Asymptotic Solution to a Class of Nonlinear Volterra Integral Equations. II
1976; Society for Industrial and Applied Mathematics; Volume: 30; Issue: 1 Linguagem: Inglês
10.1137/0130020
ISSN1095-712X
AutoresW. E. Olmstead, Richard A. Handelsman,
Tópico(s)Differential Equations and Boundary Problems
ResumoIt is known that the nonlinear Volterra integral equation \[ \varphi (t)\pi ^{( - 1 / 2)} \,\int_0^t (t - s)^{{ - 1 / 2} } [ {f(s) - \varphi ^n (s)} ]ds,\quad t\geqq 0,\geqq n\geqq 1,\] has a continuous solution $\varphi (t) \geqq 0$ which is unique for each bounded and locally lntegrable function $f(t) \geqq 0$ Our prior investigation considered the asymptotic behavior, as $t \to \infty $, of $\varphi (t)$ when $f(t) \sim \gamma _0 t^{ - a_0 } + \cdots ,\gamma _0 > 0,a_0 \geqq 0$. The goal here is to complete this analysis so that the behavior of $\varphi (t)$, as $t \to \infty $, is provided for all $a_0 \geqq 0,n \geqq 1$. In achieving this, we must treat some cases with properties that are markedly different from those previously considered.
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