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Thabet Abdeljawad

Publications and source records attributed to Thabet Abdeljawad.

At least 19 recordsLinked to original sources
arXiv2023

Hankel determinant for a general subclass of m-fold symmetric bi-univalent functions defined by Ruscheweyh operator

Making use of the Hankel determinant and the Ruscheweyh derivative, in this work, we consider a general subclass of m-fold symmetric normalized bi-univalent functions defined in the open unit disk. Moreover, we investigate the bounds for the second Hankel determinant of this class and some consequences of the results are presented. In addition, to demonstrate the accuracy on some functions and conditions, most general programs are written in Python V.3.8.8 (2021).

math.CV↗
arXiv2022

On discrete generalized nabla fractional sums and differences

This article investigates a class of discrete nabla fractional operators by using the discrete nabla convolution theorem. Inspired by this, we define the discrete generalized nabla fractional sum and differences of Riemann-Liouville and Caputo types. In the process, we give a relationship between the generalized discrete delta fractional operators introduced by Ferreira \cite{Ferreira} and the proposed discrete generalized nabla fractional operators via the dual identities. Also, we present some test examples to justify the relationship. Moreover, we prove the fundamental theorem of calculus for the defined discrete generalized nabla fractional operators. Inspired by the above operators, we define discrete generalized nabla Atangana-Baleanu-like (or Caputo-Fabrizio-like) fractional sum and differences at the end of the article.

math.CA↗
arXiv2020

On Hilfer fractional difference operator

In this article, a new definition of fractional Hilfer difference operator is introduced. Definition based properties are developed and utilized to construct fixed point operator for fractional order Hilfer difference equations with initial condition. We acquire some conditions for existence, uniqueness, Ulam-Hyers and Ulam-Hyers-Rassias stability. Modified Gronwall's inequality is presented for discrete calculus with the delta difference operator.

math.GM↗
arXiv2019

Analysis of some generalized ABC- fractional logistic models

In this article, some logistic models in the settings of Caputo fractional operators with multi-parametered Mittag-Leffer kernels (ABC) are studied. This study mainly focuses on modified quadratic and cubic logistic models in the presence of a Caputo type fractional derivative. Existence and uniqueness theorems are proved and stability analysis is discussed by perturbing the equilibrium points. Numerical illustrative examples are discussed for the studied models.

math.GM↗
arXiv2019

On more general forms of proportional fractional operators

In this article, we propose new proportional fractional operators generated from local proportional derivatives of a function with respect to another function. We present some properties of these fractional operators which can be also called proportional fractional operators of a function with respect to another function or proportional fractional operators with dependence on a kernel function.

math.GM↗
arXiv2019

Mild and classical solutions for fractional evolution differential equation

Investigating the existence, uniqueness, stability, continuous dependence of data among other properties of solutions of fractional differential equations, has been the object of study by an important range of researchers in the scientific community, especially in fractional calculus. And over the years, these properties have been investigated more vehemently, as they enable more general and new results. In this paper, we investigate the existence and uniqueness of a class of mild and classical solutions of the fractional evolution differential equation in the Banach space $Ω$. To obtain such results, we use fundamental tools, namely: Banach contraction theorem, Gronwall inequality and the $β$-times integrated $β$-times integrated $α$-resolvent operator function of an $(α,β)$-resolvent operator function.

math.CA↗
arXiv2019

On a new class of fractional difference-sum operators based on discrete Atangana-Baleanu sums

We formulate a new class of fractional difference and sum operators, study their fundamental properties, and find their discrete Laplace transforms. The method depends on iterating the fractional sum operators corresponding to fractional differences with discrete Mittag-Leffler kernels. The iteration process depends on the binomial theorem. We note in particular the fact that the iterated fractional sums have a certain semigroup property and hence the new introduced iterated fractional difference-sum operators have this semigroup property as well.

math.CA↗
arXiv2018

Rectangular metric like type spaces and fixed points

In this paper we introduce the concept of the rectangular metric like spaces, along with its topology and we prove some fixed point theorems under different contraction principles. We introduce the concept of modified metric-like space as well and prove some topological and convergence properties under the symmetric convergence. Some examples are given to illustrate the proven results and enrich the new introduced metric type spaces.

math.GN↗
arXiv2018

Sturm Liouville Equations in the frame of fractional operators with Mittag-Leffler kernels and their discrete versions

Very recently, some authors have studied new types of fractional derivatives whose kernels are nonsingular. In this article, we study Sturm-Liouville Equations ($SLEs$) in the frame of fractional operators with Mittag-Leffler kernels. We formulate some Fractional Sturm-Liouville Problems ($FSLPs$) with the diffferential part containing the left and right sided derivatives. We investigate the self-adjointness, eigenvalue and eigenfunction properties of the corresponding Fractional Sturm-Liouville Operators ($FSLOs$) by using fractional integration by parts formulas. The nabla discrete version of our results are also established.

math.CA↗
arXiv2016

Symmetric duality for left and right Riemann-Liouville and Caputo fractional differences

A discrete version of the symmetric duality of Caputo-Torres, to relate left and right Riemann-Liouville and Caputo fractional differences, is considered. As a corollary, we provide an evidence to the fact that in case of right fractional differences, one has to mix between nabla and delta operators. As an application, we derive right fractional summation by parts formulas and left fractional difference Euler-Lagrange equations for discrete fractional variational problems whose Lagrangians depend on right fractional differences.

math.CA↗
arXiv2016

Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel

In this manuscript we define the right fractional derivative and its corresponding right fractional integral for the recently introduced nonlocal fractional derivative with Mittag-Leffler kernel. Then, we obtain the related integration by parts formula. We use the $Q-$operator to confirm our results. The corresponding Euler-Lagrange equations are obtained and one illustrative example is discussed.

math.CA↗
arXiv2016

On fractional derivatives with exponential kernel and their discrete versions

In this manuscript we define the right fractional derivative and its corresponding right fractional integral with exponential kernel. Then, we provide the integration by parts formula and we use $Q-$operator to confirm our results. The related Euler-Lagrange equations were obtained and one example is analyzed. Moreover, we formulate and discus the discrete counterparts of our results.

math.DS↗
arXiv2016

A monotonicity result for the $q-$fractional operator

In this article we prove that if the $q-$fractional operator $(~_{q}\nabla_{qa}^αy)(t)$ of order $0<α\leq 1$ , $0 0$ is positive such that $y(a) \geq 0$, then $y(t)$ is $c_q(α)-$increasing, $c_q(α)=\frac{1-q^α}{1-q}q^{1-α}$. Conversely, if y(t) is increasing and $y(a)\geq 0$, then $(~_{q}\nabla_{qa}^αy)(t)\geq 0$. As an application, we proved a $q-$fractional version of the Mean-Value Theorem.

math.CA↗
arXiv2016

Monotonicity Results for Delta and Nabla Caputo and Riemann Fractional Differences via dual identities

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.

math.DS↗
arXiv2015

On Multiplicative Fractional Calculus

We set the main concepts for multiplicative fractional calculus. We define Caputo, Riemann and Letnikov multiplicative fractional derivatives and multiplicative fractional integrals and study some of their properties. Finally, the multiplicative analogue of the local conformable fractional derivative and integral is studied.

math.CA↗