**Volume 18, Issue 2, pp 216--231**

**Publication Date**: 2018-02-01

**Zead Mustafa**
- Department of Mathematics, Statistics and Physics, Qatar University, Doha-Qatar

**M. M. M. Jaradat**
- Department of Mathematics, Statistics and Physics, Qatar University, Doha-Qatar

**Arslan Ansari**
- Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

**Feng Gu**
- Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

**Hui-hui Zheng**
- Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

**Stojan Radenović**
- Faculty of Mechanical Engineering, University of Belgrade, Serbia

**M. S. Bataineh**
- Department of Mathematics, University of Sharjah, Sharjah, United Arab Emirates

The purpose of this paper is to introduce common fixed point results for two pairs of weakly compatible self-mappings in partial metric space using \(C\)-class functions on \((\psi,\varphi)\)-contractive condition. Example and application on integral equations are presented to illustrate the main result. Our results extend and generalize well know results in the literature.

(\(\psi, \varphi\))-Contractive mapping, common fixed point, coincidence point, partial metric space, weakly compatible mappings, \(C\)-class functions, integral equations

[1] T. Abdeljawad, E. Karapınar, K. Taş, Existence and uniqueness of a common fixed point on partial metric spaces, Appl. Math. Lett., 24 (2011), 1900–1904.

[2] Abdullah, M. Sarwar, Z. Mustafa, M. M. M. Jaradat, Common fixed points of (\(\phi,\psi\))-contraction on G-metric space using E.A property, J. Math. Anal., 8 (2017), 136–146.

[3] A. H. Ansari, Note on “\(\varphi-\psi\)-contractive type mappings and related fixed point, 2nd Regional Conf. Math. Appl., PNU, Iran, (2014), 377–380.

[4] H. Aydi, Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces, J. Nonlinear Anal. Optim., 2 (2011), 269–284.

[5] H. Aydi, Some fixed point results in ordered partial metric spaces, J. Nonlinear Sci. Appl., 4 (2011), 210–217.

[6] A. Bejenaru, A. Pitea, Fixed point and best proximity point theorems on partial metric spaces, J. Math. Anal., 7 (2016), 25–44.

[7] 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.

[8] A. Erduran, Z. Kadelburg, H. K. Nashine, C. Vetro, A fixed point theorem for (\(\phi, L\))-weak contraction mappings on a partial metric space, J. Nonlinear Sci. Appl., 7 (2014), 196–204

[9] D. Ilić, V. Pavlović , V. Rakočević, Some new extensions of Banach’s contraction principle to partial metric space, Appl. Math. Lett., 24 (2011), 1326–1330.

[10] M. M. M. Jaradat, Z. Mustafa, A. H. Ansari, S. Chandok, Ć. Dolićanin, Some approximate fixed point results and application on graph theory for partial (h, F)-generalized convex contraction mappings with special class of functions on complete metric space, J. Nonlinear Sci. Appl., 10 (2017), 1695–1708.

[11] M. M. M. Jaradat, Z. Mustafa, A. H. Ansari, P. S. Kumari, D. Dolicanin-Djekic, H. M. Jaradat, Some fixed point results for \(F_{\alpha,\omega \varphi}\)-generalized cyclic contractions on metric-like space with applications to graphs and integral equations, J. Math. Anal., 8 (2017), 28–45

[12] M. M. M. Jaradat, Z. Mustafa, S. U. Khan, M. Arshad, J. Ahmad, Some fixed point results on G-metric and \(G_b\)-metric spaces, Demonstr. Math., 50 (2017), 190–207.

[13] A. Kaewcharoen, T. Yuying, Unique common fixed point theorems on partial metric spaces, J. Nonlinear Sci. Appl., 7 (2014), 90–101.

[14] E. Karapınar, Ćirić types nonunique fixed point theorems on partial metric spaces, J. Nonlinear Sci. Appl., 5 (2012), 74–83.

[15] M. S. Khan, M. Swalesh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30 (1984), 1–9.

[16] V. La Rosa, P. Vetro, Fixed points for Geraghty-contractions in partial metric spaces, J. Nonlinear Sci. Appl., 7 (2014), 1–10.

[17] S. G. Matthews, Partial metric topology, Papers on general topology and applications, Flushing, NY, (1992), Ann. New York Acad. Sci., New York Acad. Sci., New York, 728 (1994), 183–197.

[18] Z. Mustafa, Common fixed points of weakly compatible mappings in G-metric spaces, Appl. Math. Sci. (Ruse), 6 (2012), 4589–4600.

[19] Z. Mustafa, T. V. An, N. V. Dung, Two fixed point theorems for maps on incomplete G-metric spaces, Appl. Math. Sci. (Ruse), 7 (2013), 2271–2281.

[20] Z. Mustafa, M. Arshad, S. U. Khan, J. Ahmad, M. M. M. Jaradat, Common fixed points for multivalued mappings in G-metric spaces with applications, J. Nonlinear Sci. Appl., 10 (2017), 2550–2564.

[21] Z. Mustafa, H. Aydi, E. Karapınar, On common fixed points in G-metric spaces using (E.A) property, Comput. Math. Appl., 64 (2012), 1944–1956.

[22] Z. Mustafa, M. M. M. Jaradat, A. H. Ansari, B. Z. Popović, H. Jaradat, C-class functions with new approach on coincidence point results for generalized \((\psi,\varphi)\)-weakly contractions in ordered b-metric spaces, SpringerPlus, 5 (2016), 1–18.

[23] Z. Mustafa, M. M. M. Jaradat, H. M. Jaradat, Some common fixed point results of graphs on b-metric space, J. Nonlinear Sci. Appl., 9 (2016), 4838–4851.

[24] Z. Mustafa, M. M. M. Jaradat, H. M. Jaradat, A remarks on the paper ”Some fixed point theorems for generalized contractive mappings in complete metric spaces”, J. Math. Anal., 8 (2017), 17–22.

[25] Z. Mustafa, M. M. M. Jaradat, E. Karapınar, A new fixed point result via property P with an application, J. Nonlinear Sci. Appl., 10 (2017), 2066–2078.

[26] Z. Mustafa, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, J. Inequal. Appl., 2013 (2013), 26 pages.

[27] Z. Mustafa, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Fixed point theorems for weakly T-Chatterjea and weakly T-Kannan contractions in b-metric spaces, J. Inequal. Appl., 2014 (2014), 14 pages.

[28] H. K. Nashine, Z. Kadelburg, S. Radenović, Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces, Math. Comput. Modelling, 57 (2013), 2355–2365.

[29] S. Oltra, O. Valero, Banach’s fixed point theorem for partial metric spaces, Rend. Istit. Mat. Univ. Trieste, 36 (2004), 17–26.

[30] S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl., 2010 (2010), 6 pages.

[31] H.-H. Zheng, F. Gu, Some results of common fixed point for four self-maps satisfying a new \(\Psi\)-contractive condition in partial metric spaces, J. Nonlinear Sci. Appl., 9 (2016), 2258–2272.