It is well known that the identity is an operator with the following property: if the operator, initially defined on an $n$-dimensional Banach space $V$, can be extended to any Banach space with norm $1$, then $V$ is isometric to ${\ell }_{\infty }^{\left(n\right)}$. We show that the set of all such operators consists precisely of those with spectrum lying in the unit circle. This result answers a question raised in [5] for complex spaces.

In this part of our paper we present several new theorems concerning the existence of common fixed points of asymptotically regular uniformly lipschitzian semigroups.

Given an $x_0\in R^n$ we study the infinite horizon problem of minimizing the expression ${\int }_0^{T}f\left(t,x\left(t\right),x^{\prime }\left(t\right)\right)dt$ as $T$ grows to infinity where $x:\left[0,\infty \right)\to R^n$ satisfies the initial condition $x\left(0\right)=x_0$. We analyse the existence and the properties of approximate solutions for every prescribed initial value $x_0$. We also establish that for every bounded set $E\subset R^n$ the $C\left(\left[0,T\right]\right)$ norms of approximate solutions $x:\left[0,T\right]\to R^n$ for the minimization problem on an interval $\left[0,T\right]$ with $x\left(0\right),x\left(T\right)\in E$ are bounded by some constant which does not depend on $T$.

This paper is devoted to the study of a stationary problem consisting of the Boussinesq approximation of the Navier–Stokes equations and two convection–diffusion equations for the temperature and concentration, respectively. The equations are considered in 3D and a velocity–pressure formulation of the Navier–Stokes equations is used. The problem is complicated by nonstandard boundary conditions for velocity on the liquid–gas interface where tangential surface forces proportional to surface gradients of temperature and concentration (Marangoni effect) and zero normal component of the velocity are assumed. The velocity field is coupled through this boundary condition and through the buoyancy term in the Navier–Stokes equations with both the temperature and concentration fields. In this paper a weak formulation of the problem is stated and the existence of a weak solution is proved. For small data, the uniqueness of the solution is established.

Let $X$ be a Banach space and $\tau$ a topology on $X$. We say that $X$ has the $\tau$-fixed point property ($\tau$-FPP) if every nonexpansive mapping $T$ defined from a bounded convex $\tau$-sequentially compact subset $C$ of $X$ into $C$ has a fixed point. When $\tau$ satisfies certain regularity conditions, we show that normal structure assures the $\tau$-FPP and Goebel-Karlovitz′s Lemma still holds. We use this results to study two geometrical properties which imply the $\tau$-FPP: the $\tau$-GGLD and $M\left(\tau \right)$ properties. We show several examples of spaces and topologies where these results can be applied, specially the topology of convergence locally in measure in Lebesgue spaces. In the second part we study the preservence of the $\tau$-FPP under isomorphisms. In order to do that we study some geometric constants for a Banach space $X$ such that the $\tau$-FPP is shared by any isomorphic Banach space $Y$ satisfying that the Banach-Mazur distance between $X$ and $Y$ is less than some of these constants.

We consider the initial-boundary value problem for second order differential-functional equations of parabolic type. Functional dependence in the equation is of the Hale type. By using Leray-Schauder theorem we prove the existence of classical solutions. Our formulation and results cover a large class of parabolic problems both with a deviated argument and integro-differential equations.

We consider a coupled PDE system arising in noise reduction problems. In a two dimensional chamber, the acoustic pressure (unwanted noise) is represented by a hyperbolic wave equation. The floor of the chamber is subject to the action of piezo-ceramic patches (smart materials). The goal is to reduce the acoustic pressure by means of the vibrations of the floor which is modelled by a hyperbolic Kirchoff equation. These two hyperbolic equations are coupled by appropriate trace operators. This overall model differs from those previously studied in the literature in that the elastic chamber floor is here more realistically modeled by a hyperbolic Kirchoff equation, rather than by a parabolic Euler-Bernoulli equation with Kelvin-Voight structural damping, as in past literature. Thus, the hyperbolic/parabolic coupled system of past literature is replaced here by a hyperbolic/hyperbolic coupled model. The main result of this paper is a uniform stabilization of the coupled PDE system by a (physically appealing) boundary dissipation.

A construction of equivariant maps based on factorization through symmetric powers of a faithful representation is presented together with several examples of related equivariant maps. Applications to differential equations are also discussed.

This paper deals with existence, uniqueness and regularity of positive generalized solutions of singular nonlinear equations of the form $-\Delta u+a\left(x\right)u=h\left(x\right)u^{-\gamma }$ in ${\mathbf{R}}^n$ where $a,h$ are given, not necessarily continuous functions, and $\gamma$ is a positive number. We explore both situations where $a,h$ are radial functions, with $a$ being eventually identically zero, and cases where no symmetry is required from either $a$ or $h$. Schauder′s fixed point theorem, combined with penalty arguments, is exploited.

We study the existence of almost periodic mild solutions of a class of partial functional differential equations via semilinear almost periodic abstract functional differential equations of the form $$(*)x^{\prime }=f\left(t,x,x_t\right).$$ To this end, we first associate with every almost periodic semilinear equation $$\left(**\right)x^{\prime }=F\left(t,x\right).$$ a nonlinear semigroup in the space of almost periodic functions. We then give sufficient conditions (in terms of the accretiveness of the generator of this semigroup) for the existence of almost periodic mild solutions of (**) as fixed points of the semigroup. Those results are then carried over to equation (*). The main results are stated under accretiveness conditions of the function $f$ in terms of $x$ and Lipschitz conditions with respect to $x_t$.

We prove the existence of nontrivial critical points with nontrivial critical groups for functionals with a local linking at $0$. Applications to elliptic boundary value problems are given.