We prove $L^2$-maximal regularity of the linear non-autonomous evolutionary Cauchy problem$$\dot{u}\left(t\right)+A\left(t\right)u\left(t\right)=f\left(t\right)\text{for}\text{a.e.}t\in [0,T],u\left(0\right)=u_0,$$ where the operator $A\left(t\right)$ arises from a time depending sesquilinear form $𝔞(t,·,·)$ on a Hilbert space $H$ with constant domain $V.$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance...

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) =...

We develolp a new method to solve an evolution equation in a non-cylindrical domain, by reduction to an abstract evolution equation..

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...

We find an exact asymptotic formula for the singular values of the integral operator of the form ${\int }_{\Omega }T(x,y)k(x-y)·\mathrm{d}y{L}^2\left(\Omega \right)\to L^2\left(\Omega \right)$ ($\Omega \subset {ℝ}^m$, a Jordan measurable set) where $k\left(t\right)=k_0\left({(t_1^2+t_2^2+...t_m^2)}^{\frac{m}{2}}\right)$, $k_0\left(x\right)=x^{\alpha -1}L\left(\frac{1}{x}\right)$, $\frac{1}{2}-\frac{1}{2m}<\alpha <\frac{1}{2}$ and $L$ is slowly varying function with some additional properties. The formula is an explicit expression in terms of $L$ and $T$.

The liner parabolic equation $\frac{y}{t}-\frac{1}{2}𝔻y+F·y={\overrightarrow{1}}_{{}_0}u$ ∂y ∂t − 1 2   Δy + F · ∇ y = 1 x1d4aa; 0 u with Neumann boundary condition on a convex open domain x1d4aa; ⊂ ℝd with smooth boundary is exactly null controllable on each finite interval if &#x1d4aa;0is an open subset of x1d4aa; which contains a suitable neighbourhood of the recession cone of $$ x1d4aa; . Here,F : ℝd → ℝd is a bounded, C1-continuous function, and F = ∇g, where g is convex and coercive.

The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $li{m}_{n\to \infty }\left|\right|Tⁿ\left|\right|/n^{1-\gamma }=\infty$. In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $limsup\left|\right|Tⁿ\left|\right|/n^{1-\gamma ₀}=0.$ In this note we construct an example of a nonpositive L¹ operator with the highest possible...

A bifurcation problem for variational inequalities $$U\left(t\right)\in K,(\dot{U}\left(t\right)-B_{\lambda }U\left(t\right)-G(\lambda ,U\left(t\right)),Z-U\left(t\right))\ge 0\text{for}\text{all}Z\in K,\text{a.a.}t\ge 0$$ is studied, where $K$ is a closed convex cone in ${ℝ}^{\kappa }$, $\kappa \ge 3$, $B_{\lambda }$ is a $\kappa \times \kappa$ matrix, $G$ is a small perturbation, $\lambda$ a real parameter. The main goal of the paper is to simplify the assumptions of the abstract results concerning the existence of a bifurcation of periodic solutions developed in the previous paper and to give examples in more than three dimensional case.