Positive Solutions And Sign-changing Solutions For Several Classes Of Problems With Integral Boundary Conditions

Differential equations with integral boundary value conditions applied math-ematics and physics in many areas are important and widely studied, such as, thermodynamic conditions, chemical energy, groundwater flow, bomb theorem and plasma flow(refer to[l-12],[21-27]). So far, the research method of integral boundary value problems is usually using the fixed point theorem. However, to the best of our knowledge, there are few articles based on the bifurcation theory, considering the existence of integral boundary value. This paper mainly using the fixed point index theorem, Leray-Schauder degree, Krein- Rutamn theory, bifurcation theory, topological degree theorem, we get the existence of sign-changing solutions and positive solutions of several classes of integral boundary value problems.Chapter 1 investigates the following two order ordinary differential equationwhere f∈C[R,R],α(s)∈ L[0,1] is non-negative, and ∫10 α2(s)ds<1/4. Under given conditions,by using the fixed point index theorem, Leray-Schauder degree, we obtained the problem exsits at least two sign-changing solutions, two positive solutions, two negative solution, and when f is an odd function,the problem exists at least eight different nontrivial solutions, including four sign-changing solutions, two positive solutions, two negative solutions.In chapter 2, we consider the nonlinear four-order ordinary differential equa-tions with integral boundary value conditionswhere f :R×Râ†'R is a continuous function,I= [0,1]. α(s) ∈L[0,1] is non-negative, and (?)10 α2(s)ds< 1. Under given conditions, using bifurcation theory,we obtained the problem exists at least 2k nontrivial solutions.In chapter 3, we study the following fractional differential equationwhere 2< α≤3,0< k< 2, cDα is caputo fractional derivative. The f:[0,1] x [0,+∞)â†'[0,+∞) is a continuous given function, satisfying some hy-pothesis conditions, η> 0 is a given constant. Under the given conditions and the application of bifurcation theory, we get the problem exsits at least two positive solutions.