In this paper we prove a regularity result for very weak solutions of equations of the type $-divA(x,u,Du)=B(x,u,Du)$, where $A$, $B$ grow in the gradient like $t^{p-1}$ and $B(x,u,Du)$ is not in divergence form. Namely we prove that a very weak solution $u\in W^{1,r}$ of our equation belongs to $W^{1,p}$. We also prove global higher integrability for a very weak solution for the Dirichlet problem $$\left\{\begin{array}{cc}-divA(x,u,Du)=B(x,u,Du)\hfill & \text{in}\Omega ,u-u_o\in W^{1,r}(\Omega ,{ℝ}^m).\hfill \end{array}\right.$$

In this paper we deal with the Cauchy problem for differential inclusions governed by $m$-accretive operators in general Banach spaces. We are interested in finding the sufficient conditions for the existence of integral solutions of the problem $x^{\text{'}}\left(t\right)\in -Ax\left(t\right)+f(t,x\left(t\right))$, $x\left(0\right)=x_0$, where $A$ is an $m$-accretive operator, and $f$ is a continuous, but non-compact perturbation, satisfying some additional conditions.

Let $D$ be a nonempty compact subset of a Banach space $X$ and denote by $S\left(X\right)$ the family of all nonempty bounded closed convex subsets of $X$. We endow $S\left(X\right)$ with the Hausdorff metric and show that there exists a set $ℱ\subset S\left(X\right)$ such that its complement $S\left(X\right)\setminus ℱ$ is $\sigma$-porous and such that for each $A\in ℱ$ and each $\tilde{x}\in D$, the set of solutions of the best approximation problem $\parallel \tilde{x}-z\parallel \to min$, $z\in A$, is nonempty and compact, and each minimizing sequence has a convergent subsequence.

We study the discrete Schrödinger operator $H$ in ${𝐙}^d$ with the surface quasi periodic potential $V\left(x\right)=g\delta \left(x_1\right)tan\pi (\alpha ·x_2+\omega )$, where $x=(x_1,x_2),x_1\in {𝐙}^{d_1},x_2\in {𝐙}^{d_2},\alpha \in {𝐑}^{d_2},\omega \in [0,1)$. We first discuss a proof of the pure absolute continuity of the spectrum of $H$ on the interval $[-d,d]$ (the spectrum of the discrete laplacian) in the case where the components of $\alpha$ are rationally independent. Then we show that in this case the generalized eigenfunctions have the form of the “volume” waves, i.e. of the sum of the incident plane wave and reflected from the hyper-plane ${𝐙}^{d_1}$ waves,...