In the current work, we study a Cauchy problem for a time-fractional pseudo-parabolic equation with a globally Lipschitz source term. We prove the unique existence of a mild solution to the problem, by the common Banach fixed point theorem. This solution is then verified that exists globally in time by Gr¨onwall’s inequality. Compare to previous works about the similar issuse, we approach in a way that does not require using weighted spaces. Although our approach share a similar spirit to previous studies, our method seems to be more precise and natural.

In this paper, we consider initial value problems for nonlinear fractional equations where source functions may discontinuous. We obtain the existence and uniqueness of maximal mild solutions of the problem. We also give some appropriate conditions such that mild solutions of the problem blow-up at a finite time. Furthermore, we discuss the continuous dependence of mild solutions of the problem with respect to fractional order.

This paper is devoted to study a fractional equation involving Caputo-Katugampola derivative with nonlocal initial condition. Unlike previous papers, in this paper, the source function of problem is assumed having a singularity. We propose some reasonable conditions such that the problem has at least one mild solution or has a unique mild solution. The desired results are proved by using the Banach, Leray-Schauder and Krasnoselskii fixed point theorems. Some examples are given to confirm our theoretical findings. Keywords: Caputo-Katugampola fractional derivative; Nonlinear integral equations; existence 2010 MSC: 26A33; 35A01; 35A02; 35R11

In this article, a class of Hindmarsh-Rose model is studied. First, all necessary conditions for the parameters of system are found in order to have one stable fixed point which presents the resting state for this famous model. After that, using the Hopf’s theorem proofs analytically the existence of a Hopf bifurcation, which is a critical point where a system’s stability switches and a periodic solution arises. More precisely, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues cross the complex plane imaginary axis. Moreover, with the suitable assumptions for the dynamical system, a small-amplitude limit cycle branches from the fixed point.