On the frequency of complex zeros of solutions of certain differential equations

[摘要] References(20)In this paper, we investigate the frequency of zeros of solutions of linear differential equations of the form w(k)+∑\limits{\jmath}=1k−1Qjw(j)+(Q0+ReP)w=0, where k{≥}2, and where Q0, …, Qk−1, R and P are arbitrary polynomials with R{¬≡}0 and P non-constant. All solutions f{¬≡}0 of such an equation are entire functions of infinite order of growth, but there are examples of such equations which can possess a solution whose zero-sequence has a finite exponent of convergence. In this paper, we show that unless a special relation exists between the polynomials Q0, …, Qk−1, and P, all solutions of such an equation have an infinite exponent of convergence for their zero-sequences. This result extends earlier results for the equation, w(k)+(Q0+ReP)w=0.

[学科分类] 数学（综合）

[关键词] Multivalued operator;monotone operator;measurable multifunction;upper and lower semicontinuous multifunctions;evolution triple;compact embedding;Shauder-Tichonov fixed point theorem;periodic trajectories