Analyzing Helmholtz phenomena for mixed boundary values via graphically controlled contractions
Abstract
Helmholtz’s problem helps us to completely understand how sound behaves in a cylinder that is closed from one of its ends and opened at another. This paper aims to employ some novel convergence results to the Helmholtz problem with mixed boundary conditions and demonstrate the existence and uniqueness of the solution by applying graph-controlled contractions. For this purpose, we enunciate graphically Reich type and graphically Ćirić type contractions in the realm of graphical-controlled metric type spaces. In our study, we showcase the existence and uniqueness of fixed point results by employing these graphical contractions. This is demonstrated through extensive examples that a graphicalcontrolled metric-type space is distinct from traditional controlled metric-type spaces. We also exhibit an example of a graphically Reich contraction that is not a classical Reich contraction. Similarly, a decent example of graphical Ćirić contraction is presented, which is distinct from the classical Ćirić contraction. Concrete illustrative examples solidify our theoretical framework.
Keyword : directed graph, Helmholtz phenomena, fixed-point, mixed boundary conditions graphic-contractions
How to Cite
Younis, M., Ahmad, H., Asmat, F., & Öztürk, M. (2025). Analyzing Helmholtz phenomena for mixed boundary values via graphically controlled contractions. Mathematical Modelling and Analysis, 30(2), 342–361. https://doi.org/10.3846/mma.2025.22546
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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S.K. Panda, A.M. Kalla, K.S. Nagy and L. Priyanka. Numerical simulations and complex valued fractional order neural networks via (ϵ-µ)uniformly contractive mappings. Chaos, Solitons & Fractals, 173:113738, 2023. https://doi.org/10.1016/j.chaos.2023.113738
S.K. Panda, V.V. Kumar and I.R. Sarma. Common fixed point theorems on weakly compatible maps on dislocated metric spaces. Mathematical Sciences, 6(71):1–5, 2012. https://doi.org/10.1186/2251-7456-6-71
S.K. Panda and D. Panthi. Connecting various types of cyclic contractions and contractive self-mappings with Hardy-Rogers self-mappings. Fixed Point Theory and Applications, 2016(15), 2016. https://doi.org/10.1186/s13663-016-0498-3
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S. Saks. Theory of the Integral. Monografje Matematyczne. Warszawa, 1935.
S. Shukla, N. Dubey and R. Shukla. Fixed point theorems in graphical cone metric spaces and application to a system of initial value problems. Journal of Inequalities and Applications, 2023(1):91, 2023. https://doi.org/10.1186/s13660-023-03002-3
S. Shukla and H.P.A. Künzi. Some fixed point theorems for multi-valued mappings in graphical metric spaces. Mathematica Slovaca, 70(3):719–732, 2020. https://doi.org/10.1515/ms-2017-0385
S. Shukla, S. Radenović and C. Vetro. Graphical metric space: a generalized setting in fixed point theory. RACSAM, 111:641–655, 2017. https://doi.org/10.1007/s13398-016-0316-0
M. Younis, H. Ahmad, M. Ozturk and D. Singh. A novel approach to the convergence analysis of chaotic dynamics in fractional order Chua’s attractor model employing fixed points. Alexandria Engineering Journal, 110:363–375, 2025. https://doi.org/10.1016/j.aej.2024.10.001
M. Younis and D. Bahuguna. A unique approach to graph-based metric spaces with an application to rocket ascension. Computational and Applied Mathematics, 42(1):44, 2023. https://doi.org/10.1007/s40314-023-02193-1
M. Younis, D. Singh, L. Chen and M. Metwali. A study on the solutions of notable engineering models. Mathematical Modelling and Analysis, 27(3):492– 509, 2022. https://doi.org/10.3846/mma.2022.15276
H. Ahmad. Analysis of fixed points in controlled metric type spaces with application. In Recent Developments in Fixed-Point Theory: Theoretical Foundations and Real-World Applications, pp. 225–240. Springer Nature Singapore, 2024. https://doi.org/10.1007/978-981-99-9546-2_9
H. Ahmad, M. Younis and M.E. Köksal. Double controlled partial metric type spaces and convergence results. Journal of Mathematics, 2021:1–11, 2021. https://doi.org/10.1155/2021/7008737
K.H. Alam, Y. Rohen, I.A. Kallel and J. Ahmad. Solution of an algebraic linear system of equations using fixed point results in C*-algebra valued extended Branciari sb-metric spaces. International Journal of Analysis and Applications, 22:139, 2024. https://doi.org/10.28924/2291-8639-22-2024-139
K.H. Alam, Y. Rohen and N. Saleem. Fixed points of (α,β,f∗) and (α,β,f∗∗)weak Geraghty contractions with an application. Symmetry, 15(1):243, 2023. https://doi.org/10.3390/sym15010243
K.H. Alam, Y. Rohen, A. Tomar and M. Sajid. On fixed point and solution to nonlinear matrix equations related to beam theory in -metric space. Journal of Nonlinear and Convex Analysis, 25(9):2149–2171, 2024.
S. Aleomraninejad, S. Rezapour and N. Shahzad. Some fixed point results on a metric space with a graph. Topology and its Applications, 159(3):659–663, 2012. https://doi.org/10.1016/j.topol.2011.10.013
M. Alfuraidan. Fixed points of multivalued mappings in modular function spaces with a graph. Fixed Point Theory and Applications, 2015(1):1–14, 2015. https://doi.org/10.1186/s13663-015-0292-7
I.A. Bakhtin. The contraction mapping principle in quasi-metric spaces. Funct. Anal. Ulianowsk Gos. Ped. Inst., 30:26–37, 1989.
S. Banach. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3:133–181, 1922. https://doi.org/10.4064/fm-3-1-133-181
I. Beg and A.R. Butt. Fixed point of set-valued graph contractive mappings. Journal of Inequalities and Applications, 2013(1):1–7, 2013. https://doi.org/10.1186/1029-242X-2013-252
N. Chuensupantharat, P. Kumam, V. Chauhan, D. Singh and R. Menon. Graphic contraction mappings via graphical b-metric spaces with applications. Bulletin of the Malaysian Mathematical Sciences Society, 42:3149–3165, 2019. https://doi.org/10.1007/s40840-018-0651-8
S. Czerwik. Contraction mappings in b-metric spaces. Acta Math. Inform., Univ. Ostrav., 1:5–11, 1993.
A. Das, B. Hazarika, S.K. Panda and V. Vijayakumar. An existence result for an infinite system of implicit fractional integral equations via generalized Darbo’s fixed point theorem. Computational and Applied Mathematics, 40:143, 2021. https://doi.org/10.1007/s40314-021-01537-z
T. Dinevari and M. Frigon. Fixed point results for multivalued contractions on a metric space with a graph. Journal of Mathematical Analysis and Applications, 405(2):507–517, 2013. https://doi.org/10.1016/j.jmaa.2013.04.014
N. Dubey, S. Shukla and R. Shukla. On graphical symmetric spaces, fixed-point theorems, and the existence of positive solution of fractional periodic boundary value problems. Symmetry, 16(2):182, 2024. https://doi.org/10.3390/sym16020182
J. Harjani and K. Sadarangani. Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications, 72(3–4):1188–1197, 2010. https://doi.org/10.1016/j.na.2009.08.003
J. Jachymski. The contraction principle for mappings on a metric space with a graph. Proceedings of the American Mathematical Society, 135(10):3241–3248, 2007.
T. Kamran, M. Samreen and Q. UL Ain. A generalization of bmetric space and some fixed point theorems. Mathematics, 5(2):19, 2017. https://doi.org/10.3390/math5020019
K. Mirmostafaee and S. Alireza. Coupled fixed points for mappings on a b-metric space with a graph. Matematiˇcki Vesnik, 69(1):49–56, 2017.
N. Mlaiki, H. Aydi, N. Souayah and T. Abdeljawad. Controlled metric type spaces and the related contraction principle. Mathematics, 6(10), 2018. https://doi.org/10.3390/math6100194
J. Nieto, R.L. Pouso and R. Rodríguez-López. Fixed point theorems in ordered abstract spaces. Proceedings of the American Mathematical Society, 135(4):1095–1103, 2007. https://doi.org/10.1090/S0002-9939-07-08729-1
D. O’Regan and A. Petru¸sel. Fixed point theorems for generalized contractions in ordered metric spaces. Journal of Mathematical Analysis and Applications, 341(2):1241–1252, 2008. https://doi.org/10.1016/j.jmaa.2007.11.026
S.K. Panda, A.M. Kalla, K.S. Nagy and L. Priyanka. Numerical simulations and complex valued fractional order neural networks via (ϵ-µ)uniformly contractive mappings. Chaos, Solitons & Fractals, 173:113738, 2023. https://doi.org/10.1016/j.chaos.2023.113738
S.K. Panda, V.V. Kumar and I.R. Sarma. Common fixed point theorems on weakly compatible maps on dislocated metric spaces. Mathematical Sciences, 6(71):1–5, 2012. https://doi.org/10.1186/2251-7456-6-71
S.K. Panda and D. Panthi. Connecting various types of cyclic contractions and contractive self-mappings with Hardy-Rogers self-mappings. Fixed Point Theory and Applications, 2016(15), 2016. https://doi.org/10.1186/s13663-016-0498-3
S.K. Panda, V. Velusamy, I. Khan and S. Niazai. Computation and convergence of fixed-point with an RLC-electric circuit model in an extended b-suprametric space. Scientific Reports, 14:9479, 2024. https://doi.org/10.1038/s41598-024-59859-x
S. Saks. Theory of the Integral. Monografje Matematyczne. Warszawa, 1935.
S. Shukla, N. Dubey and R. Shukla. Fixed point theorems in graphical cone metric spaces and application to a system of initial value problems. Journal of Inequalities and Applications, 2023(1):91, 2023. https://doi.org/10.1186/s13660-023-03002-3
S. Shukla and H.P.A. Künzi. Some fixed point theorems for multi-valued mappings in graphical metric spaces. Mathematica Slovaca, 70(3):719–732, 2020. https://doi.org/10.1515/ms-2017-0385
S. Shukla, S. Radenović and C. Vetro. Graphical metric space: a generalized setting in fixed point theory. RACSAM, 111:641–655, 2017. https://doi.org/10.1007/s13398-016-0316-0
M. Younis, H. Ahmad, M. Ozturk and D. Singh. A novel approach to the convergence analysis of chaotic dynamics in fractional order Chua’s attractor model employing fixed points. Alexandria Engineering Journal, 110:363–375, 2025. https://doi.org/10.1016/j.aej.2024.10.001
M. Younis and D. Bahuguna. A unique approach to graph-based metric spaces with an application to rocket ascension. Computational and Applied Mathematics, 42(1):44, 2023. https://doi.org/10.1007/s40314-023-02193-1
M. Younis, D. Singh, L. Chen and M. Metwali. A study on the solutions of notable engineering models. Mathematical Modelling and Analysis, 27(3):492– 509, 2022. https://doi.org/10.3846/mma.2022.15276