This book applies generalized fractional differentiation techniques of Caputo, Canavati and Conformable types to a great variety of integral inequalities e.g. of Ostrowski and Opial types, etc. Some of these are extended to Banach space valued functions. These inequalities have also great impact in numerical analysis, stochastics and fractional differential equations. The book continues with generalized fractional approximations by positive sublinear operators which derive from the presented Korovkin type inequalities and also includes abstract cases. It presents also multivariate complex Korovkin quantitative approximation theory. It follows M-fractional integral inequalities of Ostrowski and Polya types. The results are weighted so they provide a great variety of cases and applications. The second part of the book deals with the quantitative fractional Korovkin type approximation of stochastic processes and lays there the foundations of stochastic fractional calculus. The book considers both Caputo and Conformable fractional directions and derives regular and trigonometric results. The positive linear operators can be expectation operator commutative or not. This book results are expected to find applications in many areas of pure and applied mathematics and stochastics. As such this monograph is suitable for researchers, graduate students, and seminars of the above disciplines, also to be in all science and engineering libraries.
Table of Content
Caputo ψ-fractional Ostrowski inequalities.- Caputo ψ-fractional Ostrowski and Gruss inequalities involving several functions.- Weighted Caputo fractional Iyengar type inequalities.- Generalized Canavati g-fractional Iyengar and Ostrowski inequalities.- Generalized Canavati g-fractional Polya inequalities.- Caputo generalized ψ-fractional integral type inequalities.- Generalized ψ-fractional Quantitative Approximation by Sub-linear Operators.- Generalized g–iterated fractional Quantitative Approximation by Sublinear Operators.- Generalized g–Fractional vector Representation Formula and Bochner integral type inequalities for Banach space valued functions.- Iterated g–Fractional vector Bochner integral Representation Formulae and inequalities for Banach space valued functions.- Vectorial generalized g–fractional direct and iterated Quantitative Approximation by linear operators.- Quantitative Multivariate Complex Korovkin Approximation Theory.- M-fractional integral type inequalities.- Principles of Stochastic Caputo Fractional Calculus with Fractional Approximation of Stochastic Processes.- Trigonometric Caputo Fractional Approximation of Stochastic Processes.- Trigonometric Conformable Fractional Approximation of Stochastic Processe.- Commutative Caputo Fractional Korovkin Approximation for Stochastic Processes.- Trigonometric Commutative Caputo Fractional Korovkin Approximation for Stochastic Processes.- Commutative Conformable Fractional Korovkin Approximation for Stochastic Processes.- Trigonometric Commutative Conformable Fractional Korovkin Approximation for Stochastic Processes.- Concluding Remarks.
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