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Abstract In this paper, we consider a class of infinite-point boundary value problems of fractional differential equations on the infinite interval [ 0 , + ∞ ) $[0,+\infty)$ with a disturbance parameter. By using the method of upper and lower solutions, fixed point index theory and some fixed point theorems, the existence, multiplicity and nonexistence for the positive solution of the boundary value problem are obtained, respectively. The impact of the disturbance parameters on the existence of positive solutions is also given. Finally, some examples are presented to illustrate the wide range of potential applications of our main results.