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Nonlocal Differential Equations with Convolution Coefficients and Applications to Fractional Calculus

Christopher S. Goodrich ORCID logo

Abstract

The existence of at least one positive solution to a large class of both integer- and fractional-order nonlocal differential equations, of which one model case is

-A((b*uq)(1))u′′(t)=λf(t,u(t)),t(0,1),q1,

is considered. Due to the coefficient A((b*uq)(1)) appearing in the differential equation, the equation has a coefficient containing a convolution term. By choosing the kernel b in various ways, specific nonlocal coefficients can be recovered such as nonlocal coefficients equivalent to a fractional integral of Riemann–Liouville type. The results rely on the use of a nonstandard order cone together with topological fixed point theory. Applications to fractional differential equations are given, including a problem related to the (n-1,1)-conjugate problem.


Communicated by Julián López Gómez


Acknowledgements

I would like to thank the anonymous referee for his or her very useful suggestions and careful reading of the original manuscript. In particular, I thank him or her for bringing to my attention references [4, 20, 58].

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Received: 2020-12-25
Revised: 2021-08-20
Accepted: 2021-08-21
Published Online: 2021-09-25

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