TY - JOUR

T1 - Convergence of the Successive Approximation Method in the Cauchy Problem for an Integro-Differential Equation with Quadratic Nonlinearity

AU - Vaskevich, V. L.

AU - Shcherbakov, A. I.

PY - 2019/4/1

Y1 - 2019/4/1

N2 - The equations considered in this article have the form in which the time derivative of the unknown function is expressed as a double integral over the space variables of a weighted quadratic expression of the sought function. The domain of integration is unbounded and does not depend on time but depends on the space variable. We study the Cauchy problem in the function classes accompanying the equation with initial data on the positive half-line. In application to this problem, the convergence of the successive approximation method is justified. An estimate is given of the quality of the approximation depending on the number of the iterated solution. It is proved that, on any finite time interval, the posed Cauchy problem has at most one solution in the accompanying function class. An existence theorem is proved in the same class.

AB - The equations considered in this article have the form in which the time derivative of the unknown function is expressed as a double integral over the space variables of a weighted quadratic expression of the sought function. The domain of integration is unbounded and does not depend on time but depends on the space variable. We study the Cauchy problem in the function classes accompanying the equation with initial data on the positive half-line. In application to this problem, the convergence of the successive approximation method is justified. An estimate is given of the quality of the approximation depending on the number of the iterated solution. It is proved that, on any finite time interval, the posed Cauchy problem has at most one solution in the accompanying function class. An existence theorem is proved in the same class.

KW - a priori estimate

KW - Cauchy problem

KW - existence theorem

KW - nonlinear integro-differential equation

KW - quadratic nonlinearity

KW - successive approximation method

UR - http://www.scopus.com/inward/record.url?scp=85067646515&partnerID=8YFLogxK

U2 - 10.3103/S1055134419020032

DO - 10.3103/S1055134419020032

M3 - Article

AN - SCOPUS:85067646515

VL - 29

SP - 128

EP - 136

JO - Siberian Advances in Mathematics

JF - Siberian Advances in Mathematics

SN - 1055-1344

IS - 2

ER -