A study on the solutions of notable engineering models
Abstract
In the commenced work, we establish some novel results concerning graph contractions in a more generalized setting. Furthermore, we deliver some examples to elaborate and explain the usability of the attained results. By virtue of nontrivial examples, we show our results improve, extend, generalize, and unify several noteworthy results in the existing state-of-art. We adopt computer simulation validating our results. To arouse further interest in the subject and to show its efficacy, we devote this work to recent applications which emphasize primarily the applications for the existence of the solution of various models related to engineering problems viz. fourth-order two-point boundary value problems describing deformations of an elastic beam, ascending motion of a rocket, and a class of integral equations. This approach is entirely new and will open up some new directions in the underlying graph structure.
Keyword : deformations of elastic beam, ascending motion of rocket, graphic contraction, fixed point
How to Cite
Younis, M., Singh, D., Chen, L., & Metwali, M. (2022). A study on the solutions of notable engineering models. Mathematical Modelling and Analysis, 27(3), 492–509. https://doi.org/10.3846/mma.2022.15276
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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P. Baradol, J. Vujaković, D. Gopal and S. Radenović. On some new results in graphical rectangular b-metric spaces. Mathematics, 8(4):488, 2020. https://doi.org/10.3390/math8040488
F. Bojor. Fixed points of Kannan mappings in metric spaces endowed with a graph. Analele Universitatii” Ovidius” Constanta-Seria Matematica, 20(1):31– 40, 2012. https://doi.org/10.2478/v10309-012-0003-x
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S.M. El-Sayed and A.C.M. Ran. On an iteration method for solving a class of nonlinear matrix equations. SIAM Journal on Matrix Analysis and Applications, 23(3):632–645, 2002. https://doi.org/10.1137/S0895479899345571
J. Górnicki. Fixed point theorems for Kannan type mappings. Journal of Fixed Point Theory and Applications, 19(3):2145–2152, 2017. https://doi.org/10.1007/s11784-017-0402-8
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M.A. Khamsi and W.A. Kirk. An introduction to metric spaces and fixed point theory. John Wiley & Sons, 2011.
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S. Radenović, K. Zoto, N. Dedović, V. Šešum-Cavic and A.H. Ansari. Bhaskar-Guo-Lakshmikantam-Ćirić type results via new functions with applications to integral equations. Applied Mathematics and Computation, 357:75–87, 2019. https://doi.org/10.1016/j.amc.2019.03.057
A.C.M. Ran and M.C.B. Reurings. A fixed point theorem in partially ordered sets and some applications to matrix equations. Proceedings of the American Mathematical Society, 132:1435–1443, 2004. https://doi.org/10.1090/S0002-9939-03-07220-4
S. Reich. Some remarks concerning contraction mappings. Canadian Mathematical Bulletin, 14(1):121–124, 1971. https://doi.org/10.4153/CMB-1971-024-9
W.C. Xie. Differential equations for engineers. Cambridge University Press, 2010.
M. Younis and D. Singh. On the existence of the solution of Hammerstein integral equations and fractional differential equations. Journal of Applied Mathematics and Computing, 68:1087–1105, 2022. https://doi.org/10.1007/s12190-021-01558-1
M. Younis, D. Singh and A.A.N. Abdou. A fixed point approach for tuning circuit problem in dislocated b-metric spaces. Mathematical Methods in the Applied Sciences, 45(4):2234–2253, 2020. https://doi.org/10.1002/mma.7922
M. Younis, D. Singh, I. Altun and V. Chauhan. Graphical structure of extended b-metric spaces: an application to the transverse oscillations of a homogeneous bar. International Journal of Nonlinear Sciences and Numerical Simulation, 2021. https://doi.org/10.1515/ijnsns-2020-0126
M. Younis, D. Singh and A. Goyal. A novel approach of graphical rectangular b-metric spaces with an application to the vibrations of a vertical heavy hanging cable. Journal of Fixed Point Theory and Applications, 21(33):1–17, 2019. https://doi.org/10.1007/s11784-019-0673-3
M. Younis, D. Singh and L. Shi. Revisiting graphical rectangular bmetric spaces. Asian-European Journal of Mathematics, 15(4):2250072, 2022. https://doi.org/10.1142/S1793557122500723
A. Aghanians and K. Nourouzi. Fixed points for Kannan type contractions in uniform spaces endowed with a graph. Nonlinear Analysis: Modelling and Control, 21(1):103–113, 2016. https://doi.org/10.15388/NA.2016.1.7
C. Alegre, H. Da˘g, S. Romaguera and P. Tirado. Characterizations of quasi-metric completeness in terms of Kannan-type fixed point theorems. Hacettepe Journal of Mathematics and Statistics, 46(1):67–76, 2017. https://doi.org/10.15672/HJMS.2016.395
M.R. Alfuraidan. Remarks on Caristis fixed point theorem in metric spaces with a graph. Fixed Point Theory and Applications, 2014(1):1–8, 2014. https://doi.org/10.1186/1687-1812-2014-240
P. Baradol, D. Gopal and S. Radenović. Computational fixed points in graphical rectangular metric spaces with application. Journal of Computational and Applied Mathematics, 375:112805, 2020. https://doi.org/10.1016/j.cam.2020.112805
P. Baradol, J. Vujaković, D. Gopal and S. Radenović. On some new results in graphical rectangular b-metric spaces. Mathematics, 8(4):488, 2020. https://doi.org/10.3390/math8040488
F. Bojor. Fixed points of Kannan mappings in metric spaces endowed with a graph. Analele Universitatii” Ovidius” Constanta-Seria Matematica, 20(1):31– 40, 2012. https://doi.org/10.2478/v10309-012-0003-x
L.B. Ćirić and B. Fisher. Fixed Point Theory: contraction mapping principle. FMS Press, Beograd, 2003.
S.M. El-Sayed and A.C.M. Ran. On an iteration method for solving a class of nonlinear matrix equations. SIAM Journal on Matrix Analysis and Applications, 23(3):632–645, 2002. https://doi.org/10.1137/S0895479899345571
J. Górnicki. Fixed point theorems for Kannan type mappings. Journal of Fixed Point Theory and Applications, 19(3):2145–2152, 2017. https://doi.org/10.1007/s11784-017-0402-8
C.P. Gupta. Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis, 26(4):289–304, 1988. https://doi.org/10.1080/00036818808839715
J. Jachymski. The contraction principle for mappings on a metric space with a graph. Proceedings of the American Mathematical Society, 136(4):1359–1373, 2008. https://doi.org/10.1090/S0002-9939-07-09110-1
S.B. Kaliaj. A Kannan-type fixed point theorem for multivalued mappings with application. The Journal of Analysis, 27(3):837–849, 2019. https://doi.org/10.1007/s41478-018-0135-0
R. Kannan. Some results on fixed points. Bull. Cal. Math. Soc., 60:71–76, 1968.
M.A. Khamsi and W.A. Kirk. An introduction to metric spaces and fixed point theory. John Wiley & Sons, 2011.
J.J. Nieto and R. Rodríguez-López. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order, 22(3):223–239, 2005. https://doi.org/10.1007/s11083-005-9018-5
S. Radenović, T. Dŏsenović, T. Aleksić Lampert and Z. Golubovíć. A note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equations. Applied Mathematics and Computation, 273:155–164, 2016. https://doi.org/10.1016/j.amc.2015.09.089
S. Radenović, K. Zoto, N. Dedović, V. Šešum-Cavic and A.H. Ansari. Bhaskar-Guo-Lakshmikantam-Ćirić type results via new functions with applications to integral equations. Applied Mathematics and Computation, 357:75–87, 2019. https://doi.org/10.1016/j.amc.2019.03.057
A.C.M. Ran and M.C.B. Reurings. A fixed point theorem in partially ordered sets and some applications to matrix equations. Proceedings of the American Mathematical Society, 132:1435–1443, 2004. https://doi.org/10.1090/S0002-9939-03-07220-4
S. Reich. Some remarks concerning contraction mappings. Canadian Mathematical Bulletin, 14(1):121–124, 1971. https://doi.org/10.4153/CMB-1971-024-9
W.C. Xie. Differential equations for engineers. Cambridge University Press, 2010.
M. Younis and D. Singh. On the existence of the solution of Hammerstein integral equations and fractional differential equations. Journal of Applied Mathematics and Computing, 68:1087–1105, 2022. https://doi.org/10.1007/s12190-021-01558-1
M. Younis, D. Singh and A.A.N. Abdou. A fixed point approach for tuning circuit problem in dislocated b-metric spaces. Mathematical Methods in the Applied Sciences, 45(4):2234–2253, 2020. https://doi.org/10.1002/mma.7922
M. Younis, D. Singh, I. Altun and V. Chauhan. Graphical structure of extended b-metric spaces: an application to the transverse oscillations of a homogeneous bar. International Journal of Nonlinear Sciences and Numerical Simulation, 2021. https://doi.org/10.1515/ijnsns-2020-0126
M. Younis, D. Singh and A. Goyal. A novel approach of graphical rectangular b-metric spaces with an application to the vibrations of a vertical heavy hanging cable. Journal of Fixed Point Theory and Applications, 21(33):1–17, 2019. https://doi.org/10.1007/s11784-019-0673-3
M. Younis, D. Singh and L. Shi. Revisiting graphical rectangular bmetric spaces. Asian-European Journal of Mathematics, 15(4):2250072, 2022. https://doi.org/10.1142/S1793557122500723