S. M. Robinson published in 1980 a powerful theorem about solutions to certain “generalized equations” corresponding to parameterized variational inequalities which could represent the first-order optimality conditions in nonlinear programming, in particular. In fact, his result covered much of the classical implicit function theorem, if not quite all, but went far beyond that in ideas and form...

In a standard free end nonsmooth control problem in finite dimensional state space, a nonsmooth maximum principle is proved, in which the adjoint inclusion is sharper than the usual one. For end constrained problems, the same result holds, provided conditions ensuring local controllability are satisfied. The adjoint inclusion is expressed by means of a type of generalized gradient of the pseudo...

Consider a mean-reverting equation, generalized in the sense it is driven by a 1-dimensional centered Gaussian process with Hölder continuous paths on [0, T ] (T > 0). Taking that equation in rough paths sense only gives local existence of the solution because the non-explosion condition is not satisfied in general. Under natural assumptions, by using specific methods, we show the global existe...

Let Lθ be the circular cone in IR which includes second-order cone as a special case. For any function f from IR to IR, one can define a corresponding vector-valued function fLθ on IR by applying f to the spectral values of the spectral decomposition of x ∈ IR with respect to Lθ. The main results of this paper are regarding the Hdifferentiability and calmness of circular cone function fLθ . Spe...

We construct two counter-examples related to Fréchet differentiability in infinite dimensions. The first one gives a convex Lipschitzian function on a Banach space such that its convolution with a given measure is Fréchet differentiable only on a measure zero set. The second one gives a Borel function on a space with a Gaussian measure such that it is Lipschitzian along the Cameron–Martin subsp...

in this study, we introduce the classes of $phi$-strongly pseudocontractive mappings in the intermediate sense and generalized $phi$-pseudocontractive mappings in the intermediate sense and prove the existence of fixed points for those maps. the results generalise the results of several authors in literature including xiang [chang he xiang, fixed point theorem for generalized $phi$-pseudocontra...

Here we consider the ordinary and fractional approximation of functions by sublinear positive operators with applications to generalized convolution type expressed integrals such as Choquet Shilkret ones. The is under differentiability Caputo, Canavati Iterated-Caputo types. We produce Jackson inequalities basic initial conditions. So our way quantitative producing their right hand sides involv...

For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties.We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous  differentiability. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.