Unbounded Solutions for Functional Problems on the Half-Line

Abstract

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.

1. Introduction

The authors consider the following boundary value problem composed by the differential equation:where is continuous and bounded by some function, and the functional boundary conditions on the half-line are as follows:with and a continuous function verifying some monotone assumption:

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 [1]. The functional dependence on the boundary conditions allows that problem (1), (2) covers a huge variety of boundary value problems such as separated, multipoint, nonlocal, integrodifferential, with maximum or minimum arguments, as it can be seen, for instance, in [2–11] and the references therein. However, to the best of our knowledge, it is the first time where this type of functional boundary conditions are applied to the half-line.

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 [12–14]). Some results are concerned with the existence of bounded or positive solutions, as in [15, 16], and the references therein. For problem (1), (2) we prove the existence of two types of solution, depending on : if the solution is unbounded and if the solution is bounded. In this way, we gather different strands of boundary value problems and types of solutions in a single method.

The paper is organized as follows. In Section 2 some auxiliary results are defined such as the adequate space functions, some weighted norms, a criterion to overcome the lack of compactness, and the definition of lower and upper solutions. Section 3 contains the main result: an existence and localization theorem, which proof combines lower and upper solution technique with the fixed-point theory. Finally, last two sections contain, to illustrate our results, an example and an application to some problem composed by a discontinuous Emden-Fowler-type equation with a infinite multipoint conditions, which are not covered by the existent literature.

2. Definitions and Auxiliary Results

Consider the space with the norm , where In this way is a Banach space.

Solutions of the linear problem associated to (1) and usual boundary conditions are defined with Green’s function, which can be obtained by standard calculus.

Lemma 1. Let , . Then the linear boundary value problem composed byfor , has a unique solution in , given by where

Proof. If is a solution of problem (6), then the general solution for the differential equation is where , are constants. Since should satisfy the boundary conditions, we get The solution becomes And by computation with given by (8).
Conversely, if is a solution of (7), it is easy to show that it satisfies the differential equation in (6). Also and .

The lack of compactness of is overcome by the following lemma which gives a general criterion for relative compactness, referred to in [1].

Lemma 2. A set is relatively compact if the following conditions hold:(1)all functions from are uniformly bounded;(2)all functions from are equicontinuous on any compact interval of ;(3)all functions from are equiconvergent at infinity; that is, for any given , there exists a such that

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

Theorem 3 (see [17]). Let be a nonempty, closed, bounded, and convex subset of a Banach space , and suppose that is a compact operator. Then is at least one fixed point in .

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

Definition 4. Given , a function is a lower solution of problem (1), (2) if A function is an upper solution if it satisfies the reverse inequalities.

3. Existence and Localization Results

In this section we prove the existence of at least one solution for the problem (1), (2), and, moreover, some localization data.

Theorem 5. Let be a continuous function, verifying that, for each , there exists a positive function with such that for with ,Moreover, if is nondecreasing on and and there are , , lower and upper solutions of (1), (2), respectively, such thatthen problem (1), (2) has at least one solution , with , for .

Proof. Let , be, respectively, lower and upper solutions of (1), (2) verifying (16). Consider the modified problemwhere is given by For clearness, the proof will follow several steps.
Step 1 (if is a solution of (17), then , ). Let be a solution of the modified problem (17) and suppose, by contradiction, that there exists such that . Therefore If there is such that we have and . By Definition 4 we get the contradiction So , .
If the infimum is attained at then As is solution of (17), by the definition of , the following contradiction is achieved Ifthen . As is solution of (17), by Definition 4, this contradiction holds Therefore , .
In a similar way we can prove that , .
Step 2 (problem (17) has at least one solution). Let and define the operator with , and is the Green function given by (8).
Therefore, problem (17) becomesand if , , by Lemma 1 it is enough to prove that has a fixed point.
Step 2.1 ( is well defined). As is a continuous function, and, by (15), for any with That is and By Lebesgue Dominated Convergence Theorem, and analogously for Therefore .
Step 2.2 ( is continuous). Consider a convergent sequence in ; there exists such that .
With , we have as .
Step 2.3 ( is compact). Let be any bounded subset. Therefore there is such that , .
For each , and for , So ; that is, is uniformly bounded in .
is equicontinuous, because, for and , we have, as ,So is equicontinuous.
Moreover is equiconvergent at infinity, because, as ,So, by Lemma 2, is relatively compact.
Step 2.4. Let be a nonempty, closed, bounded, and convex subset. Then
Let defined by with with given by Step 2.1.
For and , we get Then ; that is,
Then, by Schauder’s Fixed Point Theorem, has at least one fixed point .
Step 3 ( is a solution of (1), (2)). By Step , as is a solution of (17) then , So, the differential equation (1) is obtained. It remains to prove that .
Suppose, by contradiction, that . Then and by the monotony of and Definition 4, the following contradiction holds So and in a similar way we can prove that .
Therefore, is a solution of (1), (2).

A similar result can be obtained if is a -Carathéodory function and

Definition 6. A function is said to be -Carathéodory if it verifies the following:(1)for each , is measurable on ;(2)for almost every , is continuous in ;(3)for each , there exists a positive function with such that, for with ,

However in this case an extra assumption on must be assumed.

Theorem 7. Let be a -Carathéodory function such that is monotone on .
If there are , , lower and upper solutions of (42), (2), respectively, such that and is nondecreasing on and , then problem (42), (2) has at least one solution with , .

Proof. The proof is similar to Theorem 5 except the first step.
Let be a solution of the modified problem composed by and the boundary conditions If, by contradiction, there is such that then , , and there exists an interval where , , .
By Definition 4 and if is nondecreasing on , this contradiction holds for : The same remains valid if is nonincreasing, considering an interval where , , .
So in both cases , .
The remaining steps are identical to the proof of Theorem 5, and we omit them.

4. Example

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

Remark that the above problem is a particular case of (1), (2) with

is continuous in , and, for , assumption (15) holds with , for some and .

As is not decreasing in and , and the functions and are lower and upper solutions for (49), respectively, then, by Theorem 5, there is at least an unbounded solution of (49) such that

5. Application

Emden-Fowler-types equations (see [18]) can model, for example, the heat diffusion perpendicular to parallel planes by where means the nonlinear heat source and is the temperature.

In the steady-state case, and with , last equation becomes

If and , (53) is called the Lane-Emden equation of the first kind, whereas in the second kind one has Both cases are used in the study of thermal explosions. For more details see [19].

In the literature, Emden-Fowler-types equations are associated to Dirichlet or Neumann boundary conditions (see [20, 21]).

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 equationcoupled with the infinite multipoint conditionswhere and are nonnegative sequences such that , , and are convergent with , .

This is a particular case of (42), (2), where

As thus is -Carathéodory. Also is monotone on ; more precisely is nondecreasing on . As is not decreasing in and , and functions and are lower and upper solutions for problem (54), (55), respectively, then, by Theorem 7, there is at least an unbounded and nonnegative solution of (54), (55) such that

Conflict of Interests

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

Acknowledgment

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