| Nonlinear boundary value problem originates from various of parts of appliedmathematics and physics. It is one of the most active fields that are studied in nonlinear functional analysis. The systems of differential equations are important parts in differential equations. The structures they present have the profound significance of physical backgrounds and mathematical models. So, the study of differential equation and furthermore the systems of differential equations has profoundly intrinsic value. Nonlinear boundary value problems of (systems of) differential equations are the combination of the two. They are both new and vital branches. They play important parts in applied mathematicsand engineering, especially in the pneumatics and the biochemistry aspacts. Therefore, it becomes of great importance to study the existence and multiple solutions of them. Furthermore, the study of the nature of the solutions is also important.The present paper employs the nonlinear functional analysis methods such as the cone theory, fixed point theory and so on, to investigate the existence of solutions of boundary value problem of nonlinear (systems of) differential equations. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into four sections according to contents.Chapter 1 Preference, we introduce the main contents of this paper.Chapter 2 We consider the following singular boundary value problems for 2n-order differential equations where n≥2, the nonlinear items f: (0,1)×[0,+∞)→[0,+∞),and canbe singular at t = 0,1. By applying the value of the corresponding Green function and the theory of fixed point index and constructing a special cone, this chapter gives the existence of two positive solutions and moreover a third one for (2.1.1).Chapter 3 We study the following boundary value problems for systems of nonlinear differential equationsthe nonlinear items f∈C([0,1]×[0,∞),[0,∞)),g∈C([0,1]×[0,∞),[0,∞)),and g(t,0)≡0. By constructing a special cone and using the method offixed point theorems of cone expansion and cone compression of norm type in different intervals, this chapter gives the existence and multiplicity of (3.1.1).Chapter 4 At the basis of the former two chapters, in this section we consider the higher order systems of differential equations of boundary value problems. And discussing the following 2n-order differential equationswhere n≥2;k=0,1,2,…,n-1, the nonlinear items f: (0,1)×[0,∞)→[0,∞),g:(0,1)×[0,∞)→[0,∞),and f, g can be singular at t = 0,1. By usingthe fixed point theorems of cone expansion and cone compression of norm type in different intervals, this chapter gives the existence and multiplicity of (4.1.1). |
