Nonlinear boundary conditions for nonlinear second order differential operators on C[0,1]

Of concern is the operator A defined formally by (Au)(x) - phi((x,u ' (x))u " (x) + psi (x,u(x),u ' (x)), with certain assumptions on phi, psi. Realizations A of A will be defined o n C[0, 1] and determined by boundary conditions. Tn earlier work [7] we too k psi equivalent to 0 and used mixed Wentzell-Robin boundary conditions of the form alpha (j)(Au)(j) + b(j)u ' (j) + c(j)u(j) = 0, j = 0, 1 where: (a(j) b(j), c(j)) + (0, 0, 0). In this paper we consider more genera l boundary conditions of the form alpha (j)(Au)(j) + beta (j)u ' (j) is an element of gamma (j)(u(j)) where gamma (j) is a maximal monotone graph in R-2 and (alpha (j),beta (j)) double dagger (0,0) The conclusion (under suitable hypotheses) is that A is In-dissipative and the Cauchy problem du/dt = Au, u(0) = f is well-posed.