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

We present a proof of Hadamard Inverse Function Theorem by the methods of Variational Analysis, adapting an idea of I. Ekeland and E. Séré. [ABSTRACT FROM AUTHOR]

This document is a preface to a special issue of the Serdica Mathematical Journal dedicated to the memory of Asen Lyubomirov Dontchev, a renowned mathematician who specialized in variational analysis, optimization, and optimal control. Dontchev passed away in 2021 at the age of 73. The preface provides a brief overview of Dontchev's career, including his education, research contributions, and publications. It also mentions his role as a mentor and his involvement in various mathematical journals. The special issue contains papers written by Dontchev's colleagues, friends, former students, and collaborators as a tribute to his work and influence. [Extracted from the article]

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 present second-order subdifferentials of Clarke's type of C functions, defined in Banach spaces with separable duals. One of them is an extension of the generalized Hessian matrix of such functions in ℝ, considered by J. B. H.-Urruty, J. J. Strodiot and V. H. Nguyen. Various properties of these subdifferentials are proved. Second-order optimality conditions (necessary, sufficient) for constrained minimization problems with C data are obtained. [ABSTRACT FROM AUTHOR]