Abstract

For an integro-differential equation of the convolution type with a power nonlinearity and a variable coefficient defined on the half-line \([0,\infty )\), we use the method of weight metrics in the cone of the space \(C^1(0,\infty ) \) formed by functions positive on \((0,\infty ) \) and vanishing at the origin to prove a global theorem on the existence and uniqueness of a solution belonging to the indicated cone. It is shown that this solution can be found by the successive approximation method of the Picard type, and an estimate is established for the rate of convergence of the approximations. Examples are given to illustrate the results obtained.

中文翻译：

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具有幂非线性和非线性系数的卷积型积分微分方程

摘要

对于具有幂非线性和在半线上\（[0，\ infty）\）上定义的可变系数的卷积型积分微分方程，我们使用空间圆锥\ {中的权重度量方法由在\（（0，\ infty）\）上为正的函数形成的C ^ 1（0，\ infty）\）在原点处消失，以证明关于属于所示锥的解的存在和唯一性的全局定理。结果表明，该解决方案可以通过Picard类型的逐次逼近方法找到，并为逼近的收敛速率建立一个估计。举例说明所获得的结果。