半闭原理及其在不动点理论中的应用
The Demiclosedness Principle and Its Applications in Fixed Point Theory
摘要: 半闭原理是非线性泛函分析中连接弱收敛与不动点性质的核心工具，广泛应用于各类迭代算法的收敛性分析。本文系统梳理了半闭原理的定义、基本性质及其在各种映射类与空间结构中的推广。首先在Hilbert空间中建立非扩张映射的Browder半闭原理，进而推广到满足Opial条件或一致凸性的Banach空间。然后讨论了该原理对严格伪压缩映射、伪压缩映射、渐近非扩张映射、集值非扩张映射以及满足边界条件的单调算子的适用性，每种情形均给出严格的数学表述与证明。最后探讨了半闭原理在不动点迭代算法收敛性分析和变分不等式与图像恢复中的应用。
Abstract: The demiclosedness principle is a core tool in nonlinear functional analysis that bridges weak convergence and fixed point properties, and it is widely applied in the convergence analysis of various iterative algorithms. This paper systematically reviews the definition and fundamental properties of the demiclosedness principle, as well as its generalizations to different classes of mappings and spatial structures. First, we present Browder’s demiclosedness principle for nonexpansive mappings in Hilbert spaces, and then discuss its extension to Banach spaces satisfying Opial’s condition or uniform convexity. The applicability of the principle is subsequently discussed for strictly pseudocontractive mappings, pseudocontractive mappings, asymptotically nonexpansive mappings, set-valued nonexpansive mappings, and monotone operators satisfying certain range conditions; rigorous mathematical formulations and proofs are provided in each case. Finally, the applications of the demiclosedness principle in the convergence analysis of fixed point iterative algorithms, variational inequalities, and image restoration are explored.
文章引用：贺龙. 半闭原理及其在不动点理论中的应用[J]. 理论数学, 2026, 16(9): 83-96. https://doi.org/10.12677/pm.2026.169201

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