The asymptotic behavior of the solution of a Volterra equation
The asymptotic behavior of the solution of a Volterra equation
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DOI：
10.1090/s0002-9939-1963-0152852-8
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发表时间：
1963-04
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通讯作者：
J. J. Levin-J.
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文献类型：
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作者：
J. J. Levin-J.
虽然我们在这里感兴趣的是（1.1）的解在t-* oo时的行为，但以下关于存在性和唯一性的观察是相关的。设g（x）在每个区间x < X < oo上也满足Lipschitz条件。通常的逐次逼近法，加上在本证明中得到的某些先验界，则很容易地意味着对于每个初始值（1.1）有唯一的解，并且该解存在于0< t < oo。即使在目前的假设下，从这些先验界和Nohel [2]的最近结果也可以很容易地得出，（1.1）的每个解都可以在0 < t < oo上连续（尽管不一定唯一）。微分（1.1）收益率
While our interest here is with the behavior of the solutions of (1.1) as t-* oo, the following observations on existence and uniqueness are relevant. Suppose g(x) also satisfies a Lipschitz condition on every interval x < X < oo. The usual method of successive approximations, together with certain a priori bounds obtained in the present proof, then easily imply that for each initial value (1.1) has a unique solution and that this solution exists on 0< t < oo. Even under the present hypothesis it follows readily from these a priori bounds and recent results of Nohel [2] that every solution of (1.1) can be continued (though not necessarily uniquely) over 0 < t < oo. Differentiating (1.1) yields