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Asymptotic behavior and nonoscillation of Volterra integral equations and functional differential equations


Authors: A. F. Izé and A. A. Freiria
Journal: Proc. Amer. Math. Soc. 52 (1975), 169-177
MSC: Primary 34K15; Secondary 45M10
DOI: https://doi.org/10.1090/S0002-9939-1975-0377233-1
MathSciNet review: 0377233
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Abstract: It is proved that if $ {q_{ij}}(t,s){\rho _j}(s){[{\rho _i}(t)]^{ - 1}}$ is bounded, $ i,j = 1,2, \ldots ,n$, and $ f(t,x,x(u(s)))$ is ``small",

$\displaystyle x(u(s)) = ({x_1}({u_1}(s)),{x_2}({u_2}(s)), \ldots ,{x_n}({u_n}(s)))$

with $ {u_i}(t) \leqslant t$ and $ {\lim _{t \to \infty }}{u_i}(t) = \infty $, the solutions of the integral equation

$\displaystyle x\left( t \right) = h(t) + \int_0^t {q(t,s)f(s,x(s),x(u(s)))ds} $

satisfy the conditions $ x(t) = h(t) + \rho (t)a(t),{\lim _{t \to \infty }}a(t) = $ constant where $ \rho (t)$ is a nonsingular diagonal matrix chosen in such a way that $ {\rho ^{ - 1}}(t)h(t)$ is bounded. The results contain, in particular, some results on the asymptotic behavior, stability and existence of nonoscillatory solutions of functional differential equations.

References [Enhancements On Off] (What's this?)

  • [1] T. G. Hallam, Asymptotic behavior of the solutions of an $ n$th order nonhomogeneous ordinary differential equation, Trans. Amer. Math. Soc. 122 (1966), 177-194. MR 32 #6000. MR 0188562 (32:6000)
  • [2] A. F. Izé, On an asymptotic property of a Volterra integral equation, Proc. Amer. Math. Soc. 28 (1971), 93-99. MR 43 #836. MR 0275078 (43:836)
  • [3] -, Asymptotic integration of a nonhomogeneous singular linear system of ordinary differential equation, J. Differential Equations 8 (1970), 1-15. MR 41 #3898. MR 0259256 (41:3898)
  • [4] A. A. Freiria, Sobre comportamento assintótico e existência de soluções não oscilatórias de uma classe de sistema de equações diferencias com retardamento, São Carlos, 1972.
  • [5] G. Ladas, Oscillation and asymptotic behavior of solutions of differential equations with retarded argument, J. Differential Equations 10 (1971), 281-290. MR 45 #681. MR 0291590 (45:681)
  • [6] P. Marušiak, Note on the Ladas' paper on oscillation and asymptotic behavior of solutions of differential equations with retarded argument, J. Differential Equations 13 (1973), 150-156. MR 0355266 (50:7742)
  • [7] R. K. Miller, Nonlinear Volterra integral equations, Benjamin, New York, 1971. MR 0511193 (58:23394)
  • [8] J. A. Nohel, Some problems in nonlinear Volterra integral equations, Bull. Amer. Math. Soc. 68 (1962), 323-329. MR 26 #2838. MR 0145307 (26:2838)
  • [9] P. Waltman, On the asymptotic behavior of solutions of a nonlinear equation, Proc. Amer. Math. Soc. 15 (1964), 918-923. MR 31 #445. MR 0176170 (31:445)
  • [10] J. A. Yorke, Selected topics in differential delay equations, Japan-U.S.A. Seminar on Ordinary Differential and Functional Equations, Lecture Notes in Math., vol. 143, Springer-Verlag, Berlin and New York, 1971, pp. 16-28. MR 0435554 (55:8513)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0377233-1
Keywords: Volterra integral equations, asymptotic properties, almost all, kernel, asymptotic behavior, uniform stability, uniform asymptotic stability, nonoscillation, functional differential equation globally bounded
Article copyright: © Copyright 1975 American Mathematical Society

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