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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 $\limsup_{t\to0}M(t)/t^p >0$ for some $p\ge1$, there exists a positive sequence $T=(t_k)_{k=1}^\infty$ tending to zero and such that $$ \sup_{k\in\mathbb{N}}\frac{M(ct_k)}{M(t_k)} 1, $$ 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 G\^ateaux differentiability of a continuous bump function on the Orlicz space $h_M(\Gamma)$ for $\Gamma$ uncountable.

We consider the classical Inverse Function 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 function, produce very general surjectivity result and, if injectivity can be ensured, Inverse Function Theorem with the expected Lipschitz-like continuity of the inverse. We also present a brief application to differential equations., Comment: This is revised version. Implicit Mapping Theorem and an application to ODE were added

[Donchev Tzanko; Дончев Цанко]; [Krastanov Mikhail; Кръстанов Михаил]; [Ribarska Nadezhda; Рибарска Надежда]; [Tsachev Tsvetomir; Цачев Цветомир]; [Zlateva Nadia; Златева Надя]

We prove surjectivity result in Fr\'echet spaces of Nash-Moser type. That is, with uniform estimates over all semimorms. Our method works for functions which are only continuous and G\^ateaux differentiable like in the recent result of Ekeland. We present the results in multi-valued setting exploring the relevant notions of map regularity., Comment: The main correction in this version is clarifying that our definition of G\^ateaux differentiability is stronger than the standard one

We provide a new proof for maximal monotonicity of the subdifferential of a convex function.

∗ The work is partially supported by NSFR Grant No MM 409/94., We develop an abstract subdifferential calculus for lower semicontinuous functions and investigate functions similar to convex functions. As application we give sufficient conditions for the integrability of a lower semicontinuous function.

[Ivanov Milen; Иванов Милен]; [Zlateva Nadia; Златева Надя] We prove that Clarke-Ledyaev multidirectional mean value inequality holds for lower semicontinuous function on smooth Banach space. As application we establish a formula for the Clarke-Rockafellar directional derivative of lower semicontinuous function.

[Georgiev Pando Gr.; Георгиев Пандо Гр.]; [Zlateva Nadia P.; Златева Надя П.] We present second-order subdifferentials of Clarke’s type of C^1,1 functions, defined in separable Banach spaces with separable duals, i.e. of functions whose gradient mapping is locally Lipschitz. One of them is an extension of the generalized Hessian matrix of such functions in R^n, 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^1,1 data are obtained.