Semigroups generated by pseudo-contractive mappings under the Nagumo condition
Semigroups generated by pseudo-contractive mappings under the Nagumo condition
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Nagumo 条件下伪收缩映射生成的半群
DOI：
10.1016/j.jde.2008.04.023
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发表时间：
2008
影响因子：
2.4
通讯作者：
C. Morales
中科院分区：
文献类型：
--
作者：
Anthony Hester;C. Morales
设X是一个Banach空间，其对偶空间X是一致凸的.证明了对任意半连续的弱Nagumo k-伪压缩映射T：D（T）<$X→X，A= T-I在D（T）上弱生成半群.在本文中，我们项目的后果，这一结果的不动点理论。特别地，我们证明了如果k<1（id est，如果T是强伪压缩的），则T有唯一的不动点。这意味着，如果T是伪压缩的（k=1），D（T）是闭的，有界的，凸的，那么T至少有一个不动点。因此，任何半连续伪压缩映射T：C→C（对于适当的C）都有一个不动点，这一直是不动点理论中一个重要的公开问题。在随后的论文中，我们探讨了后果的半群结果的存在性解决某些偏微分方程。半群结果直接暗示了u′=Au形式的时间发展方程整体解的存在性，其中A是导数的组合.本文的不动点结果暗示了Lu=f型偏微分方程解的存在性。
Let X be a Banach space whose dual space X∗is uniformly convex. We demonstrate that, for any demicontinuous, weakly Nagumo, k-pseudo-contractive mapping T:D(T)⊆X→X with closed domain, A=T−I weakly generates a semigroup on D(T). In this paper, we project the consequences of this result on fixed point theory. In particular, we show that if k<1 (id est, if T is strongly pseudo-contractive), then T has a unique fixed point. This implies that, if T is pseudo-contractive (k=1) and D(T) is closed, bounded, and convex, then T has at least one fixed point. Consequently, any demicontinuous pseudo-contractive mapping T:C→C (for an appropriate C) has a fixed point, which has been an important open question in fixed point theory for quite some time. In a subsequent paper, we explore the consequences of the semigroup result on the existence of solutions to certain partial differential equations. The semigroup result directly implies the existence of unique global solutions to time evolution equations of the form u′=Au where A is a combination of derivatives. The fixed point results from this paper imply the existence of solutions to partial differential equations of the form Lu=f.
DOI：
10.1007/978-3-662-00547-7
发表时间：
1985
期刊：
--
影响因子：
--
作者：
K. Deimling
通讯作者：
K. Deimling