A nonlinear eigenvalue problem

We reduce the problem of the zeros of an entire function related to the Painlevé I equation to the eigenvalue problem for the Dirichlet problem for a fourth-order bilinear equation. In a simplified case, this problem reduces to the problem of the position of the zeros of theta functions. We then apply the developed general method to the inverse scattering problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 175–187, November, 2008.     

On a nonlinear eigenvalue problem

In the present paper we consider a selfadjoint and nonsmooth operator-valued function on (c, d)R ¹. We suppose that the equation (L()x, x)=0,x0, has exactly one rootp(x) (c, d) and the functionf()=(L()x, x) is increasing at the pointp(x). We discuss questions of the variational theory of the spectrum. Some theorems on the variational properties of the spectrum are proved.     

A linear eigenvalue algorithm for the nonlinear eigenvalue problem

The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted ${\mathcal {B}}$ . We consider the Arnoldi method for the operator ${\mathcal {B}}$ and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator ${\mathcal {B}}$ in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples.     

On a nonlinear elliptic eigenvalue problem

Let N(λ) be the number of the solutions of the equation: , where Ω is a bounded domain in with smooth boundary. Under suitable conditions on f, we proved that N(λ)→+∞ as λ→+∞.     

A completeness theorem for a nonlinear multiparameter eigenvalue problem

In this paper we study the linked nonlinear multiparameter system

$\text{y}{}_{\mathrm{r}}{}^{\mathrm{n}}\text{(X}{}_{\mathrm{r}}\text{) + M}{}_{\mathrm{r}}\text{Y}{}_{\mathrm{r}}\text{+}\text{∑}\text{s=1}\text{k}\text{λ}{}_{\mathrm{s}}\text{(a}{}_{\mathrm{rs}}\text{(X}{}_{\mathrm{r}}\text{) + P}{}_{\mathrm{rs}}\text{) Y}{}_{\mathrm{r}}\text{(X}{}_{\mathrm{r}}\text{) = 0, r = l,…, k}$

, where x_r? [a_r, b_r], y_r is subject to Sturm-Liouville boundary conditions, and the continuous functions a_rs satisfy ¦ $\text{A ¦ (x) =}\text{det}\text{a}{}_{\mathrm{rs}}\text{(x}{}_{\mathrm{r}}\text{) > 0}$. Conditions on the polynomial operators M_r, P_rs are produced which guarantee a sequence of eigenfunctions for this problem yⁿ(x) = Π_r=1^ky_rⁿ(x_r), n ? 1, which form a basis in $\text{L}{}^2\text{([a, b], ¦ A ¦)}$. Here [a, b] = [a₁, b₁ × … × [a_k, b_k].     

A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

Given \(1\le q \le 2\) and \(\alpha \in \mathbb {R}\), we study the properties of the solutions of the minimum problem

$$\begin{aligned} \lambda (\alpha ,q)=\min \left\{ \dfrac{\displaystyle \int _{-1}^{1}|u'|^{2}dx+\alpha \left| \int _{-1}^{1}|u|^{q-1}u\, dx\right| ^{\frac{2}{q}}}{\displaystyle \int _{-1}^{1}|u|^{2}dx}, u\in H_{0}^{1}(-1,1),\,u\not \equiv 0\right\} . \end{aligned}$$

In particular, depending on \(\alpha \) and q, we show that the minimizers have constant sign up to a critical value of \(\alpha =\alpha _{q}\), and when \(\alpha >\alpha _{q}\) the minimizers are odd.

     

On solving a nonlinear matrix eigenvalue problem

Journal of Mathematical Sciences -     

On a nonlinear eigenvalue problem in ODE

In this paper we shall study the following variant of the logistic equation with diffusion:

−du^″(x)=g(x)u(x)−u²(x)     

A note on backward perturbations for the hermitian eigenvalue problem

Through different orthogonal decompositions of computed eigenvectors we can define different Hermitian backward perturbations for a Hermitian eigenvalue problem. Certain optimal Hermitian backward perturbations are studied. The results show that not all the optimal Hermitian backward perturbations are small when the computed eigenvectors have a small residual and are close to orthonormal.Dedicated to Åke Björck on the occasion of his 60th birthdayThis work was supported by the Swedish Natural Science Research Council under Contract F-FU 6952-302 and the Department of Computing Science, Umeå University.     

A nonlinear eigenvalue problem arising in a nanostructured quantum dot

In this paper we investigate a minimization problem related to the principal eigenvalue of the s-wave Schrödinger operator. The operator depends nonlinearly on the eigenparameter. We prove the existence of a solution for the optimization problem and the uniqueness will be addressed when the domain is a ball. The optimized solution can be applied to design new electronic and photonic devices based on the quantum dots.     

A nonlinear boundary eigenvalue problem for TM-polarized electromagnetic waves in a nonlinear layer

We consider the propagation of TM-polarized electromagnetic waves in a nonlinear dielectric layer located between two linear media. The nonlinearity in the layer is described by the Kerr law. We reduce the problem to a nonlinear boundary eigenvalue problem for a system of ordinary differential equations. We obtain a dispersion relation and a first approximation for eigenvalues of the problem. We compare the results with those obtained for the case of a linear medium in the layer.     

A discontinuous nonlinear eigenvalue/Free boundary problem

We study the problem (H is the Heaviside unit step function) in spherical domains Ω of arbitrary dimension. When g = 0, there are two branches of radial solutions; for small nonzero g there are solutions near the corresponding radial solution. Moreover, the set where μ = 1 is in all cases an analytic hypersurface.     

Nodal solutions for a nonlinear fourth-order eigenvalue problem

We are concerned with determining the values of λ, for which there exist nodal solutions of the fourth-order boundary value problem y″″=λa（x）f（y）,0〈x〈1,y（0）=y（1）=y″（0）=y″（1）=0where λ is a positive parameter, a ∈ C（[0, 1], （0, ∞）, f ∈C（R,R） satisfies f（u）u 〉 0 for all u ≠ 0. We give conditions on the ratio f（s）/s, at infinity and zero, that guarantee the existence of nodal solutions.The proof of our main results is based upon bifurcation techniques.     

On a nonhomogeneous,nonlinear Dirichlet eigenvalue problem

We consider a nonlinear eigenvalue problem for the Dirichlet $(p,q)$-Laplacian with a sign-changing Carath$\acute{\mathrm{e}}$odory reaction. Using variational tools, truncation and comparison techniques, and critical groups, we prove an existence and multiplicity result which is global in the parameter $\lambda >0$ (bifurcation-type theorem). Our work here complements the recent one by Papageorgiou–Qin–Rădulescu, Bull. Sci. Math. 172 (2021).     

On a second-order nonlinear discrete multi-point eigenvalue problem

We study the existence and non-existence of positive solutions of a system of nonlinear second-order difference equations with eigenvalues subject to multi-point boundary conditions.     

Principal vectors of a nonlinear finite-dimensional eigenvalue problem

In a finite-dimensional linear space, consider a nonlinear eigenvalue problem analytic with respect to its spectral parameter. The notion of a principal vector for such a problem is examined. For a linear eigenvalue problem, this notion is identical to the conventional definition of principal vectors. It is proved that the maximum number of linearly independent eigenvectors combined with principal (associated) vectors in the corresponding chains is equal to the multiplicity of an eigenvalue. A numerical method for constructing such chains is given.