Some Classes of Generalized Entire Sequences of Fuzzy Number

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 2 Ver. IV (Mar. - Apr. 2017), PP 11-18 www.iosrjournals.org

Some Classes of Generalized Entire Sequences of Fuzzy Number A.Padiarani Department of Mathematics, G.Venkata Swamy Naidu College, Kovilpatti – 628502, Tamil Nadu, India.

Abstracta: The object of the present paper is to introduce the sequence spaces o (M,, P) defined by the sequence of fuzzy numbers and p = (pk) be any bounded sequence of positive real numbers. We study their different algebraic and topological properties. We also obtain some inclusion relations between these spaces.

Keywords and phrases: Fuzzy number, entire sequence spaces, completeness, convergence ,boundedness.

This work is supported by UGC MRP(5647) I. Introduction The concept of fuzzy set theory was introduced by zadeh [14 ] and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming.

Matloka [7] introduced bounded and convergent sequences of fuzzy numbers and studied some of their properties. Later on sequences of fuzzy numbers have been discussed by Nanda [9] savas [12], Mursaleen [ 8 ], Thripathy[13 ] and Dutta [ 2] and many others. Ganapathy Iyer [1 ] studied the space of all entire functions.

Maddox [5] generalised the space of all entire functions as a special class of sequences of complex numbers co(P).

Let p = (pk) be any sequences of strictly positive real numbers. The class of sequences defined by Maddox [6] was co(p) = { x :

| xk | pk  o }

2000 Mathematics subject classification 46A45 When all the terms of (pk) are constants and all equal to p > o we have co(p) = co, the space all null sequences.

The special FK – space co(1/k) was studied by Ganapathy Iyer [ 1] who indentified it with the space of all entire functions.

Kampthan [3] studied the class, X=

f : f (z)   a z , | n!a n

n

n



| n  o as n   consisting of all entire functions of order 1 1

and type o and showed that X is a Frechet space with the total paranorm || f || = sup

| n!a



| n , (n  1) 1

n

Recently Nuray and Savas [ ] have defined the following space of sequences of fuzzy numbers l(p) =





Pk    x  ( xk ) :  d ( xk , o )   k  

where (pk) is a bounded sequence of strictly positive real numbers. Mursaleen and Metin Basarir [8] studied the spaces Fo(p), F(p) and F(p).

Our aim is to introduce the space o(M,, p). We establish the condition for o(M,, P) to be identical with co(M,, P). Also we give the necessary and sufficient condition for o(M,, P) to be included in o(M,, P).

II. Definitions and Preliminaries We begin with giving some required definitions and statements of theorems, propositions and lemmas. A fuzzy number is a fuzzy set on the real axis i.e. a mapping u:R  [0,1] which satisfies the following four conditions. (i) u is normal i.e. there exists an x0R such that u (x0) =1 ii) u is fuzzy convex i.e. u [x + (1- )y ]  min {u(x) , u(y)} for all x, y  R and for all  [0,1] iii) u is upper semi continuous iv) The set [u]0 = closure of the set

x  R : u( x)  0

x  R : u (t )  0

DOI: 10.9790/5728-1302041118

is compact (Zadeh [14] ) where

x  R : u( x)  0 denotes the

in the usual topology of R. We denote the set of all fuzzy numbers on R

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Some Classes of Generalized Entire Sequences of Fuzzy Number

by E and called it as the space of fuzzy numbers. The ď Ź -level set [u]ď Ź of u ďƒŽ E is defined by

ď ťt ďƒŽ R : u(t ) ď‚ł ď Źď ˝ , 0  ď Ź ď‚Ł 1 ď ›uď ?ď Ź  ďƒŹďƒ­

ď ť

ď ˝

ďƒŽ t ďƒŽ R : u(t )  ď Ź ,

ď Ź 0

The set [u ]ď Ź is a closed bounded and non-empty interval for each ď ŹďƒŽ [ 0,1 ] which is defined by [u]Îť=[u-(Îť),u+(Îť)] can be embedded in E. Since each rďƒŽ

ďƒŹ1, r ( x)  ďƒ­ ďƒŽ0,

can be regarded as a fuzzy number

x  r  x  r 

defined by

Let u, v, wďƒŽ E and kďƒŽ .The operations addition , scalar multiplication and product defined on E by u + v = w ďƒ› [ w] ď Ź = [ u] ď Ź + [ v] ď Ź for all ď ŹďƒŽ[ 0, 1 ] ďƒ› w-(Îť) =[u-(Îť), v-(Îť)] and w+(Îť) =[u+(Îť), v+(Îť)] and for all ď ŹďƒŽ[ 0, 1 ] [ ku]ď Ź = k [ u]ď Ź for all ď ŹďƒŽ[ 0, 1 ] and u v = w ďƒ› [ w] ď Ź = [ u] ď Ź [ v] ď Ź for all ď ŹďƒŽ [ 0, 1 ] where it is immediate that ,

ď ť w (ď Ź )  max ď ťu (ď Ź )v (ď Ź ),

ď ˝ u (ď Ź )v (ď Ź )ď ˝

w (ď Ź )  min u  (ď Ź )v (ď Ź ), u  (ď Ź )v (ď Ź ), u  (ď Ź )v (ď Ź ), u  (ď Ź )v (ď Ź )

and





u  (ď Ź )v (ď Ź ), u  (ď Ź )v (ď Ź ),







for all ď ŹďƒŽ[ 0, 1 ] . đ?‘˘ = đ?‘¤ďƒ› [ đ?‘¤ ] = [ u] ď Ą / [ v] ď Ą for all ď ĄďƒŽ[ 0, 1 ] đ?›ź đ?‘Ł đ?&#x;? đ?&#x;? = [ u− ( ď Ą) , u+ ( ď Ą) ] . − , + = min

u − đ?›ź v + đ?›ź

,

đ?‘˘âˆ’ đ?›ź đ?‘Łâˆ’ đ?›ź

đ?’— (đ?œś) đ?’— đ?œś đ?‘˘+ đ?›ź đ?‘˘+ đ?›ź

,

�+ �

,

đ?‘Łâˆ’ đ?›ź

,Max

[ u]âˆ’ď Ą [ v]+ ď Ą

,

đ?‘˘ − (đ?›ź ) đ?‘Ł − (đ?›ź )

,

�+ � �+ �

,

đ?‘˘ + (đ?›ź ) đ?‘Ł −(đ?›ź)

Let W be the set of all closed and bounded intervals A of real numbers with endpoints

A and A ie)

A= [ , ] Define the relation d on W by

ď ť A B , A B ď ˝

d ( A, B ) = max

Then it can be observed that d is a metric on W and (W, d ) is a complete metric space (Nanda [9]) . Now we can define the metric D on E by means of a Hausdroff metric d as

ď ť

sup max u  (ď Ź)  v (ď Ź) , u  (ď Ź)  v (ď Ź)

D ( u , v) = sup d([u]Îť,[v]Îť) =

ď ŹďƒŽ[ 0,1]

ď ˝

(E’, D) is a complete metric space .One can extend the natural order relation on the real line to intervals as follows .

A  B if and only if A  B and A  B The partial order relation on E is defined as follows.

ď ›ď ? ď ›ď ?

u ď‚Ł v ďƒ› u ď Ź ď‚Ł v ď Ź ďƒ› u (ď Ź ) ď‚Ł v (ď Ź ) and u An absolute value ďƒŻuďƒŻ of a fuzzy number u is defined by

ď ›u ď ?

ď Ź -level set

ď Ź

where 





ďƒŹmax ď ťu(t ), u(t )ď ˝, u (t )  ďƒ­ 0 , ďƒŽ

(ď Ź ) ď‚Ł v (ď Ź )

t  0 t  0

of the absolute value of uďƒŽE is in the form

ď ť

ď ť

u (ď Ź ))  max u  (ď Ź ) , u  (ď Ź ) The absolute value ďƒŻuvďƒŻ of u , v ďƒŽ E

DOI: 10.9790/5728-1302041118

for all ď ŹďƒŽ[ 0, 1 ]

ď ›u ď ?

ď Ź

ď ˝

u (ď Ź )  max 0, u  (ď Ź ),u  (ď Ź ) 



ď ˝

satisfies the inequalities (Talo [8])

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Some Classes of Generalized Entire Sequences of Fuzzy Number 





uv ( )  uv ( )  max u  ( ) v  ( ) ,

u  ( ) v  (  ) ,

u  ( ) v  (  ) , u  (  ) v  ( )



u E is a non-negative fuzzy number if and only if u(x)=0 for all x<0. It is immediate that u  0 if u is a non negative fuzzy number.

One can see that







D(u, 0) sup max u  () , u  ()  max u  () , u  () [ 0,1]

.

Proposition 2.1 Let u,v,wE and kR . Then (i) (E, D) Is a complete Metric space (ii) D(ku,kv)= k D(u,v) (iii) D(u+v,w+v)=D(u,w) (iv) D(u+v,w+z)  D(u,w)+D(v,z)

D(u, 0)  D(v, 0)  D(u, v)  D(u, 0)  D(v, 0) .

(v)

Lemma 2.2 The following statements hold (Talo [11])



   

D uv, 0  D u, 0 D v, 0

(i)

for all u , v  E

(ii) If uk  u as k then D ( uk, )  D ( u , 0 ) as k where (uk )  w ( F ) .

In the sequel, we require the following Definitions and lemmas.

Definition: 2.3 A sequence u = ( uk ) of fuzzy numbers is a function u from the set N into the set E .The fuzzy number uk denotes the value of the function at k and is called the kth term of the sequence. Let w(F) denote the set of all sequences of fuzzy numbers.

Definition : 2.4 A sequence ( uk ) w( F ) is called convergent with limit uE if and only if for every  > 0 there exists an n0 = n0 ()  N such that D ( uk ,u ) <  for all k  n0 .

Theorem 2.5 [7]. Let ( uk ),(vk)  w ( F ) with u k  a , v k  b as k   .Then,

u k  v k  a  b as k   ii. u k  v k  a  b as k   iii. u k v k  ab as k   . iv. u k u k vk  a b as k   where 0 [vk ]0 for all k  N and 0 [b]0 . i.

Definition : 2.6 A sequence ( uk )  w ( F ) is called bounded if and only if the set of all fuzzy numbers consisting of the terms of the sequence (uk) is a bounded set.

That is to say that a sequences ( uk )  w ( F ) is said to be bounded if and only if there exist two fuzzy numbers m and M such that m  uk  M for all kN .

III. The space o(M,,p)

Let  = (k) be a sequence of non-zero fuzzy numbers and let u = (uk) be any sequence of fuzzy numbers. Put u = (k uk) and (u)k = kuk.

Then define.

  o(M,,p) =  u  (u k )  co ( F ) : sup sup 1 lim D(u ) k , o   o s 1  cs  

    

Theorem 3.1. o(M,,p) is a complete metric space if and only if

lim inf D(k , o )  o

(3.1.1)

k 

Proof. The Metric on o(F) is given by do (u, v) = sup D (u) k , (v) k







=

k



sup sup max | (k uk )  ( )  (k vk )  ( ) | , | (k uk )  ( )  (k vk )  ( ) | k

[ 0,1]

Let {u(i)} be any Cauchy sequence in o(F) .

Then given  > 0 there exists a positive integer no such that DOI: 10.9790/5728-1302041118

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Some Classes of Generalized Entire Sequences of Fuzzy Number





sup D (u (i ) ) k , (u (i ) ) k  

(3.1.2)

k

for all i, j  no and for all k i.e.





sup sup max | (k uki )  ( )  (k ukj )  ( ) | , | (k uki )  ( )  (k ukj )  ( ) | [ 0,1]

k

no and for all k.

Let L

for all i, j 

(3.1.3)

=

lim inf {D (k , o )}

=

  lim inf  sup max | k ( ) | , | k ( ) |  [0,1] 





Using (3.1.3) and (3.1.4)



( j ) | u (i) ( ) |  k ( )  uk

L

for all I, j  no

(3.1.4)

and | uk(i ) ( )  uk( j ) ( ) | 



(3.1.5)

L

 u  is a Cauchy sequence in E’ and since (E’, D) is complete  u  u as i  Thus D u , u   for all i, j  n for all k.

L (i ) k

Hence

(i ) k

k

(i ) k

o

k

Letting j  in (3.1.5)

| uk(i ) ( )  uk ( ) | 

Then

 L

and | uk(i ) ( )  uk ( ) | 

 L

Now

| k ( ) | | uk(i ) ( )  uk ( ) | <  and

| k ( ) | | uk(i ) ( )  uk ( ) |   Hence





sup sup max | k ( ) | | uk(i ) ( )  uk ( ) | , | k ( ) | | uk(i ) ( )  uk ( ) | k

Thus

[ 0,1]



uk(i )  u in o(F) . (i )

Since each (ui) is in o(F) we have D (uk

, 0) <

 L

(3.1.6)

Using (3.1.4) and (3.1.6) D[(u)k, 0 ]

D[kuk, 0 ]

=



0 ) D(uk, 0 ) (Talo [ ])    L    L L  D(k,

 Hence u o(F) Thus o(F) is complete.

Conversely suppose (3.1.1) is not true. Then {D(k, 0 )} contains a subsequence {D(nk, 0 )} which is steadily

 

(n)  n 1

decreasing and tends to zero. Consider the sequence uk

where

1 if k  k1 , k 2 , k3 ,............k n uk( n )   0 Otherwise  Then

uk(n )  o(F) for all n = 1, 2, 3, ………..

For n > m we have

o uk( m) , uk( n) 

=



sup D nk uk( m) , nk uk( n )



k

DOI: 10.9790/5728-1302041118

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Some Classes of Generalized Entire Sequences of Fuzzy Number

0 ) Which tends to o as n, m   ( 1, 1, ............)

=

lim u

But

n

D(k(n+1),

( m) k

which is not in o(F) Thus (3.1.1) must hold whenever o(F) is complete.

IV. The space o(F, p) Throughout, let p = (pk) be a bounded sequence of strictly positive real numbers and The space o(F, p) is defined as follows . o(F, p)=



M = max (1, sup pk).



n pk   u  ( u )  c ( F , p ) : lim D (u)k , 0  0   k o n k 1  

Now o(F, p) is endowed with two topologies one is the metric topology  given by metric d, where d (u,v) =

sup Duk , vk 

pk M

, u, vo(p, F).

k

The metric d is induced by the paranorm



sup Duk , 0  k

g(u) =

k

P



1

M

The other is the topology  whose metric d is given by dλ (u,v) =



sup D[(u )k , (v)k ]Pk



1

M

, u,vo(F,p)

k

o(F,p)

Theorem 4.1

= co(F,p) if and only if (k)    ( F , p )

Proof. suppose that (k)    ( F , p ) Then (kuk)  co, (F,p) for every (uk)  co(F,p).

Hence co(F,P)  o(F,P).Always o(F,P)  co(F,P).

Therefore o(F,P) = co(F,P).

On the other hand suppose that o(F,P) = co(F,P).

If

(k )   ( F , P)

then there exist a positive integer r such that for each r, there exist a k(r) such that

k

D(k ( r ) , 0)  1 Define u by

1  uk = 0

for k  k (r ) otherwise

and take (pk)=(1) Then u co(F,P).

But

D(u, 0)  1 showing that u  o ( F , P) .

This contradiction shows that (k)    ( F , P) .

Corollary 4.2 o(F) = co(F) if and only if (k)    (F ) .

Theorem 4.3 In order that o(F,P)  µo(F,P), it is necessary and sufficient that pk      k , 0  , D( k , 0) p k  is bounded. min  D   k  

( 4.3.1)

Proof. Let A denote the set of positive integers k for which D(k , 0) Let B denote the set of positive integer k for which D(k , 0) DOI: 10.9790/5728-1302041118

pk

pk

1 .

1 .

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15 | Page

Some Classes of Generalized Entire Sequences of Fuzzy Number pk pk        pk  k k , 0  , D( k , 0)  = D , 0  .

If kA then min  D   k    k  pk      k , 0  , D( k , 0) p k  = D( k , 0) pk .

If kB then  D   k   pk      k , 0   is bounded for kA and Hence (4.3.1) is equal to the assertion that  D   k  

D  , 0  pk

k

is

bounded for kB. Suppose (4.3.1) holds and let uo(F, P) .

If kA Write uk k = (ukk)

k k

If kB uk k

In either case

= (uk) k

D u  , 0  is arbitrarily small for sufficiently large k. pk

k

k

Hence 0(F, P)  o(F,P) On the other hand if (4.3.1) is false, we can find an increasing sequence of positive integers {k(r)} such that

and D



D



, 0 k (r )  r P

k

, 0  k ( r )  r. for P

k (r )

If D(k(r),

r = 1, 2, 3, ………….

0)1

Define

r p k ( r ) , k  k (r ), ? uk    0 Otherwise pk ( r )  k ( r ) 0   1 1



Then D u k(r) If D



k(r)

0 

pk ( r )

 1

   r pk ( r ) k  k (r ) Define uk =   Otherwise 0 pk ( r ) Then D uk(r)  k ( r ) 0   1.

1

Thus in either case, u 0 (F, P) but u o(F,P).

This contradicts the hypothesis that o(F,P)  o(F,P) This completes the proof.

  k   , 0  D( k , 0) D   k  

Corollary 4.3.3 o(F, P)  o (F, P) if and only if min Theorem 4.4

is bounded

(o (F,P), ) is a complete metric space if and only if



lim inf [ D(k , 0 ) pk ]

1

M



Proof. The proof is similar to 3.1 Theorem 4.5  is finer than λ if and only if



lim sup [ D(k , 0) pk ] M

k 

1

<0

 <

(4.5.1)

Suppose that (4.5.1) holds then

DOI: 10.9790/5728-1302041118

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Some Classes of Generalized Entire Sequences of Fuzzy Number



lim sup [ D(k , 0) pk ] M

k 

That is

1



= L <  for some positive real numberL>0





pk pk  lim sup  sup max | k ( ) | m | k ( ) | M  L   k   [0,1]

(4.5.2)

Let  > 0 be any real number Let {u(n)} be any sequence converging to zero in o(F,P) with respect to  .

Then there exist some no such that D Consequently ,



1

sup [ D (uk( n) , 0 ) pk ] M k

ie)



 



u

(n) k



, 0   for all L for all n  no

L

pk pk  sup  sup max (uk( n)  ( ) M , | uk( n)  ( )] M k  [0,1]

using (4.5.2) and (4.5.3)



  D(u

d (uk( n) , o)  sup Duk( n) , o pk k



 sup D(k , o) pk R

<L Therefore



L

=



n  no



 

L

_______ (4.5.3)

1/ M

1/ M

( n) k

, o) pk

for all n n0



1/ M

.

u  converges to zero with respect to . In other words the identity map on ( (F,P), ) (n ) k

0

onto (o(F,P), ) is continuous. Hence  >  Theorem 4.3.6. Let (0(F,P) ) be a complete metric space and let q=(qk) be a bounded sequence of strictly positive real numbers. Then the following are equivalent i. 0(F,P) 0(F,q) ii.

q lim  k k  p  k

   0 

assume that (ii) is not true.

Then we can determine an increasing sequence of positive integers k(1) < k(2) < ………. Such that qk(i)<

1   pk (i ) i Define

  1  pk (i )  uk(i) =  i ( D( , 0 )) k (i )    0 

0)

pk (i )

Also D(uk(i),

0)

Then D (k(i) uk(i),

for k  k (i ) for k  k (i)

1    0 as i   i

= pk (i )

 0 as i  

But D( (k (i )

uk (i ) , 0))

qk ( i )

 1  exp[  (log i) i ]  exp    2 

This shows that u does not belong to o(F,q) which contradicts (i) and (ii) must hold. (ii)  (i) Suppose (ii) holds Then there exists r > 0 such that qk > rpk for all sufficiency large k.

Therefore D(kuk, 0 )qk  [D(kuk, 0 )pk ]r Thus o(F, P)  o (F,q).

DOI: 10.9790/5728-1302041118

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