The Growth Condition Guaranteeing Small Solutions for a Linear Oscillator with an Increasing Elasticity Coefficient
László Hatvani · Georgian Mathematical Journal · 2007
Abstract The second order linear differential equation is considered, where 𝑞 : [0, ∞) → (0, ∞) is continuous, piecewise continuously differentiable, non-decreasing, and lim𝑡→∞ 𝑞(𝑡) = ∞. A solution 𝑥0 is called small if lim𝑡→∞ 𝑥0(𝑡) = 0. It is known that the equation always has at least one nontrivial small solution, but, in general, it can have also solutions not small. The Armellini–Tonelli–Sansone Theorem says that if the function 𝑞 grows “regularly” in some sense, then all solutions are small. A generalization of this growth condition is given in terms only of the integral of 𝑞 and . It is proved that the result is a real generalization of the earlier theorems.