Several Classes Of Nonlinear Term Change, The Existence Of The Solution Of The Differential Equation Boundary Value Problem

Along with the development of modern physics and applied mathematics,many kinds of non-linear problems have emerged, which greatly urged the non-linear functional analysis to be better improved. The nonlinear functional analysisis a subject of profound theories and broad applications. It bases on nonlinearproblems of math and science, constructs general theories and methods, and playsan important role in dealing with all kinds of nonlinear integral equations, non-linear diferential equations and partial diferential equations. So the nonlinearanalysis has become one important research direction in modern mathematics.The nonlinear functional analysis is an important branch in nonlinear analysis,because it can explain well various the natural phenomenon. The boundary valueproblem of nonlinear diferential equation stems from the applied mathematics,the physics, the cybernetics and several kinds of application discipline. It is oneof the most active domains of functional analysis at present. The diferentialequations with sign-changing nonlinearity and the boundary value problem foran increasing homeomorphism and homomorphism have been discussed in recentyears. So they become a very important domain of diferential equation researchat present.In this paper, we use the nonlinear alternative of Leray-Schauder type andthe fixed point index theory to study some kinds of boundary value problems fornonlinear diferential equations with sign-changing nonlinearity.We obtain theexistence of solutions and positive solutions for these boundary value problemsand apply the main results to the boundary value problems for the diferentialequations.The thesis is divided into three chapters.In Chapter1, we investigate the existence of solution and positive solution forboundary value problems with an increasing homeomorphism and homomorphismof impulsive diferential equations on the half-line Where φ is an increasing homeomorphism and homomorphism,the boundarycondtion with nonlinear function and the nonlinear term f may change sign.We use the nonlinear alternative of Leray-Schauder type and the fixed pointindex theory to obtain the existence of solution and positive solution for bound-ary value problem (1.1.1). Finally, an example is given to demonstrate the ap-plication of our main result. The corresponding theory for impulsive boundaryvalue problem with an increasing homeomorphism and homomorphism on thehalf-line is intvestigated less now. In this chapter, we combine the increasinghomeomorphism and homomorphism with the impulsive diferential equations,the nonlinear term f may change sign and the boundary condtions with nonlinearfunction.We generalize and improve the results of[1]-[4](compare with the pas-sage [1],this chapter investigate the boundary value problems with an increasinghomeomorphism and homomorphism of impulsive diferential equations; comparewith the passage [2][3][4],the nonlinear term f of this chapter may change sign).In Chapter2, we talk about the existence of many positive solutions forboundary value problem with integral boundary conditions of impulsive diferen-tial equationswhere φ is an increasing homeomorphism and homomorphism, the nonlinearterm f may change sign.We use the fixed point index theory to obtain the existence of many positivesolutions for boundary value problem (2.1.1) with sign-changing nonlinearity,andgive an example. Because of the usage of the concavity of the function in thepast articles,we suppose that the nonlinear term f take nonnegative, which bringssome limitations to the practical problems. To solve the difcult,the nonlinearterm f may change sign in this chapter.Moreover, the boundary value prob-lem with integral boundary conditions is investigated in this chapter,it includetwo,three,multipoint and nonlocal boundary value problems as special cases.In Chapter3, we study the existence of positive soluion of a multi-point boundary problem for a class of2nth-order nonlinear singular semipositone dif-ferential equationsThis chapter generalize and improve the results of[13]-[17](compare with thepassage[13]and[16],the nonlinear term f of this chapter is semipositone; comparewith the passage[15][16][17], the are involved in the non-linear term f in this chapter).We use the fixed point index theory to obtain theexistence of positive soluion of the boundary value problem(3.1.1).