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We provide new proofs of two intrinsic properties of prox-regular sets in Hilbert spaces. [ABSTRACT FROM AUTHOR]

We consider the classical inverse mapping theorem of Nash and Moser from the angle of some recent development by Ekeland and the authors. Geometrisation of tame estimates coupled with certain ideas coming from variational analysis when applied to a directionally differentiable mapping produces very general surjectivity result and, if injectivity can be ensured, inverse mapping theorem with the expected Lipschitz-like continuity of the inverse. We also present a brief application to differential equations. [ABSTRACT FROM AUTHOR]

We study Orlicz functions that do not satisfy the \Delta _2-condition at zero. We prove that for every Orlicz function M such that \limsup _{t\to 0}M(t)/t^p \!>0 for some p\ge 1, there exists a positive sequence T=(t_k)_{k=1}^\infty tending to zero and such that \begin{equation*} \sup _{k\in \mathbb {N}}\frac {M(ct_k)}{M(t_k)} <\infty,\text { for all }c>1, \end{equation*} that is, M satisfies the \Delta _2 condition with respect to T. Consequently, we show that for each Orlicz function with lower Boyd index \alpha _M < \infty there exists an Orlicz function N such that: (a) there exists a positive sequence T=(t_k)_{k=1}^\infty tending to zero such that N satisfies the \Delta _2 condition with respect to T, and (b) the space h_N is isomorphic to a subspace of h_M generated by one vector. We apply this result to find the maximal possible order of Gâteaux differentiability of a continuous bump function on the Orlicz space h_M(\Gamma) for \Gamma uncountable. [ABSTRACT FROM AUTHOR]

We prove surjectivity result in Fréchet spaces of Nash–Moser type, that is, with uniform estimates over all seminorms. Our method works for functions, which are only continuous and strongly Gâteaux differentiable. We present the results in multi-valued setting exploring the relevant notions of map regularity. The key to our method is in geometrizing the tameness estimates and thus reducing the problem to a spectrum of problems on suitable Banach spaces. For solving the latter problems, we employ an abstract iteration scheme developed by the authors. [ABSTRACT FROM AUTHOR]

We study Lipschitz continuity with respect to the parameter of the set of solutions of a parameterized minimax problem on a product Banach space. We present a sufficient condition, ensuring that the map which to any value of the parameter assigns the set of solutions of the problem (possibly multi-valued, and unbounded) possesses Aubin property. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the optimal solutions of a calculus of variations problem are Lipschitz continuous. The result is obtained without assuming that the domain of the Lagrangian is the whole space as usually stated in the literature. So, the contribution of this Note is in giving a new sufficient condition for the nonexistence of a Lavrentiev phenomenon. To cite this article: M. Quincampoix, N. Zlateva, C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

We study integrability properties of bivariate functions defined on the product of Banach spaces. [Copyright &y& Elsevier]

We generalize Ekeland's Variational Principle for cyclic maps. We present applications of this version of the variational principle for proving of existence and uniqueness of best proximity points for different classes of cyclic maps. [ABSTRACT FROM AUTHOR]