Some fixed point theorems in fuzzy normed linear spaces☆
Introduction
Studies on fuzzy normed linear spaces are relatively recent in the field of fuzzy functional analysis. In 1984, Katsaras [15] first introduced a definition of fuzzy norm on a linear space. In 1992, Felbin [11] defined a fuzzy norm by assigning a non-negative fuzzy real number to each element of a linear space, the induced fuzzy metric of which is of Osmo Kaleva type [16]. From a different approach Cheng and Mordeson [9] in 1994 defined another type of fuzzy norm on a linear space whose associated fuzzy metric is of Kramosil and Michalek type [18]. On the other hand in studying topological properties of fuzzy real numbers, Gähler and Gähler [12] defined fuzzy norm of a fuzzy real number as a difference of its positive and negative parts.
Following Cheng and Mordeson [9], we redefined fuzzy norm on a linear space [1] and established a suitable decomposition theorem. Based on this decomposition theorem, it has been able to establish in [2], four fundamental theorems of functional analysis in fuzzy setting.
We have started proving fixed point theorems in [4], where Kirk type fixed point theorem [17] is established in fuzzy normed linear spaces with respect to the fuzzy norm introduced by us [1]. In [5], a Browder-Kirk type [7] fixed point theorem in Felbin’s type fuzzy normed linear spaces has been proved.
Although some papers have appeared in fuzzy fixed point theory by other authors [10], [13], [14], [19], [21], [23], but all these are on fuzzy metric spaces and studies on fixed point theory in fuzzy normed linear spaces is a very recent development [4], [5], [22].
The main objective of this paper is to introduce concepts of ‘asymptotic fuzzy normal structure’ and ‘strong uniformly convex fuzzy normed linear space’ and to use these geometrical properties in proving some fixed point theorems for fuzzy nonexpansive mappings. Schauder [20] fixed point theorem for fuzzy continuous-type mappings is also established in a fuzzy normed linear space. The technique that has been used is to find decomposition of these fuzzy concepts into their corresponding crisp counterpart and then the classical results on fixed point theory are extended to fuzzy setting.
The organization of the paper is as follows:
Section 1 comprises of some preliminary results and useful definitions.
In Section 2, sectional fuzzy continuous mapping on fuzzy normed linear spaces has been defined and its relations with other types of fuzzy continuous mappings are studied.
Section 3 is devoted to establish Schauder-type [20] fixed point theorems in a fuzzy normed linear space.
In Section 4, an idea of asymptotic fuzzy normal structure is introduced and a Baillon and Schoneberg-type [6] fixed point theorem in a fuzzy normed linear space is established.
In Section 5, a definition of strong uniformly convex fuzzy normed linear space is given and Browder-type [7], [8] fixed point theorems on such spaces are established. Finally, relation between a uniformly convex fuzzy normed linear space and a strong uniformly convex fuzzy normed linear space has been studied.
Section snippets
Some preliminary results
In this section some definitions and preliminary results are given which will be used in this paper. Definition 1.1 Let U be a linear space over the real or complex field F. A fuzzy subset N of is called a fuzzy norm on U iff and , the following conditions are satisfied: with ; () iff ; if ; is a non-decreasing function of R and The pair (U, N) will be referred to as asee [1]
Some results on fuzzy continuous mappings
In this section, we give a definition of sectional fuzzy continuous mapping and study its relation with other types of fuzzy continuous mappings. Definition 2.1 A mapping is said to be sectional fuzzy continuous at if such that for each such that, . If T is sectional fuzzy continuous at each point of U, then T is said to be sectional fuzzy continuous on U. Proposition 2.1 Let (U, N1) and (V, N2) be two fuzzy normed linear spaces and be a
Schauder fixed point theorem
In this section, we establish Schauder fixed point theorems in fuzzy normed linear spaces. Definition 3.1 Let (U, N) be a fuzzy normed linear space. A subset A of U is said to be l-fuzzy compact if for any sequence {xn} and for each a subsequence of {xn} and (both depending on {xn} and α) such that Lemma 3.1 Let (U, N) be a fuzzy normed linear space satisfying N(6). Then a subset A of U is l-fuzzy compact iff A is compact w.r.t. -norm of N) for each Proof First suppose
Asymptotic fuzzy normal structure
In this section, we introduce an idea of asymptotic fuzzy normal structure and establish a Baillon and Schoneberg-type [6] fixed point theorem in fuzzy normed linear space. Definition 4.1 A convex subset K of a fuzzy normed linear space (U, N) is said to have asymptotic fuzzy normal structure if for each l-fuzzy bounded convex subset H of K and each sequence {xn} in H for which there exists such that,where denotes the fuzzy
Strong fuzzy uniformly convex normed linear space
Definition 5.1 A fuzzy normed linear space (U, N) is said to be strong fuzzy uniformly convex if for each such thatwhere . Theorem 5.1 Let (U, N) be a fuzzy normed linear space satisfying (N6). If (U, N) is strong fuzzy uniformly convex then such that is uniformly convex normed linear space where is an -norm of N. Proof Since (U, N) is strong fuzzy uniformly convex, thus for each such
Conclusion
So far most of the results on geometric fuzzy fixed point theory were proved on fuzzy metric space setting only, the limitation of which is that the underlying mapping is to be restricted to satisfy some kind of stringent property, viz. contraction property. On the other hand, the classical theory of fixed points on normed linear spaces is very rich in which the intricate geometric structure of normed linear space is used to establish the existence of fixed points for much more general type of
Acknowledgments
Authors are grateful to the Editor-in-Chief for his valuable comments in rewriting the paper in the present form. They are also thankful to the Reviewers for their constructive suggestions.
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The present work is partially supported by special assistance programme (SAP) of UGC, New Delhi, India [Grant No. F.510/8/DRS/2004 (SAP-I)].