2019
Том 71
№ 5

Zernov A. E.

Articles: 10
Article (Russian)

Qualitative investigation of a singular Cauchy problem for a functional differential equation

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1344–1358

We consider the singular Cauchy problem $$txprime(t) = f(t,x(t),x(g(t)),xprime(t),xprime(h(t))), x(0) = 0,$$ where $x: (0, τ) → ℝ, g: (0, τ) → (0, + ∞), h: (0, τ) → (0, + ∞), g(t) ≤ t$, and $h(t) ≤ t, t ∈ (0, τ)$, for linear, perturbed linear, and nonlinear equations. In each case, we prove that there exists a nonempty set of continuously differentiable solutions $x: (0, ρ] → ℝ$ ($ρ$ is sufficiently small) with required asymptotic properties.

Brief Communications (Russian)

Qualitative Investigation of the Singular Cauchy Problem F(t, x, x′) = 0, x(0) = 0

Ukr. Mat. Zh. - 2003. - 55, № 12. - pp. 1720-1723

We prove the existence and uniqueness of a continuously differentiable solution with required asymptotic properties.

Brief Communications (Russian)

Qualitative Investigation of the Singular Cauchy Problem $\sum\limits_{k = 1}^n {(a_{k1} t + a_{k2} x)(x')^k = b_1 t + b_2 x + f(t,x,x'),x(0) = 0}$

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1419-1424

We prove the existence of continuously differentiable solutions $x:(0,ρ] → R$ with required asymptotic properties as $t → +0$ and determine the number of these solutions.

Brief Communications (Ukrainian)

Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1698-1703

We prove the existence of continuously differentiable solutions $x:(0,{\rho ]} \to \mathbb{R}^n$ such that $$\left\| {x\left( t \right) - {\xi }\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)} \right),{ }\left\| {x'\left( t \right) - {\xi '}\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)/t} \right),{ }t \to + 0$$ or $$\left\| {x\left( t \right) - S_N \left( t \right)} \right\| = O\left( {t^{N + 1} } \right),{ }\left\| {x'\left( t \right) - S'_N \left( t \right)} \right\| = O\left( {t^N } \right),{ }t \to + 0,$$ where $${\xi }:\left( {0,{\tau }} \right) \to \mathbb{R}^n ,{ \eta }:\left( {0,{\tau }} \right) \to \left( {0, + \infty } \right),{ }\left\| {{\xi }\left( t \right)} \right\| = o\left( 1 \right),$$ $${\eta }\left( t \right) = o\left( t \right),{ \eta }\left( t \right) = o\left( {\left\| {{\xi }\left( t \right)} \right\|} \right),{ }t \to + 0,{ }S_N \left( t \right) = \sum\limits_{k = 2}^N {c_k t^k ,}$$ $$c_k \in \mathbb{R}^n ,k \in \left\{ {2,...,N} \right\},{ }0 < {\rho } < {\tau },{ \rho is sufficiently small}{.}$$

Article (Russian)

On the Asymptotic Behavior of Solutions of a Singular Cauchy Problem

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1194-1203

We consider a singular Cauchy problem for a nonlinear differential equation unsolved with respect to the derivative of the unknown function. We prove the existence of continuously differentiable solutions, investigate their asymptotic behavior near the initial point, and determine their number.

Article (Russian)

On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 455-465

We prove the existence of continuously differentiable solutions with required asymptotic properties as t → +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation: $$\alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0,$$ where α: (0, τ) → (0, +∞), g: (0, τ) → (0, +∞), and h: (0, τ) → (0, +∞) are continuous functions, 0 < g(t) ≤ t, 0 < h(t) ≤ t, t ∈ (0, τ), $\begin{gathered} \alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0, \\ \mathop {\lim }\limits_{t \to + 0} \alpha \left( t \right) = 0 \\ \end{gathered}$ , and the function ϕ is continuous in a certain domain.

Article (Russian)

Qualitative Analysis of an Implicit Singular Cauchy Problem

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 302-310

We consider a singular Cauchy problem for a first-order ordinary differential equation unsolved with respect to the derivative of the unknown function. We prove the existence of continuously differentiable solutions with required asymptotic properties.

Brief Communications (Russian)

On the Solution of a Singular Cauchy Problem for a First-Order Differential Equation Unsolved with Respect to the Derivative of an Unknown Function

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 258-262

For a first-order ordinary differential equation, we establish conditions under which a singular Cauchy problem has a unique continuously differentiable solution with required asymptotic behavior.

Article (Ukrainian)

Solution of a singular Cauchy problem of implicit form

Ukr. Mat. Zh. - 1991. - 43, № 6. - pp. 755-760

Article (Ukrainian)

Asymptotics of solutions of a cauchy problem

Ukr. Mat. Zh. - 1991. - 43, № 2. - pp. 187-193