Mathematica Bohemica, Vol. 148, No. 2, pp. 223-236, 2023


Fixed point theorems for hybrid pair of weak compatible mappings in partial metric spaces

Santosh Kumar, Johnson Allen Kessy

Received December 19, 2020.   Published online June 6, 2022.

Abstract:  The notions of compatible mappings play a crucial role in metrical fixed point theory. Partial metric spaces are a generalization of the notion of a metric space in the sense that distance of a point from itself is not necessarily zero. In this paper, we prove coincidence and fixed point theorems for a pair of single-valued and multi-valued weak compatible mappings on a complete partial metric space. Our main results generalize, in particular, the results of Kaneko and Sessa (1989), Pathak (1995) and Kessy, Kumar and Kakiko (2017). Examples that illustrate the generality of our results are also provided.
Keywords:  partial metric space; weak compatible mapping; hybrid pair of mapping
Classification MSC:  47H10, 54H25


References:
[1] M. Abbas, B. E. Rhoades: Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. Appl. Math. Comput. 215 (2009), 262-269. DOI 10.1016/j.amc.2009.04.085 | MR 2568327 | Zbl 1185.54037
[2] M. A. Al-Thagafi, N. Shahzad: Generalized $I$-nonexpansive selfmaps and invariant approximations. Acta Math. Sin., Engl. Ser. 24 (2008), 867-876. DOI 10.1007/s10114-007-5598-x | MR 2403120 | Zbl 1175.41026
[3] I. Altun, S. Romaguera: Characterizations of partial metric completeness in terms of weakly contractive mappings having fixed point. Appl. Anal. Discrete Math. 6 (2012), 247-256. DOI 10.2298/AADM120322009A | MR 3012674 | Zbl 1289.54112
[4] I. Altun, H. Simsek: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 1 (2008), 1-8. MR 2498882 | Zbl 1172.54318
[5] H. Aydi, M. Abbas, C. Vetro: Partial Hausdorff metric and Nadler's fixed point theorem on partial metric spaces. Topology Appl. 159 (2012), 3234-3242. DOI 10.1016/j.topol.2012.06.012 | MR 2948281 | Zbl 1252.54027
[6] H. Aydi, A. Felhi, S. Sahmim: A Suzuki fixed point theorem for generalized multivalued mappings on metric-like spaces. Glas. Mat., III. Ser. 52 (2017), 147-161. DOI 10.3336/gm.52.1.11  | MR 3662609 | Zbl 06826012
[7] H. Aydi, A. Felhi, S. Sahmim: Ćirić-Berinde fixed point theorems for multi-valued mappings on $\alpha$-complete metric-like spaces. Filomat 31 (2017), 3727-3740. DOI 10.2298/FIL1712727A | MR 3703868 | Zbl 07418388
[8] H. Bouhadjera, A. Djoudi: General common fixed point theorems for weakly compatible maps. Gen. Math. 16 (2008), 95-107. MR 2439229 | Zbl 1235.54025
[9] M. Bukatin, R. Kopperman, S. Matthews, H. Pajoohesh: Partial metric spaces. Am. Math. Mon. 116 (2009), 708-718. DOI 10.4169/193009709X460831 | MR 2572106 | Zbl 1229.54037
[10] L. Ćirić, B. Samet, H. Aydi, C. Vetro: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput. 218 (2011), 2398-2406. DOI 10.1016/j.amc.2011.07.005 | MR 2838150 | Zbl 1244.54090
[11] R. H. Haghi, S. Rezapour, N. Shahzad: Be careful on partial metric fixed point results. Topology Appl. 160 (2013), 450-454. DOI 10.1016/j.topol.2012.11.004 | MR 3010350 | Zbl 1267.54044
[12] G. Jungck: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 9 (1986), 771-779. DOI 10.1155/S0161171286000935 | MR 0870534 | Zbl 0613.54029
[13] H. Kaneko, S. Sessa: Fixed point theorems for compatible multi-valued and single-valued mappings. Int. J. Math. Math. Sci. 12 (1989), 257-262. DOI 10.1155/S0161171289000293 | MR 0994907 | Zbl 0671.54023
[14] J. Kessy, S. Kumar, G. Kakiko: Fixed points for hybrid pair of compatible mappings in partial metric spaces. Adv. Fixed Point Theory 7 (2017), 489-499.
[15] T. Kubiak: Fixed point theorems for contractive type multivalued mappings. Math. Jap. 30 (1985), 89-101. MR 0828906 | Zbl 0567.54030
[16] S. G. Matthews: Metric Domains for Completeness: PhD Thesis. University of Warwick, Warwick (1985) .
[17] S. G. Matthews: Partial metric topology. Papers on General Topology and Applications. Annals of the New York Academy of Sciences 728. New York Academy of Sciences, New York (1994), 183-197. DOI 10.1111/j.1749-6632.1994.tb44144.x | MR 1467773 | Zbl 0911.54025
[18] P. P. Murthy, S. S. Chang, Y. J. Cho, B. K. Sharma: Compatible mappings of type $(A)$ and common fixed point theorems. Kyungpook Math. J. 32 (1992), 203-216. MR 1203935  | Zbl 0771.54039
[19] S. B. Nadler, Jr.: Multi-valued contraction mappings. Pac. J. Math. 30 (1969), 475-488. DOI 10.2140/pjm.1969.30.475  | MR 0254828 | Zbl 0187.45002
[20] H. K. Pathak: Fixed point theorems for weak compatible multi-valued and single-valued mappings. Acta Math. Hung. 67 (1995), 69-78. DOI 10.1007/BF01874520  | MR 1316710 | Zbl 0821.54027
[21] H. K. Pathak, M. S. Khan: A comparison of various types of compatible maps and common fixed points. Indian J. Pure Appl. Math. 28 (1997), 477-485. MR 1448037 | Zbl 0872.54033
[22] S. Sessa: On a weak commutativity condition of mappings in fixed point considerations. Publ. Inst. Math., Nouv. Sér. 32 (1982), 149-153. MR 0710984 | Zbl 0523.54030
[23] R. E. Smithson: Fixed points for contractive multifunctions. Proc. Am. Math. Soc. 27 (1971), 192-194. DOI 10.1090/S0002-9939-1971-0267564-4  | MR 0267564 | Zbl 0213.24501
[24] C. Vetro, F. Vetro: Common fixed points of mappings satisfying implicit relations in partial metric spaces. J. Nonlinear Sci. Appl. 6 (2013), 152-161. DOI 10.22436/jnsa.006.03.01 | MR 3010868 | Zbl 1432.54086

Affiliations:   Santosh Kumar, Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, 35065 Dar es Salaam, Tanzania, e-mail: drsengar2002@gmail.com; Johnson Allen Kessy, Department of Mathematics, College of Natural and Mathematical Sciences, University of Dodoma, P.O. Box 259 Dodoma, Tanzania, e-mail: johnsonkessy@ymail.com


 
PDF available at: