Feyzi Başar

Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations

The convergence of a series of fuzzy sets was examined via Zadehs Extension Principle by M. Stojakovic and Z. Stojakovic [M. Stojakovic, Z. Stojakovic, Addition and series of fuzzy sets, Fuzzy Sets and Systems 83 (1996), 341-346]. Since the utilization of this approach is quite difficult in practice, we prefer the idea of using the sum of the series of @l-level sets. The main purpose of the present paper is to determine the @a-, @b- and @c-duals of the classical sets of sequences of fuzzy numbers and is to give the necessary and sufficient conditions on an infinite matrix of fuzzy numbers transforming one of the classical sets to the another one.

Certain Sequence Spaces over the Non-Newtonian Complex Field

It is known from functional analysis that in classical calculus, the sets , , , and of all bounded, convergent, null and -absolutely summable sequences are Banach spaces with their natural norms and they are complete according to the metric reduced from their norm, where . In this study, our main goal is to construct the spaces , , , and over the non-Newtonian complex field and to obtain the corresponding results for these spaces, where .

On the Slowly Decreasing Sequences of Fuzzy Numbers

We introduce the slowly decreasing condition for sequences of fuzzy numbers. We prove that this is a Tauberian condition for the statistical convergence and the Cesaro convergence of a sequence of fuzzy numbers.

Some new results on sequence spaces with respect to non-Newtonian calculus

As an alternative to classical calculus, Grossman and Katz (Non-Newtonian Calculus, 1972) introduced the non-Newtonian calculus consisting of the branches of geometric, anageometric and bigeometric calculus etc. Following Grossman and Katz, we construct the field R(N) of non-Newtonian real numbers and the concept of non-Newtonian metric. Also, we give the triangle and Minkowski’s inequalities in the sense of non-Newtonian calculus. Later, we respectively define the sets ω(N), ℓ∞(N), c(N), c0(N) and ℓp(N) of all, bounded, convergent, null and p-absolutely summable sequences in the sense of non-Newtonian calculus and show that each of the sets forms a vector space on the field R(N) and a complete metric space.MSC:26A06, 11U10, 08A05.

Generalized Difference Spaces of Non-Absolute Type of Convergent and Null Sequences

The aim of the present paper is to introduce the spaces and of generalized difference sequences which generalize the paper due to Mursaleen and Noman (2010). These spaces are the BK-spaces of non-absolute type and norm isomorphic to the spaces and , respectively. Furthermore, we derive some inclusion relations determine the , , and duals of those spaces, and construct their Schauder bases. Finally, we characterize some matrix classes from the spaces , and to the spaces , , and c.

Certain Spaces of Functions over the Field of Non-Newtonian Complex Numbers

This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following Grossman and Katz, (Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972), we construct the field of -complex numbers and the concept of -metric. Also, we give the definitions and the basic important properties of -boundedness and -continuity. Later, we define the space of -continuous functions and state that it forms a vector space with respect to the non-Newtonian addition and scalar multiplication and we prove that is a Banach space. Finally, Multiplicative calculus (MC), which is one of the most popular non-Newtonian calculus and created by the famous exp function, is applied to complex numbers and functions to investigate some advance inner product properties and give inclusion relationship between and the set of -differentiable functions.

POWER SERIES OF FUZZY NUMBERS

Following Talo and Basar [Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations, Comput. Math. Appl. 58(2009), 717–733], we essentially deal with the power series of fuzzy numbers with real or fuzzy coefficients. We consider the different cases of power series of fuzzy numbers to be convergent and give the theorem on the term by term differentiation of power series of fuzzy numbers. Finally, we present a result on the Taylor expansion of a fuzzy valued function.

Certain spaces of sequences of fuzzy numbers defined by a modulus function

The main purpose of the present paper is to introduce the spaces £oo(F, /), c(F, / ) , co(F, / ) and ip{F, f , s) of sequences of fuzzy numbers defined by a modulus function. Furthermore, some inclusion theorems related to these sets are given and shown that 4o (F, f), co(F, f) and £P(F, f , s) are solid. 1. Preliminaries, background and notation Let W be the set of all closed bounded intervals A of real numbers with endpoints A and A, i.e. A :— [̂ 4, A]. The operations addition and multiplication by a real number on W are defined, as follows: A + B-.= {a + b:a<=A,beB}:= [A + B,A + B], aA := {aa : a € A}. One can extend the natural order relation on the real line to intervals as follows: A < B if and only if A < B and A<B. Define the relation d on W by d(A, B) : = max{|A B\, \A Then it can easily be observed that d is a metric on W (cf. Diamond and Kloeden [3]) and (W,d) is a complete metric space, (cf. Nanda [8]). A fuzzy number is a fuzzy set on the real axis, i.e. a mapping u : R —> [0,1] associating with each real number t its grade of membership u(t) which satisfies the following four conditions. (i) u is normal, i.e. there exists an to G R such that u(to) = 1. 2000 Mathematics Subject Classification: Primary 46S40; Secondary 40A05.

On the Domain of the Triangle on the Spaces of Null, Convergent, and Bounded Sequences

We introduce the spaces of -null, -convergent, and -bounded sequences. We examine some topological properties of the spaces and give some inclusion relations concerning these sequence spaces. Furthermore, we compute -, -, and -duals of these spaces. Finally, we characterize some classes of matrix transformations from the spaces of -bounded and -convergent sequences to the spaces of bounded, almost convergent, almost null, and convergent sequences and present a Steinhaus type theorem.

Some Geometric Properties of the Domain of the Double Sequential Band Matrix in the Sequence Space

The sequence space was introduced by Maddox (1967). Quite recently, the sequence space of nonabsolute type has been introduced and studied which is the domain of the double sequential band matrix in the sequence space by Nergiz and Başar (2012). The main purpose of this paper is to investigate the geometric properties of the space , like rotundity and Kadec-Klee and the uniform Opial properties. The last section of the paper is devoted to the conclusion.