Ci a ion: Hammad, H.A.; Rehman,
H.u.; De la Sen, M. A New
Fou -S ep I e a i e P ocedu e o
App oxima ing Fixed Poin s wi h
Applica ion o 2D Vol e a In eg al
Equa ions. Ma hema ics 2022,10, 4257.
h ps://doi.o g/10.3390/ma h
10224257
Academic Edi o : Paul B acken
Recei ed: 21 Oc obe 2022
Accep ed: 9 No embe 2022
Published: 14 No embe 2022
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ma hema ics
A icle
A New Fou -S ep I e a i e P ocedu e o App oxima ing Fixed
Poin s wi h Applica ion o 2D Vol e a In eg al Equa ions
Hasanen A. Hammad 1,2,* , Habib u Rehman 3and Manuel De la Sen 4
1Depa men o Ma hema ics, Unaizah College o Sciences and A s, Qassim Uni e si y,
Bu aydah 52571, Saudi A abia
2Depa men o Ma hema ics, Facul y o Science, Sohag Uni e si y, Sohag 82524, Egyp
3Depa men o Ma hema ics, Monglku ’s Uni e si y o Technology, Bangkok 10140, Thailand
4Ins i u e o Resea ch and De elopmen o P ocesses, Depa men o Elec ici y and Elec onics,
Facul y o Science and Technology, Uni e si y o he Basque Coun y, 48940 Leioa, Bizkaia, Spain
*Co espondence: [email p o ec ed] o [email p o ec ed]
Abs ac :
This wo k is de o ed o p esen ing a new ou -s ep i e a i e scheme o app oxima ing
ixed poin s unde almos con ac ion mappings and Reich–Suzuki- ype nonexpansi e mappings
(RSTN mappings, o sho ). Addi ionally, we demons a e ha o almos con ac ion mappings, he
p oposed algo i hm con e ges as e han a a ie y o o he cu en i e a i e schemes. Fu he mo e,
he new i e a i e scheme’s
ω2−
s abili y esul is es ablished and a co obo a ing example is gi en o
cla i y he concep o
ω2−
s abili y. Mo eo e , weak as well as a numbe o s ong con e gence esul s
a e demons a ed o ou new i e a i e app oach o ixed poin s o RSTN mappings. Fu he , o
demons a e he e ec i eness o ou new i e a i e s a egy, we also conduc a nume ical expe imen .
Ou majo inding is applied o demons a e ha he wo-dimensional (2D) Vol e a in eg al equa ion
has a solu ion. Addi ionally, a comp ehensi e example o alida ing he ou come o ou applica ion
is p o ided. Ou esul s expand and gene alize a numbe o ele an esul s in he li e a u e.
Keywo ds:
RSTN mapping; almos con ac ion mapping;
ω2−
s abili y; ixed poin me hodology;
nonlinea in eg al p oblem
MSC: 47H05; 39B82; 47H09
1. P elude and Basic No ions
Nowadays, a e he huge amoun o aluable pape s ha include he ixed poin (FP)
me hod, hese poin s ha e become he mains ay o nonlinea analysis due o he ease
and smoo hness o his me hod, in addi ion o he nume ous and exci ing applica ions in
economics, biology, chemis y, game heo y, enginee ing, physics, e c. [1–5].
A e y impo an b anch is he in ol emen o FPs in app oxima ion by algo i hms.
Nume ous p oblems such as con ex easibili y p oblems, con ex op imiza ion p oblems,
mono one a ia ional inequali ies, and image es o a ion p oblems can be hough o as FP
p oblems o nonexpansi e mappings, hence app oxima ing hem has a ange o specialized
applica ions, see [
6
–
12
]. I e a ion app oaches o FP issues o nonexpansi e mappings ha e
ecei ed a lo o a en ion in he li e a u e, o example, see [13–17].
F om now on, he symbols
R
,
N
,
Ξ(=)
,
∆
, and
Π
, deno e he se o eal numbe s,
na u al numbe s, FPs o he mapping
=
, and a nonemp y subse o a Banach space
(BS) Π, espec i ely.
Assume ha =:∆→∆is a sel -mapping, hen o each ,υ∈∆,
•=is called a con ac ion i he e is `∈[0, 1)so ha k= − =υk≤`k −υk.
•=is called nonexpansi e i k= − =υk≤k −υk, i.e., i is a con ac ion wi h `=1.
•=owns an FP , i == .
Ma hema ics 2022,10, 4257. h ps://doi.o g/10.3390/ma h10224257 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2022,10, 4257 2 o 26
The e a e wo main ca ego ies ha can be used o g oup he main concep s o FP
heo y. Finding he p e equisi es and equi emen s necessa y o an ope a o o admi ixed
poin s is he i s s ep. Ano he op ion is o loca e hese ixed poin s using ce ain schema ic
me hods. The i s ca ego y is known o mally as he exis ence pa , while he second
ca ego y is known as he compu a ion o app oxima ion pa . S udying he beha io s o
FPs, such as s abili y and da a dependence, is an essen ial bu less well-known opic o
FP heo y.
The class o weak con ac ions ha app op ia ely co e s he class o Zam i escu
ope a o s [
18
] was supplied by Be inde in [
19
]. Many au ho s also e e o his class o
mappings as “almos con ac ion mappings (ACM)”.
De ini ion 1. I he e a e `∈[0, 1)and δ≥0, he inequali y below holds
k= − =υk≤`k −υk+δk − = k, o all ,υ∈∆. (1)
Then =:∆→∆is called ACM.
Via he concep o s ic ly inc easing con inuous unc ions (SIC unc ions), he condi-
ion (1) gene alized by Imo u and Olan iwo [20] as ollows:
De ini ion 2.
I he e is a cons an
`∈[
0, 1
)
and a SIC unc ion
ξ:[
0,
∞)→[
0,
∞)
wi h
ξ(0) = 0such ha
k= − =υk≤`k −υk+ξ(k − = k), o all ,υ∈∆. (2)
Then =:∆→∆is called con ac i e-like.
Clea ly, he inequali y (2) educes o (1), i ξ(τ) = δτ.
Due o i s signi icance in e ms o applica ions, nume ous w i e s ha e s udied nonex-
pansi e mappings ex ensions and gene aliza ions in ecen yea s. Suzuki [
20
] p esen ed an
in iguing gene aliza ion o nonexpansi e mappings and a ained some esul s o exis-
ence and con e gence. These mappings a e equen ly e e ed o as mappings sa is ying
condi ion (C).
De ini ion 3. I he inequali y below is ue
1
2k − = k≤k −υk⇒k= − =υk≤k −υk, o all ,υ∈∆. (3)
Then =:∆→∆is said o sa is y condi ion (C).
In 2019, he class o RSTN mappings was conside ed by Pan and Pandey [
21
] as he
ollowing:
De ini ion 4. I he e is a cons an `∈[0, 1)so ha
1
2k − = k≤k −υk⇒k= − =υk≤`k − = k+`kυ− =υk+ (1−2`)k −υk, (4)
o all ,υ∈∆.Then =:∆→∆is called an RSTN mapping.
Su ely, e e y mapping sa is ying condi ion
(C)
is an RSTN mapping wi h
`=
0. The
con e se, howe e , is alse, as demons a ed in [21].
The analysis o he pe o mance and beha io o algo i hms ha make signi ican
con ibu ions o eal-wo ld applica ions is one o he key ends in FP echniques. The e-
o e, in o de o enhance he unc ionali y and con e gence beha io o algo i hms o
nonexpansi e mappings, se e al au ho s ended o de elop nume ous i e a i e schemes o
app oxima ing FPs, o example Mann [
22
], Ishikawa [
23
], Noo [
24
],
A gawal e al. [25]
,
Ma hema ics 2022,10, 4257 3 o 26
Abbas and Nazi [
26
], CR [
27
], No mal-S [
28
], Pica d-
S
[
29
], Thaku e al. [
30
], and M-
i e a i e [31] schemes.
Recen ly, Ahmad e al. [
32
] p esen ed a good i e a i e me hod known as he JK-
i e a i e p ocedu e:
z1∈∆,
= (1−η )z +η =z ,
ϑ == ,
z +1==((1−γ )= +γ =ϑ ),
o all ≥1, (5)
whe e
η
and
γ
a e sequences in
(
0, 1
)
. Fo he mappings sa is ying condi ion
(C)
, he au-
ho s gene a ed se e al weak and s ong con e gence esul s and also showed nume ically
ha he i e a i e me hod (5) con e ges quicke han he i e a ion [25,30].
Ve y ecen ly, Hasanen e al. [
33
] p esen ed a no el ou -s ep i e a i e scheme known
as he HR-i e a ion:
z0∈∆,
= (1−η )z +η =z ,
ω ==((1−α ) +α = ),
ϑ ==((1−γ )=ω +γ =ω )
z +1==ϑ ,
o all ≥1, (6)
whe e
αi
,
ηi
, and
γi
a e sequences in
[
0, 1
]
. Addi ionally, he au ho s p o ed ha his
algo i hm con e ges as e han he me hods p esen ed in [27,29–31] nume ically.
Acco ding o he abo e wo ks, we build a new ou -s ep i e a i e p ocedu e called
HR*-i e a ion o ob aining a no el app oxima ion o FPs o ACMs and RSTN mappings
as ollows:
$0∈∆,
ρ = (1−s )$ +s =$ ,
ω ==((1− )ρ + =ρ ),
ϑ ==(=(ω )),
$ +1= (1−e )ϑ +e =ϑ ,
o all ≥1, (7)
whe e s , , and e a e sequences in (0, 1).
The goal o his manusc ip is o show ha he i e a ion (7) con e ges as e han i e a-
ions (5), (6), and Thaku e al.’s [
30
] i e a i e scheme. Hence, i is as e han many sobe
i e a i e me hods in his di ec ion o ACMs. Addi ionally, he p ope y o
ω2−
s abili y o
he p oposed algo i hm is shown wi h a suppo ed example. Mo eo e , weak and s ong
con e gence esul s o he conside ed me hod a e ob ained o RSTN mappings. Ul ima ely,
we p o e ha a 2D Vol e a in eg al equa ion has a solu ion in BSs using ou main indings.
2. De ini ions and Auxilia y Lemmas
In his pa , we p o ide some basic de ini ions and concep s ha help us in ou desi ed
goal and also acili a e he eade o unde s and ou manusc ip .
Assume ha
Π∗
is a dual o a BS
Π
,
h
., .
i
e e s o he gene alized duali y pai ing
be ween
Π
and
Π∗
,
−→
deno es s ong con e gence, and
*
deno es weak con e gence.
Fo
∈Π
, he no malized duali y mapping
Θ:Π→
2
Π∗
is a mul i alued mapping
de ined as
Θ( ) = nυ∈Π∗:h ,υi=k k2=kυk2o.
A BS Πis called smoo h i he limi below exis s o all ,υ∈P
lim
a→0
k +aυk−k k
a, (8)
whe e
P={υ∈Π:kυk=1}
. He e, he no m o
Π
is called Gâ eaux di e en iable. Clea ly,
i
Π
is smoo h, hen
Θ
is a single- alued mapping. Fu he , i he limi (8) exis s and is
Ma hema ics 2022,10, 4257 4 o 26
a ained uni o mly o
υ∈Z
, hen he no m o
Π
is called F éche di e en iable o
∈P
and he ollowing inequali y is ue
h ,Θ( )i+1
2k k2≤1
2k +υk2≤ hυ,Θ( )i+1
2k k2+z(υ),
whe e z:[0, ∞)→[0, ∞)is an inc easing unc ion so ha limυ↓0z(υ)
υ=0.
De ini ion 5.
I o each
e∈(
0, 2
]
, he e exis s
δ>
0so ha
kυk≤
1,
kυk≤
1and
kυ− k>e
,
we ge
υ+
2
<1−δ o υ, ∈Π.Then a BS Πis called a uni o mly con ex.
De ini ion 6. I o any sequence {νi}in Πso ha υi*υ∈Π,implies
lim sup
→∞
kυ −υk<lim sup
→∞
kυ − k, o all ∈Πwi h ν6= .
Then a BS Πis said o sa is y Opial’s condi ion.
De ini ion 7. Assume ha {υ }is a bounded sequence in a BS Π.Fo υ∈∆⊂Π,pu
<(υ,{υ }) = lim sup
→∞
kυ −υk.
•The asymp o ic adius o {υi} ela i e o Πis desc ibed as
<(Π,{υ }) = in {<(υ,{υ }):υ∈Π}.
•The asymp o ic cen e o {υi} ela i e o Πis gi en by
Z(Π,{υ }) = {υ∈Π:<(υ,{υ }) = <(Π,{υ })}.
Clea ly, Z(Π,{υ })consis s o exac ly one poin in a uni o mly con ex BS.
De ini ion 8.
Assume ha
∆6=∅
is a closed con ex subse o a BS
Π
.A sel -mapping
=:∆→∆
is called demiclosed wi h espec o
∈Π
,i o all a sequence
{ }*∆
and
{= } −→ υ
implies = =υ.
De ini ion 9
([
34
])
.
Suppose ha
{s }
and
{ }
a e wo sequences o eal numbe s ha , espec-
i ely, con e ge o s and .I he e is α=lim →∞ks −sk
k − k.Then
(i)
{s }is con e ges o s as e han { }does o , i α=0,
(ii)
he wo sequences {s }and { }ha e he same a e o con e gence, i α∈(0, ∞).
De ini ion 10
([
34
])
.
Assume ha
{ϕ }
and
{φ }
a e wo FP i e a ion p ocedu es which con e ge
o he same poin e
υ, he e o es ima es
kϕ −e
υk≤s and kφ −e
υk≤ , ∈N
a e accessible, whe e
{s }
and
{ }
a e de ined in De ini ion 9and con e ging o 0. Then,
{ϕ }
con e ges as e o e
υ han {φ }i {s }con e ges as e han { }.
De ini ion 11. Fo a mapping =:∆→∆,i
lim
→∞k=υ −υ k=0. (9)
Then he sequence {υ }in ∆is called an app oxima e FP sequence o a mapping =.
Ma hema ics 2022,10, 4257 5 o 26
De ini ion 12
([
35
])
.
Assume ha
κ:(
0,
∞)→(
0,
∞)
is a nondec easing unc ion wi h
κ(
0
) =
0
and o each
τ>
0, i
κ(τ)>
0so ha
k= − k≥κ(d( ,Ξ(=)))
, o all
∈∆
,whe e
d(
,
Ξ(=)) = in ∗∈Ξ(=)k − ∗k
, hen he mapping
=:∆→∆
is said o sa is y he condi ion
(I).
Lemma 1
([
36
])
.
Assume ha
{ξ }
and
{ζ }
a e wo non-nega i e eal sequences e i ying he
inequali y below
ξ +1≤(1−θ )ξ +ζ ,∀ ∈N,
whe e θ ∈(0, 1),
∞
∑
=0
θ =∞and lim →∞ζ
θ =0, hen lim →∞ξ =0.
Lemma 2
([
28
])
.
Suppose ha
{ }
and
{υ }
a e any sequences o a uni o mly con ex BS
Π
such
ha he ollowing inequali ies hold
lim sup
→∞
k k≤h, lim sup
→∞
kυ k≤h and lim sup
→∞
kς + (1−ς )υ k=h,
o some
h≥
0, whe e
{ς }
is any sequence sa is ying 0
< ≤ς ≤υ<
1. Then
lim →∞k −υ k
= 0.
Lemma 3
([
32
])
.
Assume ha
=:∆→∆
is a gi en mapping. I
=
is an RSTN mapping wi h
Ξ(=)6=∅
, hen o a bi a y poin
∈∆
and
∗∈Ξ(=)
,we ha e
k= − = ∗k≤k − ∗k
.
Mo eo e , i =sa is ies condi ion (C), hen =is an RSTN mapping.
Lemma 4
([
37
])
.
Suppose ha
=:∆→∆
is an RSTN mapping, hen o all
,
υ∈∆
and some
`∈(0, 1), he inequali y below holds
k − =υk≤3+`
1−`k − = k+k −υk. (10)
We now p o ide a nume ical example ha mee s he inequali y (10) bu does no
sa is y condi ion (C).
Example 1.
Assume ha
R
endowed wi h a usual no m
k.k
is a BS and
−
1
≤∆≤
1. De ine a
mapping =:∆→∆by
= =
−
4,i −1≤ <0,
− ,i ∈[0, 1] {1
4},
0, i ∈ {1
4}.
I we se =1
4and υ=1, we ha e
1
2k − = k=1
2
1
4− =1
4
=1
8≤3
4=k −υk.
Howe e ,
k= − =υk=
=1
4− =(1)
=1>3
4=k −υk.
The e o e, he mapping =:∆→∆does no sa is y condi ion (C).
On he o he hand, w p o e ha
=
ul ills he inequali y (10). To each his esul , we sugges
he ollowing posi ions:
(p1)i −1≤ ,υ<0, we ge
| − =υ|≤| − = |+|= − =υ|=| − = |+1
4| −υ|
≤3+
1− | − = |+| −υ|.
Ma hema ics 2022,10, 4257 6 o 26
(p2)i ,υ∈[0, 1] {1
4}, hen
| − =υ|≤| − = |+|= − =υ|=| − = |+| −υ|.
(p3)i −1≤ <0and υ∈[0, 1] {1
4},we ha e
| − =υ|=| +υ|≤| |+|υ|
≤5
4| |+| −υ|(since <0and υ≥0)
= −−
4+| −υ|
=| − = |+| −υ|.
(p4)i −1≤ <0and υ=1
4,one can w i e
| − =υ|=| |≤5
4| |+ −1
4=| − = |+| −υ|.
(p5)i ∈[0, 1] {1
4}and υ=1
4,we ob ain
| − =υ|=| |≤2| |+ −1
4=| − = |+| −υ|.
Based on he abo e cases, we conclude ha = ul ills he inequali y (10) wi h 3+
1− ≥1.
3. Ra e o he Con e gence
In his pa , we demons a e analy ically ha o ACMs, ou i e a i e me hod (7)
con e ges as e han he i e a i e me hod in (5).
Theo em 1.
Le
∆6=∅
be a closed con ex subse o a BS
Π
and
=:∆→∆
be ACM. I
{$ }
is a
sequence i e a ed by (7). Then {$ } −→ $,whe e $is a unique FP o =.
P oo . Conside $∈Ξ(=). Based on (1) and (7), we ha e
kρ −$k=k(1−s )$ +s =$ − =$k
≤(1−s )k$ −$k+s k=$ − =$k(11)
≤(1−s )k$ −$k+s [`k$ −$k+δk$− =$k]
= (1−s (1−`))k$ −$k.
F om (7) and (12), we ge
kω −$k=k=((1− )ρ +e =ρ )− =$k
≤`k(1− )ρ +e =ρ −$k
≤`[(1− )kρ −$k+e k=ρ − =$k](12)
≤`[(1− (1−`))kρ −$k]
≤`[(1−s (1−`))(1− (1−`))]k$ −$k.
Using (7) and (13), we ob ain ha
kϑ −$k=k=(=ω )− =$k
≤`k=ω −$k(13)
≤`2kω −$k
≤`3[(1−s (1−`))(1− (1−`))]k$ −$k.
Ma hema ics 2022,10, 4257 7 o 26
Finally, om (7) and (14), one can w i e
k$ +1−$k=k(1−e )ϑ +e =ϑ − =$k
≤(1−e )kϑ −$k+e k=ϑ − =$k(14)
≤(1−e (1−`))kϑ −$k
≤`3(1−e (1−`))(1−s (1−`))(1− (1−`))k$ −$k.
As
`∈(
0, 1
)
and 0
<e
,
s
,
<
1, i ollows ha
(1−e (1−`))<
1,
(1−s (1−`)) <
1
and (1− (1−`)) <1, hence
(1−e (1−`))(1−s (1−`))(1− (1−`)) <1.
Thus, (15) educes o
k$i+1−$k≤`3k$ −$k.
By induc ion, one can w i e
k$ +1−$k≤`3( +1)k$0−$k→0 as →∞. (15)
Hence,
$ −→ $
. The uniqueness
$
ollows immedia ely by he de ini ion o
=
. This
inishes he p oo .
Theo em 2.
Le
∆6=∅
be a closed con ex subse o a BS
Π
and
=:∆→∆
be ACM. I
{$ }
is
a sequence i e a ed by (7). Then
{$ }
con e ges as e han
{z }
, which is made by he i e a i e
scheme (5).
P oo . Keeping in mind (15) o Theo em 1, we ge
k$ +1−$k≤`3( +1)k$0−$k, ∈N.
Addi ionally, using (5), one can ob ain
k −$k=k(1−η )z +η =z − =$k
≤(1−η )kz −$k+η k=z − =$k(16)
≤(1−η (1−`))kz −$k.
F om (5) and (17), we ha e
kϑ −$k=k= − =$k
≤`k −$k(17)
≤`(1−η (1−`))kz −$k.
Again, using (5), (17), and (18), one has
kz +1−$k=k=((1−γ )= +γ =ϑ )− =$k
≤`k(1−γ )= +γ =ϑ −$k
≤`((1−γ )k= − =$k+γ k=ϑ − =$k)
≤`2((1−γ )k −$k+γ kϑ −$k)
≤`2[(1−γ )(1−η (1−`))kz −$k+γ `(1−η (1−`))kz −$k]
≤`2[(1−γ (1−`))(1−η (1−`))kz −$k]
≤`2kz −$k.
Ma hema ics 2022,10, 4257 8 o 26
By induc ion, we ha e
kz +1−$k≤`2( +1)kz −$k. (18)
Di iding (15) by (18), we ind ha
k$ +1−$k
kz +1−$k≤`3( +1)k$0−$k
`2( +1)kz −$k=`( +1)k$0−$k
kz −$k→0, as →∞,
which implies ha {$ }con e ges as e han {z } o $.
Example 2.
Assume ha
Π=R3
and
∆= = ( 1, 2, 3):( 1, 2, 3)∈[0, 6]3
,whe e
[0, 6]3= [0, 6]×[0, 6]×[0, 6]
is a subse o
Π
equipped wi h he no m
k k=k( 1, 2, 3)k
=
| 1|+| 2|+| 3|.De ine a mapping =:∆→∆by
= = 1
3, 2
3, 3
3,i ( 1, 2, 3)∈[0, 3)3,
1
6, 2
6, 3
6,i ( 1, 2, 3)∈[3, 6]3.
I is clea ha
=
owns a unique FP, i is
(
0, 0, 0
)
. Now, we shall show ha
=
is a con ac i e-
like mapping and, hence, ACM. Fo his, we de ine he unc ion
ξ:[
0,
∞)→[
0,
∞)
by
ξ( ) =
4
.
Ob iously, ξis a SIC unc ion wi h ξ(0) = 0. I ∈[0, 3)3,we ha e
k − = k=
( 1, 2, 3)− 1
3, 2
3, 3
3
=
2 1
3,2 2
3,2 3
3
,
and
ξ(k − = k)=ξ
2 1
3,2 2
3,2 3
3
=
1
6, 2
6, 3
6
= 1
6+ 2
6+ 3
6. (19)
Analogously, i ∈[3, 6]3,one has
k − = k=
( 1, 2, 3)− 1
6, 2
6, 3
6
=
5 1
6,5 2
6,5 3
6
,
and
ξ(k − = k)=ξ
5 1
6,5 2
6,5 3
6
=
5 1
24 ,5 2
24 ,5 3
24
=
5 1
24 +
5 2
24 +
5 3
24 . (20)
A e ha , we discuss he cases below:
Ma hema ics 2022,10, 4257 9 o 26
(I)
I ,υ∈[0, 3)3, hen by (19), we ge
k= − =υk=
1
3, 2
3, 3
3−υ1
3,υ2
3,υ3
3
= 1
3−υ1
3+ 2
3−υ2
3+ 3
3−υ3
3
=1
3[| 1−υ1|+| 2−υ2|+| 3−υ3|]
=1
3k( 1, 2, 3)−(υ1,υ2,υ3)k=1
3k −υk
≤1
3k −υk+ 1
6+ 2
6+ 3
6
=1
3k −υk+ξ(k − = k).
(II)
I ,υ∈[3, 6]3, hen by (20), we ha e
k= − =υk=
1
6, 2
6, 3
6−υ1
6,υ2
6,υ3
6
= 1
6−υ1
6+ 2
6−υ2
6+ 3
6−υ3
6
=1
6[| 1−υ1|+| 2−υ2|+| 3−υ3|]
=1
6k( 1, 2, 3)−(υ1,υ2,υ3)k=1
6k −υk
≤1
6k −υk+
5 1
24 +
5 2
24 +
5 3
24
≤1
3k −υk+ξ(k − = k).
(III)
I ∈[0, 3)3and υ∈[3, 6]3, hen by (19), we ob ain ha
k= − =υk=
1
3, 2
3, 3
3−υ1
6,υ2
6,υ3
6
=
1
3−υ1
6, 2
3−υ2
6, 3
3−υ3
6
=
1
6+ 1
6−υ1
6, 2
6+ 2
6−υ2
6, 3
6+ 3
6−υ3
6
≤ 1
6+ 1
6−υ1
6+ 2
6+ 2
6−υ2
6+ 3
6+ 3
6−υ3
6
≤ 1
6+ 2
6+ 3
6+ 1
6−υ1
6+ 2
6−υ2
6+ 3
6−υ3
6
=1
6[| 1−υ1|+| 2−υ2|+| 3−υ3|]+ξ(k − = k)
≤1
3k( 1, 2, 3)−(υ1,υ2,υ3)k+ξ(k − = k)
=1
3k −υk+ξ(k − = k).
Ma hema ics 2022,10, 4257 16 o 26
The s ong con e gence esul s ha we now es ablish a e as ollows:
Theo em 5.
Le
∆
,
=
, and
Π
be as in Lemma 6. The sequence
{$ }
p oduced by HR
∗
i e -
a i e p ocedu e (7) con e ges o an elemen o
Ξ(=)
i
lim in →∞d($
,
Ξ(=)) =
0, whe e
d($ ,Ξ(=)) = in {k$ − ∗k: ∗∈Ξ(=)}.
P oo .
P o e he necessi y is clea . Con a iwise, assume ha
lim in →∞d($
,
Ξ(=)) =
0
and
∗∈Ξ(=)
. F om Lemma 5,
lim →∞k$ − ∗k
exis s o any
∗∈Ξ(=)
. I is enough
o demons a e ha he sequence
{$ }
is Cauchy in
∆
. As
lim →∞d($
,
Ξ(=)) =
0, hen o
gi en ε>0, he e is θ0∈Nso ha
d($ ,Ξ(=)) <ε
2and in {k$ − ∗k: ∗∈Ξ(=)}<ε
2, o all ≥θ0.
Pa icula ly, in
$θ0− ∗
: ∗∈Ξ(=)<ε
2. Hence, he e is ∗∈Ξ(=)so ha
$θ0− ∗
<ε
2.
Now, o θ, ≥θ0, we ge
k$θ+ −$ k≤k$θ+ − ∗k+k$ − ∗k
≤
$θ0− ∗
+
$θ0− ∗
=2
$θ0− ∗
<ε.
This p o es ha he sequence
{$ }
is Cauchy in
∆
. The closedness o
∆
implies ha
he e is an elemen
q∈∆
so ha
lim →∞$ =q
. Addi ionally,
lim →∞d($
,
Ξ(=)) =
0
leads o d(q,Ξ(=)) = 0, ha is q∈Ξ(=).
I we ake he se
∆
as nonemp y compac con ex (NCC, o sho ), we ha e he
ollowing heo em:
Theo em 6.
Le
=
and
Π
be as in Lemma 6. Assume ha
∆
is a NCC subse o
Π
.I
{$ }
is an
i e a i e sequence gene a ed by HR∗i e a i e scheme (7), hen {$ } −→ q∈Ξ(=).
P oo .
Based on Lemma 6,
lim →∞k=$ −$ k=
0. Because
∆
is a NCC, hen he e is
a con e gen subsequence
{$ i}
o
{$ }
so ha
{$ i} −→ q∈Ξ(=)
. Se ing
$ i=υ
in
Lemma 4, we ha e
k$ i− =qk≤3+`
1−`k$ i− =$ ik+k$ i−qk.
As
i→∞
, one can ind ha
$ i→ =q
, his implies ha
q==q
, i.e.,
q∈Ξ(=)
. We
conclude om Lemma 5 ha lim →∞k$ −qkexis s, hence {$ } −→ q∈Ξ(=).
The ollowing heo em is ob ained in he s ong con e gence o he sequence
{$ }
i
he ope a o =mee s condi ion (I):
Theo em 7.
Le
∆
,
=
, and
Π
be as in Lemma 6. I
{$ }
is an i e a i e sequence gene a ed by HR
∗
i e a i e scheme (7), hen {$ } −→ q∈Ξ(=)i =sa is ies condi ion (I).
P oo . Acco ding o Lemma 6, lim →∞k=$ −$ k=0. Using De ini ion 12, we ge
0≤lim
→∞κ(d($ ,Ξ(=)))≤lim
→∞k$ − =$ kimplies lim
→∞κ(d($ ,Ξ(=)))=0.
Since
κ:(
0,
∞)→(
0,
∞)
is a nondec easing unc ion wi h
κ(
0
) =
0 and o all
w>
0,
κ(w)>
0, we ge
lim →∞d($
,
Ξ(=)) =
0. Because all o he p e equisi es o Theo em 5
ha e been demons a ed, hen one can in e ha he sequence {$ } −→ q∈Ξ(=).
Ma hema ics 2022,10, 4257 17 o 26
6. Nume ical Example
In his pa , we p o ide an illus a i e example o an RSTN mapping ha does no mee
condi ion
(C)
. We also assess he con e gence o he HR
∗
i e a i e scheme in compa ison
o some o he mos popula i e a i e schemes in he li e a u e.
Example 4.
Conside
(R,k.k)
as a BS equipped wi h he usual no m and
∆= [
3, 5
]
.De ine a
mapping =:∆→∆by
= = +6
3,i <5,
2, i =5.
In o de o p o e ha =does no sa is y condi ion (C), we ake =4and υ=5, hence
1
2| − = |=1
2|4− =4|=1
3<1=| −υ|.
Howe e ,
|= − =υ|≤|=4− =5|=
10
3−6
3=4
3>1=| −υ|.
Now, o show ha =is an RSTN mapping, we conside he cases below:
(I)
I ,υ<5, we ge
`| − = |+`|υ− =υ|+ (1−2`)| −υ|
=1
2 − +6
3+1
2υ−υ+6
3
=1
2
2 −6
3+1
2
2υ−6
3
≥1
22 −6
3−2υ−6
3
=1
2
2
3−2υ
3=1
3| −υ|=|= − =υ|.
(II)
I <5and υ=5, we ob ain
`| − = |+`|υ− =υ|+ (1−2`)| −υ|
=1
2 − +6
3+1
2|5−2|
=1
2
2 −6
3+3
2=
3+1
2
≥
3=|= − =υ|.
(III)
I υ<5and =5, we ha e
`| − = |+`|υ− =υ|+ (1−2`)| −υ|
=1
2|5−2|+1
2υ−υ+6
3
=3
2+1
2
2υ−6
3=1
2+υ
3
≥υ
3=|= − =υ|.
Ma hema ics 2022,10, 4257 18 o 26
•I υ= =5, we can w i e
`| − = |+`|υ− =υ|+ (1−2`)| −υ|
=3>0=|2− =υ|=|= − =υ|.
Hence, =is RSTN mapping and has a unique FP 3.
Nume ically, by using MATLAB R2015a, we ound ha ou i e a i e scheme con e ges as e
han bo h i e a ions (5) and (6) acco ding o Tables 1and 2and Figu es 1–6as ollows:
Table 1. Nume ical compa ison o esul s o Algo i hms (5)–(7).
Numbe o I e a ions
Ini ial Poin (z1) Algo i hm (5) Algo i hms (6) Algo i hms (7)
3.00 16 13 7
3.82 23 18 10
4.44 25 20 10
Table 2. Nume ical compa ison o esul s o Algo i hms (5)–(7).
Execu ion Time in Seconds
Ini ial Poin (z1) Algo i hm (5) Algo i hms (6) Algo i hms (6)
3.00 0.00483290000000000 0.00595750000000000 0.000157200000000000
3.82 0.00236760000000000 0.00755520000000000 0.00779860000000000
4.44 0.00705930000000000 0.00946030000000000 0.00744400000000000
0 2 4 6 8 10 12 14 16
Numbe o I e a ions
10-12
10-10
10-8
10-6
10-4
10-2
100
102
Figu e 1. A g aphical compa ison o Algo i hms (5)–(7), whe e z1=3.00.
Ma hema ics 2022,10, 4257 19 o 26
0123456
Elapsed ime [sec] 10-3
10-12
10-10
10-8
10-6
10-4
10-2
100
102
Figu e 2. A g aphical compa ison o Algo i hms (5)–(7), whe e z1=3.00.
0 5 10 15 20 25
Numbe o I e a ions
10-12
10-10
10-8
10-6
10-4
10-2
100
102
Figu e 3. A g aphical compa ison o Algo i hms (5)–(7), whe e z1=3.82.
Ma hema ics 2022,10, 4257 20 o 26
12345678
Elapsed ime [sec] 10-3
10-12
10-10
10-8
10-6
10-4
10-2
100
102
Figu e 4. A g aphical compa ison o Algo i hms (5)–(7), whe e z1=3.82.
0 5 10 15 20 25
Numbe o I e a ions
10-12
10-10
10-8
10-6
10-4
10-2
100
102
Figu e 5. A g aphical compa ison o Algo i hms (5)–(7), whe e z1=4.44.
Ma hema ics 2022,10, 4257 21 o 26
1 2 3 4 5 6 7 8 9 10
Elapsed ime [sec] 10-3
10-12
10-10
10-8
10-6
10-4
10-2
100
102
Figu e 6. A g aphical compa ison o Algo i hms (5)–(7), whe e z1=4.44.
7. Sol ing 2D Vol e a In eg al Equa ion
In his sec ion, we in es iga e how ou main esul s can be applied o he nonlinea
2D Vol e a in eg al equa ion o he o m:
κ(λ,δ) = β(λ,δ) +
λ
Z0
δ
Z0
Ω1( ,u,κ( ,u))d du
+η
λ
Z0
Ω2(δ,u,κ(λ,u))du +γ
δ
Z0
Ω3(λ, ,κ(δ, ))d , (34)
o all
λ
,
δ
,
,
u∈[
0, 1
]
, whe e
κ∈Λ×Λ
,
β:[
0, 1
]×[
0, 1
]→R2
,
Ωi(i=
1, 2, 3
):[
0, 1
]×
[0, 1]×R2→R2,η,γ≥0 and Λ=C([0, 1])is a BS wi h he maximum no m
k −υk∞=max
τ∈[0,1]| (τ)−υ(τ)|, o all ,υ∈C([0, 1]).
Now, ou main heo em he e is as ollows:
Theo em 8.
Assume ha
is a nonemp y closed con ex subse o
Λ
and
=: →
desc ibed as
=κ(λ,δ) = β(λ,δ) +
λ
Z0
δ
Z0
Ω1( ,u,κ( ,u))d du
+η
λ
Z0
Ω2(δ,u,κ(λ,u))du +γ
δ
Z0
Ω3(λ, ,κ(δ, ))d .
Assume also he asse ions below a e ue
(A1) he unc ion κ:Λ×Λ→R2is con inuous;
Ma hema ics 2022,10, 4257 22 o 26
(A2)
he unc ions
Ωi(i=
1, 2, 3
):[
0, 1
]×[
0, 1
]×R2→R2
a e con inuous and he e a e he
cons an s `1,`2,`3>0so ha
|Ω1( ,u, 1( ,u))−Ω1( ,u, 2( ,u))|≤`1| 1− 2|,
|Ω2( ,u, 1( ,u))−Ω2( ,u, 2( ,u))|≤`2| 1− 2|,
|Ω3( ,u, 1( ,u))−Ω3( ,u, 2( ,u))|≤`3| 1− 2|,
o 1, 2∈R2;
(A3) o η,γ≥0, `1+η`2+γ`3≤ξ,whe e ξ∈(0, 1).
Then, he 2D Vol e a in eg al Equa ion (34) has a solu ion in
×
p o ided ha
=
has an FP.
P oo . Le κ,κ∗∈Λ×Λ, hen
kκ− =κ∗k∞=max
τ∈[0,1]|κ(λ,δ)(τ)− =κ∗(λ,δ)|
=max
τ∈[0,1]
κ(λ,δ)(τ)−β(λ,δ)(τ)−
λ
Z0
δ
Z0
Ω1( ,u,κ∗( ,u))d du
−η
λ
Z0
Ω2(δ,u,κ∗(λ,u))du −γ
δ
Z0
Ω3(λ, ,κ∗(δ, ))d
≤max
τ∈[0,1]
κ(λ,δ)(τ)−β(λ,δ)(τ)−
λ
Z0
δ
Z0
Ω1( ,u,κ( ,u))d du
−η
λ
Z0
Ω2(δ,u,κ(λ,u))du −γ
δ
Z0
Ω3(λ, ,κ(δ, ))d
+
λ
Z0
δ
Z0
Ω1( ,u,κ( ,u))d du −
λ
Z0
δ
Z0
Ω1( ,u,κ∗( ,u))d du
+η
λ
Z0
Ω2(δ,u,κ(λ,u))du −
λ
Z0
Ω2(δ,u,κ∗(λ,u))du
+γ
δ
Z0
Ω3(λ, ,κ(δ, ))d −γ
δ
Z0
Ω3(λ, ,κ∗(δ, ))d
≤max
τ∈[0,1]|κ(λ,δ)(τ)− =κ(λ,δ)|
+`1max
τ∈[0,1]
λ
Z0
δ
Z0
|κ( ,u)−κ∗( ,u)|d du +η`2max
τ∈[0,1]
λ
Z0
|κ( ,u)−κ∗( ,u)|du
+γ`3max
τ∈[0,1]
δ
Z0
|κ( ,u)−κ∗( ,u)|d ,
Ma hema ics 2022,10, 4257 23 o 26
which implies ha
kκ− =κ∗k∞≤max
τ∈[0,1]|κ(λ,δ)(τ)− =κ(λ,δ)|
+max
τ∈[0,1]`1|κ( ,u)−κ∗( ,u)|+η`2max
τ∈[0,1]|κ( ,u)−κ∗( ,u)|
+γ`3max
τ∈[0,1]|κ( ,u)−κ∗( ,u)|
≤kκ− =κ∗k∞+ (`1+η`2+γ`3)max
τ∈[0,1]|κ( ,u)−κ∗( ,u)|
≤kκ− =κ∗k∞+ξkκ−κ∗k∞
≤kκ− =κ∗k∞+kκ−κ∗k∞.
Hence, by Lemma 4,
=
is an RSTN mapping because i ul ills he condi ion (10) on
wi h
3+`
1−`=
1. Se
=∆
and
Λ=Π
, we ind ha all equi emen s o Lemma 6a e
sa is ied. The e o e,
=
has a leas one FP. Thus, p oblem (33) has a solu ion on
×
.
The ollowing example suppo Theo em 8:
Example 5. Conside he ollowing 2D Vol e a in eg al equa ion
κ(λ,δ) = π
2λ−δ2
7π+
λ
Z0
δ
Z0
cos κ( u)
2d du +2
7
λ
Z0
cos κ(λu)
2du +1
7
δ
Z0
cos κ(δ )
2d . (35)
I is clea ha p oblem (35) is a special case o (34) wi h
β(λ,δ) = π
2λ−δ2
7π,Ω1( ,u,κ( ,u))=cos κ( u)
2,
Ω2(δ,u,κ(λ,u))=cos κ(λu)
2,Ω3(λ, ,κ(δ, ))=cos κ(δ )
2,η=2
7and γ=1
7.
Then, o any ,u∈[0, 1]and 1, 2∈R2, we ind ha
|Ω1( ,u, 1( ,u))−Ω1( ,u, 2( ,u))|≤1
2|cos 1−cos 2|,
|Ω2( ,u, 1( ,u))−Ω2( ,u, 2( ,u))|≤1
2|cos 1−cos 2|, (36)
|Ω3( ,u, 1( ,u))−Ω3( ,u, 2( ,u))|≤1
2|cos 1−cos 2|,
Acco ding o he mean- alue heo em, o any
1
,
2∈R2
wi h
1< 2
he e is
b∈[ 1, 2]so ha
cos 1−cos 2
1− 2
=−sin(b), implies |cos 1−cos 2|
| 1− 2|=|−sin(b)|≤1.
Hence, |cos 1−cos 2|≤| 1− 2|and (36) educes o
|Ω1( ,u, 1( ,u))−Ω1( ,u, 2( ,u))|≤1
2| 1− 2|,
|Ω2( ,u, 1( ,u))−Ω2( ,u, 2( ,u))|≤1
2| 1− 2|,
|Ω3( ,u, 1( ,u))−Ω3( ,u, 2( ,u))|≤1
2| 1− 2|,
whe e
`1=`2=`3=1
2
and
`1+η`2+γ`3=ξ=5
7<
1. I is easy o see ha
β(λ
,
δ)
is
con inuous on [0, 1].
Ma hema ics 2022,10, 4257 24 o 26
Consequen ly, all condi ions o Theo em 8a e sa is ied. The e o e, he e exis s a
solu ion o he p oblem (36).
8. Conclusions and Fu u e Wo ks
In his s udy, a ou -s ep i e a i e scheme known as he
HR∗−
i e a i e scheme (7)
is p esen ed o app oxima ing he ixed poin s o con ac i e-like mappings and RSTN
mappings. Analy ically, i has been demons a ed ha he new i e a i e scheme con e ges
as e han he i e a i e me hod (5) o con ac i e-like mappings. Fu he mo e, we ha e
shown nume ically ha o con ac i e-like mappings, ou no el i e a i e me hod con-
e ges as e han se e al popula i e a i e schemes in he li e a u e. Addi ionally, he
ω2−
s abili y esul o he
HR∗−
i e a i e scheme (7) has also been ob ained. To cla i y
he idea o
ω2−
s abili y o he conside ed algo i hm wi h ega d o
=
, we ha e gi en an
example. Addi ionally, we ha e demons a ed a numbe o weak and s ong con e gence
heo ems o RSTN mappings in uni o mly con ex BSs. In o de o compa e he con e -
gence beha io o he p oposed algo i hm (7) wi h ce ain well-known i e a i e schemes, a
no el example o RSTN mappings has been supplied. As a p ac ical applica ion, we p o ed
ha a 2D Vol e a in eg al equa ion has a solu ion. Addi ionally, we p o ided an engaging
example o explain he ou come o ou applica ion. Finally, as u u e wo k o his pape ,
we sugges he ollowing:
(1)
I we de ine a mapping
=
in a Hilbe space
∆
endowed wi h inne p oduc space,
we can ind a common solu ion o he a ia ional inequali y p oblem by using ou
i e a ion (7). This p oblem can be s a ed as ollows: ind ℘∗∈∆such ha
h=℘∗,℘−℘∗i ≥ 0 o all ℘∈∆,
whe e
=:∆→∆
is a nonlinea mapping. Va ia ional inequali ies a e an impo an and
essen ial modeling ool in many ields such as enginee ing mechanics, anspo a ion,
economics, and ma hema ical p og amming, see [45–47].
(2)
We can gene alize ou algo i hm o g adien and ex a-g adien p ojec ion me h-
ods, hese me hods a e e y impo an o inding saddle poin s and sol ing many
p oblems in op imiza ion, see [6].
(3)
We can accele a e he con e gence o he p oposed algo i hm by adding sh inking
p ojec ion and CQ e ms. These me hods s imula e algo i hms and imp o e hei
pe o mance o ob ain s ong con e gence, o mo e de ails, see [7].
(4)
I we conside he mapping
=
as an
α−
in e se s ongly mono one and he ine ial
e m is added o ou algo i hm, hen we ha e he ine ial p oximal poin algo i hm.
This algo i hm is used in many applica ions such as mono one a ia ional inequali ies,
image es o a ion p oblems, con ex op imiza ion p oblems, and spli con ex easi-
bili y p oblems, see [
48
–
50
]. Fo mo e accu acy, hese p oblems can be exp essed as
ma hema ical models such as machine lea ning and he linea in e se p oblem.
(5)
We can y o de e mine he e o o ou p esen i e a ion.
Au ho Con ibu ions:
H.A.H. con ibu ed in concep ualiza ion, in es iga ion, me hodology, al-
ida ion and w i ing he heo e ical esul s; H.u.R. con ibu ed in concep ualiza ion, in es iga ion
and w i ing he nume ical esul s; M.D.l.S. con ibu ed in unding acquisi ion, me hodology, p ojec
adminis a ion, supe ision, alida ion, isualiza ion, w i ing and edi ing. All au ho s ha e ead and
ag eed o he published e sion o he manusc ip .
Funding: This wo k was suppo ed in pa by he Basque Go e nmen unde G an IT1555-22.
Da a A ailabili y S a emen :
The da a used o suppo he indings o his s udy a e a ailable om
he co esponding au ho upon eques .
Acknowledgmen s: The au ho s hank he Basque Go e nmen o G an IT1555-22.
Con lic s o In e es : The au ho s decla e ha hey ha e no compe ing in e es s.
Ma hema ics 2022,10, 4257 25 o 26
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