Existence Of Positive Solutions For Several Changing-Sign Nonlinear Differential Equations On Time Scales

In this dissertation,we mainly study the existence of positive solutions for different order boundary value problems changing-sign nonlinear differential equations on time scales.By using a translation transformation and fixed point theory of cone,and we establish several sufficient conditions for the existence of positive solutions.In the first chapter,we first give the symbols and some basic functions involved in this paper,and introduce the research background of the above boundary value problems of nonlinear differential equations on time scales,which can lay a foundation for the existence of positive solutions.And we list some assumptions that the boundary value problem needs to meet on time scales.Then,we give different boundary conditions,which makes the existence results play a key role in solving the different order boundary value problems.In the second chapter,some preparatory knowledge involved in this paper are introduced,including some properties and definitions of time scales,the fixed point theory of cone,etc.In the third chapter,by using the method of translation transformation,it is analyzed that the problem of changing-sign second order differential equation is transformed into a positive definite problem.Then we give two sufficient conditions for the existence of the positive solution by using the fixed point theorem of cone and the asymptotic estimation of the lower bound of the positive solution.Our innovation is that the nonlinear term contains two parts,namely,the semi-positive term and the perturbation term,in which the semi-positive term has a negative lower bound,the perturbation term can changing-sign and tends to negative infinity.In addition,we only consider the case that the weight function and the perturbation term belong to Lebesgue integrable functions.We study the nonlinear differential equation on time scales can have infinitely many singular points.In the fourth chapter,we study the existence of positive solutions of boundary value problem in third-order nonlinear differential equations with semipositive terms and suitable growth conditions and we construct a special cone.Based on the theoretical results we have obtained,with further research,we will continue to expand the research on the existence of positive solutions for high-order nonlinear dynamic equations with changing-sign including derivative terms on different time scales,and find more suitable examples.