J. Differential Equations 255 (2013) 3098–3126

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Journal of Differential Equations

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Non-periodic averaging principles for measure functionaldifferential equations and functional dynamic equations ontime scales involving impulses

M. Federson a,1, J.G. Mesquita b,∗,2

a Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-970São Carlos, SP, Brazilb Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto, Universidade de São Paulo,14040-901, Ribeirão Preto, SP, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 February 2012Revised 30 March 2013Available online 13 August 2013

We present a non-periodic averaging principle for measure func-tional differential equations and, using the correspondence be-tween solutions of measure functional differential equations andsolutions of functional dynamic equations on time scales (see Fed-erson et al., 2012 [8]), we obtain a non-periodic averaging resultfor functional dynamic equations on time scales. Moreover, usingthe relation between measure functional differential equations andimpulsive measure functional differential equations, we get a non-periodic averaging theorem for these equations. Also, it is a knownfact that we can relate impulsive measure functional differentialequations and impulsive functional dynamic equations on timescales (see Federson et al., 2013 [9]). Therefore, applying this cor-respondence to our averaging principle, we obtain a non-periodicaveraging theorem for impulsive functional dynamic equations ontime scales.

© 2013 Elsevier Inc. All rights reserved.

* Corresponding author.E-mail addresses: federson@icmc.usp.br (M. Federson), jgmesquita@ffclrp.usp.br (J.G. Mesquita).

1 Supported by FAPESP, grant 2010/09139-3, and by CNPq, grant 304646/2008-3.2 Supported by FAPESP, grant 2010/12673-1, and by CAPES, grant 6829-10-4.

0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jde.2013.07.026

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3099

1. Introduction

Consider the ordinary differential equation

{x′(t) = ε f

(x(t), t

) + ε2 g(x(t), t, ε

),

x(t0) = x0,(1.1)

where ε > 0 is a small parameter. Consider, also, the autonomous averaged differential equation

{y′(t) = ε f0

(y(t)

),

y(t0) = x0,(1.2)

where

f0(y) = limT →∞

1

T

t0+T∫t0

f (y, t)dt.

An averaging principle for (1.1) establishes conditions under which one can obtain an approxima-tion between the solutions of problem (1.1) and the solutions of the averaged problem (1.2). Therefore,averaging principles are powerful tools in the study of perturbations of differential equations.

The first justifications of non-periodic averaging principles for nonlinear system were presented byN.N. Bogolyubov and A. Mitropolskii [3], and by N.N. Bogolyubov and N.N. Krylov [2]. Their results areknow as KBM averaging method.

Throughout the years, several authors have published papers on averaging methods for differentkinds of differential systems. In the 60’s, authors as, for instance, V.I. Fodcuk [10], A. Halanay [11],J.K. Hale [12], G.N. Medvedev [22] and G.N. Medvedev et al. [23] developed methods of averaging forcertain functional differential equations (we write FDEs, for short), with a small parameter, approxi-mating them by autonomous ordinary differential equations (ODEs).

It was only in the 70’s that the investigations on averaging for FDEs indicated that the classic ap-proximation of solutions by solutions of an autonomous ODE could be replaced by an approximationby solutions of an autonomous FDE. In this direction, we can mention the papers by V. Strygin [28]and by B. Lehman and S.P. Weibel [20]. See also [17], [18] and [19]. In the late 80’s, D.D. Bainov andS.D. Milusheva [1] considered an FDE of neutral type with impulses, approximating it by autonomousODEs. Recently, in [15] and [16], the authors stated averaging results for FDEs employing the tools ofnon-standard analysis. Also, based on such papers, the authors of [6] improved the results from [15]and [16] using standard analysis though. After that, non-periodic averaging result for impulsive func-tional differential equations was obtained (see [7]).

While the theory of averaging for functional differential equations is well developed, the literatureinvolving averaging principles for measure functional differential equations is scarce. As a matter offact, we do not know any result in this direction. Also, because the theory of functional dynamicequations on time scales is very recent (see [8]), the literature concerning averaging principles forthese equations is also scarce.

It is known that one can relate measure functional differential equations to functional dynamicequations on time scales and, also, to impulsive RFDEs and functional dynamic equations on timescales. See [8] and [9]. Due to these relations, it is possible to investigate the properties of functionaldynamic equations on time scales with or without impulses and through the analysis of the propertiesof measure functional differential equations. Our aim in this paper is, therefore, to obtain an averagingprinciple for measure functional differential equations and then, using some results from [8] and [9],we obtain an averaging principle for functional dynamic equations on time scales and for impulsivefunctional dynamic equations on time scales.

3100 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

We focus our attention on a measure functional differential equation of the form⎧⎪⎪⎨⎪⎪⎩ x(t) = φ(t0) + ε

t∫t0

f (xs, s)dh1(s) + ε2

t∫t0

g(xs, s, ε)dh2(s), t � t0,

xt0 = φ,

(1.3)

where the initial condition is a left-continuous and regulated function and the integrals are inthe sense of Perron–Stieltjes with respect to nondecreasing functions h1 and h2. Here, we obtainan averaging principle for (1.3) and then, using some results from [8] and [9], we also obtain an aver-aging principle for the following three systems:⎧⎪⎪⎪⎨⎪⎪⎪⎩

y(t) = φ(t0) + ε

t∫t0

f (ys, s)dh(s) + ε2

t∫t0

g(ys, s, ε)dh(s) + ε∑k∈N,

t0�tk<t

Ik(

y(tk)),

yt0 = φ,

(1.4)

⎧⎪⎪⎨⎪⎪⎩ x(t) = φ(t0) + ε

t∫t0

f(xσ

s , s)�s + ε2

t∫t0

g(xσ

s , s, ε)�s,

x(t) = φ(t), t ∈ [t0 − r, t0]T(1.5)

and ⎧⎪⎪⎨⎪⎪⎩x(t) = x(t0) + ε

t∫t0

f(xσ

s , s)�s + ε2

t∫t0

g(xσ

s , s, ε)�s + ε

∑k∈N

t0�tk<t

Ik(x(tk)

),

x(t) = φ(t), t ∈ [t0 − r, t0]T,

(1.6)

where xσs = (xσ )s , where σ (t) = inf{s ∈ T: s � t}.

In particular, the results we present here remain true when we consider Lebesgue–Stieltjes inte-gration instead of Perron–Stieltjes integration.

The present paper is organized as follows. The second section is concerned with the Perron–Stieltjes integration theory, where we include the basic concepts and a few results. In the thirdsection, we present an averaging principle for Eq. (1.3), using some ideas borrowed from [6]. Inthe fourth section, we present an averaging principle for impulsive measure functional differentialequations, using the correspondence between these equations and measure functional differentialequations (see [9]). The fifth section is devoted to basic facts of the theory of dynamic equations ontime scales. In the sixth section, we present an averaging principle for functional dynamic equationson time scales. Finally, in the final section, we present an averaging principle for impulsive functionaldynamic equations on time scales.

2. Perron–Stieltjes integral

In this section, we introduce some definitions and properties about the Perron–Stieltjes integral.Let [a,b] be a compact interval of R. A division of [a,b] is any finite set of closed non-overlapping

intervals [ti−1, ti] ⊂ [a,b] such that a = t0 < t1 < · · · < tn = b and⋃n

i=0[ti−1, ti] = [a,b]. Moreover,ξi ∈ [ti−1, ti] for each i, then we write d = (ξi, [ti−1, ti]) ∈ T D[a,b] and we say that d is a tagged divisionof [a,b], with tags ξi of each subinterval [ti−1, ti].

A gauge of [a,b] is any function δ : [a,b] → (0,∞). Given a gauge δ of [a,b], d = (ξi, [ti−1, ti]) ∈T D[a,b] is called δ-fine, whenever [ti−1, ti] ⊂ {t ∈ [a,b]; |t − ξi | < δ(ξi)}, for every i, where | · | denotesany norm in R.

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3101

Let ‖ · ‖ denote any norm in Rn and g : [a,b] → R be a given function. We define the Perron–

Stieltjes integral of a function f : [a,b] → Rn with respect to g .

Definition 2.1. Let g : [a,b] → R be a given function. We say that a function f : [a,b] → Rn is Perron

integrable with respect to g and we say that I = ∫ ba f (t)dg(t) ∈ R

n is its Perron–Stieltjes integral, if forevery ε > 0, there is a gauge δ of [a,b] such that for every δ-fine d = (ξi, [ti−1, ti]) ∈ T D[a,b] , we have

∥∥∥∥∥b∫

a

f (t)dg(t) −∑

i

f (ξi)(

g(ti) − g(ti−1))∥∥∥∥∥ < ε.

It is a known fact that the Perron–Stieltjes integral has the usual properties of linearity, additivityover non-overlapping integrals, integrability on subintervals, etc.

Since we are interested in conditions under which the Perron–Stieltjes integral exists, we recallthe concept of a regulated function. We consider a general setting of Banach space-valued functions.

Let X be a Banach space and I ⊂ R be any interval. We denote by G(I, X) the space of regulatedfunctions f : I → X , that is, for each compact interval [a,b] ⊂ I , the one-sided limits

f (t+) = limρ→0+ f (t + ρ), t ∈ [a,b),

and

f (t−) = limρ→0− f (t + ρ), t ∈ (a,b]

exist and are finite. In particular, when I = [a,b], −∞ < a < b < ∞, we write G([a,b], X) which isa Banach space when endowed with the usual supremum norm, ‖‖∞ . See [14] for a proof of this fact.In G(I, X) we consider the topology of locally uniform convergence. Then, we denote by G−(I, X), thesubspace of G(I, X) of left-continuous functions.

We recall that functions f : I → X of locally bounded variation, that is, functions which are ofbounded variation on every compact interval I also belong to G(I, X). See [14].

The next result can be found in [13, Theorem 2.1]. The inequality below follows from the definitionof Perron–Stieltjes integral.

Theorem 2.1. If f , g ∈ G([a,b],Rn) and at least one of functions f , g has a bounded variation on [a,b], then

the integral∫ b

a f dg exists. Furthermore,

∣∣∣∣∣b∫

a

f dg

∣∣∣∣∣ � ‖ f ‖∞ varba g

if f ∈ G([a,b],Rn) and g ∈ B V ([a,b],R).

As a particular case, we obtain the next result. It can be found in [25, Corollary 1.34].

Corollary 2.1. If f : [a,b] → Rn is a regulated function and g : [a,b] → R is a nondecreasing function, then

the integral∫ b

a f (t)dg(t) exists and

∥∥∥∥∥b∫

a

f (s)dg(s)

∥∥∥∥∥ � ‖ f ‖∞(

g(b) − g(a)).

3102 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

The next result describes the right and left limits of the indefinite integral of a Perron–Stieltjesintegrable function with respect to a given function g : [a,b] → R. A proof of it can be found in [29,Proposition 2.3.16].

Theorem 2.2. Let f : [a,b] → R and g : [a,b] →R be such that∫ b

a f dg exists. Then, the function

h(t) =b∫

a

f dg

is defined on [a,b] and

(i) if g ∈ G([a,b],R), then h ∈ G([a,b],Rn) and

�+h(t) = f (t)�+g(t) and �−h(t) = f (t)�−g(t)

on [a,b], where �+ g(t) = g(t+) − g(t) and �− g(t) = g(t) − g(t−).(ii) if g ∈ B V ([a,b],R) and f is bounded on [a,b], then h ∈ B V ([a,b],Rn).

Also, we have the following result which is a special case of Theorem 1.16 in [25].

Corollary 2.2. Let f : [a,b] → Rn and g : [a,b] → R be a pair of functions such that g is regulated and∫ b

a f (t)dg(t) exists. Then the function

h(t) =t∫

a

f (s)dg(s), t ∈ [a,b],

is regulated and satisfies

h(t+) = h(t) + f (t)�+g(t), t ∈ [a,b),

h(t−) = h(t) − f (t)�−g(t), t ∈ (a,b],where �+ g(t) = g(t+) − g(t) and �− g(t) = g(t) − g(t−).

The next result describes a substitution for Perron–Stieltjes integrals. See [21], for instance.

Theorem 2.3. Assume that φ : [a,b] → [α,β] is increasing and maps [a,b] onto [α,β]. Moreover, letg : [α,β] → R and h : [α,β] → R. Then, both the integrals

β∫α

g(τ )d[h(τ )

],

b∫a

g(φ(s)

)d[h(φ(s)

)]exists whenever there exists at least one of them and in such a case, we have

β∫α

g(τ )d[h(τ )

] =b∫

a

g(φ(s)

)d[h(φ(s)

)].

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3103

The next theorem is an analogue of Gronwall’s inequality theorem for Perron–Stieltjes integrals.A proof of it can be found in [25, Corollary 1.43].

Theorem 2.4. Let h : [a,b] → [0,∞) be a nondecreasing left-continuous function, k > 0 and l � 0. Assumethat ψ : [a,b] → [0,∞) is bounded and satisfies

ψ(ξ) � k + l

ξ∫a

ψ(τ )dh(τ ), ξ ∈ [a,b].

Then, we have

ψ(ξ) � kel(h(ξ)−h(a)), ξ ∈ [a,b].3. Averaging principle for measure functional differential equations

Let r be a given real positive number. The theory of functional differential equations is usuallyconcerned with the initial value problem{

x′(t) = f (xt , t),x0 = φ,

(3.1)

where φ ∈ C([−r,0],Rn). The equivalent integral form is⎧⎪⎨⎪⎩ x(t) = x(0) +t∫

0

f (xs, s)ds,

x0 = φ,

(3.2)

where the integral is usually taken in Riemann’s or Lebesgue’s senses.In this section and throughout this paper, we consider left-continuous and regulated functions.

Thus, instead of C([−r,0],Rn), we consider G−([−r,0],Rn). Let σ > 0 be a given real number, thenfor each x ∈ G−([−r, σ ],Rn) and each t ∈ [0, σ ], the function xt : [−r,0] →R

n usually defined by

xt(θ) = x(t + θ), θ ∈ [−r,0],is now an element of G−([−r,0],Rn).

Consider the functions h1,h2 : [0,∞) →R which are left-continuous and nondecreasing.Let ε > 0 be given. Now, we focus our attention on a measure functional differential equation of

the form ⎧⎪⎨⎪⎩ x(t) = φ(0) + ε

t∫0

f (xs, s)dh1(s) + ε2

t∫0

g(xs, s, ε)dh2(s),

x0 = φ,

(3.3)

where we consider that the initial function φ is an element of G−([−r,0],Rn) and we consider thePerron–Stieltjes integral in (3.3) taken with respect to nondecreasing functions h1 and h2.

In order to obtain an averaging principle for (3.3), we consider the auxiliary initial value problem⎧⎪⎨⎪⎩ x(t) = x(0) + ε

t∫0

f

(xs,ε,

s

ε

)dh1

(s

ε

)+ ε2

t∫0

g

(xs,ε,

s

ε, ε

)dh2

(s

ε

), t ∈ [0, M],

x = φ,

(3.4)

0,ε

3104 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

where xt,ε(θ) = x(t + εθ), for θ ∈ [−r,0], φ ∈ G−([−r,0],Rn) and ε ∈ (0, ε0], where ε0 > 0. Weassume that f maps any pair (ψ, t) ∈ G−([−r,0],Rn) × [0,∞) into R

n and that the mappingt → f (yt , t) is Perron–Stieltjes integrable with respect to the function h1, for all t ∈ [0,∞). Fur-thermore, we consider that g maps (ψ, t, ε) ∈ G−([−r,0],Rn) × [0,∞) × (0, ε0] into R

n and thatthe mapping t → g(yt , t, ε) is Perron–Stieltjes integrable with respect to the function h2, for allt ∈ [0,∞).

We point out that if xt ∈ G−([−r,0],Rn), then obviously xt,ε ∈ G−([−r,0],Rn).

Remark 3.1. Note that, by a change of variables, we can transform system (3.4) into system (3.3).Indeed, let M > 0, if x is a solution of (3.4) on [0, M], then

x(t) = x(0) + ε

t∫0

f

(xs,ε,

s

ε

)dh1

(s

ε

)+ ε2

t∫0

g

(xs,ε,

s

ε, ε

)dh2

(s

ε

).

Denote ψ(s) = s/ε for s ∈ [0, M] and m(τ ) = f (xετ ,ε, τ ) for τ ∈ [0, M/ε]. Now, we want to substi-tute τ = ψ(s) = s/ε into the integral

t∫0

f

(xs,ε,

s

ε

)[dh1

(s

ε

)]=

t∫0

f

(xε(s/ε),ε,

s

ε

)[dh1

(s

ε

)]

=t∫

0

m

(s

ε

)[dh1

(s

ε

)]=

t∫0

m(ψ(s)

)[dh1

(ψ(s)

)]

=t/ε∫0

m(τ )[dh1(τ )

] =t/ε∫0

f (xετ ,ε, τ )[dh1(τ )

],

by Theorem 2.3. Now, denote n(τ ) = g(xετ ,ε, τ , ε) for τ ∈ [0, M/ε] and similarly, by Theorem 2.3, weobtain

t∫0

g

(xs,ε,

s

ε, ε

)[dh2

(s

ε

)]=

t∫0

g

(xε(s/ε),ε,

s

ε, ε

)[dh2

(s

ε

)]

=t∫

0

n

(s

ε

)[dh2

(s

ε

)]=

t∫0

n(ψ(s)

)[dh2

(ψ(s)

)]

=t/ε∫0

n(τ )[dh2(τ )

] =t/ε∫0

g(xετ ,ε, τ , ε)[dh2(τ )

].

Furthermore, put y(t) = x(tε). Then, we have

xετ ,ε(θ) = x(ετ + εθ) = x(ε(τ + θ)

) = y(τ + θ) = yτ (θ),

for τ ∈ [0, M/ε] and θ ∈ [−r,0].

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3105

Therefore, we obtain

y(t) − y(0) = x(εt) − x(0)

= ε

εt∫0

f (xs,ε, s/ε)dh1(s/ε) + ε2

εt∫0

g(xs,ε, s/ε, ε)dh2(s/ε)

= ε

εt/ε∫0

f (xεs,ε, s)dh1(s) + ε2

εt/ε∫0

g(xεs,ε, s, ε)dh2(s)

= ε

t∫0

f (ys, s)dh1(s) + ε2

t∫0

g(ys, s, ε)dh2(s),

for t ∈ [0, Mε ]. Moreover, for τ = 0, we obtain

x0,ε(θ) = x(ε0 + εθ) = x(ε(0 + θ)

) = y(0 + θ) = y0(θ). (3.5)

Therefore, if x is a solution of (3.4) and using Eq. (3.5), we obtain

φ(θ) = x0,ε(θ) = y0(θ),

which implies that y satisfies the initial condition of (3.3).On the other hand, if y is a solution of (3.3) and by Eq. (3.5), we get

φ(θ) = y0(θ) = x0,ε(θ),

which implies that x satisfies the initial condition of (3.4).Thus, if we have a solution of the measure functional differential equation (3.4) on [0, M], then,

by a change of variables, we can obtain a solution of (3.3) on [0, M/ε] and vice versa. We are theninterested in establishing an averaging result for (3.4). Then an averaging principle for (3.3) will follownaturally.

Assume that f : G−([−r,0],Rn) × [0,∞) → Rn satisfies the following conditions:

(H1) For every ϕ ∈ G−([−r,0],Rn) and for every t ∈ [0,∞), the integral∫ t

0 f (ϕ, s)dh1(s) exists;(H2) There exists a constant L > 0 such that, for every ϕ,ψ ∈ G−([−r,0],Rn) and every u1, u2 ∈

[0,∞), ∥∥∥∥∥u2∫

u1

[f (ϕ, s) − f (ψ, s)

]dh1(s)

∥∥∥∥∥ � L

u2∫u1

‖ϕ − ψ‖dh1(s).

Consider the following assumptions on g : G−([−r,0],Rn) × [0,∞) × (0, ε0] →Rn:

(H3) For every ϕ ∈ G−([−r,0],Rn), t ∈ [0,∞) and ε ∈ (0, ε0], the integral∫ t

0 g(ϕ, s, ε)dh2(s) exists.(H4) There is a constant C > 0 such that, for all ϕ ∈ G−([−r,0],Rn), ε ∈ (0, ε0] and u1, u2 ∈ [0,∞),∣∣∣∣∣

u2∫u1

g(ϕ, s, ε)dh2(s)

∣∣∣∣∣ � C

u2∫u1

dh2(s),

where h1,h2 : [0,∞) →R are left-continuous and nondecreasing functions.

3106 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

Assume further that the following conditions are fulfilled:

(H5) There exists a constant K > 0 such that, for every β � 0,

lim supT →∞

h1(T + β) − h1(β)

T� K .

(H6) There exists a constant N > 0 such that, for every β � 0,

lim supT →∞

h2(T + β) − h2(β)

T� N.

Suppose, for each ψ ∈ G−([−r,0],Rn), the limit

f0(ψ) = limT →∞

1

T

T∫0

f (ψ, s)dh1(s) (3.6)

exists, where the integral has to be understood in the sense of Perron–Stieltjes, and consider theaveraged FDE {

y = f0(yt,ε), t ∈ [0, M],y0,ε = φ,

(3.7)

where f0 is given by (3.6).

Remark 3.2. We point out that, the same way as before, we can obtain an equivalence between (3.7)and the following averaged FDE {

y = ε f0(yt), t ∈ [0, M/ε],y0 = φ.

(3.8)

Indeed, denote again ψ(s) = s/ε = τ for s ∈ [0, M] and m(τ ) = f0(yετ ,ε) for τ ∈ [0, M/ε]. We wantto substitute τ = ψ(s) = s/ε into the integral

t∫0

f0(ys,ε)ds =t∫

0

f0(yε(s/ε),ε)ds =t∫

0

m(ψ(s)

)ds

= ε

t/ε∫0

m(τ )dτ = ε

t/ε∫0

f0(yετ ,ε)dτ .

Moreover, putting u(t) = y(tε), we get

yετ ,ε(θ) = y(ετ + εθ) = y(ε(τ + θ)

) = u(τ + θ) = uτ (θ),

for τ ∈ [0, Mε ] and θ ∈ [−r,0].

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3107

Therefore, we obtain

u(t) − u(0) = y(εt) − y(0)

=εt∫

0

f0(ys,ε)ds = ε

ε(t/ε)∫0

f0(yεs,ε)ds

= ε

t∫0

f0(us)ds

for every t ∈ [0, Mε ] and we have the desired equivalence between the systems (3.7) and (3.8).

Similarly as before, we can prove that the initial conditions of systems (3.7) and (3.8) coincide.

Let O be a subset of G−([−r, σ ],Rn). We say that O has the prolongation property if, for everyy ∈ O and every t ∈ [−r, σ ], the function y given by

y(t) ={

y(t), −r � t � t,y(t), t < t � σ ,

is also an element of O .Let us denote P = {yt; y ∈ O , t ∈ [0, σ ]}, where O ⊂ G−([−r, σ ],Rn) has the prolongation prop-

erty.In what follows, notice that if conditions (H2) and (H5) are satisfied, then

∥∥ f0(φ) − f0(ψ)∥∥ =

∥∥∥∥∥ limT →∞

1

T

T∫0

f (φ, s)dh1(s) − limT →∞

1

T

T∫0

f (ψ, s)dh1(s)

∥∥∥∥∥� lim

T →∞1

T

T∫0

L‖φ − ψ‖dh1(s)

� L‖φ − ψ‖∞ lim supT →∞

h1(T ) − h1(0)

T� LK‖φ − ψ‖∞.

In particular, for every y ∈ O and every t, s ∈ [0,∞), we have∥∥ f0(ys) − f0(yt)∥∥� LK‖ys − yt‖∞, (3.9)

which implies that, if y ∈ O is a solution of (3.7) and s, t ∈ [0, M], with t � s, then, for θ ∈ [−r,0], wehave

∥∥y(s + εθ) − y(t + εθ)∥∥ =

∥∥∥∥∥s+εθ∫

t+εθ

f0(yσ ,ε)dσ

∥∥∥∥∥�

s+εθ∫t+εθ

∥∥ f0(yσ ,ε) − f0(0)∥∥dσ +

s+εθ∫t+εθ

∥∥ f0(0)∥∥dσ

� LK

s+εθ∫sup

σ∈[t−r,s]‖yσ ,ε‖∞ dσ + (s − t)

∥∥ f0(0)∥∥

t+εθ

3108 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

and, hence,

‖ys,ε − yt,ε‖∞ = supθ∈[−r,0]

∥∥y(s + εθ) − y(t + εθ)∥∥

� LK (s − t) supσ∈[t−εr,s]

‖yσ ,ε‖∞ + (s − t)∥∥ f0(0)

∥∥. (3.10)

Thus, as s−t → 0+ , ‖ys,ε − yt,ε‖∞ → 0, that is, yt,ε as a function of t , with t ∈ [0,∞), is a continuousfunction.

A proof of the next lemma can be carried out by following the ideas of [6, Lemma 3.1] withobvious adaptations for the Perron–Stieltjes integral.

Lemma 3.1. For every t > 0 and every α > 0, we have

limε→0+

1

α/ε

t/ε+α/ε∫t/ε

f (ψ, s)dh1(s) = f0(ψ), ψ ∈ G−([−r,0],Rn),

where f0 is given by (3.6).

The next corollary follows easily from Lemma 3.1.

Corollary 3.1. Let t > 0 and α > 0. Then, for every y ∈ O , we have

limε→0+

ε

α

t/ε+α/ε∫t/ε

f (yt, s)dh1(s) = f0(yt).

The proof of the next lemma follows similarly as in [6, Lemma 3.2], but we include it here in orderto illustrate the use of condition (H5).

Lemma 3.2. Suppose [0, M] is the maximal interval of existence of (3.7), where M > 0 is finite, and y isa maximal solution of (3.7). Moreover, suppose the function f : P × [0,∞) → R

n satisfies conditions (H1)

and (H2) and the nondecreasing function h1 : [0,∞) → R satisfies condition (H5). Then, given ε > 0, thereexists μ > 0 such that

∥∥∥∥∥εt∫

0

f

(ys,ε,

s

ε

)dh1

(s

ε

)−

t∫0

f0(ys,ε)ds

∥∥∥∥∥ < μ, t ∈ [0, M],

where μ tends to zero as ε → 0+ .

Proof. Given ε > 0, let δ be a gauge corresponding to ε > 0 in the definition of the Perron–Stieltjesintegral

t∫0

f

(yσ ,ε,

σ

ε

)dh1

(σ

ε

)

and consider a δ-fine d = (τi, [si, si+1]) ∈ T D[0,t] , i = 0,1,2, . . . ,m − 1. Then,

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3109

∥∥∥∥∥εt∫

0

f

(ys,ε,

s

ε

)dh1

(s

ε

)−

t∫0

f0(ys,ε)ds

∥∥∥∥∥�

m−1∑i=0

∥∥∥∥∥εsi+1∫si

[f

(ys,ε,

s

ε

)− f

(ysi ,ε,

s

ε

)]dh1

(s

ε

)∥∥∥∥∥ +m−1∑i=0

∥∥∥∥∥si+1∫si

[f0(ys,ε) − f0(ysi ,ε)

]ds

∥∥∥∥∥+

m−1∑i=0

∥∥∥∥∥si+1∫si

f

(ysi ,ε,

s

ε

)dh1

(s

ε

)−

si+1∫si

f0(ysi ,ε)ds

∥∥∥∥∥. (3.11)

We can suppose, without loss of generality, that the gauge δ is such that δ(ξ) < ε2 , for every

ξ ∈ [0, t]. Then, as in (3.10), for each i = 0,1,2, . . . ,m − 1 and each s ∈ [si, si+1],

‖ys,ε − ysi ,ε‖∞ < LK (s − si) supσ∈[si−r,s]

‖yσ ,ε‖∞ + (s − si)∥∥ f0(0)

∥∥< LK 2δ(τi) sup

σ∈[si−εr,si+1]‖yσ ,ε‖∞ + 2δ(τi)

∥∥ f0(0)∥∥

< LKε supσ∈[si−εr,si+1]

‖yσ ,ε‖∞ + ε∥∥ f0(0)

∥∥.

Therefore, for each i = 0,1,2, . . . ,m − 1,

sups∈[si ,si+1]

‖ys,ε − ysi ,ε‖∞ � ε(

LK supσ∈[−εr,M]

‖yσ ,ε‖∞ + ∥∥ f0(0)∥∥)

.

Then, taking

D = LK supσ∈[−εr,M]

‖yσ ,ε‖∞ + ∥∥ f0(0)∥∥,

we obtain

sups∈[si ,si+1]

‖ys,ε − ysi ,ε‖∞ � εD, i = 0,1,2, . . . ,m − 1. (3.12)

Now, (3.12) and conditions (H2) and (H5) imply

m−1∑i=0

∥∥∥∥∥si+1∫si

ε

[f

(ys,ε,

s

ε

)− f

(ysi ,ε,

s

ε

)]dh1

(s

ε

)∥∥∥∥∥� εL

m−1∑i=0

supσ∈[si ,si+1]

‖yσ ,ε − ysi ,ε‖∞

si+1∫si

dh1

(s

ε

)� ε2 DL

m−1∑i=0

[h1

(si+1

ε

)− h1

(si

ε

)]

= εDLm−1∑i=0

ε

si+1 − si(si+1 − si)

[h1

(si+1 − si

ε+ si

ε

)− h1

(si

ε

)]

� εDLm∑

K (si+1 − si) = εDLKt � εDLK M.

i=0

3110 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

Then, using (3.9) and (3.12), for each i = 0,1,2, . . . ,m − 1 and each s ∈ [si, si+1], we have

m−1∑i=0

∥∥∥∥∥si+1∫si

[f0(ys,ε) − f0(ysi ,ε)

]ds

∥∥∥∥∥ � LKm−1∑i=0

supσ∈[si ,si+1]

‖yσ ,ε − ysi ,ε‖∞(si+1 − si)

� εDLKm−1∑i=0

(si+1 − si) = εDLKt � εDLK M.

Finally, we assert that the sum

m−1∑i=0

∥∥∥∥∥εsi+1∫si

f

(ysi ,ε,

s

ε

)dh1

(s

ε

)−

si+1∫si

f0(ysi ,ε)ds

∥∥∥∥∥can be made arbitrarily small by Corollary 3.1. Indeed, for each αi = si+1 − si , i = 0,1,2, . . . ,m − 1,we have

m−1∑i=0

∥∥∥∥∥εsi+1∫si

f

(ysi ,ε,

s

ε

)dh1

(s

ε

)−

si+1∫si

f0(ysi ,ε)ds

∥∥∥∥∥=

m−1∑i=0

∥∥∥∥∥εsi+αi∫si

f

(ysi ,ε,

s

ε

)dh1

(s

ε

)−

si+αi∫si

f0(ysi ,ε)ds

∥∥∥∥∥=

m−1∑i=0

αi

∥∥∥∥∥ ε

αi

si/ε+αi/ε∫si/ε

f (ysi ,ε, s)dh1(s) − f0(ysi ,ε)

∥∥∥∥∥.

For each i = 0,1,2, . . . ,m − 1, define

βi = ε

αi

si/ε+αi/ε∫si/ε

f (ysi ,ε, s)dh1(s) − f0(ysi ,ε)

and let β = max{‖βi‖; i = 0,1,2, . . . ,m − 1}. Then

m−1∑i=0

αi‖βi‖� β

m−1∑i=0

αi = β

m−1∑i=0

(si+1 − si) = βt � βM,

and, hence,

∥∥∥∥∥εt∫

0

f

(ys,ε,

s

ε

)dh1

(s

ε

)−

t∫0

f0(ys,ε)ds

∥∥∥∥∥ � 2εDLK M + βM.

By Corollary 3.1, β tends to zero as ε tends to zero. Also, β depends on ε. Thus, taking μ =2εDLK M + βM , then μ tends to zero as ε → 0+ and we obtain the inequality

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3111

∥∥∥∥∥εt∫

0

f

(ys,ε,

s

ε

)dh1

(s

ε

)−

t∫0

f0(ys,ε)ds

∥∥∥∥∥ < μ,

which completes the proof. �In what follows, we present our first main result.

Theorem 3.1. Consider the measure functional differential equation (3.4), where φ ∈ G−([−r,0],Rn) andthe function f : P × [0,∞) → R

n satisfies the conditions (H1) and (H2). Moreover, assume that the nonde-creasing functions h1,h2 : [0,∞) → R satisfy the conditions (H5) and (H6), respectively, and the functiong : P × [0,∞) × (0, ε0] → R

n satisfies conditions (H3) and (H4). Consider the averaged functional differen-tial equation (3.7), where the function f0 : P → R

n is given by (3.6). Suppose [0,b) is the maximal intervalof existence of the solution of (3.4) and [0,b) is the maximal interval of existence of the solution of (3.7).Assume that xε is a maximal solution of (3.4) and y is a maximal solution of (3.7). Let M > 0 be such thatM < min(b,b). Then, for every η > 0, there exists ε0 > 0 such that for ε ∈ (0, ε0], the inequality∥∥xε(t) − y(t)

∥∥ < η

holds for every t ∈ [0, M].

Proof. We follow the ideas of [6, Theorem 3.1]. By hypothesis, for each t ∈ [0, M], the equalities

xε(t) = φ(0) + ε

t∫0

f

((xε

)s,ε,

s

ε

)dh1

(s

ε

)+ ε2

t∫0

g

((xε

)s,ε,

s

ε, ε

)dh2

(s

ε

)

and

y(t) = φ(0) +t∫

0

f0(ys,ε)ds

hold. Then, using conditions (H2), (H4) and (H5) and by the previous lemma, we obtain∥∥xε(t) − y(t)∥∥

=∥∥∥∥∥ε

t∫0

f

((xε

)s,ε,

s

ε

)dh1

(s

ε

)+ ε2

t∫0

g

((xε

)s,ε,

s

ε, ε

)dh2

(s

ε

)−

t∫0

f0(ys,ε)ds

∥∥∥∥∥� ε

t∫0

∥∥∥∥ f

((xε

)s,ε,

s

ε

)− f

(ys,ε,

s

ε

)∥∥∥∥dh1

(s

ε

)+

∥∥∥∥∥εt∫

0

f

(ys,ε,

s

ε

)dh1

(s

ε

)−

t∫0

f0(ys,ε)ds

∥∥∥∥∥+ ε2

t∫0

∥∥∥∥g

((xε

)s,ε,

s

ε, ε

)∥∥∥∥dh2

(s

ε

)

� L

t∫ ∥∥(xε

)s,ε − ys,ε

∥∥∞ dh1

(s

ε

)+

∥∥∥∥∥εt∫

f

(ys,ε,

s

ε

)dh1

(s

ε

)−

t∫f0(ys,ε)ds

∥∥∥∥∥

0 0 0

3112 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

+ ε2C

(h2

(t

ε

)− h2(0)

)< L

t∫0

∥∥(xε

)s,ε − ys,ε

∥∥∞ dh1

(s

ε

)+ μ + εCtN

� L

t∫0

∥∥(xε

)s,ε − ys,ε

∥∥∞ dh1

(s

ε

)+ μ + εC MN,

where we used the fact that condition (H6) implies

ε2C

(h2

(t

ε

)− h2(0)

)= εtC

ε

t

(h2

(t

ε

)− h2(0)

)� εtC N.

Since (xε)0 = φ = y0, we have∥∥(xε

)s,ε − ys,ε

∥∥∞ = supθ∈[−r,0]

∥∥xε(s + εθ) − y(s + εθ)∥∥

= supσ∈[−εr,s]

∥∥xε(σ ) − y(σ )∥∥ � sup

σ∈[0,s]∥∥xε(σ ) − y(σ )

∥∥since

supσ∈[−εr,0]

∥∥xε(σ ) − y(σ )∥∥ = 0,

assuming, without loss of generality, ε < 1 and using the definition of the initial condition of (3.3)and (3.7). Therefore, we obtain

∥∥xε(t) − y(t)∥∥� L

t∫0

supσ∈[0,s]

∥∥xε(σ ) − y(σ )∥∥ dh1

(s

ε

)+ μ + εC MN. (3.13)

Also, since the right-hand side of (3.13) is increasing, we have

supτ∈[0,t]

∥∥xε(τ ) − y(τ )∥∥ � L

t∫0

supσ∈[0,s]

ε∥∥xε(σ ) − y(σ )

∥∥ dh1

(s

ε

)+ μ + εC MN.

Then, by Gronwall’s inequality for the Perron–Stieltjes integral (Theorem 2.4), we obtain

supτ∈[0,t]

∥∥xε(τ ) − y(τ )∥∥� eεL(h1(t/ε)−h1(0))(μ + εC MN).

Finally,

supτ∈[0,t]

∥∥xε(τ ) − y(τ )∥∥� eK LM(μ + εC MN),

since, by hypothesis (H5), we have

εL(h1(t/ε) − h1(0)

) = tL(h1(t/ε) − h1(0)) � tLK � MLK .

t/ε

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3113

Let η = eK LM(μ + εC M K ). Then,

supτ∈[0,t]

∥∥xε(τ ) − y(τ )∥∥� η,

where η can be taken sufficiently small by the arbitrariness of ε and the theorem is proved. �Another main averaging result for system

⎧⎪⎪⎨⎪⎪⎩ y(t) = φ(0) + ε

t∫0

f (ys, s)dh1(s) + ε2

t∫0

g(ys, s, ε)dh2(s),

y0 = φ,

(3.14)

follows next as an immediate consequence of Theorem 3.1.

Corollary 3.2. Consider the measure functional differential equation (3.14) where φ ∈ G−([−r,0],Rn) andthe function f : P × [0,∞) → R

n satisfies conditions (H1) and (H2). Moreover, assume that the nonde-creasing functions h1,h2 : [0,∞) → R satisfy the conditions (H5) and (H6), respectively, and the functiong : P × [0,∞) × (0, ε0] → R

n satisfies the conditions (H3) and (H4). Consider the averaged functional dif-ferential equation

{x = ε f0(xt),

x0 = φ,(3.15)

where the function f0 : P →Rn is given by

f0(ψ) = limT →∞

1

T

T∫0

f (ψ, s)dh1(s).

Let xε and y be solutions of (3.14) and (3.15) on [0, Mε ] respectively. Then, for every η, there exists an ε0 > 0

such that

∥∥xε(t) − y(t)∥∥ < η

holds for every t ∈ [0, Mε ] and ε ∈ (0, ε0].

4. Averaging for impulsive measure functional differential equations

In this section, we present an averaging result for measure functional differential equations withimpulses. In order to prove such a result, we will employ an averaging result for measure functionaldifferential equations without impulses and a correspondence between these last ones and impulsivemeasure functional differential equations. See [9].

Consider, for each k = 1,2, . . . , the sequence of impulse operators Ik : Rn → Rn at preassigned

times t1, . . . , tm , where tk → ∞ when k → ∞. Let B ⊂ Rn be an open set.

The next lemma is essential to prove our averaging result for impulsive measure functional differ-ential equations. A proof of it can be found in [9, Lemma 2.3].

3114 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

Lemma 4.1. Let m ∈ N, a � t1 < t2 < · · · < tm � b. Consider a pair of functions f : [a,b] → R andh : [a,b] → R, where h is regulated, left-continuous on [a,b], and continuous at t1, . . . , tm. Let f : [a,b] →R

and h : [a,b] → R be such that f (t) = f (t) for every t ∈ [a,b] \ {t1, . . . , tm} and h − h is constant on each

of the intervals [a, t1], (t1, t2], . . . , (tm−1, tm], (tm,b]. Then∫ b

a f dh exists if and only if∫ b

a f dh exists; in thatcase, we have

b∫a

f dh =b∫

a

f dh +∑

k∈{1,...,m},tk<b

f (tk)�+h(tk).

The following theorem shows that an impulsive measure functional differential equation can bealways transformed to a measure functional differential equation without impulses. For more details,see [9].

Theorem 4.1. Let m ∈ N, t0 � t1 < · · · < tm < t0 + σ , I1, . . . , Im : Rn → Rn, G1 ⊂ G([−r,0],Rn), f : G1 ×

[t0, t0 +σ ] → Rn. Assume that h : [t0, t0 +σ ] → R is a regulated left-continuous function which is continuous

at t1, . . . , tm. For every y ∈ G1 , define

f (y, t) ={

f (y, t), t ∈ [t0, t0 + σ ] \ {t1, . . . , tm},Ik

(y(0)

), t = tk for some k ∈ {1, . . . ,m}. (4.1)

Moreover, let c1, . . . , cm ∈R be constants such that the function h : [t0, t0 + σ ] → R given by

h(t) =⎧⎨⎩

h(t), t ∈ [t0, t1],h(t) + ck, t ∈ (tk, tk+1] for some k ∈ {1, . . . ,m − 1},h(t) + cm, t ∈ (tm, t0 + σ ]

satisfies �+h(tk) = 1 for every k ∈ {1, . . . ,m}. Then x : [t0 − r, t0 + σ ] →Rn is a solution of the system

⎧⎪⎪⎨⎪⎪⎩x(t) = x(t0) +

t∫t0

f (xs, s)dh(s) +∑

k∈{1,...,m},tk<t

Ik(x(tk)

), t ∈ [t0, t0 + σ ],

xt0 = φ

(4.2)

if and only if x satisfies ⎧⎪⎪⎨⎪⎪⎩ x(t) = x(t0) +t∫

t0

f (xs, s)dh(s), t ∈ [t0, t0 + σ ],

xt0 = φ.

(4.3)

Consider, also, the following conditions on impulse operators Ik : Rn →Rn:

(A∗) There exists a constant K1 > 0 such that

∥∥Ik(x)∥∥ � K1

for every k ∈ {1,2, . . . ,m} and x ∈Rn .

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3115

(B∗) There exists a constant K2 > 0 such that∥∥Ik(x) − Ik(y)∥∥ � K2‖x − y‖

for every k = {1,2, . . . ,m} and x, y ∈ Rn .

The next lemma can be found in [9, Lemma 3.3] and it describes how the conditions ofCarathèodory and Lipschitz-type concerning the function f and the Lipschitz and boundedness condi-tions for the impulse operators can be transferred to f , when it is defined the same way as describedin Theorem 4.1.

Lemma 4.2. Let m ∈ N, t0 � t1 < · · · < tm < t0 + σ , B ⊂ Rn, I1, . . . , Im : B → R

n, P = G([−r,0], B), O =G([t0 − r, t0 + σ ], B). Assume that g : [t0, t0 + σ ] → R is a left-continuous nondecreasing function which iscontinuous at t1, . . . , tm. Let f : P ×[t0, t0 +σ ] → R

n be a function such that the integral∫ t0+σ

t0f (yt , t)dg(t)

exists for every y ∈ O . For every y ∈ P , define

f (y, t) ={

f (y, t), t ∈ [t0, t0 + σ ] \ {t1, . . . , tm},Ik

(y(0)

), t = tk for some k ∈ {1, . . . ,m}.

Moreover, let c1, . . . , cm ∈ R be constants such that the function g : [t0, t0 + σ ] →R given by

g(t) =⎧⎨⎩

g(t), t ∈ [t0, t1],g(t) + ck, t ∈ (tk, tk+1] for some k ∈ {1, . . . ,m − 1},g(t) + cm, t ∈ (tm, t0 + σ ]

satisfies �+ g(tk) = 1 for every k ∈ {1, . . . ,m}.

1. Assume there exists a constant M > 0 such that∥∥∥∥∥u2∫

u1

f (yt, t)dg(t)

∥∥∥∥∥ � M(

g(u2) − g(u1))

whenever t0 � u1 � u2 � t0 + σ , y ∈ O , and a constant K1 > 0 such that∥∥Ik(x)∥∥ � K1

for every k ∈ {1, . . . ,m} and x ∈ B. Then

∥∥∥∥∥u2∫

u1

f (yt, t)dg(t)

∥∥∥∥∥� (M + K1)(

g(u2) − g(u1))

whenever t0 � u1 � u2 � t0 + σ and y ∈ O .2. Assume there exists a constant L > 0 such that∥∥∥∥∥

u2∫u1

(f (yt , t) − f (zt, t)

)dg(t)

∥∥∥∥∥ � L

u2∫u1

‖yt − zt‖∞ dg(t)

whenever t0 � u1 � u2 � t0 + σ , y, z ∈ O , and a constant K2 > 0 such that

3116 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

∥∥Ik(x) − Ik(y)∥∥ � K2‖x − y‖

for every k ∈ {1, . . . ,m} and x, y ∈ B. Then

∥∥∥∥∥u2∫

u1

(f (yt, t) − f (zt, t)

)dg(t)

∥∥∥∥∥� (L + K2)

u2∫u1

‖yt − zt‖∞ dg(t)

whenever t0 � u1 � u2 � t0 + σ and y, z ∈ O .

Now, we are able to present an averaging result for impulsive measure functional differential equa-tions.

Theorem 4.2. Consider the impulsive measure functional differential equation

⎧⎪⎪⎨⎪⎪⎩y(t) = y(t0) + ε

t∫t0

f (ys, s)dh(s) + ε2

t∫t0

g(ys, s, ε)dh(s) + ε∑k∈N,tk<t

Ik(

y(tk)),

y0 = φ,

(4.4)

where φ ∈ G−([−r,0],Rn) and the function f : G−([−r,0],Rn) × [0,∞) → Rn satisfies conditions (H1)

and (H2). Moreover, assume that the nondecreasing function h : [0,∞) → R satisfies condition (H5) and thefunction g : G−([−r,0],Rn) × [0,∞) × (0, ε0] → R

n satisfies condition (H3) and is bounded. Also, sup-pose the impulse operators Ik : Rn → R

n satisfy conditions (A∗) and (B∗), with the same Lipschitzian andboundedness constant for all k ∈ N. Further, suppose the integral

f0(ψ) = limT →∞

1

T

T∫0

f (ψ, s)dh(s)

exists for every ψ ∈ G−([−r,0],Rn) and denote

I0(y) = limT →∞

1

T

∑0�tk<T

Ik(y), y ∈ B.

Consider the averaged differential equation

{x = ε f0(xt) + ε I0(y),

x0 = φ.(4.5)

Let xε and y be solutions of (4.4) and (4.5) on [0, M/ε] respectively. Then, for every η, there exists ε0 > 0 suchthat the inequality

∥∥xε(t) − y(t)∥∥ < η

holds for every t ∈ [0, M/ε] and every ε ∈ (0, ε0].

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3117

Proof. Define the function h : [0,∞) →R by

h(t) ={

h(t), t ∈ [0, t1],h(t) + ck, t ∈ (tk, tk+1] for some k ∈N,

where the sequence {ck}∞k=1 is chosen in such a way that �+h(tk) = 1 for every k ∈N.

Note that the function h is nondecreasing and left-continuous. Also, by condition (H5) and by thedefinition of h, there exists a constant C > 0 such that, for every β � 0, the following inequality

lim supT →∞

h(T + β) − h(β)

T� C

holds.According to the assumptions, we have

xε(t) = xε(0) +t∫

0

(ε f

(xε

s , s) + ε2 g

(xε

s , s, ε))

dh(s) +∑

0�tk<t

ε Ik(xε(tk)

)for every ε ∈ (0, ε0] and every t ∈ [0, M/ε]. Let

F ε(y, t) ={

ε f (y, t) + ε2 g(y, t, ε), t /∈ {t1, t2, . . .},ε Ik

(y(0)

), t = tk for some k ∈N

for every y ∈ G−([−r,0],Rn) and every t � 0. By Theorem 4.1,

xε(t) = xε(0) +t∫

0

F ε(xε

s , s)

dh(s) (4.6)

for every ε ∈ (0, ε0] and every t ∈ [0, M/ε]. For y ∈ G−([−r,0],Rn) and t � 0, define

F ε(y, t) = ε f (y, t) + ε2 g(y, t, ε), (4.7)

where

f (y, t) ={

f (y, t), t /∈ {t1, t2, . . .},Ik

(y(0)

), t = tk for some k ∈ N

and

g(y, t, ε) ={

g(y, t, ε), t /∈ {t1, t2, . . .},0, t = tk for some k ∈N.

It follows from (4.6) and (4.7) that, for every ε ∈ (0, ε0], the function xε : [−r, M/ε] → B is a solu-tion of the initial value problem⎧⎪⎪⎨⎪⎪⎩ x(t) = x(0) + ε

t∫0

f (xs, s)dh(s) + ε2

t∫0

g(xs, s, ε)dh(s),

x = φ.

(4.8)

0

3118 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

By the definition of function f and hypotheses, follows from Lemma 4.2 that the function f satis-fies condition (H2).

Using Lemma 4.1, we have

T∫0

f (x, s)dh(s) =T∫

0

f (x, s)dh(s) +∑

k; 0�tk<T

f (x, tk)�+h(tk)

=T∫

0

f (x, s)dh(s) +∑

k; 0�tk<T

Ik(x(0)

)(4.9)

for every x ∈ G−([−r,0],Rn). Consequently, the function f0 : G−([−r,0],Rn) →Rn defined by

f0(x) = limT →∞

1

T

T∫0

f (x, s)dh(s), x ∈ G−([−r,0],Rn),

satisfies

f0(x) = limT →∞

1

T

T∫0

f (x, s)dh(s) + limT →∞

1

T

∑k; 0�tk<T

Ik(x(0)

) = f0(x) + I0(x(0)

),

for x ∈ G−([−r,0],Rn).Finally, by Corollary 3.2, for every η, there exists ε0 > 0 such that

∥∥xε(t) − yε(t)∥∥� η

holds for every ε ∈ (0, ε0] and every t ∈ [0, M/ε]. �5. Dynamic equations on time scales: basic definitions

In this section, we present some basic concepts concerning the theory of dynamic equations ontime scales. For more details, the reader may want to consult [4] or [5].

A time scale is a closed and nonempty subset of the real line R and we denote this by T. Consider,further, that T has the topology that it inherits from the standard topology on R.

Let T be a time scale. For t ∈ T, the forward jump operator σ : T → T is given by

σ(t) := inf{s ∈ T; s > t},

while the backward jump operator ρ : T → T is given by

ρ(t) := sup{s ∈ T; s < t},

where inf∅ := supT and sup∅ = infT, where ∅ denotes the empty set. If σ(t) > t , then t is calledright-scattered, while if ρ(t) < t , t is called left-scattered.

If t < supT and σ(t) = t , then t is called right-dense and if t > infT and ρ(t) = t , then t is calledleft-dense.

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3119

The function μ : T → [0,∞) defined by

μ(t) := σ(t) − t

is called graininess function.We say that a function f : T → R is right-dense continuous (we write rd-continuous, for short)

provided f is continuous at right-dense points of T and the left-sided limits exist and are finite atleft-dense points of T. The set of all rd-continuous functions is denoted by Crd(T).

We denote the intersection [a,b]∩T by [a,b]T . We will use here the definitions introduced in [27]and the same notations as in [24].

Given a real number t � supT, let

σ (t) = inf{s ∈ T; s � t}.

Note that σ(t) and σ (t) are not necessarily equal. Since T is a closed set, we have σ (t) ∈ T.Further, denote by

T∗ =

{(−∞, supT], if supT< ∞,

(−∞,∞), otherwise.

Then, given a function x : T →Rn , we consider its extension xσ : T∗ → R

n given by

xσ (t) = x(σ (t)

), t ∈ T

∗.

Also, note that

xσt = (

xσ)

t, t ∈ T∗.

According to the next theorem, which can be found in [9], for instance, the �-integral of a timescale function f : T → R

n can be viewed as the Perron–Stieltjes integral of the extended functionf σ : T∗ → R

n .

Theorem 5.1. Let f : T → Rn be a function such that the delta Perron integral

∫ ba f (s)�s exists for every

a,b ∈ T, a < b. Choose an arbitrary a ∈ T and define

F1(t) =t∫

a

f (s)�s, t ∈ T and F2(t) =t∫

a

f σ (s)dg(s), t ∈ T∗, (5.1)

where g(s) = σ (s) for every s ∈ T∗ . Then F2 = F σ

1 . In particular, F2(t) = F1(t) for every t ∈ T.

A proof of the next theorem can be found in [8, Theorem 4.2].

Theorem 5.2. Let T be a time scale, g(s) = σ (s) for every s ∈ T∗ , [a,b] ⊂ T

∗ . Consider a pair of func-

tions f1, f2 : [a,b] → Rn such that f1(t) = f2(t) for every t ∈ [a,b] ∩ T. If

∫ ba f1(s)dg(s) exists, then∫ b

a f2(s)dg(s) also exists and both integrals coincide.

The next theorem describes the correspondence between measure functional differential equationsand functional dynamic equations on time scales. Its proof can be found in [8].

3120 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

Theorem 5.3. Let r, η > 0, [t0 −r, t0 +η]T be a time scale interval, t0 ∈ T, B ⊂ Rn, C = C([t0 −r, t0 +η]T, B),

P = {xσ (t); x ∈ C, t ∈ [t0, t0 + η]}, f : P × [t0, t0 + η]T → Rn, φ ∈ C([t0 − r, t0]T, B). Assume that for

every x ∈ C , the function t → f (xσ (t), t) is rd-continuous on [t0, t0 + η]T . Define g(s) = σ (s) for every s ∈[t0, t0 + η]. If x : [t0 − r, t0 + η]T → B is a solution of the functional dynamic equation

{x�(t) = f

(xσ

t , t), t ∈ [t0, t0 + η]T,

x(t) = φ(t), t ∈ [t0 − r, t0]T,(5.2)

then xσ : [t0 − r, t0 + η] → B satisfies

⎧⎪⎪⎪⎨⎪⎪⎪⎩xσ (t) = xσ (t0) +

t∫t0

f(xσ

s , σ (s))

dg(s), t ∈ [t0, t0 + η],

xσt0

= φσ .

(5.3)

Conversely, if y : [t0 − r, t0 + η] → B is a solution of the measure functional differential equation

⎧⎪⎪⎨⎪⎪⎩y(t) = y(t0) +

t∫t0

f(

ys, σ (s))

dg(s), t ∈ [t0, t0 + η],

yt0 = φσ ,

(5.4)

then y = xσ , where x : [t0 − r, t0 + η]T → B satisfies (5.2).

Now, consider the next result that will be essential to our purposes. A proof of it can be found in[9, Lemma 5.3].

Lemma 5.1. Let [t0 −r, t0 +η]T be a time scale interval, t0 ∈ T, O = G([t0 −r, t0 +η], B), P = G([−r,0], B),f : P × [t0, t0 + η]T → R

n an arbitrary function. Define g(t) = σ (t) and f σ (y, t) = f (y, σ (t)) for everyy ∈ P and t ∈ [t0, t0 + η].

1. If the integral∫ t0+η

t0f (yt , t)�t exists for every y ∈ O , then the integral

∫ t0+ηt0

f σ (yt , t)dg(t) exists forevery y ∈ O .

2. Assume there exists a constant M > 0 such that

∥∥∥∥∥u2∫

u1

f (yt, t)�t

∥∥∥∥∥� M(u2 − u1)

for every y ∈ O and u1, u2 ∈ [t0, t0 + η]T , u1 � u2 . Then

∥∥∥∥∥u2∫

u1

f σ (yt , t)dg(t)

∥∥∥∥∥ � M(

g(u2) − g(u1))

whenever t0 � u1 � u2 � t0 + η and y ∈ O .

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3121

3. Assume there exists a constant L > 0 such that∥∥∥∥∥u2∫

u1

(f (yt, t) − f (zt, t)

)�t

∥∥∥∥∥� L

u2∫u1

‖yt − zt‖∞�t

for every y, z ∈ O and u1, u2 ∈ [t0, t0 + η]T , u1 � u2 . Then

∥∥∥∥∥u2∫

u1

(f σ (yt , t) − f σ (zt, t)

)dg(t)

∥∥∥∥∥� L

u2∫u1

‖yt − zt‖∞ dg(t)

whenever t0 � u1 � u2 � t0 + η and y, z ∈ O .

6. Averaging principle for functional dynamic equations on time scales

In this section, we prove a non-periodic averaging principle for functional dynamic equations ontime scales.

The following result will be useful. A proof of it can be found in [26, Corollary 5.6].

Lemma 6.1. If supT = ∞, limt→∞ μ(t)/t = 0 and h : [t0,∞)T → Rn is �-integrable on every compact

subinterval of [t0,∞)T , then

limT →∞

1

T

t0+T∫t0

hσ (s)dη(s) = limT →∞, t0+T ∈T

1

T

t0+T∫t0

h(s)dη(s)

where η(t) = σ (t), provided the limit on the right-hand side exists.

Theorem 6.1. Let T be a time scale with supT = ∞, [t0 − r,∞)T a time scale interval, t0 ∈ T, ε0 > 0, L > 0,B ⊂ R

n and limt→∞ μ(t)/t = 0. Consider the function f : G−([−r,0], B) × [t0,∞)T → Rn and a bounded

function g : G−([−r,0], B) × [t0,∞)T × (0, ε0] → Rn such that the following conditions are satisfied:

(i) For every b > t0 and y ∈ G([t0 − r,b], B), the functions t → f (yt , t) and t → g(yt , t, ε) are regulatedon [t0,b]T .

(ii) For every b > t0 and y ∈ C([t0 − r,b]T, B), the functions t → f (yσt , t) and t → g(yσ

t , t, ε) arerd-continuous on [t0,b]T .

(iii) There is a constant C > 0 such that for x, y ∈ G−([−r,0], B) and u1, u2 ∈ [t0,∞)T ,

∣∣∣∣∣u2∫

u1

[f (x, s) − f (y, s)

]�s

∣∣∣∣∣ � C

u2∫u1

‖x − y‖�s.

(iv) If y : [−r,0] → B is a regulated function, then the integral

f0(y) = limT →∞, t0+T ∈T

1

T

t0+T∫t0

f (y, s)�s

exists.

3122 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

Let φ ∈ G−([t0 − r, t0]T, B). Suppose for every ε ∈ (0, ε0], the �-integral equation⎧⎪⎪⎨⎪⎪⎩ x(t) = x(t0) + ε

t∫t0

f(xσ

s , s)�s + ε2

t∫t0

g(xσ

s , s, ε)�s,

x(t) = φ(t), t ∈ [t0 − r, t0]T(6.1)

has a solution xε : [t0 − r, t0 + M/ε]T → Rn, and the averaged FDE{

y = ε f0(

yσt

),

y(t) = φ(t), t ∈ [t0 − r, t0]T(6.2)

has a solution y : [t0 −r, t0 + M/ε]T → Rn. Then, for every η > 0, there exists ε0 > 0 such that for ε ∈ (0, ε0],∥∥xε(t) − y(t)

∥∥ < η,

for every t ∈ [t0, t0 + Mε ]T .

Proof. This proof is inspired in the proof of Theorem 5.3 from [26].Without loss of generality, we assume that t0 = 0. Otherwise, consider the shifted problem with

the time scale T = {t − t0; t ∈ T} and f (x, t) = f (x, t + t0) and g(x, t, ε) = g(x, t + t0, ε).For t ∈ [0,∞), y ∈ G−([−r,0], B) and ε ∈ (0, ε0], define

f σ (y, t) = f(

y, σ (t))

and gσ (y, t, ε) = g(

y, σ (t), ε).

Since limt→∞ μ(t)/t = 0, there are numbers D > 0 and τ ∈ T such that μ(t)/t � D for everyt ∈ [τ ,∞)T . Note that

μ(ρ(σ (t)

)) = σ(ρ(σ (t)

)) − ρ(σ (t)

) = σ (t) − ρ(σ (t)

),

which implies

σ (t) − μ(ρ(σ (t)

)) = ρ(σ (t)

)� ρ(t) � t.

Then, if t ∈R is such that ρ(σ (t)) � τ , we have

σ (t) � t + μ(ρ(σ (t)

))� t + Dρ

(σ (t)

)� t + Dt = t(D + 1).

For every t ∈ [0,∞), let h(t) = σ (t). Then, for sufficiently large T and for every a � 0, we have

h(a + T ) − h(a)

T= σ (a + T ) − σ (a)

T� (a + T )(D + 1) − σ (a)

T,

and, consequently,

lim supT →∞

h(a + T ) − h(a)

T� lim sup

T →∞(a + T )(D + 1) − σ (a)

T= D + 1.

Therefore condition (H5) is satisfied for the function h.

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3123

By Theorems 5.1, 5.2 and 6.1, we have

f0(y) = limT →∞

1

T

T∫0

f (y, s)�s = limT →∞

1

T

T∫0

f(

y, σ (s))

dh(s)

= limT →∞

1

T

T∫0

f(

y, σ (s))

dh(s) = limT →∞

1

T

T∫0

f σ (y, s)dh(s)

for every y ∈ G−([−r,0], B). Since xε is a solution of (6.1), Theorem 5.3 implies xσε is a solution of

the system {Dxσ = ε f σ

(xσ

t , t)

Dh(t) + ε2 gσ(xσ

t , t, ε)

Dh(t),

xσ0 = φσ .

(6.3)

By Lemma 5.1, the hypotheses (H1) and (H2) concerning the function f and the hypotheses (H3)

and (H4) concerning g are fulfilled.Thus all hypotheses of Theorem 3.2 are satisfied. Then, for every η > 0, there exists ε0 > 0 such

that, for every ε ∈ (0, ε0] and every t ∈ [0, M/ε], the inequality

∥∥xσε (t) − y(t)

∥∥� η

holds, where y is a solution of (6.2). The proof is complete after observing that xσε (t) = xε(t) for

t ∈ [0, M/ε]T . �7. Averaging for impulsive functional dynamic equations on time scales

In this section, we prove an averaging principle for impulsive functional dynamic equations ontime scales.

Consider the next theorem which presents a correspondence between impulsive measure func-tional differential equations and impulsive functional dynamic equations on time scales. A proof of itcan be found in [9].

Theorem 7.1. Let [t0 − r, t0 + η]T be a time scale interval, t0 ∈ T, B ⊂ Rn, f : G([−r,0], B) × [t0,

t0 +η]T → Rn, φ ∈ G([t0 −r, t0]T, B). Define g(s) = σ (s) for every s ∈ [t0, t0 +η]. If x : [t0 −r, t0 +η]T → B

is a solution of the impulsive functional dynamic equation⎧⎪⎪⎨⎪⎪⎩x(t) = x(t0) +

t∫t0

f(xσ

s , s)�s +

∑k∈{1,...,m},

tk<t

Ik(x(tk)

), t ∈ [t0, t0 + η]T,

x(t) = φ(t), t ∈ [t0 − r, t0]T,

(7.1)

then xσ : [t0 − r, t0 + η] → B is a solution of the impulsive measure functional differential equation⎧⎪⎪⎪⎨⎪⎪⎪⎩y(t) = y(t0) +

t∫t0

f(

ys, σ (s))

dg(s) +∑

k∈{1,...,m},tk<t

Ik(

y(tk)), t ∈ [t0, t0 + η],

y = φσ .

(7.2)

t0

3124 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

Conversely, if y : [t0 − r, t0 + η] → B satisfies (7.2), then it must have the form y = xσ , where x : [t0 − r,t0 + η]T → B is a solution of (7.1).

The following result is our averaging theorem for impulsive functional dynamic equations on timescales. Its proof is similar to the proof of Theorem 6.1 from the previous section. We include it herehowever.

Theorem 7.2. Let T be a time scale with supT = ∞, [t0 − r,∞)T a time scale interval, t0 ∈ T, ε0 > 0, L > 0,B ⊂ R

n and limt→∞ μ(t)/t = 0. Let {tk}k∈N ∈ T be a sequence of points of impulses such that t0 � t1 <

t2 < · · · < tm < · · · . Also, consider the impulse operators Ik : B →Rn which satisfy conditions (A∗) and (B∗).

Suppose the function f : G−([−r,0], B)×[t0,∞)T → Rn satisfies conditions (H1) and (H2) and the function

g : G−([−r,0], B) × [t0,∞)T × (0, ε0] → Rn is bounded. Moreover, suppose the following conditions are

satisfied:

(i) For b > t0 and y ∈ G−([t0 − r,b], B), the integrals∫ b

t0f (ys, s)�s and

∫ bt0

g(ys, s, ε)�s exist.(ii) If y : [−r,0] → B is a regulated function, then the integral

f0(y) = limT →∞, t0+T ∈T

1

T

t0+T∫t0

f (y, s)�s

exists.(iii) Denote

I0(y) = limT →∞

1

T

∑k; 0�tk<T

Ik(y), y ∈ B.

Let φ ∈ G−([t0 − r, t0]T, B). Suppose for every ε ∈ (0, ε0], the �-integral equation

⎧⎪⎪⎨⎪⎪⎩x(t) = x(t0) + ε

t∫t0

f(xσ

s , s)�s + ε2

t∫t0

g(xσ

s , s, ε)�s + ε

∑k∈N

t0�tk<t

Ik(x(tk)

),

x(t) = φ(t), t ∈ [t0 − r, t0]T(7.3)

has a solution xε : [t0 − r, t0 + Mε ]T → R

n and the averaged differential equation

{y = ε f0

(yσ

t

) + I0(y),

y(t) = φ(t), t ∈ [t0 − r, t0]T(7.4)

has a solution y : [t0 − r, t0 + Mε ]T → R

n. Then, for every η > 0, there exists ε0 > 0 such that for everyε ∈ (0, ε0], ∥∥xε(t) − y(t)

∥∥ < η,

for t ∈ [t0, t0 + Mε ]T .

Proof. Using the same argument as in proof of Theorem 6.1, we assume, without loss of generality,that t0 = 0.

M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126 3125

Given t ∈ [0,∞), y ∈ G−([t0 − r, t0]T, B) and ε ∈ (0, ε0], define the following functions:

f σ (y, t) = f(

y, σ (t))

and gσ (y, t, ε) = g(

y, σ (t), ε).

Since limt→∞ μ(t)/t = 0, there are numbers D > 0 and τ ∈ T such that μ(t)/t � D for everyt ∈ [τ ,∞)T . Moreover,

μ(ρ(σ (t)

)) = σ(ρ(σ (t)

)) − ρ(σ (t)

) = σ (t) − ρ(σ (t)

),

which implies

σ (t) − μ(ρ(σ (t)

)) = ρ(σ (t)

)� ρ(t) � t.

Then, if t ∈R is such that ρ(σ (t)) � τ , we have

σ (t)� t + μ(ρ(σ (t)

))� t + Dρ

(σ (t)

)� t + Dt = t(D + 1).

For every t ∈ [0,∞), let h(t) = σ (t). Then, for sufficiently large T and for every a � 0, we have

h(a + T ) − h(a)

T= σ (a + T ) − σ (a)

T� (a + T )(D + 1) − σ (a)

T, (7.5)

and, therefore, applying the lim sup as T → ∞ in both sides of Eq. (7.5), follows immediately that thecondition (H5) is satisfied for the function h.

By Theorems 5.1, 5.2 and 6.1, we obtain

f0(y) = limT →∞

1

T

T∫0

f (y, s)�s = limT →∞

1

T

T∫0

f(

y, σ (s))

dh(s)

= limT →∞

1

T

T∫0

f(

y, σ (s))

dh(s) = limT →∞

1

T

T∫0

f σ (y, s)dh(s)

for every y ∈ G−([−r,0], B). Then, since xε is a solution of (7.3), follows from Theorem 5.3 that xσε is

a solution of⎧⎪⎪⎪⎨⎪⎪⎪⎩xσε (t) = xσ

ε (0) + ε

t∫0

f σ((

xσε

)s, s

)dh(s) + ε2

t∫0

gσ((

xσε

)s, s, ε

)dh(s) + ε

∑k∈Ntk<t

Ik(xσε (tk)

),

xσ0 = φ.

(7.6)

By Lemma 5.1, follows that the function f σ satisfies conditions (H1) and (H2) and the function gσ

satisfies boundedness condition. Thus, all hypotheses of Theorem 4.2 are satisfied and we obtain thatfor every η > 0, there exists ε0 > 0 such that, for every ε ∈ (0, ε0] and t ∈ [0, M/ε], the inequality∥∥xσ

ε (t) − yε(t)∥∥� η

holds, where yε is a solution of (7.4). Since xσε (t) = xε(t) for t ∈ [0, M/ε]T , the result follows. �

3126 M. Federson, J.G. Mesquita / J. Differential Equations 255 (2013) 3098–3126

Acknowledgments

We acknowledge the important contribution and helpful discussions with Prof. Antonín Slavík.Also, the authors thank the anonymous referee for valuable comments which helped to improve thispaper.

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