Document Type : Original Article


Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria


In this work, a collection of various fixed point results of $\Theta$-contractions are examined. Important results from when the concept was introduced up to the recent developments are discussed. Hence, the aim of this paper is to collate and analyze the advances of fixed point results in the setting of $\Theta$-contractions which will be helpful and handy for researchers in fixed point theory and related domains.


[1] T. Abdeljawad, R. P. Agarwal, E. Karapinar, and P. S. Kumari, Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-Metric space. Symmetry, 11(5) (2019), 686.
[2] I. Altun, N. Hussain, M. Qasim, and H. H. Al-Sulami, A new fixed point result of Perov type and its application to a semilinear operator system, Mathematics, 7(11) (2019), 1019.
[3] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrale, Fund. Math., 3 (1922), 133-181.
[4] R.K. Bisht and R.P. Pant, A remark on discontinuity at fixed point, J. Math. Anal. Appl., 445 (2017), 1239–1241.
[5] A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Pub. Math. Debrecen, 57(1-2) (2000), 31-37.
[6] S. Chaipornjareansri, Fixed point theorems for generalized θ-ϕ-contractions in S-metric spaces, Ann. Meet. Math. 2018, Thai J. Math., Special Issue (2019), 46-59.
[7] T. Hu, D. Zheng, and J. Zhou, Some new fixed point theorems on partial metric spaces, Int. J. Math. Anal., 12 (2018), 343-352.
[8] N. Hussain, A. E. Al-Mazrooei, and J. Ahmed, Fixed point results for generalized (α-η)-θ contractions with applications, J. Nonlinear Sci. Appl., 10(8) (2017), 4197-4208.
[9] A. Hussain and M. Adeel, Remark on new fixed point theorems for α-Hθ-contractions in ordered metric spaces, J. Fixed Point Theory Appl., 21(2) (2019), 63.
[10] M. Imdad, W. A. Alfaqih, and I. A. Khan, Weak θ-contractions and some fixed point results with applications to fractal theory, Adv. Differ. Equ., 2018(1) (2018), 1-18.
[11] M. Imded, B. Ali, W. M. Alfaqih, S. Sessa, and A. Aldurayhim, New fixed Point results via (θ, ψ)R-weak contractions with an application, Symmetry, 12(6) (2020), 887.
[12] K. Javed, M. Arshad, A. S. Baazeem, and N. Mlaiki, Solving a fractional differential equation via θ-contractions in R-complete metric spaces, Aims Math. 7(9) (2022), 16869-16888.
[13] J. A. Jiddah, M. Alansari, O. M. Mohamed, M. S. Shagari, and A. A. Bakery, Fixed point results of Jaggi-type hybrid contraction in generalized metric spaces, J. Funct. Spaces, 2022 (2022).
[14] J. A. Jiddah, M. Noorwali, M. S. Shagari, S. Rashid, and F. Jarad, Fixed point results of a new family of hybrid contractions in generalized metric space with applications, Aims Math., 7(10) (2022), 17894-17912.
[15] M. Jleli, E. Karapinar, and B. Samet, Further generalizations of the Banach contraction principle, J. Inequal. Appl., 2014(1) (2014), 1-9.
[16] M. Jleli and B. Samet, B. (2014). A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), 1-8.
[17] S. Kanwal, A. Ali, A. Al Mazrooei, and G. Santos-Garcia, Existence of fuzzy fixed points of set-valued fuzzy mappings in metric and fuzzy metric spaces, Aims Math., 8(5) (2022), 10095–10112.
[18] S. Kanwal and A. Azam, Bounded lattice fuzzy coincidence theorems with applications, J. Intell. Fuzzy Syst. 36 (2019), 1-15.
[19] S. Kanwal, A. Azam, and F. A. Shami, On coincidence theorem in intuitionistic fuzzy b-metric spaces with application, J. Funct. Spaces, 2022 (2022), Article ID 5616824, 10 pages.
[20] S. Kanwal, U. Hanif, M.E. Noorwali, and M.A. Alam, On fixed-point results of generalized contractions, J. Funct. Spaces, 2022 (2022), Article ID 9167716, 11 pages.
[21] S. Kanwal, M. S. Shagari, H. Aydi, A. Mukheimer, and T. Abdeljawad, Common fixed-point results of fuzzy mappings and applications on stochastic Volterra integral equations, J. Inequal. Appl. 2022(1) (2022), 1-15.
[22] A. Kari, M. Rossafi, E. Marhrani, and M. Aamri, New fixed point theorems for θ-ϕ-contraction on rectangular b-metric spaces, Int.J. Math. Math. Sci., 2020 (2020), 12 pages.
[23] X. Liu, S. Chang, Y. Xiao, and L. Zhao, Existence of fixed points for θ-type contraction and θ-type Suzuki contraction in complete metric spaces, Fixed Point Theory Appl., 2016 (2016), 1-12.
[24] S. G. Matthews, Partial metric topology, Proc. 8th Summer Conf. General Topology Appl., Ann. New York Acad. Sci., 728(1) (1994), 183-197.
[25] G. Minak and I. Altun, Ordered θ-contractions and some fixed point results, J. Nonlinear Funct. Anal., 2017 (2017), Article ID 41, 1-9.
[26] B. Mohammadi, F. Golkarmanesh, and V. Parvaneh, Common fixed point results via implicit contractions for multi-valved mappings on b-metric like spaces, Cogent Math. Statist., 5(1) (2018), Article ID 1493761.
[27] Z. Ma, A. Hussain, M. Adeel, N. Hussain, and E. Savas, Best proximity point results for generalized θ-contractions and application to matrix equations, Symmetry 11 (2019), no. 1, 93.
[28] W. Onsod, P. Kumam, and T. Saleewong, Fixed point of Suzuki-Geraghty type θ-contractions in partial metric spaces with some applications, Sci. Technol. Asia, 28(1) (2023), 17-25.
[29] W. Onsod, T. Saleewong, J. Ahmad, A.E. Al-Mazrooei, and P. Kumam, Fixed Points of a θ-contraction on metric spaces with a graph, Commun. Nonlinear Anal., 2 (2016), page 139-149.
[30] V. Parvaneh, Some J-S type fixed Point Theorems for α-θ-Generalzed Contraction in b-Metric spaces, J. Math. Anal., 8(3) (2017), 135-147.
[31] V. Parvaneh, F. Golkarmanesh, N. Hussain, and P. Salimi, New Fixed point theorems for α-Hθ-contractions in ordered metric spaces, J. Fixed Point Theory Appl., 18(4) (2016), 905-925.
[32] V. Parvaneh, N. Hussain, A. Mukheimer, and H. Aydi, On fixed point results for modifies θ-contractions with applications, Axioms, 8(3) 2019, 13 pages.
[33] A. Perveen, W.M. Alfaqih, S. Sessa, and M. Imdad, θ∗-weak contractions and discontinuity at the fixed point with applications to matrix and integral equations, Axioms 10(3) (2021), 209.
[34] M. Rossafi, A. Kari, E. Marhrani, and M. Aamri, fixed point theorems for generalized θ-ϕ-Expansive mapping in rectangular metric spaces, Abstr. Appl. Anal., 2021 (2021), Article ID 6642723, 10 pages.
[35] M. Rossafi, A. Kari, C. Park, J.R. Lee, New fixed point theorems for θ-ϕ-contraction on bmetric spaces, J. Math. Comput. Sci., 29(1) (2023), 12-27.
[36] I.A. Rus, Picard operators and applications, Sci. Math. Jap., 58(1) (2003), 191–219.
[37] M. S. Shagari, S. Rashid, K. M. Abualnaja, and M. Alansari, On nonlinear fuzzy set-valued θ-contractions with applications, AIMS Math., 6(10) (2021), 10431-10448.
[38] T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal.: Theory Meth. Appl. 71(11) (2009), 5313-5317.
[39] S.S. Yesilkaya and C. Aydin, Fixed point results of expansive mappings in metric spaces, Mathematics 8 (2020), 1800.
[40] J. Zhang, Y. Su, and Q. Cheng A note on ’A best proximity point theorem for Geraghtycontractions, Fixed Point Theory Appl., 2013 (2013), 1-4.
[41] D. Zheng, Fixed point theorems for generalized θ-ϕ-contractions in G-metric spaces, J. Funct. Spaces, 2018 (2018), Article ID 1418725, 8 pages.
[42] D. Zheng, Z. Cai, and P. Wang, New fixed point theorems for θ-ϕ contraction in complete metric spaces, J. Nonlinear Sci. Appl., 10(5) (2017), 2662–2670.
[43] D. Zheng, X. Liu, and G. Zhang, Some fixed point theorems for θ-ϕ C-contractions, J. Nonlinear Sci. Appl., 10 (2017), 5723–5733.
[44] D. Zheng and P.Wang, Weak θ-ϕ-contraction and discontinuity, J. Nonlinear Sci. Appl., 10(5) (2017), 2318-2323.