Existence And Uniqueness Of Solutions To Singular Differential Equations

The singular boundary problems of nonlinear differential equations are important subjects in the theory of differential equations. The most results are devoted on the existence of one solution or multiple solutions. On the uniqueness of the solution, Zhao zengqin[34] has studied the singular boundary problem to the second ordinary differential equations, but the method is complicated. To prove the uniqueness, the general method is that we must prove the two solutions are equivalent. In this paper, we will prove the existence and uniqueness of solutions for singular boundary value problems of differential equations by using the mixed monotone method.The whole contents is divided into five chapters.In chapter 1, as the beginning of this paper, we offered some relative knowledge, such as the cone theory and semi-order method in nonlinear functional analysis. Moreover, we established a new fixed point theorem of the mixed monotone iterative operator.In chapter 2, as the second part of this thesis, by using the mixed monotone method, we gave the existence and uniqueness of singular elliptic differential equations which including singular higher order ordinary differential equations, singular p-Laplace equations, singular second delay differential equations .In chapter 3, as one of the main parts of this thesis, by using the mixed monotone method, we discussed the existence and uniqueness of singular elliptic boundary value problems. About singular elliptic equation with the non-increasing monotonicity assumption on the nonlinearity , the author has studied the uniqueness of solutions in the reference[87]. But when the elliptic equation without the non-increasing monotonicity assumption on the nonlinearity, few author have studied the uniqueness. Of course, the equation we discussed has special nonlinearity.In chapter 4, we first discussed the existence of multiple positive solutions of the second order nonsingular Dirichlet boundary value problem for impulsive differential equations by using the fixed point index theorem in cones . Next, we presented some new existence results for singular boundary value problems for second order impulsive differential equations by using fixed point theorem in cones and Leray-Schauder nonlinear alternative theorem. At last, we discussed the uniqueness of solutions for singular second impulsive differential equations by using mixed monotone method.