Linear openness of multifunctions in metric spaces

We discuss linear openness results for multifunctions with closed graphs in metric spaces

Second order tangency conditions and differential inclusions: a counterexample and a remedy

In this paper we show that second order tangency conditions are superfluous not to say useless while discussing the existence condition for certain second order differential inclusions. In this regard, a counterexample is provided even in the simpler setting of second order differential equations, where a substitute condition is propound. In the setting of differential inclusions, the corresponding substitute condition allows for us to prove existence of sufficiently many approximate solutions without the use of any convexity, measurability, or upper semicontinuity assumption. Accordingly, some proofs in the related literature are greatly simplified

Computation of the tangent cone to the image of a mapping at a singular point

The paper concerns the calculus of the tangent cone to the set F(N) at the point F(p) for sufficiently small neighborhoods N of the singular point p, which means F'(p)(X)subset Y. The general inclusion F'(p)(X)subseteq Tsb{F(N)}(F(p)) is improved to the special equality F'(p)(X)+K=Tsb{F(N)}(F(p)) for a cone K provided that F'(p)(X)+L=Y for a complementing space L. The construction of K inside L depends on the behaviour of F"(p) towards the kernel of F'(p). Here X and Y are Banach spaces, the function Fcolon Xto Y is twice continuously Fréchet differentiable at the point p, F' and F" denote Fréchet differentials, and T stands for the tangent cone