非线性算子方程变号解的存在性与多解性及其应用

Existence and Multisolvability of Sign-Inversing Solution to Nonlinear Operator Equations and Their Applications

利用Banach空间中的锥理论和不动点理论讨论了非线性算子方程变号解的存在性和多解性,通过一个上解给出了非线性算子方程变号解的存在性定理,进而又在一个上解和一个下解的条件下得到了四解存在定理,同时还针对一种重要的非线性算子方程即一类Sturm-Liouville两点边值问题,具体讨论了其变号解的存在性及四解的存在性,相应得到了变号解存在定理和四解存在定理.最后通过一个具体的例子给出定理的应用.

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.