- Research Article
- Open access
- Published:
Existence of Positive Solutions of Fourth-Order Problems with Integral Boundary Conditions
Boundary Value Problems volume 2011, Article number: 297578 (2011)
Abstract
We study the existence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions, ,
,
,
,
,
, where
is continuous,
are nonnegative. The proof of our main result is based upon the Krein-Rutman theorem and the global bifurcation techniques.
1. Introduction
The deformations of an elastic beam in an equilibrium state, whose both ends are simple supported, can be described by the fourth-order boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ2_HTML.gif)
where is continuous; see Gupta [1, 2]. In the past twenty more years, the existence of solutions and positive solutions of these kinds of problems and the Lidstone problem has been extensively studied; see [3–9] and the references therein. In [3], Ma was concerned with the existence of positive solutions of (1.1) and (1.2) under the assumptions:
(H1) is continuous and there exist constants
, with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ3_HTML.gif)
uniformly for , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ4_HTML.gif)
uniformly for , where
;
(H2) for
and
;
(H3) there exist constants satisfying
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ5_HTML.gif)
Ma proved the following.
Theorem A (see [3, Theorem ]).
Let (H1), (H2), and (H3) hold. Assume that either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ6_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ7_HTML.gif)
where denotes the first generalized eigenvalue of the generalized eigenvalue problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ8_HTML.gif)
Then (1.1) and (1.2) have at least one positive solution.
At the same time, we notice that a class of boundary value problems with integral boundary conditions appeared in heat conduction, chemical engineering underground water flow, thermoelasticity, and plasma physics. Such a kind of problems include two-point, three-point, multipoint and nonlocal boundary value problems as special cases and attracting the attention of a few readers; see [10–13] and the references therein. For example, In particular, Zhang and Ge [10] used Guo-Krasnoselskii fixed-point theorem to study existence and nonexistence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ9_HTML.gif)
where may be singular at
and (or)
;
is continuous, and
are nonnegative.
Motivated by [3, 10], in this paper, we consider the existence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ10_HTML.gif)
under the assumption
(H4) are nonnegative, and
. The main result of this paper is the following.
Theorem 1.1.
Let (H1), (H2), (H3), and (H4) hold. Assume that either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ11_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ12_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ15_HTML.gif)
Then (1.9) has at least one positive solution.
Remark 1.2.
Theorem 1.1 generalizes [3, Theorem ] where the special case
and
was treated.
Remark 1.3.
Zhang and Ge [10] proved existence and nonexistence of positive solutions via Guo-Krasnoselskii fixed-point theorem under some conditions which do not involve the eigenvalues of (1.12)–(1.14). While our Theorem 1.1 is established under (1.10) or (1.11) which is related to the eigenvalues of (1.12)–(1.14). Moreover, (1.10) and (1.11) are optimal. Let us consider the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ17_HTML.gif)
In this case, and the corresponding eigenfunction is
. However, (1.15) and (1.16) has no positive solution. (In fact, suppose on the contrary that (1.15) and (1.16) has a positive solution
. Multiplying (1.15) with
and integrating from
to
, we get a desired contradiction!).
Suppose that is a real Banach space with norm
. Let
be a cone in
. A nonlinear mapping
is said to be positive if
. It is said to be
-completely continuous if
is continuous and maps bounded subsets of
to precompact subset of
. Finally, a positive linear operator
on
is said to be a linear minorant for
if
for
. If
is a continuous linear operator on
, denote
the spectral radius of
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ18_HTML.gif)
The following lemma will play a very important role in the proof of our main results, which is essentially a consequence of Dancer [14, Theorem ].
Lemma 1.4.
Assume that
-
(i)
has nonempty interior and
;
-
(ii)
is
-completely continuous and positive,
for
,
for
and
(1.18)
where is a strongly positive linear compact operator on
with the spectral radius
,
satisfies
as
locally uniformly in
.
Then there exists an unbounded connected subset of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ20_HTML.gif)
such that .
Moreover, if has a linear minorant
and there exists a
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ21_HTML.gif)
such that and
, then
can be chosen in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ22_HTML.gif)
Proof.
Since is a strongly positive compact endomorphism of
and
has nonempty interior, we have from Amann [15, Theorem
] that the set
in [14, Theorem
] reduces to a single point
. Now the desired result is a consequence of Dancer [14, Theorem
].
The rest of the paper is arranged as follows. In Section 2, we state and prove some preliminary results about the spectrum of (1.12)–(1.14). Finally, in Section 3, we proved our main result.
2. Generalized Eigenvalues
Lemma 2.1 (see [10]).
Assume that (H4) holds. Then for any , the boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ23_HTML.gif)
has a unique solution which is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ24_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ25_HTML.gif)
Lemma 2.2 (see [10]).
Assume that (H4) holds. Then for any , the boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ26_HTML.gif)
has a unique solution which is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ27_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ28_HTML.gif)
Lemma 2.3 (see [10]).
Assume that (H4) holds. Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ29_HTML.gif)
Let
(H5) be two given constants with
.
Definition 2.4.
One says that
is a generalized eigenvalue of linear problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ31_HTML.gif)
if (2.8) and (2.9) have nontrivial solutions.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ32_HTML.gif)
Let with the norm
. Let
with the norm
.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ33_HTML.gif)
For , from Lemma 2.1, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ34_HTML.gif)
By simple calculations, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ35_HTML.gif)
Combining this with (H4), we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ36_HTML.gif)
This together with (2.12) and the fact that imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ37_HTML.gif)
Since , we may define the norm of
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ38_HTML.gif)
We claim that is a Banach space.
In fact, let be a Cauchy sequence, that is,
as
. From the definition of
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ39_HTML.gif)
where is a normal in
defined by
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ40_HTML.gif)
By the completeness of , there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ41_HTML.gif)
From the fact that , we have that for arbitrary
, there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ42_HTML.gif)
and subsequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ43_HTML.gif)
Fixed and let
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ44_HTML.gif)
This is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ45_HTML.gif)
Therefore, is a Banach space.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ46_HTML.gif)
Then the cone is normal and nonempty interior
and
.
In fact, for any , it follows from the definition of
that
(1) there exist real number , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ47_HTML.gif)
(2) ,
.
From and (H4), we obtain that
for some
. Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ48_HTML.gif)
and subsequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ49_HTML.gif)
for some . We may take
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ50_HTML.gif)
Now, let us define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ51_HTML.gif)
Then , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ52_HTML.gif)
Thus, . Obviously,
.
Lemma 2.5.
Assume that holds. Then for any
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ53_HTML.gif)
Proof.
In fact, for , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ54_HTML.gif)
From , we have that
, and so
, and accordingly
.
We have from the fact that ,
, that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ55_HTML.gif)
which implies that , and consequently
.
For , define a linear operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ56_HTML.gif)
Theorem 2.6.
Assume that (H4) and (H5) hold. Let be the spectral radius of
. Then (2.8) and (2.9) has an algebraically simple eigenvalue,
, with a positive eigenfunction
. Moreover, there is no other eigenvalue with a positive eigenfunction.
Remark 2.7.
If , then
can be explicitly given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ57_HTML.gif)
and the corresponding eigenfunction .
Proof of Theorem 2.6.
From Lemma 2.2, it is easy to check that (2.8) and (2.9) is equivalent to the integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ58_HTML.gif)
We claim that .
In fact, for , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ59_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ60_HTML.gif)
and for some constant , it concludes that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ61_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ62_HTML.gif)
where , it follows that
, and accordingly
.
If , then
on
, and accordingly
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ63_HTML.gif)
Thus , and accordingly
.
Now, since , and
is compactly embedded in
, we have that
is compact.
Next, we show that is positive.
For , if
, from Lemma 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ64_HTML.gif)
Combining this with (2.39), there exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ65_HTML.gif)
For , if
, applying a similar proof process of (2.43), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ66_HTML.gif)
Combining this with (2.39), there exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ67_HTML.gif)
This together with (2.9) and (H4) imply on
.
Therefore, it follows from (2.43) and (2.45) that .
Now, by the Krein-Rutman theorem ([16, Theorem C]; [17, Theorem
]),
has an algebraically simple eigenvalue
with an eigenfunction
. Moreover, there is no other eigenvalue with a positive eigenfunction. Correspondingly,
with a positive eigenfunction of
, is a simple eigenvalue of (2.8) and (2.9). Moreover, for (2.8) and (2.9), there is no other eigenvalue with a positive eigenfunction.
3. The Proof of the Main Result
Before proving Theorem 1.1, we denote by setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ68_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ69_HTML.gif)
It is easy to check that is compact.
Let be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ70_HTML.gif)
Obviously,()impliesthat
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ71_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ72_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ73_HTML.gif)
then is nondecreasing and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ74_HTML.gif)
Let us consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ75_HTML.gif)
as a bifurcation problem from the trivial solution . It is to easy to check that (3.8) can be converted to the equivalent equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ76_HTML.gif)
From the proof process of Theorem 2.6, the operator ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ77_HTML.gif)
is compact and strongly positive. Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ78_HTML.gif)
then we have from (3.4) and Lemma 2.5 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ79_HTML.gif)
locally uniformly in . From
and Theorem 2.6 (with obvious changes), it follows that if
is a nontrivial solution of (3.8) with
, then
. Combining this with Lemma 1.4, we conclude that there exists an unbounded connected subset
of the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ80_HTML.gif)
such that .
Proof of Theorem 1.1.
It is clear that any solution of the form yields a solution
of (1.9). We will show that
crosses the hyperplane
. To do this, it is enough to show that
joins
to
. Let
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ81_HTML.gif)
we note that for all
since
is the only solution of (3.8) for
and
.
Case 1 ().
In this case, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ82_HTML.gif)
We divide the proof into two steps.
Step 1.
We show that if there exists a constant number such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ83_HTML.gif)
then joins
to
.
From (3.16), we have that . We divide the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ84_HTML.gif)
by and set
. Since
is bounded in
, choosing a subsequence and relabeling if necessary, we see that
for some
with
. Moreover, we have from (3.7) and Lemma 2.5 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ85_HTML.gif)
Since Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ86_HTML.gif)
where , again choosing a subsequence and relabeling if necessary. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ87_HTML.gif)
This together with Theorem 2.6 imply that . Therefore,
joins
to
.
Step 2.
We show that there exists a constant be such that
for all
.
By Lemma 1.4, we only need to show that has a linear minorant
and there exists a
such that
and
.
By,there exist constants
satisfying
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ88_HTML.gif)
For , let
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then is a linear minorant of
. Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ90_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ91_HTML.gif)
Combining this with (2.39), we conclude that , here,
. Therefore, we have that from Lemma 1.4 that
.
Case 2 ().
In this case, if such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ92_HTML.gif)
and then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ93_HTML.gif)
and, moreover, .
Assume that there exists , such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ94_HTML.gif)
Applying a similar argument to that used in Step 1 of Case 1, after taking a subsequence and relabeling if necessary, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ95_HTML.gif)
Again joins
to
and the result follows.
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Acknowledgments
The authors are very grateful to the anonymous referees for their valuable suggestions. This paper was supported by the NSFC (no. 11061030), the Fundamental Research Funds for the Gansu Universities.
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Ma, R., Chen, T. Existence of Positive Solutions of Fourth-Order Problems with Integral Boundary Conditions. Bound Value Probl 2011, 297578 (2011). https://doi.org/10.1155/2011/297578
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DOI: https://doi.org/10.1155/2011/297578