Some New Localization Sets For Nonlinear Eigenvalues

Nonlinear eigenvalue problem is widely used in dynamic analysis of damped structures,stability analysis of time-delay systems,numerical simulation of quantum dots,vibration analysis of fluid-solid structures and many other fields,and its eigenvalues play an important role in practical applications.In this thesis,I mainly study the localization problem of nonlinear eigenvalues,and give three new localization sets for its eigenvalues.More specifically,we first give Brauer-type localization set,DZ-type localization set and S-SDD localization set of nonlinear eigenvalues by using two row elements and the relationship between the nonsingular matrix class and the inclusion region of nonlinear eigenvalues.Secondly,it is proved that the above results are better than those in the previous studies of [Kosti(?) V,Garda(?)evi(?) D.On the Ger(?)gorin-type localizations for nonlinear eigenvalue problems.Applied Mathematics and Computation,2018,337:179-189] and [Michailidou C,Psarrakos P.Gershgorin type sets for eigenvalues of matrix polynomials.Electronic Journal of Linear Algebra,2018,34(1):652-674].Further,an example is also given to illustrate the effectiveness of the results.Finally,the obtained results are applied to study the stability of single mass spring systems and time-delay systems.