Existence of a solution for a nonlinear integral equation by nonlinear contractions involving simulation function in partially ordered metric space

Nonlinea Analysis: Modelling and Con ol, Vol. 28, No. 3, 578–596
h ps://doi.o g/10.15388/namc.2023.28.32119
P ess
Exis ence o a solu ion o a nonlinea in eg al equa ion
by nonlinea con ac ions in ol ing simula ion unc ion
in pa ially o de ed me ic space*
Pa aneh Lo0lo0a, Ma yam Shamsb,1, Manuel De la Senc,d,2
aDepa men o Ma hema ics,
Behbahan Kha am Alanbia Uni e si y o Technology,
Behbahan 6361647189, I an
[email p o ec ed]
bDepa men o Pu e Ma hema ics,
Uni e si y o Shah eko d,
Shah eko d 88186-34141, I an
[email p o ec ed]
cIns i u e o Resea ch and De elopmen o P ocesses,
Uni e si y o Basque Coun y, Campus o Leioa, Bizkaia
dDepa men o Elec ici y and Elec onics,
Facul y o Science and Technology,
Uni e si y o The Basque Coun y (UPV/EHU),
48080 Bilbao, Spain
[email p o ec ed]
Recei ed: June 28, 2022 / Re ised: Feb ua y 24, 2023 / Published online: Ap il 26, 2023
Abs ac . In a ecen pape , Khojas eh e al. p esen ed a new collec ion o simula ion unc ions, said
Z-con ac ion. This o m o con ac ion gene alizes he Banach con ac ion and makes di e en
ypes o nonlinea con ac ions. In his a icle, we discuss a pai o nonlinea ope a o s ha applies
o a nonlinea con ac ion including a simula ion unc ion in a pa ially o de ed me ic space. Fo
his pai o ope a o s wi h and wi hou con inui y, we de i e some esul s abou he coincidence and
unique common ixed poin . In he ollowing, many known and dependen consequences in ixed
poin heo y in a pa ially o de ed me ic space a e deduced. As well, we u nish wo in e es ing
examples o explain ou main consequences, so ha one o hem does no apply o he p inciple o
Banach con ac ion. Finally, we use ou consequences o c ea e a solu ion o a pa icula ype o
nonlinea in eg al equa ion.
Keywo ds: simula ion unc ions, coincidence poin , compa ible, pa ially o de ed me ic space,
in eg al equa ion.
*This esea ch was suppo ed by Basque Go e nmen , g an No. 1555-22.
1The au ho was suppo ed by Shah eko d Uni e si y and he Cen e o Excellence o Ma hema ics.
2Co esponding au ho .
© 2023 The Au ho (s). Published by Vilnius Uni e si y P ess
This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion Licence, which
pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal au ho and sou ce
a e c edi ed.
Exis ence o a solu ion o a nonlinea in eg al equa ion 579
1 In oduc ion
Fixed poin heo y is a clea subjec , which a o ds bene icial echniques and senses o
dealing wi h di e se p oblems. Pa icula ly, we men ion he being o solu ions o ma h-
ema ical ques ions diminishable o equi alen ixed poin p oblems. Thus, we emembe
ha he Banach con ac ion p inciple [5] is based on his heo y. Ne e heless, he ixed
poin heo y has been able o a ac many esea che s. In 2018, Ve o [31] p o ed he
exis ence and uniqueness o a ixed poin in he se ing o o de ed me ic spaces by
in oducing he no ion o o de ed S-G-con ac ion. Hoc and his colleagues [13] p o ided
some new ixed poin heo ems in compac me ic space. In 2022, Kim [18] s udied he
exis ence o a coupled ixed poin in Hilbe space. Also, Gau am e al. [11] in oduced
he no ion o in e pola i e Ma kowski- ype con ac ion, and hey ob ained he solu ion o
he nonlinea ma ix equa ions. The e o e, he e a e many achie emen s o en husias s,
look, o example, [6,8–10,19–22, 28, 29, 33].
Recen ly, lo s o conclusions became appa en linked o ixed poin heo ems in an
o de ed me ic space. Run and Reu ings [27] exp essed he i s conclusion in his o ien a-
ion, whe e hey expanded he Banach con ac ion p inciple in me ic space equipped wi h
a pa ial o de . Subsequen ly, Nie o and Rod íguez-López [24] gene alized he p e ious
esul s and used hem o ind a unique solu ion o a speci ic ype o o dina y di e en ial
equa ion. Mo e p og ess in he abo e-a gued esul s is de ec ed in [2,3,12, 23, 25, 30].
Recen ly, he concep o simula ion unc ion was in oduced and s udied by Khojas eh
e al. [17]. By using he simula ion unc ions Ve o [32] in es iga ed he exis ence o
a common ixed poin and coincidence poin in bo h me ic space and pa ial me ic
space. In his a icle, we p esume a pai o nonlinea ope a o s sa is ying in nonlinea
con ac ions including a simula ion unc ion in a me ic space wi h a pa ial o de . We
gene alize some esul s Khojas e e al. [17] o ob ain coincidence and common ixed
poin esul s o his pai o ope a o s wi h and wi hou con inui y. Also, we p ocess wo
in e es ing examples o explain ou main esul s, so ha one o hem does no apply o he
p inciple o Banach con ac ion. Then we exploi ou achie emen s o c ea e a solu ion
o a pa icula ype o nonlinea in eg al equa ion.
2 P elimina ies
The ollowing de ini ion was gi en by A goubi e al. [4].
De ini ion 1. Le (X, d)be a me ic space, and le ζ: [0,∞)×[0,∞)→Rsa is ies he
ollowing condi ions:
(ζ1)ζ(p, q)< q −p o all p, q > 0;
(ζ2) I {pn}and {qn}a e sequences in (0,∞)such ha limn→∞ pn= limn→∞ qn=
l > 0, hen
lim sup
n→∞
ζ(pn, qn)<0.
Then ζis a simula ion unc ion.
Nonlinea Anal. Model. Con ol, 28(3):578–596, 2023
580 P. Lo0lo0e al.
Rema k 1. Ini ially, Khojas eh e al. [17] de ined he simula ion unc ion as a mapping ζ:
[0,∞)×[0,∞)→Rsa is ying ζ(0,0) = 0 and condi ions (ζ1) and (ζ2) o De ini ion 1.
In he ollowing, we will use he modi ied de ini ion by A goubi e al. [4].
Be o e s a ing he main esul s o his esea ch, we ende many examples ha high-
ligh hei possible applicabili y o he ield o ixed poin heo y.
Example 1. (See [17].) Le ζi: [0,∞)×[0,∞)→R,i= 1,2,...,6, be de ined by
(i) ζ1(p, q) = ψ(q)−φ(p) o all p, q ∈[0,∞), whe e ψ, φ : [0,∞)→[0,∞)a e
wo con inuous unc ions such ha ψ( ) = φ( ) = 0 i only i = 0 and ψ( )<
⩽φ( ) o all > 0.
(ii) ζ2(p, q) = αq −p o all p, q ∈[0,∞)is a pa icula case o ζ1wi h φ( ) =
and ψ( ) = α o all ⩾0and α∈[0,1).
(iii) ζ3(p, q) = q−ϕ(q)−p o all p, q ∈[0,∞), whe e ϕ: [0,∞)→[0,∞)is
a lowe semicon inuous unc ion such ha φ−1(0) = {0}.
(i ) ζ4(p, q) = qϕ(q)−p o all p, q ∈[0,∞), whe e ϕ: [0,∞)→[0,1) is a unc ion
such ha lim sup → +ϕ( )<1 o all > 0.
( ) ζ5(p, q) = q−( (p, q)/g(p, q))p o all p, q ∈[0,∞), whe e , g : [0,∞)→
(0,∞)a e wo con inuous unc ions wi h espec o each a iable such ha
(p, q)> g(p, q) o all p, q > 0.
( i) ζ6(p, q) = q−Rp
0φ(u) du o all p, q ∈[0,∞), whe e φ: [0,∞)→[0,∞)is
a unc ion such ha R
0φ(u) duexis s, and R
0φ(u) du> o each  > 0.
De ini ion 2. Le (X, d)be a me ic space and S, T :X→X. I =Su =Tu o
some uin X, hen uis called a coincidence poin o Sand T.
De ini ion 3. (See [15].) Le (X, d)be a me ic space. The mappings S, T :X→X
a e compa ible i and only i o any sequence {un}in Xsuch ha limn→∞ Sun=
limn→∞ Tun,limn→∞ d(STun, T Sun)=0.
De ini ion 4. (See [16].) Le (X, d)be a me ic space. The mappings S, T :X→Xa e
weakly compa ible i and only i Su =Tu o some u∈Ximplies ha STu =T Su o
Sand Tcommu e a hei coincidence poin s.
I Sand Ta e compa ible, hen Sand Ta e weakly compa ible.
De ini ion 5. (See [7].) Le (X, 4)is a pa ially o de ed se and S, T :X→X.Sis
said o be T-nondec easing i o u, ∈X,
Tu 4T =⇒Su 4S .
3 Main esul
The ollowing main heo em is a gene alized coincidence poin heo em o maps ha a e
no necessa ily con inuous.
h ps://www.jou nals. u.l /nonlinea -analysis
Exis ence o a solu ion o a nonlinea in eg al equa ion 581
Theo em 1. Le (X, 4)be a pa ially o de ed se and suppose ha he e exis s a me ic
don Xsuch ha (X, d)is comple e me ic space. Suppose ha he e exis a simula ion
unc ion ζand S, T :X→Xsuch ha
ζd(Sx, Sy), d(Tx, Ty)⩾0∀x, y ∈X:Tx 4T y, (1)
and suppose he ollowing hypo heses:
(i) SX ⊆TX and TX is closed;
(ii) Sis T-nondec easing;
(iii) I {Txn} ⊂ Xis a nondec easing sequence, which con e ges o Tu in TX, hen
Txn4Tu o all n⩾0.
I he e exis s x0∈Xsuch ha Tx04Sx0, hen Sand Tha e a coincidence poin , ha
is, he e exis s ∈Xsuch ha S =T .
P oo . Using he heo em condi ion, we ha e x0∈Xsuch ha T x04Sx0. Since
SX ⊆TX, hen he e exis s x1∈Xsuch ha Tx1=Sx0and Tx04Sx0=Tx1.
Since Sis T-nondec easing, we ha e Sx04Sx1. Con inuing his p ocess, we cons uc
he sequence {xn}wi h he ollowing condi ions:
Sxn=Txn+1 ∀n⩾0,(2)
and
(3)
Tx04Sx0=Tx14Sx1=T x24Sx24· · ·
4Sxn−1=Txn4Sxn=Txn+1 4· · · .
I wo consecu i e membe s o he sequences {Sxn}o {Txn}a e equal, hen he con-
clusion o he heo em ollows. So we ha e
d(Sxn, Sxn+1)6= 0, d(Txn, Txn+1)6= 0 ∀n⩾0.(4)
I o some n∈N, we assume ha d(Txn−1, Txn)< d(T xn, Txn+1), hen by p ope y
(ζ1) o simula ion unc ion and (2)–(4) we ha e
0⩽ζd(Sxn−1, Sxn), d(Txn−1, Txn)
=ζd(Txn, Txn+1), d(Txn−1, Txn)
< d(Txn−1, Txn)−d(Txn, T xn+1)<0.
This con adic ion shows ha
d(Txn, Txn+1)⩽d(Txn−1, Txn).
This implies ha he sequence {d(Txn−1, Txn)}is a mono one dec easing sequence o
nonnega i e eal numbe s, and consequen ly, he e exis s ⩾0such ha he sequence
{d(Txn−1, Txn)}con e ges o .
Nonlinea Anal. Model. Con ol, 28(3):578–596, 2023
582 P. Lo0lo0e al.
Suppose > 0. By (3) we know ha he elemen s Txnand T xn+1 a e compa able,
so using p ope y (ζ2) o a simula ion unc ion wi h pn=d(Sxn, Sxn+1)and qn=
d(Sxn−1, Sxn), we ha e
0⩽lim sup
n→∞
ζd(Sxn−1, Sxn), d(Txn−1, Txn)
= lim sup
n→∞
ζd(Txn, Txn+1), d(Txn−1, Txn)<0,
which is a con adic ion, and hence,
lim
n→∞ d(Txn−1, Txn) = 0.
The nex s ep is o show ha he sequence {T xn}is Cauchy. By con adic ion and
by Lemma 2.1 o [14] he e exis an  > 0and {Txm(k)},{Txn(k)}⊂{Txn}wi h
n(k)> m(k)⩾k o all k∈Nsuch ha
lim
k→∞ d(Txm(k), Txn(k)) = lim
k→∞ d(Txm(k)+1, Txn(k)+1) = , (5)
d(Txm(k), Txn(k))⩾. (6)
Then we can assume ha
d(Txm(k)+1, Txn(k)+1)>0∀k∈N.(7)
Again, by (3) we know ha he elemen s Txm(k)and Txn(k)a e compa able, so using
(5)–(7) and p ope y (ζ2) o a simula ion unc ion wi h pn=d(Txm(k)+1, Txn(k)+1)and
qn=d(Txm(k), Txn(k)), we ha e
0⩽lim sup
k→∞
ζd(Sxm(k), Sxn(k)), d(Txm(k), Txn(k))
= lim sup
k→∞
ζd(Txm(k)+1, Txn(k)+1), d(Txm(k), Txn(k))<0,
which is a con adic ion. We conclude ha he sequence {Txn}is a Cauchy sequence,
and hence, {Txn}is con e gen in he comple e me ic space (X, d).TX is closed,
he e o e, by (2) he e exis s u∈Xsuch ha
lim
n→∞ Sxn= lim
n→∞ Txn=Tu. (8)
F om (3) and (8) we know ha {T xn}is a nondec easing sequence in TX such ha
Txn→Tu, hen by condi ion (iii) and (4) we ha e
Txn≺Tu. (9)
Again, by (3), (4) and since Sis T-nondec easing, we ha e
Sxn≺Su. (10)
h ps://www.jou nals. u.l /nonlinea -analysis

Exis ence o a solu ion o a nonlinea in eg al equa ion 583
Using p ope y (ζ1) o a simula ion unc ion, (9) and (10), we ha e
0⩽ζd(Sxn, Su), d(Txn, Tu)< d(T xn, Tu)−d(Sxn, Su)∀n∈N.
Taking n→ ∞ in he abo e inequali y, we ha e limn→∞ Sxn=Su. Then
Su = lim
n→∞ Sxn= lim
n→∞ Txn=Tu. (11)
This comple es he p oo .
Now, we will p o e he exis ence and uniqueness heo em o a common ixed poin .
Theo em 2. I in Theo em 1, i is addi ionally assumed ha Sand Ta e weakly com-
pa ible and Tu 4TTu, whe e uis a coincidence poin o Sand T, hen Sand Tha e
a common ixed poin in X. Mo eo e , i a se o ixed poin s o Tis o ally o de ed, hen
Sand Tha e a unique common ixed poin .
P oo . We p o e =Su =Tu. Since Sand Ta e weakly compa ible, by (11) we ha e
STu =TSu. Then
T =TTu=TSu =STu =SSu =S . (12)
I T = o S = , hen is a common ixed poin . O he wise, i.e., i T 6= and
S 6= , by p ope y (ζ1) o a simula ion unc ion wi h Tu 4TTu
0⩽ζd( , S ), d( , T )=ζd(Su, SSu), d(Tu, TTu)
< d(Tu, TT u)−d(Su, SSu).
Using (11) and (12) in he abo e inequali y, we ha e
d(Su, SSu)< d(Tu, T Tu) = d(Su, SSu),
which is a con adic ion. The e o e, T = o S = , and we conclude ha =S =
T .
Now, suppose ha he se o ixed poin s o Tis o ally o de ed. Assume on he
con a y ha =S =T and 0=S 0=T 0bu 6= 0. Since and 0con ain a se
o ixed poin s o T, wi hou loss o gene ali y, we assume ha T 4T 0. I S =S 0
o T =T 0, hen = 0, which is a con adic ion. O he wise, i.e., i S 6=S 0and
T 6=T 0, by p ope y (ζ1) o a simula ion unc ion we ha e
0⩽ζd(S , S 0), d(T , T 0)=ζd( , 0), d( , 0)
< d( , 0)−d( , 0) = 0,
which is a con adic ion. The e o e, Sand Tha e a unique common ixed poin .
In he nex heo em, we will omi condi ion (iii) o Theo em 1, and we will assume
ha S, T :X→Xa e con inuous and compa ible.
Nonlinea Anal. Model. Con ol, 28(3):578–596, 2023
584 P. Lo0lo0e al.
Theo em 3. Le (X, 4)be a pa ially o de ed se , and le he e exis s a me ic don X
such ha (X, d)is comple e me ic space. Suppose ha he e exis a simula ion unc ion
ζand S, T :X→Xsuch ha
ζd(Sx, Sy), d(Tx, Ty)⩾0∀x, y ∈X:Tx 4T y.
We suppose he ollowing hypo heses:
(i) SX ⊆TX;
(ii) Sis T-nondec easing;
(iii) Sand Ta e con inuous;
(i ) The pai {S, T }is compa ible.
I he e exis s x0∈Xsuch ha Tx04Sx0, hen Sand Tha e a coincidence poin , ha
is, he e exis s u∈Xsuch ha Su =Tu. Fu he , i Tu 4TTu and he se o ixed
poin s o Tis o ally o de ed, hen Sand Tha e a unique common ixed poin .
P oo . Following he p oo o Theo em 1, we ha e ha {T xn}is a Cauchy sequence in
he comple e me ic space (X, d). Then he e exis s u∈Xsuch ha
lim
n→∞ Sxn= lim
n→∞ Txn=u. (13)
Since Sand Ta e compa ible, his implies ha
lim
n→∞ S(Txn), T(Sxn)= 0.(14)
F om (13) and he con inui y o Sand Twe ha e
lim
n→∞ T(Txn) = Tu, lim
n→∞ S(Txn) = Su. (15)
By he iangula inequali y we ha e
d(Su, Tu)⩽Su, S(T xn)+dS(T xn), T(Sxn)+dT(Txn+1), Tu.
By (14) and (15) and le ing n→ ∞, we ob ain:
d(Su, Tu)⩽0,
he e o e, Su =Tu, ha is, uis he coincidence poin o Sand T.
Finally, because Sand Ta e compa ible ( he e o e, hey a e weakly compa ible) and,
on he o he hand, Tu 4TTu and se o ixed poin s o Tis o ally o de ed, hen by
Theo em 2, Sand Tha e a unique common ixed poin .
I T:X→Xis he iden i y mapping, we can deduce easily he ollowing ixed poin
esul s. I is an immedia e consequence o Theo em 1.
h ps://www.jou nals. u.l /nonlinea -analysis
Exis ence o a solu ion o a nonlinea in eg al equa ion 585
Theo em 4. Le (X, 4)be a pa ially o de ed se and suppose ha he e exis s a me ic
don Xsuch ha (X, d)is comple e me ic space. Suppose ha he e exis a simula ion
unc ion ζand S:X→Xsuch ha
ζd(Sx, Sy), d(x, y)⩾0∀x, y ∈X:x4y.
We suppose he ollowing hypo heses:
(i) Sis a nondec easing unc ion;
(ii) I {un}is a nondec easing sequence, which con e ges o uin X, hen un4u
o all n⩾0.
I he e exis s x0∈Xsuch ha x04Sx0, hen Shas a ixed poin .
The ollowing esul is an immedia e consequence o Theo em 3.
Theo em 5. Le (X, 4)be a pa ially o de ed se and suppose ha he e exis s a me ic
don Xsuch ha (X, d)is comple e me ic space. Suppose ha he e exis a simula ion
unc ion ζand S:X→Xsuch ha
ζd(Sx, Sy), d(x, y)⩾0∀x, y ∈X:x4y.
We suppose he ollowing hypo heses:
(i) Sis a nondec easing unc ion;
(ii) Sis con inuous.
I he e exis s x0∈Xsuch ha x04Sx0, hen Shas a ixed poin .
4 Consequences
In his sec ion, as applica ions, we ob ain some esul s o Theo em 1 in ixed poin heo y
in pa ially o de ed me ic space ia speci ic choices o simula ion unc ions.
Le (X, 4)be a pa ially o de ed se and suppose ha he e exis s a me ic don X
such ha (X, d)is a comple e me ic space.
Co olla y 1. Le S, T :X→Xbe mappings such ha he e exis wo con inuous
unc ions φ, ψ : [0,∞)→[0,∞) e i ying ψ( ) = φ( )=0i and only i = 0,
ψ( )< ⩽φ( ) o all > 0, and
φd(Sx, Sy)⩽ψd(Tx, Ty)∀x, y ∈X:Tx 4T y.
We suppose he ollowing hypo heses:
(i) SX ⊆TX and TX is closed;
(ii) Sis T-nondec easing;
(iii) I {Txn}⊂Xis a nondec easing sequence con e ges o Tu in TX, hen Txn4
Tu o all n⩾0.
Nonlinea Anal. Model. Con ol, 28(3):578–596, 2023
586 P. Lo0lo0e al.
I he e exis s x0∈Xsuch ha Tx04Sx0, hen Sand Tha e a coincidence poin , ha
is, he e exis s u∈Xsuch ha Su =T u. Fu he , i Sand Ta e weakly compa ible,
Tu 4TTu, and he se o ixed poin s o Tis o ally o de ed, hen Sand Tha e a unique
common ixed poin .
P oo . The esul ollows om Theo ems 1 and 2 by aking as simula ion unc ion
ζ1(p, q) = ψ(q)−φ(p)∀p, q ⩾0,
which was in oduced in Example 2.
Co olla y 2 [Banach ype]. Le S, T :X→Xbe mappings such ha he e exis s
α∈[0,1) e i ying
d(Sx, Sy)⩽αd(Tx, Ty)∀x, y ∈X:T x 4Ty.
We suppose he ollowing hypo heses:
(i) SX ⊆TX and TX is closed;
(ii) Sis T-nondec easing;
(iii) I {Txn}⊂Xis a nondec easing sequence con e ges o Tu in TX, hen Txn4
Tu o all n⩾0.
I he e exis s x0∈Xsuch ha Tx04Sx0, hen Sand Tha e a coincidence poin , ha
is, he e exis s u∈Xsuch ha Su =T u. Fu he , i Sand Ta e weakly compa ible,
Tu 4TTu, and he se o ixed poin s o Tis o ally o de ed, hen Sand Tha e a unique
common ixed poin .
P oo . The esul ollows om Theo ems 1 and 2 by aking as simula ion unc ion
ζ2(p, q) = αq −p∀p, q ⩾0,
which was in oduced in Example 2.
Co olla y 3. Le S, T :X→Xbe mappings such ha he e exis s a lowe semicon in-
uous unc ion ϕ: [0,∞)→[0,∞) e i ying ϕ−1({0}) = {0}and
d(Sx, Sy)⩽d(Tx, Ty)−ϕd(Tx, T y)∀x, y ∈X:Tx 4Ty.
We suppose he ollowing hypo heses:
(i) SX ⊆TX and TX is closed;
(ii) Sis T-nondec easing;
(iii) I {Txn} ⊂ Xis a nondec easing sequence, which con e ges o Tu in T X, hen
Txn4Tu o all n⩾0.
I he e exis s x0∈Xsuch ha Tx04Sx0, hen Sand Tha e a coincidence poin , ha
is, he e exis s u∈Xsuch ha Su =T u. Fu he , i Sand Ta e weakly compa ible,
Tu 4TTu, and he se o ixed poin s o Tis o ally o de ed, hen Sand Tha e a unique
common ixed poin .
h ps://www.jou nals. u.l /nonlinea -analysis
Exis ence o a solu ion o a nonlinea in eg al equa ion 593
which implies
kx−yk⩽1
1− |δ|P2N2
kTx −Tyk.(18)
F om (17) and (18) we ge
kSx −Syk⩽||P1N1
1− |δ|P2N2
kTx −Tyk,
and, since α=||P1N1/(1 − |δ|P2N2)<1, i we de ine ζ(p, q) = αq −p o all
p, q ∈[0,∞), hen we ha e
ζd(Sx, Sy), d(Tx, Ty)⩾0
o all x, y ∈C(I)wi h Tx 4Ty. Thus, condi ion (1) is i ially sa is ied. Nex , we can
show ha S(C(I)) ⊆T(C(I)). Indeed, by (iii) o x( )∈C(I)we ha e
TSx( ) + 1( )
=Sx( ) + 1( )− 1( )−δ
T
Z
0
n2( , s)k2s, Sx(s) + 1(s)ds
=Sx( )−δ
T
Z
0
n2( , s)k2 s, 
s
Z
0
n1(s, )k1 , x( )d + 1(s)− 2(s)!ds
=Sx( ).
Clea ly, hypo hesis (i ) means ha Sis T-nondec easing. Nex , by ( ) we ge
x0− 1( )−δ
T
Z
0
n2( , s)k2s, x0(s)ds⩽− 2( ) + 
Z
0
n1( , s)k1s, x0(s)ds,
ha is, Tx04Sx0. Thus, all he cases o Theo em 1 a e sa is ied, and hence, i s
esul holds, ha is, Sand Tha e a las a coincidence poin . Consequen ly, he in eg al
equa ion (16) has a solu ion in C(I).
7 Conclusion
In his wo k, we conside a pai o nonlinea ope a o s sa is ying a nonlinea con ac-
ion in ol ing a simula ion unc ion in a me ic space endowed wi h a pa ial o de . Fo
his pai o ope a o s wi h and wi hou con inui y, we es ablish coincidence and unique
common ixed poin esul s. Mo eo e , an applica ion o ou esul s ob ained o p o e he
exis ence o a solu ion o an in eg al equa ion is p esen ed.
Nonlinea Anal. Model. Con ol, 28(3):578–596, 2023

594 P. Lo0lo0e al.
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