Some new common fixed point theorems for Geraghty contraction type maps in partial metric spaces

Document Type: Original Article

Author

Abstract

In this paper, we prove some new common fixed point theorems for Geraghtys type contraction mappings on partial metric spaces. Theorems presented are
generalizations of fixed point theorems of Altun et al. [Generalized Geraghty type mappings on partial metric spaces and fixed point results, Arab. J. Math. 2, (2013), no. 3, 247-253]. We also give some examples to illustrate the usability of the obtained results

Keywords

[1] M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341(1) (2008), 416-420.
[2] T. Abdeljawad, E. Karapinar, K. Tas, Existence and uniqueness of a common fixed point on partial metric spaces, Appl. Math. Lett. 24 (2011), no. 11, 1900-1904.
[3] R. P. Agarwal, P. Ravi, M. A. Alghamdi, N. Shahzad, Fixed point theory for cyclic generalized contractions in partial metric spaces, Fixed Point Theory Appl. 2012 (2012), 40, 11 Pgs.
[4] A. Aghanians, K. Fallahi, K. Nourouzi, D. O'Regan, Some coupled coincidence point theorems in partially ordered uniform spaces. Cubo 16 (2014), no. 2, 121-134.
[5] I. Altun, K. Sadarangani, Generalized Geraghty type mappings on partial metric spaces and fixed point results, Arab. J. Math. 2 (2013), no. 3, 247-253. 
[6] I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology Appl. 157 (2010), no. 18, 2778-2785.
[7] H. Aydi, M. Abbas and C. Vetro, Partial Hausdorff metric and Nadler's fixed point theorem on partial metric spaces. Topology. Appl. 159 (2012), no. 14, 3234-3242.
[8] P. Charoensawan, Common fixed point theorems for Geraghty's type contraction mapping with two generalized metrics endowed with a directed graph in JS-metric spaces, Carpathian J. Math. 34 (2018), no. 3, 305-312.
[9] Lj. Ciric, Some recent results in metrical fixed point theory, Beograd, (2003). 
[10] B. Deshpande, A. Handa, Utilizing isotone mappings under Geraghty-type contraction to prove multidimensional fixed point theorems with application. J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 25 (2018), no. 4,279-295.
[11] H. Faraji, D. Savic and S. Radenovic, Fixed point theorems for Geraghty contraction type mappings in b-metric spaces and applications, Axioms, 8(34), (2019), 12 pages.
[12] M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604-608.
[13] R. H. Haghi, SH. Rezapour, N. Shahzad, Be careful on partial metric fixed point results. Topology Appl. 160 (2013), no. 3, 450-454.
[14] H. Huang, L. Paunovic, S. Radenovic, On some new fixed point results for rational Geraghty contractive mappings in ordered b-metric spaces. J. Nonlinear Sci. Appl. 8 (2015), no. 5, 800-807.
[15] Z. Kadelburg, H. K. Nashine, S. Radenovic. Fixed point results under various contractive conditions in partial metric spaces. Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat. RACSAM 107 (2013), no. 2, 241-256.
[16] W. Kirk, N. Shahzad, Fixed point theory in distance spaces, Springer, (2014). 
[17] W. Long, S. Khaleghizadeh, P. Salimi, S. Radenovic, S. Shukla, Some new fixed point results in partial ordered metric spaces via admissible mappings. Fixed Point Theory Appl. 2014 (2014):117, 18 Pgs.
[18] S. G. Matthews, Partial metric topology, Research Report 212, Dept. of Computer Science, University of Warwick, (1992). 
[19] S. G. Matthews, Partial metric topology, Papers on general topology and applications (Flushing, NY, 1992),183-197, Ann. New York Acad. Sci., 728, New York Acad. Sci., New York, (1994).
[20] S. Oltra, O. Valero, Banach's fixed point theorem for partial metric spaces, Rend. Instit. Mat. Univ. Trieste 36 (2004), no. 1-2, 17-26.
[21] S. Radenovic, Coincidence point results for nonlinear contraction in ordered partial metric spaces, J. Indian Math. Soc. (N.S.) 81 (2014), no. 3-4, 319-333. 
[22] B. Samet, Existence and uniqueness of solutions to a system of functional equations and applications to partial metric spaces, Fixed Point Theory 14 (2013), no. 2, 473-481. 
[23] O. Valero, On Banach fixed point theorems for partial metric spaces. Appl. Gen. Topol. 6 (2005), no. 2, 229-240.