## 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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