The estimates for the number of the eigenvalues of abstract and differential operator functions

We consider an operator function (F(\lambda)) for (\lambda\in(\sigma,\tau)\subseteq\mathbb R) whose values are semibounded selfadjoint operators in Hilbert space (\mathfrak H). Our main goal is to estimate the number (\mathcal N_F(\alpha,\beta)) of the eigenvalues of (F(\lambda)) on a segment ([\alpha,\beta)\Subset(\sigma,\tau)). In particular, we prove the estimates (\mathcal N_F(\alpha,\beta)\geqslant \nu_F(\beta)-\nu_F(\alpha)) and (\mathcal N_F(\alpha,\beta)= \nu_F(\beta)-\nu_F(\alpha)) where (\nu(\xi)) is the number of the negative eigenvalues of the operator (F(\xi)), (\xi\in(\sigma,\tau)). The obtained results are applied for the functions of ordinary differential operators on a finite interval.