Existence And Uniqueness Of Solutions For Boundary Value Problems Of Nonlinear Differential Equations

Along with science’s and technology’s development, various non-linear problemhas aroused people’s widespread interest day by day, and so the nonlinear analysishas become one important research directions in modern mathematics. The nonlinearfunctional analysis is an important branch in nonlinear analysis. The boundary valueproblem of nonlinear diferential equation stems from the applied mathematics, thephysics, the cybernetics and each kind of application discipline is one of most activedomains of functional analysis at present. The singular and integral boundary valueproblems are the hot spot which have been discussed in recent years, and becometwo very important domains of diferential equation research at present. In this paperusing the fixed point index theorem, Leray-Schauder degree theory, the fixed pointtheorem as well lower and upper solutions, we discuss several kinds of singular andintegral problems and give some conditions of the existence and uniqueness of solutions.Meanwhile we apply the main results to the existence and uniqueness of solutions forthe singular and integral diferential equations.The thesis is divided into four sections according to contents.Chapter1Preference, we introduce the main contents of this paper.Chapter2We use he fixed point index theorem to investigate the following2p-order and2q-order systems of singular semipositone boundary value problems withintegral boundary conditions where are continuous;are nonnegative,Some new results on the existence of C2p-2[0,1]×C2q-2[0,1] positive solutions and C2p-1[0,1]×C2q-1[0,1] positive solutions for this class of differential equations are derived, and an example is given to demonstrate the application of our main results.Chapter3We will study the following fourth-order p-Laplacian differential equa-tions with integral boundary conditions and a sign-changing nonlinear term and where f:[0,1]×R4→R are continuous;(?) is an increasing homeomorphism with (?)(0)=0,(?)(R)=R; hi:R3→R,gi:R→R (i=1,2)are continuous; ki>0(i1,2);R=(-∞,+∞). In this chapter, by the method of upper and lower solutions and Leray-Schauder degree theory, we investigate the existence and uniqueness of a solutions for (3.1.1) and (3.1.2).Chapter4In this chapter, using the methods of lower and upper solution and the fixed point theorem, we study the existence of positive solutions for the following fourth order singular m-point boundary value problems where are constants,[0,∞)) and f(t,μ) is decreasing on u. In this chapter, a sufficient condition for the existence of C2[0,1] as well as C3[0,1] positive solutions is given by constructing lower and upper solutions and with the comparison theorem.