For an integro-differential equation of the convolution type defined on the half-line [0, ∞) with a power nonlinearity and variable coefficient, we use the weight metrics method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions in the space C [0, ∞). It is shown that the solution can be found by a successive approximation method of the Picard type; an estimate for the rate of convergence of the approximations is produced. We present sharp two-sided a-priori estimates for the solutions. These estimates enable us to construct a complete metric space which is invariant under the nonlinear convolution operator considered here and to prove that the equation induced by this operator has a unique solution in this space as well as in the class of all non-negative continuous functions vanishing at the origin. Such equations with operators of fractional calculus as the Riemann-Liouville, Erdélyi-Kober, Hadamard fractional integrals arise, in particular, when describing the process of propagation of shock waves in gas-filled pipes, solving the problem about heating a half-infinite body in a nonlinear heat-transfer process, considering models of population genetics, and elsewhere.

中文翻译：

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变系数非线性卷积积分微分方程

对于定义在半线 [0, ∞) 上的具有幂非线性和可变系数的卷积类型的积分微分方程，我们使用权重度量方法来证明关于解的存在性和唯一性的全局定理空间 C [0, ∞) 中的非负函数锥。结果表明，该解可以通过Picard型逐次逼近法求得；产生近似收敛速度的估计。我们对解决方案提出了尖锐的两侧先验估计。这些估计使我们能够构建一个完整的度量空间，该空间在此处考虑的非线性卷积算子下是不变的，并证明由该算子导出的方程在该空间以及所有非负连续函数的类中具有唯一解消失在原点。尤其是在描述激波在充气管道中的传播过程、解决有关加热半无限体的问题时，出现了诸如 Riemann-Liouville、Erdélyi-Kober、Hadamard 分数积分等分数阶微积分算子的方程在非线性传热过程中，考虑群体遗传学模型和其他地方。