Multiple positive solutions for a coupled system of $p$-Laplacian fractional order three-point boundary value problems

Abstract

In this paper, we attempt to establish the existence of at least three positive solutions for a coupled system of $p$-Laplacian fractional order boundary value problems \begin{aligned}\begin{cases} D_{a^+}^{\beta _1}(\phi _{p}(D_{a^+}^{\alpha _1}u(t)))=f_1(t, u(t), v(t)) & \quad a\lt t\lt b,\\ D_{a^+}^{\beta _2}(\phi _{p}(D_{a^+}^{\alpha _2}v(t)))=f_2(t, u(t), v(t)) & \quad a\lt t\lt b, \end{cases}\end{aligned} with the boundary conditions \begin{aligned}\begin{cases} u^{(j)}(a)=0,\ j=0,1,2,\ldots , \ u{''}(b)=\delta u{''}(\xi ),\\ \phi _{p}(D_{a^+}^{\alpha _1}u(a))=0, \ \phi _{p}(D_{a^+}^{\alpha _1} u(b)) =\vartheta \phi _{p}( D_{a^+}^{\alpha _1}u(\eta )),\\ v^{(j)}(a)=0,\ j=0,1,2,\ldots , \ v{''}(b)=\delta v{''}(\xi ),\\ \phi _{p}(D_{a^+}^{\alpha _2}v(a))=0, \ \phi _{p}(D_{a^+}^{\alpha _2} v(b))=\vartheta \phi _{p}( D_{a^+}^{\alpha _2}v(\eta )), \end{cases}\end{aligned} by applying the five functional fixed point theorem.