Positive Solutions to Fractional Boundary Value Problems with Nonlinear Boundary Conditions

Abstract

We consider a system of boundary value problems for fractional differential equation given by $D_{0^{+}}^{\beta }{\varphi }_p(D_{0^{+}}^{\alpha }u)(t)={\lambda }_1{a}_1(t)f_1(u(t),v(t))$, $t\in (\mathrm{0,1})$, $D_{0^{+}}^{\beta }{\varphi }_p(D_{0^{+}}^{\alpha }v)(t)={\lambda }_2{a}_2(t)f_2(u(t),v(t))$, $t\in (\mathrm{0,1})$, where , $\beta \le 2$, , ${\lambda }_1$, ${\lambda }_2$ are eigenvalues, subject either to the boundary conditions $D_{0^{+}}^{\alpha }u(0)=D_{0^{+}}^{\alpha }u(1)=0$, $u(0)=0$, $D_{0^{+}}^{{\beta }_1}u(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{1\mathrm{i\hspace{0.17em}}}{D}_{0^{+}}^{{\beta }_1}u({\xi }_{1i})=0$, $D_{0^{+}}^{\alpha }v(0)=D_{0^{+}}^{\alpha }v(1)=0$, $v(0)=0$, $D_{0^{+}}^{{\beta }_1}v(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{2\mathrm{i\hspace{0.17em}}}{D}_{0^{+}}^{{\beta }_1}v({\xi }_{2i})=0$ or $D_{0^{+}}^{\alpha }u(0)=D_{0^{+}}^{\alpha }u(1)=0$, $u(0)=0$, $D_{0^{+}}^{{\beta }_1}u(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{1\mathrm{i\hspace{0.17em}}}{D}_{0^{+}}^{{\beta }_1}u({\xi }_{1i})={\psi }_1(u)$, $D_{0^{+}}^{\alpha }v(0)=D_{0^{+}}^{\alpha }v(1)=0$, $v(0)=0$, $D_{0^{+}}^{{\beta }_1}v(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{2\mathrm{i\hspace{0.17em}}}{D}_{0^{+}}^{{\beta }_1}v({\xi }_{2i})={\psi }_2(v)$, where , $\alpha -{\beta }_1-1\ge 0$ and ${\psi }_1$, ${\psi }_2:C([\mathrm{0,1}])\to [0$, $\mathrm{\infty })$ are continuous functions. The Krasnoselskiis fixed point theorem is applied to prove the existence of at least one positive solution for both fractional boundary value problems. As an application, an example is given to demonstrate some of main results.