In this paper, we investigate the existence of solutions for nonlinear delay Caputo q—fractional difference equations. The main result is proved by means of Krasnoselskii’s fixed point theorem. As an application, we link the conclusion of the main theorem to an existence result for Lotka—Volterra model.

We discuss existence, and uniqueness of solutions of nonlinear differential equations of fractional order. The differential operators are taken in the Caputo sense and the initial condition is a fuzzy number. To this aim, contraction mapping principle and the fixed point theorem are used and finally an example is given to more illustration of obtained results.

Enza Maria Valente, Patrick M. Abou-Sleiman, Viviana Caputo, Miratul M. K. Muqit, Kirsten Harvey, Suzana Gispert, Zeeshan Ali, Domenico Del Turco, Anna Rita Bentivoglio, Daniel G Healy, Alberto Albanese, Robert Nussbaum, Rafael González-Maldonado, Thomas Deller, Sergio Salvi, Pietro Cortelli, William P. Gilks, David S. Latchman, Robert J. Harvey, Bruno Dallapiccola, Georg Auburger, Nicholas W. ...

In this paper we have used the homotopy analysis method (HAM) to obtain solution of space-time fractional advectiondispersion equation. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented. The obtained results using homotopy analysis method demonstrate the reliability and efficiency of the proposed algorithm.

In this paper, the Caputo time varying singular fractional differential systems with delay and the Riemann-Liouville time varying singular fractional differential systems with delay are considered . By the D− inverse matrix and α − δ function, two fundamental solutions are given. The variation formulae for time varying singular fractional differential systems with delay are obtained. Mathematic...

The Weierstrass–Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor continuous-time linear systems described by the Caputo–Fabrizio derivative. A method for computing solutions of continuous-time systems is presented. Necessary and sufficient conditions for the positivity and stability of these systems are established. The discussion is illustrated ...

Abstract. The study of fractional variational problems in terms of a combined fractional Caputo derivative is introduced. Necessary optimality conditions of Euler–Lagrange type for the basic, isoperimetric, and Lagrange variational problems are proved, as well as transversality and sufficient optimality conditions. This allows to obtain necessary and sufficient Pareto optimality conditions for ...

Inspired by a number of recent investigations, we introduce the new analogues of the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials, the Apostol-Genocchi polynomials based on Mittag-Leffler function. Making use of the Caputo-fractional derivative, we derive some new interesting identities of these polynomials. It turns out that some known results are derived as special cases.

In this paper, an anomalous advection-dispersion model involving a new general Liouville–Caputo fractional-order derivative is addressed for the first time. The series solutions of the general fractional advection-dispersion equations are obtained with the aid of the Laplace transform. The results are given to demonstrate the efficiency of the proposed formulations to describe the anomalous adv...

In this paper, we study the controllability of linear and nonlinear fractional damped dynamical systems, which involve fractional Caputo derivatives, with different order in finite dimensional spaces using the Mittag-Leffler matrix function and the iterative technique. A numerical example is provided to illustrate the theory.