Approximate Fixed Point Theorems in Fuzzy Norm Spaces for an Operator

We define approximate fixed point and fuzzy diameter in fuzzy norm spaces. We prove theorems for various types of well-known generalized contractions on fuzzy norm spaces with the use of two general lemmas that are given regarding approximate fixed points of operators on fuzzy norm spaces. 1. Introduction In this paper, starting from the article of Berinde [1], we study some well-known types of operators on fuzzy norm spaces, and we give some fuzzy approximate fixed points of such operators. Fuzzy set was defined by Zadeh [2]. Katsaras [3], while studying fuzzy topological vector spaces, was the first to introduce in 1984 the idea of fuzzy norm on a linear space. In 1992, Felbin [4] defined a fuzzy norm on a linear space with an associated metric of the Kaleva and Seikkala type [5]. A further development along this line of inquiry took place when, in 1994, Cheng and Mordeson [6] evolved the definition of a further type of fuzzy norm having a corresponding metric of the Kramosil and Michálek type [7]. Chitra and Mordeson [8] introduce a definition of norm fuzzy, and thereafter the concept of fuzzy norm space has been introduced and generalized in different ways by Bag and Samanta in [9–11]. Throughout this paper, the symbols and mean the and the , respectively. 2. Some Preliminary Results We start our work with the following definitions. Definition 1. Let be a linear space on . A function is called fuzzy norm if and only if for every and for every , the following properties are satisfied: : for every ,？ : if and only if for every ,？ : for every and ,？ : for every ,？ : the function is nondecreasing on , and .A pair is called a fuzzy norm space. Sometimes, we need two additional conditions as follows:？ : for all .？ : function is continuous for every and on subset is strictly increasing. Let be a fuzzy norm space. For all , we define norm on as follows: Then is an ascending family of normed on and they are called -norm on corresponding to the fuzzy norm on . Some notation, lemmas, and examples which will be used in this paper are given in the following. Lemma 2 (see [9]). Let be a fuzzy norm space such that it satisfies conditions and . Define the function as follows: Then(a) is a fuzzy norm on .(b) . Lemma 3 (see [9]). Let be a fuzzy norm space such that it satisfies conditions and , and . Then if and only if for every . Note that the sequence converges if there exists a such that In this case, is called the limit of . Example 4 (see [9]). Let be the real or complex vector space and let be defined on as follows: for all and . Then is a fuzzy norm space,