scholarly journals On the One-Leg Methods for Solving Nonlinear Neutral Differential Equations with Variable Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Wansheng Wang ◽  
Shoufu Li
Keyword(s):  
Differential Equations ◽  
Lipschitz Condition ◽  
Error Bounds ◽  
Natural Question ◽  
Variable Delay ◽  
Neutral Differential Equations ◽  
The One ◽  
Error Behavior ◽  
Lipschitz Conditions

Based onA-stable one-leg methods and linear interpolations, we introduce four algorithms for solving neutral differential equations with variable delay. A natural question is which algorithm is better. To answer this question, we analyse the error behavior of the four algorithms and obtain their error bounds under a one-sided Lipschitz condition and some classical Lipschitz conditions. After extensively numerically experimenting, we give a positive conclusion.

COMPARISON THEOREMS FOR CAPUTO–HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2019 ◽  
Vol 27 (03) ◽  
pp. 1950036 ◽  
Author(s):  
LI MA
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Fractional Derivatives ◽  
Comparison Theorems ◽  
Hadamard Fractional Differential Equations ◽  
Fractional Differential ◽  
The One ◽  
Lipschitz Conditions ◽  
Theoretical Results

The main purpose of this paper is to investigate the comparison theorems for fractional differential equations involving Caputo–Hadamard fractional derivatives. First, we indicate the continuous dependence on parameters of solutions for Caputo–Hadamard fractional differential equations (C-HFDEs). Then, the first and second comparison theorems for C-HFDEs are proposed and proved, respectively. In addition, we establish the generalized comparisons for C-HFDEs under the one-side Lipschitz conditions. At last, the corresponding examples are also provided to verify the theoretical results.

scholarly journals Mixed Caputo Fractional Neutral Stochastic Differential Equations with Impulses and Variable Delay

2021 ◽  
Vol 5 (4) ◽  
pp. 239
Author(s):  
Mahmoud Abouagwa ◽  
Rashad A. R. Bantan ◽  
Waleed Almutiry ◽  
Anas D. Khalaf ◽  
Mohammed Elgarhy
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Calculus ◽  
Variable Delay ◽  
Existence And Uniqueness Theorem ◽  
Poisson Jumps ◽  
New Class ◽  
Special Cases ◽  
Lipschitz Conditions ◽  
Theoretical Results

In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and non-Lipschitz conditions as special cases. Some existing results are generalized and enhanced. Finally, an application is offered to illustrate the obtained theoretical results.

Stability in measure for uncertain delay differential equations based on new Lipschitz conditions

2021 ◽  
pp. 1-13
Author(s):  
Yin Gao ◽  
Lifen Jia
Keyword(s):  
Control System ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Delay Time ◽  
Delay Differential Equations ◽  
Special Class ◽  
Liu Process ◽  
The Stability ◽  
Delay Differential ◽  
Lipschitz Conditions

Uncertain delay differential equations (UDDEs) charactered by Liu process can be employed to model an uncertain control system with a delay time. The stability of its solution always be a significant matter. At present, the stability in measure for UDDEs has been proposed and investigated based on the strong Lipschitz condition. In reality, the strong Lipschitz condition is so strictly and hardly applied to judge the stability in measure for UDDEs. For the sake of solving the above issue, the stability in measure based on new Lipschitz condition as a larger scale of applications is verified in this paper. In other words, if it satisfies the strong Lipschitz condition, it must satisfy the new Lipschitz conditions. Conversely, it may not be established. An example is provided to show that it is stable in measure based on the new Lipschitz conditions, but it becomes invalid based on the strong Lipschitz condition. Moreover, a special class of UDDEs is verified to be stable in measure without any limited condition. Besides, some examples are investigated in this paper.

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