Versions Of Ekeland Research Articles

Nonstandard Calculus of Variations

We review approaches to the calculus of variations and optimal control by methods of nonstandard analysis. We present Young’s generalized curves, relaxed controls, versions of Ekeland’s variational principle and of the maximum principle, and nonstandard discretizations of the Euler–Lagrange equations within the Robinson model-theoretic framework (RA) and the Nelson internal set theory (IST). The reader can find a nonstandard treatment of the so-called stochastic calculus of variations in [10].

Parametric Ekeland's variational principle

A parametrized version of Ekeland's variational principle is proved, showing that under suitable conditions, the minimum point of the perturbed function can be chosen to depend continuously on a parameter. Applications of this result are given.

A version of Zhong′s coercivity result for a general class of nonsmooth functionals

A version of Zhong′s coercivity result (1997) is established for nonsmooth functionals expressed as a sum Φ + Ψ, where Φ is locally Lipschitz and Ψ is convex, lower semicontinuous, and proper. This is obtained as a consequence of a general result describing the asymptotic behavior of the functions verifying the above structure hypothesis. Our approach relies on a version of Ekeland′s variational principle. In proving our coercivity result we make use of a new general Palais‐Smale condition. The relationship with other results is discussed.

On an extension of ekeland's principle for vector-valued functions

In this Paper we Give a Version of Ekeland's Variational Principle Together with a Qualitative Complement for Vector Valued functions. One Section is Devoted to Some Applications.

Mapping theorems for gateaux differentiable and accretive operators

LET P BE a nonlinear operator mapping the Banach space X into the Banach space Y. There are many approaches to studying solvability of the equation Px = y for y E Y, a considerable number of which involve local or infinitesimal assumptions on the operator P. This is due in part to the fact that P frequently arises from a differential or integral operator which has monotonicity or smoothness properties for which local apriori estimates are readily obtainable. In this paper we derive mapping theorems for operators P satisfying two different kinds of local assumptions: differentiability assumptions and monotonicity assumptions. Our approach in the former case relies on an abstract Newton-Kantorovich scheme which was pioneered by M. Altman (see, for example, [l]) and further developed in [19]; in the latter case we extend some results of F. E. Browder using a combination of both new and standard techniques. In each case our basic technique involves an application of a new version of Ekeland’s Theorem, which we derive from a fundamental maximal principle of H. Brezis and F. E. Browder. In Section 2 of this paper we derive our new version of Ekeland’s Theorem and in Section 3 apply this result to nonlinear operators. Using different (and, by now, standard) techniques we prove a domain invariance result in Section 4 needed for our result an accretive operators.