相关论文: A nonlinear singular perturbation problem

We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the $\alpha$-stable operator and the second one (possibly degenerate) corresponds to…

We present several operator and norm inequalities for Hilbert space operators. In particular, we prove that if $A_{1},A_{2},...,A_{n}\in {\mathbb B}({\mathscr H})$, then…

Denote by w(A) the numerical radius of a bounded linear operator A acting on Hilbert space. Suppose that A is invertible and that the numerical radius of A and of its inverse are no greater than 1+e for some non-negative e. It is shown that…

We consider the indefinite Sturm-Liouville differential expression \[\mathfrak{a}(f) := - \frac{1}{w}\left( \frac{1}{r} f' \right)',\] where $\mathfrak{a}$ is defined on a finite or infinite open interval $I$ with $0\in I$ and the…

Consider two inverse problems for ZS-operators problems on the unit interval. It means that there are two corresponding mappings $F, f$ from a Hilbert space of potentials $H$ into their spectral data. They are called isomorphic if $F$ is a…

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive…

Consider the equation $-\ve^2\Delta u_\ve+q(x)u_\ve=f(u_\ve)$ in $\R^3$, $|u(\infty)|<\infty$, $\ve=const>0$. Under what assumptions on $q(x)$ and $f(u)$ can one prove that the solution $u_\ve$ exists and $\lim_{\ve\to 0} u_\ve=u(x)$, where…

We prove sufficient conditions for Hausdorff convergence of the spectra of sequences of closed operators defined on varying Hilbert spaces and converging in norm-resolvent sense, i.e. $\|J_\varepsilon(1+A_\varepsilon)^{-1} -…

In this paper, we consider discrete Schr\"odinger operators of the form, \begin{equation*} (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n). \end{equation*} We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n-1})$.…

Consider an operator equation $F(u)=0$ in a real Hilbert space. The problem of solving this equation is ill-posed if the operator $F'(u)$ is not boundedly invertible, and well-posed otherwise. A general method, dynamical systems method…

Let $L_0$ be a closed symmetric positive definite operator with nonzero defect indices $n_\pm(L_0)$ in a separable Hilbert space ${\mathscr H}$. It determines a family of dynamical systems $\alpha^T$, $T>0$, of the form \begin{align*} &…

Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…

Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and…

We prove the Weyl-von Neumann-Berg theorem for quaternionic right linear operators (not necessarily bounded) in a quaternionic Hilbert space: Let $N$ be a right linear normal (need not be bounded) operator in a quaternionic separable…

We consider additive perturbations of the type $K_t=K_0+tW$, $t\in [0,1]$, where $K_0$ and $W$ are self-adjoint operators in a separable Hilbert space $\mathcal{H}$ and $W$ is bounded. In addition, we assume that the range of $W$ is a…

Let $F$ be a nonlinear Frechet differentiable map in a real Hilbert space. Condition sufficient for existence of a solution to the equation $F(u)=0$ is given, and a method (dynamical systems method, DSM) to calculate the solution as the…

Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies…

We consider skew-symmetric operators $A_{0}$ on a Hilbert space $H$ and characterise all (nonlinear) m-accretive restrictions of $A:=-A_{0}^{\ast}$ in terms of the "deficiency spaces" $\ker(1\pm A)$. The results are illustrated by several…

We consider a self-adjoint operator $T$ on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of $T$ and on the bounded perturbation $V$…

In this work, we consider some relationships between a closed range operator $T$ and a fusion frame $\mathcal{W}=(W_i,w_i)_{i\in I}$ for a Hilbert space $\mathcal{H}$ that provides that the sequence $(\overline{T(W_i)},v_i)_{i\in I}$ is a…