Branching Solutions of the Cauchy Problem for Nonlinear Loaded Differential Equations with Bifurcation Parameters

Abstract

The Cauchy problem for a nonlinear system of differential equations with a Stieltjes integral (loads) of the desired solution is considered. The equation contains bifurcation parameters where the system has a trivial solution for any values. The necessary and sufficient conditions are derived for those parameter values (bifurcation points) in the neighborhood of which the Cauchy problem has a non-trivial real solution. The constructive method is proposed for the solution of real solutions in the neighborhood of those points. The method uses successive approximations and builds asymptotics of the solution. The theoretical results are illustrated by example. The Cauchy problem with loads and bifurcation parameters has not been studied before.

The linear Fredholm integral equations with functionals and parameters

“…In [10] the branching solutions of the Cauchy problem for nonlinear loaded differential equations with bifurcation parameters were studied. The purpose of this 84 DREGLEA SIDOROV, SIDOROV AND SIDOROV study is to prove the properties of the resolvent integral operator as applied to the second kind Fredholm integral equations with local and integral loads, and to formulate and prove constructive theorems of existence and convergence to the desired solution of successive approximations.…”