Finite difference schemes with monotone operators
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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 [0, T], T > 0,
u (T) ∈ −β u(T) − b ,
(1.1) (1.2)
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 ) ∈ ki Aui + 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 + pi ui−1 ∈ ki Aui + gi ,
u1 − u 0 ∈ α u 0 − a ,
i = 1, N,
uN+1 − uN ∈ −β uN+1 − b ,
(1.3) (1.4)
where a,b ∈ H are given, (pi )i=1,N , (ri )i=1,N , and (ki )i=1,N are sequences of positive numbers, and (gi )i=1,N ∈ H N . Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:1 (2004) 11–22 2000 Mathematics Subject Classification: 39A12, 39A70, 47H05 URL: http://dx.doi.org/10.1155/S1687183904310046
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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 on A, α, and β. The case pi ≡ 1, ri ≡ 0, gi ≡ 0 was discussed in [14] for the boundary conditions u0 = a and uN+1 = b. These boundary conditions can be seen as a particular case of (1.4) with α = β = ∂ j (the subdifferential of j), where j : H → R is the lower-semicontinuous, convex, and proper function: 0,
x = 0, j(x) = +∞, otherwise.
(1.5)
In [6, 8, 13, 14], one studies the existence, uniqueness, and asymptotic behavior of the solution of the difference equation
pi + ri ui+1 − 2pi + ri ui + pi ui−1 ∈ ki Aui + gi ,
i ≥ 1,
(1.6)
(pi ≡ 1, ri ≡ 0 in [13, 14] and the general case in [6, 8]), subject to the boundary conditions u0 = a,
sup ui < ∞. i≥0
(1.7)
Here · is the norm of H. In [7], the author establishes the existence for problem (1.3), (1.4) under the hypothesis that A is also strongly monotone. Other classes of difference or differential inclusions in abstract spaces are presented in [3, 4, 5]. In Section 2
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