Existence Of Solutions And Multiple Positive Solutions For Nonlinear Boundary Value Problems

In recent years, the boundary value problem of nonlinear differential equations has become an important branch in the domain of differential equations. Besides, it has also comprehensively applied background and important meaning of theory-oriented in some research fields, such as physics,astronomy,biology and sociology. The research concerning the very problem has already began in the Sturm-Liouville's period before 100 years. Upon to now, the depth,span,research ways and tools of researching on the problem have made great strides. This thesis consists of three chapters, which study the existence of solutions of several classes of nonlinear boundary value problems. The results given in this thesis extend and improve the corresponding ones in the literatures.In Chapter 1, we consider the periodic boundary value problem for first order ordinary differential equationswhere B(t) = diag[b₁(t),b₂(t),…,b_n(t)], f∈C([0,1]×Rⁿ,Rⁿ),b_i∈C([0,1],R) ,b_i(t)≠0, t∈[0,1], i = 1,2,…, n. By using Schaefer's fixed point theorem a sufficient condition for existence of solution of the problem is obtained, which extend the corresponding results in [J Math Anal Appl, 2006, 323: 1325-1332].In Chapter 2, we consider the PBVP for first order impulsive differential equationswhere b∈C(J, R) with b(t)≠0 for t∈J, J = [0, T], 0 = t₀ < t₁ < t₂ <…< t_p < t_p+1 = T, I_k∈C(Rⁿ,Rⁿ), k = 1,…,p , and f : J×Rⁿ→Rⁿ is a L¹-Caratheodory function,f(t_k⁺,u)and f(t_k^-,u) exist, f(t_k^-,u)= f(t_k,u). By using Schaefer's fixed point theorem asufficient condition for existence of solution of the problem is obtained, which extends the corresponding results in [J Math Anal Appl, 2007, 331: 902-912], [J Math Anal Appl, 2007, 325: 226-236] and so on.In Chapter 3, by using fixed point theorem of Krasonselskii and Leggett-Williams theorem, we study the three point boundary value problem of second order nonlinear differential equationswhere 0<η<1,αis a positive constant, p, q, f satisfy:The results obtained improve and extend the corresponding results of [J Math Anal Appl, 2003, 279: 216-227], [Appl Math Comput, 2006, 182: 258-268] and so on.