@FMI c Kyung Moon Sa Co.

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1 Annals of Fuzzy Mahemaics Infomaics Volume, No. 2, (Apil 20), pp. 9-3 ISSN c Kyung Moon Sa Co. hp:// On lacunay saisical convegence in inuiionisic fuzzy n-nomed linea spaces N. Thillaigovindan, S. Ania Shanhi, Young Bae Jun Received 3 Novembe 200; Acceped Decembe 200 Absac. We inoduce he concep of lacunay saisical convegence in inuiionisic fuzzy n-nomed linea space. Some inclusion elaions beween he ses of saisically convegen lacunay saisically convegen sequences ae esablished in an inuiionisic fuzzy n-nomed linea space. We also define lacunay saisical Cauchy sequence in an inuiionisic fuzzy n-nomed linea space pove ha i is equivalen o lacunay saisically convegen sequence. 200 AMS Classificaion: 40A05, 60B99 Keywods: Lacunay sequence, Lacunay saisical convegence, Lacunay saisical Cauchy sequence. Coesponding Auho: N. Thillaigovindan (hillai n@sify.com). Inoducion Moivaed by he heoy of n-nomed linea space [2, 3, 6] fuzzy nomed linea space [, 2, 4, 6, 7] he noions of fuzzy n-nomed linea space [8] inuiionisic fuzzy n-nomed linea space [9] have been developed. The concep of saisical convegence fo eal numbe sequences was fis inoduced by Fas [5] Schonebeg [22] independenly. Lae, i was fuhe invesigaed fom sequence poin of view linked wih summabiliy heoy by Fidy [9], Sala [2] many ohes. Fidy [0] inoduced he concep of lacunay saisical convegence. Some wok on lacunay saisical convegence can be found in [3, 8,, 5, 20]. Kaakus [4] e al. invesigaed he saisical convegence on inuiionisic fuzzy nomed spaces gave a chaaceizaion fo saisically convegen sequences on hese spaces. Musaleen [7] e al. exended he esuls of Kaakus fom single sequence o double sequences on inuiionisic fuzzy nomed spaces. Ou aim in his pape is o inoduce he noions of lacunay saisical convegence lacunay saisical Cauchy sequence in inuiionisic fuzzy n-nomed

2 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 linea spaces esablish some inclusion elaions beween saisical convegence lacunay saisical convegence in inuiionisic fuzzy n-nomed linea spaces. 2. Peinaies In his secion we ecall some useful definiions esuls. Definiion 2. ([23]). A binay opeaion : [0, ] [0, ] [0, ] is a coninuous -nom if saisfies he following condiions: (i) is commuaive associaive, (ii) is coninuous, (iii) a = a, fo all a [0, ], (iv) a b c d wheneve a c b d a, b, c, d [0, ]. Definiion 2.2 ([23]). A binay opeaion : [0, ] [0, ] [0, ] is coninuous -co-nom if saisfies he following condiions: (i) is commuaive associaive, (ii) is coninuous, (iii) a 0 = a fo all a [0, ], (iv) a b c d wheneve a c b d a, b, c, d [0, ]. Definiion 2.3 ([9]). An inuiionisic fuzzy n-nomed linea space o in sho i-f-n-nls is an objec of he fom A = {(X, N(x, ), M(x, )) x = (x, x 2,, x n ) X n } whee X is a linea space ove a field F, is a coninuous -nom, is a coninuous -co-nom N, M ae fuzzy ses on X n (0, ); N denoes he degee of membeship M denoes he degee of non-membeship of (x, x 2,, x n, ) X n (0, ) saisfying he following condiions: () N(x, ) + M(x, ), (2) N(x, ) > 0, (3) N(x, x 2,, x n, ) = if only if x, x 2,, x n ae linealy dependen, (4) N(x, x 2,, x n, ) is invaian unde any pemuaion of x, x 2,, x n, ) if c 0,c F, (6) N(x, x 2,, x n, s) N(x, x 2,, x n, ) N(x, x 2,, x n + x n, s + ), (7) N(x, ) : (0, ) [0, ] is coninuous in, (8) M(x, ) > 0, (9) M(x, x 2,, x n, ) = 0 if only if x, x 2,, x n ae linealy dependen, (0) M(x, x 2,, x n, ) is invaian unde any pemuaion of x, x 2,, x n, (5) N(x, x 2,, cx n, ) = N(x, x 2,, x n, c () M(x, x 2,, cx n, ) = M ( x, x 2,, x n, c ), if c 0, c F, (2) M(x, x 2,, x n, s) M(x, x 2,, x n, ) M(x, x 2,, x n + x n, s + ), (3) M(x, ) : (0, ) [0, ] is coninuous in. Example 2.4 ([24]). Le (X,,, ) be a n-nomed linea space, whee X = R. Define a b = min{a, b} a b = max{a, b} fo all a, b [0, ], N(x, x 2,, x n, ) = e x,x 2,,x n 20

3 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 M(x, x 2,, x n, ) = e x,x2,,xn. Then A = {X, N(x, x 2,, x n, ), M(x, x 2,, x n, ) (x, x 2,, x n ) X n } is an i-f-n-nls. Remak 2.5. Fo convenience we denoe he inuiionisic fuzzy n-nomed linea space by A = (X, N, M,, ). Definiion 2.6 ([2]). Le (U, N) be a fuzzy nomed linea space. We define a se B(x, α, ) as B(x, α, ) = {y : N(x y, ) > α}. Definiion 2.7 ([9]). A sequence {x nk } in an i-f-n-nls A is said o convege o ξ X wiespec o he inuiionisic fuzzy n-nom (N, M) if fo each ɛ > 0, > 0 hee exiss a posiive inege n 0 such ha N(x, x 2,, x n, x nk ξ, ) > ɛ M(x, x 2,, x n, x nk ξ, ) < ɛ fo all k n 0. The elemen ξ is called he i of he sequence {x nk } wiespec o he inuiionisic fuzzy n-nom (N, M) is denoed as (N, M) x nk = ξ. Definiion 2.8 ([9]). A sequence {x nk } in an i-f-n-nls A is said o be Cauchy wiespec o he inuiionisic fuzzy n-nom (N, M) if fo each ɛ > 0 > 0 hee exiss a posiive inege m 0 such ha N(x, x 2,, x n, x np x nq, ) > ɛ M(x, x 2,, x n, x np x nq, ) < ɛ wheneve p, q m 0. Definiion 2.9 ([9]). A lacunay sequence is an inceasing inege sequence θ = {k } of posiive ineges such ha k 0 = 0 = k k as. Thoughou his pape he inevals deemined by θ will be denoed by I = (k, k ] q = k k. Definiion 2.0 ([9]). Fo a lacunay sequence θ = {k }, he numbe sequence {x k } is said o be lacunay saisically convegen o a numbe ξ povided ha fo each ɛ > 0, {k I : x k ξ ɛ} = 0, whee he veical ba denoes he cadinaliy of he enclosed se. In his case we wie S θ x k = ξ. k 3. Lacunay saisical convegence in inuiionisic fuzzy n-nomed linea space Definiion 3.. Le A be an i-f-n-nls. We define an open ball B(x,, ) wih cene x on he n h coodinae of X n adius 0 < <, as fo > 0. B(x,, ) = {y X : N(x, x 2,, x n, x y, ) > M(x, x 2,, x n, x y, ) < } Definiion 3.2. Le A be an i-f-n-nls. A sequence {x nk } of elemens in X is said o be saisically convegen o ξ X wiespec o he i-f-n-nom (N, M) if fo each ɛ > 0 > 0, hee exiss p N such ha p p {k p : N(x, x 2,, x n, x nk ξ, ) ɛ M(x, x 2,, x n, x nk ξ, ) ɛ} = 0. 2

4 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 The elemen ξ is called he saisical i of he sequence {x nk } wiespec o he inuiionisic fuzzy n-nom (N, M) is denoed as S(N, M) x nk = ξ o x nk ξ(s (N,M) ). Definiion 3.3. Le A be an i-f-n-nls θ be a lacunay sequence. A sequence {x nk } of elemens in X is said o be lacunay saisically convegen o ξ X wih espec o he i-f-n-nom (N, M) if fo each ɛ > 0 > 0, {k I : N(x, x 2,, x n, x nk ξ, ) ɛ M(x, x 2,, x n, x nk ξ, ) ɛ} = 0. The elemen ξ is called he lacunay saisical i of he sequence {x nk } wiespec o he inuiionisic fuzzy n-nom (N, M) is denoed as S(N,M) θ x n k = ξ o x nk ξ(s(n,m) θ ). We denoe by S(N,M) θ (X), he se of all lacunay saisically convegen sequences in an i-f-n-nls A. Nex we show ha fo any fixed θ, S(N,M) θ i is unique povided i exiss. Theoem 3.4. Le A be an i-f-n-nls θ be a fixed lacunay sequence. If {x nk } is a sequence in X such ha S(N,M) θ x n k = ξ exiss, hen i is unique. Poof. Suppose ha hee exis elemens ξ, η wih (ξ η) X such ha S θ (N,M) x n k = ξ S θ (N,M) x n k = η. Le ɛ > 0 be abiay. Choose s > 0 such ha (3.) ( s) ( s) > ɛ s s < ɛ. Fo > 0, we ake A = {k I N(x, x 2,, x n, x nk ξ, ) > s, M(x, x 2,, x n, x nk ξ, ) < s} B = {k I N(x, x 2,, x n, x nk η, ) > s, M(x, x 2,, x n, x nk η, ) < s}. We shall fis show ha fo ξ η > 0, A B = φ. Fo, if m A B hen N(x, x 2,, x n, ξ η, ) N(x, x 2,, x n, x m ξ, 2 ) N(x, x 2,, x n, x m η, 2 ) > ( s) ( s) > ɛ. by (3.). Since ɛ > 0 is abiay we have N(x, x 2,, x n, ξ η, ) = fo evey > 0. Similaly M(x, x 2,, x n, ξ η, ) = 0 fo evey > 0. This implies ha ξ η = 0, a conadicion o ξ η. Thus A B = φ heefoe A B c. Hence 22

5 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 we have {k I : N(x, x 2,, x n, x nk ξ, ) > s M(x, x 2,, x n, x nk ξ, ) < s} {k I : N(x, x 2,, x n, x nk η, ) s M(x, x 2,, x n, x nk η, ) s}. Since S(N,M) θ x n k = η, i follows ha {k I : N(x, x 2,, x n, x nk ξ, ) > s Since his canno be negaive we have M(x, x 2,, x n, x nk ξ, ) < s} 0. {k I : N(x, x 2,, x n, x nk ξ, ) > s M(x, x 2,, x n, x nk ξ, ) < s} = 0. This conadics he fac ha S θ (N,M) x n k = ξ. Hence ξ = η. Theoem 3.5. S(N,M) θ (X) is a linea space. Poof. Le {x nk } be a sequence in X. (i) If S(N,M) θ x n k = ξ α 0 R, hen we need o pove ha Le ξ > 0 > 0. If we ake S θ (N,M) αx n k = α ξ. A = {k I : N(x, x 2,, x n, x nk ξ, ) > ɛ M(x, x 2,, x n, x nk ξ, ) < ɛ} B = {k I : N(x, x 2,, x n, αx nk αξ, ) > ɛ M(x, x 2,, x n, αx nk αξ, ) < ɛ}. Le m A. Then we have N(x, x 2,, x n, αx m αξ, ) = N(x, x 2,, x n, x m ξ, α ) N(x, x 2,, x n, x m ξ, ) N(0, α ) N(x, x 2,, x n, x m ξ, ) N(x, x 2,, x n, x m ξ, ) > ɛ M(x, x 2,, x n, αx m αξ, ) = M(x, x 2,, x n, x m ξ, α ) M(x, x 2,, x n, x m ξ, ) M(0, α ) M(x, x 2,, x n, x m ξ, ) 0 M(x, x 2,, x n, x m ξ, ) < ɛ 23

6 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 which implies ha m B. Hence we have A B heefoe B c A c. I follows ha {k I : N(x, x 2,, x n, αx nk αξ, ) ɛ M(x, x 2,, x n, αx nk αξ, ) ɛ} {k I : N(x, x 2,, x n, x nk ξ, ) ɛ M(x, x 2,, x n, x nk ξ, ) ɛ}. Since S(N,M) θ x n k = ξ, i follows ha S(N,M) θ αx n k = αξ. (ii) Le {x nk } {y nk } be wo sequences in X. If S(N,M) θ x n k = ξ S(N,M) θ y n k = η, hen we have o pove ha S(N,M) θ (x n k + y nk ) = ξ + η. Le ɛ > 0 be given. Choose s > 0 as in (3.). Fo > 0, we define he following ses: A = {k I : N(x, x 2,, x n, (x nk + y nk ) (ξ + η), ) > ɛ M(x, x 2,, x n, (x nk + y nk ) (ξ + η), ) < ɛ} B = {k I : N(x, x 2,, x n, x nk ξ, ) > s M(x, x 2,, x n, x nk ξ, ) < s}, C = {k I : N(x, x 2,, x n, y nk η, ) > s Le m B C. Then we have by (3.) M(x, x 2,, x n, y nk η, ) < s}. N(x, x 2,, x n, (x m + y m ) (ξ + η), ) = N(x, x 2,, x n, (x m ξ) + ((y m η), ) N(x, x 2,, x n, (x m ξ), 2 ) N(x, x 2,, x n, ((y m η), 2 ) > ( s) ( s) > ɛ M(x, x 2,, x n, (x m + y m ) (ξ + η), ) M(x, x 2,, x n, x m ξ, 2 ) M(x, x 2,, x n, y m η, 2 ) < s s < ɛ. This shows ha m A hence (B C) A. Theefoe we have A c (B c C c ). I follows ha {k I : N(x, x 2,, x n, (x nk + y nk ) (ξ + η), ) ɛ M(x, x 2,, x n, (x nk + y nk ) (ξ + η), ) ɛ} {k I : N(x, x 2,, x n, x nk ξ, ) s M(x, x 2,, x n, x nk ξ, ) s} + {k I : N(x, x 2,, x n, y nk η, ) s M(x, x 2,, x n, y nk η, ) s}. 24

7 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 Since S θ (N,M) x n k = ξ S θ (N,M) y n k = η we have S θ (N,M) (x n k + y nk ) = ξ + η. This complees he poof. Theoem 3.6. Le A be an i-f-n-nls. Fo any lacunay sequence θ, S (N,M) (X) S(N,M) θ (X) if only if inf q >. Poof. Sufficien pa: Suppose ha inf q >. Then hee exiss a δ > 0 such ha q + δ fo sufficienly lage which implies ha h k δ +δ. If {x n k } is saisically convegen o ξ wiespec o i-f-n-nom (N, M), hen fo each ɛ > 0, > 0 sufficienly lage, we have δ +δ {k I : N(x, x 2,, x n, x nk ξ, ) ɛ M(x, x 2,, x n, x nk ξ, ) ɛ} k {k I : N(x, x 2,, x n, x nk ξ, ) ɛ M(x, x 2,, x n, x nk ξ, ) ɛ} k {k k : N(x, x 2,, x n, x nk ξ, ) ɛ M(x, x 2,, x n, x nk ξ, ) ɛ}. I follows ha x nk ξ(s θ (N,M) ). Hence S (N,M)(X) S(N,M) θ (X). Necessay pa: Suppose ha inf q =. Then we can selec a subsequence k {k (j) } of he lacunay sequence θ such ha (j) k (j) < + j k (j) k (j ) > j, whee (j) (j ) + 2. Le ξ 0 X. We define a sequence {x nk } as follows: x nk = { ξ, if k I (j) fo some j =, 2, 0, ohewise. We shall show ha {x nk } is saisically convegen o ξ wiespec o he i-f-nnom (N,M). Le ɛ > 0 > 0. Choose ɛ (0, ) such ha B(0, ɛ, ) B(0.ɛ, ) ξ / B(0, ɛ, ). Also fo each p we can find a posiive numbe j p such ha 25

8 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 k (jp ) < p k (jp )+. Then we have {k p : N(x, x 2,, x n, x nk ξ, ) ɛ, p k (j p) M(x, x 2,, x n, x nk ξ, ) ɛ} {k p : N(x, x 2,, x n, x nk ξ, ) ɛ, M(x, x 2,, x n, x nk ξ, ) ɛ } k (j p) [ {k k(jp ) : N(x, x 2,, x n, x nk ξ, ) ɛ, k (j p) M(x, x 2,, x n, x nk ξ, ) ɛ } + {k < k p : N(x (j p), x 2,, x n, x nk ξ, ) ɛ, M(x, x 2,, x n, x nk ξ, ) ɛ } ] {k k (jp ) : N(x, x 2,, x n, x nk ξ, ) ɛ, M(x, x 2,, x n, x nk ξ, ) ɛ } + k (j p) (k (j p )+ k (jp )) < j p + j p + + = j p + j p + fo each p. I follows ha S(N, M) x nk = ξ. Nex we shall show ha {x nk } is no lacunay saisically convegen wiespec o he i-f-n-nom(n, M). Since ξ 0 we choose ɛ > 0 such ha ξ / B(0, ɛ, ) fo > 0. Thus j = j (j) {k (j) < k k (j) : N(x, x 2,, x n, x nk 0, ) ɛ, (j),j=,2, (j) (k (j) k (j ) ) = j M(x, x 2,, x n, x nk 0, ) ɛ} (j) ((j) ) = {k < k k : N(x, x 2,, x n, x nk ξ, ) ɛ, M(x, x 2,, x n, x nk ξ, ) ɛ} = 0. Hence neihe ξ no 0 can be lacunay saisical i of he sequence {x nk } wih espec o he inuiionisic fuzzy n-nom (N, M). No ohe poin of X can be lacunay saisical i of he sequence as well. Hence {x nk } / S(N,M) θ (X) compleing he poof. The following example esablishes ha lacunay saisical convegence need no imply saisical convegence. Example 3.7. Le (X,,, ) be a n-nomed linea space, whee X = R. Define a b = ab a b = min{a + b, } fo all a, b [0, ], N(x, x 2,, x nk, ) = M(x, x 2,, x nk, ) = 26 + x,x 2,,x nk x,x2,,xn k + x,x 2,,x nk.

9 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 Then A = (X, N, M,, ) is an i-f-n-nls. We define a sequence {x nk } by { nk fo k ( ) + k k, N x nk = 0 ohewise. Fo ɛ > 0 > 0, Then so, we ge K (ɛ, ) = {k N : N(x, x 2,, x n, x nk, ) ɛ K (ɛ, ) = {k N : M(x, x 2,, x n, x nk, ) ɛ}. + x,x 2,,x n,x nk ɛ x,x 2,,x nk + x,x 2,,x n,x nk ɛ} = {k N : x, x 2,, x n, x nk ɛ ɛ > 0} = {k N : x nk = nk} = {k N : k ( ) + k k, N} K (ɛ, ) = {k N : k ( ) + k k, N} K (ɛ, ) = 0 x nk 0 (S θ (N,M) ). On he ohe h x nk 0 (S (N,M) ), since + x,x 2,,x n,x nk N(x, x 2,, x n, x nk, ) = { = +nk, fo k ( ) + k k, N, ohewise x,x 2,,x nk + x,x 2,,x nk M(x, x 2,, x n, x nk, ) = { nk = +nk, fo k ( ) + k k, N 0, ohewise 0. Hence x nk 0 (S (N,M) ). Theoem 3.8. Le A be an i-f-n-nls. Fo any lacunay sequence θ, S(N,M) θ (X) S (N,M) (X) if only if sup q <. Poof. Sufficien pa: If sup q <, hen hee is a H > 0 such ha q < H fo all. Suppose ha x nk ξ (S(N,M) θ ), le P = {k I : N(x, x 2,, x n, x nk ξ, ) ɛ M(x, x 2,, x n, x nk ξ, ) ɛ}. 27

10 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 By definiion of a lacunay saisical convegen sequence, hee is a posiive numbe 0 such ha (3.2) P < ɛ foall > 0. Now le K = max {P : 0 } p be any inege saisfying k < p k. Then we have {k p : N(x, x 2,, x n, x nk ξ, ) ɛ p M(x, x 2,, x n, x nk ξ, ) ɛ} k {k k : N(x, x 2,, x n, x nk ξ, ) ɛ M(x, x 2,, x n, x nk ξ, ) ɛ} k {P + P P 0 + P P } K k 0 + P k {h h P + } ( ) 0 K k + P k sup { } > 0 0 K k + ɛ k k 0 k (by (3.2)) 0 K k + ɛ q 0 K k + ɛ H, so {x nk } is saisically convegen. Hence S(N,M) θ (X) S (N,M)(X). Necessay pa: Suppose ha sup q =. Le ξ 0 X. Selec a subsequence {k (j) } of he lacunay sequence θ = {k } such ha q (j) > j, k (j) > j + 3. Define a sequence {x nk } as follows: { ξ, if k (j) < k 2k (j) fo some j =, 2, x nk = 0, ohewise. Since ξ 0 we can choose ɛ > 0 such ha ξ / B(0, ɛ, ) fo > 0. Now fo j >, (j) {k k (j) : N(x, x 2,, x n, x nk 0, ) ɛ M(x, x 2,, x n, x nk 0, ) ɛ} < (j) (k (j) ) < (k (j) k (j) ) (k (j) ) < j. Theefoe we have {x nk } S θ (N,M) (X). Bu {x n k } / S (N,M) (X). Fo 2k (j) {k 2k (j) : N(x, x 2,, x n, x nk 0, ) ɛ M(x, x 2,, x n, x nk 0, ) ɛ} = 2k (j) {k () + k (2) + + k (j) } > 2. This shows ha{x nk } canno be saisically convegen wiespec o he inuiionisic fuzzy n-nom (N, M). Theoems immediaely give he following coollay. 28

11 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 Coollay 3.9. Le A be an i-f-n-nls. Fo any lacunay sequence θ, S(N,M) θ (X) = S (N,M) (X) if only if < inf q sup q <. Definiion 3.0. Le A be an i-f-n-nls θ be a lacunay sequence. A sequence {x nk } in X is said o be lacunay-θ-saisically Cauchy povided hee is a subsequence {x nk () } of he sequence {x nk } such ha k () I fo each, (N, M) x n k () = ξ fo each ɛ > 0 > 0, {k I : N(x, x 2,, x n, x nk x nk (), ) ɛ M(x, x 2,, x n, x nk x nk (), ) ɛ} = 0. Theoem 3.. Le A be an i-f-n-nls θ be lacunay sequence. A sequence {x nk } in X is lacunay-θ-saisically convegen if only if i is lacunay-θsaisically Cauchy. Poof. We fis assume ha S(N,M) θ x n k = ξ. Fo > 0 j N, le K(j, ) = {k N : N(x, x 2,, x n, x nk ξ, ) > j Then we have he following: (i) K(j +, ) K(j, ) (ii) K(j, ) I as. M(x, x 2,, x n, x nk ξ, ) < j }. This implies ha we can choose a posiive inege m() such ha fo m(), we have K(, ) I > 0. i.e., K(, ) I. Nex we can choose m(2) > m() so ha m(2) implies K(2, ) I. Thus fo eac saisfying m() m(2), choose k () I such ha k () I K(, ), i.e., N(x, x 2,, x n, x nk () ξ, ) > 0 M(x, x 2,, x n, x nk () ξ, ) <. In geneal, we can choose m(p + ) > m(p) such ha > m(p + ) implies I K(p +, ). Thus fo all saisfying m(p) m(p + ), choose k () I K(p, ), i.e., (3.3) N(x, x 2,, x n, x nk () ξ, ) > p M(x, x 2,, x n, x nk () ξ, ) < p. Thus k () I fo eac ogehe wih (3.3) implies ha (N, M) x n k () = ξ. Fo ɛ > 0, choose s > 0 such ha ( s) ( s) > ɛ s s < ɛ. Fo > 0, if 29

12 N. Thillaigovindan e al./annals of Fuzzy Mahemaics Infomaics (20), No. 2, 9 3 we ake A = {k I : N(x, x 2,, x n, x nk x nk (), ) > ɛ M(x, x 2,, x n, x nk x nk (), ) < ɛ} B = {k I : N(x, x 2,, x n, x nk ξ, ) > s M(x, x 2,, x n, x nk ξ, ) < s} C = {k I : N(x, x 2,, x n, x nk () ξ, ) > s M(x, x 2,, x n, x nk () ξ, ) < s}, hen we find ha (B C) A heefoe A c (B c C c ). Thus we have {k I : N(x, x 2,, x n, x nk x nk (), ) ɛ M(x, x 2,, x n, x nk x nk (), ) ɛ} {k I : N(x, x 2,, x n, x nk ξ, ) s M(x, x 2,, x n, x nk ξ, ) s} + {k I : N(x, x 2,, x n, x nk () ξ, ) s M(x, x 2,, x n, x nk () ξ, ) s}. Since S(N,M) θ x n k = ξ (N, M) n k () = ξ i follows ha {x nk } is S(N,M) θ - Cauchy. Convesely, suppose ha {x nk } is a lacunay-θ-saisically Cauchy sequence wih espec o he i-f-n-nom (N, M). By definiion hee is a subsequence {x nk () } of he sequence {x nk }, such ha k () I fo eac, (N, M) = ξ fo each ɛ > 0 > 0, (3.4) x n k () {k I : N(x, x 2,, x n, x nk x nk (), ) ɛ As befoe we have he following inequaliy M(x, x 2,, x n, x nk x nk (), ) ɛ} = 0. {k I : N(x, x 2,, x n, x nk ξ, ) ɛ M(x, x 2,, x n, x nk ξ, ) ɛ} {k I : N(x, x 2,, x n, x nk x nk (), ) s M(x, x 2,, x n, x nk x nk () ξ, ) s} + {k I : N(x, x 2,, x n, x nk () ξ, ) s M(x, x 2,, x n, x nk () ξ, ) s}. Since (N, M) x n k () = ξ i follows fom (3.4) ha x nk ξ(s θ (N,M) ). Acknowledgemens. The auhos ae hankful o he eviewes fo hei valuable commens suggesions owads he qualiy impovemen of he pape. 30

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