scholarly journals New interpolation spaces and strict Hölder regularity for fractional abstract Cauchy problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Md Mansur Alam ◽  
Shruti Dubey ◽  
Dumitru Baleanu
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Analytic Semigroup ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Interpolation Spaces ◽  
Holder Regularity

AbstractWe know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

scholarly journals Space ofω-Periodic Limit Functions and Its Applications to an Abstract Cauchy Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Rui Xie ◽  
Chuanyi Zhang
Keyword(s):  
Cauchy Problem ◽  
Function Space ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Type Function ◽  
Inclusion Relations ◽  
Asymptotically Periodic ◽  
New Space ◽  
Study Inclusion

We introduce a new space consisting of what we callω-periodic limit functions. We investigate some properties of the new function space. In particular, we study inclusion relations among asymptotically periodic type function spaces. Finally, we apply theω-periodic limit functions to investigate the existence and uniqueness of asymptoticallyω-periodic mild solutions of an abstract Cauchy problem.

scholarly journals Stability of mild solutions of the fractional nonlinear abstract Cauchy problem

2021 ◽  
Vol 30 (1) ◽  
pp. 272-288
Author(s):  
J. Vanterler da C. Sousa ◽  
◽  
Kishor D. Kucche ◽  
E. Capelas de Oliveira ◽  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Banach Fixed Point Theorem ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

<abstract><p>Since the first work on Ulam-Hyers stabilities of differential equation solutions to date, many important and relevant papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional calculus, in particular, involving fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of mild solutions for fractional nonlinear abstract Cauchy problem in the intervals $ [0, T] $ and $ [0, \infty) $ using Banach fixed point theorem.</p></abstract>

Regularity theory of Kolmogorov operator revisited

2020 ◽  
pp. 1-12
Author(s):  
Damir Kinzebulatov
Keyword(s):  
Elliptic Equation ◽  
Vector Fields ◽  
Regularity Theory ◽  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Kolmogorov Operator ◽  
Holder Regularity ◽  
Critical Order

Abstract We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

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