scholarly journals Monotonicity results for delta and nabla caputo and Riemann fractional differences via dual identities

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3671-3683 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Bahaaeldin Abdalla
Keyword(s):  
Difference Operators ◽  
Fractional Difference ◽  
Monotonicity Properties ◽  
Delta Type ◽  
Dual Identities ◽  
The Right ◽  
Monotonicity Results

Recently, some authors have proved monotonicity results for delta and nabla fractional differences separately. In this article, we use dual identities relating delta and nabla fractional difference operators to prove shortly the monotonicity properties for the (left Riemann) nabla fractional differences using the corresponding delta type properties. Also, we proved some monotonicity properties for the Caputo fractional differences. Finally, we use the Q??operator dual identities to prove monotonicity results for the right fractional difference operators.

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 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.

scholarly journals On Delta and Nabla Caputo Fractional Differences and Dual Identities

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Thabet Abdeljawad
Keyword(s):  
Difference Equations ◽  
Fractional Difference ◽  
Integration By Parts ◽  
Fractional Difference Equations ◽  
Delta Type ◽  
Integration By Parts Formula ◽  
Left And Right ◽  
Dual Identities

We investigate two types of dual identities for Caputo fractional differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by theQ-operator relates left and right fractional sums and differences. Two types of Caputo fractional differences are introduced; one of them (dual one) is defined so that it obeys the investigated dual identities. The relation between Riemann and Caputo fractional differences is investigated, and the delta and nabla discrete Mittag-Leffler functions are confirmed by solving Caputo type linear fractional difference equations. A nabla integration by parts formula is obtained for Caputo fractional differences as well.

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

Monotonicity results for delta fractional differences revisited

2017 ◽  
Vol 67 (4) ◽  
Author(s):  
Lynn Erbe ◽  
Christopher S. Goodrich ◽  
Baoguo Jia ◽  
Allan Peterson
Keyword(s):  
Recent Result ◽  
Fractional Difference ◽  
Monotonicity Results

AbstractIn this paper, by means of a recently obtained inequality, we study the delta fractional difference, and we obtain the following interrelated theorems, which improve recent results in the literature.Theorem A: ℕTheorem B: ℕTheorem C: ℕIn addition, we obtain the following result, which extends a recent result due to Atici and Uyanik.Theorem D: ℕ

Monotonicity results for nabla fractional h ‐difference operators

2020 ◽  
Vol 44 (2) ◽  
pp. 1207-1218
Author(s):  
Xiang Liu ◽  
Feifei Du ◽  
Douglas Anderson ◽  
Baoguo Jia
Keyword(s):  
Difference Operators ◽  
Monotonicity Results
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