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This work is devoted to the study of uniqueness and existence of positive solutions for a second-order boundary value problem with integral condition. The arguments are based on Banach contraction principle, Leray Schauder nonlinear alternative, and Guo-Krasnosel’skii fixed point theorem in cone. Two examples are also given to illustrate the main results.

Boundary value problem with integral boundary conditions is a mathematical model for of various phenomena of physics, ecology, biology, chemistry, and so forth. Integral conditions come up when values of the function on the boundary are connected to values inside the domain or when direct measurements on the boundary are not possible. The presence of an integral term in the boundary condition leads to great difficulties. Our aim, in this work, is the study of existence, uniqueness, and positivity of solution for the following second-order boundary value problem:

Various types of boundary value problems with integral boundary conditions were studied by many authors using different methods see [

This paper is organized as follows. In Section

In this section, we introduce notations, definitions, and preliminary facts that will be used in the sequel.

A mapping defined on a Banach space is completely continuous if it is continuous and maps bounded sets into relatively compact sets.

Let

If a sequence

Now we state the nonlinear alternative of Leray-Schauder.

Let

A function

the map

the map

for each

We recall the definition of positive solution.

A function

We expose the well-known Guo-Krasnosel'skii fixed point Theorem on cone [

Let

Then

Throughout this paper, let

Let

Integrating two times the equation

We have the following result which is useful in what follows.

This section deals with the existence and uniqueness of solutions for the problem (

Suppose that the following hypotheses hold.

There exist two nonnegative functions

Then the problem (

Transform the

Then for

Applying hypothesis H3 again gives

Suppose that the following hypotheses hold:

There exist three nonnegative functions

Then the BVP (

First we show that

(1)

Due to (P1)

(2)

(3)

Since

Secondly, we apply the nonlinear alternative of Leray-Schauder to prove the existence of solution. Let us make the following notations:

Second, if

In this section the existence results for positive solutions for problem (

(

(

The following result gives a priori estimates for solutions of problem (

Assume that hypotheses (Q1)-(Q2) hold then

for

for

(i) Let

(ii) Let

by similar ideas, it yields

The proof is complete.

Assume that hypotheses (Q1)-(Q2) hold, then the solution of the problem (

It follows from Lemma

Further, according to inequalities (

ending the proof Lemma

Define the quantities

Assume that hypotheses (Q1)-(Q2) hold, then problem (

To prove this theorem we apply the well-known Guo-Krasnosel'skii fixed point Theorem in cone.

Denote

It is easy to check that

Using Lemma

Let us consider the superlinear case. Since

Taking hypothesis (Q_{2}) into account, one can choose

Second, since

Let

Let us choose

To illustrate the main results, we consider the following examples.

Consider the following boundary value problem (P1):

The following boundary value problem (P2):