Positive solutions for the system of higher order singular nonlinear boundary value problems

Abstract

In this paper, by using  Krasnosel'skii fixed point theorem and under suitable conditions, we present the existence of  single and multiple positive solutions to the following systems
$$
\begin{aligned}
(-1)^mu^{(2m)}&=\lambda f(t, u(t), v(t))=0,~~~~ t\in[a, b],\\
(-1)^nv^{(2n)}&=\mu g(t, u(t), v(t))=0,~~~~ t\in[a, b],\\
u^{(2i)}(a)&=u^{(2i)}(b)=0,~~~~0\leq i\leq m-1,\\
v^{(2j)}(a)&=v^{(2j)}(b)=0,~~~~0\leq j\leq n-1,
\end{aligned}
$$
where  $\lambda, \mu>0, m,n\in \N$. We derive two explicit eigenvalue intervals of  $\lambda$  and $\mu$ for  the existence of at least one
positive solution  and the existence of at least two  positive solutions for the above higher order two-point boundary value problem.