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1 International Journal of Mathematical Archive-3(8), 01, Available online through ISSN COMMONFIXED POINT THEOREM IN BANACH SPACE Neena Vijaywargiand & Shyam Patkar* *Department of Mathematics, Truba Institute of Engineering & Information Technology, Bhopal, India (Received on: ; Accepted on: ) ABSTRACT In the present paper we establish a fixed point theorem in Banach space taking new rational expression, which satisfies the well-known results. Key words: Self mapping, Continuous mappings, Non contraction mappings, Banach Space. AMS subject classification: 47H10, 54H5, INTRODUCTION&PRELIMINARIES The study of Non-Contraction mapping concerning the existence of fixed point draws attention of various authors in non linear analysis dealing with the study of Non expansive mapping and the existence of fixed points. It is well known that the differential and integral equations that arise in the physical problems are generally non linear, therefore the fixed point methods specially Banach contraction Principle ([1],19) provides a powerful tool for obtaining the solution of their equations which were very difficult to solve by any other methods. It is also true that some qualitative properties of the solution of related equations are proved by functional analysis approach. Many authors have presented valuable results with non contraction mapping ([7], 007) in Banach space. Definition.1: (Banach space) A Banach space (XX,. )is a normed vector space such that X is complete under the metric induced by the norm. Example.1: The set of continuous functions on closed interval of real line with the norm. of function f given by ff = sup x X ff(xx) is a Banach space,where sup denotes the supermom. Definition.: (Normed linear space) let. denotes a function from a linear space X into R that satisfies the following axioms i) xx XX, xx 0, xx = 0 iff xx = 0 ii) xx, yy XX, xx + yy xx + yy iii) xx XX, RR, αααα = αα xx xxis called the norm of x and (XX,.. ) is called a Normed linear space. Example -..1: RR nn,. pp, xx RR nn, xx = max nn ii = 1 xx ii Example -..: ll pp,. pp, 1 pp, xx ll pp = {xx: xx RR, ii=1 xx pp < }, xx = ( xx ii pp 1 ) 1 pp Definition.3: Asequence {x n } in a normed space is said to be a Cauchy sequence ifxx nn xx mm 0 ii. ee. given εε > 0,there exist an integer NN such that xx nn xx mm < εε,for all mm, nn > NN as mm, nn Corresponding author: Shyam Patkar* *Department of Mathematics, Truba Institute of Engineering & Information Technology, Bhopal, India International Journal of Mathematical Archive- 3 (8), August 01 39

2 3. MAIN RESULT Theorem 3.1: Let f be a mapping of a Banach space X into itself. If f satisfies the following conditions; f = I, where I is identity mapping. (3.1.1) xx ff(xx)+xx ff(xx)+ yy ff(xx) ff(x) ff(y)(3.1.) α +yy ff(yy) For every x, y Є X, where α, δ, η > 0 and 5α +4γ+δ+η <, then f has a fixed point. If α +δ+η < 1 then f has an unique fixed point. Proof: Suppose x is a point in the Banach space X. Taking y = 1 (f + I) (x),z = f(y) and u = y - z we have zz xx= ff(yy) ff (xx) = ff(yy) ff(ff(xx)) α yy ff(xx)yy ff(yy)+yy ff(yy)yy ff (xx)+ yy ff (xx)ff(xx) ff(yy ) yy ff(xx)+f(x) f (x) + γ [yy ff(yy) + ff(x) ff (x)] +δ [yy ff (xx) + ff(x) ff(y)] +η yy ff(xx) zz xx α yy ff(xx)yy ff(yy)+yy ff(yy)+ ff(x) ff(y) yy ff(xx)+xx ff(xx) + γ [yy ff(yy) + xx ff(xx)] +δ [xx yy + f(x) f(y)] +η yy ff(xx) zz xx α yy ff(yy)(yy ff(xx)+)+f(x) f(y) + γ [yy ff(yy) + xx ff(xx)] +δ [xx yy + f(x) f(y)] +η yy ff(xx) zz xx α yy ff(yy)xx ff(xx)+ff(x) ff(y) + γ [yy ff(yy) + xx ff(xx)] +δ [xx yy + ff(x) ff(y)] +η yy ff(xx) zz xx α + ff(x) ff(y) + γ [yy ff(yy) + xx ff(xx)] +δ [xx yy + ff(x) ff(y)] +η yy ff(xx) zz xx α xx 1 + (ff + I) (x) ff(x) ff(1 (ff + I) (x))+γ [yy ff(yy) + xx ff(xx)] +δ xx 1 (ff + I) (x) + ff(x) ff(1 (ff + I) (x)) +η 1 (ff + I) (x) ff(xx) zz xx α xx ff(xx) xx ff(xx) + γ [yy ff(yy) + xx ff(xx)] +δ 1 xx ff(xx) + 1 xx ff(xx) +η 1 xx ff(xx) zz xx αyy ff(yy) + 1 xx ff(xx)+ β 1 xx ff(xx) + γ [ yy ff(yy) +xx ff(xx) ] +δ [xx ff(xx)] +η 1 xx ff(xx) zz xx ( αα + +γγ + δδ + ηη )xx ff(xx) + (α+ γ) yy ff(yy)(3.1.3) Also uu xx = yy zz xx = 1 (ff + I)(x) z x= ff(xx) zz=ff(xx) ff(yy) 01, IJMA. All Rights Reserved 330

3 uu xx α xx ff(xx)+xx ff(xx)+ yy ff(xx) +yy ff(yy) uu xx α xx ff(xx)(+)+ yy ff(xx) uu xx α xx ff(xx)yy ff(yy) + yy ff(xx) uu xx α xx ff(xx)yy ff(yy) + yy ff(xx) γ[xx ff(xx) + yy ff(yy)] +δ [xx ff(yy) + yy ff(xx)] +η xx yy xx ff(xx)yy ff(yy) uu xx α xx ff 1 + (f + I) (x) 1 (f + I) (x) ff(xx) + γ [xx ff(xx) + yy ff(yy)]+δ xx ff 1 (f + I) (x) + 1 +η xx 1 (f + I) (x) uu xx (f + I) (x) ff(xx) α xx ff(xx)yy ff(yy) xx ff(xx) xx ff(xx) + γ [xx ff(xx) + yy ff(yy)] +δ 1 xx ff(xx) + 1 xx ff(xx) +η 1 xx ff(xx) uu xx α [yy ff(yy)+ 1 xx ff(xx) ] + γ [xx ff(xx) + yy ff(yy)] + δxx ff(xx)+ η1 xx ff(xx) uu xx ( αα + γγ + δδ + ηη )xx ff(xx) + (α +γ) yy ff(yy)(3.1.4) Now, zz uu= (zz xx) (xx uu) (zz xx+xx uu) [( αα + γγ + δδ + ηη )xx ff(xx) + (α +γ )yy ff(yy)] + [(αα γγ + δδ + ηη )xx ff(xx) + (α +γ) yy ff(yy)] zz uu (α + γγ + δδ + η)xx ff(xx)+(4α +γ)yy ff(yy) Also zz uu= ff(yy) (yy zz) From (3.1.5) = ff(yy) yy ff(yy)) =yy ff(yy)(3.1.5) yy ff(yy) (α + β + λ + μ + γγ + δδ + η)xx ff(xx) + (4α + λ +γ) yy ff(yy) [ - (4α +γ + λ )] yy ff(yy) (α + β + λ + μ + γγ + δδ + η)xx ff(xx) yy ff(yy) q xx ff(xx) Where q = (α+β + λ+μ+γγ+δδ+ η) [ (4α + λ +γ) ] < 1 Since 5α + 4γγ + δδ + η< 01, IJMA. All Rights Reserved 331

4 Let g = 1 (f + I) then for every x Є X gg (xx) gg(xx) = g(y) y = 1 (ff + I) y y = 1 yy ff(yy) qq xx ff(xx) By the definition of q, we claim that {g n (x)} is a Cauchy sequence in X. By the completeness, {g n (x)} converges to some element x 0 in X i.e. lim nn gg nn (x) = x 0 which implies that g(x 0 ) = x 0 hence f(x 0 ) = x 0 i.e. x 0 is fixed point of f For the uniqueness: If possible let y 0 ( x 0 ) be another fixed point of f then = ff(xx 0 ) ff(yy 0) α xx 0 yy 0 xx 0 ff(xx 0) +xx 0 ff(xx 0) xx 0 ff(yy 0) + xx 0 ff(yy 0) yy 0 ff(xx 0 ) xx 0 yy 0 +yy 0 ff(yy 0 ) + γ xx 0 ff(xx 0) + yy 0 ff(yy 0 ) +δ [xx 0 ff(yy 0 ) + yy 0 ff(xx 0 )]+η α xx 0 yy 0 xx 0 yy 0 +δ +η α +δ +η α +δ +η = (α +δ+η) Since α +δ+η < 1 =0 xx 0 = yy 0 This complete the proof REFERENCES 1. Ahmad A. and Shakil, M. Some fixed point theorems in Banach spaces Nonlinear Funct. Anal. And Appl. 11(006) Banach S. Surles operation dans les ensembles abstraitsetleur application aux equations intergrals Fund. Math. 3(19) Badshah V.H. and Gupta, O.P. Fixed point theorems in Banach and Banach spaces Jnanabha 35(005) Browder F.E. Non-expansive non-linear operators in Banach spaces Proc. Nat. Acad. Sci. U.S.A. 54 (1965) Datson W.G. Jr. Fixed point of quasi non-expansive mappings J, Austral. Math. Soc. 13 (197) Gohde D. Zumprinzipdevkntraktivenabbilduing Math. Nachr 30 (1965) , IJMA. All Rights Reserved 33

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