Generalized Suzuki $(\psi,\phi)$-contraction in complete metric spaces

Document Type : Research Paper


1 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

2 Lagos State University, Ojo Campus, Lagos, Nigeria


In this paper, we introduce the concept of $(\psi, \phi)$-Suzuki and $(\psi, \phi)$-Jungck-Suzuki contraction type mappings and we establish the existence, uniqueness and coincidence results for $(\psi, \phi)$-Suzuki and $(\psi, \phi)$-Jungck-Suzuki contraction mappings in the frame work of complete metric spaces. As an application, we apply our result to find the existence and uniqueness of solutions of a differential equation.


[1] C.T. Aage and J.N. Salunke, Fixed points for weak contractions in G-metric spaces, Appl. Math. E-Notes. 12 (2012) 23–28.
[2] Y. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert space, New results in operator theory and its applications, 7–22, Oper. Theory Adv. Appl., 98, Birkh¨auser, Basel, 1997.
[3] S. Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math. 3 (1922) 133–181.
[4] B.S. Choudhury and C. Bandyopadhyay, Suzuki type common fixed point theorem in complete metric space and partial metric space, Filomat 29(6) (2015) 1377–1387.
[5] D. Doric, Common fixed point for generalized (ψ, φ)-weak contractions, Appl. Math. Lett. 22 (2009) 1896–1900.
[6] P.N. Dutta and B.S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl. 2008 (2008), Article ID 406368, 1–8.
[7] G. Jungck, Commuting mappings and fixed points, Amer. Math. Month. 83(4) (1976) 261–263.
[8] G. Jungck and N. Hussain, Compatible maps and invariant approximations, J. Math. Anal. Appl. 325(2) (2007) 1003–1012.
[9] M. Geraghty, On contractive mappings, Proc. Am. Math. Soc. 40 (1973) 604–608.
[10] K. Goebel, A coincidence theorem, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 16 (1968) 733–735.
[11] J. Harjani and K. Sadarangni, Fixed point theorems for weakly contraction mappings in partially ordered sets, Nonlinear Anal. 71 (2009) 3403–3410.
[12] J. Harjani and K. Sadarangni, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (2010) 1188–1197.
[13] R. Kannan, Some results on fixed point theorems, Bull. Calcutta Math. Soc. 10 (1968) 71–76.
[14] E. Karapinar and K. Tas, Generalized (C)-conditions and related fixed point theorems, Comput. Math. Appl. 61 (2011) 3370–3380.
[15] M. S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30(1) (1984) 1–9.
[16] A.A. Mebawondu, C. Izuchukwu, K.O. Aremu and O.T. Mewomo, Some fixed point results for a generalized TAC-Suzuki-Berinde type F-contractions in b-metric spaces, Appl. Math. E-Notes 19 (2019) 629–653.
[17] A.A. Mebawonduand O.T. Mewomo, Some fixed point results for TAC-Suzuki contractive mappings, Commun. Korean Math. Soc. 34(4) (2019) 1201–1222.
[18] A.A. Mebawondu and O.T. Mewomo, Some convergence results for Jungck-AM iterative process in hyperbolic spaces, Aust. J. Math. Anal. Appl. 16(1) (2019), Art. 15, 20 pp.
[19] A. A. Mebawondu and O.T. Mewomo, Suzuki-type fixed point results in Gb-metric spaces, Asian-Eur. J. Math. (2020). DOI: 10.1142/S1793557121500704
[20] S. Moradi, Kannan fixed point theorem on complete metric spaces and on generalized metric spaces depended on another function, arXiv:0903.1577v1 [math.FA].
[21] J. Morales and E. Rojas, Some generalizations of Jungck’s fixed point theorem, Int. J. Math. Math. Sci. 19 (2012) 1–19.
[22] J. Morales and E. Rojas, Some results on T-Zamfirescu operators, Revista Notas Mat. 5(1) (2009) 64–71.
[23] J. Morales and E. Rojas, Cone metric spaces and fixed point theorems of T-Kannan contractive mappings, Int. J. Math. Anal. 4(4) (2010) 175–184.
[24] J. Morales and E. Rojas, T-Zamfirescu and T-weak contraction mappings on cone metric spaces, arXiv:0909.1255v1 [math.FA].
[25] P.P. Murthy, L.N. Mishra and U.D. Patel, n-tupled fixed point theorems for weak-contraction in partially ordered complete G-metric spaces, New Trends Math. Sci. 3(4) (2015) 50–75.
[26] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47(4) (2001) 2683–2693.
[27] W. Shatanawi and M. Postolache, Some fixed-point results for a G-weak contraction in G-metric spaces, Abstr. Appl. Anal. 2012 (2012) 1–19.
[28] P. Salimi and V. Pasquale, A result of Suzuki type in partial G-metric spaces, Acta Math. Sci. Ser. B (Engl. Ed.) 34(2) (2014) 274–284.
[29] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340(2) (2008) 1088–1095.
[30] T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal. 71(11) (2009) 5313–5317.
[31] M. Urmila, N.K. Hemant Kumar and Rashmi, Fixed point theory for generalized (ψ, φ)-weak contractions involving f − g reciprocally continuity, Indian J. Math. 56(2) (2014) 153–168.
[32] Z. Qingnian and S. Yisheng, Fixed point theory for generalized φ-weak contractions, Appl. Math. Lett. 22(1) (2009) 75–78.
[33] F. Xiaoming and W. Zhigang, Some fixed point theorems in generalized quasi-partial metric spaces, J. Nonlinear Sci. Appl. 9(4) (2016) 1658–1674.
Volume 12, Issue 1
May 2021
Pages 963-978
  • Receive Date: 18 September 2018
  • Accept Date: 28 September 2020