Volume 11, Issue 1

We prove that if ℚ is a nw-nep forcing then it cannot add a dominating real. We also show that amoeba forcing cannot be P ( X )/ I if I is an ℵ 1 -complete ideal. Furthermore, we generalize the results of [Shelah, Israel J. Math. 88: 159–174, 1994].

A multiobjective nonlinear programming problem is considered. Sufficiency theorems are derived for efficient and properly efficient solutions under generalized ( F, ρ )-convexity assumptions. Weak, strong and strict converse duality theorems are established for a general Mond–Weir type dual relating properly efficient solutions of the primal and dual problems.

This paper presents new existence results for singular discrete boundary value problems for the one-dimension p -Laplacian. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign. Our results are new even for p = 2.

A function ƒ from a countable product X = Π i X i of Polish spaces into a Polish space is separately nowhere constant provided it is nowhere constant on every section of X . We show that every continuous separately nowhere constant function is one-to-one on a product of perfect subsets of X i 's. This result is used to distinguish between n -cube density notions for different n ≤ ω , where ω -cube density is a basic notion behind the Covering Property Axiom CPA formulated by Ciesielski and Pawlikowski. We will also distinguish, for different values of α < ω 1 , between the notions of α -prism densities — the more refined density notions used also in CPA.

In this paper, a generalization of convexity, namely V - r -invexity, is considered in the case of nonlinear multiobjective programming problems where the functions involved are differentiable. The assumptions on Pareto solutions are relaxed by means of V - r -invex functions. Also some duality results are obtained for such optimization problems.

We prove the existence of weak solutions for some quasilinear elliptic reaction-diffusion systems with Dirichlet boundary conditions and satisfying to the two main properties: the positivity of the solutions and the balance law. The nonlinearity we consider here has critical growth with respect to the gradient and the data are in L 1 .

We study the existence of solutions to a class of problems u ″ + ƒ( t, u ) = 0, u (0) = u (1) = 0, where ƒ( t , ·) is allowed to be singular at t = 0, t = 1.

This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω 2 q of . The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L 2 (Ω 2 q ). As a bypass, we study the inner product over the space of polynomials in the variables z , , in which ƒ( D ̅) is the differential operator with symbol ƒ( z ̄). On the spaces of spherical harmonics, it is shown that the inner product [·, ·] reduces to a multiple of the L 2 (Ω 2 q ) inner product. Bi-orthogonality in (, [·, ·]) is fully investigated.

We consider the wave equation with a fractional damping of order between 0 and 1 and a polynomial source. Introducing a new functional and using an argument due to Georgiev and Todorova [J. Differential Equations 109: 295–308, 1994] together with some appropriate estimates, it is proved that some solutions blow up in finite time.

In this paper we consider the Hadamard product ∗ of regular functions using the concept of subordination. Let P ( A, B ) denote the class of regular functions subordinated to the linear fractional transformation (1 + Az )/(1 – Bz ), where A + B ≠ 0 and | B | ≤ 1. By P ( A, B ) ∗ P ( C, D ) we denote the set {ƒ ∗ g : ƒ ∈ P ( A, B ), g ∈ P ( C, D )}. It is known ([London, Math. Japon. 43: 23–29, 1996], [Stankiewicz, Stankiewicz, Folia Sci. Univ. Tech. Resov. Math. 7: 93-101, 1988]), that for some complex numbers A, B, C, D there exist X and Y such that P ( A, B ) ∗ P ( C, D ) ⊂ P ( X, Y ). The purpose of this note is to find the necessary and sufficient conditions for the equality of the classes P ( A, B ) ∗ P ( C, D ) and P ( X, Y ).