Document Type : Research Paper


Department of Mathematics‎, ‎Faculty of Science and Arts‎, ‎Giresun University‎, ‎Turkey


‎In this paper‎, ‎we introduce more general contractions called $\varphi $-fixed‎ ‎point point for $(F,\varphi‎ ,‎\alpha )_{s}$ and $(F,\varphi‎ ,‎\alpha )_{s}$-‎weak contractions‎. ‎We prove the existence and uniqueness of $\varphi $-‎fixed point point for $(F,\varphi‎ ,‎\alpha )_{s}$ and $(F,\varphi‎ ,‎\alpha‎‎)_{s}$-weak contractions in complete $b$-metric spaces‎. ‎Some examples are‎ ‎supplied to support the usability of our results‎. ‎As applications‎, ‎necessary‎ ‎conditions to ensure the existence of a unique solution for a nonlinear‎ ‎inequality problem are also discussed‎. ‎Also‎, ‎some new fixed point results in‎ ‎partial metric spaces are proved.


Main Subjects

[1] A. Akbar, M. Gabeleh, Global optimal solutions of noncyclic mappings in metric spaces, J. Optim. Theory Appl. 153 (2012), 298-305.
[2] A. Ansari, H. Isık, S. Radenovic, Coupled fixed point theorems for contractive mappings involving new function classes and applications, Filomat. 31 (7) (2017), 1893-1907.
[3] I. A. Bakhtin, The contraction principle in quasi-metric spaces, Funct. Anal. 30 (1989), 26-37.
[4] S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math. 3 (1922), 133-181.
[5] V. Berinde, Contract ¸ii generalize ¸ si applicat ¸ii, Editura Club Press 22, Baia Mare, 1997.
[6] V. Berinde, Generalized contractions in quasimetric spaces, Seminar on Fixed Point Theory. 3 (1993), 3-9.
[7] V. Berinde, Sequences of operators and fixed points in quasimetric spaces, Stud. Univ. Babe¸ s-Bolyai Math. 16 (4) (1996), 23-27.
[8] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1 (1993), 5-11.
[9] H. Huang, G. Deng, Z. Chen, S. Radenovic, On some recent fixed point results for α-admissible mappings in b-metric spaces, J. Comp. Anal. Appl. 25 (2) (2018), 255-269.
[10] N. Hussain, E. Karapınar, P. Salimi, F. Akbar, α-admissible mappings and related fixed point theorems, Fixed Point Theory Appl. (2013), 2013:114.
[11] H. Isık, N. Hussain, M. A. Kutbi, Generalized rational contractions endowed with a graph and an application to a system of integral equations, J. Comput. Anal. Appl. 22 (6) (2017), 1158-1175.
[12] H. Isık, P. Kumam, Fixed points under new contractive conditions via cyclic (α,β,r)-admissible mappings, J. Advanced. Math. Studies. 11 (1) (2018), 17-23.
[13] H. Isık, S. Radenovic, A new version of coupled fixed point results in ordered metric spaces with applications. UPB. Sci. Bull. (Series A). 79 (2) (2017), 131-138.
[14] H. Isık, D. Turkoglu, Generalized weakly α-contractive mappings and applications to ordinary differential equations. Miskolc. Math. Notes. 17 (1) (2016), 365-379.
[15] M. Jleli, B. Samet, C. Vetro, Fixed point theory in partial metric spaces via φ-fixed point’s concept in metric spaces, J. Inequl. Appl. (2014), 2014:426.
[16] A. Latif, J. R. Roshan, V. Parvaneh, N. Hussain, Fixed point results via α−admissible mappings and cyclic contractive mappings in partial b-metric spaces, J. Inequal. Appl. (2014), 2014:345.
[17] S. G. Matthews, Partial metric topology, Ann. N.Y. Acad. Sci. 728 (1994), 183-197.
[18] R. Miculescu, A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl. 19 (2017), 2153-2163.
[19] Z. Mustafa, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, J. Inequal. Appl. (2013), 2013:562.
[20] M. Nazam, M. Arshad, C. Park, A common fixed point theorem for a pair of generalized contraction mappings with applications, J. Comp. Anal. Appl. 25 (3) (2018), 552-564.
[21] M. Pacurar, A fixed point result for φ-contractions on b-metric spaces without the boundedness assumption,
Fasc. Math. 43 (2010), 127-137.
[22] S. Phiangsungnoen, W. Sintunavarat, P. Kumam, Fixed point results, generalized Ulam-Hyers stability and well-posedness via α-admissible mappings in b-metric spaces, Fixed Point Theory and Appl. (2014), 2014:188.
[23] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α−φ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154-2165.
[24] S. Shukla, Partial b-metric spaces and fixed point theorems, Mediterr. J. Math. 11 (2014), 703-711.
[25] D. Singh, V. Chauhan, R. Wangkeeree, Geraghty type generalized F-contractions andrelated applications in partial b-metric spaces, Inte. J. Analysis. (2017), 2017:8247925.
[26] W. Sintunavarat, Nonlinear integral equations with new admissibility types in b-metric spaces, J. Fixed Point Theory Appl. 18 (2) (2016), 397-416.
[27] C. Zhu, W. Xu, C. Chen, X. Zhang, Common fixed point theorems for generalized expansive mappings in partial b-metric spaces and an application, J. Inequal. Appl. (2014), 2014:475.