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

Along with science's and technology's development, various non-linear problem has aroused people's widespread interest day by day, and so the nonlinear analysis has become one important research directions 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 value problem of nonlinear differ-ential equation stems from the applied mathematics, the physics, the cybernetics and each kind of application discipline. It is one of most active domains of functional anal-ysis at present. The singular and integral boundary value problems are the hot spot which have been discussed in recent years, and become two very important domains of differential equation research at present. In this paper using the cone theory, the fixed point theory as well lower and upper solutions, we discuss several kinds of sin-gular and integral 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 for the singular and integral differential equations.The thesis is divided into four sections according to contents.Chapter 1 Preference, we introduce the main contents of this paper.Chapter 2 We use the fixed point theory. Nagumo condition and upper and lower solution to investigate the following second-order integral boundary value problem where f∈([0, 1]×R2,R),ξ(s),η(3)∈C1[0,1] are nondecreasing on [0,1], and/u(s)dη(s) denote the Riemann-Stieltjes integrals of u with respect toξandη, respectively,α,γ> 0,β,δ> 0. We obtain the existence and uniqueness of solution for nonlinear Sturm-Liouville integral boundary value problem (2.1.1), and an example is given to demonstrate the application of our main results. Chapter 3 We will study the following fourth-order nonlinear singular boundary value problems where f∈C((0,1)×(0,+∞)×(0,+∞), (0,+∞)), and f(t,u,v) may be singular at t=0,t=1, u=0 and v=0,λis a positive parameter andα∈[π2/4,π2). By constructing a special cone and using the well-known fixed point theorem, we can get the existence and uniqueness of positive solutions for the singular boundary value problems. The dependence of positive solutions on the parameterλis also studied.Chapter 4 By employing the fixed point theory, we study the existence of positive solutions for the fourth-order integral boundary value problems where a(t)∈C[0,1],α<π2,f∈C([0,1]×[0,+∞) x (-∞,0], [0,+∞)), p, q∈L[0,1], p(s), q(s)≥0, We generalize the results in [21].