Existence and approximation of solutions of second-order nonlinear four point boundary value problems (Q2581729)

Consider the boundary value problem \[ x''=f(t,x,x'),\;t\in[a,b], \quad x(a)=x(c),\;x(b)=x(d),\tag{*} \] with \(a<c\leq d<b\). The authors use the method of lower and upper solutions to prove the existence of a solution to (*). They apply a generalized method of quasilinearization to obtain monotone sequences converging quadratically to a solution of (*).