Abstract
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.
Highlights
Fractional calculus is the emerging mathematical field which is devoted to studying convolution-type pseudo-differential operators, integrals and derivatives of any arbitrary real or complex order
The fractional calculus has been considered as the best tool for the generalization of fractional differential equations
The existence of solutions to the fractional differential equations with anti-periodic boundary value conditions has been studied by many authors
Summary
Fractional calculus is the emerging mathematical field which is devoted to studying convolution-type pseudo-differential operators, integrals and derivatives of any arbitrary real or complex order. The existence of solutions to the fractional differential equations with anti-periodic boundary value conditions has been studied by many authors (see [16,17,18,19,20,21]). In [27], Hu and Zhang gained the existence of positive solutions of fractional differential equation with periodic boundary value conditions of the form:
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