2016 xó ADVANCES IN MATHEMATICS(CHINA) xxx., 2016
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1 µ45 µx ½ Ù Vol.45, No.x 206 xó ADVANCES IN MATHEMATICS(CHINA) xxx., 206 doi: 0.845/sxjz.2050b ²Â» µ ¼ Ulam È ( Ų¼ Ò¼ Ã,,, ) : ÉÐ Ì Õ ÎÏÓ, ÊÔ Í - Í Ë 6f(x+y) 6f(x y)+4f(3y) = 3f(x+2y) 3f(x 2y)+9f(2y) Î Ulam ÑÏÒ. : Õ ; Í - Í Ë; Ulam ÑÏÒ MR(200) Æ : 39B72; 47H5??? / Å : O77.3 ƽ : A À : (206)0x-0xxx-08 0 ÝÅ»ÓÍ 2] ÖÒ Hilbert ÓÍ «ÈÎÓÍ ÌÎ, Õ Ý»ÓÍ, ÓÍ Æ ÓÍ ÛÝÄ«Í ÓÍ, Â Ü ØĐ, ÓÍØĐÕ»ØĐ ³ 2]. 940, Ulam 4] Þ»½ ÎÁ, Þ Î, Hyers 4] Õ 94 Þ Banach ÓÍ ¾ÑÊ ÎÁ. 978, Rassias 0] Ü Î Ulam Î. Ö Þ Ulam Î, 3, 7, 9]. Lee ² 6] Þ Ý»ÓÍ ÑÊ»½ º»½ Hyers-Ulam-Rassias Î, Park 8] Þ Ý»ÓÍ Cauchy Ʋµ Cauchy ½ Î. Skof 3] º Ð Þº»½ f(x+y)+f(x y) = 2f(x)+2f(y) Ulam Î. º»½, º. Jun Kim 5] Þ»½ f(2x+y)+f(2x y) = 2f(x+y)+2f(x y)+2f(x) Ð Þ Î. ³,»½. Chang Jung ] Õ Ü Ú Â ÞÁ º»½ 6f(x+y)+6f(x y)+4f(3y) = 3f(x+2y) 3f(x 2y)+9f(2y) () Õ Banach ÓÍ ĐÞ Ulam Î. Õ Æ 6, 8] Â, Þ ÝÅ»ÓÍ, Õ ĐÞ»½ () Ulam Î. ¹ : ¹ : Ã Ê : Ų¼ ÒÒ Ã (No. 2 20). songai-min@63.com
2 2 ¼ Ø 44 ľ. ± X Õ C ÈÎÓÍ,» : X R : () x 0, x = 0 x = 0; (2) λx = λ x ; (3) x+y K( x + y ) (ŠƲµ), λ C, x,y X, K», Ö Å»», (X, ) Å»ÓÍ. (3) Ñ, 2n i= 2n x i K n x i, i= 2n+ i= 2n+ x i K n+ ß» n Æ x,x 2,,x 2n+ X Ú. ŠƲµ Ä K Å»», Å»ÓÍ Å Banach ÓÍ. Å»» - ÑÊÎ (0 ), Ö -»», Ç x,y X, x+y x + y, Ü ÕÄ Å Banach ÓÍ -Banach ÓÍ. Aoki-Rolewicz Ø ] Đ, Å»»²Ì -»». Ð -»» Đ Ê½Å, Å»» Đ Ç Õ -»». Ä È» «: M n (X) «X n n Ý Å ; e j M,n (C) «j Ý, Ð Ý 0; E i,j M,n (C) «(i,j) Ý, Ð Ý 0; E i,j x M,n (X) «(i,j) Ý, Ð Ý 0; θ M m (X) «Ñ X Ñ Ý..2 2] ± X ËÝÓÍ, (M n (X), n )»ÓÍ. () (X, n }) Ý»ÓÍ,»» : (i) x+θ n+m = x n, (ii) Bx n B x n, xb n B x n, ÜÙ x M n (X), B M n (C). (2) (X, n }) Ý Banach ÓÍ, X Banach ÓÍ (X, n }) Ý»ÓÍ. Ç M n,m (C) = B(C n,c m ), «C n C m, C n Hilbert ÓÍ. Å, A = a ij ] C n, A = A, ÜÙ A = ā ji ](Ï Æ 2])..3 ± X ËÝÓÍ, (M n (X), n ) Å»ÓÍ. () B M n (C) x M n (X), (i) x+θ n+m = x n, (ii) xb n B x n, Bx n B x n, Ö (X, n }) ÝÅ»ÓÍ. (2) (X, n }) ÝÅ»ÓÍ, X Å»ÓÍ (X, n }) ÝÅ»ÓÍ. ¾³, Â Ñ Ý -»»ÓÍ É. i= x i
3 x : Þ«¼ÔÎ ¼¾ ± Ulam À Ï 3 ± E,F ËÝÓÍ, h : E F, h n : M n (E) M n (F) h n (x ij ]) = h(x ij )], x ij ] M n (E).. ± (X, n }) Õ C Ý -»»ÓÍ, Ö () x kl X, E kl x kl n = x kl ; (2) x ij x ij ] n n i,j= x ij }, x ij ] M n (X), 0 < ; (3) x m = x mij ] x = x ij ] M k (X), lim m x m = x lim m x mij = x ij. ù () E kl x kl = e k x kle l, M n,m (C)»» Ñ E kk = E kk Ekk = E kk 2, Ñ E kk =, ¹ e k e k = E kk = e k 2, Ç e k = e k =, ¹ ÝÅ»ÓÍ E kl x kl n e k x kl e l = x kl, e k (E kl x kl )e l = x kl, Ö x kl e k E kl x kl n e l = E kl x kl n x kl, ¹ E kl x kl n = x kl. (2) x = x ij ] M n (X),e k xe l = x kl, e l = e k =, ¹ x kl x ij ] n, x ij ] = n i,j= E ij x ij, ¹Ñ x ij ] n = n i,j= E ij x ij n n i,j= x ij }. (3) x mij x ij x mij x ij ] n = x mij ] x ij ] n n i,j= x mij x ij }, ¹ Ð. 2 () ±Áº Ulam Ë X Ý»ÓÍ, Y -Banach ÝÓÍ, f : X Y, Df : X 2 Y Æ Df n : M n (X) 2 M n (Y). É Df(a,b) = 6f(a+b) 6f(a b)+4f(3b) 3f(a+2b)+3f(a 2b) 9f(2b); Df n (x ij ],y ij ]) = 6f n (x ij ]+y ij ]) 6f n (x ij ] y ij ])+4f n (3y ij ]) 3f n (x ij ]+2y ij ]) +3f n (x ij ] 2y ij ]) 9f n (2y ij ]), a,b X, x = x ij ], y = y ij ] M n (X). 2. ± ϕ : X 2 0, ) Φ (a,b) = ϕ(0,2 i b)+4ϕ(2 i a,2 i b)] 4 i <, a,b X, 0 <, (2) f : X Y Æ, f(0) = 0, x = x ij ], y = y ij ] M n (X), Df n (x ij ],y ij ]) n n ϕ(x ij,y ij ). (3) i,j= Ö Õ º Q : X Y x = x ij ] M n (X), f n (x ij ]) Q n (x ij ]) n K n Φ (x ij,x ij ) 2 i,j= ù Õ (3) (i,j) (s,t) ³, Ð x ij = 0, y ij = 0. Ð x st = 0, x st «y st, 4f(3x st ) 9f(2x st ) ϕ(0,x st ). Ð y st = x st, f(3x st )+3f(x st ) 3f(2x st ) ϕ(x st,x st ). ]. (4)
4 4 ¼ Ø 44 Ö ÜµÂ 3f(2x st ) 2f(x st ) = 4f(3x st ) 9f(2x st ) 4f(3x st ) 2f(x st )+2f(2x st ) K4 f(3x st )+3f(x st ) 3f(2x st ) + 4f(3x st ) 9f(2x st ) ] K4ϕ(x st,x st )+ϕ(0,x st )], 4 f(2x st) f(x st ) K 2 4ϕ(x st,x st )+ϕ(0,x st )]. Õ µ, 2 l x st x st, Ü 4 l, 4 l+f(2l+ x st ) 4 lf(2l x st ) K 4ϕ(2 l x st,2 l x st )+ϕ(0,2 l ] x st ) 2 4 l, ¹ ß» m < l, 4 l+f(2l+ x st ) 4 mf(2m x st ) l 4 i+f(2i+ x st ) 4 if(2i x st ) l K 4ϕ(2 i x st,2 i x st )+ϕ(0,2 i ] x st ) 2 4 i, i=m i=m (2) Æ µñ f(2 l x 4 l st )} Y Cauchy ß, Y. ¹ f(2 l x 4 l st )} Û. Ð Q : X Y, Q( = lim l 4 f(2 l, a X. Õ (5) Ð m = 0,l, a À x l st, Q( f( K 4ϕ(2 i a,2 i +ϕ(0,2 i ] } 2 4 i, (6) f Æ», Q Ñ Q( = Q(, (2) Ñ Ñ (3) ¹Ñ 4ϕ(2 i a,2 i b)+ϕ(0,2 i b)] lim i 4 i = 0, ϕ(2 i a,2 i b) lim i 4 i = 0; Df(2 i a,2 i b) ϕ(2 i a,2 i b)), DQ(a,b) = lim l 4 l Df(2l a,2 l b) lim l 4 lϕ(2l a,2 l b) = 0, 3, Ø 2.] Ñ Q º. ÄÐ Q Î, Õ Q : X Y º, Ʋµ (6), Ö a X Q( Q ( = lim lim lim 4 k Q(2k Q (2 k 4 k Q(2k f(2 k + f(2 k Q (2 k ] 2 K 4ϕ(2 i+k a,2 i+k +ϕ(0,2 i+k ] i+k = 0. (5)
5 x : Þ«¼ÔÎ ¼¾ ± Ulam À Ï 5 ¹ Q Ʋµ (6) º. Ø. ÆƲµ (6),  f n (x ij ]) Q n (x ij ]) n n K 2 i,j= n f(x ij ) Q(x ij ) ] i,j= = K n Φ (x ij,x ij ) 2 i,j= x = x ij ] M n (X) Ú, Ø Ð. 2.2 ± ϕ : X 2 0, ) Φ 2 (a,b) = ϕ(0,2 i b)+4ϕ(2 i a,2 i b)] ] } 4ϕ(2 m x ij,2 m x ij )+ϕ(0,2 m x ij ) 4 m ] 8 i <, a,b X, 0 <, (7) f : X Y, f(0) = 0, x = x ij ], y = y ij ] M n (X), Df n (x ij ],y ij ]) n n ϕ(x ij,y ij ). (8) i,j= Ö Õ C : X Y x = x ij ] M n (X), Ö f n (x ij ]) C n (x ij ]) n K n Φ 2 (x ij,x ij ) 24 i,j= ù Õ (8) (i,j) (s,t) ³, Ð x ij = 0, y ij = 0. Ð x st = 0, x st «y st, 2f(x st )+4f(3x st ) 5f(2x st ) ϕ(0,x st ). Ð y st = x st, 3f(x st ) f(3x st )+3f(2x st ) ϕ(x st,x st ). ܵ 24f(x st ) 3f(2x st ) ]. (9) K 2f(x st )+4f(3x st ) 5f(2x st ) +4 3f(x st ) f(3x st )+3f(2x st ) ] K4ϕ(x st,x st )+ϕ(0,x st )], 8 f(2x st) f(x st ) K 24 4ϕ(x st,x st )+ϕ(0,x st )]. Õ µ, 2 l x st x st, Ü 8 l, 8 l+f(2l+ x st ) 8 lf(2l x st ) K 4ϕ(2 l x st,2 l x st )+ϕ(0,2 l ] x st ) 24 8 l,
6 6 ¼ Ø 44 ¹ ß» m < l, 8 l+f(2l+ x st ) 8 mf(2m x st ) l 8 i+f(2i+ x st ) 8 if(2i x st ) l K 4ϕ(2 i x st,2 i x st )+ϕ(0,2 i ] x st ) 24 8 i, i=m i=m (0) (7) Æ µñ 8 l f(2 l x st )} Y Cauchy ß, Y. ¹ 8 l f(2 l x st )} Û. Ð C : X Y, C( = lim l 8 l f(2 l, a X. Õ (0) Ð m = 0,l, a À x st, C( f( K 4ϕ(2 i a,2 i +ϕ(0,2 i ] } 24 8 i, () f», C Ñ C( = C(, Ô (7) (8) DC(a,b) = lim l 8 l Df(2l a,2 l b) lim l 8 lϕ(2l a,2 l b) = 0, 3, Ø 2.2] Ñ C. ÄÐ C Î, Õ C : X Y, Ʋµ (), Ö a X, C( C ( = lim lim lim 8 k C(2k C (2 k 8 k C(2k f(2 k + f(2 k C (2 k ] 2 K 4ϕ(2 i+k a,2 i+k +ϕ(0,2 i+k ] i+k = 0, ¹ C Ʋµ (). Ø. ÆƲµ (), f n (x ij ]) C n (x ij ]) n n i,j= n f(x ij ) C(x ij ) ] 4ϕ(2 m x ij,2 m x ij )+ϕ(0,2 m ] } x ij ) K 24 i,j= = K n Φ 2 (x ij,x ij ) 24 i,j= ]. 8 m x = x ij ] M n (X) Ú, Ø Ð. 2.3 ± ϕ : X 2 0, ) Φ (a,b) = ϕ(0,2 i b)+4ϕ(2 i a,2 i b)] 4 i <, a,b X, 0 <, (2)
7 x : Þ«¼ÔÎ ¼¾ ± Ulam À Ï 7 f : X Y f(0) = 0, x = x ij ],y = y ij ] M n (X), Df n (x ij ],y ij ]) n n ϕ(x ij,y ij ), (3) Ö Õ º Q : X Y Æ C : X Y x = x ij ] M n (X), n K 3 4 ϕ(2 m x ij,2 m x ij )+ ϕ(0,2 m ] } x ij ) f n (x ij ]) Q n (x ij ]) C n (x ij ]) n 2 4 m. ϕ(a,b) = ϕ(a,b)+ϕ( a, b). i,j= i,j= ù Ð f e ( = 2 (f(+f( ), Ö f e(0) = 0, f e ( = f e (, Df e (a,b) = 2 Df(a,b)+ 2 Df( a, b) K 2 ϕ(a,b)+ϕ( a, b)], Ð f o ( = 2 (f( f( ), Ö f o(0) = 0, f o ( = f o (, Df o (a,b) = 2 Df(a,b) 2 Df( a, b) K 2 ϕ(a,b)+ϕ( a, b)]. Ø Ñ, Õ º Q Æ C f e ( Q( K2 4 ϕ(2 m a,2 m + ϕ(0,2 m ] } 24 4 m ; ¹ f o ( C( K ϕ(2 m a,2 m + ϕ(0,2 m 8 m ] }. f( Q( C( K f e ( Q( + f o ( C( ] K3 ( 4 ϕ(2 m a,2 m + ϕ(0,2 m 24 4 m ( 4 ϕ(2 m a,2 m + ϕ(0,2 m + ¹ Ø. Æ µñ K3 2 f n (x ij ]) Q n (x ij ]) C n (x ij ]) n 8 m ( 4 ϕ(2 m a,2 m + ϕ(0,2 m n i,j= n i,j= 4 m ) ] ) ] } f(x ij ) Q(x ij ) C(x ij ) ] ( K 3 2 ) ] }, (4) 4 ϕ(2 m x ij,2 m x ij )+ ϕ(0,2 m ) ] x ij ) 4 m
8 8 ¼ Ø 44 x = x ij ] M n (X) Ú, Ø Ð. ³ ] Chang, I.S. and Jung, Y.S., Stability of a functional equation deriving from cubic and quadratic functions, J. Math. Anal. Al., 2003, 283(2): ] Effros, E.G. and Ruan, Z.J., On aroximation roerties for oerator saces, Internat. J. Math., 990, (2): ] Eskandani, G.Z., Gavruta, P., Rassias, J.M. and Zarghami, R., Generalized Hyers-Ulam stability for a general mixed functional equation in quasi-β-normed saces, Mediterr. J. Math., 20, 8(3): ] Hyers, D.H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 94, 27(4): ] Jun, K.W. and Kim, H.M., The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Al., 2002, 274(2): ] Lee, J.R., Shin, D.Y. and Park, C.K., Hyers-Ulam stability of functional equations in matrix normed saces, J. Inequal. Al., 203, 203: Article ID 22. 7] Najati, A. and Ranjbari, A., Stability of homomorhisms for a 3D Cauchy-Jensen tye functional equation on C -ternary algebras, J. Math. Anal. Al., 2008, 34(): ] Park, C.K., Lee, J.R. and Shin, D.Y., Functional equations and inequalities in matrix aranormed saces, J. Inequal. Al., 203, 203: Article ID ] Poa, D., Hyers-Ulam-Rassias stability of a linear recurrence, J. Math. Anal. Al., 2005, 309(2): ] Rassias, T.M., On the stability of the linear maing in Banach saces, Proc. Amer. Math. Soc., 978, 72(2): ] Rolewicz, S., Metric Linear Saces, Warsaw: Polish Scientific Publishers PWN, ] Ruan, Z.J., Subsaces of C -algebras, J. Funct. Anal., 988, 76(): ] Skof, F., Prorieta locali e arossimazione di oeratori, Milan J. Math., 983, 53: 3-29 (in Italian). 4] Ulam, S.M, Problems in Modern Mathematics, New York, John Wiley & Sons, 940. The Ulam Stability Of Functional Equation On Matrix Quasi-normed Saces SONG Aimin (College of Mathematics, Gansu Normal University for Nationalities, Gannan, Gansu, , P. R. Chin Abstract: In this aer, we defined the matrix quasi-normed saces,and roved the Ulam stability of functional equation deriving from cubic and quadratic functions 6f(x+y) 6f(x y)+4f(3y) = 3f(x+2y) 3f(x 2y)+9f(2y). Keywords: matrix quasi-normed saces; functional equation deriving from cubic and quadratic function; Ulam Stability
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