Application on Inner Product Space with. Fixed Point Theorem in Probabilistic

Transcription

1 Journl of Applied Mhemics & Bioinformics, vol.2, no.2, 2012, 1-10 ISSN: prin, online Scienpress Ld, 2012 Applicion on Inner Produc Spce wih Fixed Poin Theorem in Probbilisic Rjesh Shrivsv 1, B. Krishn Reddy 2, K. Shshidhr Reddy 3, M. Vijy Kumr 4 nd P. Devids 5 Absrc In his presen pper we obin fixed poin heorem in complee probbilisic - Inner produc spce. To sudy he exisence nd uniqueness of soluion for liner vlerr inegrl equion. Mhemics Subjec Clssificion : 46S50 Keywords: Fixed poin, vlerr Inegrl Equion Probbilisic Inner Produc Spce 1 Inroducion This pper is o obin new fixed poin heorem in probbilisic -inner produc spce, where is -norm of h-ype, o sudy he exisence nd 1 e-mil: rjeshrj0101@redeffmil.com 2 e-mil: bkrbkr 07@gmil.com 3 e-mil: shshidhrko9@yhoo.co.in 4 e-mil: greengirl is2001@yhoo.com 5 e-mil: devidsjlph@gmil.com Aricle Info: Received : Jnury 12, Revised : Mrch 23, 2012 Published online : Sepember 15, 2012

2 2 Applicion on Inner Produc Spce wih Fixed Poin Theorem uniqueness of soluion for liner Vlerr inegrl equion in complee - PIP-spce. Throughou his pper, le R =,+, R+ = 0,+, D denoes he se of ll disribuion funcions. 2 Preliminries Definiion 2.1. A mpping F : R R + is clled disribuion funcion if i is nondecresing nd lef-coninuous wih inf R F = 0,supF = 1. R Definiion 2.2. A probbilisic -inner produc spce briefly, -PIPspce is riple X,F,, where X is rel liner spce, is -norm nd F is mpping of X X DF x,y will denoe he disribuion funcion Fx,y nd F x,y will represen he vlue of F x,y R sisfying he following condiions: PI 1 f x,x 0 = 0 PI 2 f x,y = F y,x PI 3 f x,x = H > 0 x = 0 where 0 0 H = 1 > 0 PI 4 f x,y, > 0 F x,y = H, = 0 1 F x,y +, < 0 PI 5 F x+y,z = sup F x,z s,f y,z s,f y,z r, R,S +r =,S R,r R, Definiion 2.3. A -norm is h-ype if he fmily of funcions { m } m=1 is equi-coninuous = 1, where 1 =,, m =, m 1, [0,1],m = 2,3,...

3 R. Shrivsv e l 3 Definiion 2.4. Le X,F, be -PIP-spce. 1. A sequence x n X is sid o converge o x X if ɛ > 0, α 0,1], N,whenn N,F xn x,x n xε > 1 α. 2. A sequence x n X is clled Cuchy sequence if ɛ > 0, α 0,1], N, when m,n N, F xn x m,x n x m ε > 1 α. 3 Min Resuls Theorem 3.1. Le X,F, be complee PIP spce nd be -norm of h-ype. Le T : X,F, X,F, be liner mpping sisfying he following condiion. F xy +F y.z kα,β kβ.γ F Tx,y,z 1 F x,y kα.β For ll x,y,z X, 0,α,γ 0,+ nd kα.γ : 0,+ 0,+ 0.1 is funcion. Then T hs excly one fixed poin x X. Furher more for ny x 0 X The ierive sequence {T n x 0 } T converges o X. Proof. Firsly we prove h ny x 0 X. The sequence {x m } m=0 is τ - cuchy sequence where {x m } m=0 = {x 0,x 1 = Tx 0...x m = T m x 0...} Le us consider F Tx,y,z F xy +F y.z kα.β F xy kβ.γ kα.β

4 4 Applicion on Inner Produc Spce wih Fixed Poin Theorem F xyz F x,z F xy kα.γ kα.β or F x,z F x,z F xy kα.γ kα.β By - PI 5 we hve = F x0 Tx 0 +T kα.γ m x 0,z kα.γ 1 F x0 Tx 0,z,F Tx0 T m x 0,z 1 F x0 Tx 0,z,F Tx0 T m 1 x 0,z 1 = F x0 Tx 0,z,F Tx0 Tx 0 +Tx 0 T m 1 x 0,z 1 1 F x0 Tx 0,z, F x0 Tx 0,z, kα,β F Tx0 T m 1 x 0,z 1 kα,β 1 F x0 Tx 0,z, F x0 Tx 0,y, F Tx0 T m 2 x 0,z F x0 Tx 0,z, F x0 Tx 0,y, 1..., F x0 Tx 0,z,F x0 Tx 0,y... F x0 T m x 0.z

5 R. Shrivsv e l 5 Becuse of k α,γ 0,1, herefore we ge 1 By he propery of disribuion funcion, we hve 1 F x0 Tx 0,z F x0 Tx 0,z By he propery of -norm, we obin F x0 T m x 0,z 1 F x0 Tx 0,z, 1..., F x0 Tx 0,z = F m 1 1 x0 Tx 0,z F x0 Tx 0,z,F x0 Tx 0,z So for ny posiive ineger m, n we hve F T n x 0 T m+n x 0,T n x 0 T m+n x 0 F x0 T m x 0,T n x 0 T m+n x 0 k n+1 α,γ F x0 T m x 0,x 0 T m x 0 k 2n+1 α,γ m 1 1 F x0 T m x 0,x 0 T m x 0 k 2n+1 α,γ m 1 m F x0 Tx 0,x 0 Tx 0 k 2n+1 α,γ 2m F x0 Tx 0,x 0 Tx 0. k 2n+1 α,γ Noe h is norm of h-ype, he fmily of funcions m m=1 is equiconinuous = 1, nd he disribuion funcion F is nondecreseing wih

6 6 Applicion on Inner Produc Spce wih Fixed Poin Theorem sup R F = 1, hen we hve lim F T 1 n x 0 T m+n x 0,T n x 0 T m+n x 0 F x0 Tx0,x0 Tx0 lim 2m k 2n+1 α,γ By - P1-3, we hve lim T n x 0 T n+m x 0 = 0. so {T m x 0 } m=0 is Cuchy sequence in X. By he compleeness of X, le x m x Xm. Secondly, we prove h X is fixed poin of T. Becuse of F xi Tx i,x Tx F xi 1 Txi 1,x Tx... F x0 Tx 0,x Tx k i α,γ Then we hve lim i F x i Tx i,x Tx lim i F x0 Tx 0,x Tx k i α,γ = 1 F x Tx,x Tx i F x xi,x Tx 1,F xi Tx i,x Tx Becuse x i x when i,... is equi-coninous 3.1 nd F θ,x Tx = 1, we hve lim i F x Tx i,x Tx = 1, > 0. Hence F x Tx,x Tx 1 F x Txi,x Tx 1 F x Txi,x Tx,F Txi Tx,x Tx,F Txi x,x Tx

7 R. Shrivsv e l 7 So F x Tx,x Tx hve x = Tx. 1i, > 0. By PI 3, we If here exiss poin y X such h y = Ty, hen F x y,x y = F Tx Ty,Tx Ty F x y,x y k 2 α,γ In he sme wy, we obin F x y,x y F x y,x y k 2 α,γ... f x y,x y k 2n α,γ So F x y,x y 1n, > 0. By - PI - 3, we hve x = y. Therefore x* is he unique fixed poin in X. Finlly, we prove h he sequence {T n x 0 } T-converges o x for ny x 0 X. Becuse of F x T n x 0,x T n x 0 l = F Tx T n x 0,x T n 1 x 0 F x T n 1 x 0,x T n 1 x 0 k 2 α,γ = F Tx T n 1 x 0,Tx T n 1 x 0 k 2 α,γ F x T n 2 x 0,x T n 2 x 0 k 4 α,γ... F x x 0,x x 0 k 2n α,γ We hve lim F x T n x 0,x T n x 0 lim F x x 0,x x 0 k 2n α,γ = 1

8 8 Applicion on Inner Produc Spce wih Fixed Poin Theorem Similrly lim F x T n x 0 T n x 0 lim F x x 0,x x 0 lim F x T n x 0 x 0 T n x 0 lim F x x 0,x x 0 lim F x x 0,x x 0 So T n x 0 x 0 n This complees he proof. k 2n α,γ k 2n α,γ k 2n α,γ = 1 = 1 1 = 1 4 Applicion Theorem 3.1, we uilize his heorem o sudy he exisence nd uniqueness of soluion of liner Vlerr inegrl equion in complee - PIP spce. Le [, b] be fixed rel inervl. We define liner operion in L 2 [,b], x+y = x+y,αx = αx Then L 2 [,b] is liner spce. We led inner produc ino L 2 [,b],x,y = c xyd. Hencex,yisfinienumber,.,. sisfiesllcondiionsofinnerproducnd L 2 [,b] is inner produc spce by.,. Becuse L 2 [,b] is infinie dimension nd compleeness. Define spce L 2 [, b,], F,, where F : L3[,b,c] L3[,b,c] D,Fx, y, z = H x y z. Then L 2 [, b], F, is - PIP spce. In fc, le {xn} be Cuchy sequence in L 2 [, b], F,. Then for ny ɛ > 0, λ 0, 1], N, when m, n N, we hve F xm x n,x m x n ε > 1 λ becuse of F xm x n,x m x n ε = Hε x m x n,x m x n φµ x m x n = H = H ε ε c b x m x n x m x n dφµ [x m x n ] 2 dφµ > 1 λ,

9 R. Shrivsv e l 9 We hve c [x m x n ] 2 d 0, And hen x m x n 0, So x n is Cuchy sequence in L 2 [,b]. By he compleeness of L 2 [,b] we hve xn x L 2 [,b]. Hence x L 2 [,b],f,. So L 2 [,b],f, is complee - PIP spce. Theorem 4.1. Le L 2 [, b],f, be complee - PIP spce. Then he following condiions re sisfied: i xs zs,ds x z, x L2 [,b] ii Le T be liner mpping nd defined s follows Tx = f + k,sxsds Where f L 2 [,b] is given funcion k,s is coninuous funcion defined on b c, s, λ is consn, we denoe mx k,s = M. b, s Then when λm 0, 1, T hs unique fixed poin in L 2 [,b]. Furhermore, for ny x 0 L 2 [,b], he ierive sequence {T n x 0 } T - converges o he fixed poin.

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