk ≤ Hψ(x)
for all x ∈ P (ψ, c). Suppose there exist a completely continuous operator A :
P (ψ, c) → P and 0 < a < b < c such that
θ(λx) ≤ λθ(x)
for 0 ≤ λ ≤ 1andx ∈ ∂P (θ, b)
and
(i) ψ(Ax) > c for all x ∈ ∂P (ψ, c);
(ii) θ(Ax) < b for all x ∈ ∂P (θ, b);
(iii) P (α, a) 6= ∅ and α(Ax) > a for x ∈ ∂P (α, a).
Then, A has at least two fixed points, x1 and x2 belonging to P (ψ, c) satisfying
a < α(x1 ) with θ(x1 ) < b and b < θ(x2 ) with ψ(x2 ) < c.
Let 0 < a < b be given and let α be a nonnegative continuous concave functional
on the cone P . Define the convex sets Pa , P (α, a, b) by
Pa = {x ∈ P : kxk < a},
P (α, a, b) = {x ∈ P : a ≤ α(x), kxk ≤ b}.
Then we state the Leggett-Williams fixed point theorem [19].
Lemma 2.6. Let P be a cone in a real Banach space B, A : P c → P c be completely
continuous and α be a nonnegative continuous concave functional on P with α(x) ≤
kxk for all x ∈ P c . Suppose there exists 0 < d < a < b ≤ c such that
(i) {x ∈ P (α, a, b) : α(x) > a} =
6 ∅ and α(Ax) > a for x ∈ P (α, a, b);
(ii) kAxk < d for kxk ≤ d;
(iii) α(Ax) > a for x ∈ P (α, a, c) with kAxk > b.
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EXISTENCE OF POSITIVE SOLUTIONS
5
Then A has at least three fixed points x1 , x2 , x3 satisfying kx1 k < d, a < α(x2 ),
kx3 k > d, and α(x3 ) < a.
3. Existence of One Positive Solution
For convenience, we define some important constants
Z
C1 = (T + β + γ)−1 ϕp (
T
h(s)∇s),
(3.1)
h(s)∇s).
(3.2)
0
Z T
C2 = (η + β + γ)−1 ϕp (
η
Theorem 3.1. Assume there exist positive numbers a 6= b such that the conditions
(H1) There is a > 0 such that f (t, u) ≤ ϕp (aC1 ) for t ∈ [0, T ]T and 0 ≤ u ≤ a;
(H2) There is b > 0 such that f (t, u) ≥ ϕp (bC2 ) for t ∈ [η, T ]T and Tη b ≤ u ≤ b.
Then (1.1)–(1.2) has at least one positive solution u such that kuk lies between a
and b.
Proof. Without loss of generality, we may suppose that 0 < a < b. Define the
bounded open ball centered at the origin by
Ωa = {u ∈ B : kuk ≤ a},
T
Then 0 ∈ Ωa ⊂ Ωb . For u ∈ P
kAuk = sup
t
hZ
ϕq
t∈[0,T ]
0
Z
+ βϕq (
T
Ωb = {u ∈ B : kuk ≤ b}.
∂Ωa so that kuk = a, by (H1) and (3.1), we have
T
Z
h(τ )f (τ, u(τ ))∇τ ∆s
s
h(s)f (s, u(s))∇s) + γϕq
Z
0
≤ (T + β + γ)ϕq
T
i
h(s)f (s, u(s))∇s
η
T
Z
h(s)f (s, u(s))∇s
0
≤ (T + β + γ)ϕq
T
Z
h(s)ϕp (aC1 )∇s
0
≤ aC1 (T + β + γ)ϕq
Z
T
h(s)∇s ≤ a.
0
Hence, kAuk ≤ kuk for u ∈ P
Then
T
∂Ωa . Similarly, let u ∈ P
min u(t) ≥
t∈[η,T ]
η
b
T
T
∂Ωb so that kuk = b.
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H.-R. SUN, Y.-H. WANG
EJDE-2008/92
and
kAuk ≥ Au(η)
Z η Z
=
ϕq
0
+ βϕq
Z
T
s
T
h(τ )f (τ, u(τ ))∇τ ∆s
Z
h(s)f (s, u(s))∇s + γϕq (
0
T
h(s)f (s, u(s))∇s)
η
Z
≥ (η + β + γ)ϕq (
T
h(s)f (s, u(s))∇s)
η
Z
≥ (η + β + γ)bC2 ϕq
T
h(s)∇s
η
= b = kuk
T
by (H2) and (3.2). Consequently,
kAuk ≥ kuk for u ∈ P ∂Ωb . By Lemma 2.4, A
T
has a fixed point u ∈ P (Ωb \ Ωa ), which is a positive solution of (1.1), (1.2), such
that a ≤ kuk ≤ b.
For t ∈ [0, T ]T , we define
f (t, u)
,
ϕp (u)
f (t, u)
f 0 (t) = lim supu→0+
,
ϕp (u)
f0 (t) = lim inf u→0+
f (t, u)
,
ϕp (u)
f (t, u)
f ∞ (t) = lim supu→∞
.
ϕp (u)
f∞ (t) = lim inf u→∞
(3.3)
(3.4)
Corollary 3.2. The boundary-value problem (1.1), (1.2) has at least one positive
solution provided either
(H3) f 0 (t) < ϕp (C1 ) for t ∈ [0, T ]T and f∞ (t) > ϕp T ηC2 for t ∈ [η, T ]T or
(H4) f0 (t) > ϕp T ηC2 for t ∈ [η, T ]T and f ∞ (t) < ϕp (C1 ) for t ∈ [0, T ]T ,
where C1 , C2 , f0 , f∞ , f 0 , f ∞ are as in (3.1), (3.2), (3.3), (3.4), respectively. In
particular, if f is superlinear in ϕp (u) (f 0 (t) = 0 and f∞ (t) = ∞) or sublinear
in ϕp (u) (f0 (t) = ∞ and f ∞ (t) = 0), then (1.1), (1.2) has at least one positive
solution.
Proof. First suppose (H3) holds. Then, there are sufficiently small a > 0 and
sufficiently large b > 0 such that
f (t, u)
≤ ϕp (C1 ) for t ∈ [0, T ]T , u ∈ (0, a],
ϕp (u)
f (t, u)
ηb
T C2 ≥ ϕp
for t ∈ [η, T ]T , u ∈ [ , +∞).
ϕp (u)
η
T
Then
f (t, u) ≤ ϕp (uC1 ) ≤ ϕp (aC1 ),
f (t, u) ≥ ϕp
T C2 u
) ≥ ϕp (C2 b),
η
t ∈ [0, T ]T , u ∈ [0, a],
t ∈ [η, T ]T , u ∈ [
ηb
, b].
T
In particular, both (H1) and (H2) hold, so that by Theorem 3.1, (1.1), (1.2) has at
least one positive solution.
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EXISTENCE OF POSITIVE SOLUTIONS
7
Next assume (H4) holds. Then there exist 0 < a < b such that
f (t, u)
T C2
≥ ϕp
) for t ∈ [η, T ]T , u ∈ (0, a],
(3.5)
ϕp (u)
η
f (t, u)
≤ ϕp (C1 ) for t ∈ [0, T ]T , u ∈ [b, +∞).
(3.6)
ϕp (u)
From (3.5) we have f (t, u) ≥ ϕp T Cη2 u ≥ ϕp (C2 a) for t ∈ [η, T ]T , u ∈ [ ηa
T , a]
satisfying (H2) with respect to a. Now consider (3.6), we wish to show that (H1)
holds. To that end, we consider the two cases: (1) f (t, u) is bounded or (2) f (t, u)
is unbounded.
Case 1: Suppose there exists C > 0 such that f (t, u) ≤ C for t ∈ [0, T ]T and
ϕ (C)
u ∈ [0, ∞). By (3.6), there is r ≥ max{b, qC1 } such that f (t, u) ≤ C ≤ ϕp (C1 r)
for t ∈ [0, T ]T , u ∈ [0, r]. Thus (H1) is satisfied with respect to r.
Case 2: If f is unbounded, there exist t0 ∈ [0, T ]T and r0 ≥ b such that
f (t, u) ≤ f (t0 , r0 ) ≤ ϕp (C1 r0 )
for t ∈ [0, T ]T and u ∈ [0, r0 ].
and (H1) is satisfied with respect to r0 . Thus in both cases condition (H1) hold and
Theorem 3.1 yields the conclusion.
4. Twin Solutions
In this section, we fix c ∈ T such that η < c < T , and denote
Z T
C3 = (c + β + γ)−1 ϕp
h(s)∇s .
c
Define the nonnegative, increasing and continuous functionals ψ, θ, and α on P by
ψ(u) = min u(t) = u(η),
θ(u) = max u(t) = u(η),
t∈[η,c]T
t∈[0,η]T
α(u) = max u(t) = u(c).
t∈[0,c]T
We observe that, for each u ∈ P,
ψ(u) = θ(u) ≤ α(u).
In addition, for each u ∈ P , ψ(u) = u(η) ≥
kuk ≤
T
ψ(u),
η
η
T
kuk. Thus
u ∈ P.
Finally, we also note that
θ(λu) = λθ(u),
(4.1)
0 ≤ λ ≤ 1 and u ∈ ∂P (θ, b0 ).
We now present the results in this section.
Theorem 4.1. Assume that there are positive numbers a0 < b0 < c0 such that
C1 0
ηC1 0
0 < a0 <
b <
c.
C3
T C3
Assume further that f (t, u) satisfies the following conditions:
(i) f (t, u) > ϕp (c0 C2 ), (t, u) ∈ [η, T ]T × [c0 , Tη c0 ],
(ii) f (t, u) < ϕp (b0 C1 ), (t, u) ∈ [0, T ]T × [0, Tη b0 ],
(iii) f (t, u) > ϕp (a0 C3 ), (t, u) ∈ [c, T ]T × [a0 , Tc a0 ].
(4.2)
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H.-R. SUN, Y.-H. WANG
EJDE-2008/92
Then (1.1)-(1.2) has at least two positive solutions u1 and u2 such that
a0 < max u1 (t)
with max u1 (t) < b0 ,
t∈[0,c]T
t∈[0,η]T
0
b < max u2 (t)
min u2 (t) < c0 .
with
t∈[0,η]T
t∈[η,c]T
Proof. By the definition of operator A and its properties, it suffices to show that
the conditions of Lemma 2.5 hold with respect to A.
We first show that if u ∈ ∂P (ψ, c0 ), then ψ(Au) > c0 . Indeed, if u ∈ ∂P (ψ, c0 ),
then ψ(u) = mint∈[η,c]T u(t) = u(η) = c0 . Since u ∈ P , kuk ≤ Tη ψ(u) = Tη c0 , we have
c0 ≤ u(t) ≤ Tη c0 , t ∈ [η, T ]T . As a consequence of (i), f (t, u(t)) > ϕp (c0 C2 ), t ∈
[η, T ]T . Also, Au ∈ P implies
ψ(Au) = Au(η)
Z η Z
=
ϕq
0
+ βϕq
Z
T
s
T
h(τ )f (τ, u(τ ))∇τ ∆s
T
Z
h(s)f (s, u(s))∇s + γϕq
0
h(s)f (s, u(s))∇s
η
≥ (η + β + γ)ϕq
T
Z
h(s)f (s, u(s))∇s
η
0
> (η + β + γ)
ηc C2 ϕq
T
T
Z
h(s)∇s = c0 .
η
Next, we verify that θ(Au) < b0 for u ∈ ∂P (θ, b0 ). Let us choose u ∈ ∂P (θ, b0 ),
then θ(u) = maxt∈[0,η]T u(t) = u(η) = b0 . This implies 0 ≤ u(t) ≤ b0 , t ∈ [0, η]T .
Since u ∈ P , we also have b0 ≤ u(t) ≤ kuk ≤ Tη u(l) = Tη b0 for t ∈ [η, T ]T . So
T 0
b,
η
0 ≤ u(t) ≤
t ∈ [0, T ]T .
Using (ii), we get
f (t, u(t)) < ϕp (b0 C1 ),
t ∈ [0, T ]T .
Also, Au ∈ P implies that
θ(Au) = Au(η) ≤ Au(T )
Z T
Z T
=
ϕq (
h(τ )f (τ, u(τ ))∇τ )∆s
0
+ βϕq
s
Z
T
Z
h(s)f (s, u(s))∇s + γϕq (
0
T
h(s)f (s, u(s))∇s)
η
≤ (T + β + γ)ϕq
Z
T
h(s)f (s, u(s))∇s
0
0
< (T + β + γ)b C1 ϕq
Z
T
h(s)∇s = b0 .
0
Finally, we prove that P (α, a0 ) 6= ∅ and α(Au) > a0 for all u ∈ ∂P (α, a0 ). In
0
fact, the constant function a2 ∈ P (α, a0 ). Moreover, for u ∈ ∂P (α, a0 ), we have
α(u) = maxt∈[0,c]T u(t) = u(c) = a0 . This implies a0 ≤ u(t) ≤ Tc a0 , t ∈ [c, T ]T .
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EXISTENCE OF POSITIVE SOLUTIONS
Using assumption (iii), f (t, u(t)) > ϕp (a0 C3 ),
obtain
α(Au) = (Au)(c)
Z
Z c
=
ϕq
T
9
t ∈ [c, T ]T . As before Au ∈ P , we
h(τ )f (τ, u(τ ))∇τ ∆s
s
0
+ βϕq
T
Z
Z
h(s)f (s, u(s))∇s + γϕq
0
T
h(s)f (s, u(s))∇s
η
≥ (c + β + γ)ϕq
Z
T
h(s)f (s, u(s))∇s
c
0
> (c + β + γ)
a
ϕq
L
Z
T
h(s)∇s = a0 .
c
Thus, by Lemma 2.5, there exist at least two fixed points of A which are positive
solutions u1 and u2 , belonging to P (ψ, c0 ), of (1.1)-(1.2) such that
a0 < α(u1 )
with θ(u1 ) < b0 ,
b0 < θ(u2 )
with ψ(u2 ) < c0 .
In analogy to Theorem 4.1, we have the following result.
Theorem 4.2. Assume that there are positive numbers a0 < b0 < c0 such that
0 < a0 <
c 0
cC2 0
b <
c.
T
T C1
Assume further that f (t, u) satisfies the following conditions:
(i) f (t, u) < ϕp (c0 C1 ) for (t, u) ∈ [0, T ]T × [0, Tη c0 ],
(ii) f (t, u) > ϕp (b0 C2 ) for (t, u) ∈ [η, T ]T × [b0 , Tη b0 ],
(iii) f (t, u) < ϕp (a0 C1 ) for (t, u) ∈ [c, T ]T × [0, Tc a0 ].
Then (1.1)-(1.2) has at least two positive solutions u1 and u2 such that
a0 < max u1 (t)
t∈[0,c]T
0
b < max u2 (t)
t∈[0,η]T
with max u1 (t) < b0 ,
t∈[0,η]T
with max u2 (t) < c0 .
t∈[η,c]T
Corollary 4.3. Assume that f satisfies conditions
(i) f0 (t) > ϕp (C2 ), t ∈ [η, T ]T and f∞ (t) = lim inf u→∞ fϕ(t,u)
> ϕp (C3 ), t ∈
p (u)
[c, T ]T ;
(ii) there exists a0 > 0 such that f (t, u) < ϕp (a0 C1 ) for (t, u) ∈ [0, T ]T ×[0, Tη a0 ].
Then (1.1)-(1.2) has at least two positive solutions.
Corollary 4.4. Suppose that f satisfies conditions
(i) f0 (t) < ϕp ( Tη C1 ), t ∈ [0, T ]T and f∞ (t) < ϕp ( Tc C1 ), t ∈ [c, T ]T ;
(ii) there exists b0 > 0 such that f (t, u) > ϕp (b0 C2 ), for (t, u) ∈ [η, T ]T ×[b0 , Tη b0 ].
Then (1.1)-(1.2) has at least two positive solutions.
By applying Theorems 4.1 and 4.2, it is easy to prove that Corollaries 4.3 and
4.4 hold, respectively.
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H.-R. SUN, Y.-H. WANG
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5. Existence of three solutions
Let the nonnegative continuous concave functional Ψ : P → [0, ∞) be defined by
Ψ(u) = min u(t) = u(η),
u ∈ P.
t∈[η,T ]T
Note that for u ∈ P , Ψ(u) ≤ kuk.
Theorem 5.1. Suppose that there exist constants 0 < d0 < a0 such that
(i) f (t, u) < ϕp (d0 C1 ), (t, u) ∈ [0, T ]T × [0, d0 ];
(ii) f (t, u) ≥ ϕp (a0 C2 ), (t, u) ∈ [η, T ]T × [a0 , Tη a0 ];
(iii) one of the following conditions holds:
< ϕp (C1 );
(D1) lim supu→∞ maxt∈[0,T ]T fϕ(t,u)
p (u)
(D2) there exists a number c0 >
(t, u) ∈ [0, T ]T × [0, c0 ].
T
η
a0 such that f (t, u) < ϕp (c0 C1 ) for
Then (1.1)-(1.2) has at least three positive solutions.
Proof. By the definition of operator A and its properties, it suffices to show that
the conditions of Lemma 2.6 hold with respect to A.
We first show that if (D1) holds, then there exists a number l0 > Tη a0 such that
A : P l0 → Pl0 . Suppose that
lim sup max
u→∞ t∈[0,T ]T
f (t, u)
< ϕp (C1 )
ϕp (u)
holds, then there are τ > 0 and δ < C1 such that if u > τ, then
max
t∈[0,T ]T
f (t, u)
≤ ϕp (δ).
ϕp (u)
That is to say,
f (t, u) ≤ ϕp (δu),
(t, u) ∈ [0, T ]T × [τ, ∞).
Set λ = max{f (t, u) : (t, u) ∈ [0, T ]T × [0, τ ]}, then
f (t, u) ≤ λ + ϕp (δu),
(t, u) ∈ [0, T ]T × [0, ∞).
(5.1)
Take
T
λ
l0 > max{ a0 , ϕq (
)}.
η
ϕp (C1 ) − ϕp (δ)
(5.2)
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EXISTENCE OF POSITIVE SOLUTIONS
11
If u ∈ P l0 , then by (3.1), (5.1) and (5.2), we obtain
kAuk = Au(T )
Z T
Z
=
ϕq (
0
Z
+ γϕq (
T
Z
h(τ )f (τ, u(τ ))∇τ )∆s + βϕq (
s
T
T
h(s)f (s, u(s))∇s)
0
h(s)f (s, u(s))∇s)
η
Z
≤ (T + β + γ)ϕq (
T
h(s)f (s, u(s))∇s)
0
Z
≤ (T + β + γ)ϕq (
T
h(s)(λ + ϕp (δu(s))∇s)
0
T
Z
≤ (T + β + γ)ϕq (λ + ϕp (δl ))ϕq (
0
h(s)∇s)
0
= ϕq (λ + ϕp (δl0 ))
1
< l0 .
C1
Next we verify that if there is a positive number r0 such that if f (t, u) < ϕp (r0 /N )
for (t, u) ∈ [0, T ]T ×[0, r0 ], then A : P r0 → Pr0 .
Indeed, if u ∈ P r0 , then
kAuk = Au(T )
≤ (T + β + γ)ϕq
Z
T
h(s)f (s, u(s))∇s
0
r0
< (T + β + γ)ϕq
N
Z
T
h(s)∇s = r0 ,
0
thus, Au ∈ Pr0 . Hence, we have shown that either (D1 ) or (D2 ) holds, then there
exists a number c0 with c0 > Tη a0 such that A : P c0 → Pc0 . It is also note from (i)
that A : P d0 → Pd0 .
Now, we show that {u ∈ P (Ψ, a0 , Tη a0 ) : Ψ(u) > a0 } 6= ∅ and Ψ(Au) > a0 for all
u ∈ P (Ψ, a0 , Tη a0 ). In fact,
u=
T
(η + T )a0
∈ {u ∈ P (Ψ, a0 , a0 ) : Ψ(u) > a0 }.
2η
η
For u ∈ P (Ψ, a0 , Tη a0 ), we have
a0 ≤ min u(t) = u(η) ≤ u(t) ≤
t∈[η,T ]T
T 0
a,
η
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H.-R. SUN, Y.-H. WANG
EJDE-2008/92
for all t ∈ [η, T ]T . Then, in view of (ii), we know that
Ψ(Au) = min Au(t) = Au(η)
t∈[η,T ]T
η
Z
=
ϕq
0
+ γϕq
T
Z
Z
τ
T
Z
h(τ )f (τ, u(τ ))∇τ ∆s + βϕq (
T
h(s)f (s, u(s))∇s)
0
h(s)f (s, u(s))∇s
η
≥ (η + β + γ)ϕq
Z
T
h(s)f (s, u(s))∇s
η
≥ (η + β + γ)a0 C2 ϕq
T
Z
h(s)∇s = a0 .
η
Finally, we assert that if u ∈ P (Ψ, a0 , c0 ) and kAuk >
Suppose u ∈ P (Ψ, a0 , c0 ) and kAuk > Tη a0 . Then
T
η
a0 , then Ψ(Au) > a0 .
Ψ(Au) = min Au(t) = Au(η)
t∈[η,T ]T
≥
η
η
Au(T ) = kAuk > a0 .
T
T
To sum up, the hypotheses of Lemma 2.6 are satisfied, hence (1.1)–(1.2) has at
least three positive solutions u1 , u2 , u3 such that
ku1 k < d0 , a0 < min u2 (t) and
t∈[η,T ]T
ku3 k > d0 with
min u3 (t) < a0 .
t∈η,T ]T
From Theorem 5.1, we see that, when assumptions such as (i), (ii), (iii) are
imposed appropriately on f , we can establish the existence of an arbitrary odd
number of positive solutions of (1.1), (1.2).
Theorem 5.2. If
0 < d01 < a01 <
T 0
T
a1 < d02 < a02 < a02 < d03 < . . . < d0n ,
η
η
n ∈ N,
(i) f (t, u) < ϕp (d0i C1 ), (t, u) ∈ [0, T ]T × [0, d0i ];
(ii) f (t, u) ≥ ϕp (a0i C2 ), (t, u) ∈ [η, T ]T × [a0i , Tη a0i ];
then (1.1)-(1.2) has at least 2n − 1 positive solutions.
Proof. When n = 1, it is immediate from condition (i) that A : P d01 → Pd01 ⊂ P d01 ,
which means that A has at least one fixed point u1 ∈ P d01 by the Schauder fixed
point theorem. When n = 2, it is clear that Theorem 4.1 holds (with c1 = d02 ).
Then we can obtain at least three positive solutions u1 , u2 , and u3 satisfying
ku1 k < d01 , min u2 (t) > a01
t∈[η,T ]T
and
ku3 k > d01 with
Following this way, we complete the proof by induction.
min u3 (t) < a01 .
t∈[η,T ]T
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EXISTENCE OF POSITIVE SOLUTIONS
13
6. Example
Let T = {1 − (1/2)N0 } ∪ {1}, where N0 denote the set of nonnegative integers.
31
Take T = 1, p = 32 , β = γ = 14 , η = 15
16 , c = 32 . If we let h(s) ≡ 1, then by (3.1)
and (3.2) we have
Z T
2
−1
C1 = (T + β + γ) ϕp
1∇s = ,
3
0
Z T
4
C2 = (η + β + γ)−1 ϕp
1∇s =
.
23
η
Suppose
f (t, u) = f (u) :=
√
6 + 9u
(2 + sin u) u,
20(1 + u)
t ∈ [0, 1]T , u ≥ 0,
9
then f0 = f 0 = 53 , f∞ = 20
, f ∞ = 27
20 .
Firstly, by easy calculation, it is easy to get
r
r
2
T
64
ϕp (C1 ) =
≈ 1.224, ϕp ( C2 ) =
≈ 0.431,
3
η
345
So the condition (H3) of Corollary 3.2 holds. Thus by Corollary 3.2, the boundaryvalue problem
p
∇
1
6 + 9u(t)
(2 + sin u(t)) u(t) = 0,
|u∆ (t)|− 2 u∆ (t) +
(6.1)
20(1 + u(t))
1
1
15 u(0) − u∆ (0) = u∆
, u∆ (1) = 0,
(6.2)
2
2
16
has at least one positive solution.
Secondly, since
r
r
3
5
11
η 9
c f0 = < ϕp C1 =
≈ 0.791, f∞ =
< ϕp C1 =
≈ 0.479.
5
T
8
20
T
48
If we choose b0 = 0.7, then
min
0
t∈[0,T ]T ,u∈[b0 , 16
15 b ]
f (t, u) ≈ 0.800 > ϕp (b0 C1 ) ≈ 0.683.
So all the assumptions of Corollary 4.4 are satisfied. Therefore by Corollary 4.4
the boundary value problem (5.1)–(5.2) has at lest two solutions u1 and u2 with
0 < ku1 k ≤ 0.7 < ku2 k.
References
[1] R. P. Agarwal, M. Bohner, W.T. Li; Nonoscillation and Oscillation Theory for Functional
Differential Equations, Pure and Applied Mathematics Series, Vol. 267, Marcel Dekker, 2004.
[2] D. R. Anderson; Solutions to second-order three-point problems on time scales, J. Differ.
Equations Appl. 8(2002) 673-688.
[3] D. R. Anderson; Nonlinear triple point problems on time scales, Electron. J. Differential
Equations 47(2004) 1-12.
[4] D. R. Anderson, R. Avery, J. Henderson; Existence of solutions for a one-dimensional plaplacian on time scales, J. Differ. Equations Appl. 10(2004) 889-896.
[5] F. M. Atici, G. Sh. Guseinov; On Green’s functions and positive solutions for boundary value
problems on time scales, J. Comput. Appl. Math. 141(2002) 75-99.
[6] F. M. Atici, S. G. Topal; The generalized quasilinearization method and three point boundary
value problems on time scales, Appl. Math. Lett. 18(2005) 577-585.
14
H.-R. SUN, Y.-H. WANG
EJDE-2008/92
[7] R. I. Avery, J. Henderson; Two positive fixed points of nonlinear operator on ordered Banach
spaces, Comm. Appl. Nonlinear Anal., 8(2001), 27-36.
[8] M. Bohner, A. Peterson; Dynamic Equation On Time Scales: an Introduction with Applications. Boston: Birkhauser., 2001.
[9] M. Bohner, A. Peterson; Advances in Dynamic Equations on Time Scales, Birkhauser Boston,
2003.
[10] J. J. DaCunha, J. M. Davis, P. K. Singh; Existence results for singular three point boundary
value problems on time scales, J. Math. Anal. Appl, 295(2004), 378-391.
[11] D. Guo, V. Lakshmikantham; Nonlinear Problems in Abstract Cones, Academic press, San
Diego, 1988.
[12] Z. He; Double positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales, J. Comput. Appl. Math. 182(2005) 304-315.
[13] S. Hilger; Analysis on measure chains-a unified approach to continuous and discrete calculus,
Results Math. 18(1990) 18-56.
[14] M. A. Jones, B. Song, D. M. Thomas; Controlling wound healing through debridement, Math.
Comput. Modelling 40 (2004) 1057-1064.
[15] W. T. Li, H. R. Sun; Positive solutions for second order m-point boundary value problems
on time scales, Acta. Math. Sinica. 22(6)(2006) 1797-1804.
[16] M. Krasnoselskii; Positive Solutions of Operator Equations, Noordhoff Groningen, 1964.
[17] V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan; Dynamic Systems on Measure
Chains, Kluwer Academic Publishers, Boston, 1996.
[18] E. R. Kaufmann; Positive solutions of a three-point boundary value problem on a time scale,
Electron. J. Differential Equations 2003(2003), No. 82, 1-11.
[19] R. Leggett, L. Williams; Multiple positive fixed points of nonlinear operators on ordered
Banach spaces, Indiana Univ. Math. J., 28(1979), 673-688.
[20] V. Spedding; Taming Nature’s Numbers, New Scientist, July (2003) 28-32.
[21] D. M. Thomas, L. Vandemuelebroeke, K. Yamaguchi; A mathematical evolution model for
phytoremediation of metals, Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 411-422.
[22] H. R. Sun, W. T. Li; Existence of positive solutions for nonlinear three-point boundary value
problems on time scales, J. Math. Anal. Appl. 299(2004) 508-524.
[23] H. R. Sun, W. T. Li; Positive solutions for p-Laplacian m -point boundary value problems
on time scales, Tainwanese J. Math. 12(2008), 93-115.
[24] H. R. Sun, W. T. Li; Multiple positive solutions for p-Laplacian m-point boundary value
problems on time scales, Appl. Math. Comput. 182(2006), 478-491.
[25] H. R. Sun, L.T. Tang, Y. H. Wang; Eigenvalue problem for p-Laplacian three-point boundary
value problems on time scales, J. Math. Anal. Appl. 331(2007), 248-262.
Hong-Rui Sun
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000,
China
E-mail address: hrsun@lzu.edu.cn
Ying-Hai Wang
School of Physical Science and Technology, Lanzhou University, Lanzhou, Gansu 730000,
China
E-mail address: yhwang@lzu.edu.cn