Existence Of Solutions For Several Kinds Of Nonlinear Differential Equations

With the development of science technology and applied mathematics,nonlinear functional analysis is paid much attention as an important part of applied mathematics for a long time.Many scholars study nonlinear functional analysis.Nonlinear di?erential equations is an important direction of nonlinear functional analysis.It can be used for many kinds of natural phenomenons,so it is valued in mathematics and natural science.Nonlinear di?erential equations stem from the applied mathematics,the physics, the chemistry, the cybernetics and each kind of application discipline. It is one of the most popular domains of nonlinear functional analysis at present.Impulsive di?erential equations and fractional di?erential equations are the hot spots which have been discussed in recent years, and become two important domains of di?erential equations at present.In this paper using the Avery-Peterson ?xed point theory, upper and lower solutions as well as the variational method, we study several kinds of nonlinear di?erential equations and give some conditions of the existence of solutions. Meanwhile we apply the main results to the existence of solutions for the nonlinear di?erential 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 Avery-Peterson ?xed point theory and discuss the following existence of positive solutions to third-order impulsive di?erential equations with advanced arguments and nonlocal boundaryconditions denote the right and left limits of x′at tk,where λ denotes a linear functional on C(J) given byλ[x] =∫10x(t)dΛ(t) involving a Stieltjes integral with a suitable functionΛ of bounded variation. λ[x] is positive to all positive x. The measure dΛ can be a signed measure.We obtain the multiple existence of solution for third-order impulsive di?erential equations and an example is given to demonstrate the application of our main results.Chapter 3 We will study the following the eigenvalue for a class of singular p-Laplacian fractional di?erential equations involving the Riemann-Stieltjes integral boundary condition where Dβt, Dαtare standard Riemann-Liouville derivatives with 1 < α ≤2, 0 < β ≤ 1. A is a function of bounded variation and∫10x(s)d A(s)denotes the Riemann-Stieltjes integral of x with respect to A, the pLaplacian operator φpis de?ned as φp(s) = |s|p-2s, p > 1, f(t, x, x) :(0, 1) ×(0, +∞) ×(0, +∞) â†'(0, +∞) is continuous and may be singular at t = 0, 1; x = 0.We can get the eigenvalue for the existence of solutions for the singular p-Laplacian fractional di?erential equations involving the Riemann-Stieltjes integral boundary condition. An example is given to demonstrate the application of our main results.Chapter 4 By employing the variational method, we study the existence of sign-changing solution for a fourth-order asymptotically linear elliptic problemwhere ?2:= ?(?) denotes the biharmonic operator, ? ? ?Nis a bounded domain with smooth boundary, and c < λ1(λ1is the ?rst eigenvalue of-?in H10(?)) is a parameter.We can receive a positive solution,a negative solution and a sign-changing solution.