Electronic Journal of Differential Equations, Vol. 2007(2007), No. 23, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
MULTIPLE POSITIVE SOLUTIONS FOR FOURTH-ORDER
THREE-POINT p-LAPLACIAN BOUNDARY-VALUE PROBLEMS
HANYING FENG, MEIQIANG FENG, MING JIANG, WEIGAO GE
Abstract. In this paper, we study the three-point boundary-value problem
for a fourth-order one-dimensional p-Laplacian differential equation
`
´00
`
´
φp (u00 (t)) + a(t)f u(t) = 0, t ∈ (0, 1),
subject to the nonlinear boundary conditions:
u0 (1) = ηu0 (0),
u(0) = ξu(1),
(φp (u00 (0))0 = α1 (φp (u00 (δ))0 ,
u00 (1) =
p
β1 u00 (δ),
p−1
where φp (s) = |s|p−2 s, p > 1. Using the five functional fixed point theorem
due to Avery, we obtain sufficient conditions for the existence of at least three
positive solutions.
1. Introduction
This paper concerns the existence of three positive solutions for the fourth-order
three-point boundary-value problem (BVP for short) consisting of the p-Laplacian
differential equation
00
φp (u00 (t)) + a(t)f (u(t)) = 0, t ∈ (0, 1),
(1.1)
with the nonlinear boundary conditions
u0 (1) = ηu0 (0),
u(0) = ξu(1),
(φp (u00 (0))0 = α1 (φp (u00 (δ))0 ,
u00 (1) =
p
β1 u00 (δ),
p−1
(1.2)
where f : R → [0, +∞) and a : (0, 1) → [0, +∞) are continuous functions, φp (s) =
|s|p−2 s, p > 1, α1 , β1 ≥ 0, ξ 6= 1, η 6= 1 and 0 < δ < 1.
Two-point boundary-problems for differential equation are used to describe a
number of physical, biological and chemical phenomena. For additional background
and results, we refer the reader to the monograph by Agawarl, O’Regan and Wong
[1] as well as to the recent contributions by [2, 9, 13, 14, 20].
Boundary-value problems for n-th order differential equation [15, 16, 22] and
even-order can arise, especially for fourth-order equations, in applications, see [4,
5, 6, 7, 8] and references therein.
2000 Mathematics Subject Classification. 34B10, 34B15, 34B18.
Key words and phrases. Fourth-order boundary-value problem; one-dimensional p-Laplacian;
five functional fixed point theorem; positive solution.
c
2007
Texas State University - San Marcos.
Submitted November 30, 2006. Published February 4, 2007.
Supported by grants 10671012 from NNSF, and 20050007011 from SRFDP of China.
1
2
H. FENG, M. FENG, M. JIANG, W. GE
EJDE-2007/23
Recently, three-point boundary-value problems of the differential equations were
presented and studied by many authors, see [10, 11, 12, 21] and the references cite
there. However, three-point BVP (1.1), (1.2) have not received as much attention
in the literature as Lidstone condition BVP
u0000 (t) = a(t)f (u(t)),
t ∈ (0, 1),
00
u(0) = u(1) = u (0) = u00 (1) = 0,
(1.3)
and the three-point BVP for the second-order differential equation
u00 (t) + a(t)f (u(t)) = 0,
u(0) = 0,
t ∈ (0, 1),
u(1) = αu(η),
(1.4)
that were extensively considered, in [13, 14, 20] and [21], respectively. The results
of existence of positive solutions of BVP (1.1), (1.2) are relatively scarce.
Most recently, Liu and Ge studied two class of four-order four-point BVPs successively in [17, 18]. They proved that existence of at least two or three positive
solutions. To the best of our knowledge, existence results of multiple positive solutions for fourth-order three-point BVP (1.1), (1.2) have not been found in literature.
Motivated by the works in [17, 18], the purpose of this paper is to establish the
existence of at least three positive solutions of (1.1), (1.2).
For the remainder of the paper, we assume that:
R1
(i) 0 < 0 a(s)ds < ∞;
(ii) q satisfies p1 + 1q = 1 and φq (z) = |z|q−2 z.
2. Background and definitions
For the convenience of the reader, we provide some background material from
the theory of cones in Banach spaces. We also state in this section a fixed point
theorem by Avery.
Definition 2.1. Let X be a real Banach space. A nonempty closed set P ⊂ X is
said to be a cone provided that
(i) x ∈ P and λ ≥ 0 implies λx ∈ X, and
(ii) x ∈ P and −x ∈ P implies x = 0.
Every cone P ⊂ X induces an ordering in X given by x ≤ y if and only if
y − x ∈ P.
Definition 2.2. The map ψ is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E provided that ψ : P → [0, ∞) is
continuous and
ψ(tx + (1 − t)y) ≥ tψ(x) + (1 − t)ψ(y)
for all x, y ∈ P and 0 ≤ t ≤ 1. Similarly, we say the map β is a nonnegative
continuous convex functional on a cone P of a real Banach space E provided that
β : P → [0, ∞) is continuous and
β(tx + (1 − t)y) ≤ tβ(x) + (1 − t)β(y)
for all x, y ∈ P and 0 ≤ t ≤ 1.
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MULTIPLE POSITIVE SOLUTIONS
3
Let γ, β, θ be nonnegative, continuous, convex functionals on P and α, ψ be
nonnegative, continuous, concave functionals on P . Then for nonnegative numbers
h, a, b, d and c we define the following sets:
P (γ, c) = {x ∈ P : γ(x) < c},
P (γ, α, a, c) = {x ∈ P : a ≤ α(x), γ(x) ≤ c},
Q(γ, β, d, c) = {x ∈ P : β(x) ≤ d, γ(x) ≤ c},
P (γ, θ, α, a, b, c) = {x ∈ P : a ≤ α(x), θ(x) ≤ b, γ(x) ≤ c},
Q(γ, β, ψ, h, d, c) = {x ∈ P : h ≤ ψ(x), β(x) ≤ d, γ(x) ≤ c}.
To prove our results, we need the following Five Functionals Fixed Point Theorem due to Avery [3] which is a generalization of the Leggett-Williams fixed point
theorem.
Theorem 2.1. Suppose X is a real Banach space and P is a cone of X, γ, β, θ
are three nonnegative, continuous, convex functionals and α, ψ are nonnegative,
continuous, concave functionals such that
α(x) ≤ β(x),
||x|| ≤ M γ(x)
for x ∈ P (γ, c) and some positive numbers c, M . Again, assume that
T : P (γ, c) → P (γ, c)
be a completely continuous operator and there are positive numbers h, d, a, b with
0 < d < a such that
(i) {x ∈ P (γ, θ, α, a, b, c) : α(x) > a} 6= ∅ and x ∈ P (γ, θ, α, a, b, c) implies
α(T x) > a.
(ii) {x ∈ Q(γ, β, ψ, h, d, c) : β(x) < d} 6= ∅ and x ∈ Q(γ, β, ψ, h, d, c) implies
β(T x) < d.
(iii) x ∈ P (γ, α, a, c) with θ(T x) > b implies α(T x) > a.
(iv) x ∈ Q(γ, β, d, c) with ψ(T x) < h implies β(T x) < d.
Then T has at least three fixed points x1 , x2 , x3 ∈ P (γ, c) such thah
β(x1 ) < d,
a < α(x2 ),
d < β(x3 ),
with
α(x3 ) < a.
3. Related lemmas
Lemma 3.1 ([17]). Suppose f ∈ C(R, R), then the three-point BVP
−y 00 = f (t),
0
0
y (0) = α1 y (δ),
t ∈ (0, 1)
y(1) = β1 y(δ)
has a unique solution
Z
y(t) =
1
g(t, s)f (s)ds,
0
t ∈ (0, 1),
(3.1)
4
H. FENG, M. FENG, M. JIANG, W. GE
where M = (1 − α1 )(1 − β1 ) 6= 0 and


1 − β1 δ − t + β1 t,










1 − β1 δ + (1 − β1 )(α1 s − s − α1 t),
1 
g(t, s) =
1 − α1 − β1 s + α1 β1 s − t
M



+α1 t + β1 t + α1 β1 t,






(1 − s)(t − α1 ),



EJDE-2007/23
if 0 ≤ s ≤ t < δ < 1
or 0 ≤ s ≤ δ ≤ t ≤ 1,
if 0 ≤ t ≤ s ≤ δ < 1,
if 0 ≤ δ ≤ s ≤ t ≤ 1,
if 0 < δ ≤ t ≤ s ≤ 1
or 0 ≤ t < δ ≤ s ≤ 1.
Lemma 3.2 ([19]). Suppose f ∈ C(R, R), then the two-point BVP
−y 00 = f (t),
t ∈ (0, 1)
0
y (1) = ηy 0 (0)
y(0) = ξy(1),
(3.2)
has a unique solution
Z
y(t) =
1
h(t, s)f (s)ds,
t ∈ [0, 1],
0
where M1 = (1 − ξ)(1 − η) 6= 0 and
(
s + η(t − s) + ξη(1 − t),
1
h(t, s) =
M1 t + ξ(s − t) + ξη(1 − s),
0 ≤ s ≤ t ≤ 1,
0 ≤ t ≤ s ≤ 1.
Remark 3.3. It is easy to check that if α1 < 1, and 0 ≤ β1 < 1, then g(t, s) ≥ 0
for (t, s) ∈ [0, 1] × [0, 1]. If ξ, η ≥ 0 and M1 = (1 − ξ)(1 − η) ≥ 0, then h(t, s) ≥ 0
for (t, s) ∈ [0, 1] × [0, 1]
If u(t) is a solution of BVP (1.1) and (1.2). By Lemma 3.1 and (3.1), one has
Z 1
00
φp (u (t)) = −
g(t, s)a(s)f (u(s))ds.
(3.3)
0
By Lemma 3.2 and (3.2), one obtains
Z 1
Z
u(t) =
h(t, s)φq
0
1
g(s, τ )a(τ )f (u(τ ))dτ ds.
(3.4)
0
Lemma 3.4 ([19]). Suppose 0 ≤ ξ, η < 1, 0 < t1 < t2 < 1 and δ ∈ (0, 1). Then,
for s ∈ [0, 1],
t1
h(t1 , s)
≥ ,
h(t2 , s)
t2
h(1, s)
1
≤ .
h(δ, s)
δ
(3.5)
(3.6)
Lemma 3.5 ([19]). Suppose ξ, η > 1, 0 < t1 < t2 < 1 and δ ∈ (0, 1). Then, for
s ∈ [0, 1],
1 − t2
h(t2 , s)
≥
,
h(t1 , s)
1 − t1
h(0, s)
1
≤
.
h(δ, s)
1−δ
(3.7)
(3.8)
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MULTIPLE POSITIVE SOLUTIONS
5
4. Triple positive solutions to (1.1), (1.2)
Now let the Banach space E = C[0, 1] be endowed with the maximum norm
kuk = maxt∈[0,1] |u(t)|. Let the two cones P1 , P2 ⊂ X defined by
P1 = {u ∈ X : u(t) is nonnegative, concave, nondecreasing on (0, 1)} ,
P2 = {u ∈ X : u(t) is nonnegative, concave, nonincreasing on (0, 1)} .
Next, choose t1 , t2 , t3 ∈ (0, 1) and t1 < t2 . Define nonnegative, continuous, concave
functions α, ψ and nonnegative, continuous, convex functions β, θ, γ on P1 by
γ(u) = max u(t) = u(t3 ),
u ∈ P1 ,
ψ(u) = min u(t) = u(δ),
u ∈ P1 ,
t∈[0,t3 ]
t∈[δ,1]
u ∈ P1 ,
β(u) = max u(t) = u(1),
t∈[δ,1]
α(u) = min u(t) = u(t1 ),
u ∈ P1 ,
θ(u) = max u(t) = u(t2 ),
u ∈ P1 .
t∈[t1 ,t2 ]
t∈[t1 ,t2 ]
It is easy to verify that α(u) = u(t1 ) ≤ u(1) = β(u) and kuk = u(1) ≤
1
t3 γ(u) for u ∈ P1 .
1
t3 u(t3 )
=
Theorem 4.1. Suppose 0 ≤ ξ, η < 1, α1 < 1, 0 ≤ β1 < 1 and there exist numbers
0 < a < b < c such that 0 < a < b < tt21 b ≤ c and f (w) satisfies the following
conditions:
a
f (w) < φp ( ), 0 ≤ w ≤ a,
(4.1)
C
t2
b
(4.2)
f (w) > φp ( ), b ≤ w ≤ b,
B
t1
c
1
f (w) ≤ φp ( ), 0 ≤ w ≤ c,
(4.3)
A
t3
where
1
Z
A=
h(t3 , s)φq
Z
0
1
B=
h(t1 , s)φq
Z
Z
0
Z
C=
1
h(1, s)φq
Z
0
1
g(s, τ )a(τ )dτ ds,
0
t2
t1
1
g(s, τ )a(τ )dτ ds,
g(s, τ )a(τ )dτ ds.
0
Then (1.1), (1.2) has at least three positive solutions u1 , u2 , u3 ∈ P1 (γ, c) such that
u1 (t1 ) > b,
u2 (1) < a,
u3 (t1 ) < b
(4.4)
with u3 (1) > a, ui (δ) ≤ c for i = 1, 2, 3.
Proof. We begin by defining the completely continuous operator T : P1 → X by
(3.4) as
Z 1
Z 1
h(t, s)φq
g(s, τ )a(τ )f (u(τ ))dτ ds
(T u)(t) =
0
0
6
H. FENG, M. FENG, M. JIANG, W. GE
EJDE-2007/23
for u ∈ P1 . It is easy to prove that (1.1), (1.2) has a positive solution u = u(t) if
and only if the operator T has a fixed point on P1 .
Firstly, we prove T : P1 (γ, c) ⊂ P1 (γ, c). For u ∈ P1 , by Remark 3.1, it is easy
to check that T u ≥ 0. Moreover,
Z t Z
(T u)0 (t) = (1 − ξ) η
φq
0
1
Z
+
φq
Z
t
1
g(s, τ )a(τ )f (u(τ ))dτ ds
0
1
g(s, τ )a(τ )f (u(τ ))dτ ds ≥ 0
0
and
(T u)00 (t) = −φq
1
Z
g(t, s)a(s)f (u(s))ds ≤ 0.
0
So, we have T P1 ⊂ P1 .
For u ∈ P1 (γ, c), 0 ≤ u(t) ≤ kuk ≤
1
t3 γ(u)
≤
1
t3 c.
By (4.3), it follows that
γ(T u) = max (T u)(t) = (T u)(t3 )
0≤t≤t3
Z 1
Z 1
=
h(t3 , s)φq
g(s, τ )a(τ )f (u(τ ))dτ ds
0
Z
≤
0
1
h(t3 , s)φq
Z
0
1
g(s, τ )a(τ )dτ φp (c/A) ds
0
Z
Z 1
c 1
h(t3 , s)φq
g(s, τ )a(τ )dτ ds
=
A 0
0
c
= A = c.
A
Thus, T : P1 (γ, c) ⊂ P1 (γ, c).
Secondly, by taking
t2
b − b,
t1
u2 (t) = a − ε2 for 0 < ε2 < a − δa,
u1 (t) = b + ε1
for 0 < ε1 <
It is immediate that
t2
b, c) : α(u) > b} =
6 ∅,
t1
u2 (t) ∈ {Q(γ, β, ψ, δa, a, c) : β(u) < a} =
6 ∅.
u1 (t) ∈ {P (γ, θ, α, b,
In the following steps, we verify the remaining conditions of Theorem 2.1. Now the
proof is divided into four steps.
Step 1: We prove that
u ∈ P (γ, θ, α, b,
t2
b, c)
t1
implies
α(T u) > b.
(4.5)
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MULTIPLE POSITIVE SOLUTIONS
In fact, u(t) ≥ u(t1 ) = α(u) ≥ b for t1 ≤ t ≤ t2 , and u(t) ≤ u(t2 ) = θ(u) ≤
t1 ≤ t ≤ t2 . Thus using (4.2), one gets
7
t2
t1 b
for
α(T u) = min (T u)(t) = (T u)(t1 )
t1 ≤t≤t2
Z 1
Z 1
=
h(t1 , s)φq
g(s, τ )a(τ )f (u(τ ))dτ ds
0
0
1
Z
≥
h(t1 , s)φq
Z
0
t1
t2
1
Z
>
t2
h(t1 , s)φq
Z
0
g(s, τ )a(τ )f (u(τ ))dτ ds
g(s, τ )a(τ )dτ φp (b/B) ds
t1
Z
Z t2
b 1
=
h(t1 , s)φq
g(s, τ )a(τ )dτ ds
B 0
t1
b
= B = b.
B
Step 2: We show that
u ∈ Q(γ, β, ψ, δa, a, c)
implies
β(T u) < a.
(4.6)
In fact, 0 ≤ u(t) ≤ u(1) = β(u) ≤ a for 0 ≤ t ≤ 1, Thus using (4.1), one arrives at
β(T u) = max (T u)(t) = (T u)(1)
δ≤t≤1
Z 1
Z 1
=
h(1, s)φq
g(s, τ )a(τ )f (u(τ ))dτ ds
0
Z
0
1
<
h(1, s)φq
Z
0
1
g(s, τ )a(τ )dτ φp (a/C) ds
0
Z
Z 1
a 1
h(1, s)φq
g(s, τ )a(τ )dτ ds
=
C 0
0
a
= C = a.
C
Step 3: We verify that
u ∈ Q(γ, β, a, c)
with ψ(T u) < δa implies β(T u) < a.
(4.7)
By Lemma 3.4,
β(T u) = max (T u)(t) = (T u)(1)
δ≤t≤1
Z 1
Z 1
=
h(1, s)φq
g(s, τ )a(τ )f (u(τ ))dτ ds
0
0
Z 1
h(1, s)
h(δ, s)φq
g(s, τ )a(τ )f (u(τ ))dτ ds
0 h(δ, s)
0
1
1
≤ (T u)(δ) = ψ(T u) < a.
δ
δ
Step 4: We prove that
Z
1
=
u ∈ P (γ, α, b, c)
with θ(T u) >
t2
b
t1
implies
α(T u) > b.
(4.8)
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H. FENG, M. FENG, M. JIANG, W. GE
EJDE-2007/23
By Lemma 3.4,
α(T u) = min (T u)(t) = (T u)(t1 )
t1 ≤t≤t2
Z 1
Z 1
=
h(t1 , s)φq
g(s, τ )a(τ )f (u(τ ))dτ ds
0
0
1
Z 1
h(t1 , s)
=
h(t2 , s)φq
g(s, τ )a(τ )f (u(τ ))dτ ds
0 h(t2 , s)
0
t1
t1
≥ (T u)(t2 ) = θ(T u) > b.
t2
t2
Therefore, the hypotheses of Theorem 2.1 are satisfied and there exist three positive
solutions x1 , x2 , x3 for BVP (1.1), (1.2) satisfying (4.4).
Z
Similarly, choose t1 , t2 , t3 ∈ (0, 1) and t1 < t2 . Define nonnegative, continuous,
concave functions α, ψ and nonnegative, continuous, convex functions β, θ, γ on P2
by
γ(u) = max u(t) = u(t3 ),
x ∈ P2 ,
ψ(x) = min u(t) = u(δ),
u ∈ P2 ,
t∈[t3 ,1]
t∈[0,δ]
u ∈ P2 ,
β(u) = max u(t) = u(0),
t∈[0,δ]
α(u) = min u(t) = u(t2 ),
u ∈ P2 ,
θ(u) = max u(t) = u(t1 ),
u ∈ P2 .
t∈[t1 ,t2 ]
t∈[t1 ,t2 ]
It is easy to verify that α(u) = u(t2 ) ≤ u(0) = β(u) and kuk = u(0) ≤
1
t3 γ(u) for u ∈ P2 . So, we obtain the following result.
1
t3 u(t3 )
=
Theorem 4.2. Suppose ξ, η > 1, α1 < 1, 0 ≤ β1 < 1 and there exist numbers
1
0 < a < b < c such that 0 < a < b < 1−t
1−t2 b ≤ c and f (w) satisfy the following
conditions:
a
f (w) < φp ( ), 0 ≤ w ≤ a,
(4.9)
C
b
1 − t1
f (w) > φp ( ), b ≤ w ≤
b,
(4.10)
B
1 − t2
c
1
f (w) ≤ φp ( ), 0 ≤ w ≤ c,
(4.11)
A
t3
where
Z 1
Z 1
h(t3 , s)φq
g(s, τ )a(τ )dτ ds,
A=
Z
0
1
B=
h(t2 , s)φq
Z
0
Z
1
h(0, s)φq
C=
0
Z
0
t2
t1
1
g(s, τ )a(τ )dτ ds,
g(s, τ )a(τ )dτ ds.
0
Then (1.1), (1.2) has at least three positive solutions u1 , u2 , u3 ∈ P1 (γ, c) such that
u1 (t2 ) > b,
u2 (0) < a,
u3 (t2 ) < b,
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MULTIPLE POSITIVE SOLUTIONS
9
with u3 (0) > a and ui (δ) ≤ c for i = 1, 2, 3.
Since the proof of the above theorem is similar to that of Lemma 3.5, we omit
it.
Acknowledgments. The authors wish to thank the referee for his (or her) valuable
corrections to the original manuscript.
References
[1] R. P. Agarwal, D. O. Regan, and P. J. Y. Wong, Positive solutions of Differential, Difference,
and Integral Equations, Kluwer Academic, Dordrecht, 1999.
[2] R. I. Avery, J. Henderson, Three symmetric positive solutions for a second-order boundaryvalue problem, Appl. Math. Lett. 13 (2000) 1-7.
[3] R. I. Avery, A generalization of the Leggett-Williams fixed-point theorem, Math. Sci. Res.
Hot-line, 2 (1998) 9-14.
[4] L. E. Bobisud, D. O’Regan, W. D. Royalty, Existence and nonexistence for a singular
boundary-value problem, Appl. Anal. 28 (1998), 245-256.
[5] L. E. Bobisud, D. O’Regan, W. D. Royalty, Solbability of some nonlinear boundary-value
problems, Nonl. Anal. 12 (1998), 855-869.
[6] L. E. Bobisud, Existence of solutions for a singular boundary-value problem, Appl. Anal. 35
(1990), 43-57.
[7] L. E. Bobisud, Y. S. Lee, Existence for a class of nonliear boundary-value problems, Appl.
Anal. 38 (1990), 45-67.
[8] C. J. Chyan, J. Henderson, Positive solutions of 2m−th boundary-value problems, Appl.
Math. Letters 15 (2002), 767-774.
[9] P. W. Eloe, J. Henderson, Positive solutions and nonlinear (k, n − k) conjugate eigenvalue
problem, Diff. Equ. Dynam. Syst. 6 (1998), 309-317.
[10] W. Feng, J. R. L. Webb, Solvability of a three-point boundary-value problem at resonance,
Nonl. Appl. TMA 30(6) (1997) 3227-3238.
[11] C. P. Gupta, Solvability of a three-point nonlinear boundary-value problem for a second order
ordinary differential equation, J. Math. Anal. Appl. 168 (1992) 540-551.
[12] C. P. Gupta, Trofimchuk, Existence of a solution to a three-point boundary-value problem
and the spectral radius of o related linear operator, Nonl. Anal. TMA 34 (1998) 498-507.
[13] J. R. Graef, B. Yang, On a nonlinear boundary-value problem for fourth-order equations,
Appl. Anal. 72 (1999), 439-448.
[14] J. R. Graef, B. Yang, Existence and non-existence of positive solutions of fourth-order nonlinear boundary-value problems, Appl. Anal. 74(2000), 201-204.
[15] J. Henderson, H. B. Thompson, Existence of multiple positive solutions for some nth order
boundary-value problems, Nonlinear Anal., 7 (2000) 55-62.
[16] B. Liu, J. Yu, Solvability of multi-point boundary-value problems at resonance(II), Appl.
Math. Comput., 136 (2003) 353-377.
[17] Y. Liu, W. Ge, Solbablity a nonlinear four-point boundary problem for a fouth order differential equation, Taiwanes J. Math. (7) (2003) 591-604.
[18] Y. Liu, W. Ge, Existence theorems of positive solutions for fourth-order a Solbablity a nonlinear four-point boundary problems, Anal. Appl. (2) (2004) 71-85.
[19] D. Ma, The existence of solution for boundary-value problem with a p-Laplacian operator,
Doctoral Thesis, 2006.
[20] R. Y. Ma, H. Y. Wang, On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal. 59 (1995), 225-231.
[21] R. Y. Ma, Positive solutions of nonlinear three point boundary-value problem, Electronic J.
Diff. Equs. 1998 (1998), No. 34, 1-8.
[22] P. K. Palamides, Multi point boundary-value problems at resonance for n-order differential
equations. positive and monotone solutions, Electron. J. Diff. Eqns., Vol. 2004 (2004), No 25
pp. 1-14.
10
H. FENG, M. FENG, M. JIANG, W. GE
EJDE-2007/23
Hanying Feng
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China.
Department of Mathematics, Shijiazhuang, Mechanical Engineering College, Shijiazhuang
050003, China
E-mail address: fhanying@yahoo.com.cn
Meiqiang Feng
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
Department of Fundamental Sciences, Beijing Information Technology Institute, Beijing, 100101, China
E-mail address: meiqiangfeng@sina.com
Ming Jiang
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang
050003, China
E-mail address: jiangming27@163.com
Weigao Ge
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
E-mail address: gew@bit.edu.cn