Positive solutions for generalized Caputo fractional differential equations using lower and upper solutions method

Abstract

The existence and uniqueness of positive solutions are investigated for a new class of boundary value problems for a fractional differential equation involving generalized Caputo fractional derivative of order $\vartheta$ ($1<\vartheta\leq2$). Our approach
relies on the properties of a green function, Banach's contraction principle, and Schauder's fixed point technique on a cone. Moreover, we use building the upper and lower control functions for analysis of the results of our suggested problem. In the end, two examples
are given to justify our acquired results.