PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 91(105) (2012), 125–135 DOI: 10.2298/PIM1205125N

ORDINARY DIFFERENTIAL EQUATIONS WITH DELTA FUNCTION TERMS

Marko Nedeljkov and Michael Oberguggenberger

Abstract. This article is devoted to nonlinear ordinary differential equations with additive or multiplicative terms consisting of Dirac delta functions or derivatives thereof. Regularizing the delta function terms produces a family of smooth solutions. Conditions on the nonlinear terms, relating to the order of the derivatives of the delta function part, are established so that the regularized solutions converge to a limiting distribution.

Introduction

This paper is devoted to ordinary differential equations (and systems) of the form

(0.1)  y′(t) = f(t, y(t)) + g(y(t))δ^{(s)} , y(t_{0}) = y_{0},

where δ^{(s)} denotes the s-th derivative of the Dirac delta function. The case of constant g(y) ≡ α will be referred to as the additive case, the general case with s = 0 will be called the multiplicative case. We shall replace the delta function by a family of regularizations ϕ_{ε}(t) = ε^{−1} ϕ(t/ε) and ask under what conditions on f, g and s the family of regularized solutions admits a limit as ε → 0.

In the case of partial differential equations, such weak limits have been termed delta waves and studied in various situations, see e.g. [15, 16, 21]. The interest in problem (0.1) comes also from the fact that such equations have been proven to admit solutions in the Colombeau algebra of generalized functions [5, 8, 10].

Equations of the type (0.1) with s = 0 arise in nonsmooth mechanics [1, 3, 7, 14] and are referred to as measure differential equations or impulsive differential equations, often considered under the form

y′(t) = f(t, y(t)) + ∑_{i=1}^{N} g_{i}(y(t))δ(t − t_{i}).

2010 Mathematics Subject Classification: Primary 46F10; Secondary 34A37.

The first author was partially supported by OAED and Ministry of Science, Serbia (III-44006 and OI-174024).

Such an equation can be interpreted piecewise: the function g_{i}(y(t)) is fixed during the action of the delta distribution, i.e., g_{i}(y(t)) is substituted by lim_{t→t_{i} ,t<t_{i}} g_{i}(y(t)) and this value depends on the classical solution up to the point t_{i} only. In the additive case one can alternatively consider the delta function term as the derivative of a function of bounded variation. We refer to the vast literature on such equations [2, 4, 6, 12, 13, 17]. The point of view of distribution theory and regularizing sequences is sometimes taken as well. Here the work [18, 19, 20] is relevant, in which higher order linear differential equations with measures as coefficients were considered and unique solutions in the space of Borel measures with primitives being normalized to being right-continuous functions were obtained. There it was also shown that the regularized solutions converge to a solution of a measure differential equation of the same type.

Higher derivatives (with s = 1 or s = 2) in nonlinear differential equations arise, e.g., in geodesic equations and geodesic deviation equations for impulsive gravitational waves [11, 22] and in the calculus of variations with strongly singular potential [9].

In this article, we adopt the approach of regularization and taking limits. The admissible order s of the delta function term is arbitrary in principle, but may be restricted by the type of sublinearity of the nonlinear function f (no restriction if f is bounded).

The paper is organized as follows. In the first section we establish the existence of a limiting function in the additive case when f is a sublinear function of order r with respect to y, r < 1/s, or f is globally Lipschitz in the case s = 1. In both cases the limit is the sum of a function continuous in [t_{0}, 0) ∪ (0, ∞) and a multiple of δ^{(s−1)} . Depending on the case, the function part may be continuous across t = 0 or suffer a jump (whose value we compute). Similar assertions can also be obtained for systems of differential equations.

The second section addresses the multiplicative case with s = 0. If f and g are globally Lipschitz or f is arbitrary and g has a sufficiently small Lipschitz constant, the existence of a limiting functions with a jump at t = 0 is derived. In both sections we illuminate the required conditions by means of a number of examples and counterexamples. The limiting functions do not depend on the regularization if non-negative mollifiers are used. This may or may not be the case if the nonnegativity condition is violated, as is shown by various examples.

1. The additive case

In this section, we study the equation

y′(t) = f(t, y(t)) + αδ^{(s)} (t), y(−1) = y_{0}

on some interval [−1, T] with T > 0. Here α ∈ ℝ and s ⩾ 1 is an integer (s = 0 will arise as a special case in the next section). Throughout, f is assumed to be continuous in (t, y) and locally Lipschitz with respect to y, uniformly on compact time intervals. We suppose that the free equation y′(t) = f(t, y(t)) is uniquely solvable on the whole interval [−1, T] for whatever data y(−1) = y_{0} ∈ ℝ, and is also uniquely solvable on [0, T] for arbitrary data y(0) = y_{1}.

We are going to find limits of the family of regularized solutions y_{ε}(t) as ε → 0 when the δ-distribution is substituted by some mollifier ϕ_{ε}(t) = ε^{−1} ϕ(t/ε). We shall suppose that ϕ ∈ C^{∞} (ℝ) has integral one, and suppϕ = [−a, b], a, b ⩾ 0.

The supports of ϕ_{ε} and its derivatives are contained in [−aε, bε]. Our first concern will be that y_{ε} does not blow up in this interval; note that y_{ε}(t) coincides with ȳ(t), the classical solution to y′(t) = f(t, y(t)), y(−1) = y_{0}, up to t = −aε, and lim_{ε→0} y_{ε}(−aε) = ȳ(0). To have convergence of y_{ε}(t) on the whole interval [−1, T] we will have to verify that the limit lim_{ε→0} y_{ε}(bε) exists, thus providing the initial data for the limiting solution for t > 0.

A function g : ℝ → ℝ will be called sublinear of order r, 0 ⩽ r < 1, if |g(u)| ⩽ C(1 + |u|^{r}) for some C > 0 and all u ∈ ℝ. In this terminology, a function which is sublinear of order 0 is bounded. In the following, ϕ_{±} ⩾ 0 denotes the positive and negative part of ϕ, respectively, so that ϕ = ϕ_{+} − ϕ_{−} .

Theorem 1.1. (a) Let 0 ⩽ r < 1 and assume that f is sublinear of order r with respect to y, uniformly on compact intervals with respect to t. Let 0 < s < 1/r (i.e., s is arbitrary if f is bounded). Then the solutions to

(1.1)  y′_{ε}(t) = f(t, y_{ε}(t)) + αϕ_{ε}^{(s)} (t), y_{ε}(−1) = y_{0}

converge to y(t) = ȳ(t)+αδ^{(s−1)} (t), where ȳ(t), t ∈ [−1, T], is the classical solution to

y′(t) = f(t, y(t)), y(−1) = y_{0}.

(b) Let f be globally Lipschitz with respect to y, uniformly on compact intervals with respect to t and assume that the double limits

lim_{η→0, y→±∞} \frac{f(η, y)}{y} = M_{±}

exist. Let s = 1. Then the solutions to (1.1) converge to y(t) = ȳ_{1}(t)+αδ(t), where ȳ_{1}(t) is equal to ȳ(t) for t ∈ [−1, 0] and ȳ_{1}(t) is the classical solution to

y′(t) = f(t, y(t)), y(0) = ȳ(0) + β,

t ∈ [0, ∞) with

β = α( M_{+} ∫_{−a}^{b} ϕ_{+} (u) du − M_{−} ∫_{−a}^{b} ϕ_{−} (u) du) , if α > 0,

β = α( M_{−} ∫_{−a}^{b} ϕ_{+} (u) du − M_{+} ∫_{−a}^{b} ϕ_{−} (u) du) , if α < 0.

Proof. We shall rewrite (1.1) in the following way. Let y_{ε}(t) = y_{1ε}(t) + αϕ_{ε}^{(s−1)} (t), where y_{1ε} is the solution to

y_{1ε}′(t) = f(t, y_{1ε}(t) + αϕ_{ε}^{(s−1)} (t)), y_{1ε}(−1) = y_{0},

i.e.,

y_{1ε}(t) = y_{0} + ∫_{−1}^{t} f(u, y_{1ε}(u) + αϕ_{ε}^{(s−1)} (u)) du.

Obviously, y_{1ε}(t) = ȳ(t) for t ∈ [−1, −aε], and y_{0ε} := y_{1ε}(−aε) → ȳ(0) as ε → 0.

First we shall show that y_{1ε}(t) is bounded for t ∈ [−aε, bε]. By using the sublinearity we have

|y_{1ε}(t) − y_{0ε}| ⩽ ∫_{−aε}^{t} |f(u, y_{1ε}(u) + αϕ_{ε}^{(s−1)} (u))| du
⩽ ∫_{−aε}^{t} C(1 + |y_{1ε}(u) + αϕ_{ε}^{(s−1)} (u) − y_{0ε} + y_{0ε} |^{r}) du
⩽ Cd_{ε} + C|y_{0ε}|^{r}d_{ε} + C ∫_{−aε}^{bε} |α|^{r}|ϕ_{ε}^{(s−1)} (u)|^{r}du
+ C ∫_{−aε}^{t} |y_{1ε}(u) − y_{0ε}|^{r}du,

where d_{ε} = aε + bε. After the change of the variables u/ε ↦ u we have

∫_{−aε}^{bε} |ϕ_{ε}^{(s−1)} (u)|^{r}du = ε^{1−rs} ∫_{−a}^{b} |ϕ^{(s−1)} (u)|^{r}du.

That means that for 0 < s < 1/r the above integral converges to zero as ε → 0. When f is globally Lipschitz and s = 1, a similar estimate holds with r = 1, and the integral is seen to remain bounded as ε → 0. Using |u|^{r} ⩽ max{1, |u|} we get from Gronwall’s inequality that

max{1, |y_{1ε}(t) − y_{0ε}|} ⩽ C_{1}e^{Cd_{ε}} .

For 0 < s < 1/r, this in turn implies that

|y_{1ε}(t) − y_{0ε}| ⩽ C_{2}d_{ε} + C_{3}(ε^{1−rs} + d_{ε}e^{rCd_{ε}} ) → 0

as ε → 0 and hence y_{1ε}(bε) → ȳ(0). Thus the limiting solution has no jump at 0 and y_{1ε}(t) → ȳ(t) for t > 0 as ε → 0, thereby proving (a).

Let f be globally Lipschitz and s = 1. As mentioned, an estimate as above holds with r = 1 and so |y_{1ε}(t) − y_{0ε}| ⩽ β̄ for some β̄ > 0 on [−aε, bε]. We shall prove that y_{1ε}(t), which is the solution to

y_{1ε}′(t) = f(t, y_{1ε}(t) + αϕ_{ε}(t)), y_{1ε}(−aε) = y_{0ε},

and the solution z_{ε}(t) to

z_{ε}′(t) = f(t, αϕ_{ε}(t)), z_{ε}(−aε) = y_{0ε}

differ at most by d_{ε}C(|y_{0ε}| + β̄) on [−aε, bε], where C is the Lipschitz constant for f on [−a, b], y ∈ ℝ. It holds that

|y_{1ε}(t) − z_{ε}(t)| ⩽ ∫_{−aε}^{t} |f(u, y_{1ε}(u) + αϕ_{ε}(u)) − f(u, αϕ_{ε}(u))|du
⩽ ∫_{−aε}^{t} C|y_{1ε}(u)|du ⩽ d_{ε}C(|y_{0ε}| + β̄).

In particular, y_{1ε}(bε) − z_{ε}(bε) → 0 as ε → 0.

In the case when the limits M_{±} = lim_{η→0, y→±∞} \frac{f(η,y)}{y} exist, we can compute the value of the limiting jump. Indeed,

z_{ε}(bε) = y_{0ε} + ∫_{−aε}^{bε} f(u, αϕ(u/ε)ε^{−1} ) du
= y_{0ε} + ε ∫_{−a}^{b} f(εu, αϕ(u)ε^{−1} ) du
= y_{0ε} + ε ∫_{[−a,b]∩{|ϕ(u)|⩽ε}} f(εu, αϕ(u)ε^{−1} ) du

+ ∫_{[−a,b]∩\{|ϕ(u)|>ε\}} \frac{f(εu, αϕ(u)ε^{−1} )}{α ϕ(u) ε^{−1}} αϕ(u) du.

The first integral is less than or equal to

ε(b − a) sup_{t∈[−εa,εb],|y|⩽|α|} |f(t, y)|

and converges to 0 as ε → 0; the second integral converges to α(M_{+} ∫_{−a}^{b} ϕ_{+} (u) du−

M_{−} ∫_{−a}^{b} ϕ_{−} (u) du) by Lebesgue’s dominated convergence theorem, if α > 0, and similarly for α < 0. □

Remark 1.1. If the mollifier ϕ is nonnegative, then the above jump does not depend on it, i.e., it equals αM_{+} .

Example 1.1. Let f(y) = y sin(log(1 + y^{2})). Then f is globally Lipschitz, but does not have limits M_{+} and M_{−} required in (b). We let α = 1 and choose the mollifier ϕ ⩾ 0 such that ϕ ≡ 1 on an interval I ⊂ [−a, b] and

∫_{[−a,b]∖I} ϕ(u) du = 1/4, length(I) = 3/4.

Then

z_{ε}(bε)= ∫_{−aε}^{bε} \frac{1}{ε}ϕ( \frac{u}{ε}) sin( log( 1 + \frac{1}{ε^{2}}ϕ^{2}( \frac{u}{ε}) ) ) du

= ∫_{−a}^{b} ϕ(u) sin( log( 1 + \frac{1}{ε^{2}}ϕ^{2}(u)) ) du

= \frac{3}{4} sin( log( 1 + \frac{1}{ε^{2}}) ) + ∫_{[−a,b]∖I} ϕ(u) sin( log( 1 + \frac{1}{ε^{2}}ϕ^{2}(u)) ) du.

We can choose a subsequence ε_{k} → 0 so that sin( log(1 + 1/ε_{k}^{2})) = (−1)^{k}. Since the second term is less then or equal to 1/4 in absolute value, we see that the sequence z_{ε}(bε) is oscillating with a jump of height at least 1. So there does not exist a limiting solution on the right-hand side of t = 0.

The assumption that f is globally Lipschitz is not necessary for the existence of a limit of the regularized solutions, as can be seen from the following example (even in the case s = 1).

Example 1.2. The dissipative case f(t, y) = −y^{3}. We analyze the case y_{0} > 0 only and assume here that ϕ ⩾ 0. Obviously any solution to y′ = −y^{3} decreases for positive y and increases when y is negative since f is decreasing with respect to y. By using the comparison theorem one can see that the solution y_{1ε} to

y_{1ε}′(t) = f(t, y_{1ε}(t) + ϕ_{ε}(t)), y_{0ε} := ȳ(−aε) > 0

is less or equal to the solution to

v′(t) = f(t, v + g_{ε}(t)), v(−aε) = y_{0ε},

where g_{ε}(t) ⩽ ϕ_{ε}(t) and

g_{ε}(t) = \{ \begin{matrix}0, & t < −ā_{ε} \\ ξ̄_{ε}, & t ∈ [−ā_{ε}, b̄_{ε}] \\ 0, & t > b̄_{ε}, \end{matrix}

for some 0 < ā_{ε} ⩽ aε, 0 < b̄_{ε} ⩽ bε such that b̄_{ε} + ā_{ε} = ξ̄_{ε}^{−1} /2 with ξ̄_{ε} → ∞ as ε → 0. This means that y_{1ε}(t) ⩽ v(t), where

v′(t) = \{ \begin{matrix}−v^{3}, v(−aε) = y_{0ε}, & t ∈ [−aε, −ā_{ε}] \\ −(v + ξ̄_{ε})^{3}, v(−ā_{ε}) = ȳ_{0ε}, & t ∈ [−ā_{ε}, b̄_{ε}] \\ −v^{3}, v(b̄_{ε}) = ỹ_{0ε}, & t ∈ [b̄_{ε}, bε]. \end{matrix}

Observe that y_{0ε} ≈ ȳ_{0ε}; here ȳ_{0ε}, ỹ_{0ε} are the terminal values of the solutions on the preceding intervals. The solution on the second interval is given by

v(t) = (ȳ_{0ε} + ξ̄_{ε})(1 + 2(t + ā_{ε})(ȳ_{0ε} + ξ̄_{ε})^{2})^{−1/2} − ξ̄_{ε}.

This means that v(b̄_{ε}) behaves like \sqrt{ξ̄_{ε}} − ξ̄_{ε}, which goes to −∞ as ε → 0. (This is then true of v(t) for b̄_{ε} ⩽ t ⩽ bε as well.) The same is true for y_{1ε}(bε), i.e., y_{1ε}(bε) = ξ̃_{ε} → −∞ as ε → 0. The function ϕ_{ε}(t) = 0 for t ⩾ bε, and y_{1ε} is the solution to

y_{1ε}′(t) = −y_{1ε}^{3}, y_{1ε}(bε) = ξ̃_{ε},

that is,

y_{1ε}(t) = −(ξ̃_{ε}^{−2} + 2(t − bε))^{−1/2}

and y_{1ε}(t) → −(2t)^{−1/2} , t > 0. That gives that the solution to

y_{ε}′(t) = −y_{ε}^{3}(t) + ϕ′_{ε}(t)

converges to ȳ_{1}(t) + δ(t) where

ȳ_{1}(t) = \{ \begin{matrix}y_{0}(1 + 2(t + 1)y_{0}^{2})^{−1/2} , & t < 0 \\ −(2t)^{−1/2} , & t > 0. \end{matrix}

It is not true that if y′ = f(t, y), y(−1) = y_{0} has a unique global solution in [−1, ∞), then the solutions to y_{ε}′ = f(t, y_{ε}) + ϕ′′_{ε}with the same initial data have a limiting function. This can be seen from the following example.

Example 1.3. The solutions to

y′_{ε}(t) = (1 + y_{ε}^{2})^{1/2} + ϕ′′_{ε}(t), y(−1) = y_{0}

do not converge to any function defined for t > 0 as ε → 0; in fact, they diverge to ∞, uniformly on every interval [t_{0}, ∞), t_{0} > 0. Indeed, the solution to

y_{1ε}′(t) = (1 + (y_{1ε} + ϕ_{ε}′)^{2})^{1/2} , y_{1ε}(−1) = y_{0}

is increasing and its derivative is greater than Cε^{−2} on an interval of length O(ε), on which ϕ′_{ε} is strictly different from 0. That means that the value of the function at t = bε is greater or equal to a constant times ε^{−1} . After this point the function y_{1ε} continues to increase, and that implies blow up to ∞, uniformly on each interval [t_{0}, ∞) bounded away from zero.

Remark 1.2. Theorem 1.1 can be easily extended to the case of (n × n)- systems, where f : ℝ^{n+1} → ℝ^{n} and α ∈ ℝ^{n}. One just has to replace the absolute value in the estimates by the norm ∥y∥ = max{|y_{1}|, …, |y_{n}|} in ℝ^{n}. The function f : ℝ^{n+1} → ℝ is sublinear of order r with respect to y if there exists a constant C such that ∥f(t, y)∥ ⩽ C(1 + ∥y∥^{r}). With these modifications, part (a) of Theorem 1.1 remains valid. As for part (b), suitable limiting behavior on f has to be imposed as the components y_{i} tend to infinity. Various versions of part (b) can be obtained in this way; we leave out the details.

2. The multiplicative case

This section is devoted to the equation

y′(t) = f(t, y(t)) + g(y(t))δ(t), y(−1) = y_{0}

where f satisfies the same general assumptions as in Section 1 and g ∈ C^{1}(ℝ).

Proposition 2.1. Let f and g be globally Lipschitz (uniformly on compact time intervals) and assume that G(y) = ∫ dy/g(y) is invertible. Then the solutions to

(2.1)  y_{ε}′(t) = f(t, y_{ε}(t)) + g(y_{ε}(t))ϕ_{ε}(t), y_{ε}(−1) = y_{0}

converge to the limiting function

ȳ(t) = \{ \begin{matrix}ȳ_{1}(t), & t ⩽ 0 \\ ȳ_{2}(t), & t > 0 \end{matrix}

where ȳ_{1} is the solution to

ȳ′_{1}(t) = f(t, ȳ_{1}(t)), ȳ_{1}(−1) = y_{0}

and ȳ_{2} is the solution to

ȳ′_{2}(t) = f(t, ȳ_{2}(t)), ȳ_{2}(0) = G^{−1} (G(ȳ_{1}(0)) + 1).

Proof. We have only to check what is going on for t ∈ [−aε, bε]. Let y_{0ε} := y_{ε}(−aε) = ȳ_{1}(−aε). Then

y_{ε}(t) = y_{0ε} + ∫_{−aε}^{t} (f(s, y_{ε}(s)) + g(y_{ε}(s)) ϕ_{ε}(s))ds,

|y_{ε}(t)| ⩽ |y_{0ε}| + ∫_{−aε}^{t} (L_{f} + L_{g}|ϕ_{ε}(s)|)|y_{ε}(s)|ds + ∫_{−aε}^{t} (|f(s, 0)| + |g(0)||ϕ_{ε}(s)|)ds,

where L_{f} and L_{g} are the Lipschitz constants for f and g, respectively. Letting C_{ϕ} = ∫_{−a}^{b} |ϕ(t)|dt, Gronwall’s inequality gives

|y_{ε}(t)| ⩽ (|y_{0ε}| + d_{ε}C_{f} + C_{g}C_{ϕ}) exp(d_{ε}L_{f} + L_{g}C_{ϕ}) ⩽ C < ∞,

i.e., |y_{ε}(t)| remains bounded for t ∈ [−aε, bε]. (Here C_{f} and C_{g} are bounds on |f| and |g| in the appropriate regions.) Using this fact we will prove that the solution z_{ε}(t) to

(2.2)  z_{ε}′(t) = g(z_{ε}(t))ϕ_{ε}(t), z_{ε}(−aε) = y_{0ε}

satisfies

sup_{t∈[−aε,bε]} |y_{ε}(t) − z_{ε}(t)| → 0, ε → 0.

In particular, this will be true for t = bε. Indeed,

|y_{ε}(t) − z_{ε}(t)| ⩽ ∫_{−aε}^{t} |f(s, y_{ε}(s))|ds + ∫_{−aε}^{t} L_{g}|y_{ε}(s) − z_{ε}(s)||ϕ_{ε}(s)|ds.

By Gronwall’s inequality,

|y_{ε}(t) − z_{ε}(t)| ⩽ ∫_{−aε}^{t} |f(s, y_{ε}(s))|ds exp(L_{g}C_{ϕ}) ⩽ d_{ε}(C_{f} + CL_{f} ) exp(L_{g}C_{ϕ}) → 0

as ε → 0. This means that, in the limit, the jump of y_{ε} is equal to the jump of z_{ε} at zero. But the latter can be easily found. By integrating (2.2) we obtain

∫_{z_{ε}(−aε)}^{z_{ε}(bε)} g(z)^{−1} dz = ∫_{−aε}^{bε} ϕ_{ε}(t) dt = 1

and G(z_{ε}(bε)) = G(z_{ε}(−aε)) + 1. This gives the desired result. □

The next result shows that the conditions on f can be relaxed to the local Lipschitz case, provided that g has a sufficiently small (global) Lipschitz constant. As in Section 1, we require again that the free equation y′(t) = f(t, y(t)) is uniquely solvable on the whole interval [−1, T] for whatever data y(−1) = y_{0} ∈ ℝ, and is also uniquely solvable on [0, T] for arbitrary data y(0) = y_{1}.

Proposition 2.2. Let f be locally Lipschitz, uniformly on compact time intervals, let g be globally Lipschitz with constant L_{g}. Set C_{ϕ} = ∫_{−a}^{b} |ϕ(t)|dt and assume that 1/g has an invertible primitive G. If L_{g}C_{ϕ} < 1, then the assertions of Proposition 2.1 remain valid, at least on the interval [−1, T].

Proof. We keep the notations from Proposition 2.1; in particular, we let y_{0ε} = ȳ_{1}(−aε). Choose η such that η > |g(ȳ_{1}(0))|C_{ϕ}/(1 − L_{g}C_{ϕ}). Let B_{ε} be the set of continuous functions on [−aε, bε] such that

sup_{t∈[−aε,bε]} |u(t) − y_{0ε}| ⩽ η,

and define the operator M on B_{ε} by

Mu(t) = y_{0ε} + ∫_{−aε}^{t} (f(s, u(s)) + g(u(s))ϕ_{ε}(s))ds.

We are going to show that M is a contraction on B_{ε} for all sufficiently small ε. First, if u ∈ B_{ε}, then

|Mu(t) − y_{0ε}| ⩽ |∫_{−aε}^{t} (f(s, u(s)) + g(u(s))ϕ_{ε}(s)) ds|
⩽ d_{ε}C_{f} + L_{g}C_{ϕ}|y_{0ε} − ȳ_{1}(0)| + L_{g}C_{ϕ}η + |g(ȳ_{1}(0))|C_{ϕ} ⩽ η

for sufficiently small ε. Here the constant C_{f} denotes the maximum of |f| on [−aε_{0}, bε_{0}] × [y_{0ε} − η, y_{0ε} + η] for some fixed ε_{0} and ε ⩽ ε_{0}. We have used the decomposition

g(u(s)) = g(u(s)) − g(y_{0ε}) + g(y_{0ε}) − g(ȳ_{1}(0)) + g(ȳ_{1}(0))

and that L_{g}C_{ϕ}η + |g(ȳ_{1}(0))|C_{ϕ} < η by definition. Next, if u, v ∈ B_{ε}, then

|Mu(t) − Mv(t)| ⩽ |∫_{−aε}^{t} (f(s, u(s)) − f(s, v(s)))ds|

+ |∫_{−aε}^{t} (g(u(s)) − g(v(s)))ϕ_{ε}(s) ds|

⩽ (d_{ε}L_{f} + L_{g}C_{ϕ}) sup_{s∈[−aε,bε]} |u(s) − v(s)|.

By assumption, the constant on the right-hand side is less than 1 for sufficiently small ε, thus M is a contraction on B_{ε}.

But the solution y_{ε}(t) of equation (2.1) is the unique fixed point. This shows that y_{ε}(t) is bounded by η on [−aε, bε]. The rest of the proof is the same as in Proposition 2.1. □

Let us remark that if ϕ is nonnegative, then C_{ϕ} = 1 and the condition of Proposition 2.2 reduces to L_{g} < 1.

Example 2.1. The case s = 0 in Section 1 can be seen as the special case of Proposition 2.2, where g(y) ≡ α. It follows that under the assumptions on f above, the family of regularized solutions y_{ε} to

y_{ε}′(t) = f(t, y_{ε}(t)) + αϕ_{ε}, y_{ε}(−1) = y_{0}

converges to

ȳ(t) = \{ \begin{matrix}ȳ_{1}(t), & t ⩽ 0 \\ ȳ_{2}(t), & 0 < t ⩽ T \end{matrix}

where ȳ_{1} is the solution to y′ = f(t, y), y(−1) = y_{0} and ȳ_{2} is the solution to y′ = f(t, y), y(0) = y_{0} + α.

We note that here the limiting function is also a distributional solution; this cannot be asserted in the situation of Theorem 1.1 in general.

Remark 2.1. For completeness, we remark that if the singularity is in the same point where the initial condition is given, then the solution depends on the regularization as a rule. This can be seen from simplest linear equations. In fact, the solutions to

y′_{ε}(t) = y_{ε}(t)ϕ_{ε}(t), y_{ε}(0) = y_{0}

are given by and converge to

y_{ε}(t) = y_{0} exp( ∫_{0}^{t/ε} ϕ(s) ds) → y_{0} exp( ∫_{0}^{∞} ϕ(s) ds)

for t > 0. Depending on the support of ϕ, the integral ∫_{0}^{∞} ϕ(s) ds can have any real value (any value between 0 and 1 for non-negative ϕ). The same holds in the additive case

y′_{ε}(t) = y_{ε}(t) + ϕ_{ε}(t), y_{ε}(0) = y_{0}

with

y_{ε}(t) = y_{0}e^{t} + ∫_{0}^{t/ε} e^{t−εs} ϕ(s) ds → e^{t} ( y_{0} + ∫_{0}^{∞} ϕ(s) ds) .

Acknowledgements. We are grateful to Uladzimir Hrusheuski for carefully reading the manuscript and making a number of valuable suggestions for improvement.

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Department of Mathematics (Received 29 08 2011) University of Novi Sad (Reivised 13 12 2011) Novi Sad Serbia

    marko@dmi.uns.ac.rs

Unit of Engineering Mathematics

University of Innsbruck

Innsbruck Austria

    michael.oberguggenberger@uibk.ac.at