Continuous dependence on the initial condition and a parameter of the solutions of a class of differential equations with variable structure and impulses

1991 ◽  
Vol 22 (4) ◽  
pp. 641-658
Author(s):  
A. B. DISHLIEV ◽  
D. D. BAINOV
Keyword(s):  
Differential Equations ◽  
Continuous Dependence ◽  
Variable Structure ◽  
Initial Condition

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.

scholarly journals Existence and Uniqueness Results for Two-Dimensional Stochastic Linearised Boussinesq Equation

2020 ◽  
Vol 6 (1) ◽  
pp. 20
Author(s):  
Sofije Hoxha ◽  
Fejzi Kolaneci
Keyword(s):  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Boussinesq Equation ◽  
Initial Condition ◽  
Two Dimensional ◽  
Boussinesq Problem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Drainable Porosity

The water flow in saturated zones of the soil is described by two-dimensional Boussinesq equation. This paper is devoted to investigating the linearised stochastic Boussinesq problem in the presence of randomness in hydraulic conductivity, drainable porosity, recharge, evapotranspiration, initial condition and boundary condition. We use the Sabolev spaces and Galerkin method. Under some suitable assumptions, we prove the existence and uniqueness results, as well as, the continuous dependence on the data for the solution of linearised stochastic Boussinesq problem. Keywords: linearised stochastic Boussinesq equation, Galerkin method, existence and uniqueness results, and continuous dependence on the data.

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