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The existence and uniqueness of the solutions to the Dirichlet boundary value problem in the Banach spaces is discussed by using the fixed point theory of condensing mapping, doing precise computation of measure of noncompactness, and calculating the spectral radius of linear operator.

This paper is mainly concerned with the following second-order Dirichlet boundary value problem:

In the last several decades, there has been much attention focused on the boundary value problems for various nonlinear ordinary differential equations, difference equations, and functional differential equations, see [

In particular, the authors in [

We would like to mention the results due to [

The authors in [

Motivated by the above-mentioned work [

This paper is organized as follows. In Section

Let

Let

Assume that

The mapping

The following lemmas are of great importance in the proof of our main results.

Suppose that

Note that

if

if

if

After direct computations, it is easy to see that

We now claim that solution of the boundary value problem (

It is easy to see that

If

Assume that

the spectral radius

If

(1) Define operator

Let

(2) If

In particular. If

Let

If

We take

Suppose that

Let

Set

Setting

If

For any

Let

For any

Assume that

Suppose that

Assume that

For any

For any

Thus, we get

If

Assume that

In this section, we present and prove our main results.

Let

there exist two positive numbers

for any a bounded set

there exist two positive numbers

Define the integral operator

We now show that

For any

Using the properties of the noncompactness measure together with

By Lemma

Hence,

Let

Indeed. If there exists a constant

Let

By (

Thus,

By the Gelfand theorem [

Set

Denote

Take

In [

In [

Let

for all

there exist two positive numbers

Assume that the operator

By Lemma

Next, we establish a uniqueness of solution for the problem (

Let

Assume that operator

We will prove that for sufficiently large

By induction, we have

Moreover, we can choose

Further, take

By the direct application of the Banach contraction mapping principle, the conclusion of Theorem

The following theorem is concerned with the existence of positive solutions for problem (

Let

there exists a constant

for any a bounded set

Consider the linear boundary value problem

Set

Multiply by

Next we show that

However, on the other hand, we have

Thus, For any

Hence,

Further, we obtain

This means that

By

Wen-Xue Zhou's work was supported by NNSF of China (10901075), Program for New Century Excellent Talents in University (NCET-10-0022), the Key Project of Chinese Ministry of Education (210226), and NSF of Gansu Province of China (1107RJZA091).