Exact Number Of Solutions For Two Classes Of Boundary Value Problems Involving Nonlinear Differential Operators

In this thesis, by using Time-map method, we study the existence and multi-plicity of solutions of two classes of differential boundary value problems involving one-dimensional p-Laplacian operator and one-dimensional mean curvature opera-tor, respectively. The main results are described as follows.1. By using Time-map method, we discuss the Dirichlet boundary value problem involving one-dimensional p-Laplacian operator where p∈(1,2], A>0,f∈C1((-r,0)∪(0,r),R) for some constant r>0, f(s)s>0,s≠0. We show the existence of sign-changing solutions under the assumptions Further, we also show that the problem has exactly one solution having specified nodal properties for λ∈(0,λ*) for some λ∈G (0,∞). The main results extend and improve the corresponding ones of Ma Ruyun et al [Abs. Appl. Anal., Article ID492026,6pages,2013],2. When the nonlinearity f satisfies f(0)<0, by using Time-map method, we consider the Sturm-Liouville boundary value problem involving one-dimensional p-Laplacian operatorwhere p∈(1,2],λ>0, α≥0,β≥0,f: R→R is continuous. We obtain that there exists λ*G (0,∞) such that problem has exactly one solution having specified nodal properties for λ∈(0, λ*). The main results extend and improve the corre-sponding ones of A. Castro, R. Shivaji [Proc. Roy. Soc. Edinburgh Sect.,1988], V. Anuradha, R. Shivaji [Results Math.,1992], V. Anuradha, R. Shivaji [Nonlinear Anal.,1994] and A. Lakmeche, A. Hammoudi [J. Math. Anal. Appl.,2006],3. By using Time-map method, we consider the existence of positive solution for one-dimensional prescribed mean curvature equation where λ>0is a parameter,f:[0,∞)→[0,∞) is continuous. Further, when f sat-isfies max{up,uq}≤f(u)≤up+uq,0<p≤q<+∞, we obtain the exact number of positive solution. The main results in some sense extend and improve the corre-sponding ones of P. Habets, P. Omari [Commun. Contemp. Math.,2007] and W. Li, Z. Liu [J. Math. Anal. Appl.,2010].4. By using Time-map method, we study the existence of positive solution of one-dimensional prescribed mean curvature problem with singular nonlinearitywhere λ>0, L>0are parameters,0<p≤q<+∞. We obtain the exact number of positive solutions. The main results in some sense extend and improve the corresponding ones of N. D. Brubaker, J. A. Pelesko [Nonlinear Anal.,2012], H. Pan, R. Xing [Nonlinear Anal. RWA,2012] and Y. H. Cheng, K. C. Hung and S. H. Wang [Nonlinear Anal.,2013],...