Generalized quasilinearization method for nonlinear functional differential equations
The authors investigate a nonlinear initial value problem for the functional-differential equation \(x'= f(t,x_t)\), \(t\in [0,T]\), with \(x_t(s)= x(s+ t)\), \(-\tau\leq s\leq 0\), \(t\in [-\tau,T]\), \(\tau> 0\), \(f\in C[J\times\Gamma, \mathbb{R}]\) and \(\Gamma= C[[-\tau, 0],\mathbb{R}]\). They develop a method called ``quasilinearization'' for this equation without requiring the function involved to be convex concave. A monotone sequence of approximate solutions converging monotonically to a solution to the problem, with convergence higher then quandratic, is obtained.
scientific article; zbMATH DE number 1932173