A New Generalized Gronwall Inequality with a Double Singularity and Its Applications to Fractional Stochastic Differential Equations

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.

Keywords:

• Gronwall inequality
• fractional stochastic differential equations
• mild solutions

Additional information

Funding

The work of X.L.D. was supported by the Science and Technology Planning Project (2018JM1038) of Shaanxi Province and Innovative Talent Promotion Project of Shaanxi, China (2019KJXX-032). The research of D. Cao and J.J. Nieto has been partially supported by the Agencia Estatal de Investigacion (AEI) of Spain under Grant MTM2016-75140-P and co-financed by European Community fund FEDER.