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(1)Kuwait J. Sci. 41 73: ) 84+30 508/. On pointwise statistical convergence of order of sequences of fuzzy mappings   

(2)    

(3)        3. 33. 33. Department of Mathematics, F{rat University, 23119 Elaz{g, Turkey, e-mail: [email protected] 33 Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, Garchuk, Guwahati-781035, Assam India. E-mail: [email protected], [email protected] 3. ABSTRACT          !   " #$  !  ! %  ! !&&' ($) *(  $#    ! + ""'  ' %  ! %  ! !&&' ($  #    %#"    "

(4) #$  !  ! %  ! !&&' ($) " ( " , S F†+ " #$   $ wp F†+((,"'  $#) Keywords: -  ((,"'.   " #$ . %  !. !&&' ($). AMS Subject Classi®cation: /001 /0 01 /2/1 0345). INTRODUCTION.   !  " #$   $# ,' 6'$( 78949:   ;  !  ($ ,"  <  8931)    !  " #$     ,' = 78918:  * 78918:  "   ,' = ,$ 78919: "') #  '   > (  " #$   ,     ' ! * "' $ ' (, '  ' $(  ? '     )     ! #$ !(  %     ! #  "?  =((,"' ' ,'  789@@:.  A ' 75008:.  75003:.  et al. 75002:. *' 789@1:. BC ? A BC $C  75005:. BC $C  et al) 7500/:. D E? 75088:.  A ' 7899/:. =" 789@0:. ' 78994:  (' )  F$ "   " #$    # ,    "  ("F %  , "E et al. 750025004:. "E et al. 75004:. "E? et al. 75009:. $ 75000:. "? et al. 75009:. BC ? et al..

(5) 18. Mikail Et Binod Chandra Tripathy and Amar Joyti Dutta. 75009:. " A D E 75003:. " A D  75009:. ' A =( 75088:  ' A  750025004: F    "'  %  ! !&&' (,)           !   " #$  !  ! %  ! !&&' ($)    5  $#  ,! ## ,  " #$  pÿ -  ((,"'  !&&' (,)    3  $#    !   " #$  !      ÿ ""'  ' %  ! %  ! !&&' ($  #    %#"    " #$  !  ! %  ! !&&'

(6) ($) < " ," (  ". " , wp F†  S F†  , S F†  S F†) DEFINITIONS AND PRELIMINARIES.  ; !  " #$   $ pÿ -  #$  !  %  ! " (,      " "' !    !"" > " ! #"(    ;  )    #    ;  , ("' "     $"   %#" ! , % )   !  " #$     ' ! , !   N ! " (,)  ' !  , E ! N  ; ,' 8 Xn E k† provided the limit exists  E† ˆ lim n!1. n kˆ8.  E     !  ! E)   "  ' ; , ! N  & " '   Ec† ˆ 8 ÿ  E†)  ÿ density !  , E ! N  ; ,' D "? 75080:)  , " (,    0 <  8)  ÿ density !  , E ! N  ; ,' 8  E† ˆ lim jfk  n : k 2 Egj; provided the limit exists; n. n.  jfk  n : k 2 Egj   (, ! "( ! E  F $ n: ! x ˆ xk†   %     xk ; ' P k† ! "( "" k F    ! ÿ density &   '  xk ; ' P k† ! Halmost all k according to H   ,,#  ,' H a:a:k † H)   "  ' ; , ! N  & ÿ density   Ec† ˆ 8 ÿ  E†   " ! 0 < < 8  $"  %"' " "' ! ˆ 8:    ÿ density ! '      " ' !     ˆ 8:.

(7) On pointwise statistical convergence of order ofsequences of fuzzy mappings. 19.   !  " #$  !  %  ! (,  $# ,' BI# A  75005:  !   " #$  !   $ pÿ -  ((,"' !   ,' D "? 75080:) *&&'         (' ,  X ! "( ! )  "      "( x 2 X  $  ((, $ u x† ?$ #"  ‰0; 8Š  u x† ˆ 0 $  ((, 0 < u x† < 8  " ((,  u x† ˆ 8  !"" ((,)  $  6 78921:  !&&' , ! X   (' , f x; u x†† : x 2 Xg ! X 2 ‰0; 8Š ! ( !  u : X ! ‰0; 8Š)  !  u "!  !  !  !&&' )  C Rn†   !("' ! "" (' (  #F , ! Rn: n ! ;

(8) 2 R  A; B 2 C R †  A ‡ B† ˆ A ‡ B;

(9) †A ˆ

(10) A†; 8A ˆ A  ! ;

(11)  0;  ‡

(12) †A ˆ A ‡

(13) A:    , A  B  ; ,' 

(14) > ( 1 A; B† ˆ maxfsup inf ka ÿ bk ; sup inf ka ÿ bkg; b2B a2A. a2A b2B. n k : k   "  " (  Rn:   "" ?  C R †;  †   (" (  ).  L Rn† ˆ fu : Rn ! ‰0; 8Š : u satisfies i† ÿ iv† belowg;  i† u  ("    F  x0 2 Rn    u x0 † ˆ 8; ii† u  !&&' #F   ! x; y 2 Rn  0    8; u x ‡ 8 ÿ †y† 1.  min‰u x†; u y†Š;. iii† u   ( . iv†  " ! fx 2 Rn : u x† > 0g;  ,' ‰uŠ0;  ( ) ! u 2 L Rn†;  u  ""  !&&' (,  L Rn†    ,  !&&'. (,  ) * 0 <  8  +"#"  ‰uŠ  ; ,' ‰uŠ ˆ fx 2 Rn. : u x†  g.  !( i† ÿ iv†  !""   +"#"  ‰uŠ 2 C Rn†:.

(15) 20. Mikail Et Binod Chandra Tripathy and Amar Joyti Dutta. =( (  ! ÿ"#"   ;  !""J  u; v 2 L R†   ÿ"#"  , ‰uŠ ˆ 2u 8 ; u 5 3 ; ‰vŠ ˆ 2v 8 ; v 5 3 ; 2 ‰0; 8Š :   #. 2u8 v8 ; u5 v5 3 28u28 v5 ; u35 v8 3 < ku8 ; ku5 ; if k 0 : : 2ku5 ; ku8 3 ; otherwise:. ‰u ‡ v Š ˆ. ‡ . ‰u ÿ v Š ˆ. ÿ ÿ . ‰kuŠ ˆ. ; !   8  q < 1;. dq u; v† ˆ.  d. 0. . 0Z8 @  ‰. ‡ . 1. 18 A. q ‰uŠ ; ‰vŠ †Š d. =q. u; v† ˆ sup 1 ‰uŠ ; ‰vŠ †;  1.  

(16) > ( ) ""' d u; v† ˆ qlim dq u; v†  dq  ds ! q  s; ( A " 78990:. ?(?( A  75003:) * (" '   $   d ""    dq  8  q  1:  !&&' ($ X   ($ !(   T  Rn†    ! "" !&&' (,)  %  ! !&&' ($   !   (    ! # $   $    ! !&&' ($) <   %  ! !&&' ($ ,' Xk†: ! Xk†   %  ! !&&' ($  Xk t††   %  ! !&&' (, ! #' t 2 T: $   (, t   ( !   ! ( !  %  ! !&&' ($ Xk † ;    %  ! !&&' (, Xk t†† : ! Xk t†† #$ !   (, t    T   $ limk Xk t† ˆ X t† ;   '  Xk† #$   X  T "? 789@4:) 1 . 1. 0 8. !1. MAIN RESULTS.          !   " #$  !      ! ÿ ""'  ' %  ! %  ! !&&' ($) < $#  " ,   " #$  !     " #$  ! 

(17) 

(18) † ! %  ! !&&' ($  " , $ pÿ -  ((,"' !   $ pÿ -  ((,"' ! 

(19) 

(20) †   " ,.

(21) On pointwise statistical convergence of order ofsequences of fuzzy mappings. 21. $ pÿ -  ((,"' !     " #$  ! 

(22) 

(23) † %  ! !&&' ($) ! $#$   " "  "" $#   ;) De®nition 3.1  0 <  8 , $#)  %  ! !&&' ($ Xk†    ,   ""' #$ !    +  ""' #$  !&&' (, X    T ! ! #' " > 0; lim 8 jfk  n : d Xk t†; X t††  " for every t 2 Tg j ˆ 0; n. n. i:e: ! #' t2 T;. 8      S ÿ lim Xk t† ˆ X t†  T:  (  ! #'  > 0;    $ N    8 jfk  n : d Xk t†; X t††  " for every t 2 Tg j < ; d Xk t†; X t†† < " for a:a:k † :. †. n. ! "" n > N ˆ N "; ; t††  !   " > 0:   "  !  %"' 78: " ! "" , ;"' (' k;  lim Xk t† ˆ X t†  T:   ,   lim Xk t† ˆ X t† (" S ÿ lim Xk t† ˆ X t†  T:   ! ""   ""' #$ %  ! !&&' ($ !  "" ,  ,' S F† : * ˆ 8;  ""  S F†  ! S F†     "  X ˆ 0;  ""  S0 F†  ! S F†: De®nition 3.2  , ' " (,    0 <  8  " Xk† ,  %  ! !&&' ($    T:  %  Xk†    ""'  ' %  !  ;  +  ""'  ' %  #  ! #' " > 0  F  (, N ˆ N "; x††    d Xk t†; XN t†† < " for a:a:k † and every t 2 T. ) lim. 8 jfk  n. n!1 n. : d Xk t†; XN t††  " for every t 2 Tg j ˆ 0:. De®nition 3.3  !&&' %    E F†  ( d    , (" 7 ": ! Xk† 2 E F†  Yk†     d Yk; 0†  d Xk; 0† (". Yk † 2 E F†:.

(24) 22. Mikail Et Binod Chandra Tripathy and Amar Joyti Dutta. De®nition 3.4  !&&' %    E F†    , ( ! X ˆ Xk† 2 E F† (" :X  E F†     " ! "" %  !. &  )       (  ) Remark 1. *(  ,# ;  !""  !  !&&' %    E F†  "    () De®nition 3.5  !&&' %    E F†    , '(( ! Xk † 2 E F† (" X k † 2 E F†     ( ! N: De®nition 3.6  !&&' %    E F†    , #$  ! ! Xk † 2 E F† (" Yk † 2 E F†  Yk ˆ 0 # Xk ˆ 0;   0 t† ˆ 8; for t ˆ 0; 0; 0; :::; 0† : 0; otherwise Theorem 3.7  0 <  8  Xk †  Yk † , %  ! !&&' ($: i† ! S ÿ lim Xk t† ˆ X0 t†  c 2 R;  S ÿ lim cXk t† ˆ cX0 t†; ii† ! S ÿ lim Xk t† ˆ X0 t†  S ÿ lim Yk t† ˆ Y0 t†;  S ÿ"( Xk t† †. ‡Yk t†† ˆ X0 t† ‡ Y0 t† :. Proof.  ! . "   c ˆ 0: =  c 6ˆ 0   ! ! i† !"" !(  !""$ %"' 8 jfk  n : d cXk t†; cX0 t††  "gj n

(25)

(26)

(27) 

(28)  8 "

(29)

(30)  k  n : d Xk t†; X0 t††  :. n. ii†.

(31). =  S ÿ lim Xk t† ˆ X0 t†  $" %"'  $. jcj.

(32). S ÿ lim Yk t† ˆ Y0 t† :. d Xk t† ‡ Yk t†; X0 t† ‡ Y0 t††  d Xk t†; X0 t†† ‡ d Yk t†; Y0 t†† :. '. ! $# " > 0  # 8 j fk  n : d Xk t† ‡ Yk t†; X0 t† ‡ Y0 t††  "g j n

(33) n 8

(34) "o

(35)

(36) 8

(37)

(38) n " o

(39)

(40) 

(41) k  n : d Xk t†; X0 t†† 

(42) ‡

(43) k  n : d Yk t†; Y0 t††  n 5 n 5

(44) :

(45)   S ÿ"( Xk t† ‡ Yk t†† ˆ X0 t† ‡ Y0 t† : Theorem 3.8  Xk† ,  %  ! !&&' ($ ;    T: .

(46) On pointwise statistical convergence of order ofsequences of fuzzy mappings. 23. !""$ (  %#"J i† Xk †    ÿ ""' #$ %   T; ii† Xk †   ÿ ""'  ' %   T; iii† Xk †   %  ! !&&' ($ !       #$ %  ! !&&' ($ Yk†    Xk t† ˆ Yk t† ! a:a:k †  ! #' t 2 T: Proof. i ) ii†"=  S ÿ"(Xk t† ˆ X t†  T  " " > 0) " d Xk t†; X t†† < ! a:a:k †  ! N     d XN t†; X t†† < 5;   # 5 d Xk t†; XN t††  d Xk t†; X t†† ‡ d XN t†; X t†† < 5" ‡ 5" for a:a:k †. ! #' t 2 T:

(47)   Xk t††   ÿ ""'  ' % ) ii ) iii† F (  Xk t†† , ÿ ""'  '    " ,"" B ˆ BÿXN 8 t†; 81  Xk t† ! a:a:k † ! ( # (, N 8†  ! #' t 2 T: " "' '   M    8  B ˆ B XM t†; 5  Xk t† ! a:a:k †  ! #' t 2 T:   "  B8 ˆ B T B  Xk t† ! a:a:k †  ! #' t 2 T: ! B8   "  ! ( "   %"  8   Xk t† ! a:a:k † 8 ! #' t 2 T:     ,' $ N 5†   B ˆ B XN 5 tT†; /  Xk t† ! a:a:k †  ,'   $ $( B5 ˆ B8 B  X8 k t† ! a:a:k †  ! #' t 2 T;  B5  ( "   %"  5: $        %  fBmgm 8 ! " ,""    !   m; Bm  Bm 8;  ( ! Bm   $  8  X t† 2 B ! a:a:k †  ! #' t 2 T: '   ! "  k m 5m 8 T (  # m 8 Bm 6ˆ  T F "'  "() =    !&&' ($ X t†    X t† 2 m 8 Bm ! #' t 2 T: $  !   Xk t† 2 Bm ! a:a:k †  ! #' t 2 T;     $ # $ %  fHmgm 8    lim 8 jfk  n : Xk t†2= Bm for every t 2 Tg j < 8 if n > Hm : 5† †. 0. 0. 00. †. 00. 1 ˆ. ‡. ÿ. 1 ˆ. 1 ˆ. 1 ˆ. n. n. m.      ,%  Zk t†† ! Xk t†† $ ! ( ! Xk t†    ! Hm < k  Hm‡8  Xk t†2= Bm ! #' t 2 T  Xk t†   ( ! Zk t† :.

(48) 24. Mikail Et Binod Chandra Tripathy and Amar Joyti Dutta. F ;  %  Yk t†† ,' Yk t† ˆ. (Xt;. if Xk t† is a term of Zk t† Xk t†; otherwise; †. ! #' t 2 T:  klim Yk t† ˆ X t†  T; ! ! " > m8 > 0  k > Hm   Xk t†   ( ! Zk t†† ;   ( Yk t† ˆ X t8†  T  Yk t† ˆ Xk t† 2 Bm  T  d Yk t†; X t†† ( ! Bm  5m 8 ! #' t 2 T: < "   Xk t† ˆ Yk t† ! a:a:k †  ! #' t 2 T:  #!'   ,#  ! Hm < n  Hm 8  !1. ÿ. ‡. fk  n. : Yk t† 6ˆ Xk t† for every t 2 Tg  fk  n : Xk t†2= Bm for every t 2 Tg.  ,' 75:. 8 jfk  n : n 8 jfk  n  n. Yk t† 6ˆ Xk t† for every t 2 Tgj. : Xk t†2= Bm for every t 2 Tgj < m8.

(49)    "(  0  n ! 1  Xk t† ˆ Yk t† ! a:a:k †  ! #' t 2 T) ! ii† (" iii†) *""' (  iii† " ' Xk t† ˆ Yk t† ! a:a:k †  ! #'t 2 T;  klim Yk t† ˆ X t†  T)  " > 0  !   n; !1. fk  n . S. : d Xk t†; X t††  " for every t 2 Tg fk  n : Xk t† 6ˆ Yk t† for every t 2 Tg fk  n : d Yk t†; X t††  " for every t 2 Tg. =  klim Yk t† ˆ X t†  T;  "    ;F (, ! $ ' L ˆ L "; t† : ! lim 8 j fk  n : d Xk t†; X t††  " for every t 2 Tg j n n 8 j fk  n : Xk t† 6ˆ Yk t† for every t 2 Tg j ‡ lim L ˆ 0  lim !1. !1. n!1 n. n!1 n. ,  Xk t† ˆ Yk t† a:a:k † ! #' t 2 T:

(50)   d Xk t†; X t†† < " ! a:a:k †  ! #' t 2 T;  i† "  !  (")   (( %  ! ( 3)@  #  !""$ ") Corollary 3.9 ! Xk †   %  ! !&&' ($   .

(51) S ÿ lim Xk t† ˆ Xk  T; lim Xk i† t† ˆ X t†  T:. 25.  Xk†   ,%  ÿXk i t†1   . On pointwise statistical convergence of order ofsequences of fuzzy mappings †. Theorem 3.10  0 < 

(52)  8 ;  S F†  S

(53) F†) Proof. ! 0 < 

(54)  8; . 8 j fk  n : d Xk t†; X t††  " for every t 2 Tg j n

(55) 8 j fk  n : d Xk t†; X t††  " for every t 2 Tg j  n. ! #' " > 0   $#  S F†  S

(56) F†:  !""$ "   %  !  ,# () Corollary 3.11 !  %  ! !&&' ($ Xk†   ""' #$ !  ;  X; ! ( 0 <  8;     ""' #$  X: De®nition 3.12  , ' " (,    0 <  8  " p ,  # " (,)  %  ! !&&' ($ Xk†    , $"' pÿ -  ((," !  !    !&&'+#" !  X    n X 8 d Xk t†; X t†††p ˆ 0: lim n!1 n. kˆ8.      w lim Xk t† ˆ X t†  T:   ! "" $"' pÿ -  ((," %  ! !&&' ($ !  "" ,  ,' w p F†:    "  X ˆ 0;  ""  w p0 F†  ! w p F†: Theorem 3.13  0 < 

(57)  8  p ,  # " (,  w p F†  w

(58) p F† : Proof.   %  Xk † , $"' pÿ -  ((," !  )  $# 

(59)    0 < 

(60)  8   # " (, p  ÿ p. (' . 8 Xn d Xk t†; X t†††p 8 Xn d Xk t†; X t†††p. n

(61). kˆ8. n. kˆ8.   $#  w p F†  w

(62) p F†) Corollary 3.14  0 < 

(63)  8  p ,  # " (,)  i† ! ˆ

(64) ;  w p F† ˆ w

(65) p F†; ii† w p F†  wp F† !   2 0; 8Š  0 < p < 1:.

(66) 26. Mikail Et Binod Chandra Tripathy and Amar Joyti Dutta. Theorem 3.15  0 <  8  0 < p < q < 1:  w q F†  w p F†: Proof. () Theorem 3.16  

(67) , ;F " (,    0 < 

(68)  8 . 0 < p < 1) !  %  ! !&&' ($ Xk†  $"' pÿ -  ((," !  ;  X;     ""' #$ ! 

(69) ;  X: Proof. * ' %  ! !  Xk † ;  T;   . XdX t;Xt n. kˆ8. k †. †††p  j fk  n. : d Xk t†; X t††p  " for every t 2 Tg j "p.    8 Xn d Xk t†; X t†††p 8 j fk  n n. kˆ8. 8. 

(70) j fk  n n. n. : d Xk t†; X t††p  " for every t 2 Tg j "p. : d Xk t†; X t††p  " for every t 2 Tg j "p.  ,  ;F " (,    0 <  8  0 < p < 1) !  %  ! !&&' ($ Xk†  $"' pÿ -  ((," !  ;  X;     ""' #$ !  ;  X: Theorem 3.18 i†    S 0 F†  w p0 F†  "      () ii†    S F†  w p F†   "  () Proof. i† < "" # "' ! S 0 F†     ,  (""')  Xk† ,  %  ! !&&' ($  S 0 F† :  d Yk t†; 0†  d Xk t†; 0† ! "" k 2 N)  ! !"" !(  !""$  "J fk 2 N : d Xk t†; 0†  "g  fk 2 N : d Yk t†; 0†  "g:   !  ! !"" !(  (? 8) ii†  ! !"" !(  !""$ F(") Example 1  ˆ 8    %  Xk†  Yk† ;  !""J  8; 8; 8; :::; 8† Xk t† ˆ 08;; for t ˆotherwise Corollary 3.17.

(71) 8> 8; >< 0; >>: 8; 0;. On pointwise statistical convergence of order ofsequences of fuzzy mappings. Yk t† ˆ. for all k odd and t ˆ ÿ8; ÿ8; ÿ8; :::; ÿ8† for all k odd and t 6ˆ ÿ8; ÿ8; ÿ8; :::; ÿ8† for all k even and t ˆ 8; 8; 8; :::; 8† for all k even and t 6ˆ 8; 8; 8; :::; 8†. 27.  Xk† ,"$  S F† , Yk†   ,"$)

(72)      S F†   ") Example 2  ˆ 8    %  fXk g ; ,' Xk ˆ 8; for all k 2 N:   Jth    ZJ ;  Yk† 2 ZJ ) Yk ˆ Xk ! "" kˆ 5i ‡ 8;i 2 N  Yk ˆ 0 )  fXkg 2 S F†; , Yk†   ,"$  S F†: Theorem 3.19    S 0 F† ; w p0 F† ; S F†  w p F†   '(( ) Proof .  ! !"" !(  !""$ F(") Example 3  ˆ 8    %  Xk† ; ,' 8> 8; k ˆ i3; i 2 N and t ˆ 8; 8; 8; :::; 8† > < 3 Xk t† ˆ > 0; k ˆ i3 ; i 2 N and t 6ˆ 8; 8; 8; :::; 8† and t ˆ 0; 0; 0; :::; 0† >: 08;; kk 6ˆ6ˆ ii3;; forfor any any and t 6ˆ 0; 0; 0; :::; 0†   $ %  Yk† ! Xk† ;  !"" J Yk † ˆ X8 ;. X5; X@ ; X3 ; X54; X/ ; X2/; X1 ; X851 ; X2; X582; X4; X3/3 ; X9 ; :::†:.     S F† ; , Yk†   ,"$  S F† ;  Xk† ,"$.      S F†   '(( )    ,  (""') REFERENCES Altõn Y., Et, M. & CËolak, R. 2006.  '  "  " ' $"' #$  ! $"& >  %  ! !&&' (,) (  (   "  5272+4:J 8088+8050) Altõn, Y., Et, M. & Tripathy, B. C. 2007.    " #$  ! %  ! !&&' ($) " ! *&&' (  1575:J /51+133) Altõn, Y., Et, M. & Basarõr, M. 2007)  ( $"& >  %  ! !&&' (,)  " ! =    $$ 3478:J 8+8/) Altõnok, H., CËolak, R. & Et, M. 2009. +>  %    ! !&&' (,) *&&' =  ='( 160758:J 385@+3839).

(73) 28. Mikail Et Binod Chandra Tripathy and Amar Joyti Dutta. Burgin, M. 2000. ' ! !&&' "() *&&' =  ='( 115: /33+//3) CËolak, R., Altõnok, H. & Et, M. 2009. B"& >  %  ! !&&' (,)  ="  * " 4073:J 8802+8884) CËolak, R. 2010. = " #$  !  :    "'.   "   " J (' ," 858+859) Connor, J. S. 1988.  = "  $ pÿ -  #$  ! % ) "' 8: /4+23) Diamond, P. & Kloeden, P. 1990.    ! !&&' ) *&&' =  ='( 3575:J 5/8+5/9) Et, M. & Nuray, F. 2001. 1m ÿ " #$ )  " !   " (  3272:J 928+929) Et, M. 2003. =$"' "( ((," >  %  !  m ; ,'  (") = = ( ( (

(74) $  407/:J /23+ /42) Et, M., Altõn, Y., Choudhary B. & Tripathy B. C. 2006.  ( " ! %  ; ,' %  ! " & ! ) ( " %"  "  975:J 331+3/5) Fast, H. 1951. = " #$  %) ""%( ( ( 2J 5/8+5//) Fridy, J. A. 1985.   " #$ ) "' 5J 308+383) Gadjiev, A. D. & Orhan, C. 2002. =( F( ( #  " #$ )  ?'  " ! (  3278:J 859+83@) GoÈ khan, A., Et, M. & Mursaleen, M. 2009. "( " '  "  $"' "( " ' #$  ! %  ! !&&' (,) ( "  ( ""$ 4973+/:J 1/@+111) GoÈ khan, A. & GuÈ ngoÈ r, M. 2002.    " #$ )  " !   " (  3379:J 8349+83@/) GuÈ ngoÈ r, M., Et, M. & Altõn, Y. 2004. =$"' V ; ; q†ÿ((," %  ; ,' " & ! ) " (   ( 15775:J 128+148) IsË õk, M. 2011. =$"' "( w; ; q†ÿ((," % ) (  ="#  6171:J 449+4@@) Lakshmikantham, V. & Mohapatra, R. N. 2003. ' ! *&&' >" %   " '"  * )  ?) Matloka, M. 1987. *&&' ($+%   ) =* 30J 8@+51) Mursaleen, M. & BasË arõr, M. 2003.  (  %    ! !&&' (,)  " !   " (  34J 8318+8314).

(75) On pointwise statistical convergence of order ofsequences of fuzzy mappings. Rath, D. & Tripathy, B. C. 1994.   ""'. 29. #$   ""'  ' % )  " !   " (  257/:J 3@8+3@2) SÆla t, T. 1980.   ""' #$ %  ! " (,) (  ="#  30J 839+810) Schoenberg, I. J. 1959)  $,"' !  !   " ((,"' () (  ( " "' 66J 328+341) Steinhaus, H. 1951. = " #$    " #$  '(%) ""%( ( ( 2J 43+4/) Talo, OÈ. & BasË ar, F. 2009. ( !  " ! " "  ! %  ! !&&' (,  " (F !() (  (   "  587/:J 484+433) Tripathy, B. C. & Sarma, B. 2011. =( ," %    ! !&&' (, ; ,' " & ! )   (  =  =  $"  3178:J 83/+8/0) Tripathy, B. C. 1997. F !( , ( " ! % ) " ! ( " "'  "  20675:J //@+/10) Tripathy, B. C. & Dutta, A. J. 2007.  !&&' "+#" ," %    5 `pF ) ( "  ( ""$ 4679+80:J 859/+8599) Tripathy, B. C. & Dutta, A. J. 2006.  # ," %    ! !&&' (,) (  (   "  5975:J 8038+8034) Zadeh, L. A. 1965. *&&' ) !(  " 8J 33@+313) Zygmund, A. 1979. $( =) (,$ #' ) (,$ ) Submitted : Revised : Accepted :. 11/06/2013 27/12/2013 24/03/2014.

(76) 30. On pointwise statistical convergence of order ofsequences of fuzzy mappings. + !,. k. 9 at. 9@hO. 781035 -. , *( A. F Q9}f. **. h. F9ZI. , *. G HQ9tAyG. !. C9= Q@ GQGO 9V O. ("+. <. *. ,. ,. * +. *. @ jE { 9w |. - H R; G 23119 - JGQqyG ?g|9F - J9 \9 QyG ~Ts* G F - 9 F y wAyGh e zgyG p ?|OtA}yG J9SGQOyG O g| - ? \9 QyG e 9+vQ@. *. +. ( (". ,@9$ (. *. +. (. ,. *. %. +. *. G ?=gV**. (zgy. ?Y;L. Ph ?=@Q}yG | F9ZI G HQ9tAyG e q| IQ| dh h ,EJ=yG GP p eOt vV vv ?v vy9vAvAv| e v vqv| vavgv ,xvyP vzvf Ih;vfh .?vV vWv}vyG J9vtv v=vavAvyG J9v vy9vAvAv}vy k k ^g< xyPv Xs9 9}v .9 at h 9 F9ZIE ?<Q9tA}yG ? y9AA}zy ?8p9w| 9 C B=D h ? F9ZI G .? tyG ? g }GAyGh F9ZI G HQ9tAyG < J9s;gyG xy. ,. #. (. +. (% "!. ,. (%. ,. !. +. }. ). !. (. +. +. *(. $ ,. + +. +. %!. ,. !. +. !. #+. +.

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