Nonpivot and Implicit Projected Dynamical Systems on Hilbert Spaces

Abstract

This paper presents a generalization of the concept and uses of projected dynamical systems to the case of nonpivot Hilbert spaces. These are Hilbert spaces in which the topological dual space is not identified with the base space. The generalization consists of showing the existence of such systems and their relation to variational problems, such as variational inequalities. In the case of usual Hilbert spaces these systems have been extensively studied, and, as in previous works, this new generalization has been motivated by applications, as shown below.

On the Price Vector Fields of a Differentiated Oligopoly

“…This corresponds to a situation where X and Y simultaneously make an identical price increase by a, then gravitating toward a 2 , a 2 with potential rotations around a 2 , a 2 , indeed a plausible event to a duopoly (cf. [3]). For bookdeeping we have for the vector field V (R ∩ T clockwise ) a divergence equal to 0, hence the flows being incompressible, and curl (V) = (0, 0, 4) T .…”

Evolution Solutions of Equilibrium Problems: A Computational Approach

“…Given that we are not computing equilibrium states of TDEP we have no need for such restrictions, and in fact we highlight (as we did in previous works in related contexts [7,24]) that dropping monotonicity-type conditions makes the dynamics of a problem much richer, but not complex enough to be intractable. Using the computational method which we present below in section "Computing Evolution Trajectories" completely removes the need to impose uniqueness of point-wise steady states and opens up the possibilities of finding perhaps several curves of steady states (i.e., non-unique points in the solution set of an EVI problem), periodic behaviour (evolution trapped in a periodic cycle, which has no counterpart in any VI or EVI model), or simply finding an estimate of the evolution of a particular initial state of TDEP into a later one.…”

“…(1) has unique solutions x 2 AC.R C ; K/ for each initial point , 12, 13, 17]. Moreover, this equation has the property (see [1,12,13,17] for proofs):…”

Contractive properties of truncated operators through Schauder bases with some applicationsc

“…The relevance of the theory in properties has been studied concerning theoretical aspects and their applications such as the existence and uniqueness of solutions in differential, difference and hybrid equations as well as in continuous-time, discrete -time, digital and hybrid dynamic systems stability theory in the above problems [5,[9][10][11]17], [21][22][23] and related problems connected with fixed point theory, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. A lot of importance has also been paid to the investigation of the existence and uniqueness of coupled and common fixed points and best proximity points for several mappings and related properties as well as to the relevance of fixed points in the context of variational inequalities and fuzzy metric spaces [5-7, 8, 13, 20].…”