Zernov A. E.

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.

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

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.

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 $${\xi }:\left( {0,{\tau }} \right) \to \mathbb{R}^n ,{ \eta }:\left( {0,{\tau }} \right) \to \left( {0, + \infty } \right),{ }\left\| {{\xi }\left( t \right)} \right\| = o\left( 1 \right),$$ $${\eta }\left( t \right) = o\left( t \right),{ \eta }\left( t \right) = o\left( {\left\| {{\xi }\left( t \right)} \right\|} \right),{ }t \to + 0,{ }S_N \left( t \right) = \sum\limits_{k = 2}^N {c_k t^k ,}$$ $$c_k \in \mathbb{R}^n ,k \in \left\{ {2,...,N} \right\},{ }0 < {\rho } < {\tau },{ \rho is sufficiently small}{.}$$

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.

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.

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.

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.