Finite difference schemes with monotone operators

Nc Apreutesei

doi:10.1155/s1687183904310046

Abstract

Several existence theorems are given for some second-order difference equations associated with maximal monotone operators in Hilbert spaces. Boundary conditions of monotone type are attached. The main tool used here is the theory of maximal monotone operators.

Highlights

In [1, 2], the authors proved the existence of the solution of the boundary value problem p(t)u (t) + r(t)u (t) ∈ Au(t) + f (t), a.e. on [0, T], T > 0, (1.1)
We study the existence and uniqueness of the solution of problem (1.3), (1.4) under various conditions on A, α, and β
We show that the difference of the two solutionsi=1,N and (vi)i=1,N of (3.7) is a constant

Summary

Introduction

In [1, 2], the authors proved the existence of the solution of the boundary value problem p(t)u (t) + r(t)u (t) ∈ Au(t) + f (t), a.e. on [0, T], T > 0,. Where A : D(A) ⊆ H → H, α : D(α) ⊆ H → H, and β : D(β) ⊆ H → H are maximal monotone operators in the real Hilbert space H (satisfying some specific properties), a, b are given elements in the domain D(A) of A, f ∈ L2(0,T; H), and p, r : [0,T] → R are continuous functions, p(t) ≥ k > 0 for all t ∈ [0,T] Particular cases of this problem were considered before in [9, 10, 12, 15, 16].

12 Finite difference schemes with monotone operators

Preliminary results

Existence theorems