- Research Article
- Open Access

# Limit Properties of Solutions of Singular Second-Order Differential Equations

- Irena Rachůnková
^{1}Email author, - Svatoslav Staněk
^{1}, - Ewa Weinmüller
^{2}and - Michael Zenz
^{2}

**2009**:905769

https://doi.org/10.1155/2009/905769

© Irena Rachůnková et al. 2009

**Received:**23 April 2009**Accepted:**28 May 2009**Published:**29 June 2009

## Abstract

We discuss the properties of the differential equation , a.e. on , where , and satisfies the -Carathéodory conditions on for some . A full description of the asymptotic behavior for of functions satisfying the equation a.e. on is given. We also describe the structure of boundary conditions which are necessary and sufficient for to be at least in . As an application of the theory, new existence and/or uniqueness results for solutions of periodic boundary value problems are shown.

## Keywords

- Periodic Problem
- Associate Boundary Condition
- Singular Equation
- Compact Subinterval
- Euler Transformation

## 1. Motivation

where , , and the function is defined for a.e. and for all . The above equation is singular at because of the first term in the right-hand side, which is in general unbounded for . In this paper, we will also alow the function to be unbounded or bounded but discontinuous for certain values of the time variable . This form of is motivated by a variety of initial and boundary value problems known from applications and having nonlinear, discontinuous forcing terms, such as electronic devices which are often driven by square waves or more complicated discontinuous inputs. Typically, such problems are modelled by differential equations where has jump discontinuities at a discrete set of points in , compare [1].

This study serves as a first step toward analysis of more involved nonlinearities, where typically,
has singular points also in
and
. Many applications, compare [2–12], showing these structural difficulties are our main motivation to develop a framework on existence and uniqueness of solutions, their smoothness properties, and the structure of boundary conditions necessary for
to have at least continuous first derivative on
. Moreover, using new techniques presented in this paper, we would like to extend results from [13, 14] (based on ideas presented in [15]) where problems of the above form but with *appropriately smooth data function*
have been discussed.

Here, we aim at the generalization of the existence and uniqueness assertions derived in those papers for the case of smooth . We are especially interested in studying the limit properties of for and the structure of boundary conditions which are necessary and sufficient for to be at least in .

where and . Clearly, this means that .

*necessary and sufficient*for the solution of (1.6) to be unique and at least continuously differentiable, ? To answer this question, we can use techniques developed in the classical framework dealing with boundary value problems, exhibiting a singularity of the first and second kind; see [15, 16], respectively. However, in these papers, the analytical properties of the solution are derived for nonhomogeneous terms being at least continuous. Clearly, we need to rewrite problem (1.6) first and obtain its new form stated as,

Here, may become unbounded for , the condition , or equivalently is not the correct condition for the solution to be continuous on

From the above remarks, we draw the conclusion that a new approach is necessary to study the analytical properties of (1.1).

## 2. Introduction

The following notation will be used throughout the paper. Let be an interval. Then, we denote by the set of functions which are (Lebesgue) integrable on . The corresponding norm is . Let . By , we denote the set of functions whose th powers of modulus are integrable on with the corresponding norm given by .

Moreover, let us by and denote the sets of functions being continuous on and having continuous first derivatives on , respectively. The norm on is defined as .

Finally, we denote by and the sets of functions which are absolutely continuous on and which have absolutely continuous first derivatives on , respectively. Analogously, and are the sets of functions being absolutely continuous on each compact subinterval and having absolutely continuous first derivatives on each compact subinterval , respectively.

specified in the following definition.

Definition 2.1.

Let
. A function
satisfies *the*
*-Carathéodory conditions* on the set
if

(i) is measurable for all ,

(ii) is continuous for a.e. ,

(iii) for each compact set there exists a function such that for a.e. and all .

We will provide a full description of the asymptotical behavior for of functions satisfying (2.1) a.e. on . Such functions will be called solutions of (2.1) if they additionally satisfy the smoothness requirement ; see next definition.

Definition 2.2.

In Section 3, we consider linear problems and characterize the structure of boundary conditions necessary for the solution to be at least continuous on . These results are modified for nonlinear problems in Section 4. In Section 5, by applying the theory developed in Section 4, we provide new existence and/or uniqueness results for solutions of singular boundary value problems (2.1) with periodic boundary conditions.

## 3. Linear Singular Equation

where and .

Now, let , . Without loss of generality, we may assume that . For , we choose , and we have and .

which means that . We now use the properties of to represent all functions satisfying (3.1) a.e. on . Remember that such function does not need to be a solution of (3.1) in the sense of Definition 2.2.

Lemma 3.1.

Let , , and let be given by (3.2).

is the set of all functions satisfying (3.1) a.e. on .

is the set of all functions satisfying (3.1) a.e. on .

Proof.

Moreover, the function is a particular solution of (3.1) on . Therefore, the first statement follows. Analogous argument yields the second assertion.

We stress that by (3.8), the particular solution of (3.1) belongs to . For , we can see from (3.9) that it is useful to find other solution representations which are equivalent to (3.10) and (3.11), but use instead of , if .

Lemma 3.2.

Let and let be given by (3.2).

is the set of all functions satisfying (3.1) a.e. on .

is the set of all functions satisfying (3.1) a.e. on .

Proof.

which completes the proof.

of (3.1) for satisfies . Main results for the linear singular equation (3.1) are now formulated in the following theorems.

Theorem 3.3.

Moreover, can be extended to the whole interval in such a way that .

Proof.

Therefore , and consequently .

where are real matrices, and is an arbitrary vector. Then the following result follows immediately from Theorem 3.3.

Theorem 3.4.

is nonsingular.

Proof.

into the boundary conditions (3.26b).

Theorem 3.5.

Let and let a function satisfy equation (3.1) a.e. on . For , only one of the following properties holds:

(i) , ,

(ii) , .

For , satisfies only one of the following properties:

(i) , ,

(ii) , .

In particular, can be extended to the whole interval with if and only if .

Proof.

Therefore , and consequently .

where are real constants. Then the following result follows immediately from Theorem 3.5.

Theorem 3.6.

Let , . Then for any and any there exists a unique solution of the boundary value problem (3.38a) and (3.38b) if and only if .

Proof.

into the boundary conditions (3.38b).

To illustrate the solution behaviour, described by Theorems 3.3 and 3.5, we have carried out a series of numerical calculations on a MATLAB software package bvpsuite designed to solve boundary value problems in ordinary differential equations. The solver is based on a collocation method with Gaussian collocation points. A short description of the code can be found in [17]. This software has already been used for a variety of singular boundary value problems relevant for applications; see, for example, [18].

subject to initial or boundary conditions specified in the following graphs. All solutions were computed on the unit interval .

## 4. Limit Properties of Functions Satisfying Nonlinear Singular Equations

In this section we assume that the function satisfying differential equation (2.1) a.e. on is given. The first derivative of such a function does not need to be continuous at and hence, due to the lack of smoothness, does not need to be a solution of (2.1) in the sense of Definition 2.2. In the following two theorems, we discuss the limit properties of for .

Theorem 4.1.

and can be extended on in such a way that .

Proof.

Let for a.e. . By (2.2), there exists a function such that for a.e. . Therefore, . Since the equality holds a.e. on , the result follows immediately due to Theorem 3.3.

Theorem 4.2.

and can be extended on in such a way that .

Proof.

Let be as in the proof of Theorem 4.1. According to Theorem 3.5 and (4.1), satisfies (4.3) both for and .

## 5. Applications

Results derived in Theorems 4.1 and 4.2 constitute a useful tool when investigating the solvability of nonlinear singular equations subject to different types of boundary conditions. In this section, we utilize Theorem 4.1 to show the existence of solutions for periodic problems. The rest of this section is devoted to the numerical simulation of such problems.

Periodic Problem

Definition 5.1.

A function
is called *a solution of the boundary value problem* (5.1a) and (5.1b), if
satisfies equation (5.1a) for a.e.
and the periodic boundary conditions (5.1b).

and we see that it is singular. Consequently, the assumption of Theorem 3.4 is not satisfied, and the linear periodic problem (3.26b) subject to (5.1b) is not uniquely solvable. However this is not true for nonliner periodic problems. In particular, Theorem 5.6 gives a characterization of a class of nonlinear periodic problems (5.1a) and (5.1b) which have only one solution. We begin the investigation of problem (5.1a) and (5.1b) with a uniqueness result.

Theorem 5.2 (uniqueness).

for a.e. and all . Then problem (5.1a) and (5.1b) has at most one solution.

Proof.

We consider two cases.

Case 1.

- (i)
Let . We can assume that . (Otherwise we choose .) Then we can find satisfying for and . Let be the first zero of . Then, if we set , we see that satisfies (5.5). Let have no zeros on . Then on , and, due to (5.4), . Since , we can find and such that satisfies (5.5).

- (ii)
Let on . By (5.4), , and . We may again assume that . It is possible to find such that , , on . Since , has at least one zero in . If is the first zero of , then satisfies (5.5).

Case 2.

- (i)

- (ii)
Let for some . If , then we can find an interval satisfying (5.5). If and on , then and, by (5.4), , . Hence, there exists an interval satisfying (5.5).

To summarize, we have shown that in both, the case of intersecting solutions and and the case of separated and , there exists an interval satisfying (5.5).

which contradicts . Consequently, we have shown that , and the result follows.

Under the assumptions of Theorem 4.1 any solution of (5.1a) satisfies Therefore, we can investigate (5.1a) subject to the auxiliary conditions (5.10) instead of the equivalent original problem (5.1a) and (5.1b). This change of the problem setting is useful for obtaining of a priori estimates necessary for the application of the Fredholm-type existence theorem (Lemma 5.5) during the proof.

Theorem 5.3 (existence).

Proof.

Step 1 (existence of auxiliary solutions ).

for a.e. . It can be shown that and which satisfy the -Carathéodory conditions on are nondecreasing in their second argument and a.e. on ; see [19]. Therefore, also satisfies the -Carathéodory conditions on , and there exists a function such that for a.e. and all .

Since the homogeneous problem , has only the trivial solution, we conclude by the Fredholm-type Existence Theorem (see Lemma 5.5) that there exists a solution of problem (5.21).

Step 2 (estimates of ).

which contradicts (5.23), and thus on . The inequality on can be proved in a very similar way.

Step 3 (estimates of ).

Denote . If , then .

Case 1.

where is given by (5.15). Therefore .

Case 2.

Hence, according to (5.15), we again have .

Step 4 (convergence of ).

Hence (4.1) is satisfied. Applying Theorem 4.1, we conclude that and . Therefore satisfies the periodic conditions on . Thus is a solution of problem (5.1a) and (5.1b) and on .

Example 5.4.

In order to show the existence of solutions to the periodic boundary value problem (5.1a) and (5.1b), the Fredholm-type Existence Theorem is used, see for example, in [20, Theorem ], [11, Theorem ] or [21, page 25]. For convenience, we provide its simple formulation suitable for our purpose below.

Lemma 5.5 (Fredholm-type existence theorem).

has a solution .

If we combine Theorems 5.2 and 5.3, we obtain conditions sufficient for the solution of (5.1a) and (5.1b) to be unique.

Theorem 5.6 (existence and uniqueness).

Let all assumptions of Theorems 5.2 and 5.3 hold. Then problem (5.1a) and (5.1b) has a unique solution . Moreover satisfies (5.14).

Example 5.7.

for .

Estimated convergence order for the periodic boundary value problem (5.43)-(5.1a) and *a* = 1.

| Error estimate | Conv. order |
---|---|---|

1 | 5.042446e–003 | — |

2 | 2.850171e–003 | 0.823075 |

3 | 1.681410e–003 | 0.761377 |

4 | 1.029876e–003 | 0.707200 |

5 | 6.514046e–004 | 0.660845 |

6 | 4.231359e–004 | 0.622433 |

7 | 2.807926e–004 | 0.591616 |

8 | 1.894611e–004 | 0.567604 |

9 | 1.294654e–004 | 0.549335 |

10 | 8.930836e–005 | 0.535699 |

## Declarations

### Acknowledgments

This research was supported by the Council of Czech Goverment MSM6198959214 and by the Grant no. A100190703 of the Grant Agency of the Academy of Sciences of the Czech Republic.

## Authors’ Affiliations

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