桥梁工程中高阶常微分方程组广义Lidstone问题的正解

The positive solution of generalized Lidstone problem of higher order ordinary differential equations in bridge engineerings

研究桥梁工程中出现的如下高阶常微分方程组广义Lidstone边值问题正解的存在性{(-1)^mu^(2m)=f1(t,u,-u″,…,(-1)^m-1u^(2m-2),v,-v″,…,(-1)^n-1v^(2n-2),(-1)^nv^2n)=f2(t,u,-u″,…,(-1)^m-1u^(2m-2),v,-v″,…,(-1)^n-1v^(2n-2),αiu^(2i)(0)-βiu^(2i+1)(0)=γiu^(2i)(1)+δiu^(2i+1)(1),(i=0,1,…,m-1),αjv^(2j)(0)-βjv^(2j+1)(0)=γjv^(2j)(1)+δjv^(2j+1)(1),(j=0,1,…,n-1).其中f1,f2∈C([0,1]×R^m+n+,R+)(R+=[0,∞)),所研究的方程组中两个方程可以有不同的阶数,且各阶导数满足不同的边界条件.在先验估计的基础上,利用不动点指数理论证明了以上边值问题正解的存在性.该研究在桥梁等领域都具有重要的实用价值和研究意义.

This paper mainly deals with the existence of positive solutions for the following generalized Lidstone boundary value problem of higher order ordinary differential equations in bridge engineerings{(-1)^mu^(2m)=f1(t,u,-u″,…,(-1)^m-1u^(2m-2),v,-v″,…,(-1)^n-1v^(2n-2),(-1)^nv^2n)=f2(t,u,-u″,…,(-1)^m-1u^(2m-2),v,-v″,…,(-1)^n-1v^(2n-2),αiu^(2i)(0)-βiu^(2i+1)(0)=γiu^(2i)(1)+δiu^(2i+1)(1),(i=0,1,…,m-1),αjv^(2j)(0)-βjv^(2j+1)(0)=γjv^(2j)(1)+δjv^(2j+1)(1),(j=0,1,…,n-1).Among them,the two equations studied in f1,f2∈C([0,1]×R^m+n+,R+)(R+=[0,∞))can have different orders and their derivatives satisfy different boundary value conditions.Based on a priori estimate,the existence of positive solutions for the above boundary value problems is proved by using the fixed-point index theory.The research in this paper has important practical value and significance in bridge engineerings.