Newton-Like Methods on Generalized Banach Spaces and Fractional Calculus

2015 ◽  
pp. 1-21
Author(s):  
George A. Anastassiou ◽  
Ioannis K. Argyros
Keyword(s):  
Fractional Calculus ◽  
Banach Spaces

scholarly journals Newton-Type Methods on Generalized Banach Spaces and Applications in Fractional Calculus

Algorithms ◽  
2015 ◽  
Vol 8 (4) ◽  
pp. 832-849 ◽  
Author(s):  
George Anastassiou ◽  
Ioannis Argyros
Keyword(s):  
Fractional Calculus ◽  
Banach Spaces

scholarly journals Controllability of Impulsive Fractional Functional Integro-Differential Equations in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
C. Ravichandran ◽  
J. J. Trujillo
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Banach Spaces ◽  
Mixed Type ◽  
Sufficient Conditions ◽  
Controllability Problem ◽  
Controllability Result ◽  
Fractional Functional

This paper is concerned with the controllability problem for a class of mixed type impulsive fractional integro-differential equations in Banach spaces. Sufficient conditions for the controllability result are established by using suitable fixed point theorem combined with the fractional calculus theory and solution operator under some weak conditions. The example is given in illustrate the theory. The results of this article are generalization and improved of the recent results on this issue.

scholarly journals Existence of solutions for impulsive fractional integrodifferential equations involving Gronwall’s inequality in Banach spaces

2012 ◽  
Vol 21 (2) ◽  
pp. 115-122
Author(s):  
A. AL-OMARI ◽  
◽  
M. H. M. RASHID ◽  
K. KARTHIKEYAN ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fractional Calculus ◽  
Banach Spaces ◽  
Caputo Derivative ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Fixed Point Method ◽  
Integrodifferential Equations ◽  
Gronwall’S Inequality ◽  
Mild Conditions

In this paper, we study boundary value problems for impulsive fractional integrodifferential equations involving Caputo derivative in Banach spaces. A generalized singular type Gronwall inequality is given to obtain an important priori bounds. Some sufficient conditions for the existence solutions are established by virtue of fractional calculus and fixed point method under some mild conditions.

scholarly journals On the Theory of Fractional Calculus in the Pettis-Function Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Hussein A. H. Salem
Keyword(s):  
Integral Equation ◽  
Fractional Calculus ◽  
Boundary Value Problem ◽  
Banach Spaces ◽  
Fractional Order ◽  
Continuous Solution ◽  
Usual Definition ◽  
Weakly Continuous ◽  
Definition Of ◽  
Fractional Order Integral

Throughout this paper, we outline some aspects of fractional calculus in Banach spaces. Some examples are demonstrated. In our investigations, the integrals and the derivatives are understood as Pettis integrals and the corresponding derivatives. Our results here extended all previous contributions in this context and therefore are new. To encompass the full scope of our paper, we show that a weakly continuous solution of a fractional order integral equation, which is modeled off some fractional order boundary value problem (where the derivatives are taken in the usual definition of the Caputo fractional weak derivative), may not solve the problem.

Decay integral solutions for a class of impulsive fractional differential equations in Banach spaces

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Tran Ke ◽  
Do Lan
Keyword(s):  
Fixed Point ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Banach Spaces ◽  
Fractional Differential Equations ◽  
Impulsive Fractional Differential Equations ◽  
Fixed Point Approach ◽  
Fractional Differential ◽  
Integral Solutions

AbstractThe aim of this paper is to establish the existence of decay integral solutions to a class of retarded fractional differential equations involving impulsive effects. The results are obtained by using the fixed point approach and fractional calculus tools in Banach spaces. Applications to both ordinary and partial differential equations are presented.

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