Monotone and Concave Positive Solutions to Three-Point Boundary Value Problems of Higher-Order Fractional Differential Equations

Abstract

We study the three-point boundary value problem of higher-order fractional differential equations of the form ${Dc}_{0+}^{\rho }u\left(t\right)+f\left(t, u\left(t\right)\right)=0$, , , $u^{\prime }(0)=u^{\prime \prime }(0)=\cdots =u^{\left(n-1\right)}(0)=0$, $u(1)+p{u}^{\prime }(1)=q{u}^{\prime }(\mathrm{\xi })$, where ${\mathrm{ }}^c{D}_{0+}^{\rho }$ is the Caputo fractional derivative of order $\rho$, and the function $f:[\mathrm{0,1}]\times [0,\mathrm{\infty })\mapsto [0,+\mathrm{\infty })$ is continuously differentiable. Here, $0⩽q⩽p$, , . By virtue of some fixed point theorems, some sufficient criteria for the existence and multiplicity results of positive solutions are established and the obtained results also guarantee that the positive solutions discussed are monotone and concave.