Positive Solutions For High Order P-laplacian Differential Equations Boundary Value Problems

The impulsive differential equation is an important branch of ordinarydifferential equation. It shows the phenomenon of things which distinguished atcertain instants that their state is subjected to a sudden change. This kind ofdifferential equations arises concrete mathematical modelling, such as theoreticalphysics, cybernetics, population dynamics, biotechnologies, industrial robotics,economics, etc. Their theories can truly reflect the phenomenon of the objectiveworld. Many researchers show great interest in impulsive differential equations. Nowimpulsive differential equations constitute one of most active domains of nonlinearfunctional analysis at present.The theory of boundary value problems with integral boundary conditions forordinary differential equations is from various physical and applied-mathematicsfields. We note that many problems in heat conduction, chemical engineering,underground water flow, theory of elasticity, plasma physics can be reduced to thenonlocal problems with integral boundary conditions. The existence and multiplicityof positive solutions is one of the most important thesis of such problems. Recently,by means of the cone expansion and compression theory, the fixed point indextheory as well as the topological degree theory, many authors had obtained theexistence of positive solutions to several kinds of boundary value problems fordifferential equations with integral boundary condition. However, when thep-Laplacian operator is absent, to my best knowledge, there are few worksconsidering the relative results.In this paper, by using the fixed point index theory and cone theory, we mainlyconsider the existence and multiple results for positive solutions of the boundaryvalue problems for higher-order p-Laplacian and impulsive differential equations, andseveral corresponding applications are also given. The problems we discuss are moreprofound, so we can generalize and extend our resluts of previous papers to somedegree.The paper is divided into four chapters. In chapter1, the historical background and current development of boundary value problems for ordinary differentialequations is introduced. Besides, the topic of this paper is generally illustrated. Inchapter2, we give the related concepts, theorem and symbols for application in thesequel. In chapter3, based on the fixed point index and some theories in cone, westudy the existence of positive solutions for the fourth-order p-Laplacian problemwith integral boundary conditions. Under the different linearity, several existenceresults are established. In chapter4, the main results are about the existence andmultiplicity of positive solutions for higher-order impulsive differential equation withp-Laplacian operator and integral boundary conditions. Finally, an example is alsoincluded to illustrate our main results.