Abstract
In this paper, we propose sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions. The Banach contraction principle and the Schauder’s fixed point theorem are used to prove the existence and uniqueness results of the problem. The different orders in one fractional delta differences, one fractional nabla differences, two fractional delta sum, and two fractional nabla sum are considered. Finally, we present an illustrative example.
Highlights
Nowaday, fractional calculus is attractive knowledge for many reseachers in many fields.In particular, the fractional calculus has been used in many research works related to biological, biomechanics, magnetic fields, echanics of micro/nano structures, and physical problems.We can find fractional delta difference calculus and fractional nabla difference calculus in [8–24]and [25–36], respectively
The fractional calculus has been used in many research works related to biological, biomechanics, magnetic fields, echanics of micro/nano structures, and physical problems
Definitions and properties of fractional difference calculus are presented in the book [37]
Summary
Fractional calculus is attractive knowledge for many reseachers in many fields. The fractional calculus has been used in many research works related to biological, biomechanics, magnetic fields, echanics of micro/nano structures, and physical problems (see [1–7]). Malinowska and Torres [38] presented the delta-nabla calculus of variations. Jin and Hou [42] investigated existence of positive solutions for discrete delta-nabla fractional boundary value problems with p-Laplacian. We aim to extend the study of delta-nabla calculus that has appeared in discrete fractional boundary value problems. We have found that the research works related to delta-nabla calculus were presented as above. The boundary value problem for sequential fractional delta-nabla difference equation has not been studied before. Our problem is sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions as given by Mathematics 2020, 8, 476; doi:10.3390/math8040476 www.mdpi.com/journal/mathematics.
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