Existence Of Positive Solutions For A Class Of Nonlinear Boundary Value Problems

With the increasingly rapid development of society and the progress of science and tech-nology, social science, natural science and engineering technology and many other fields, such as physics, economics and ecology, etc., have put forward many nonlinear problems. Nonlinear functional analysis-one of the most important branch of modern mathematics has came into being by generations of scholars during the development of solving such problems. As a research discipline, nonlinear functional analysis not only has the deep theoretical foundations but also has a wide range of practical applications. It appears in mathematics and the natural sciences, setting up some general theories and methods to deal with all kinds of nonlinear problems, such as topological degree theory, cone theory, critical point theory and monotone operator theory, etc. Because it can explain all kinds of natural phenomena, its rich theory and advanced method have provided us many effective theoretic tools for solving the current nonlinear problems which emerge in an endless stream in the field of science and technology. It plays an irreplaceable role in dealing with practical problems corresponding to the various nonlinear integral equations, differential equations and partial differential equations. And the results of their research are also widely used in computational mathematics, control theory, optimization theory, the power system as well as in economics.The theory of boundary value problems for nonlinear ordinary and partial differential equations is among the most active and fruitful fields. Such problems have find roots in applied mathematics, physics, control theory, and other applied sciences. Therefore, the research of boundary value problems is not only of great theoretical significance and but also of wide applicability. Because of the positive solutions for the boundary value problems are generally the most practical solutions to a class solutions, In the research process, it is the positive solutions that many scholars pay attention to researches. In the research process, the scholars converted the existence of positive solutions of the differential equation to the existence of the fixed point for Cone-Preserving operators. And the fixed point index theory and topological degree theory can be also applied to the existence of fixed point problem. The Schauder fixed point theorem, Krasnosel’skii fixed point theorem and Leggett-Williams fixed point theorem and its generalization5functional fixed point theorem are the most commonly used tools.Although there are many scholars have studied the existence of multiplicity of two point and multi-point boundary value problems especially on the existence of positive solutions, and their research results are also very rich, but because of those most common fixed point theorem have many conditions:first, the nonlinear term must be continuous, then the Green function also need to meet the given assumptions, so that the applicability of them has certain limitations. Therefore, until now there are still many challenging problems to be solved. This paper is on the basis of the previous studies to broaden some restrictions, and let the conditions are more general. So that, this article improved and generalized their results.This paper consists of four chapters. In the four chapters, we mainly use the fixed point index theory to discussed the existence and multiplicity of positive solutions as well as the uniqueness of the positive solution for a class of generalized Lidstone equation boundary val-ue problem; the existence of positive solutions for a class of four order ordinary differential equations boundary value problem; the existence of positive solutions for a class of four order p-Laplace boundary value problem and the existence of positive solutions for a system of four order p-Laplacian equations boundary value problem.In Chapter One, we study the existence, multiplicity and uniqueness of positive solutions for the generalized Lidstone boundary value problem: wheref€C([0,1] x R+n,R+),αi,βi,γi,δi≥0,(i=l,2,…,n),Ai=αiδi+βiγi+αiγi>0. Krein-Rutman theorem and Krasnoselskii-Zabreiko fixed point theorem are the main tools to complete our work in this paper. Under more general conditions,Our main results extend and improve the results in the existing literatures.It must be pointed out that the coefficients αi,βi,γi,δi vary with i. Literature with such a complex boundary conditions is rarely appear, the main idea in this chapter comes from the [5], but the conditions are more general than [5], therefore,the results of this paper further generalize some results in [5]. As an application, we then apply the result to the second chapter of four order boundary value problems for ordinary differential equations and the third chapter of the four order p-Laplace boundary value problem.In the second chapter, we study the existence of positive solutions for a system of fourth-order boundary value problem: wheref1,f2∈C([0,1] x R+4,R+)(R+=[0,+∞)),αi,βi,γi,δi≥0,(i=1,2),and△i=αiδi+βiγi+αiγi>0. In this chapter, on the basis of the first chapter, we used the processing method of [8], namely the use of linear and nonnegative matrix to describe the nonlinear growth and coupling action. This chapter is different from the first chapter, in order to reduce the difficulty of the order reduction problem, each equation here which we discussed is only fourth-order. Because the integral-integral equations are equivalent to the original fourth-order equations, so what we need is only to study the reduced integral-integral equations and can obtain the existence of positive solutions of the above fourth-order boundary value problem.In the third chapter, we study the existence of positive solutions for the following class of fourth-order p-Laplace boundary value problem: wherep>0,f∈C([0,1]×R2,R+),ai,bi,ci,di>0,(i=1.2),δi=aidi+bici+aici>0. The idea of this problem comes from the first chapter and [14]. we note, the problem is a disturbance with the problem of the fourth-order boundary value problem when p=1corresponding to. It is the Jensen’s integral inequalities that establish some relations between the two issues.This method in the recent literature rarely see. Based on the priori estimates by use of the Jensen integral inequalities, we use the fixed point index theory to prove the existence of positive solutions for the problem. Under more general conditions,Our main results extend and improve the results in the existing literatures.In Chapter Four, we study the existence of positive solutions for the following system of fourth-order p-Laplacian boundary value problem: Where p, q>0and f1,f2∈G([0,1]×R+,R+). The idea of this problem comes from [8,14],[14] studied the existence of positive solutions for a single fourth-order p-Laplacian boundary value problem. But our research is a system of fourth-order p-Laplacian boundary value problems, this research is rarely seen. Although boundary value conditions are same with [14], but because of u and v which in the equations are influence each other, so compared to the [14] our study is more complicated. We regard our problem as a perturbation of the semilinear coeuterpart of the problem, i.e. the case with p=q=1. It is the Jensen’s integral inequalities that establish some relations between the two issues.we establish a priori estimates by use of the Jensen integral inequality for the positive solutions of the varieties of adjuvant problem of the above boundary value problem, the problem is changed into a system of equivalent integral-integral equations, then use the methods of using linear function and nonnegative matrix that describes nonlinear growth and coupling behavior which put forward in [8], prove the existence of positive solution of the problem.