On The Existence Of Multiple Solutions Of Boundary Value Problems (for Systems) Of Differential Equations

The systems of differential equations are very important parts in differential equa-tions. Nonlinear boundary value problem often in applied mathematics and physics. It is one of the most active fields that are studied in nonlinear functional analysis. So, the study of differential equation and furthermore the systems of differential equations has profoundly intrinsic value. Nonlinear multiple-point boundary value problems of (systems of) differential equations are both new and vital branches. They play very important parts in applied mathematics and engineering, especially in the pneumatics and the biochemistry aspacts. So, it take the more important parts in studying the existence and multiple solutions of them. Furthermore, Studying the nature of the solutions is also takes an very important part.The present paper employs the nonlinear functional analysis methods such as the cone theory,Leggett-Williams'fixed point theory Krasnosels'kii's fixed points theorem and so on, to investigate the existence of countable solutions of multipoint 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 some 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 n-points boundary value problems for p-Laplacian differential equations on infinite intervalsWhereφp(s)=|s|p-2s, p> 1,(φp)-1=φq,1/p+1/q=1, with 0<ξ1<ξ2<…<ξn< 1, and h:R+→R+, f(t, u, v):R+3→R+is continuous function, R+=[0,+∞),αi≥0. and can be singular at t=0,1. By applying the value of the theory of fixed point index, constructing a special cone, Leggett-Williams'fixed point theory and this chapter gives the existence of multiple positive solutions for (2.1.1).Chapter 3 We study the following boundary value problems for systems of non-linear differential equationsWhereφp(s)=|s|,p-2s,p> 1, (φP)-1=<φq,1/p+1/q=1,ξi∈(0,1) with 0<ξ1<ξ2<…<ξn<1.By constructing a special cone and using the method of fixed 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 In this section we consider the differential equations of boundary value problems depend on a parameterWhereλ> 0 is a parameter, (β> 0,0<η< 1,0<αη< 1,△:= (1-αη)+β(1-α)> 0, a∈C((0,1), (0,+∞)), a(t) may be singular at t= 0 and or t= 1. f∈C([0,1]×(0,+∞), (0,+∞)). By applying the value of the corresponding Green function and using the 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).