The existence and nonexistence of positive solutions for fractional differential equations with nonhomogeneous boundary conditions

Abstract

In this paper, we study the existence and nonexistence of the positive solutions for a class of fractional differential equations with nonhomogeneous boundary conditions and the impact of the disturbance parameters a, b on the existence of positive solutions. By using the upper and lower solutions method and the Schauder fixed point theorem, we obtain the sufficient conditions for the boundary value problem to have at least one positive solution, two positive solutions, and no positive solution, respectively. Moreover, under certain conditions, we prove that there exists a bounded and continuous curve L dividing $[0,+\infty)\times[0,+\infty)$ into two separate subsets $\Lambda^{E}$ and $\Lambda^{N}$ with $L\subseteq\Lambda^{E}$ such that the boundary value problem has at least two positive solutions for each $(a,b)\in\Lambda^{E}\setminus L$ , one positive solution for each $(a,b)\in L$ , and no positive solution for any $(a,b)\in\Lambda ^{N}$ . Finally, we give some examples to illustrate our main results.