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This paper presents an existence and localization result of unbounded solutions for a second-order differential equation on the half-line with functional boundary conditions. By applying unbounded upper and lower solutions, Green’s functions, and Schauder fixed point theorem, the existence of at least one solution is shown for the above problem. One example and one application to an Emden-Fowler equation are shown to illustrate our results.

The authors consider the following boundary value problem composed by the differential equation:

Boundary value problems on the half-line arise naturally in the study of radially symmetric solutions of nonlinear elliptic equations, and many works have been done in this area; see [

Lower and upper solutions method is a very adequate technique to deal with boundary value problems as it provides not only the existence of bounded or unbounded solutions but also their localization and, from that, some qualitative data about solutions, their variation and behavior (see [

The paper is organized as follows. In Section

Consider the space

Solutions of the linear problem associated to (

Let

If

Conversely, if

The lack of compactness of

A set

all functions from

all functions from

all functions from

The existence tool will be Schauder’s fixed point theorem.

Let

The functions considered as lower and upper solutions for the initial problem are defined as follows.

Given

A function

In this section we prove the existence of at least one solution for the problem (

Let

Let

For clearness, the proof will follow several steps.

If there is

So

If the infimum is attained at

If

Therefore

In a similar way we can prove that

Therefore, problem (

Therefore

With

For each

So

So

Moreover

So, by Lemma

Let

For

Then

Then, by Schauder’s Fixed Point Theorem,

Suppose, by contradiction, that

So

Therefore,

A similar result can be obtained if

A function

for each

for almost every

for each

However in this case an extra assumption on

Let

If there are

The proof is similar to Theorem

Let

By Definition

The same remains valid if

So in both cases

The remaining steps are identical to the proof of Theorem

Consider the second-order problem in the half-line with one functional boundary condition:

Remark that the above problem is a particular case of (

As

Emden-Fowler-types equations (see [

In the steady-state case, and with

If

In the literature, Emden-Fowler-types equations are associated to Dirichlet or Neumann boundary conditions (see [

To the best of our knowledge, it is the first time where some Emden-Fowler is considered together with functional boundary conditions on the half-line.

Consider that we are looking for nonnegative solutions for the problem composed by the discontinuous differential equation

This is a particular case of (

As

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by national founds of FCT Fundação para a Ciência e a Tecnologia, in the project UID/MAT/04674/2013 (CIMA).