运用上下解方法及不动点指数理论,讨论非齐次边界条件下四阶微分方程四点边值问题{u(4)(t)-f(t,u(t),u"(t))=0,t∈[0,1],u(0)=λ1,u(1) =λ2,au"(ξ1)-bu”'(ξ1)=-λ 3,cu"(ξ2)+du”'(ξ2)=-λ4.得到正解存在的充分条件.给出该非齐次边界条件下,四阶微分方程四点边值问题至少存在一个正解、两个正解及无正解时,参数(λ1,λ2,λ3,λ4)的取值范围.其中:(λ1,λ2,λ3,λ4)∈R4+｛(0,0,0,0)}为参数,0≤ξ1≤ξ2≤1,a,b,c,d为非负常数,f∈C([0,1]×[0,+∞)×(-∞,0],[0,+∞)).%By using the lower and upper solutions method and fixed point index theory,the sufficient conditions for the existence of positive solutions of the following nonlinear boundary value problem with nonhomogeneous four-point boundary condition are discussed.{u(4)(t)-f(t,u(t),u”(t)) =0,t∈[0,1],u(0) =λ1,u(1) =λ2,au”(ξ1)-bu”'(ξ1) =-λ3,cu”(ξ2) +du”'(ξ2) =-λ4.Where (λ1,λ2,λ3,λ4) ∈R4+ { (0,0,0,0)} are parameters,0≤ξ ≤ξ2≤1.a,b,c,d are nonnegative constants,f∈C([0,1] × [0,+ ∞) × (-∞,0],[0,+ ∞)).The regions of (λ1,λ2,λ3,λ4),in which the fourth-order differential equation with four-point boundary condition with nonhomogeneous boundary conditions has at least one positive solution,two positive solutions and no positive solution,are determined.
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