Positive solutions of four-point singular boundary value problems

Existence of C1 positive solutions for a class of second-order nonlinear singular equations of the type-x″(t)+λx′(t)=f(t,x(t)),t∈(0,1),subjectto four-point boundary conditions of the typex(0)=ax(η),x(1)=bx(δ),0<η⩽δ<1,is established. Existence of C1 positive solution is proved with the upper and lower solutions method. Examples are included to show the validity of our results. Finally, the method of quasilinearization is developed to approximate the solution. We show that under suitable conditions on f, there exists a sequence of solutions of linear problems that converges monotonically and quadratically to the solution of the original nonlinear problem.