Volume 46, Issue 3

Numerical stability of two main direct methods for solving the symmetric saddle point problem are analyzed. The first one is a generalization of Golub’s method for the augmented system formulation (ASF) and uses the Householder QR decomposition. The second method is supported by the singular value decomposition (SVD). Numerical comparison of some direct methods are given.

Let R be a semiprime ring. An additive mapping F : R → R is called a generalized derivation of R if there exists a derivation d : R → R such that F ( xy ) = F ( x ) y + xd ( y ) holds, for all x , y ∈ R . The objective of the present paper is to study the following situations: (1) [ d ( x ), F ( y )] = ±[ x , y ]; (2) [ d ( x ), F ( y )] = ± x ο y ; (3) [ d ( x ), F ( y )] = 0; (4) d ( x )ο F ( y ) = ± x ο y ; (5) d ( x )ο F ( y ) = ±[ x , y ]; (6) d ( x )ο F ( y ) = 0; (7) d ( x ) F ( y )± xy ∈ Z ( R ), for all x , y in some appropriate subset of R .

Some variants of the Lagrange and Cauchy mean-value theorems lead to the conclusion that means, in general, are not symmetric. They are symmetric iff they coincide (respectively) with the Lagrange and Cauchy means. Under some regularity assumptions, we determine the form of all the relevant symmetric means.

In this paper, two new integral inequalities of Hadamard-type for product of m and (α, m )-convex functions and their applications for special means are given.

In this paper, we present a double inequality for the gamma function by estimating bounds of Г p function. Later, we also give a new inequality of gamma function.

In this article, we study the L p , 1 ≤ p < ∞ approximation properties of general multivariate singular integral operators over R N , N ≥ 1. We establish their convergence to the unit operator with rates. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss–Weierstrass, Poisson Cauchy and trigonometric singular integral operators where this theory can be applied directly.

In the present paper, we derive some subordination and superordination results for p-valent functions in the open unit disk by using certain fractional derivative operator. Relevant connections of the results, which are presented in the paper, with various known results are also considered.

K. I. Noor (2007 Appl. Math. Comput. 188, 814–823) has defined the classes 𝒬 k ( a , b , λ , γ) and 𝒯 k ( a , b , λ, γ) of analytic functions by means of linear operator connected with incomplete beta function. In this paper, we have extended some of the results and have given other properties concerning these classes.

Let ℳ n ( a , b , c ) denote a class of functions of the form f ( z ) = z − a n +1 z n +1 − … − a k z k − … which are analytic in open unit disk U = { z : | z | < 1} and satisfy the condition ∑k=n+1∞k(k+a)k+bak<c,   ak≥0,   a≥b≥0,   0<c≤1,   n∈N={1,2,…}.$$\sum\limits_{k = n + 1}^\infty {{{k(k + a)} \over {k + b}}a_k < c,\;\;\;a_k \ge 0,\;\;\;a \ge b \ge 0,\;\;\;0 < c \le 1,\;\;\;n \in N = \{ 1,2, \ldots \} .} $$ In this paper, we obtain the extreme points and support points of the class ℳ n ( a , b , c ) of functions.

This paper deals with theorems and formulas using the technique of Laplace and Steiltjes transforms expressed in terms of interesting alternative logarithmic and related integral representations. The advantage of the proposed technique is illustrated by logarithms of integrals of importance in certain physical and statistical problems.

In this paper, we discuss and present various results about acting and boundedness conditions of the autonomous Nemitskij operator on certain function spaces related to the space of all real valued Lipschitz (of bounded variation, absolutely continuous) functions defined on a compact interval of ℝ. We obtain a result concerning the integrability of products of the form ψ ο f · f′ · f ( k ) and a generalized version of the chain rule for functions a.e differentiable, in the sense of Lebesgue. As an application, we get a generalization of a theorem due to V. I. Burenkov for the case of functions of bounded Riesz- p -variation.

In this paper, we establish the strong convergence for a modified three-step iterative scheme with errors associated with three mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest. Remark at the end simplifies many known results.

In this paper, we discuss some properties of the weighted Hankel operator Hψβ$H_\psi ^\beta $ and describe the conditions on which the weighted Hankel operator Hψβ$H_\psi ^\beta $ and weighted Toeplitz operator Tϕβ$T_\phi ^\beta $, with ϕ , ψ ∈ L ∞ ( β ) on the space H 2 ( β ), β = { β n } n∈Ƶ being a sequence of positive numbers with β 0 = 1, commute. It is also proved that if a non-zero weighted Hankel operator Hψβ$H_\psi ^\beta $ commutes with Tϕβ$T_\phi ^\beta $, which is not a multiple of the identity, then Hψβ=μTϕβ$H_\psi ^\beta = \mu T_\phi ^\beta $, for some μ ∈ ℂ.

We consider an elastic thin film as a bounded open subset ω of ℝ 2 . First, the effective energy functional for the thin film ω is obtained, by Г-convergence and 3 D -2 D dimension reduction techniques applied to the sequence of re-scaled total energy integral functionals of the elastic cylinders ω×(-ε2,ε2)$\omega \times \left( { - {\varepsilon \over 2},{\varepsilon \over 2}} \right)$ as the thickness ε goes to 0. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function for the elastic cylinders has the growth prescribed by an Orlicz convex function M . Here M is assumed to be non-power-growth-type and to satisfy the conditions ∇ 2 and ∇ 2 (that is equivalent to the reflexivity of Orlicz and Orlicz–Sobolev spaces generated by M ). These results extend results of H. Le Dret and A. Raoult for the case M ( t ) = | t | p for some p ∈ (1, ∞).

Some results concerning translative coverings of squares and triangles by two, three and four unit squares are presented.

In this paper, we introduce and study the concept of [ γ , γ′ ]-preopen sets in topological space.