The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus

By Kai Diethelm. Fractional Calculus and Applied Analysis, 2012

84 citations recorded by the source corpus.

We generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (firstorder) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.

Subfields: Applied Mathematics, Mechanics of Materials, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/9a2b0c9a31e9338b729a84225e378edb85665556

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In Applied Mathematics

1. Ravi P. Agarwal
2. Donal O'Regan
3. Vasile Lupulescu
4. Sadia Arshad
5. C. Sin
6. Martin J. Bohner
7. Tongxing Li
8. Samir H. Saker

In Mechanics of Materials

1. Donal O'Regan
2. Vasile Lupulescu
3. Ravi P. Agarwal
4. Sadia Arshad
5. C. Sin

In Modeling and Simulation

1. Donal O'Regan
2. Ravi P. Agarwal
3. Vasile Lupulescu
4. Sadia Arshad
5. C. Sin
6. Yuri Luchko
7. Yong Zhou
8. Jinrong Wang

Citations in the corpus

• Fuzzy fractional integral equations under compactness type condition (2012, background): “A systematic study of the fractional differential equations can be found in [2], [3], [4], [13], [14], [15], [18], [22], [29], [31].”
• Monotonicity of functions and sign changes of their Caputo derivatives (2015, unknown): “cannot be extended in this manner. Our goal here is to perform a similar investigation for possible fractional versions of Theorem 1.1. In a certain sense, it thus continues the author’s earlier work [3] that transferred other classical calculus results, MONOTONICITY AND SIGN OF DERIVATIVE ... 3 inc”
• Well-posedness of general Caputo-type fractional differential equations (2018, background): “The new uniqueness results for solutions of fractional ordinary differential equations involving the Caputo derivative were obtained in [5, 6].”