Volume 24, Issue 2

An antiperiodic boundary value problem is considered for systems of linear generalized differential equations. A Green-type theorem on the unique solvability of the problem and representation of its solution are established. Effective necessary and sufficient conditions (of spectral type) are given for the unique solvability of the problem as well.

For higher-order Emden–Fowler type equations with regular nonlinearity, the asymptotic behavior of their blow-up solutions is investigated. It is proved that for weakly superlinear differential equations, all such solutions have power-law asymptotic behavior.

Efficient conditions guaranteeing the solvability of multi-point boundary value problems for linear functional-differential equations are established in this paper. The results are proved using the theorems on functional-differential inequalities.

We answer a conjecture and an open problem concerning integral sequences of the form ∫01f⁢(x)⁢f⁢(x2)⁢⋯⁢f⁢(xn)⁢dxn,n≥2.\sqrt[n]{\int_{0}^{1}f(x)f(x^{2})\cdots f(x^{n})\,\mathrm{d}x},\quad n\geq 2.

A characterization is given of all sets in an uncountable commutative group (G,+){(G,+)} which contain at least one absolutely nonmeasurable subset of G .

Sufficient conditions are found for the solvability of the following boundary value problem: u(n)(t)=f(u)(t),u(i-1)(0)=φi(u(n-1)(0)) (i=1,…,n-1),lim inft→+∞|u(n-2)(t)|<+∞,u^{(n)}(t)=f(u)(t),\qquad u^{(i-1)}(0)=\varphi_{i}(u^{(n-1)}(0))\quad(i=1,% \dots,n-1),\qquad\liminf_{t\to+\infty}\lvert u^{(n-2)}(t)|<+\infty, where f:Cn-1⁢(ℝ+)→Lloc⁢(ℝ+){f\colon C^{n-1}(\mathbb{R}_{+})\to L_{\mathrm{loc}}(\mathbb{R}_{+})} is a continuous Volterra operator, and φi:ℝ→ℝ{\varphi_{i}\colon\mathbb{R}\to\mathbb{R}} (i=1,…,n{i=1,\dots,n}) are continuous functions.

In this paper, we establish sharp weighted bounds (Buckley-type theorems) for one-sided maximal and fractional integral operators in terms of one-sided Ap{A_{p}} characteristics.

Sufficient conditions for the existence of periodic, bounded, and unbounded solutions of the equation u′′=f⁢(t,u){u^{\prime\prime}=f(t,u)} are proved.

In this article we consider the two-point boundary value problem {u(4)⁢(t)=p⁢(t)⁢u⁢(t)+h⁢(t) for ⁢a≤t≤b,u(i)(a)=c1⁢i,u(i)(b)=c2⁢i (i=0,1),\left\{\begin{aligned} &\displaystyle u^{(4)}(t)=p(t)u(t)+h(t)\quad\text{for }% a\leq t\leq b,\\ &\displaystyle u^{(i)}(a)=c_{1i},\quad u^{(i)}(b)=c_{2i}\quad(i=0,1),\end{% aligned}\right. where c1⁢i,c2⁢i∈R{c_{1i},c_{2i}\in R}, h,p∈L⁢([a,b];R){h,p\in L([a,b];R)}. Here we study the question of dimension of the space of nonzero solutions and oscillatory behaviors of nonzero solutions on the interval [a,b]{[a,b]} for the corresponding homogeneous problem, and establish efficient sufficient conditions of solvability for the nonhomogeneous problem.

We consider the following two-dimensional system of non-linear equations: u′=g⁢(t)⁢|v|1α⁢sgn⁡v,v′=-p⁢(t)⁢|u|α⁢sgn⁡u,u^{\prime}=g(t)|v|^{\frac{1}{\alpha}}\operatorname{sgn}v,\quad v^{\prime}=-p(t% )|u|^{\alpha}\operatorname{sgn}u, where α>0{\alpha>0}, and g:[0,+∞[→[0,+∞[{g\colon{[0,+\infty[}\rightarrow{[0,+\infty[}} and p:[0,+∞[→ℝ{p\colon{[0,+\infty[}\rightarrow\mathbb{R}} are locally integrable functions. Moreover, we assume that the coefficient g is non-integrable on [0,+∞]{[0,+\infty]}. We establish new non-oscillation criteria for the considered system, which generalize known results for the corresponding linear system and for second order differential equations. In particular, the presented criteria are in compliance with the results of Hille and Nehari.

We describe a reduction technique allowing one to combine an analysis of the existence of solutions with an efficient construction of approximate solutions for a state-dependent multi-impulsive boundary value problem which consists of non-linear system of differential equations u′⁢(t)=f⁢(t,u⁢(t)) for a.e. ⁢t∈[a,b],u^{\prime}(t)=f(t,u(t))\quad\text{for a.e. }t\in[a,b], subject to the state-dependent impulse condition u⁢(t+)-u⁢(t-)=γt⁢(u⁢(t-)) for ⁢t∈(a,b)⁢ such that ⁢g⁢(t,u⁢(t-))=0,u(t+)-u(t-)=\gamma_{t}(u(t-))\quad\text{for }t\in(a,b)\text{ such that }g(t,u(% t-))=0, and the non-linear two-point boundary condition V⁢(u⁢(a),u⁢(b))=0.V(u(a),u(b))=0.