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FINITE DIFFERENCE SCHEMES WITH MONOTONE OPERATORS N. C. APREUTESEI Received 13 October 2003 and in revised form 10 December 2003 To the memory of my mother, Liliana 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. 1. 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), u (0) ∈ α u(0) − a , a.e. on. | FINITE DIFFERENCE SCHEMES WITH MONOTONE OPERATORS N. C. APREUTESEI Received 13 October 2003 and in revised form 10 December 2003 To the memory of my mother Liliana 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. 1. 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 e Au t f t a.e. on 0 T T 0 u 0 e a u 0 - a u T e-f u T - b 1.1 1.2 where A D A Q H H a D a Q H H and f D f Q 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 e L2 0 T H and p r 0 T R are continuous functions p t k 0 for all t e 0 T . Particular cases of this problem were considered before in 9 10 12 15 16 . If p 1 r 0 f 0 T TO and the boundary conditions are u 0 a and sup u t II t 0 TO instead of 1.2 the solution u t of 1.1 1.2 defines a semigroup of nonlinear contractions S1 2 t t 0 on the closure D A of D A see 9 10 . This semigroup and its infinitesimal generator A1 2 have some important properties see 9 10 11 12 . A discretization of 1.1 is pi ui 1 - 2ui ui 1 ri ui 1 - uì e kiAui gi i 1 N where N is a given natural number pi ri ki 0 gi e H. This leads to the finite difference scheme pi r ui 1 - flpi rfjui piui-1 e kiAui gi i 1 N ui u0 e a u0 a un 1 un e f un 1 b 1.3 1.4 where a b e H are given pi i 1N ri i TN and ki i 1 N are sequences of positive numbers and gi i 1 N e HN. Copyright 2004 Hindawi Publishing Corporation Advances inDifferenceEquations 2004 1 2004 11-22 2000 Mathematics Subject Classification 39A12 39A70 47H05 URL http dx.doi.org 10.1155 S1687183904310046 12 Finite difference schemes with monotone operators In this paper we study the existence and uniqueness of the solution of problem 1.3 1.4 under various conditions
