Перегляд за Автор "Kaveh Eftekharinasab"

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  • Документ
    A Generalized Palais-Smale Condition in the Fr'{e}chet space setting
    (2018) Kaveh Eftekharinasab
    The Palais-Smale condition was introduced by Palais and Smale in the mid-sixties and applied to an extension of Morse theory to infinite dimensional Hilbert spaces. Later this condition was extended by Palais for the more general case of real functions over Banach-Finsler manifolds in order to obtain Lusternik-Schnirelman theory in this setting.   Despite the importance of Fr'{e}chet spaces, critical point theories have not been developed yet in these spaces.Our aim in this paper is to extend the Palais-Smale condition to the cases of $C^1$-functionals on Fr'{e}chet spaces and Fr'{e}chet-Finsler manifolds of class  $C^1$.    The difficulty in the Fr'{e}chet  setting is the  lack of a general solvability theory for differential equations. This restricts us to adapt the deformation results (which are essential tools to locate critical points) as they appear as solutions of Cauchy problems. However,  Ekeland proved the result, today is known as Ekleand’s variational principle, concerning the existence of almost-minimums for a wide class of real functions on complete metric spaces. This principle can be used to obtain minimizing Palais-Smale sequences.  We use this principle along with the introduced conditions to obtain some customary results concerning the existence of minima in the Fr'{e}chet setting.Recently it has been developed the projective limit techniques to overcome problems (such as  solvability theory for differential equations) with Fr'{e}chet spaces. The idea of this approach is to represent a Fr'{e}chet space as the projective limit of Banach spaces. This approach provides solutions for a wide class of differential equations and every Fr'{e}chet space and therefore can be used to obtain deformation results.  This method would  be the proper framework for further development of critical point theory in the Fr'{e}chet setting.
  • Документ
    On the generalization of the Darboux theorem
    (2019) Kaveh Eftekharinasab
    Darboux theorem to more general context of Frechet manifolds we face an obstacle:  in general vector fields do not have local flows. Recently, Fr'{e}chet geometry has been developed in terms of projective limit of Banach manifolds. In this framework under an appropriate Lipchitz condition The Darboux theorem asserts that a symplectic  manifold $(M^{2n},omega)$ is locally symplectomorphic to $(R^{2n}, omega_0)$, where $omega_0$  is the standard symplectic form on  $R^{2n}$. This theorem was proved by Moser in 1965, the idea of proof, known as the Moser’s trick, works in many situations. The Moser tricks is to construct an appropriate isotopy $ ff_t $  generated by a time-dependent vector field $ X_t  $ on $M$ such that $ ff_1^{*} omega = omega_0$. Nevertheless,  it was showed by Marsden that Darboux theorem is not valid for weak symplectic Banach manifolds. However, in 1999 Bambusi showed that if we  associate to each point of a Banach manifold a suitable Banach space (classifying space) via a given symplectic form then the Moser trick can be applied to obtain the theorem if the  classifying space does not depend on the point of the manifold and a suitable smoothness condition holds.  If we want to try to generalize the local flows exist and with some restrictive conditions the Darboux theorem was proved by Kumar.  In this paper we consider the category of so-called bounded Fr'{e}chet manifolds and prove that in this category vector fields have local flows and following the idea of Bambusi we associate to each point of a manifold a Fr'{e}chet space independent of the choice of the point and with the assumption of bounded smoothness on vector fields  we prove the Darboux theorem.
  • Документ
    Some applications of transversality for infinite dimensional manifolds
    (2021) Kaveh Eftekharinasab
    We present some transversality results for a category of Frechet manifolds, the so-called MCk - Frechet manifolds. In this context, we apply the obtained transversality results to construct the degree of nonlinear Fredholm mappings by virtue of which we prove a rank theorem, an invariance of domain theorem and a Bursuk-Ulam type theorem.
  • Документ
    Some applications of transversality for infinite dimensional manifolds
    (2021) Kaveh Eftekharinasab
    We present some transversality results for a category of Frechet manifolds, the so-called MCk - Frechet manifolds. In this context, we apply the obtained transversality results to construct the degree of nonlinear Fredholm mappings by virtue of which we prove a rank theorem, an invariance of domain theorem and a Bursuk-Ulam type theorem.