THE ASYMPTOTIC BOUNDS OF SOLUTIONS OF A GENERALIZED PANTOGRAPH EQUATION

This paper studies the asymptotic behavior of solutions of the generalized pantograph equation $ y'(t)=Ay(qt)+By(t)+Cy'(qt) $, where $ A, B, C $ are $ n×n $ complex matrices. By considering two cases of the coefficient matrix $ B $: Diagonalizable and non-diagonalizable, the asymptotic boundaries of solutions are discussed, respectively. When $ B $ is diagonalizable, the asymptotic boundary of solutions is dominated by the largest positive real part of the eigenvalues: If the smallest positive real part of eigenvalues exceeds the product of the delay parameter and the largest positive real part, then the components of solutions grow exponentially according to the corresponding eigenvalues, otherwise, all solutions are constrained by the largest positive real part of eigenvalues. When $ B $ cannot be diagonalized, the asymptotic boundary of solutions depends on the distribution of eigenvalues: If $ B $ has a unique multiple eigenvalue, then the real part of this eigenvalue determines the growth rate of solutions, otherwise, the components of solutions grow exponentially according to the corresponding eigenvalues in the Jordan blocks. Hence, every solution has an exponential asymptotic boundary, which depends on the eigenvalues of the coefficient matrix $ B $.