Fractional inequalities In Banach algebras (Q2144790)

In this book the author presents his odyssey into the study of fractional inequalities in Banach algebras. The book which is made up of thirteen Chapters focuses on different research of the author in fractional inequalities in Banach algebras. In Chapter 1, the author gives a brief introduction into vectorial fractional calculus and Banach algebras motivated by the previous results of \textit{A. Ostrowski} [Comment. Math. Helv. 10, 226--227 (1938; JFM 64.0209.01)], the author uses the concept of generalized Caputo left and right vectorial Taylor formulae to derive and prove a mixed fractional Ostrowski and Grüss-type inequalities involving several Banach algebra valued functions with applications. In Chapter 2, the author employs the concept of iterated generalized Caputo left and right vectorial Taylor formulae to establish mixed sequential generalized fractional Ostrowski and Grüss-type inequalities involving several Banach algebra valued functions with applications. The uniform and $L^{p}$ estimates of the results obtained with respect to all norms $\left\Vert .\right\Vert_{p},1\leq p\leq \infty$ are similarly derived and proved. In Chapter 3, the author introduces the generalized Canavati fractional left and right vectorial Taylor formulae to derive and prove fractional Ostrowski, Opial and Grüss-type inequalities involving several Banach algebra valued functions with applications. The estimates of the results obtained are with respect to all norms $\left\Vert .\right\Vert_{p},1\leq p\leq \infty$. Applications of the results obtained are also provided and discussed. In Chapter 4, the author uses the generalized Canavati fractional left and right vectorial Taylor formulae to prove the left and right fractional Hilbert-Pachpatte inequalities for Banach algebra valued functions. The Chapter ends with two applications of the results obtained. Furthermore, the derived estimates are with respect to all norms $\left\Vert .\right\Vert_{p},1\leq p\leq \infty$. In Chapter 5, the author uses the generalized vectorial Taylor formulae involving ordinary vector derivatives to obtain mixed Ostrowski, Opial and Hilbert-Pachpatte for several Banach algebra valued functions with applications. The estimates obtained are with respect to all norms $\left\Vert .\right\Vert_{p},1\leq p\leq \infty$. Chapter 6 is concerned with several smooth functions from a compact convex set of $\mathbb{R}^{k},k\geq 2$ to a Banach algebra. For this case, the author establishes the multivariate Ostrowski inequalities with estimates and give applications. In Chapter 7, the author employs the concept of generalized Caputo left and right vectorial Taylor formulae to derive and prove generalized fractional Ostrowski and Grüss inequalities involving several functions with values in the von Neumann-Schatten class $\mathcal{B}_{p}(H),1\leq p\leq \infty$. Also estimates of the results obtained with respect to all $p$-Schatten norms, $1\leq p\leq \infty$ are provided. In Chapter 8, the author uses iterated generalized Caputo left and right vectorial Taylor formulae to establish sequential generalized fractional Ostrowski and Grüss-type inequalities for several functions with values in the von Neumann-Schatten class $\mathcal{B}_{p}(H),1\leq p\leq \infty $. Estimates of the results obtained with respect to all $p$-Schatten norms, $1\leq p\leq \infty$ are derived and proved. Applications of the results are given. Chapter 9, the author uses generalized Canavati fractional left and right vectorial Taylor formulae to establish fractional Ostrowski, Opial and Grüss-type inequalities for several functions that take values in the von Neumann-Schatten class $\mathcal{B}_{p}(H),1\leq p\leq \infty$. The estimates of the results obtained with respect to all $p$-Schatten norms, $1\leq p\leq \infty$ are included with applications. In Chapter 10, the author employs the generalized Canavati fractional left and right vectorial Taylor formulae to prove the corresponding left and right fractional Hilbert-Pachpatte inequalities for von Neumann-Schatten class $\mathcal{B}_{\gamma}(H)$ valued functions. The sequential fractional case of the above results is also established. Applications are also included. In Chapter 11, the author uses the generalized vectorial Taylor formulae involving ordinary vector derivatives to obtain mixed Ostrowski, Opial and Hilbert-Pachpatte-type inequalities for several von Neumann-Schatten class $\mathcal{B}_{\gamma}(H)$ valued functions. The estimates derived are with respect to all norms $\left\Vert .\right\Vert_{p},1\leq p\leq \infty$. Applications are also provided. In Chapter 12, the author uses several smooth functions from a compact convex set of $\mathbb{R}^{k},k\geq 2$ to a Neumann-Schatten class $\mathcal{B}_{\gamma}(H),\gamma \geq 1$, which is a Banach algebra to derive and prove general multivariate Ostrowski inequalities with estimates in norms $\left\Vert .\right\Vert_{p},1\leq p\leq \infty$ for all $1\leq p\leq \infty $. Applications of the results obtained are provided. Chapter 13, the conclusion, is a summary of the various techniques used in the book to derive and prove a variety of mixed regular and iterated generalized fractional inequalities. The book is written in such a way that each Chapter contains a preliminary introduction of the basic concepts needed to understand the subject presented. This book will definitely be of interest to graduate students and researchers into the field of fractional calculus in Banach algebras, particularly due to its wide applications in pure and applied mathematics, applied sciences like geophysics, physics, chemistry, economics and engineering.