A Class Of Semi-linear Differential Equations, Existence And Uniqueness,

In this paper we mainly use Initial Value Problem (IVP) method to study the existence and uniqueness of the solution to the equation, which is a kind of semi-linear differential one. First we discuss the equation Lu + N[u] = f .We transfer the problem whether the equation has solution to initial value problem of ordinary differential equation by homotopy deformation, and then we obtain a necessary and sufficient condition for the existence of the solution of the original equation, a necessary condition for the uniqueness of the solution, and an estimation of the solution. We also obtain different conditions for the problem whether the semi-linear differential equation has solution according to different homotopy deformations. Furthermore, we discuss the equation Lu + N[u] = F(u) by using similar method and present conditions for the existence and uniqueness of the solution to the equation. At last we apply the results to a kind of fourth order elliptic boundary value problem and Duffing equation.