On non-linear Markov operators: surjectivity vs orthogonal preserving property

In the present paper, we consider non-linear Markov operators, namely polynomial stochastic operators. We introduce a notion of orthogonal preserving polynomial stochastic operators. The purpose of this study is to show that surjectivity of non-linear Markov operators is equivalent to their orthogonal preserving property.     

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

Nanoflower-like composites of ZnO/SiO2 synthesized using bamboo leaves ash as reusable photocatalyst

In this study, the synthesis of ZnO/SiO₂ nanocomposites using bamboo leaf ash (BLA) and tested their photocatalytic activity for rhodamine B decolorization have been conducted. The nanocomposites were prepared by the sol–gel reaction of zinc acetate dihydrate, which was used as a zinc oxide precursor, with silica gel obtained from the caustic extraction of BLA. The effect of the Zn content (5, 10, and 20 wt%) on the physicochemical characteristics and photocatalytic activity of the nanocomposites was investigated. The results of X-ray diffraction, scanning electron microscopy, gas sorption, and transmission electron microscopy characterization confirmed the mesoporous structure of the composites containing nanoflower-like ZnO (wurtzite) nanoparticles of 10–30 nm in size dispersed on the silica support. Further, the nanocomposites were confirmed to be composed of ZnO/SiO₂ by X-ray photoelectron spectroscopy analysis. Meanwhile, diffuse-reflectance UV–visible spectrophotometry analysis of the nanocomposites revealed band gap energies of 3.38–3.39 eV. Of the tested nanocomposites, that containing 10 wt% Zn exhibited the highest decolorization efficiency (99%) and fastest decolorization rate. In addition, the degradation efficiencies were not reduced significantly after five repeated runs, demonstrating the reusability of the nanocomposite catalysts. Therefore, the ZnO/SiO₂ nanocomposite obtained from BLA is a promising reusable photocatalyst for the degradation of dye-polluted water.     

On surjective second order non-linear Markov operators and associated nonlinear integral equations

It was known that orthogonality preserving property and surjectivity of nonlinear Markov operators, acting on finite dimensional simpleces, are equivalent. It turns out that these notions are no longer equivalent when such kind of operators are considered over on infinite dimensional spaces. In the present paper, we find necessary and sufficient condition to be equivalent of these notions, for the second order nonlinear Markov operators. To do this, we fully describe all surjective second order nonlinear Markov operators acting on infinite dimensional simplex. As an application of this result, we provided some sufficient conditions for the existence of positive solutions of nonlinear integral equations whose domain are not compact.