Abstract
We consider the singular Cauchy problem
, 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] → (ρ is sufficiently small) with required asymptotic properties.
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References
N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations [in Russian], Nauka, Moscow (1991).
N. V. Azbelev, “Contemporary state and trends of development of the theory of functional differential equations,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 6, 8–19 (1999).
N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Elements of the Modern Theory of Functional Differential Equations. Methods and Applications [in Russian], Institute of Computer Studies, Moscow (2002).
R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, et al., “Theory of equations of neutral type,” in: VINITI Series in Mathematical Analysis [in Russian], VINITI, Moscow (1981), pp. 55–126.
G. P. Pelyukh and A. N. Sharkovskii, Introduction to the Theory of Functional Equations [in Russian], Naukova Dumka, Kiev (1974).
J. Hale, Theory of Functional Differential Equations, Springer, Berlin (1977).
N. P. Erugin, A Reader for a General Course in Differential Equations [in Russian], Nauka i Tekhnika, Minsk (1972).
I. T. Kiguradze, Some Singular Boundary-Value Problems for Ordinary Differential Equations with Singularity [in Russian], Tbilisi University, Tbilisi (1975).
V. A. Chechik, “Investigation of systems of ordinary differential equations with singularity,” Tr. Mosk. Mat. Obshch., No. 8, 155–198 (1959).
N. V. Azbelev, M. Zh. Alvesh, and E. I. Bravyi, “On singular boundary-value problems for a linear functional differential equation of the second order,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 3–11 (1999).
M. Zh. Alvesh, “On the solvability of a two-point boundary-value problem for a singular functional differential equation,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 12–19 (1999).
E. I. Bravyi, “On the solvability of a boundary-value problem for a singular functional differential equation,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 5, 17–23 (1993).
E. I. Bravyi, Linear Functional Differential Equations with Internal Singularities [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Perm (1996).
A. I. Shindyapin, “On a boundary-value problem for a singular equation,” Differents. Uravn., 20, No. 3, 450–455 (1984).
L. J. Grimm, “Analytic solutions of a neutral differential equation near a singular point,” Proc. Amer. Math. Soc., 36, No. 1, 187–190 (1972).
L. J. Grimm and L. M. Hall, “Holomorphic solutions of singular functional differential equations,” J. Math. Anal. Appl., 50, No. 3, 627–638 (1975).
A. E. Zernov, “On the solvability and asymptotics of solutions of one functional differential equation with singularity,” Ukr. Mat. Zh., 53, No. 4, 455–465 (2001).
B. P. Demidovich, Lectures on Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).
A. E. Zernov, “On the solvability and asymptotic properties of one singular Cauchy problem,” Differents. Uravn., 28, No. 5, 756–760 (1992).
A. E. Zernov, “Qualitative analysis of an implicit singular Cauchy problem,” Ukr. Mat. Zh., 53, No. 3, 302–310 (2001).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1344–1358, October, 2005.
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Zernov, A.E., Chaichuk, O.R. Qualitative investigation of a singular Cauchy problem for a functional differential equation. Ukr Math J 57, 1571–1589 (2005). https://doi.org/10.1007/s11253-006-0015-7
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DOI: https://doi.org/10.1007/s11253-006-0015-7