Corrigendum to "$\eta$-admissible mappings in $C^*$-algebra-valued $\mathcal{MP}$-metric spaces with an application"

Document Type : Research Paper


1 Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran

2 Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran

3 Department of Mathematics, Azadshahr Branch, Islamic Azad University, Azadshahr, Iran



This article is a revision and correction of the chapter book [S. Hadi Bonab, V. Parvaneh, Z. Bagheri, $\eta_{\mathcal{A}}$-Admissible Mappings for Four Maps in $C*$-Algebra-Valued MP-Metric Spaces with an Application, In: P. Debnath, Delfim F. M. Torres, Yeol Je Cho, Advanced Mathematical Analysis and its Applications, CRC Press, 2023, 97-113.]. In this article, we first introduce the concept of $\eta$-admissible mapping in $C^*$-algebra valued $\mathcal{MP}$-metric spaces, which is a generalization and combination of "modular metric spaces", "parametric metric spaces" and "$C^*$-algebra-valued metric spaces". Then, for four mappings in these spaces, we prove several fixed-point theorems. We give an example and an application regarding the solvability of operator equations and integral equations, respectively, to support the new findings.


[1] M. Asim and M. Imdad, C-algebra valued symmetric spaces and fixed point results with an application, Korean J. Math. 28 (2020), no. 1, 17–30.
[2] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations int`egrales, Fund. Math. 3 (1922), 133–181.
[3] L. Ciric, V. Parvaneh, and N. Hussain, Fixed point results for weakly α-admissible pairs, Filomat 30 (2016), no. 14, 3697–3713.
[4] P. Debnath, Banach, Kannan, Chatterjea, and Reich-type contractive inequalities for multivalued mappings and their common fixed points, Math. Methods Appl. Sci. 45 (2022), no. 3, 1587–1596.
[5] P. Debnath, Optimization through best proximity points for multivalued F-contractions, Miskolc Math. Notes 22 (2021), no. 1, 143–151.
[6] P. Debnath, A new extension of Kannan’s fixed point theorem via F-contraction with application to integral equations, Asian-Eur. J. Math. 15 (2022), no. 07, 2250123.
[7] M.E. Ege and C. Alaca, Some results for modular b-metric spaces and an application to system of linear equations, Azerb. J. Math. 8 (2018), 3–14.
[8] A.D. Filip and A. Petrusel, Fixed point theorems on spaces endowed with vector-valued metrics, J. Fixed Point Theory Appl. 2010 (2010), 1–15.
[9] S. Hadi Bonab, R. Abazari, and A. Bagheri Vakilabad, Partially ordered cone metric spaces and coupled fixed point theorems via α-series, Math. Anal. Contemp. Appl. 1 (2019), no. 1, 50–61.
[10] S. Hadi Bonab, R. Abazari, A. Bagheri Vakilabad, and H. Hosseinzadeh, Coupled fixed point theorems on G-metric spaces via α-series, Glob. Anal. Discrete Math. 6 (2021), no. 1, 1–12.
[11] S. Hadi Bonab, R. Abazari, A. Bagheri Vakilabad, and H. Hosseinzadeh, Generalized metric spaces endowed with vector-valued metrics and matrix equations by tripled fixed point theorems, J. Inequal. Appl. 2014 (2020), 1–16.
[12] S. Hadi Bonab, V. Parvaneh, and Z. Bagheri, ηA-admissible mappings for four maps in C-algebra-valued MP-metric spaces with an application, P. Debnath, Delfim F. M. Torres, Yeol Je Cho, Advanced Mathematical Analysis and its Applications, CRC Press, 2023, pp. 97–113.
[13] S. Hadi Bonab, V. Parvaneh, H. Hosseinzadeh, H. Aydi, and S.J. Hosseini Ghoncheh, n-tuple fixed point theorems via α-series in C-algebra-valued metric spaces with an application in integral equations, Int. J. Ind. Math. 15 (2023), no. 2, 95–105.
[14] H. Hosseinzadeh, S. Hadi Bonab, and Kh. Amini Sefidab, Some common fixed point theorems for four mapping in generalized metric spaces, Thai J. Math. 20 (2022), no. 1, 425–437. [15] H. Hosseinzadeh, H. Isik, and S. Hadi Bonab, n-tuple fixed point theorems via α-series on partially ordered cone metric spaces, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 2, 3115–3126.
[16] N. Hussain, S. Khaleghizadeh, P. Salimi, and A.A.N. Abdou, A new approach to fixed point results in triangular intuitionistic fuzzy metric spaces, Abstr. Appl. Anal. 2014 (2014), 16 pages.
[17] T. Kamran, M. Samreen, and Q.U. Ain, A generalization of b-metric space and some fixed point theorems, Mathematics 5 (2017), no. 19.
[18] M.A. Kutbi and A. Latif, Fixed points of multivalued mappings in modular function spaces, Fixed Point Theory Appl., 2009 (2009), Article ID 786357.
[19] Z.H. Ma, L.N. Jiang, C∗-algebra valued b-metric spaces and related fixed point theorems, Fixed Point Theory Appl. 2015 (2015), no. 222, 1–12.
[20] Z. Ma, L. Jiang, and H. Sun, C∗-algebra-valued metric spaces and related fixed point theorems, Fixed Point Theory Appl. 2014 (2014), no. 1.
[21] B. Moeini, A.H. Ansari, and C. Park, JHR-operator pairs in C∗-algebra-valued modular metric spaces and related fixed point results with application, Numer. Funct. Anal. Optim. 39 (2018), no. 16, 1785–1805.
[22] J.V. Morales, Subordinate semi-metric spaces and fixed point theorems, J. Math. 12 (2018), 1–5.
[23] G.J. Murphy, C∗-Algebras and Operator Theory, Academic Press, Inc., Boston, MA, USA, 1990.
[24] J. Musielak and W. Orlicz, On modular spaces, Stud. Math. 18 (1959), 591–597.
[25] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Company, 1950.
[26] V. Parvaneh, S. Hadi Bonab, H. Hosseinzadeh, and H. Aydi, A tripled fixed point theorem in C-algebra-valued metric spaces and application in integral equations, Adv. Math. Phys. 2021 (2021), 1–6.
[27] V. Parvaneh, N. Hussain, H. Hosseinzadeh, and P. Salimi, Coupled fixed point results for α-admissible Mizoguchi-Takahashi contractions in b-metric spaces with applications, Sahand Commun. Math. Anal. 7 (2017), no. 1, 85–104.
[28] V. Parvaneh, N. Hussain, M.A. Kutbi, and M. Khorshdi, Some fixed point results in extended modular b-metric spaces with application to integral equations, J. Math. Anal. 10 (2019), no. 5, 14–33.
[29] J.R. Roshan, V. Parvaneh, Z. Kadelburg, and N. Hussain, New fixed point results in b-rectangular metric spaces, Nonlinear Anal. Model. Control 21 (2016), no. 5, 614–634.
[30] B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for α − ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154–2165.
[31] T. Stephen, Y. Rohen, N. Mlaiki, M. Bina, N. Hussain, and D. Rizk, On fixed points of rational contractions in generalized parametric metric and fuzzy metric spaces, J. Inequal. Appl. 2021 (2021), no. 125, 1–15.
[32] Y. Talaei, S. Micula, H. Hosseinzadeh, and S. Noeiaghdam, A novel algorithm to solve nonlinear fractional quadratic integral equations, AIMS Math. 7 (2022), no. 7, 13237–13257.
[33] J. Uma Maheswari, A. Anbarasan, M. Gunaseelan, V. Parvaneh, and S. Hadi Bonab. Solving an integral equation via C-algebra-valued partial b-metrics, Fixed Point Theory Algorithms Sci. Eng. 18 (2022), 1–14.

Articles in Press, Corrected Proof
Available Online from 03 January 2024
  • Receive Date: 12 September 2023
  • Revise Date: 24 October 2023
  • Accept Date: 02 November 2023