2020 MATHEMATICA BOHEMICA 17 pp
Online i s
THE PERIODIC PROBLEM FOR THE SECOND ORDER
INTEGRO-DIFFERENTIAL EQUATIONS
WITH DISTRIBUTED DEVIATION
Sulkhan Mukhigulash ili,Ve onika No o ná, B no
Recei ed Ap il 19, 2019. Published online June 12, 2020.
Communica ed by Leonid Be ezansky
Abs ac . We s udy he ques ion o he unique sol abili y o he pe iodic ype p oblem
o he second o de linea in eg o-di e en ial equa ion wi h dis ibu ed a gumen de ia ion
u′′( ) = p0( )u( ) + Zω
0
p( , s)u(τ( , s)) ds+q( ),
and on he basis o he ob ained esul s by he a p io i boundedness p inciple we p o e
he new esul s on he sol abili y o pe iodic ype p oblem o he second o de nonlinea
unc ional di e en ial equa ions, which a e close o he linea in eg o-di e en ial equa ions.
The p o ed esul s a e op imal in some sense.
Keywo ds: linea in eg o-di e en ial equa ion; pe iodic p oblem; dis ibu ed de ia ion;
sol abili y
MSC 2020: 34K06, 34K13, 34B15
1. In oduc ion
On he in e al I= [0, ω], conside he second o de linea in eg o-di e en ial
equa ion
(1.1) u′′( ) = p0( )u( ) + Zω
0
p( , s)u(τ( , s)) ds+q( ),
Fo S. Mukhigulash ili, he esea ch was suppo ed by he g an RVO: 67985840. Fo
V. No o ná, he esea ch was suppo ed by he Czech Science Founda ion, p ojec
No. GA16-03796S.
c
The au ho (s) 2020. This is an open access a icle unde he CC BY-NC-ND licence cbnd
DOI: 10.21136/MB.2020.0061-19 1
and nonlinea unc ional di e en ial equa ion
(1.2) u′′( ) = F(u)( ) + q( )
wi h he pe iodic ype wo poin bounda y condi ions
(1.3) u(i−1)(ω)−u(i−1)(0) = ci, i = 1,2,
whe e c1, c2∈R,p0, , q ∈L∞(I, R),p∈L∞(I2,R),τ:I2→Iis a measu able unc-
ion, and F:C′(I, R)→L∞(I, R)is a con inuous ope a o . (The spaces C′(I, R)
and L∞(I, R)a e de ined below.)
We will say ha a unc ion u:I→Ris a solu ion o p oblem (1.2), (1.3) i i is
absolu ely con inuous oge he wi h i s i s de i a i e, sa is ies equa ion (1.2) almos
e e ywhe e on Iand sa is ies condi ions (1.3).
I is well-known ha he e a e many subjec s in physics and echnology using
ma hema ical me hods ha depend on he in eg o-di e en ial equa ions. Fo hese
and o pu ely heo e ical easons ample in e es ing li e a u e is de o ed o he pe-
iodic p oblem o he in eg o-di e en ial equa ions (see, e.g., [4], [6], [3], [10] and
he e e ences he ein). Ou wo k is mo i a ed by some o iginal esul s o he
unc ional di e en ial equa ions wi h a gumen de ia ion (see [1], [2], [9]), and he
esul s o Nie o (see [10]), E be and Guo (see [4]), and Kuo-Shou Chiu (see [3]).
Nie o in [10] s udied linea equa ion (1.1) on he in e al I= [0,2π]when p0≡M,
p( , s) = Nk( , s)and τ( , s)≡s, i.e. he equa ion o he o m
(1.4) u′′( ) = Mu( ) + N[Ku]( ) + q( )
unde condi ions (1.3) wi h c1=c2= 0,whe e [Ku]( ) = R2π
0k( , s)u(s) ds, k ∈
L2(I×I),M > 0and N∈R. In his pape , di e en su icien e icien condi ions
o he unique sol abili y o linea p oblem (1.4), (1.3) a e es ablished, and one o
hem, he condi ion kτk2<1, is op imal, whe e τ( , s) = R2π
0G( , )k( , s) d , and G
is he G een’s unc ion o he pe iodic p oblem o he equa ion ′′( ) = M ( ).On
he basis o hese esul s, he pe iodic p oblem o he nonlinea equa ion u′′( ) =
( , u( ),[Ku]( )) is s udied e en in he case when he ke nel kchanges i s sign.
In [6] he au ho s de elop he mono one i e a i e me hod based on compa ison e-
sul s, which is applicable o p oblem (1.4), (1.3) only i Kis Vol e a ope a o wi h
nonnega i e ke nel. A mo e gene al case is conside ed in pape [4], he e he op-
e a o Kis o he o m [Kx] = N[Tx] + N1[Sx], whe e Tis he in eg al ope a o
o Vol e a ype and Sis he in eg al ope a o o F edholm ype wi h nonnega i e
ke nels. Chiu in [3] in es iga es he exis ence o pe iodic solu ions o he sys ems
o in eg o-di e en ial equa ions wi h piecewise al e na ely e a ded and ad anced
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a gumen o gene alized ype. In he men ioned pape he au ho p o es in e es ing
esul s o he sol abili y and unique sol abili y, bu hese esul s do no ake in o
accoun he e ec o a gumen de ia ion.
In his pape we es ablish he heo ems which in some sense comple e and gene -
alize he esul s o he wo ks ci ed abo e as well as some o he known esul s. We
i s desc ibe some classes o unique sol abili y o linea p oblem (1.1), (1.3), and
on he basis o hese esul s, by he a p io i boundedness p inciple, we p o e he
exis ence heo ems o nonlinea p oblem (1.2), (1.3). The condi ions we ob ain ake
in o accoun he e ec o a gumen de ia ion, and in some sense a e op imal (see
Rema ks 2.1, 2.3).
In e es ing esul s ollow om ou main p oposi ion o such special cases o equa-
ions (1.1) and (1.2) as a e linea in eg o-di e en ial equa ions wi h dis ibu ed delay
(see Co olla y 2.2), linea di e en ial equa ions wi h a gumen de ia ion (see Co ol-
la y 2.3), o he nonlinea equa ion
(1.5) u′′( ) =  , u( ),Zω
0
V(u)( , s)u(τ( , s)) ds+q( ),
whe e :I×R2→Ris om he Ca a héodo y class, and V:C′(I, R)→L∞(I2,R)
is a con inuous bounded ope a o .
Also ou esul s allow o ob ain condi ions o unique sol abili y o a la ge class
o he wo poin BVP o highe o de unc ional di e en ial equa ions. He e as an
example o such p oblems we conside n h o de linea unc ional di e en ial equa ion
wi h a gumen de ia ion
(1.6) u(n)( ) = p1( )u(τ( )) + q( )
unde he wo poin bounda y condi ions
(1.7) u(i−1)(ω)−u(i−1)(0) = ci, u(j−1)(0) = cj, i = 1,2, j = 3,...,n,
i n⩾3,whe e ck∈R,k=1, n,p1∈L∞(I, R),and τ:I→Iis a measu able
unc ion.
Th oughou he pape we use he ollowing no a ions: R= ]−∞,∞[,R+= [0,∞[;
C(I;R)is he Banach space o con inuous unc ions u:I→Rwi h he no m kukC=
max{|u( )|: ∈I};C′(I;R)is he Banach space o he unc ions u:I→Rwhich a e
con inuous oge he wi h hei i s de i a i es wi h he no m kukC′= max{|u( )|+
|u′( )|: ∈I};L(I;R)is he Banach space o he Lebesgue in eg able unc ions
p:I→Rwi h he no m kpkL=Rω
0|p(s)|ds;L∞(I, R)is he space o he essen ially
bounded measu able unc ions p:I→Rwi h he no m kpk∞= esssup{|p( )|:
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∈I};L∞(I2,R)is he se o such unc ions p:I2→R, ha o any ixed ∈I,
p( , ·)∈L(I, R)and Rω
0|p(·, s)|ds∈L∞(I, R).Also o a bi a y p0, p1∈L∞(I, R),
p∈L∞(I2,R),and measu able τ:I2→Iwe will use he no a ions
l0(p0, p)( ) = |p0( )|+Zω
0
|p( , s)|ds,
l1(p, τ) = 2π
ωZω
0Zω
0
|p(ξ, s)||τ(ξ, s)−ξ|dsdξ1/2
.
De ini ion 1.1. Le σ∈ {−1,1}and τ:I2→Ibe measu able unc ion. We
will say ha he pai o unc ions (h0, h), whe e h0∈L∞(I, R+)and h∈L∞(I2,R+)
belong o he se Pσ
τi o a bi a y measu able unc ions p0:I→Rand p:I2→R
such ha
0⩽σp0( )⩽h0( ),0⩽σp( , s)⩽h( , s) o , s ∈I,(1.8)
p0( ) + Zω
0
p( , s) ds6≡ 0,(1.9)
he homogeneous p oblem
′′( ) = p0( ) ( ) + Zω
0
p( , s) (τ( , s)) ds,(1.10)
(i−1)(ω)− (i−1)(0) = 0, i = 1,2,(1.11)
has no non i ial solu ion.
2. S a emen o he main esul s
2.1. Linea p oblem.
P oposi ion 2.1. Le σ∈ {−1,1},
(2.1) h0∈L∞(I, R+), h ∈L∞(I2,R+), h0( ) + Zω
0
h( , s) ds6≡ 0,
and o almos all ∈I he inequali y
(2.2) 1−σ
2l0(h0, h)( ) + l1(h, τ)l1/2
0(h0, h)( )<4π2
ω2
holds. Then
(2.3) (h0, h)∈Pσ
τ.
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Theo em 2.1. Le σ∈ {−1,1}, σp0∈L∞(I, R+),σp ∈L∞(I2,R+)and condi-
ion (1.9) be ul illed. Mo eo e , le o almos all ∈I he inequali y
(2.4) 1−σ
2l0(p0, p)( ) + l1(p, τ)l1/2
0(p0, p)( )<4π2
ω2
hold. Then p oblem (1.1),(1.3) is uniquely sol able.
R e m a k 2.1. Condi ion (2.4) is op imal in he sense ha o he equa ion
(2.5) u′′( ) = p0( )u( ) o ∈[0,2π]
when p0( )⩽0,condi ion (2.4) ans o ms in o he condi ion |p0( )|<1,which is
op imal, because i p0≡ −1, hen sin is a nonze o solu ion o p oblem (2.5), (1.3)
wi h c1=c2= 0.
F om he las heo em i also ollows he well known ac ha i p0( )⩾0, hen
p oblem (2.5), (1.3) wi h c1=c2= 0,has only he ze o solu ion.
When in equa ion (1.1) he coe icien s p0and pa e nonnega i e, hen 1−σ= 0,
and om Theo em 2.1 i ollows:
Co olla y 2.1. Le
(2.6) p0∈L∞(I, R+), p ∈L∞(I2,R+), p0( ) + Zω
0
p( , s) ds6≡ 0,
and o almos all ∈Ile he inequali y
Zω
0Zω
0
p(ξ, s)|τ(ξ, s)−ξ|dsdξp0( ) + Zω
0
p( , s) ds<4π2
ω2
hold. Then p oblem (1.1),(1.3) is uniquely sol able.
Le now p0≡0,τ( , s)≡ −ν( , s),and
(2.7) 0⩽ν( , s)⩽ o , s ∈I.
Then equa ion (1.1) ans o ms in o he in eg o-di e en ial equa ion wi h dis ibu ed
delay
(2.8) u′′( ) = Zω
0
p( , s)u( −ν( , s)) ds+q( ),
and om Co olla y 2.1 i ollows:
Co olla y 2.2. Le p∈L∞(I2,R+),Rω
0p( , s) ds6≡ 0and o almos all ∈Ile
he inequali y Zω
0Zω
0
p(ξ, s)ν(ξ, s) dsdξZω
0
p( , s) ds < 4π2
ω2
hold. Then p oblem (2.8),(1.3) is uniquely sol able.
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I p0≡0and τ( , s) = τ( ) o , s ∈I, hen equa ion (1.1) ans o ms in o
equa ion (1.6) wi h n= 2,p1( ) = Rω
0p( , s) ds, and hen om Co olla y 2.1 i
ollows:
Co olla y 2.3. Le p1∈L∞(I, R+)be such ha o almos all ∈I he inequali y
p1( )Zω
0
p1(s)|τ(s)−s|ds < 4π2
ω2
holds. Then p oblem (1.6),(1.3) when n= 2, is uniquely sol able.
Co olla y 2.4. Le n⩾3and he unc ion p1∈L∞(I, R+)be such ha o
almos all ∈I he inequali y
Zω
0Z
0
p1(s)|τ(s)− |dsd Zω
0
p1(s) ds⩽4π2((n−3)!)2
ω2(n−2)
holds. Then p oblem (1.6),(1.7) is uniquely sol able.
R e m a k 2.2. I in Co olla ies 2.1–2.3 we assume ha σp0=h0and σp =h,
we ge he su icien e icien condi ions which gua an ee inclusion (2.3).
2.2. Nonlinea p oblem. Now we conside he heo ems on he sol abili y o
nonlinea p oblem (1.2), (1.3). Fi s we will in oduce he e he de ini ions.
De ini ion 2.1. We will say ha he ope a o Fbelongs o Ca a héodo y’s local
class and w i e F∈K(C′, L∞)i F:C′(I, R)→L∞(I, R)is con inuous ope a o ,
and o an a bi a y > 0
sup{|F(x)( )|:kxkC′⩽ , x ∈C′(I, R)} ∈ L∞(I, R+).
De ini ion 2.2. Le σ∈ {−1,1},inclusion (2.3) hold and he ope a o s V0:
C′(I, R)→L∞(I, R), V :C′(I, R)→L∞(I2,R)be con inuous. Then we will say
ha (V0, V )∈E(h0, h, Pσ
τ)i o all x∈C′(I, R) he condi ions
(2.9) 0⩽σV0(x)( )⩽h0( ),0⩽σV (x)( , s)⩽h( , s) o , s ∈I
hold, and
(2.10) in {kL(x, 1)kL:x∈C′(I, R)}>0,
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whe e
(2.11) L(x, y)( ) = V0(x)( )y( ) + Zω
0
V(x)( , s)y(τ( , s)) ds.
Also h oughou he pape we assume ha
(2.12) sgn x=(1i x⩾0,
−1i x < 0.
Then he nex heo em is ue:
Theo em 2.2. Le σ∈ {−1,1}and
(2.13) (V0+e
V0, V )∈E(h0, h, Pσ
τ),
whe e σV0(x)( )⩾0,σe
V0(x)( )⩾0on I o all x∈C′(I, R).
Mo eo e , le he cons an 0>0, he ope a o F∈K(C′, L∞)and he unc ion
g0∈L(I, R+)be such ha he condi ions
(2.14) g0( )⩽σ(F(x)( )−L(x, x)( )) sgn x( )
⩽|e
V0(x)( )x( )|+η( , kxkC′) o ∈I, kxkC′⩾ 0,
and
(2.15) |c2|⩽Zω
0
g0(s) ds−Zω
0
q(s) ds
hold, whe e he unc ion η:I×R+→R+is summable in he i s a gumen , non-
dec easing in he second one, and admi s o he condi ion
(2.16) lim
→∞
1
Zω
0
η(s, ) ds= 0.
Then p oblem (1.2),(1.3) has a leas one solu ion.
R e m a k 2.3. Inequali y (2.15) canno be eplaced by he inequali y
(2.17) |c2|⩽Zω
0
g0(s) ds−Zω
0
q(s) ds+ε,
no ma e how small ε > 0would be. Indeed, i F≡0,q( )≡εω−1,g0≡0,c2= 0,
hen ins ead o (2.15), inequali y (2.17) holds and all o he condi ions o Theo em 2.2
a e ul illed wi h L(x, y)≡0,η≡0,e
V0≡h0≡1,σ= 1.Ne e heless, in ha case,
p oblem (1.2), (1.3) is no sol able.
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R e m a k 2.4. Le σ∈ {−1,1},(h0, h)∈Pσ
τ,
(2.18) V0(x)( ) = p0( ),e
V0(x)( ) = ep0( ),
whe e σp0, σep0∈L∞(I, R+), and V:C′(I, R)→L∞(I2,R)be he con inuous ope -
a o . Then due o De ini ion 2.2 i is ob ious ha inclusion (2.13) holds i
(2.19) 0<|p0( )|, σ(p0( ) + ep0( )) ⩽h0( ) o ∈I,
0⩽σV (y)( , s)⩽h( , s) o , s ∈I, y ∈C′(I, R).
Co olla y 2.5. Le σ∈ {−1,1},inclusion (2.3) hold, he unc ions g0, σp0, σep0∈
L∞(I, R+),and he con inuous ope a o V:C′(I, R)→L∞(I2,R)be such ha
inequali ies (2.15),(2.19) a e ul illed. Mo eo e , le
(2.20) g0( )⩽σ( ( , x1, x2)−p0( )x1−x2) sgn x1
⩽|ep0( )x1|+η( , |x1|) o ∈I, x1, x2∈R,
whe e η:I×R+→R+be summable in he i s a gumen , nondec easing in he
second one and admi s o condi ion (2.16). Then p oblem (1.5),(1.3) has a leas
one solu ion.
E x a m p l e 2.1. The in eg o-di e en ial equa ion wi h dis ibu i e delay
(2.21) u′′( ) = αu( ) + β
1 + kukC′Z1
0
|u′(s)|u −
1 + sds+q( ) o ∈[0,1],
whe e α, β ∈R+and α6= 0,unde condi ions (1.3) wi h ω= 1,c2= 0,has a
leas one solu ion i R1
0q(s) ds= 0 and β(α+β)<8π2/ln 2 ≈113,91. Indeed, in
iew o Co olla y 2.2 he las inequali y gua an ees he alidi y o inclusion (2.3),
and hen all he assump ions o Co olla y 2.5 wi h σ= 1,p0≡h0≡α,V(y)( ) =
β|y′( )|/(1 + kykC′),h≡β,g0≡ep0≡q( , )≡0a e ul illed. The sol abili y o
p oblem (2.21), (1.3) does no ollow om he p e iously known esul s.
3. Auxilia y P oposi ions
Now conside he modi ica ion o he well known Wi inge ’s inequali y (see The-
o em 258 in [5]).
P oposi ion 3.1. Le ′′ ∈L∞(I, R)and condi ions (1.11) hold. Then
(3.1) Zω
0
′2(s) ds⩽ω
2π2Zω
0
( ′′(s))2ds.
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Lemma 3.1. Le all he condi ions o P oposi ion 2.1and condi ions (1.8),(1.9)
hold. Then p oblem (1.10),(1.11) has only he i ial solu ion.
P o o . On he con a y, assume ha p oblem (1.10), (1.11) has nonze o solu-
ion . I ≡c(ob iously c6= 0), hen ′′ ≡0and in iew o (1.10) we ge he con a-
dic ion wi h (1.9), i.e. 6≡ cons . The e o e due o (1.11) he inequali y ′6≡ cons
holds and hen he e exis ∗, ∗∈Isuch ha ∗< ∗and ′( ∗)− ′( ∗)6= 0.
The e o e om (1.10) by (1.8), Schwa z and Cauchy-Schwa z inequali ies i ollows
ha
0<| ′( ∗)− ′( ∗)|⩽Zω
0p0(ξ) (ξ) + Zω
0
p(ξ, s) (τ(ξ, s)) dsdξ
⩽Zω
0
|p0(ξ)|dξZω
0
|p0(ξ)| 2(ξ) dξ1/2
+Zω
0Zω
0
|p(ξ, s)|dsdξZω
0Zω
0
|p(ξ, s)| 2(τ(ξ, s)) dsdξ1/2
⩽δ1/2Zω
0
|p0(ξ)|dξ+Zω
0Zω
0
|p(ξ, s)|dsdξ1/2
,
whe e δ=Rω
0δ0(ξ) dξ,δ0(ξ) = σRω
0p(ξ, s) 2(τ(ξ, s)) ds+p0(ξ) 2(ξ), and hen
(3.2) δ > 0.
Analogously om (1.10) by (1.8), Schwa z and Cauchy-Schwa z inequali ies we ge
(3.3) Zω
0
( ′′(ξ))2dξ⩽Zω
0|p0(ξ)|1/2(|p0(ξ)| 2(ξ))1/2
+Zω
0
|p(ξ, s)|ds1/2Zω
0
|p(ξ, s)| 2(τ(ξ, s)) ds1/22
dξ
⩽Zω
0
l0(h0, h)(ξ)δ0(ξ) dξ.
Now no e ha in iew o (1.10), o δ he ep esen a ion is ue:
(3.4) δ=σZω
0
(ξ) ′′(ξ) dξ+Zω
0Zω
0
|p(ξ, s)| (τ(ξ, s))Zτ(ξ,s)
ξ
′(η) dηdsdξ.
Due o (3.1) and (3.3), by in eg a ion by pa s and bounda y condi ions (1.11) we
ind ha
(3.5) σZω
0
(ξ) ′′(ξ) dξ=−σZω
0
′2(ξ) dξ⩽1−σ
2Zω
0
′2(ξ) dξ
⩽1−σ
2ω
2π2Zω
0
l0(h0, h)(ξ)δ0(ξ) dξ.
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condi ion (2.13) and nonnega i i y o he unc ion η, we ge ha uis a solu ion o
he equa ion
(4.5) u′′( ) = L(u, u)( ) + (λ+ (1 −λ)ν( ))e
V0(u)( )u( ) + η1( , kukC′),
whe e
η1( , kukC′) = σ(1 −λ)(1 + η( , kukC′))ν( ) sgn u( ) + (1 −λ)q( ),
ν( ) = σ(F(u)( )−L(u, u)( )) sgn u( )
|e
V0(u)( )u( )|+η( , kukC′) + 1 ,
and due o condi ion (2.14) he es ima ions
(4.6) 0⩽ν( )<1,|η1( , kukC′)|⩽1 + η( , kukC′) + |q( )|
a e alid on I. Now no e ha acco ding o condi ions (2.13), (4.6) and he nonneg-
a i i y o he ope a o s σe
V0(u)( )and σV0(u)( ), he es ima ion
0⩽σ(V0(u)( ) + (λ+ (1 −λ)ν( ))e
V0(u)( )) ⩽σ(V0(u)( ) + e
V0(u)( )) ⩽h0( )
is sa is ied on I. Consequen ly, due o inclusion (2.13), o a bi a y λ∈(0,1) he
inclusion (V1, V )∈E(h0, h, Pσ
τ),whe e V1(x)( ) = V0(x)( )+(λ+(1−λ)ν( ))e
V0(x)( ),
is alid oo. Then om (4.5) by Lemma 3.2 due o inequali y (4.6) we ge he
es ima ion
kukC′⩽0µ(u) + |c1|+|c2|+Zω
0
(|q(s)|+η(s, kukC′) + 1) ds,
which in iew o (4.3) con adic s wi h inequali y (4.4), i.e. ou assump ion is in alid
and es ima ion (3.27) holds.
On he o he hand, om Lemma 3.4 due o inclusion (2.13) i ollows ha he pai
o he ope a o e
Land condi ions (1.11) belongs o he Opial class O2
0,and he e o e
all he assump ions o Lemma 3.5 a e ul illed, om which he sol abili y o p oblem
(1.1), (1.3) ollows. 
P o o o Co olla y 2.5. Assume ha he ope a o s V0,e
V0a e de ined by (2.18).
Then due o Rema k 2.4 in iew o (2.19), inclusion (2.13) holds. Also, om (2.20)
he alidi y o condi ions (2.14) ollows, wi h
F(x)( ) =  , x( ),Zω
0
V(x)( , s)x(τ( , s)) ds,
L(x, y)( ) = p0( )y( ) + Zω
0
V(x)( , s)y(τ( , s)) ds.
The e o e all he assump ions o Theo em 2.2 a e ul illed om which alidi y o ou
co olla y immedia ely ollows. 
16 Online i s

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Au ho s’ add esses:Sulkhan Mukhigulash ili, Ins i u e o Ma hema ics o he Czech
Academy o Sciences, Žižko a 22, 616 62 B no, Czech Republic, e-mail: [email p o ec ed];
Ve onika No o ná, Facul y o Business and Managemen , B no Uni e si y o Technology,
Kolejní 2906/4, 612 00 B no, Czech Republic, e-mail: [email p o ec ed].
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