Research On Two Kinds Of Nonlocal Boundary Value Problem For Ordinary Diferential Equation

The boundary value problems for ordinary diferential equations are very important.With the development of science and technology, all kinds of natural problems in thefields of engineering, mechanics, astronomy, economics, cybernetics, biology etc can bedescribed into the ordinary diferential equations boundary value problems. As we allknow, it is quite difcult to find out the solutions of the ordinary diferential equations. Sothe theoretical study of the existence of the solutions and their characters from the theoryattract a great attention. With the development of the ordinary diferential equations,the study of the multi-point boundary value problems become increasingly active.The ordinary diferential equations with nonlocal boundary value problems meanthat the boundary conditions not only depend on the solution in the interval points, butalso depend on the solution in the range of interval.For the reason, it can describe many important physics phenomenons more clearly;on the other hand, it can consider many discrete multi-point boundary value problemsinto the continuous interval. Therefore, it has important theoretical significance andapplication value.The research of the ordinary diferential equations with nonlocal boundary condi-tions is becoming an important subject. For example, heat conduction, chemical en-gineering, thermo-elasticity, and blood flow can be reduced to the ordinary diferentialequations with nonlocal boundary conditions. Because the boundary value problem withintegral boundary conditions has widespread application background, they have impor-tant research value. We refer the readers to [19-24] and the references therein. Withsolving these problems, most methods are applied. This paper makes use of Rabinowitz’sglobal bifurcation theorem to get the existence of multiply solutions.Feng, Ji and Ge [2] consider the following BVPwhere f∈C[R, R], a(s)∈L[0, 1] is nonnegative with120a(s)ds < 1. We obtain theexistence of 2 solutions by using fixed point theory under some suitable conditions. Re-cently, utilizing the fixed point index theory and computing eigenvalues and algebraic multiplicity of the corresponding linear operator, Zhang and Sun [1] obtain the existenceof 8 diferent nontrivial solutions when f satisfies certain conditions for the followingintegral boundary value problem0a(s)x(s)ds,where f∈C[R, R], a(s)∈L[0, 1] is nonnegative with10a2(s)ds < 1.The existence of the solutions for the integral boundary value problem has beenstudied and the methods usually used are some fixed point theorems. However, to ourknowledge, there are few papers considering the existence of solutions for integral bound-ary value problem by using bifurcation theorem.Motivated by these, in this paper, by using Rabinowitz’s global bifurcation theorem,we obtain the existence of at least 2n nontrivial solutions under some suitable conditions.The dissertation contains two chapters.Chapter 1. Consider the existence of multiple solutions for nonlinear second-orderordinary diferential equation with integral boundary value condition is nonnegative with10a2(s)ds < 1. By using Ra-binowitz’s global bifurcation theorem, we obtain the existence of at least 2n nontrivialsolutions for BVP (0.1) under some suitable conditions. Six concrete examples are givento illustrate the application of the main results.Chapter 2. Consider the existence of multiple solutions for nonlinear fourth-orderordinary diferential equation with integral boundary value conditionwhere f :R×Râ†'Ris a continuous function, a(s)∈L[0, 1] is nonnegative with10a2(s)ds < 1. By using Rabinowitz’s global bifurcation theorem, we obtain the exis-tence of at least 2n nontrivial solutions for BVP (0.2) under some suitable conditions.