کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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5774622 | 1413563 | 2017 | 13 صفحه PDF | دانلود رایگان |
We consider nonlinear eigenvalue problems of the form (â)Tx+ϵB(x)=λx, where T is a self-adjoint bounded linear operator acting in a real Hilbert space H, and B:HâH is a (possibly) nonlinear continuous perturbation term. Assuming that λ0 is an isolated eigenvalue of finite multiplicity of T, we ask if for ϵâ 0 and small there are “eigenvalues” of (â) near λ0, that is, numbers λϵ for which (â) is satisfied by some normalized “eigenvector” xϵ of T+ϵB. In this paper we recall some recent results giving an affirmative answer to this question, and for these cases we prove - assuming in addition Lipschitz continuity on B - upper and lower bounds for the perturbed eigenvalues λϵ which are determined by those for the nonlinear Rayleigh quotient ãB(v),vã/ãv,vã with v in the eigenspace Ker(Tâλ0I). This yields in particular information on the rate of convergence of λϵ to λ0 as ϵâ0. Applications are given in the sequence space l2, and in the Sobolev space H01 to deal with some nonlinearly perturbed ordinary or partial differential equations.
Journal: Journal of Mathematical Analysis and Applications - Volume 455, Issue 2, 15 November 2017, Pages 1720-1732