Existence of Positivity of the Solutions for Higher Order Three-Point Boundary Value Problems involving p-Laplacian

Abstract

The present study focusses on the existence of positivity of the solutions to the higher order three-point boundary value problems involving $p$-Laplacian $$[\phi_{p}(x^{(m)}(t))]^{(n)}=g(t,x(t)),~~t \in [0, 1],$$ $$ \begin{aligned} x^{(i)}(0)=0, &\text{~for~} 0\leq i\leq m-2,\\ x^{(m-2)}(1)&-\alpha x^{(m-2)}(\xi)=0,\\ [\phi_{p}(x^{(m)}(t))]^{(j)}_{\text {at} ~ t=0}&=0, \text{~for~} 0\leq j\leq n-2,\\ [\phi_{p}(x^{(m)}(t))]^{(n-2)}_{\text {at} ~ t=1}&-\alpha[\phi_{p}(x^{(m)}(t))]^{(n-2)}_{\text {at} ~ t=\xi}=0, \end{aligned} $$ where $m,n\geq 3$, $\xi\in(0,1)$, $\alpha\in (0,\frac{1}{\xi})$ is a parameter. The approach used by the application of Guo--Krasnosel'skii fixed point theorem to determine the existence of positivity of the solutions to the problem.