Difference inequalities generated by impulsive parabolic differential-functional problems (Q1372085)

An abstract parabolic differential-functional initial-boundary value problem with impulses is considered. Together with this problem, the author formulates a corresponding finite difference problem. The finite difference equations are of explicit type. The main goal of this paper is the convergence theorem for a given finite difference approximation of the original problem. The proof of the convergence is based on certain monotonicity property of the finite difference approximation considered here. This property is expressed in the form of some finite difference recurrent inequality (see for example Theorem 1). Under a number of hypotheses concerning the original problem and its finite difference approximation (including stability of the approximate problem) the author proves her convergence result. In fact, this kind of finite difference approximation could be only conditionally stable. This problem is not discussed here -- the author simply assumes stability. The special section of the paper is devoted to the almost linear problem. One numerical example is given.