On the extremal solutions of 𝑛th-order linear differential equations

Abstract

Distribution of zeros of extremal solutions of linear nth-order differential equations is discussed. Existence and nonexistence of extremal solutions with certain zero distributions are established. For instance, it is proved that every extremal solution for [ α , η 1 ( α ) ] [\alpha , {\eta _1}(\alpha )] of the equation y ( n ) + p n − 1 y ( n − 1 ) + ⋯ + p 0 y = 0 {y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y = 0 has a zero of order 2 at η 1 ( α ) {\eta _1}(\alpha ) and has no more than n − 2 n - 2 zeros on [ α , η 1 ( α ) ) if p i ≦ 0 , i = 0 , 1 , ⋯ , n − 2 [\alpha , {\eta _1}(\alpha ))\;{\text {if}}\;{p_i} \leqq 0,i = 0,1, \cdots , n - 2 .