A kind of stricter Hyers-Ulam stability of second order linear differential equations of Carathéodory type
In this work, the authors study Hyers-Ulam stability conditions for second order linear differential equations \[ \phi'' + b_1 \phi' + b_0 \phi = f(t), \quad t\in (\sigma, \tau) \tag{1} \] \((-\infty \le\sigma < \tau \le +\infty),\) where \(f(t)\) is a real-valued locally integrable function on \((\sigma,\tau)\), \(b_1, b_0 \in \mathbb{R}\). Let \(\lambda_1, \lambda_2\) be the characteristic numbers of the corresponding homogeneous equation \[ \phi'' + b_1 \phi' + b_0 \phi = 0, \quad t\in (\sigma, \tau). \tag{1} \] The main result of this work (Theorem 1) states that if under some restrictions on \(\lambda_1\) and \(\lambda_2\) Eq. (1) has a Hyers-Ulam stricter \(\widetilde{\phi}(t)\) (in some sense an approximate solution to Eq. (1); see Definition 2) then Eq. (1) has an exact (Carateodory) solution \(\phi(t)\) close to \(\widetilde{\phi}(t)\) on \((\sigma, \tau)\) by sup-norm: \(\sup\limits_{t\in (\sigma, \tau)} |\phi(t) - \widetilde{\phi}(t)| \le \varepsilon\), for every \(\varepsilon >0\), corresponding to the Hyers-Ulam stricter \(\widetilde{\phi}(t)\).
scientific article; zbMATH DE number 7317515