Thabet Abdeljawad

On conformable fractional calculus

Recently, the authors Khalil et?al. (2014) introduced a new simple well-behaved definition of the fractional derivative called conformable fractional derivative. In this article we proceed on to develop the definitions there and set the basic concepts in this new simple interesting fractional calculus. The fractional versions of chain rule, exponential functions, Gronwalls inequality, integration by parts, Taylor power series expansions, Laplace transforms and linear differential systems are proposed and discussed.

On Riemann and Caputo fractional differences

In this paper, we define left and right Caputo fractional sums and differences, study some of their properties and then relate them to Riemann-Liouville ones studied before by Miller K. S. and Ross B., Atici F.M. and Eloe P. W., Abdeljawad T. and Baleanu D., and a few others. Also, the discrete version of the Q-operator is used to relate the left and right Caputo fractional differences. A Caputo fractional difference equation is solved. The solution proposes discrete versions of Mittag-Leffler functions.

Existence and uniqueness of a common fixed point on partial metric spaces

Abstract In this work, a general form of the weak ϕ -contraction is considered on partial metric spaces, to get a common fixed point. It is shown that self-mappings S , T on a complete partial metric space X have a common fixed point if it is a generalized weak ϕ -contraction.

Fixed points for generalized weakly contractive mappings in partial metric spaces

Abstract Partial metric spaces were introduced by S. G. Matthews in 1994 as a part of the study of denotational semantics of dataflow networks. In this article, we prove fixed point theorems for generalized weakly contractive mappings on partial metric spaces. These theorems generalize many previously obtained results. An example is given to show that our generalization from metric spaces to partial metric spaces is real.

A generalized contraction principle with control functions on partial metric spaces

Partial metric spaces were introduced by Matthews in 1994 as a part of the study of denotational semantics of data flow networks. In this article, we prove a generalized contraction principle with control functions @f and @j on partial metric spaces. The theorems we prove generalize many previously obtained results. We also give some examples showing that our theorems are indeed proper extensions.

Caputo q-fractional initial value problems and a q-analogue Mittag–Leffler function

Abstract Caputo q-fractional derivatives are introduced and studied. A Caputo -type q-fractional initial value problem is solved and its solution is expressed by means of a new introduced q-Mittag–Leffler function. Some open problems about q-fractional integrals are proposed as well.

Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay

Fractional calculus started to play an important role for analysis of the evolution of the nonlinear dynamical systems which are important in various branches of science and engineering. In this line of taught in this paper we studied the stability of fractional order nonlinear time-delay systems for Caputos derivative, and we proved two theorems for Mittag-Leffler stability of the fractional nonlinear time delay systems.

Caputo-type modification of the Hadamard fractional derivatives

Generalization of fractional differential operators was subjected to an intense debate in the last few years in order to contribute to a deep understanding of the behavior of complex systems with memory effect. In this article, a Caputo-type modification of Hadamard fractional derivatives is introduced. The properties of the modified derivatives are studied.

Dual identities in fractional difference calculus within Riemann

We investigate two types of dual identities for Riemann fractional sums and differences. The first type relates nabla- and delta-type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. These dual identities insist that in the definition of right fractional differences, we have to use both nabla and delta operators. The solution representation for a higher-order Riemann fractional difference equation is obtained as well.

On Caputo modification of the Hadamard fractional derivatives

This paper is devoted to the study of Caputo modification of the Hadamard fractional derivatives. From here and after, by Caputo-Hadamard derivative, we refer to this modified fractional derivative (Jarad et al. in Adv. Differ. Equ. 2012:142, 2012, p.7). We present the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting. Also, several new related results are presented.