Nonlinear Operator Equations with a Functional Perturbation of the Argument of Neutral Type

摘要

We construct small solutions x(t) → 0 as t → 0 of nonlinear operator equations F(x(t), x(a(t)), t) = 0 with a functional perturbation a(t) of the argument. By the Newton diagram method, we reduce the problem to quasilinear operator equations with a functional perturbation of the argument. We show that the solutions of such equations can have not only algebraic but also logarithmic branching points and contain free parameters. The number of free parameters and the form of the solution depend on the properties of the Jordan structure of the operator coefficients of the equation.