A new approach for the sequence spaces of fuzzy level sets with the partial metric

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Available online at www.ispacs.com/jfsva Volume 2014, Year 2014 Article ID jfsva-00179, 13 Pages

doi:10.5899/2014/jfsva-00179 Research Article

A new approach for the sequence spaces of fuzzy level sets with

the partial metric

U˘gur Kadak1,2∗_{, Muharrem ¨}_Ozl¨uk1

(1)Department of Mathematics, Faculty of Science, Gazi University, Ankara, Turkey

(2)Department of Mathematics, Faculty of Science, Bozok University, Yozgat, Turkey

Copyright 2014 c⃝U˘gur Kadak and Muharrem ¨Ozl¨uk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we investigate the classical sets of sequences of fuzzy numbers by using partial metric which is based on a partial ordering. Some elementary notions and concepts for partial metric and fuzzy level sets are given. In addition, several necessary and sufficient conditions for partial completeness are established by means of fuzzy level sets. Finally, we give some illustrative examples and present some results between fuzzy and partial metric spaces.

Keywords:Sequence space, partial metric space, fuzzy level sets, complete metric space, fuzzy completeness

1 Introduction

Introduced in 1992, a partial metric space is a generalization of the notion of metric space defined in 1906 by Maurice Frechet such that the distance of a point from itself is not necessarily zero. This notion has a wide array of applications not only in many branches of mathematics, but also in the field of computer domain and semantics. Motivated by the needs of computer science for non Hausdorff Scott topology, one show that much of the essential structure of metric spaces, such as Banach’s contraction mapping theorem, can be generalised to allow for the possi-bility of non zero self-distancesd(x,x).

The discipline of mathematics has traditionally taken zero self distance for granted because, before computer science, there was little reason to consider the computability of a metric distanced(x,y). More precisely, mathematics has un-derstandably assumed that each metric distance is a totally defined structure. To assert thatd(x,x) =0 for eachx∈X

is in computational terms a useful means to specify thatxis totally computed. On the other hand; the effectiveness of level sets comes from not only their required memory capacity for fuzzy sets, but also their two valued nature. This nature contributes to an effective derivation of the fuzzy-inference algorithm based on the families of the level sets. Besides this, the definition of fuzzy sets by level sets offers advantages over membership functions, especially when the fuzzy sets are in universes of discourse with many elements. This definition considerably reduces the required memory capacity for the fuzzy sets and the processing time for fuzzy inference. In this study, the relations between fuzzy level sets and partial metric are introduced and discussed think about distance function.

A partially ordered set (or poset) is a pair(X,⊑)such that⊑is a partial order onX. For each partial metric space (X,p)let⊑pbe the binary relation overXsuch thatx⊑py(to be read,xis part ofy) if and only ifp(x,x) =p(x,y). For the partial metric max(min){a,b}over the nonnegative reals,⊑max(⊑min)is the usual≥(≤)ordering. For intervals,

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http://www.ispacs.com/journals/jfsva/2014/jfsva-00179/ Page 2 of 13

[a,b]⊑p[c,d]if and only if[c,d]is a subset of[a,b].

Byω(F)andE1, we denote the set of all sequences of fuzzy numbers and the set of all fuzzy numbers onR_,

respec-tively. We define the classical setsℓ∞(H),c(H),c0(H)andℓp(H)consisting of the bounded, convergent, null and p-summable sequences of fuzzy numbers with the partial metricHs, as follows:

ℓ∞(H) :=

{

u= (uk)∈ω(F): sup k∈N

Hs(uk,0)<∞, 0∈E1

} ,

c(H) :=

{

u= (u_k)∈ω(F): lim k→∞H

s₍_u

k,u) =0 for some u∈E1

} ,

c0(H) :=

{

u= (uk)∈ω(F): lim k→∞H

s₍_u

k,0) =0

} ,

ℓp(H) :=

{

u= (uk)∈ω(F): ∞

∑

k=0

Hs(uk,0)p<∞

}

, (1≤p<∞),

where the distance functionHsdenote the partial metric of fuzzy numbers with the level sets defined by

H(u,v) = sup λ∈[0,1]

p([u]λ,[v]λ) = sup λ∈[0,1]

{

max{u+_λ,v+_λ} −min{u−_λ,v−_λ})

=max{u+₀,v+₀} −min{u−₀,v−₀}

and

Hs(u,v) =2H(u,v)−H(u,u)−H(v,v),

for anyu,v∈E1with the partial ordering⊑H. One can show thatℓ∞(H),c(H)andc0(H)are complete metric spaces with the partial meticH∞defined by

H_∞(u,v):=sup k∈N

{Hs(u_k,v_k)},

whereu= (uk)andv= (vk)are the elements of the setsc(H),c0(H)orℓ∞(H). Also, the spaceℓp(H)is complete metric space with the partial metricHpdefined by

Hp(u,v):=

{_∞

∑

k=0

Hs(uk,vk)p

}1/p

,(1≤p<∞),

whereu= (uk)andv= (vk)are the points ofℓp(H).

The main purpose of the present paper is to study the corresponding setsℓ∞(H),c(H),c0(H)andℓp(H)of sequences of fuzzy numbers by using partial metric to the classical sequence spaces. The rest of this paper is organized, as follows:

Section 2 is devoted to the partial distance functions of fuzzy numbers with the level sets. Prior to stating and proving the main results concerning the fuzzy level sets, we give some lemmas about the different types of partial metric. The final section is devoted to the completeness of the setsℓ∞(H),c(H),c0(H)andℓp(H)by taking into account the partially ordering and some related examples.

2 Preliminaries, Background and Notation

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Proposition 2.1. [14](Nonzero self-distance) LetSωbe the set of all infinite sequencesx= (x0,x1,x2, . . .)over a set

S. For all such sequencesxandyletds(x,y) =2−k_{, where}_k_{is the largest number (possibly}_∞_{) such that}_x

i=yifor eachi<k. Thusds(x,y)is defined to be 1 over 2 to the power of the length of the longest initial sequence common to bothxandy. It can be shown that(Sω,ds)is a metric space.

To be interested in an infinite sequencexthey would want to know how to compute it, that is, how to write a computer program to print out the valuesx0, thenx1, thenx2, and so on. Asxis an infinite sequence, its values cannot be printed out in any finite amount of time, and so computer scientists are interested in how the sequencexis formed from its parts, the finite sequences(x0),(x0,x1),(x0,x1,x2)and so on. After each valuexk is printed, the finite sequence

x= (x0,x1,x2, . . . ,xk)represents that part of the infinite sequence produced so far. Each finite sequence is thus thought of in computer science as being a partially computed version of the infinite sequencex, which is totally computed. Suppose now that the above definition ofds is extended toS∗, the set of all finite sequences overS. If xis a finite sequence thends(x,x) =2−kfor some numberk<∞, which is not zero, sincexj=xjcan only hold ifxjis defined.

Definition 2.1. [10] LetX be a non-empty set and p be a function fromX×X to the setR+_{of non-negative real}

numbers. Then the pair(X,p)is called apartial metric spaceandpis apartial metricforX, if the following partial metric axioms are satisfied for allx,y,z∈X:

(P1) x=yif and only ifp(x,x) =p(x,y) =p(y,y),

(P2) 0≤p(x,x)≤p(x,y), (P3) p(x,y) =p(y,x),

(P4) p(x,z)≤p(x,y) +p(y,z)−p(y,y).

Each partial metric space thus gives rise to a metric space with the additional notion of nonzero self-distance intro-duced. Also, a partial metric space is a generalization of a metric space; indeed, if an axiomp(x,x) =0 is imposed, then the above axioms reduce to their metric counterparts. Thus, a metric space can be defined to be a partial metric space in which each self-distance is zero.

It is clear thatp(x,y) =0 impliesx=yfrom (P1) and (P2). But,x=ydoes not implyp(x,y) =0, in general. A basic example of a partial metric space is the pair(R+_,p), wherep(x,y) =max{x,y}for allx,y∈R+_.

Each partial metricponX generates aT0topologyτponX which has as a base the family openp-balls{Bp(x,ε):

x∈X,ε>0}, whereBp(x,ε) ={y∈X:p(x,y)<p(x,x) +ε}for allx∈Xandε>0.

Remark 2.1. [9] Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function

p(., .) need not be continuous in the sense that xn→x andyn→y implies p(xn,yn)→p(x,y). For example, if

X= [0,+∞)andp(x,y) =max{x,y}forx,y∈X, then for{xn}={1},p(xn,x) =x=p(x,x)for eachx≥1 and so, e.g.,xn→2 andxn→3 whenn→∞.

Proposition 2.2. [15] Ifpis a partial metric onX, then the functionpsdefined by

ps : X×X −→ R+

(x,y) 7−→ ps(x,y) =2p(x,y)−p(x,x)−p(y,y)

is a usual metric onX. For example, in(R−_,p)wherepis the usual partial metric onR−_{, we obtain the usual distance}

inR−_{since for any}x,y∈R−_,ps(x,y) =2p(x,y)−p(x,x)−p(y,y) =x+y−2 min{x,y}=|x−y|.

Definition 2.2. [14] A partial order onX is a binary relation⊑onXsuch that (i) x⊑x(reflexivity)

(ii) Ifx⊑yandy⊑xthenx=y(antisymmetry) (iii) Ifx⊑yandy⊑zthenx⊑z(transitivity).

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Definition 2.3. [9] Let(X,≼)be a partially ordered set. Then: (a) elementsx,y∈X are called comparable ifx≼yory≼xholds;

(b) a subsetK _ofXis said to be well ordered if every two elements ofK _{are comparable;}

(c) a mapping f:X→Xis called nondecreasing with respect to≼ifx≼yimplies f(x)≼ f(y).

For the partial metric max{a,b}over the nonnegative reals,⊑maxis the usual≥ordering. For intervals,[a,b]⊑p [c,d]if and only if[c,d]is a subset of[a,b]. Thus the notion of a partial metric extends that of a metric by introducing nonzero self-distance, which can then be used to define the relation is part of, which, for example, can be applied to model the output from a computer program.

Definition 2.4. (cf. [11, 15, 16, 1]) Let(X,p)be a partial metric space and(xn)be a sequence in(X,p). Then, we say that

(a) A sequence(xn)converges to a pointx∈Xif and only ifp(x,x) =limn→∞p(xn,x). (b) A sequence(xn)is aCauchy sequenceif there exists (and is finite) limm,n→∞p(xn,xm).

(c) A partial metric space(X,p)is said to becompleteif every Cauchy sequence(xninX converges, with respect to the topologyτp, to a pointx∈X such thatp(x,x) =limm,n→∞p(xm,xn). It is easy to see that, every closed subset of a complete partial metric space is complete.

(d) A mapping f :X→X is called to be continuousatx0∈X if for everyε>0, there existsδ >0 such that

f(Bp(x0,δ))⊂Bp(f(x0),ε).

(e) A sequence(xn)in a partial metric space(X,p)converges to a pointx∈X, for anyε>0 such thatx∈Bp(x,ε), there existsn0≥1 so that for anyn≥n0,xn∈Bp(x,ε).

Lemma 2.1. [15] Let(X,p)be a partial metric space. Then,

(i) (xn)is a Cauchy sequence in(X,p)if and only if it is a Cauchy sequence in the metric space(X,ps),

(ii) A partial metric space (X,p) is complete if and only if the metric space(X,ps)is complete. Furthermore, limn→∞ps(xn,x) =0 if and only ifp(x,x) =limn→∞p(xn,x) =limm,n→∞p(xn,xm).

In the partial metric space(R−_,p), the limit of the sequence(−1/n)is 0 since one has limn→∞ps(−1/n,0)whereps is the usual metric induced byponR−_.

Lemma 2.2. [9] Let(X,p)be a partial metric space, f:X→Xbe a given mapping. Suppose that f is continuous at

x0∈Xand for each sequencexn∈X, ifxn→x0inτpthenf(xn)→f(x0)inτpholds.

Definition 2.5. [1] Suppose that (X1,p1) and(X2,p2)are partial metric spaces with induced metrics ps1 and ps2 respectively. Then the function f :(X1,p1)→(X2,p2)is said to be continuous if both f :(X1,τp1)→(X2,τp2)and

f:(X1,ps₁)→(X2,ps₂)are respectively continuous in the sense of topological spaces and metric spaces continuity.

Definition 2.6. [9] LetXbe a nonempty set. Then(X,p,≼)is called an ordered (partial) metric space if: (i) (X,p)is a (partial) metric space,

(ii) (X,≼)is a partially ordered set.

Definition 2.7. [14] A sequencex= (xn)of points in a partial metric space(X,p)is Cauchy if there existsa≥0 such that for eachε>0 there existsksuch that for alln,m>k,|p(xn,xm)−a|<ε. In other words,xis Cauchy if the numbersp(xn,xm)converge to someaasnandmapproach infinity, that is, if limn,m→∞p(xn,xm) =a. Note that then limn,m→∞p(xn,xn) =a, and so if(X,p)is a metric space thena=0.

Definition 2.8. [14] A sequencex= (xn)of points in a partial metric space(X,p)converges toyinXif

p(y,y) = lim

n→∞p(xn,xn) =nlim→∞p(xn,y).

Thus if a sequence converges to a point then the self-distances converge to the selfdistance of that point.

Lemma 2.3. [13] Assume thatxn→zasn→∞a partial metric space(X,p)such thatp(z,z) =0. Then limn→∞p(xn,y) =

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2.1 _{The Level sets of fuzzy numbers}

Afuzzy numberis a fuzzy set on the real axis, i.e., a mappingu:R_→_[0,1]which satisfies the following four conditions:

(i) uis normal, i.e., there exists anx0∈Rsuch thatu(x0) =1.

(ii) uis fuzzy convex, i.e.,u[λx+ (1−λ)y]≥min{u(x),u(y)}for allx,y∈R_{and for all}_λ _∈_[₀_,₁_]_.

(iii) uis upper semi-continuous.

(iv) The set[u]0={x∈R_:u(x)>0}is compact, (cf. Zadeh [4]), where{x∈R_:u(x)>0}denotes the closure of the set{x∈R_:u(x)>0}in the usual topology ofR_.

We denote the set of all fuzzy numbers onR_byE1and call it asthe space of fuzzy numbers.λ-level set[u]λ ofu∈E1 is defined by

[u]λ =

 



{x∈R_:u(x)≥λ} , 0<λ≤1,

{x∈R_:u(x)>λ} , λ =0.

The set[u]λ is closed, bounded and non-empty interval for eachλ∈[0,1]which is defined by[u]λ= [u−(λ),u+(λ)].

R_{can be embedded in}E1, since eachr∈R_{can be regarded as a fuzzy number}rdefined by

r(x) =

{

1 , x=r, 0 , x̸=r.

Theorem 2.1. [7] Let[u]λ = [u−(λ),u+(λ)]for u∈E1and for eachλ∈[0,1]. Then the following statements hold:

(i) u−is a bounded and non-decreasing left continuous function on]0,1].

(ii) u+is a bounded and non-increasing left continuous function on]0,1].

(iii) The functions u−and u+are right continuous at the pointλ=0.

(iv) u−(1)≤u+(1).

Conversely, if the pair of functions u−and u+satisfies the conditions (i)-(iv), then there exists a unique u∈E1such that[u]λ = [u−(λ),u+(λ)]for eachλ ∈[0,1]. The fuzzy number u corresponding to the pair of functions u−and u+

is defined by u:R_→_[₀_,₁_], u(x) =sup{λ :u−(λ)≤x≤u+(λ)}.

Now we give the definitions of triangular fuzzy numbers with theλ-level set.

Definition 2.9. [3, Definition, p. 137](Triangular fuzzy number) The membership functionµ(u)of a triangular fuzzy

numberurepresented by(u1,u2,u3)is interpreted, as follows:

µ₍u)(x) =

 



x−u1

u2−u1 , u1≤x≤u2,

u3−x

u3−u2 , u2≤x≤u3,

0 , x<u1, x>u3.

Then, the result[u]λ := [u−(λ),u+(λ)] = [(u2−u1)λ+u1,(u2−u3)λ+u3]holds for eachλ ∈[0,1].

Letu,v,w∈E1andα∈R_{. Then the operations addition, scalar multiplication and product defined on}E1byu+v=

w⇔[w]λ= [u]λ+[v]λfor allλ∈[0,1]thenw−(λ) =u−(λ)+v−(λ) andw+(λ) =u+(λ)+v+(λ) for all λ∈[0,1]. LetW be the set of all closed bounded intervalsAof real numbers with endpointsAandA, i.e.,A:= [A,A]. Define the relationdonW byd(A,B):=max{|A−B|,|A−B|}. Then it can easily be observed thatd is a metric onW (cf. Diamond and Kloeden [6]) and(W,d)is a complete metric space, (cf. Nanda [8]). Now, we can define the metricD

onE1by means of the Hausdorff metricdas

D(u,v):= sup λ∈[0,1]

d([u]λ,[v]λ):= sup λ∈[0,1]

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Proposition 2.3. [2] Let u,v,w,z∈E1_and_α_∈_R_{. Then, the following statements hold:}

(i) (E1,D)is a complete metric space,(cf. Puri and Ralescu [17]). (ii) D(αu,αv) =|α|D(u,v).

(iii) D(u+v,w+v) =D(u,w).

(iv) D(u+v,w+z)≤D(u,w) +D(v,z).

(v) |D(u,0)−D(v,0)| ≤D(u,v)≤D(u,0) +D(v,0).

Definition 2.10. [5] u∈E1 is said to be a non-negative fuzzy number if and only if u(x) =0for all x<0. It is immediate that u≽0if u is a non-negative fuzzy number. By E₁+, we denote the set of non-negative fuzzy numbers. Similarly, u∈E1is said to be a non-positive fuzzy number if and only if u(x) =0for all x>0. It is immediate that u≼0if u is a non-positive fuzzy number. By E₁−, we denote the set of non-positive fuzzy numbers. By E_∼1, we denote the set of neither non-positive fuzzy numbers nor non-negative fuzzy numbers.

Then, it is trivial that the following statements hold:

(i) u≽0⇔u−(λ)≥0 andu+(λ)≥0 for allλ∈[0,1]. (ii) u≼0⇔u−(λ)≤0 andu+(λ)≤0 for allλ∈[0,1]. (iii) u̸∼0⇔u−(λ)<0 andu+(λ)>0 for someλ ∈[0,1].

Definition 2.11. [5] The following statements hold:

(a) A sequence u= (uk)of fuzzy numbers is a function u from the setNinto the set E1. The fuzzy number ukdenotes

the value of the function at k∈Nand is called as the general term of the sequence.

(b) A sequence(un)∈ω(F)is called convergent to u∈E1, if and only if for everyε>0 there exists an n0=

n0(ε)∈Nsuch that D(un,u)<εfor all n≥n0.

(c) A sequence(un)∈ω(F)is called bounded if and only if the set of its terms is a bounded set. That is to say that a sequence(un)∈ω(F)is said to be bounded if and only if there exist two fuzzy numbers m and M such that m≼un≼M for all n∈N. This means that m−(λ)≤u−n(λ)≤M−(λ)and m+(λ)≤un+(λ)≤M+(λ)for all

λ∈[0,1].

The boundedness of the sequence(un)∈ω(F)is equivalent to the fact that sup

n∈N

D(un,0) =sup n∈N

sup λ∈[0,1]

max{|u−_n(λ)|,|u+_n(λ)|}<∞.

If the sequence(u_k)∈ω(F)is bounded then the sequences of functions{u_k−(λ)}and{u+_k(λ)}are uniformly bounded in[0,1].

Definition 2.12. [20] Let{un}be a sequence of fuzzy-valued functions defined on a set A⊆E1. We say that{un}

converges pointwise on A if for each x∈A the sequence{un(x)}of fuzzy-valued functions converges. If the sequence

{un}converges pointwise on a set A, then we can define u:A→E1by lim

n→∞un(x) =u(x) for each x∈A.

Thus, in terms ofε−Nnotation,{un}converges to a fuzzy numberuonA⊆E1iff for eachx∈Aand for an arbitrary

ε>0, there exists an integerN=N(ε,x)such thatD(un(x),u(x))<εwhenevern>N. The integerNin the definition of pointwise convergence may, in general, depend on bothε>0 andx∈A. If, however, one integer can be found that works for all points inA, then the convergence is said to be uniform. That is, a sequence of fuzzy-valued functions

{un}converges uniformly touon a setAif for eachε>0, there exists an integerN0=N(ε)such that

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Obviously the sequence(un)∈ω(F)of fuzzy-valued functions converges to a fuzzy numberuif and only if{u−_n(λ)}

and{u+_n(λ)}converge uniformly tou−(λ)andu+(λ)on[0,1], respectively. Often, we say thatuis the uniform limit of the sequence{un}onAand writeun→uuniformly onA.

We emphasize that uniform convergence on a set implies (pointwise) convergence on that set. But the converse is not true, as we shall soon see in a number of examples. Thus, uniform convergence is a stronger form of convergence. Finally, we remark that it is apparent that if a sequence of fuzzy-valued functions converges uniformly on a setA, then it converges on every compact subset ofA.

Definition 2.13. [18] A sequence{un(x)}of fuzzy valued functions converges uniformly to u(x)on a set I if for each ε>0there exists a number n0(ε)such that D(un(x),u(x))<εfor all x∈I and n>n0(ε).

3 Completeness of the sequence spaces of fuzzy level sets with the partial metric

Proposition 3.1. [1] Let x,y∈X and define the partial distance function p by p: X×X −→ R+

(x,y) 7−→ p(x,y) =max{x,y} and

p: X×X −→ R+

(x,y) 7−→ p(x,y) =−min{x,y},

for X=R+and X =R−, respectively. Then,(R+_,p)is complete partial metric space where the self-distance for any point x∈R+is its value itself. The pair(R−_,p)is complete partial metric space for which p is called the usual partial metric onR−_{, and where the self-distance for any point x}_∈R−_{is its absolute value.}

The open balls are of the formBp(x,ε) ={y∈R+_{: max}_{x,y}<ε}= (0,ε)for allx∈R+_and_ε_>_{0 with}x≤ −ε

otherwise, if x>ε thenBp(x,ε) = /0. Suppose that y∈Bp(x,ε), then max{x,y}<ε which implies thaty<ε. Similarly, the open balls are of the formBp(x,ε) ={y∈R−_:₋_min_{x,y}<ε}= (−ε,0)for allx∈R−_and_ε_>₀

withx≥ −εotherwise, ifx<−εthenBp(x,ε) =/0. Suppose thaty∈Bp(x,ε), then−min{x,y}<εwhich implies that min{x,y}>−ε, hencey>−ε.

Lemma 3.1. Let us consider the set I={[u−_λ,u+_λ]:u−_λ,u+_λ ∈R_{} ⊂}E1 of the level sets and the distance function H:I×I→R+_{by means of the Hausdorff partial metric p with respect to the usual ordering on}R_{defined by}

H(u,v) = sup λ∈[0,1]

p([u]λ,[v]λ) = sup λ∈[0,1]

(

max{u+_λ,v+_λ} −min{u−_λ,v−_λ})

=max{u+₀,v+₀} −min{u−₀,v−₀}

for any u,v∈E1. Then the pair(E1,H)is a fuzzy partial metric space.

The partial ordering relation onE1is defined byu≼v⇔[u]λ ≼[v]λ ⇔u−(λ)≤v−(λ)andu+(λ)≤v+(λ)for all

λ ∈[0,1]. Here,[u]λ ⊑H[v]λ if, and only if, [v]λ ⊆[u]λ. Indeed, p([u]λ,[u]λ) = p([u]λ,[v]λ)impliesu+0 −u

−

0 = max{u+₀,v+₀} −min{u−₀,v−₀}. Suppose thatv+₀ >u+₀, thenu+₀−u−₀ =v+₀ −min{u−₀,v₀−} and min{u−₀,v−₀} −u+₀ =

v+₀ −u+₀ >0, hence min{u−₀,v−₀}>u−₀ which is impossible, thenv+₀ ≤u₀+. Similarly, one proves thatu−₀ ≤v−₀. Consequently,[v]λ ⊆[u]λ if, and only if,p([u]λ,[u]λ) =p([u]λ,[v]λ).

Theorem 3.1. The spaces(E1,H)and(E1,Hs)are complete.

Definition 3.1. [12] Let(X,p)be a partial metric space. For any x∈X andε>0, respectively the open and closed ball for the partial metric p by setting

Bε(x) ={y∈X:p(x,y)<ε} , Bε(x) ={y∈X:p(x,y)<ε}

Remark 3.1. [12] Contrary to the metric space case, some open balls may be empty. As an example, in a partial metric space (X, p), the open balls B_p₍_x_,_x₎(x)are empty for any x∈X .

Now, we can give the relationship of the fuzzy partial metricsH_±s and fuzzy Hausdorff metricDwhere the fuzzy distance

D(u,v):= sup λ∈[0,1]

d([u]λ,[v]λ):= sup λ∈[0,1]

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If we takeλ=0, and for allu,v∈E1∗

± then

D(u,v):=max{|u−₀ −v−₀|,|u+₀ −v+₀|}:=H_±s(u,v). (3.1)

Proposition 3.2. Define H∞on the spaceγ(H)by

H∞ : γ(H)×γ(H) −→ R+

(u,v) 7−→ H∞(u,v) =sup k∈N

{Hs(uk,vk)}

where u= (uk),v= (vk)∈γ(H)whereγ(H)denotes any of the spacesℓ∞(H), c(H)or c0(H). Then,(γ(H),H∞)is complete metric space with the partial metric.

Proof. Since the proof is similar for the spacesc(H)andc0(H), we prove the theorem only for the spaceℓ∞(H). Let

u= (u_k),v= (v_k)andw= (w_k)∈ℓ∞(H). Then,

(i) By using the axiom (P1) in Definition 2.1, it is trivial that

u=v ⇔ H∞(u,v) =sup k∈N

{Hs(uk,vk)}=sup k∈N

{2H(uk,vk)−H(uk,uk)−H(vk,vk)}=0

⇔ sup k

{Hs(uk,uk):k∈N}=sup k

{Hs(vk,vk):k∈N}=0

⇔ H∞(u,v) =H∞(u,u) =H∞(v,v).

(ii) By using the axiom (P2) in Definition 2.1, it folllows that

H∞(u,u) =sup k

{Hs(uk,uk):k∈N} ≤sup k

{Hs(uk,vk):k∈N}=H∞(u,v)

(iii) By using the axiom (P3) in Definition 2.1, it is clear that

H∞(u,v) =sup k

{Hs(uk,vk):k∈N}=sup k

{Hs(vk,uk):k∈N}=H∞(v,u).

(iv) By using the axiom (P4) in Definition 2.1 with the inequalitiesH(uk,wk)≤H(uk,vk) +H(vk,wk)−H(vk,vk) andHs(uk,uk) =0, we have

H_∞(u,w) ≤ sup k

{Hs(u_k,w_k):k∈N_}₌_sup

k

{2H(u_k,w_k)−H(u_k,u_k)−H(w_k,w_k)} ≤ sup

k

{2[H(uk,vk) +H(vk,wk)−H(vk,vk)]−H(uk,uk)−H(wk,wk)} = sup

k

{2H(uk,vk)−H(uk,uk)−H(vk,vk) +2H(vk,wk)−H(vk,vk)−H(wk,wk)}

≤ sup k

{2H(uk,vk)−H(uk,uk)−H(vk,vk)}+sup k

{2H(vk,wk)−H(vk,vk)−H(wk,wk)} = sup

k

{Hs(uk,vk)}+sup k

{Hs(vk,wk)} −sup k

{Hs(vk,vk)} = H∞(u,v) +H∞(v,w)−H∞(v,v).

Therefore, one can conclude that(ℓ∞(H),H∞)is a partial metric space onℓ∞(H). It remains to prove the completeness of the spaceℓ∞(H). Let(um)be any Cauchy sequence onℓ∞(H)whereum=

{

u₁(m),u₂(m), . . .}. Then, for anyε>0, there existsN∈N_{for all}m,r>Nsuch that

H_∞(um,ur) =sup k∈N

{

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A fortiori, for every fixedk∈N_{and for}m,r>N {

Hs(u(_km),u_k(r)):k∈N}_<_ε_. _(3.2)

Hence for every fixedk∈N_{, by using the completeness of}₍E1,Hs)in Theorem 3.1, we say the sequence(u(_km)) =

{

u(_k1),u(_k2), . . .} is a Cauchy sequence and is uniformly convergent. Now, we suppose that limm→∞u(km) =uk and

u= (u1,u2, . . .). We must show that lim

m→∞H∞(um,u) =0 and u∈ℓ∞(H). (3.3) The constantN∈N_{for all}m>N, taking the limit forr→∞in (3.2), we obtain

Hs(u(_km),uk

)

<ε (3.4)

for allk∈N_{. Since}₍u_k(m))∈ℓ∞(H), there exists a fuzzy numberM≻0 such thatHs(u(km),0)≤Mfor allk∈N. Thus, (3.4) gives together with the triangle inequality of partial metric form>Nthat

Hs(uk,0) ≤ Hs

( uk,u(km)

)

+Hs(u(_km),0)−Hs(u_k(m),u(_km))≤ε+M. (3.5) It is clear that (3.5) holds for everyk∈N_{whose right-hand side does not involve}k. This leads us to the consequence thatu= (u1,u2, . . .)is bounded sequence of fuzzy numbers henceu∈ℓ∞(H). Also, from (3.4) we obtain form>N that

H∞(um,u) =sup k∈N

{

Hs(u(_km),uk

)}

≤ε.

This shows that (3.3) holds and limmH∞(um,u) =0. Since(um)is an arbitrary Cauchy sequence,(ℓ∞(H)is complete.

Example 3.1. Consider the membership functions uk(t)and vk(t)defined by the triangular fuzzy numbers as

uk(t) =

 



(k+1)t−1 , 1 k+1≤t≤

2 k+1, 3−(k+1)t , _k₊2₁<t≤_k₊3₁,

0 , otherwise

and vk(t) =

 



(k+1)(t−1) , 1≤t≤1+ 1 k+1, 2−(k+1)(t−1) , 1+_k₊1₁<t≤1+_k₊2₁,

0 , otherwise

for all k∈N.

It is trivial that u_k−(λ) = λ_k₊+₁1 andu+_k(λ) =3_k−₊λ₁ for all λ ∈[0,1]. Therefore we see that (uk)−₀ =1/(k+1)and (uk)+₀ =3/(k+1). Then, it is conclude that

sup k

{

Hs(uk,0)} = sup k

{

2H(uk,0)−H(uk,uk)−H(0,0)}=sup k

{

(uk)+₀ + (uk)−₀}

=sup k

{4/(k+1)}<∞

and(uk)∈ℓ∞(H). Similarly,v−_k(λ) =1+_kλ₊₁ andv+_k(λ) =1+2_k−₊λ₁ for allλ ∈[0,1]. It is clear that(vk)−₀ =1 and (vk)+₀ =1+ 2

k+1. We observe that sup

k

{

Hs(vk,0)} = sup k

{

2H(vk,0)−H(vk,vk)−H(0,0)}=sup k

{

2+ 2

k+1

} <∞

Then,(vk)∈ℓ∞(H). Now we can calculate the partial distance between the sequences of fuzzy numbersu= (uk)and

v= (vk)∈ℓ∞(H)that

H∞(u,v) = sup k∈N

{Hs(uk,vk)}=sup k∈N

{2H(uk,vk)−H(uk,uk)−H(vk,vk)}

= sup k∈N

{2[max{u+_k(0),v+_k(0)} −min{u−_k(0),v−_k(0)}]−[u+_k(0)−u−_k(0)]−[v+_k(0)−v−_k(0)]

= sup k∈N

{

2

(

1+ 1

k+1

)

−

(

2

k+1

)

−

(

2

k+1

)}

=sup k∈N₁

{(

2− 2

k+1

)}

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Proposition 3.3. Define the distance function Hpby

Hp : ℓp(H)×ℓp(H) −→ R+ (u,v) 7−→ Hp(u,v) =

{ _∞ ∑

k=0

Hs(uk,vk)p

}1/p

, (1≤p<∞)

where u= (uk), v= (vk)∈ℓp(H). Then,(ℓp(H),Hp)is complete metric space with the partial metric.

Proof. It is obvious thatHpsatisfies the axioms (P1), (P2) and (P3). To prove (P4), we use Minkowski’s inequality on real andHp(v,v) =0. Letu= (uk),v= (vk)andw= (wk)∈ℓp(H). Then,

Hp(u,w) =

{ ∞

∑

k=0

(Hs(uk,wk))p

}1/p

=

{ ∞

∑

k=0

[2H(uk,wk)−H(uk,uk)−H(wk,wk)]p

}1/p

≤

{ ∞

∑

k=0

[2(H(uk,vk) +H(vk,wk)−H(vk,vk))−H(uk,uk)−H(wk,wk)]p

}1/p

≤

{ ∞

∑

k=0

[2H(uk,vk)−H(uk,uk)−H(vk,vk)]p

}1/p

+

{ ∞

∑

k=0

[2H(vk,wk)−H(vk,vk)−H(wk,wk)]p

}1/p

= Hp(u,v) +Hp(v,w)−Hp(v,v).

Therefore, one can conclude that(ℓp(H),Hp)is a partial metric space and complete. Since the proof is analogous for the casesp=1 andp=∞we omit their detailed proof and we consider only case 1<p<∞. It remains to prove the completeness of the space(ℓp(H),Hp).

Let(um)be any Cauchy sequence onℓp(H)whereum=

{

u(₁m),u₂(m), . . .}. Then for everyε>0, there existsN∈N

such that

Hp(um,ur) =

{ ∞

∑

k=0

Hs(u(_km),u(_kr))p

}1/p

<ε (3.6)

for allm,r>N. We obtain for each fixedk∈N_{from (3.6) that}

Hs(u(_km),u(_kr))<ε. (3.7)

For allm,r>Nand the constantk∈N_{, by using the completeness}₍_E1_,_Hs₎_{in Theorem 3.1, we say the sequence} (u(_km))is a Cauchy sequence of fuzzy numbers and is uniformly convergent.

Now, we suppose that limm→∞u(km)=ukandu= (u1,u2, . . .). We must show that lim

m→∞Hp(um,u) =0 and u∈ℓp(H). We have from (3.7) for each j∈N_andm,r>Nthat

j

∑

k=0

Hs(u(_km),u_k(r))p<εp. (3.8)

Take anym>N. Let us pass to limit firstlyr→∞and next j→∞in (3.8) to obtainHp(um,u)<εfor allm>Nit follows that limm→∞Hp(um,u) =0. By the using Minkowski’s inequality for each j∈Nthat

{ j

∑

k=0

Hs(uk,0)p

}1/p

≤

{ j

∑

k=0

Hs(uk,u(km)) p

}1/p

+

{ j

∑

k=0

Hs(u(_km),0)p

}1/p

−

{ j

∑

k=0

Hs(u(_km),u(_km))p

}1/p <∞

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Example 3.2. Consider the membership functions u_k(t)and v_k(t)defined by the trapezoidal fuzzy numbers as

uk(t) =

  

 

k(k+1)t−1 , _k₍_k1₊₁₎≤t≤_k₍_k2₊₁₎, 3−k(k+1)t , 2

k(k+1)<t≤

3 k(k+1),

0 , otherwise,

and

vk(t) =

        

6k(k+1)t−3 , 1

2k(k+1)≤t≤3k(k2+1),

1, , 2

3k(k+1)<t≤

3 4k(k+1),

4−4k(k+1)t , 3

4k(k+1)<t≤k(k1+1),

0 , otherwise

for all k∈N.

It is obvious thatu−_k(λ) =_k₍λ_k+₊1₁₎ andu+_k(λ) =_k₍3_k−₊λ₁₎,v−_k(λ) =_6kλ₍_k+₊3₁₎ andv+_k(λ) = _4k4₍−_k₊λ₁₎ for allλ ∈[0,1]. Then, takingp=1

∞

∑

k=0

Hs(uk,0)p= ∞

∑

k=0

{

u+_k(0)−u−_k(0)}

= ∞

∑

k=0

{

3

k(k+1)− 1

k(k+1)

}

= ∞

∑

k=0

{

2

k(k+1)

}

=2

andu∈ℓp(H)and similarly one can conclude thatv∈ℓp(H). Now we can calculate the partial distance between the sequences of fuzzy numbersu= (uk)andv= (vk)∈ℓp(H)that

Hp(u,v) =

{ ∞

∑

k=0

Hs(uk,vk)p

}1/p

= ∞

∑

k=0

{

2(

max{u+_k(0),v+_k(0)} −min{u−_k(0),v−_k(0)})

−(u+_k(0)−u−_k(0))−(v+_k(0)−v−_k(0))

}

= ∞

∑

k=0

{

5

k(k+1)− 3

k(k+1)− 1 2k(k+1)

}

= ∞

∑

k=0

{

5 2k(k+1)

}

=5/2.

Acknowledgements

We have benefited a lot from the referee’s report. So, the authors would like to express their gratitude for their constructive suggestions which improved the presentation and readability of the paper.

4 Conclusion

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