In this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory....

In this paper, we use a modification of Krasnoselskii’s fixed point theorem introduced by Burton (see [Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem. Nonlinear Stud., 9 (2002), 181–190.] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equation with variable delay $$x^{\text{'}}\left(t\right)=-a\left(t\right)h\left(x\left(t\right)\right)+\frac{d}{dt}Q\left(t,x\left(t-\tau \left(t\right)\right)\right)+G\left(t,x\left(t\right),x\left(t-\tau \left(t\right)\right)\right).$$ The stability of the zero solution of this eqution provided that $h\left(0\right)=Q\left(t,0\right)=G\left(t,0,0\right)=0$. The Caratheodory condition is used for the functions $Q$ and $G$.

We construct time quasi-periodic solutions and prove almost global existence for the energy supercritical nonlinear Schrödinger equations on the torus in arbitrary dimensions. The main new ingredient is a geometric selection in the Fourier space. This method is applicable to other nonlinear equations.

In this paper we study existence and uniqueness of solutions for a system consisting from fractional differential equations of Riemann-Liouville type subject to nonlocal Erdélyi-Kober fractional integral conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. Examples illustrating our results are also presented.