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On quadratically integrable solutions of the second order linear equation. (English) Zbl 1090.34537

The authors investigate the linear differential equation \[ u''+p(t)u=0, \tag{(E)} \] where \(p:[0,+\infty [\to ]-\infty ,+\infty [\) is locally integrable function. The sets \(V_i(p)\), \(i\in \{0,1\}\), of solutions of Eq. (E) satisfying \(\int ^{+\infty }{u^2(s)\over s^i}\,\text{d}s<+\infty \) and the set \(Z(p)\) of solutions of Eq. (E) satisfying \(\lim _{t\to +\infty }u(t)=0\) are studied. The authors presents integral criteria for \(\dim V_i(p)=0\), \(\dim V_i(p)=1\) and \(V_i(p)=Z(p)\). The criteria are formulated in the terms of numbers \(p_*=\liminf _{t\to +\infty }\Bigl (c_p-\int _1^t p(s)\,\text{d}s\Bigr )\), \(p^*=\limsup _{t\to +\infty }\Bigl (c_p-\int _1^t p(s)\,\text{d}s\Bigr )\), where \(c_p=\lim _{t\to +\infty }{1\over t}\int _1^t\int _1^s p(\eta )\,\text{d}\eta \,\text{d}s\).

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations
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