Finite Difference Schemes with Monotone Operators

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]. If p ≡ 1, r ≡ 0, f ≡ 0, T = ∞, and the boundary conditions are u(0) = a and sup{‖u(t)‖, t ≥ 0} <∞ 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 − ui) ∈ kiAui + gi, i = 1,N , where N is a given natural number, pi,ri,ki > 0, gi ∈H . This leads to the finite difference scheme ( pi + ri ) ui+1 − ( 2pi + ri ) ui + piui−1 ∈ kiAui + gi, i= 1,N , (1.3) u1 −u0 ∈ α ( u0 − a ) , uN+1 −uN ∈−β ( uN+1 − b ) , (1.4)