On first and second order linear Stieltjes differential equations

J. Ma h. Anal. Appl. 511 (2022) 126010
Con en s lis s a ailable a ScienceDi ec
Jou nal o Ma hema ical Analysis and Applica ions
www.else ie .com/loca e/jmaa
On fi s and second o de linea S iel jes diffe en ial equa ions
F ancisco J. Fe nández ∗, Ignacio Ma quéz Albés, F. Ad ián F. Tojo
CITMAga, 15782, San iago de Compos ela, Spain
Depa amen o de Es a ís ica, Análise Ma emá ica e Op imización, Uni e sidade de San iago de
Compos ela, 15782, Facul ade de Ma emá icas, Campus Vida, San iago, Spain
a i c l e i n o a b s a c
A icle his o y:
Recei ed 21 Sep embe 2021
A ailable online 13 Janua y 2022
Submi ed by S. Hencl
Keywo ds:
S iel jes de i a i e
Second o de
Uniqueness
Exis ence
G een’s unc ion
This wo k deals wi h he ob aining o solu ions o fi s and second o de S iel jes
diffe en ial equa ions. We define he no ion o S iel jes de i a i e on he whole
domain o he unc ions in ol ed, p o ide a no ion o n- imes con inuously S iel jes-
diffe en iable unc ions and p o e exis ence and uniqueness esul s o S iel jes
diffe en ial equa ions in he space o such unc ions. We also p esen he G een’s
unc ions associa ed o he diffe en p oblems and an applica ion o he S iel jes
ha monic oscilla o .
© 2022 The Au ho (s). Published by Else ie Inc. This is an open access a icle
unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
1. In oduc ion
The e has been a ecen su ge in he s udy o S iel jes diffe en ial equa ions ocused on ob aining ap-
plicable esul s compa able o hose a ailable o classical de i a i es [2–14,16,17,19]. These wo ks cen e
hei a en ion in he p ocu ing o solu ions o fi s o de diffe en ial equa ions and sys ems. The heo y
de eloped s a s wi h he ob aining o simple solu ions, like he solu ion o he fi s o de linea p oblem
[3,4], which is iden ified wi h he exponen ial, in o de o, la e , p o e exis ence and uniqueness esul s in
mo e gene al se ings [7,13,16]. Some o hese wo ks also p o ide in e es ing p ac ical applica ions [5,9]and
o he s gene alize he amewo k in se e al ways, such as allowing o sign changing de i a o s [4], conside ing
se e al diffe en de i a o s [16]o gene alizing he concep o S iel jes de i a i e [15].
In any case, all o he a o emen ioned wo ks es ic hemsel es o he fi s o de case. The eason behind
his is ha , in o de o s udy highe o de p oblems, he no ion o highe o de S iel jes de i a i e has o be
co ec ly defined, which is no ob ious. In ac , he fi s difficul y lies on he me e defini ion o he S iel jes
de i a i e, which, o he bes o ou knowledge, is nowhe e defined in he li e a u e on he whole domain o
defini ion o he unc ion, some hing which impedes aking a second de i a i e.
*Co esponding au ho .
E-mail add esses: ja[email p o ec ed] (F.J. Fe nández), [email p o ec ed] (I. Ma quéz Albés),
[email p o ec ed] (F.A.F. Tojo).
h ps://doi.o g/10.1016/j.jmaa.2022.126010
0022-247X/© 2022 The Au ho (s). Published by Else ie Inc. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
2F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
In his wo k we p o ide his defini ion, which enables us o s udy second o de p oblems. Fi s , we conside
he S iel jes de i a i e in he whole o he domain o he unc ion, which allows us o alk abou he space
o con inuously S iel jes-diffe en iable unc ions in he same way we speak o he space o con inuously
diffe en iable unc ions wi h he usual de i a i e. We can hen explo e he fi s o de p oblem in his space,
ob aining exis ence and uniqueness esul s ha mi o hose o he p e ious wo ks. In ac , we p ofi om
he oppo uni y o e isi ing he solu ion o he fi s o de linea p oblem o p o ide a cons uc i e way
o ob aining i s solu ion. All o hese s eps a e also aken wi h a u he gene aliza ion: ou unc ions a e
allowed o ake eal o complex alues. Fu he mo e, we ob ain he explici exp ession o he G een’s unc ion
o he fi s o de linea p oblem wi h ini ial condi ions and we cons uc he S iel jes e sions o he sine
and cosine unc ions using he complex e sion o he S iel jes exponen ial.
Once we ha e s udied he fi s o de p oblem wi h a ious deg ees o egula i y (some hing he subsequen
spaces o n- imes con inuously S iel jes-diffe en iable unc ions allow), we mo e on o s udy second o de
p oblems. Fi s , we p esen exis ence and uniqueness esul s o he homogeneous second o de p oblem
wi h cons an coefficien s and hen we s udy he non homogeneous case wi h a ying deg ees o egula i y.
He e we also ob ain he explici exp ession o he G een’s unc ion o he second o de linea p oblem wi h
ini ial condi ions. All his wo k is hen illus a ed wi h an applica ion o he S iel jes ha monic oscilla o
o which we also analyze he esonance effec . Finally, in o de o alida e he explici solu ions ob ained,
we compa e hem wi h he nume ical app oxima ion o he co esponding fi s o de linea sys em using
he nume ical scheme in oduced in [2].
The s uc u e o his wo k is as ollows: In Sec ion 2we p esen some p elimina y concep s and we
p o e se e al esul s ela ed o Lebesgue-S iel jes in eg al. In Sec ion 3we in oduce he space o bounded
S iel jes diffe en iable unc ions and analyze some o i s p ope ies. We s udy he fi s o de linea S iel jes
diffe en ial equa ion in Sec ion 4, including in he complex case. In his sec ion we also define he complex
S iel jes exponen ial and he S iel jes e sion o he sine and cosine unc ions. In Sec ion 5we s udy he
homogeneous S iel jes second o de p oblem wi h cons an coefficien s, he non homogeneous case and we
also ob ain an explici solu ion o bo h si ua ions. Finally, in Sec ion 6we p esen an applica ion o he
S iel jes ha monic oscilla o . We ob ain he explici solu ion o he o e damped, c i ically damped and
unde damped cases, and p o ide an example in which he esonance effec appea s. In o de o alida e he
explici solu ion ob ained, we compa e i wi h he nume ical solu ion o he co esponding fi s o de linea
sys em.
2. P elimina ies
Le [a, b] ⊂Rbe an in e al, F he field Ro Cand g:R →Ra le -con inuous non-dec easing
unc ion. We will e e o such unc ions as de i a o s. Fo hese unc ions, we define he se Dg={dn}n∈Λ
(whe e Λ ⊂N) as he se o all discon inui y poin s o g, namely, Dg={ ∈R :Δ
+g( ) >0}whe e
Δ+g( ) := g( +) −g( ), ∈R, and g( +) deno es he igh hand side limi o ga . We also define
Cg:= { ∈R:gis cons an on ( −ε, +ε) o someε>0}.
Obse e ha Cgis open in he usual opology o R, so we can w i e
Cg=
n∈
Λ{(an,b
n)}(2.1)
whe e 
Λ⊂Nand (ak, bk) ∩(aj, bj) =∅ o k=j. Wi h his no a ion, we deno e N−
g:= {an}n∈
Λ Dg,
N+
g:= {bn}n∈
Λ Dgand Ng:= N−
g∪N+
g.
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 3
Rema k 2.1. Fo he aims o his pape , we will assume wi hou loss o gene ali y ha g(a) =0. Fu he mo e,
we will also assume ha gis con inuous a x =a. As poin ed ou in [3, p. 21] and [12, P oposi ion 4.28], he
con inui y assump ion has no impac in he s udy o diffe en ial equa ions, which is ou final goal. Finally,
in o de o p ope ly define he S iel jes de i a i e in he whole [a, b], we will also ask ha [a, b] Cg=∅.
We define gB:R →Ras:
gB( )=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩

s∈[a, )∩Dg
Δ+g(s), >a,
−
s∈[ ,a)∩Dg
Δ+g(s), ≤a.
I is clea ha gBis a le -con inuous and non-dec easing unc ion. Mo eo e , he map gC:R →Rgi en
by
gC( ):=g( )−gB( ),
is also non-dec easing and con inuous. We say gC ha is he con inuous pa o gand gBis he jump pa
o g. Obse e ha bo h gBand gCa e con inuous a x =aand gC(a) =gB(a) =0.
Th oughou his wo k we conside he Lebesgue–S iel jes measu e space (R, Mg, μg), whe e Mgand μg
a e he σ-algeb a and measu e cons uc ed in an analogous ashion o he classical Lebesgue measu e, whe e
he leng h o [c, d)is gi en by μg([c, d)) =g(d) −g(c). The in e es ed eade may e e o [11] o de ails
conce ning his measu e space. We mus emphasize ha , in he case o conside ing g( ) = , we eco e he
classic Lebesgue measu e space ha we will deno e by (R, L, μ) ≡(R, MId, μId) whe e Id is he iden i y
unc ion. Fu he mo e, we can define he measu e space associa ed wi h he con inuous pa , (R, MgC, μgC),
he jump pa , (R, MgB, μgB), and he one associa ed wi h he de i a o i sel , (R, Mg, μg). I we deno e
by B(τu) he Bo el σ-algeb a associa ed o τu, he usual opology o R, we ha e ha B(τu) ⊂M
gand also
B(τu) ⊂M
gM, wi h M=C, B. We mus men ion ha i E⊂R, μ∗
gM(E) ≤μ∗
g(E), o M=C, B, being
μ∗
gMand μ∗
g he ou e measu es associa ed o gMand g espec i ely, M=C, B. We also ha e ha i E⊂R
is a bounded se , hen μ∗
g(E) <∞.
We ha e he ollowing lemma ha , in pa icula , p o ides us wi h a ela ionship be ween he σ-algeb as
Mg, MgCand MgB.
Lemma 2.2. The ollowing p ope ies hold o he maps g, gCand gB:
1. Gi en an elemen E∈M
g he e exis s H∈Gδ( ha is, His a coun able in e sec ion o open se s) and
N∈M
gsuch ha E⊂H, N⊂H, μg(N) =0and E=H N.
2. Gi en an elemen E∈M
g he e exis s F∈Fσ( ha is, Fis a coun able union o closed se s) and
N∈M
gsuch ha μg(N) =0, F∩N=∅and E=F∪N.
3. Mg⊂M
gC.
4. MgB=P(R).
P oo . Since B(τu) ⊂M
gand gis le -con inuous, we ha e ha
μg(E)=in 
n∈N
μg([an,b
n)) : E⊂
n∈N
[an,b
n)=in 
n∈N
μg((an,b
n)) : E⊂
n∈N
(an,b
n).
Indeed on he one hand gi en {(an, bn)}n∈Nsuch ha E⊂n∈N(an, bn), we ha e ha
4F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
μ∗
g(E)≤μ∗
g
n∈N
(an,b
n)≤
n∈N
μ∗
g((an,b
n)),
he e o e, μg(E) ≤in n∈Nμg((an,b
n)) : E⊂n∈N(an,b
n). On he o he hand, gi en ε >0and
{[an, bn)}n∈Nsuch ha E⊂n∈N[an, bn), we ha e, hanks o he le -con inui y o g, ha he e ex-
is s {(˜an, bn)}n∈Nsuch ha [an, bn) ⊂(˜an, bn)and μ∗
g((˜an, bn)) ≤μ∗
g([an, bn)) +ε/2n, n ∈N. Thus,
n∈Nμ∗
g(˜an, bn) ≤n∈Nμ∗
g([an, bn)) +εand we conclude, aking he infimum in bo h sides o inequali y,
ha in n∈Nμg((an,b
n)) : E⊂n∈N(an,b
n)≤μg(E). Now, we can p oceed as in [1, Co olla ies 15.5
and 15.8] o ob ain 1 and 2, espec i ely.
Now, o 3, gi en an elemen E∈M
g, he e exis s F∈Fσand N∈M
gsuch ha μg(N) =0, F∩N=∅
and E=F∪N. Now, F∈B(τu) ⊂M
cand μ∗
gC(N) ≤μ∗
g(N) =0so we ha e ha N∈M
gCsince
(R, MgC, μgC)is a comple e measu e space. The e o e, E⊂M
gC.
Finally, o E∈P(R), we ha e ha E=(E DgB) ∪(E∩DgB). Now, (E DgB) ⊂CgBand hen
μ∗
gB(E DgB) =0, so E DgB∈M
gB. Finally E∩DgB∈B(τu) ⊂M
gB. The e o e E∈M
gB, which
finishes he p oo o 4. 
We deno e by L1
g([a, b); F) he se o unc ions :[a, b) →Fsuch ha hei eal and imagina y pa s,
ha is, Re( )and Im( ) espec i ely, a e measu able and [a,b)| | dμg<∞. Fo his class o unc ions we
define

[a,b)
dμg=
[a,b)
Re( )dμg+i
[a,b)
Im( )dμg.
Lemma 2.3. Gi en a unc ion ∈L
1
g([a, b); F),

[a, )
dμg=
[a, )
dμgC+
s∈[a, )∩Dg
(s)Δ+g(s),∀ ∈[a, b].
P oo . Gi en a unc ion ∈L
1
g([a, b); F), hanks o Lemma 2.2 and he ac ha μgC(E) ≤μg(E) o all
E∈M
g, we ha e ha ∈L
1
gC([a, b); F). Now hanks o [18, Theo ems 6.3.13, 6.12.3 and 6.12.7], and
sepa a ing he eal and imagina y pa i necessa y, we ha e he desi ed esul . 
Co olla y 2.4. Gi en E∈M
gand aking =χE( he cha ac e is ic unc ion associa ed o E) in Lemma 2.3
we ha e ha
μg(E)=μgC(E)+ 
s∈E∩Dg
Δ+g(s).
We now in oduce a ool ha will allow us o ans o m Lebesgue-S iel jes in eg als wi h espec o gC
in o he usual Lebesgue ones. In pa icula , in ligh o Lemma 2.3, his means ha we will ha e a way o
ans o ming any Lebesgue-S iel jes in eg al in o a Lebesgue one.
Defini ion 2.5 (Pseudo-in e se o gC). Gi en an in e al [a, b]and a de i a o g:R →R, we define he
pseudo-in e se o he con inuous pa gCin he in e al [0, gC(b)] by:
γ:x∈[0,gC(b)] →γ(x)=min ∈[a, b]:gC( )=x∈[a, b].(2.2)
In [7, P oposi ion 5.1] we can find some o he p ope ies o he pseudo-in e se o a con inuous de i a o
mapping he eal line on o he eal line. Fo ou con ex , by ex ending linea ly he map gou side o he
in e al [a, b]we can ob ain he equi ed p ope y, which leads o he ollowing esul .
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 5
P oposi ion 2.6. We ha e he ollowing p ope ies o he pseudo-in e se o he con inuous pa gCin he
in e al [0, gC(b)]:
•Fo all x ∈[0, gC(b)], gC(γ(x)) =x.
•Fo all ∈[a, b], γ(gC( )) ≤ .
•Fo all ∈[a, b], /∈CgC∪N+
gC, γ(gC( )) = .
•The map γis s ic ly inc easing.
•The map γis le -con inuous e e ywhe e and con inuous a e e y x ∈[0, gC(b)], x /∈gC(Cg).
Now we a e eady o p o e he ollowing esul .
P oposi ion 2.7. Gi en an in e al [a, b]and a de i a o g:R →R:
1. The con inuous pa
gC:([a, b],MgC)→([0,gC(b)],L)
is a measu able mo phism.1
2. The pseudo-in e se o he con inuous pa gC
γ:([0,gC(b)],L)→([a, b],MgC)
is a measu able mo phism.
P oo . Le us p o e he wo s a emen s sepa a ely.
1. Le us conside a subse E⊂[0, gC(b)] such ha E∈L. We ha e ha he e exis s F∈Fσand N∈L
wi h μ(N) =0such ha F∩N=∅and E=F∪N. I is clea ha (gC)−1(F) ∈B(τu)so, i we p o e ha
μ∗
gC((gC)−1(N)) =0, whe e μ∗
gCis he ou e Lebesgue-S iel jes measu e, we will ha e finished.
Now, since μ(N) =0, gi en ε >0, he e exis s a coun able disjoin amily {[cn, 
dn)}n∈Nsuch ha
N⊂n∈N[cn, 
dn)and n∈N(
dn−cn) <ε. We ha e ha (gC)−1([ck, 
dk)) =[γ(ck), γ(
dk)), o all k∈N,
hus (gC)−1(N) ⊂n∈N[γ(cn), γ(
dn)). Finally,
μ∗
gC((gC)−1N)≤
n∈N
μ∗
gC[γ(cn),γ(
dn)) = 
n∈NgC(γ(
dn)) −gC(γ(cn))=
n∈N
(
dn−cn)<ε.
Since ε >0was a bi a ily chosen, we ha e ha μ∗
gC((gC)−1N) =0, which finishes he p oo o 1.
2. Le us conside a subse E⊂[a, b]such ha E∈M
gC. We ha e ha he e exis s F∈Fσand N∈M
gC
such ha μg(N) =0, F∩N=∅and E=F∪N. Thus, we conclude ha γ−1(E) =γ−1(F) ∪γ−1(N).
Now, since γis s ic ly inc easing, i is a Bo el map, so we ha e ha γ−1(F) ∈B(τu) ⊂L. Hence, i
we p o e ha μ∗(γ−1(N)) =0, whe e μ∗is he ou e Lebesgue measu e, we a e done. The p oo in his
case is analogous o he p e ious one, he only diffe ence lies in ha , gi en an in e al [c, d), we ha e ha
γ−1([c, d)) ⊂[gC(c), gC(d)], hus μ∗(γ−1([c, d))) ≤gC(d) −gC(c) =μ∗
gC([c, d)). 
The ollowing Co olla y is in he line o [2, Lemma 1].
1Gi en wo measu able spaces (X, ΣX)and (Y, ΣY), we say ha a unc ion :X→Yis a measu able mo phism i −1(F) ∈ΣX,
o all F∈ΣY.

6F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
Co olla y 2.8. Gi en a unc ion ∈L
1
g([a, b); F), o e e y ∈[a, b],

[a, )
dμg=
[a, )
dμgC+
s∈[a, )∩Dg
(s)Δ+g(s)= 
[0,gC( )) 
dμ+
s∈[a, )∩Dg
(s)Δ+g(s),
whe e 
= ◦γ, γ: ∈[0, gC(b)] →γ( )is gi en by (2.2)and μdeno es he Lebesgue measu e.
P oo . We w i e (X, ΣX) =([a, ), MgC)and (Y, ΣY) =([0, gC( )), L). We ha e, hanks o P oposi ion 2.7,
ha 
:Y→Fis a measu able unc ion and gC:(X, ΣX) →(Y, ΣY)is a measu able mo phism, which (c .
[20, Exe cise 1.4.38]) ensu es ha

Y
dgC
∗μgC=
X
(
◦gC)dμgC,
whe e
gC
∗μgC:E∈ΣY→gC
∗μgC(E)=μgC((gC)−1(E))
is he push o wa d measu e in (Y, ΣY). Howe e , gi en an elemen (c, d) ⊂[0, gC( )), i is clea ha
gC
∗μgC(c, d) =μgC((gC)−1(c, d)) =d −c. In pa icula , (c . [1, Theo em 13.8]) gC
∗μgC=μ. The e o e,

Y
dgC
∗μgC=
[0,gC( )) 
dμ.
Finally, since μgC(CgC∪N+
gC) =0and γ(gC(s)) =s o all s ∈[a, b] (CgC∪N+
gC), we ha e ha

[a, )
(
◦gC)dμgC=
[a, )
dμgC.
Finally, we ecall a concep o con inui y in oduced in [3]as well as some o i s p ope ies. To ha end
we define he g- opology, τg, as he amily o hose se s U⊂Rsuch ha o e e y x ∈U he e exis s δ>0
such ha i y∈Rsa isfies |g(y) −g(x)| <δ hen y∈U. Then, he ollowing defini ion can be unde s ood
as he con inui y o a unc ion :(I, τg) →(F, τu), see [15, Lemma 6].
Defini ion 2.9 (g-con inuous unc ion). A unc ion :[a, b] →Fis g-con inuous a a poin ∈[a, b], o
con inuous wi h espec o ga , i o e e y ε >0, he e exis s δ>0such ha | ( ) − (s)| <ε, o e e y
s ∈[a, b]wi h |g( ) −g(s)| <δ. I is g-con inuous a e e y poin ∈[a, b], we say ha is g-con inuous
on [a, b].
P oposi ion 2.10 ([3, P oposi ion 3.2]). I :[a, b] →Ris g-con inuous on [a, b], hen
1. is con inuous om he le a e e y 0∈(a, b];
2. i gis con inuous a 0∈[a, b), hen so is ;
3. i gis cons an on some [α, β] ⊂[a, b], hen so is .
In pa icula , g-con inuous unc ions on [a, b]a e con inuous on [a, b]when gis con inuous on [a, b).
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 7
3. The space o bounded g-diffe en iable unc ions
In he li e a u e –see, o ins ance, [3,4,11,15]– au ho s use he ollowing defini ion o S iel jes de i a i e.
Defini ion 3.1. We define he S iel jes de i a i e, o g–de i a i e, o unc ion :[a, b] →Ra a poin
∈[a, b] Cgas

g( )=⎧
⎪
⎪
⎨
⎪
⎪
⎩
lim
s→
(s)− ( )
g(s)−g( ), /∈Dg,
lim
s→ +
(s)− ( )
g(s)−g( ), ∈Dg,
p o ided he co esponding limi s exis and, in ha case, we say ha is g–diffe en iable a . In pa icula ,
o ∈N+
g∪N−
g, he g-de i a i e a mus be unde s ood in he ollowing sense:

g( )=⎧
⎪
⎪
⎨
⎪
⎪
⎩
lim
s→ +
(s)− ( )
g(s)−g( ), ∈N+
g,
lim
s→ −
(s)− ( )
g(s)−g( ), ∈N−
g.
(3.1)
Rema k 3.2. Obse e ha he poin s o Cga e excluded om he defini ion o g–de i a i e. This is because
he co esponding limi canno be conside ed a hose poin s since hey a e in a neighbo hood whe e he
co esponding unc ion is no defined. Obse e also ha he p e ious defini ion is also alid o unc ions
wi h alues in C.
Rema k 3.3. Taking in o accoun Defini ion 3.1 and gi en a unc ion :[a, b] →R, he ollowing condi ions
will be necessa y o he exis ence o he g-de i a i e in all o he poin s o [a, b] Cg:
•I a ∈[a, b] Cg, hen a /∈N−
g. Indeed i a ∈N−
g, o calcula e he g-de i a i e a awe need o know
he alues o o he le o a, which a e no defined. Obse e ha Cg∩Ng=∅ he e o e he p e ious
condi ion is equi alen o a /∈N−
g.
•I b ∈[a, b] Cg, hen b /∈N+
g∪Dg. Indeed i b ∈N+
g∪Dg, o calcula e he g-de i a i e a bwe need
o know he alues o o he igh o b, which a e no defined. Obse e ha Cg∩Ng=Cg∩Dg=∅
he e o e he p e ious condi ion is equi alen o b /∈N+
g∪Dg.
• The e exis s ( +) o e e y ∈(a, b) ∩Dg(which is also a sufficien condi ion o he exis ence o he
g-de i a i e a ha poin ).
•Gi en ∈(a, b) ∩N−
gand ε >0, he e exis s δ>0such ha , i s < wi h g( ) −g(s) <δ hen,
| (s) − ( )| <ε. We say, in ha case, ha is g-con inuous om he le a . To check his ac i
is enough o obse e ha gis le con inuous (in he usual sense) a . The unc ion migh no be
g-con inuous a . Indeed, ake o ins ance
g: ∈R→g( )=⎧
⎪
⎨
⎪
⎩
, ≤1,
1,1≤ ≤2,
−1, ≥2.
(3.2)
Then,
: ∈[0,3] → ( )= , 0≤ ≤1,
+1,1< ≤3,
8F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
is g-diffe en iable a =1since
lim
s→1−
(s)− (1)
g(s)−g(1) = lim
s→1−
s−1
s−1=1.
Obse e ha gis con inuous a =1, bu is no , so canno be g-con inuous a ha poin .
•Gi en ∈(a, b) ∩N+
g, and ε >0, he e exis s δ>0such ha , i s > wi h g(s) −g( ) <δ hen,
| (s) − ( )| <ε. We say, in ha case, ha is g-con inuous om he igh a . To check his ac i
is enough o obse e ha gis igh con inuous (in he usual sense) a . Obse e ha , once again, he
migh no be g-con inuous a such poin s. Indeed, ake o ins ance gas in (3.2)and
: ∈[0,3] → ( )= , 0≤ <2,
+1,2≤ ≤3.
In his case, is g-diffe en iable a = 2 bu is no g-con inuous a such poin .
•Gi en ∈(a, b) (Cg∪Dg∪Ng), is g-con inuous a . In pa icula , is con inuous a since gis
con inuous a hose poin s.
We conclude ha , in e es ingly enough, he g-diffe en iabili y o a unc ion a a poin o Ngdoes no
imply he g-con inui y o he unc ion a he poin . The g-diffe en iabili y o a unc ion only gua an ees he
g-con inui y a he poin s o (a, b) (Cg∪Dg∪Ng).
Defini ion 3.4 (C1
g([a, b]; F)space). Le g:R →Rbe such ha a /∈N−
gand b /∈N+
g∪Dg. We say ha
:[a, b] →Fbelongs o C1
g([a, b]; F)i he ollowing condi ions a e me :
1. ∈C
g([a, b]; F),
2. ∃ 
g(x), o e e y x ∈[a, b] Cg,
3. ∃h ∈C
g([a, b]; F)such ha h(x) = 
g(x), o e e y x ∈[a, b] Cg.
Unless necessa y, we will w i e C1
g([a, b]) ins ead o C1
g([a, b]; F) o b e i y.
Le us show now ha i we assume ha b /∈Cg(obse e ha , in ha case, he hypo hesis [a, b] Cg=∅
is i ially sa isfied) he p e ious defini ion is consis en inso a as he unc ion gi en by 3, i i exis s, i is
unique.
P oposi ion 3.5. Le [a, b] ⊂Rbe a closed in e al, g:R →Ra de i a o such ha a /∈N−
gand b /∈
Cg∪N+
g∪Dgand ∈C
g([a, b]; F)be g-diffe en iable a e e y x ∈[a, b] Cg. I h1, h2∈C
g([a, b]) a e such
ha h1(x) =h2(x) = 
g(x), o e e y x ∈[a, b] Cg, hen h1=h2.
P oo . Le us show ha h1(x) =h2(x) o e e y x ∈Cg. Gi en x∈Cg, he e exis s a unique connec ed
componen o Cg, (an, bn), such ha x∈(an, bn). Le us see ha h1(x) =h2(x) = 
g(bn). Indeed, since h1
is g-con inuous, we ha e, by P oposi ion 2.10, ha h1is cons an on (an, bn)and le -con inuous, he e o e,
h1(x) =h1(bn) = 
g(bn) o e e y x ∈(an, bn). The case o h2is p o en analogously. 
Rema k 3.6. Obse e ha i b ∈Cg, gi en a unc ion ∈C
1
g([a, b]) he unc ion h ∈C
g([a, b]) such ha

g(x) =h(x) o e e y x ∈[a, b] Cgis no uniquely defined in a neighbo hood o poin bsince we can no
compu e he g-de i a i e a x =bn, wi h b ∈(an, bn) ⊂Cg.
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 9
A possible way o defining he g-de i a i e a he poin s o Cg, which is cohe en wi h he defini ion
o he space C1
g, ollows om he p e ious p oo . Indeed, we can gene alize Defini ion 3.1 in he ollowing
e ms.
Defini ion 3.7. Le [a, b] ⊂Rbe a closed in e al and g:R →Ra de i a o such ha a /∈N−
gand
b /∈Cg∪N+
g∪Dg. We define he S iel jes de i a i e, o g–de i a i e, o a unc ion :[a, b] →Fa a poin
∈[a, b]as

g( )=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
lim
s→
(s)− ( )
g(s)−g( ), /∈Dg∪Cg,
lim
s→ +
(s)− ( )
g(s)−g( ), ∈Dg,
lim
s→b+
n
(s)− (bn)
g(s)−g(bn), ∈(an,b
n)⊂Cg,
(3.3)
wi h an, bnas in (2.1); p o ided he co esponding limi s exis . In ha case, we say ha is g–diffe en iable
a . The g-de i a i e in he poin s Ngmus be unde s ood as in (3.1).
Rema k 3.8. I ollows om he Defini ion 3.7 ha , o ∈Dg, 
g( )exis s i and only i ( +)exis s and,
in ha case,

g( )= ( +)− ( )
Δ+g( ).
Simila ly, o any ∈(an, bn) ⊂Cg, we ha e ha 
g( )exis s i and only i 
g(bn)exis s and, in ha case,

g( ) = 
g(bn).
The ollowing esul which includes some basic p ope ies o he S iel jes de i a i e is a gene aliza ion o
[12, P oposi ion 3.13].
P oposi ion 3.9. Le [a, b] ⊂Rbe a closed in e al and g:R →Ra de i a o such ha a /∈N−
gand
b /∈Cg∪N+
g∪Dg. Gi en an elemen ∈[a, b]we deno e by:
∗= , /∈Cg,
bn, ∈(an,b
n)⊂Cg,
wi h an, bnas in (2.1). I 1, 2a e wo g-diffe en iable unc ions a , hen:
•The unc ion λ1 1+λ2 2is g-diffe en iable a o any λ1, λ2∈Rand
(λ1 1+λ2 2)
g( )=λ1( 1)
g( )+λ2( 2)
g( ).
•The p oduc 1 2is g-diffe en iable a and
( 1 2)
g( )=( 1)
g( ) 2( ∗)+( 2)
g( ) 1( ∗)+( 1)
g( )( 2)
g( )Δ+g( ∗).(3.4)
•I 2( ∗)( 2( ∗)+( 2)
g( ) Δ+g( ∗)) =0, he quo ien 1/ 2is g-diffe en iable a and
 1
2
g
( )= ( 1)
g( ) 2( ∗)−( 2)
g( ) 1( ∗)
2( ∗)( 2( ∗)+( 2)
g( )Δ
+g( ∗)) (3.5)
16 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
( )= B( ) C( ),
whe e B∈ACgB([0, T]; F)is he unique solu ion o he p oblem
 
gB( )−β( ) ( )=0,g
B−a.e. ∈[0,T),
(0) = 1,(4.5)
gi en by
B( )=
s∈[0, )∩Dg!1+β(s)Δ+g(s)"; (4.6)
and C∈ACgC([0, T]; F)is he unique solu ion o
 
gC( )−β( ) ( )=0,g
C−a.e. ∈[0,T),
(0) = 0,(4.7)
gi en by
C( )=u(gC( )),(4.8)
whe e u ∈AC([0, T]; F)is he unique solu ion o
u( )=
β( )u( ),a.e. ∈[0,gC(T)),
u(0) = 0,(4.9)
whe e 
β=β◦γand γis p o ided by Defini ion 2.5.
Fu he mo e, can be w i en as
( )= 0exp ⎛
⎜
⎝
[0, )
β(s)dμg⎞
⎟
⎠,(4.10)
wi h

β( )=⎧
⎪
⎨
⎪
⎩
β( ), ∈[0,T) Dg,
ln (1 + β( )Δ+g( ))
Δ+g( ), ∈[0,T)∩Dg.
P oo . Exis ence and uniqueness: I sol es (4.1), hen (x, y) whe e x := Re and y:= Im sol es he eal
sys em
⎧
⎪
⎨
⎪
⎩
x
g( )−Re β( )x( )+Imβ( )y( )=0,g−a.e. ∈[0,T),
y
g( )−Im β( )x( )−Re β( )y( )=0,g−a.e. ∈[0,T),
x(0) = Re 0,y(0) = Im 0,
(4.11)
and ice- e sa, ha is, he unc ion x +iy, whe e (x, y)is a solu ion o (4.11), sol es (4.1). Now, i is easy
o see ha (4.11)sa isfies he condi ions o [6, Theo em 4.3] wi h L =| Re(β)| +| Im(β)|, so i has a unique
solu ion on [0, T]. Hence, (4.1)has a unique solu ion he e as well.

F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 17
Exp ession o he solu ion: Gi en he na u e o p oblem (4.1) whe e he g-de i a i e has o be a mul iple
o i sel i is only na u al o use an ansa z o he o m
( )= 0exp ⎛
⎜
⎝
[0, )
β(s)dμg⎞
⎟
⎠,
wi h 
β∈L
1
g([0, T); F). Gi en ha Mg⊂M
gCand Mg⊂M
gB, i is clea ha i 
β∈L
1
g([0, T); F), hen

β∈L
1
gC([0, T); F)and 
β∈L
1
gB([0, T); F). Fu he mo e,
( )= 0exp ⎛
⎜
⎝
[0, )
β(s)dμgB+
[0, )
β(s)dμgC⎞
⎟
⎠
= 0exp ⎛
⎜
⎝
[0, )
β(s)dμgB⎞
⎟
⎠exp ⎛
⎜
⎝
[0, )
β(s)dμgC⎞
⎟
⎠= B( ) C( ),
whe e B( ) := exp [0, )
β(s)dμB
gand C( ) := 0exp [0, )
β(s)dμC
g. F om he defini ion we deduce
ha B∈AC
gB([0, T]; F)and C∈AC
gC([0, T]; F). Hence, gi en he gB-con inui y o B, we ha e ha
( B)
g( ) =0 o e e y ∈[0, T) (Dg∪Cg) and, hanks o he gC-con inui y o C, i holds ha ( C)
g( ) =0
o e e y ∈[0, T) ∩Dg. Thus, by P oposi ion 3.9,

g( )=( B)
g( ) C( ), ∈[0,T)∩Dg,
B( )( C)
g( ),g−a.e. ∈[0,T) (Dg∪Cg).
This implies ha we will ha e a diffe en equa ion o each o he componen s o he solu ion:
( B)
g( )=β( ) B( ), ∈[0,T)∩Dg,(4.12)
( C)
g( )=β( ) C( ),g−a.e. ∈[0,T) (Dg∪Cg).(4.13)
We will s a s udying equa ion (4.12). Fo ∈[0, T) ∩Dgwe ha e ha
( B)
g( )= B( +)− B( )
Δ+g( )=( B)
gB( ).
Now, i we de elop equa ion (4.12):
B( +)− B( )
Δ+g( )=β( ) B( )
and we ge ha
B( +)= B( )(1 + β( )Δ
+g( )), ∈[0,T)∩Dg.(4.14)
In o de o ge a solu ion candida e o equa ion (4.12), define h( ) =ln(1+β( )Δ+g( ))/Δ+g( )i
∈Dg, h( ) =0i /∈Dg. Then, aking in o accoun ha μgB( ) =0 o e e y /∈Dg, we define
18 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
H( ):=exp⎛
⎜
⎝
[0, )
h( )dμgB⎞
⎟
⎠=exp⎛
⎜
⎝
[0, )
ln(1 + β(s)Δ+g(s))
Δ+g(s)dμgB⎞
⎟
⎠
=exp⎛
⎝
s∈[0, )∩Dg
ln(1 + β(s)Δ+g(s))⎞
⎠
=exp⎛
⎝
s∈[0, )∩Dg!ln 1+β(s)Δ+g(s)+iA g(1 + β(s)Δ+g(s))"⎞
⎠.
To show ha His well defined, le us check ha he se ies

s∈[0, )∩Dg
ln 1+β(s)Δ+g(s)and 
s∈[0, )∩Dg
A g(1 + β(s)Δ+g(s))
a e absolu ely con e gen . We ha e ha

s∈[0,T )∩Dgln 1+β(s)Δ+g(s)=
s∈Aln 1+β(s)Δ+g(s)+
s∈Bln 1+β(s)Δ+g(s),
whe e
A=s∈[0,T)∩Dg:1+β(s)Δ+g(s)≥1=s∈[0,T)∩Dg:ln
1+β(s)Δ+g(s)≥0,
B=s∈[0,T)∩Dg:1+β(s)Δ+g(s)<1=s∈[0,T)∩Dg:ln
1+β(s)Δ+g(s)<0.
In o de o bound he sum on Ai is enough o ake in o accoun ha 0 ≤ln(1 +x) ≤x o e e y x ∈[0, ∞):

s∈Aln 1+β(s)Δ+g(s)=
s∈A
ln 1+β(s)Δ+g(s)≤
s∈A
ln !1+|β(s)|Δ+g(s)|"≤
s∈A|β(s)|Δ+g(s)<∞,
because β∈L
1
gB([0, T), F).
Now, le us ocus on he sum on B. Fo any s ∈B, aking in o accoun ha 1 +β(s)Δ+g(s) =0, we
ha e ha
0<1+β(s)Δ
+g(s)2=[1+Re(β(s)) Δ+g(s)]2+[Im(β(s)) Δ+g(s)]2
=1+2Re(β(s)) Δ+g(s)+|β(s)Δ
+g(s)|2<1.
In pa icula , 2 Re(β(s)) Δ+g(s) +|β(s) Δ+g(s)|2<0which yields Re(β(s)) <0. Now, we can conside he
ollowing se s:
B1=#s∈B:0<1+2Re(β(s))Δ+g(s)+|β(s)Δ+g(s)|2<1
2$,
B2=#s∈B:1
2≤1+2Re(β(s))Δ+g(s)+|β(s)Δ+g(s)|2<1$.
Obse e ha B=B1∪B2. The defini ion o B1implies ha
1>2|Re(β(s))|Δ+g(s)−|β(s)Δ+g(s)|2>1
2,∀s∈B1.
The e o e,
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 19
|Re(β(s))|Δ+g(s)>1
4,∀s∈B1.
Hence, we ha e ha B1is fini e since, o he wise, we would ha e ha β/∈L
1
gB([0, T), F), which is a
con adic ion. Fo he elemen s in he se B2we ha e ha :
1
2≥2|Re(β(s))|Δ+g(s)−|β(s)Δ+g(s)|2>0,∀s∈B2.
Thus, i we ake in o accoun ha ln(1/(1 −x)) ≤2x, o e e y x ∈[0, 1/2],
ln 1+β(s)Δ+g(s)=1
2ln !1+2Re(β(s))Δ+g(s)+|β(s)Δ+g(s)|2"
=1
2ln !1/!1+2Re(β(s))Δ+g(s)+|β(s)Δ+g(s)|2""
=1
2ln !1/!1−!2|Re(β(s))|Δ+g(s)−|β(s)Δ+g(s)|2"""
≤2!2|Re(β(s))|Δ+g(s)−|β(s)Δ+g(s)|2"≤4|Re(β(s))|Δ+g(s).
Hence,

s∈Bln 1+β(s)Δ+g(s)<∞.
Le us now bound he e m associa ed wi h he a gumen . Taking in o accoun ha |a an(x)| ≤|x| o e e y
x ∈R, we ha e ha

s∈[0,T )∩DgA g(1 + β(s)Δ+g(s))≤
s∈[0,T )∩Dg
|Im(β(s))Δ+g(s)|
|1+Re(β(s))Δ+g(s)|.
Le us di ide he se [0, T) ∩Dgin o he subse s

B1=s∈[0,T)∩Dg:|Re(β(s))|Δ+g(s)>1/2,

B2=([0,T)∩Dg) 
B1.
Obse e ha 
B1mus be o fini e ca dinali y. On he o he hand, gi en ∈
B2,
1+Re(β(s))Δ+g(s)≥1
2.
Thus,

s∈B2
|Im(β(s))Δ+g(s)|
|1+Re(β(s))Δ+g(s)|≤2
s∈B2Im(β(s))Δ+g(s)<∞.
Hence, we conclude ha His well defined. In o de o p o e ha His a solu ion o (4.12), we obse e ha ,
gi en ∈[0, T) ∩Dg,
H( +) = lim
s→ +exp ⎛
⎜
⎝
[0,s)
ln(1 + β(s)Δ+g(s))
Δ+g(s)dμgB⎞
⎟
⎠
20 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
= lim
s→ +exp ⎛
⎜
⎝
[0, )
ln(1 + β(s)Δ+g(s))
Δ+g(s)dμgB+ln(1+β( )Δ+g( )) + 
( ,s)
ln(1 + β(s)Δ+g(s))
Δ+g(s)dμgB⎞
⎟
⎠
=(1 + β( )Δ+g( )) exp ⎛
⎜
⎝
[0, )
ln(1 + β(s)Δ+g(s))
Δ+g(s)dμgB⎞
⎟
⎠=(1+β( )Δ+g( ))H( ),
so equa ion (4.14)holds and B:= His a solu ion o (4.12). Obse e ha , gi en any se A ⊂[0, T) Dg,
we ha e A =(A Dg) ∪(A ∩(Dg Dg)) hus μ∗
gB(A) ≤μ∗
gB(A Dg) +μ∗
gB(A ∩(Dg Dg)) ≤μ∗
gB(A Dg) +
μ∗
g(Dg Dg) =0. The e o e Bsa isfies (4.5) and, mo eo e ,
B( )=exp⎛
⎝
s∈[0, )∩Dg
ln(1 + β(s)Δ+g(s))⎞
⎠=
s∈[0, )∩Dg!1+β(s)Δ+g(s)".
Le us now s udy equa ion (4.13). Fi s , obse e ha , gi en an elemen ∈[0, T) (Dg∪Cg), he e exis s
δ>0such ha gis con inuous on ( −δ, +δ). In he case ∈N−
gwe u he know ha gis s ic ly
inc easing on he in e al ( −δ, ], and cons an on ( , +δ). In he case ∈N+
g, gwould be cons an on
( −δ, )and s ic ly inc easing on [ , +δ). In any case (obse e ha , i ∈N−
gwe ha e o ake he limi
om he le and in he case ∈N+
g he limi om he igh , espec i ely):
( C)
g( ) = lim
s→
C(s)− C( )
g(s)−g( )= lim
s→
C(s)− C( )
gC(s)−gC( )=( C)
gC( ).
Hence, aking in o accoun ha μ∗
g(A) =0 ⇔μ∗
gC(A) =0 o any A ⊂[0, T) Dg, oge he wi h he ac
ha Cg=CgC, we see ha equa ion (4.13)is equi alen o
( C)
gC( )=β( ) C( ),g
C−a.e. ∈[0,T) (Dg∪CgC).(4.15)
Le us obse e ha μgC(Dg∪CgC) ≤μgC(Dg Dg) +μgC(Dg) +μgC(CgC) =0, since μgC(Dg Dg) ≤
μg(Dg Dg) =0by hypo hesis. The e o e, (4.15)is equi alen o:
( C)
gC( )=β( ) C( ),g
C−a.e. ∈[0,T).(4.16)
Now we will see ha C( ) := u(gC( )), wi h u ∈AC([0, gC(T)]; F) he solu ion o (4.9)sa isfies equa-
ion (4.16). On he one hand, we ha e ha 
β=β◦γ∈L
1([0, gC(T)]; F). Indeed, he measu abili y is a
consequence o P oposi ion 2.7. Now, using a simila a gumen as he one in he p oo o Co olla y 2.8:

[0,gC(T))
|
β|dμ=
[0,T )
|β|dμgC≤
[0,T )
|β|dμg<∞.
Thus, (4.9)admi s a unique solu ion
u( )= 0exp ⎛
⎜
⎝
[0, )
β(s)dμ⎞
⎟
⎠∈AC([0,gC(T)]; F).
In pa icula , C( ) =u(gC( )) is such ha ( C)
g( ) =0 o e e y ∈Dg. Indeed,
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 21
( C)
g( ) = lim
s→ +
C(s)− C( )
g(s)−g( )= lim
s→ +
u(gC(s)) −u(gC( ))
g(s)−g( )=0,
hanks o he con inui y o he composi ion u ◦gC. On he o he hand, since u ∈AC([0, gC(T)]; F)is he
solu ion o (4.9), he e exis s a Lebesgue-null se N⊂[0, gC(T)] such ha
u( )=
β( )u( ),∀ ∈[0,gC(T)] N.
In pa icula ,
u(gC( )) = 
β(gC( )) u(gC( )),∀ ∈[0,T] (gC)−1(N),
whence, by P oposi ion 4.1,
( C)
gC( )=u(gC( )) = 
β(gC( )) u(gC( )),∀ ∈[0,T] (gC)−1(N).
Taking in o accoun ha γ(gC( )) = o e e y ∈[0, T] (CgC∪N+
gC), ha μgC(CgC∪N+
gC) =0and ha
μgC((gC)−1(N)) =0(see he p oo o P oposi ion 2.7), we deduce ha
( C)
gC( )=β( ) C( ),g
C-a.e. ∈[0,T).
Las , in ega d o C, using a easoning simila o he one used in he p oo o he Co olla y 2.8, we ha e
ha
C( )=u(gC( )) = 0exp ⎛
⎜
⎝
[0,gC( )) 
β(s)dμ⎞
⎟
⎠= 0exp ⎛
⎜
⎝
[0, )
β(s)dμgC⎞
⎟
⎠.
Finally, le us check ha := C Bis in he space ACg([0, T]; F). To show his, le us define

β( )=⎧
⎪
⎨
⎪
⎩
β( ), ∈[0,T) Dg,
ln (1 + β( )Δ+g( ))
Δ+g( ), ∈[0,T)∩Dg,
and check ha
( )= 0exp ⎛
⎜
⎝
[0, )
β(s)dμg⎞
⎟
⎠= 0exp ⎛
⎜
⎝
[0, ) Dg
β(s)dμg+
s∈[0, )∩Dg
β(s)Δ+g(s)⎞
⎟
⎠.
Indeed, on he one hand,
( )= 0⎡
⎣
s∈[0, )∩Dg!1+β(s)Δ+g(s)"⎤
⎦exp ⎛
⎜
⎝
[0,gC( )) 
β(s)dμ⎞
⎟
⎠
= 0exp ⎛
⎜
⎝
[0,gC( )) 
β(s)dμ+
s∈[0, )∩Dg
ln (1 + β(s)Δ+g(s))
Δ+g(s)Δ+g(s)⎞
⎟
⎠.
Now, hanks o he ac ha μ(Dg) =0as i is a coun able se , we see ha

22 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010

[0,gC( ))
(β◦γ)(s)dμ=
[0,gC( ))
(
β◦γ)(s)dμ.
Thus, by Co olla y 2.8,

[0, )
β(s)dμg=
[0,gC( )) 
β(s)dμ+
s∈[0, )∩Dg
ln (1 + β(s)Δ+g(s))
Δ+g(s)Δ+g(s).
Finally, i is clea ha 
β∈L
1
g([0, T); F), he e o e ∈ACg([0, T); F). 
Rema k 4.3. We mus ake in o accoun he ollowing ema ks:
1. I gCis cons an , hen he solu ion o (4.1)is educed o 0 Band he hypo hesis μg(Dg Dg) =0is
no necessa y.
2. The hypo hesis μg(Dg Dg) =0 ha appea s in he s a emen o Theo em 4.2 has been used o exp ess
he solu ion o (4.1)as he p oduc o he solu ions o he p oblems (4.5)and (4.7). This hypo hesis is no
essen ial o gua an ee he exis ence o a solu ion o p oblem (4.1). E en in he case μg(Dg Dg) =0, we will
ha e (4.10)is well defined and a alid solu ion o p oblem (4.1). Indeed, since 
β∈L
1
g([0, T); F), we ha e
ha
⎛
⎜
⎝
[0, )
β(s)dμg⎞
⎟
⎠

g
( )=
β( ),g-a.e. ∈[0,T).
The e o e, (4.2) ensu es ha
⎛
⎜
⎝exp ⎛
⎜
⎝
[0, )
β(s)dμg⎞
⎟
⎠⎞
⎟
⎠

g
( )=β( )exp
⎛
⎜
⎝
[0, )
β(s)dμg⎞
⎟
⎠,g-a.e. ∈[0,T) Dg.(4.17)
Now, gi en ∈[0, T) ∩Dg,
lim
s→ +exp ⎛
⎜
⎝
[0,s)
β(s)dμg⎞
⎟
⎠= lim
s→ +exp ⎛
⎜
⎝
[0, )
β(s)dμg+ln(1+β( )Δ+g( )) + 
( ,s)
β(s)dμg⎞
⎟
⎠
=(1 + β( )Δ+g( )) exp ⎛
⎜
⎝
[0, )
β(s)dμg⎞
⎟
⎠,
so equa ion (4.17)is also sa isfied o he poin s o Dg.
Rema k 4.4. The p e ious esul is a gene aliza ion o he esul s in [3, Sec ion 6] o se e al easons.
1. The solu ion ob ained is alid in he complex case, whe eas in [3]i is only applied o he eal case.
The gene aliza ion o he complex case is immedia e conside ing he complex exponen ial and he p incipal
b anch o he complex loga i hm.
2. We ha e p o en ha he hypo hesis

s∈[0,T )∩Dgln 1+β(s)Δ+g(s)<∞
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 23
occu ing in [3, Defini ion 6.1 and Lemma 6.5] is no necessa y, i being a di ec consequence o β∈
L1
g([0, T); F)and g(T) <∞. This was also p o en in [13, Lemma 3.1] o he eal case.
3. The solu ion ob ained gene alizes ha in [3, Lemma 6.5]. Indeed, in he pa icula case β∈L
1
g([0, T); R)
and gi en ha 1 +β( )Δ+g( ) =0 o e e y ∈[0, T) ∩Dg, we ha e ha , o e e y ∈[0, T) ∩Dg,
A g(1 + β( )Δ+g( )) = π, 1+β( )Δ+g( )<0,
0,1+β( )Δ+g( )>0.
Hence, i we w i e T−
β:= { ∈[0, T) ∩Dg:1 +β( )Δ+g( ) <0}and T+
β={ ∈[0, T) ∩Dg:1 +β( )Δ+g( ) >
0}(obse e ha T−
βis o fini e ca dinali y), we ha e ha
ln !1+β( )Δ+g( )"=ln 1+β( )Δ+g( ), ∈T+
β,
ln 1+β( )Δ+g( )+iπ, ∈T−
β.
Taking in o accoun he p e ious obse a ions,
( )=exp
⎛
⎜
⎝
[0, ) Dg
β(s)dμg+
∈[0, )∩Dg
ln 1+β( )Δ+g( )+i
s∈[0, )∩T−
β
π⎞
⎟
⎠
=cos
⎛
⎜
⎝
s∈[0, )∩T−
β
π⎞
⎟
⎠exp ⎛
⎜
⎝
[0, ) Dg
β(s)dμg+
∈[0, )∩Dg
ln 1+β( )Δ+g( )⎞
⎟
⎠.
Hence, i T−
β={ 1, ..., k}and k+1 := T, we ge
( )=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
exp ⎛
⎜
⎝
[0, ) Dg
β(s)dμg+
∈[0, )∩Dg
ln 1+β( )Δ+g( )⎞
⎟
⎠, ∈[0,
1],
cos(jπ)exp⎛
⎜
⎝
[0, ) Dg
β(s)dμg+
∈[0, )∩Dg
ln 1+β( )Δ+g( )⎞
⎟
⎠, ∈( j,
j+1],
j=1,...,k,
which is p ecisely he solu ion in [3, Lemma 6.5].
4. In he case ha he e exis s some elemen ∈[0, T) ∩Dgsuch ha 1 +β( )Δ+g( ) =0, he se
T0
β:= { ∈[0,T)∩Dg:1+β( )Δ+g( )=0}
is o fini e ca dinali y and, he e o e, i we deno e by 0
β:= min T0
βi T0
β=∅, 0
β:= To he wise, we ha e
ha
( )=⎧
⎨
⎩
u(gC( ))
s∈[0, )∩Dg!1+β(s)Δ+g(s)", ∈[0,
0
β],
0, ∈( 0
β,T].
Taking in o accoun ha we a e assuming ha gis con inuous a =0, we ha e ha 0
β=minT0
β>0.
Thus, ( ) =0 o e e y ∈[0, 0
β].
24 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
Defini ion 4.5. Gi en an elemen β∈L
1
g([0, T); F)and 0=1, we deno e he solu ion o p oblem (4.1)
cons uc ed in Theo em 4.2 by expg(β; 0, ) ∈AC
g([0, T]; F)and call i he complex g-exponen ial map o
jus g-exponen ial map.
In he ollowing esul we p esen some impo an p ope ies o he complex g-exponen ial unc ion.
P oposi ion 4.6. Le β, β1, β2∈L
1
g([0, T); F). The ollowing p ope ies hold:
1. I a =Reβand b =Imβ hen
expg(β;0, )=
u∈[0, )∩Dg!1+a(u)Δ+g(u)+ib(u)Δ+g(u)"exp ⎛
⎜
⎝
[0,gC( ))
(a◦γ)dμ⎞
⎟
⎠
·⎡
⎢
⎣cos ⎛
⎜
⎝
[0,gC( ))
(b◦γ)dμ⎞
⎟
⎠+isin ⎛
⎜
⎝
[0,gC( ))
(b◦γ)dμ⎞
⎟
⎠⎤
⎥
⎦.
(4.18)
2. expg(β;0, )=exp
g(β; 0, ), o e e y ∈[0, T].
3. Gi en n ∈N, expg(β; 0, )n=exp
g(pn(β); 0, ) ∈ACg([0, T]; F), whe e
pn(β)( )=nβ( )+
n

k=2 n
kβ( )kΔ+g( )k−1,n∈N.
4. Gi en n ∈N, expg(β; 0, )−n=exp
g(qn(β); 0, ) ∈ACg([0, 0
β]; F), whe e
qn(β)( )=−pn(β)( )
1+pn(β)( )Δ
+g( ),n∈N.
Obse e ha expg(β; 0, )−nis no well defined in ( 0
β, T]since expg(β; 0, ·) =0in ha se .
5. Fo all ∈[0, T),
expg(β1;0, )exp
g(β2;0, )=exp
g(β1+β2+β1β2Δ+g;0, ).(4.19)
P oo . 1. Indeed,
expg(a+bi;0, )
=exp⎛
⎜
⎝
[0, ) Dg
a(s)dμg+i
[0, ) Dg
b(s)dμg⎞
⎟
⎠exp ⎛
⎝
u∈[0, )∩Dg
ln(1 + (a(u)+ib(u))Δ+g(u))⎞
⎠
=exp⎛
⎜
⎝
[0, ) Dg
a(s)dμg⎞
⎟
⎠
u∈[0, )∩Dg!1+a(u)Δ+g(u)+ib(u)Δ+g(u)"
·⎡
⎢
⎣cos ⎛
⎜
⎝
[0, ) Dg
b(s)dμg⎞
⎟
⎠+isin ⎛
⎜
⎝
[0, ) Dg
b(s)dμg⎞
⎟
⎠⎤
⎥
⎦.
Now he o mula is ob ained easoning as in Co olla y 2.8.
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 25
2. This p ope y is clea om he defini ion o he complex conjuga e.
3. Obse e ha pn(β) ∈L
1
g([0, T); F)since
pn(β)L1
g([0,T );F)≤nβL1
g([0,T )+
n

k=2 n
k
∈[0,T )∩Dg!|β( )|Δ+g( )"k<∞.
Thus he solu ion o p oblem (4.1) whe e we conside pn(β)ins ead o βis gi en by = B Cwhe e
B( )=
s∈[0, )∩Dg1+nβ(s)+
n

k=2 n
kβ(s)kΔ+g(s)k−1Δ+g(s),
=
s∈[0, )∩Dg1+nβ(s)Δ+g(s)+
n

k=2 n
kβ(s)kΔ+g(s)k,
=
s∈[0, )∩Dgn

k=0 n
kβ(s)kΔ+g(s)k=
s∈[0, )∩Dg!1+β(s)Δ+g(s)"n
=⎛
⎝
s∈[0, )∩Dg!1+β(s)Δ+g(s)"⎞
⎠
n
,
C( )=exp⎛
⎜
⎝
[0, )nβ(s)+
n

k=2 n
kβ(s)kΔ+g(s)k−1dμgC⎞
⎟
⎠=exp⎛
⎜
⎝n
[0, )
β(s)dμgC⎞
⎟
⎠
=⎡
⎢
⎣exp ⎛
⎜
⎝
[0, )
β(s)dμgC⎞
⎟
⎠⎤
⎥
⎦
n
.
Hence, expg(β; 0, )n=exp
g(pn(β); 0, ).
4. Obse e ha qn(β) ∈L
1
g([0, T); F)since
qnL1
g([0, 0
βT);F)≤nβL1
g([0,T )+
∈[0,T )∩Dg
|pn(β)( )Δ
+g( )|
|1+pn(β)( )Δ
+g( )|<∞,
because pn(β) ∈L
1
g([0, T); F). The e o e, he solu ion o p oblem (4.1), whe e we conside qn(β)ins ead o
β, is gi en by = B Cwhe e
B( )=
s∈[0, )∩Dg1−nβ(s)+n
k=2 !n
k"β(s)kΔ+g(s)k−1
1+!nβ(s)+n
k=2 !n
k"β(s)kΔ+g(s)k−1"Δ+g(s)Δ+g(s),
=
s∈[0, )∩Dg1
1+!nβ(s)+n
k=2 !n
k"β(s)kΔ+g(s)k−1"Δ+g(s)
=⎛
⎝
s∈[0, )∩Dg!1+β(s)Δ+g(s)"⎞
⎠
−n
,
C( )=exp⎛
⎜
⎝
[0, )
−nβ(s)+n
k=2 !n
k"β(s)kΔ+g(s)k−1
1+!nβ(s)+n
k=2 !n
k"β(s)kΔ+g(s)k−1"Δ+g(s)dμgC⎞
⎟
⎠
32 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
expg(x;0,s)−1expg(z;0,s)=exp
g−x
1+xΔ+g;0,s
expg(z;0,s)
=exp
g−x
1+xΔ+g+z−xzΔ+g
1+xΔ+g;0,s
=exp
gz−x
1+xΔ+g;0,s
.
The e o e,

[0, )
expg(x;0,s)−1expg(z;0,s)
1+xΔ+g(s)dμg(s)=
[0, )
1
1+xΔ+g(s)expgz−x
1+xΔ+g;0,s
dμg(s)
=(z−x)−1
[0, )expgz−x
1+xΔ+g;0,·
g
(s)dμg(s)
=(z−x)−1+expgz−x
1+xΔ+g;0,
−expgz−x
1+xΔ+g;0,0,
=(z−x)−1-expg(x;0, )−1expg(z;0, )−1..
Finally,
( )=exp
g(x;0, )+exp
g(x;0, )(z−x)−1-expg(x;0, )−1expg(z;0, )−1.
=exp
g(x;0, )+(z−x)−1-expg(z;0, )−expg(x;0, )..
Obse e ha , diffe en ia ing 
gagain, we ob ain ha 
g−(x +z) 
g+x z =0, so, o any alues P, Q ∈C,
aking x =(−P+/P2−4Q)/2, z=P−x, sol es he equa ion 
g+P

g+Q =0.
This ac illus a es how we can ob ain a solu ion o a second o de p oblem om a fi s o de p oblem.
In he nex sec ion we s udy his ype o p oblems.
5. Linea g-diffe en ial p oblems o second o de wi h cons an coefficien s
In his sec ion we conside g-diffe en ial p oblems o second o de wi h cons an coefficien s. Since we
will assume ha he coefficien s a e cons an , we will look o solu ions in he space BC2
g([0, T]; F). Once
again, we assume ha 0 /∈N−
gand T/∈N+
g∪Dg∪Cg.
5.1. The homogeneous case
Le us conside he second o de homogeneous linea Cauchy p oblem
⎧
⎪
⎨
⎪
⎩

g( )+P

g( )+Q ( )=0,∀ ∈[0,T],
(0) = x0,

g(0) = 0,
(5.1)
whe e P, Q, x0, 0∈F. We s a by defining wha we unde s and as a solu ion o p oblem (5.1).
Defini ion 5.1. We say ∈BC2
g([0, T]; F)is a solu ion o (5.1)i i sa isfies he equa ion

g( )+P

g( )+Q ( )=0,∀ ∈[0,T]
and he ini ial condi ions (0) =x0and 
g(0) = 0.
We ha e he ollowing lemma, whose p oo is s aigh o wa d om he linea i y o he g-de i a i e.

F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 33
Lemma 5.2. Le x0, 0∈Fand 1, 2∈BC2
g([0, T]; F)be such ha
( k)
g( )+P( k)
g( )+Q
k( )=0,∀ ∈[0,T],k=1,2.(5.2)
I 1(0) ( 2)
g(0) − 2(0) ( 1)
g(0) =0, hen =c1 1+c2 2is a solu ion o (5.1), whe e
c1=( 2)
g(0) x0− 0 2(0)
1(0) ( 2)
g(0) − 2(0) ( 1)
g(0),
c2= 0 1(0) −( 1)
g(0) x0
1(0) ( 2)
g(0) − 2(0) ( 1)
g(0).
Theo em 5.3. Fo (5.1), he ollowing hold:
•I P2−4 Q =0, hen, defining λ1=(−P+/P2−4Q)/2and λ2=(−P−/P2−4Q)/2, we ha e
ha
( )= 0−λ2x0
λ1−λ2expg(λ1;0, )− 0−λ1x0
λ1−λ2expg(λ2;0, )
is a solu ion o (5.1). Fu he mo e, ∈BC∞
g([0, T]; F)and i is he unique solu ion in ha space.
•I P2−4 Q =0, hen, aking λ =−P/2,
( )=x0expg(λ;0, )+( 0−λx
0)exp
g(λ;0, )
[0, )
1
1+λΔ+g(s)dμg(s)
is a solu ion o (5.1). Fu he mo e, ∈BC∞
g([0, T]; F)and i is he unique solu ion in ha space.
P oo . We conside he cha ac e is ic equa ion o p oblem (5.1),
λ2+Pλ+Q=0.
I P2−4 Q =0, le 1=exp
g(λ1; 0, )and 2( ) =exp
g(λ2; 0, ). By Co olla y 4.19 we ha e ha
1, 2∈BC
∞
g([0, T]; F) ⊂BC
2
g([0, T]; F). Fu he mo e, i can be checked ha bo h unc ions sa is y (5.2).
On he o he hand,
1(0) ( 2)
g(0) − 2(0) ( 1)
g(0) = λ2−λ1=0.
Hence, by Lemma 5.2, he e exis s a solu ion o p oblem (5.1)gi en by
( )= 0−λ2x0
λ1−λ2expg(λ1;0, )− 0−λ1x0
λ1−λ2expg(λ2;0, ).
I P2−4 Q =0we ge he double oo λ =−P/2o he cha ac e is ic equa ion. Obse e ha he le
hand side o he equa ion occu ing in (5.1)can be w i en as (∂g+P/2)2 whe e ∂gdeno es he g-de i a i e
ope a o . Hence, we define 1( ) := expg(λ; 0, ), which is a solu ion o (∂g+P/2) =0and conside he
unique solu ion o
( 2)
g( )=λ
2( )+ 1( ),g−a.e. ∈[0,T),
2(0) = 0.
34 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
Since (∂g+P/2) 1=0, i is clea ha (∂g+P/2)2 2=0. Fu he mo e, 2(0) =0and ( 2)
g(0) =1, so
2is he solu ion we a e looking o . By Co olla y 4.19, 1∈BC
∞
g([0, T]; F) and, applying Co olla y 4.21,
2∈BC∞
g([0, T]; F)as well. Now, hanks o P oposi ion 4.12, we ha e ha :
2( )=exp
g(λ;0, )
[0, )
1
1+λΔ+g(s)dμg.
Since 2(0) =0, ( 2)
g(0) =1we ha e ha :
1(0) ( 2)
g(0) − 2(0) ( 1)
g(0) = 1 =0.
Thus, by Lemma 5.2, he e exis s a solu ion o p oblem (5.1)gi en by
( )=x0expg(λ;0, )+( 0−λx
0)exp
g(λ;0, )
[0, )
1
1+λΔ+g(s)dμg(s).
Finally, i we define u( ) = 
g( ) ∈BC∞
g([0, T]) we ha e ha he pai o unc ions (u, ) ∈[ACg([0, T]; F)]2
sa isfies he ollowing sys em o diffe en ial equa ions:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
u
g
( )=01
−Q−P ( )
u( ),
(0) =x0,u(0) = 0.
Thanks o [3, Theo em 7.3] we ha e ha he p e ious sys em has a unique solu ion in [ACg([0, T]; F)]2,
he e o e is he unique solu ion o (5.2)in he space BC∞
g([0, T]; F). 
5.2. The non homogeneous case
In his sec ion we ocus on he non homogeneous e sion o he second o de linea p oblem, namely,
⎧
⎪
⎪
⎨
⎪
⎪
⎩

g( )+P

g( )+Q ( )= ( ),∀ ∈[0,T],
(0) = x0,

g(0) = 0,
(5.3)
whe e P, Q, x0, 0∈Fa e cons an alues and ∈ACg([0, T]; F). Since he coefficien s a e cons an , i will
be he egula i y o he e m ha de e mines he addi ional egula i y o he solu ion. As in he p e ious
sec ions, we will see ha i is possible o p o e he uniqueness o solu ion when we conside he solu ion in
he space BC2
g([0, T]; F).
Theo em 5.4. Le ∈BCn
g([0, T]; F)and assume 1 +λ Δ+g( ) =0, o all ∈[0, T) ∩Dgand λ ∈Fsuch
ha λ2+Pλ +Q =0. Then, p oblem (5.3)has a unique solu ion ∈BCn+2
g([0, T], F)gi en by
( )=x0expg(λ2;0, )+( 0−λ2x0)exp
g(λ2;0, )·
[0, )
expg(λ2;0,s)−1
1+λ2Δ+g(s)expg(λ1;0,s)dμg(s)
+exp
g(λ2;0, )
[0, )
expg(λ2;0,s)−1
1+λ2Δ+g(s)expg(λ1;0,s)·⎛
⎜
⎝
[0,s)
expg(λ1;0, )−1
1+λ1Δ+g( ) ( )dμg( )⎞
⎟
⎠dμg(s),
(5.4)
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 35
whe e (x −λ1)(x −λ2) =x2+Px +Q.
P oo . Le be λ1, λ2 he wo complex eigen alues o he cha ac e is ic polynomial x2+Px +Q =0. Assume
2∈C
n+2
g([0, T], F)is a solu ion o p oblem (5.3). Obse e ha , i we define 1=( 2)
g−λ2 2, i is clea
ha 1∈Cn+1
g([0, T], F)and ( 1)
g−λ1 1= , 1(0) =( 2)
g(0) −λ2 2(0) = 0−λ2x0, so 1has o sol e
he p oblem
( 1)
g( )=λ1 1( )+ ( ),g−a.e. ∈[0,T)
(0) = 0−λ2x0.(5.5)
By Co olla y 4.21, p oblem (5.5)has a unique solu ion in Cn+1
g([0, T], F), so 1is ha unique solu ion.
Fu he mo e, by defini ion, 1=( 2)
g−λ2 2and 2(0) =x0, so 2sol es he p oblem
( 2)
g( )=λ2 2( )+ 1( ),g−a.e. ∈[0,T)
(0) = x0.(5.6)
By Co olla y 4.21, p oblem (5.6)has a unique solu ion in Cn+2
g([0, T], F), so 2is ha unique solu ion. This
implies ha , i a solu ion in Cn+2
g([0, T], F)o p oblem (5.3) exis s, i has o be unique.
In o de o ob ain ha unique solu ion i is enough o e ace he s eps we ha e aken o p o e he
uniqueness. Le 1be he unique solu ion o p oblem (5.5)in Cn+1
g([0, T], F)and le 2be he unique
solu ion o p oblem (5.6)in Cn+2
g([0, T], F). Clea ly 2is a solu ion o p oblem (5.3).
In o de o ob ain he explici exp ession o he solu ion, obse e ha , by P oposi ion 4.12, 1is o he
o m
1( )=( 0−λ2x0)exp
g(λ1;0, )+exp
g(λ1;0, )
[0, )
expg(λ1;0,s)−1 (s)
1+λ1Δ+g(s)dμg(s),
and 2o he o m
2( )=x0expg(λ2;0, )+exp
g(λ2;0, )
[0, )
expg(λ2;0,s)−1 1(s)
1+λ2Δ+g(s)dμg(s)
=x0expg(λ2;0, )+( 0−λ2x0)exp
g(λ2;0, )
[0, )
expg(λ2;0,s)−1expg(λ1;0,s)
1+λ2Δ+g(s)dμg(s)
+exp
g(λ2;0, )
[0, )
expg(λ2;0,s)−1expg(λ1;0,s)
1+λ2Δ+g(s)
·
[0,s)
expg(λ1;0, )−1 ( )
1+λ1Δ+g( )dμg( )dμg(s).
Now, hanks o P oposi ion 4.6,
expg(λ2;0, )−1expg(λ1;0, )=exp
g(−λ2/(1 + λ2Δ+g( )); 0, )exp
g(λ1;0, )
=exp
g((λ1−λ2)/(1 + λ2Δ+g( )); 0, ).
Thus,
36 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
2( )=x0expg(λ2;0, )+( 0−λ2x0)exp
g(λ2;0, )
[0, )
1
1+λ2Δ+g(s)expgλ1−λ2
1+λ2Δ+g;0,s
dμg(s)
+exp
g(λ2;0, )
[0, )
1
1+λ2Δ+g(s)expgλ1−λ2
1+λ2Δ+g;0,s

·
[0,s)
expg(λ1;0, )−1 ( )
1+λ1Δ+g( )dμg( )dμg(s).
Rema k 5.5. No e ha , o λ ∈Fsuch ha λ2+Pλ +Q =0, he condi ion 1 +λ Δ+g( ) =0can only
happen o a fini e numbe o ∈[0, T) ∩Dg.
Rema k 5.6. F om he p e ious exp ession we can de i e he exp ession o G een’s unc ion o p oblem (5.3)
jus by equa ing

R
G( , ) ( )dμg( )=exp
g(λ2;0, )
[0, )
1
1+λ2Δ+g(s)expgλ1−λ2
1+λ2Δ+g;0,s

·
[0,s)
expg(λ1;0, )−1 ( )
1+λ1Δ+g( )dμg( )dμg(s).
(5.7)
Now, i we conside he p oduc measu e space ([0, T], Mg·Mg, μg·μg), we ha e, by Fubini’s Theo em [1,
Theo em 10.10],

R
G( , ) ( )dμg( )=exp
g(λ2;0, )
[0,T ]·[0,T ]
1
1+λ2Δ+g(s)expgλ1−λ2
1+λ2Δ+g;0,s

·expg(λ1;0, )−1 ( )
1+λ1Δ+g( )χ[0,s)( )χ[0, )(s)dμg·dμg
=exp
g(λ2;0, )
[0,T ]
[0,T ]
expg(λ1;0, )−1expgλ1−λ2
1+λ2Δ+g;0,s

·(1 + λ1Δ+g( ))−1(1 + λ2Δ+g(s))−1 ( )χ( , )(s)χ[0, )( )dμg(s)dμg( )
=exp
g(λ2;0, )
[0, )
expg(λ1;0, )−1 ( )
1+λ1Δ+g( )
·⎛
⎜
⎝
( , )
1
1+λ2Δ+g(s)expgλ1−λ2
1+λ2Δ+g;0,s
dμg(s)⎞
⎟
⎠dμg( ).
The e o e, o , ∈[0, T],
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 37
G( , )=exp
g(λ2;0, )exp
g(λ1;0, )−1(1 + λ1Δ+g( ))−1χ[0, )( )
·
( , )
1
1+λ2Δ+g(s)expgλ1−λ2
1+λ2Δ+g;0,s
dμg(s)
=exp
g(λ2;0, )exp
g(λ1;0, )−1(1 + λ1Δ+g( ))−1χ[0, )( )
·
[ , )
1
1+λ2Δ+g(s)expgλ1−λ2
1+λ2Δ+g;0,s
dμg(s)
−expg(λ2;0, )exp
g(λ2;0, )−1(1 + λ1Δ+g( ))−1(1 + λ2Δ+g( ))−1Δ+g( )χ[0, )( ).
Obse e ha :
•I λ1=λ2,
(s)=exp
gλ1−λ2
1+λ2Δ+g;0,s
∈ACg([ , T ]; F)
is he solu ion o
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩

g(s)= λ1−λ2
1+λ2Δ+g(s) (s),g−a.e. s ∈[ , T ),
( )=exp
gλ1−λ2
1+λ2Δ+g;0, .
The e o e,

[ , )
λ1−λ2
1+λ2Δ+g(s)expgλ1−λ2
1+λ2Δ+g;0,s
dμg(s)
=exp
gλ1−λ2
1+λ2Δ+g;0,
−expgλ1−λ2
1+λ2Δ+g;0, 
=exp
g(λ1;0, )exp
g(λ2;0, )−1−expg(λ1;0, )exp
g(λ2;0, )−1.
Thus, he G een’s unc ion in he case λ1=λ2has he ollowing exp ession:
G( , )=exp
g(λ2;0, )exp
g(λ1;0, )−1(1 + λ1Δ+g( ))−1χ[0, )( )
·(λ1−λ2)−1!expg(λ1;0, )exp
g(λ2;0, )−1−expg(λ1;0, )exp
g(λ2;0, )−1"
−expg(λ2;0, )exp
g(λ2;0, )−1(1 + λ1Δ+g( ))−1(1 + λ2Δ+g( ))−1Δ+g( )χ[0, )( )
=+(λ1−λ2)−1expg(λ1;0, )exp
g(λ1;0, )−1(1 + λ1Δ+g( ))−1χ[0, )( )
−(λ1−λ2)−1expg(λ2;0, )exp
g(λ2;0, )−1(1 + λ1Δ+g( ))−1χ[0, )( )
−expg(λ2;0, )exp
g(λ2;0, )−1(1 + λ1Δ+g( ))−1(1 + λ2Δ+g( ))−1Δ+g( )χ[0, )( )
=+(λ1−λ2)−1expg(λ1;0, )exp
g(λ1;0, )−1(1 + λ1Δ+g( ))−1χ[0, )( )
−(λ1−λ2)−1expg(λ2;0, )exp
g(λ2;0, )−1(1 + λ2Δ+g( ))−1χ[0, )( ).
(5.8)
•I λ1=λ2, we ha e he ollowing exp ession o he G een’s unc ion:
G( , )=exp
g(λ;0, )exp
g(λ;0, )−1(1 + λΔ+g( ))−1χ[0, )( )·
( , )
1
1+λΔ+g(s)dμg(s)

38 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
=exp
g(λ;0, )exp
g(λ;0, )−1(1 + λΔ+g( ))−1χ[0, )( )
·⎛
⎜
⎝
[0, )
1
1+λΔ+g(s)dμg(s)−
[0, )
1
1+λΔ+g(s)dμg(s)−Δ+g( )
1+λΔ+g( )⎞
⎟
⎠(5.9)
=+exp
g(λ;0, )⎛
⎜
⎝
[0, )
1
1+λΔ+g(s)dμg(s)⎞
⎟
⎠expg(λ;0, )−1(1 + λΔ+g( ))−1χ[0, )( )
−expg(λ;0, )⎛
⎜
⎝
[0, )
1
1+λΔ+g(s)dμg(s)⎞
⎟
⎠expg(λ;0, )−1(1 + λΔ+g( ))−1χ[0, )( )
−expg(λ;0, )exp
g(λ;0, )−1(1 + λΔ+g( ))−2Δ+g( )χ[0, )( ).
Rema k 5.7. Obse e ha we can a i e o exp essions (5.8)and (5.9)using an in eg a ion by pa s a gumen
in o mula (5.7). Indeed, gi en wo elemen s h1, h2∈ACg([0, T]; F)we ha e ha h1h2∈ACg([0, T]; F)and
(h1h2)
g( )=(h1)
g( )h2( )+h1( )(h2)
g( )+(h1)
g( )(h2)( )Δ
+g( ),g−a.e. ∈[0,T].
Obse e ha we a e explici ly excluding he poin s o Cgin he abo e o mula. In pa icula , o ∈[0, T],
h1( )h2( )−h1(0) h2(0) = 
[0, )
(h1)
g(s)h2(s)dμg(s)+ 
[0, )
h1(s)(h2)
g(s)dμg(s)
+
[0, )
(h1)
g(s)(h2)(s)Δ
+g(s)dμg(s).
(5.10)
Now we s udy wo cases:
•Case λ1=λ2. Le us conside
h1( )=(λ1−λ2)−1expg(λ1;0, )exp
g(λ2;0, )−1,
h2( )= 
[0, )
expg(λ1;0, )−1
1+λ1Δ+g( ) ( )dμg( ).
We ha e ha h1, h2∈ACg([0, T]; F), so

[0, )
expg(λ2;0,s)−1
1+λ2Δ+g(s)expg(λ1;0,s)⎛
⎜
⎝
[0,s)
expg(λ1;0, )−1
1+λ1Δ+g( ) ( )dμg( )⎞
⎟
⎠dμg(s)
=(λ1−λ2)−1expg(λ1;0, )exp
g(λ2;0, )−1
[0, )
expg(λ1;0, )−1
1+λ1Δ+g( ) ( )dμg( )
−
[0, )
(λ1−λ2)−1expg(λ1;0,s)exp
g(λ2;0,s)−1expg(λ1;0,s)−1
1+λ1Δ+g(s) (s)dμg(s)
−
[0, )
expg(λ2;0,s)−1
1+λ2Δ+g(s)expg(λ1;0,s)expg(λ1;0,s)−1
1+λ1Δ+g(s) (s)Δ
+g(s)dμg(s),
(5.11)
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 39
and we eco e he exp ession o G een’s unc ion in (5.8). Obse e ha by subs i u ing (5.11)in (5.4)
we ob ain
2( )= 0−λ2x0
λ1−λ2expg(λ1;0, )− 0−λ1x0
λ1−λ2expg(λ2;0, )
+(λ1−λ2)−1expg(λ1;0, )
[0, )
expg(λ1;0,s)−1
1+λ1Δ+g(s) (s)dμg
−(λ1−λ2)−1expg(λ2;0, )
[0, )
expg(λ2;0,s)−1
1+λ2Δ+g(s) (s)dμg.
(5.12)
We ha e ha
h( )= 0−λ2x0
λ1−λ2expg(λ1;0, )− 0−λ1x0
λ1−λ2expg(λ2;0, )∈BC∞
g([0,T]; F)
is he solu ion o he homogeneous equa ion (5.1)and
p( )=+(λ1−λ2)−1expg(λ1;0, )
[0, )
expg(λ1;0,s)−1
1+λ1Δ+g(s) (s)dμg
−(λ1−λ2)−1expg(λ2;0, )
[0, )
expg(λ2;0,s)−1
1+λ2Δ+g(s) (s)dμg
is a pa icula solu ion in he space BCn+2
g([0, T], F)o he non homogeneous equa ion (5.3) ha sa isfies
p(0) =( p)
g(0) =0.
•Case λ1=λ2. We ha e ha exp ession (5.4) educes o
2( )=x0expg(λ;0, )+( 0−λx0)exp
g(λ;0, )
[0, )
1
1+λΔ+g(s)dμg(s)
+exp
g(λ;0, )
[0, )
1
1+λΔ+g(s)⎛
⎜
⎝
[0,s)
expg(λ;0, )−1
1+λΔ+g( ) ( )dμg( )⎞
⎟
⎠dμg(s).
(5.13)
Define
h1( )= 
[0, )
1
1+λΔ+g(s)dμg(s),
h2( )= 
[0, )
expg(λ;0,s)−1
1+λΔ+g(s) (s)dμg(s).
We ha e ha h1, h2∈ACg([0, T]; F). Hence, by o mula (5.10),

[0, )
1
1+λΔ+g(s)⎛
⎜
⎝
[0,s)
expg(λ;0, )−1
1+λΔ+g( ) ( )dμg( )⎞
⎟
⎠dμg(s)
40 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
=
[0, )
1
1+λΔ+g(s)dμg(s)
[0, )
expg(λ;0,s)−1
1+λΔ+g(s) (s)dμg(s) (5.14)
−
[0, )
⎛
⎜
⎝
[0,s)
1
1+λΔ+g( )dμg( )⎞
⎟
⎠expg(λ;0,s)−1
1+λΔ+g(s) (s)dμg(s)
−
[0, )
expg(λ;0,s)−1
(1 + λΔ+g(s))2 (s)Δ
+g(s)dμg(s).
Subs i u ing exp ession (5.14)in(5.13)we ob ain
2( )=x0expg(λ;0, )+( 0−λx0)exp
g(λ;0, )
[0, )
1
1+λΔ+g(s)dμg(s)
+exp
g(λ;0, )
[0, )
1
1+λΔ+g(s)dμg(s)
[0, )
expg(λ;0,s)−1
1+λΔ+g(s) (s)dμg(s)
−expg(λ;0, )
[0, )
⎛
⎜
⎝
[0,s)
1
1+λΔ+g( )dμg( )⎞
⎟
⎠expg(λ;0,s)−1
1+λΔ+g(s) (s)dμg(s)
−expg(λ;0, )
[0, )
expg(λ;0,s)−1
(1 + λΔ+g(s))2 (s)Δ
+g(s)dμg(s).
Obse e ha
h( )=x0expg(λ;0, )+( 0−λx0)exp
g(λ;0, )
[0, )
1
1+λΔ+g(s)dμg(s)∈BC∞
g([0,T]; F)
is he solu ion o he homogeneous equa ion (5.3)and
p( )=exp
g(λ;0, )
[0, )
1
1+λΔ+g(s)dμg(s)
[0, )
expg(λ;0,s)−1
1+λΔ+g(s) (s)dμg(s)
−expg(λ;0, )
[0, )
⎛
⎜
⎝
[0,s)
1
1+λΔ+g( )dμg( )⎞
⎟
⎠expg(λ;0,s)−1
1+λΔ+g(s) (s)dμg(s)
−expg(λ;0, )
[0, )
expg(λ;0,s)−1
(1 + λΔ+g(s))2 (s)Δ
+g(s)dμg(s)
is a pa icula solu ion o he non homogeneous equa ion (5.3)in he space BCn+2
g([0, T], F) ha sa isfies
p(0) =( p)
g(0) =0.
6. The S iel jes ha monic oscilla o
In his sec ion we p esen an applica ion ela ed o he eal solu ion o he S iel jes ha monic oscilla o
(g-ha monic oscilla o ). Le g:R →Rbe a de i a o such ha 0 /∈N−
gand T/∈N+
g∪Dg∪Cgand deno e
by gCi s con inuous pa . We conside he ollowing equa ion:
F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010 41
⎧
⎪
⎪
⎨
⎪
⎪
⎩

g( )+2ζω
0 
g( )+ω2
0 ( )=0,g−a.e. ∈[0,T),
(0) = x0,

g(0) = 0,
(6.1)
whe e x0and 0a e eal numbe s and:
•ω0is he undamped angula equency o he oscilla o ,
ω0=0k
m,
whe e m >0is he mass o he oscilla o and k>0is a measu e o he s iffness o he sp ing;
•ζis he damping a io,
ζ=c
2√mk ,
wi h c >0, he iscous damping coefficien ( esis ance o he medium). I ζ>1we ha e an o e damped
oscilla o , i ζ=1 he oscilla o is c i ically damped and, i ζ<1, he oscilla o is unde damped.
Obse e ha he solu ions o he cha ac e is ic equa ion a e gi en by:
λ=1
2−2ζω
0±14ζ2ω2
0−4ω2
0=−ζω
0±ω0/ζ2−1.
Assume ha 1 +λ Δ+g( ) =0 o all ∈[0, T) ∩Dgand o all λsolu ion o he cha ac e is ic equa ion.
We ha e he ollowing eal solu ion o he g-ha monic oscilla o in e ms o he damping a io:
•I ζ>1, we ha e wo eal solu ions o he cha ac e is ic equa ion, λ1=−ζω
0−ω0/ζ2−1and
λ2=−ζω
0+ω0/ζ2−1. Thus, he solu ion o (6.1)is gi en by
( )= 0−λ2x0
λ1−λ2expg(λ1;0, )− 0−λ1x0
λ1−λ2expg(λ2;0, ),
= 0−λ2x0
λ1−λ2exp(λ1gC( ))
s∈[0, )∩Dg!1+λ1Δ+g(s)"
− 0−λ1x0
λ1−λ2expg(λ2gC( ))
s∈[0, )∩Dg!1+λ2Δ+g(s)".
•I ζ=1, we ha e one eal solu ion o he cha ac e is ic equa ion, λ =−ζω
0. Thus, he solu ion o (6.1)
is gi en by
( )=x0expg(λ;0, )+( 0−λx
0)exp
g(λ;0, )
[0, )
1
1+λΔ+g(s)dμg,
=exp(λg
C( ))
s∈[0, )∩Dg!1+λΔ+g(s)"·⎡
⎣x0+( 0−λx
0)⎛
⎝gC( )+ 
s∈[0, )∩Dg
Δ+g(s)
1+λΔ+g(s)⎞
⎠⎤
⎦.
•I ζ<1, we ha e a pai o conjuga e complex solu ions, λ1=−ζω
0+i ω0/1−ζ2and λ2=−ζω
0−
i ω0/1−ζ2. I we deno e by a =−ζω
0and b =ω0/1−ζ2, we ha e ha he solu ion is gi en by
48 F.J. Fe nández e al. / J. Ma h. Anal. Appl. 511 (2022) 126010
Fig. 6.6. Compa ison be ween he exac solu ion and he nume ical app oxima ion ( e ical lines ha e o be unde s ood as jumps
and no as a mul i alued unc ion).
Finally, in Fig. 6.6, we can see he compa ison be ween he exac solu ion and he nume ical app oxima ion
o h =1.e −1and h =1.e −2.
Acknowledgmen s
The au ho s would like o hank he anonymous e e ee o hei commen s, sugges ions and co ec ions,
as hey ha e g ea ly con ibu ed o imp o e he quali y o he manusc ip .
The au ho s we e pa ially suppo ed by Xun a de Galicia, p ojec ED431C 2019/02, and by he Agen-
cia Es a al de In es igación (AEI) o Spain unde g an MTM2016-75140-P, co-financed by he Eu opean
Communi y und FEDER. Ignacio Má quez Albés was pa ially suppo ed by Xun a de Galicia unde g an
ED481B-2021-074.
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