Vol. 8, No. 1 (2019): Regular Issue

In this paper, we study the existence and uniqueness of mild solutions of impulsive neutral stochastic functional differential equations with infinite delay and Poisson jumps under non-Lipschitz conditionwith Lipschitz condition being considered as a special case by means of the successive approximation. Further, We study the continuous dependence of solutions on the initial value by means of a corollary of the Bihari inequality.

In this article, the authors have studied the problem of synchronization of time-delay chaotic systems with uncertainties and external disturbances. The effectiveness of the problem statement is visualized through synchronization of time-delay advanced Lorenz system and double time-delay R¨ossler system with parametric uncertainties and disturbances using active control method. Numerical simulations are carried out using Runge-Kutta algorithm for delay differential equations (DDEs) and the results are depicted through graphs. The physical meaning of time-delay system is that a signal is transmitted and received at a later time, which is found to occur in active sensing problems. The salient feature of the article is the demonstration of the efficiency of the considered method during synchronization of time-delay chaotic systems even in presence of uncertainties and external disturbances.

The paper deals with a spatiotemporal dynamics study of criminality city-by-city in attempt to model mechanisms which give rise to synchronisation outbreak of crimes in different cities and villages. The study gives sharp relations between social and penal pressions.

A study on the existence and uniqueness of a solution for nonlinear neutral pantograph equations with Hilfer-Hadamard fractional derivative is the main aim of this paper. The existence result is obtained by using Schauder fixed point theorem. Also, we discuss Ulam stability for these equations based on Banach contraction principle.

Random walk related to diffusion driven transport is widely used to describe phenomena in the context of physics, biology, statistics, and econometrics. However, contrary to the normal (Gaussian) diffusion process, anomalous diffusion takes place in many natural phenomena ranging from turbulence to animal foraging. In situations where spreading process is more rapid than the normal diffusion, the distribution of steps comprises many short movements broken episodically by less frequent long excursions, commonly known as Lévy flight. Recent quantification of high resolution foraging data of many free-ranging animals suggest that they display Lévy flight behavior in environments where prey is scarce. This article describes the fundamental properties of Lévy flight and how it encodes the statistical characteristics of the animal foraging patterns.

In this article, we cultivate the existence theory for the following boundary value problem of fractional integro-differential equations Dα u(t) = f(t,u(t), (φu)(t)), t ∈ [0,T], 1 <α ≤ 2, (φ u)(t)) = γ(t, s)u(s)ds, together with fractional integro-differential boundary conditions Dα−2u(0+) = 0, Dα−1u(0+) =νIα−1u(η), 0 <η < T. By using the coincidence degree theory, we will obtain a new criteria for the existence of the solutions of the given boundary value problems. We present an example to illustrate our main results.

In this work, we establish a set of sufficient conditions for the approximate controllability for a class of Sobolev-type fractional neutral evolution inclusions in Banach spaces. We use Bohnenblust-Karlin’s fixed point theorem to prove our main results. Further, we extend our result to study the approximate controllability for nonlinear fractional control system with nonlocal conditions. An example is also given to illustrate our main results.

In this paper, we first derive the solution representation of fractional Langevin differential equation represented by the fractional differential coefficient in the sense of Caputo fractional derivative in terms of Mittag-Leffler function. Based on this solution representation, controllability of linear fractional Langevin dynamical systems is studied by using Grammian matrix. Sufficient conditions for the controllability of the nonlinear system are established by using the Schauder’s fixed point theorem. An example is given to verify the results.

In this work we present a new derivative of q-uniform type, which contains several definitions of known q-derivatives. Some examples and properties similar to those of the ordinary calculus are also presented.