This paper is mainly devoted to the study of implicit multifunction theorems in terms of Clarke coderivative in general Banach spaces. We present new sufficient conditions for the local metric regularity, metric regularity, Lipschitz-like property, nonemptiness, and lower semicontinuity of implicit multifunctions in general Banach spaces. The basic tools of our analysis involve the Ekeland variational principle, the Clarke subdifferential, and the Clarke coderivative.
This chapter describes smooth variational principles (of Ekeland type) as infinite two-player games. These variational principles are based on a simple but careful recursive choice of points where certain functions that change during the process have values close to their infima. Like many other recursive constructions, the choice has a natural description using the language of infinite two-player games with perfect information. The chapter first considers the perturbation game used in Theorem 7.2.1 to formulate an abstract version of the variational principle before showing how to specialize it to more standard formulations. It then examines the bimetric variant of the smooth variational principle, along with the perturbation functions that are relatively simple. It concludes with an assessment of cases when completeness and lower semicontinuity hold only in a bimetric sense.
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 ( · ) .
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.
Implicit Multifunction Theorems in Banach Spaces
