THE SPACE L_q OF DOUBLE SEQUENCES

The spaces BS, BS(t), CSp, CSbp, CSr and BV of double sequences have recently been studied by Altay and Ba¸sar [J. Math. Anal. Appl. 309(1)(2005), 70–90]. In this work, following Altay and Ba¸sar [1], we introduce the Banach space Lq of double sequences corresponding to the well-known space &#8467;q of single sequences and examine some properties of the space Lq. Furthermore, we determine the &#946;(&#965;)-dual of the space and establish that the &#945;- and &#947;-duals of the space Lq coincide with the &#946;(&#965;)-dual; where 1 &#8804; q &#60; &#8734; and &#965; 2 {p, bp, r}.</p

On strongly I-Lacunary Cauchy sequences of sets

In this study, we examinate the ideas of Wijsman strongly lacunary Cauchy, Wijsman strongly I-lacunary Cauchy and Wijsman strongly I∗-lacunary Cauchy sequences of sets and investigate the relationship between them

Multipliers for bounded convergent double sequences

In this paper, we investigate multipliers for bounded convergence of double sequences and study some properties and relations between l2 ∞, c2(b) and c20(b)

The space Lq of double sequences

The spaces BS, BS(t), CSp, CSbp, CSr and BV of double sequences have recently been studied by Altay and Başar [J. Math. Anal. Appl. 309(1)(2005), 70–90]. In this work, following Altay and Başar [1], we introduce the Banach space Lq of double sequences corresponding to the well-known space ℓq of single sequences and examine some properties of the space Lq. Furthermore, we determine the β(υ)-dual of the space and establish that the α- and γ-duals of the space Lq coincide with the β(υ)-dual; where 1 ≤ q < ∞ and υ ∈ {p, bp, r}

On Kuratowski i-convergence of sequences of closed sets

In this paper we extend the concepts of statistical inner and outer limits (as introduced by Talo, Sever and Bas¸ar) to I− inner and I− outer limits and give some I− analogue of properties of statistical inner and outer limits for sequences of closed sets in metric spaces, where I is an ideal of subsets of the set N of pos- itive integers. We extend the concept of Kuratowski statistical convergence to Kuratowski I− convergence for a sequence of closed sets and get some properties for Kuratowski I− convergent sequences. Also, we examine the relationship between Kuratowski I− convergence and Hausdorff I− convergence

On some classes of four dimensional regular matrices

In 1981, Rath and Tripathy presented some classes of regular matrices such that every bounded sequence is limitable by some member of each class of ordinary conservative matrices using ordinary sequences. The goal of this paper is to present multidimensional analogues of their results

Multipliers for bounded statistical convergence of double sequences

Multipliers and factorizations for bounded statistically convergent sequences were studied in μ-density by Connor et al. [J. Connor, K.Demirci, C. Orhan, Multipliers and factorizations for bounded statistically convergent sequences, Analysis 22 (2002), 321-333]. In this paper we get analogous results of multipliers for bounded statistically convergent double sequences

On statistical convergence of double sequences of closed sets

In this paper, we introduce the concepts of statistical inner and statistical outer limits for double sequences of closed sets and give some formulas for finding these limits. Also, we give the Kuratowski statistical convergence of double sequences of sets by means of the statistical inner and statistical outer limits of a double sequence of closed sets

Γ-CONVERGENCE OF DOUBLE SEQUENCES OF FUNCTIONS, AND MINIMIZERS

In the present paper, we introduce the concept of Γ-convergence of a double sequence of functions defined from a metric space into real numbers. This convergence is useful as it is a convenient concept of convergence for approximating minimization problems in the field of mathematical optimization. First, we compare this convergence with pointwise and uniform convergence and obtain some properties of Γ-convergence. Later we deal with the problem of minimization. We prove that, under some additional assumptions, the Γ-convergence of a double sequence (f_{kl}) to a function f implies the convergence of the minimum values of f_{kl} to the minimum value of f. Moreover, we prove that each limit point of the double sequence of the minimizers of f_{kl} is a minimizer of f

E-core of double sequences

Boos, Leiger and Zeller [1,2] defined the concept of e-convergence. In this paper we introduce the concepts of e-limit superior and inferior for real double sequences and prove some fundamental properties of e-limit superior and inferior. In addition to these results we define e-core for double sequences. Also, we show that that if A is a nonnegative C e -regular matrix then the e-core of Ax is contained in e-core of x, provided that Ax exists