On the Faedo–galerkin Approximation of Mild Solution for Neutral Nonlinear Delay Differential Equations

We present new results concerning the existence of an approximate mild solution to a neutral differential equation involving state-dependent delay by using the Faedo–Galerkin method within an arbitrary separable Hilbert space ℍ. Initially, the associated integral equation of the state-dependent delay neutral differential equation is analyzed, and then a sequence of approximate integral equations by projecting the associated integral equation onto a finite-dimensional space is derived. We establish sufficient conditions for the existence and uniqueness of the solution to each approximate integral equation utilizing the Banach fixed-point theorem and analytic semigroup. Furthermore, we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Additionally, we explore the Faedo–Galerkin approximations of the solution and provide some convergence results. Finally, we include an example in this manuscript to illustrate the theoretical results discussed herein.