A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation

Abstract

A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered, $Lx=f(t,x,x^{\prime })$, $t\in (a,b)$, $g\left(x\right(a),x(b),x^{\prime }(a),x^{\prime }(b\left)\right)=0$, $x\left(b\right)=x\left(a\right)$ 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.