Solutions of Systems with Two-Terms Fractional Difference Operators

2013 ◽  
pp. 183-189
Author(s):  
Ewa Girejko ◽  
Dorota Mozyrska ◽  
Małgorzata Wyrwas
Keyword(s):  
Difference Operators ◽  
Fractional Difference

scholarly journals Discrete fractional order two-point boundary value problem with some relevant physical applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
R. Dhineshbabu ◽  
S. Rashid ◽  
M. Rehman
Keyword(s):  
Boundary Value Problem ◽  
Fractional Order ◽  
Contraction Mapping ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Difference Operators ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Fractional Difference ◽  
Rassias Stability

Abstract The results reported in this paper are concerned with the existence and uniqueness of solutions of discrete fractional order two-point boundary value problem. The results are developed by employing the properties of Caputo and Riemann–Liouville fractional difference operators, the contraction mapping principle and the Brouwer fixed point theorem. Furthermore, the conditions for Hyers–Ulam stability and Hyers–Ulam–Rassias stability of the proposed discrete fractional boundary value problem are established. The applicability of the theoretical findings has been demonstrated with relevant practical examples. The analysis of the considered mathematical models is illustrated by figures and presented in tabular forms. The results are compared and the occurrence of overlapping/non-overlapping has been discussed.

Orlicz lacunary sequence spaces of 𝑙-fractional difference operators

2020 ◽  
Vol 26 (2) ◽  
pp. 173-183
Author(s):  
Kuldip Raj ◽  
Kavita Saini ◽  
Anu Choudhary
Keyword(s):  
Difference Operator ◽  
Lacunary Sequence ◽  
Sequence Spaces ◽  
Difference Operators ◽  
Normed Spaces ◽  
Fractional Difference ◽  
New Approach ◽  
Inclusion Relations ◽  
Difference Sequence Spaces ◽  
Bounded Sequences

AbstractRecently, S. K. Mahato and P. D. Srivastava [A class of sequence spaces defined by 𝑙-fractional difference operator, preprint 2018, http://arxiv.org/abs/1806.10383] studied 𝑙-fractional difference sequence spaces. In this article, we intend to make a new approach to introduce and study some lambda 𝑙-fractional convergent, lambda 𝑙-fractional null and lambda 𝑙-fractional bounded sequences over 𝑛-normed spaces. Various algebraic and topological properties of these newly formed sequence spaces have been explored, and some inclusion relations concerning these spaces are also established. Finally, some characterizations of the newly formed sequence spaces are given.

A monotonicity result for discrete fractional difference operators

2014 ◽  
Vol 102 (3) ◽  
pp. 293-299 ◽  
Author(s):  
Rajendra Dahal ◽  
Christopher S. Goodrich
Keyword(s):  
Difference Operators ◽  
Fractional Difference ◽  
Monotonicity Result

scholarly journals TheZ-Transform Method and Delta Type Fractional Difference Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Dorota Mozyrska ◽  
Małgorzata Wyrwas
Keyword(s):  
Matrix Function ◽  
Initial Value Problems ◽  
Stability Problem ◽  
Difference Operators ◽  
Fractional Order Systems ◽  
Initial Value ◽  
Fractional Difference ◽  
Transform Method ◽  
Delta Type ◽  
The Stability

The Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type difference initial value problems for linear fractional-order systems are discussed. We take under our consideration the possible solutions via the classicalZ-transform method. We stress the formula for the image of the discrete Mittag-Leffler matrix function in theZ-transform. We also prove forms of images in theZ-transform of the expressed fractional difference summation and operators. Additionally, the stability problem of the considered systems is studied.

scholarly journals Multiparameter Fractional Difference Linear Control Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Dorota Mozyrska
Keyword(s):  
Control Systems ◽  
Fractional Order ◽  
Initial Conditions ◽  
Linear Control Systems ◽  
Difference Operators ◽  
Fractional Operators ◽  
Fractional Difference ◽  
Order Difference ◽  
Time Control ◽  
Controllability And Observability

The Riemann-Liouville-, Caputo-, and Grünwald-Letnikov-type fractional order difference operators are discussed and used to state and solve the controllability and observability problems of linear fractional order discrete-time control systems with multiorder and multistep. It is shown that the obtained results do not depend on the type of fractional operators and steps. The comparison of systems is made under the number of steps needed, firstly to achieve a final point, and secondly to distinguish initial conditions for particular operator.

SOME REMARKS ABOUT THE CONNECTION BETWEEN FRACTIONAL DIVIDED DIFFERENCES, FRACTIONAL B-SPLINES, AND THE HERMITE–GENOCCHI FORMULA

2008 ◽  
Vol 06 (02) ◽  
pp. 279-290 ◽  
Author(s):  
PETER MASSOPUST ◽  
BRIGITTE FORSTER
Keyword(s):  
Natural Extension ◽  
Divided Differences ◽  
Short Paper ◽  
Difference Operators ◽  
Fractional Difference ◽  
B Splines ◽  
Definition Of

Fractional B-splines are a natural extension of classical B-splines. In this short paper, we show their relations to fractional divided differences and fractional difference operators, and present a generalized Hermite–Genocchi formula. This formula then allows the definition of a larger class of fractional B-splines.

scholarly journals Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Qasem M. Al-Mdallal ◽  
Mohamed A. Hajji
Keyword(s):  
Type Inequality ◽  
Difference Operators ◽  
Fractional Difference ◽  
Positive Order ◽  
Contraction Theorem ◽  
Banach Contraction ◽  
Sturm Liouville ◽  
Initial Difference ◽  
Arbitrary Positive Order ◽  
Banach Contraction Theorem

Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order0<α≤1with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order2<α≤3is proved and the ordinary difference Lyapunov inequality then follows asαtends to2from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.

scholarly journals On the Definitions of Nabla Fractional Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Ferhan M. Atici
Keyword(s):  
Difference Operator ◽  
Difference Operators ◽  
Fractional Operators ◽  
Initial Value ◽  
Fractional Difference ◽  
Power Rule ◽  
The One ◽  
Definition Of ◽  
The Right ◽  
Commutative Property

We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced.

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