The Existence of Positive Solutions for a New Coupled System of Multiterm Singular Fractional Integrodifferential Boundary Value Problems

Abstract

We discuss the existence of positive solutions for the coupled system of multiterm singular fractional integrodifferential boundary value problems $D_{0^{+}}^{\alpha }u(t)+f_1(t,u(t),v(t),({\varphi }_{1}u)(t),({\psi }_{1}v)(t),D_{0^{+}}^{p}u(t),D_{{\mathrm{0}}^{+}}^{{\mu }_{\mathrm{1}}}v(t),D_{{\mathrm{0}}^{+}}^{{\mu }_{\mathrm{2}}}v(t),\dots ,D_{{\mathrm{0}}^{+}}^{{\mu }_m}v(t))=\mathrm{0},D_{{\mathrm{0}}^{+}}^{\beta }v(t)+f_{\mathrm{2}}(t,u(t),v(t),({\varphi }_{\mathrm{2}}u)(t),({\psi }_{\mathrm{2}}v)(t),D_{{\mathrm{0}}^{+}}^{q}v(t),D_{{\mathrm{0}}^{+}}^{{\nu }_{\mathrm{1}}}u(t),D_{{\mathrm{0}}^{+}}^{{\nu }_{\mathrm{2}}}u(t),\dots ,D_{{\mathrm{0}}^{+}}^{{\nu }_m}u(t))=\mathrm{0}$, $u^{(i)}(\mathrm{0})=\mathrm{0}$ and $v^{(i)}(\mathrm{0})=\mathrm{0}$ for all $\mathrm{0}\le i\le n-\mathrm{2}$, $[D_{{\mathrm{0}}^{+}}^{{\delta }_{\mathrm{1}}}u(t){]}_{t=\mathrm{1}}=\mathrm{0}$ for and $\alpha -{\delta }_{\mathrm{1}}\ge \mathrm{1}$, $[D_{{\mathrm{0}}^{+}}^{{\delta }_{\mathrm{2}}}v(t){]}_{t=\mathrm{1}}=\mathrm{0}$ for and $\beta -{\delta }_{\mathrm{2}}\ge \mathrm{1}$, where $n\ge \mathrm{4}$, , , $(i=\mathrm{1,2},\dots ,m)$, ${\gamma }_j,{\lambda }_j:[\mathrm{0,1}]\times [\mathrm{0,1}]\to (\mathrm{0},\mathrm{\infty })$ are continuous functions $(j=\mathrm{1,2})$ and $({\varphi }_{j}u)(t)={\int }_{\mathrm{0}}^t\mathrm{‍}{\gamma }_j(t,s)u(s)ds,\mathrm{}\mathrm{}\mathrm{}({\psi }_{j}v)(t)={\int }_{\mathrm{0}}^t\mathrm{‍}{\lambda }_j(t,s)v(s)ds.$ Here $D$ is the standard Riemann-Liouville fractional derivative, $f_j$ $(j=\mathrm{1,2})$ is a Caratheodory function, and $f_j(t,x,y,z,w,v,u_{\mathrm{1}},u_{\mathrm{2}},\dots ,u_m)$ is singular at the value 0 of its variables.