Existence of fixed point in c-contraction
MetadataShow full item record
CitationAzhdari, P. (2009). Existence of fixed point in c-contraction. Maltepe Üniversitesi. s. 323.
Kramosil and Michalek introduced the notion of fuzzy metric space which is similar to generalized Menger space. Then George and Veeramani imposed some stronger conditions on fuzzy metric space in order to obtain a Hausdorff topology. Many authors have extended fixed point theorem to different type of contraction in both probabilistic and fuzzy metric space. Mihet also showed fixed point theorem for fuzzy contractive mappings by using point convergence. In this paper we use the concept of point convergence for showing the existence of fixed point for B-contractions and C-contractions mapping. We notice that the condition of point convergency is weaker than convergency. Definition 1: A B-contraction on a probabilistic space (X, F) is a sel fmapping f of X for which Ff(p)f(q)(kt) Fpq(t) 8p, q 2 X, 8t ¿ 0, k 2 (0, 1). A mapping f : X ! X is called a C-contraction if there exists k 2 (0, 1) such that for all Fxy(t) ¿ 1 - t ) Ff(x)f(y)(kt) ¿ 1 - kt 8x, y 2 X , t ¿ 0. Definition 3: Let (X,M, T) be a fuzzy metric space. A sequence xn in X is said to be point convergent to x 2 X if there exists t ¿ 0 such that limn!1M(xn, x, t) = 1 Theorem 1: : Let (X,M, T) be George and Veeramani fuzzy metric space and sup 0 a ¡1 T(a, a) = 1 and A : X ! X be a B-contraction. Suppose that for some x 2 X the sequence of An(x) has a p-convergent subsequence. Then A has a unique fixed point. Theorem 2 : Let (X,M, T) be a George and Veermani fuzzy metric space and A : X ! X be a C-contraction and sup 0 a¡1T(a, a) = 1. Suppose that for some x 2 X the sequence of An(x) has a p-convergent subsequence. Then A has a unique fixed point. Existence of fixed point when the subsequence satisfied in p-convergency condition can extended to generalized C-contraction. Theorem : Let (X,M, T) be a George and Veermani fuzzy metric space and A : X ! X be a generalized C-contraction and sup 0 a¡1T(a, a) = 1. Suppose that for some x 2 X the sequence An(x) has a p-convergent subsequence. Then, A has a fixed point.
SourceInternational Conference of Mathematical Sciences
- Makale Koleksiyonu 
The following license files are associated with this item: