Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations

Abstract

We study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type ${}^c𝒟{}_{0+}^{\delta }u\left(t\right)+f(t,u(t\left)\right)=0$, $t\in (0,1)$, $0<t<1$. $u\left(1\right)=\beta u\left(\eta \right)+{\lambda }_2$, $u^{\prime }\left(0\right)=\alpha u^{\prime }\left(\eta \right)-{\lambda }_1$, $u^{\prime \prime }\left(0\right)=0$, $u^{\prime \prime \prime }\left(0\right)=0\cdots u^{(n-1)}\left(0\right)=0$, where, $n-1<\delta <n$, $n(\ge 3)\in \mathbb{N}$, $0<\eta ,\alpha ,\beta <1$, the boundary parameters ${\lambda }_1,{\lambda }_2\in {ℝ}^{+}$ and ${}^{c}D{}_{0+}^{\delta }$ is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results.