In this paper, we consider the Maxwell-Stokes type system containing $p$-curlcurl equation in a bounded multiply connected domain. We demonstrate the existence of a weak solution. Moreover, we show the continuous dependence of the solution on known data.

We consider the family of $3 \times 3$ operator matrices ${\bf H}(K)$, $K \in {\Bbb T}^3:=(-\pi; \pi]^3$ associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set $\Lambda \subset {\Bbb T}^3$ to prove the existence of infinitely many eigenvalues of ${\bf H}(K)$ for all $K \in \Lambda$ when the associated Friedrichs model has a zero energy resonance. It is found that for every $K \in \Lambda$, the number $N(K, z)$ of eigenvalues of ${\bf H}(K)$ lying on the left of $z$, $z\lt0$, satisfies the asymptotic relation $\lim\limits_{z \to -0} N(K, z) |\log|z||^{-1}={\mathcal U}_0$ with $0\lt{\mathcal U}_0\lt\infty$, independently on the cardinality of $\Lambda$. Moreover, we prove that for any $K \in \Lambda$ the operator ${\bf H}(K)$ has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model.

The work deals with a study of a nonlinear parabolic equation with hysteresis, containing a nonlinear monotone operator in the diffusion term. The well-posedness of the model equation is addressed by using an implicit time discretization scheme in conjunction with the piecewise monotonicity of the hysteresis operator, and a fundamental inequality due to M. Hilpert. A characterization of the $\omega$-limit set of the solution is then given through the study of the long-time behaviour of the solution of the equation in which we investigate the convergence of trajectories to limit points.

Let $\mathcal{B}$ be a unital Banach algebra, $a\in \mathcal{B}$, $G$ be a convex domain of $\mathbb{C}$ with $\sigma \left( a\right) \subset G$ and $\gamma \subset G$ is a piecewise smooth path parametrized by $\lambda\left( t\right)$, $t\in \left[ 0,1\right]$ from $\lambda \left( 0\right)=\alpha$ to $\lambda \left( 1\right) =\beta$, with $\beta \neq \alpha$. If $f:G\rightarrow \mathbb{C}$ is analytic on $G$, then by using the analytic functional calculus we obtain among others the following result $\begin{multline*} \left\Vert f\left( a\right) -\int_{0}^{1}f\left( \left( 1-t\right) \lambda +ta\right) dt\right\Vert \leq \left\Vert a-\lambda \right\Vert \int_{0}^{1}t\left\Vert f^{\prime }\left( ta+\left( 1-t\right) \lambda \right) \right\Vert dt \\ \leq \left\Vert a-\lambda \right\Vert \left\{ \begin{array}{l} \frac{1}{2}\sup_{t\in \left[ 0,1\right] }\left\Vert f^{\prime }\left( ta+\left( 1-t\right) \lambda \right) \right\Vert , \\ \\ \frac{1}{\left( q+1\right) ^{1/q}}\left( \int_{0}^{1}\left\Vert f^{\prime }\left( ta+\left( 1-t\right) \lambda \right) \right\Vert ^{p}dt\right) ^{1/p}, \\ \\ \int_{0}^{1}\left\Vert f^{\prime }\left( ta+\left( 1-t\right) \lambda \right) \right\Vert dt,% \end{array} \right. \end{multline*}$ for all $\lambda \in G$. Some example for the exponential function of elements in Banach algebras are also provided.