Common Fixed Point Theorems for Weakly Subsequentially Continuous Mappings in Modified Intuitionistic Fuzzy Metric Spaces

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Common Fixed Point Theorems for Weakly

Subsequentially Continuous Mappings in Modified

Intuitionistic Fuzzy Metric Spaces

Naeem Salem

1,∗

,

Said Beloul

2

1_{Department of Mathematics, University of Management and Technology, Lahore, Pakistan}

2_{Department of Mathematics, Faculty of Exact Sciences, El-Oued University, P.O.Box789, El-Oued 39000, Algeria}

Abstract

The aim of this paper is to establish some common fixed point results for two weakly subsequentially continuous and compatible of type (E) pairs of self mappings via implicit relation in modified intuitionistic fuzzy metric spaces, also we give an example to illustrate our results.

Keywords

Common Fixed Point, Weakly Subsequentially Continuous, Compatible of Type (E), Modified IFMS

1 Introduction

The notion of intuitionistic fuzzy sets was been introduced by Atanassov [1], it can be considered as a generalization to fuzzy sets concept due to Zadeh [30]. Later C¸ oker [6] introduced a topology on intuitionistic fuzzy sets. Park [21] introduced the notion of intuitionistic fuzzy metric spaces as a generalization to fuzzy metric spaces, which is a combination between intuitionistic fuzzy sets and the concept of a fuzzy metric space given by George and Veeramani [9], many authors established some results concerning fixed point in such spaces, see for example [2, 5, 12, 14, 19, 29].

Meanwhile, jungck[16] defined the concept of compatible mappings, Jungck and Roadhes [17] generalized the last concept to the weakly compatible mappings, which is weaken than the compatible ones. Mishra et al [20] generalized the concept of compatibility in the setting of fuzzy metric spaces, he obtained some common fixed point theorems for compatible mappings in such spaces. Recently Bouhadjera and Godet Tobie [4] introduced the concept of subsequential continuity and utilized it with the concept of subcompatible mappings to establish a common fixed point, later Imdad et al.[13] improved these results and replaced subcompatibility by compatibility and subsequential continuity by reciprocal continuity, more recently, Gopal and Imdad [11] combined subsequential continuous maps with compatible maps concept to obtain some results in fuzzy metric spaces. In present work, we will generalize certain definitions to intuitionistic fuzzy metric spaces in order to obtain some common fixed point theorems by combining the concept of weakly subsequentially continuous mappings due to second author [3] with compatible of type (E) mappings given by Singh et al.[27, 28].

2 Preliminaries

Lemma 2.1. [7] Let tL∗be a non empty set such:

L∗={(x1, x2)∈[0,1]2, x1+x2≤1},

also define an operation≤L∗ as follows:

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for all(x1, x2),(y1, y2)∈[0,1]2. So(L∗,≤L∗)is a complete lattice.

Definition 2.1. [1] We called n intuitionistic fuzzy setAζ,ηin universeU every objet

Aζ,η = {(ζA(u), ηA(u)), u ∈ U}, where u ∈ U, for all u ∈ U, ζA(u) ∈ [0,1]and ηA(u) ∈ [0,1]are said to be the

membership degree and the non-membership degree, respectively, ofu∈Aζ,ηand furthermore they satisfyζA(u) +ηA(u)≤

1.

For everyzi = (xi, yi)∈L∗, ifci∈[0,1]such that n

X

j=1

cj= 1, then we have:

z1(x1, y1) +...+zn(xn, yn) = n

X

j=1

zj(xj, yj) =

_Xn

j=1

cjxj, n

X

j=1

cjyj

∈L∗.

Its units are given by0L∗= (0,1)and1_L∗= (1,0). A triangular normT =∗on[0,1]is considered as an commutative,

associative and increasing mappingT : [0,1]2→[0,1]such thatT(1, x) = 1∗x=x, for allx∈[0,1]. The same, a triangular co-normS=is a commutative, associative and increasing mappingS: [0,1]2→[0,1]such thatS(0, x) = 0x=x, for allx∈[0,1].

Definition 2.2. [7, 8] LetT be a continuous t-norm onL∗, thenT is a continuous t-representable if and only if there exist a continuous t-norm∗and a continuous t-conormon[0,1]satisfy:

T(x, y) = (x1∗y1, x2y2),

for allx= (x1, x2), y=y1, y2)∈L∗.

Now, let{Tn}be a sequence such that{T1=T }and forn >1andx(i)∈L∗,

Tn₍_x(1)_{, ..., x}(n+1)₌_T₍_Tn−1₍_x(1)_{, ..., x}(n)₎_{, x}(n+1)₎_.

Definition 2.3. [8] A mappingT : (L∗)2→L∗is called a triangular norm onL∗if for all(x, y, z)∈(L∗)3, the following conditions hold:

1. T(x,1L∗) =x.

2. T(x, y) =T(y, x).

3. T(x,T(y, z)) =T((T(x, y)), z).

4. x≤L∗x0, y≤L∗y0=⇒ T(x, y)≤mathcalT(x0, y0), (monotonicity).

Definition 2.4. [8] An decreasing mappingN : L∗ → L∗ is called a negator onL∗ if it satisfying: N(0L∗ = 1_L∗ and

N(1L∗= 0_L∗. Further,N is called an involutive negator if for allx∈L∗we haveN(N(x)) =x.

Aa decreasing mappingN : [0,1]→[0,1]is called a negator on[0,1]if it satisfyingN(0) = 1andN(1) = 0. We denote byNsfor the standard negator on[0,1]which is defined for allx∈[0,1]by:Ns(x) = 1−x.

Definition 2.5. [24] LetM, N : X2_×₍₀_,_∞₎ _→ _[0_,_1]_{be two fuzzy sets satisfying}_M₍_{x, y, t}_{) +}_N₍_{x, y, t}₎ _≤ _{1, for all}

x, y∈X andt >0. The triplet(X,MM,N,T)is called to be an intuitionistic fuzzy metric space ifX is an arbitrary and

non-empty set,T is a continuous t-representable andMM,N is a mapping (an intuitionistic fuzzy set)fromX2×(0,∞)into

L∗satisfying the following conditions for everyx, y∈X andt, s >0:

(i) MM,N((x, y, .) : (0,1)→L∗is continuous,

(ii) MM,N(x, y, t) = 1L∗if and only ifx=y,

(iii) MM,N(x, y, t)>L∗0_L∗,

(iv) MM,N(x, y, t) =MM,N(x, y, t)(y, x, t),

(v) MM,N(x, y, t+s)≥L∗T(M_M,N(x, z, t)(x, z, t),M_M,N(z, y, t)(z, y, s)).

Example 2.1. [24] Let(X, d)be a metric space andT(a, b) = (a1, b1),min(a2+b2,1))for alla = (a1, a2)and b =

(b1, b2)∈L∗.M andN are to fuzzy sets onX2×(0,∞)defined by:

MM,N(x, y, t) = (M(x, y, t), N(x, y, t)) = (

htn

htn₊_md₍_{x, y}₎,

md(x, y)

htn₊_md₍_{x, y}₎),

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Example 2.2. [24]

Let X = _Nand T(a, b) = (max(0, a1 +b1 −1), a2+b2−a2b2))for all a = (a1, a2)and b = (b1, b2) ∈ L∗.

M, N :X2_×₍₀_,_∞₎_→_L∗_{are two fuzzy sets such that:}

MM,N(x, y, t) = (M(x, y, t), N(x, y, t)) =

(x_y,y−_yx), ifx≤y

(_xy,x−_xy), ify≤x

Then(X,MM,N,T)is modified intuitionistic fuzzy metric space.

Definition 2.6. [2] Let(X, M, N,∗,)be an intuitionistic fuzzy metric space and let{xn}be a sequence inX,

1. {xn}is convergent to some pointxinX,if lim

n→∞MM,N(xn, x, t) = 1L∗, for allt >0,

2. {xn}is a Cauchy sequence if, for each0< ε <1andt >0, there existsn0such that

MM,N(xn, ym, t)>(Nε, ε),

and for eachn, m≥n0, whereNsis the standard negator.

3. (X, M, N,∗,)is complete if every Cauchy sequence in it is convergent.

Lemma 2.2. [23] LetMM,N be an intuitionistic fuzzy metric.Then, for anyt > 0,MM,N(x, y, t)is non-decreasing with

respect totin(L∗,≤L∗), for allx, y∈X.

Definition 2.7. [24] Let(X,MM,N,T)be a modified IFMS. ThenMM,Nis said to be continuous onX×X×(0,∞), if

lim

n→∞MM,N(xn, yn, tn) =MM,N(x, y, t),

whenever a sequence(xn, yn, tn)inX×X×(0,∞)converges to a point(x, y, t)∈X×X×(0,∞), that is lim

n→∞MM,N(xn, x, t) =

lim

n→∞MM,N(yn, y, t) = 1L

∗, and lim

n→∞MM,N(x, y, tn) =MM,N(x, y, t).

Lemma 2.3. [24] Let(X,MM,N,T)be a modified IFMS. ThenMM,Nis a continuous function onX×X×(0,∞).

Definition 2.8. MappingsAandSon a modified IFMS(X,MM,N,T)are:

1. compatible [24] if

lim

n→∞MM,N(ASxnn, SAxn, t) = 1L

∗, for allt >0, whenever{xn}is a sequence inXsuch that lim

n→∞Axn= limn→∞Axn=z∈X.

2. non compatible [29] if there exists at least one sequence{xn}inXsuch that

lim

n→∞Axn= limn→∞Sxn=x∈X,

but

lim

n→∞MM,N(ASxn, SAxn, t)6= 1L∗,

or non-existent for at least onet >0.

3. weakly compatible [24] if they commute at their coincidence points, i.e, ifAu =Sufor someu∈X, thenASu=

SAu.

4. satisfy the property (E.A) [24] if there exists a sequence{xn}inXsuch that for allt >0

lim

n→∞MM,N(Axn, z, t) = limn→∞MM,N(Sxn, z, t) = 1L∗,

for somez∈X.

Definition 2.9. [19] Two mappingsAandSof a modified IFMS(X,MM,NT)are reciprocally continuous if

lim

n→∞MM,N(ASxn, Az) = limn→∞MM,N(SAxn, Sz, t) = 1L

∗

whenever{xn}is a sequence inXsuch that

lim

n→∞Axn = limn→∞Sxn=z,

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Motivated by [3], we get:

Definition 2.10. Mappings(AandSare weakly subsequentially continuous (wsc), if there exists a sequence{xn}satisfying

lim

n→∞Axn= limn→∞Sxn=z,

for somez∈Xand

lim

n→∞MM,N(ASxn, Az, t) = 1L

∗, or

lim

n→∞MM,N(SAxn, Sz, t) = 1L∗.

In above definition if we have only

lim

n→∞MM,N(ASxn, Az, t) = 1L

∗,

then(A, S)is called to beA-subsequentially continuous.

Example 2.3. LetX = [0,2]andMM,N(x, y, t) = (_t₊_|_xt₋_y_|, | x−y|

t+|x−y|)withT(a, b) = (a1b1,min(a2+b2,1)) for all

a= (a1, a2)andb= (b1, b2)∈L∗. DefineA, Sas follows:

Ax=

1 +x, 0≤x≤1

x+1

2 , 1< x≤2

, Sx=

1−x, 0≤x≤1 2−x, 1< x≤2

It is clear the discontinuity ofAandSat 1. Let{xn}be a sequence inXdefined by:xn=

1

n, for eachn≥1.

Clearly that

lim

n→∞MM,N(Axn,1, t) = limn→∞MM,N(Sxn,1, t) = 1Last,

we have again:

lim

n→∞MM,N(ASxn, A(1), t) = 1L

∗.

Hence(A, S)isA-subsequentially continuous.

Motivated by [27, 28], define:

Definition 2.11. Self mapsAandSof a modified IFMS(X,MM,N,T)are said to be compatible of type (E), if

lim

n→∞MM,N(S

2_x

n, Az, t) == lim

n→∞MM,N(SAxn, Az, t) = 1Last,

and

lim

n→∞MM,N(A

2_x

n, Sz, t) = lim

n→∞MM,N(ASxn, Sz, t) = 1L∗,

whenever{xn}is a sequence inX satisfying:

lim

n→∞Sxn== limn→∞Axn=z,

for somez∈X.

Definition 2.12. Two self mappingsAandSof a modified IFMS(X,M,∗T)into itself are said to beA-compatible of type (E), if

lim

n→∞MM,N(A

2_x

n, Sz, t) = lim

n→∞MM,N(ASxn, Sz, t) = 1L

∗.

Remark that if the pair (A, S)is compatible of type (E), so it isS-compatible andA-compatible of type (E), but the converse may be not true.

Example 2.4. Let X = R+ and MM,N(x, y, t) = (_t₊_|_xt₋_y_|,_t₊|x_|−_x₋y|_y_|)with T(a, b) = (a1b1,min(a2+b2,1)) for all

a= (a1, a2)andb= (b1, b2)∈L∗. We defineA, Sas follows:

Ax=

2, 0≤x≤2

x+ 1, x >2 Sx=

x+2

2 , 0≤x≤2

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Consider the sequence{xn}which defined by:xn= 2−

1

n, for all n≥1.

lim

n→∞MM,N(Sxn,2, t) = 1L

∗

lim

n→∞MM,N(A

2_x

n, S(2), t) = lim

n→∞MM,N(ASxn, S(2), t) = 1L

∗

lim

n→∞MM,N(S

2_x

n, A(2), t) = lim

n→∞MM,N(SAxn, A(2), t) = 1L

∗,

then(A, S)is compatible of type (E).

LetΨbe the set of all continuous functionsF : (L∗)6_→_L∗_{satisfying the conditions:}

(F1) : For allu, v≥L∗0_L∗, F(u, v, u, v, v, u)≥_L∗0_L∗,orF(u, v, v, u, u, v)≥_L∗ 0_L∗implies thatu≥_L∗v. (F2) : For allu, v≥L∗0_L∗, F(u, u,1,1, u, u)≥_L∗0_L∗implies thatu≥_L∗1,

whereu= (u1, u2), v= (v1, v2)and1 = 1L∗= (1,0).

Example 2.5.

F(t1, t2, t3, t4, t5, t6) = 15t1−13t2+ 5t3−7t4+t5−t6

Example 2.6.

F(t1, t2, t3, t4, t5, t6) =t1−

1 2t2−

5 6t3+

1

3t4+t5−t6

3 Main results

Theorem 3.1. Let(X,MM,N,T)be a modified intuitionistic fuzzy metric space,A, B, Sare self mappings onX, if

1. the pair(A, S)is compatible of type (E) and weakly subsequentially continuous and ,

2. the pair(B, T)is compatible of type (E) and weakly subsequentially continuous.

Hence(A, S)as well as(B, T)have a coincidence point.

Moreover, the mappingsA, B, SandT have a unique common fixed point provided for allx, y∈Xandt >0we have:

F MM,N(Ax, By, t),MM,N(Sx, T y, t),MM,N(Ax, Sx, t),

MM,N(By, T y, t),MM,N(Sx, By, t),MM,N(T y, Ax, t)

≥L∗0_L∗, (1)

Proof. Since the pair(A, S)is wsc, there exists a sequence{xn}inXsuch that lim

n→∞Axn= limn→∞Sxn=zfor somez∈X

and lim

n→∞ASxn=Az,nlim→∞SAxn=Sz. Again(A, S)is compatible of type (E) implies that

lim

n→∞ASxn= limn→∞A

2_x

n=Sz

and

lim

n→∞SAxn= limn→∞S

2_x

n=Az,

consequently we obtainAz = Sz andz is a coincidence point forA andS. Similarly forB andT, since(B, T)is wsc (suppose that it isB-subsequentially continuous) there exists a sequence{yn}such

lim

n→∞Byn= limn→∞T yn=w

for somew∈Xand

lim

n→∞BT yn=Bw.

The pair(B, T)is compatible of type (E) implies that

lim

n→∞BT yn= limn→∞B

2_y

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lim

n→∞T Byn= limn→∞T

2_y

n=Bw,

so we haveBw=T w.

We claimAz=Bw, if not by using (1)) we get:

F(MM,N(Az, Bw, t),MM,N(Az, Bw, t),1,1,MM,N(Az, Bw, t),MM,N(Az, Bw, t))≥L∗0_L∗

and so from(F2), we getMM,N(Az, Bw, t) = 1. HenceAz=Bw.

Now we will provez=Az, if not by using (1)) we get:

F(MM,N(Axn, Bw, t)M(Sxn, T w, t),1,

MM,N(Axn, Sxn, t),MM,N(Sxn, Bw, t),MM,N(Axn, T w, t)))≥L∗0_L∗

lettingn→ ∞we get

F(MM,N(z, Bw, t)M(z, T w, t),1,1,MM,N(z, Bw, t),MM,N(z, T w, t)

=F(MM,N(z, Az, t),M(z, Az, t),1,1,MM,N(z, Az, t),MM,N(z, Az, t)≥L∗ 0_L∗

which implies thatMM,N(z, Az, t) = 1L∗. Hencez=Az.

Now, We will show thatz=w, if not by using (1), we get:

F(MM,N(Axn, Byn, t)M(Sxn, T yn, t),MM,N(Axn, Sxn, t),

MM,N(Byn, T yn, t),MM,N(Sxn, Byn, t),MM,N(Axn, T yn, t))≥L∗0_L∗,

F(MM,N(z, w, t)M(z, w, t),1,1,MM,N(z, w, t),MM,N(z, w, t)≥L∗0_L∗

thenMM,N(z, w, t) = 1L∗. Hencez=w.

For the uniqueness, supposeqanother fixed point, by using (1) we get:

F(MM,N(z, q, t)M(z, q, t),1,1,MM,N(z, q, t),MM,N(z, q, t)geqL∗0_L∗

thenMM,N(z, q, t) = 1. Hencezis unique.

IfA=BandS =T we get the following corollary:

Corollary 3.1. Let(X,MM,N,T)be a modified IFMS and letA, Stwo self mappings onXinto itself such the pair(A, S)

is wsc and compatible of type (E), thenAandShave a coincidence point.

AandShave a unique common fixed point provided, for allx, y∈Xwe have:

F MM,N(Ax, Ay, t),MM,N(Sx, Sy, t),MM,N(Ax, Sx, t),

MM,N(Ay, Sy, t),MM,N(Sx, Ay, t),MM,N(Ax, Sy, t)

≥L∗0_L∗.

Corollary 3.2. The results of Theorem 3.1 remain true if the conditions (1) and (2) are replaced by the following:

(10) the pair(A, S)isA-subsequentially continuous andA-compatible of type (E) (orS-subsequentially continuous and

S-compatible of type (E)),

(20) the pair(B, T)isB-subsequentially continuous andB-compatible of type (E) (orT-subsequentially continuous and

T-compatible of type (E)).

Theorem 3.2. The results of Theorem 3.1 remain true if the conditions (1) and (2) are replaced by the following: (100) the pair(A, S)is subsequentially continuous andA-compatible(orS-compatible) of type (E),

(200) the pair(B, T)is subsequentially continuous andB-compatible(orT-compatible) of type (E).

Theorem 3.3. Let(X,MM,N,T)be a modified IMS and letA, B, Sbe self mappings onX satisfying(1), suppose

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2. A(X)⊂T(X)( orB(X)⊂S(X)),

3. (B, T)is weakly compatible (or(A, S)is weakly compatible).

ThenA, B, SandThave unique common fixed point inX.

Proof. Since the pair (A, S) is wsc and compatible of type (E), there exists a sequence {xn} in X such lim

n→∞Axn =

lim

n→∞Sxn=zfor somez∈XandAz=Sz.

Moreover the subsequence{Axn}converges tozinAXand sinceA(X)⊂T(X), so there existsv∈Xsuchz=T v, we

showz=Bv, if not by using (1) we get:

F MM,N(Axn, Bv, t),MM,N(Sxn, T v, t),MM,N(Axn, Sxn, t),

M(Bv, T v, t),MM,N(Sxn, Bv, t),MM,N(Axn, T v, t)≥L∗0_L∗,

lettingn→ ∞, we get:

F MM,N(z, Bv, t),1,1,M(Bv, z, t),MM,N(z, Bv, t),1

≥L∗0_L∗,

using(F2), we get

MM,N(T v, Bv, t) = 1,

thenT v=Bvand sovis a coincidence point forBandT, the pair(B, T)is weakly compatible implies thatBz=T z. We show thatAz=Bz, if not by using (1) we get

F MM,N(Az, Bz, t),M(Sz, T z, t),1,1,MM,N(Sz, Bz, t),MM,N(Az, T z, t)≥L∗0_L∗,

from(F2), we get

MM,N(Az, Bz, t)≥L∗1L∗,

for allt >0, which impliesMM,N(Az, Bz, t) = 1and soAz=Bz.

Now, we claimz=Az, if not by using (1), we get:

F MM,N(Az, Bv, t),M(Sz, T v, t),1,1,MM,N(Sz, Bv, t),MM,N(Az, T v, t)

=

F MM,N(Az, z, t),M(Az, z, t),1,1,MM,N(Az, z, t),MM,N(Az, z, t)

≥L∗0_L∗,

using(F2), we get

MM,N(Az, z, t)≥L∗1L∗,

for allt >0, thenMM,N(Az, z, t) = 1andzis a common fixed point forA, B, SandT.

For the uniqueness, it is similar as in proof of Theorem 3.1.

Theorem 3.4. Let(X,MM,N,T)be a modified IFMS and letA, B, Sbe self mappings a onX satisfying(1), suppose

1. the pair(A, S)( or(B, T)) is wsc and compatible of type (E),

2. A(X)⊂T(X)( orB(X)⊂S(X)),

3. {Byn}converges for every sequence{yn}inX, whenever{T yn}converges( or{Axn}converges for every sequence{xn}

inX, whenever{Sxn}converges,

4. (B, T)is weakly compatible (or(A, S)is weakly compatible).

ThenA, B, SandThave unique common fixed point inX.

Proof. As in proof of Theorem 3.1, if the pair(A, S)is wsc and compatible of type (E), then there exists a sequence{xn}in

Xsatisfying lim

n→∞Axn = limn→∞Sxn=zfor somez∈X andAz=Sz.

The inclusionA(X)⊂T(X)implies that there exists a sequence{yn}inXsuch lim

n→∞Axn= limn→∞T yn=z.

We will show lim

n→∞Byn=z, if not by using (1) we get:

F MM,N(Axn, Byn, t),M(Sxn, T yn, t)MM,N(Axn, Sxn, t),

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F MM,N(z, lim

n→∞Byn, t),1,1,MM,N( limn→∞Byn, z, t),MM,N( limn→∞Byn, z, t),1

≥L∗ 0_L∗,

from(F1), we obtainMM,N(z, lim

n→∞Byn, t) = 1. Hencenlim→∞Byn=z.

Now, we will provez=Az=Sz, if not by using 1 we get:

F MM,N(Az, Byn, t),M(Sz, T yn, t),1,M(Byn, T yn, t),

MM,N(Sz, Byn, t),MM,N(Az, T yn, t)

≥L∗0_L∗

lettingn→ ∞we get:

F MM,N(Az, z, t),M(Az, z, t),1,1,MM,N(Az, z, t),MM,N(Az, z, t)

≥L∗0_L∗,

from(F2), we getMM,N(Az, z, t)≥L∗= 1_L∗, for allt >0. Hencezis a common fixed point forAandS.

SinceA(X)⊂T(X), there existsw∈Xsuchz=Az=T w, we will sowT w=Bw, by puttingx=zandy=win(1), we get:

F MM,N(T w, Bw, t),1,1,M(Bw, T w, t),MM,N(T w, Bw, t),1=

F MM,N(Az, Bw, t),1,1,M(Bw, T w, t),MM,N(Sz, Bw, t),MM,N(T w, Az, t)

≥L∗0_L∗,

using(F2), we getMM,N(T w, Bw, t)≥L∗1_L∗for allt >0, then:

MM,N(T w, Bw, t) = 1L∗,

which implies thatT w=Bw.

The pair(B, T)is weakly compatible, soz=Bz=T zand consequentlyzis a common fixed point forA, B, SandT. For the uniqueness, it is similar as in proof of Theorem 3.1.

Example 3.1. Let(X,MM,N,T)be a modified intuitionistic fuzzy metricspace, withX = [0,1].For alla= (a1, a2)and

b= (b1, b2)∈L∗,and eacht∈(01)letT(a, b) = (a1b1),min(a2+b2,1)), define:

M(Az, z, t) = ( t

t+|x−y|,

|x−y|

t+|x−y|)

. Define self mappingsA, B, SandT by:Ax=Bx= 1,

Sx=T x=

1, xis rational

1

4, xis irrational

Consider a sequence{xn}such for eachn≥1we have:

xn=_n1, clearly that lim

n→∞Axn= limn→∞Sxn= 1, also we have:

lim

n→∞ASxn=A(1) = 1

lim

n→∞A

2_x

n=S(1) = 1,

then(A, S)isA-subsequentially continuous andA-compatible of type (E).

TakingF(t1, t2, t3, t4, t5, t6) =t1− −1₂t2−5₆t3+1₃t4+t5−t6, then we will check that the inequality(1)is satisfied:

forx, y∈[0,1], we have:

F(t1, t2, t3, t4, t5, t6) = (1,0)−

1 2(1,0) +

5 6(1,0)−

1

3(1,0)−(1,0) + (1,0) = (0,0)≥L∗0_L∗.

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