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Positive Solutions for Integral Boundary Value Problem with ϕ-Laplacian Operator
Boundary Value Problems volume 2011, Article number: 827510 (2011)
Abstract
We consider the existence, multiplicity of positive solutions for the integral boundary value problem with -Laplacian
,
,
,
, where
is an odd, increasing homeomorphism from
onto
. We show that it has at least one, two, or three positive solutions under some assumptions by applying fixed point theorems. The interesting point is that the nonlinear term
is involved with the first-order derivative explicitly.
1. Introduction
We are interested in the existence of positive solutions for the integral boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ1_HTML.gif)
where , and
satisfy the following conditions.
(H1) is an odd, increasing homeomorphism from
onto
, and there exist two increasing homeomorphisms
and
of
onto
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ2_HTML.gif)
Moreover, , where
denotes the inverse of
.
(H2) is continuous.
are nonnegative, and
,
.
The assumption (H1) on the function was first introduced by Wang [1, 2], it covers two important cases:
and
. The existence of positive solutions for two above cases received wide attention (see [3–10]). For example, Ji and Ge [4] studied the multiplicity of positive solutions for the multipoint boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ3_HTML.gif)
where ,
. They provided sufficient conditions for the existence of at least three positive solutions by using Avery-Peterson fixed point theorem. In [5], Feng et al. researched the boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ4_HTML.gif)
where the nonlinear term does not depend on the first-order derivative and
,
. They obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of
by applying Krasnoselskii fixed point theorem.
As for integral boundary value problem, when is linear, the existence of positive solutions has been obtained (see [8–10]). In [8], the author investigated the positive solutions for the integral boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ5_HTML.gif)
The main tools are the priori estimate method and the Leray-Schauder fixed point theorem. However, there are few papers dealing with the existence of positive solutions when satisfies (H1) and
depends on both
and
. This paper fills this gap in the literature. The aim of this paper is to establish some simple criteria for the existence of positive solutions of BVP(1.1). To get rid of the difficulty of
depending on
, we will define a special norm in Banach space (in Section 2).
This paper is organized as follows. In Section 2, we present some lemmas that are used to prove our main results. In Section 3, the existence of one or two positive solutions for BVP(1.1) is established by applying the Krasnoselskii fixed point theorem. In Section 4, we give the existence of three positive solutions for BVP(1.1) by using a new fixed point theorem introduced by Avery and Peterson. In Section 5, we give some examples to illustrate our main results.
2. Preliminaries
The basic space used in this paper is a real Banach space with norm
defined by
, where
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ6_HTML.gif)
It is obvious that is a cone in
.
Lemma 2.1 (see [7]).
Let , then
,
.
Lemma 2.2.
Let , then there exists a constant
such that
.
Proof.
The mean value theorem guarantees that there exists , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ7_HTML.gif)
Moreover, the mean value theorem of differential guarantees that there exists , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ8_HTML.gif)
So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ9_HTML.gif)
Denote ; then the proof is complete.
Lemma 2.3.
Assume that (H1), (H2) hold. If is a solution of BVP(1.1), there exists a unique
, such that
and
,
.
Proof.
From the fact that , we know that
is strictly decreasing. It follows that
is also strictly decreasing. Thus,
is strictly concave on [0, 1]. Without loss of generality, we assume that
. By the concavity of
, we know that
,
. So we get
. By
, it is obvious that
. Hence,
,
.
On the other hand, from the concavity of , we know that there exists a unique
where the maximum is attained. By the boundary conditions and
, we know that
or 1, that is,
such that
and then
.
Lemma 2.4.
Assume that (H1), (H2) hold. Suppose is a solution of BVP(1.1); then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ10_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ11_HTML.gif)
Proof.
First, by integrating (1.1) on , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ12_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ13_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ14_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ15_HTML.gif)
According to the boundary condition, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ16_HTML.gif)
By a similar argument in [5], ; then the proof is completed.
Now we define an operator by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ17_HTML.gif)
Lemma 2.5.
is completely continuous.
Proof.
Let ; then from the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ18_HTML.gif)
So is monotone decreasing continuous and
. Hence,
is nonnegative and concave on [0, 1]. By computation, we can get
. This shows that
. The continuity of
is obvious since
is continuous. Next, we prove that
is compact on
.
Let be a bounded subset of
and
is a constant such that
for
. From the definition of
, for any
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ19_HTML.gif)
Hence, is uniformly bounded and equicontinuous. So we have that
is compact on
. From (2.13), we know for
,
, such that when
, we have
. So
is compact on
; it follows that
is compact on
. Therefore,
is compact on
.
Thus, is completely continuous.
It is easy to prove that each fixed point of is a solution for BVP(1.1).
Lemma 2.6 (see [1]).
Assume that (H1) holds. Then for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ20_HTML.gif)
To obtain positive solution for BVP(1.1), the following definitions and fixed point theorems in a cone are very useful.
Definition 2.7.
The map is said to be a nonnegative continuous concave functional on a cone of a real Banach space
provided that
is continuous and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ21_HTML.gif)
for all and
. Similarly, we say the map
is a nonnegative continuous convex functional on a cone of a real Banach space
provided that
is continuous and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ22_HTML.gif)
for all and
.
Let and
be a nonnegative continuous convex functionals on
,
a nonnegative continuous concave functional on
, and
a nonnegative continuous functional on
. Then for positive real number
, and
, we define the following convex sets:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ23_HTML.gif)
Theorem 2.8 (see [11]).
Let be a real Banach space and
a cone. Assume that
and
are two bounded open sets in
with
,
. Let
be completely continuous. Suppose that one of following two conditions is satisfied:
(1),
, and
,
;
(2),
, and
,
.
Then has at least one fixed point in
.
Theorem 2.9 (see [12]).
Let be a cone in a real Banach space
. Let
and
be a nonnegative continuous convex functionals on
,
a nonnegative continuous concave functional on
, and
a nonnegative continuous functional on
satisfying
for
, such that for positive number
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ24_HTML.gif)
for all . Suppose
is completely continuous and there exist positive numbers
, and
with
such that
and
for
;
() for
with
;
() and
for
with
.
Then has at least three fixed points
, such that
for
,
,
with
,
.
3. The Existence of One or Two Positive Solutions
For convenience, we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ25_HTML.gif)
where denotes 0 or
.
Theorem 3.1.
Assume that (H1) and (H2) hold. In addition, suppose that one of following conditions is satisfied.
-
(i)
There exist two constants
with
such that
(a) for
and
(b) for
;
(ii);
(iii).
Then BVP(1.1) has at least one positive solution.
Proof.
-
(i)
Let
,
.
For , we obtain
and
, which implies
. Hence, by (2.12) and Lemma 2.6,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ26_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ27_HTML.gif)
Next, for , we have
. Thus, by (2.12) and Lemma 2.6,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ28_HTML.gif)
From (2.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ29_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ30_HTML.gif)
Therefore, by Theorem 2.8, it follows that has a fixed point in
. That is BVP(1.1) has at least one positive solution such that
.
-
(ii)
Considering
, there exists
such that
(3.7)
Choosing such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ32_HTML.gif)
then for all , let
. For every
, we have
. In the following, we consider two cases.
Case 1 ().
In this case,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ33_HTML.gif)
Case 2 ().
In this case,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ34_HTML.gif)
Then it is similar to the proof of (3.6); we have for
.
Next, turning to , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ35_HTML.gif)
Let . For every
, we have
. So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ36_HTML.gif)
Then like in the proof of (3.3), we have for
. Hence, BVP(1.1) has at least one positive solution such that
.
-
(iii)
The proof is similar to the (i) and (ii); here we omit it.
In the following, we present a result for the existence of at least two positive solutions of BVP(1.1).
Theorem 3.2.
Assume that (H1) and (H2) hold. In addition, suppose that one of following conditions is satisfied.
(I),
, and there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ37_HTML.gif)
(II),
, and there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ38_HTML.gif)
Then BVP(1.1) has at least two positive solutions.
4. The Existence of Three Positive Solutions
In this section, we impose growth conditions on which allow us to apply Theorem 2.9 of BVP(1.1).
Let the nonnegative continuous concave functional , the nonnegative continuous convex functionals
,
, and nonnegative continuous functional
be defined on cone
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ39_HTML.gif)
By Lemmas 2.1 and 2.2, the functionals defined above satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ40_HTML.gif)
for all . Therefore, the condition (2.19) of Theorem 2.9 is satisfied.
Theorem 4.1.
Assume that (H1) and (H2) hold. Let and suppose that
satisfies the following conditions:
for
;
for
.
for
;
Then BVP(1.1) has at least three positive solutions , and
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ41_HTML.gif)
where defined as (3.1),
.
Proof.
We will show that all the conditions of Theorem 2.9 are satisfied.
If , then
. With Lemma 2.2 implying
, so by (
), we have
when
. Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ42_HTML.gif)
This proves that .
To check condition () of Theorem 2.9, we choose
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ43_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ44_HTML.gif)
Then and
, so
. Hence, for
, there is
,
when
. From assumption (
), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ45_HTML.gif)
It is similar to the proof of assumption (i) of Theorem 3.1; we can easily get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ46_HTML.gif)
This shows that condition () of Theorem 2.9 is satisfied.
Secondly, for with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ47_HTML.gif)
Thus condition () of Theorem 2.9 holds.
Finally, as , there holds
. Suppose that
with
; then by the assumption (
),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ48_HTML.gif)
So like in the proof of assumption (i) of Theorem 3.1, we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ49_HTML.gif)
Hence condition () of Theorem 2.9 is also satisfied.
Thus BVP(1.1) has at least three positive solutions , and
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ50_HTML.gif)
5. Examples
In this section, we give three examples as applications.
Example 5.1.
Let ,
. Now we consider the BVP
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ51_HTML.gif)
where for
.
Let ,
. Choosing
. By calculations we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ52_HTML.gif)
For ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ53_HTML.gif)
for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ54_HTML.gif)
Hence, by Theorem 3.1, BVP(5.1) has at least one positive solution.
Example 5.2.
Let ,
. Consider the BVP
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ55_HTML.gif)
where for
.
Let ,
. Then
. It easy to see
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ56_HTML.gif)
Choosing , for
,
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ57_HTML.gif)
Hence, by Theorem 3.2, BVP(5.5) has at least two positive solutions.
Example 5.3.
Let ,
; consider the boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ58_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ59_HTML.gif)
Choosing ,
,
,
, then by calculations we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ60_HTML.gif)
It is easy to check that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ61_HTML.gif)
Thus, according to Theorem 4.1, BVP(5.8) has at least three positive solutions , and
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ62_HTML.gif)
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Acknowledgments
The research was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNU-KJCXGC-3-47.
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Ding, Y. Positive Solutions for Integral Boundary Value Problem with ϕ-Laplacian Operator. Bound Value Probl 2011, 827510 (2011). https://doi.org/10.1155/2011/827510
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DOI: https://doi.org/10.1155/2011/827510