Volume 11, Issue 1

In this paper we define a subclass of uniformly quasiconvex functions and show that this class is preserved under the Alexander and Bernardi integral operators.

We establish some new oscillation criteria for higher order nonlinear differential equations. The main idea involves comparing our equation with the related linear second order differential equations so that the known oscillation theorems from the literature can be employed directly.

The canonical form of elements of a 𝐺-free nilpotent group of step 3 is defined assuming that the group 𝐺 contains no elements of order 2.

The Banach fixed point theorem is used to investigate the existence of fuzzy solutions for first and second order neutral functional differential equations with nonlocal conditions.

Let V be a complex localizing Banach space with countable unconditional basis and E a rank r holomorphic vector bundle on P ( V ). Here we study the holomorphic embeddings of P ( E ) into products of projective spaces and the holomorphic line bundles on P ( E ). In particular we prove that if r ≥ 3, then H 1 ( P ( E ), L ) = 0 for every holomorphic line bundle L on P ( E ).

Using the complex representation formulae of regular solutions of equations of statics of the theory of elastic mixtures, we construct the explicit solutions of the Dirichlet and Neumann type boundary value problems for an annulus in the form of absolutely and uniformly convergent series.

Particular solutions of a class of higher order ordinary differential equations, with non-constant coefficients, are determined by using the properties of the Laguerre exponentials functions introduced by G. Dattoli and P. E. Ricci in [Georgian Math. J. 10: 481–494, 2003]

We study the Hyers–Ulam stability problem for the Cauchy and Wilson integral equations where 𝐺 is a topological group, 𝑓, 𝑔 : 𝐺 → ℂ are continuous functions, μ is a complex measure with compact support and σ is a continuous involution of 𝐺. The result obtained in this paper are natural extensions of the previous works concerning the Hyers–Ulam stability of the Cauchy and Wilson functional equations done in the particular case of μ = δ e : The Dirac measure concentrated at the identity element of 𝐺.

Let 𝐸 be a uniformly convex Banach space which satisfies Opial's condition or its dual 𝐸* has the Kadec–Klee property, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑇 𝑗 : 𝐶 → 𝐶 an asymptotically nonexpansive mapping for each 𝑗 = 1, 2, . . . , 𝑟. Suppose {𝑥 𝑛 } is generated iteratively by where 𝑈 𝑛(0) = 𝐼, 𝐼 is the identity map and { α 𝑛(𝑗) } is a suitable sequence in [0, 1]. If the set of common fixed points of is nonempty, then weak convergence of {𝑥 𝑛 } to some is obtained.

In this paper we characterize varieties of universal algebras in which every fibre Pt( B ) of the fibration of points has normal product projections, i.e., ( X × Y )/ X ≈ Y holds in every fibre Pt( B ).

New modified sine sums are introduced and a criterion for the L 1 -convergence of these modified sine sums under a new class K is obtained. Also a necessary and sufficient condition for the L 1 -convergence of the cosine series is deduced as a corollary.

The notion of a homogeneous covering of a given set is introduced and examined. Some homogeneous coverings of a Euclidean space, consisting of pairwise congruent geometric figures (spheres, hyperplanes, etc.), are constructed using the method of transfinite induction.

Some entire modular forms of weight on the congruence subgroup Γ 0 (4 N ) are constructed if s is odd > 11. The constructed type of entire modular forms is useful for revealing the arithmetical meaning of additional terms in formulas for the number of representations of positive integers by positive quadratic forms with integral coefficients if the number s of variables is odd > 11.

We derive a backward stochastic differential equation and a Bellman equation characterizing the minimal entropy martingale measure for market models, where asset prices are driven by Markov diffusion processes. A relation between these equations is established.

Sufficient conditions are established for the oscillation and nonoscillation of the system 𝑢′ = 𝑞(𝑡)𝑣, 𝑣′ = –𝑝(𝑡)𝑢, where 𝑝, 𝑞 : [0, +∞[→] – ∞, +∞[ are locally summable functions, 𝑞(𝑡) ≥ 0 for 𝑡 ≥ 0, and .

Formulas of variation are proved for a solution of quasi-linear controlled neutral differential equations with variable delays, when at the initial moment of time the value of the initial function always coincides with the value of the trajectory.

A neutral difference equation with positive and negative coefficients is a natural model for a population with depletion of the aged. In this paper, we obtain conditions which are sufficient for one such equation to support an eventually positive population.

Two sharp inequalities of Simpson type are established. Applications in numerical integration are also given.

We extend Rothberger's theorem (on the equivalence between CH and the existence of Luzin and Sierpiński-sets having power 𝔠) and certain paradoxical constructions due to Erdös. More precisely, by employing a suitable σ -ideal associated to the ( α, β )-games introduced by Schmidt, we prove that the Continuum Hypothesis holds if and only if there exist subgroups of (ℝ, +) having power 𝔠 and intersecting every “absolutely losing” (respectively, every meager and null) set in at most countably many points.