In this paper we employ the Monotone Iteration Method and the Leray-Schauder Degree Theory to study an ${ℝ}^2$-parametrized system of elliptic equations. We obtain a curve dividing the plane into two regions. Depending on which region the parameter is, the system will or will not have solutions. This is an Ambrosetti-Prodi-type problem for a system of equations.

Some strong convergence theorems are established for the Ishikawa iteration processes for accretive operators in uniformly smooth Banach spaces.

We consider a model for diffusive phase transitions, for instance, the component separation in a binary mixture. Our model is described by two functions, the absolutete temperature $\theta :=\theta \left(t,x\right)$ and the order parameter $w:=w\left(t,x\right)$, which are governed by a system of two nonlinear parabolic PDEs. The order parameter $w$ is constrained to have double obstacles ${\sigma }_{*}\le w\le {\sigma }^{*}$ (i.e., ${\sigma }_{*}$ and ${\sigma }^{*}$ are the threshold values of $w$). The objective of this paper is to discuss the semigroup $\left\{S\left(t\right)\right\}$ associated with the phase separation model, and construct its global attractor.

In this paper we discuss several operator ideal properties for so called Carleson embeddings of tent spaces into specific $L^q\left(\mu \right)$-spaces, where $\mu$ is a Carleson measure on the complex unit disc. Characterizing absolutely $q$-summing, absolutely continuous and $q$-integral Carleson embeddings in terms of the underlying measure is our main topic. The presented results extend and integrate results especially known for composition operators on Hardy spaces as well as embedding theorems for function spaces of similar kind.

We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE’s which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the interior of a bounded domain $\Omega$, coupled to a “parabolic–like” beam equation holding on $\partial \Omega$, and wherein the coupling is accomplished through velocity terms on the boundary. Our result is an analog of a recent result by Lasiecka and Triggiani which shows the exponential stability of the wave equation via Neumann feedback control, and like that work, depends upon a trace regularity estimate for solutions of hyperbolic equations.

This paper is concerned with the approximate and exact controllability properties of the wave equation with interior point controls entering via the concentrated force, the velocity of the displacement and the moment. The emphasis is given to the moving point controls and their dual observations whose advantages and disadvantages, versus the static ones, are analyzed with respect to the space dimension, the duration of the control time interval and the function spaces involved.