We study a new Jerk equation involving three fractional derivatives and anti periodic conditions. By Banach contraction principle, we present an existence and uniqueness result for the considered problem. Then, by applications of Krasnoselskii fixed point theorem, another result for the existence of at least one solution is established. Also, An illustrative example is discussed. At the end, an approximation for Caputo derivaitive is proposed and some chaotic behaviours are discussed by means of the Runge Kutta 4th order method.

We study a new Jerk equation involving three fractional derivatives and anti periodic conditions. By Banach contraction principle, we present an existence and uniqueness result for the considered problem. Then, by applications of Krasnoselskii fixed point theorem, another result for the existence of at least one solution is established. Also, An illustrative example is discussed. At the end, an approximation for Caputo derivaitive is proposed and some chaotic behaviours are discussed by means of the Runge Kutta 4th order method.

We focus on a new type of nonlinear integro-differential equations with nonlocal integral conditions. The considered problem has one nonlinearity with time variable singularity. It involves also some convergent series combined to Riemann-Liouville integrals. We prove a uniqueness of solutions for the proposed problem, then, we provide some examples to illustrate this result. Also, we discuss the Ulam-Hyers stability for the problem. Some numerical simulations, using Rung Kutta method, are discussed too. At the end, a conclusion follows.

In this work, we are concerned with a sequential nonlinear random differential equation of fractional order with nonlocal conditions. This is the first time in the literature where sequential problems and random ones are combined and considered. An existence and uniqueness of solutions for the problem is obtained by means of an appropriate random fixed point theorem. Then, new concepts on the sequential continuous and fractional derivative dependence are introduced. At the end, some results of stability on random, as well for deterministic, data dependence are discussed.