Existence of Positive Solutions for a Third-Order Multi-Point Boundary Value Problem ()
where 
for 
The interesting point lies in the fact that the nonlinear term is allowed to depend on the first order derivative
.
1. Introduction
It shows that problems related to nonlocal conditions have many applications in many problems such as in the theory of heat conduction, thermoelasticity, plasma physics, control theory, etc. The current analysis of these problems has a great interest and many methods are used to solve such problems. Recently certain three point boundary value problems for nonlinear ordinary differential equations have been studied by many authors [1-9]. The literature concerning these problems is extensive and application of theorems of functional analysis has attracted more interest. Recently, the study of existence of positive solution to third-order boundary value problems has gained much attention and is a rapidly growing field see [1,2,6,8-11]. However the approaches used in the literature are usually topological degree theory and fixed-point theorems in cone. We are interested in the existence, uniqueness and positivity of solution to the third-order multi-point nonhomogeneous boundary value problem (BVP1):
where 
for 
The organization of this paper is as follows. In Section 2, we present some preliminaries that will be used to prove our results. In Section 3, we discuss the existence and uniqueness of solution for the BVP1 by using Leray-Schauder nonlinear alternative and Banach contraction theorem. Finally, in Section 4 we study the positivity of solution by applying the Guo-Krasnosel’skii fixed point theorem.
2. Preliminary Lemmas
We first introduce some useful spaces. we will use the classical Banach spaces,
. We also use the Banach space
, equipped with the norm
where 
.
Firstly we state some preliminary results.
Lemma 1 Let 
and 
then the problem
(2.1)
(2.2)
has a unique solution
(2.3)
where
(2.4)
(2.5)
Proof Integrating the Equation (2.1), it yields
From the boundary condition 
we deduce that 
and
.
And the boundary condition 
implies
Therefore we have
Now it is easy to have
which achieves the proof of Lemma 1.
We need some properties of functions
.
Lemma 2 For all t, s, such that 
we have
Proof It is easy to see that, if 
If 
then
Lemma 3 For all t, s such that
, 
, we have
Proof For all 
if 
it follows from (2.4) that
and
If 
it follows from (2.4) that
Therefore
Lemma 4 (See [6]) We define an operator 
by
Lemma 5 (See [5]) The function 
is a solution of the (BVP1) if and only if T has a fixed point in X, i.e.
.
3. Existence Results
Now, we give some existence results for the BVP1 Theorem 6 Assume that 
and there exist nonnegative functions 
such that 
we have
and
then, the (BVP1) has a unique solution in 
Proof We shall prove that T is a contraction. Let 
then
So we can obtain
Similarly, we have
From this we deduce
Then T is a contraction. From Banach contraction principe we deduce that T has a unique fixed point which is the unique solution of (BVP1).
We will employ the following Leray-Schauder nonlinear alternative [12].
Lemma 7 Let Fbe Banach space and 
be a bounded open subset of F,
. 
be a completely continuous operator. Then, either there exists
, 
such that
, or there exists a fixed point 
Theorem 8 We assume that 
and there exist nonnegative functions 
such that
Then the (BVP1) has at least one nontrivial solution
.
Proof Setting
Remarking that 
and 
then there exists an interval 
such that
and 
a.e. 
Le 
With the help of Ascoli-Arzela Theorem we show that 
is a completely continuous mapping. We assume that 
such that 
then 
we have
and
This shows that 
From this we get
this contradicts 
By applying Lemma 7, T has a fixed point 
and then the BVP1 has a nontrivial solution 
4. Positive Results
In this section, we discuss the existence of positive solutions for (BVP1). We make the following additional assumptions.
(Q1) 
where 
and 
(Q2) 
We need some properties of functions 
Lemma 9 For all
, we have
where
.
Proof It is easy to see that.
If 
If 
Lemma 10 Let 
and assume that 
then the unique solution u of the (BVP1) is nonnegative and satisfies
Proof Let 
it is obvious that 
is nonnegative. For any 
by (2.3) and Lemmas 2 and 3, it follows that
(4.1)
On the other hand, (2.4) and Lemma 11 imply that, for any 
we have
From (4.1) it yields
(4.2)
Therefore, we have
Similarly, we get
On the other hand, for 
and using Lemma 10 and (4.1) we obtain
(4.3)
Therefore,
Finally, regrouping (4.2) and (4.3) we have
Definition 11 Let use introduce the following sets
K is a non-empty closed and convex subset of X.
Lemma 12 (See [5]) The operator T is completely continuous and satisfies 
To establish the existence of positive solutions of (BVP1), we will use the following Guo-Krasnosel’skii fixed point theorem [13].
Theorem 13 Let E be a Banach space and let 
be a cone. Assume that
, 
are open subsets of E with 
and let
be a completely continuous operator. In addition suppose either 1) 
and 
or 2) 
and 
holds. Then 
has a fixed point in 
The main result of this section is the following Theorem 14 Let (Q1) and (Q2) hold, 
and assume that
Then the problem (BVP1) has at least one positive solution in the case 1) 
and 
or 2) 
and 
Proof We prove the superlinear case. Since 
then for any 
such that 
for
. Let 
be an open set in X defined by
then, for any 
it yields
Therefore
So
If we choose
then it yields
Now from 
we have 
such that 
for
. Let
Denote by 
the open set 
If 
then
then 
Let 
then
And
Choosing
we get 
By the first part of Theorem13, T has at least one fixed point in 
such that 
This completes the superlinear case of the theorem 14. Proceeding as above we proof the sublinear case. This achieves the proof of Theorem 14.
Example 15 Consider the following boundary value problem
(E1)
Set 
and 
where 
and and 
. One can choose
It is easy to prove that 
are nonnegative functions, and
Hence, by Theorem 6, the boundary value problem (E1) has a unique solution in X.
2) Now if we estimate 
as
then one can choose
. So 
are nonnegative functions. Hence, by Theorem 8, the boundary value problem (E1) has at least one nontrivial solution, 
Example 16 Consider the following boundary value problem
(E2)
where, 
and
Then 
We put, 
and 
, when 
and when 
Then
By theorem 13 1) the BVP (E2) has at least one positive solution.