Multiple Positive Solutions for a Coupled System of p-Laplacian Fractional Order Two-Point Boundary Value Problems

Abstract

This paper establishes the existence of at least three positive solutions for a coupled system of $p$-Laplacian fractional order two-point boundary value problems, $D_{0^{+}}^{{\beta }_1}({\varphi }_p(D_{0^{+}}^{{\alpha }_1}u(t)))=f_1(t,u(t),v(t))$, $t\in (\mathrm{0,1})$, $D_{0^{+}}^{{\beta }_2}({\varphi }_p(D_{0^{+}}^{{\alpha }_2}v(t)))=f_2(t,u(t),v(t))$, $t\in (\mathrm{0,1})$, $u(0)=D_{0^{+}}^{q_1}u(0)=0$, $\gamma u(1)+\delta D_{0^{+}}^{q_2}u(1)=0$, $D_{0^{+}}^{{\alpha }_1}u(0)=D_{0^{+}}^{{\alpha }_1}u(1)=0$, $v(0)=D_{0^{+}}^{q_1}v(0)=0$, $\gamma v(1)+\delta D_{0^{+}}^{q_2}v(1)=0$, $D_{0^{+}}^{{\alpha }_2}v(0)=D_{0^{+}}^{{\alpha }_2}v(1)=0$, by applying five functionals fixed point theorem.