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This article aims to construct the presence and diversity principles of minimum one or two positive solutions for a Caputotype fractional-order nonlinear differential equation (CFONLDE for short) with an advanced argument under three-point boundary value conditions (BVCs for short). Guo-Krasnoselskii’s fixed point theorem and Fixed-point index theory in cone spaces are used to analyze this article. First, the Green’s function of the corresponding boundary value problem for a linear fractional differential equation with an advanced argument has been established. Next, several essential properties of that Green’s function have been proved. Finally, in cone spaces, some novel presence and diversity principles of minimum of one or two positive solutions for a CFONLDE with an advanced argument are obtained. To support the analytic proof, some particular examples are included.

Presence and diversity of positive solutions for a Caputo-type fractional order nonlinear differential equation with an advanced argument