Positive Solutions For A Third-order Three-point BVP With Sign-changing Green’s Function

Recently, third-order boundary value problems (BVPs for short) have attracteda lot of attention for their wide practical background．Especially, the theoreticalstudy of positive solutions of third-order three-point BVP has developed rapidly inthe last30years because of its important practical signifcance, but most of the workare achieved when the corresponding Green’s functions are nonnegative. There havebeen very few researches on the existence of positive solutions to the BVPs whenthe Green’s functions are sign-changing, which provides a broad space for the studyof positive solutions to the BVPs with sign-changing Green’s functions.Therefore, in chapter2, we discuss the existence of positive solution of third-order three-point BVP with sign-changing Green’s function. Although the corre-sponding Green’s function is sign-changing, the existence of positive solution is stillobtained under some suitable conditions on nonlinear term. The tool used is theGuo-Krasnoselskii fxed point theorem.In chapter3, by using the Leggett-Williams fxed point theorem, we continueto study the BVP in chapter2and obtain the existence of three positive solutionsto the problem. Furthermore, for arbitrary positive integer, existence results forat least21positive solutions are obtained.In chapter4, by applying monotone iteration method, we not only obtain theexistence of monotone positive solution to the BVP with sign-changing Green’sfunction, but also the iterative sequence of monotone positive solution is given. Itis worth mentioning that the iterative scheme starts of with zero function, which isvery useful and feasible for computational purpose. Finally, we give an example toillustrate the main results.