Generalized metric spaces: A survey

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• 作者：M. A. Khamsi
• 关键词：Primary 47H09 ; Secondary 46B20 ; 47H10 ; 47E10 ; Banach contraction principle ; cone metric spaces ; fixed point ; generalized metric spaces ; Menger spaces ; b ; metric spaces ; G ; metric spaces ; modular metric spaces ; partially ordered metric spaces
• 刊名：Journal of Fixed Point Theory and Applications
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：17
• 期：3
• 页码：455-475
• 全文大小：735 KB
• 参考文献：1.A. A. Abdou and M. A. Khamsi, Fixed point results of pointwise contractions in modular metric spaces. Fixed Point Theory Appl. 2013 (2013), doi:10.-186/-687-1812-2013-163 .
  2.A. A. Abdou and M. A. Khamsi, On the fixed points of nonexpansive mappings in modular metric spaces. Fixed Point Theory Appl. 2013 (2013), doi:10.-186/-687-1812-2013-229 .
  3.Agarwal R. P.: Contraction and approximate contraction with an application to multipoint boundary value problems. J. Comput. Appl. Math. 9, 315-25 (1983)MATH MathSciNet CrossRef
  4.Aronszajn N., Panitchpakdi P.: Extensions of uniformly continuous transformations and hyperconvex metric spaces. Pacific J. Math. 6, 405-39 (1956)MATH MathSciNet CrossRef
  5.Banaschewski B., Bruns G.: Categorical characterization of the MacNeille completion. Arch. Math. (Basel) 18, 369-77 (1967)MATH MathSciNet CrossRef
  6.S. R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems. Academic Press, New York, 1974.
  7.V. V. Chistyakov, Modular metric spaces. I. Basic concepts. Nonlinear Anal. 72 (2010), 1-4.
  8.V. V. Chistyakov, Modular metric spaces. II. Application to superposition operators. Nonlinear Anal. 72 (2010), 15-0.
  9.H. Covitz and S. B. Nadler, Jr., Multi-valued contraction mappings in generalized metric spaces. Israel J. Math. 8 (1970), 5-1.
  10.S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces. Atti Semin. Mat. Fiz. Univ. Modena 46 (1998), 263-76.
  11.H. Dehghan, M. Eshaghi Gordji and A. Ebadian, Comment on ‘Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory and Applications, doi:10.-186/-687-1812-2011-93 , 20 pages-/em>. Fixed Point Theory Appl. 2012 (2012), doi:10.-186/-687-1812-2012-144 .
  12.Dhage B.C.: Generalised metric spaces and mappings with fixed point. Bull. Calcutta Math. Soc. 84, 329-36 (1992)MATH MathSciNet
  13.B. C. Dhage, Generalized metric spaces and topological structure. I. An. ?tiin?. Univ. Al. I. Cuza Ia?i. Mat. (N.S.) 46 (2000), 3-4.
  14.B. C. Dhage, On generalized metric spaces and topological structure. II. Pure. Appl. Math. Sci. 40 (1994), 37-1.
  15.B. C. Dhage, On continuity of mappings in D-metric spaces. Bull. Calcutta Math. Soc. 86 (1994), 503-08.
  16.B. C. Dhage, Generalized D-metric spaces and multi-valued contraction mappings. An. ?tiin?. Univ. Al. I. Cuza Ia?i. Mat. (N.S.) 44 (1998), 179-00.
  17.L. Diening, Theoretical and numerical results for electrorheological fluids. Ph.D. Thesis, University of Freiburg, Germany, 2002.
  18.W.-S. Du, A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 72 (2010), 2259-261.
  19.M. Escardó, Synthetic topology: of data types and classical spaces. Electron. Notes Theor. Comput. Sci. 87 (2004), 21-56.
  20.M. Fitting, Metric methods: Three examples and a theorem. J. Logic Programming 21 (1994), 113-27.
  21.M. Fitting, Fixpoint semantics for logic programming: A survey. Theoret. Comput. Sci. 278 (2002), 25-1.
  22.M. Fréchet, Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo 22 (1906), 1-2.
  23.S. G?hler, 2-metriche r?ume und ihre topologische struktur. Math. Nachr. 26 (1963), 115-48.
  24.S. G?hler, Zur geometric 2-metriche raüme. Rev. Roumaine Math. Pures Appl. 40 (1966), 664-69.
  25.K. S. Ha, Y. J. Cho and A. White, Strictly convex and strictly 2-convex 2- normed spaces. Math. Japon. 33 (1988), 375-84.
  26.P. Harjulehto, P. H?st?, M. Koskenoja and S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values. Potential Anal. 25 (2006), 205-22.
  27.F. Hausdorff, Grundzüge der Mengenlehre. Leipzig, 1914.
  28.J. Heinonen, T. Kilpel?inen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. The Clarendon Press, Oxford University Press, New York, 1993.
  29.L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332 (2007), 1468-476.
  30.S. Jankovi?, Z. Kadelburg and S. Radenovi?, On cone metric spaces: A survey. Nonlinear Anal. 74 (2011), 2591-601.
  31.E. Jawhari, M. Pouzet and D. Misane, Retracts: Graphs and ordered sets from the metric point of view. In: Combinatorics and Ordered Sets (Arcata, Calif., 1985), Contemp. Math. 57, Amer. Math. Soc., Providence, RI, 1986, 175-26.
  32.Z. Kadelburg, S. Radenovi? and V. Rako?evi?, A note on the equivalence of some metric and cone metric fixed point results. Appl. Math. Lett. 24 (2011), 370-74.
  33.M. A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010 (2010), doi:10.-155/-010/-15398 .
  34.M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory. John Wiley, New York, 2001.
  35.M. A. Khamsi, W. K. Kozlowski and S. Reich, Fixed point theory in modular function spaces. Nonlinear Anal. 14 (1990), 935-53.
  36.M.
• 作者单位：M. A. Khamsi (1) (2)
  
  1. Department of Mathematical Science, The University of Texas at El Paso, El Paso, TX, 79968, USA
  2. Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, P.O. Box 411, Dhahran, 31261, Saudi Arabia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Mathematical Methods in Physics
  
• 出版者：Birkh盲user Basel
• ISSN：1661-7746

文摘

Banach’s contraction mapping principle is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis with special applications to the theory of differential and integral equations. Because the underlined space of this theorem is a metric space, the theory that developed following its publication is known as the metric fixed point theory. Over the last one hundred years, many people have tried to generalize the definition of a metric space. In this paper, we survey the most popular generalizations and we discuss the recent uptick in some generalizations and their impact in metric fixed point theory. Mathematics Subject Classification Primary 47H09 Secondary 46B20 47H10 47E10