一类具非局部边界条件的四阶非线性微分方程的对称正解

Symmetric Positive Solutions to a Fourth-Order Nonlinear Differential Equation with Nonlocal Boundary Conditions

本文考虑形如的非线性四阶微分方程非局部边值问题,这里a,b∈L^1[0,1],g:(0,1)→[0,∞)在(0,1)上连续、对称,且可能在t=0和t=1处奇异．f:[0,1]×[0,∞)→[0,∞)连续且对所有x∈[0,∞],f(·,x)在[0,1]上对称．在某些适当的增长性条件下,应用Krasnoselskii不动点定理证明了对称正解的存在性和多重性．

We consider the nonlocal boundary value problem for a nonlinear fourthorder ordinary differential equation of the form{u″″（t）=g（t）f（t,u（t））,0〈t〈1,u（0）=u（1）=∫0^1a（s）u（s）ds,u″（0）=u″（1）=∫0^1b（s）u″（s）ds where a,b ∈ L^1[0,1],g：（0,1）→[0,∞） is continuous,symmetric on （0,1） and maybe singular at t=0 and t=1.f：[0,1]×[0,∞）→[0,∞） is continuous and f（·,x） is symmetric on [0,1] for all x E [0,∞）.Under some suitable growth conditions,we show the existence and multiplicity of symmetric positive solutions of that above problem by applying Krasnoselskii＇s fixed point theorem in a come.