ABSTRACT

By using Leray-Schauder nonlinear alternative, Banach contraction theorem and Guo-Krasnosel’skii theorem, we discuss the existence, uniqueness and positivity of solution to the third-order multi-point nonhomogeneous boundary value problem (BVP1):

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.

REFERENCES