The Banach-Cacciopoli and Kantorovich principles for the Cauchy problem in theory of nonlinear systems with partial derivatives (Q2761386)

The authors consider the first-order partial matrix equation with quadratic nonlinearity. It is shown that the existence conditions for all \(t\), the solution to the Cauchy problem, and the convergence of the corresponding successive approximations can be obtained via the classical principle of fixed point [see \textit{L.~V.~Kantorovich} and \textit{G.~P.~Akilov}, ``Functional analysis'', Moscow, Nauka (1984; Zbl 0555.46001) and \textit{P.~P.~Zabrejko}, Dokl. Akad. Nauk BSSR 35, No. 11, 975-978 (1991; Zbl 0747.45003)]. This principle is applied to the stationary nonlinear integral operator in the Banach space, which is constructed in the explicit form.