A nonlinear singular perturbation problem

A new continuous regularized Gauss-Newton-type method with simultaneous updates of the operator $(F^{\pr*}(x(t))F'(x(t))+\ep(t) I)^{-1}$ for solving nonlinear ill-posed equations in a Hilbert space is proposed. A convergence theorem is proved. An attractive and novel feature of the proposed method is the absence of the assumptions about the location of the spectrum of the operator $F'(x)$. The absence of such assumptions is made possible by a source-type condition.

Consider an operator equation (*) $B(u)+\ep u=0$ in a real Hilbert space, where $\ep>0$ is a small constant. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the equation $B(u)=0$. Existence of the unique solution is proved by the DSM for eq...

For the first time, some hypersingular nonlinear boundary-value problems with a small parameter~$\varepsilon$ at the highest derivative are described. These problems essentially (qualitatively and quantitatively) differ from the usual linear and quasilinear singularly perturbed boundary-value problems and have the following unusual properties: (i) in hypersingular boundary-value problems, super thin boundary layers arise, and the derivative at the boundary layer can have ve...

The method is proposed for the study of many-point boundary value problems for systems of nonlinear ODE, by reducing them to special equivalent integral equations, and allows us [in contrast with the known method [1]] to consider boundary and initial value problems. Here we avoid the mechanism of Green's function whose construction is quite nontrivial, especially in case of many-point boundary value problems. The proposed algorithm [2, 3] makes it easier to write out the cond...

We present recent advances in the analysis of nonlinear equations with singular operators and nonlinear optimization problems with constraints given by singular mappings. The results are obtained within the framework of $p$-regularity theory, which has developed successfully over the last forty years. We illustrate the theory with its applications to degenerate problems in various areas of mathematics. In particular, we address the problem of describing the tangent cone to th...

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 generating (i.e., cyclic) subspace for $K_0$. If $\lambda_0$ is an eigenvalue of $K_0$, then under the additional assumption that $W$ is nonnegative, the Lebesgue measure of the set of all $t\in [0,1]$ for which $\lambda_0$ is an eigenvalue of $...

In this article, we consider the linear operator equation in a Banach space. The relative perturbation of the solution x corresponding to the perturbation of y, the perturbation of A and the perturbation of both A, y are characterized from the pseudospectrum and the condition pseudospectrum of A. Certain examples are given to illustrate the results. A relation between the pseudospectrum and the condition pseudospectrum of an operator are established. The distance to instabili...

Singularly perturbed problems present inherent difficulty due to the presence of a thin boundary layer in its solution. To overcome this difficulty, we propose using deep operator networks (DeepONets), a method previously shown to be effective in approximating nonlinear operators between infinite-dimensional Banach spaces. In this paper, we demonstrate for the first time the application of DeepONets to one-dimensional singularly perturbed problems, achieving promising results...

In \cite{Os} a general spectral approximation theory was developed for compact operators on a Banach space which does not require that the operators be self-adjoint and also provides a first order correction term. Here we extend some of the results of that paper to nonlinear eigenvalue problems. We present examples of its application that arise in electromagnetics.

We consider the nonlinear eigenvalue problem $Lx + \varepsilon N(x) = \lambda Cx$, $\|x\|=1$, where $\varepsilon,\lambda$ are real parameters, $L, C\colon G \to H$ are bounded linear operators between separable real Hilbert spaces, and $N\colon S \to H$ is a continuous map defined on the unit sphere of $G$. We prove a global persistence result regarding the set $\Sigma$ of the solutions $(x,\varepsilon,\lambda) \in S \times \mathbb R\times \mathbb R$ of this problem. Namely, ...