Existence of positive periodic solutions of some nonlinear fractional differential equations

•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 char...

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Vydáno v:Communications in nonlinear science & numerical simulation Ročník 50; s. 51 - 67
Hlavní autoři: Cabada, Alberto, Kisela, Tomáš
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier B.V 01.09.2017
Elsevier Science Ltd
Témata:
Differential equations
Fixed points (mathematics)
Green's functions
Green’s function
Iterative methods
Krasnosel’skiĭ fixed point
Mathematical analysis
Monotone Iterative Methods
Nonlinear equations
Ordinary differential equations
Periodic Fractional Equation
Singularities
34B27
Green’s function
Periodic Fractional Equation
Krasnosel’skiĭ fixed point
Monotone Iterative Methods
34A08
ISSN:1007-5704, 1878-7274
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Shrnutí:•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.
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ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2017.02.010