Picard Type Iterative Scheme with Initial Iterates in Reverse Order for a Class of Nonlinear Three Point BVPs

We consider the following class of three point boundary value problem , , where , , the source term is Lipschitz and continuous. We use monotone iterative technique in the presence of upper and lower solutions for both well-order and reverse order cases. Under some sufficient conditions, we prove some new existence results. We use examples and figures to demonstrate that monotone iterative method can efficiently be used for computation of solutions of nonlinear BVPs. 1. Introduction In recent years, multipoint boundary value problems have been extensively studied by many authors ([1–5] and the references there in). Multipoint BVPs have lots of applications in various branches of science and engineering; for example, Webb [6] studied a second-order nonlinear boundary value problem subject to some nonlocal boundary conditions, which models a thermostat, and Zou et al. [7] studied the design of a large size bridge with multipoint supports. It is well known that one of the most important tools for dealing with existence results for nonlinear problems is the method of upper and lower solutions. The method of upper and lower solutions has a long history and some of its ideas can be traced back to Picard [8]. Later, it was extensively studied by Dragoni [9]. Recently, there have been numerous results in the presence of an upper solution and a lower solution with . But, in many cases, the upper and lower solutions may occur in the reversed order also, that is, . Cabada et al. [10] considered the monotone iterative method for the following BVP: with reversed ordered upper and lower solutions. So far, there have been some results in the presence of reverse ordered upper and lower solutions [10–13]. Xian et al. [14] considered the following second-order three point BVP: where , , , . They used the fixed point index theory with non-well-ordered upper and lower solutions. Recently, Li et al. [15] studied the existence and uniqueness of solutions of second-order three point BVP with upper and lower solutions in the reversed order via the monotone iterative method in Banach space. The present work proves some new existing results for three point BVPs. Our technique is based on Picard-type iterative scheme and is quite simple and efficient from computational point of view. We believe that it can be very well adapted for this type of problem. In this paper we consider the following three point BVP: where , , , . We have allowed to take both negative and positive values. The paper is divided into 4 sections. In Section 2, we construct Green's function and establish