Limit Properties of Solutions of Singular Second-Order Differential Equations

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.

In this paper, we study the analytical properties of the differential equation

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

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

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

To clarify the aims of this paper and to show that it is necessary to develop a new technique to treat the nonstandard equation given above, let us consider a model problem which we designed using the structure of the boundary value problem describing a membrane arising in the theory of shallow membrane caps and studied in [10]; see also [6, 9],

subject to boundary conditions

where Note that (1.2) can be written in the form

which is of form (1.1) with

Function is not defined for and for if . We now briefly discuss a simplified linear model of (1.4),

where and . Clearly, this means that .

The question which we now pose is the role of the boundary conditions (1.3), more precisely, are these boundary conditions necessary and sufficient for the solution of (1.6) to be unique and at least continuously differentiable, ? To answer this question, we can use techniques developed in the classical framework dealing with boundary value problems, exhibiting a singularity of the first and second kind; see [15, 16], respectively. However, in these papers, the analytical properties of the solution are derived for nonhomogeneous terms being at least continuous. Clearly, we need to rewrite problem (1.6) first and obtain its new form stated as,

which suggest to introduce a new variable, . In a general situation, especially for the nonlinear case, it is not straightforward to provide such a transformation, however. We now introduce and immediately obtain the following system of ordinary differential equations:

where or equivalently,

where . According to [16], the latter system of equations has a continuous solution if and only if the regularity condition holds. This results in

compare conditions (1.3). Note that the Euler transformation, which is usually used to transform (1.6) to the first-order form would have resulted in the following system:

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

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

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

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

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

As already said in the previous section, we investigate differential equations of the form

where . For the subsequent analysis we assume that

specified in the following definition.

Definition 2.1.

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

(i) is measurable for all ,

(ii) is continuous for a.e. ,

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

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

Definition 2.2.

A function is called a solution of (2.1) if and satisfies

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.

First, we consider the linear equation, ,

where and .

As a first step in the analysis of (3.1), we derive the necessary auxiliary estimates used in the discussion of the solution behavior. For , let us denote by

Assume that . Then

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

First, let . Then , , and

Now, let . Then , , and

Hence, for , ,

Consequently, (3.3), (3.6), and the Hölder inequality yield, ,

Therefore

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

Lemma 3.1.

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

(i)If , then

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

(ii)If , then

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

Proof.

Let . Note that (3.1) is linear and regular on . Since the functions and are linearly independent solutions of the homogeneous equation on , the general solution of the homogeneous problem is

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

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

Lemma 3.2.

Let and let be given by (3.2).

(i)If , then

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

(ii)If , then

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

Proof.

Let us fix and define

In order to prove (i) we have to show that for , where . This follows immediately from (3.9), since

and hence we can define as follows:

For we have

which completes the proof.

Again, by (3.9), the particular solution,

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

Theorem 3.3.

Let and let satisfy equation (3.1) a.e. on . Then

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

Proof.

Let a function be given. Then, by (3.10), there exist two constants such that for ,

Using (3.8), we conclude

For and , we have . Furthermore, for a.e. ,

By the Hölder inequality and (3.6) it follows that

where

Therefore , and consequently .

It is clear from the above theorem, that given by (3.21) is a solution of (3.1) for . Let us now consider the associated boundary value problem,

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

Theorem 3.4.

Let , . Then for any and any there exists a unique solution of the boundary value problem (3.26a) and (3.26b) if and only if the following matrix,

is nonsingular.

Proof.

Let be a solution of (3.1). Then satisfies (3.21), and the result follows immediately by substituting the values,

into the boundary conditions (3.26b).

Theorem 3.5.

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

(i), ,

(ii), .

For , satisfies only one of the following properties:

(i), ,

(ii), .

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

Proof.

Let , and let be given. Then, by (3.13), there exist two constants such that

Hence

Let , then it follows from (3.9) . Also, by (3.29), . Let . Then (3.9), (3.29), and (3.30) imply that

Let . Then, by (3.14), for any ,

If , then by (3.9), and it follows from (3.32) that . Let . Then we deduce from (3.9), (3.32), and (3.33) that

Let . Then on , satisfies (3.29) and (3.30), with . If , then, by (3.9), and . Let . Then

In particular, for , can be extended to in such a way that if and only if . Then, the associated boundary conditions read and . Finally, for a.e. ,

and by the Hölder inequality, (3.3), and (3.25),

Therefore , and consequently .

Again, it is clear that given by (3.29) for and , and given by (3.32) for is a solution of (3.1), and if and only if . Let us now consider the boundary value problem

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

Theorem 3.6.

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

Proof.

Let be a solution of (3.1). Then satisfies (3.29) for and , and (3.32) for . We first note that, by (3.9), for all ,

Therefore, in both, (3.29) and (3.32), and the result now follows by substituting the values,

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

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

Finally, we expect , and therefore we solve (3.41) subject to the terminal conditions . See Figures 1, 2, and 3.

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

Theorem 4.1.

Let us assume that (2.2) holds. Let and let satisfy equation (2.1) a.e. on . Finally, let us assume that that

Then

and can be extended on in such a way that .

Proof.

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

Theorem 4.2.

Let us assume that condition (2.2) holds. Let and let satisfy equation (2.1) a.e. on . Let us also assume that (4.1) holds. Then

and can be extended on in such a way that .

Proof.

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

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:

Definition 5.1.

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

Conditions (5.1b) can be written in the form (3.26b) with , , and . Then, matrix (3.27) has the form

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

Theorem 5.2 (uniqueness).

Let and let us assume that condition (2.2) holds. Further, assume that for each compact set there exists a nonnegative function such that

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

Proof.

Let and be different solutions of problem (5.1a) and (5.1b). Since , there exists a compact set such that for . Let us define the difference function for . Then

First, we prove that there exists an interval such that

We consider two cases.

Case 1.

Assume that and have an intersection point, that is, there exists such that . Since and are different, there exists , , such that .

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

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

Case 2.

Assume that and have no common point, that is, on . We may assume that on . By (5.4), there exists a point satisfying .

(i) Let on . Then, by (5.1a) and (5.3),

for a.e. , which is a contradiction to on .

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

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

Now, by (5.1a), (5.3), and (5.5), we obtain

Denote by . Then , and for a.e. . Consequently,

Integrating the last inequality in , we obtain

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

In the following theorem we formulate sufficient conditions for the existence of at least one solution of problem (5.1a) and (5.1b) with . In the proof of this theorem, we work also with auxiliary two-point boundary conditions:

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

Theorem 5.3 (existence).

Let and let (2.2) hold. Further, assume that there exist , , , and such that ,

for a.e. ,

for a.e. and all , where

Then problem (5.1a) and (5.1b) has a solution such that

Proof.

Step 1 (existence of auxiliary solutions ).

By (5.13), there exists such that

For , let

Motivated by [19], we choose , , and, for a.e. , all , and , we define the following functions:

Due to (5.11),

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

We now investigate the auxiliary problem

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

Step 2 (estimates of ).

We now show that

Let us define for and assume

By (5.21), we can assume that . Since , we can find such that

Then, by (5.19), (5.20), and (5.21), we have

for a.e. . Hence,

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

Step 3 (estimates of ).

We now show that

By (5.19) and (5.22) we have for a.e. , and so, due to (5.17) and (5.21), we have for a.e. ,

Denote . If , then .

Case 1.

Let . Then there exists such that on , . By (5.12), (5.22), (5.28), and , it follows for a.e. ,

Consequently,

where is given by (5.15). Therefore .

Case 2.

Let . Then there exists such that on , . By (5.12), (5.13), (5.22), (5.28), and , we obtain for a.e.

Consequently,

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

Step 4 (convergence of ).

Consider the sequence of solutions of problems (5.21), , . It has been shown in Steps 2 and 3 that (5.22) and (5.27) hold, which means that the sequences and are bounded in . Therefore is equicontinuous on . According to (5.17), (5.19), and (5.21), we obtain for ,

Let us now choose an arbitrary compact subinterval . Then there exists such that for each . By (5.33), the sequence is equicontinuous on . Therefore, we can find a subsequence such that converges uniformly on , and converges uniformly on . By the diagonalization theorem; see [11], we can find a subsequence such that there exists with

Therefore and . For in (5.33), Lebesgue's dominated convergence theorem yields

Consequently, satisfies equation (5.1a) a.e. on . Moreover, due to (5.22) and (5.27), we have

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

Example 5.4.

Let , , , for some , and . Moreover, let be nonnegative, and let be bounded on . Then in Theorem 5.3 the following class of functions is covered:

for a.e. and all , provided if and if . In particular, for ,

or

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

Lemma 5.5 (Fredholm-type existence theorem).

Let satisfy (2.2), let matrices , vector be given, and let . Let us denote by , and assume that the linear homogeneous boundary value problem

has only the trivial solution. Moreover, let us assume that there exists a function such that

Then the problem

has a solution .

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

Theorem 5.6 (existence and uniqueness).

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

Example 5.7.

Functions satisfying assumptions of Theorem 5.6 can have the form

for .

We now illustrate the above theoretical findings by means of numerical simulations. Figure 4 shows graphs of solutions of problem (5.43), (5.1a). In Figure 5 we display the error estimate for the global error of the numerical solution and the so-called residual (defect) obtained from the substitution of the numerical solution into the differential equation. Both quantities are rather small and indicate that we have found a solution to the analytical problem (5.43)-(5.1a).

We now pose that question about the values of the first derivative at the end points of the interval of integration, and . According to the theory, it holds that . Therefore, we approximate the values of the first derivative of the numerical solution and show these values in Figure 6. One can see that indeed . Also, to support this observation, we plotted in Figure 7 the numerical solutions obtained for the step size tending to zero, or equivalently, grids becoming finer.

We finally observe experimentally the order of convergence of the numerical method (collocation). Clearly, we do not expect very hight order to hold, since the analytical solution has nonsmooth higher derivatives. However, the method is convergent and, according to Table 1, we observe that its order tends to .

The results of the numerical simulation for the boundary value problem (5.44)-(5.1a), can be found in Figures 8, 9, 10, and 11.

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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