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The bilinear from a(·,·)on the real Hilbert space is said to be coercive if there is a α>0) such that α(v,v)≥α‖v‖_H v∈H. (0.1)Assume that α(·,·)is a bilinear continuous and coercive form on H.Then, giving a f∈H′arbitrarily,There exists a unique u∈H such that α(u,v) = f(v). (0.2)Furthermore, u depends continuously on f. This is the Lax-Milgrarm Theorem obtained in 1954.In 1972, A. K. Aziz[1] improved the condition (0.1) and obtained the sufficient condition of Eq. (0.2) being Well-posed.In this paper,by different method,we proved, more simply, this sufficient condition in §1. Furthermore, we proved that this sufficient condition is also the necessary one. In §2 we improved the error estimate of approximate solution of (0.2). which was obtained by Céa[3] in 1964. In §3 we discussed the Well-posed problem of Eq.(2), when a(·, ·) is monotonic. This result may be considered as a generalization of the dffinition of coercive,and the Lax-Milgram Theorem as a special example. In §4 we exte 相似文献
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实Hilbert空间H上双线性型a(·,·)称作正定的:如果■α>O,使(0.1)■设a(·,·)是H上双线性连续正定型,则对任意给定的f∈H′,存在唯一的u∈H,使a(u,v)=f(v),并且u连续地依赖于f.这就是于1954年得到的Lax-Milgram定理。 A. K. Aziz~([1])改进条件(0.1)于1972年得到了(0.2)有适定解的充分条件。本文■1用不同方法更简单地证明了这个充分条件,并且证明了这个充分条件是必要条件。■2改进了Cea~([3])于1964年得到的关于(0.2)的近似解的一个误差估计。■3对于G(.,.)是单调情形时讨论了方程(0.2)的解的适定性,这个结果的特例。■4把变分不等式的一个结果推广到积空间的情形,这个结果包含了J.L.Lions~([2])于1967年证明的关于变分不等式的定理。用这里的方法有希望推广郭友中~([4])的有关定理。 相似文献
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