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Asymptotic behavior of solutions of a singular Cauchy problem for a functional-differential equation

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Abstract

For the singular Cauchy problem

$$ \alpha (t)x\prime(t) = f\left( {t,x\left( {g(t)} \right)} \right),x\prime\left( {h(t)} \right),\quad x(0) = 0, $$

where α: (0, τ) → (0, +∞) is a continuous function and \( \mathop {\lim }\limits_{t \to + 0} \alpha (t) = 0 \), the authors prove the existence of a nonempty set of continuously differentiable solutions x: (0, ρ] → ℝ (ρ ∈ (0, τ) is sufficiently small) with the known asymptotic as t → +0.

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Authors and Affiliations

  1. K. D. Ushinskii South Ukraine State Pedagogical University, Odessa, Ukraine

    A. E. Zernov & O. R. Chaichuk

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  1. A. E. Zernov
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  2. O. R. Chaichuk
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Correspondence to A. E. Zernov.

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.

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Zernov, A.E., Chaichuk, O.R. Asymptotic behavior of solutions of a singular Cauchy problem for a functional-differential equation. J Math Sci 160, 123–127 (2009). https://doi.org/10.1007/s10958-009-9491-2

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