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By using the coincidence degree theorem, we obtain a new result on the existence of solutions for a class of fractional differential equations with periodic boundary value conditions, where a certain nonlinear growth condition of the nonlinearity needs to be satisfied. Furthermore, we study another class of differential equations of fractional order with periodic boundary conditions at resonance. A new result on the existence of positive solutions is presented by use of a Leggett–Williams norm-type theorem for coincidences. Two examples are given to illustrate the main result at the end of this paper.

Periodic boundary value problems for two classes of nonlinear fractional differential equations