三阶周期边值问题的正解

Positive Solution to the Third-order Periodicboundary Value Problem

由于一般的线性周期边值问题的解可以由相应的齐次线性周期边值问题的解表示出来,则只需要求得该齐次线性周期边值问题的解即可。通过求得该齐次线性周期边值问题中微分方程的特征根再联立其周期边界条件就可以得到该齐次线性周期边值问题的唯一解。然后在满足一定的条件下使得该唯一解大于零。再定义一个算子A和锥K,则周期边值问题的解等价于A的非零不动点,运用锥上的不动点指数理论可以得到三阶周期边值问题u-(a+2b)u″+(b 2+c 2+2ab)u′-a(b 2+c 2)u=f(t,u),t∈[0,2π]u(i)(0)=u(i)(2π),i=0,1,2正解的存在性,其中:f∈C([0,2π]×[0,+∞),[0,+∞));a,b,c∈R且满足a<0,b≥920,-625≤c<0。

Since the solution to thegeneral linear periodic boundary value problem can be expressed by the corresponding solution to the homogeneous linear periodic boundary value problem,it’s applicable to only obtain the solution to the latter problem.The above-mentioned only solution can be obtained by gaining the characteristic root of the differential equation in this problem and then attaching it to its periodic boundary condition.Certain requirements are met to make sure the only solution greater than zero.Another operator A and cone K are defined,and then the solution to the periodic boundary value problem is equivalent to the non-zero fixed point of A.The fixed point index theory on the cone can be used to obtain the existence of the third-order periodic boundary value problem,u -(a+2b)u″+(b2+c2+2ab)u′-a(b2+c2)u=f(t,u),t∈[0,2π]u(i)(0)=u(i)(2π),i=0,1,2where,f∈C([0,2π]×[0,+∞),[0,+∞));a,b,c∈R,and a<0,b≥920,-625≤c<0.