scholarly journals On the Definitions of Nabla Fractional Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Ferhan M. Atici

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.

2020 ◽  
Vol 26 (2) ◽  
pp. 173-183
Author(s):  
Kuldip Raj ◽  
Kavita Saini ◽  
Anu Choudhary

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.


2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
Mary Jacintha ◽  
Abdullah Özbekler

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form Δψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μ λ−s−1−μys=0, where zλ=aλ+bλΔμyλ, Δμ stands for the fractional difference operator in Riemann-Liouville settings and of order μ, 0<μ≤1, and η≥1 is a quotient of odd positive integers and λ∈ℕλ0+1−μ. New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples.


1946 ◽  
Vol 9 (2) ◽  
pp. 159-170
Author(s):  
Kopel Kagan

No satisfactory definition of Dominium in Roman Law has yet been achieved. Amongst English writers Austin many years ago found great difficulty in this question while in modern times Professor Buckland has written ‘it is thus difficult to define Dominium precisely.’ Again, Poste, dealing with Gaius' discussion of dominium, says that his opening statements are ‘deplorably confused.’ These examples are enough to indicate the condition, of uncertainty which prevails. In my submission this uncertainty exists mainly because the conception of ususfructus has never yet been explained adequately. Of Possessio it has been said ‘the definition of Possessio to give the results outlined is a matter of great difficulty. No perfectly correct solution may be possible,’ and this statement is generally accepted as a correct assessment of the present position in juristic literature. But here, too, in my opinion, the reason is again connected with usufruct, for the possessio of the usufructuary has not yet been adequately determined. Gaius (2.93) tells us ‘usufructuarius vero usucapere non potest; primuum quod non possidet, sed habet ius utendi et fruendi.’ Ulpian holds that he had possessio in fact (‘Naturaliter videtur possidere is qui usum fructum habet’ D.41.2.12). On this subject Roby says ‘the fructuary was not strictly a possessor and therefore if he was deprived from enjoying he had not a claim to the original interdict de vi but in virtue of his quasi-possessio a special interdict was granted him.’ Austin saw difficulty in the whole problem of possessio. He wrote ‘by Savigny in his treatise on possessio it is remarked that the possessio of a right of usufruct … resembles the possessio of a thing, by the proprietor, or by an adverse possessor exercising rights of property over the thing. And that a disturbance of the one possession resembles the disturbance of the other. Now this must happen for the reason I have already stated:—namely, that the right of usufruct or user, like that of property, is indefinite in point of user. For what is possession (meaning legal possession not mere physical handling of the subject) but the exercise of a right ?’


2005 ◽  
Vol 18 (1) ◽  
pp. 95-117 ◽  
Author(s):  
Marjorie E. Kornhauser

Publicity of information is a fundamental principle of American democracy. Not only is it instrumental in increasing compliance with the laws, a necessity of any government, but also it is an essential element of the right to know-which itself is an aspect of the first amendment right to free speech. Unfortunately, publicity often conflicts with another fundamental right-the right to privacy. In regards to taxes, citizens essentially have two rights to know: a right to know what the tax laws are, and a right to know that these laws are being administered fairly. Publicity in the tax context traditionally means making tax return information public records in an attempt to ensure the fair administration of the tax laws. This type of publicity, however, generates intense hostility because taxpayers perceive it as a huge invasion of their privacy.After examining the pros and cons of traditional publicity of tax information, this Essay suggests that tax publicity be reconceived more broadly. Redefined in the dictionary sense of simply the transmission of information, tax publicity can include a wide array of communications, varying as to content and audience, which can better achieve publicity’s underlying goals with minimal invasions of privacy. A large portion of publicity in this broad sense can be-and should be-educational.The Essay outlines four publicity proposals to stimulate discussion. Three use the expanded definition of publicity and focus on individual taxpayers: an annual tax statement, a short booklet to accompany the 1040, called Know Your Taxes, and an annual W-4. These essentially educational programs should deliver tax information to taxpayers more effectively than currently occurs. The fourth, more controversial, proposal suggests partial publicity-in the traditional sense. It attempts, however, to minimize the customary objections to publicizing tax return information by reducing invasions of privacy.All the proposals will cost money, but probably less than the costs of enforcing compliance only through increased audits and litigation. They may also have psychic and political costs. Although recent studies show that more informed taxpayers are often more compliant, some of the information may trigger negative attitudes which would decrease compliance and/or create pressure for lower taxes.Regardless of whether taxpayer reactions to the increased information are positive or negative, the greater publicity proposed in the Essay could have salutary effects, especially if it occurred in the context of a rational debate by elected officials about tax policy (instead of the current inflammatory rhetorical sound bites). On the one hand, if taxpayers respond positively to publicity, compliance will increase. If they act negatively, and their hostility to taxes increase, at least the publicity will arm them with more precise information that will allow them to focus their objections to the income tax and thereby lobby more effectively for real tax reform.


2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Dorota Mozyrska ◽  
Małgorzata Wyrwas

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.


2017 ◽  
Vol 1 (36) ◽  
Author(s):  
Morton Luiz Faria de Medeiros

RESUMOO artigo busca analisar o conceito e a natureza jurídica da denúncia anônima, a partir dos métodos de abordagem hipotético-dedutivo e hermenêutico, por intermédio da interpretação de textos jurídicos extraídos de documentos legislativos, jurisprudenciais e doutrinários. Para tanto, principia-se pela definição de anonimato, investigando-se se a denúncia anônima constitui manifestação da liberdade de expressão ou de manifestação de pensamento, ideia ou opinião, a partir de depuração desses conceitos elementares que a circundam. Embora não haja uniformidade normativa ou doutrinária quanto a delimitações terminológicas para distinguir entre as liberdades de expressão, de manifestação de pensamento, de manifestação de opinião e de imprensa, conclui-se que o direito fundamental de liberdade de manifestação de pensamento é mais abrangente, possuindo dimensões individual (liberdade de expressão) e social (liberdade de prestação de informação), além da modalidade de liberdade de provocação de autoridade pública – em que se enquadram a denúncia anônima e os direitos de petição e de ação, por exemplo. Contudo, se, por um lado, a denúncia anônima não se confunde com o direito de petição – em face de este ostentar caráter político e estar atrelado ao direito a ser informado (o que é dificultado pelo anonimato) – tampouco pode ser equiparado ao direito de ação, que para instaurar processo judicial já demanda um mínimo de elementos de prova.ABSTRACTThis paper analyzes the concept and legal basis of anonymous reporting, using hypothetical-deductive and hermeneutics methods, through the interpretation of legal texts extracted from legislative, doctrine and jurisprudence documents. It begins with the definition of anonymity, investigating whether anonymous report constitutes manifestation of freedom of expression or manifestation of thought, idea or opinion, from the depuration of the elementary concepts that surround it. Although there is no normative or doctrinal uniformity regarding terminological delimitations to distinguish between the freedoms of expression, of expression of thought, of expression of opinion and of the press, it is concluded that the fundamental right of freedom of expression of thought is wider, including individual (Freedom of expression) and social (freedom to provide information) dimensions, in addition to the modality of freedom of provocation of public authority - which include anonymous reporting and petition and action rights, for example. However, if, on the one hand, the anonymous reporting is not the same as the right of petition - because the latter has a political character and is linked to the right to be informed (which is hampered by anonymity), it cannot be the same as the right of action, which in order to institute legal proceedings already requires a minimum of evidence.


2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Dorota Mozyrska

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.


Author(s):  
PETER MASSOPUST ◽  
BRIGITTE FORSTER

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.


2015 ◽  
Vol 9 (1) ◽  
pp. 139-149 ◽  
Author(s):  
Ferhan Atici ◽  
Meltem Uyanik

In this paper, we introduce two new monotonicity concepts for a nonnegative or nonpositive valued function defined on a discrete domain. We give examples to illustrate connections between these new monotonicity concepts and the traditional ones. We then prove some monotonicity criteria based on the sign of the fractional difference operator of a function f, ??f with 0 < ? < 1. As an application, we state and prove the mean value theorem on discrete fractional calculus.


Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1979
Author(s):  
Ravi P. Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan ◽  
Ricardo Almeida

The main aim of the paper is to present an algorithm to solve approximately initial value problems for a scalar non-linear fractional differential equation with generalized proportional fractional derivative on a finite interval. The main condition is connected with the one sided Lipschitz condition of the right hand side part of the given equation. An iterative scheme, based on appropriately defined mild lower and mild upper solutions, is provided. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. The approximate method is based on the application of the method of lower and upper solutions combined with the monotone-iterative technique.


Sign in / Sign up

Export Citation Format

Share Document