scholarly journals Weak Topologies for Carathéodory Differential Equations: Continuous Dependence, Exponential Dichotomy and Attractors

2018 ◽  
Vol 31 (3) ◽  
pp. 1617-1651
Author(s):  
Iacopo P. Longo ◽  
Sylvia Novo ◽  
Rafael Obaya
Keyword(s):  
Differential Equations ◽  
Continuous Dependence ◽  
Exponential Dichotomy ◽  
Weak Topologies

scholarly journals Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations

2021 ◽  
pp. 1-27
Author(s):  
Tomás Caraballo ◽  
Alexandre N. Carvalho ◽  
José A. Langa ◽  
Alexandre N. Oliveira-Sousa
Keyword(s):  
Differential Equations ◽  
Continuous Dependence ◽  
Exponential Dichotomy ◽  
Nonlinear Evolution ◽  
Nonuniform Hyperbolicity ◽  
Exponential Dichotomies ◽  
Infinite Dimensional ◽  
Linear Evolution ◽  
Nonuniform Exponential Dichotomies ◽  
Nonuniform Exponential Dichotomy

In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturbation. Moreover, we provide conditions to obtain uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. We also present an example of evolution process in a Banach space that admits nonuniform exponential dichotomy and study the permanence of the nonuniform hyperbolicity under perturbation. Finally, we prove persistence of nonuniform hyperbolic solutions for nonlinear evolution processes under perturbations.

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

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