## Existence of fixed point in c-contraction

#### Citation

Azhdari, P. (2009). Existence of fixed point in c-contraction. Maltepe Üniversitesi. s. 323.#### Abstract

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: [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 [1]: 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 [2]: 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.

#### Source

International Conference of Mathematical Sciences#### Collections

- Makale Koleksiyonu [586]

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