We consider some discrete and continuous dynamics in a Banach space involving a non expansive operator J and a corresponding family of strictly contracting operators Φ (λ, x): = λ J($\frac{1-\lambda }{\lambda }$x) for λ ∈ ] 0,1] . Our motivation comes from the study of two-player zero-sum repeated games, where the value of the n-stage game (resp. the value of the λ-discounted game) satisfies the relation vn = Φ($\frac{1}{n}$, $v_{n-1}$) (resp. $v_{\lambda }$ = Φ(λ, $v_{\lambda }$)) where J is the Shapley operator of the game. We study the evolution equation u'(t) =...

Similarly to quasidifferential equations of Panasyuk, the so-called mutational equations of Aubin provide a generalization of ordinary differential equations to locally compact metric spaces. Here we present their extension to a nonempty set with a possibly nonsymmetric distance. In spite of lacking any linear structures, a distribution-like approach leads to so-called right-hand forward solutions. These extensions are mainly motivated by compact subsets of the Euclidean space...

The purpose of this paper is to give theorems on continuity and differentiability with respect to (h,t) of the solution of the initial value problem du/dt = A(h,t)u + f(h,t), u(0) = u₀(h) with parameter $h\in \Omega \subset {ℝ}^m$ in the “hyperbolic” case.

We study a Cauchy problem for non-convex valued evolution inclusions in non separable Banach spaces under Filippov type assumptions. We establish existence and relaxation theorems.

In this paper we study evolution inclusions generated by time dependent convex subdifferentials, with the orientor field $F$ depending on a parameter. Under reasonable hypotheses on the data, we show that the solution set $S\left(\lambda \right)$ is both Vietoris and Hausdorff metric continuous in $\lambda \in \Lambda$. Using these results, we study the variational stability of a class of nonlinear parabolic optimal control problems.

This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space $H$$$\left\{\begin{array}{c}{u}^{}\hfill \end{array}\right.$$

We prove existence and asymptotic behaviour of a weak solutions of a mixed problem for $$\begin{array}{cc}& u^{\text{'}\text{'}}+Au-\Delta u^{\text{'}}+{\left|v\right|}^{\rho +2}{\left|u\right|}^{\rho }u=f_1\hfill \\ & v^{\text{'}\text{'}}+Av-\Delta v^{\text{'}}+{\left|u\right|}^{\rho +2}{\left|v\right|}^{\rho }v=f_2*\hfill \end{array}$$ where $A$ is the pseudo-Laplacian operator.

In this paper we study Cauchy problems for retarded evolution inclusions, where the Fréchet subdifferential of a function F:Ω→R∪{+∞} (Ω is an open subset of a real separable Hilbert space) having a φ-monotone subdifferential of oder two is present. First we establish the existence of extremal trajectories and we show that the set of these trajectories is dense in the solution set of the original convex problem for the norm topology of the Banach space C([-r, T₀], Ω) ("strong relaxation theorem")....

Si studiano esistenza, unicità e regolarità delle soluzioni strette, classiche e forti dell’equazione di evoluzione non autonoma $u^{\prime }(t)\mathrm{—}A(t)u(t)=f(t)$, con il dato iniziale $u(0)=x$, in spazi di Banach. I dominii degli operatori $A(t)$ variano in $t$ e non sono necessariamente densi in $E$. Si danno condizioni necessarie e sufficienti per l'esistenza e la regolarità holderiana della soluzione e della sua derivata.