椭圆变分不等式的小扰动

Small perturbation of elliptic variational inequalities

研究Ｂａｎａｃｈ空间中椭圆变分不等式的扰动问题，得到了扰动问题存在唯一解的一个充分条件；并用它处理了一类半线性微分积分方程的边值问题．设Ｖ是可分自反Ｂａｎａｃｈ空间，Ｖ′是Ｖ的对偶空间，Ｋ是Ｖ中非空闭凸子集，则有定理１设Ｔ：Ｖ→Ｖ′，Ａ：Ｋ→Ｖ′，且满足（ｉ）Ｔ是有界线性算子，存在常数α＞０，使得（Ｔｖ，ｖ）≥αｖ２，ｖ∈Ｖ；（ｉ）Ａ是伪单调算子，存在常数λ＞０，使得（Ａｕ－Ａｖ，ｕ－ｖ）≤λｕ－ｖ２，ｕ，ｖ∈Ｋ；（ｉｉ）α＞λ．则存在唯一的ｕ∈Ｋ，使得（Ｔｕ，ｖ－ｕ）＋（Ａｕ，ｖ－ｕ）≥（ｆ，ｖ－ｕ）。

A perturbation problem of elliptic variational inequalities in Banach space is studied and a sufficient condition on existence and uniquenss of solutions of the perturbation problem are obtained.As an example,a boundary value problem of semilinear differential and integral enquations has been discussed.Let V be a countable reflexive Banach space V′ the dual space, K the closed convex set in V ,then one has Theorem 1\ Let T:VV′,A:KV′ ,the following condition is satisffied (i) T is a bounded linear operator,there is some α>0 such that(Tv,v) ≥α‖v‖ 2,\ for all v∈V; (ii) A is a pseddo monotone operator,there is some λ>0 such that(Au-Av,u-v)≤λ‖u-v‖,\ for all u,v∈K; (iii) α>λ . Then there is only u∈K such that(Tu,v-u)+(Au,v-u)≥(f,v-u),\ for all v∈K.