Volume 14, Issue 4

In this paper we obtain some new oscillation criteria for the second order nonlinear neutral delay dynamic equation (𝑥(𝑡) – 𝑝(𝑡)𝑥(𝑡 – τ 1 )) ΔΔ + 𝑞(𝑡)𝑓(𝑥(𝑡 – τ 2 )) = 0, on a time scale 𝕋. Moreover, a new sufficient condition for the oscillation sublinear equation (𝑥(𝑡) – 𝑝(𝑡)𝑥(𝑡 – τ 1 )) ″ + 𝑞(𝑡)𝑓(𝑥(𝑡 – τ 2 )) = 0, is presented, which improves other conditions and an example is given to illustrate our result.

For some class of nonuniformly degenerated elliptic equations of second order, a necessary and sufficient condition for boundary points to be regular is found. This condition is an analogue of Wiener's criterion for the Laplace equation.

The goal of this paper is to present some extensions of the method of quasi-reversibility applied to an ill-posed Cauchy problem associated with an unbounded linear operator in a Hilbert space. The key point to our proof is the use of a new perturbation to construct a family of regularizing operators for the considered problem. We show the convergence of this method, and we estimate the convergence rate under a priori regularity assumptions on the problem data.

We study the pointwise convergence of Walsh–Fourier series on the class of functions of generalized 𝑝-bounded fluctuation.

We estimate the rate of convergence of the Bézier variant of Durrmeyer type Meyer–König and Zeller operators for functions with derivatives of bounded variation defined on [0, 1].

This paper contains some results connected with topologies generated by lower and semi-lower density operators. We show that in some measurable spaces (𝑋, 𝑆, 𝐽) there exists a semi-lower density operator which does not generate a topology. We investigate some properties of nowhere dense sets, meager sets and σ -algebras of sets having the Baire property, associated with the topology generated by a semi-lower density operator.

The strong asymptotics of monic extremal polynomials with respect to the norm 𝐿𝑝( σ ) are studied. The measure σ is concentrated on the segment [–1, 1] plus a finite set of mass points in a region of the complex plane exterior to the segment [–1, 1].

We consider finite systems of straight lines in the Euclidean plane 𝐑 2 with some of their combinatorial characteristics. Euler's formula is applied for obtaining results of combinatorial type for such systems. In particular, a lower estimate for the number of two-sided and three-sided domains determined by a given finite line-system in 𝐑 2 is presented and it is shown that this estimate is precise in a certain sense.

Let 𝑋 be a completely regular Hausdorff space, 𝐸 a quasi-complete locally convex space, 𝐶(𝑋) (resp. 𝐶 𝑏 (𝑋)) the space of all (resp. all, bounded), scalar-valued continuous functions on 𝑋, and 𝐵(𝑋) and 𝐵 0 (𝑋) be the classes of Borel and Baire subsets of 𝑋. We study the spaces 𝑀 𝑡 (𝑋,𝐸), 𝑀 τ (𝑋,𝐸), 𝑀 σ (𝑋,𝐸) of tight, τ -smooth, σ -smooth, 𝐸-valued Borel and Baire measures on 𝑋. Using strict topologies, we prove some measure representation theorems of linear operators between 𝐶 𝑏 (𝑋) and 𝐸 and then prove some convergence theorems about integrable functions. Also, the Alexandrov's theorem is extended to the vector case and a representation theorem about the order-bounded, scalar-valued, linear maps from 𝐶(𝑋) is generalized to the vector-valued linear maps.

Some “sharp” conditions are established for a kind of linear neutral difference equations with continuous arguments not to possess eventually positive solutions. The existence and asymptotic behavior are obtained for positive solutions of the kind of equations. The results for linear cases are further extended to nonlinear ones. A comparison principle, which is a necessary and sufficient condition, for linear equations not to possess eventually positive solutions is also presented.

The main purpose of this paper is to prove that the class of near locally connected continua contains no non-metrizable continuum 𝑋 which admits a Whitney map for 𝐶(𝑋). In particular, each near locally connected continuum 𝑋 which admits a Whitney map for 𝐶(𝑋) is metrizable.

This paper is concerned with the rate of convergence of recursive estimation procedures for parameters of discrete time stochastic processes. Applications to estimation of parameters in several models are presented to illustrate the theory.

We are concerned with the oscillatory and nonoscillatory behavior of solutions of differential equations involving an even order nonlinear Sturm–Liouville operator of the form where α and β are distinct positive constants. We first give the criteria for the existence of nonoscillatory solutions with specific asymptotic behavior on infinite intervals, and then derive necessary and sufficient conditions for all solutions of (∗) to be oscillatory by eliminating all nonoscillatory solutions of (∗).

Let 𝑓 be a 2 π periodic function in 𝐿 1 [0,2 π ] and , be its Fourier coefficients. Extending the classical result of Zygmund, Schramm and Waterman obtained the sufficiency conditions for the absolute convergence of Fourier series of functions of ∧𝐵𝑉 (𝑝) and φ ∧𝐵𝑉. Here we have generalized these results by obtaining certain sufficiency conditions for the convergence of the series , where is a strictly increasing sequence of natural numbers and 𝑛 –𝑘 = –𝑛 𝑘 for all 𝑘, for such functions.

In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

We consider a generalized von Foerster equation in one dimensional spatial variable and construct finite difference schemes for the initial value problem. The stability of finite difference schemes on irregular meshes generated by characteristics is studied.

Corrections to [Oniani, Georgian Math. J. 13: 501–514, 2006] are listed.