Pa ana Jou nal o Science and Educa ion, .11, n.5, (21-27), Oc obe 12, 2025
PJSE, ISSN 2447-6153, © 2015-2025
h ps://si es.google.com/si e/pjsciencea/
Recei ed: Sep embe 24, 2025; Accep ed: Oc obe 11, 2025; Published: Oc obe 12, 2025.
On Ekeland Va ia ional P inciple in Asymme ic
b-Me ic Spaces
Anas Yusu 1* and Junaidu Na iu Mallam2
Abs ac
This pape ex ends he Ekeland Va ia ional P inciple (EVP) o he se ing o asymme ic b-me ic spaces,
which gene alize bo h asymme ic me ic and b-me ic spaces. By dis inguishing o wa d, backwa d and bi-
comple eness, we es ablish co esponding o wa d, backwa d and bi-comple e e sion o EVP. Examples a e
p o ided o demons a e ha o wa d and backwa d p inciples may yield di e en minimize s while he bi-comple e
EVP educes o he classical EVP when symme y holds. As an applica ion, we ob ain a o wa d and backwa d
Ca isi- ype ixed poin heo em. These esul s b oaden he scope o EVP and o e a new ools o op imiza ion
and ixed-poin heo y in asymme ic amewo ks.
Keywo ds
ixed poin , b-me ic space, quasi-me ic space, Ekeland a ia ional p inciple, Ca isi ixed poin .
1,2Depa men o Ma hema ics, Fede al Uni e si y, Bi nin Kebbi, Nige ia.
*Co esponding au ho : [email p o ec ed]
Con en s
1In oduc ion 21
2P elimina ies 22
3Resul s 23
4Conclusion 27
Re e ences 27
1. In oduc ion
Since i s in oduc ion in he 1970s by I a Ekeland [
1
], he
Ekeland Va ia ional P inciple (EVP) has become a co ne -
s one o mode n op imiza ion heo y and nonlinea analysis.
The p inciple no only p o ides a a ia ional cha ac e iza ion
o app oxima e minimize s bu also se es as a uni ying ool
o de i ing ixed poin heo ems, Ca is i’s heo em, and c i -
ical poin esul s in Banach space se ings. I s applica ions
span di e se ields including op imal con ol, equilib ium
p oblems, con ex analysis, and pa ial di e en ial equa ions.
Following Ekeland’s o iginal wo k, many gene aliza ions o
he p inciple ha e been es ablished. Ki k [
2
] and Takahashi
[
3
] ex ended EVP o comple e me ic spaces and no med lin-
ea spaces, while Bo wein and P eiss [
4
] de eloped smoo h
a ia ional p inciples applicable in in ini e-dimensional se -
ings. Wilson [
11
] in oduced asymme ic me ic spaces as
quasi-me ic spaces and Cobza
s¸
[
5
] s udied quasi-me ic e -
sions o he Ekeland a ia ional p inciple and i s connec ions
wi h comple eness p ope ies o he unde lying quasi-me ic
space. Fa kas e al. [
13
] es ablished a gene alized a ia ional
On Ekeland Va ia ional P inciple in Asymme ic b-Me ic Spaces — 22/27
p inciple o b-me ic spaces.
On ano he on , Bakh in [
6
] in oduced he concep o a
b-me ic space, whe e he iangle inequali y is elaxed by
a cons an
s≥1
. This gene aliza ion has a ac ed consid-
e able in e es in ixed poin heo y, as shown in he wo ks
o Cze wik [
7
] and subsequen au ho s, leading o nume ous
esul s on con ac i e mappings, Ca is i- ype heo ems, and
a ia ional inequali ies in b-me ic spaces. Bo a, Moln
´
a and
Va ga [
8
] in es iga ed EVP in b-me ic spaces and showed
ha he p inciple con inues o hold unde sui able comple e-
ness assump ions. Fixed poin esul s in asymme ic me ic
spaces we e s udied by Kho shid andpou e al. [
14
]. Recen
de elopmen s in b-me ic and asymme ic me ic spaces can
be ound in [
15
,
16
]. S udies in asymme ic b-me ic (quasi-
b-me ic) spaces appea in [17,18,19].
Despi e hese ad ances, ela i ely li le a en ion has been paid
o asymme ic b-me ic spaces, which combine bo h asym-
me y and he elaxed iangle inequali y. These spaces a ise
na u ally in applica ions whe e dis ances a e no symme ic
and iangula es ima es a e app oxima e, such as compu e
science, decision heo y, and models o di ec ed ne wo ks. To
he bes o ou knowledge, no sys ema ic ea men o EVP in
asymme ic b-me ic spaces has ye been gi en.
Ou wo k ills his gap by o mula ing and p o ing o wa d,
backwa d, and bi-comple e e sions o he Ekeland Va ia-
ional P inciple in asymme ic b-me ic spaces. In doing so,
we uni y and ex end he p e ious lines o esea ch: when he
coe icien
s=1
, ou esul s educe o he asymme ic me -
ic se ing, and when symme y is imposed, we eco e he
classical EVP in b-me ic and me ic spaces.
2. P elimina ies
De ini ion 2.1 (b-me ic [
6
,
7
]).Le
X
be a non-emp y se and
s≥1
a eal numbe . A unc ion
d:X×X→R+
is called a
b-me ic p o ided ha o all x,y,z∈X:
(i) d(x,y) = 0i and only i x =y,
(ii) d(x,y) = d(y,x),
(iii) d(x,z)≤sd(x,y)+d(y,z).
A pai (X,d)is called a b-me ic space.
I is clea ha he de ini ion o b-me ic space ex ends he
usual me ic space; indeed i
s=1
we eco e he classical
me ic space.
Example 2.1 ([9]).The space `p(R)wi h 0<p<1,
`p(R) = {(xn)⊂R:
∞
∑
n=1
|xn|p<∞},
oge he wi h
d(x,y) = ∞
∑
n=1
|xn−yn|p1/p,
is a b-me ic space. By an elemen a y calcula ion we ob ain
ha
d(x,z)≤21/pd(x,y)+d(y,z),
so one may ake s =21/p.
De ini ion 2.2 (Asymme ic me ic [
10
]).A unc ion
d:X×
X→R+
is called an asymme ic me ic and
(X,d)
is an
asymme ic me ic space i o all x,y,z∈X:
(i) d(x,y) = 0i and only i x =y,
(ii) d(x,z)≤d(x,y)+d(y,z).
Example 2.2 ([10]).Le α>0. De ine d :R×R→R+by
d(x,y) = (y−x,y≥x,
α(x−y),y<x.
Then d is an asymme ic me ic on R.
De ini ion 2.3. Le
ρ
be an asymme ic me ic on
X
. The
conjuga e o ρis he mapping ¯
ρde ined by
¯
ρ(x,y) = ρ(y,x),x,y∈X.
The mapping
ds(x,y) = max{ρ(x,y),¯
ρ(x,y)}=max{ρ(x,y),ρ(y,x)}
is called he symme ized me ic on
X
, and
ds
is a me ic on
X i and only i ρis an asymme ic me ic on X.
De ini ion 2.4 (Fo wa d/backwa d opology [
10
]).The o -
wa d opology
τ+
induced by
d
is gene a ed by o wa d open
balls
B+(x,ε) = {y∈X:d(x,y)<ε},x∈X,ε>0.
Likewise, he backwa d opology
τ−
induced by
d
is gene a ed
by backwa d open balls
B−(x,ε) = {y∈X:d(y,x)<ε}.
De ini ion 2.5 (Fo wa d/backwa d con e gence [
10
]).A se-
quence
(xn)n∈N
is said o be o wa d con e gen o
x0∈X
i
lim
n→∞d(x0,xn) = 0,
and backwa d con e gen o x0i
lim
n→∞d(xn,x0) = 0.
We deno e o wa d con e gence by
xn
→x0
and backwa d
con e gence by xn
b
→x0.
On Ekeland Va ia ional P inciple in Asymme ic b-Me ic Spaces — 23/27
De ini ion 2.6 (Fo wa d/backwa d comple eness [
10
]).A
space
(X,d)
is called o wa d comple e ( esp. backwa d com-
ple e) i e e y o wa d ( esp. backwa d) Cauchy sequence in
X o wa d( esp.backwa d) con e ges o a poin in X.
De ini ion 2.7 (Fo wa d/backwa d Cauchy [
10
]).A sequence
(xk)k∈N⊂X
is o wa d Cauchy i o e e y
ε>0
he e exis s
N
such ha o all
m≥n≥N
,
d(xn,xm)<ε
. The sequence is
backwa d Cauchy i o all
m≥n≥N
we ha e
d(xm,xn)<ε
.
De ini ion 2.8 (Asymme ic b-me ic / quasi-b-me ic [
17
]).
Le
X
be a non-emp y se . A unc ion
ρ:X×X→[0,∞)
is
called an asymme ic b-me ic (also called quasi-b-me ic) on
X i he e exis s a cons an s ≥1such ha o all x,y,z∈X:
(i) ρ(x,y) = 0i x =y,
(ii) ρ(x,z)≤sρ(x,y)+ρ(y,z).
He e s is called he coe icien o he asymme ic b-me ic.
Rema k 2.1.
•
I symme y is added,
ρ(x,y) = ρ(y,x)
, hen
ρ
educes
o a s anda d b-me ic.
•I s =1, we ge an asymme ic me ic space.
So he hie a chy is
me ic ⊂b-me ic ⊂asymme ic b-me ic.
Example 2.3. De ine ρ:R×R→[0,∞)by
ρ(x,y) = max{0,y−x}2= (y−x)2
+.
Then
ρ(x,x) = 0
, and in gene al
ρ(x,y)6=ρ(y,x)
. One can
check he b- iangle inequali y wi h s =2:
ρ(x,z)≤2ρ(x,y)+ρ(y,z),
by aking
ρ(x,y)=(max{0,y−x})2= (y−x)2
+
. This can
easily be seen as
((z−x)+)2≤2((z−y)2
++(y−x)2
+)
, hence
(R,ρ)
is an asymme ic b-me ic space wi h coe icien
s=2
.
Theo em 2.1 (Classical EVP [
1
]).Le
(X,d)
be a comple e
me ic space. Le
:X→R
be p ope , lowe semicon inuous
and bounded below. Then o e e y
ε>0
and e e y
x∈X
wi h
(x)≤in
X +ε
and e e y λ>0 he e exis s y ∈X such ha
(y)≤ (x),d(x,y)≤λ,
and o all z 6=y,
(z)> (y)−ε
λd(y,z).
Theo em 2.2 (Ca is i- ype ixed poin heo em [
12
]).Le
(X,d)
be a comple e me ic space and le
T:X→X
be a
mapping such ha
d(x,Tx)≤ (x)− (Tx) o all x ∈X,
whe e
:X→[0,+∞)
is lowe semicon inuous. Then
T
has
a leas one ixed poin .
3. Resul s
De ini ion 3.1 (Fo wa d/backwa d lowe semicon inui y).A
unc ion :X→Ris o wa d-lsc i xn
→x implies
(x)≤limin
n→∞ (xn),
and backwa d-lsc i xn
b
→x implies
(x)≤limin
n→∞ (xn),
.
Theo em 3.1 (Fo wa d Ekeland Va ia ional P inciple).Le
(X,ρ)
be a o wa d comple e asymme ic b-me ic space wi h
coe icien
s≥1
. Le
:X→R
be p ope , bounded below
and o wa d-lsc. Fix ε>0. Suppose x0∈X sa is ies
(x0)≤in
X +ε.
Fo e e y
λ>0
se
α:=ε/λ>0
. Then he e exis s
x∗∈X
such ha :
(1) (x∗)≤ (x0),
(2) ρ(x0,x∗)≤sλ,
(3) Fo e e y y ∈X wi h y 6=x∗,
(y)> (x∗)−α ρ(x∗,y).
Equi alen ly,
(y)+α ρ(x∗,y)≥ (x∗) o all y ∈X.
P oo .
Fix
λ>0
and se
α=ε/λ
. We cons uc induc i ely
a sequence
(xn)
as ollows. Choose a summable sequence
(εn)
wi h
εn↓0
and
∑∞
n=0εn<∞
. De ine o each
n≥0
he
pe u bed unc ional
ϕn(y):= (y)+α ρ(xn,y).
Choose xn+1∈Xsuch ha
(xn+1)+α ρ(xn,xn+1)≤in
y∈Xϕn(y)+εn.(1)
Pu ing y=xnin (1) gi es
(xn+1)+α ρ(xn,xn+1)≤ (xn)+ εn,
hence
α ρ(xn,xn+1)≤ (xn)− (xn+1)+ εn.
The e o e
( (xn))
is noninc easing and bounded below, so i
con e ges o some
L≥in X
. Summing he inequali y om
n=0 o N−1 yields
α
N−1
∑
n=0
ρ(xn,xn+1)≤ (x0)− (xN)+
N−1
∑
n=0
εn.
On Ekeland Va ia ional P inciple in Asymme ic b-Me ic Spaces — 24/27
Le N→∞. Since (xN)→Land ∑εn<∞we ob ain
α
∞
∑
n=0
ρ(xn,xn+1)≤ (x0)−L+
∞
∑
n=0
εn≤ε+
∞
∑
n=0
εn<∞.
Thus ∑∞
n=0ρ(xn,xn+1)<∞and ρ(xn,xn+1)→0.
Fo wa d Cauchy p ope y. Fo
m>n
, epea ed use o he
b- iangle inequali y gi es
ρ(xn,xm)≤s(ρ(xn,xm−1)+ρ(xm−1,xm))
≤s(ρ(xn,xn+1)+ρ(xn+1,xn+2)+...+
ρ(xm−1,xm))
≤s
m−1
∑
k=n
ρ(xk,xk+1)
Thus, o e e y
η>0
he e exis s
N
such ha o all
m>n≥
N
we ha e
ρ(xn,xm)<η
. Hence
(xn)
is o wa d Cauchy. By
o wa d comple eness he e exis s x∗∈Xwi h ρ(x∗,xn)→0
(i.e. xn
→x∗).
Dis ance bound. Fo e e y m,
ρ(x0,xm)≤s(ρ(x0,x1)+ρ(x1,x2)+...+ρ(xm−1,xm))
=s
m−1
∑
k=0
ρ(xk,xk+1)
Le ing m→∞and using he summabili y bound gi es
ρ(x0,x∗)≤s
∞
∑
k=0
ρ(xk,xk+1)≤sε+∑εn
α.
Choosing
∑εn
a bi a ily small and ecalling
α=ε/λ
yields
ρ(x0,x∗)≤sλ. Thus (2) holds.
Showing
(x∗)≤ (x0)
. Since
is o wa d-lsc and
xn
→x∗
we ge
(x∗)≤limin
n→∞ (xn) = L≤ (x0),
so (1) holds.
Va ia ional inequali y. F om
(1)
wi h an a bi a y
y∈X
we
ha e
(xn+1)+α ρ(xn,xn+1)≤ (y)+ α ρ(xn,y)+εn.(2)
Rea ange:
(y)≥ (xn+1)+αρ(xn,xn+1)−ρ(xn,y)−εn.
Le ing
n→∞
, using
ρ(xn,xn+1)→0
,
ρ(xn,y)→ρ(x∗,y)
(by
o wa d con e gence o
(xn)
o
x∗
), and
(xn+1)→ (x∗)
, we
ob ain
(y)≥ (x∗)−α ρ(x∗,y).
Hence
(y)+ α ρ(x∗,y)≥ (x∗)
o all
y
. I equali y occu s
o some
y6=x∗
hen by using inequali y 3and le ing
n→∞
gi es equali y in he limi . Tha is
lim
n→∞(( (y)+α ρ(xn,y)+εn)−( (xn+1)+α ρ(xn,xn+1))) = 0.
(3)
Because each e m
xn+1
is an app oxima e minimize o
ϕn(y)
,
he only way he alues a
xn+1
and a
y
can be a bi a ily close
is i he poin s hemsel es become a bi a ily close in he
o wa d sense:
ρ(xn+1,y)→0
, bu we al eady ha e
xn+1→x∗
.
So
xn+1
would be con e ging o bo h
x∗
and
y
, o cing
y=x∗
,
a con adic ion. The e o e s ic inequali y holds o
y6=x∗
,
p o ing (3).
Co olla y 3.1 (Backwa d EVP).Le
(X,ρ)
be backwa d com-
ple e asymme ic b-me ic space wi h coe icien
s≥1
. Le
:X→R
be p ope , bounded below and backwa d-lsc. Fix
ε>0and suppose x0∈X sa is ies
(x0)≤in
X +ε.
Fo e e y
λ>0
, se
α:=ε/λ>0
. Then he e exis s
x∗∈X
such ha :
(1) (x∗)≤ (x0),
(2) ρ(x∗,x0)≤sλ,
(3) Fo e e y y ∈X wi h y 6=x∗,
(y)> (x∗)−α ρ(y,x∗).
Equi alen ly, (y)+α ρ(y,x∗)≥ (x∗) o all y.
P oo .
The p oo pa allels ha o Theo em 3.1 wi h he oles
o a gumen s in
ρ
e e sed: a each s ep one minimizes he
backwa d-pe u bed unc ionals and uses backwa d-Cauchy
and backwa d comple eness. The o de o limi s uses backwa d-
lsc o .
Theo em 3.2 (Bi-comple e (symme ized) EVP).Le
(X,ρ)
be an asymme ic b-me ic space wi h coe icien
s≥1
. De ine
he symme ized me ic
ds(x,y):=max{ρ(x,y),ρ(y,x)},x,y∈X.
Assume
(X,ρ)
is bi-comple e (i.e.
(X,ds)
is comple e). Le
:
X→R
be p ope , bounded below and lowe semicon inuous
wi h espec o ds. Fix ε>0and suppose x0∈X sa is ies
(x0)≤in
X +ε.
Then o e e y λ>0 he e exis s x∗∈X such ha :
(1) (x∗)≤ (x0),
(2) ds(x0,x∗)≤sλ,
On Ekeland Va ia ional P inciple in Asymme ic b-Me ic Spaces — 25/27
(3) Fo e e y y ∈X wi h y 6=x∗,
(y)> (x∗)−ε
λds(x∗,y).
P oo . Pu α:=ε/λ>0. Fo each z∈Xde ine
Gz(w):= (w)+αds(w,z),w∈X.
Each
Gz
is
ds
-lsc because
is
ds
-lsc and
w7→ ds(w,z)
is con-
inuous. By comple eness o
(X,ds)
any minimizing sequence
o
Gz
is
ds
-Cauchy and has a
ds
-limi whe e he in imum is
a ained. Thus we may induc i ely choose a sequence
(xn)
wi h x0gi en and, o each n≥0,
Gxn(xn+1) = in
w∈XGxn(w).
Pu ing w=xnin he minimiza ion yields
(xn+1)+αds(xn+1,xn)≤ (xn).
Hence
αds(xn+1,xn)≤ (xn)− (xn+1),
so
( (xn))
is noninc easing and bounded below and he e o e
con e gen . Summing he inequali y gi es
α
∞
∑
n=0
ds(xn+1,xn)≤ (x0)−lim
n→∞ (xn)≤ε,
so
∑ds(xn+1,xn)≤ε/α=λ(4)
.
ds
-Cauchy and limi . Fo
m>n
, he he b- iangle inequali y
gi es
ds(xn,xm)≤s
m−1
∑
k=n
ds(xk+1,xk)
Since he se ies o inc emen s con e ges, he igh handside
ends o
0
as
n→∞
. Hence
(xn)
is
ds
-Cauchy. By comple e-
ness he e exis s x∗∈Xwi h ds(xn,x∗)→0 as n→∞.
F om lowe semi-con inui y o
and mono onici y o
(xn),we ge
(x∗)≤limin
n→∞ (xn) = L≤ (x0),
so p ope y (1) holds.
To show
ds(x0,x∗)≤sλ
. Use he b- iangle inequali y
epea edly: o e ey m,
ds(x0,xm)≤s(ds(x0,x1)+ds(x1,x2)+...+ds(xm−1,xm))
=s
m−1
∑
k=0
ds(xk,xk+1)
Le ing m→∞and equa ion (4) we ob ain
ds(x0,x∗)≤s
∞
∑
k=0
ds(xk,xk+1)s·ε
α=sλ.
Thus (2) holds.
Va ia ional inequali y. Fo a bi a y
x∈X
, he minimize
p ope y yields o each n
(xn+1)+αds(xn+1,xn)≤ (x)+αds(x,xn).(5)
Rea ange:
(x)≥ (xn+1)+αds(xn+1,xn)−ds(x,xn).
Le ing
n→∞
. Since
xn→x∗
in
ds
we ha e
ds(xn+1,xn)→0
,
ds(x,xn)→ds(x,x∗)
, while
(xn+1)→ (x∗)
. Passing o he
limi in (5) we ob ain
(x)≥ (x∗)−αds(x,x∗).
Hence (x)+αds(x,x∗)≥ (x∗).
S ic inequali y.
To see s ic inequali y when
y6=x∗
, suppose
(y)+αds(y,x∗)≥
(x∗)
o some
y6=x∗
. Then om (5) wi h
x=y
equali y mus
hold in he limi , o cing
ds(xn+1,y)→0
, bu we al eady ha e
xn+1→x∗
in
ds
. So
ds(y,x∗) = 0
, hence
y=x∗
which is a
con adic ion. The e o e s ic inequali y holds o
y6=x∗
,
p o ing (3).
Co olla y 3.2 (Reduc ion o classical EVP).Le
(X,ρ)
be an
asymme ic b-me ic space wi h coe icien
s=1
. Suppose
ρ
is symme ic (so
ρ(x,y) = ρ(y,x)
o all
x,y
). Then
ds=ρ
is
a comple e me ic on
X
. Le
:X→R
be p ope , bounded
below and lowe semicon inuous wi h espec o
ρ
. Fix
ε>0
and suppose x0∈X sa is ies
(x0)≤in
X +ε.
Then o e e y λ>0 he e exis s x∗∈X such ha :
(x∗)≤ (x0),ρ(x0,x∗)≤λ,
and o e e y y 6=x∗,
(y)> (x∗)−ε
λρ(x∗,y).
P oo .
I
ρ
is symme ic and
s=1
hen
ds(x,y) = ρ(x,y)
. Ap-
plying Theo em 3.2 wi h
ds=ρ
yields he classical Ekeland
a ia ional p inciple; he ac o
s
disappea s om he dis ance
bound.
Example 3.1 (Fo wa d s Backwa d EVP selec ing di e en
minimize s).Take X = [0,3]and de ine
ρ(x,y) = (y−x,y≥x,
2(x−y),y<x.
On Ekeland Va ia ional P inciple in Asymme ic b-Me ic Spaces — 26/27
Then
ρ
is asymme ic and sa is ies he b- iangle inequali y
wi h s =2:
ρ(x,z)≤2ρ(x,y)+ρ(y,z).
De ine
(x)=(x−0.5)2
o
x∈[0,3]
and choose EVP pa am-
e e s x0=2,ε=2.25,λ=2.25 so ha ε/λ=1.
Fo wa d penalized unc ion. The o wa d penalized unc ion
is
g (y) = (y)+ ε
λρ(x0,y) = (y−0.5)2+ρ(2,y).
Compu e ρ(2,y):
ρ(2,y) = (4−2y,y≤2,
y−2,y≥2.
On [0,2],
g (y)=(y−0.5)2+4−2y,g0 (y) = 2(y−0.5)−2=2y−3,
so he c i ical poin is
y=1.5
and
g00
(y) = 2>0
, hence
y=
1.5
is he minimize on
[0,2]
. On
[2,3]
he unc ion inc eases.
Thus he global minimize o g on [0,3]is x∗=1.5.
Check he dis ance bound:
ρ(x0,x∗) = ρ(2,1.5) = 2(2−
1.5) = 1≤sλ=2(2.25) = 4.5
. Also
(1.5) = 1≤ (2) =
2.25 so (x∗)≤ (x0).
The o wa d a ia ional inequali y equi es o
x∗=1.5
and any y 6=1.5,
(y)> (1.5)−ρ(1.5,y).
o equi alen ly
∆(y):= (y)−( (1.5)−ρ(1.5,y)).
Compu ing ∆(y)sepe a ely on he wo egions.
1.
I
y>1.5;
hen,
ρ(1.5,y) = y−1.5
. So
∆(y):= (y−
0.5)2−1+ (y−1.5) = y2−y+0.25 −1+y−1.5=
y2−2.25, so om y >1.5, y2>2.25,so ∆(y)>0
2.
Fo
y<1.5;
hen,
ρ(1.5,y) = 2(1.5−y) = 3−2y
. So
∆(y):= (y−0.5)2−1+(3−2y) = y2−y+0.25−1+
3−2y=y2−3y+2.25 = (y−0.5)2>0
Thus, he a ia ional inequali y holds s ic ly o e e y
y6=x∗
. So he o wa d EVP conclusion holds and
x∗=1.5
is
he o wa d EVP poin .
Backwa d penalized unc ion. The backwa d penalized unc-
ion is
gb(y) = (y)+ρ(y,2).
Compu e ρ(y,2):
ρ(y,2) = (2−y,y≤2,
2(y−2),y≥2.
On [0,2],
gb(y)=(y−0.5)2+2−y,g0
b(y) = 2(y−0.5)−1=2y−2,
so he c i ical poin is
y=1
(wi h
g00
b=2>0
). On
[2,3]
he unc ion inc eases. Hence he global minimize is
xb=1
.
Checking he backwa d a ia ional inequali y shows
xb=1
sa is ies i s ic ly. The e o e he o wa d EVP and backwa d
EVP selec di e en minimize s (
1.5
s
1
) o he same s a ing
da a.
Theo em 3.3 (Fo wa d Ca is i’s heo em).Le
(X,ρ)
be an
asymme ic b-me ic space wi h coe icien
s≥1
which is
o wa d comple e. Le
:X→R
be p ope , bounded below
and o wa d-lsc. Le
T:X→X
be a map sa is ying he
Ca is i- ype condi ion
ρ(x,Tx)≤ (x)− (Tx) o e e y x ∈X.(6)
Then T has a ixed poin .
P oo .
Fix a bi a y
x0∈X
. Since
is bounded below,
in X
is ini e; choose
ε>0
and
λ>0
such ha
(x0)≤in X +ε
.
Se
α=ε/λ
and apply he o wa d EVP (Theo em 3.1) o
ob ain x∗∈Xsa is ying
(x∗)≤ (x0),ρ(x0,x∗)≤sλ,
and o e e y y6=x∗,
(y)> (x∗)−α ρ(x∗,y).
I
Tx∗=x∗
we a e done. Suppose
Tx∗6=x∗
. Pu
y=Tx∗
in
he a ia ional inequali y o ge
(Tx∗)> (x∗)−α ρ(x∗,Tx∗).
Rea ange:
(Tx∗)+ α ρ(x∗,T x∗)> (x∗).
On he o he hand, he Ca is i condi ion (6) a x∗implies
ρ(x∗,Tx∗)≤ (x∗)− (Tx∗),
i.e.
(Tx∗)+ ρ(x∗,Tx∗)≤ (x∗).
Because
α>0
and
ρ(x∗,Tx∗)≥0
, he wo inequali ies con-
adic each o he , hence
Tx∗=x∗
. The e o e
T
has a ixed
poin .
On Ekeland Va ia ional P inciple in Asymme ic b-Me ic Spaces — 27/27
Co olla y 3.3 (Backwa d Ca is i’s heo em).Le
(X,ρ)
be
an asymme ic b-me ic space wi h coe icien s ≥1which is
o wa d comple e. Le
:X→R
be p ope , bounded below
and o wa d-lsc. Le T :X→X be a map sa is ying
ρ(Tx,x)≤ (x)− (Tx) o e e y x ∈X.
Then T has a ixed poin .
P oo .
The p oo ollows by applying he backwa d EVP
o he backwa d-pe u bed unc ionals and duplica ing he
con adic ion a gumen used in Theo em 3.3.
4. Conclusion
In his pape we es ablished new ex ensions o he Ekeland
Va ia ional P inciple wi hin he amewo k o asymme ic
b-me ic spaces. By in oducing o wa d, backwa d, and bi-
comple e e sions o EVP, we showed ha he asymme y o
he dis ance leads o dis inc a ia ional inequali ies, which in
u n may yield di e en minimize s. An illus a i e example
was p o ided o highligh he di e ences be ween o wa d
and backwa d EVP.
As an applica ion, we de i ed a o wa d and backwa d Ca is i-
ype ixed poin heo em in asymme ic b-me ic spaces. These
indings no only ex end he scope o EVP beyond he exis ing
esul s in me ic, asymme ic-me ic, and b-me ic spaces bu
also p o ide new ools o op imiza ion, equilib ium p oblems,
and nonlinea analysis in asymme ic amewo ks.
Fu u e wo k may ocus on de eloping mul i alued and se -
alued e sions o EVP in asymme ic b-me ics, s udying
s abili y esul s, and explo ing applica ions o a ia ional in-
equali ies in non-symme ic en i onmen s.
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