This paper concerns a new kind of fractional differential equation of arbitrary order by combining a multi-point boundary condition with an integral boundary condition. By solving the equation which is equivalent to the problem we are going to investigate, the Green's functions are obtained. By defining a continuous operator on a Banach space and taking advantage of the cone theory and some fixed point theorems, the existence of multiple positive solutions for the BVPs is proved based on some properties of Green's functions and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are provided to illustrate the results.
Fluopyram is commonly used to control banana leaf spot, anthracnose, and scab in tropical agricultural areas. To explore its behaviour in tropical agricultural environments, dissipation, adsorption, and leaching behaviours of fluopyram in three typical banana planting soils were studied. Also, its dissipation and migration capabilities in different regions and different soil types were evaluated. The results showed that the dissipation of fluopyram was in accordance with the first-order kinetic equation in the three banana soils, but the degradation rates were quite different. The degradation half-lives in the Hainan latosol, Yunnan sandy soil, and Fujian Plain alluvial soil were 46.21 days, 36.48 days and 57.76 days, respectively. Fluopyram also exhibited high adsorption and low leachability in the three soils. The Fujian Plain alluvial soil had the highest adsorption capacity for fluopyram, while fluopyram had the low leachability in the Yunnan sandy soil.
We concentrate on investigating the existence of positive solutions for fractional-order differential equations with integral conditions in this article. The problem is issued by applying Avery-Peterson fixed-point theorem and the properties of Green's function. At the same time, we provide an example to make our results clear and easy for readers' to understand the multiplicity of solutions.
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