四阶两点非齐次边值问题正解的唯一性及对参数依赖性

Uniqueness and parameter dependence of positive solutions of fourth-order second-point nonhomogeneous BVPs

利用单调混合算子理论,研究了四阶两点非齐次边值问题:x′′′′t+f(t,x)=0,t∈0,1,x0=α,x′0=β,x1=λ,x′1=-μ正解的存在性与唯一性问题,其中,α>0,β>0,λ>0,μ>0,f(t,x)∈C([0,1]×[0,∞),[0,∞)),f(t,x)对于x单调递增,并且存在0≤θ<1使得f(t,kx)≥kθf(t,x),t∈[0,1],k∈[0,1],x∈[0,∞)成立.给参数α,β,λ,μ赋予一定的条件,证明了上述问题存在唯一正解,并且研究了解对参数的依赖性.

In this paper,by using the mixed monotone operator theory in cones,the fourth-order second-point nonhomogeneous boundary-value problem is investigated:x′′′′t+f(t,x)=0,t∈0,1,x0=α,x′0=β,x1=λ,x′1=-μ,whereα>0,β>0,λ>0,μ>0,f(t,x)∈C([0,1]×[0,∞),[0,∞))is continuous and monotonically increasing for x.There exists 0≤θ<1 such that f(t,kx)≥kθf(t,x),t∈[0,1],k∈[0,1],x∈[0,∞).The existence and uniqueness of a positive solution and for the above problem is proved and the dependence of this solution on the parametersα,β,λandμis studied.