Fixed point theorems in generalized M fuzzy metric spaces

Abstract

The most important result in the FPT is the famous theorem of L.E.J.Brouwer which says that every continuous self-mapping of the closed unit Rn, the n- dimensional Euclidean space possesses a fixed point. This result was published by Brouwer (1910). If f : Rn and#8594; Rn is be a continuous mapping and suppose that, for some r gt 0 and all and#955; gt 0 , f(x) + and#955;x and#8800; 0, for any x with and#9553;x and#9553; = r. Then there exists a point x0, and#9553; x0and#9553;and#8804; r such that f(x0) = 0. Now it is known that this assertion is equivalent to BFPTH. Also, A.L.Cauchy s was the first mathematician to give a proof for the existence and uniqueness of the solution of the differential equation dy/dx = f (x, y); y (x0) = y0 when f is a continuous differentiable function. newlineBrouwer proved his famous theorem in 1910. Where the spaces are subsets of Rn are not of much use in functional analysis where one is generally concerned with infinite dimensional subset of some function spaces. There exist many contraction mapping theorem in difference spaces. C.D Birhoff and O.D.Kellogg gave one proof of BFPT with the assumption about convexity and compactness. In 1927 P.J. Schauder extended the Birkhoff- Kellogg theorem to metric linear spaces newline newline newlineand in 1930, Schauder extended BFPT to the result that every compact convex set in a Banach space has the fixed point property for continuous mapping, as well as that every weakly compact convex set in a separable Banach space has the fixed point property for weakly continuous mappings. The condition of compactness in SFPT was a very strong condition. As many problems in analysis do not have compact setting, it was natural to modify this theorem by relaxing the condition of compactness. A.N.Tychonoff (1935) had proved generalization of SFPT for the case of compact operators on locally convex linear spaces, and M.Hukuhara in 1950. An interesting extension was obtained by Browder (1959) under some deep conditions for the iterations of the mappings. newline newline