The Study Of Existence Of Solutions For Two Classes Of Fourth-order Two-point Nonlinear Differential Equations Under Sturm-Liouville Condition

Our work in this paper is mainly on two classes of nonlinear fourth-order boundary value problems under Sturm-Liouville condition. On the hypothesis of weaker monotonicity or singular property to the nonlinear functions, we respectively obtain the existence and uniqueness and even the multiplicity results for the fourth-order differential boundary value problems.In Chapter 1, we assume the nonlinear function f is a Caratheodory function, and satisfies weaker monotonicity, then with the weak maximum principle, we get the existence and uniqueness result for the first kind of fourth-order BVP. In Chapter 2, our attention is mainly paid to another Sturm-Liouville fourth-order differential BVP with a nonnegative parameter A, under the hypothesis of the singularity to the linearity and nonlinearity, we arrive at the existence results of at least one and two C_[0,1]³ positive solutions applying the approximation theory of operators and the fixed point index theory. Meanwhile, we provide the exact interval for the parameter A either. Later, we go ahead with making further investigation on the second singular BVP, here, we construct special convex and concave functional, and make use of the Leggett—Williams fixed point theory attaining its at least three and even multiple solutions. Similarly, we also acquire the range for the parameter A.Worth mentioning, the existence results we gain in our paper is under the Sturm-Liouville boundary condition, which can be precisely transformed into much simpler boundary value problems as attaching to constants asα,β,γandδsome particular values . Correspondingly, we can see our research with much more generality.