二阶方程+随机参数+稳定性.pdfVIP

J. Differential Equations 248 (2010) 21–49 Contents lists available at ScienceDirect Journal of Differential Equations /locate/jde Stability properties of solutions of linear second order differential equations with random coefficients ˝ ∗,1 Sándor Csörgo, László Hatvani University of Szeged, Bolyai Institute, Aradi vértanúk tere 1, Szeged, H-6720 Hungary a r t i c l e i n f o a b s t r a c t Article history: The equation Received 20 March 2009 Available online 28 August 2009 2 x + a (t)x = 0, a(t) := ak 0 if tk−1 t tk (k ∈ N) MSC: primary 34F05, 34C11 is considered where {a }∞ is given and {t }∞ is a random k k=1 k k=1 secondary 70L05 sequence. Sufficient conditions are proved which guarantee either stability or instability for the zero solution. Stability means that all Keywords: solutions almost surely tend to zero as t → ∞. By instability we Small solution mean that the sequence of the expected values of the amplitudes Random oscillations of every solution tends to infinity as k → ∞. It turns out that Impulses ak ∞ (k → ∞) implies stability for all absolutely continuous P