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作 者:何文魁
机构地区:[1]西北师范大学数学与统计学院,兰州730070
出 处:《黑龙江大学自然科学学报》2014年第1期46-50,共5页Journal of Natural Science of Heilongjiang University
基 金:国家自然科学基金资助项目(10671158);甘肃省自然科学基金资助项目(3ZS051-A25-016)
摘 要:运用锥上的不动点定理,考虑二阶奇异Neumann边值问题{x″(t)+a(t)x(t)=f(t,x(t)),t∈(0,1),x'(0)=x'(1)=0,正解的存在性,其中0<a(t)<(π2)/4,f∈C((0,1)×(0,+∞),[0,+∞)),且在t=0,t=1和x=0处允许有奇性。考虑对应问题的格林函数及其正性的估计,将其转化为等价的积分方程,即将问题正解的存在性问题转化为判断一个算子方程不动点的存在性问题进行求解。讨论算子的全连续性,最后证明问题(2)正解的存在性。By using the fixed-point theorem in cones, the existence of positive solutions of the fol- lowing singular second-order Neumann boundary value problem is considered: x"(t) +a(t)x(t) =f(t, x(t)), tE(0,1), x'(0) =x'(1) =0. 2 'IT where 0 〈 a (t) 〈 -^- , andf~ C( (0,1) ^(0, +^),[0, +~)), may be singular at t=0, t=l and x =0. By constructing Green's function of the corresponding problem and giving a positive estimates for the Green's function, the boundary value problem is converted to an equivalent integral equation, i. e., the existence of positive solution for the problem is transformed into the existence of a fixed point of the operator equation. Finally, the existence of positive the completely continuous of operator. solution of the problem is investigated by employing
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