ACTA issues

On small solutions of second order linear differential equations with non-monotonous random coefficients

László Hatvani

Acta Sci. Math. (Szeged) 68:3-4(2002), 705-725

Abstract. The equation $$x''+a^2(t)x=0, a(t):=a_k\ \hbox{ if }\ t_{k-1} \le t< t_k, \ \hbox{ for }\ k=1,2,\ldots $$ is considered, where the sequence $\{a_k\} ^\infty_{k=1}$ $(a_k>0, k=1,2,\ldots )$ is given, and $t_{k+1}-t_k$, $k=1,2,\ldots $ are totally independent random variables uniformly distributed on interval $[0,1]$. The probability of events $\gamma =0$, $\Gamma =0$, and $\Gamma >0$ are studied, where $$\gamma :=\liminf_{t\to\infty }\left(x^2(t)+{(x'(t))^2\over a(t)}\right ),\qquad \Gamma :=\limsup_{t\to\infty }\left(x^2(t)+{(x'(t))^2\over a(t)}\right ).$$

AMS Subject Classification (1991): 34D20, 34F05

Received April 3, 2002, and in revised form October 21, 2002. (Registered under 2864/2009.)