Existence of positive periodic solutions of some nonlinear fractional differential equations - Cabada, Alberto; Kisela, Tomáš (Communications in nonlinear science & numerical simulation, 2017)
•Are considered nonlinear fractional differential equations coupled to periodic boundary value conditions.•The right-hand side of the equation contains certain singularities.•Our approach is based on Krasnoselskii fixed point theorem and monotone iterative techniques.•The discussed problems are characterized by a Green's function which has integrable singularities.•Due to the type of singularities contained in right-hand side, the approaches used by other authors cannot be utilized.•The numerical algorithms related to lower and upper solutions do not seem to be used for these kind of problems in the literature.•Illustrative examples are showed on the paper. The paper is devoted to study of existence and uniqueness of periodic solutions for a particular class of nonlinear fractional differential equations admitting its right-hand side with certain singularities. Our approach is based on Krasnosel’skiĭ and Schauder fixed point theorems and monotone iterative technique which enable us to extend some previously known results. The discussed problems are characterized by a Green’s function which has integrable singularities disallowing a direct use of classical techniques known from theory of ordinary differential equations, therefore proper modifications are proposed. Furher, the paper presents simple numerical algorithms directly built on the iterative technique used in theoretical proofs. Illustrative examples conclude the paper.