In this paper, we introduce the concepts of I2-Cauchy, I∗ 2 -Cauchy double sequence of fuzzy numbers and study their some properties and relations, where I2 denotes the ideal of subsets of N× N.

We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H2 we get a formula that appears to be new. In the projective model of H2 we have P = (1/2) ∫ w dφ. Here w is the Euclidean length of the projection of K from the ideal boundary point R = (cosφ, sinφ) onto...

In the present paper we study lightlike submanifolds of almost paracontact metric manifolds. We define invariant lightlike submanifolds. We study radical transversal lightlike submanifolds of para-Sasakian manifolds and investigate the geometry of distributions. Also we introduce a general notion of paracontact Cauchy-Riemann (CR) lightlike submanifolds and we derive some necessary and sufficie...

The notion of a strongly summing sequence is introduced. Such a sequence is weak-Cauchy, a basis for its closed linear span, and has the crucial property that the dual of this span is not weakly sequentially complete. The main result is: Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either a strongly summing sequence or a convex block basis equivalent t...

The notion of a strongly summing sequence is introduced. Such a sequence is weak-Cauchy, a basis for its closed linear span, and has the crucial property that the dual of this span is not weakly sequentially complete. The main result is: Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either a strongly summing sequence or a convex block basis equivalent t...

A singular Cauchy-Nicoletti problem for a system of three ordinary differential equations is considered. An approach which combines topological method of T. Ważewski and Schauder’s principle is used. Theorem concerning the existence of a solution of this problem (a graph of which lies in a given domain) is proved. Moreover, an estimation of its coordinates is obtained.

We use τK to denote the topology of DK (Ω) equipped with such metric. The topology of D (Ω) can be defined precisely. Let β be the collection of all convex balanced sets W ⊂ D (Ω) such that DK (Ω) ∩ W ∈ τK for every compact K ⊂ Ω. Let τ be the collection of all unions of sets of the form φ+W with φ ∈ D (Ω) and W ∈ β. Theorem 1. τ is a topology in D (Ω) and β is a local base for τ . The topology...

Recently, K. Kamano studied sums of products of Bernoulli numbers including poly-Bernoulli numbers. A relation among these sums was given, and an explicit expression of sums of two products was also given, reduced to the famous Euler’s formula. The concept of poly-Cauchy numbers is given by the author as a generalization of the classical Cauchy number and an analogue of poly-Bernoulli number. I...

and Applied Analysis 3 Lemma 3. Let {x k } and {z k } be any sequence generated by Algorithm 1. Suppose that g is monotone and that the solution set of (1) is not empty, then {x k } and {z k } are both bounded. Furthermore, it holds that lim k→∞ 󵄩󵄩󵄩󵄩xk − zk 󵄩󵄩󵄩󵄩 = 0, (13) lim k→∞ 󵄩󵄩󵄩󵄩xk+1 − xk 󵄩󵄩󵄩󵄩 = 0. (14) Proof. From (6), we have ⟨g (z k ) , x k − z k ⟩ = −α k ⟨g (z k ) , d k ⟩ ≥ σα 2 k 󵄩󵄩󵄩󵄩...