Iterative method for a class of nonlinear eigenvalue problems

Let H Be a complex and separable Hilbert space and consider in H the nonlinear eigenvalue problem where A, B, and C belong to the class of unbounded nonsymmetric operators, which are K- positive K-symmetric. Sufficient conditions insuring the existence of the eigenvalues of (i) are investigated. An iterative method for approximating the eigenvalues of (i) is developed and its convergence proved. Some numerical examples are given to illustrate the theory.     

A Ulm-like method for inverse eigenvalue problems

We propose a Ulm-like method for solving inverse eigenvalue problems, which avoids solving approximate Jacobian equations comparing with other known methods. A convergence analysis of this method is provided and the R-quadratic convergence property is proved under the assumption of the distinction of given eigenvalues. Numerical experiments as well as the comparison with the inexact Newton-like method are given in the last section.     

Multiple nontrivial solutions for nonlinear eigenvalue problems

In this paper we study a nonlinear eigenvalue problem driven by the -Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposite constant sign solutions. The approach relies on a combination of variational and minimization methods coupled with the construction of upper-lower solutions. The framework of the paper incorporates problems with concave-convex nonlinearities.

     

A moment problem for one-dimensional nonlinear pseudoparabolic equation

We are concerned with a moment problem for a nonlinear pseudoparabolic equation with one space dimension on an interval. The boundary conditions are imposed in terms of the zero-order moment and the first-order moment. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in the usual Sobolev space. We are able to get regularity of the solution so that both solution and its derivative with respect to the time variable belong to the same Sobolev space with respect to the space variable. This feature is different from problems with parabolic equations, where the regularity order of solution is higher than that of the time derivative with respect to the space variable. Previous results reflected only this parabolic nature for the pseudoparabolic equation.     

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→+∞.     

Algorithms for hyperbolic quadratic eigenvalue problems

We consider the quadratic eigenvalue problem (or the QEP) , where and are Hermitian with positive definite. The QEP is called hyperbolic if 4(x^*Ax)(x^*Cx)$"> for all nonzero . We show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the QEP. A hyperbolic QEP is overdamped if is positive definite and is positive semidefinite. We show that a hyperbolic QEP (whose eigenvalues are necessarily real) is overdamped if and only if its largest eigenvalue is nonpositive. For overdamped QEPs, we show that all eigenpairs can be found efficiently by finding two solutions of the corresponding quadratic matrix equation using a method based on cyclic reduction. We also present a new measure for the degree of hyperbolicity of a hyperbolic QEP.

     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

Exponentially convergent parallel algorithm for nonlinear eigenvalue problems

A. V. Klimenko, V. L. Makarov ^{A new algorithm for nonlinear eigenvalue problems is proposed.The numerical technique is based on a perturbation of the coefficientsof differential equation combined with the Adomian decompositionmethod for the nonlinear part. The approach provides an exponentialconvergence rate with a base which is inversely proportionalto the index of the eigenvalue under consideration. The eigenpairscan be computed in parallel. Numerical examples are presentedto support the theory. They are in good agreement with the spectralasymptotics obtained by other authors.}     

On Ricceri’s conjecture for a class of nonlinear eigenvalue problems

In this work we give a negative answer to a conjecture proposed by B. Ricceri in the reference [B. Ricceri, A remark on a class of nonlinear eigenvalue problems, Nonlinear Anal. 69 (2008) 2964–2967] for a class of nonlinear elliptic eigenvalue problems.     

Existence of three solutions for a class of elliptic eigenvalue problems

In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rⁿ is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W₀^1,2(Ω).     

A rank‐exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem M(λ)x = 0, where M(λ) is a large parameter‐dependent matrix. In several applications, M(λ) has a structure where the higher‐order terms of its Taylor expansion have a particular low‐rank structure. We propose a new Arnoldi‐based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector‐valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169–195]. In this paper, the low‐rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large‐scale problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of solutions for a class of third-order nonlinear boundary value problems

In this paper, we are concerned with the following third-order ordinary differential equation:

     

A branch and search algorithm for a class of nonlinear knapsack problems

This paper discusses a class of nonlinear knapsack problems where the objective function is quadratic. The method is a branch and search procedure which includes an efficient algorithm to find the continuous (relaxed) solution and a reduction rule which computes tight lower and upper bounds on the integer variables.     

Asymptotic analysis of some nonlinear problems using Hopf-Cole transform and spectral theory

We consider initial boundary value problems for certain nonlinear scalar parabolic equations. A formula for the unique classical solution by Hopf-Cole transformations is obtained and the asymptotic behaviour of the solution as time goes to ∞ is studied.