On The Existence Of Positive Solutions For Some Differential Equation And Differential Equations

The nonlinear analysis is one of the research directions in modern mathematics,which has both profound theoretical significance and widely used value. It based onnonlinear problems in various fields of mathematics and natural sciences, and it hasestablished a number of general theories to deal with many nonlinear problems. Thetheory of diferential equations is an important branch of the theory of diferential e-quations, and diferential equations theory is an important application of nonlinear, Itpresents the structure not only has profound physical background and practical sig-nificance, but also has important research value and research significance. Nonlinearboundary value problem rises from many aspects of applied mathematics and physics,while nonlinear diferential equation and diferential equations’ boundary value prob-lem is one of important branch in nonlinear boundary value problems and it has avery important role in applied mathematics, physics and engineering. So, researchthe existence of solutions of nonlinear diferential equation and diferential equation-s’ boundary value problems, and then study the property of the countable positivesolutions becomes very meaningful. In this paper, using the cone theory, cone expan-sion and compression fixed point theorem, mixed monotone method and fixed pointtheorem to study the existence of countable positive solutions of nonlinear diferen-tial equation and diferential equations’ boundary value problem, and we get someencouraging results.Based on the inherent relationship between the structure and content of study,this paper includes the following chapters:In Chapter1, introduction, introduces the research topic and research significance.In Chapter2, using the cone expansion and compression fixed point theorem, wediscuss the positive solutions of a second order three-point diferential system, where λ,μ,κis a constant,f∈C((a,b))×R+,R+),that is to say,f may be singular at t=a,t=b.Using the cone theory,fixed point theorem we study the existence of countable positive solutions of the second order three-point differential system,and we give two examples.In Chapter3,we study the existence of positive solutions of the following differ-ential equations’ boundary probleem, where.f,g∈C((0,1)×R+,R+),f(χ,0)三0,g(z,0)三0α,β>0,△=(1一βη)+α(1-β)>0,f,g may be singular at χ=0,χ=1,0<η<1,α(χ)may be singulaar at χ=0.In Chapter4,by using the mixed monotone metethod,we discuss the existence of positive solutions of a second order three-point differential equation, m>0is a constant,0<η<1,κ>0,f(t,u,u)is continuous.