Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 31, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
MULTIPLE POSITIVE SOLUTIONS FOR
INTEGRO-DIFFERENTIAL EQUATIONS WITH INTEGRAL
BOUNDARY CONDITIONS AND SIGN CHANGING
NONLINEARITIES
MEI JIA, PINGYOU WANG
Abstract. In this article, we show the existence of multiple positive solutions
for integro-differential equations with one-dimensional p-Laplacian operator,
sign changing nonlinearities, and integral boundary conditions. By using the
Schauder fixed point theorem and the Krasnosel’skii fixed point theorem, we
obtain sufficient conditions for the existence of at least two positive solutions.
1. Introduction
In this article, we study the existence of positive solutions for the following
integro-differential equation with integral boundary conditions, and sign changing
nonlinearities:
(ϕp (u0 (t)))0 + w(t)f (t, u(t), Au(t), Bu(t)) = 0,
u(1) = −Bu(η),
u(0) = Au(ξ),
0 < t < 1,
(1.1)
Rt
R1
where Au(t) = 0 g(t, s)u0 (s) ds, Bu(t) = t h(t, s)u0 (s) ds, 0 < ξ ≤ η < 1, ϕp (u) =
|u|p−2 u is the one-dimensional p-Laplacian operator wiht p > 1, ϕq = (ϕp )−1 , and
1
1
q + p = 1. By using the Schauder fixed point theorem and the Krasnosel’skii
fixed point theorem, we obtain sufficient conditions for the existence of at least two
positive solutions under suitable conditions assumed on the nonlinear terms f and
w.
The theory of boundary-value problems for integro-differential equations arises
in different areas of applied mathematics, fluid dynamics, plasma physics, biological
sciences and chemical kinetics (for details, see [4, 3, 22] and the references therein).
Since boundary-value problems with integral boundary conditions include two,
2000 Mathematics Subject Classification. 34B15, 34B18.
Key words and phrases. Boundary value problems; p-Laplacian operator;
integral boundary conditions; fixed point theorem; multiple positive solutions.
c
2012
Texas State University - San Marcos.
Submitted November 16, 2011. Published February 23, 2012.
Supported by grants 10ZZ93 from the Innovation Program of Shanghai Municipal Education
Commission, and 11171220 from the National Natural Sciences Foundation of China.
1
2
M. JIA, P. WANG
EJDE-2012/31
three, multi-point and nonlocal boundary-value problems as special cases, the existence and multiplicity of positive solutions for such problems have been put emphasis on continuously (see [8, 20, 23, 24, 25, 14, 17, 13, 18] and references therein). Because of the wide mathematical and physical background, the existence of positive
solutions for nonlinear boundary-value problems with p-Laplacian has also received
wide attention. For details, we can refer to see [5, 16, 7, 6, 19, 21, 15, 17, 13, 12, 9].
The main tools for such problems are various kinds of fixed-point theorem in cones
(see [5, 16, 7, 21, 15, 17, 13, 12]), the monotone iterative technique (see [19]) and
the fixed point index theory (see [6, 12]). If the nonlinear term is nonnegative,
we can apply the concavity of solutions in the proofs. Under the assumption that
the nonlinear term is nonnegative, authors obtained the existence of at least one
positive solutions or multiple positive solutions, see [5, 16, 7, 19, 21, 15, 13].
By using the upper and lower solution approach and the growth restriction approach, in [1] the author presented some general existence theorems second-order
boundary-value problems with sign changing nonlinearities:
y 00 + q(t)f (t, y) = 0,
0 < t < 1,
y(0) = 0 = y(1),
(1.2)
and
y 00 + q(t)f (t, y) = 0,
0 < t < 1,
y(0) = 0, θ(y(1)) + y(1) = 0,
where the nonlinear term f is allowed to change sign and θ may be nonlinear. Moreover, in [2], the authors discussed the singular Dirichlet boundary-value problem
(1.2) and established existence results, where nonlinearity f is allowed to change
sign and may be singular at y = 0.
Guo [11] established a new fixed point theorem in double cones and discussed
the existence of positive solutions for the second-order three-point boundary-value
problem
x00 + f (t, x) = 0,
0
x(0) − βx (0) = 0,
0 ≤ t ≤ 1,
x(1) = αx(η),
where f is allowed to change sign. Sufficient conditions of the existence of at
least two positive solutions for the boundary-value problems above are obtained
by imposing growth conditions on f . By applications of fixed point index theory,
Cheung and Ren [6] proved the existence of two positive solutions for the problem
(Φp (u0 ))0 + h(t)f (t, u) = 0,
0 < t < 1,
with each of the following two sets of boundary conditions
u0 (0) = 0,
u(1) =
m−2
X
αi u(ξi );
i=1
u(0) =
m−2
X
αi u(ξi ),
u0 (1) = 0,
i=1
+
where h : [0, 1] → R and f : [0, 1] × [0, ∞) → R are continuous functions. Ji
[12] studied the existence of positive solutions for the one-dimensional p-Laplacian
EJDE-2012/31
MULTIPLE POSITIVE SOLUTIONS
3
equation
(Φp (u0 ))0 + f (t, u, u0 ) = 0,
u0 (0) =
m−2
X
αi u0 (ξi ),
u(1) =
0 < t < 1,
m−2
X
i=1
βi u(ξi ),
i=1
where f may change sign. They show that it has at least one or two positive solutions under some assumptions by applying the fixed point theorem and fixed point
index theory. Liu, Jia and Tian [17] studied the existence of positive solutions for
the boundary-value problem, with integral boundary conditions and sign changing
nonlinearities of one-dimensional p-Laplacian
(Φp (u0 ))0 + f (t, u) = 0,
au(0) − bu0 (0) =
m−2
X
0 < t < 1,
Z 1
u(1) =
g(s)u(s) ds,
ai u(ξi ),
0
i=1
where a, b ∈ [0, +∞), ai ∈ (0, +∞), i = 1, 2, . . . , m, 0 < ξ1 < ξ2 · · · < ξm−2 < 1,
m ≥ 3. The sufficient conditions for the existence of at least two positive solutions
were obtained by using a fixed point theorem in double cones given in [11].
Recently, by using the expansion and compression fixed point theorem of norm
in cone under suitable conditions imposed on the nonlinear term f and w, Jia and
Wang [13] established sufficient conditions for the existence of at least one positive
solutions for (1.1), where the nonlinear term f and w are nonnegative. However,
there are a few works devoted to the integro-differential boundary-value problems
with integral boundary conditions, one-dimensional p-Laplacian operator and sign
changing nonlinearities.
Motivated by the above, we obtain some meaningful conclusions by considering the existence of multiply positive solutions for (1.1), with integral boundary
conditions and sign changing nonlinearities of one-dimensional p-Laplacian.
The following hypotheses will be assumed throughout this paper:
(H1) f : [0, 1] × [0, +∞) × R2 → R is continuous;
(H2) f (t, 0, ·, ·) ≥ 0, w ∈ L1 [0, 1], f (t, 0, ·, ·) 6= 0, w(t) ≥ 0 and w 6= 0 a.e. on
[0,1];
(H3) g, h ∈ C([0, 1] × [0, 1], [0, +∞)), g(ξ, s) is monotone decreasing with respect
to s ∈ [0, 1] and h(η, s) is monotone increasing with respect to s ∈ [0, 1].
2. Preliminaries
1
For any y ∈ L [0, 1], y(t) ≥ 0 and y(t) 6≡ 0 for t ∈ [0, 1], we denote
Z ξ
Z s
g(ξ, s)ϕq C −
y(τ ) dτ ds
H(C) =
0
0
Z
1
Z
h(η, s)ϕq C −
+
η
s
Z
Z
ϕq C −
y(τ ) dτ ds +
0
1
0
s
y(τ ) dτ ds.
0
1
Lemma 2.1. Suppose that (H3) holds. Then for each y ∈ L [0, 1], y(t) ≥ 0 and
y(t) 6≡ 0 for t ∈ [0, 1], the equation H(C) = 0 has a unique solution in (−∞, +∞)
R1
and the solution Cy ∈ (0, 0 y(τ ) dτ ). Moreover, there exists σ ∈ (0, 1) such that
Rσ
Cy = 0 y(τ ) dτ .
4
M. JIA, P. WANG
EJDE-2012/31
The proof of Lemma 2.1 is similar to that of [13, Lemma 2.2]. In this article, we
take
Z σ
σy = inf σ ∈ (0, 1) : Cy =
y(τ ) dτ .
(2.1)
0
For the convenience, we recall the following results (see [13]).
Lemma 2.2 ([13, Lemma 2.3]). Assume (H3) holds. Then for each y ∈ L1 [0, 1],
y(t) ≥ 0 and y(t) 6≡ 0 for t ∈ [0, 1], the boundary-value problem
(ϕp (u0 (t)))0 + y(t) = 0,
u(0) = Au(ξ),
0 < t < 1,
(2.2)
u(1) = −Bu(η),
has a unique solution of the form
Z
Z s
Z t Z ξ
y(τ ) dτ ds +
ϕq Cy −
u(t) =
g(ξ, s)ϕq Cy −
0
0
s
y(τ ) dτ ds,
(2.3)
0
0
or
Z
u(t) = −
1
Z
s
h(η, s)ϕq Cy −
η
Z
1
y(τ ) dτ ds −
Z
ϕq Cy −
0
t
s
y(τ ) dτ ds, (2.4)
0
where Cy satisfies H(Cy ) = 0.
Remark 2.3. By Lemma 2.1 and Lemma 2.2, (2.3) and (2.4) can be changed into
(2.5) and (2.6), respectively.
Z t Z σy
Z ξ
Z σy
y(τ ) dτ ds,
(2.5)
y(τ ) dτ ds +
ϕq
u(t) =
g(ξ, s)ϕq
s
0
s
0
and
Z
u(t) =
1
h(η, s)ϕq
η
Z
s
Z
y(τ ) dτ ds +
σy
1
ϕq
Z
t
s
y(τ ) dτ ds.
(2.6)
σy
Lemma 2.4 ([13, Lemma 2.4]). Suppose (H3) holds. If y ∈ L1 [0, 1], y(t) ≥ 0 and
y(t) 6≡ 0 for t ∈ [0, 1]. Then the solution of boundary-value problem (2.2) has the
following properties:
(1) u(t) is a concave function;
(2) u(t) ≥ 0, t ∈ [0, 1];
(3) u(σy ) = max0≤t≤1 u(t) and u0 (σy ) = 0, where σy is defined in (2.1).
Lemma 2.5 ([9]). If u ∈ C[0, 1] and u(t) ≥ 0 is a concave function. Then for each
γ ∈ (0, 21 ), we have
min u(t) ≥ γkuk.
γ≤t≤1−γ
Let X = C[0, 1] and kuk = max0≤t≤1 |u(t)|, take 0 < δ < min{ 21 , ξ, η} and
denote
P = {u ∈ X : u(t) ≥ 0, t ∈ [0, 1]}
and
K = {u ∈ P : u(t) is a concave function on [0, 1] and
Obviously, P, K ⊂ X are two cones of X with K ⊂ P .
min
δ≤t≤1−δ
u(t) ≥ δkuk}.
EJDE-2012/31
MULTIPLE POSITIVE SOLUTIONS
5
Denote B + = max{B, 0}. For u ∈ K, we define T ∗ : K → K by
R
R
ξ
σu
+

g(ξ,
s)ϕ
w(τ
)f
(τ,
u(τ
),
Au(τ
),
Bu(τ
))
dτ
ds

q
s

R
 0R

t
σu

+
+ ϕq
w(τ )f (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds,
0 ≤ t ≤ σu ,
0
s
R
T ∗ u(t) = R 1
s


h(η, s)ϕq σu w(τ )f + (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds

 ηR


+ 1 ϕq R s w(τ )f + (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds,
σu ≤ t ≤ 1,
t
σu
where σu is defined in (2.1). It follows T ∗ is well definition from (H1)–(H3), Lemma
2.4 and Lemma 2.5. Define T : P → P by
h R
R
ξ
σu

g(ξ,
s)ϕ
w(τ
)f
(τ,
u(τ
),
Au(τ
),
Bu(τ
))
dτ
ds

q

s
 0
R
i+

R

σu

+ t ϕq
w(τ )f (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds ,
0 ≤ t ≤ σu ,
0
s
h
R
T u(t) =
R
1
s


h(η, s)ϕq σu w(τ )f (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds

η


i+

R
Rs

+ 1 ϕq
w(τ
)f
(τ,
u(τ
),
Au(τ
),
Bu(τ
))
dτ
ds ,
σu ≤ t ≤ 1 .
t
σu
Define S : P → X by
R
R
ξ
σ

g(ξ, s)ϕq s u w(τ )f (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds

0

R

R

σu

+ t ϕq
w(τ
)f
(τ,
u(τ
),
Au(τ
),
Bu(τ
))
dτ
ds,
0 ≤ t ≤ σu ,
0
s
R
Su(t) = R 1
s


h(η, s)ϕq σu w(τ )f (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds

η



+ R 1 ϕq R s w(τ )f (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds,
σu ≤ t ≤ 1 .
t
σu
From Lemma 2.2, we have the following result.
Lemma 2.6. Suppose that (H1)–(H3) hold. Then a function u(t) is a solution of
boundary-value problem (1.1) if and only if u(t) is a fixed point of the operator S.
We can easily prove that the following lemma holds.
Lemma 2.7. Suppose that (H1)–(H3) hold. Then T ∗ : K → K is completely
continuous.
For u ∈ X, denote θ : X → P by (θu)(t) = max{u(t), 0}, then T = θ ◦ S.
Lemma 2.8 ([6, Lemma 2.2]). If S : K → X is completely continuous, then
T = θ ◦ S : K → K is also completely continuous.
Lemma 2.9. Suppose (H1)–(H3) hold. If u is a fixed point of operator T , then u
is also a fixed point of operator S.
Proof. Let u be a fixed point of the operator T , if we prove Su(t) ≥ 0 for t ∈ [0, 1],
then u(t) is a fixed point of operator S.
Suppose Su(t) ≥ 0 for t ∈ [0, 1] is not true, then there exists a t0 ∈ (0, 1) such
that u(t0 ) = 0 > Su(t0 ). Let (t1 , t2 ) be the maximal interval which contains t0 and
such that Su(t) < 0, t ∈ (t1 , t2 ). It follows [t1 , t2 ] 6= [0, 1] from (H2).
Case 1: If t2 < 1, we have u(t) = 0 for t ∈ [t1 , t2 ], Su(t) < 0 for t ∈ (t1 , t2 )
and Su(t2 ) = 0. Thus (Su)0 (t2 ) ≥ 0. From (H2), we know [ϕp ((Su)0 (t))]0 =
−w(t)f (t, 0, Au(t), Bu(t)) ≤ 0 for t ∈ [t1 , t2 ] and we can get (Su)0 (t) is monotone
decreasing on [t1 , t2 ]. So t1 = 0, Su(t1 ) < 0 and
(Su)0 (t) ≥ (Su)0 (t2 ) ≥ 0, t ∈ [0, t2 ].
6
M. JIA, P. WANG
EJDE-2012/31
On the other hand, if ξ ≤ t2 , we have
Z ξ
0 > Su(0) =
g(ξ, s)(Su)0 (s) ds ≥ 0,
0
which is a contradiction.
If ξ > t2 , by using mean value theorem of integral, we have
Z t2
Z ξ
0
Su(0) =
g(ξ, s)(Su) (s) ds +
g(ξ, s)(Su)0 (s) ds
0
t2
Z
= g(ξ, ξ1 )
t2
(Su)0 (s) ds + g(ξ, t2 )
Z
0
ξ2
(Su)0 (s) ds
t2
= g(ξ, ξ1 )(Su(t2 ) − Su(0)) + g(ξ, t2 )(Su(ξ2 ) − Su(t2 ))
= g(ξ, ξ1 )(−Su(0)) + g(ξ, t2 )Su(ξ2 ),
and
0 > (1 + g(ξ, ξ1 ))Su(0) = g(ξ, t2 )Su(ξ2 ),
(2.7)
where ξ1 ∈ [0, t2 ] and ξ2 ∈ [t2 , ξ].
It follows that (2.7) is a contradiction if Su(ξ2 ) ≥ 0.
If Su(ξ2 ) < 0, let (t3 , t4 ) be the maximal interval which contains ξ2 and such that
Su(t) < 0, t ∈ (t3 , t4 ). It is obvious that [t3 , t4 ] ⊂ [t2 , 1]. If t4 < 1, we have u(t) = 0
for t ∈ [t3 , t4 ], Su(t) < 0 for t ∈ (t3 , t4 ) and Su(t3 ) = 0. Thus (Su)0 (t3 ) ≤ 0. From
(H2), we know [ϕp ((Su)0 (t))]0 = −w(t)f (t, 0, Au(t), Bu(t)) ≤ 0 and ϕp ((Su)0 (t)) is
monotone decreasing on [t3 , t4 ], we can obtain (Su)0 (t) is monotone decreasing on
[t3 , t4 ]. It is easy to show that
(Su)0 (t) ≤ (Su)0 (t3 ) ≤ 0, t ∈ [t3 , t4 ].
Hence, t4 = 1 and Su(1) < 0. Since ξ ≤ η, we have (Su)0 (t) ≤ 0, t ∈ [η, 1] and
Z 1
0 > Su(1) = −
h(η, s)(Su)0 (s) ds ≥ 0,
η
which is a contradiction.
Therefore, t2 < 1 is not true. We have t2 = 1.
Case 2: If t1 > 0, we have Su(t) = 0 for t ∈ [t1 , 1], Su(t) < 0 for t ∈ (t1 , 1) and
Su(t1 ) = 0. Thus (Su)0 (t1 ) ≤ 0. We have [ϕp ((Su)0 (t))]0 = −f (t, 0, Au(t), Bu(t)) ≤
0 by (H2). This implies (Su)0 (t) ≤ 0 and Su(t) < 0 for t ∈ (t1 , 1] and Su(1) =
mint∈[t1 ,1] Su(t).
We can prove that
Su(t) ≥ 0 for t ∈ [0, t1 ].
(2.8)
If there exists a t5 ∈ [0, t1 ] such that Su(t5 ) < 0 and there is a maximal interval
[t6 , t7 ] which contains t5 such that Su(t) < 0 for t ∈ (t6 , t7 ). Obviously [t6 , t7 ) ∩
[t1 , 1] = ∅, so 1 6∈ (t6 , t7 ); i.e., t7 < 1, this is a contradiction with the above
discussion. Thus we can show Su(t) ≥ 0 for t ∈ [0, t1 ].
For Su(1) < 0, we have
Z 1
Su(1) = −
h(η, s)(Su)0 (s) ds.
η
Then, if η ≥ t1 , we have
Z
0 > Su(1) = −
η
1
h(η, s)(Su)0 (s) ds ≥ 0,
EJDE-2012/31
MULTIPLE POSITIVE SOLUTIONS
7
which is a contradiction.
If η < t1 , by using mean value theorem of integral, there exist η1 ∈ [η, t1 ] ⊂ [0, t1 ]
and η2 ∈ [t1 , 1] such that
Z t1
Z 1
Su(1) = −
h(η, s)(Su)0 (s) ds −
h(η, s)(Su)0 (s) ds
η
t1
t1
Z
= −h(η, t1 )
(Su)0 (s) ds − h(η, η2 )
Z
η1
1
(Su)0 (s) ds
t1
= −h(η, t1 )(Su(t1 ) − Su(η1 )) − h(η, η2 )(Su(1) − Su(t1 ))
= h(η, t1 )Su(η1 )) − h(η, η2 )Su(1),
and
0 > (1 + h(η, η2 ))Su(1) = h(η, t1 )Su(η1 ).
(2.9)
By (2.8), we have Su(η1 ) ≥ 0. Hence, (2.9) is a contradiction. Therefore t1 = 0.
The above also contradicts [t1 , t2 ] 6= [0, 1]. Thus Su(t) ≥ 0 for t ∈ [0, 1]. That is
u(t) is a fixed point of operator S.
Next we state the Krasnosel’skii Fixed Point Theorem [10].
Lemma 2.10. Let E be a Banach space and K ⊂ E be a cone in E. Assume Ω1
and Ω2 are open subsets of E with 0 ∈ Ω1 and Ω1 ⊂ Ω2 and A : K ∩ (Ω2 \Ω1 ) → K
be a completely continuous operator. In addition, suppose either
kAuk ≤ kuk,
u ∈ K ∩ ∂Ω1 , and kAuk ≥ kuk,
u ∈ K ∩ ∂Ω2 ;
kAuk ≥ kuk,
u ∈ K ∩ ∂Ω1 , and kAuk ≤ kuk,
u ∈ K ∩ ∂Ω2 .
or
hold. Then A has a fixed point in K ∩ (Ω2 \Ω1 ).
3. Main result
Denote
M = min
nZ
δ
1/2
ϕq
Z
1/2
Z
w(τ ) dτ ds,
s
1−δ
ϕq
1
g(ξ, s) ds)ϕq
(1 +
o
w(τ ) dτ ds ,
w(τ ) dτ ,
0
1
h(η, s) ds)ϕq
0
1
Z
0
Z
s
1/2
1/2
Z
n
N = max (1 +
Z
Z
1
w(τ ) dτ
o
.
0
For the next theorem, we assume that f satisfies the following growth conditions:
(H4) f (t, u, x, y) ≥ 0 for (t, u, x, y) ∈ [0, 1] × [c1 , c3 ] × R2 ;
(H5) f (t, u, x, y) < ϕp ( cN2 ) for (t, u, x, y) ∈ [0, 1] × [0, c2 ] × R2 ;
c3
(H6) f (t, u, x, y) ≥ ϕp ( M
) for (t, u, x, y) ∈ [δ, 1 − δ] × [δc3 , c3 ] × R2 .
Theorem 3.1. Suppose (H1)–(H6) hold. There exist constants c1 , c2 , c3 such that
n g(ξ, ξ)
h(η, η) o
0 < c1 ≤ min
,
δc2 , c2 < δc3 < c3 .
1 + g(ξ, 0) 1 + h(η, 1)
Then (1.1) has at least two positive solutions u1 and u2 such that
0 < ku1 k < c2 ≤ ku2 k ≤ c3 .
8
M. JIA, P. WANG
EJDE-2012/31
Proof. Let Ω1 = {u ∈ K : kuk < c2 }. For any u ∈ Ω1 , we have u ∈ K and kuk ≤ c2 .
Denote
kT uk = max |T u(t)| = T u(t).
0≤t≤1
If t < σu , it follows from (H5 ) that
hZ ξ
Z σu
g(ξ, s)ϕq
w(τ )f (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds
T u(t) =
0
s
t̄
Z
+
ϕq
0
Z
σu
Z
i+
w(τ )f (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds
s
1
Z
1
c2
<
g(ξ, s)ϕq
w(τ )ϕp ( ) dτ ds +
N
0
0
Z 1
Z 1
h
i
c2
w(τ ) dτ ) 1 +
g(ξ, s) ds
= ϕq
N
0
0
≤ c2 ,
and if t̄ > σu , we have
hZ 1
Z
T u(t) =
h(η, s)ϕq
η
Z
+
ϕq
0
1
Z
w(τ )ϕp (
0
c2
) dτ ds
N
w(τ )f (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds
σu
1
ϕq
Z
t̄
Z
s
1
Z
s
i+
w(τ )f (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds
σu
1
Z
1
c2
<
h(η, s)ϕq
w(τ )ϕp ( ) dτ ds +
N
0
0
Z 1
Z 1
h
i
c2
w(τ ) dτ 1 +
h(η, s) ds
= ϕq
N
0
0
≤ c2 .
Z
1
ϕq
0
Z
0
1
w(τ )ϕp (
c2
) dτ ds
N
We have
kT uk < c2 = kuk for u ∈ Ω1 .
(3.1)
By using Schauder fixed point theorem, we can get the T has at least one fixed
point u1 in Ω1 . That is, T u1 = u1 and ku1 k < c2 . If ku1 k = 0, we have u1 ≡ 0,
t ∈ [0, 1], which is a contradiction with (H2). Hence, 0 < ku1 k < c2 .
It follows that (1.1) has at least one positive solutions u1 such that 0 < ku1 k < c2
from Lemma 2.9. Let
Ω2 = {u ∈ K : kuk < c3 }.
For any u ∈ ∂Ω2 , we have u ∈ K and kuk = c3 and δc3 ≤ u(t) ≤ c3 for δ ≤ t ≤ 1 − δ.
By Lemma 2.4, we have
1
.
kT ∗ uk = T ∗ u(σu ) ≥ T ∗ u
2
If σu ≥ 1/2, it follows from (H6) that
Z ξ
Z σu
kT ∗ uk ≥
g(ξ, s)ϕq
w(τ )f + (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds
0
Z
+
1/2
ϕq
0
Z
s
s
σu
w(τ )f + (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds
EJDE-2012/31
MULTIPLE POSITIVE SOLUTIONS
1/2
Z
≥
ϕq
δ
w(τ )f + (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds
s
1/2
Z
1/2
Z
9
1/2
Z
c3
) dτ ds
M
δ
s
Z 1/2 Z 1/2
c3
=
ϕq
w(τ ) dτ ds
M δ
s
≥ c3 .
≥
ϕq
w(τ )ϕp (
If σu < 1/2, it follows from (H6) that
Z 1
Z s
kT ∗ uk ≥
h(η, s)ϕq
w(τ )f + (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds
η
σu
1
Z
+
ϕq
Z
1−δ
≥
ϕq
Z
1−δ
≥
ϕq
Z
1/2
c3
=
M
≥ c3 .
s
w(τ )f + (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds
1/2
1/2
Z
w(τ )f + (τ, u(τ ), Au(τ ), Bu(τ )) dτ ds
σu
1/2
Z
s
s
1/2
Z
1−δ
ϕq
1/2
c3
) dτ ds
M
w(τ ) dτ ds
w(τ )ϕp (
Z
s
1/2
Hence, we can show that
kT ∗ uk ≥ c3 = kuk
foru ∈ ∂Ω2 .
(3.2)
As in the proof of (3.1), we obtain
kT ∗ uk < c2 = kuk
for u ∈ ∂Ω1 .
Therefore, by using Krasnosel’skii fixed point theorem, T ∗ has at least one fixed
point u2 ∈ Ω2 \Ω1 with c2 ≤ ku2 k ≤ c3 . Subsequently, we prove u2 is also a fixed
point of S.
If u ∈ Ω2 \Ω1 and T ∗ u = u, then u ∈ K, c2 ≤ kuk ≤ c3 and mint∈[δ,1−δ] u(t) ≥
δkuk ≥ δc2 . By Lemma 2.4, Lemma 2.7 and the definition of T ∗ , we obtain
Z σu
(T ∗ u)0 (t) = ϕq
w(τ )f + (τ, u(τ ), Au(τ ), Bu(τ )) dτ ) ds , t ∈ [0, 1]
t
(T ∗ u)0 (σu ) = 0, (T ∗ u)0 (t) ≥ 0 if t ∈ [0, σu ], (T ∗ u)0 (t) ≤ 0 if t ∈ [σu , 1]
and
min u(t) = min{u(0), u(1)} ≥ 0.
0≤t≤1
If min0≤t≤1 u(t) = u(0), when ξ ≤ σu , by (H3) and mean value theorem of integral,
there exists a ξ1 ∈ [0, ξ] such that
Z ξ
Z ξ
u(0) = Au(ξ) =
g(ξ, s)u0 (s) ds = g(ξ, ξ1 )
u0 (s) ds ≥ g(ξ, ξ)(u(ξ) − u(0))
0
0
and
u(0) ≥
g(ξ, ξ)u(ξ)
δg(ξ, ξ)
δg(ξ, ξ)c2
δg(ξ, ξ)c2
≥
kuk ≥
≥
.
1 + g(ξ, ξ)
1 + g(ξ, ξ)
1 + g(ξ, ξ)
1 + g(ξ, 0)
(3.3)
10
M. JIA, P. WANG
EJDE-2012/31
When ξ > σu , by (H3) and mean value theorem of integral, there exist ξ2 ∈ [0, σu ]
and ξ3 ∈ [σu , ξ] such that
Z σu
Z ξ
u(0) =Au(ξ) =
g(ξ, s)u0 (s) ds +
g(ξ, s)u0 (s) ds
0
σu
σu
Z
= g(ξ, ξ2 )
Z
0
ξ
u (s) ds + g(ξ, ξ3 )
0
u0 (s) ds
σu
= g(ξ, ξ2 )(u(σu ) − u(0)) + g(ξ, ξ3 )(u(ξ) − u(σu ))
≥g(ξ, ξ)u(ξ) − g(ξ, 0)u(0)
and
δg(ξ, ξ)
δg(ξ, ξ)c2
g(ξ, ξ)u(ξ)
≥
kuk ≥
.
(3.4)
1 + g(ξ, 0)
1 + g(ξ, 0)
1 + g(ξ, 0)
If min0≤t≤1 u(t) = u(1), when η ≤ σu , by (H3) and mean value theorem of
integral, there exist η1 ∈ [η, σu ] and η2 ∈ [σu , 1] such that
Z σu
Z 1
u(1) = −Bu(η) = −
h(η, s)u0 (s) ds −
h(η, s)u0 (s) ds
u(0) ≥
η
Z
= −h(η, η1 )
σu
σu
u0 (s) ds − h(η, η2 )
η
Z
1
u0 (s) ds
σu
= −h(η, η1 )(u(σu ) − u(η)) − h(η, η2 )(u(1) − u(σu ))
= h(η, η2 )u(η) − h(η, η1 )u(1) + (h(η, η2 ) − h(η, η1 ))u(σu )
≥ h(η, η)u(η) − h(η, 1)u(1)
and
δh(η, η)
δh(η, η)c2
h(η, η)u(η)
≥
kuk ≥
.
(3.5)
1 + h(η, 1)
1 + h(η, 1)
1 + h(η, 1)
When η > σu , by (H3) and mean value theorem for integrals, there exists η3 ∈ [η, 1]
such that
Z 1
u(1) = −Bu(η) = −
h(η, s)u0 (s) ds
u(1) ≥
η
= h(η, η3 )(u(η) − u(1)) ≥ h(η, η)(u(η) − u(1))
and
u(1) ≥
h(η, η)u(η)
δh(η, η)
δh(η, η)c2
δh(η, η)c2
≥
kuk ≥
≥
.
1 + h(η, η)
1 + h(η, η)
1 + h(η, η))
1 + h(η, 1)
(3.6)
Hence, it follows
min u(t) ≥ min
0≤t≤1
n g(ξ, ξ)
h(η, η) o
,
δc2 ≥ c1
1 + g(ξ, 0) 1 + h(η, 1)
from (3.3)–(3.6).
Therefore, if u ∈ Ω2 \ Ω1 and T ∗ u = u, we have
c1 ≤ u(t) ≤ kuk ≤ c3 .
It follows f (t, u(t), Au(t), Bu(t)) ≥ 0, t ∈ [0, 1] from (H4). Thus, T ∗ u = Su.
That is, the fixed point of T ∗ on Ω2 \ Ω1 is also a fixed point of S. We can get
the boundary-value problem (1.1) has at least one positive solutions u2 such that
c2 ≤ ku2 k ≤ c3 . The proof is complete.
EJDE-2012/31
MULTIPLE POSITIVE SOLUTIONS
11
For the next theorem, we have a new assumption:
(H7) For each t ∈ [0, 1], g(t, s) and h(t, s) are monotone with respect to s.
We denote
Mg = max{ max g(t, 0), max g(t, t)},
t∈[0,1]
t∈[0,1]
Mh = max{ max h(t, 1), max h(t, t)}.
t∈[0,1]
t∈[0,1]
If (H7) holds, for each t ∈ [0, 1], when g(t, s) is decreasing with respect to s, there
exists a ξ¯1 ∈ [0, t] such that
Z t
Z ξ̄1
0
|(Au)(t)| = |
g(t, s)u (s) ds| = g(t, 0)|
u0 (s) ds| ≤ 2g(t, 0)kuk ≤ 2Mg kuk,
0
0
and when g(t, s) is monotone creasing with respect to s, there exists a ξ¯2 ∈ [0, t]
such that
Z t
Z t
0
g(t, s)u (s) ds| = g(t, t)|
u0 (s) ds| ≤ 2g(t, t)kuk ≤ 2Mg kuk.
|(Au)(t)| = |
0
ξ̄2
As above, if (H7) holds, we can show that
|(Bu)(t)| ≤ 2Mh kuk,
for each t ∈ [0, 1].
As in the proof of Theorem 3.1, we obtain the following theorem under the assumption taht f satisfies the following growth conditions:
(H8) f (t, u, x, y) ≥ 0 for (t, u, x, y) ∈ [0, 1] × [c1 , c3 ] × [−2Mg c3 , 2Mg c3 ]
× [−2Mh c3 , 2Mh c3 ];
(H9) f (t, u, x, y) < ϕp ( cN2 ) for (t, u, x, y) ∈ [0, 1] × [0, c2 ] × [−2Mg c2 , 2Mg c2 ] ×
[−2Mh c2 , 2Mh c2 ];
c3
) for (t, u, x, y) ∈ [δ, 1−δ]×[δc3 , c3 ]×[−2Mg c3 , 2Mg c3 ]×
(H10) f (t, u, x, y) ≥ ϕp ( M
[−2Mh c3 , 2Mh c3 ].
Theorem 3.2. Suppose (H1)–(H3) and (H7)–(H10) hold. Then there exist constants c1 , c2 , c3 such that such that
n g(ξ, ξ)
h(η, η) o
0 < c1 ≤ min
,
δc2 , and c2 < δc3 < c3
1 + g(ξ, 0) 1 + h(η, 1)
Then (1.1) has at least two positive solutions u1 and u2 such that
0 < ku1 k < c2 ≤ ku2 k ≤ c3 .
We have a new assumption:
(H11) For each t, s ∈ [0, 1], gs (t, s) and hs (t, s) are bounded.
We denote
Ng = max{ max g(t, 0), max g(t, t), sup gs (t, s)},
t∈[0,1]
t∈[0,1]
t,s∈[0,1]
Nh = max{ max h(t, 1), max h(t, t), sup hs (t, s)}
t∈[0,1]
t∈[0,1]
t,s∈[0,1]
If (H11) holds, for each t ∈ [0, 1], we have
Z t
|(Au)(t)| = |
g(t, s)u0 (s) ds|
0
Z
= |(g(t, t)u(t) − g(t, 0)u(0)) −
t
gs (t, s)u(s) ds| ≤ 3Ng kuk.
0
12
M. JIA, P. WANG
EJDE-2012/31
As above, if (H11) holds, we can show that
|(Bu)(t)| ≤ 3Nh kuk,
for each t ∈ [0, 1].
As in the proof of Theorem 3.1, we can get the following theorem, and f satisfies
the following growth conditions:
(H12) f (t, u, x, y) ≥ 0 for (t, u, x, y) ∈ [0, 1] × [c1 , c3 ] × [−3Ng c3 , 3Ng c3 ]
× [−3Nh c3 , 3Nh c3 ];
(H13) f (t, u, x, y) < ϕp ( cN2 ) for (t, u, x, y) ∈ [0, 1] × [0, c2 ] × [−3Ng c2 , 3Ng c2 ] ×
[−3Nh c2 , 3Nh c2 ];
c3
) for (t, u, x, y) ∈ [δ, 1−δ]×[δc3 , c3 ]×[−3Ng c3 , 3Ng c3 ]×
(H14) f (t, u, x, y) ≥ ϕp ( M
[−3Nh c3 , 3Nh c3 ].
Theorem 3.3. Suppose (H1)–(H3) and (H11–(H14)) hold. Then there exist nonnegative constants c1 , c2 , c3 such that
n g(ξ, ξ)
h(η, η) o
0 < c1 ≤ min
,
δc2 , and c2 < δc3 < c3
1 + g(ξ, 0) 1 + h(η, 1)
Then (1.1) has at least two positive solutions u1 and u2 such that
0 < ku1 k < c2 ≤ ku2 k ≤ c3 .
References
[1] R.P. Agarwal, D. O’Regan; Some new results for singular problems with sign changing nonlinearities, J. Comput. Appl. Math. 113 (2000), 1-15.
[2] R .P. Agarwal, D. O’Regan, V. Lakshmikantham, S. Leela, Nonresonant singular boundaryvalue problems with sign changing nonlinearities, Appl. Math. Comput. 167 (2005), 12361248.
[3] B. Ahmad, S. Sivasundaram; Existence and approximation of solutions of forced duffing
type integro-differential equations with three-point nonlinear boundary conditions, Nonlinear
Anal.: Real World Applications 11 (2010), 2905-2912.
[4] Qasem M. Al-Mdallal; Monotone iterative sequences for nonlinear integro-differential equations of second order, Nonlinear Anal.: Real World Applications 12 (2011), 3665-3673
[5] Z. Bai, Z. Gui, W. Ge; Multiple positive solutions for some p-Laplacian boundary-value
problems, J. Math. Anal. Appl. 300 (2004), 477-490.
[6] W. Cheung, J. Ren; Twin positive solutions for quasi-linear multi-point boundary-value problems, Nonlinear Anal. 62 (2005), 167-177.
[7] H. Feng, W. Ge; Existence of three positive solutions for m-point boundary-value problems
with one-dimensional p-Laplacian, Nonlinear anal. 68 (2008), 2017-2026.
[8] J. M. Gallardo; Second order differential operators with integral boundary conditions and
generation of semigroups, Rocky Mountain J. Math. 30 (2000), 1265-1292.
[9] W. Ge. The boundary-value problems for nonlinear differential equations. Science Press, Beijing. 2007, 147- 228. (In Chinese)
[10] D. Guo, V. Lakshmikantham; Nonlinear problems in abstract cones, Academic Press, San
Diego, 1988.
[11] Y. Guo, W. Ge, S. Dong; Two positive solutions for second order three point value problems
with sign change nonlinearities, Acta. Math. Appl. Sinica. 27 (2004), 522-529. (In Chinese)
[12] D. Ji, Y. Tian, W. Ge; Positive solutions for one-dimensional p-Laplacian boundary-value
problems with sign changing nonlinearity, Nonlinear Anal., 71 (2009) 5406-5416.
[13] M. Jia, P. Wang; Existence of positive solutions for boundary-value problems with onedimensional p-Laplacian, Proceedings of the 5th International Congress on Mathematical
Biology (ICMB2011, Nanjing) Vol. 1, 125-130.
[14] G. L. Karakostas, P. Ch. Tsamatos; Multiple positive solutions of some Fredholm integral
equations arisen from nonlocal boundary-value problems, Electron. J. Differential Equations
30 (2002), 1-17.
[15] A. Lakmeche, A. Hammoudi; Multiple positive solutions of the one-dimensional p-Laplacian,
J. Math. Anal. Appl. 317 (2006), 43-49.
EJDE-2012/31
MULTIPLE POSITIVE SOLUTIONS
13
[16] H. Lian, W. Ge; Positive solutions for a four-point boundary-value problem with the pLaplacian, Nonlinear Anal., 68 (2008), 3493-3503
[17] S. Liu, M. Jia, Y. Tian; Existence of positive solutions for boundary-value problems with integral boundary conditions and sign changing nonlinearities, Electron. J. Differential Equations,
Vol. 2010(2010), No. 163, pp. 1-12.
[18] A. Lomtatidze, L. Malaguti; On a nonlocal boundary-value problems for second order nonlinear singular differential equations, Georgian Math. J. 7 (2000), 133-154.
[19] D. Ma, Z. Du, W. Ge; Existence and iteration of monotone positive solutions for multipoint
boundary-value problem with p-Laplacian operator, Comput. Math. Appl. 50 (2005), 729-739.
[20] R. Ma; Positive solutions for second-order functional differential equations, Dynam. Systems
Appl. 10 (2001), 215–224.
[21] Y. Wang, C. Hou; Existence of multiple positive solutions for one-dimensional p-Laplacian,
J. Math. Anal. Appl. 315 (2006), 144-153.
[22] A. Wazwaz; A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equations, Appl. Math. Comput. 181 (2006),
1703-1712.
[23] J .R. L. Webb, G. Infante; Positive solutions of nonlocal boundary-value problems involving
integral conditions, NoDEA Nonlinear Differential Equations Appl., 15(2008) 45–67.
[24] J. R .L. Webb, G. Infante; Positive solutions of nonlocal boundary-value problems: a unified
approach, J. London Math. Soc. 74(2006) 673–693.
[25] S. Xi, M. Jia, H. Ji; Positive solutions of boundary-value problems for systems of secondorder differential equations with integral boundary condition on the half-line, Electron. J.
Qual. Theory Differ. Equ. 1 (2009), 1-12.
Mei Jia
College of Science, University of Shanghai for Science and Technology, Shanghai
200093, China
E-mail address: jiamei-usst@163.com
Pingyou Wang
College of Science, University of Shanghai for Science and Technology, Shanghai
200093, China
E-mail address: wpingy2008@163.com