In the present paper a concept of topological essentiality for a large class of multivalued mappings is introduced. This concept is strictly related to the Leray-Schauder topological degree theory but is simpler and also more general. Applying the above concept to boundary value problems for differential inclusion with both upper semi-continuous and lower semi-continuous right hand sides, several new results are obtained.

Consider the following neutral delay differential equation

where P ∈ ℝ, T ∈ (0, ∞), σ ∈ (0, ∞) and Q ∈ C[(t0, ∞), [0, ∞)]. We obtain a sufficient condition for the oscillation of all solutions of Equation (*) with P = −1, which does not require that

But, for the cases −1 < P < 0 and P < −1, we show that (**) is a necessary condition for the oscillation of all solutions of Equation (*). These new results solve some open problems in the literature.

It is shown that the set of all orthodox subsemigroups of an orthodox semigroup forms a lattice. This lattice is a join-sublattice of the lattice of all semigroups, but is not in general a meet-sublattice. We obtain results concerning lattice isomorphisms between orthodox semigroups, several of which include known results for inverse semigroups as special cases.

We show that trigonometric Lagrange interpolating approximation with arbitrary real distinct nodes in Lp space for 1 ≤ p < ∞, as that with equally spaced nodes in Lp space for 1 < p < ∞ in an earlier paper by T.F. Xie and S.P. Zhou, may also be arbitrarily “bad”. This paper is a continuation of this earlier work by Xie and Zhou, but uses a different method.

The set functions associated with Schrödinger's equation are known to be unbounded on the algebra of cylinder sets. However, there do exist examples of scalar valued set functions which are unbounded, yet σ-additive on the underlying algebra of sets. The purpose of this note is to show that the set functions associated with Schrödinger's equation are not σ-additive on cylinder sets. In the course of the proof, general conditions implying the non-σ-additivity of unbounded set functions are given.

The minimum area a(v) of a v–sided convex lattice polygon is known to satisfy . We conjecture that a(v) = cv3 + o(v3), for c a constant; we prove that , and that for some positive constant c, .

Bishop-Gromov type comparison theorems for the volume of a tube about a sub-manifold of a complete Riemannian manifold whose Ricci curvature is bounded from below are proved. The Kähler analogue is also proved.

Let F be a multifunction from [a, b] × ℝn × ℝn into ℝn, with non-empty closed convex values. In this paper we prove that, under suitable assumptions, the multivalued boundary value problem

has at least one solution u ∈ W2, p([a, b], ℝn). Next we point out some particular cases.

In a Minkowski plane with unit circle E, the product of the positive circumference of a plane convex body K and that of its polar dual is greater than or equal to the square of the Euclidean length of the polar dual of E. Equality holds if and only if K is a Euclidean unit circle.

The solution of the linear boundary layer problem is presented as a sum of the reduced solution and the layer function, which is approximated by a truncated orthogonal series. The layer subinterval is determined through the numerical layer length, which depends on the perturbation parameter and the degree of the spectral approximation. Modified Hermite polynomials are used as an orthogonal basis. Systematic methods are presented to obtain the recurrence relations for the coefficients in modified Hermite series solution of linear differential equations with polynomial coefficients. The upper bound for the error function is constructed and the numerical example is included.

For a linearly independent sequence in a normed linear space the moment space is defined. Basic properties of moment spaces are discussed as well as a necessary and sufficient condition for the moment space to be a closed subspace of l∞.

We study some smoothness properties of a Banach space X that are related to the weak* asymptotic-norming properties of the dual space X*. These properties imply that X is an Asplund space and are related to the duality mapping of X.

Certain elliptic equations of higher order are studied and a sufficient condition is given that every solution is oscillatory in an exterior domain. The principal tool is an averaging technique which enables one to reduce the n–dimensional problem to a one-dimensional problem.

Optimal control problem of semilinear evolutionary distributed parameter systems with impulse controls is considered. Necessary conditions of optimal controls are derived. The result generalises the usual Pontryagin's maximum principle.

We prove that there exist skew type Hadamard matrices of order 4n for n = 67, 113, 127, 157, 163, 181 and 241 which have not been constructed so far. In particular there exists a Hadamard matrix of order 4 · 163, which was unknown until now. We mention that very recently we have constructed skew type Hadamard matrices of orders 4n for n = 37 and 43.

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.

In this paper, we deal with the following question: What kind of open covers are normal if they have cushioned open refinements? For this, we prove that an open cover consisting of members with compact closure is a desired one.