Some Variants of the Ekeland Variational Principle for a Set-Valued Map

2005 ◽  
Vol 124 (1) ◽  
pp. 187-206 ◽  
Author(s):  
T. X. D. Ha
Keyword(s):  
Variational Principle ◽  
Ekeland Variational Principle

scholarly journals Ekeland Variational Principle in the Variable Exponent Sequence Spaces ℓp(·)

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 375
Author(s):  
Monther R. Alfuraidan ◽  
Mohamed A. Khamsi
Keyword(s):  
Fixed Point ◽  
Variational Principle ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Successful Treatment ◽  
Variable Exponent ◽  
Triangle Inequality ◽  
Sequence Spaces ◽  
Ekeland Variational Principle ◽  
The Core

In this work, we investigate the modular version of the Ekeland variational principle (EVP) in the context of variable exponent sequence spaces ℓ p ( · ) . The core obstacle in the development of a modular version of the EVP is the failure of the triangle inequality for the module. It is the lack of this inequality, which is indispensable in the establishment of the classical EVP, that has hitherto prevented a successful treatment of the modular case. As an application, we establish a modular version of Caristi’s fixed point theorem in ℓ p ( · ) .

scholarly journals Existence of Exponentialp-Stability Nonconstant Equilibrium of Markovian Jumping Nonlinear Diffusion Equations via Ekeland Variational Principle

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Ruofeng Rao ◽  
Shouming Zhong
Keyword(s):  
Variational Principle ◽  
Nonlinear Diffusion ◽  
Lyapunov Stability Theory ◽  
Ekeland Variational Principle ◽  
Diffusion Equations ◽  
Nonlinear Diffusion Equations ◽  
Markovian Jumping ◽  
Existence Criterion ◽  
Delay Dependent ◽  
First Time

The authors obtained a delay-dependent exponentialp-stability criterion for a class of Markovian jumping nonlinear diffusion equations by employing the Lyapunov stability theory and some variational methods. As far as we know, it is the first time to apply Ekeland variational principle to obtain the existence of exponential stability equilibrium ofp-Laplacian dynamic system so that some methods used in this paper are different from those methods of many previous related literatures. In addition, the obtained existence criterion is only involved in the activation functions so that the criterion is simpler and easier than other existence criteria to be verified in practical application. Moreover, a numerical example shows the effectiveness of the proposed methods owing to the large allowable variation range of time-delay.

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