Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 31, pp. 1-16.
Title: Approximations of solutions to nonlinear Sobolev type evolution equations
Authors: Dhirendra Bahuguna (Indian Institute of Technology Kanpur, India)
Reeta Shukla (Indian Institute of Technology Kanpur, India)
Abstract:
In the present work we study the approximations of solutions to a
class of nonlinear Sobolev type evolution equations in a Hilbert space.
These equations arise in the analysis of the partial neutral functional
differential equations with unbounded delay. We consider an associated
integral equation and a sequence of approximate integral equations.
We establish the existence and uniqueness of the solutions to every
approximate integral equation using the fixed point arguments.
We then prove the convergence of the solutions of the approximate
integral equations to the solution of the associated integral equation.
Next we consider the Faedo-Galerkin approximations of the solutions and
prove some convergence results. Finally we demonstrate some of the
applications of the results established.
Submitted October 10, 2001. Published March 17, 2003.
Math Subject Classifications: 34A12, 34A45, 34G20, 47D06, 47J35.
Key Words: Faedo-Galerkin approximation; analytic semigroup;
mild solution; contraction mapping theorem; fixed points.