The Existence and Continuous Dependence Theorem for Solutions of Differential Equations

1978 ◽  
pp. 53-77
Author(s):  
R. V. Gamkrelidze
Keyword(s):  
Differential Equations ◽  
Continuous Dependence ◽  
Continuous Dependence Theorem

scholarly journals Stochastic PDEs and Infinite Horizon Backward Doubly Stochastic Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Bo Zhu ◽  
Baoyan Han
Keyword(s):  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Continuous Dependence ◽  
Infinite Horizon ◽  
Stochastic Pdes ◽  
Sufficient Condition ◽  
Probabilistic Interpretation ◽  
Doubly Stochastic ◽  
Terminal Values ◽  
Continuous Dependence Theorem

We give a sufficient condition on the coefficients of a class of infinite horizon BDSDEs, under which the infinite horizon BDSDEs have a unique solution for any given square integrable terminal values. We also show continuous dependence theorem and convergence theorem for this kind of equations. A probabilistic interpretation for solutions to a class of stochastic partial differential equations is given.

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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