Finite difference schemes with monotone operators (Q2388384)

Using the functional-analytical theory of maximal monotone operators, the author studies second-order difference-inclusions of the form \[ (p_i+r_i)u_{i+1}-(2p_i+r_i)u_i+p_iu_{i-1} \in k_iAu_i+g_i \] for \(i=1,\dots,N\) under the boundary conditions \[ u_1-u_0\in\alpha(u_0-a), \qquad u_{N+1}-u_N\in-\beta(u_{N+1}-b). \] Here, \(N\) is some positive integer, \(p_i,r_i,k_i\) are positive reals, \(a,b,g_i\) are elements of some given Hilbert space \(H\), and \(A,\alpha,\beta\) denote maximal monotone operators in \(H\). Under some additional conditions on \(\alpha,\beta\) and on the Yosida approximation of \(A\), sufficient criteria for the existence of solutions for the above boundary value problem are given. Stronger assumptions (e.g., strong monotonicity of \(A\) or \(\alpha\) being one-to-one) also yield uniqueness.