| In this thesis, by using the fixed point index theory, the existence of connected sets of solutions to parameterized families of equations and lower and upper solutions method, we study the existence and multiplicity of solutions to several classes of boundary value problems for nonlinear first-order differential equations. The main results are as follows:1. We study the existence, multiplicity and nonexistence of positive ω-periodic solutions for the first-order differential equation with delay by applying the fixed point index theory and lower and upper solutions method, where λ>0is a parameter; a∈C(R,[0,∞)), b∈C(R,(0,∞)), τ∈C(R, R) are ω-periodic functions and∫0ω a(t)dt>0; f, g∈C([0,∞),[0,∞)). Under some certain growth conditions on the nonlinearity∫, the existence, multiplicity and nonexistence of positive periodic solutions are established in terms of different value of parameters λ. These results extend and improve the corresponding ones of G. Zhang, S. Cheng [Appl. Anal.,2002] and obtain more precise parameter range than the corresponding results of Y. Wu [Nonlinear Anal.,2008]. In the proof of our main results, delay plays a very important role, but no impact on the proof of the corresponding results in G. Zhang, S. Cheng [Appl. Anal.,2002] and Y. Wu [Nonlinear Anal.,2008].2. Existence and multiplicity of solutions for the first-order multi-pint boundary value problem at resonance are established, by using the existence of connected sets of solutions to parameterized families of equations, where f:J×R→Ris a continuous function, e∈C(J,R),0<η1<η2<…<ηm<1and ak>0(k=1,2,…, m) are constants. We firstly develop the method of lower and upper solutions for the first-order boundary value problem (P2), and then by discussing the cases that the lower and upper solutions are well or opposite order, we establish the existence and multiplicity of solutions for (P2). |
