Properties Of Connected Component Of Solutions Set Of Nonlinear Problems

It is well known that many problems in the fields of physics,chemistry,biology,geography and economics can be described by nonlinear differential equations with parameters.When a parameter exceeds a critical value,the problem will change from the original state to a new state,which is a common characteristic of these problems.It is called the phenomenon of bifurcation.Because bifurcation problems have important theoretical significance and a wide range of applications in mathematics and other subjects,the bifurcation theory and its applications have always been an important part of mathematics research.Three basic problems of bifurcation theory are the local bifurcation problems,the global structure problems and the application problems.The partial bifurcation theorem of Krasnosel’skii has been studied for the first time in infinite dimensional space by the theory of topology degree for the bifurcation problems,a sufficient condition for the bifurcation points is established.P.H.Rabinowitz used the topological degree methods to discuss several possibilities for the connected components diverging from the eigenvalues of odd numbers,the global bifurcation theorem of Rabinowitz was established.Under the conditions of the univalent eigenvalues and the appropriate smoothness of the equations,the partial bifurcation theorem of the Crandall-Rabinowitz is proved that the set connected component is a smooth curve near the bifurcation point.In order to depict the direction of the connected components near the bifurcation points more meticulously,a natural problem is how to solve the set of unbounded connected components from the points of bifurcation or asymptotically to the bifurcation points or the asymptotic bifurcation points.That is to say,the connected component of solutions set is supercritical(from the right side of the bifurcation point or the asymptotic bifurcation point)or subcritical(from the left side of the bifurcation point or the asymptotic bifurcation point)or infinitely from one side to the other side oscillating towards the bifurcation point or the asymptotic bifurcation point.In this paper,we mainly study the asymptotic oscillation of the connected components.In this paper,we use the bifurcation theory,especially the Rabinowitz global structure theorem,Leray-Schauder degree method and detailed analysis.Under the condition of nonlinear term oscillation,we prove that the positive solution set of a nonlocal boundary value problem is derived from the existence of infinite connected components.The results show that this connected component infinitely oscillates around a parameter interval to infinity.To obtain such a result,we need to overcome the difficulties caused by the non-differentiability of the nonlinear term and establish an existence result of the solution set of the non-differentiable abstract operator equation from an infinitely connected components.We also need to analyze the properties of the positive solution of the nonlocal problem and establish an important equation for the parameter correlation.Using the main results of this paper,we can get the result of the existence of infinite solutions of the nonlocal boundary value problem when the parameter takes some values.