Open Access

Limit Properties of Solutions of Singular Second-Order Differential Equations

  • Irena Rachůnková1Email author,
  • Svatoslav Staněk1,
  • Ewa Weinmüller2 and
  • Michael Zenz2
Boundary Value Problems20092009:905769

DOI: 10.1155/2009/905769

Received: 23 April 2009

Accepted: 28 May 2009

Published: 29 June 2009

Abstract

We discuss the properties of the differential equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq1_HTML.gif , a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq2_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq3_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq4_HTML.gif satisfies the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq5_HTML.gif -Carathéodory conditions on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq6_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq7_HTML.gif . A full description of the asymptotic behavior for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq8_HTML.gif of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq9_HTML.gif satisfying the equation a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq10_HTML.gif is given. We also describe the structure of boundary conditions which are necessary and sufficient for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq11_HTML.gif to be at least in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq12_HTML.gif . 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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq14_HTML.gif , and the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq15_HTML.gif is defined for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq16_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq17_HTML.gif . The above equation is singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq18_HTML.gif because of the first term in the right-hand side, which is in general unbounded for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq19_HTML.gif . In this paper, we will also alow the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq20_HTML.gif to be unbounded or bounded but discontinuous for certain values of the time variable https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq21_HTML.gif . This form of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq22_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq23_HTML.gif has jump discontinuities at a discrete set of points in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq24_HTML.gif , compare [1].

This study serves as a first step toward analysis of more involved nonlinearities, where typically, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq25_HTML.gif has singular points also in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq26_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq27_HTML.gif . Many applications, compare [212], 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq28_HTML.gif to have at least continuous first derivative on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq29_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq30_HTML.gif have been discussed.

Here, we aim at the generalization of the existence and uniqueness assertions derived in those papers for the case of smooth https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq31_HTML.gif . We are especially interested in studying the limit properties of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq32_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq33_HTML.gif and the structure of boundary conditions which are necessary and sufficient for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq34_HTML.gif to be at least in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq35_HTML.gif .

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],
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ2_HTML.gif
(1.2)
subject to boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ3_HTML.gif
(1.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq36_HTML.gif Note that (1.2) can be written in the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ4_HTML.gif
(1.4)
which is of form (1.1) with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ5_HTML.gif
(1.5)
Function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq37_HTML.gif is not defined for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq38_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq39_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq40_HTML.gif . We now briefly discuss a simplified linear model of (1.4),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ6_HTML.gif
(1.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq42_HTML.gif . Clearly, this means that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq43_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq44_HTML.gif of (1.6) to be unique and at least continuously differentiable, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq45_HTML.gif ? 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq46_HTML.gif 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,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ7_HTML.gif
(1.7)
which suggest to introduce a new variable, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq47_HTML.gif . In a general situation, especially for the nonlinear case, it is not straightforward to provide such a transformation, however. We now introduce https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq48_HTML.gif and immediately obtain the following system of ordinary differential equations:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ8_HTML.gif
(1.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq49_HTML.gif or equivalently,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ9_HTML.gif
(1.9)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq50_HTML.gif . According to [16], the latter system of equations has a continuous solution if and only if the regularity condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq51_HTML.gif holds. This results in
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ10_HTML.gif
(1.10)
compare conditions (1.3). Note that the Euler transformation, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq52_HTML.gif which is usually used to transform (1.6) to the first-order form would have resulted in the following system:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ11_HTML.gif
(1.11)

Here, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq53_HTML.gif may become unbounded for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq54_HTML.gif , the condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq55_HTML.gif , or equivalently https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq56_HTML.gif is not the correct condition for the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq57_HTML.gif to be continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq58_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq59_HTML.gif be an interval. Then, we denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq60_HTML.gif the set of functions which are (Lebesgue) integrable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq61_HTML.gif . The corresponding norm is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq62_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq63_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq64_HTML.gif , we denote the set of functions whose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq65_HTML.gif th powers of modulus are integrable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq66_HTML.gif with the corresponding norm given by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq67_HTML.gif .

Moreover, let us by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq69_HTML.gif denote the sets of functions being continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq70_HTML.gif and having continuous first derivatives on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq71_HTML.gif , respectively. The norm on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq72_HTML.gif is defined as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq73_HTML.gif .

Finally, we denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq75_HTML.gif the sets of functions which are absolutely continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq76_HTML.gif and which have absolutely continuous first derivatives on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq77_HTML.gif , respectively. Analogously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq79_HTML.gif are the sets of functions being absolutely continuous on each compact subinterval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq80_HTML.gif and having absolutely continuous first derivatives on each compact subinterval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq81_HTML.gif , respectively.

As already said in the previous section, we investigate differential equations of the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ12_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq82_HTML.gif . For the subsequent analysis we assume that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ13_HTML.gif
(2.2)

specified in the following definition.

Definition 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq83_HTML.gif . A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq84_HTML.gif satisfies the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq85_HTML.gif -Carathéodory conditions on the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq86_HTML.gif if

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq87_HTML.gif is measurable for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq88_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq89_HTML.gif is continuous for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq90_HTML.gif ,

(iii) for each compact set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq91_HTML.gif there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq92_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq93_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq94_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq95_HTML.gif .

We will provide a full description of the asymptotical behavior for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq96_HTML.gif of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq97_HTML.gif satisfying (2.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq98_HTML.gif . Such functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq99_HTML.gif will be called solutions of (2.1) if they additionally satisfy the smoothness requirement https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq100_HTML.gif ; see next definition.

Definition 2.2.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq101_HTML.gif is called a solution of (2.1) if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq102_HTML.gif and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ14_HTML.gif
(2.3)

In Section 3, we consider linear problems and characterize the structure of boundary conditions necessary for the solution to be at least continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq103_HTML.gif . 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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq104_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ15_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq105_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq106_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq107_HTML.gif , let us denote by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ16_HTML.gif
(3.2)
Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq108_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ17_HTML.gif
(3.3)

Now, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq109_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq110_HTML.gif . Without loss of generality, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq111_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq112_HTML.gif , we choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq113_HTML.gif , and we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq115_HTML.gif .

First, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq116_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq117_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq118_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ18_HTML.gif
(3.4)
Now, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq119_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq120_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq121_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ19_HTML.gif
(3.5)
Hence, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq123_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ20_HTML.gif
(3.6)
Consequently, (3.3), (3.6), and the Hölder inequality yield, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq124_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ21_HTML.gif
(3.7)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ22_HTML.gif
(3.8)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ23_HTML.gif
(3.9)

which means that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq125_HTML.gif . We now use the properties of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq126_HTML.gif to represent all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq127_HTML.gif satisfying (3.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq128_HTML.gif . Remember that such function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq129_HTML.gif does not need to be a solution of (3.1) in the sense of Definition 2.2.

Lemma 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq130_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq131_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq132_HTML.gif be given by (3.2).

(i) If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq133_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ24_HTML.gif
(3.10)

is the set of all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq134_HTML.gif satisfying (3.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq135_HTML.gif .

(ii) If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq136_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ25_HTML.gif
(3.11)

is the set of all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq137_HTML.gif satisfying (3.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq138_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq139_HTML.gif . Note that (3.1) is linear and regular on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq140_HTML.gif . Since the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq142_HTML.gif are linearly independent solutions of the homogeneous equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq143_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq144_HTML.gif , the general solution of the homogeneous problem is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ26_HTML.gif
(3.12)

Moreover, the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq145_HTML.gif is a particular solution of (3.1) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq146_HTML.gif . Therefore, the first statement follows. Analogous argument yields the second assertion.

We stress that by (3.8), the particular solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq147_HTML.gif of (3.1) belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq148_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq149_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq150_HTML.gif instead of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq151_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq152_HTML.gif .

Lemma 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq153_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq154_HTML.gif be given by (3.2).

(i) If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq155_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ27_HTML.gif
(3.13)

is the set of all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq156_HTML.gif satisfying (3.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq157_HTML.gif .

(ii) If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq158_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ28_HTML.gif
(3.14)

is the set of all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq159_HTML.gif satisfying (3.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq160_HTML.gif .

Proof.

Let us fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq161_HTML.gif and define
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ29_HTML.gif
(3.15)
In order to prove (i) we have to show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq162_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq163_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq164_HTML.gif . This follows immediately from (3.9), since
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ30_HTML.gif
(3.16)
and hence we can define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq165_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ31_HTML.gif
(3.17)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq166_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ32_HTML.gif
(3.18)

which completes the proof.

Again, by (3.9), the particular solution,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ33_HTML.gif
(3.19)

of (3.1) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq167_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq168_HTML.gif . Main results for the linear singular equation (3.1) are now formulated in the following theorems.

Theorem 3.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq169_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq170_HTML.gif satisfy equation (3.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq171_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ34_HTML.gif
(3.20)

Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq172_HTML.gif can be extended to the whole interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq173_HTML.gif in such a way that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq174_HTML.gif .

Proof.

Let a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq175_HTML.gif be given. Then, by (3.10), there exist two constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq176_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq177_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ35_HTML.gif
(3.21)
Using (3.8), we conclude
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ36_HTML.gif
(3.22)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq178_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq179_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq180_HTML.gif . Furthermore, for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq181_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ37_HTML.gif
(3.23)
By the Hölder inequality and (3.6) it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ38_HTML.gif
(3.24)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ39_HTML.gif
(3.25)

Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq182_HTML.gif , and consequently https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq183_HTML.gif .

It is clear from the above theorem, that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq184_HTML.gif given by (3.21) is a solution of (3.1) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq185_HTML.gif . Let us now consider the associated boundary value problem,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ40_HTML.gif
(3.26a)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ41_HTML.gif
(3.26b)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq186_HTML.gif are real matrices, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq187_HTML.gif is an arbitrary vector. Then the following result follows immediately from Theorem 3.3.

Theorem 3.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq188_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq189_HTML.gif . Then for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq190_HTML.gif and any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq191_HTML.gif there exists a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq192_HTML.gif of the boundary value problem (3.26a) and (3.26b) if and only if the following matrix,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ42_HTML.gif
(3.27)

is nonsingular.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq193_HTML.gif be a solution of (3.1). Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq194_HTML.gif satisfies (3.21), and the result follows immediately by substituting the values,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ43_HTML.gif
(3.28)

into the boundary conditions (3.26b).

Theorem 3.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq195_HTML.gif and let a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq196_HTML.gif satisfy equation (3.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq197_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq198_HTML.gif , only one of the following properties holds:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq199_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq200_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq201_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq202_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq203_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq204_HTML.gif satisfies only one of the following properties:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq205_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq206_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq207_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq208_HTML.gif .

In particular, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq209_HTML.gif can be extended to the whole interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq210_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq211_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq212_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq213_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq214_HTML.gif be given. Then, by (3.13), there exist two constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq215_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ44_HTML.gif
(3.29)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ45_HTML.gif
(3.30)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq216_HTML.gif , then it follows from (3.9) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq217_HTML.gif . Also, by (3.29), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq218_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq219_HTML.gif . Then (3.9), (3.29), and (3.30) imply that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ46_HTML.gif
(3.31)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq220_HTML.gif . Then, by (3.14), for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq221_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ47_HTML.gif
(3.32)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ48_HTML.gif
(3.33)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq222_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq223_HTML.gif by (3.9), and it follows from (3.32) that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq224_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq225_HTML.gif . Then we deduce from (3.9), (3.32), and (3.33) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ49_HTML.gif
(3.34)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq226_HTML.gif . Then on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq227_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq228_HTML.gif satisfies (3.29) and (3.30), with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq229_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq230_HTML.gif , then, by (3.9), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq231_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq232_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq233_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ50_HTML.gif
(3.35)
In particular, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq234_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq235_HTML.gif can be extended to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq236_HTML.gif in such a way that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq237_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq238_HTML.gif . Then, the associated boundary conditions read https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq239_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq240_HTML.gif . Finally, for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq241_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ51_HTML.gif
(3.36)
and by the Hölder inequality, (3.3), and (3.25),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ52_HTML.gif
(3.37)

Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq242_HTML.gif , and consequently https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq243_HTML.gif .

Again, it is clear that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq244_HTML.gif given by (3.29) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq245_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq246_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq247_HTML.gif given by (3.32) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq248_HTML.gif is a solution of (3.1), and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq249_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq250_HTML.gif . Let us now consider the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ53_HTML.gif
(3.38a)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ54_HTML.gif
(3.38b)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq251_HTML.gif are real constants. Then the following result follows immediately from Theorem 3.5.

Theorem 3.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq252_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq253_HTML.gif . Then for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq254_HTML.gif and any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq255_HTML.gif there exists a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq256_HTML.gif of the boundary value problem (3.38a) and (3.38b) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq257_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq258_HTML.gif be a solution of (3.1). Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq259_HTML.gif satisfies (3.29) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq260_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq261_HTML.gif , and (3.32) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq262_HTML.gif . We first note that, by (3.9), for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq263_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ55_HTML.gif
(3.39)
Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq264_HTML.gif in both, (3.29) and (3.32), and the result now follows by substituting the values,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ56_HTML.gif
(3.40)

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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ57_HTML.gif
(3.41)

subject to initial or boundary conditions specified in the following graphs. All solutions were computed on the unit interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq265_HTML.gif .

Finally, we expect https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq266_HTML.gif , and therefore we solve (3.41) subject to the terminal conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq267_HTML.gif . See Figures 1, 2, and 3.
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig1_HTML.jpg
Figure 1

Illustrating Theorem 3. 3: solutions of differential equation (3.41) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq268_HTML.gif , subject to boundary conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq269_HTML.gif . See graph legend for the values of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq270_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq271_HTML.gif . According to Theorem 3.3 it holds that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq272_HTML.gif for each choice of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq273_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq274_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig2_HTML.jpg
Figure 2

Illustrating Theorem 3. 5 for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq275_HTML.gif : solutions of differential equation (3.41) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq276_HTML.gif , subject to boundary conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq277_HTML.gif . See graph legend for the values of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq278_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq279_HTML.gif . According to Theorem 3.5 a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq280_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq281_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq282_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq283_HTML.gif in dependence of values https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq284_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq285_HTML.gif . In order to precisely recover a solution satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq286_HTML.gif , the respective simulation was carried out as an initial value problem with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq287_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq288_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig3_HTML.jpg
Figure 3

Illustrating Theorem 3. 5 for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq289_HTML.gif : solutions of differential equation (3.41) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq290_HTML.gif , subject to boundary conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq291_HTML.gif . See graph legend for the values of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq292_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq293_HTML.gif . Here, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq294_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq295_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq296_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq297_HTML.gif .

4. Limit Properties of Functions Satisfying Nonlinear Singular Equations

In this section we assume that the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq298_HTML.gif satisfying differential equation (2.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq299_HTML.gif is given. The first derivative of such a function does not need to be continuous at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq300_HTML.gif and hence, due to the lack of smoothness, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq301_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq302_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq303_HTML.gif .

Theorem 4.1.

Let us assume that (2.2) holds. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq304_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq305_HTML.gif satisfy equation (2.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq306_HTML.gif . Finally, let us assume that that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ58_HTML.gif
(4.1)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ59_HTML.gif
(4.2)

and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq307_HTML.gif can be extended on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq308_HTML.gif in such a way that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq309_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq310_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq311_HTML.gif . By (2.2), there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq312_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq313_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq314_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq315_HTML.gif . Since the equality https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq316_HTML.gif holds a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq317_HTML.gif , the result follows immediately due to Theorem 3.3.

Theorem 4.2.

Let us assume that condition (2.2) holds. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq318_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq319_HTML.gif satisfy equation (2.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq320_HTML.gif . Let us also assume that (4.1) holds. Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ60_HTML.gif
(4.3)

and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq321_HTML.gif can be extended on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq322_HTML.gif in such a way that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq323_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq324_HTML.gif be as in the proof of Theorem 4.1. According to Theorem 3.5 and (4.1), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq325_HTML.gif satisfies (4.3) both for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq326_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq327_HTML.gif .

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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ61_HTML.gif
(5.1a)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ62_HTML.gif
(5.1b)

Definition 5.1.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq328_HTML.gif is called a solution of the boundary value problem (5.1a) and (5.1b), if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq329_HTML.gif satisfies equation (5.1a) for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq330_HTML.gif and the periodic boundary conditions (5.1b).

Conditions (5.1b) can be written in the form (3.26b) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq331_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq332_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq333_HTML.gif . Then, matrix (3.27) has the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ63_HTML.gif
(5.2)

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq334_HTML.gif and let us assume that condition (2.2) holds. Further, assume that for each compact set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq335_HTML.gif there exists a nonnegative function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq336_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ64_HTML.gif
(5.3)

for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq337_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq338_HTML.gif . Then problem (5.1a) and (5.1b) has at most one solution.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq339_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq340_HTML.gif be different solutions of problem (5.1a) and (5.1b). Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq341_HTML.gif , there exists a compact set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq342_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq343_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq344_HTML.gif . Let us define the difference function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq345_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq346_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ65_HTML.gif
(5.4)
First, we prove that there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq347_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ66_HTML.gif
(5.5)

We consider two cases.

Case 1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq348_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq349_HTML.gif have an intersection point, that is, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq350_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq351_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq352_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq353_HTML.gif are different, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq354_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq355_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq356_HTML.gif .
  1. (i)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq357_HTML.gif . We can assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq358_HTML.gif . (Otherwise we choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq359_HTML.gif .) Then we can find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq360_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq361_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq362_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq363_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq364_HTML.gif be the first zero of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq365_HTML.gif . Then, if we set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq366_HTML.gif , we see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq367_HTML.gif satisfies (5.5). Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq368_HTML.gif have no zeros on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq369_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq370_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq371_HTML.gif , and, due to (5.4), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq372_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq373_HTML.gif , we can find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq374_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq375_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq376_HTML.gif satisfies (5.5).

     
  2. (ii)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq377_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq378_HTML.gif . By (5.4), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq379_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq380_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq381_HTML.gif . We may again assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq382_HTML.gif . It is possible to find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq383_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq384_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq385_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq386_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq387_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq388_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq389_HTML.gif has at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq390_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq391_HTML.gif is the first zero of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq392_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq393_HTML.gif satisfies (5.5).

     

Case 2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq394_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq395_HTML.gif have no common point, that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq396_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq397_HTML.gif . We may assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq398_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq399_HTML.gif . By (5.4), there exists a point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq400_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq401_HTML.gif .
  1. (i)
    Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq402_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq403_HTML.gif . Then, by (5.1a) and (5.3),
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ67_HTML.gif
    (5.6)
     
for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq404_HTML.gif , which is a contradiction to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq405_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq406_HTML.gif .
  1. (ii)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq407_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq408_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq409_HTML.gif , then we can find an interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq410_HTML.gif satisfying (5.5). If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq411_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq412_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq413_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq414_HTML.gif and, by (5.4), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq415_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq416_HTML.gif . Hence, there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq417_HTML.gif satisfying (5.5).

     

To summarize, we have shown that in both, the case of intersecting solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq418_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq419_HTML.gif and the case of separated https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq420_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq421_HTML.gif , there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq422_HTML.gif satisfying (5.5).

Now, by (5.1a), (5.3), and (5.5), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ68_HTML.gif
(5.7)
Denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq423_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq424_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq425_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq426_HTML.gif . Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ69_HTML.gif
(5.8)
Integrating the last inequality in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq427_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ70_HTML.gif
(5.9)

which contradicts https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq428_HTML.gif . Consequently, we have shown that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq429_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq430_HTML.gif . In the proof of this theorem, we work also with auxiliary two-point boundary conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ71_HTML.gif
(5.10)

Under the assumptions of Theorem 4.1 any solution of (5.1a) satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq431_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq432_HTML.gif and let (2.2) hold. Further, assume that there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq433_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq434_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq435_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq436_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq437_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ72_HTML.gif
(5.11)
for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq438_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ73_HTML.gif
(5.12)
for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq439_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq440_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ74_HTML.gif
(5.13)
Then problem (5.1a) and (5.1b) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq441_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ75_HTML.gif
(5.14)

Proof.

Step 1 (existence of auxiliary solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq442_HTML.gif ).

By (5.13), there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq443_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ76_HTML.gif
(5.15)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq444_HTML.gif , let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ77_HTML.gif
(5.16)
Motivated by [19], we choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq445_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq446_HTML.gif , and, for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq447_HTML.gif , all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq448_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq449_HTML.gif , we define the following functions:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ78_HTML.gif
(5.17)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ79_HTML.gif
(5.18)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ80_HTML.gif
(5.19)
Due to (5.11),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ81_HTML.gif
(5.20)

for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq450_HTML.gif . It can be shown that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq451_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq452_HTML.gif which satisfy the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq453_HTML.gif -Carathéodory conditions on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq454_HTML.gif are nondecreasing in their second argument and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq455_HTML.gif a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq456_HTML.gif ; see [19]. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq457_HTML.gif also satisfies the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq458_HTML.gif -Carathéodory conditions on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq459_HTML.gif , and there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq460_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq461_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq462_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq463_HTML.gif .

We now investigate the auxiliary problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ82_HTML.gif
(5.21)

Since the homogeneous problem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq464_HTML.gif , has only the trivial solution, we conclude by the Fredholm-type Existence Theorem (see Lemma 5.5) that there exists a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq465_HTML.gif of problem (5.21).

Step 2 (estimates of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq466_HTML.gif ).

We now show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ83_HTML.gif
(5.22)
Let us define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq467_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq468_HTML.gif and assume
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ84_HTML.gif
(5.23)
By (5.21), we can assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq469_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq470_HTML.gif , we can find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq471_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ85_HTML.gif
(5.24)
Then, by (5.19), (5.20), and (5.21), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ86_HTML.gif
(5.25)
for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq472_HTML.gif . Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ87_HTML.gif
(5.26)

which contradicts (5.23), and thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq473_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq474_HTML.gif . The inequality https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq475_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq476_HTML.gif can be proved in a very similar way.

Step 3 (estimates of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq477_HTML.gif ).

We now show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ88_HTML.gif
(5.27)
By (5.19) and (5.22) we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq478_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq479_HTML.gif , and so, due to (5.17) and (5.21), we have for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq480_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ89_HTML.gif
(5.28)

Denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq481_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq482_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq483_HTML.gif .

Case 1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq484_HTML.gif . Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq485_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq486_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq487_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq488_HTML.gif . By (5.12), (5.22), (5.28), and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq489_HTML.gif , it follows for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq490_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ90_HTML.gif
(5.29)
Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ91_HTML.gif
(5.30)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq491_HTML.gif is given by (5.15). Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq492_HTML.gif .

Case 2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq493_HTML.gif . Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq494_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq495_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq496_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq497_HTML.gif . By (5.12), (5.13), (5.22), (5.28), and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq498_HTML.gif , we obtain for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq499_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ92_HTML.gif
(5.31)
Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ93_HTML.gif
(5.32)

Hence, according to (5.15), we again have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq500_HTML.gif .

Step 4 (convergence of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq501_HTML.gif ).

Consider the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq502_HTML.gif of solutions of problems (5.21), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq503_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq504_HTML.gif . It has been shown in Steps 2 and 3 that (5.22) and (5.27) hold, which means that the sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq505_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq506_HTML.gif are bounded in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq507_HTML.gif . Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq508_HTML.gif is equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq509_HTML.gif . According to (5.17), (5.19), and (5.21), we obtain for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq510_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ94_HTML.gif
(5.33)
Let us now choose an arbitrary compact subinterval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq511_HTML.gif . Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq512_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq513_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq514_HTML.gif . By (5.33), the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq515_HTML.gif is equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq516_HTML.gif . Therefore, we can find a subsequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq517_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq518_HTML.gif converges uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq519_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq520_HTML.gif converges uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq521_HTML.gif . By the diagonalization theorem; see [11], we can find a subsequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq522_HTML.gif such that there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq523_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ95_HTML.gif
(5.34)
Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq524_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq525_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq526_HTML.gif in (5.33), Lebesgue's dominated convergence theorem yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ96_HTML.gif
(5.35)
Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq527_HTML.gif satisfies equation (5.1a) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq528_HTML.gif . Moreover, due to (5.22) and (5.27), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ97_HTML.gif
(5.36)

Hence (4.1) is satisfied. Applying Theorem 4.1, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq529_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq530_HTML.gif . Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq531_HTML.gif satisfies the periodic conditions on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq532_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq533_HTML.gif is a solution of problem (5.1a) and (5.1b) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq534_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq535_HTML.gif .

Example 5.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq536_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq537_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq538_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq539_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq540_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq541_HTML.gif . Moreover, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq542_HTML.gif be nonnegative, and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq543_HTML.gif be bounded on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq544_HTML.gif . Then in Theorem 5.3 the following class of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq545_HTML.gif is covered:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ98_HTML.gif
(5.37)
for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq546_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq547_HTML.gif , provided https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq548_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq549_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq550_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq551_HTML.gif . In particular, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq552_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq553_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ99_HTML.gif
(5.38)
or
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ100_HTML.gif
(5.39)

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq554_HTML.gif ], [11, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq555_HTML.gif ] or [21, page 25]. For convenience, we provide its simple formulation suitable for our purpose below.

Lemma 5.5 (Fredholm-type existence theorem).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq556_HTML.gif satisfy (2.2), let matrices https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq557_HTML.gif , vector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq558_HTML.gif be given, and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq559_HTML.gif . Let us denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq560_HTML.gif , and assume that the linear homogeneous boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ101_HTML.gif
(5.40)
has only the trivial solution. Moreover, let us assume that there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq561_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ102_HTML.gif
(5.41)
Then the problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ103_HTML.gif
(5.42)

has a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq562_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq563_HTML.gif . Moreover https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq564_HTML.gif satisfies (5.14).

Example 5.7.

Functions satisfying assumptions of Theorem 5.6 can have the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ104_HTML.gif
(5.43)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Equ105_HTML.gif
(5.44)

for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq565_HTML.gif .

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).
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig4_HTML.jpg
Figure 4

Illustrating Theorem 5. 6: solutions of differential equation (5.43), subject to periodic boundary conditions (5.1a). See graph legend for the values of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq566_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig5_HTML.jpg
Figure 5

Error estimate (a) and residual (b) for (5. 43)-(5.1a), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq567_HTML.gif .

We now pose that question about the values of the first derivative at the end points of the interval of integration, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq568_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq569_HTML.gif . According to the theory, it holds that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq570_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq571_HTML.gif . Also, to support this observation, we plotted in Figure 7 the numerical solutions obtained for the step size https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq572_HTML.gif tending to zero, or equivalently, grids becoming finer.
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig6_HTML.jpg
Figure 6

First derivative of the numerical solution to (5. 43)-(5.1a) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq573_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig7_HTML.jpg
Figure 7

Numerical solutions of (5. 43)-(5.1a) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq574_HTML.gif in the vicinity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq575_HTML.gif (a) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq576_HTML.gif (b). The step size is decreasing according to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq577_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq578_HTML.gif .
Table 1

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq579_HTML.gif

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

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.
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig8_HTML.jpg
Figure 8

Illustrating Theorem 5. 6: solutions of the boundary value problem (5.44)-(5.1a). See graph legend for the values of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq580_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig9_HTML.jpg
Figure 9

Error estimate (a) and residual (b) for (5. 44)-(5.1a), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq581_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig10_HTML.jpg
Figure 10

First derivative of the numerical solution to (5. 44)-(5.1a) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq582_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_Fig11_HTML.jpg
Figure 11

Numerical solutions of (5. 44)-(5.1a) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq583_HTML.gif in the vicinity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq584_HTML.gif (a) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq585_HTML.gif (b). The step size is decreasing according to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F905769/MediaObjects/13661_2009_Article_888_IEq586_HTML.gif .

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

(1)
Department of Mathematical Analysis, Faculty of Science, Palacký University
(2)
Institute for Analysis and Scientific Computing, Vienna University of Technology

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© Irena Rachůnková et al. 2009

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