Limit Properties of Solutions of Singular Second-Order Differential Equations
© Irena Rachůnková et al. 2009
Received: 23 April 2009
Accepted: 28 May 2009
Published: 29 June 2009
We discuss the properties of the differential equation , a.e. on , where , and satisfies the -Carathéodory conditions on for some . A full description of the asymptotic behavior for of functions satisfying the equation a.e. on is given. We also describe the structure of boundary conditions which are necessary and sufficient for to be at least in . As an application of the theory, new existence and/or uniqueness results for solutions of periodic boundary value problems are shown.
where , , and the function is defined for a.e. and for all . The above equation is singular at because of the first term in the right-hand side, which is in general unbounded for . In this paper, we will also alow the function to be unbounded or bounded but discontinuous for certain values of the time variable . This form of is motivated by a variety of initial and boundary value problems known from applications and having nonlinear, discontinuous forcing terms, such as electronic devices which are often driven by square waves or more complicated discontinuous inputs. Typically, such problems are modelled by differential equations where has jump discontinuities at a discrete set of points in , compare .
This study serves as a first step toward analysis of more involved nonlinearities, where typically, has singular points also in and . Many applications, compare [2–12], showing these structural difficulties are our main motivation to develop a framework on existence and uniqueness of solutions, their smoothness properties, and the structure of boundary conditions necessary for to have at least continuous first derivative on . Moreover, using new techniques presented in this paper, we would like to extend results from [13, 14] (based on ideas presented in ) where problems of the above form but with appropriately smooth data function have been discussed.
Here, we aim at the generalization of the existence and uniqueness assertions derived in those papers for the case of smooth . We are especially interested in studying the limit properties of for and the structure of boundary conditions which are necessary and sufficient for to be at least in .
From the above remarks, we draw the conclusion that a new approach is necessary to study the analytical properties of (1.1).
The following notation will be used throughout the paper. Let be an interval. Then, we denote by the set of functions which are (Lebesgue) integrable on . The corresponding norm is . Let . By , we denote the set of functions whose th powers of modulus are integrable on with the corresponding norm given by .
Finally, we denote by and the sets of functions which are absolutely continuous on and which have absolutely continuous first derivatives on , respectively. Analogously, and are the sets of functions being absolutely continuous on each compact subinterval and having absolutely continuous first derivatives on each compact subinterval , respectively.
specified in the following definition.
We will provide a full description of the asymptotical behavior for of functions satisfying (2.1) a.e. on . Such functions will be called solutions of (2.1) if they additionally satisfy the smoothness requirement ; see next definition.
In Section 3, we consider linear problems and characterize the structure of boundary conditions necessary for the solution to be at least continuous on . These results are modified for nonlinear problems in Section 4. In Section 5, by applying the theory developed in Section 4, we provide new existence and/or uniqueness results for solutions of singular boundary value problems (2.1) with periodic boundary conditions.
3. Linear Singular Equation
which means that . We now use the properties of to represent all functions satisfying (3.1) a.e. on . Remember that such function does not need to be a solution of (3.1) in the sense of Definition 2.2.
We stress that by (3.8), the particular solution of (3.1) belongs to . For , we can see from (3.9) that it is useful to find other solution representations which are equivalent to (3.10) and (3.11), but use instead of , if .
which completes the proof.
of (3.1) for satisfies . Main results for the linear singular equation (3.1) are now formulated in the following theorems.
into the boundary conditions (3.26b).
Let and let a function satisfy equation (3.1) a.e. on . For , only one of the following properties holds:
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 . This software has already been used for a variety of singular boundary value problems relevant for applications; see, for example, .
4. Limit Properties of Functions Satisfying Nonlinear Singular Equations
In this section we assume that the function satisfying differential equation (2.1) a.e. on is given. The first derivative of such a function does not need to be continuous at and hence, due to the lack of smoothness, does not need to be a solution of (2.1) in the sense of Definition 2.2. In the following two theorems, we discuss the limit properties of for .
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.
A function is called a solution of the boundary value problem (5.1a) and (5.1b), if satisfies equation (5.1a) for a.e. and the periodic boundary conditions (5.1b).
and we see that it is singular. Consequently, the assumption of Theorem 3.4 is not satisfied, and the linear periodic problem (3.26b) subject to (5.1b) is not uniquely solvable. However this is not true for nonliner periodic problems. In particular, Theorem 5.6 gives a characterization of a class of nonlinear periodic problems (5.1a) and (5.1b) which have only one solution. We begin the investigation of problem (5.1a) and (5.1b) with a uniqueness result.
Theorem 5.2 (uniqueness).
We consider two cases.
Let . We can assume that . (Otherwise we choose .) Then we can find satisfying for and . Let be the first zero of . Then, if we set , we see that satisfies (5.5). Let have no zeros on . Then on , and, due to (5.4), . Since , we can find and such that satisfies (5.5).
Under the assumptions of Theorem 4.1 any solution of (5.1a) satisfies Therefore, we can investigate (5.1a) subject to the auxiliary conditions (5.10) instead of the equivalent original problem (5.1a) and (5.1b). This change of the problem setting is useful for obtaining of a priori estimates necessary for the application of the Fredholm-type existence theorem (Lemma 5.5) during the proof.
Theorem 5.3 (existence).
for a.e. . It can be shown that and which satisfy the -Carathéodory conditions on are nondecreasing in their second argument and a.e. on ; see . Therefore, also satisfies the -Carathéodory conditions on , and there exists a function such that for a.e. and all .
In order to show the existence of solutions to the periodic boundary value problem (5.1a) and (5.1b), the Fredholm-type Existence Theorem is used, see for example, in [20, Theorem ], [11, Theorem ] or [21, page 25]. For convenience, we provide its simple formulation suitable for our purpose below.
Lemma 5.5 (Fredholm-type existence theorem).
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).
Estimated convergence order for the periodic boundary value problem (5.43)-(5.1a) and a = 1.
This research was supported by the Council of Czech Goverment MSM6198959214 and by the Grant no. A100190703 of the Grant Agency of the Academy of Sciences of the Czech Republic.
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