Some generic properties in fixed point theory

AbstractIt is shown that, in the sense of the Baire category, almost all continuous single valued α-nonexpansive mappings T: C → C do have fixed points. Here C is a nonempty closed convex and bounded subset of an infinite dimensional Banach space. A similar result holds for upper semicontinuous α-nonexpansive mappings which are compact convex valued. Corresponding results for single valued and set-valued nonexpansive mappings are reviewed

Non-convex-valued differential inclusions in Banach spaces

Digitalitzat per Artypla

Relaxation Methods for Mixed-Integer Optimal Control of Partial Differential Equations

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments

Regularity of a kind of marginal functions in Hilbert spaces

We study well-posedness of some mathematical programming problem depending on a parameter that generalizes in a certain sense the metric projection onto a closed nonconvex set. We are interested in regularity of the set of minimizers as well as of the value function, which can be seen, on one hand, as the viscosity solution to a Hamilton-Jacobi equation, while, on the other, as the minimal time in some related optimal time control problem. The regularity includes both the Fréchet differentiability of the value function and the Hölder continuity of its (Fréchet) gradient

Hukuhara's topological degree for non compact valued multifunctions

We present a direct construction of a topological degree for multivalued vector fields I - F in a Banach space, where F takes closed, bounded, convex (or non convex) values and the set-valued range of F is precompact in the Pompeiu-Hausdorff metric. Some useful properties of our topological degree are established. Applications to fixed point theory including a Borsuk's type result are considered

Hukuhara's topological degree for non compact valued multifunctions

We present a direct construction of a topological degree for multivalued vector fields I - F in a Banach space, where F takes closed, bounded, convex (or non convex) values and the set-valued range of F is precompact in the Pompeiu-Hausdorff metric. Some useful properties of our topological degree are established. Applications to fixed point theory including a Borsuk's type result are considered