A second-order boundary value problem with nonlinear and mixed boundary conditions: existence, uniqueness, and approximation (Q1957568)

Summary: A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered \[ Lx=f(t,x,x'),\quad t\in (a,b), \] \[ g(x(a),x(b),x'(a),x'(b)) = 0,\quad x(b)=x(a) \] in which \(L\) is a formally self-adjoint second-order differential operator. Under appropriate assumptions on \(L\), \(f\), and \(g\), existence and uniqueness of solutions is established by the method of upper and lower solutions and Leray-Schauder degree theory. The general quasilinearization method is then applied to this problem. Two monotone sequences converging quadratically to the unique solution are constructed.