Volume 18, Issue 4

Recently, Huang, Li and O'Regan introduced a class of (scalar) generalized -complementarity problems and three classes of variational inequalities in real Banach spaces for a fixed cone. Our aim is to introduce a vector case of their problems in the real topological vector spaces setting for a moving cone and to generalize their results even in the scalar case.

We show how to construct a free or a totally free quadratic module on suitable construction data by using free simplicial groups. Similar freeness results are also explored for quadratic chain complexes.

We consider an initial boundary-value problem for the regularized long-wave equation. A three level one-parameter family of conservative difference schemes is studied. Two level schemes are used to find the values of the unknown functions on the first level. A numerical method for selection of artificial boundary conditions is proposed. It is proved that the finite difference scheme converges at rate O ( 2 h 2 ) when an exact solution belongs to the Sobolev space .

In this paper, we investigate the existence, the number and the growth of periodic meromorphic solutions of the Riccati differential equation u A ( z ) u 2 , where A ( z ) is a periodic function with period . In addition, we give many examples to illustrate our results.

We prove that { : L ( R s ), { x : M ( x )>1} M dx < } V ( R s ) L (1 ln L ) s ( R s ) V ( R s ), where M ( x ) is a maximal function with respect to parallelepipeds and V ( R s ) is a set of those functions of s variables ( x ) for which in every octant , i 1, . . . , 2 s , one can find s 1 variables such that |( x )| does not increase with respect to those variables which are positive in this octant and does not decrease with respect to those variables which are negative.

A defining equation of a curve of genus five with at least two Weierstrass points with gap sequence {1, 2, 3, 5, 9} determining the same pencil of degree 4 is obtained. As an application, a defining equation of such a curve with an involution is obtained as well as all the Weierstrass points and their weights.

We show that if a plane convex curve of constant width is sufficiently smooth and has strictly positive curvature at all of its points, then it can be approximated by algebraic convex curves of constant width.

Criteria of the conditional well-posedness of nonlocal problems for linear partial differential equations of hyperbolic type and for higher order linear ordinary differential equations with singular coefficients are established.

The main purpose of this paper is to establish a necessary condition and two sufficient conditions ensuring that the shift-invariant system is a frame for . As some applications, the results are used to obtain some known conclusions about wavelet frames and Gabor frames.

For a double array of blockwise M -dependent random elements { V mn : m 1, n 1} taking values in a real separable Rademacher type p (1 p 2) Banach space, we provide conditions to obtain the almost sure convergence for double sums , m 1, n 1. The paper treats two cases: (i) { V mn : m 1, n 1} is block-wise M -dependent with EV mn 0, m , n 1, and (ii) { V mn : m 1, n 1} is block-wise p -orthogonal. The conditions for case (i) are shown to provide exact characterizations of Rademacher type p and stable type p Banach spaces. Examples are given showing that the conditions cannot be removed or weakened. It is also demonstrated that some of the well-known theorems in the literature are special cases of our results.

In this paper we obtain the boundedness, 0 < p 1, of integral operators of the form T ( x ) | x a 1 y | 1 | x a m y | m ( y ) dy , 1 m n and a i \ {0}, a i a j for i j , 1 i , j m . We also show that these operators are not bounded on H p ().