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

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Kai Diethelm
Keyword(s):  
Fractional Calculus ◽  
Uniqueness Theorem ◽  
Mean Value ◽  
Mean Value Theorems ◽  
The Mean

AbstractIn some of the statements of the author’s paper “The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus” (

scholarly journals On Opial-Type Inequalities for Superquadratic Functions and Applications in Fractional Calculus

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Waqas Nazeer ◽  
Ghulam Farid ◽  
Zabidin Salleh ◽  
Ayesha Bibi
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Mean Value ◽  
Mean Value Theorems ◽  
Superquadratic Functions

We have studied the Opial-type inequalities for superquadratic functions proved for arbitrary kernels. These are estimated by applying mean value theorems. Furthermore, by analyzing specific functions, the fractional integral and fractional derivative inequalities are obtained.

Some weighted quadrature methods based upon the mean value theorems

2020 ◽  
Author(s):  
Herbert H. H. Homeier ◽  
Hari M. Srivastava ◽  
Mohammad Masjed‐Jamei ◽  
Zahra Moalemi
Keyword(s):  
Mean Value ◽  
Mean Value Theorems ◽  
Quadrature Methods ◽  
The Mean ◽  
Weighted Quadrature

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

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Kai Diethelm
Keyword(s):  
Uniqueness Theorem ◽  
Differential Operators ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Initial Value ◽  
First Order ◽  
Fractional Differential ◽  
The Mean ◽  
Fractional Differential Operators ◽  
First Order Differential Equations

AbstractWe 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.

scholarly journals Analysis of discrete fractional operators

2015 ◽  
Vol 9 (1) ◽  
pp. 139-149 ◽  
Author(s):  
Ferhan Atici ◽  
Meltem Uyanik
Keyword(s):  
Fractional Calculus ◽  
Difference Operator ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Fractional Operators ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
The Mean ◽  
Discrete Domain

In this paper, we introduce two new monotonicity concepts for a nonnegative or nonpositive valued function defined on a discrete domain. We give examples to illustrate connections between these new monotonicity concepts and the traditional ones. We then prove some monotonicity criteria based on the sign of the fractional difference operator of a function f, ??f with 0 < ? < 1. As an application, we state and prove the mean value theorem on discrete fractional calculus.

Mean value theorems for polynomial solutions of linear elliptic equations with constant coefficients in the complex plane

2020 ◽  
Vol 17 (4) ◽  
pp. 594-600
Author(s):  
Olga Trofymenko
Keyword(s):  
Elliptic Equation ◽  
Complex Plane ◽  
Elliptic Equations ◽  
Regular Polygon ◽  
Mean Value ◽  
Linear Elliptic Equation ◽  
Mean Value Theorems ◽  
Polynomial Solutions ◽  
Constant Coefficients ◽  
The Mean

We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon.

71. The Mean Value Theorems

1961 ◽  
Vol 45 (352) ◽  
pp. 141 ◽  
Author(s):  
A. J. Moakes
Keyword(s):  
Mean Value ◽  
Mean Value Theorems ◽  
The Mean
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