# Fixed point results of \((\phi,\psi)\)-weak contractions in ordered \(b\)-metric spaces

### Abstract

The purpose of this paper is to prove some results on fixed point, coincidence point, coupled coincidence point and coupled common fixed point for the mappings satisfying generalized \((\phi, \psi)\)-contraction conditions in complete partially ordered \(b\)-metric spaces. Our results generalize, extend and unify most of the fundamental metrical fixed point theorems in the existing literature. A few examples are illustrated to support our findings.

### References

M. Abbas, B. Ali, B. Bin-Mohsin, N. Dedović, T. Nazir and S. Radenović, “Solutions and Ulam-Hyers stability of differential inclusions involving Suzuki type multivalued mappings on b-metric spaces”, Vojnotehnički Glasnik/Military Technical Courier, vol. 68, no. 3, pp. 438–487, 2020.

A. Aghajani and R. Arab, “Fixed points of (ψ, φ, θ)-contractive mappings in partially ordered b-metric spaces and applications to quadratic integral equations”, Fixed Point Theory Appl., Art. ID 245, 20 pages, 2013.

A. Aghajani, M. Abbas and J. R. Roshan,“Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces”, Math. Slovaca, vol. 64, no. 4, pp. 941–960, 2014.

M. Akkouchi, “Common fixed point theorems for two selfmappings of a b-metric space under an implicit relation”, Hacet. J. Math. Stat., vol. 40, no. 6, pp. 805–810, 2011.

S. Aleksić, Z. Mitrović and S. Radenović, “On some recent fixed point results for single and multi-valued mappings in b-metric spaces”, Fasc. Math., no. 61, pp. 5–16, 2018.

S. Aleksić, H. Huang, Z. D. Mitrović and S. Radenović, “Remarks on some fixed point results in b-metric spaces”, J. Fixed Point Theory Appl., vol. 20, no. 4, Paper No. 147, 17 pages, 2018.

S. Aleksić, Z. D. Mitrović and S. Radenović, “Picard sequences in b-metric spaces”, Fixed Point Theory, vol. 21, no. 1, pp. 35–45, 2020.

R. Allahyari, R. Arab and A. Shole Haghighi, “A generalization on weak contractions in partially ordered b-metric spaces and its application to quadratic integral equations”, J. Inequal. Appl., Art. ID 355, 15 pages, 2014.

A. Amini-Harandi, “Fixed point theory for quasi-contraction maps in b-metric spaces”, Fixed Point Theory, vol. 15, no. 2, pp. 351–358, 2014.

H. Aydi, M.-F. Bota, E. Karapinar and S. Moradi, “A common fixed point for weak φ- contractions on b-metric spaces”, Fixed Point Theory, vol. 13, no. 2, pp. 337–346, 2012.

H. Aydi, A. Felhi and S. Sahmim, “On common fixed points for (α, ψ)-contractions and generalized cyclic contractions in b-metric-like spaces and consequences”, J. Nonlinear Sci. Appl., vol. 9, no. 5, pp. 2492–2510, 2016.

I. A. Bakhtin, “The contracting mapping principle in an almost metric space”, (Russian) Funkts. Anal., No. 30, pp. 26–37, 1989.

V. Berinde and M. Păcurar, “The early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects”, Carpathian J. Math., vol. 38, no. 3, pp. 523–538, 2022.

N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 5 à 10, Paris: Hermann, 1974.

B. S. Choudhury, N. Metiya and A. Kundu, “Coupled coincidence point theorems in ordered metric spaces”, Ann. Univ. Ferrara Sez. VII Sci. Mat., vol. 57, no. 1, pp. 1–16, 2011.

S. Czerwik, “Contraction mappings in b-metric spaces‘”, Acta Math. Inform. Univ. Ostraviensis, vol. 1, pp. 5–11, 1993.

S. Czerwik, “Nonlinear set-valued contraction mappings in b-metric spaces”. Atti Sem. Mat. Fis. Univ. Modena, vol. 46, no. 2, pp. 263–276, 1998.

L. J. Ćirić, N. Cakić, M. Rajović and J. S. Ume, “Monotone generalized nonlinear contractions in partially ordered metric spaces”, Fixed Point Theory Appl., Art. ID 131294, 11 pages, 2008.

L. J. Ćirić, “Some recent results in metrical fixed point theory; Fixed point theorems for mixed monotone operators and applications to integral equations”, Belgrade: University of Belgrade, 2003.

P. Debnath, Z. D. Mitrović and S. Radenović, “Interpolative Hardy-Rogers and Reich-Rus-Ćirić type contractions in b-metric and rectangular b-metric spaces”, Mat. Vesnik, vol. 72, no. 4, pp. 368–374, 2020.

D. Ðorić, “Common fixed point for generalized (ψ, φ)-weak contractions”, Appl. Math. Lett., vol. 22, no. 12, pp. 1896–1900, 2009.

H. Faraji, D. Savić, S. Dragana and S. N. Radenović, “Fixed point theorems for Geraghty contraction type mappings in b-metric spaces and applications”, Axioms, vol. 8, no. 1, Article ID 34, 12 pages, 2019.

T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications”, Nonlinear Anal., vol. 65, no. 7, pp. 1379–1393, 2006.

E. Graily, S. M. Vaezpour, R. Saadati and Y. J. Cho, “Generalization of fixed point theorems in ordered metric spaces concerning generalized distance”, Fixed Point Theory Appl., 30, 8 pages, 2011.

J. Harjani, B. López and K. Sadarangani, “Fixed point theorems for mixed monotone operators and applications to integral equations”, Nonlinear Anal., vol. 74, no. 5, pp. 1749–1760, 2011.

N. T. Hieu and N. V. Dung, “Some fixed point results for generalized rational type contraction mappings in partially ordered b-metric spaces”, Facta Univ. Ser. Math. Inform., vol. 30, no. 1, pp. 49–66, 2015.

H. Huang, S. Radenović and J. Vujaković, “On some recent coincidence and immediate con- sequences in partially ordered b-metric spaces”, Fixed Point Theory and Appl., 2015:63, 18 pages, 2015

H. Huang and S. Xu, “Fixed point theorems of contractive mappings in cone b-metric spaces and applications”, Fixed Point Theory Appl., 2013:112, 10 pages, 2013.

N. Hussain, D. Ðorić, Z. Kadelburg and S. Radenović, “Suzuki-type fixed point results in metric type spaces”, Fixed Point Theory Appl., 2012:126, 12 pages, 2012.

N. Hussain, Z. D. Mitrović and S. Radenović, “A common fixed point theorem of Fisher in b-metric spaces”, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM., vol. 113, no. 2, pp. 949–956, 2019.

J. Jachymski, “Equivalent conditions for generalized contractions on (ordered) metric spaces”, Nonlinear Anal., vol. 74, no. 3, pp. 768–774, 2011.

M. Jovanović, Z. Kadelburg and S. Radenović, “Common fixed point results in metric-type spaces”, Fixed Point Theory Appl., Article ID 978121, 15 pages, 2010.

K. Kalyani, N. Seshagiri Rao and L. N. Mishra, “Coupled fixed points theorems for generalized weak contractions in ordered b-metric spaces”, Asian-Eur. J. Math., vol. 15, no. 3, Paper No. 2250050, 22 pages, 2022.

E. Karapınar, Z. D. Mitrović, A. Öztürk and S. Radenović, “On a theorem of Ćirić in b-metric spaces”, Rend. Circ. Mat. Palermo (2), vol. 70, no. 1, pp. 217–225, 2021.

M. A. Khamsi and N. Hussain, “KKM mappings in metric type spaces”, Nonlinear Anal., vol. 73, no. 9, pp. 3123–3129, 2010.

W. A. Kirk and N. Shahzad, Fixed point theory in distance spaces, Cham: Springer, 2014.

V. Lakshmikantham and L. Ćirić, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces”, Nonlinear Anal., vol. 70, no. 12, pp. 4341–4349, 2009.

N. V. Luong and N. X. Thuan, “Coupled fixed point theorems in partially ordered metric spaces”, Bull. Math. Anal. Appl., vol. 2, no. 4, pp. 16–24, 2010.

B. Mitiku, K. Kalyani and N. Seshagiri Rao, “Some fixed point results of generalized (φ, ψ)- contractive mappings in ordered b-metric spaces”, BMC Research Notes, vol. 13, Article No. 537, 2020.

Z. D. Mitrović, S. Radenović, F. Vetro and J. Vujaković, “Some remark on TAC-contractive mappings in b-metric spaces”, Mat. Vesnik, vol. 70, no. 2, pp. 167–175, 2018.

N. Mlaiki, N. Dedovic, H. Aydi, M. Gardašević-Filipović, B. Bin-Mohsin and S. Radenović, “Some new observations on Geraghty and Ćirić type results in b-metric spaces”, Mathematics, vol. 7, Article ID 643, 11 pages, 2019.

J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations”, Order, vol. 22, no. 3, pp. 223–239, 2005.

M. V. Pavlović and S. N. Radenović, “A note on the Meir-Keeler theorem in the context of b-metric spaces”, Vojnotehnički Glasnik/Military Technical Courier, vol. 67, no. 1, pp. 1–12, 2019.

O. Popescu, “Fixed points for (ψ, φ)-weak contractions”, Appl. Math. Lett., vol. 24, no. 1, pp. 1–4, 2011.

A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations”, Proc. Amer. Math. Soc., vol. 132, no. 5, pp. 1435–1443, 2004.

J. R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei and W. Shatanawi, “Common fixed points of almost generalized (ψ,φ)s-contractive mappings in ordered b-metric spaces”, Fixed Point Theory Appl., 2013:159, 23 pages, 2013.

J. R. Roshan, V. Parvaneh and I. Altun, “Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations”, Appl. Math. Comput., vol. 226, pp. 725–737, 2014.

B. Samet and H. Yazidi, “Coupled fixed point theorems in partially ordered e-chainable metric spaces”, J. Math. Comput. Sci., vol. 1, no. 3, pp. 142–151, 2010.

N. Seshagiri Rao and K. Kalyani, “Unique fixed point theorems in partially ordered metric spaces”, Heliyon, vol. 6, e05563, 7 pages, 2020.

N. Seshagiri Rao and K. Kalyani, “Generalized contractions to coupled fixed point theorems in partially ordered metric spaces”, Zh. Sib. Fed. Univ. Mat. Fiz., vol. 13, no. 4, pp. 492–502, 2020.

N. Seshagiri Rao and K. Kalyani, “Coupled fixed point theorems with rational expressions in partially ordered metric spaces”, J. Anal., vol. 28, no. 4, pp. 1085–1095, 2020.

N. Seshagiri Rao, K. Kalyani and K. Khatri, “Contractive mapping theorems in partially ordered metric spaces”, CUBO, vol. 22, no. 2, pp. 203–214, 2020.

N. Seshagiri Rao, K. Kalyani and B. Mitiku, “Fixed point theorems for nonlinear contractive mappings in ordered b-metric space with auxiliary function”, BMC Research Notes, vol. 13, Paper No. 451, 2020.

N. Seshagiri Rao, K. Kalyani and B. Mitiku, “Fixed point results of almost generalized (φ, ψ, θ)s-contractive mappings in ordered b-metric spaces”, Afr. Mat., vol. 33, no. 2, Paper No. 64, 19 pages, 2022.

I. M. Vul’pe, D. Ostraĭh and F. Hoĭman, “The topological structure of a quasimetric space”, (Russian) in Investigations in functional analysis and differential equations, Math. Sci., Interuniv. Work Collect., Shtiintsa, pp. 14–19, 1981.

*CUBO*, vol. 24, no. 2, pp. 343–368, Aug. 2022.

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