On some known fixed point results in the complex domain: survey
Abstract
In this survey paper, we consider some known results from the fixed point theory with complex domain. The year 1926 is very significant for this subject. This is the beginning of the research and application of the fixed point theory in complex analysis. The Denjoy-Wolf theorem, together with the Banach contraction principle, is one of the main tools in the mathematical analysis.
References
Ahlfors, L. 1953. Complex Analysis. New York: McGraw-Hill, 1st ed.
Anderson, J.M., & Vasil’ev, A. 2008. Lower Schwarz-Pick Estimates and Angular Derivatives. Ann. Acad. Sci. Fennicae Math., 33, pp.101–110. Available at: http://www.acadsci.fi/mathematica/Vol33/AndersonVasilev.html. Accessed: 10.03.2018.
Beardon, A.F. 1990. Iteration of contractions and analytic maps. J. Lond. Math. Soc., s2-41(1), pp.141–150. Available at: https://doi.org/10.1112/jlms/s2-41.1.141.
Beardon, A.F. 1997. The dynamics of contractions. Ergodic Theory and Dynamical Systems, 17(6), pp.1257-1266. Available at: https://doi.org/10.1017/s0143385797086434.
Budzynska, M., Kuczumow, T., & Reich, S. 2012. A Denjoy-Wolff theorem for compact holomorphic mappings in reflexive Banach spaces. J. Math. Anal. Appl. 396(2), pp.504–512. Available at: https://doi.org/10.1016/j.jmaa.2012.06.044.
Budzynska, M., Kuczumow, T., & Reich, S. 2013a. Theorems of Denjoy-Wolff type. Annali di Matematica Pura ed Applicata, 192(4), pp.621–648. Available at: https://doi.org/10.1007/s10231-011-0240-z.
Budzynska, M., Kuczumow, T., & Reich, S. 2013b. A Denjoy-Wolff theorem for compact holomorphic mappings in complex Banach spaces. Annales Academiae Scientiarum Fennicae Mathematica, 38, pp.747-756. Available at: https://doi.org/10.5186/aasfm.2013.3846.
Burckel, R.B. 1981. Iterating analytic self-maps of discs. Am. Math. Monthly, 88(6), pp.396–407. Available at: https://doi.org/10.2307/2321822.
Caratheodory, M. 1960. Theory of functions of a complex variable. Chelsea, New York. Vol. 2, English edition.
Contreras, M.D., Díaz-Madrigal, S., & Pommerenke, C. 2006. On boundary critical points for semigroups of analytic functions. Mathematica Scandinavica, 98(1). Available at: https://doi.org/10.7146/math.scand.a-14987.
Cowen, C.C. 1981. Iteration and the Solution of Functional Equations for Functions Analytic in the Unit Disk. Trans. Amer. Math. Soc., 265, pp.69–95. Available at: https://doi.org/10.1090/S0002-9947-1981-0607108-9.
Cowen, C.C. 2010. Fixed points of functions analytic in the unit disk. In: Conference on complex analysis, University of Illinois, May 22. University of Illinois.
Cowen, C.C., & Pommerenke, Ch. 1982. Inequalities for the Angular Derivative of an Analytic Function in the Unit Disk. J. London Math. Soc., s2-26(2), pp.271–289. Available at: https://doi.org/10.1112/jlms/s2-26.2.271.
Denjoy, A. 1926. Sur l’itération des fonctions analytiques, C.R. Acad. Sci. Paris, Serie 1, 182, pp.255–257 (in French).
Earle, C.J., & Hamilton, R.S. 1970. A fixed point theorem for holomorphic mappings. In S. Chern & S. Smale Eds., Proceeding of Symposia on Pure Mathematics. Providence, Rhode Island: American Mathematical Society (AMS), pp.61-65. Available at: https://doi.org/10.1090/pspum/016.
Goebel, K., Sekowski, T., & Stachura, A. 1980. Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball. Nonlinear Analysis: Theory, Methods & Applications, 4(5), pp.1011–1021. Available at: https://doi.org/10.1016/0362-546X(80)90012-7.
Goebel, K. 1982. Fixed points and invariant domains of holomorphic mappings of the Hilbert ball. Nonlinear Analysis: Theory, Methods & Applications, 6(12), pp.1327–1334. Available at: https://doi.org/10.1016/0362-546X(82)90107-9.
Harris, L.A. 2003. Fixed points of holomorphic mappings for domains in Banach spaces. Abstract and Applied Analysis, 2003(5), pp.261-274. Available at: https://doi.org/10.1155/S1085337503205042.
Hayden, T.L., & Suffridge, T.J. 1971. Biholomorphic maps in Hilbert space have a fixed point. Pacific J. Math., 38(2), pp.419–422. Available at: https://doi.org/10.2140/pjm.1971.38.419.
Hayden, T.L., & Suffridge, T.J. 1976. Fixed points of holomorphic maps in Banach spaces. Proceedings of the American Mathematical Society, 60(1), pp.95-105. Available at: https://doi.org/10.1090/s0002-9939-1976-0417869-3.
Henrici, P. 1969. Fixed points of analytic functions. Technical report NO. CS 137.
Julia, G. 1918. Mémoire sur l’itération des fonctions rationelles. J. Math. Pures Appl., 8, pp.47–295 (in French).
Khatskevich, V., Reich, S., & Shoikhet, D. 1995. Fixed point theorems for holomorphic mappings and operator theory in indefinite metric spaces. Integral Equations and Operator Theory, 22(3), pp.305-316. Available at: https://doi.org/10.1007/bf01378779.
Kuczumow, T. 1984. Common fixed points of commuting holomorphic mappings in Hilbert ball and polydisc. Nonlinear Analysis: Theory, Methods & Applications, 8(5), pp.417-419. Available at: https://doi.org/10.1016/0362-546x(84)90081-6.
Kuczumow, T., Reich, S., & Shoikhet, D. 2001. Fixed Points of Holomorphic Mappings: A Metric Approach. In W.A. Kirk& B. Sims Eds., Handbook of Metric Fixed Point Theory. Dordrecht: Springer Nature, pp.437-515. Available at: https://doi.org/10.1007/978-94-017-1748-9_14.
Lemmens, B., Lins, B., Nussbaum, R., & Wortel, M. 2016. Denjoy-Wolff theorems for Hilbert’s and Thompson’s metric spaces, arXiv: 1410.1056v4 [math. DS].
Mateljević, M. 1998. Holomorphic fixed point theorem on Riemann surfaces. Math. Balkanica, 12(1–2), pp.1–4.
Reich, S., & Shoikhet, D. 1996. Generation theory for semigroups of holomorphic mappings in Banach spaces. Abstract and Applied Analysis, 1(1), pp.1-44. Available at: https://doi.org/10.1155/s1085337596000012.
Rudin, W. 1978. The fixed-point sets of some holomorphic maps. Bull. Malays. Math. Soc., 1, pp.25–28.
Suffridge, T.J. 1974. Common fixed points of commuting holomorphic maps of the hyperball. The Michigan Mathematical Journal, 21(4), pp.309–314. Aalable at: https://doi.org/10.1307/mmj/1029001354.
Wolff, J. 1926. Sur une généralization d'un théorème de Schwartz. C.R. Hebd. Seanc. Acad., . 182, pp.918–920 and 183, pp.500–502 (in French).
Xu, Q., Tang, Y., Yang, T., & Srivastava, H.M. 2016. Schwarz lemma involving the boundary fixed point. Fixed Point Theory and Applications, 2016(1). Available at: https://doi.org/10.1186/s13663-016-0574-8.
Proposed Creative Commons Copyright Notices
Proposed Policy for Military Technical Courier (Journals That Offer Open Access)
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).