Research article

A relation theoretic m-metric fixed point algorithm and related applications

  • Received: 14 January 2023 Revised: 02 May 2023 Accepted: 08 May 2023 Published: 09 June 2023
  • MSC : 34A08, 47H10, 54H25

  • In this article, we introduce the concept of generalized rational type $ F $ -contractions on relation theoretic m-metric spaces (denoted as $ F_{R}^{m} $-contractions, where $ R $ is a binary relation) and some related fixed point theorems are provided. Then, we achieve some fixed point results for cyclic rational type $ F_{R}^{m} $- generalized contraction mappings. Moreover, we state some illustrative numerically examples to show our results are true and meaningful. As an application, we discuss a positive definite solution of a nonlinear matrix equation of the form $ \Lambda = S+\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Lambda \right) Q_{i} $.

    Citation: Muhammad Tariq, Muhammad Arshad, Mujahid Abbas, Eskandar Ameer, Saber Mansour, Hassen Aydi. A relation theoretic m-metric fixed point algorithm and related applications[J]. AIMS Mathematics, 2023, 8(8): 19504-19525. doi: 10.3934/math.2023995

    Related Papers:

  • In this article, we introduce the concept of generalized rational type $ F $ -contractions on relation theoretic m-metric spaces (denoted as $ F_{R}^{m} $-contractions, where $ R $ is a binary relation) and some related fixed point theorems are provided. Then, we achieve some fixed point results for cyclic rational type $ F_{R}^{m} $- generalized contraction mappings. Moreover, we state some illustrative numerically examples to show our results are true and meaningful. As an application, we discuss a positive definite solution of a nonlinear matrix equation of the form $ \Lambda = S+\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Lambda \right) Q_{i} $.



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    [1] S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations integrales, Fund. Math., 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181
    [2] E. Ameer, H. Aydi, M. Arshad, H. Alsamir, M. S. Noorani, Hybrid multivalued type contraction mappingsin $\alpha K$-complete partial b-metric Spaces and applications, Symmetry, 11 (2019), 86. https://doi.org/10.3390/sym11010086 doi: 10.3390/sym11010086
    [3] A. Latif, R. F. Subaie, M. O. Alansari, Fixed points of generalized multi-valued contractive mappings in metric type spaces, J. Nonlinear Var. Anal., 6 (2022), 123–138. https://doi.org/10.23952/jnva.6.2022.1.07 doi: 10.23952/jnva.6.2022.1.07
    [4] X. Kang, N. Fang, Some common coupled fixed point results for the mappings with a new contractive condition in a Menger PbM-metric space, J. Nonlinear Funct. Anal., 2023, (2023), 9. https://doi.org/10.23952/jnfa.2023.9 doi: 10.23952/jnfa.2023.9
    [5] H. Aydi, M. Abbas, C. Vetro, Partial hausdorff metric and Nadler's fixed point theorem on partial metric spaces, Topol. Appl., 159 (2012), 3234–3242. https://doi.org/10.1016/j.topol.2012.06.012 doi: 10.1016/j.topol.2012.06.012
    [6] I. Beg, A. R. Butt, Common fixed point for generalized set valued contractions satisfying an implicit relation in partially ordered metric spaces, Math. Commun., 15 (2010), 65–76.
    [7] A. Baklouti, M. Mabrouk, Essential numerical ranges of operators in semi-Hilbertian spaces, Ann. Funct. Anal., 13 (2022), 16. https://doi.org/10.1007/s43034-021-00161-6 doi: 10.1007/s43034-021-00161-6
    [8] A. Alam, M. Imdad, Relation-theoretic metrical coincidence theorems, Filomat, 31 (2017), 4421–4439. https://doi.org/10.2298/FIL1714421A doi: 10.2298/FIL1714421A
    [9] M. Imdad, Q. H. Khan, W. M. Alfaqih, R. Gubrana, A relation-theoretic $(F, R)$-contraction principle with applications to matrix equations, Bul. Math. Anal. Appl., 10 (2018), 1–12.
    [10] S. Reich, A. J. Zaslavski, Convergence of inexact iterates of strict contractions in metric spaces with graphs, J. Appl. Numer. Optim., 4 (2022), 215–220. https://doi.org/10.23952/jano.4.2022.2.07 doi: 10.23952/jano.4.2022.2.07
    [11] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435–1443.
    [12] J. J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223–239. https://doi.org/10.1007/s11083-005-9018-5 doi: 10.1007/s11083-005-9018-5
    [13] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 94. https://doi.org/10.1186/1687-1812-2012-94 doi: 10.1186/1687-1812-2012-94
    [14] M. B. Zada, M. Sarwar, Common fixed point theorems for rational $F_{R}$-contractive pairs of mappings with applications, J. Inequal. Appl., 2019 (2019), 11. https://doi.org/10.1186/s13660-018-1952-z doi: 10.1186/s13660-018-1952-z
    [15] S. G. Matthews, Partial metric topology, Ann. N. Y. Acad. Sci., 728 (1994), 183–197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x doi: 10.1111/j.1749-6632.1994.tb44144.x
    [16] R. Jain, H. K.Nashine, Z. Kadelburg, Some fixed point results on relational quasi partial metric spaces and application to nonlinear matrix equations, Symmetry, 13 (2021), 993. https://doi.org/10.3390/sym13060993 doi: 10.3390/sym13060993
    [17] A. Baklouti, Quadratic Hom-Lie triple systems, J. Geom. Phys., 121 (2017), 166–175. https://doi.org/10.1016/j.geomphys.2017.06.013 doi: 10.1016/j.geomphys.2017.06.013
    [18] C. Vetro, F. Vetro, A homotopy fixed point theorem in $0$-complete partial metric space, Filomat, 29 (2015), 2037–2048. https://dx.doi.org/10.2298/FIL1509037V doi: 10.2298/FIL1509037V
    [19] N. Hussain, G. Ali, I. Iqbal, B. Samet, The existence of solutions to nonlinear matrix equations via fixed points of multivalued F-contractions, Mathematics, 8 (2020), 212. https://doi.org/10.3390/math8020212 doi: 10.3390/math8020212
    [20] S. Kumar, S. Luambano, On some fixed point theorems for multivalued F-contractions in partial metricspaces, Demonstr. Math., 54 (2021), 151–161. https://doi.org/10.1515/dema-2021-0012 doi: 10.1515/dema-2021-0012
    [21] M. Asadi, E. Karapinar, P. Salimi, New extension of $p$-metric spaces with fixed points results on M-metric spaces, J. Inequal. Appl., 2014 (2014), 18. https://doi.org/10.1186/1029-242X-2014-18 doi: 10.1186/1029-242X-2014-18
    [22] A. Ali, H. Işík, H. Aydi, E. Ameer, J. R. Lee, M. Arshad, On multivalued SU-type $\theta $-contractions and related applications, Open Math., 18 (2020), 386–399. https://doi.org/10.1515/math-2020-0139 doi: 10.1515/math-2020-0139
    [23] A. Ali, F. Uddin, M. Arshad, M. Rashid, Hybrid fixed point results via generalized dynamic process for F-HRS type contractions with application, Physica A, 538 (2020), 122669. https://doi.org/10.1016/j.physa.2019.122669 doi: 10.1016/j.physa.2019.122669
    [24] M. Tariq, M. Arshad, E. Ameer, A. Aloqaily, S. S. Aiadi, N. Mlaik, On Relational Weak $\left(F_{R}^{m}, \eta \right) $-Contractive Mappings and Their Applicationons, Symmetry, 15 (2023), 922. https://doi.org/10.3390/sym15040922 doi: 10.3390/sym15040922
    [25] M. Tariq, M. Abbas, A. Hussain, M. Arshad, A. Ali, H. Sulami, Fixed points of non-linear set-valued $(\alpha _{\ast }\phi _{M})$-contraction mappings and related applications, AIMS Math., 7 (2022), 8861–8878. https://doi.org/10.3934/math.2022494 doi: 10.3934/math.2022494
    [26] M. Tariq, E. Ameer, A. Ali, H. A. Hammad, F. Jarad, Applying fixed point techniques for obtaining a positive definite solution to nonlinear matrix equations, AIMS Math., 8 (2022), 3842–3859. https://doi.org/10.3934/math.2023191 doi: 10.3934/math.2023191
    [27] A. Ali, E. Ameer, S. S. Aiadi, M. Tariq, M. Arshad, N. Mlaiki, et al., New extension to fuzzy dynamic system and fuzzy fixed point results with an application, AIMS Math., 8 (2023), 1208–1229. https://doi.org/10.3934/math.2023061 doi: 10.3934/math.2023061
    [28] H. M. Srivastava, A. Ali, A. Hussain, M. Arshad, H. Al-Sulami, A certain class of $\theta _{L}$-type non-linear operatorsand some related fixed point results, J. Nonlinear Var. Anal., 6 (2022), 69–87. https://doi.org/10.23952/jnva.6.2022.1.05 doi: 10.23952/jnva.6.2022.1.05
    [29] A. Ali, A. Hussain, M. Arshad, H. A. Sulami, M. Tariq, Certain new development to the orthogonal binary relations, Symmetry, 14 (2022), 1954. https://doi.org/10.3390/sym14101954 doi: 10.3390/sym14101954
    [30] A. Baklouti, J. Schutz, S. Dellagi, A. Chelbi, Selling or leasing used vehicles considering their energetic type, the potential demand for leasing, and the expected maintenance costs, Energy Rep., 8 (2022), 1125–1135. https://doi.org/10.1016/j.egyr.2022.07.074 doi: 10.1016/j.egyr.2022.07.074
    [31] I. Altun, M. Asim, M. Imdad, W. M. Alfaqih, Fixed point results for $F_{R}$-generalized contractive mappings in parial metric space, Math. Slovaca, 69 (2019), 1413–1424. https://doi.org/10.1515/ms-2017-0318 doi: 10.1515/ms-2017-0318
    [32] M. Asadi, M. Azhini, E. Karapinar, H. Monfared, Simulation functions over m-metric spaces, East. Asian Math. J., 33 (2017), 559–570. https://doi.org/10.7858/eamj.2017.039 doi: 10.7858/eamj.2017.039
    [33] H. Monfared, M. Azhini, M. Asadi. Fixed point results on $m$-metric spaces, J. Math. Anal., 7 (2016), 85–101.
    [34] E. Karapínar, M. Abbas, S. Farooq, A discussion on the existence of best proximity points that belong to the zero set, Axioms, 9 (2020), 19. https://doi.org/10.3390/axioms9010019 doi: 10.3390/axioms9010019
    [35] I. Altun, G. Minak, H. Dag, Multivalued F-contractions on complete metric space, J. Nonlinear Convex Anal., 16 (2015), 659–666.
    [36] A. Alam, M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl., 17 (2015), 693–702. https://doi.org/10.1007/s11784-015-0247-y doi: 10.1007/s11784-015-0247-y
    [37] W. A. Kirk, P. S. Srinivasan, P. Veeramani, Fixed Points for mapping satsifying cyclic contractive conditions, Fixed Point Theor., 4 (2003), 79–89.
    [38] S. Bose, S. M. Hossein, K. Paul, Positive definite solution of a nonlinear matrix equation, J. Fixed Point Theory Appl., 18 (2016), 627–643. https://doi.org/10.1007/s11784-016-0291-2 doi: 10.1007/s11784-016-0291-2
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