Positive solutions for fractional differential equations with variable coefficients

In this article, we study the existence of the positive solutions for a class of differential equations of fractional order with variable coefficients. The equation of this type plays an important role in the description and modeling of control systems, such as $PD^{mu}$-controller. The differential operator is taken in the Riemann-Liouville sense. Our analysis relies on the Leggett-Williams fixed point theorem

Solvability for a class of evolution equations of fractional order with nonlocal conditions on the half-line

Abstract In this paper, we get a new form equivalent integral equation for a class of evolution equations of fractional order with nonlocal conditions on the half-line. With the aid of it, the uniqueness of the mild solution is obtained by the Banach contraction theorem. Also, we present the existence and uniqueness theorem of positive mild solutions by the monotone iterative method without assumption of lower and upper solutions

Existence of positive mild solutions for a class of fractional evolution equations on the half line

Abstract The equivalent integral equation of a new form for a class of fractional evolution equations is obtained by the method of Laplace transform, which is different from those given in the existing literature. By the monotone iterative method without the assumption of lower and upper solutions, we present some new results on the existence of positive mild solutions for the abstract fractional evolution equations on the half-line

Existence and uniqueness of positive mild solutions for a class of fractional evolution equations on infinite interval

Abstract Based on an equivalent integral equation of a new type for a class of fractional evolution equations, which is different from those obtained in the existing literature, the paper investigates a class of fractional evolution equations with nonlocal conditions on infinite interval. Without the assumption of lower and upper solutions, we present a new result on the existence and uniqueness of positive mild solutions for the abstract fractional evolution equations by using the monotone iterative method

Existence and uniqueness for a kind of nonlocal fractional evolution equations on the unbounded interval

Abstract By means of a Laplace transform and its inverse transform, we obtain a correct equivalent integral equation for some kind of nonlocal abstract differential equations (fractional order) on the right half-axis. Based on it, the existence result is established by Knaster’s theorem, and the uniqueness of the mild solution is obtained using the Banach contraction principle

Multiplicity and uniqueness for a class of discrete fractional boundary value problems

summary:The paper deals with a class of discrete fractional boundary value problems. We construct the corresponding Green's function, analyse it in detail and establish several of its key properties. Then, by using the fixed point index theory, the existence of multiple positive solutions is obtained, and the uniqueness of the solution is proved by a new theorem on an ordered metric space established by M. Jleli, et al. (2012)

Synthesis of Halopyrazole Matrine Derivatives and Their Insecticidal and Fungicidal Activities

Matrine is a traditional botanical pesticide with a broad-spectrum biological activity that is widely applied in agriculture. Halopyrazole groups are successfully introduced to the C13 of matrine to synthesize eight new derivatives with a yield of 78–87%. The insecticidal activity results show that the introduction of halopyrazole groups can significantly improve the insecticidal activity of matrine on Plutella xylostella, Mythimna separata and Spodoptera frugiperda with a corrected mortality rate of 100%, which is 25–65% higher than matrine. The fungicidal activity results indicate that derivatives have a high inhibitory effect on Ceratobasidium cornigerum, Cibberella sanbinetti, Gibberrlla zeae and Collectot tichum gloeosporioides. Thereinto, 4-Cl-Pyr-Mat has the best result, with an inhibition rate of 23–33% higher than that of matrine. Therefore, the introduction of halogenated pyrazole groups can improve the agricultural activity of matrine