Limit Properties of Solutions of Singular Second-Order Differential Equations

We discuss the properties of the differential equation u′′ t a/t u′ t f t, u t , u′ t , a.e. on 0, T , where a ∈ R\{0}, and f satisfies the Lp-Carathéodory conditions on 0, T × R2 for some p > 1. A full description of the asymptotic behavior for t → 0 of functions u satisfying the equation a.e. on 0, T is given. We also describe the structure of boundary conditions which are necessary and sufficient for u to be at least in C1 0, T . As an application of the theory, new existence and/or uniqueness results for solutions of periodic boundary value problems are shown.


Motivation
In this paper, we study the analytical properties of the differential equation and for all (x, y) ∈ D ⊂ R × R. The above equation is singular at t = 0 because of the first term in the right-hand side, which is in general unbounded for t → 0. In this paper, we will also alow the function f to be unbounded or bounded but discontinuous for certain values of the time variable t ∈ [0, T ].
This form of f 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 f has jump discontinuities at a discrete set of points in (0, T ), cf. [16].
This study serves as a first step towards analysis of more involved nonlinearities, where typically, f has singular points also in u and u ′ . Many applications, cf.
[1]- [7], [13], [17]- [19], 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 u to have at least continuous first derivative on [0, T ]. Moreover, using new techniques presented in this paper, we would like to extend results from [21] and [14] (based on ideas presented in [8]) where problems of the above form but with appropriately smooth data function f have been discussed.
Here, we aim at the generalization of the existence and uniqueness assertions derived in those papers for the case of smooth f . We are especially interested in studying the limit properties of u for t → 0 and the structure of boundary conditions which are necessary and sufficient for u to be at least in C 1 [0, T ].
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 [17]; see also [5], and [13], subject to boundary conditions where a 0 ≥ 0, b 0 < 0, γ > 1. Note that (1.2) can be written in the form which is of form (1.1) with Function f is not defined for u = 0 and for t = 0 if γ ∈ (1, 2). We now briefly discuss a simplified linear model of the equation (1.4), where β = 2γ − 4 and γ > 1. Clearly, this means that β > −2.
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 u of (1.5) to be unique and at least continuously differentiable, u ∈ C 1 [0, 1]? 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 [8], and [9], respectively. However, in these papers, the analytical properties of the solution u are derived for nonhomogeneous terms being at least continuous. Clearly, we need to rewrite problem (1.5) first and obtain its new form stated below, which suggests 1 to introduce a new variable, v(t) := t 3 u ′ (t). We now introduce z(t) := (u(t), v(t)) T , and immediately obtain the following system of ordinary differential equations, where β + 3 > 1, or equivalently, where g ∈ C[0, 1]. According to [9], the latter system of equations has a continuous solution if and only if the regularity condition M z(0) = 0 holds. This results cf. conditions (1.3). Note that the Euler transformation, ζ(t) := (u(t), tu ′ (t)) T which is usually used to transform (1.5) to the first order form would have resulted in the following system: (1.9) Here, w may become unbounded for t → 0, the condition N ζ(0) = 0, or equivalently lim t→0+ tu ′ (t) = 0 is not the correct condition for the solution u to be continuous on [0, 1].
From the above remarks, we draw the conclusion that a new approach is necessary to study the analytical properties of equation (1.1).

Introduction
The following notation will be used throughout the paper. Let J ⊂ R be an interval. Then, we denote by L 1 (J) the set of functions which are (Lebesgue) integrable on J. The corresponding norm is u 1 := J |u(t)|dt. Let p > 1. By L p (J), we denote the set of functions whose p-th powers of modulus are integrable on J with the corresponding norm given by u p := J |u(t)| p dt 1/p . Finally, we denote by AC(J) and AC 1 (J) the sets of functions which are absolutely continuous on J, and which have absolutely continuous first derivatives on J, respectively. Analogously, AC loc (J) and AC 1 loc (J) are the sets of functions being absolutely continuous on each compact subinterval I ⊂ J, and having absolutely continuous first derivatives on each compact subinterval I ⊂ J, respectively.
As already said in the previous section, we investigate differential equations of the form such that |f (t, x, y)| ≤ m K (t) for a.e. t ∈ [0, T ] and all (x, y) ∈ K.
We will provide a full description of the asymptotical behavior for t → 0+ of functions u satisfying (2.1) a.e. on (0, T ]. Such functions u will be called solutions of (2.1) if they additionally satisfy the smoothness requirement u ∈ AC 1 [0, T ], 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 [0, 1]. 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.
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 c ∈ [0, T ], let us denote by Assume that a < 0. Then Now, let a > 0, c > 0. Without loss of generality, we may assume that 1 p = 1 − a. For 1 p = 1 − a, we choose p * ∈ (1, p), and have h ∈ L p * [0, T ] and 1 Consequently, (3.3), (3.4) and the Hölder inequality yield, t ∈ (0, T ], which means that ϕ a ∈ C[0, 1]. We now use the properties of ϕ a to represent all functions u ∈ AC 1 loc (0, T ] satisfying (3.1) a.e. on [0, T ]. Remember that such function u does not need to be a solution of (3.1) in the sense of Definition 2.2.
is the set of all functions u ∈ AC 1 is the set of all functions u ∈ AC 1 loc (0, T ] satisfying (3.1) a.e. on (0, T ].
Since the functions u 1 h (t) = 1 and u 2 h (t) = t a+1 are linearly independent solutions of the homogeneous equation u ′′ (t) − a t u ′ (t) = 0 on (0, T ], the general solution of the homogeneous problem is, Moreover, the function u p (t) = c t ϕ a (c, s)ds is a particular solution of (3.1) on (0, T ]. Therefore, the first statement follows. Analogous argument yields the second assertion. We stress that by (3.5), the particular solution u p = c t ϕ a (c, s)ds of equation For a < 0, we can see from (3.6), that it is useful to find other solution representations which are equivalent to (3.7) and (3.
is the set of all functions u ∈ AC 1 is the set of all functions u ∈ AC 1 loc (0, T ] satisfying (3.1) a.e. on (0, T ].
Proof. Let us fix c ∈ (0, T ] and define In order to prove (i) we have to show that This follows immediately from (3.6), since and hence we can define d i as follows For a = −1 we have which completes the proof. Again, by (3.6), the particular solution, Main results for the linear singular equation (3.1) are now formulated in the following theorems.
Proof. Let a function u be given. Then, by (3.7), there exist two constants c 1 , c 2 ∈ R such that for t ∈ (0, T ], By the Hölder inequality and (3.4) it follows, It is clear from the above theorem, that u ∈ AC 1 [0, T ] given by (3.11) is a solution of (3.1) for a > 0. Let us now consider the associated boundary value problem, where B 0 , B 1 ∈ R 2×2 are real matrices, and β ∈ R 2 is an arbitrary vector. Then the following result follows immediately from Theorem 3.3. if and only if the following matrix, is nonsingular.
Proof. Let u be a solution of equation (3.1). Then u satisfies (3.11) and the result follows immediately by substituting the values into the boundary conditions (3.13b). 2 Theorem 3.5. Let a < 0 and let a function u ∈ AC 1 loc (0, T ] satisfy equation (3.1) a.e. on (0, T ]. For a ∈ (−1, 0), only one of the following properties holds, For a ∈ (−∞, −1], u satisfies only one of the following properties: In particular, u can be extended to the whole interval Proof. Let a ∈ (−1, 0) and let u be given. Then, by (3.9), there exist two constants c 1 , c 2 ∈ R such that Let a = −1. Then, by (3.10), for any c 1 , c 2 ∈ R, and where b 0 , b 1 , b 2 , β ∈ R are real constants. Then the following result follows immediately from Theorem 3.5. 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 TM 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 [24]. This software has already been used for a variety of singular boundary value problems relevant for applications, e.g. [23].
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 [0, 1].

Limit properties of functions satisfying nonlinear singular equations
In this section we assume that the function u ∈ AC 1 loc (0, T ] satisfying differential equation (2.1) a.e. on [0, T ] is given. The first derivative of such a function does not need to be continuous at t = 0 and hence, due to the lack of smoothness, u 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 u for t → 0.

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: and we see that it is singular. Consequently, the assumption of Theorem 3.4 is not satisfied and the linear periodic problem (3.13a) 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 (5.1) which have only one solution. We begin the investigation of problem (5.1) with a uniqueness result.
Theorem 5.2. (Uniqueness) Let a > 0 and let us assume that condition (2.2) holds. Further, assume that for each compact set K ⊂ R × R there exists a nonnegative function h K ∈ L 1 [0, T ] such that for a.e. t ∈ [0, T ] and all (x 1 , y 1 ), (x 2 , y 2 ) ∈ K. Then problem (5.1) has at most one solution.

Integrating the last inequality in
which contradicts v ′ (β) = 0. Consequently, we have shown that u 1 ≡ u 2 and the result follows.
Then problem (5.1) has a solution u such that Proof.
We now investigate the auxiliary problem Since the homogeneous problem u ′′ (t) = 1 n u(t), u(0) = u(T ), u ′ (T ) = 0, has only the trivial solution, we conclude by the Fredholm-type Existence Theorem (see Lemma 5.5) that there exists a solution u n ∈ AC 1 [0, T ] of problem (5.14).
Step 3. Estimates of u ′ n . We now show that In order to show the existence of solutions to the periodic boundary value problem (5.1), the Fredholm-type Existence Theorem is used, see e.g. in [15] (Theorem 4), [18] (Theorem 2.1) or [20] (p.25). For convenience, we provide its simple formulation suitable for our purpose below. Then the problem If we combine Theorem 5.2 and Theorem 5.3, we obtain conditions sufficient for the solution of (5.1) to be unique. Example 5.7. Functions satisfying assumptions of Theorem 5.6 can have the form for t ∈ (0, 1], x, y ∈ R. We now illustrate the above theoretical findings by means of numerical simulations. In the following graph 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 (5.23)-(5.1a). We now pose that question about the values of the first derivative at the end points of the interval of integration, t = 0 and t = 1. According to the theory, it holds u ′ (0) = u ′ (1) = 0. 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 u ′ (0) ≈ 0, u ′ (1) ≈ 0. Also, to support this observation, we plotted in Figure 7 the numerical solutions obtained for the step-size h 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 1 2 .
The results of the numerical simulation for the boundary value problem (5.24)-(5.1a), can be found in Figures 8-11.    Figure 11: Numerical solutions of (5.24)-(5.1a) and a = 1 in the vicinity of t = 0 (left) and t = 1 (right). The step-size is decreasing according to h = 1 2 n .