Vol. 21 (2020), No. 2, pp. 781–790 DOI: 10.18514/MMN.2020.3197
FUZZY BARRELS ON LOCALLY CONVEX FUZZY TOPOLOGICAL VECTOR SPACES
B. DARABY, N. KHOSRAVI, AND A. RAHIMI Received 16 January, 2020
Abstract. In this paper, we introduce the concept of fuzzy barrels on locally convex fuzzy to- pological vector spaces. We present some characterizations of the fuzzy locally convex spaces, which are fuzzy barrelled. Using fuzzy barrels, we prove that the same fuzzy sets are bounded in any fuzzy topology of dual pair. Finally, we prove the Banach-Steinhaus theorem for the fuzzy topological vector spaces.
2010Mathematics Subject Classification: 03E72; 54A40 Keywords: polar fuzzy sets, fuzzy barrel, fuzzy barrelled spaces
1. INTRODUCTION
In [13], the concept of fuzzy topological vector space was introduced by Katsaras and Liu. In [11], Katsaras changed the definition of fuzzy topological vector space and he considered the linear fuzzy topology on the scalar field K(R orC), which consists of all lower semi-continuous functions fromKintoI= [0,1]. At the first, the idea of fuzzy norm on linear spaces introduced by Katsaras. After then, in different approach many authors like Cheng and Mordeson [2], Felbin [10], Bag and Samanta [1], etc. introduced the notion of fuzzy normed linear space. Later, Xiao and Zhu [16], Fang [9], Daraby et. al. [5,6], redefined the notion of Felbin’s [10] definition of fuzzy norm. Das [7] introduced a fuzzy topology generated by fuzzy norm and studied some properties of this topology.
The concept of weak linear fuzzy topology on a fuzzy topological vector space as a generalization of usual weak topology was studied in [4]. Also, in [4] the authors have proved that this fuzzy topology consists of all weakly lower semi-continuous fuzzy sets on a given vector space whenKendowed with its usual fuzzy topology.
In [3], the concept of polar fuzzy sets on fuzzy dual spaces was studied. Also, in [3]
the polar linear fuzzy topologies on fuzzy dual spaces have introduced by the help of polar fuzzy sets and a Mackey-Arens type theorem on fuzzy topological vector spaces has investigated.
c
2020 Miskolc University Press
One of the most powerful theorems of functional analysis is the Banach-Steinhaus theorem. This theorem asserts that the set of continuous linear mapping that is point- wise bounded is bounded uniformly on barrelled spaces. This shows the importance of the barrelled spaces in functional analysis. In this paper we are going to study the fuzzy barrelled spaces.
2. PRELIMINARIES
Let X be a non-empty set. A fuzzy set in X is an element of the set IX of all functions fromX intoI.
Definition 1([7]). LetXandY be any two non-empty sets, f:X→Y be a mapping andµbe a fuzzy subset ofX. Then f(µ)is a fuzzy subset ofY defined by
f(µ)(y) = (
supx∈f−1(y)µ(x) f−1(y)6=∅,
0 else,
for ally∈Y, where f−1(y) ={x: f(x) =y}. Ifηis a fuzzy subset ofY, then the fuzzy subset f−1(η)ofX is defined by f−1(η)(x) =η(f(x))for allx∈X.
Definition 2([11]). A fuzzy topology on a setXis a subsetτf ofIX satisfying the following conditions:
(i) τf contains every constant fuzzy set inX, (ii) ifµ1,µ2∈τf, thenµ1∧µ2∈τf,
(iii) ifµi∈τf for eachi∈A, then supi∈Aµi∈τf. The pair(X,τf)is called a fuzzy topological space.
The elements ofτf are called fuzzy open sets inX.
Definition 3([7]). A fuzzy topological space(X,τf)is said to be fuzzy Hausdorff if forx,y∈X andx6=ythere existη,β∈τf withη(x) =β(y) =1 andη∧β=0.
A mapping f from a fuzzy topological space X into a fuzzy topological spaceY is called fuzzy continuous if f−1(µ)is fuzzy open inX for each fuzzy open setµin Y. SupposeX is a fuzzy topological space andx∈X. A fuzzy setµinX is called a neighborhood ofx∈Xif there is a fuzzy open setηwithη≤µandη(x) =µ(x)>0.
Warren [15] has proved that a fuzzy set µ inX is fuzzy open if and only if µ is a neighborhood ofxfor each x∈X withµ(x)>0. The collectionO of fuzzy sets is called a fuzzy open covering ofµwhenµ≤supν∈Oν. The fuzzy setµis called fuzzy compact if any fuzzy open covering ofµhas finite subcovering.
Definition 4([7]). Ifµ1andµ2are two fuzzy subsets of a vector spaceE, then the fuzzy setµ1+µ2is defined by
(µ1+µ2)(x) = sup
x=x1+x2
(µ1(x1)∧µ2(x2)).
Ift∈K,we define the fuzzy setsµ1×µ2andtµas follows:
(µ1×µ2)(x1,x2) =min{µ1(x1),µ2(x2)}
and
(i) fort6=0,(tµ)(x) =µ(xt) for all x∈E, (ii) fort=0,(tµ)(x) =
(0 x6=0, supy∈Eµ(y) x=0.
A fuzzy setµin vector spaceE is called balanced iftµ≤µfor each scalartwith
|t| ≤1. As, it has been shown in [13],µis balanced if and only ifµ(tx)≥µ(x)for each x∈E and each scalartwith|t| ≤1. Also, whenµis balanced, we haveµ(0)≥µ(x) for eachx∈E. The fuzzy setµis called absorbing if and only if supt>0tµ=1. Then a fuzzy setµ is absorbing wheneverµ(0) =1. We shall say that the fuzzy setµ is convex if and only if for allt∈I,tµ+ (1−t)µ≤µ, [11]. Also a fuzzy setµis called absolutely convex if it is balanced and convex.
Definition 5([8]). A fuzzy topologyτf on a vector spaceE is said to be a fuzzy linear topology, if the mappings
f :E×E→E,(x,y)→x+y, g:K×E→E,(t,x)→tx,
are continuous when Kis equipped with the fuzzy topology induced by the usual topology,E×EandK×Eare the corresponding product fuzzy topologies. A vector spaceE with a fuzzy linear topologyτf, denoted by the pair(E,τf), is called fuzzy topological vector space (abbreviated to FTVS).
Definition 6([11]). Let(E,τf)be a fuzzy topological vector space, the collection ν⊂τf of neighborhoods of zero is a local base whenever for each neighborhoodµof zero and eachθ∈(0,µ(0))there isγ∈νsuch thatγ≤µandγ(0)>θ.
Definition 7([12]). A fuzzy seminorm onEis a fuzzy setµinE which is abso- lutely convex and absorbing.
We say that a fuzzy setµ, in a vector spaceE, absorbs the fuzzy setηifµ(0)>0 and for everyθ<µ(0) there existst >0 such thatθ∧(tη)≤µ. A fuzzy setµ in a fuzzy linear spaceE is called bounded if it is absorbed by every neighborhood of zero.
The Katsaras norm was defined as follows:
Definition 8([12]). A fuzzy norm on vector spaceE is an absolutely convex and absorbing fuzzy setρwith inft>0(tρ)(x) =0, forx6=0.
Definition 9([12]). A fuzzy topological vector space(E,τf)is called locally con- vex if it has a neighborhood base at zero consisting of convex fuzzy sets.
Let(E,τ)be a topological vector space. The collection of all lower semicontinu- ous functions fromEin to[0,1]is a fuzzy linear topology onE denoted byw(τ).
Proposition 1([3]). Let(E,τf)be a fuzzy topological vector space. Then there is a linear topologyτon E such thatτf =w(τ).
In [14], the concept of dual pair defined as follows:
We call (E,E0) a dual pair, wheneverE andE0 are two vector spaces over the sameKscaler field andhx,x0iis a bilinear form onEandE0satisfying the following conditions:
(D) For eachx6=0 inE, there isx0∈E0such thathx,x0i 6=0.
(D0) For eachx06=0 inE0there isx∈E such thathx,x0i 6=0.
Let(E,E0)be a dual pair. We denote byσf(E,E0), the weakest linear fuzzy topology on E which makes all x0 ∈E0 fuzzy continuous as linear functionals from E into (K,w(τ)), whereτis the usual topology on K. We call σf(E,E0), the weak fuzzy topology (see [4]).
Lemma 1([4]). Let(E,E0)be a dual pair. If we consider the usual fuzzy topology ω(τ)onK, then
σf(E,E0) =ω(σ(E,E0)), whereσ(E,E0)usual weak topology on E.
LetEbe a vector space andµ∈IE. Theθ-level set ofµis defined by [µ]θ={x∈E:µ(x)≥θ},
where 0<θ≤1. It is clear that forθ1,θ2∈(0,1]withθ1<θ2we have[µ]θ1⊇[µ]θ2. Therefore{[µ]θ:θ∈(0,1]}is a decreasing collection of subsets ofE.
Proposition 2([3]). Let(E,τf)be a fuzzy topological vector space andτf =w(τ).
Then the following hold.
(a) The fuzzy set µ∈IE is fuzzy compact if and only if[µ]θ is compact in(E,τ) for each0<θ≤1,
(b) The fuzzy set µ∈IE is fuzzy closed if and only if[µ]θ is closed in(E,τ) for each0<θ≤1,
(c) The fuzzy set µ∈IE is fuzzy absolutely convex if and only if[µ]θis absolutely convex in(E,τ)for each0<θ≤1.
3. FUZZY BARRELLED SPACES
The concept of polar of fuzzy sets was introduced in [3] as follows:
Let(E,E0)be a dual pair. For the non-empty subsetAofE, its polarA◦ defined by:
A◦={x0∈E0: sup
x∈A
|hx,x0i| ≤1}.
Definition 10([3]). Let(E,E0)be a dual pair,µ∈IE. Forx0∈E0andµ∈IE, we setAµ,x0 ={θ∈(0,1]:x0∈[µ]◦1−θ}. Forµ6=0, we define the fuzzy setµ◦onE0as
µ◦(x0) =
(supAµ,x0 , Aµ,x06=∅, 0, Aµ,x0=∅, and call it the fuzzy polar ofµinE.
Remark 1 ([3]). We note that ifµ=1, then for θ∈(0,1]we have [µ]1−θ=E.
Therefore[µ]◦1−θ=E◦={0}. This shows that forx06=0,Aµ,x0 =∅. Thenµ◦(x0) =0 but clearly we haveµ◦(0) =1.
Lemma 2([3]). Let(E,E0)be a dual pair and µ∈IE. Then we have µ◦= sup
θ∈(0,1]
θ∧χ[µ]◦ 1−θ.
Theorem 1([3]). If(E,τf)is a Hausdorff locally convex fuzzy topological vector space and µ is a fuzzy neighborhood of origin, then µ◦isσf(E0,E)-compact.
Definition 11. A fuzzy subsetµin locally convex fuzzy topological vector space Eis said to be a fuzzy barrel if it is absolutely convex, absorbent and fuzzy closed.
In every Katsaras normed space(E,ρ), the fuzzy setρ(the closure ofρ) is abso- lutely convex, absorbent and fuzzy closed. Thereforeρis a fuzzy barrel.
Proposition 3. Let (E,τf) be a locally convex fuzzy topological vector space.
Then there is a linear fuzzy topologyτBon E such that all the fuzzy barrels of(E,τf) are fuzzy neighborhoods in(E,τB).
Proof. Let(E,τf)be a locally convex fuzzy topological vector space andBbe a fuzzy barrel inE. ThenBis a Katsaras seminorm inEand the collection
UB={θ∧(tB) :t>0,0<θ≤1}
is a neighborhood base for a fuzzy linear topology onE, denoted byτB. Now, we set τB= \
B is a f uzzy barrel
τB.
The fuzzy topologyτB is a linear fuzzy topology on E, since for each fuzzy bar- rel B, τB is a linear fuzzy topology. Also, it is clear that all the barrels are fuzzy
neighborhoods in(E,τB).
It is clear that for the locally convex fuzzy topological vector space (E,τf) the fuzzy topology τB is weaker than τf, since for each absolutely convex absorbent fuzzy neighborhoodµinτf, the closure ofµis a fuzzy barrel. But the converse is not true in general.
Definition 12. A locally convex fuzzy topological vector space(E,τf)is said to be a fuzzy barrelled space if every fuzzy barrel inEis a fuzzy neighborhood. In other words, for each fuzzy barrelµinEthere is a fuzzy neighborhoodνsuch thatν≤µ.
Corollary 1. Let(E,τf)be a fuzzy barrelled locally convex space. Then the fuzzy topologiesτf andτBare equivalent, since in the fuzzy barrelled spaces every fuzzy barrel is a fuzzy neighborhood. In fact for each neighborhood B inτBthere is a fuzzy neighborhoodνinτf such thatν≤B.
Lemma 3. Let (E,τ) be a locally convex topological vector space, and B be a barrel in E. ThenχB is a fuzzy barrel in fuzzy topological vector space(E,ω(τ)).
Proof. SinceBis a barrel, then it is absolutely convex and closed. Now Proposi- tion2shows thatχBis fuzzy absolutely convex and closed. Also sinceBis absorbing then for eachx∈Bthere isλ>0 such thatx∈λB. Then xλ∈B. This shows that χB(λx) =1.Then supt>0(tχB)(x) =supt>0χB(xt) =1. SoχBis absorbing.
Proposition 4. Let E be a Hausdorff locally convex fuzzy topological vector space with dual E0. Then the fuzzy subset µ of E is a fuzzy barrel if and only if µ is the fuzzy polar of aσf(E0,E)-bounded fuzzy subset of E0.
Proof. Letµbe a fuzzy polar of aσf(E0,E)-bounded fuzzy subset ofE0. We prove that the fuzzy subsetµofE is a fuzzy barrel. Firstly, we prove that µ◦ is balanced.
So it is enough to show thatµ◦(tx0)≥µ◦(x0)for allx0∈Eandt∈Rwith|t| ≤1. We have
µ◦(tx0) =supAµ,tx0 =sup{θ∈(0,1]:tx0∈[µ]◦1−θ}
=sup{θ∈(0,1]:x0∈(t[µ]1−θ)◦}=sup{θ∈(0,1]:x0∈[tµ]◦1−θ}
=supAtµ,x0 = (tµ)◦(x0) = 1
|t|µ◦(x0)
≥1×µ◦(x0) =µ◦(x0).
The convexity ofµ◦is obvious, since[µ]◦1−θis a convex set for eachθ∈(0,1]. We note that for eachx∈E;ϕx(x0) =<x,x0>is aσf(E0,E)−continuous linear functional on E0.Now, since(θ∧χ[µ]◦
1−θ)is a closed subset ofKandµ◦=Vµ(x)>θϕ−1x (θ∧χ[µ]◦ 1−θ) then µ◦ is σf(E0,E)−closed. Conversely, suppose µ is a fuzzy barrel. Then it is absolutely convex and absorbent. Now, we prove thatµ◦isσf(E0,E)−bounded. We have
µ◦◦(x) =sup{θ:x∈[µ◦]◦1−θ}.
Letτf =w(τ). Sinceµis fuzzy closed and absolutely convex, so[µ]θ is closed and absolutely convex with respect toτ. Then, we have
[µ◦]◦1−θ= ([µ]◦θ)◦= [µ]◦◦θ = [µ]θ.
This shows that
µ◦◦(x) =sup{θ:x∈[µ◦]◦1−θ}
=sup{θ:x∈[µ]◦◦θ }
=sup{θ:x∈[µ]θ}
=µ(x).
The fuzzy setµ◦isσf(E0,E)-compact by Theorem1. Henceµ◦isσf(E0,E)-bounded.
Lemma 4. Let(E,τf) be a locally convex fuzzy topological vector space. Then fuzzy barrel absorbs every convex compact fuzzy set.
Proof. Letµbe a fuzzy barrel andνbe a convex compact fuzzy set. By Proposition 1there is a linear locally convex topologyτonE such thatτf =ω(τ).Now, sinceν is a compact fuzzy set,[ν]θ is compact in(E,τ) for each 0<θ≤1 by Proposition 2. Also for each 0<θ≤1,[µ]θ is a usual barrel. Then there is λ>0 such that [ν]θ⊆λ[µ]θ for all 0<θ≤1. So ifν(x)≥θ, then µ(xλ)≥θ.That is,(λµ)(x)≥θ.
Thusν≤λµ.
Theorem 2. The same fuzzy sets are bounded in every linear fuzzy topology of dual pair.
Proof. Let τf be any fuzzy topology of the dual pair (E,E0). Then τf is obvi- ously finer thanσf(E,E0).Thenτf-bounded fuzzy sets areσf(E,E0)-bounded. Con- versely, letνbe aσf(E,E0)-bounded fuzzy set andµbe a closed absolutely convex τf-neighborhood. Thenν◦ be a fuzzy neighborhood inE0 underσf(E0,E)andµis absolutely convex andσf(E0,E)-compact by Theorem 1. Hence by Lemma 4, ν◦ absorbsµ◦.Thenµ◦◦absorbsν◦◦.Sinceµ≤µ◦◦andν◦◦=ν.Thenνabsorbsµ. This
shows thatνisτf-bounded.
Definition 13. Let(E,τf)be a fuzzy topological vector space. The familyF ⊆E0 is called fuzzy equicontinuous if for each fuzzy neighborhoodµinK, there is a fuzzy neighborhoodν∈τf such that f(ν)≤µfor all f ∈F.
Corollary 2. Let E be Hausdorff locally convex fuzzy topological vector space with dual E0 then E is fuzzy barrelled if and only if everyσf(E0,E)-bounded fuzzy subset of E0is fuzzy equicontinuous.
Corollary 3. Let E be Hausdorff locally convex fuzzy topological vector space with dual E0, then E has the fuzzy topologyτf(E,E0).
Definition 14. LetE andFbe locally convex fuzzy topological vector spaces and P be a collection of linear mappings from E into F. We say that P is pointwise
bounded if for everyx∈Eand every absolutely convex closed fuzzy neighborhood νinFwe have
_
t>0
^
T∈P
(tν)(T(x)) =1.
One of the powerful theorems of functional analysis, the Banach-Steinhaus the- orem, asserts that the set of continuous linear mapping that is bounded at each point of a Banach space is bounded uniformly on the unit ball. In the following theorem we prove the Banach-Steinhaus theorem for fuzzy barrelled spaces.
Theorem 3. Let E be a fuzzy barrelled space and F a fuzzy convex space. Then any pointwise bounded fuzzy set of fuzzy continuous linear mappings of E into F is fuzzy equicontinuous.
Proof. LetPbe a pointwise fuzzy bounded set of continuous linear operators from EintoF.Supposeνis a fuzzy closed absolutely convex fuzzy neighborhood inF.We setB=VT∈PT−1(ν).ThenBabsolutely convex, and closed. We prove thatBis also absorbing. Sinceνis absorbing forx∈Ewe have
sup
t>0
(tB)(x) =_
t>0
^
T∈P
T−1(ν)(x t)
=_
t>0
^
T∈P
ν(T(x t))
= _
T∈P
^
t>0
ν(T(x) t )
=1.
HenceBis a fuzzy barrel inE.Now, sinceEis barrelled, thenBis a fuzzy neighbor- hood. Also for eachT0∈Pwe have
T0(B)(y)6T0((T0)−1(ν))(y)
= sup
x∈(T0)−1(y)
((T0)−1(ν))(x)
= sup
x∈(T0)−1(y)
ν(T0(x)) 6ν(y).
This shows thatPis equicontinuous.
Example1. Consider the fuzzy locally convex spaceRnendowed with the fuzzy Katsaras norm
ρ(x1, ....xn) = (
1 Σni=1x2i <1, 0 else.
It is clear thatRnis complete (a fuzzy Banach space). We prove thatRnis barrelled.
We have
ρ(x1, ....xn) =
(1 Σni=1x2i ≤1, 0 else.
This show thatρis compact (sinceRnis complete). Now, supposeBis a fuzzy barrel inRn. By Lemma4,Babsorbsρi.e. for each 0<θ≤B(0)there ist>0 such that θ∧(tρ)≤B.This yields thatθ∧(tρ)≤θ∧(tρ)≤B. ThenBabsorbsρ. ThusBis a fuzzy neighborhood inRnandRnis barrelled.
Example2. Consider the fuzzy locally convex spacesE=RnandF=Rendowed with the Katsaras fuzzy norms
ρ(x1, ....xn) = (
1 Σni=1x2i <1, 0 else, and
ν(x) = (
1 |x|<1, 0 else.
For i=1,2,· · ·,nlet Ti(x1,x2,· · ·,xn) =xi. Then the collectionP={Ti:i=1,2,
· · ·,n}is pointwise bounded since _
t>0
^
i=1,···,n
(tν)(Ti(x1,· · ·,xn)) =_
t>0
^
i=1,···,n
(tν)(xi)
=_
t>0
^
i=1,···,n
ν(xi t )
=ν(0)
=1.
Now, sinceE is fuzzy barrelled, by Theorem3,Pis fuzzy equicontinuous.
4. CONCLUSION
The Banach-Steinhaus theorem is one of the important theorems in functional analysis, which asserts that the set of continuous linear mapping that is pointwise bounded is bounded uniformly on barrelled spaces. Theorem3 is an extension of Banach-Steinhaus theorem to the more general case of fuzzy locally convex spaces.
ACKNOWLEDGMENT
The authors are grateful to the referees for their valuable comments and sugges- tions, which have greatly improved the paper.
REFERENCES
[1] T. Bag and S. K. Samanta, “Finite dimensional fuzzy normed linear spaces.”The Journal of Fuzzy Mathematics., vol. 11, no. 3, pp. 687–705, 2003.
[2] S. C. Cheng and J. N. Mordeson, “Fuzzy linear operators and fuzzy normed linear spaces.”Bulletin of the Calcutta Mathematical Society., vol. 86, no. 5, pp. 429–436, 1994.
[3] B. Daraby, N. Khosravi, and A. Rahimi, “Polar fuzzy topologies on fuzzy topological vector spaces.”arXiv preprint arXiv:1910.04152., 2019.
[4] B. Daraby, N. Khosravi, and A. Rahimi, “Weak fuzzy topology on fuzzy topological vector spaces.”arXiv preprint arXiv:1910.03405., 2019.
[5] B. Daraby, Z. Solimani, and A. Rahimi, “A note on fuzzy Hilbert spaces.”Journal of Intelligent
& Fuzzy Systems., vol. 31, pp. 313–319, 2016, doi:10.3233/IFS-162143.
[6] B. Daraby, Z. Solimani, and A. Rahimi, “Some properties of fuzzy Hilbert spaces.”Complex Analysis and Operator Theory., vol. 11, pp. 119–138, 2017, doi:10.1007/s11785-016-0581-0.
[7] N. R. Das and P. Das, “Fuzzy topology generated by fuzzy norm.”Fuzzy Sets and Systems., vol.
107, pp. 349–354, 1999, doi:10.1016/S0165-0114(97)00302-3.
[8] J. X. Fang, “On local bases of fuzzy topological vector spaces.”Fuzzy Sets and Systems., vol. 87, pp. 341–347, 1997, doi:10.1016/0165-0114(95)00364-9.
[9] J. X. Fang, “On I-topology generated by fuzzy norm.”Fuzzy Sets and Systems., vol. 157, pp.
2739–2750, 2006, doi:10.1016/j.fss.2006.03.024.
[10] C. Felbin, “Finite dimensional fuzzy normed linear space.”Fuzzy Sets and Systems., vol. 48, pp.
239–248, 1992, doi:10.1016/0165-0114(92)90338-5.
[11] A. K. Katsaras, “Fuzzy topological vector spaces I.”Fuzzy Sets and Systems., vol. 6, pp. 85–95, 1981, doi:10.1016/0165-0114(81)90082-8.
[12] A. K. Katsaras, “Fuzzy topological vector spaces II.”Fuzzy Sets and Systems., vol. 12, no. 2, pp.
143–154, 1984, doi:10.1016/0165-0114(84)90034-4.
[13] A. K. Katsaras and D. B. Liu, “Fuzzy vector spaces and fuzzy topological vector spaces.”Journal of Mathematical Analysis and Applications., vol. 58, pp. 135–146, 1977, doi: 10.1016/0022- 247X(77)90233-5.
[14] A. P. Robertson and W. Robertson,Topological vector spaces, Cambridge University Press, 1980.
[15] R. H. Warren, “Neighborhoods bases and continuity in fuzzy topological spaces.”Rocky Mountain Journal of Mathematics., vol. 8, pp. 459–470, 1978.
[16] J. Xiao and X. Zhu, “On linearly topological structure and property of fuzzy normed linear space.”
Fuzzy Sets and Systems., vol. 125, pp. 153–161, 2002, doi:10.1016/S0165-0114(00)00136-6.
Authors’ addresses
B. Daraby
Department of Mathematics, University of Maragheh, P. O. Box 55136-553, Maragheh, Iran E-mail address:bdaraby@maragheh.ac.ir
N. Khosravi
Department of Mathematics, University of Maragheh, P. O. Box 55136-553, Maragheh, Iran E-mail address:nasibeh−khosravi@yahoo.com
A. Rahimi
Department of Mathematics, University of Maragheh, P. O. Box 55136-553, Maragheh, Iran E-mail address:rahimi@maragheh.ac.ir
