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2022 | Buch

# Fractional Inequalities In Banach Algebras

verfasst von: George A. Anastassiou

Buchreihe: Studies in Systems, Decision and Control

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### Über dieses Buch

This book presents generalized Caputo fractional Ostrowski and Grüss-type inequalities involving several Banach algebra valued functions. Furthermore, the author gives generalized Canavati fractional Ostrowski, Opial, Grüss, and Hilbert-Pachpatte-type inequalities for multiple Banach algebra valued functions. By applying the p-Schatten norms over the von Neumann–Schatten classes, the author produces the analogous refined and interesting inequalities. The author provides many applications. This book’s results are expected to find applications in many areas of pure and applied mathematics, especially in fractional inequalities and fractional differential equations. Other interesting applications are in applied sciences like geophysics, physics, chemistry, economics, and engineering. This book is appropriate for researchers, graduate students, practitioners, and seminars of the above disciplines, also to be in all science and engineering libraries.

### Inhaltsverzeichnis

##### Chapter 1. Generalized Fractional Ostrowski and Grüss Inequalities with Multiple Banach Algebra Valued Functions
Abstract
Using generalized Caputo fractional left and right vectorial Taylor formulae we establish mixed fractional Ostrowski and Grüss type inequalities involving several Banach algebra valued functions. The estimates are with respect to all norms $$\left\| \cdot \right\| _{p}$$, $$1\le p\le \infty .$$ It follows [6].
George A. Anastassiou
##### Chapter 2. Iterated Generalized Fractional Ostrowski and Grüss Inequalities with Multiple Banach Algebra Valued Functions
Abstract
Employing iterated generalized Caputo fractional left and right vectorial Taylor formulae we establish mixed sequential generalized fractional Ostrowski and Grüss type inequalities for several Banach algebra valued functions. The estimates are with respect to all norms $$\left\| \cdot \right\| _{p}$$, $$1\le p\le \infty .$$ We finish with applications. It follows [5].
George A. Anastassiou
##### Chapter 3. Generalized Canavati Fractional Ostrowski, Opial and Grüss Inequalities with Multiple Banach Algebra Valued Functions
Abstract
Using generalized Canavati fractional left and right vectorial Taylor formulae we establish mixed fractional Ostrowski, Opial and Grüss inequalities involving several Banach algebra valued functions. The estimates are with respect to all norms $$\left\| \cdot \right\| _{p}$$, $$1\le p\le \infty .$$ We provide also applications. It follows [6].
George A. Anastassiou
##### Chapter 4. Generalized Canavati Fractional Hilbert–Pachpatte Inequalities for Banach Algebra Valued Functions
Abstract
Using generalized Canavati fractional left and right vectorial Taylor formulae we prove corresponding left and right fractional Hilbert–Pachpatte inequalities for Banach algebra valued functions. We cover also the sequential fractional case. We finish with applications. It follows [3]
George A. Anastassiou
##### Chapter 5. Generalized Ostrowski, Opial and Hilbert-Pachpatte Inequalities for Banach Algebra Valued Functions Involving Integer Vectorial Derivatives
Abstract
Using a generalized vectorial Taylor formula involving ordinary vector derivatives we establish mixed Ostrowski, Opial and Hilbert-Pachpatte inequalities for several Banach algebra valued functions. The estimates are with respect to all norms $$\left\| \cdot \right\| _{p}$$, $$1\le p\le \infty$$. We finish with applications. It follows [3].
George A. Anastassiou
##### Chapter 6. Multivariate Ostrowski Inequalities for Several Banach Algebra Valued Functions
Abstract
Here we are dealing with several smooth functions from a compact convex set of $$\mathbb {R}^{k}$$, $$k\ge 2$$ to a Banach algebra. For these we prove general multivariate Ostrowski inequalities with estimates. It follows [2].
George A. Anastassiou
##### Chapter 7. p-Schatten Norm Generalized Fractional Ostrowski and Grüss Inequalities for Multiple Functions
Abstract
Employing generalized Caputo fractional left and right vectorial Taylor formulae we establish generalized fractional Ostrowski and Grüss inequalities involving several functions that take values in the von Neumann-Schatten class $$\mathcal {B}_{p}\left( H\right)$$, $$1\le p<\infty$$. The estimates are with respect to all p-Schatten norms, $$1\le p<\infty$$.
George A. Anastassiou
##### Chapter 8. p-Schatten Norm Iterated Generalized Fractional Ostrowski and Grüss Inequalities for Multiple Functions
Abstract
Using iterated generalized Caputo fractional left and right vectorial Taylor formulae we establish sequential generalized fractional Ostrowski and Grüss inequalities for several functions that take values in the von Neumann–Schatten class $$\mathcal {B}_{p}\left( H\right)$$, $$1\le p<\infty$$. The estimates are given for all p-Schatten norms, $$1\le p<\infty$$.
George A. Anastassiou
##### Chapter 9. p-Schatten Norm Generalized Canavati Fractional Ostrowski, Opial and Grüss Inequalities for Multiple Functions
Abstract
Using generalized Canavati fractional left and right vectorial Taylor formulae we establish generalized fractional Ostrowski, Opial and Grüss inequalities for several functions that take values in the von Neumann-Schatten class $$\mathcal {B}_{p}\left( H\right)$$, $$1\le p<\infty$$.
George A. Anastassiou
##### Chapter 10. -Schatten Norm Generalized Canavati Fractional Hilbert–Pachpatte Inequalities with von Neumann–Schatten Class Valued Functions
Abstract
Employing generalized Canavati fractional left and right vectorial Taylor formulae we prove corresponding left and right fractional Hilbert–Pachpatte inequalities for von Neumann–Schatten class $$\mathcal {B}_{\gamma }\left( H\right)$$ valued functions. We cover also the sequential fractional case.
George A. Anastassiou
##### Chapter 11. -Schatten Norm Generalized Ostrowski, Opial and Hilbert–Pachpatte Inequalities with von Neumann–Schatten Class Valued Functions Using Ordinary Vectorial Derivatives
Abstract
Using a generalized vectorial Taylor formula involving ordinary vector derivatives we establish mixed Ostrowski, Opial and Hilbert–Pachpatte type inequalities for several von Neumann–Schatten class $$\mathcal {B}_{\gamma }\left( H\right)$$ valued functions. The estimates are with respect to all norms $$\left\| \cdot \right\| _{p}$$, $$1\le p\le \infty$$. We finish with applications. It follows [4].
George A. Anastassiou
##### Chapter 12. -Schatten Norm Multivariate Ostrowski Inequalities for Multiple Neumann–Schatten Class Valued Functions
Abstract
Here we are dealing with several smooth functions from a compact convex set of $$\mathbb {R}^{k}$$, $$k\ge 2$$ to a Neumann–Schatten class $$\mathcal {B}_{\gamma }\left( H\right)$$, $$\gamma \ge 1$$, which is a Banach algebra. For these we prove general multivariate Ostrowski inequalities with estimates in norms $$\left\| \cdot \right\| _{p},$$ for all $$1\le p\le \infty$$. We provide also interesting applications. It follows [2].
George A. Anastassiou
##### Chapter 13. Conclusion
Abstract
During the last 50 years fractional calculus due to its wide applications to many applied sciences has become a main trend in mathematics.
George A. Anastassiou
Titel
Fractional Inequalities In Banach Algebras
verfasst von
George A. Anastassiou