Gronwall Inequality And The Hyers-Ulam Stability Of Differential Equations

Gronwall inequality is an important tool to study the continuity, uniqueness,Hyers-Ulam stability about the solution of the boundary value problem and so on.Therefore, the study of Gronwall inequalities has caused many scholars’ interest, and it has been developed rapidly. The fractional Gronwall inequalities are proposed based on the integer order Gronwall inequality in the traditional sense. From 2007,By solving the differential equation, H Ye gave a class of Gronwall-Bellman type inequality, the fractional order Gronwall inequality theory is gradually developed in various fields of science, In particular, the theory of viscoelasticity, control theory,chemical process, complex network and other fields.At the same time, the results of Gronwall inequalities have greatly promoted the development of Hyers-Ulam stability of differential equations. Hyers-Ulam stability is different from the traditional equation stability, which is to find an approximate solution near the exact solution of the equation. so it is very important in engineering mechanics and other disciplines. On the basis of previous studies, I mainly do the following two aspects of the work:(1) By means of iterative method and mathematical induction, two kinds of Gronwall inequalities have been considered. At the same time, the concrete examples are given and the boundedness of solutions and the Hyers-Ulam stability of a class of differential equations are proved.(2) Using Gronwall inequalities as a tool, the Hyers-Ulam stability of nonlinear non-homogeneous three-order differential equations are considered, and the conclusion is extended to differential equations of n-order. In this paper we give a detailed process.