Volume 16, Issue 4

Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

This paper is concerned with the existence of extreme solutions of nonlinear three-point boundary value problems for a class of first order impulsive functional differential equations. In the presence of a lower solution α and an upper solution β with the classical condition α ≤ β or the reversed ordering condition β ≤ α , some sufficient conditions for the existence of extreme solutions are obtained by using the method of upper and lower solutions coupled with the monotone iterative technique.

We study three-dimensional unilateral contact problems with friction for hemitropic elastic solids. We give their mathematical formulation in the form of spatial variational inequalities and show the equivalence to the corresponding minimization problem. Based on our variational inequality approach, we prove existence and uniqueness theorems. We prove also that solutions continuously depend on the data of the original problem and on the friction coefficient. Our investigation includes the special particular case of only traction-contact boundary conditions without prescribing displacement and microrotation vectors along some part of the boundary of a hemitropic elastic body. Then the problems are not unconditionally solvable and we derive the necessary and sufficient conditions of their solvability.

On the interval [0, ω ], consider the periodic boundary value problem where 𝑛 ≥ 2, 𝑙 𝑖 : 𝐶([0, ω ];𝑅) → 𝐿([0, ω ];𝑅) (𝑖 = 0,…,𝑛 – 1) are linear bounded operators, 𝑞 ∈ 𝐿([0, ω ];𝑅), 𝑐 𝑗 ∈ 𝑅 (𝑗 = 0,…,𝑛 – 1). The effective sufficient conditions guaranteeing the unique solvability of the considered problem are established.

We give a proof of the Jawerth embedding for function spaces with dominating mixed smoothness of Besov and Triebel–Lizorkin type where 0 < 𝑝 0 < 𝑝 1 ≤ ∞ and 0 < 𝑞 0 ,𝑞 1 ≤ ∞ and with If 𝑝 1 < ∞, we prove also the Franke embedding Our main tools are discretization by a wavelet isomorphism and multivariate rearrangements.

We study the stability of the zero solution of nonlinear delay difference equations by fixed point theory. An example is given to illustrate our theory.

We consider the Bézier variant of Chlodovsky–Durrmeyer operators 𝐷 𝑛, α for functions 𝑓 measurable and locally bounded on the interval [0,∞). By using the Chanturia modulus of variation we estimate the rate of pointwise convergence of (𝐷 𝑛, α 𝑓) (𝑥) at those 𝑥 > 0 at which the one-sided limits 𝑓(𝑥+), 𝑓(𝑥–) exist. In the special case α = 1 the recent result of [Ibikli, Karsli, J. Inequal. Pure Appl. Math. 6: 12, 2005] concerning the Chlodovsky–Durrmeyer operators 𝐷 𝑛 is essentially improved and extended to more general classes of functions.

Following the paper of Pkhakadze [Trudy Tbiliss. Mat. Inst. Razmadze 20: 167–209, 1954], we consider some properties of real-valued functions of two variables, which are not assumed to be measurable with respect to the two-dimensional Lebesgue measure on the plane 𝐑 2 , but for which the corresponding iterated integrals exist and are equal to each other. Close connections of these properties with certain set-theoretical axioms are emphasized.

For systems of nonlinear nonautonomous ordinary differential equations, the conditions, optimal in a certain sense, are established, which guarantee the solvability and well-posedness of the problem on bounded solutions, the vanishing at infinity of all bounded solutions and the global asymptotic stability of a trivial solution.

The functional equations of the form are considered. They are connected with quadrature rules of the approximate integration. We show that such equations characterize polynomials in the class of continuous functions. It is also shown that if the number of components is sufficiently small, then the continuity is forced by the equation itself. Unique solvability of the considered problem are established.

The Riemann–Hilbert problem for an analytic function is solved in weighted classes of Cauchy type integrals in a simply connected domain not containing 𝑧 = ∞ and having a density from variable exponent Lebesgue spaces. It is assumed that the domain boundary is a piecewise smooth curve. The solvability conditions are established and solutions are constructed. The solution is found to essentially depend on the coefficients from the boundary condition, the weight, space exponent values at the angular points of the boundary curve and also on the angle values. The non-Fredholmian case is investigated. An application of the obtained results to the Neumann problem is given.

We give an example of a function which belongs only to the largest space in the chain of embedded spaces, important in various problems of analysis.

In this work, we use a functional analysis approach to prove the well posedness of a mixed problem with a nonclassical condition for a viscoelastic equation with dissipation. The proof is mainly based on some a priori bounds and on some density arguments.

We prove that for any noncomputable c.e. set 𝐴 there is a non-computable c.e. set 𝐵 ⊆ 𝐴 such that for every noncomputable c.e. set 𝑊 we have 𝑊 ≰⃥ 𝑄 𝐴 – 𝐵 < 𝑄 𝐴. We show that if c.e. 𝑄-degrees 𝐚 and 𝐛 form a minimal pair in the c.e. 𝑄-degrees, then 𝐚 and 𝐛 form a minimal pair in the 𝑄-degrees.

It turns out that, as different from a strong symmetrical differentiation basis J ∗, unilateral strong differentiation bases in are not equivalent to a complete strong differentiation basis throughout the space of locally integrable functions.