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- W2652845868 abstract "Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mn mathvariant=normal>0</mml:mn><mml:mo><</mml:mo><mml:mi>α</mml:mi><mml:mo>≤</mml:mo><mml:mn mathvariant=normal>1</mml:mn></mml:math>with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mi mathvariant=normal>C</mml:mi><mml:mi mathvariant=normal>F</mml:mi><mml:mi mathvariant=normal>C</mml:mi></mml:math>) and Riemann (<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mi mathvariant=normal>C</mml:mi><mml:mi mathvariant=normal>F</mml:mi><mml:mi mathvariant=normal>R</mml:mi></mml:math>) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mn mathvariant=normal>2</mml:mn><mml:mo><</mml:mo><mml:mi>α</mml:mi><mml:mo>≤</mml:mo><mml:mn mathvariant=normal>3</mml:mn></mml:math>is proved and the ordinary difference Lyapunov inequality then follows as<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math>tends to<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mrow><mml:mn mathvariant=normal>2</mml:mn></mml:mrow></mml:math>from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given." @default.
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- W2652845868 date "2017-01-01" @default.
- W2652845868 modified "2023-10-14" @default.
- W2652845868 title "Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications" @default.
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- W2652845868 doi "https://doi.org/10.1155/2017/4149320" @default.
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