Generalization of Quasi-Differentiable Maps

Transcription

1 Globl Journl of Mtheticl Sciences: Theory nd Prcticl. ISSN Volue 4, Nuber 3 (0), Interntionl Reserch Publiction House htt:// Generliztion of Qusi-Differentible Ms Sushil Kur Alied Science Dertent, Bhrti Vidhyeeth s College of Engineering, New Delhi-0063, Indi E-il: sushilkur6n@gil.co Abstrct The concet of usi differentibility ws introduced in 995 by A.Byoui nd this Differentibility is stronger thn Frechet differentibility. In this er, new concet of differentibility Wek Qusi Differentible Ms hs been introduced nd it s soe chrcteriztions like linerity, Lischitzin roerty chin rule nd etc. hve been derived. 000 Mthetics Subject Clssifiction: 46A6, 46E50 Keywords: Qusi differentible, Chin rule, Frechet Differentible Introduction Differentibility of function on nored sces is the ost iortnt concet in nlysis. Frechet nd Gteux differentibility of function on nored sces soe kind of differentibility [,, 3]. In 995, A. Byoui [4, 5] introduced concet of differentibility, is known s usi-differentibility in F-sces. Let E nd F be -nored sce nd -nored sce resectively (0 <, ) ndu, oen subset of E. A ing f : U F is sid to be -usi differentible or differentible t U, if there exists liner T L( E, F), such tht f( x) f( ) T ( x ) li = 0 x (.) T is clled -differentil or usi differentil of function f t oint If =

2 50 Sushil Kur Then this ing is known s suer differentible ing t. i.e. for every ε> 0, there exists δ> 0 such tht 0 < x <δ f( x) f( ) T ( x ) ε i.e. f( + h) f( ) T ( h) ε h (.) Furtherore, A. Byoui [6, 7] discussed the vrious roerties of - differentible ings. In 006, A. Byoui [8] shows tht Qusi differentibility of s y not be Frechet differentibility of s by soe exles. The definition of wek holoorhic [9] is Let E nd F be two colex nored sce, nd let U be oen subset of E. A ing f : U F is sid to be wek holoorhic on U if λ f : U C is holoorhic for every λ F In this note the wek usi differentible between loclly bounded sces nd soe chrcteriztions of this hve been derived. In section the concet of Wek usi tngent nd wek usi differentible s is introduced. In section 3 soe roerties of this is derived, nd t lst in section 4 result relted ing into roduct sce is derived. Wek usi differentible s Let E nd F be -nored sce nd -nored sce resectively (0 <, ) nd U, be oen in E.. Mings f,g: U F re wek usi tngent to ech other if ( ψ f ) nd ( ψ g ) re usi tngent to ech other t i.e. ( ψ f)( x) ( ψ g)( x) li = 0, ψ L( F) x x (.). A ing: f : U F is sid to be wek -usi differentible if ψ f is -usi differentible for everyψ F. i.e. t U, if continuous liner ing ψ T t ( T L( E, F) nd by the definition of usi differentible ) such tht ( f)( x) ( f)( ) ( T) ( x ) li = 0 x ψ ψ ψ (.)

3 Generliztion of Qusi-Differentible Ms 5 If =, f is clled wek usi differentible t. Hence, for every > 0, δ > 0 : 0 < <δ. ( )( ) ( )( ) ( ) ( ) f x f T x x ψ ψ ψ ε (.3) If f is wek usi differentible t ech oint of U, then f is sid to be wek usi differentible on U. Proerties of wek usi differentible Theore 3.: (Linerity) The set of wek usi differentible ings t U for vector sce. Proof: Let S be the set of wek usi differentible ings. If f, g S, then by definition of wek usi differentible, there exist liner ings ( ψ T) nd such tht ψ for ll ψ ( T ) L( F) F ε ( f )( x) ( f)( ) ( T) ( x ) ψ ψ ψ (3.) ε ( g)( x) ( g)( ) ( T ) ( x ) ψ ψ ψ (3.) For given ε > 0, since is wek usi nor. ( ψ f + ψ g)( x) ( ψ f + ψ g)( ) + ( ψ T ψ T ) ( ) x σ( ( ψ f )( x) ( ψ f )( ) ( ψ T) ( x ) + ( ψ g)( x) ( ψ g)( ) ( ψ T ) ( x ) ) Dividing on both sides by x nd tking liit x ( ψ f + ψ g)( x) ( ψ f + ψ g)( ) ( ψ T + ψ T ) ( x ) li x ( ψ f )( x) ( ψ f )( ) ( ψ T) ( x ) σ [li x ( ψ g)( x) ( ψ g)( ) ( ψ T ) ( x ) x + li ] x D( ψ f + ψ g)() = D( ψ f)() + D( ψ g)() (3.3)

4 5 Sushil Kur Let λ 0 be sclr Now, we hve to rove D( λ( ψ f )( )) = λd( ψ f )( ) λ 0 For D( λ ψ of )( ) λ[( ψ f)( x) ( ψ f)( ) ( ψ T) ( x )] li x λψ ( f)( x) λψ ( f)( ) λψ ( T) ( x ) = li x D( λ ( ψ f)( ) = λd( ψ f( )) (3.4) Theore 3.: (Lischitzin roerty) Let E nd F be -nored nd -nored sces resectively, (0 <, ) nd U, oen in E. If f : U F is wek usi differentible t U, then there exist c > 0 nd δ > 0 such tht ( ψ f )( x) ( ψ f)( ) c for x U, <δ. Proof: Let T = D( ψ f )( ) Δ ( x) = ( ψ f )( x) ( ψ f )( ) T( x ) for x U Then ( ψ f )( x) ( ψ f )( ) = T( x ) +Δ ( x) T + Δ x ( T( x ) T ) Since (3.5) Δ( x) li = 0 x nd Δ ( ) = 0 Given =, δ > 0 such tht x δ, x U, then

5 Generliztion of Qusi-Differentible Ms 53 Δ( x) Therefore ( ψ f )( x) ( ψ f )( ) ( T + ) ( ψ f )( x) ( ψ f)( ) c (3.6) where c= T +. Theore 3.3: (Chin Rule). Let f : U F be wek usi differentible t U nd let T L( F, G) where E, F re -nored sce nd -nored sce nd U is oen in E, then ( T f ) is wek usi differentible t U. Proof: We hve to rove, ( T f ) is wek usi differentible t ψ ( T f ) is usi differentible t. Let S = T f Therefore ( ψ S)( x) ( ψ S)( ) ψ D( S( x ) = ψ ( Sx ( ) S ( ) DSx ( ) ψ Sx ( ) S ( ) DS ( )( x ) Dividing on both sides by x nd Tking the li x, we get ( ψ S)( x) ( ψ S)( ) ψ DS( x ) ψ Sx ( ) S ( ) DSx ( ) ψ ( T f )( x) ( T f )( ) D( T f )( x ) = x ψ ( T f)( x) ( T f)( ) ( T Df)( x ) = ψ T f ( x) f ( ) Df ( x ) 0 = x

6 54 Sushil Kur T f is wek usi differentible. Ming into roduct sce Theore 4: Let U E be oen in -nored sce E nd Fn -nored sces ( n =, ). A ing f : U F F F3 Fn is wek usi differentible t U if nd only if ech coordinte fn = πn f is wek usi differentible t Further Df ( ) = ( Df( ), Df( ) Df( )) where πn is the rojection fro F F F3 F onto F n. Proof: Let F n be n -nored sces for n =,. And let F = F F F3 Fn Then F is n F -sce nd toology induced by n F -nor on F, is the roduct toology. Let π n: F Fn be the rojection ings onto the n th fctor F n nd let un: Fn F be nturl ebedding defined by u ( x ) = (0,0, 0, x,0,0 0) n n n Then both π n nd u n re continuous liner s. π nou n = F n (the identify on F n ) uoπ = (the identify on F ) n n F Let U be n oen subset of E nd let f : U F nd let th f = π of : U F be the n coordinte. Then n n n f = u π of = u f = ( f, f f ) η n η η n = n = So, if f is wek usi differentible t, then then by chin rule (Theore 3.3) f ( ) u Df ( ) = n = ( Df ( ), Df ( ) Df ( )) n uη π is usi differentible t, n Conversely If ech f n is wek usi differentible t then f is clerly wek usi differentible t.

7 Generliztion of Qusi-Differentible Ms 55 References [] H.Crtn, 97, Differentil Clculus, Hernn, Pris. [] Wlter Rudin, 973, Functionl Anlysis, McGrhil. [3] J Muzic, 986, Colex Anlysis in Bnch Sces, North Hollnd Mth. Studies. [4] A. Byoui, 003, Foundtions of colex nlysis in Non loclly convex sces, North Hollnd Publiction, Mthetics Studies. [5] A. Byoui, 006, Suer differentil clculus in F -sces, Reserch reort in thetics, no 7. [6] A. Byoui, 997, Men vlue theore for colex bounded loclly sce, Couniction in Alied Non-liner Anlysis (4), [7] A. Byoui, 005, Bolzno s Interedite-vlue theore for usiholoorhic s, Centrl Euroen Journl of Mthetics, 3(), [8] Soo Bong Che, Holoorhy nd clculus in Nored sces, Mrcel Dekker, Inc., New York. [9] A. Byoui, 006, Byoui differentil is different fro Frechet Differentil, Centrl Euroen Journl of Mthetics, 4(4),.,

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