A Kirk type characterization of completeness for partial metric spaces. (English) Zbl 1193.54047
Let \((X,d)\) a metric space. A mapping \(f:X\to X\) is called a Caristi’s mapping if there exists a lower semicontinuous function \( \phi :X\to [0,+\infty )\) satisfying
\[ d(x,f x)\leq \phi (x)-\phi (f x), \]
for all \(x\in X\). W. A. Kirk [Colloq. Math. 36, 81–86 (1976; Zbl 0353.53041)] proved that a metric space \((X,D)\) is complete if and only if every Caristi’s mapping has a fixed point. A map \(p:X\times X\to [0,+\infty )\) is said to be a partial metric if for all \(x,y,x \in X\): (i) \(x=y \Leftrightarrow p(x,x)=p(x,y)=p(y,y)\); (ii) \(p(x,x)\leq p(x,y)\); (iii) \(p(x,y)=p(y,x)\), (iv)\( p(x,z)\leq p(x,y)+ p(y,z)-p(y,y).\) In this paper the author studies the relationship between the existence of fixed points for Caristi’s type mappings and completeness in a partial metric space.
\[ d(x,f x)\leq \phi (x)-\phi (f x), \]
for all \(x\in X\). W. A. Kirk [Colloq. Math. 36, 81–86 (1976; Zbl 0353.53041)] proved that a metric space \((X,D)\) is complete if and only if every Caristi’s mapping has a fixed point. A map \(p:X\times X\to [0,+\infty )\) is said to be a partial metric if for all \(x,y,x \in X\): (i) \(x=y \Leftrightarrow p(x,x)=p(x,y)=p(y,y)\); (ii) \(p(x,x)\leq p(x,y)\); (iii) \(p(x,y)=p(y,x)\), (iv)\( p(x,z)\leq p(x,y)+ p(y,z)-p(y,y).\) In this paper the author studies the relationship between the existence of fixed points for Caristi’s type mappings and completeness in a partial metric space.
Reviewer: Genaro López Acedo (Sevilla)
Citations:
Zbl 0353.53041References:
| [1] | Caristi J: Fixed point theorems for mappings satisfying inwardness conditions.Transactions of the American Mathematical Society 1976, 215: 241-251. · Zbl 0305.47029 · doi:10.1090/S0002-9947-1976-0394329-4 |
| [2] | Beg I, Abbas M: Random fixed point theorems for Caristi type random operators.Journal of Applied Mathematics & Computing 2007,25(1-2):425-434. 10.1007/BF02832367 · Zbl 1148.47044 · doi:10.1007/BF02832367 |
| [3] | Downing D, Kirk WA: A generalization of Caristi’s theorem with applications to nonlinear mapping theory.Pacific Journal of Mathematics 1977,69(2):339-346. · Zbl 0357.47036 · doi:10.2140/pjm.1977.69.339 |
| [4] | Feng Y, Liu S: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings.Journal of Mathematical Analysis and Applications 2006,317(1):103-112. 10.1016/j.jmaa.2005.12.004 · Zbl 1094.47049 · doi:10.1016/j.jmaa.2005.12.004 |
| [5] | Jachymski JR: Caristi’s fixed point theorem and selections of set-valued contractions.Journal of Mathematical Analysis and Applications 1998,227(1):55-67. 10.1006/jmaa.1998.6074 · Zbl 0916.47044 · doi:10.1006/jmaa.1998.6074 |
| [6] | Khamsi MA: Remarks on Caristi’s fixed point theorem.Nonlinear Analysis: Theory, Methods & Applications 2009,71(1-2):227-231. 10.1016/j.na.2008.10.042 · Zbl 1175.54056 · doi:10.1016/j.na.2008.10.042 |
| [7] | Kirk WA: Caristi’s fixed point theorem and metric convexity.Colloquium Mathematicum 1976,36(1):81-86. · Zbl 0353.53041 |
| [8] | Latif, A., Generalized Caristi’s fixed point theorems, No. 2009, 7 (2009) · Zbl 1171.54031 |
| [9] | Kirk WA, Caristi J: Mappings theorems in metric and Banach spaces.Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 1975,23(8):891-894. · Zbl 0313.47041 |
| [10] | Suzuki T: Generalized Caristi’s fixed point theorems by Bae and others.Journal of Mathematical Analysis and Applications 2005,302(2):502-508. 10.1016/j.jmaa.2004.08.019 · Zbl 1059.54031 · doi:10.1016/j.jmaa.2004.08.019 |
| [11] | Park S: Characterizations of metric completeness.Colloquium Mathematicum 1984,49(1):21-26. · Zbl 0556.54021 |
| [12] | Reich S: Kannan’s fixed point theorem.Bollettino dell’Unione Matematica Italiana 1971, 4: 1-11. · Zbl 0219.54042 |
| [13] | Subrahmanyam PV: Completeness and fixed-points.Monatshefte für Mathematik 1975,80(4):325-330. 10.1007/BF01472580 · Zbl 0312.54048 · doi:10.1007/BF01472580 |
| [14] | Suzuki T, Takahashi W: Fixed point theorems and characterizations of metric completeness.Topological Methods in Nonlinear Analysis 1996,8(2):371-382. · Zbl 0902.47050 |
| [15] | Dhompongsa, S.; Yingtaweesittikul, H., Fixed points for multivalued mappings and the metric completeness, No. 2009, 15 (2009) · Zbl 1179.54055 |
| [16] | Suzuki T: A generalized Banach contraction principle that characterizes metric completeness.Proceedings of the American Mathematical Society 2008,136(5):1861-1869. · Zbl 1145.54026 · doi:10.1090/S0002-9939-07-09055-7 |
| [17] | Matthews, SG, Partial metric topology, No. 728, 183-197 (1994), New York, NY, USA · Zbl 0911.54025 |
| [18] | Heckmann R: Approximation of metric spaces by partial metric spaces.Applied Categorical Structures 1999,7(1-2):71-83. · Zbl 0993.54029 · doi:10.1023/A:1008684018933 |
| [19] | O’Neill, SJ, Partial metrics, valuations, and domain theory, No. 806, 304-315 (1996), New York, NY, USA · Zbl 0889.54018 |
| [20] | Romaguera S, Schellekens M: Partial metric monoids and semivaluation spaces.Topology and Its Applications 2005,153(5-6):948-962. 10.1016/j.topol.2005.01.023 · Zbl 1084.22002 · doi:10.1016/j.topol.2005.01.023 |
| [21] | Romaguera S, Valero O: A quantitative computational model for complete partial metric spaces via formal balls.Mathematical Structures in Computer Science 2009,19(3):541-563. 10.1017/S0960129509007671 · Zbl 1172.06003 · doi:10.1017/S0960129509007671 |
| [22] | Schellekens M: The Smyth completion: a common foundation for denotational semantics and complexity analysis.Electronic Notes in Theoretical Computer Science 1995, 1: 535-556. · Zbl 0910.68135 · doi:10.1016/S1571-0661(04)00029-5 |
| [23] | Schellekens MP: A characterization of partial metrizability: domains are quantifiable.Theoretical Computer Science 2003,305(1-3):409-432. · Zbl 1043.54011 · doi:10.1016/S0304-3975(02)00705-3 |
| [24] | Waszkiewicz P: Quantitative continuous domains.Applied Categorical Structures 2003,11(1):41-67. 10.1023/A:1023012924892 · Zbl 1030.06005 · doi:10.1023/A:1023012924892 |
| [25] | Waszkiewicz P: Partial metrisability of continuous posets.Mathematical Structures in Computer Science 2006,16(2):359-372. 10.1017/S0960129506005196 · Zbl 1103.06004 · doi:10.1017/S0960129506005196 |
| [26] | Penot J-P: Fixed point theorems without convexity.Bulletin de la Société Mathématique de France 1979, (60):129-152. · Zbl 0454.47044 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
