Multiplicity of Positive Solutions for Fractional Differential Equation with p-Laplacian Boundary Value Problems

Abstract

We investigate the existence of multiple positive solutions of fractional differential equations with $p$-Laplacian operator , $u^{\left(j\right)}\left(a\right)=\mathrm{0},\hspace{0.17em}\hspace{0.17em}j=\mathrm{0,1},\mathrm{2},\dots ,n-\mathrm{2}$, $u^{({\alpha }_{\mathrm{1}})}(b)=\xi u^{({\alpha }_{\mathrm{1}})}(\eta )$, ${\varphi }_p(D_{a^{+}}^{\alpha }u(a))=\mathrm{0}=D_{a^{+}}^{{\beta }_{\mathrm{1}}}({\varphi }_p(D_{a^{+}}^{\alpha }u(b)))$, where $\beta \in (\mathrm{1,2}]$, $\alpha \in (n-\mathrm{1},n],\hspace{0.17em}\hspace{0.17em}n\ge \mathrm{3}$, $\xi \in (\mathrm{0},\mathrm{\infty })$, $\eta \in (a,b)$, ${\beta }_{\mathrm{1}}\in (\mathrm{0,1}]$, ${\alpha }_{\mathrm{1}}\in \{\mathrm{1,2},\dots ,\alpha -\mathrm{2}\}$ is a fixed integer, and ${\varphi }_p(s)=|s{|}^{p-\mathrm{2}}s,\hspace{0.17em}\hspace{0.17em}p>\mathrm{1},\hspace{0.17em}\hspace{0.17em}{\varphi }_p^{-\mathrm{1}}={\varphi }_q,\hspace{0.17em}\hspace{0.17em}(\mathrm{1}/p)+(\mathrm{1}/q)=\mathrm{1}$, by applying Leggett–Williams fixed point theorems and fixed point index theory.