Estimate of solutions for differential and difference functional equations with applications to difference methods q K. Kropielnicka a , L. Sapa b,⇑ a Institute of Mathematics, University of Gdan ´sk, Ul. Wita Stwosza 57, 80-952 Gdan ´sk, Poland b Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland article info Keywords: Parabolic differential and difference functional equations Parabolic solution Estimate of solution Difference method abstract The theorems on the estimate of solutions for nonlinear second-order partial differential functional equations mainly of parabolic type with Dirichlet’s condition and for the suit- able explicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent difference method given is considered without an assumption of the global generalized Perron condition on the functional variable but with local one in some sense only. It is a consequence of our estimate theorems. The functional dependence is of the Volterra type. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction The aim of the paper is to prove the theorems on the estimate of solutions for nonlinear second-order partial differential functional equations, mainly parabolic, with Dirichlet’s condition and for generated by them explicit finite difference func- tional schemes. We also give the applications of the results. More precisely, we give the theorem on the convergence of a difference method to a classical solution for the differential functional problems which, by the theorems on the estimate given, may be treated in some function subspaces B(X, R)  B(X, R), R  R is an interval. It is a new idea in the nonlinear difference methods. This considerably extends the class of problems which are solvable with the method described. It is because the Lipschitz, Perron or generalized Perron conditions on f with respect to z do not have to be assumed globally in B(X, R) as in the papers due to Malec, Ma ˛ czka, Voigt and Rosati [13,14,16], Kamont, Leszczyn ´ ski and Kropielnicka [9,11] and Sapa [21], but in B(X, R) only. Therefore, in particular, equations with the polynomial right-hand sides are admit- ted (see the examples in Section 7). Our results can be extended to implicit difference methods and weakly coupled systems. We now formulate the differential functional problem. Let D t ¼ @ @t and D i ¼ @ @x i , D ij ¼ @ 2 @x j @x i for i, j = 1, ... , n, where t 2 R, x =(x 1 , ... , x n ) 2 R n . Put D x =(D 1 , ... , D n ) and D ð2Þ x ¼½D ij  n i;j¼1 . Let functions f : D ? R and u : E 0 [ @ 0 E ? R be given (the relevant sets are defined in Section 2.1). Consider a nonlinear second-order partial differential functional equation of the form D t zðt; xÞ¼ f ðt; x; z; D x zðt; xÞ; D ð2Þ x zðt; xÞÞ ð1:1Þ with the initial condition and the boundary condition of the Dirichlet type zðt; xÞ¼ uðt; xÞ on E 0 [ @ 0 E: ð1:2Þ The equation may be nonlinear with respect to second derivatives. Such an equation is called strongly nonlinear. The func- tional dependence is of the Volterra type (e.g., delays or Volterra type integrals). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.12.106 q The research of the authors was partially supported by the Polish Ministry of Science and Higher Education. ⇑ Corresponding author. E-mail addresses: karolina.kropielnicka@math.univ.gda.pl (K. Kropielnicka), lusapa@mat.agh.edu.pl (L. Sapa). Applied Mathematics and Computation 217 (2011) 6206–6218 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc