带有参数的三阶非线性差分系统正解的存在性

Existence of positive solution to third-order nonlinear difference system with parameter

研究三阶差分系统边值问题Δ3ui(k)+λhi(k)fi(u1(k),u2(k),…,un(k))=0,k∈[0,T],ui(0)=ui(1)=ui(T+3)=0,i=1,2,…,n.若令f0=sum from i=1 to n lim‖u‖→0 fi(u)/‖u‖且f∞=sum from i=1 to n lim‖u‖→∞ fi(u)/‖u‖,则在f0=0且f∞=∞,或者f0=∞且f∞=0的情况下,运用不动点指数理论证明对于所有的λ>0,上述系统存在一个正解.

The boundary-value problem of third-order difference system △^3ui（k）＋λhi（k）fi（u1（k）,u2（k）,…,un（k））=0,k∈[0,T],ui（0）=ui（1）=ui（T＋3）=0,i=1,2,…,n was studied.Let f^0=∑i=1↑nlim｜｜u｜｜→0fi（u）/｜｜u｜｜ and f^∞=∑i=1↑nlim｜｜u｜｜→∞fi（u）/｜｜u｜｜ Then, by using the theory of fixed point index, it was verified that for all λ〉0, the above system would possess a positive solution if f^0=0 and f^∞=∞,or f^0=∞ and f^∞=0