Differential equations with generalized coefficients

Dept. of Math. University of Oslo Pure Mathematics No. 20 ISSN 0806–2439 June 2004

Differential equations with generalized coefficients

Aleh Yablonski

Department of Functional Analysis, Belarusian State University, F.Skaryna av.,4, 220050, Minsk, BELARUS

Email: yablonski@bsu.by

Department of Mathematics , University of Oslo Box 1053 Blindern , N-0316 Oslo, NORWAY , Email: alehy@math.uio.no

Abstract

Differential equations with generalized coefficients by using the algebra of new gen- eralized functions are investigated. It is shown that the different interpretations of the solutions of such equations can be described by the unique approach of the algebra of new generalized functions.

Key words and phrases: algebra of new generalized functions, differential equations with generalized coefficients, functions of finite variation.

2000 Mathematics subject classification : 34A37, 34K45, 46F30.

1 Introduction

The theory of generalized functions is one of the most powerful tools for investigating lin- ear differential equations. However, from the very beginning the distribution theory has an essential disadvantage: it is inapplicable to the solution of nonlinear problems. Therefore different interpretations of the solution of the nonlinear differential equations were proposed by many mathematicians. Unfortunately different interpretations of the same equation lead, in general, to different solutions. See, e.g., [1, 2, 3, 4, 5, 6, 7]. Usually differential equations are used to describe the dynamics of real systems or phenomena. In order to choose ade- quate interpretations of such equations one has to consider the reasons which were used for modeling the dynamics of the real systems by these equations.

In this paper we will consider the following nonlinear equation with generalized coeffi- cients

X(t) =˙ f(X(t)) ˙L(t) +g(X(t)), (1.1) wheret ∈[a;b]⊂R, ˙L(t) is a derivative of the function of finite variation in the distributional sense. In general, since ˙L(t) is a distribution and the function f(X(t)) is not smooth then the productf(X(t)) ˙L(t) is not well defined and the solution of the equation (1.1) essentially depends on the interpretation. Recall some approaches to the interpretation of the equation (1.1).

The first approach is concerned with considering the equation in the framework of dis- tribution theory. According to this approach the product of distributions from some classes are defined and then one tries to find the solution of the equation (1.1) in these classes of distributions. For example in the papers [1, 4, 5] the product of some distributions and dis- continuous functions was defined. See also monograph [7, Ch. 1, §8] for another definition.

Notice that the solutions of the equation (1.1) which can be obtained by using the products from [1, 4, 5] and [7] are different.

The second approach is to interpret the equation (1.1) as the following integral equation x(t) =x⁰+

Z t a

f(x(s))dL(s) + Z t

a

g(s)ds,

where the integrals are understood in Lebesgue-Stieltjes, Perron-Stieltjes, etc., sense. See, e.g., [2, 6]. But in this approach the solution of the integral equation depends on the interpretation of the integral and the definition of the function x(t) in the discontinuity points of L(t).

The third approach is based on the idea of the approximation of the solution of the equation (1.1) by the solutions of the ordinary differential equations, which are constructed by using the smooth approximation of the function L(t). In the monograph [7, Ch. 4] it was shown that in this case the limit of the solutions of the smoothed equations exists; it is called “approximative solution” and it is a solution of the following equation

x(t) =x⁰+ Z t

a

f(x(s))dL^c(s) + Z t

a

g(x(s))ds+X

ζi<t

S(ζ_i, x(ζ_i−),∆L(ζ_i)),

where L^c(t) is a continuous part of the function L(t), ζ_i and ∆L(ζ_i) = L(ζ_i+)− L(ζ_i−) are the epochs and the sizes of jumps of the function L respectively, and the function S(ζ_i, x(ζ_i−),∆L(ζ_i)) is defined as a solution of some auxiliary ordinary differential equa- tion. Notice that the solution of the last equation coincides with the solution of the equation (1.1) which was obtained in [7] by using the definition of the product of distributions.

In this paper we will consider the equation (1.1) by using the algebra of new generalized functions from [8]. This is to say we will interpret the equation (1.1) as the equation in the differentials in the algebra of new generalized functions. Such interpretation says that the solution of the equation (1.1) is a new generalized function.

The main purpose of the article is to show that under some conditions this new generalized function associates with some ordinary function which is natural to call the solution of the equation (1.1). Also it will be shown that the solutions of the equation (1.1) in the sense of the previous approaches can be obtained from the solution of the equation in the differentials in the algebra of new generalized functions.

The method of analogous algebra of new generalized stochastic processes was successfully used in the articles [9, 10, 11] for researching stochastic differential equations.

2 The algebra of new generalized functions

In this section we recall the definition of the algebra of new generalized functions from [8].

At first we define an extended real line Re using a construction typical for non-standard analysis.

Let R = {(x_n)^∞_n=1 : x_n ∈ R for all n ∈ N} be a set of real sequences. We will call two sequences {x_n} ∈ R and {y_n} ∈ R equivalent if there is a natural number N such that xn =yn for all n > N.

The setRe of equivalence classes will be called the extended real line and any of the classes a generalized real number.

It is easy to see thatR⊂Re because one may associate with any ordinary numberx∈R a class containing a stationary sequence withx_n =x. It is evident thatRe is an algebra. The product exeyof two generalized real numbers is defined as the class of sequences equivalent to the sequence {xnyn}, where {xn}and {yn} are the arbitrary representatives of the classesex and yerespectively.

For any segment T = [a;b] ⊂ R one can construct an extended segment Te in a similar way. Let H denote the subset of Re of nonnegative “infinitely small numbers”: H = {eh ∈ Re :eh= [{h_n}], h_n>0,limh_n= 0}.

Consider the set of sequences of infinitely differentiable functions{f_n(x)} onR. We will call two sequences {fn(x)} and {gn(x)} equivalent if for each compact set K ⊂R there is a natural number N such that f_n(x) = g_n(x) for all n > N and x ∈ K. The set of classes of equivalent functions is denoted byG(R) and its elements are called new generalized functions.

Similarly one can define the space G(T) for any interval T= [a;b].

For each distribution f we can construct a sequence f_n of smooth functions such that f_n converges to f (e.g. one can consider the convolution of f with someδ-sequence). This sequence defines the new generalized function which corresponds to the distributionf. Thus the space of distributions is a subset of the algebra of new generalized functions. However, in this case, infinitely many new generalized functions correspond to one distribution (e.g., by taking different δ-sequences). We will say that the new generalized function fe= [{fn}]

associates with the ordinary function or distribution f if f_n converges to f in some sense.

Let fe= [{f_n(x)}] and eg = [{g_n(x)}] be generalized functions. Then there is defined a composition fe◦eg = [{fn(gn(x))}]∈ G(R). In the same way one can define the value of the new generalized functionfeat the generalized real pointex= [{x_n}]∈Re asfe(ex) = [{f_n(x_n)}].

For eacheh = [{h_n}]∈ H and fe= [{f_n(x)}] ∈ G(R) we define a differential d_ehfe∈ G(R) by d_ehfe = [{f_n(x +h_n) −f_n(x)}]. The construction of the differential was proposed by Lazakovich (see [12, 9]) for the algebra of new generalized stochastic processes.

Now we can give an interpretation of the equation (1.1) using the methods of the in- troduced algebras G(R) and G(T). Let L(t), t ∈ [a;b] = T be a right-continuous function of finite variation. We replace ordinary functions in equation (1.1) by corresponding new generalized functions and then write the algebra’s differentials. So we have

d_ehX(ee t) =fe(X(ee t))d_ehL(ee t) +eg(X(ee t))d_ehet, (2.1) with initial value X|e _[

ea;eh) = Xe⁰, where eh = [{h_n}] ∈ H, ea = [{a}] ∈ T,e et = [{t_n}] ∈ T,e Xe = [{X_n(t)}], fe= [{f_n(x)}], eg = [{g_n(x)}], Xe⁰ = [{X_n⁰(t)}], Le = [{L_n(t)}], andL_n →L, bn →b, σn →σ and X_n⁰ →X(0). If Xe associates with some (generalized) function X then we say that X is a solution of the equation (1.1).

The purpose of the present paper is to investigate when the solution Xe of the equation (2.1) associates with some ordinary function and to describe possible associated solutions.

3 Main results

In this section we will formulate the main theorem. The proof of the theorem will be given in the next sections.

Let L(t), t ∈ T = [a;b] be a right-continuous function of finite variation and L(a) = 0.

We will assume that L(t) = L(b) ift > b and L(t) =L(a) if t < a. Denote by V^v_uL the total variation of the function L on the interval [u;v] ⊂ T. Suppose that f and g are Lipschitz continuous with constant M and for all x∈R

|f(x)|+|g(x)| ≤M(1 +|x|). (3.1) Consider as representatives of the new generalized functionsf,e egandLefrom the equation (2.1) the following convolutions with δ-sequence:

L_n(t) = (L∗ρ_n)(t) = Z 1/n

0

L(t+s)ρ_n(s)ds, g_n =g∗ρ_n, f_n =f ∗ρ_n, (3.2) where ρ_n∈C^∞(R), ρ_n ≥0, supp ρ_n⊆[0; 1/n] and R1/n

0 ρ_n(s)ds= 1.

By using the representatives we can rewrite the equation (2.1) in the following form:

X_n(t+h_n)−X_n(t) =f_n(X_n(t))(L_n(t+h_n)−L_n(t)) +g_n(X_n(t))h_n,

X_n|_[a;a+hn)(t) =X_n⁰(t). (3.3)

The solution Xe of the equation (2.1) associates with some function if and only if the sequence of the solutionsX_nof the equation (3.3) converges. Therefore we have to investigate the limiting behavior of the sequence X_n.

Let t be an arbitrary point of T. There exists m_t ∈ N and τ_t ∈ [a;a+h_n) such that t = τ_t+m_th_n. Set t_k = τ_t+kh_n, k = 0,1, . . . , m_t. Then the solution of the equation (3.3) can be written as

X_n(t) =X_n⁰(τ_t) +

mt−1

X

k=0

f_n(X_n(t_k))(L_n(t_k+1)−L_n(t_k)) +

mt−1

X

k=0

g_n(X_n(t_k))h_n. (3.4) Consider the function F_n(x) : [−∞; +∞]→[0; 1] given by

F_n(x) = Z 1/n

x

ρ_n(s)ds. (3.5)

Since ρ_n(s) ≥ 0, then F_n is a non-increasing function, 0 ≤ F_n(x) ≤ 1 and F_n(+∞) = 0, Fn(−∞) = 1. Denote byF_n⁻¹ the inverse function of Fn, i.e., F_n⁻¹ : [0; 1]→[−∞; +∞] and

F_n⁻¹(u) = sup{x:F_n(x) = u}. (3.6) In order to describe the limits of the sequence X_n we consider the integral equation

X(t) = x⁰+ Z t

a

g(X(s))ds+ Z t

a

f(X(s))dL^c(s)

+ X

a<s≤t

ϕ(∆L(s)f, X(s−),1)−X(s−)

, t∈T, (3.7)

where L^c is a continuous part of the function L, ∆L(s) = L(s+)−L(s−) is a size of the jump of the function L at the epoch s, and ϕ(z, x, u) denotes the solution of the integral equation

ϕ(z, x, u) = x+ Z

[0;u)

z(ϕ(z, x, v))µ(dv), (3.8)

and µ(du) is a probability measure defined on the Borel subsets of the interval [0; 1]. In order to underline the dependence ϕ(z, x, u) on the measure µ sometimes we will also use the notation ϕ(z, x, u, µ). Here and in what follows all integrals are understood in the Lebesgue-Stieltjes sense.

Remark 3.1 Let us note some particular cases of the equation (3.7).

If the measureµgives the mass one to the point0, then the equation (3.8) has the solution ϕ(z, x, u) =

x, u = 0,

x+z(x), u∈(0; 1].

Hence the equation (3.7) becomes

X(t) =x⁰+ Z

(a,t]

f(X(s−))dL(s) + Z t

a

g(X(s))ds. (3.9)

The interpretation of the equation (1.1) as the equation (3.9) was considered by Das and Sharma [2], see also Pandit and Deo [6].

If the measure µ is the Lebesgue measure, then the equation (3.8) can be written as an ordinary differential equation

∂ϕ(z, x, u)

∂u =z(ϕ(z, x, u))

and ϕ(z, x,0) = x. Then the solution of the equation (3.7) is a so-called “approximative solution” (see, e.g. [7]). It was shown by Zavalishchin and Sesekin in [7] that if we consider approximation L⁰_n of L, then the sequence X_n⁰ of the solutions of the equation (1.1), where L is replaced by L⁰_n converges to the “approximative solution”, i.e., to the solution of the equation (3.7) with the measure µ equal to the Lebesgue measure.

The solution of the equation in the sense of the papers [1, 4, 5] can be obtained from the equation (3.7) by using appropriate measure µ.

The equation (3.8) has a unique solution if, for example, the function z is Lipschitz continuous (see, e.g. [13]). Since we concerned with the case where f is globally Lipschitz, then the function ϕ(∆L(s)f, X(s−),1) is always defined.

We shall discuss the existence and some properties of the solutions of the equations (3.7) and (3.8) in Section 4.

Definition 3.2 We say that the function σ : [0; 1] → [0; 1] belongs to class G if there is a system of pairwise-disjoint intervals (a_i;b_i)⊆[0; 1], i∈I such that

σ(u) =

b_i, u∈(a_i;b_i], u, u /∈S

i∈I(a_i;b_i]. (3.10)

Remark 3.3 There are two most important examples of the functions from the class G:

1. σ(u) =1_(0;1](u), i.e. σ(u) = 1 if u > 0 and σ(u) = 0 if u = 0. In this case I = {1}

and (a₁;b₁] = (0; 1]. The measure µ generated by this function gives the mass one to the point 0, and it follows from Remark 3.1 that the equation (3.7) with this measure µ can be written in the form (3.9).

2. σ(u) = u. In this case I = ∅. From Remark 3.1 we deduce that the solution of the equation (3.7) with the measure generated by σ is an “approximative solution”.

The following theorem describes the limits of the sequenceX_n. Theorem 3.4 Let f and g be Lipschitz functions satisfying (3.1), R

t∈T|X_n⁰(τ_t)−x⁰|dt→0 andF_n(F_n⁻¹(u)−δh_n)→σ(u)asn → ∞andh_n→0for allδ∈(0; 1)and for any continuity point u∈[0; 1] of σ. Then σ belongs to class G and

Z

T

|Xn(t)−X(t)|dt →0

as n → ∞ and h_n → 0, where X(t) is a solution of the equation (3.7) with the measure µ generated by the function σ.

Ifδ-sequence ρ_n has the form ρ_n(u) = nρ(nu), whereρ ∈C^∞(R), ρ ≥0, supp ρ⊆ [0; 1]

and R1

0 ρ(u)du= 1, then we will say that δ-sequence ρ_n is of the simplest type.

Corollary 3.5 Let δ-sequence ρ_n be of the simplest type. Then the sequence F_n(F_n⁻¹(u)− δhn) converges weakly for any δ ∈ (0; 1) and the limit does not depend on δ if and only if either 1/n = o(h_n) or h_n = o(1/n). Moreover, suppose that f and g are global Lipschitz functions and R

t∈T|X_n⁰(τ_t)−x⁰|dt→0 as n → ∞ and h_n →0. Then Z

T

|Xn(t)−X(t)|dt→0, where Xn is a solution of the equation (3.3) and

1. X(t) is a solution of the equation (3.9) if 1/n =o(h_n),

2. X(t) is a solution of the equation (3.7) with the measure µ is equal to the Lebesgue measure, if h_n=o(1/n).

Notice thatF_n(F_n⁻¹(u)−δh_n)→1_(0;1](u) if and only if 1/n =o(h_n). And F_n(F_n⁻¹(u)− δh_n) → u if and only if h_n = o(1/n). Hence Remark 3.3 and Corollary 3.5 imply that for the δ-sequence of the simplest type the solutions X_n of the equation (3.3) converge either to the solution of the equation (1.1) in the sense of papers [2, 6] if 1/n = o(h_n) or to the

“approximative solution” of the equation (1.1) in the sense of monograph [7] if h_n=o(1/n).

For a continuous functionLequation (3.7) has not the last term, and hence is equivalent to the equation (3.9). Therefore we have the following result

Corollary 3.6 Let L be a continuous function of finite variation on T. Suppose that f and g are Lipschitz continuous and R

t∈T|X_n⁰(τt)−x⁰|dt →0 as n→ ∞ and hn→0. Then Z

T

|X_n(t)−X(t)|dt→0,

where X_n is a solution of the equation (3.3), and X(t) is a solution of the equation (3.9).

Remark 3.7 It will follow from the proof of Theorem 3.4 that L¹-norm can be replaced by sup-norm if the initial condition X_n⁰ converges in sup-norm.

4 Discussing of the equations

In this section we consider the equation (3.7) and show that it has the unique solution. We will prove also some inequalities for solutions of the equations (3.7) and (3.8).

The following modification of Gronwall’s inequality from [13] will be very useful in the sequel.

Lemma 4.1 ([13]) Let ν be a locally bounded measure on R, and let Y(t), t ≥ 0 be a function of bounded variation on each bounded interval such that for all t ≥0

0≤Y(t)≤A+B Z

[0;t)

Y(s)ν(ds), where A and B are nonnegative constants.

Then for each t≥0

Y(t)≤Ae^Bν([0;t))Y

s<t

(1 +Bν({s}))e^−Bν({s}) ≤Ae^Bν([0;t)). We will also need the stronger inequality in the discrete case (see, [14]).

Lemma 4.2 ([14]) Let a_n, b_n and y_n (n = 1,2, . . .) are nonnegative real numbers and if x_n ≤a_n+Pn−1

i=1 b_ix_i for all n = 1,2, . . ., then x_n≤a_n+ 1

pn−1 n−1

X

i=1

a_ib_ip_i ≤ max

1≤i≤na_ie^Pn−1j=1bj

for all n = 1,2, . . ., where p_k =Qk

i=1(1 +b_i)⁻¹ (k = 1,2, . . .).

By using Lemma 4.1 it is easy to show from the definition of ϕ(z, x, u) the following inequalities.

Lemma 4.3 Let ϕ(z, x, u) be a solution of the equation (3.8) with real-valued function z such that |z(x)−z(y)| ≤ K|x−y| and |z(x)| ≤ K(1 + |x|) for all x, y ∈ R. Then the following inequalities hold for all x, y∈R, u, v∈[0; 1], u > v:

1. |ϕ(z, x, u)−ϕ(z, y, u)| ≤ |x−y|e^K. 2. |ϕ(z, x, u)| ≤(K+|x|)e^K.

3. |ϕ(z, x, u)−x| ≤K(1 + (|x|+K)e^K).

4. |ϕ(z, x, u)−x−ϕ(z, y, u) +y| ≤ |x−y|Ke^K.

5. |ϕ(z, x, u)−ϕ(z, x, v)| ≤K(1 + (K+|x|)e^K)µ([v, u)) For a given function of finite variation L, we define

h(s, x) = ϕ(∆L(s)f, x,1)−x

∆L(s) =

Z

[0;1)

f(ϕ(∆L(s)f, x, u))µ(du).

By using this function we can rewrite the equation (3.7) in the following form:

X(t) = x⁰+ Z t

a

g(X(s))ds+ Z t

a

f(X(s))dL^c(s) +

Z

(a;t]

h(s, X(s−))dL^d(s), t∈T, (4.1)

where L^d is the discontinuous part of the functionL, i.e. L^d =L−L^c.

Since ϕ(0, x, u) = x for all x ∈ R and u ∈ [0; 1], then h(s, x) = f(x) if ∆L(s) = 0 and the equation (4.1) can be written as follows:

X(t) =x⁰+ Z t

a

g(X(s))ds+ Z

(a;t]

h(s, X(s−))dL(s), t∈T. (4.2) Lemma 4.4 For f Lipschitz continuous with constant M, the following inequality holds:

|h(s, x)−h(s, y)| ≤M|x−y|e^M∆L(s).

Proof. From the definitions of the functions h and ϕ we have

|h(s, x)−h(s, y)| ≤ Z

[0;1)

|f(ϕ(∆L(s)f, x, v))−f(ϕ(∆L(s)f, y, v))|µ(dv)

≤M Z

[0;1)

|ϕ(∆L(s)f, x, v)−ϕ(∆L(s)f, y, v)|µ(dv)≤M|x−y|e^M∆L(s),

where we have used Lemma 4.3 for the last inequality. 2

It was shown by Groh [13] that the homogeneous equation (4.2), i.e., g = 0, has the unique solution for a Lipschitz continuous function h. Since by Lemma 4.4 the function h satisfies a uniform Lipschitz condition provided f is Lipschitz continuous, then one can use an evident modification of the method proposed in [13] in order to prove the existence and the uniqueness of the solution of the equation (4.2) if g is Lipschitz continuous, too. Since the equations (4.2) and (3.7) are equivalent, then the equation (3.7) has a unique solution.

Lemma 4.5 Let functionsf andg be Lipschitz continuous with constantM satisfying (3.1).

Then for solutions X and Xn of the equations (3.7) and (3.3), respectively, the following inequalities hold for all t, s∈T, t > s and l, n ∈N:

1. |X(t)| ≤C(1 +|x⁰|);

|X_n(t)| ≤C(1 +|X_n⁰(τ_t)|),

where the constant C depends only on M, |T| and V^b_aL.

2. |X(t)−X(s)| ≤M(1 +C(1 +|x⁰|))((t−s) +V^t_s+L).

3. |X_n(t +lh_n) −X_n(t)| ≤ M(1 +C(1 + |X_n⁰(τ_t)|))(lh_n +V_t^t+lhnL_n) ≤ M(1 +C(1 +

|X_n⁰(τt)|))(lhn+V^t+lh_t+ ^n+1/nL).

Proof. The proof of these inequalities is standard and it uses the definitions of X and X_n, Lemma 4.1, inequality (3.1), and Lipschitz continuity of f and g. 2

5 Auxiliary statements

Let us introduce some notations. The function of bounded variationL admits the following unique representation L = L^c +L^d, where L^c is a continuous function and L^d piecewise constant function. Since L is a right continuous function, then L^d is a right continuous, too. Denote by ζ_i, i ∈ N the epochs of jumps of the function L. It is evident, that L^d(t) = P

ζi≤t∆L(ζ_i). Moreover V^v_uL = V^v_uL^c +V^v_uL^d for any a ≤ u < v ≤ b and V_u^vL^d = P

u≤ζi≤v|∆L(ζ_i)|. We assume that if L has only q jumps, then ζ_q+1 = ζ_q+2 = · · · = b+ 1 and ∆L(ζ_q+1) = ∆L(ζ_q+2) = · · ·= 0.

Fix an arbitrary ε > 0. Since V^b_aL <∞, then V_a^bL^d =P∞

i=1|∆L(ζ_i)| <∞. Hence there is a number N ∈ N such that P∞

i=N+1|∆L(ζ_i)| ≤ ε. We can represent L^d in the following way:

L^d =L^≤N +L^>N, (5.1)

where L^≤N(t) = PN

i=11{ζ_i≤t}∆L(ζ_i) and L^>N(t) = P∞

i=N+11{ζ_i≤t}∆L(ζ_i). From this it follows readily that V^v_uL^>N ≤ε and V_u^vL^≤N ≤V^v_uL^d<∞ for all a≤u < v ≤b.

Set L^c_n = L^c ∗ρ_n, L^d_n = L^d∗ρ_n, L^≤N_n =L^≤N ∗ρ_n and L^>N_n =L^>N ∗ρ_n, where ρ_n from the formula (3.2). Then we have L_n =L^c_n+L^d_n and L^d_n =L^≤N_n +L^>N_n . Furthermore for all t ∈Twe have

|L^c_n(t)−L^c(t)| ≤V^t+1/n_t L^c. (5.2) From now on we will denote by C the constant which does not depend on n, h_n and t ∈T, and its value can change from one formula to another.

Lemma 5.1 Suppose that f and g are Lipschitz continuous functions. Let X_n(t) and X(t) be solutions of the equations (3.3) and (3.7) respectively. Then for all t∈T

mt−1

X

k=0

g_n(X_n(t_k))h_n− Z t

a

g(X(s))ds

≤

Ch_n+C/n+Ch_n

mt−1

X

k=0

|X_n(t_k)−X(t_k)|.

Proof. The proof of the lemma is standard and, therefore, is omitted. 2 Lemma 5.2 Suppose that f and g are Lipschitz continuous functions. Let X_n(t) and X(t) be solutions of the equations (3.3) and (3.7) respectively. Then for all t∈T

mt−1

X

k=0

f_n(X_n(t_k))(L^c_n(t_k+1)−L^c_n(t_k))− Z t

a

f(X(s))dL^c(s)

≤Ch_n+C/n

+C(1 +|X_n⁰(τ_t)|) sup

|u−v|≤hn+1/n

V^v_uL^c+C

mt−1

X

k=0

|X_n(t_k)−X(t_k)||L^c(t_k+1)−L^c(t_k)|.

Proof. The following representation is evident.

mt−1

X

k=0

f_n(X_n(t_k))(L^c_n(t_k+1)−L^c_n(t_k))− Z t

a

f(X(s))dL^c(s)

=

mt−1

X

k=0

(fn(Xn(tk))−f(X(tk)))(L^c(tk+1)−L^c(tk))

+

mt−1

X

k=0

f_n(X_n(t_k))

(L^c_n(t_k+1)−L^c_n(t_k))−(L^c(t_k+1)−L^c(t_k))

+

mt−1

X

k=0

Z tk+1

tk

(f(X(t_k))−f(X(s)))dL^c(s)

− Z t0

a

f(X(s))dL^c(s) = I1+I2+I3−I4. Sincef is Lipschitz continuous we have from (3.2)

|I1| ≤C/n+C

mt−1

X

k=0

|Xn(tk)−X(tk)||L^c(tk+1)−L^c(tk)|. (5.3) By using the “summation by parts” formula we get forI2

I2 =

mt−1

X

k=1

fn(Xn(tk−1))−fn(Xn(tk))

(L^c_n(tk)−L^c(tk)) +fn(Xn(tmt−1))(L^c_n(tmt)−L^c(tmt))−fn(Xn(t0))(L^c_n(t0)−L^c(t0)).

It follows from Lemma 4.5, the Lipschitz continuity off, and from the inequalities (3.1) and (5.2), that

|I₂| ≤C(1 +|X_n⁰(τ_t)|) sup

|u−v|≤1/n

V^v_uL^c. (5.4)

Also, Lemma 4.5 implies the following estimation for I3:

|I₃| ≤Ch_n+C sup

|u−v|≤hn

V_u^vL^c. (5.5)

By using inequality (3.1) it easy to show that

|I₄| ≤CV^t_a⁰L^c ≤C sup

|u−v|≤hn

V^v_uL^c. (5.6)

The inequalities (5.3) – (5.6) imply the statement of the lemma. 2 Suppose that for anyt∈Tandn ∈Nthe partition 0 =ξ₀ⁿ(t)≤ξ₁ⁿ(t)≤. . .≤ξ_p+2ⁿ (t) = 1, where p depends on n, of the interval [0,1] is given. Consider the recurrent sequence ϕⁿ_k(t), t ∈T,k = 0,1. . . p+ 2 of the functions given by

ϕⁿ_k+1(t) = ϕⁿ_k(t) +z(ϕⁿ_k(t))(ξ_k+1ⁿ (t)−ξ_kⁿ(t))

ϕⁿ₀(t) = x, (5.7)

where x∈R, and z is a Lipschitz continuous function with constant K, such that

|z(y)| ≤K(1 +|y|), y ∈R. (5.8) Denote for allu∈[0; 1], t∈T and n = 1,2, . . .

σⁿ(u, t) =

ξ_kⁿ(t), if ξ_k−1ⁿ (t)< u≤ξ_kⁿ(t),

0, if u= 0. (5.9)

and

φⁿ(u, t) =

ϕⁿ_k(t), if ξⁿ_k−1(t)< u≤ξ_kⁿ(t),

x, if u= 0. (5.10)

Then

φⁿ(u, t) =x+ Z

[0;u)

z(φⁿ(s, t))d_sσⁿ(s, t).

Lemma 5.3 Suppose that there exists a nondecreasing left continuous function σ(u), u ∈ [0; 1] such that

Z

T

|σⁿ(u, t)−σ(u)|dt →0

as n→ ∞ for any continuity point u of σ. Then σ belongs to class G and Z

T

|φⁿ(u, t)−φ(u)|dt →0

as n→ ∞ for any continuity point u of φ, where φ(u) is a solution of the equation

φ(u) = x+ Z

[0,u)

z(φ(s))dσ(s). (5.11)

Remark 5.4 Notice that φ(u) = ϕ(z, x, u, µ), where ϕ(z, x, u, µ) is a solution of the equa- tion (3.8) with measure µ generated by σ. Furthermore φⁿ(u, t) = ϕ(z, x, u, µⁿ(t)), where ϕ(z, x, u, µⁿ(t)) is a solution of the equation (3.8) with the measure µⁿ(du, t) generated by σⁿ(u, t).

Proof. From the definition of σⁿ we have for any u ∈ [0; 1] that σⁿ(u, t) ≥ u for all t ∈ T. Hence σ(u) ≥ u for all u ∈ [0; 1]. Set A = {w ∈ [0; 1] : σ(w) = w} and B = {w ∈ [0; 1] : σ(w) > w}. Let u ∈ [0; 1] be a continuity point of σ and u ∈ B. Then R

T|σⁿ(u, t)−d|dt → 0, where d = σ(u) > u. For any subsequence {n⁰} of {n} we can choose a subsequence {n⁰⁰}of{n⁰}such thatσⁿ⁰⁰(u, t)→d for almost allt∈T. By Egorov’s theorem (see, i.g., [15, Ch. 2, §9, Th. 2, p.55]) for any δ > 0 there exists a measurable set Tδ ⊂Tsuch that |T\Tδ|< δ and on the setTδ the sequenceσⁿ⁰⁰(u, t) converges uniformly to d. Here and in what follows |D| denotes Lebesgue measure of the set D. Therefore for any v ∈ (u, d) and large enough n⁰⁰ ∈ N we have u < v < σⁿ⁰⁰(u, t) for all t ∈ T_δ. Hence σⁿ⁰⁰(u, t) =σⁿ⁰⁰(v, t) for all t∈Tδ. Moreover

Z

T

|σⁿ⁰⁰(v, t)−d|dt≤ Z

Tδ

|σⁿ⁰⁰(u, t)−d|dt+ 2δ.

Letting n→ ∞since δ is an arbitrary small we haveR

T|σⁿ⁰⁰(v, t)−d|dt→0. It means that for eachv ∈(u;d) and any subsequence{n⁰}of{n}we can choose a subsequence{n⁰⁰}of{n⁰} such that R

T|σⁿ⁰⁰(v, t)−d|dt→ 0. HenceR

T|σⁿ(v, t)−d|dt →0 and σ(v) =σ(u) =d > v.

Therefore (u;d)⊂B.

It follows from the left-continuity of σ that there exists ξ > 0 such that for any w ∈ (u−ξ;u) we haved =σ(u)≥σ(w)> u > w. Hence (u−ξ;u)⊂B. Therefore (u−ξ;d)⊂B andB is an open set. Consequently we can writeB =S

i∈I(a_i;b_i), where (a_i;b_i)∩(a_j;b_j) = ∅ if i6=j.

Let x ∈ (ai;bi) for some i ∈ I. If σ(x) > bi then x < bi < σ(x) and, as was shown above, σ(b_i) =σ(x) > b_i. Hence b_i ∈ B, but by definition b_i ∈/ B. Therefore σ(x) ≤ b_i. If d =σ(x) < b_i, then a_i < x < σ(x) = d < b_i and σ(d) = d, i.e. d /∈ B. This contradiction shows that σ(x) = bi. Since σ(x) = xif x /∈B then the function σ belongs to class G.

Suppose σ has the form (3.10). For any > 0 there is a number N ∈ N such that P∞

i=N+1(b_i−a_i)< . Consider the function σ^N(u, t) =

σ(u), if u /∈S∞

i=N+1(a_i;b_i], u, if u∈S∞

i=N+1(a_i;b_i].

It is evident that V₀¹|σ−σ^N| ≤. Let φ^N be a solution of the following equation:

φ^N(u) =x+ Z

[0;u)

z(φ^N(s))dσ^N(s), u∈[0; 1]. (5.12) Lipschitz continuity forz and Lemma 4.3 imply

|φ^N(u)−φ(u)| ≤K Z

[0;u)

|φ^N(s)−φ(s)|dσ(s) +K₁V₀¹|σ−σ^N|, where K₁ = (|x|+K)e^K. Applying Lemma 4.1 we get

|φ^N(u)−φ(u)| ≤K₁V¹₀|σ−σ^N|e^Kσ(u) ≤K₁e^K. (5.13) Denote by k_i, i = 1,2, . . . , N the number such that ξ_kⁿ

i(t) < (a_i +b_i)/2 ≤ ξ_kⁿ

i+1(t). We assume that a₀ =b₀ = 0 and a_i =b_i = 0, i= 1,2, . . . if N = 0. Setk₀ =−1,k_N+1 =p+ 3 and N_k = max{i:k_i ≤k}, k = 0,1, . . . , p+ 2. For any n ∈N, l = 0,1, . . . , p+ 2 and t ∈T we define

ξe_lⁿ(t) =







a_i, if a_i < ξ_lⁿ(t)≤ξ_kⁿ_i(t), b_i, if b_i > ξ_lⁿ(t)≥ξ_kⁿ

i+1(t), ξ_lⁿ(t), in the other cases.

It follows from (5.7) that we can write ϕⁿ_k+1(t) in the following form:

ϕⁿ_k+1(t) = x+

k

X

l=0

z(ϕⁿ_l(t))(ξ_l+1ⁿ (t)−ξ_lⁿ(t)) =x+

Nk

X

i=1

z(ϕⁿ_k

i(t))(ξ_kⁿ

i+1(t)−ξⁿ_k

i(t))

+

Nk+1

X

i=1

(ki−1)∧k

X

l=ki−1+1

z(ϕⁿ_l(t))(ξ_l+1ⁿ (t)−ξ_lⁿ(t)). (5.14)

From the equations (5.12) and (5.14) we have for anyn ∈N,k = 0,1, . . . , p+ 2 and t∈T

|ϕⁿ_k+1(t)−φ^N(ξe_k+1ⁿ (t))| ≤

Nk

X

i=1

|z(ϕⁿ_k

i(t))(ξ_kⁿ

i+1(t)−ξ_kⁿ

i(t))−z(φ^N(a_i))(b_i−a_i)|

+

Nk+1

X

i=1

(ki−1)∧k

X

l=ki−1+1

|z(ϕⁿ_l(t))(ξ_l+1ⁿ (t)−ξ_lⁿ(t))−z(φ^N(eξ_lⁿ(t)))(ξ_l+1ⁿ (t)−ξ_lⁿ(t))|

+

Nk

X

i=1

ki−1

X

l=ki−1+1

z(φ^N(eξ_lⁿ(t)))(ξ_l+1ⁿ (t)−ξ_lⁿ(t))− Z ai

bi−1

z(φ^N(s))ds

+

k

X

l=k_Nk+1

z(φ^N(eξ_lⁿ(t)))(ξ_l+1ⁿ (t)−ξ_lⁿ(t))−

Z ξ^en_k+1(t) b_Nk

z(φ^N(s))ds

=

Nk

X

i=1

I₁ⁱ +

Nk+1

X

i=1

I₂ⁱ +

Nk

X

i=1

I₃ⁱ +I4. (5.15)

Consider I₃ⁱ, i = 1,2, . . . , N_k. Ifbi−1 =a_i, then ξeⁿ_l(t) = a_i for ki−1+ 1 ≤l ≤ k_i−1 and from Lemma 4.3 and inequality (5.8) we have

I₃ⁱ ≤

ki−1

X

l=ki−1+1

z(φ^N(a_i))(ξ_l+1ⁿ (t)−ξⁿ_l(t))

≤K₂|b_i−1−ξ_kⁿ_i−1+1(t)|+K₂|a_i−ξ_kⁿ

i(t)|, (5.16) where K₂ =K(1 + (|x|+K)e^K).

If bi−1 < a_i, then ξe_lⁿ(t) = bi−1 for ξ_kⁿ_i−1+1(t) ≤ ξⁿ_l(t) < σⁿ(bi−1, t) and ξe_lⁿ(t) = a_i for ξ_kⁿ_i(t)≥ξⁿ_l(t)≥σⁿ(ai, t). Hence

I₃ⁱ ≤K₂|bi−1−ξ_kⁿ_i−1+1(t)|+K₂|a_i−ξ_kⁿ

i(t)|+ Z ai

bi−1

|z(φ^N(σeⁿ(s, t)))−z(φ^N(s))|ds, (5.17) where eσⁿ(s, t) = ξe_lⁿ(t) if ξ_lⁿ(t)< s≤ξ_l+1ⁿ (t), l= 0, . . . , p+ 2.

Sincez is Lipschitz continuous with constantK, then Lemma 4.3 and inequalities (5.16) and (5.17) yield

I₃ⁱ ≤K₂|bi−1−ξ_kⁿ_i−1+1(t)|+K₂|a_i−ξ_kⁿ_i(t)|+KK₁ Z ai

bi−1

|eσⁿ(s, t)−s|ds. (5.18) Operating in the same way one can obtain the following estimation forI₄:

I4 ≤K2|bN_k −ξ_kⁿ

Nk+1(t)|+KK1

Z a_Nk+1

b_Nk

|eσⁿ(s, t)−s|ds. (5.19) ForI₁ⁱ by using Lemma 4.3 we have

I₁ⁱ ≤K₂|bi−1−ξ_kⁿ_i−1+1(t)|+K₂|a_i−ξ_kⁿ

i(t)|+K|ϕⁿ_k

i(t)−φ^N(a_i)||ξ_kⁿ

i+1(t)−ξ_kⁿ

i(t)|. (5.20)

Forz Lipschitz continuous we get I₂ⁱ ≤K

(ki−1)∧k

X

l=ki−1+1

|ϕⁿ_l(t)−φ^N(eξ_lⁿ(t))||ξ_l+1ⁿ (t)−ξ_lⁿ(t)|. (5.21)

Collecting formulas (5.15), (5.18), (5.19), (5.20) and (5.21) we have

|ϕⁿ_k+1(t)−φ^N(eξ_k+1ⁿ (t))| ≤2K2 Nk

X

i=1

(|bi−ξ_kⁿ_i+1(t)|+|ai−ξ_kⁿ_i(t)|)

+KK₁

Nk+1

X

i=1

Z ai

bi−1

|eσⁿ(s, t)−s|ds+K

k

X

l=0

|ϕⁿ_l(t)−φ^N(eξ_lⁿ(t))||ξ_l+1ⁿ (t)−ξ_lⁿ(t)|.

Applying Lemma 4.2 to the inequality above we obtain

|ϕⁿ_k+1(t)−φ^N(eξ_k+1ⁿ (t))| ≤ 2K2 Nk

X

i=1

(|bi−ξ_kⁿ_i+1(t)|+|ai−ξ_kⁿ_i(t)|)

+KK₁

Nk+1

X

i=1

Z ai

bi−1

|σeⁿ(s, t)−s|ds

!

e^Kξnk+1(t). (5.22) Fix an arbitrary continuity pointu ∈ [0; 1] of the function φ and let ξ_kⁿ(t) be such that ξ_kⁿ(t) < u ≤ ξⁿ_k+1(t). From the definition of φⁿ(u, t) and inequalities (5.13) and (5.22) we have

|φⁿ(u, t)−φ(u)| ≤ |ϕⁿ_k+1(t)−φ^N(eξ_k+1ⁿ (t))|+|φ^N(ξe_k+1ⁿ (t))−φ(eξⁿ_k+1(t))|

+|φ(eξ_k+1ⁿ (t))−φ(u)| ≤2K₂e^K

N

X

i=1

(|b_i−ξ_kⁿ

i+1(t)|+|a_i−ξⁿ_k

i(t)|) +KK₁e^K

N

X

i=1

Z ai

bi−1

|σeⁿ(s, t)−s|ds+KK₁e^K Z 1

bN

|eσⁿ(s, t)−s|ds+K₁e^K

+ Z

[^{u; ξe}k+1ⁿ (t))

|z(φ(s))|dσ(s).

Taking the integral on t∈T in both sides of the above inequality yields Z

T

|φⁿ(u, t)−φ(u)|dt ≤2K₂e^K

N

X

i=1

Z

T

|b_i−ξ_kⁿ

i+1(t)|dt+ Z

T

|a_i−ξ_kⁿ

i(t)|dt

+KK₁e^K Z

T

Z

[0,1]\S∞ i=1(ai,bi]

|eσⁿ(s, t)−s|dsdt +K₁(1 +K)e^K|T|+

Z

T

Z

[^{u; ξe}k+1ⁿ (t))

|z(φ(s))|dσ(s)dt

= 2K₂e^K

N

X

i=1

(J₁ⁱ +J₂ⁱ) +KK₁e^KJ₃+K₁(1 +K)e^K|T|+J₄. (5.23)

Ifu∈[a_i;b_i] for somei= 1, . . . , N, then ξe_k+1ⁿ (t) = a_i orξe_k+1ⁿ (t) = b_i. Hence J₄ = 0.

If u /∈ SN

i=1[a_i;b_i], then ξe_k+1ⁿ (t) = ξ_k+1ⁿ (t) = σⁿ(u, t). Therefore from Lemma 4.3 and inequality (5.8) we have if u /∈SN

i=1[ai;bi] J₄ ≤K₁

Z

T

|σⁿ(u, t)−u|dt≤K₁ Z

T

|σⁿ(u, t)−σ(u)|dt+K₁|T|.

Consequently for continuity point u of φ we get lim sup

n→∞

J₄ ≤K₁|T|. (5.24)

Since (a_i +b_i)/2, i= 1, . . . , N is a continuity point of σ, then Z

T

|σⁿ((ai+bi)/2, t)−σ((ai+bi)/2)|dt→0.

But σ((a_i+b_i)/2) = b_i and σⁿ((a_i+b_i)/2, t) =ξ_kⁿ_i+1(t). Hence Z

T

|b_i−ξ_kⁿ_i+1(t)|dt→0.

Therefore, for all i= 1, . . . , N

n→∞lim J₁ⁱ = 0. (5.25)

Consider J₂ⁱ. For anyδ > 0 set A^δ,i_n ={t∈T :ξ_kⁿ

i(t)> a_i +δ}. Then forδ < (b_i−a_i)/2 we have that v = a_i +δ < (a_i+b_i)/2, σ(v) = b_i and (a_i+b_i)/2 > ξ_kⁿ

i(t) ≥ σⁿ(v, t) for all t ∈A^δ,i_n . Hence, sincev is a continuity point of σ

|A^δ,i_n |(b_i−a_i)/2≤ Z

A^δ,in

|ξ_kⁿ_i(t)−b_i|dt ≤ Z

T

|σⁿ(v, t)−σ(v)|dt→0 as n→ ∞. Thus |A^δ,i_n | →0 for all δ <(b_i −a_i)/2 and i= 1, . . . , N.

Set B_n^δ,i = {t ∈ T : ξ_kⁿ

i(t) < a_i −δ}. Then for w = a_i −δ we have σ(w) ≤ a_i and σⁿ(w, t) = ξⁿ_ki+1(t)≥(ai+bi)/2 fort∈B_n^δ,i. Hence, ifw is a continuity point of σ, then

|B_n^δ,i|(b_i−a_i)/2≤ Z

Bn^δ,i

|σⁿ(w, t)−σ(w)|dt≤ Z

T

|σⁿ(w, t)−σ(w)|dt →0 as n→ ∞. Therefore |B_n^δ,i| →0 for all i= 1, . . . , N and almost all δ >0.

Thus for alli= 1, . . . , N and almost all (b_i−a_i)/2> δ >0 lim sup

n→∞

Z

T

|ξ_kⁿ

i(t)−a_i|dt= lim sup

n→∞

Z

T\(^Aδ,in

SBn^δ,i)

|ξ_kⁿ

i(t)−a_i|dt ≤δ|T|.

Since δ can be arbitrary small then for all i= 1, . . . , N

n→∞lim J₂ⁱ = lim

n→∞

Z

T

|ξⁿ_ki(t)−ai|dt= 0. (5.26)

Consider J₃. For any s ∈ [0; 1]\S∞

i=1(a_i, b_i] and δ > 0 denote by C_n^δ(s) the following subset of T: C_n^δ(s) = {t ∈ T : eσⁿ(s, t) < s−δ}. For all v = s −δ we have σⁿ(v, t) = σⁿ(s, t)≥s =σ(s)> σ(v)≥v for t∈C_n^δ(s). Hence, if v is a continuity point of σ, then

(s−σ(v))|C_n^δ(s)| ≤ Z

C^δ_n(s)

|σⁿ(v, t)−σ(v)|dt≤ Z

T

|σⁿ(v, t)−σ(v)|dt→0.

Thus |C_n^δ(s)| → 0 as n → ∞ for s /∈ S∞

i=1(a_i, b_i] and almost all δ > 0. Therefore, since σeⁿ(s, t)< s we have

lim sup

n→∞

Z

T

|σeⁿ(s, t)−s|dt= lim sup

n→∞

Z

T\C_n^δ(s)

|σeⁿ(s, t)−s|dt≤δ|T|.

Hence

n→∞lim Z

T

|eσⁿ(s, t)−s|dt = 0.

It follows from Lebesgue’s theorem that

n→∞lim J₃ = lim

n→∞

Z

[0,1]\S∞ i=1(ai,bi]

Z

T

|eσⁿ(s, t)−s|dtds= 0. (5.27)

Thus from equalities (5.23), (5.25), (5.26), (5.27) and (5.24) we obtain lim sup

n→∞

Z

T

|φⁿ(u, t)−φ(u)|dt≤K₁((1 +K)e^K+ 1)|T|

for any continuity point u∈[0; 1] of φ and for all >0. Hence

n→∞lim Z

T

|φⁿ(u, t)−φ(u)|dt= 0,

which completes the proof of the lemma. 2

Denote by j_i, i = 1, . . . the number such that t_ji ≤ ζ_i − 1/n < t_ji+1. Set ξ_k^n,i(t) = F_n(ζ_i−t_ji+k), i= 1, . . .,t∈T, n∈Nand k = 0,1, . . . , p+ 2, wherep= [1/(nh_n)] (brackets denote the integer part of the number). Notice that ξ_k^n,i(t) depends on t ∈ T, since t_ji+k depends on t. Denote by ϕ^n,i_k (z, x, t) the sequence which is defined by the formula (5.7) for ξ_kⁿ(t) = ξ^n,i_k (t). Let σⁿ_i(u, t) be the sequence of functions which is given by the formula (5.9) with ξ_n(t) =ξ_k^n,i(t).

The following lemma from [16] (see also [17]) will give the necessary and sufficient con- ditions for σⁿ_i(u, t) to satisfy the statement of Lemma 5.3.

Lemma 5.5 ([16]) Suppose that F_n(F_n⁻¹(u)−δh_n)→ σ(u) as n → ∞ and h_n →0 for all δ ∈(0; 1) and all continuity points u∈[0; 1] of σ. Then

Z

T

|σ_iⁿ(u, t)−σ(u)|dt →0

as n→ ∞ and h_n →0 for i= 1,2, . . . and all continuity points u∈[0; 1] of σ.