Topological Methods in Nonlinear Analysis

The role of equivalent metrics in fixed point theory

Adrian Petruşel, Ioan A. Rus, and Marcel-Adrain Şerban

Full-text: Open access

Abstract

Metrical fixed point theory is accomplished by a wide class of terms:

$\bullet$ operators (bounded, Lipschitz, contraction, contractive, nonexpansive, noncontractive, expansive, dilatation, isometry, similarity, Picard, weakly Picard, Bessaga, Janos, Caristi, pseudocontractive, accretive, etc.),

$\bullet$ convexity (strict, uniform, hyper, etc.),

$\bullet$ deffect of some properties (measure of noncompactness, measure of nonconvexity, minimal displacement, etc.),

$\bullet$ data dependence (stability, Ulam stability, well-posedness, shadowing property, etc.),

$\bullet$] attractor,

$\bullet$ basin of attraction$\ldots$

The purpose of this paper is to study several properties of these concepts with respect to equivalent metrics.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 41, Number 1 (2013), 85-112.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461253857

Mathematical Reviews number (MathSciNet)
MR3086535

Zentralblatt MATH identifier
1278.54044

Citation

Petruşel, Adrian; Rus, Ioan A.; Şerban, Marcel-Adrain. The role of equivalent metrics in fixed point theory. Topol. Methods Nonlinear Anal. 41 (2013), no. 1, 85--112. https://projecteuclid.org/euclid.tmna/1461253857


Export citation

References

  • R.P. Agarwal, M. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge Univ. Press (2001) \ref\key 2
  • J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht (2003) \ref\key 3
  • J. Appell, E. De Pascale and A. Vignoli, Nonlinear Spectral Theory, W. de Gruyter, Berlin (2004) \ref\key 4
  • M. Arov, S. Reich and A.J. Zaslavski, Uniform convergence of iterates for a class asymptotic contraction , Fixed Point Theory, 8 (2007), 3–9 \ref\key 5
  • C.E. Aull and R. Lowen, Handbook of the History of General Topology, Kluwer Acad. Publ., Dordrecht (2001) \ref\key 6
  • P.B. Bailey, L.F. Shampine and P.E. Waltman, Nonlinear Two Point Boundary Values Problems, Academic Press, New York (1968) \ref\key 7
  • A.I. Ban and S.G. Gal, Defects of Properties in Mathematics, World Scientific, New Jersey (2002) \ref\key 8
  • A.F. Beardon, Iteration of Rational Functions, Springer (1991) \ref\key 9
  • G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer (1993) \ref\key 10
  • V. Berinde, Iterative Approximations of Fixed Points, Springer, Berlin (2007) \ref\key 11
  • S.R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York (1974) \ref\key 12
  • C. Bessaga, On the converse of the Banach fixed point principle , Colloq. Math., 7 (1959), 41–43 \ref\key 13
  • E. Bohl, Linear operator equations on a partially ordered vector space , Aequat. Math., 4(1970), 89–98 \ref\key 14
  • J. Borsík and J. Doboš, Functions whose composition with every metric is a metric , Math. Slovaca, 31 (1981), 3–12, (in Russian) \ref\key 15
  • J. Borsík and J. Doboš, On metric preserving functions , Real Anal. Exchange, 13(1987/88, no. 1), 285–293 \ref\key 16
  • N. Bourbaki, Topologie générale, Herman, Paris (1961) \ref\key 17
  • A. Brown and C. Pearcy, An Introduction to Analysis, Springer, New York (1995) \ref\key 18
  • P.J. Bushell, The Cayley–Hilbert metric and positive operators , Linear Algebra Appl., 84(1986), 271–280 \ref\key 19
  • T. Cardinali and P. Rubbioni, A generalization of Caristi's fixed point theorem in metric spaces , Fixed Point Theory, 11(2010, no. 1), 3–10 \ref\key 20
  • A. Chiş-Novac, R. Precup and I.A. Rus, Data dependence of fixed points for non-self generalized contractions , Fixed Point Theory, 10(2009, no. 1), 73–87 \ref\key 21
  • P. Collaco and J.C.E. Silva, A complete comparison of \rom25 contraction conditions, Nonlinear Anal., 30(1997), 471–476 \ref\key 22
  • P. Corazza, Introduction to metric preserving functions , Amer. Math. Monthly, 104(1999, no. 4), 309–323 \ref\key 23
  • C. Corduneanu, Bielecki method in the theory of integral equations , Ann. Univ. Marie Curie Sklodowska, 38 (1984, no. 2), 23–40 \ref\key 24
  • C. Costantini and S. Levi, Metrics that generate the same hyperspace convergence , Set-Valued Anal., 1(1993, no. 2), 141–157 \ref\key 25
  • F.S. De Blasi and J. Myjak, Sur la porosité des contractions sans point fixe , C.R. Acad. Sci. Paris, 308(1989), 51–54 \ref\key 26
  • M.M. Deza and E. Deza, Encyclopedia of Distances, Springer, Berlin(2009) \ref\key 27
  • J. Doboš, A survey of metric preserving functions , Questions Answers Gen. Topology, 13(1995, no. 2), 129–134 \ref\key 28
  • J. Doboš and Z. Piotrowski, Some remarks on metric preserving functions , Real Anal. Exchange, 19(1993-1994), 317–320 \ref\key 29 ––––, A note on metric preserving functions , Internat. J. Math. Math. Sci., 19 , no. 1 (1996), 199–201 \ref\key 30
  • T. Dominguez, M.A. Japón and G. Lopez, Metric fixed point results concerning measures of noncompactness , Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht (2001), 239–268 \ref\key 31
  • P.N. Dowling, C.J. Lennard and B. Turett, Renorming of $l^{1}$ and $c_{0}$ and fixed point properties , Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht (2001), 269–297 \ref\key 32
  • J. Dugundji, Topology, Allyn and Bacon, Boston (1966) \ref\key 33
  • T. Eirolo, O. Nevalinna and S. Yu. Pilyugin, Limit shadowing property , Numer. Funct. Anal. Optim., 18(1977), 75–92 \ref\key 34
  • J. Eisenfeld, V. Lakshmikantham and S. Bernfeld, On the construction of a norm associated with the measure of noncompactness , Nonlinear Anal., 1(1976, no. 1), 49–54 \ref\key 35
  • R. Engelking, General Topology, PWN, Warszawa (1977) \ref\key 36
  • M. Fréchet, Les éspaces abstraits, Gauthier–Villars, Paris (1928) \ref\key 37
  • M. Furi and M. Martelli, On the minimal displacement of point under $\alpha $-Lipschitz maps in normed spaces , Boll. Un. Mat. Ital., 9(1974), 791–799 \ref\key 38
  • V. Glăvan and V. Guţu, Shadowing in affine iterated function systems , Fixed Point Theory, 10(2010), 229–243 \ref\key 39
  • K. Goebel, Metric environment of the topological fixed point theorems , Handbook of Metric Fixed Point Theory (W.A. Kirk and B. Sims, eds.), Kluwer Acad. Publ., Dordrecht (2001), 577–611 \ref\key 40 ––––, On the minimal displacement of points under Lipschitzian mappings , Pacific J. Math., 48(1973), 151–163 \ref\key 41
  • K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, London (1990) \ref\key 42
  • L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Kluwer Acad. Publ., Dordrecht (1999) \ref\key 43
  • A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York (2003) \ref\key 44
  • J.K. Hale and O. Lopes, Fixed point theorems and dissipative processes , J. Differential Equations, 13(1973), 391–402 \ref\key 45
  • S. Heikkilä and G.F. Roach, On equivalent norms and the contraction mapping principle , Nonlinear Anal., 8(1984), 1241–1252 \ref\key 46
  • M.C. Irwin, Smooth Dynamical Systems, Acad. Press (1980) \ref\key 47
  • A.A. Ivanov, Fixed Points of Metric Space Mapping, LOMI, Leningrad (1976), (in Russian) \ref\key 48
  • J. Jachymski, An extension of A. Ostrowski's theorem on the round-off stability of iterations , Aequat. Math., 53 (1997), 242–253 \ref\key 49 ––––, A short proof of the converse to the contraction principle and some related results , Topol. Methods Nonlinear Anal., 15(2000), 179–186 \ref\key 50 ––––, Converses to fixed point theorems of Zermelo and Caristi , Nonlinear Anal., 52(2003), 1455–1463 \ref\key 51
  • L. Janos, A converse of Banach's contraction theorem , Proc. Amer. Math. Soc., 18(1967), 287–289 \ref\key 52 ––––, On mappings contractive in the sense of Kannan , Proc. Amer. Math. Soc., 61(1976, no. 1), 171–175 \ref\key 53
  • E. Jawari, D. Misane and M. Pouzet, Retracts: graph and ordered set from the metric point of view , Contemp. Math., 57 (1986), 175–226 \ref\key 54
  • S. Kasahara, A remark on the converse of Banach contraction theorem , Amer. Math. Monthly, 75(1968), 775–776 \ref\key 55
  • J.L. Kelley, General Topology, Van Nostrand, New York (1955) \ref\key 56
  • M.A. Khamsi and W.A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Wiley–Interscience, New York (2001) \ref\key 57
  • W.A. Kirk, Metric fixed point theory: old problems and new directions , Fixed Point Theory, 11(2010), 45–58 \ref\key 58 ––––, Approximate fixed points of nonexpansive maps , Fixed Point Theory, 10(2009), 275–288 \ref\key 59 ––––, Caristi's fixed point theorem and metric convexity , Colloq. Math., 36(1976), 81–86 \ref\key 60
  • W.A. Kirk and B. Sims \rom(eds.), Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht (2001) \ref\key 61
  • V. Klee, Stability of the fixed point property , Colloq. Math., 8(1961), 41–46 \ref\key 62
  • M.A. Krasnosel'skiĭ and P. Zabreĭko, Geometrical Methods in Nonlinear Analysis, Springer, Berlin (1984) \ref\key 63
  • T. Kuczumov, S. Reich and D. Shoikhet, Fixed point of holomorphic mappings: a metric approach , Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht (2001), 437–515 \ref\key 64
  • M. Kwapisz, An extension of Bielecki's method of proving global existence and uniqueness results for functional equations , Univ. of Gdańsk, preprint(1984) \ref\key 65
  • B.K. Lahiri, M.K. Chakrabarty and A. Sem, Converse of Banach's contraction principle and stor operation , Proc. Nat. Acad. Sci. India Sect. A, 79(2009), 367–374 \ref\key 66
  • A. Lechicki and S. Levi, Wijsman convergence in the hyperspace of a metric space, Boll. Un. Mat. Ital. B, 1 (1987), 439–451 \ref\key 67
  • N. Levine, Remarks on uniform continuity in metric spaces , Amer. Math. Monthly, 67(1960), 562–563 \ref\key 68
  • Z.Q. Liu, Fixed point theorems for condensing and compact maps , Kobe J. Math., 11(1994), 129–135 \ref\key 69
  • E. Llorens Fuster, The fixed point property for renormings of $l_{2}$ , Seminar Math. Anal., Univ. Seville, 2006 , 121–159 \ref\key 70
  • P.R. Meyers, A converse to Banach's contraction theorem , J. Res. Nat. Bur Standards Sect. B, 71B(1967), 73–76 \ref\key 71 ––––, On contractive semigroups and uniform asymptotic stability , J. Res. Nat. Bur Standards Sect. B, 74B (1970), 115–120 \ref\key 72
  • V.I. Opoitsev, A converse to the principle of contracting maps , Russian Math. Surveys, 31(1976), 175–204 \ref\key 73
  • K. Palmer, Shadowing in Dynamical Systems, Kluwer Acad. Publ., Dordrecht (2000) \ref\key 74
  • M. Păcurar, An approximate fixed point proof of the Banach–Ghöhde–Kirk fixed point theorem , Creative Math. Inform., 17(2008), 43–47 \ref\key 75
  • M. Păcurar and R.V. Păcurar, Approximate fixed point theorems for weak contractions on metric spaces , Carpathian J. Math., 23(2007), 149–155 \ref\key 76
  • A. Petruşel, Multivalued weakly Picard operators and applications , Sci. Math. Japon., 59(2004), 169–202 \ref\key 77
  • A. Petruşel and I.A. Rus, Well-posedness of the fixed point problem for multivalued operators , Applied Analysis and Differential Equations (O. Cârjă and I.I. Vrabie, eds.), World Scientific (2007), 295–306 \ref\key 78 ––––, The theory of a metric fixed point theorem for multivalued operators , Proc. of the Ninth International Conference on Fixed Point Theory and its Applications, Changhua, Taiwan-2009 (L.J. Lin, A. Petruşel and H.K. Xu, eds.), Yokohama Publ. (2010), 161–175 \ref\key 79
  • A. Petruşel, I.A. Rus and J.-C. Yao, Well-posedness in the generalized sense of the fixed point problems , Taiwanese J. Math., 11(2007, no. 3), 903–914 \ref\key 80
  • S.Yu. Pilyugin, Shadowing in Dynamical Systems, Springer, Berlin (1999) \ref\key 81
  • V. Radu, The fixed point alternative and the stability of functional equations , Fixed Point Theory, 4(2003), 91–96 \ref\key 82 ––––, Some suitable metrics on fuzzy metric spaces , Fixed Point Theory, 5(2004), 323–347 \ref\key 83
  • S. Reich and A.J. Zaslavski, Well-posedness of fixed point problems , Far East J. Math. Sci., Special Volume, Part III, 393–401 (2001) \ref\key 84
  • B.E. Rhoades, A comparison of various definitions of contractive mappings , Trans. Amer. Math. Soc., 226(1977), 257–290 \ref\key 85 ––––, On weighted norms and the contraction mapping principle , NAM-Bericht, 10 (1974, Univ. of Göttingen) \ref\key 86
  • W. Robert and L. Janos, Constructing metrics with the Heine–Borel property , Proc. Amer. Math. Soc., 100(1987), 567–573 \ref\key 87
  • I. Rosenholtz, Evidence of a conspiracy among fixed theorems , Proc. Amer. Math. Soc., 53(1975, no. 1), 213–218 \ref\key 88
  • I.A. Rus, Picard operators and applications , Sci. Math. Japon., 58(2003), 191–219 \ref\key 89 ––––, Picard operators and well-posedness of fixed point problems , Studia Univ. Babeş–Bolyai Math., 52(2007, no. 3), 147–157 \ref\key 90 ––––, Weakly Picard operators and applications, Sem. on Fixed Point Theory, Cluj-Napoca, 2(2001), 41–58 \ref\key 91 ––––, Picard operators and well-posedness of fixed point problems , Studia Univ. Babeş–Bolyai, Mathematica, 52 (2007, no. 3), 147–156 \ref\key 92 ––––, The theory of a metrical fixed point theorem: theoretical and applicative relevances , Fixed Point Theory, 9(2008), 541–559 \ref\key 93 ––––, Principles and Applications of Fixed Point Theory, Dacia, Cluj-Napoca (1979), (in Romanian) \ref\key 94 ––––, Generalized Contractions and Applications, Cluj Univ. Press (2001) \ref\key 95 ––––, Remarks on Ulam stability of the operatorial equations , Fixed Point Theory, 10(2009), 305–320 \ref\key 96 ––––, Some nonlinear functional differential and integral equations via weakly Picard operator theory: a survey , Carpathian J. Math., 26(2010), 230–258 \ref\key 97 ––––, Fixed Point Structure Theory, Cluj Univ. Press, Cluj-Napoca (2006) \ref\key 98 ––––, Weakly Picard mappings, Comment. Math. Univ. Caroline, 34(1993, no. 4), 769–773 \ref\key 99
  • I.A. Rus, A. Petruşel and G. Petruşel, Fixed Point Theory, Cluj University Press (2008) \ref\key 100
  • I.A. Rus and M.-A. Şerban, Some generalizations of a Cauchy lemma and applications , Topics in Mathematics, Computer Science and Philosophy, A Festschrift for Wolfgang W. Breckner, Cluj Univ. Press (2008), 173–181 \ref\key 101
  • M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston(1999) \ref\key 102
  • T. Sekowski, On normal structure stability of fixed point property and the modulus of noncompact convexity , Rend. Sem. Mat. Fis. Milano, 56(1986), 147–153 \ref\key 103
  • I. Singer, Abstract Convex Analysis, John Wiley and Sons, Toronto (1997) \ref\key 104
  • T.K. Sreenivasan, Some properties of distance functions , J. Indian Math. Soc., 11(1947), 38–43 \ref\key 105
  • M.A. Şerban, Spaces with perturbed metrics and fixed point theorems , Automat. Comput. Appl. Math., 17 (2008, no. 2), 323–334 \ref\key 106 ––––, Abstract fixed point principles via perturbed metrics , to appear \ref\key 107
  • G. Targonski, Topics on Iteration Theory, Vandenhoeck and Ruprecht (1981) \ref\key 108
  • F. Terpe, Metric preserving functions , Proc. Conf. Topology and Measure \romIV, Greifswald (1984), 189–197 \ref\key 109
  • M. Turinici, Volterra functional equations via projective techniques , J. Math. Anal. Appl., 103(1984), 211–229 \ref\key 110
  • R.W. Vallin, On metric preserving functions and infinite derivatives , Acta Math. Univ. Comen., . 67v, no. 2(1989), 373–376 \ref\key 111
  • A.I. Vasiliev, Introducing an equivalent metric in a linear space by a family of its subnets , Proc. Steklov Inst. Math. (2001), 235–242 \ref\key 112
  • W. Walter, A note on contraction, SIAM Review, 18(1976), 107–111 \ref\key 113
  • G.T. Whyburn, Analytic Topology, Amer. Math. Soc. Collog. Publ., 28(1942) \ref\key 114
  • W.A. Wilson, On certain types of continuous transformations of metric spaces , Amer. J. Math., 57(1935), 62–68 \ref\key 115
  • Z.B. Xu and L.S. Wang, Quantitative properties of nonlinear Lipschitz operators I. The Lip number , Acta Math. Appl. Sinica, 19(1996), 175–184, (in Chinese) \ref\key 116
  • S.V. Zhestkov and P.P. Zabreĭko, On a converse theorem to the fixed point principle in the theory of the Cauchy problem for linear normal partial differential systems , Dokl. Akad. Nauk. Belarusi, 45(2001), 12–16, (in Russian) \ref\key 117
  • Y.-H. Zhou, J. Yu, H. Yang and S.-W. Xiang, Hadamard types of well-posedness of non-self set-valued mappings for coincide points , Nonlinear Anal., 63(2005), e2427–e2436