A New Generalized Gronwall Inequality with a Double Singularity and Its Applications to Fractional Stochastic Differential Equations
A New Generalized Gronwall Inequality with a Double Singularity and Its Applications to Fractional Stochastic Differential Equations
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DOI：
10.1080/07362994.2019.1640612
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发表时间：
2019-07
影响因子：
1.3
通讯作者：
Xiao-Li Ding;C. Daniel;J. Nieto
中科院分区：
文献类型：
--
作者：
Xiao-Li Ding;C. Daniel;J. Nieto
在许多情况下，利用不动点理论证明了一类微分方程解的存在唯一性。本文利用算子理论和巧妙的技巧研究了半线性分数次随机微分方程温和解的适定性。首先讨论了一类Volterra积分算子的一些性质，然后建立了一个新的具有双重奇异性的广义Gronwall积分不等式。最后，利用这些性质和积分不等式研究了半线性分数次随机微分方程温和解的适定性。可以看出，利用前人的结果来研究温和解的适定性是简洁和有效的。
Abstract In many cases, the existence and uniqueness of the solution of a differential equation are proved using fixed point theory. In this paper, we utilize the theory of operators and ingenious techniques to investigate the well-posedness of mild solution to semilinear fractional stochastic differential equations. We first discuss some properties of a class of Volterra integral operators and then establish a new generalized Gronwall integral inequality with a double singularity. Finally, we use the properties and integral inequality to study the well-posedness of mild solution to the semilinear fractional stochastic differential equations. One sees that it is concise and effectiveness using the previous results to investigate the well-posedness of the mild solution.