On almost λ-statistical convergence of fuzzy numbers

The purpose of this paper is to introduce the concepts of almost λ-statistical convergence and strongly almost λconvergence of fuzzy numbers. We obtain some results related to these concepts. It is also shown that almost λstatistical convergence and strongly almost λ-convergence are equivalent for almost bounded sequences of fuzzy numbers.


Introduction
The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh (1965) and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming.Matloka (1986) introduced bounded and convergent sequences of fuzzy numbers and studied some of their properties.Later on sequences of fuzzy numbers have been discussed by Diamond and Kloeden (1994), Nanda (1989), Savaş (2000Savaş ( , 2006)), Esi (2006), Tripathy andBaruah (2009, 2010a andb), Tripathy andBorgogain (2008, 2011), Tripathy andDutta (2007, 2010), Tripathy and Sarma (2011) and many others.Let

   
: The space   It is well-known that Throughout this paper by a fuzzy number we mean a function X from R n to [0,1] which is normal, fuzzy convex, upper semicontinuous and the closure of is compact.These properties imply that for each on R, and we have Let L(R) denote the set of all fuzzy numbers.The linear structure of L(R) induces the addition X + Y and the scalar multiplication λX in terms of   level sets, by The additive identity and multiplicative identity of L(R) are denoted by 0 and 1, respectively.
For r in R and X in L(R), the product rX is defined as follows: is complete metric space (MATLOKA, 1986).
of fuzzy numbers is a function X from the set N of natural numbers into L(R).The fuzzy number X k denotes the value of the function at , 1986).
We denote by w F denotes the set of all sequences of fuzzy numbers is bounded (MATLOKA, 1986).We denote by F   the set of all bounded sequences of fuzzy numbers is said to be convergent to a fuzzy number X 0 if for every , 1986).We denote by c F the set of all convergent sequences Nanda (1989) studied the classes of bounded and convergent sequences of fuzzy numbers and showed that these are complete metric spaces.
The metric d has the following properties: L(R).The notion of statistical convergence for a sequence of complex numbers was introduced by Fridy (1985) and many others.Over the years and under different names statistical convergence has been discussed in the different theories such as the theory of Fourier analysis, ergodic theory and number theory.Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy (1985), Salat (1980), Connor (1999) and many others.This concept extends the idea to apply to sequences of fuzzy numbers with Kwon and Shim (2001), Et et al. (2005), Nuray and Savaş (1995) and many others.
Savaş ( 2006) defined almost convergence for fuzzy numbers as follows: The sequence of fuzzy numbers is said to be almost convergent to a fuzzy number where This means that for every and for all m.
A sequence of fuzzy numbers is said to be statistically convergent to a fuzzy number X 0 if for every The set of all statistically convergent sequences of fuzzy numbers is denoted by F S .We note that if a sequence of fuzzy numbers converges to a fuzzy number o X , then it is statistically converges to X 0 .But the converse statement is not necessarily valid. Let be a non-decreasing sequence of positive real numbers tending to infinity and λ 1 = 1 and , for all n N  .
The generalized de la Vallee-Poussin means is defined by Leindler (1965).

Let
be sequence of fuzzy numbers.
In this case, we write . The set of all almost λstatistically convergent sequences is denoted by F S  ˆ.
In the special case λ n = n for all In this case, we write . The set of all strongly almost   convergent sequences is denoted by   p w F , ˆ .In the special case λ n = n for all and we said that of fuzzy numbers is bounded.By F l  ˆ, we shall denote the set of all almost bounded sequences of fuzzy numbers.
In this section we give some inclusion relations between strongly almost λconvergence and almost λ-statistically convergence and show that they are equivalent for almost bounded sequences of fuzzy numbers.We also study the inclusion under certain restrictions on the sequence . Then the proof follows from the following inequality; . By Minkowski's inequality, we get Hence, we obtain the result.The following theorem shows that almost λ-statistical convergence and strongly almost λ-convergence are equivalent for almost bounded sequences of fuzzy numbers.
Theorem 2.2.Let the sequence where n G is the same as given in Theorem 2.2.(b).Thus,

Conclusion
In this paper, we constructed the concepts of almost λ-statistical convergence of fuzzy numbers and then we obtained effectiveness results in connection with these concepts.Moreover, we showed that almost λ-statistical convergence and strongly almost λ-convergence are equivalent for almost bounded sequences of fuzy numbers.
1, let u k = 0 and v k = w k .Then it is clear that for all N k  , we have w


for all k and m .So, e.    