We consider a model operator ${\mathbf H}_{\mu\lambda},$ $\mu,\lambda \geq 0$ associated with the energy operator of a lattice system describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. The existence of infinitely many negative eigenvalues of ${\mathbf H}_{0\lambda}$ is proved for the case where the associated Friedrichs model have a zero energy resonance and an asymptotics of the form ${\mathcal U}_0 |\log|z||$ for the number of eigenvalues of ${\mathbf H}_{0\lambda}$ lying below $z<0,$ is obtained. We find the conditions for the infiniteness of the number of eigenvalues located inside (in the gap, in the below of the bottom) of the essential spectrum of ${\mathbf H}_{\mu\lambda}$.

In this paper, we give an $L^p$ version of Hardy's uncertainty principles for a large class of integral and $q$-integral transforms. As an applications, we discuss an $L^p$ version of Hardy's theorem for the generalized Fourier transform associated with the Sturm-Liouville operator and for the Jacobi-Dunkl transform.

We study the existence and the asymptotic behavior of positive solutions of the nonlinear equation $$\Delta u\;+\; \varphi(.,u)+\psi(.,u)=0,$$ in NTA- cones in $\mathbb{R}^n (n\geq 3)$.

The purpose of this paper is to search for periodic solutions to a system of nonlinear difference equations of the form \[\Delta x(t) = f(\epsilon,t,x(t)).\] The corresponding linear homogeneous system has an $n$-dimensional kernel, i.e. the system is at full resonance. We provide sufficient conditions for the existence of periodic solutions based on asymptotic properties of the nonlinearity $f$ when $\epsilon=0$. To this end, we employ a projection method using the Lyapunov-Schmidt procedure together with Brouwer's fixed point theorem.

In this paper we study the approximate positive controllability and positive observability of linear systems. The characterization of the above concepts is given. The duality between the approximate positive controllability and positive observability is obtained. Necessary and sufficient conditions are formulated and proved. The abstract results are illustrated by several examples.

In this paper, we give many refinements and generalizations of Hardy-type inequalities on time scales for convex functions, nonnegative convex functions, monotone convex functions and nonnegative monotone convex functions. Further we give refinements for power and exponential functions. Finally we present several examples of these inequalities on different time scales.