Abstract
In this paper, we discuss the existence of positive solutions to the singular Dirichlet boundary value problems (BVPs) for ordinary differential equations (ODEs) of the form
where
MSC: 34B18, 34B16, 34A12.
Keywords:
singular ordinary differential equation of the second order; time singularities; space singularities; positive solutions; existence of solutions; polynomial collocation1 Introduction
In the present work, we analyze the existence of positive solutions to the singular Dirichlet BVP,
Here, we assume that
(i)
(ii)
(iii) for each compact set
For such functions, we use the notation
The differential operator on the lefthand side of Eq. (1a) can be equivalently written
as
The aim of this paper is to study the case
In the sequel, we introduce the basic notation and state the preliminary results required
in the analysis of problem (1a) and (1b). Here, we focus our attention on the case
Let u be a positive solution of problem (1a) and (1b), where
The investigation of singular Dirichlet BVPs has a long history and a lot of methods for their analysis are available. One of the most important ones is the topological degree method providing various fixed point theorems and existence alternative theorems; see, e.g., Lemma 1. For more information on the topological degree method and its application to numerous BVPs, including Dirichlet problems, we refer the reader to the monographs by Mawhin [1315].
Throughout this paper, we work with the following conditions on the function f in (1a):
(H_{1})
(H_{2}) There exists an
(H_{3}) For a.e.
holds, where
By
we denote the norms in
Definition 1 We say that a function
Remark 1 Let a function g have the properties specified in (H_{3}). Then for each
that
In order to prove that the singular problem (1a) and (1b) has a positive solution,
we use regularization and sequential techniques. To this end, for
and
respectively. Then it follows from (H_{1}) that
Hence,
and
As a first step in the analysis, we investigate auxiliary regular BVPs of the form
To show the solvability of problem (7a) and (7b), we use the following alternative of LeraySchauder type which follows from [[16], Theorem 5.1].
Lemma 1LetEbe a Banach space, Ube an open subset ofEand
A1: ℱ has a fixed point in
A2: there exists an element
In limit processes, we apply the following Vitali convergence theorem; cf.[1719].
Lemma 2Let
(i)
(ii) the sequence
We recall that a sequence
The paper is organized as follows. In Section 2, we collect auxiliary results used in the subsequent analysis. Section 3 is devoted to the study of limit properties of solutions to Eq. (7a). In Section 4, we investigate auxiliary regular problems associated with the singular problem (1a) and (1b). We show their solvability and describe properties of their solutions. An existence result for the singular problem (1a) and (1b) is given in Section 5. Finally, in Section 6, we illustrate the theoretical findings by means of numerical experiments.
Throughout the paper
2 Preliminaries
In this section, auxiliary statements necessary for the subsequent analysis are formulated.
Lemma 3Let
Then
(i) rcan be extended on
(ii) Hcan be extended on
holds for a.e.
Proof (i) It is clear that
we have
(ii) Let
Then
for
Let
In particular,
Hence, cf. (10),
and therefore,
Lemma 4Let
Proof It follows from Lemma 2 that
Since
which contradicts (13). Let
we conclude from the relation
that
The last equality contradicts (13). Consequently, (12) holds and the result follows. □
Lemma 5Let
Proof Since (cf. (10))
for
3 Limit properties of solutions to Eq. (7a)
Here, we investigate asymptotic properties of solutions of (7a). We also provide a related integral equation this solution satisfies.
Lemma 6Let (H_{1}) hold. Let
holds for
Proof Choose
and
Then
We now discuss the linear Euler differential equation
Let H be the function given in Lemma 3. By Lemma 3(ii), H can be extended on
Corollary 1Let (H_{1}) hold. Let
Proof The result holds by Lemma 6 with
Remark 2 Corollary 1 says that the set of all solutions
4 Auxiliary regular problems
In order to prove the solvability of problem (7a) and (7b), we first have to investigate the problem
depending on the parameter λ. Here,
The following result shows that the solvability of problem (17a) and (17b) is equivalent
to the solvability of an integral equation in the set
Lemma 7Let (H_{1}) hold. Thenuis a solution of problem (17a) and (17b) if and only ifuis a solution of the integral equation
in the set
Proof Let u be a solution of Eq. (17a). Then
holds for
Let u be a solution of Eq. (18) in
The following results provide bounds for solutions of problem (17a) and (17b).
Lemma 8Let (H_{1})(H_{3}) hold. Then there exists a positive constantS (independent of
hold. Moreover, for any solutionuof problem (17a) and (17b), there exists
Proof Let u be a solution of problem (17a) and (17b). Then by Lemma 7, equality (18) holds for
follows from (18). Hence,
It is clear that L is independent of the choice of solution u to problem (17a) and (17b) and independent of
We now show that inequality (21) holds for some
Since
Hence,
Hence,
In particular,
Let
It follows from
and therefore, we have
where
Consequently, there exists
We are now in the position to prove the existence result for problem (7a) and (7b).
Lemma 9Let (H_{1})(H_{3}) hold. Then for each
Proof Let S be a positive constant in Lemma 8 and let us define
Then Ω is an open and bounded subset of the Banach space
By Lemma 7, any fixed point of the operator
(i)
(ii)
We first verify that
It follows from Lemma 5 and (9) that
for
Since
we have
Then by Lemma 5 and (9), the inequalities
are satisfied for
holds for a.e.
Due to the fact that by Lemma 7 any fixed point u of the operator
Let
Lemma 10Let (H_{1})(H_{3}) hold. Let
Proof
By Lemma 9, the inequalities
hold, where S is a positive constant and
for a.e.
5 The existence result for BVP (1a) and (1b)
This section is devoted to the main result on the existence of positive solutions to the original BVP (1a) and (1b).
Theorem 1Let (H_{1})(H_{3}) hold. Then problem (1a) and (1b) has at least one positive solution.
Proof By Lemma 9, for each
holds for a.e.
Let
and
follows. By Lemma 4 (for
In addition,
it follows from Lemma 2 that
and
that
Taking the limit
we have
Hence,
and
6 Numerical simulations
For the numerical simulation, we choose
where
The reason for changing the boundary conditions from (1b) to (36b) is that the differential equation (36a) subject to (1b) is not well posed; see [23]. However, to enable successful numerical treatment, wellposedness of the model is crucial. This property means that Eq. (36a) subject to proper boundary conditions has at least a locally unique solution,^{a} and this solution depends continuously on the problem data. The wellposedness of the problem is important for two reasons. First of all, it allows to express errors in the solution of the analytical problem in terms of modeling errors and data errors (all measured via appropriate norms). Therefore, when the errors in the data become smaller due to more precise modeling or smaller measurement inaccuracies, the errors in the solution will decrease. The second reason is that the wellposedness decides if the numerical simulation will be at all successful. If the analytical problem is illposed, then the inevitable roundoff errors can become extremely magnified and fully spoil the accuracy of the approximation.
In what follows, we work with
By [23], problem (37), (36b) is well posed and therefore it is suitable for the numerical treatment. To see this, we need to look at a general solution of the homogeneous equation
If we set
whose roots
6.1 MATLAB Code bvpsuite
To illustrate the analytical results discussed in the previous section, we solved numerically examples of the form (36a) and (36b) using a MATLAB™ software package bvpsuite designed to solve BVPs in ODEs and differential algebraic equations. The solver routine is based on a class of collocation methods whose orders may vary from two to eight. Collocation has been investigated in the context of singular differential equations of first and second order in [24,25], respectively. This method could be shown to be robust with respect to singularities in time and retains its high convergence order in the case that the analytical solution is appropriately smooth. The code also provides an asymptotically correct estimate for the global error of the numerical approximation. To enhance the efficiency of the method, a mesh adaptation strategy is implemented, which attempts to choose grids related to the solution behavior in such a way that the tolerance is satisfied with the least possible effort. Error estimate procedure and the mesh adaptation work dependably provided that the solution of the problem and its global error are appropriately smooth.^{b} The code and the manual can be downloaded from http://www.math.tuwien.ac.at/~ewa webcite. For further information, see [26]. This software proved useful for the approximation of numerous singular BVPs important for applications; see, e.g., [3,9,27,28].
6.2 Preliminaries
Before dealing with two nonlinear models specified in Sections 6.3 and 6.4, we have to compute numerical solutions for a simpler linear^{c} model of the form
where a was chosen as
Figure 1. Problem (39a) and (39b): Numerical solutions for different values ofa.
6.3 Example 1
We first investigate the following problem:
The nonlinearity f in (40a) has the form
and it satisfies (H_{1})(H_{3}) with
It follows from Theorem 1 that there exists at least one value of
Let u be a solution of problem (1a) and (1b) with f from (41). Then by (35), we obtain
Therefore,
Let
Then (42) implies
Consequently,
In order to solve the nonlinear problem (40a) and (40b), we first have to solve a series of auxiliary problems for parameterdependent differential equations
We begin the calculations with
In the case of Example 1 and
1. Numerical approximation of BVP (39a) and (39b) is used as an initial guess for
ODE (45) with
2. Use the above approximation as an initial guess for ODE (45) with
3. Use the above approximation as an initial guess for ODE (45) with
4. Use the above approximation as an initial guess for ODE (45) with
After the last step, we have solved problem (40a) and (40b) subject to boundary conditions
Figure 2. Problem (40a) and (40b): Numerical solutions for different values ofa. Values of
Table 1. Problem (40a) and (40b): Complete data of the numerical simulation for different values ofa
6.4 Example 2
The above approach has been also accordingly applied for Example 2. Here, we consider the problem
The righthand side f in Eq. (46a) now reads
and has a singularity at
Theorem 1 guarantees the existence of at least one
and hence by (35),
Consequently,
For Example 2, the auxiliary ODE is constructed using ODE (39a),
For all values of a, we choose
1. Numerical approximation of BVP (39a) and (39b) is used as an initial guess for
ODE (49) with
2. Use the above approximation as an initial guess for ODE (49) with
3. Use the above approximation as an initial guess for ODE (49) with
4. Use the above approximation as an initial guess for ODE (49) with
5. Use the above approximation as an initial guess for ODE (49) with
After the last step, we have solved BVP (46a) and (46b) subject to boundary conditions
7 Conclusions
In the present article, we deal with the existence of positive solutions to the singular Dirichlet problem of the form
where
Here, it is only known that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
IR and SS contributed to the analytical part of the work and AS and EBW contributed to its numerical part. All authors read and approved the final version of the manuscript.
Acknowledgements
Dedicated to Jean Mawhin on the occasion of his 70th birthday.
This research was supported by the grant Matematické modely a struktury, PrF2012017. The authors thank the referees for suggestions which improved the paper.
End notes
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