nonautonomous equations
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2022 ◽  
Vol 6 (1) ◽  
pp. 34
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan

In this paper, nonlinear nonautonomous equations with the generalized proportional Caputo fractional derivative (GPFD) are considered. Some stability properties are studied by the help of the Lyapunov functions and their GPFDs. A scalar nonlinear fractional differential equation with the GPFD is considered as a comparison equation, and some comparison results are proven. Sufficient conditions for stability and asymptotic stability were obtained. Examples illustrating the results and ideas in this paper are also provided.


2015 ◽  
Vol 139 (5) ◽  
pp. 538-557 ◽  
Author(s):  
Jifeng Chu ◽  
Fang-Fang Liao ◽  
Stefan Siegmund ◽  
Yonghui Xia ◽  
Weinian Zhang

2014 ◽  
Vol 12 (02) ◽  
pp. 131-160
Author(s):  
LUIS BARREIRA ◽  
CLAUDIA VALLS

We establish the existence of stable manifolds under sufficiently small perturbations of a linear impulsive equation. Our results are optimal, in the sense that for vector fields of class C1 outside the jumping times, the invariant manifolds are also of class C1 outside these times. We also consider the case of C1 parameter-dependent perturbations and we establish the C1 dependence of the stable manifolds on the parameter. The proof uses the fiber contraction principle. We emphasize that we consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy.


2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Chao Yue ◽  
Chengming Huang

This paper is concerned with the convergence analysis of numerical methods for stochastic delay differential equations. We consider the split-step theta method for nonlinear nonautonomous equations and prove the strong convergence of the numerical solution under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. In particular, these conditions admit that the diffusion coefficient is highly nonlinear. Furthermore, the obtained results are supported by numerical experiments.


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