- Research Article
- Open access
- Published:
Limit Properties of Solutions of Singular Second-Order Differential Equations
Boundary Value Problems volume 2009, Article number: 905769 (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.
1. Motivation
In this paper, we study the analytical properties of the differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ1_HTML.gif)
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
.
To clarify the aims of this paper and to show that it is necessary to develop a new technique to treat the nonstandard equation given above, let us consider a model problem which we designed using the structure of the boundary value problem describing a membrane arising in the theory of shallow membrane caps and studied in [10]; see also [6, 9],
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ2_HTML.gif)
subject to boundary conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ3_HTML.gif)
where Note that (1.2) can be written in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ4_HTML.gif)
which is of form (1.1) with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ5_HTML.gif)
Function is not defined for
and for
if
. We now briefly discuss a simplified linear model of (1.4),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ6_HTML.gif)
where and
. Clearly, this means that
.
The question which we now pose is the role of the boundary conditions (1.3), more precisely, are these boundary conditions 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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ7_HTML.gif)
which suggest to introduce a new variable, . In a general situation, especially for the nonlinear case, it is not straightforward to provide such a transformation, however. We now introduce
and immediately obtain the following system of ordinary differential equations:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ8_HTML.gif)
where or equivalently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ9_HTML.gif)
where . According to [16], the latter system of equations has a continuous solution if and only if the regularity condition
holds. This results in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ10_HTML.gif)
compare conditions (1.3). Note that the Euler transformation, which is usually used to transform (1.6) to the first-order form would have resulted in the following system:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ11_HTML.gif)
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.
As already said in the previous section, we investigate differential equations of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ12_HTML.gif)
where . For the subsequent analysis we assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ13_HTML.gif)
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.
A function is called a solution of (2.1) if
and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ14_HTML.gif)
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
First, we consider the linear equation, ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ15_HTML.gif)
where and
.
As a first step in the analysis of (3.1), we derive the necessary auxiliary estimates used in the discussion of the solution behavior. For , let us denote by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ16_HTML.gif)
Assume that . Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ17_HTML.gif)
Now, let ,
. Without loss of generality, we may assume that
. For
, we choose
, and we have
and
.
First, let . Then
,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ18_HTML.gif)
Now, let . Then
,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ19_HTML.gif)
Hence, for ,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ20_HTML.gif)
Consequently, (3.3), (3.6), and the Hölder inequality yield, ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ21_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ23_HTML.gif)
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).
(i) If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ24_HTML.gif)
is the set of all functions satisfying (3.1) a.e. on
.
(ii) If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ25_HTML.gif)
is the set of all functions satisfying (3.1) a.e. on
.
Proof.
Let . Note that (3.1) is linear and regular on
. Since the functions
and
are linearly independent solutions of the homogeneous equation
on
, the general solution of the homogeneous problem is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ26_HTML.gif)
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).
(i) If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ27_HTML.gif)
is the set of all functions satisfying (3.1) a.e. on
.
(ii) If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ28_HTML.gif)
is the set of all functions satisfying (3.1) a.e. on
.
Proof.
Let us fix and define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ29_HTML.gif)
In order to prove (i) we have to show that for
, where
. This follows immediately from (3.9), since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ30_HTML.gif)
and hence we can define as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ31_HTML.gif)
For we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ32_HTML.gif)
which completes the proof.
Again, by (3.9), the particular solution,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ33_HTML.gif)
of (3.1) for satisfies
. Main results for the linear singular equation (3.1) are now formulated in the following theorems.
Theorem 3.3.
Let and let
satisfy equation (3.1) a.e. on
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ34_HTML.gif)
Moreover, can be extended to the whole interval
in such a way that
.
Proof.
Let a function be given. Then, by (3.10), there exist two constants
such that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ35_HTML.gif)
Using (3.8), we conclude
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ36_HTML.gif)
For and
, we have
. Furthermore, for a.e.
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ37_HTML.gif)
By the Hölder inequality and (3.6) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ38_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ39_HTML.gif)
Therefore , and consequently
.
It is clear from the above theorem, that given by (3.21) is a solution of (3.1) for
. Let us now consider the associated boundary value problem,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ40_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ41_HTML.gif)
where are real matrices, and
is an arbitrary vector. Then the following result follows immediately from Theorem 3.3.
Theorem 3.4.
Let ,
. Then for any
and any
there exists a unique solution
of the boundary value problem (3.26a) and (3.26b) if and only if the following matrix,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ42_HTML.gif)
is nonsingular.
Proof.
Let be a solution of (3.1). Then
satisfies (3.21), and the result follows immediately by substituting the values,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ43_HTML.gif)
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.
Let , and let
be given. Then, by (3.13), there exist two constants
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ44_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ45_HTML.gif)
Let , then it follows from (3.9)
. Also, by (3.29),
. Let
. Then (3.9), (3.29), and (3.30) imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ46_HTML.gif)
Let . Then, by (3.14), for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ47_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ48_HTML.gif)
If , then
by (3.9), and it follows from (3.32) that
. Let
. Then we deduce from (3.9), (3.32), and (3.33) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ49_HTML.gif)
Let . Then on
,
satisfies (3.29) and (3.30), with
. If
, then, by (3.9),
and
. Let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ50_HTML.gif)
In particular, for ,
can be extended to
in such a way that
if and only if
. Then, the associated boundary conditions read
and
. Finally, for a.e.
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ51_HTML.gif)
and by the Hölder inequality, (3.3), and (3.25),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ52_HTML.gif)
Therefore , and consequently
.
Again, it is clear that given by (3.29) for
and
, and
given by (3.32) for
is a solution of (3.1), and
if and only if
. Let us now consider the boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ53_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ54_HTML.gif)
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.
Let be a solution of (3.1). Then
satisfies (3.29) for
and
, and (3.32) for
. We first note that, by (3.9), for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ55_HTML.gif)
Therefore, in both, (3.29) and (3.32), and the result now follows by substituting the values,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ56_HTML.gif)
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].
The equations being dealt with are of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ57_HTML.gif)
subject to initial or boundary conditions specified in the following graphs. All solutions were computed on the unit interval .
Finally, we expect , and therefore we solve (3.41) subject to the terminal conditions
. See Figures 1, 2, and 3.
Illustrating Theorem 3. 3: solutions of differential equation (3.41) with , subject to boundary conditions
. See graph legend for the values of
and
. According to Theorem 3.3 it holds that
for each choice of
and
.
Illustrating Theorem 3. 5 for : solutions of differential equation (3.41) with
, subject to boundary conditions
. See graph legend for the values of
and
. According to Theorem 3.5 a solution
satisfies
or
or
in dependence of values
and
. In order to precisely recover a solution satisfying
, the respective simulation was carried out as an initial value problem with
and
.
Illustrating Theorem 3. 5 for : solutions of differential equation (3.41) with
, subject to boundary conditions
. See graph legend for the values of
and
. Here,
, and
, or
,
.
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.
Let us assume that (2.2) holds. Let and let
satisfy equation (2.1) a.e. on
. Finally, let us assume that that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ58_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ59_HTML.gif)
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.
Let us assume that condition (2.2) holds. Let and let
satisfy equation (2.1) a.e. on
. Let us also assume that (4.1) holds. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ60_HTML.gif)
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
We deal with a problem of the following form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ61_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ62_HTML.gif)
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).
Conditions (5.1b) can be written in the form (3.26b) with ,
, and
. Then, matrix (3.27) has the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ63_HTML.gif)
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).
Let and let us assume that condition (2.2) holds. Further, assume that for each compact set
there exists a nonnegative function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ64_HTML.gif)
for a.e. and all
. Then problem (5.1a) and (5.1b) has at most one solution.
Proof.
Let and
be different solutions of problem (5.1a) and (5.1b). Since
, there exists a compact set
such that
for
. Let us define the difference function
for
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ65_HTML.gif)
First, we prove that there exists an interval such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ66_HTML.gif)
We consider two cases.
Case 1.
Assume that and
have an intersection point, that is, there exists
such that
. Since
and
are different, there exists
,
, such that
.
-
(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.
Assume that and
have no common point, that is,
on
. We may assume that
on
. By (5.4), there exists a point
satisfying
.
-
(i)
Let
on
. Then, by (5.1a) and (5.3),
(5.6)
for a.e. , which is a contradiction to
on
.
-
(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).
Now, by (5.1a), (5.3), and (5.5), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ68_HTML.gif)
Denote by . Then
, and
for a.e.
. Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ69_HTML.gif)
Integrating the last inequality in , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ70_HTML.gif)
which contradicts . Consequently, we have shown that
, and the result follows.
In the following theorem we formulate sufficient conditions for the existence of at least one solution of problem (5.1a) and (5.1b) with . In the proof of this theorem, we work also with auxiliary two-point boundary conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ71_HTML.gif)
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).
Let and let (2.2) hold. Further, assume that there exist
,
,
, and
such that
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ72_HTML.gif)
for a.e. ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ73_HTML.gif)
for a.e. and all
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ74_HTML.gif)
Then problem (5.1a) and (5.1b) has a solution such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ75_HTML.gif)
Proof.
Step 1 (existence of auxiliary solutions ).
By (5.13), there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ76_HTML.gif)
For , let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ77_HTML.gif)
Motivated by [19], we choose ,
, and, for a.e.
, all
, and
, we define the following functions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ78_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ79_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ80_HTML.gif)
Due to (5.11),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ81_HTML.gif)
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
.
We now investigate the auxiliary problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ82_HTML.gif)
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 ).
We now show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ83_HTML.gif)
Let us define for
and assume
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ84_HTML.gif)
By (5.21), we can assume that . Since
, we can find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ85_HTML.gif)
Then, by (5.19), (5.20), and (5.21), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ86_HTML.gif)
for a.e. . Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ87_HTML.gif)
which contradicts (5.23), and thus on
. The inequality
on
can be proved in a very similar way.
Step 3 (estimates of ).
We now show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ88_HTML.gif)
By (5.19) and (5.22) we have for a.e.
, and so, due to (5.17) and (5.21), we have for a.e.
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ89_HTML.gif)
Denote . If
, then
.
Case 1.
Let . Then there exists
such that
on
,
. By (5.12), (5.22), (5.28), and
, it follows for a.e.
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ90_HTML.gif)
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ91_HTML.gif)
where is given by (5.15). Therefore
.
Case 2.
Let . Then there exists
such that
on
,
. By (5.12), (5.13), (5.22), (5.28), and
, we obtain for a.e.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ92_HTML.gif)
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ93_HTML.gif)
Hence, according to (5.15), we again have .
Step 4 (convergence of ).
Consider the sequence of solutions of problems (5.21),
,
. It has been shown in Steps 2 and 3 that (5.22) and (5.27) hold, which means that the sequences
and
are bounded in
. Therefore
is equicontinuous on
. According to (5.17), (5.19), and (5.21), we obtain for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ94_HTML.gif)
Let us now choose an arbitrary compact subinterval . Then there exists
such that
for each
. By (5.33), the sequence
is equicontinuous on
. Therefore, we can find a subsequence
such that
converges uniformly on
, and
converges uniformly on
. By the diagonalization theorem; see [11], we can find a subsequence
such that there exists
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ95_HTML.gif)
Therefore and
. For
in (5.33), Lebesgue's dominated convergence theorem yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ96_HTML.gif)
Consequently, satisfies equation (5.1a) a.e. on
. Moreover, due to (5.22) and (5.27), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ97_HTML.gif)
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.
Let ,
,
,
for some
, and
. Moreover, let
be nonnegative, and let
be bounded on
. Then in Theorem 5.3 the following class of functions
is covered:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ98_HTML.gif)
for a.e. and all
, provided
if
and
if
. In particular, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ99_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ100_HTML.gif)
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).
Let satisfy (2.2), let matrices
, vector
be given, and let
. Let us denote by
, and assume that the linear homogeneous boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ101_HTML.gif)
has only the trivial solution. Moreover, let us assume that there exists a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ102_HTML.gif)
Then the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ103_HTML.gif)
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.
Functions satisfying assumptions of Theorem 5.6 can have the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ104_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ105_HTML.gif)
for .
We now illustrate the above theoretical findings by means of numerical simulations. Figure 4 shows graphs of solutions of problem (5.43), (5.1a). In Figure 5 we display the error estimate for the global error of the numerical solution and the so-called residual (defect) obtained from the substitution of the numerical solution into the differential equation. Both quantities are rather small and indicate that we have found a solution to the analytical problem (5.43)-(5.1a).
Illustrating Theorem 5. 6: solutions of differential equation (5.43), subject to periodic boundary conditions (5.1a). See graph legend for the values of .
We now pose that question about the values of the first derivative at the end points of the interval of integration, and
. According to the theory, it holds that
. Therefore, we approximate the values of the first derivative of the numerical solution and show these values in Figure 6. One can see that indeed
. Also, to support this observation, we plotted in Figure 7 the numerical solutions obtained for the step size
tending to zero, or equivalently, grids becoming finer.
We finally observe experimentally the order of convergence of the numerical method (collocation). Clearly, we do not expect very hight order to hold, since the analytical solution has nonsmooth higher derivatives. However, the method is convergent and, according to Table 1, we observe that its order tends to .
The results of the numerical simulation for the boundary value problem (5.44)-(5.1a), can be found in Figures 8, 9, 10, and 11.
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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.
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Rachůnková, I., Staněk, S., Weinmüller, E. et al. Limit Properties of Solutions of Singular Second-Order Differential Equations. Bound Value Probl 2009, 905769 (2009). https://doi.org/10.1155/2009/905769
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DOI: https://doi.org/10.1155/2009/905769