Multiple positive solutions for nonlinear singular m-point boundary value problem

Abstract

In this paper, we study the existence of multiple positive solutions for the following nonlinear m-point boundary value problem (BVP)

$$\left\{\begin{array}{c}(\phi (u^{\prime }){)}^{\prime }+a(t)f(u(t))=0\text{,}0<t<1\text{,}\hfill \\ u^{\prime }(0)=\sum _{i=1}^{m-2}{a}_i{u}^{\prime }({\xi }_i)\text{,}\hfill \\ u(1)=\sum _{i=1}^k{b}_{i}u({\xi }_i)-\sum _{i=k+1}^s{b}_{i}u({\xi }_i)-\sum _{i=s+1}^{m-2}{b}_i{u}^{\prime }({\xi }_i)\text{,}\hfill \end{array}\right.$$

where $\phi :R\to R$ is an increasing homeomorphism and homomorphism and $\phi (0)=0\text{,}1⩽k⩽s⩽m-2\text{,}{a}_i\text{,}{b}_i\in (0\text{,}+\infty )$ with $0<{\sum }_{i=1}^k{b}_i-{\sum }_{i=k+1}^s{b}_i<1\text{,}0<{\sum }_{i=1}^{m-2}{a}_i<1\text{,}0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<1\text{,}a(t)\in C((0\text{,}1)\text{,}[0\text{,}+\infty ))\text{,}f\in C([0\text{,}+\infty )\text{,}[0\text{,}+\infty ))$. Some new results are obtained for the existence of at least twin or triple positive solutions of the above problem by applying Avery–Henderson and a new fixed-point theorems, respectively. As an application, some examples are included to illustrate the main results. In particular, our results extend and improve some known results.

Introduction

In this paper, we study the existence of positive solutions for the following nonlinear m-point boundary value problem (BVP):

$$\left\{\begin{array}{c}(\phi (u^{\prime }){)}^{\prime }+a(t)f(u(t))=0\text{,}0<t<1\text{,}\hfill \\ u^{\prime }(0)=\sum _{i=1}^{m-2}{a}_i{u}^{\prime }({\xi }_i)\text{,}\hfill \\ u(1)=\sum _{i=1}^k{b}_{i}u({\xi }_i)-\sum _{i=k+1}^s{b}_{i}u({\xi }_i)-\sum _{i=s+1}^{m-2}{b}_i{u}^{\prime }({\xi }_i)\text{,}\hfill \end{array}\right.$$

where $\phi :R\to R$ is an increasing homeomorphism and homomorphism and $\phi (0)=0\text{,}1⩽k⩽s⩽m-2\text{,}{a}_i\text{,}{b}_i\in (0\text{,}+\infty )$ with $0<{\sum }_{i=1}^k{b}_i-{\sum }_{i=k+1}^s{b}_i<1\text{,}0<{\sum }_{i=1}^{m-2}{a}_i<1\text{,}0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<1\text{,}a(t)\in C((0\text{,}1)\text{,}[0\text{,}+\infty ))\text{,}f\in C([0\text{,}+\infty )\text{,}[0\text{,}+\infty ))$.

A projection $\phi :R\to R$ is called an increasing homeomorphism and homomorphism, if the following conditions are satisfied:

• if $x⩽y$, then $\phi (x)⩽\phi (y)\forall x\text{,}y\in R$;

• $\phi$ is continuous bijection and its inverse mapping is also continuous;

• $\phi (\mathit{xy})=\phi (x)\phi (y)\forall x\text{,}y\in R$.

The study of multi-point boundary value problems for linear second-order ordinary differential equations was initiated by Il’in and Moviseev [1], [2]. Motivated by the study of Il’in and Moviseev [1], [2], Gupta [3] studied certain three-point boundary value problems for nonlinear ordinary differential equations. Since then, more general nonlinear multi-point boundary value problems have been studied by several authors. We refer the reader to [4], [5], [6], [10], [12] for some references along this line. Multi-point boundary value problems describe many phenomena in the applied mathematical sciences. For example, the vibrations of a guy wire of a uniform cross-section and composed of N parts of different densities can be set up as a multi-point boundary value problems (see [7]); many problems in the theory of elastic stability can be handle by the method of multi-point boundary value problems (see [8]).

In 2001, Ma [6] studied m-point boundary value problem (BVP)

$$\left\{\begin{array}{cc}{u}^{″}(t)+h(t)f(u)=0\text{,}\hfill & 0⩽t⩽1\text{,}\hfill \\ u(0)=0\text{,}\hfill & u(1)=\sum _{i=1}^{m-2}{\alpha }_{i}u({\xi }_i)\text{,}\hfill \end{array}\right.$$

where ${\alpha }_i>0(i=1\text{,}2\text{,}\dots \text{,}m-2)\text{,}0<{\sum }_{i=1}^{m-2}{\alpha }_i<1\text{,}0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<1$, and $f\in C([0\text{,}+\infty )\text{,}[0\text{,}+\infty ))\text{,}h\in C([0\text{,}1]\text{,}[0\text{,}+\infty ))$. Author established the existence of positive solutions theorems under the condition that f is either superlinear or sublinear.

In [4], Ma and Castaneda studied the following m-point boundary value problem (BVP):

$$\left\{\begin{array}{cc}{u}^{″}(t)+h(t)f(u)=0\text{,}\hfill & 0⩽t⩽1\text{,}\hfill \\ u^{\prime }(0)=\sum _{i=1}^{m-2}{a}_i{u}^{\prime }({\xi }_i)\text{,}\hfill & u(1)=\sum _{i=1}^{m-2}{\beta }_{i}u({\xi }_i)\text{,}\hfill \end{array}\right.$$

where ${\alpha }_i>0\text{,}{\beta }_i>0(i=1\text{,}2\text{,}\dots \text{,}m-2)\text{,}0<{\sum }_{i=1}^{m-2}{\alpha }_i<1\text{,}0<{\sum }_{i=1}^{m-2}{\beta }_i<1\text{,}0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<1$, and $f\in C([0\text{,}+\infty )\text{,}[0\text{,}+\infty ))\text{,}h\in C([0\text{,}1]\text{,}[0\text{,}+\infty ))$. They showed the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed-point theorem in cones.

Recently, Ma et al. [5] used the monotone iterative technique in cones to prove the existence of at least one positive solution for m-point boundary value problem (BVP)

$$\left\{\begin{array}{cc}({\varphi }_p(u^{\prime }){)}^{\prime }+a(t)f(t\text{,}u(t))=0\text{,}\hfill & 0<t<1\text{,}\hfill \\ u^{\prime }(0)=\sum _{i=1}^{m-2}{a}_i{u}^{\prime }({\xi }_i)\text{,}\hfill & u(1)=\sum _{i=1}^{m-2}{b}_{i}u({\xi }_i)\text{,}\hfill \end{array}\right.$$

where $0<{\sum }_{i=1}^{m-2}{b}_i<1\text{,}0<{\sum }_{i=1}^{m-2}{a}_i<1\text{,}0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<1\text{,}a(t)\in L^1[0\text{,}1]\text{,}f\in C([0\text{,}1]\times [0\text{,}+\infty )\text{,}[0\text{,}+\infty ))$.

Motivated by the results mentioned above, in this paper we study the existence of positive solutions for m-point boundary value problem (1.1). We generalize the results in [4], [5], [6], [15], [16], [17], [18].

In the rest of the paper, we make the following assumptions:

• $a_i\text{,}{b}_i\in (0\text{,}+\infty )\text{,}0<{\sum }_{i=1}^k{b}_i-{\sum }_{i=k+1}^s{b}_i<1\text{,}0<{\sum }_{i=1}^{m-2}{a}_i<1\text{,}0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<1$;

• $f\in C([0\text{,}+\infty )\text{,}[0\text{,}+\infty ))\text{,}a(t)\in C((0\text{,}1)\text{,}[0\text{,}+\infty ))$.

By a positive solution of BVP (1.1), we understand a function u which is positive on (0, 1) and satisfies the differential equations as well as the boundary conditions in BVP (1.1).