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

Izé, A. F.; Freiria, A. A.

doi:10.1090/S0002-9939-1975-0377233-1

Asymptotic behavior and nonoscillation of Volterra integral equations and functional differential equations
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by A. F. Izé and A. A. Freiria PDF

Proc. Amer. Math. Soc. 52 (1975), 169-177 Request permission

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.

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Sobre comportamento assintótico e existência de soluções não oscilatórias de uma classe de sistema de equações diferencias com retardamento

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