Peano differentiability generalizes ordinary differentiability to higher order. There are two ways to define Peano differentiability for functions defined on non-open sets. For both concepts, we investigate the question under which conditions a function defined on a closed set can be extended to a Peano differentiable function on the ambient space if the sets and functions are definable in an o...

The main result is an improvement of previous results on the equation f(x) + f(y)− f(x+ y) = g[φ(x) + φ(y)− φ(x+ y)] for a given function φ. We find its general solution assuming only continuous differentiability and local nonlinearity of φ. We also provide new results about the more general equation f(x) + f(y)− f(x+ y) = g(H(x, y)) for a given function H. Previous uniqueness results required ...

We prove that if Ω ⊂ R is a bounded domain with C-boundary and g ∈ C(R), then any viscosity solution u ∈ C(Ω) of the infinity Laplacian equation (1.1) is C(Ω). The interior C and C-regularity of u in dimension two has been proved by Savin [20] and Evans-Savin [15] respectively. We also show that for any n ≥ 3, if Ω ⊂ R is a bounded domain with C-boundary and g ∈ C(R), then the solution u of equ...

This article is centered around generalizing a previous implicit function theorem of the author to be applicable for maps f : E⊓F → F which can be lifted to Keller C π –maps fi : E ⊓ Fi → Fi with Fi Banach and F = proj limFi . We prove theorems about existence and differentiability of functions g satisfying f (x, g (x)) = b constant. In addition to these abstract theorems, we give several examp...

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $left(varphi_1, varphi_2right)$-convex function $g, $ with arbitrarily small norm,  such that $f + g $ attains its strong minimum on $X. $ This result extends some of the  well-known varitional principles as that of Ekeland [On the variational principle,  J. Ma...

Adomian decomposition method has been applied to solve many functional equations so far. In this article, we have used this method to solve the fuzzy differential equation under generalized differentiability. We interpret a fuzzy differential equation by using the strongly generalized differentiability. Also one concrete application for ordinary fuzzy differential equation with fuzzy input data...

This article is centered around generalizing a previous implicit function theorem of the author to be applicable for maps f : E⊓F → F which can be lifted to Keller C π –maps fi : E ⊓ Fi → Fi with Fi Banach and F = proj limFi . We prove theorems about existence and differentiability of functions g satisfying f (x, g (x)) = b constant. In addition to these abstract theorems, we give several examp...

This article is centered around generalizing a previous implicit function theorem of the author to be applicable for maps f : E⊓F → F which can be lifted to Keller C π –maps fi : E ⊓ Fi → Fi with Fi Banach and F = proj limFi . We prove theorems about existence and differentiability of functions g satisfying f (x, g (x)) = b constant. In addition to these abstract theorems, we give several examp...

Recently, in their paper, the authors generalized notion of differentiability by defining it for all points functional domain (not only interior points) which can be considered meaningful. In this continuous is introduced differentiable function f:X→Rm with a not necessarily open X⊆Rn; i.e., continuity mapping df:X→HomRn,Rm considered. addition to introducing context approach differentiability,...

• Banach-Alaoglu: compactness of polars • Variant Banach-Steinhaus/uniform boundedness • Second polars • Weak boundedness implies boundedness • Weak-to-strong differentiability The comparison of weak and strong differentiability is due to Grothendieck, although the original sources are not widely available.