scholarly journals Solutions to the Cauchy problem for differential equations in Banach spaces with fractional order

2011 ◽  
Vol 62 (3) ◽  
pp. 1303-1311 ◽  
Author(s):  
Zhi-Wei Lv ◽  
Jin Liang ◽  
Ti-Jun Xiao
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Order ◽  
The Cauchy Problem

scholarly journals On existence theorems for differential equations in Banach spaces

1985 ◽  
Vol 32 (1) ◽  
pp. 73-82 ◽  
Author(s):  
Józef Banaś
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Banach Spaces ◽  
Existence Theorems ◽  
The Cauchy Problem

In this paper we show that a number of existence theorems for the Cauchy problem of ordinary differential equations in Banach spaces are only apparent generalizations of the previous ones.

scholarly journals Generalized Uniqueness Theorem for Ordinary Differential Equations in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Ezzat R. Hassan ◽  
M. Sh. Alhuthali ◽  
M. M. Al-Ghanmi
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Uniqueness Theorem ◽  
Banach Spaces ◽  
Nonlinear Ordinary Differential Equations ◽  
Empty Interior ◽  
Reflexive Banach Spaces ◽  
Uniqueness Criterion ◽  
The Cauchy Problem

We consider nonlinear ordinary differential equations in Banach spaces. Uniqueness criterion for the Cauchy problem is given when any of the standard dissipative-type conditions does apply. A similar scalar result has been studied by Majorana (1991). Useful examples of reflexive Banach spaces whose positive cones have empty interior has been given as well.

scholarly journals Kneser's theorem for differential equations in Banach spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 419-434 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Vector Field ◽  
Differential Equations ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Solution Set ◽  
The Cauchy Problem ◽  
Uniformly Continuous

We consider the Cauchy problem x (t) = f (t,x (t)), x (O) = xO defined in a nonreflexive Banach space and with the vector field f: T × X → X being weakly uniformly continuous. Using a compactness hypothesis that involves the weak measure of noncompactness, we prove that the solution set of the above Cauchy problem is nonempty, connected and compact in .

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 On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Ashordia ◽  
Inga Gabisonia ◽  
Mzia Talakhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
The Cauchy Problem ◽  
Generalized Ordinary Differential Equations

AbstractEffective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

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