Existence Of Positive Solutions For Two Kinds Of Nonlinear Nonlocal Differential Equations Boundary Value Problems

In recent years,nonlinear nonlocal differential equations have played a considerably prominent role in various areas,for instance,systems of particles in thermodynamic equilibrium interact through gravitational potential;mean field equations;the two-dimensional full turbulence characteristics of a real flow;and shear bands of metals deformed at high strain rates.With the development of the nonlinear nonlocal differential equations and boundary value problems with nonlocal boundary conditions,it is of great theoretical significance and application value to study the existence and multiplicity of positive solutions for the nonlinear nonlocal differential equations boundary value problems with nonlocal boundary conditions.In this dissertation,we study the existence and multiplicity of positive solutions for two kinds of nonlinear nonlocal differential equations boundary value problems,and finally obtain some meaningful new results.The dissertation is built by the following four chapters:The first chapter is introduction,and it introduces the development of the nonlinear nonlocal differential equations,the major elements,the innovation,definition and fixed point theorems of this dissertation.In chapter 2,we are concerned with the existence of positive solutions for the following nonlocal differential equation where 0<p<1≤q,the function A:R→R is continuous.We study under nonlocal boundary conditions the existence and multiplicity of positive solutions for equation(1.1)when the nonlinear terms is continuous,singular,and changes sign,respectively,by using the fixed point index theory on cones,Guo-Krasnoselskii fixed point theorem on cones and the fixed point theorem in double cones.The above problem is nonlocal due to the coefficient function A(∫01(up(s)+uq(s))ds)involving an integral,and the nonlocal boundary conditions.Then,in Chapter 3,we consider the following convolution equations with nonlocal boundary condition where λ>0 is a parameter,f:[0,1]×[0,+∞)→(0,+∞)is continuous and φ(u)=∫01u(s)dα(s)is a Stieltjes integral with the function α which is of bounded variation and monotone increasing on[0,1].There are two different nonlocal elements in problem(1.2):one is the convolution a＊u",which appears in the equation itself;the other is Stieltjes integralφ(u),which occurs in the boundary condition.Using the theory of fixed point index,the existence,multiplicity and nonexistence results of positive solutions are established according to the different values of parameter λ.Finally,the fourth chapter is the summary and prospect,and it mainly introduces the main conclusions and shortcomings of this dissertation,the progress of the follow-up work and the future research plan.