分数阶微分方程边值问题正解的存在性

Existence of Positive Solutions for Boundary Value Problem of Fractional Differential Equations

分数阶导数是整数阶导数的推广,分数阶导数有Riemann-Liouville分数阶导数、Marchaud分数阶导数、Caputo分数阶导数等。分数阶微分方程模型具有深刻的物理背景和丰富的理论内涵,在诸多领域应用广泛,如血液流动问题、化学工程、热弹性、地下水流动、人口动力学等。分数阶微分方程边值问题正解的性质是近几年研究的热点之一。在本文中,首先,构造相应线性边值问题的格林函数,其次,分析格林函数的性质,构造合适的锥,再次,利用Guo-Krasnoselskii不动点定理得到了带积分边界条件的分数阶微分方程边值问题正解的存在性结果,最后,通过一个实例说明了结果的合理性。

Fractional derivatives are generalizations for derivative of integral order. There are several kinds of fractional derivatives, such as Riemann-Liouville fractional derivative, Marchaud fractional deriva-tive Caputo fractional derivative, etc. Fractional differential equation model has profound physical background and rich theoretical connotation. It is widely used in many fields, such as blood flow problem, chemical engineering, thermoelasticity, groundwater flow, population dynamics and so on. The properties of positive solutions for boundary value problems of fractional differential equations are one of the hot topics in recent years. In this paper, firstly, the Green’s function of the corre-sponding linear boundary value problem is constructed. Secondly, the properties of the Green’s function are analyzed, a suitable cone is constructed. Thirdly, by using Guo-Krasnoselskii fixed point theorem, the existence of positive solutions for boundary value problems of fractional differential equations with integral boundary conditions is obtained. Finally, an example is given to illustrate the rationality of the results.