Search

In this paper, we investigate the well-posedness, regularity, Ulam-Hyers stability, and averaging principle for solutions of Itô–Doob stochastic Hadamard fractional pantograph equations of order α ∈ (1 / 2 , 1) in L p spaces with p ≥ 2 . We first establish the existence and uniqueness of results by applying the Banach fixed point theorem, and demonstrate the continuous dependence of the solutions on the initial values. We then prove the time regularity of the solutions to the proposed equations and show that the solutions are ( α - 1 2 )-Hölder continuous. Next, we analyze the Ulam–Hyers stability of the considered equations utilizing the Gronwall inequality. Finally, we study the averaging principle for the studied equations under a general averaging condition. Our results are illustrated with two examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the cyclic anti-periodic boundary value problem of a nonlinear tripled system of fractional Langevin equations. By means of Krasnoselskii fixed point theorem and Banach contraction mapping theorem, we obtain sufficient conditions for the existence and uniqueness of results. Moreover, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the proposed problem. Finally, two illustrative examples are presented. [ABSTRACT FROM AUTHOR]

In this study, a new definition of the fractional integral operator is first proposed, which generalizes some well-known fractional integral operators. Then, by using this newly generalized fractional integral operator, we proved several new Hermite-Hadamard-type inequalities for convex functions. Finally, we provided some corollaries to show that the current results extend and enrich the previous results in the literature. [ABSTRACT FROM AUTHOR]

In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms D a + β H C (H C D a + α + p (t)) x (t) + q (t) x (t) = 0 , 0 < a < t < b , and D a + η H C ϕ p [ (H C D a + γ + u (t)) x (t) ] + v (t) ϕ p (x (t)) = 0 , 0 < a < t < b , subject to mixed boundary conditions, respectively, where p (t) , q (t) , u (t) , v (t) are real-valued functions and 0 < β < 1 < α < 2 , 1 < γ , η < 2 , ϕ p (s) = | s | p − 2 s , p > 1 . The boundary value problems of fractional Langevin-type equations were firstly converted into the equivalent integral equations with corresponding kernel functions, and then the Lyapunov-type inequalities were derived by the analytical method. Noteworthy, the Langevin-type equations are multi-term differential equations, creating significant challenges and difficulties in investigating the problems. Consequently, this study provides new results that can enrich the existing literature on the topic. [ABSTRACT FROM AUTHOR]

The large-scale uniform triangular prism-like CeO2 nanostructures with rough surface have been purposefully fabricated by a simple and efficient hydrothermal route. X-ray diffractometer (XRD), scanning electron microscope (SEM), X-ray photoelectron spectrum, Raman spectra and photoluminescence (PL) spectra were employed to research the obtained CeO2 samples. It is found that many Ce3+ ions and oxygen vacancies in the surface of cubic fluorite structure CeO2 samples. The possible mechanism involved the dissolution of rod-like structures and followed by the recrystallization of triangular prism-like structures was discovered. Ce3+ ions and oxygen vacancies may account for the similar broad photoluminescence emission bands for all samples, while their different emission intensity is attributed to the effects of morphology and size of the samples. [ABSTRACT FROM AUTHOR]

The monodisperse 3D flower-like CeO nanospheres with many channels have been purposefully synthesized via controlling the morphology of corresponding CeCOOH precursors by a facile hydrothermal method. The X-ray diffraction results indicate that the calcined samples have a cubic fluorite structure of CeO with no crystalline impurity phase. The scanning electron microscope and transmission electron microscope images display that the flower-like nanospheres with a diameter of 200-400 nm are composed of numerous irregular nano-particles as the petals with less than 100 nm in diameter, which are found highly oriented and perpendicularly aligned with lower density to form an open porous structure. X-ray photoelectron spectroscopy and Raman analyses demonstrate that a certain amount of Ce ions and oxygen vacancies exist in the surface of CeO samples. The magnetic measurements show that the obtained samples have excellent ferromagnetism at room temperature, which can be reasonably explained for the influences of the morphology of samples, oxygen vacancies and Ce ions. [ABSTRACT FROM AUTHOR]

The CeO nanocubes have been successfully synthesized via a facile hydrothermal method. X-ray diffractometer, scanning electron microscope, transmission electron microscope, X-ray photoelectron spectroscopy (XPS), Raman scattering and photoluminescence spectra were employed to characterize the samples. The results showed that the CeO nanocubes have a cubic fluorite structure and the rugged CeO nanocubes with an edge length of about 200 nm transformed into smooth one with an edge length of about 70 nm with the change of the surfactant. XPS and Raman analyses indicate that there are Ce ions and oxygen vacancies in the surface of samples. Both of nanocubes exhibit similar emission peaks of photoluminescence at room temperature and the emission intensity increases with the increase of the concentration of oxygen vacancies. [ABSTRACT FROM AUTHOR]

Different surface morphologies of TiO films were prepared through hydrothermal synthesis method on transparent fluorine-doped tin oxide (FTO) substrates by changing reaction temperatures. The as-prepared samples were characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), selected area electron diffraction (SAED), and X-ray photoelectron spectrometer (XPS). As the hydrothermal temperatures increases, the surface morphologies of the TiO changes from nanorods (150 °C) to nanobuds (180 °C), and finally to nanoflowers (210 °C). Evolution of these structures are accompanied by great variations of optical properties and photocatalytic activities including a narrowing of band gap from 3.01 to 2.97 eV, increase of UV-visible absorption intensity and specific surface area, and photocatalytic degradation efficiencies from 88.18 to 95.56%. Under ultraviolet light (UV light) irradiation, the TiO nanoflowers exhibit significantly activity (95.56%) in degradation of methyl orange (MO) compared to commercial P25 (76.15%). The outstanding photocatalytic activity of the TiO nanoflowers can be attributed to the synergetic effect of much larger specific surface area, the larger content of oxygen vacancy, and higher intensity of absorption. These findings help to grow unique TiO films with desired structure and activities for photocatalyst applications. [ABSTRACT FROM AUTHOR]

SO4 2−/TiO2–MxOy (M = Zr, Ce, La) were prepared by the precipitation-impregnation method and characterized by X-ray powder diffraction (XRD), Fourier transform infrared (FT-IR), and temperature-programmed desorption (NH3-TPD). Catalytic activities were evaluated in the acetalization of 1,3-propanediol (1,3-PD) with acetaldehyde and hydrolysis of 2-methyl-1,3-dioxane (2MD). SO4 2−/TiO2–ZrO2 (STZ) exhibited the best catalytic activity both in the acetalization and hydrolysis. With the molar ratio of Zr4+/Ti4+ = 1:4, the highest yields were 96.45% in 3 h and 93.68% of 2MD hydrolyzed in 18 h, in contrast to the yields lower than 60% by using other superacids. These results are consistent with the strongest acidity of the superacid containing Zr4+ among prepared superacids containing other cations. [ABSTRACT FROM AUTHOR]

In this paper, the existence and multiplicity of solutions of three-point p- Laplacian boundary value problems at resonance with onesided Nagumo condition are studied by using degree theory and upper and lower solutions method. Some known results are improved. [ABSTRACT FROM AUTHOR]