What Do You Mean by “Nonlinear Eigenvalue Problems”?

Abstract

A nonlinear eigenvalue problem is generally described by an equation of the form F(λ, x) = 0, where F(λ, 0) = 0 for all λ, and contains by definition two unknowns: the eigenvalue parameter λ and the "nontrivial" vector(s) x corresponding to it. The nonlinear dependence of F can be in either of them (and of course in both), and also the research in this area seems to follow two quite different directions. In this review paper, we try to collect some points of possible common interest for both fields.

Unification Theories: New Results and Examples

“…For example, for A 3 = I n , one could search for solutions of the following type: X = αI n + βA + γA 2 . Now, (12) implies that (2αβ + γ 2 + α)A 2 + (α 2 + 2βγ + γ)A + (2αγ + β 2 + β)I n − D = 0. It can be shown that we can produce a large class of solutions in this way, if D is of a certain type.…”

“…This problem is about the properties of the eigenvalues of the matrices A, B and C, where A + B = C. A good reference is the paper [11]. Note that if A is "small" then D − AXA could be regarded as a deformation of D. 12can be interpreted as a "generalized eigenvalue problem" (see, for example, [12]). 12is a type of Yang-Baxter matrix equation (see, for example, [13,14]) if D = O n and X = −Y.…”

“…There are several methods to solve (12). For example, for A 3 = I n , one could search for solutions of the following type: X = αI n + βA + γA 2 .…”

Is the cosmological constant an eigenvalue?

“…Nonlinear eigenvalue problems of the type of Eq. ( 8) are frequently found in the mathematical literature, see [27] for a recent review. More details are provided in the Appendix A.…”

Nonlinear Rayleigh Quotients and Nonlinear Spectral Theory

“…For a recent review of some features of nonlinear operators and their eigenvalues, with applications to ordinary and partial differential equations, we refer the interested reader to [15].…”