Reduced graphene oxide supported titanium dioxide (GO/TiO2) heterojunction composites as highly active photocatalysts were synthesized via simple ultrasonic mixing and hydrothermal reaction using TiCl3 and GO as precursors. Their structure and morphology were characterized by X-ray diffraction, Fourier transform infrared spectroscopy, Raman spectra, UV-vis spectroscopy, and thermogravimetic analysis. The GO/TiO2 heterojunction composites were used to degrade methyl orange (MO). The adsorption and photocatalytic degradation rate of the prepared GO/TiO2 composites increased by nearly three times compared with that of pristine TiO2 or GO, which reached up 90%, to degrade MO after 4 h, which provides a simple method to obtain photocatalytic materials.

Let be a unital Banach algebra. Assume that a has a generalized inverse a+. Then is said to be a stable perturbation of a if . In this paper we give various conditions for stable perturbation of a generalized invertible element and show that the equation is closely related to the gap function . These results will be applied to error estimates for perturbations of the Moore-Penrose inverse in C*–algebras and the Drazin inverse in Banach algebras.

For any bounded linear operator A in a Banach space, two generalized condition numbers, k(A) and k(A) are defined in this paper. These condition numbers may be applied to the perturbation analysis for the solution of ill-posed differential equations and bounded linear operator equations in infinite dimensional Banach spaces. Different expressions for the two generalized condition numbers are discussed in this paper and applied to the perturbation analysis of the operator equation.