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Publicly Available Published by De Gruyter March 22, 2016

Theory and applications of first-order systems of Stieltjes differential equations

  • Marlène Frigon EMAIL logo and Rodrigo López Pouso ORCID logo

Abstract

We set up the basic theory of existence and uniqueness of solutions for systems of differential equations with usual derivatives replaced by Stieltjes derivatives. This type of equations contains as particular cases dynamic equations on time scales and impulsive ordinary differential equations.

MSC 2010: 34A12; 34A34; 26A24

1 Introduction

Let g: be monotone, nondecreasing and left-continuous everywhere. This paper contains the basic existence and uniqueness theory for initial value problems of the form

(1.1)xg(t)=f(t,x(t)),t[t0,t0+T],x(t0)=x0,

where xg denotes the derivative of the function x(t) with respect to g. This new notion of derivative (to be defined in Section 5) is consistent with Stieltjes integration with respect to g. In fact, the equivalent integral form of (1.1) is just the usual one with Lebesgue integrals replaced by Lebesgue–Stieltjes integrals with respect to g.

These Stieltjes differential equations with respect to g or, simply, g-differential equations, have many applications as we show in Sections 8 and 9. Probably their most important virtue is that they serve as a unified framework for dynamic equations on time scales and differential equations with impulses at fixed times, as shown in [9]. For the convenience of readers and to illustrate clearly the applications of g-differential equations, we explain in a somewhat more detailed fashion how to carry out these transformations in Section 8. In Section 9, our results on g-differential equations are applied to mathematical models. The first one describes the level of water in a cylindrical tank taking into account that the evaporation is not the same during the day and during the night. The second model concerns the variation of the concentration of salt in water under the assumption that the temperature varies as a nondecreasing function of time.

The rest of the paper is organized as follows. In Section 2, we introduce a suitable topology on the real line associated to g. In Section 3, we introduce adequate functional spaces and study their properties as a first step towards the proofs of existence results for (1.1) by means of reduction to abstract fixed point problems. More precisely, we study the concept of continuity with respect to g, which is interesting in its own right and also gives us needful information about Borel-measurability of the type of functions that we must work with. Section 4 contains compactness criteria for sets of g-continuous functions. These criteria are fundamental in the proof that (1.1) has at least one solution by means of Schauder’s fixed point theorem. Section 5 is a review of the theory on g-derivatives and integrals established in [9], with new results on absolute continuity with respect to g. In Section 6, we construct exponential functions for the g-calculus, in the sense that we solve explicitly the g-differential equation xg(t)=c(t)x(t). Those exponential functions permit us to give the solutions to nonhomogeneous linear g-differential equations. Section 7 contains the main results in this paper, namely, the Picard and Peano theorems for (1.1) in the Carathéodory sense. We also present conditions ensuring that the solution of the nonlinear g-differential equation is g-differentiable everywhere on its domain, thus getting solutions in a classical sense.

2 The g-topology in the real line

In what follows, g: is monotone, nondecreasing and continuous from the left everywhere. Let us consider the set of points around which g is constant, namely

(2.1)Cg={t:g is constant on (t-ε,t+ε) for some ε>0},

and the set of discontinuity points of g, which is at most countable and shall be denoted by

(2.2)Dg={t:g(t+)-g(t)>0},

where, as usual, g(t+) stands for the right-hand side limit of g at t.

Let us consider the pseudometric

(2.3)ρ(s,t)=|g(s)-g(t)|for all s,t.

This mapping ρ:2[0,+) is obviously reflexive, ρ(t,t)=0 for all t, and it satisfies the triangle inequality

ρ(s,t)ρ(s,r)+ρ(r,t)for all s,t,r.

However, it fails to be a genuine metric because we may have ρ(s,t)=0 for some st, and this has the effect that the induced topology may not be Hausdorff.[1] An easily accesible good introduction to topologies induced by pseudometrics can be found in [7].

The pseudometric ρ generates a topology in exactly as ordinary metrics do. Thus, basic neighborhoods of a point t are given by open balls

(2.4)(t,ε)={s:ρ(s,t)<ε}={s:|g(s)-g(t)|<ε},ε>0,

and open sets are just arbitrary unions of these balls. Let us denote this topology by τg, and let us call it the g-topology in .

Open balls in τg need not be intervals with arbitrarily small length or even open intervals in the usual sense. Notice that, if t0Cg as defined in (2.1), then there is ε0>0 such that g is constant on (t0-ε0,t0+ε0), and therefore

(t0-ε0,t0+ε0)(t0,ε)for all ε>0 (no matter how small).

This implies that we cannot separate points s,t(t0-ε0,t0+ε0), st, which means that τg is not a Hausdorff topology when Cg.

On the other hand, if t0Dg as defined in (2.2) and we take ε(0,g(t0+)-g(t0)), then

(t0,ε)=(a,t0]for some a[-,t0).

We have proven that balls in τg need not be open intervals when Dg.

The following base for the g-topology will be useful.

Proposition 2.1

A countable base of τg is given by the family of intervals of the form (a,b) with a,bQ, a<b, along with the intervals (a,b], with aQ, bDg as defined in (2.2) and a<b.

As a consequence, the g-topology is second countable.

Proof.

Every open set in the g-topology is the union of open balls as defined in (2.4). Each open ball is nothing but

g-1((α,β))={t:α<g(t)<β}for some α,β, α<β.

Since g is nondecreasing, g-1((α,β)) is an interval with endpoints

(2.5)a=inf{t:α<g(t)}[-,+)
(2.6)b=sup{t:g(t)<β}(-,+].

It is easy to prove that g-1((α,β))=(a,b] when bDg and g(b)<βg(b+); otherwise g-1((α,β))=(a,b) for some b.

Therefore, if Uτg, then there exist two not necessarily countable families of intervals {(ai,bi)}i, {(aj,bj]}j𝒥 with bjDg for all j𝒥, such that

U=i(ai,bi)j𝒥(aj,bj].

The set A=i(ai,bi) is open for the usual topology, and then it can be expressed as a countable union of intervals of the form (a,b) with a,b. The set B=j𝒥(aj,bj] can be also reduced to a countable union. First, we join together all the intervals (aj,bj], j𝒥, which correspond to the same point bj=tUDg, and we get an interval (at,t] for some at[-,t). Therefore, we have

B=tUDg(at,t],

and the union is countable because so is Dg. Now, each interval (at,t] can be expressed as a countable union of intervals of the form (a,t] with a, which completes the proof. ∎

As we will see, this topology will be related to the notion of g-continuity that we introduce in the following section.

3 The space of g-continuous functions

We define an adequate new notion of continuity which, as we shall see, does not imply continuity in the usual sense, nor is implied by it. We denote by the Euclidean norm in N.

Definition 3.1

A function f:AN is g-continuous at a point t0A (or continuous with respect to g at t0) if for every ε(0,), there exists δ(0,) such that

(3.1)[tA,|g(t)-g(t0)|<δ]f(t)-f(t0)<ε.

We say that f is g-continuous on A if it is g-continuous at every point t0A.

Let us start remarking that Definition 3.1 of g-continuous functions can be interpreted from a topological point of view as continuity with respect to the pseudometric ρ introduced in (2.3).

Clearly, constant functions are g-continuous. Nontrivial examples of g-continuous functions are the functions λg, for λ. In particular, g-continuous functions need not be continuous.

The previous remark notwithstanding, g-continuous functions are continuous to some extent, as indicated in our next proposition.

Proposition 3.2

Let a,bR be such that a<b. If f:[a,b]RN is g-continuous on [a,b], then the following hold:

  1. f is continuous from the left at every t0(a,b].

  2. If g is continuous at t0[a,b), then so is f.

  3. If g is constant on some [α,β][a,b], then so is f.

In particular, g-continuous functions on [a,b] are continuous on [a,b] when g is continuous on [a,b).

Proof.

Let t0(a,b] and ε>0 be fixed. Now, take δ>0 such that (3.1) is satisfied. Since g is continuous from the left at t0, we can find δ>0 such that

0t0-t<δ|g(t)-g(t0)|<δ,

and then, (3.1) implies that f(t)-f(t0)<ε (provided that t[a,b]). We have thus found a value δ>0 such that

[t[a,b], 0t0-t<δ]f(t)-f(t0)<ε,

which shows that f is continuous from the left at t0. This proves the first claim in the statement.

If we assume now that g is continuous at some t0[a,b), then we can prove that f is continuous from the right at t0 by means of the same argument that we used in the previous case. This, together with the first part, proves part (2).

Finally, if g is constant on [α,β][a,b], then, for each ε>0, condition (3.1) for t0=α implies that f(t)-f(α)<ε for any t[α,β]. Since ε>0 can be arbitrarily small, we deduce that f(t)=f(α) for every t[α,β]. ∎

Our next example shows that g-continuous functions need not be regulated or locally bounded at discontinuity points of g.

Example 3.3

Define g(t)=t for t0, and g(t)=t+1 for t>0. We shall prove that the following function is g-continuous on :

f(t)={0if t0,sin(1/t)tif t>0.

In this case, g-continuity at points t00 is the same as usual continuity, and therefore f is g-continuous at every t00. To prove it, notice first that, for t0>0 and t0, we have

(3.2)|g(t)-g(t0)|=|t-1-t0|=1+t0-t>1.

Since f is continuous at t0, for each fixed ε>0, we take δ(0,1) such that

|t-t0|<δ|f(t)-f(t0)|<ε.

Now, for any t such that |g(t)-g(t0)|<δ, we infer from (3.2) that t>0, hence |g(t)-g(t0)|<δ is the same as |t-t0|<δ, which implies that |f(t)-f(t0)|<ε. The proof that f is g-continuous at every t0>0 is complete, and the proof that it is g-continuous at every t0<0 is similar.

To prove that f is g-continuous at t0=0, we notice that, for any ε>0, it suffices to take δ(0,1) because in that case the relation

|g(t)-g(0)|<δ

implies that t<0, and therefore |f(t)-f(0)|=0<ε as desired.

We denote by 𝒞g([a,b]) the set of g-continuous functions on the interval [a,b], and by 𝒞g([a,b]) the subset of g-continuous functions which are also bounded on [a,b]. We remark that, according to Proposition 3.2, these two sets are equal in the case where g is continuous on [a,b), but they are different in general if g is discontinuous, as shown in Example 3.3.

The set 𝒞g([a,b]) is a real vector space with the usual operations and it can be equipped with the sup norm:

(3.3)f0=supt[a,b]f(t)for every f𝒞g([a,b]).
Theorem 3.4

(𝒞g([a,b]),0) is a Banach space.

Proof.

It suffices to prove that 𝒞g([a,b]) is a closed subspace of the Banach space of bounded functions on [a,b] equipped with the sup norm.

Let {fn}n be a sequence in 𝒞g([a,b]) which converges uniformly in [a,b] to some bounded function f:[a,b]N. We have to show that f𝒞g([a,b]). Since we already know that f is bounded, we only have to prove that f is g-continuous at every t0[a,b].

Let t0[a,b] and ε>0 be fixed. Take n0 sufficiently large so that fn0-f0<ε/3. Now, let δ>0 be given by the definition of g-continuity of fn0 at t0 with ε replaced with ε/3. We then have, for any t[a,b] such that |g(t)-g(t0)|<δ, that

f(t)-f(t0)f(t)-fn0(t)+fn0(t)-fn0(t0)+fn0(t0)-f(t0)<3ε3=ε.

The proof is complete. ∎

As a corollary of Proposition 2.1, we obtain important information on measurability of g-continuous functions.

Corollary 3.5

Let AR be a Borel set. Every g-continuous function f:ARRN is Borel-measurable and, as a consequence, g-measurable (i.e., measurable with respect to the σ-algebra corresponding to the Lebesgue–Stieltjes measure generated by g).

Proof.

Let UN be open in the usual topology. Since f is g-continuous, the set f-1(U) is open in the g-topology. Proposition 2.1 guarantees that f-1(U) is a countable union of Borel sets, hence a Borel set itself. This proves that f is Borel-measurable.

Finally, the g-measurability of f is a consequence of the fact that Lebesgue–Stieltjes measures are Borel measures. ∎

4 Relatively compact subsets of 𝒞g([a,b])

Let a,b be such that a<b. In this section, we introduce sufficient conditions for subsets of 𝒞g([a,b]) to be relatively compact with the sup-norm topology.

The main result in this section is a version of the sufficient part of the Ascoli–Arzelà theorem for spaces of g-continuous functions. Interestingly, the most general version of the Ascoli–Arzelà theorem is valid for spaces of continuous functions defined on compact Hausdorff spaces, but in this case [a,b] need not be Hausdorff or compact for the g-topology. More specifically, [a,b] is not Hausdorff if [a,b]Cg, and it is not compact if there is a single t0[a,b)Dg (recall our Example 3.3, where we exhibited a g-continuous function on [-1,1] which is not bounded; this implies that [-1,1] is not pseudocompact for the g-topology, and pseudocompactness is equivalent to compactness in pseudometric spaces, see [7, Chapter 4, Theorem 8.17]).

We shall use a deep result by Fraňková [6, Proposition 2.3], which gives us a characterization of relatively compact sets in spaces of regulated functions. Since g-continuous functions need not be regulated at discontinuity points of g, we have to introduce assumptions to control the behavior of the functions at those points.

It is worth to point out that the first characterization of compact sets in spaces of regulated functions is due to Hildebrandt [8], but Franˇková’s characterization is somewhat clearer on what we need at discontinuity points.

First, we recall some definitions used in [6]. A family 𝒮 of regulated functions f:[a,b]N has uniform one-sided limits at a point t0[a,b] if, for every ε>0, there exists δ>0 such that for every f𝒮 and t[a,b], we have

f(t)-f(t0+)<εwhenever t(t0,t0+δ)

and

f(t)-f(t0-)<εwhenever t(t0-δ,t0).

We say that the family 𝒮 is equiregulated if it has uniform one-sided limits at every point t0[a,b].

Next we state Fraňková’s Theorem.

Theorem 4.1

Theorem 4.1 ([6, Proposition 2.3])

A family S of regulated functions f:[a,b]RN is relatively compact in the sup-norm topology if and only if it is equiregulated, there exists α>0 such that f(a)α for all fS, and, for any t[a,b], there exists a number γt>0 such that, for every fS, we have

(4.1)f(t)-f(t-)γtif t(a,b],
(4.2)f(t)-f(t+)γtif t[a,b).

We are already in a position to introduce sufficient conditions for families of g-continuous functions to be relatively compact.

Theorem 4.2

A family S of elements of Cg([a,b]) is relatively compact in the sup-norm topology if it satisfies the following conditions:

  1. The family 𝒮 is g-equicontinuous, i.e. for every ε>0 and every t[a,b], there exists δ>0 such that

    f(t)-f(s)<εfor all s[a,b] such that |g(t)-g(s)|<δ, and for all f𝒮.
  2. The set {f(a):f𝒮} is bounded.

  3. The family 𝒮 has uniform right-hand side limits in [a,b)Dg, i.e. for every t0[a,b)Dg and every ε>0, there exists δ>0 such that, for every f𝒮, we have

    f(t)-f(t0+)<εwhenever t(t0,t0+δ).
  4. The family 𝒮 has uniformly bounded jump discontinuities in [a,b)Dg, i.e. for every t[a,b)Dg, there exists γt>0 such that, for every f𝒮, we have

    f(t)-f(t+)γt.

Proof.

It suffices to prove that 𝒮 satisfies all the conditions in Fraňková’s theorem 4.1.

First, condition (ii) is equivalent to the existence of some α>0 such that f(a)α for all f𝒮. Second, for each t[a,b], we have (4.1)–(4.2) for the value γt given by condition (iv) because every f𝒮 is left-continuous at every t(a,b] and right-continuous at every t[a,b)Dg.

Now, it only remains to show that 𝒮 is equiregulated as a consequence of conditions (i) and (iii). Indeed, for each t0(a,b] and each ε>0, we use condition (i) to find some δ>0 such that

(4.3)f(t)-f(t0)<εfor all t[a,b] such that |g(t)-g(t0)|<δ, and for all f𝒮.

Since g is left-continuous at t0, there exists δ>0 such that |g(t)-g(t0)|<δ if t(t0-δ,t0), and then, (4.3) implies that

f(t)-f(t0-)=f(t)-f(t0)<εfor all t(t0-δ,t0), and for all f𝒮.

The proof that 𝒮 has uniform limits from the right at every t0[a,b)Dg is similar, using condition (i) and right-continuity of g-continuous functions at points which do not belong to Dg. Finally, condition (iii) ensures that 𝒮 has uniform limits from the right at every t0[a,b)Dg. ∎

We remark that the conditions in Theorem 4.2 are not necessary for relative compactness. To see it, consider a “family” 𝒮 consisting of only one bounded g-continuous function f:[a,b] which does not have a right-hand side limit at a point t0[a,b)Dg. The set 𝒮 is trivially compact, but it does not satisfy condition (iii) in Theorem 4.2. This raises an interesting problem: to find necessary and sufficient conditions for relative compactness of subsets of 𝒞g([a,b]).

5 Theoretical background on g-derivatives and integrals

Following [9], we define the derivative with respect to g (or g-derivative) of a real-valued real function f at a point t0Cg as follows, provided that the corresponding limit exists:

(5.1)fg(t0)=limtt0f(t)-f(t0)g(t)-g(t0)if t0Dg
(5.2)orfg(t0)=limtt0+f(t)-f(t0)g(t)-g(t0)if t0Dg,

where Cg and Dg are defined in (2.1) and (2.2), respectively. Notice that the g-derivative at a point t0Dg exists if and only if f(t0+) exists and (5.2) can be rewritten as

(5.3)fg(t0)=f(t0+)-f(t0)g(t0+)-g(t0).

We say that f is g-differentiable at t0 if fg(t0) exists, and we say that f is g-differentiable in a set AR when f is g-differentiable at every t0ACg.

There will be no need to define g-derivatives for the points in Cg because, according to [9, Proposition 2.5], we have μg(Cg)=0, where μg stands for the Lebesgue–Stieltjes measure induced by g. Roughly speaking, g-differentiable functions can be recovered by integrating their g-derivatives in the Stieltjes sense with respect to g, and this is why null sets with respect to the induced Lebesgue–Stieltjes measure play no role in this theory. More precisely, we have the following Fundamental Theorem of Calculus for the Lebesgue–Stieltjes integral [9, Theorem 5.4].

Theorem 5.1

Theorem 5.1 (Fundamental Theorem of Calculus for the Lebesgue–Stieltjes integral)

Let a,bR with a<b and F:[a,b]R. The following conditions are equivalent:

  1. The function F is absolutely continuous with respect to g (or g-absolutely continuous) according to the following definition. For each ε>0, there exists some δ>0 such that, for any family {(an,bn)}n=1m of pairwise disjoint open subintervals of [a,b], the inequality

    n=1m(g(bn)-g(an))<δ𝑖𝑚𝑝𝑙𝑖𝑒𝑠n=1m|F(bn)-F(an)|<ε.
  2. The function F fulfills the following properties:

    1. There exists Fg(t) for g-almost all t[a,b) (i.e., for all t except on a set of μg measure zero).

    2. Fgg1([a,b)), the set of Lebesgue–Stieltjes integrable functions with respect to μg.

    3. For each t[a,b], we have

      (5.4)F(t)=F(a)+[a,t)Fg(s)𝑑μg.

We remark that μg({t})=g(t+)-g(t)>0 when tDg (see [9, equation (6)]). In particular, if t[a,b)Dg, then (5.4) implies that

F(t+)-F(t)={t}Fg(s)𝑑μg=Fg(t)(g(t+)-g(t)),

which may be nonzero. Therefore, g-absolutely continuous functions need not be continuous at discontinuity points of g.

However, g-absolutely continuous functions have some nice properties.

Proposition 5.2

Proposition 5.2 ([9, Proposition 5.3])

If F is g-absolutely continuous on [a,b], then it has bounded variation and it is continuous from the left at every x[a,b). Moreover, F is continuous in [a,b]Dg, where Dg is the set of discontinuity points of g, and if g is constant on some (α,β)[a,b], then F is constant on (α,β) as well.

Absolutely continuous functions with respect to g have many properties in common with absolutely continuous functions in the usual sense. For instance, it is easy to prove that linear combinations of g-absolutely continuous functions are g-absolutely continuous, and the following two results are also straightforward generalizations of the same results in the classical context.

Proposition 5.3

Let f1:[a,b]R be g-absolutely continuous. Assume that f1([a,b])[c,d] for some c,dR, c<d, and let f2:[c,d]R satisfy a Lipschitz condition on [c,d]. Then, f2f1 is g-absolutely continuous on [a,b].

Proof.

Let L>0 be a Lipschitz constant for f2. We prove that f2f1 is g-absolutely continuous by checking directly the definition given in Theorem 5.1. For each ε>0, it suffices to take δ>0 given by the definition of g-absolute continuity of f1 with ε replaced by ε/L. Now, if {(an,bn)}n=1m is a family of nonoverlapping subintervals of [a,b] such that

n=1m(g(bn)-g(an))<δ,

then

n=1m|f2(f1(bn))-f2(f1(an))|Ln=1m|f1(bn)-f1(an)|<LεL=ε.
Proposition 5.4

If f1,f2 are g-absolutely continuous on [a,b], then so is their product.

Proof.

The product can be expressed as a linear combination of g-absolutely continuous functions, namely,

f1f2=12((f1-f2)2-f12-f22).

The function rr2 is locally Lipschitzian and therefore, Proposition 5.3 ensures that (f1-f2)2 and fi2, i=1,2, are g-absolutely continuous. ∎

We say that a vector-valued function F:[a,b]N is g-absolutely continuous on [a,b] if each of its components is g-absolutely continuous. We denote by 𝒜𝒞g([a,b]) the set of g-absolutely continuous functions on [a,b] with values in N.

The following proposition follows directly from Theorem 5.1 and [9, Proposition 5.3].

Proposition 5.5

The set ACg([a,b]) is included in BCg([a,b]).

Among relatively compact subsets in 𝒞g([a,b]), we find integrably bounded subets in 𝒜𝒞g([a,b]) with bounded initial value.

Proposition 5.6

Let SACg([a,b]) be such that {F(a):FS} is bounded. Assume that there exists a function hLg1([a,b)) such that

Fg(t)h(t)for g-almost all t[a,b), and for all F𝒮.

Then, S is relatively compact in BCg([a,b]).

Proof.

We claim that 𝒮 is g-equicontinuous. Indeed, let ε>0. Since hg1([a,b)), there exists δ>0 such that

Eh𝑑μg<εfor all E[a,b), g-measurable and such that μg(E)<δ.

Let t[a,b]. By Theorem 5.1, for τ(t,b] such that |g(t)-g(τ)|<δ, one has

F(τ)-F(t)=[t,τ)Fg(s)dμg[t,τ)h(s)dμg<εfor all F𝒮,

since |g(τ)-g(t)|=μg([t,τ)). Similarly, for τ[a,t) such that |g(t)-g(τ)|<δ,

F(t)-F(τ)<εfor all F𝒮.

Moreover, 𝒮 has uniform right-hand side limits in [a,b)Dg, since, for tDg, there exists δt>0 such that

μg((t,τ))=|g(τ)-g(t+)|<δwhenever τ(t,t+δt).

Thus,

F(τ)-F(t+)=(t,τ)Fg(s)dμg(t,τ)h(s)dμg<εwhenever τ(t,t+δt) and for all F𝒮.

Let us observe that 𝒮 has uniformly bounded jump discontinuities in [a,b)Dg, since for every tDg,

F(t+)-F(t)={t}Fg(s)dμg{t}h(s)dμgfor all F𝒮.

The conclusion follows from Theorem 4.2. ∎

It is our aim to establish the basic theory on existence and uniqueness of local solutions to (1.1), where f=f(t,x) is a N-valued function which is defined in a neighborhood of (t0,x0)N+1, and xg(t) is the g-derivative of each component of the unknown x=(x1,x2,,xN) at t.

Definition 5.7

A (Carathéodory) solution of (1.1) is a g-absolutely continuous function x:[t0,t0+T]N, with T>0, such that x(t0)=x0 and

xg(t)=f(t,x(t))for g-almost all t[t0,t0+T].

In order to study existence and uniqueness of solutions of (1.1), we can assume without loss of generality that g is continuous at the initial time t0 and, as a result, the g-measure of {t0} is zero. Indeed, if g were not continuous at t0, then we could replace g by g~, defined as g~(t)=g(t) for all t>t0, g~(t)=g(t0+) for tt0. Obviously, g~ is a left-continuous nondecreasing function which is continuous at t0. Moreover, the g and the g~-derivatives coincide at points t>t0 and, using (5.3), for any g-differentiable function h=h(t) at t0, we have

hg(t0)=h(t0+)-h(t0)g(t0+)-g(t0).

Now, it is easy to prove that if y=y(t) is a solution of

(5.5)yg~(t)=f(t,y(t))for g~-almost all t[t0,t0+T],  y(t0)=x0+f(t0,x0)(g(t0+)-g(t0)),

then x(t)=y(t) for t(t0,t0+T], x(t0)=x0, is a solution of (1.1). Conversely, every solution of (1.1), say x=x(t), t[t0,t0+T], yields a solution of (5.5) which is defined by y(t)=x(t) for all t(t0,t0+T] and y(t0)=x(t0+)=x0+f(t0,x0)(g(t0+)-g(t0)).

6 An exponential

In this section, we introduce an exponential function and we study its nice properties.

Definition 6.1

Let cg1([a,b)) be such that

(6.1)c(t)(g(t+)-g(t))>-1for every t[a,b)Dg

and

(6.2)t[a,b)Dg|log(1+c(t)(g(t+)-g(t)))|<.

We define ec(,a):[a,b](0,) by

(6.3)ec(t,a)=e[a,t)c~(s)𝑑μg,

where

(6.4)c~(t)={c(t)if t[a,b]Dg,log(1+c(t)(g(t+)-g(t)))g(t+)-g(t)if t[a,b)Dg.

Notice that conditions (6.1) and (6.2) “disappear” when g is continuous. We first observe that ec(,a) is well defined.

Lemma 6.2

Let c~ be as defined in (6.4). Then, c~Lg1([a,b)).

Proof.

It is clear that c~ is g-measurable since c~=c on [a,b)Dg and Dg is countable. Moreover,

[a,b)|c~(s)|dμg=[a,b)Dg|c~(s)|dμg+[a,b)Dg|c~(s)|dμg
=[a,b)Dg|c(s)|dμg+s[a,b)Dg|c~(s)|(g(s+)-g(s))
=[a,b)Dg|c(s)|dμg+s[a,b)Dg|log(1+c(s)(g(s+)-g(s)))|<,

because cg1([a,b)) and satisfies (6.2). ∎

Lemma 6.3

Let cLg1([a,b)) satisfy (6.1) and (6.2). Then, ec(,a) is g-absolutely continuous and solves the initial value problem

(6.5)xg(t)=c(t)x(t)for g-almost all t[a,b), x(a)=1.

Equivalently,

ec(t,a)=1+[a,t)c(s)ec(s,a)𝑑μgfor every t[a,b].

Proof.

From Lemma 6.2 and [9, Proposition 5.2 and Theorem 2.4], it follows that the map t[a,t)c~(s)𝑑μg is g-absolutely continuous on [a,b], and

([a,t)c~(s)𝑑μg)g=c~(t)g-almost everywhere in [a,b).

Proposition 5.3 guarantees that ec(,a)𝒜𝒞g([a,b]), and the chain rule for g-derivatives [9, Theorem 2.3] implies that

(ec(t,a))g=ec(t,a)([a,t)c~(s)𝑑μg)g=ec(t,a)c~(t)
(6.6)=c(t)ec(t,a)g-almost everywhere in [a,b)Dg.

Also,

(6.7)(ec(t,a))g=c(t)ec(t,a)for every tDg.

Indeed, for tDg,

(ec(t,a))g=limst+ec(s,a)-ec(t,a)g(s)-g(t)
=limst+e[a,s)c~(r)𝑑μg-e[a,t)c~(r)𝑑μgg(s)-g(t)
=limst+ec(t,a)(e{t}c~(r)𝑑μge(t,s)c~(r)𝑑μg-1)g(s)-g(t)
=limst+ec(t,a)(ec~(t)(g(t+)-g(t))e(t,s)c~(r)𝑑μg-1)g(s)-g(t)
=limst+ec(t,a)((1+c(t)(g(t+)-g(t)))e(t,s)c~(r)𝑑μg-1)g(s)-g(t)
=limst+ec(t,a)((c(t)(g(t+)-g(t))e(t,s)c~(r)𝑑μgg(s)-g(t))+(e(t,s)c~(r)𝑑μg-1g(s)-g(t)))=c(t)ec(t,a),

since

limst+e(t,s)c~(r)𝑑μg=1.

We deduce from the fact that ec(,a)𝒜𝒞g([a,b]) and from (6.6) and (6.7) that

ec(t,a)=ec(a,a)+[a,t)c(s)ec(s,a)𝑑μgfor every t[a,b].

Now, we study the case where c does not satisfy (6.1). Observe that if x is a solution of (6.5), then

x(t+)=(1+c(t)(g(t+)-g(t)))x(t)for all tTc-={t[a,b)Dg:1+c(t)(g(t+)-g(t))<0}.

So, for tTc- such that x(t)0, t is a point of discontinuity of x where there is a change of sign. There could be only a finite number of them as it is shown in the following result.

Lemma 6.4

Let cLg1([a,b)). Then, Tc- has finite cardinality.

Proof.

Since cg1([a,b)),

>cLg1tTc-|c(t)(g(t+)-g(t))|>tTc-1.

If Tc-={t1,,tk} with at1<t2<<tk, we can define e^c(,a):[a,b]{0} by

(6.8)e^c(t,a)={e[a,t)c^(s)𝑑μgif att1,(-1)ie[a,t)c^(s)𝑑μgif ti<tti+1,i=1,,k,

where tk+1=b and

(6.9)c^(t)={c(t)if t[a,b]Dg,log|1+c(t)(g(t+)-g(t))|g(t+)-g(t)if t[a,b)Dg.

The following result follows directly from Lemmas 6.3 and 6.4.

Lemma 6.5

Let cLg1([a,b)) satisfy

(6.10)c(t)(g(t+)-g(t))-1for every t[a,b)Dg

and

(6.11)t[a,b)Dg|log|1+c(t)(g(t+)-g(t))||<.

Then, e^c(,a) is g-absolutely continuous and solves the initial value problem (6.5). Equivalently,

e^c(t,a)=1+[a,t)c(s)e^c(s,a)𝑑μgfor every t[a,b].

Moreover, e^c(t,a)e^c(t+,a)<0 if and only if tTc-.

In the next result, we study the case where (6.10) is not satisfied.

Lemma 6.6

Let cLg1([a,b)) be such that Tc0={t[a,b)Dg:1+c(t)(g(t+)-g(t))=0}. Let also t0=minTc0a, and assume that

(6.12)t[a,t0)Dg|log|1+c(t)(g(t+)-g(t))||<.

Then, xc(,a):[a,b]R defined by

xc(t,a)={e^c(t,a)if tt0,0if t>t0,

is g-absolutely continuous and solves the initial value problem (6.5). Equivalently,

xc(t,a)=1+[a,t)c(s)xc(s,a)𝑑μgfor every t[a,b].

It is worth to mention that if Tc0, then arguing as in Lemma 6.4 permits to deduce that this set has finite cardinality.

Let us consider the following nonhomogeneous linear g-differential equation:

(6.13){xg(t)+c(t)x(t+)=h(t)for g-almost all t[a,b),x(a)=x0,

where x0, hg1([a,b)) and cg1([a,b)) satisfies (6.10) and (6.11). An explicit solution to this problem can be given thanks to the exponential e^c.

Proposition 6.7

Let x0R, hLg1([a,b)) Let cLg1([a,b)) satisfy (6.10) and (6.11). Then, xACg([a,b]) is a solution of (6.13) if and only if

x(t)=e^c-1(t,a)(x0+[a,t)e^c(s,a)h(s)𝑑μg)for every t[a,b],

where

(6.14)e^c-1(t,a)=1e^c(t,a).

In particular, the unique g-absolutely continuous solution of

(6.15)xg(t)=-c(t)x(t+)for g-almost all t[a,b), x(a)=x0,

is x(t)=x0e^c-1(t,a) for all t[a,b].

Proof.

First of all, observe that, for x𝒜𝒞g([a,b]), one has x()e^c(,a)𝒜𝒞g([a,b]) by virtue of Proposition 5.4 and Lemma 6.5, and

(6.16)(x(t)e^c(t,a))g=e^c(t,a)xg(t)+x(t+)c(t)e^c(t,a)for g-almost all t[a,b).

Indeed, from Lemma 6.5 and the rule for g-derivatives of products of functions, it follows that

(x(t)e^c(t,a))g=e^c(t,a)xg(t)+x(t)c(t)e^c(t,a)for g-almost all t[a,b)Dg,

and for tDg,

(e^c(t,a)x(t))g=limst+e^c(s,a)x(s)-e^c(t,a)x(t)g(s)-g(t)
=limst+e^c(t,a)(x(s)-x(t))+(e^c(s,a)-e^c(t,a))x(s)g(s)-g(t)
=e^c(t,a)xg(t)+x(t+)(e^c(t,a))g
=e^c(t,a)xg(t)+x(t+)c(t)e^c(t,a).

The conclusion follows from (6.16) and Theorem 5.1. ∎

Now, let us consider the following nonhomogeneous linear g-differential equation:

(6.17){xg(t)+d(t)x(t)=h(t)for g-almost all t[a,b),x(a)=x0,

where x0 and d,hg1([a,b)).

Proposition 6.8

Let x0R, hLg1([a,b)), and let dLg1([a,b)) satisfy

(6.18)d(t)(g(t+)-g(t))1for every t[a,b)Dg

and

(6.19)t[a,b)Dg|log|1-d(t)(g(t+)-g(t))||<.

Then, (6.17) has a unique solution xACg([a,b]).

Proof.

Observe that

xg(t)+d(t)x(t)=xg(t)+d(t)x(t+)+d(t)(x(t)-x(t+))
=(1-d(t)(g(t+)-g(t)))xg(t)+d(t)x(t+).

By (6.18), we deduce that x is a solution of (6.17) if and only if it is a solution of the equation

(6.20)xg(t)+d¯(t)x(t+)=h¯(t)g-almost everywhere in [a,b),x(a)=x0,

where

d¯(t)=d(t)1-d(t)(g(t+)-g(t))andh¯(t)=h(t)1-d(t)(g(t+)-g(t)).

Let us notice that d¯,h¯g1([a,b)). Indeed, A={t[a,b):d(t)(g(t+)-g(t))>1/2} has at most finite cardinality, since dg1([a,b)) and

>dLc1tA|d(t)(g(t+)-g(t))|>tA12.

So,

|d¯(t)|2|d(t)|and|h¯(t)|2|h(t)|for all t[a,b)A.

Observe that

1+d¯(t)(g(t+)-g(t))0for all t[a,b)Dg,

and

t[a,b)Dg|log|1+d¯(t)(g(t+)-g(t))||=t[a,b)Dg|log|1-d(t)(g(t+)-g(t))||.

It follows from Proposition 6.7 that (6.20) has a unique solution, and hence, so does (6.17). ∎

To conclude this section, let us remark that if dg1([a,b)) is such that

T-d0={t[a,b):d(t)(g(t+)-g(t))=1},

then, this set has finite cardinality by an argument analogous to the proof of Lemma 6.4. Observe that a solution x of (6.17) must verify

x(t+)=h(t)(g(t+)-g(t))for all tT-d0.

In particular, x(t+) does not depend on x(t) and d(t). Hence, for T-d0={t1,,tk} with at1<t2<<tk, if d satisfies

(6.21)t([a,b)Dg)T-d0|log|1-d(t)(g(t+)-g(t))||<,

then (6.17) has a unique solution. Indeed, the conclusion follows from Proposition 6.8 applied on [a,t1] and then, piece by piece on [ti,ti+1] using h(ti)(g(ti+)-g(ti)) as a new initial condition for i=1,,k.

7 Picard and Peano type existence results

In this section, we establish existence and uniqueness results to the system of g-differential equations

(7.1){xg(t)=f(t,x(t))for g-almost all t[t0,t0+τ),x(t0)=x0.

As mentioned at the end of Section 5, we may assume without loss of generality that g is continuous at t0.

A definition analogous to the classical notion of Carathéodory functions can be given in our context.

Definition 7.1

Let X be a nonempty subset of N. We say that f:[a,b]×XN is g-Carathéodory if it satisfies the following conditions:

  1. For every xX, f(,x) is g-measurable.

  2. For g-almost all t[a,b], f(t,) is continuous on X.

  3. For every r>0, there exists hrg1([a,b)) such that

    f(t,x)hr(t)for g-almost all t[a,b), and for all xX, xr.

It is a very well-known result that compositions f(,x()) are measurable when f is Carathéodory and x is g-measurable when g is the identity and μg is the Lebesgue measure. It remains true when we consider any Borel measure, such as μg, see [2, Chapter 1, Section 4]. We include a standard proof of the next result for the convenience of readers.

Lemma 7.2

Let X be a nonempty subset of RN and f:[a,b]×XRN a g-Carathéodory function. Then, for every xBCg([a,b]), the map f(,x()) is in Lg1([a,b)).

Proof.

Let x𝒞g([a,b]). We know from Corollary 3.5 that x is Borel-measurable and, as a result, it can be expressed as a pointwise limit of a sequence of Borel-measurable simple functions, say {αn}n. Since X is separable, we assume without loss of generality that αn(t)X for all t[a,b] and all n.

For each n, the composition f(,αn()) is g-measurable thanks to condition (i) in Definition 7.1 and the fact that αn is piecewise constant on a finite family of Borel measurable subsets of [a,b].

Since αnx pointwise in [a,b], we deduce from condition (ii) that

f(t,x(t))=limnf(t,αn(t))for g-almost all t[a,b],

which proves that f(,x()) is g-measurable as an almost everywhere pointwise limit of a sequence of g-measurable functions.

Finally, we deduce from condition (iii) that f(,x())g1([a,b)). ∎

We first establish a global existence and uniqueness result.

Theorem 7.3

Let f:[t0,t0+T]×RNRN satisfy the following conditions:

  1. For every xN, f(,x) is g-measurable.

  2. f(,x0)g1([t0,t0+T)).

  3. There exists Lg1([t0,t0+T),[0,)) such that, for g-almost all t[t0,t0+T) and every x,yN, we have

    f(t,x)-f(t,y)L(t)x-y.

Then, (7.1) has a unique solution defined on the interval [t0,t0+T].

Proof.

Let us define F:𝒞g([t0,t0+T])𝒞g([t0,t0+T]) by

F(x)(t)=x0+[t0,t)f(s,x(s))𝑑μg.

Observe that the assumptions imply that f is g-Carathéodory. So, F is well defined. We consider the following norm on 𝒞g([t0,t0+T]):

xL=supt[t0,t0+T]x(t)eL-1(t,t0),

where eL-1(,t0)=1/eL(,t0) and ec(,a) is defined in (6.3). It is easy to see that this norm is equivalent to the norm 0. Hence, (𝒞g([t0,t0+T]),L) is a Banach space.

We claim that the function F is a contraction. Indeed, using Lemma 6.3 and the fact that eL(t0,t0)=1 since g is continuous at t0, we deduce, for every x,y𝒞g([t0,t0+T]), that

F(x)-F(y)L=supt[t0,t0+T]eL-1(t,t0)[t0,t)f(s,x(s))-f(s,y(s))dμg
supt[t0,t0+T]eL-1(t,t0)[t0,t)L(s)eL(s,t0)eL-1(s,t0)x(s)-y(s)𝑑μg
x-yLsupt[t0,t0+T]eL-1(t,t0)[t0,t)L(s)eL(s,t0)𝑑μg
=x-yLsupt[t0,t0+T]eL-1(t,t0)(eL(t,t0)-1)
(1-eL-1(t0+T,t0))x-yL.

The Banach contraction principle implies that F has a unique fixed point, and hence, (7.1) has a unique solution. ∎

Here is a local existence and uniqueness result.

Theorem 7.4

Let r>0 and f:[t0,t0+T]×B(x0,r)¯RN satisfying the following conditions:

  1. For every xB(x0,r)¯, f(,x) is g-measurable.

  2. f(,x0)g1([t0,t0+T)).

  3. There exists Lg1([t0,t0+T),[0,)) such that for g-almost all t[t0,t0+T) and every x,yB(x0,r)¯, we have

    f(t,x)-f(t,y)L(t)x-y.

Then, there exists τ(0,T] such that (7.1) has a unique solution defined on the interval [t0,t0+τ].

Proof.

The assumptions (ii) and (iii) imply that there exists Mg1([t0,t0+T),[0,)) such that

(7.2)f(t,x)M(t) for g-almost all t[t0,t0+T) and all xB(x0,r)¯.

Fix τ(0,T] such that

(7.3)[t0,t0+τ)M(s)𝑑μgr.

Let us define the closed subset

X={x𝒞g([t0,t0+τ]):x(t)-x0r for all t[t0,t0+τ]},

and F:XX by

F(x)(t)=x0+[t0,t)f(s,x(s))𝑑μg.

The inequalities (7.2) and (7.3) imply that F is well defined.

As in the proof of the previous theorem, if we endow 𝒞g([t0,t0+τ]) with the equivalent norm

xL=supt[t0,t0+τ]x(t)eL-1(t,t0),

we can show that F is a contraction. The conclusion follows from the Banach contraction principle. ∎

A Peano type existence result can also be obtained for g-differential equations.

Theorem 7.5

Let r>0 and f:[t0,t0+T]×B(x0,r)¯RN a g-Carathéodory function. Then, there exists τ(0,T] such that (7.1) has a solution defined on the interval [t0,t0+τ].

Proof.

Since f is g-Carathéodory, there exists hkg1([t0,t0+T),[0,)) with k=r+x0 such that

(7.4)f(t,x)hk(t) for g-almost all t[t0,t0+T) and all xB(x0,r)¯.

We fix τ(0,T] such that

[t0,t0+τ)hk(s)𝑑μgr.

Let us define X and F:XX as in the proof of the previous theorem. The set X is a nonempty, closed, convex subset of 𝒞g([t0,t0+τ]). One can deduce by means of (7.4) and Proposition 5.6 that F is compact. Hence, Schauder’s theorem guarantees that F has at least one fixed point, which is a solution of (7.1). ∎

Finally, we show that the results in this section imply the corresponding results in the classical sense of “everywhere” g-differentiable solutions when we have some g-continuity.

Proposition 7.6

Let x:[t0,t0+τ]R be a solution of (7.1). If f(,x()) is g-continuous on [t0,t0+τ], then

xg(t)=f(t,x(t))for all t[t0,t0+τ)Cg.

In particular, xg is g-continuous on [t0,t0+τ)Cg.

Proof.

Definition 5.7 ensures that xg(t)=f(t,x(t)) for all t[t0,t0+τ)Dg because each singleton {t} has positive g-measure if tDg. So, it only remains to show that xg(t)=f(t,x(t)) for all t[t0,t0+τ)(CgDg). Since x is a Carathéodory solution, we know that

x(t)=x0+[t0,t)f(s,x(s))𝑑μgfor all t[t0,t0+τ].

Let us fix a point t[t0,t0+τ)(CgDg). Since g is not constant on any neighborhood of t, we may have g(s)<g(t) for all s<t, g(s)>g(t) for all s>t, or both. If g(s)<g(t) for all s<t, and t>t0, we have, for s[t0,t), that

(g(t)-g(s))infsr<tf(r,x(r))=μg([s,t))infsr<tf(r,x(r))
[s,t)f(r,x(r))𝑑μg
=x(t)-x(s)
μg([s,t))supsr<tf(r,x(r))
=(g(t)-g(s))supsr<tf(r,x(r)).

Hence, there exists

limst-x(s)-x(t)g(s)-g(t)=f(t,x(t)),

because f(,x()) is continuous at t (see Proposition 3.2).

If g(s)=g(t) on some [t,t+δ], δ>0, then the previous limit is xg(t) and the proof is complete. If, on the other hand, g(s)>g(t) for all s>t, then we use a similar argument to prove that the g-derivative of x from the right at t exists and is equal to f(t,x(t)) too.

The case g(s)=g(t) on some [t-δ,t], δ>0, can be treated in a similar way and we omit the details. ∎

As a consequence of the previous result, we obtain easily Peano’s theorem for g-differential equations. To do it, we introduce the following definition.

Definition 7.7

Let f:AN+1N and let t0, x0N be such that (t0,x0)A. We shall say that f is (g×IN)-continuous at (t0,x0) if for every ε>0, there exists δ>0 such that

[(t,x)A,|g(t)-g(t0)|<δ,x-x0<δ]f(t,x)-f(t0,x0)<ε.

We shall say that f is (g×IN)-continuous in A if it is (g×IN)-continuous at every (t0,x0)A.

Notice that (g×IN)-continuity reduces to continuity in the usual sense when g is the identity. Notice also that (g×IN)-continuity on a set AN+1 is equivalent to continuity with respect to the product topology in A, τg×τ, where τg is the g-topology in and τ is the usual topology in N.

We are now in a position to establish the classical form of Theorem 7.5.

Theorem 7.8

Let f:[t0,t0+T]×B(x0,r)¯RN be a (g×IRN)-continuous function. If there exists a function hLg1([t0,t0+T),[0,)) such that

f(t,x)h(t)for g-almost all t[t0,t0+T), and for all xB(x0,r)¯,

then there exists τ(0,T] such that (7.1) has a solution which satisfies the g-differential equation at every t[t0,t0+τ)Cg, and its g-derivative is g-continuous on [t0,t0+τ)Cg.

Proof.

Recall that Theorem 7.5 guarantees that (7.1) has at least one g-absolutely continuous solution x:[t0,t0+τ]N. In particular, x is g-continuous on [t0,t0+τ]. Since (t,x(t))[t0,t0+T]×B(x0,r)¯ for all t[t0,t0+τ], it is just routine to prove that f(,x()) is g-continuous on [t0,t0+τ]. Now, Proposition 7.6 implies the result. ∎

8 Particular cases of g-differential equations

It is shown in [9] that g-differential equations contain as particular cases first-order dynamic equations on time scales and ordinary differential equations with impulses at fixed times. This section has a twofold purpose: first, to describe with more detail how to turn dynamic and impulsive equations into g-differential equations, and, second, to introduce a new way for transforming easily most dynamic equations into ordinary differential equations with impulses, which are much simpler. Our approach is different to that of Akhmet and Turan [1], who have shown that some dynamic equations can be studied as ODEs with impulses to obtain results on existence and stability of periodic and almost periodic solutions. However, the time scales considered in [1] can only be of the form

𝕋=i=-[t2i-1,t2i],

where {ti} is a sequence of real numbers such that ti<ti+1, |ti|+ as |i|, and

-(t2i-t2i-1)=,(t2i-t2i-1)=.

8.1 “Typical” first-order dynamic equations are just ODEs with impulses

Let 𝕋 be a time scale, i.e., a nonempty closed subset of the reals, and consider the first-order dynamic equation

(8.1)xΔ(t)=f(t,x(t)),t𝕋,

where f:𝕋×.

Equation (8.1) can be an ordinary differential equation, a difference equation, a q-difference equation, or a sort of mix of all of them depending on the time scale 𝕋 that we use. Readers are referred to [3] for the theory of time scales, Δ-derivatives, dynamic equations and their applications. In the sequel, we need the forward jump operator, which is defined by σ(t)=inf{s𝕋:t<s} for all t𝕋, t<sup𝕋. We say that a point t𝕋 is right-scattered when t<σ(t). Remember that the set

(8.2)J={t𝕋:t<σ(t)}

is at most countable; see [4, Lemma 3.1].

Next, we present a way to reduce most dynamic equations to ODEs with impulses[2] at points of J.

Let us say that the time scale 𝕋 is typical when

Examples of typical time scales are , , q (q>0), and many countable unions of closed intervals and singletons. Real world applications of the theory of time scales involve typical time scales only. A Cantor set with positive Lebesgue measure is an example of a time scale which is not typical because in that case 𝕋𝕋̊=𝕋 and it has positive measure (however, Cantor’s ternary set is a typical time scale).

Proposition 8.1

Assume that the time scale T is typical, let x:TR be a solution of (8.1), and let a,bT with a<b. Define y:[a,b]R as follows:

y(t)={x(t)if t𝕋,x(σ(s))if s<tσ(s) for some s𝕋.

Then, y is a solution of the following ordinary differential equation with impulses:

(8.3)y(t)=F(t,y(t))for almost all t[a,b]J,
(8.4)y(t+)=It(y(t))for all t[a,b)J,

where J is as in (8.2),

F(t,y)={f(t,y)if t𝕋,0if t𝕋,

and for each tJ,

It(y)=y+(σ(t)-t)f(t,y)for all y.

Conversely, if y:[a,b]R is a solution of (8.3)–(8.4), and it is left-continuous on (a,b], then its restriction x=y|[a,b]T solves (8.1) at every t[a,b)J and at almost every t[a,b]TJ. In particular, if TT̊=, then

xΔ(t)=f(t,x(t))for all t[a,b)𝕋.

Proof.

For every t[a,b)J, we have y(r)=x(σ(t)) for all r(t,σ(t)]. Hence,

f(t,x(t))=xΔ(t)=x(σ(t))-x(t)σ(t)-t=y(t+)-y(t)σ(t)-t,

which implies (8.4).

For every t𝕋̊, the Δ-derivative of x is just the usual one and, moreover, x and y agree on a neighborhood of t. Therefore,

f(t,x(t))=xΔ(t)=x(t)=y(t)for all t𝕋̊.

The remaining points in [a,b] are contained either in 𝕋𝕋̊, which has zero measure by our assumptions, or in the set [a,b]𝕋=(a,b)𝕋=t[a,b)J(t,σ(t)), where y has zero derivative everywhere because it is constant on each component (t,σ(t)), t[a,b)J. The proof of (8.3) is complete.

The proof of the converse is similar and we omit it (the left-continuity of y is needed to prove the identity y(t+)=y(σ(t)) for every t[a,b)J). ∎

8.2 ODEs with impulses are Stieltjes differential equations

Let J={tk:k} be a countable subset of an interval [a,b) and consider an impulsive ODE

(8.5)x(t)=f(t,x(t))for almost all t[a,b]J,
(8.6)x(t+)=It(x(t)),tJ.

Usually, J is assumed to be a finite set, but we can easily generalize this to

(8.7)

where J stands for the set of accumulation points of J. (Notice that this assumption is satisfied by the set J in the previous section, because J𝕋𝕋̊ and the latter was assumed to have zero measure.)

Define a nondecreasing and left-continuous function g: by

g(t)=t+{k:tk<t}2-kfor all t,

where we mean that the sum takes the value zero when {k:tk<t}=. Clearly, g is strictly increasing and, for each fixed t and s<t, we have

0<g(t)-g(s)=t-s+{k:stk<t}2-k,

which tends to zero as st-. On the other hand, at every tkJ, we have

g(tk+)-g(tk)=2-k,

and g is continuous at every tJ.

We observe that xg(t)=x(t) if tJJ. In particular, xg(t)=x(t) for almost all t[a,b]J thanks to our assumption (8.7).

Now, it is easy to check that x is a solution of (8.5)–(8.6) if, and only if, it is a solution of the Stieltjes equation

xg(t)=F(t,x(t))for all tJ and almost all t[a,b]J,

where

F(t,x)={f(t,x)if tJ,x+(g(t+)-g(t))It(x)if tJ.

8.3 First-order dynamic equations are Stieltjes differential equations

We close this section by showing that equation (8.1) is a particular case of a Stieltjes equation and no matter how 𝕋 is. To do so, we fix a,b𝕋, a<b, and we introduce the Slavík function corresponding to [a,b]𝕋, see [10], which is the function g:𝕋 defined as g(t)=a for all t<a,

g(t)=inf{s𝕋:st}for all t(a,b],

and g(t)=b for t>b. Clearly, the Slavík function g is nondecreasing and left-continuous. Another fundamental property of g is that g(t)=t for all t[a,b]𝕋.

Now, if x(t) is a solution of (8.1) on [a,b]𝕋, we construct its Slavík extension as the function y: defined by

(8.8)y(t)=x(g(t))for all t,

and it can be proved, see [9, Theorem 3.1], that if x is continuous from the left at every right-scattered point of [a,b)𝕋, then

(8.9)yg(t)=xΔ(t)=f(t,x(t))=f(t,y(t))for all t[a,b)𝕋.

The g-derivative of y(t) cannot be computed at points t𝕋 because g is constant about these points. This is not really a problem because the set 𝕋 has zero Lebesgue–Stieltjes measure with respect to g. In particular, yg(t)=f(t,y(t)) for g-almost all t[a,b).

Conversely, if y:[a,b] is a g-continuous solution of (8.9), then its restriction x=y|[a,b]𝕋 solves (8.1) on [a,b)𝕋. To prove it, we consider separately two cases.

Case 1. For every t[a,b)J, we have, thanks to (1) and (3) in Proposition 3.2, that

f(t,y(t))=yg(t)=y(t+)-y(t)g(t+)-g(t)=y(σ(t))-y(t)g(t+)-g(t)=x(σ(t))-x(t)σ(t)-t=xΔ(t).

Case 2. Let t[a,b)𝕋, tJ. We have to prove that there exists xΔ(t) which coincides with f(t,x(t)). In this case, this is equivalent to prove that the following limit exists:

(8.10)limst,s𝕋x(s)-x(t)s-t=f(t,x(t)).

Since t is right-dense, we have g(s)>g(t) whenever s[a,b]𝕋 and s>t. Hence, for every s(t,b]𝕋, we have

x(s)-x(t)s-t=y(s)-y(t)g(s)-g(t),

which implies that

(8.11)limst+,s𝕋x(s)-x(t)s-t=limst+,s𝕋y(s)-y(t)g(s)-g(t)=yg(t)=f(t,y(t))=f(t,x(t)).

If t is left-scattered, then, (8.11) is equivalent to (8.10) and we are done. If, on the contrary, t is left-dense, then we adjust what we did for (8.11) to deduce that

(8.12)limst-,s𝕋x(s)-x(t)s-t=f(t,x(t)).

Now, (8.11) and (8.12) imply (8.10).

9 Applications

In this section, we present two applications to convince the reader that some models describing chemical or physical phenomenons can be improved by considering the theory of g-differential equations. Moreover, we show that it could be convenient to model some discontinuous phenomenons by considering a nondecreasing function g having some discontinuities.

9.1 Evaporating water

An open top cylindrical tank contains an initial amount of water reaching a level of x0 meters high. We want to design a mathematical model for x(t), the water height at every time t>0, under the sole assumption that water vanishes due to evaporation.

Undoubtedly, the simplest model (probably rather unrealiable) is

(9.1)x(t)=kfor all t0, x(0)=x0,

for some constant k<0. We can refine this model by the usual way, i.e., replacing the right-hand side term in the differential equation by some function f(t,x(t)). However, it is the aim of this section to convince readers that we can also improve on the model by considering Stieltjes derivatives instead of usual ones and therefore we leave the simple constant term in the right-hand side of the equation.

A reasonable yet still simple assumption is that evaporation only occurs during the day due to sunlight and that water level does not change during the night. In this case, we have to distinguish between day and night times, and we do it by identifying days with the intervals [2k,2k+1] for k=0,1,2,, and nights with [2k+1,2k+2], k=0,1,2,. The idea is to differentiate with respect to a function g which is constant on night intervals [2k+1,2k+2] (so that they become g-null sets), and which assigns greater measure to middays than to the first and last hours of each day, when evaporation is not so strong. A possible choice is

g(t)=0tmax{sin(πs),0}𝑑sfor all t,

which is constant on night intervals and has maximum slopes at middays (see its graph in Figure 1). We remark that we have chosen this function g for simplicity and that more accurate functions could be nondifferentiable or even discontinuous, and we still could use them in our theory.

Figure 1 Graph of g.
Figure 1

Graph of g.

Figure 2 Solution of (9.2) for x0=1${x_{0}=1}$ and k=-1/10${k=-1/10}$.
Figure 2

Solution of (9.2) for x0=1 and k=-1/10.

Now, replacing the derivative by the g-derivative in (9.1) gives

(9.2)xg(t)=kfor all t0 (tCg), x(0)=x0,

whose unique solution is

x(t)=x0+[0,t)k𝑑μg=x0+kg(t)for all t0.

See the plot of the graph of x(t) in Figure 2.

Let us refine the model a little bit further. Since tank walls produce shadows over the water, then the evaporation speed is likely to be a decreasing function of x(t). The simplest model for describing this situation is the following linear model:

(9.3)xg(t)=c(t)x(t)for all t0 (tCg), x(0)=x0,

for some function c(t)0. Now, the solution of (9.3) is given by

(9.4)x(t)=x0ec(t,0)=x0e[0,t)c(s)𝑑μg.

In particular, when c(t)=k<0, we get

x(t)=x0ekg(t),

and we include its graph in Figure 3 for x0=1 and k=-1/5.

Even more complicated, assume also that every morning we refill the tank up to a level which is proportional to that before refilling. According to our previous distinction between days and nights, we identify “mornings” with times tk=2k for k and we want solutions to satisfy

(9.5)x(2k+)=λkx(2k)for some λk>1.

To incorporate this new feature in (9.3), it suffices to redefine our function g on [2,+) as

g(t)=max{k:2kt}+0tmax{sin(πs),0}𝑑sfor all t2,

which has jump discontinuities at refilling moments (we have constructed g so that g(2k+)-g(2k)=1 for all k, which simplifies notation, but we only need g(2k+)-g(2k)>0).

Figure 3 Solution of (9.3) for x0=1${x_{0}=1}$ and c⁢(t)=-1/5${c(t)=-1/5}$.
Figure 3

Solution of (9.3) for x0=1 and c(t)=-1/5.

Figure 4 Graph of g for a model of evaporation with refilling at times t=2⁢k${t=2k}$.
Figure 4

Graph of g for a model of evaporation with refilling at times t=2k.

In this case, we have to modify also the right-hand side in the differential equation of (9.3), because it implies

x(2k+)-x(2k)=c(2k)x(2k),

and we want (9.5). Therefore, we must define c(2k)=λk-1 for all k.

Once more, we use the results in Section 6 on linear equations. The solution of (9.3) with the new derivator is

(9.6)x(t)=x0ec(t,0)=x0e[0,t)c~(s)𝑑μg.

As an example, take

c(t)={110if t=2k for some k,-15otherwise.

In this case we have

[0,t)c~(s)𝑑μg=[0,t){2k:k}c~(s)𝑑μg+[0,t){2k:k}c~(s)𝑑μg
=-15μg([0,t){2k:k})+2k<tlog(1+c(2k))
=-15(g(t)-max{k:2k<t})+max{k:2k<t}log(1+110),

where we mean

max{k:2k<t}=0for t2.

Summing up, the solution given in (9.6) reduces in this case to

(9.7)x(t)=exp(-15(g(t)-max{k:2k<t}))(1+110)max{k:2k<t}.

Remark 9.1

Since g is piecewise continuously differentiable, problem (9.3) can be rewritten as an initial value problem for an ordinary differential equation with impulses. We have chosen such a well-behaved derivator g only for simplicity. Remember that there exist nondecreasing functions g (even strictly increasing and continuous) having zero derivative almost everywhere [11, Theorem 4.54], and in those cases the g-differential equation is not merely an ordinary differential equation with impulses.

Figure 5 Graph of the solution of (9.7).
Figure 5

Graph of the solution of (9.7).

9.2 Dissolving salt in water

Consider a water tank with a great amount of solid salt lying at the bottom. Let x(t) denote grams of salt dissolved in the water after t0 minutes. For the sake of simplicity, we suppose that there is a mechanical device which keeps water circulating so fast that we can assume that at every minute the mixture is saturated, i.e., no more salt can be dissolved. In this situation, x(t) is a constant function and the problem is rather uninteresting.

It is well known that we can speed up many dissolution processes by heating the mixtures. Indeed, empirical evidence shows that the maximum amount of salt which can be dissolved in a fixed volume of water is an increasing nonlinear function of the temperature (this is well documented in many easily accesible sources). Therefore, we consider now the more interesting problem of describing our function x(t) under the additional assumption that temperature varies as a nondecreasing function of time, which we denote by g:[0,+)[10,90], i.e., g(t) is the temperature in Celsius degrees at minute t.

In this case, a wrong assumption is that the variation of x(t) is proportional to the variation of time, i.e.,

(9.8)x(t+h)-x(t)=ch(t0,h>0)

for some c. To see that (9.8) is a wrong assumption we notice that if the temperature does not change during the time interval [t,t+h], then x should be constant on [t,t+h], and therefore (9.8) implies c=0. As a result, x should be constant at every moment but this is not what really happens if there is some variation of temperature. This suggests replacing the constant c by some function, but there is a better way to study the variation of x. Indeed, much more reasonable is to assume that the variation of x(t) on each time interval [t,t+h] is proportional to the variation of temperature on the same interval, i.e., for any t0 and h>0 we have

(9.9)x(t+h)-x(t)=c[g(t+h)-g(t)]

for some c. Now, the definition of g-derivative readily yields

(9.10)xg(t)=cfor all t0, tCg.

Assuming that c is constant is too simplistic, both in (9.8) and (9.9). As we mentioned before, the maximum amount of salt which can be dissolved is a nonlinear function of the temperature, so we should revise (9.9) and replace the constant c by, at least, something like c(g(t)), or even c(g(t),x(t)). In doing so, we end up with the g-differential equation

xg(t)=c(g(t),x(t))for all t0, tCg.

Since we suppose that g is a known function, then we can simply write f(t,x)=c(g(t),x), and we have

(9.11)xg(t)=f(t,x(t))for all t0, tCg.

Notice that if c is a continuous function, then f is a (g×I)-continuous function. This shows how (g×IN)-continuity naturally arises in applications of g-differential equations. Moreover, if there exists L such that |f(t,x)-f(t,y)|L|x-y| for all x,y, Theorem 7.3 ensures the existence of a unique solution.

It can be argued that one can use the Mean Value Theorem in (9.9) to get (9.8) with some function c, obtaining in this way an ordinary differential equation. However the function g need not be differentiable and it even can be discontinuous, thus making it impossible to use the Mean Value Theorem. Can temperature have jump discontinuities with respect to time? Maybe not, but sometimes it is convenient to consider discontinuities in practice. Imagine, for instance, that we pour a bucket of hot water into the tank. If the amount of added water is big and it is much hotter than the water in the tank, then assuming that temperature jumps instantaneously to a higher level is easier than using a continuous function g with a very big slope in a very small time interval. In such a model, we may expect the concentration to be discontinuous in the classical sense. However, it would be continuous with respect to g, since it is a solution to a g-differential equation.

Funding statement: Marlène Frigon was partially supported by NSERC Canada. Rodrigo López Pouso was partially supported by Ministerio de Economía y Competitividad, Spain, and FEDER projects MTM2010-15314 and MTM2013-43014-P.

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Received: 2015-11-10
Accepted: 2016-2-1
Published Online: 2016-3-22
Published in Print: 2017-2-1

© 2017 by De Gruyter

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