关键词: 特征值, 一致正线性算子, 变号解, 不动点指数

Abstract: The existence and multisolvability of the sign-inversing solution to nonlinear operator equations are discussed, based on the cone theory and fixed point theory in Banach space. An existence theorem of sign-inversing solution is proved by an upper bound solution, further, the four solutions, i.e. positive, negative, null and sign-inversing solutions, are obtained through an upper bound solution and a lower solution. Then, the existence of both the sign-inversing solution and the four solutions are discussed in detail for a kind of important nonlinear operator equations, i.e. the Sturm-Liouville two-point boundary value problems and, correspondingly the theorems of the existence of both the solutions as above are proved. An example is given to illustrate their applications.