Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. (English) Zbl 1252.54027

Aydi, Hassen; Abbas, Mujahid; Vetro, Calogero

Topology Appl. 159, No. 14, 3234-3242 (2012).

Summary: We introduce the concept of a partial Hausdorff metric. We initiate study of fixed point theory for multi-valued mappings on partial metric space using the partial Hausdorff metric and prove an analogous to the well-known Nadler’s fixed point theorem. Moreover, we give a homotopy result as application of our main result.

MSC:

54H25	Fixed-point and coincidence theorems (topological aspects)
54E35	Metric spaces, metrizability
54C60	Set-valued maps in general topology

Keywords:

partial Hausdorff metric; multi-valued mappings; Nadler’s fixed point theorem

References:

[1]	Abbas, M.; Nazir, T., Fixed point of generalized weakly contractive mappings in ordered partial metric spaces, Fixed Point Theory Appl., 2012, 1, 19 pp (2012) · Zbl 1281.54015
[2]	Altun, I.; Simsek, H., Some fixed point theorems on dualistic partial metric spaces, J. Adv. Math. Stud., 1, 1-8 (2008) · Zbl 1172.54318
[3]	Altun, I.; Sola, F.; Simsek, H., Generalized contractions on partial metric spaces, Topology Appl., 157, 2778-2785 (2010) · Zbl 1207.54052
[4]	Bukatin, M. A.; Shorina, S. Yu., Partial metrics and co-continuous valuations, (Nivat, M.; etal., Foundations of Software Science and Computation Structure. Foundations of Software Science and Computation Structure, Lecture Notes in Comput. Sci., vol. 1378 (1998), Springer), 125-139 · Zbl 0945.06006
[5]	Ćirić, L.j.; Samet, B.; Aydi, H.; Vetro, C., Common fixed points of generalized contractions on partial metric spaces and an application, Appl. Math. Comput., 218, 2398-2406 (2011) · Zbl 1244.54090
[6]	Damjanovic, B.; Samet, B.; Vetro, C., Common fixed point theorems for multi-valued maps, Acta Math. Sci. Ser. B Engl. Ed., 32, 818-824 (2012) · Zbl 1265.54166
[7]	Di Bari, C.; Vetro, P., Fixed points for weak \(φ\)-contractions on partial metric spaces, Int. J. Eng., Contemp. Math. Sci., 1, 1, 5-13 (2011)
[8]	Heckmann, R., Approximation of metric spaces by partial metric spaces, Appl. Categ. Structures, 7, 71-83 (1999) · Zbl 0993.54029
[9]	Matthews, S. G., Partial metric topology, (Proc. 8th Summer Conference on General Topology and Applications. Proc. 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci., vol. 728 (1994)), 183-197 · Zbl 0911.54025
[10]	Paesano, D.; Vetro, P., Suzukiʼs type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topology Appl., 159, 3, 911-920 (2012) · Zbl 1241.54035
[11]	Romaguera, S., A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl., 2010 (2010), Article ID 493298, 6 pp · Zbl 1193.54047
[12]	Romaguera, S.; Valero, O., A quantitative computational model for complete partial metric spaces via formal balls, Math. Structures Comput. Sci., 19, 541-563 (2009) · Zbl 1172.06003
[13]	Schellekens, M. P., The correspondence between partial metrics and semivaluations, Theoret. Comput. Sci., 315, 135-149 (2004) · Zbl 1052.54026
[14]	Nadler, S. B., Multivalued contraction mappings, Pacific J. Math., 30, 475-488 (1969) · Zbl 0187.45002