縮小写像の不動点定理とその周辺
Banach contraction principle:
its extensions and applications
完備距離空間を主舞台とする不動点定理および不動点近似定理を動画とレジュメで紹介する。予備知識は、微分積分学と距離空間論（完備距離空間を含む）の初歩である。ただし、後半の一部（第10講）では（線型代数学の教科書で説明されている）ベクトル空間の初歩的な知識があると理解の助けになる。距離空間の理解があやふやな方は、本Webサイト内の数学解析入門1，2などの該当箇所を復習したうえで再訪されたい(微分積分学と線型代数学の参考書はいくらでもあるので特に挙げない)。
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NEW！
🔸March 4, 2024
(1) The PDF notes for Lectures 4, 6, 10 were replaced by new one with additional references.
(2) New PDF note for an additional topic"cyclic contraction mappings" was up-loaded.
Basic Course
Additional Topic
1. Cyclic Contraction Mappings
References
In Japanese
🔸『距離空間と位相空間』 高橋渉（横浜図書）
🔸『非線形・凸解析学入門』 高橋渉（横浜図書）
In English
🔸S. Banach, "Sur les opérations dans les ensembles abstraits et leur applications aux éuations intérales," Fundamenta Mathematicae 3 (1922): 133-181.
🔸V. Berinde, "On the convergence of the Ishikawa iteration in the class of quasi contractive operators," Acta Mathematica Universitatis Comenianae, 73.1 (2004): 119-126.
🔸V. Berinde, "A convergence theorem for Mann iteration in the class of Zamfirescu operators," Seria Matematica Informatica XLV 1 (2007): 33-41.
🔸V. Berinde, "Iterative Approximation of Fixed Points," Springer, (2007).
🔸F.F. Bonsall and K.B. Vedak, Lectures on some fixed point theorems of functional analysis, No. 26. Bombay: Tata Institute of Fundamental Research, 1962.
🔸L.B. Ćirić, "Generalized contractions and fixed-point theorems." Publications de l'Institut Mathematique, 12.26 (1971): 19-26.
🔸L.B. Ćirić, "A generalization of Banach’s contraction principle," Proceedings of the American Mathematical Society, 45(2) (1974): 267-273.
🔸S.K. Chatterjea, "Fixed-point theorems," Dokladi na Bolgarskata Akademiya na Naukite, 25.6 (1972): 727-+.
🔸M. Edelstein, "On fixed and periodic points under contractive mappings," Journal of the London Mathematical Society 1.1 (1962): 74-79.
🔸S. Ishikawa, "Fixed points by a new iteration method," Proceedings of the American Mathematical Society 44(1) (1974): 147-150.
🔸R. Kannan, "Some Results on Fixed Points." Bulletin Calcutta Mathematical Society, 10, (1968), 71-76.
🔸R. Kannan, "Some results on fixed points—II." The American Mathematical Monthly, 76.4 (1969): 405-408.
🔸E. Karapinar, "Revisiting the Kannan type contractions via interpolation." Advances in the Theory of Nonlinear Analysis and its Application, 2.2 (2018): 85-87.
🔸A. Kondo, "Strong convergence theorems using three-step mean iteration for Zamfirescu mappings in Banach spaces," Acta Mathematica Universitatis Comenianae, 92(2), (2023) 165-178.
🔸Reich, Simeon. "Some remarks concerning contraction mappings." Canadian Mathematical Bulletin, 14.1 (1971): 121-124.
🔸W. Takahashi, "Introduction to Nonlinear and Convex Analysis", (Yokohama Publishers)
🔸D. Wardowski, "Fixed points of a new type of contractive mappings in complete metric spaces," Fixed point theory and applications 2012(1) (2012): 1-6.
🔸D. Wardowski and N. Van Dung. "Fixed points of F-weak contractions on complete metric spaces." Demonstratio Mathematica, 47.1 (2014): 146-155.
🔸T. Zamfirescu, "Fix point theorems in metric spaces," Archiv der Mathmatik 23 (1972): 292-298.
References for Additional Topics
Convex Contraction Mappings
🔸V. Berinde, "Approximating fixed points of almost convex contractions in metric spaces," Ann. Acad. Rom. Sci. Ser. Math. Appl. 12 (2020): 11-23.
🔸V.I. Istraţescu, "Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters.—I," Annali di Matematica Pura ed Applicata, 130 (1982):89-104.
Cyclical Contraction Mappings
🔸W.A. Kirk, P.S. Srinivasan, P. Veeramani, "Fixed point to mappings satisfyaing cyclical contractive conditions," Fixed Point Theory, 4.1 (2003), 79-89.
🔸P.S. Kumari and D. Panthi. "Cyclic contractions and fixed point theorems on various generating spaces," Fixed Point Theory and Applications 2015.1 (2015): 1-17.
Discontinuous Mappings
🔸R.K. Bisht, et al., "On discontinuity at fixed point via power quasi contraction." Publications de l'Institut Mathematique, 108.122 (2020): 5-11.
🔸B. Fisher, "Quasicontractions on metric spaces." Proceedings of the American Mathematical Society 75.2 (1979): 321-325.
Local Contractions
🔸L.B. Ćirić, "Locally eventually contractive Fixed-Point Mappings." Publications de l'Institut Mathematique, 41 (1980): 33-36.
🔸T. Hu and W.A. Kirk, "Local contractions in metric spaces," Proceedings of the American Mathematical Society, 68.1 (1978): 121-124.
🔸V. Filipe Martins‐da‐Rocha and Yiannis Vailakis, "Existence and uniqueness of a fixed point for local contractions," Econometrica 78.3 (2010): 1127-1141.
Meir and Keeler
🔸A. Meir and E. Keeler, "A theorem on contraction mappings," J. Math. Anal. Appl. 28 (1969): 326-329.