The Bifurcation Curves Of Positive Solutions Of Boundary Value Problem With Convex-Concave-Convex Nonlinearities

In the thesis,we discuss the following the multiplicity of the positive solutions of the following boundary value problem (?)when ?> 0 is a parameter,the nonlinear term f(u)is convex-concave-convex and positive in [0,?).By using time-map techniques,we discuss the bifurcation curves when the nonlinear term f(u)it is asymptotic superlinear,asymptotic sublinear or asymptotic linear.The thesis consists of four chapters.In Chapter 1,we briefly introduce the historical background and the recently development of boundary value problem and the main content of the thesis.In Chapter 2,firstly we prove that the positive solutions of the bifurcation curve is monotone increasing if f(u)is asymptotic sublinear or asymptotic linear and satisfy some suitable conditions.Secondly we prove that the positive solutions of the bifurcation curve is ?-shaped if f(u)is asymptotic superlinear or asymptotic linear and satisfy other some suitable conditions.Lastly we give some concrete examples to show the application of the results of this chapter.In Chapter 3,firstly we prove that the positive solutions of the bifurcation curve is S-shaped if f(u)is asymptotic sublinear or asymptotic linear and satisfy some suitable conditions.Secondly we prove that the positive solutions of the bifurcation curve is ?-liked shaped if f(u)is asymptotic superlinear or asymptotic linear and satisfy other some suitable conditions.Lastly we also give some concrete examples to show the application of the results of this chapter.Chapter 4 is the conclusion of the thesis and the following problems we are going to solve.