Celal Çakan

The σ-convergence and σ-core of double sequences

The σ-convergence and σ-core of a real bounded sequence were introduced in [R. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J. 30 (1963) 81-94] and [S.L. Mishra, B. Satapathy, N. Rath, Invariant means and σ-core, J. Indian Math. Soc. 60 (1984) 151-158], respectively. In this work, we extend these ideas to double sequences.

On the Cesàro convergence of sequences of fuzzy numbers

Abstract In this paper we have determined necessary and sufficient Tauberian conditions under which convergence follows from ( C , 1 ) -convergence of sequences of fuzzy numbers.

Some Topological and Geometrical Properties of a New Difference Sequence Space

We introduce the new difference sequence space 𝑎𝑟𝑝(Δ) . Further, it is proved that the space 𝑎𝑟𝑝(Δ) is the BK-space including the space 𝑏𝑣𝑝, which is the space of sequences of pbounded variation. We also show that the spaces 𝑎𝑟𝑝(Δ), and l𝑝 are linearly isomorphic for 1≤𝑝l∞. Furthermore, the basis and the 𝛼-, 𝛽- and 𝛾-duals of the space 𝑎𝑟𝑝(Δ) are determined. We devote the final section of the paper to examine some geometric properties of the space 𝑎𝑟𝑝(Δ).

A class of conservative four-dimensional matrices.

The concepts and for double sequences were introduced by Patterson in 1999. In this paper, we have studied some new inequalities related to these concepts by using the RH-conservative four-dimensional matrices.

Some new inequalities related to the invariant means and uniformly bounded function sequences

Abstract Agnew gave the necessary and sufficient conditions for the triangular regular matrices to yield lim sup n G ( A n ( F ) ; D ) ≤ lim sup k G ( f k ; D ) for all uniformly bounded real function sequences F = ( f k ) . Recently, Duman studied some variants of this inequality by replacing the operator limsup with st-limsup. In this note, we present some new inequalities involving invariant means like the above mentioned results.

The riesz convergence and riesz core of double sequences

In this article, we have introduced the Riesz convergence and Riesz core of double sequences and determined the necessary and sufficient conditions on a four-dimensional matrix A to yield PR- core{Ax} ⊆ P - core{x} and PR- core{Ax} ⊆ st2- core{x} for all x∈ℓ∞2.Mathematics Subject Classification 2000: 40C05; 40J05; 46A45.

Rough I-Convergence

Abstract In this work, using the concept of I-convergence and using the concept of rough convergence, we introduced the notion of rough I-convergence and the set of rough I-limit points of a sequence and obtained two rough I-convergence criteria associated with this set. Later, we proved that this set is closed and convex. Finally, we examined the relations between the set of I-cluster points and the set of rough I-limit points of a sequence

The extension of the Knopp core theorem to the sequences of fuzzy numbers

The concept of the core of a sequence of fuzzy numbers has been introduced by Aytar et al. in The core of a sequence of fuzzy numbers, Fuzzy Sets and Systems, 159(24) (2008) 3369-3379]. Quite recently, some matrix transformations on the classical sets of sequences of fuzzy numbers have been characterized by Talo and Basar in Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations, Comput. Math. Appl. 58 (2009) 717-733]. In this paper, we have proved a core theorem for sequences of fuzzy numbers which is analogous to the famous Knoop Core Theorem for real sequences. To achieve this goal, we have studied some properties of the lim sup and lim inf of sequences of fuzzy numbers. Also, we have characterized a class of positive regular matrices of fuzzy numbers such that these matrices leave the core of any bounded sequence of fuzzy numbers invariant.

Some topological and geometrical properties of the sequence space er (u,p)

Abstract In this paper, we introduce the sequence space e r ( u,p ) and investigate its some topological and geometrical properties such as basis, α-, β-, γ- duals and the uniform Opial property.

Characterization of -Core and Absolute Equivalence of Double Sequences

The -core of a double sequence has been defined and it is studied by many authors. In this paper, we have determined two permutations and on the set of natural numbers for which for all .