Volume 3, Issue 6

The concept of a strongly isolated solution of the nonlinear boundary value problem dx ( t ) = dA ( t ) · f ( t , x ( t )), h ( x ) = 0, is introduced, where A : [ a , b ] → R n×n is a matrix-function of bounded variation, f : [ a , b ] × R n → R n is a vector-function belonging to a Carathéodory class, and h is a continuous operator from the space of n -dimensional vector-functions of bounded variation into R n . It is stated that the problems with strongly isolated solutions are correct. Sufficient conditions for the correctness of these problems are given.

A characterization of a regular family of semimartingales as a maximal family of processes with respect to which one can define a stochastic line integral with natural continuity properties is given.

We consider a partition of a space X consisting of a meager subset of X and obtain a sufficient condition for the existence of a subfamily of this partition which gives a non-Baire subset of X . The condition is formulated in terms of the theory of J. Morgan [Point Set Theory, Marcel Dekker, 1990].

For a hyperbolic type model equation of third order a Darboux type problem is investigated in a dihedral angle. It is shown that there exists a real number ρ 0 such that for α > ρ 0 the problem under consideration is uniquely solvable in the Frechet space. In the case where the coefficients are constants, Bochner's method is developed in multidimensional domains, and used to prove the uniquely solvability of the problem both in Frechet and in Banach spaces.

We consider various possibilities concerning the continuous extension of continuous functions taking values in an ultrametric space. In Section 1 we consider Tietze-type extension theorems concerning continuous extendibility of continuous functions from compact and closed subsets to the whole space. In Sections 2 and 3 we consider extending “separated” continuous functions in such a way that certain continuous extensions remain separated. Functions taking values in a complete ultravalued field are dealt with in Section 2, and the real and complex cases in Section 3.

Sufficient conditions are found for the oscillation of proper solutions of the system of differential equations where f i : R + × R 2 m → R ( i = 1, 2) satisfy the local Carathéodory conditions and σ i , τ i : R + → R ( i = 1, . . . , m ) are continuous functions such that σ i ( t ) ≤ t for t ∈ R + , .

Sufficient (almost necessary) conditions are given on the weight functions u (·), v (·) for to hold when Φ 1 , Φ 2 are -functions with subadditive , and Ms (0 ≤ s < n ), is the usual fractional maximal operator.