Hyers–Ulam Stability of Differential Operators on Reproducing Kernel Function Spaces
In this paper, we investigate the Hyers–Ulam stability of the differential operators \(T_\lambda \) and D on the weighted Hardy spaces \(H_\beta ^2\) with the reproducing property. We obtain a necessary and sufficient condition in order that D is stable on \(H_\beta ^2\), and construct an example concerning the stability of \(T_\lambda \) on \(H_\beta ^2\). Moreover, we also investigate the Hyers–Ulam stability of the partial differential operators \(D_i\) on the several variables reproducing kernel space \(H_f^2(\mathbb {B}_d)\).
We present what is to our knowledge the most complete 1-D numerical analysis of the evolution and the propagation ::: dynamics of an ultrashort laser pulse in a Ti:Sapphire laser oscillator. This study confirms the dispersion ::: managed model of mode-locking, and emphasizes the role of the Kerr nonlinearity in generating mode-locked ::: spectra with a smooth and well-behaved spectral phase. A very good agreement with experimental measurements ::: of pulse energy, spectrum, and temporal width of extracavity compressed pulses is found.