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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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", \[ x(u(s)) = ({x_1}({u_1}(s)),{x_2}({u_2}(s)), \ldots ,{x_n}({u_n}(s)))\] with ${u_i}(t) \leqslant t$ and ${\lim _{t \to \infty }}{u_i}(t) = \infty$, the solutions of the integral equation \[ x\left ( t \right ) = h(t) + \int _0^t {q(t,s)f(s,x(s),x(u(s)))ds} \] 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.
- Thomas G. Hallam, Asymptotic behavior of the solutions of an $n{\rm th}$ order nonhomogeneous ordinary differential equation, Trans. Amer. Math. Soc. 122 (1966), 177–194. MR 188562, DOI https://doi.org/10.1090/S0002-9947-1966-0188562-8
- A. F. Izé, On an asymptotic property of a Volterra integral equation, Proc. Amer. Math. Soc. 28 (1971), 93–99. MR 275078, DOI https://doi.org/10.1090/S0002-9939-1971-0275078-0
- A. F. Izé, Asymptotic integration of a nonhomogeneous singular linear system of ordinary differential equations, J. Differential Equations 8 (1970), 1–15. MR 259256, DOI https://doi.org/10.1016/0022-0396%2870%2990035-5 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.
- G. Ladas, Oscillation and asymptotic behavior of solutions of differential equations with retarded argument, J. Differential Equations 10 (1971), 281–290. MR 291590, DOI https://doi.org/10.1016/0022-0396%2871%2990052-0
- Pavol Marušiak, Note on the Ladas’ paper on “Oscillation and asymptotic behavior of solutions of differential equations with retarded argument” (J. Differential Equations 10 (1971), 281–290) by G. Ladas, J. Differential Equations 13 (1973), 150–156. MR 355266, DOI https://doi.org/10.1016/0022-0396%2873%2990037-5
- Richard K. Miller, Nonlinear Volterra integral equations, W. A. Benjamin, Inc., Menlo Park, Calif., 1971. Mathematics Lecture Note Series. MR 0511193
- J. A. Nohel, Some problems in nonlinear Volterra integral equations, Bull. Amer. Math. Soc. 68 (1962), 323–329. MR 145307, DOI https://doi.org/10.1090/S0002-9904-1962-10790-3
- Paul Waltman, On the asymptotic behavior of solutions of a nonlinear equation, Proc. Amer. Math. Soc. 15 (1964), 918–923. MR 176170, DOI https://doi.org/10.1090/S0002-9939-1964-0176170-8
- James A. Yorke, Selected topics in differential delay equations, Japan-United States Seminar on Ordinary Differential and Functional Equations (Kyoto, 1971) Springer, Berlin, 1971, pp. 16–28. Lecture Notes in Math., Vol. 243. MR 0435554
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34K15, 45M10
Retrieve articles in all journals with MSC: 34K15, 45M10
Additional Information
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