RELATION-THEORETIC COMMON FIXED POINTS FOR ALMOST $\mathcal{F}_{\tilde{\mathcal{R}} _{\Im}}$-CONTRACTION TYPE MAPS IN $\mathbb{B}_{2}$-METRIC SPACES AND APPLICATION TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

Year 2025, Volume: 15 Issue: 9, 2239 - 2259, 01.09.2025

M. V. R. Kameswari Stojan Radenovıc , M. Madhuri A. Bharathi

Abstract

This paper introduces a novel class of contraction mappings called "almost $\mathcal{F}_{\tilde{\mathcal{R}} {\Im}}$-contraction type maps" in the framework of $\mathbb{B}_{2}$-metric spaces. These contractions are utilized to establish results regarding coincidence points and common fixed points furnished with a binary relation. Furthermore, the paper aims to broaden the scope of these findings by offering illustrative examples. The paper concludes with an application of these concepts to prove the existence of solutions of a nonlinear fractional differential equation. Our results broaden the scope of those reported in [16] and expand on comparable findings previously documented in the literature.

Keywords

Binary relations , Almost $\mathcal{F}_{\tilde{\mathcal{R}} _{\Im}}$-contraction type maps , $\mathbb{B}_{2}$-metric spaces , Common coincidence points , Common fixed points , Fourth-order boundary value problems

References

• Abbas, M., Jungck, G., (2008), Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. of Mathematical Analysis and Appl., 341, pp.416–420, https://doi.org/10.1016/j.jmaa.2007.09.070.
• Abbas, M., Rakocevic, V., \& Tsegaye Leyew, B., (2017), Common fixed points of $(\alpha, \beta)$-generalized rational multivalued contractions in dislocated quasi b-metric spaces and applications, Filomat, 31, pp.3263–3284, https://doi.org/10.2298/FIL1711263A.
• Alam, A., Imdad, M., (2017), Relation-theoretic metrical coincidence theorems, Filomat, 31, pp.4421–4439, https://doi.org/10.2298/FIL1714421A.
• Alam, A., Imdad, M., (2015), Relation-theoretic contraction principle, J. of Fixed Point Theory and Appl., 17, pp.693–702. https://doi.org/10.1007/s11784-015-0247-y.
• Alam, A., Imdad, M., (2018), Nonlinear contractions in metric spaces under locally T-transitive binary relations, Fixed Point Theory, 19, pp.13–24, https://doi.org/10.48550/arXiv.1512.00348.
• Berinde, V., Păcurar, M., (2022), The early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects, Carpathian Journal of Mathematics, 38(3), pp.523-538, https://www.jstor.org/stable/27150504.
• Czerwik, S., (1993), Contraction mappings in b-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1, pp.5–11, http://dml.cz/dmlcz/120469.
• Dass, B. K., Gupta, S., (1975), An extension of Banach contraction principle through rational expression, Indian Journal of Pure and Applied Mathematics, 6, pp.1455-1458.
• Fabiano, N., Kadelburg, Z., Mirkov, N., Vesna \v{S}e\v{s}um \v{C}avi\'{c}, and Radenovi\'{c}, S., (2022), On F-contractions: A Survey, Contemporary Mathematics, 3(3), pp.327, https://ojs.wiserpub.com/index.php/CM/article/view/1517.
• Fadail, Z. M. , Ahmad, A. G. B., Ozturk, V., \& Radenovi\'{c}, S., (2015), Some remarks on fixed point results of $b_2$ metric spaces, Far East Journal of Mathematical Sciences (FJMS), 97(5), pp.533-548, https://10.17654/FJMSJul2015-533-548.
• Faruk, Sk., Asik, H., Qamrul, H. K., (2021), Relation-theoretic metrical coincidence theorems under weak C-contractions and K-contractions, AIMS Mathematics, 6(12), pp.13072–13091, https://10.3934/math.2021756.
• Fisher, B., (1980), Mappings satisfying a rational inequality, Bulletin mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, 24, pp.247–251, http://www.jstor.org/stable/43680572.
• Gahler, S., (1963), 2-metrische Räume und ihre topologische Struktur, Mathematische Nachrichten, 26, pp.115–118, https://doi.org/10.1002/mana.19630260109.
• Jaggi, D. S., (1977), Some unique fixed point theorems, Indian Journal of Pure and Applied Mathematics, 8(2), pp.223–230.
• Kolman, B., Busby, R. C., Ross, S. C., (2000), Discrete mathematical structures, PHI Learning Private Limited, New Delhi.
• Kumama, P., Mitrovi\'{c}, Z. D., Pavlovi\'{c}, V.I., (2019), Some fixed point theorems in b$_2$-metric spaces, Vojnotehnički Glasnik / Military Technical Courier, 67, Issue 3.
• Moussaoui, A., Todorcevi´c, V., Pantovi´c, M., Radenovi´c, S., \& Melliani., (2023), Fixed point results via G-transitive binary relation and fuzzy L-R-contraction, Mathematics, 11, pp.1768, doi.org/10.3390/math11081768.
• Murali. A., K. Muthunagai, K., (2023), Best proximity point theorems for generalized rational type contraction conditions involving control functions on complex valued metric spaces, Advanced Fixed Point Theory, 13:31, https://doi.org/10.28919/afpt/8299.
• Mustafa, Z., Parvaneh, V., Roshan, J. R., Kadelburg, Z., (2014), b2-Metric spaces and some fixed point theorems, Journal of Fixed Point Theory and Applications, 144, https://doi.org/10.1186/1687-1812-2014-144.
• Ovidiu, P., Gabriel, S., (2020), Two Fixed Point Theorems Concerning F-Contraction in Complete Metric Spaces, Symmetry, 12, pp.58, https://doi.org/10.3390/sym12010058.
• Pachpatte, B. G., (1979), Common fixed-point theorems for mappings satisfying rational inequalities, Indian Journal of Pure and Applied Mathematics, 10, pp.1362–1368.
• Perveen, A., Khan, I. A., Imdad, M., (2019), Relation theoretic common fixed point results for generalized weak nonlinear contractions with an application, Axioms, 8, pp.1-20, https://doi.org/10.3390/axioms8020049.
• Radenovi\'{c}, S., Mirkov N., and Paunovi\'{c}, Lj., (2021), Some new results on F-contractions in 0-complete partial metric spaces and 0-complete metric-like spaces, Fractal and Fractional, 5, pp.34, https://doi.org./10.3390/fractalfract5020034.
• Ran A. C. M., Reurings, M. C. B., (2004), A fixed point theorem in partially ordered sets and some applications to matrix equations, Proceedings of the American Mathematical Society, 132, pp.1435–1443, http://dx.doi.org/10.1090/S0002-9939-03-07220-4.
• Rangamma, M., Murthy, P. R. B., Reddy, P. M., (2017), A common fixed point theorem for a family of self maps in cone b2-metric space, International Journal of Pure and Applied Mathematics, 2, pp.359–368.
• Samet, B., Turinici, M., (2012), Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications, Communications in Mathematical Analysis, 13, pp.82–97.
• Samera, M.S., Salvatore, S., Waleed M. A., and Fawzia Shaddad, (2021), Common fixed point results for almost Rg-Geraghty type contraction mappings in b$_{2}$-metric spaces with an application to integral equations, Axioms, 10, 2, 101, pp.1-19, https://www.mdpi.com/2075-1680/10/2/101.
• Shoaib, M., Sarwar, M., Shah, K., Kumam, P., (2016), Fixed Point Results and its Applications to the Systems of Non-linear Integral and Differential Equations of Arbitrary Order, Journal of Nonlinear Sciences and Applications, 9, https://doi.org/10.22436/jnsa.009.06.128.
• Shoaib, M., Sarwar, M., Kumam, P., (2018), Multi-valued Fixed Point Theorem via F- contraction of Nadler Type and Application to Functional and Integral Equations, Boletim da Sociedade Paranaense de Matemática, 39, https://doi.org/10.5269/bspm.41105.
• Singh Chouhan, V., Sharma, R., (2014), Coupled fixed point theorems for rational contractions in partially ordered metric spaces, International Journal of Modern Mathematical Sciences, 12, 3, pp.165-174.
• Mitrovi\'{c}, S., Fabiano, N., Radojevi\'{c}, S. and Radenovi\'{c}, S., (2023), Remarks on Perov Fixed-Point Results on F-contraction Mappings Equipped with Binary Relation, Axioms, 12, 518, https://doi.org/10.3390/axioms12060518.
• Turinici, M., (2011), Ran Reurings theorems in ordered metric spaces, arxiv, 27, 11, https://doi.org/10.48550/arXiv.1103.5207.
• Turinici, M., (2011), Product fixed points in ordered metric spaces, arxiv, https://doi.org/10.48550/arXiv.1110.3079.
• Vujakovi\'{c}, J., Mitrovi\'{c}, S., Mitrovi\'{c}, A. D., Radenovi\'{c}, S., (2022), On Wardowski type results within G-metric spaces, Advanced Mathematical Analysis and its Applications.
• Vulpe, I. M., Ostrajkh, D., Khojman, F., (1981), The topological structure of a quasi-metric space (Russian) in Investigations in functional analysis and differential equations, Mathematical Sciences, pp.14–19.
• Wardowski, D., (2012), Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory and Applications, 94, https://doi.org/10.1186/1687-1812-2012-94.
• Younis, M., Mirkov, N., Savi\'{c}, A., Pantovi\'{c}, M., Radenovi\'{c}, S., (2023), Some critical remarks on recent results concerning F-contractions in b-metric spaces, CUBO, A Mathematical Journal, 25, pp.57-66, https://doi.org/10.56754/0719-0646.2501.057.
• Zada, M. B., Sarwar, M., (2019), Common fixed point theorems for rational $F_R$ contractive pairs of mappings with applications, Journal of Inequalities and Applications, 11, https://doi.org/10.1186/s13660-018-1952.
• Zada, M. B., Sarwar, M., Kumam, P., (2018), Fixed Point Results of Rational Type Contractions in b-Metric Spaces, International Journal of Analysis and Applications, 16, 6, pp.904-920.
• Zada, M. B., Sarwar, M., Tunç, C., (2018), Fixed point theorems in b-metric spaces and their applications to non-linear fractional differential and integral equations, Journal of Fixed Point Theory and Applications, 20, https://doi.org/10.1007/s11784-018-0510-0.
• Zhang, M., Zhu, C., (2022), Theorems of Common Fixed Points for Some Mappings in b2 Metric Spaces, Mathematics, 10, 3320, https://doi.org/10.3390/math10183320.