Approximation and convergence rate of nonlinear eigenvalues: Lipschitz perturbations of a bounded self-adjoint operator

What Do You Mean by “Nonlinear Eigenvalue Problems”?

“…Section 3 is devoted to discuss two specific problems, one for each type, with the scope of giving samples of very recent research in either field. While for the problem discussed in Section 3.1 there are already concrete numerical examples [8], these are still missing for the problem presented in Section 3.2 [9]. The main aim of this final section is to partially fill this gap by further commenting (in Section 4.1) on the formula…”

“…Rellich's work was a main starting point for the very vast literature concerning the systematic analysis of the perturbation of eigenvalues of linear operators, both in finite and infinite dimensional spaces; see Kato's book [7] and the references therein. Our aim in this section is to indicate some partial results about similar questions for nonlinear eigenvalue problems, both of type G and of type K, recently appearing in [8,9], respectively.…”

“…where p ∈ C 1 ([a, b]), p > 0 and q ∈ C ([a, b]). The first remark on the applicability of Theorem 8 to such kind of problems, in one or more space variables, is that one must not worry of the multiplicity of the unperturbed eigenvalue because-as recalled for instance in [9]-the operator B corresponding to the nonlinear term f is a gradient operator, so that the assumption b) of Theorem 8 is satisfied. In [9] it is previously recalled that these problems can be cast in the operator form of Equation (87) on taking as Hilbert space the Sobolev space H 1 0 (Ω) ≡ W 1,2 0 (Ω), equipped with the scalar product…”

“…In both Equations (10) and (11), one common problem is-in the light of what is done for linear operators [7]-to see how the perturbed eigenvalues λ( ) (provided that they exist) will depend on near a given unperturbed eigenvalue λ 0 . To this purpose, we review the main points of the recent contributions [8,9], respectively, to Equation (10) and to Equation (11).…”

Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion

“…Bifurcation problems have a long history and there are so many results concerning the asymptotic properties of bifurcation diagrams. We refer to [1][2][3][4][5][6][7][8] and the references therein. Moreover, bifurcation problems with nonlinear diffusion have been proposed in the field of population biology, and several model equations of logistic type have been considered.…”