A Special Kind Of Quadratic Two Parameter Eigenvalue Problem And Its Application

Multi-parameter eigenvalue problems have important applications in the boundary value problem of differential equations,multi-parameter Sturm-Liouville problem,aeroelastic problem in fluid mechanics and so on.The quadratic two-parameter eigenvalue problem plays an important role in the critical time delay estimation of Time delay differential equations.In this thesis,a special kind of quadratic two parameter eigenvalue problem(quadratic two parameter eigenvalue problem without ?₁² and ?₂²terms)and its application in critical time delay estimation of time delay differential equations are studied.Firstly,the quadratic two-parameter eigenvalue problem is linearized into a linear three-parameter eigenvalue problem by introducing new variables.Then by using the equivalence between the non-singular multi-parameter eigenvalue problem and a set of coupled generalized eigenvalue problems,the matrix determinant of the coupled generalized eigenvalue problem is block triangular matrix by simultaneous unitary block triangulation.Then the norm estimators of the inverses of the matrices determinants(35)₀ are obtained by applying the properties of the obtained block triangular matrices,and the bounds of the spectral radius of such special quadratic two-parameter eigenvalue problems are obtained.Finally,the results are applied to the critical time delay estimation of time delay differential equations and a time delay interval is obtained that keeps the delay differential equation stable.