Summary: We consider nonlinear eigenvalue problems of the form \[T x + \epsilon B(x) = \lambda x \tag{\(\ast\)}\], where \(T\) is a self-adjoint bounded linear operator acting in a real Hilbert space \(H\), and \(B : H \rightarrow H\) is a (possibly) nonlinear continuous perturbation term. Assuming that \(\lambda_0\) is an isolated eigenvalue of finite multiplicity of \(T\), we ask if, for \(\epsilon \neq 0\) and small, there are “eigenvalues” of \((\ast)\) near \(\lambda_0\), that is, numbers \(\lambda_\epsilon\) for which \((\ast)\) is satisfied by some normalized “eigenvector” \(x_\epsilon\) of \(T + \epsilon B\). 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 \(\lambda_\epsilon\) which are determined by those for the nonlinear Rayleigh quotient \(\langle B(v), v \rangle / \langle v, v \rangle\) with \(v\) in the eigenspace \(\operatorname{Ker}(T - \lambda_0 I)\). This yields particular information on the rate of convergence of \(\lambda_\epsilon\) to \(\lambda_0\) as \(\epsilon \rightarrow 0\). Applications are given in the sequence space \(l^2\) and in the Sobolev space \(H_0^1\) to deal with some nonlinearly perturbed ordinary or partial differential equations.