APPLICATIONES MATHEMATICAE 35,2 (2008), pp. 155–175 K. Kropielnicka (Gda´ nsk) IMPLICIT DIFFERENCE METHODS FOR QUASILINEAR PARABOLIC FUNCTIONAL DIFFERENTIAL PROBLEMS OF THE DIRICHLET TYPE Abstract. Classical solutions of quasilinear functional differential equa- tions are approximated with solutions of implicit difference schemes. Proofs of convergence of the difference methods are based on a comparison tech- nique. Nonlinear estimates of the Perron type with respect to the functional variable for given functions are used. Numerical examples are given. 1. Introduction. For any two metric spaces X and Y we denote by C (X, Y ) the class of all continuous functions defined on X and taking values in Y . Let M n×n denote the set of all n × n real matrices. We will use vectorial inequalities, understanding that the same inequalities hold between the corresponding components. Let E = [0,a] × (−b, b), D =[−d 0 , 0] × [−d, d], where a> 0, b =(b 1 ,...,b n ), b i > 0 for 1 ≤ i ≤ n, d 0 ∈ R + , d = (d 1 ,...,d n ) ∈ R n + and R + = [0, ∞). We put c =(c 1 ,...,c n )= b + d and ∂ 0 E = [0,a] ×([−c, c] \ (−b, b)), E 0 =[−d 0 , 0] ×[−c, c], Ω = E ∪E 0 ∪∂ 0 E. For a function z : Ω → R and a point (t, x) ∈ [0,a] × [−b, b] we define a function z (t,x) : D → R as follows: z (t,x) (ξ,y)= z (t + ξ,x + y) for (ξ,y) ∈ D. The function z (t,x) is the restriction of z to the set [t − d 0 ,t] × [x − d, x + d] and this restriction is shifted to the set D. Elements of the space C (D, R) will be denoted by w, w and so on. Write Σ = E × C (D, R) and suppose 2000 Mathematics Subject Classification : 65M12, 35R10. Key words and phrases : functional differential equations, implicit difference methods, stability and convergence. [155] c  Instytut Matematyczny PAN, 2008