APPLICATIONS OF THE FRACTIONAL DIFFERENCE OPERATOR FOR STUDYING EULER STATISTICAL CONVERGENCE OF SEQUENCES OF FUZZY REAL NUMBERS AND ASSOCIATED KOROVKIN-TYPE THEOREMS Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 11 (29), No3, 2022, pp. 91-108

DOI: 10.15393/j3.art.2022.11770

91

UDC 517.521

Kuldip Raj, Kavita Saini, M. Mursaleen

APPLICATIONS OF THE FRACTIONAL DIFFERENCE OPERATOR FOR STUDYING EULER STATISTICAL CONVERGENCE OF SEQUENCES OF FUZZY REAL NUMBERS AND ASSOCIATED KOROVKIN-TYPE

THEOREMS

Abstract. The present work focuses on the statistical Euler summa-bility, Euler statistical convergence, and Euler summability of sequences of fuzzy real numbers via the generalized fractional difference operator. We make an effort to establish some relations between different sorts of Euler convergence. Further, we discuss the fuzzy continuity and demonstrate a fuzzy Korovkin-type approximation theorem. Finally, we study fuzzy rate of the convergence of approximating fuzzy positive linear operators through the modulus of continuity.

Key words: Euler mean, sequences of fuzzy real numbers, statistical convergence, rate of convergence, approximation theorem 2020 Mathematical Subject Classification: 40A05, 40A30,

46S40, 47S40

1. Introduction and Preliminaries. The theory of statistical convergence was initially presented by Fast [7] and Steinhaus [21]. It has been further designed by Connor [5], Fridy [8], Miller and Orhan [13]. For advanced developments in the field of statistical convergence and the neighbour topics, see [15], [16]. Mursaleen and Alotaibi [17] also proved an approximation theorem for a function of two variables by means of statistical A—summability. For a detailed study on summability theory and approximation results, see [1], [11], [14], [18], and many others. In 1965, Zadeh [27] introduced the concept of fuzzy numbers. Savas [22] studied statistical convergence for a sequence of fuzzy numbers. Later, Ay-tar et al. [2] expanded the concept of statistical superior limit and inferior limit to statistically bounded sequences of fuzzy real numbers. Talo and

© Petrozavodsk State University, 2022

Ba§ar ( [23], [24]) studied certain classes of sequences of fuzzy numbers; for further study, see [6], [25], [26].

A fuzzy set ii: R ^ [0,1] is called a fuzzy number if it satisfies the following criteria:

(i) u is normal, i. e., there exists an x0 E R, such that ■ii(xo) = 1;

(ii) u is convex, i. e., for x0,x E R and 0 ^ r ^ 1:

u(tx0 + (1 — r)x) ^ min{ii(x0),u(x)};

(iii) u is upper semi-continuous;

(iv) supp(ti) = cl{x E R: u(x) > 0} is compact and it is denoted by [ii]0.

Throughout the paper, RF denotes the space of all fuzzy numbers. Suppose [ii]0 = {x E R: u(x) > 0} and the ¿-level set is [u}% = {x E R: ux) ^ i}, (0 < i ^ 1). For any u,v E R^ and A E R, it is positive to define uniquely the sum u © i and the scalar multiplication to A E R of u as

[u © i]* = [£] + [if and [A © uY = A[£].

Now, the interval [u]1 is denoted by [u(^,u(^], where u^} ^ u^} and u(i),uii) E R, i E [0,1]. Then, for u,v E RF define

u ^ v & uZ' ^ v(_' and v™ ^ v™, V 0 ^ i ^ 1.

I- ^ v(a^ Now, d: Rw x R^ ^ R is given by

a(u, v) = sup max < 1 u_ — v_ 1, 1 u+ — v\ 1 >.

Here, (R^,d) is a complete metric space [20]. Let g,h: [a,b] ^ R^ be fuzzy-valued functions. Then the distance between g and h is given by

d*(g,h)= sup sup max< | g__ — h__| , | g++ — |

Let y = (y#) be a sequence of fuzzy real numbers. Then (y#) is called statistically convergent to a fuzzy number L, if, for every t > 0:

lim 1 |{tf ^ -q : d(y#, L) ^ e}| = 0. v ^

By Q, we mean the space of real-valued sequences. Let y = be any sequence in Q and h be a constant. Recently, Baliarsingh [3], [4] introduced

a new version of difference sequence space of fractional order, given by

(A"jf)"=£ ■w e n (i>

where r,s,t are real numbers and (t)# is the Pochhammer symbol of a real number , which is defined as

,, /1, W = 0)

\r(r) = [r][r + 1][r + 2] ■ ■ ■ [r + d — 1], ($ e N).

Here the series (1) is convergent for all i > r + s (see [9]).

Definition 1. [19] A sequence y = (y&) is said to be Euler statistically convergent to , if for each > 0

B£ = ^ (1+ß)" -II > £}

has zero natural density, i. e.,

lim |Bg|N = 0. (1 + ß)"

Definition 2. A sequence y = (y&) of fuzzy real numbers is said to be statistically A^'4 Euler summable ( st — Arf,s'tSß) to a fuzzy number L, if for every > 0

lim —

Tj

0.

Definition 3. A sequence y = (y#) of fuzzy real numbers is said to be Arr^i't Euler statistically (A'rr,s'tStß) convergent to a fuzzy number L, if for each > 0

{$ ^ (1+ß)": ßv-^d(ATfrs'ty^, L) > e} has zero natural density, i. e.,

lim ^ (1 + ß)" : ß"-dWye, L) > e}| = 0.

■n^tx (1 + ß)'i

Definition 4. A sequence y = (yo) of fuzzy real numbers is said to be S'f Euler summable to a fuzzy number L, if

1 "

lim --— > ß"-0 Art'S'tyi) = L as ri —oo.

(l+^fcT r y '

Also, y = (y$) is said to be strongly Ajj^ Euler summable (Ar1,S'tS^) to L, if

1 "

lim ^ 1 . dA^y», L) = 0.

(1+^"¿T ( 1 )

Definition 5. A sequence y = (yo) of fuzzy real numbers is said to be strongly Ar,S't Euler summable with order a (Ar,S'tSß)a (0 < a < x>)

to L, if

1 v

1lm TTTrV L)a = 0.

2. Relation between different convergence concepts of sequences of fuzzy real numbers.

Theorem 1. Suppose d(Arf,s'ty$ -L) ^ M, V E N. If a sequence H = ( y$) is A"*'1 Euler statistically (A" s'tSt111 ) convergent to L, then it is strongly A"hf'1 Euler summable (A^^S^) to L.

Proof. Suppose ^v-$d(Ar's'ty$,L) ^ M, V rj,ê E N. By the given condition, we have

lim 7—Tn-1{& ^ (1 + ^)v: ^v-$d(Ahs'ty,,L) > £}l = 0. (1 +

Consider

G(e) = ^ (1 + ^)v : ^v-$d(Ahs'ty$,L) > e}

and

Gc(e) = {$ ^ (1 + ^)v : ^d(Ahs'ty$, L) < e}.

Then

1 v

^^v-$d(Ahs'ty$,L) =

v

9=1

i V i V

£ ^(ArW^+TT-^ È ^dWy^L) ^

(1+ß)V ^ ' (1+ß)V 9=1

9€G(e) 9eGc(e)

< supdWy*,^ |G(e)| ^

^ (YTtFMl°(£)l+ £ ^ ° + £ = £

as r] ^ œ, which implies у = (y&) is strongly Arf,s,t Euler summable (A^SE) to L. □

Theorem 2.

„г ........l,^ .. i W) be (^ S)a

(a) A sequence of fuzzy numbers у = (y#) be (Arf,s,tSE)c to L. If

0 < a < 1 and 0 ^ d(Arl;s,yii, L) < 1

or

1 ^ a < œ and 1 ^ d(Arf,s,tyL) < œ,

r,S,t Qif

then (уф) is (Arf,s, StE) convergent to L.

(b) A sequence of fuzzy numbers у = (y#) is (Arf,s,tStE) convergent to

L and

If 0 < a < 1 and 1 ^ M < œ or 1 ^ a < œ and 0 ^ M < 1, then ( у#)

r,s,t oF\ °E )a

" ' У&) be (Ah SE )*

V

■П^ж (1 + ß)v

Consider

n(r-\ — r„q ^ (л I

and

d(Ar1,s,tyi),L) ^ M, for allé G N.

is (Arf,s,tSE)« to L. Proof. (a) Let y = (y#) be (A^^SE)« to L, i.e.,

J™, ттт^Т.^^лг = °.

■&=i

G(e) = {é ^ (1 +ßT : d(Als,tyi),L) > e]

Gc(e) = {é ^ (1 + 0)" : d(Als,tVi))L) < e]. By the given conditions, we have

d(Als,ty, - L) ^ d(A^y, - L)*

and

t^* d(Arhs,ty, -L) ^ ^ d(Arhs,ty, -L)*.

Then

1 v

■Y,ßv-*d(Aris% — L)a S

1 v

S (T^g^^

1 v 1 v

E (A'^W) + E (A'^W) S

tfÇG^) i9Ç.Gc(e)

11 (1+^)v ,=1 ^ ( r ^ ) ^ (1 + ß)v ¿1

|G(e)|

= e -у—--> О as ri ^ œ,

(1+ß)v 1 ,

which implies (у,) is (A]"s'tStß) converge to L.

(b) Let y = y$ be (A^^St^) convergent to L and ßv-i>d(Arr"s'ty,,L) ^ M, for all ■& G N. Then

1 v

1 nt^ ,v-,MAr's't

J2ßv-4A rs*w — L)

1 v 1 v

E ^(Ah'S'Wr+(TT^ E ^(^wr

,e=1e) {>еас(Е)

= U (v) + v2(v),

where

v (n) =

(1+^)v

1 v

v (v) = ttttVv E ^dA^LY

,=1 $eo(£)

1 v

v^) = (TT^ E ^dA^LY

If tf G G(e), then

) v r

1 v

v(^)=(T^ E ^d(AГ'Ш^Г ^

,e=1e)

1 1 (1 + ß)11 f- ^ \ h y^ )

9=1 9eG(e)

£ suprv-l9d(Ahs'ty9,L)IG(e)\]j £ M(j^ ^ 0 as v ^ n.

If •& e Gc(e), then

1

1

V2(v) = (Y^ E (AhT'Wr £

9=1 9£Gc(e)

1

£

1 £ ß1-9d^'W)!^ =£ as V

(1+^ 9=1 h /(1+^)1

9egc(£)

v r gt

Thus, ( 1 + ^ J2 d(Arf,s' y& — L)a ^ 0 as rj ^ Hence, (yv) is (A^Sfe)a tol □

3. Korovkin-type theorem and rates of equi-statistical convergence. In this section, we use the concept of statistical A^rf'1 — Euler summability method ( st — A r,s'tSE) to prove a Korovkin-type approximation theorem. A fuzzy number valued function g : [ a, b] ^ R p is said to be fuzzy continuous at y0 E [a, b], iff yv ^ y0; then d(y9, y0) ^ 0 as ^ <x>. In other words, we can say that on an interval [a, b] g is fuzzy continuous if it is fuzzy continuous for any u E [a, b], and we denote the space of all fuzzy continuous functions on the interval [a, b] by Cf[a, b}. In this case, Cf[a, b] is just a cone, not a vector space. Now let £ : Cf[a, b] ^ Cf [a, b] be an operator. We say that £ is fuzzy linear, if for every (i, (2 E R, g\, g2 E Cf[a, b], and u E [a, b]:

C(Ci 0J1® C2 © 92; u) = Ci © C(gi; u) © C2 © ^(92; u). Also, is called fuzzy positive linear operator, if it is fuzzy linear and

C(9i;u) < C(92;u) for any g 1, g2 E Cf5] and for any u E [a, b] with

9i(u) ^ 92(u).

In this paper, we use the test function ej, which is given by ej(u) = uj; here j = 0,1, 2.

Theorem 3. Consider the fuzzy sequence {} of positive linear operators from Cf[a, b] into itself. Suppose that there exists a corresponding sequence {£m} of positive linear operators from C [a, b] into itself s.t.

{Ug;u)}± =Ui£';«), (2)

for all u E [a, b], g E Cf[a, b] and m E N. Suppose also that

s t — lim

1

^A^Uej) — ej =0 (j = 0,1, 2). (3)

Then, for all g E Cf [a, b]

lim d*(

v^^ V(1 +

1 v

st — v^i (T^ E ^"A^M-i) = 0 (4)

n

0=1

Proof. Suppose g E Cf[a, b], u E [a, b] and i E [0,1]. Since g± E C[a, b], for every e > 0, there exists a number p > 0, such that |g±(y) —g± (x) I < £ whenever Iv — uI < p. Since g is fuzzy bounded, we get |g± (u)I ^ R± . Then, for all E [ , ], we have

\ 2

I9(?(«) — g¥(u)I ^ e + 2R«^—T1

P2

which implies

2R?(v — u)2 < (g®(v) — <£'(«)) < e + ^(v — u)2.

Using the positivity and linearity of the operators £m, we have 1 v , 2 R \

e«(1, u)( s — ^(v — u)2) <

0=1

1

(1 + »)"

< E^A^a,«)^) — ^V)) <

0=1

1 .v. / 2 R N

< E ^"Ar'&o. u)(s+(„—„)»).

Suppose u is fixed; then g± (u) is a constant number and we have: 1 v

— ---- E^A^^u) —

(1 + ^9=1

2 pi 1 1

^ (T^ £ ^1-9Ah'"'«9(0 -«)2.») <

1 1

1 1

9=1 1

2 pi 1 1

Also,

ytß1-9Ahs'tC-9(9^)(v),u) - gV(u) =

1 1

= (^1 e^ i)(-),u)

1 ^■1-9Arh'%(d)iv),u)-

9=1 1 1

- E^X'^C9(1,u)+

1 1

+ ¿l)(u)^^ gß1-9Ah^9((1,u) - 1),

which gives: 1 v

(Y^ L^Ar4£9(2i*V),u) — ^(u)

9=1

1

1

—-J2^v-9Ahs't e9(1,u)+

+

2 Ri 1

P2 (1+^

J^A^^ -u)2,u) +

9=1

+ g±(u)(—^yj ((i,u) - 1). (5)

Next, consider the second part of the above inequality:

1

, -Y -u)2,u)

= u

= \ V ^A^Uv2 + u2 - 2uv, u) =

(1 + ^9=^ r

1 V 1 V

2 Ts^A^6(1, u) - 2u(^)V ,u) +

9=1

+

1

(1+^)V

Y,^Aïs'%(v 2,u) =

9=1

--- V tiV-9Aïs%(v2,u)

(1 + ^)V9=1

22 2, u) u2

2 u

l(1 + ^)v9=1

J2vV-9Als%(v,u) -u

+

+ u2

L(1 + ^)V

Ar'^9(1,u) - 1

9=1

Using the above equality with (5), we have

J^Z^A^ÏW) - g$(u) £

£ e

(1 + ^)V

J2^-9AïS'% (1,u)+

+

2 R

±

P2

2 u

9=1 V

L(1 + ^)V9=1

J2vV-9Aïs% (V2 ,u) -u2

L(1 + ^)V9=1

Y,^-9Ar%(v,u) -u

+

1

1

1

1

1

+ u2

+ g¥(u)

1

+

0=1

l

Y,^Ars'%(i,u) -1

1

+

2 R

1

Y,^Ars'%(1,u) -1

■±

p2

2 u

+ u

L(1 + ^,)V - 1

- 1

2,u) -u2

0=1 V

J2^V-0Ars'%(v,u) -u

0=1 V

Y,^-0Ks'%(1,u) -1

+

+

0=1

+ g¥(u)

L(1 + ^)V0=T r ' )

Now,

1

(1 +P>)V

0=1

<

< £+ (£ +

+

2 R± c2

+ R

i)

^ßV-0^rS%0(1,u) - 1

4 R±c 1

P2 (1+^)V

(1+^)V0=1 it^-0ArS'%(v,u) -u

+

+

+

2 R ± 1

P2 (1 + P>)V

^ßV-0Ars'%(v2,u) -u2

0=1

where c = max{|a|, |6|}. Let N±(e) = max + + R±,,• Taking supermum over u E [a, b], we transform the above inequality to

(TT^ g^V-0Ar^0(^),u) - ^(u)

1

£ £ +К

±

E^X^e o,u)

eo

+

9=1

+

а + (el,u) - el

+

+

(l+ß)vi=l

■i _ E^X^e2,u) - e2 }. (б)

Then, from (2), we have

d*

-(l+^)v

ÉßV_l9Ar;s,%(g),g

9=1

'' 4

suP d( Tr—~^^2^V_9AhS,tC9(g;u),g)

uG[a, Ъ]

(1+ß)

9=1

sup sup max{ ——- 9ah,s,ic9(^-(^)) -g_(u)

ue[a,b] -G[o,1] l (l + ^)v 9=1

l

(l + ^)v9=1

Y.vV_9Ahs%(g + W) - (u)|}

sup max I —-- £vV_9AZ'%(gi(v)) - £-(«)

9=1

V

(1+^)'

Y^ßV_9Arrs,tÙ(g + (г;)) - (u)||}. (Т)

9=1

From (б) and (Т), we get

d*

1 V

£ e +

(1 + ^

9Ahf'* ^9(еo,u) - eo

9=1

+

+

l

(l + ^)vI]^V_9Ah,',^9(el,u) - el

+

+

(1 + ^

v л

J2^V_9Ar¡s,%9(e2,u) - e2 }

9=1

l

l

l

l

l

l

where N(e) = sup max-] IN_(e)|, |N+ (e)| >. For a given e1 > 0, take

ie [0,1] ^ J

number e > 0, such that e < e1. Consider

1

^ ={n E N: d*(Y,»"-*A?%(g),g) > ,

0=1

Sj = \n E N:

(1 + P)

^j ;u) _ 6j

0=1

e1 _ £' 3 N (£).

2 2 where j = 0,1, 2. Therefore, S C Sj. Thus, 8(S) ^ 8(Sj). Hence,

=0 =0

using (3), we obtain (4). □

Definition 6. Consider the positive non-increasing sequence (s V) of real numbers. A sequence (y0) of fuzzy number is said to be statistically Ar1's't Euler summable ( st _ Arf,s'tS^!) convergent to L with the fuzzy rate o(sV) for every £ > 0, if we have

lim —

1 V

0=1

and, therefore, we can write it as

y' _L = (st_ ATr8,tS%) o(sv). Now, the modulus of continuity of g E Cf [a, b] is given by z(g;p) = sup d(g(u), g(v))

for any 0 < p ^ b _ a that satisfies

z(g, lu _vl) ^ (l +

| u _ |

Next, we have the following result:

Theorem 4. Consider the fuzzy sequence {} of positive linear operators from CF [a, b] into itself. Suppose that there exists a corresponding sequence {£m} of positive linear operators from C [a, b] into itself, such that {£m(g; u)}± = Cm(g±; u), for all u E [a, b], g E Cf [a, b] and m E N. Consider two positive non-increasing sequences (sV) and (pV) of real numbers. Further, suppose that

a

1

0

(i) IIo) — eol =st — Ar;s'tsE 0(sv), (u) z(g, v7IlLrn((v — u)2;x)|) = st — E °(Pv). Then we have

d*(ím(g), 9) = St — Ar,rs'tsF 0(av), where a = max{sv,pv}.

Proof. Consider g E Cf [a, b] and u E [a, b]. Then, using positivity and linearity of the operator £m with the continuity of fuzzy modulus, we have

£

u) — o

+

E^A^U^;u) — g±(u)

£ 19±(u)l (1+^ ^pv-9A^%(eo;u)

+ ^^v-9Ars'%(l<4(v) — g±(u)|; u) £

1 v

»v-9Aha'%(e o;u) — eo

£ R±± 1

+

(1 + ^)v9=1

£ R

±

(i + ^)v

E^A^ Ue o;u)

u) — o

9=1

+

(1 + ^)v9=1

£ R

±

(i + ^)v

^pv-9Ars'%(e o;u)

u) — o

9=1

+

+ z(g ± ,f) + z(g Lf) +

9=1

(i+ m)"

(_

f2 V(1+m)"

9A'rlrs't^9(eo;u) — eo

+

z(9± ,f) ( 1 v-9 AT'S't ¿(f

' Ar — u) ;u

2

9=1

where R± = ¡(/±1. Taking supermum norm of both sides of the above inequality, we have

1

1

1

1

V

£

£ R

±

1

(l + p)v

o;«)

«) - eo

+

0=1

1

(l+p)v0=1

o;«) - eo

+

_L i i ^ _L Z(9¿^

+ z(g±,^) + ———

(l+^)v0=1

Now, take ^ = ; then we have

1 v

£^-0Ah>a>iU<4;«) - £(«)

V

£

£ R

±

1

(I + p)v 1

Y^V-"Ahs'%(e o;«)

«) - eo

+

0=1 V

0Ah'S,ÎC0(eo;«) - eo

+ 2z(g ^0).

Therefore, we have

1 v

£ R

±

1

(1 + p)v

YvV-0AhS%(e o;«)

«) - eo

0=1

+

+ ±,^0) {

1

(1+P)V

J]pV-0Ah'^0(eo;«) - eo + 2z(^0) £

£ H<

1

0=1 V

+% ^0)

&Ar}f,t^0(eo;u) - eo

1

(i + p)v

•I _

Y^-0ArhS'%(eo;«) - eo }

0=1

where H = max{R±, 2}. Using Definition 6 and conditions of Theorem 4, we have desired result. □

1

Acknowledgment. The author deeply appreciates the suggestions of the

reviewers and the editor that improved the presentation of the paper.

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Received April, 26, 2022.

In revised form, September 20, 2022.

Accepted September 23, 2022.

Published online October 14, 2022.

Kuldip Raj

School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J&K, India

E-mail: [email protected] Kavita Saini

School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J&K, India

E-mail: [email protected] M. Mursaleen

Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India E-mail: [email protected]