Volume 7, Issue 2

Some new criteria for the oscillation of all solutions of second order equations of the form ( a ( t )| y ′( t )| σ –1 y ′( t ))′ + q ( t )| y ( t )| σ sgn y ( t ) = 0, σ > 1, are established. The obtained results extend, unify and corollate many of the known oscillation criteria for second order equations.

The convergence of a multidimensional functional series essentially depends on the way its partial sums are formed. Different ways of definition of partial sums lead to different kinds of convergence. In the paper relations between various kinds of unconditional almost everywhere convergence of multidimensional functional series are studied.

The existence of mild solutions of first order semilinear evolution inclusions on a compact real interval is investigated in Banach spaces with nonlocal conditions. A fixed point theorem for condensing maps due to Martelli is used.

A two-dimensional crack problem in the Comninou formulation is investigated for a piecewise homogeneous plane. Applying a special integral representation formula for the displacement vector the problem is reduced to a system of singular integral equations. The system is analysed and its solvability is proved using the potential method and the theory of singular integral equations.

For a fixed sequence f . = ( f n ) of independent identically distributed symmetric random variables with , we introduce the notion of K f. -convex Banach space and the notions of ( f n )-bounding and ( f n )-converging operators acting between Banach spaces. It is shown that the dual of the space of ( f n )-converging operators between a Hilbert space and a K f. -convex Banach space admits a precise description in terms of trace duality. The obtained results recover similar formulations for almost summing and γ -Radonifying operators.

We present some recent results dealing with the qualitative behavior of solutions of the quasilinear second order differential equation where a , b are positive continuous real functions, and Φ j ( u ) = | u | j –2 u , j > 1. Following a classification of solutions, proposed in the linear case in [Marini and Zezza, J. Diff. Equat. 28: 1–17, 1978], we divide all the solutions of (∗) with respect to their asymptotic behavior into two classes. The existence and uniqueness is considered: a topological approach is employed and a fixed point result for operators associated to boundary value problems on a half-line is used. In addition, some asymptotic estimates are presented and the convergence of solutions to zero as t → ∞ is studied.

It is shown that for every f ∈ L \ L (1 + ln + L ) there exists an equimeasurable function g such that for every 0 < λ 1 < λ 2 we have μ {λ 1 < M I ( g ) < λ 2 } = ∞.

A neutral differential equation of the form ( x ( t ) + μ ( t ) x ( ρ ( t ))) ( n ) + f ( t, x ( τ 1 ( t )), . . . , x ( τ m ( t ))) = 0 is considered, where μ , ρ , τ i : R + → R ( i = 1, . . . , m ) are continuous functions, 0 ≤ μ ( t ) ≤ 1, ρ ( t ) ≤ t for t ∈ R + , and the function f : R + × R m → R satisfies the local Carathéodory conditions. Sufficient conditions are given for the considered equation to have the so-called "weak" properties A and B .

A class of quasilinear hyperbolic equations of mixed type whose characteristic roots are simultaneously characteristic invariants is found. For a special equation of this class, a general integral is constructed in terms of invariants in the closed form by using the method of characteristics. Based on the structure of families of characteristics, the initial Cauchy problem is investigated. The structure of the solution definition and regularity domains is defined using the properties of initial perturbations.

A general nonuniqueness theorem is given for ordinary differential equations with singularities. The criterion uses vector Lyapunov functions and extends the previously known criteria ones. The applicability is illustrated by several examples.

New oscillation and nonoscillation criteria are established for the equation u″ + p ( t )| u | α | u′ | 1– α sgn u = 0, where α ∈]0, 1] and the function p :]0, +∞[→] – ∞, +∞[ is locally integrable.

The properties of solutions of higher order neutral differential equations with distributed type deviating arguments are obtained.

A sum of the singular series corresponding to the number of representations of positive integers by some diagonal quadratic forms with integral coefficients is obtained.

This paper is concerned with periodic boundary value problems involving first order differential equations with functional arguments. The main feature of the paper is that the existence of maximal and minimal solutions is obtained by constructing sequences of upper and lower solutions of the initial value problems and not by establishing the comparison principle.

Assuming ( A 0 , A 1 , . . . , A n ) is a jointly symmetric α -stable random vector, 0 < α ≤ 2, a general formula for the expected number of real zeros of the random algebraic polynomial was obtained by Rezakhah in 1997. Rezakhah's formula is applied to the case where ( A 0 , . . . , A n ) is formed by consecutive observations from the Lévy stable noise or from certain harmonizable stable processes, and more explicit formulas are derived for the expected number of real zeros of .

The notions of a generalized differential and a generalized spherical derivative of an arbitrary order are introduced for a function of several variables and Fatou type theorems are proved on the boundary properties of partial derivatives of an arbitrary order of the Poisson integral for the half-space, when the integral density has a generalized differential or a generalized spherical derivative.