A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with $p$-Laplacian

Abstract

In this paper, we consider the existence of positive solutions for a class of singular fourth-order $p$-Laplacian equation with multi-point boundary conditions. By means of a monotone iterative technique, a necessary and sufficient condition for the existence of a pseudo-$C^3[0,1]$ positive solution is established. Also, the existence and uniqueness of the pseudo-$C^3[0,1]$ positive solution, the iterative sequence of the solution and an error estimation for the pseudo-$C^3[0,1]$ positive solution are obtained.

Introduction

In this paper, we consider the existence of positive solutions for the singular fourth-order multi-point boundary value problem with $p$-Laplacian

$$\{\begin{array}{c}{\left[{\phi }_p\left(u^{″}\left(t\right)\right)\right]}^{″}=f(t,u\left(t\right))\text{,}0<t<1\text{,}\\ u\left(0\right)=\sum _{i=1}^{m-2}{a}_{i}u\left({\xi }_i\right)\text{,}u\left(1\right)=0\text{,}{u}^{″}\left(0\right)=\sum _{i=1}^{m-2}{b}_i{u}^{″}\left({\eta }_i\right)\text{,}{u}^{″}\left(1\right)=0\text{,}\end{array}$$

where ${\phi }_p\left(t\right)={\left|t\right|}^{p-2}t,p\ge 2$, $0<a_i,b_i,{\xi }_i,{\eta }_i<1,i=1,2,\dots ,m-2$, are constants, ${\sum }_{i=1}^{m-2}{a}_i<1$, ${\sum }_{i=1}^{m-2}{b}_i<1$, $m\ge 3$, $f\in C((0,1)\times [0,+\infty ),[0,+\infty ))$ may be singular at $t=0$ and/or 1.

Singular differential boundary value problems (BVP) arise from many branches of applied mathematics and physics; for example, gas dynamics, Newtonian fluid mechanics, nuclear physics, engineering sciences and so on can all be described using the above problems. In particular, the study of positive solutions for multi-point (including three-point) BVPs has attracted much attention in recent years; see [4], [7], [8] and the references cited therein. Most results so far have been obtained mainly by using the fixed-point theorems in cones, such as the Krasnoselskii fixed-point theorem [5], the Leggett–Williams theorem [6], Avery and Henderson’s theorem [1], and so on. But these works are only interested in obtaining sufficient conditions for the existence of positive solutions of BVPs. However seeking necessary and sufficient conditions for solutions for singular differential equations is also important, interesting and challenging work. In the special case where $p=2$, i.e., the $p$-Laplacian operator is not involved in the equation, a small sample of works on obtaining necessary and sufficient conditions for the existence of positive solutions of BVPs can be found in the literature [2], [3], [9], [10], [11], [12], [13]. It is well known that when a $p$-Laplacian operator is involved in the equation, the differential operator ${\left[{\phi }_p\left(u^{″}\right)\right]}^{″}$ is nonlinear; thus, the Fredholm Alternative and the maximum principle cannot be extracted as in [3], [4], [10]. So in this case, there appears to be very little known about necessary and sufficient conditions for the existence of positive solutions of BVPs. So our aim in this paper is to establish a necessary and sufficient condition for the existence of a positive solution for the nonlinear singular BVP (1.1) with $p$-Laplacian. At the same time, we will also give the existence and uniqueness of the positive solution, the iterative sequence of the solution, the error estimation and the convergence rate of the positive solutions. Our main tool is the monotone iterative technique, which is essentially different from those of [2], [3], [9], [10], [11], [12], [13].

This paper is organized as follows. In Section 2, we first present some preliminaries and lemmas. Next we give some properties of Green’s functions which are used to prove our main results. Then in Section 3 the existence, uniqueness, error estimation and convergence rate for the pseudo-$C^3[0,1]$ positive solution for BVP (1.1) will be established by using the monotone iterative technique; at the end some other kinds of multi-point boundary value problems are discussed and some further results are obtained.