19 research outputs found
Existence results for fractional differential inclusions with three-point fractional integral boundary conditions
Control systems described by a class of fractional semilinear evolution hemivariational inequalities and their relaxation property
Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations
Toward a Mathematical Holographic Principle
In work started in [17] and continued in this paper our objective is to study
selectors of multivalued functions which have interesting dynamical properties,
such as possessing absolutely continuous invariant measures. We specify the
graph of a multivalued function by means of lower and upper boundary maps
and On these boundary maps we define a position
dependent random map which, at each time
step, moves the point to with probability and to
with probability . Under general conditions, for each
choice of , possesses an absolutely continuous invariant measure
with invariant density Let be a selector which has
invariant density function One of our objectives is to study conditions
under which exists such that has as its invariant density
function. When this is the case, the long term statistical dynamical behavior
of a selector can be represented by the long term statistical behavior of a
random map on the boundaries of We refer to such a result as a
mathematical holographic principle. We present examples and study the
relationship between the invariant densities attainable by classes of selectors
and the random maps based on the boundaries and show that, under certain
conditions, the extreme points of the invariant densities for selectors are
achieved by bang-bang random maps, that is, random maps for which $p(x)\in
\{0,1\}.