The Cauchy problem for the Sturm-Liouville equations and estimates of its solution with respect to the \(\lambda\) parameter
Consider the Cauchy problem \[ y''+\lambda\varrho(x) y= 0,\quad x> 0,\quad y(0)= 0,\quad y'(0)= 1,\tag{\(*\)} \] where \(\lambda\) is a numerical parameter and \(\varrho(x)\) is a function summable on \([0,e]\) and satisfying \(0< \varrho_0\leq\varrho(x)\leq \varrho_1\). The author estimates the modulus of a solution \(y(x,\lambda,\varrho)\) to \((*)\) under some additional assumptions on \(\varrho(x)\). Especially, conditions are derived that the estimate \(|y(x,\lambda, \varrho)|\leq c_1/\sqrt\lambda\) holds.
scientific article; zbMATH DE number 1557363