Authors:
Achala Mishra,Hiral Raja,DOI NO:
https://doi.org/10.26782/jmcms.2025.08.00004Keywords:
Cauchy sequence,completeness,uniqueness,Fixed point,G-cone metric space,φ-contraction,Normal cone,Perturbation function,Abstract
In this work, we provide a set of enhanced fixed-point theorems over Banach spaces with normal cones in the context of G-cone metric spaces. Our results extend and generalize existing theorems by incorporating φ-contractive mappings and perturbation functions within the contractive conditions. Specifically, we propose new fixed-point theorems using φ-difference type conditions, auxiliary control functions, and jointly lower semi-continuous metrics. We present illustrative instances to confirm that the theorems are applicable. The results obtained improve classical fixed-point theorems and offer broader applicability in nonlinear analysis. We also demonstrate the applicability of the developed theorems to fractional differential equations.Refference:
I. Abbas, M., & Rhoades, B. E. (2008). Fixed- and periodic-point results in cone metric spaces. Applied Mathematics Letters, 21(5), 521–526.
II. Afshari, H., Alsulami, H. H., & Karapinar, E. (2016). On the extended multivalued Geraghty type contractions. Journal of Nonlinear Sciences and Applications, 9, 4695–4706. 10.22436/jnsa.009.06.108.
III. Amini-Harandi, A_r., & Emami, H. (2010). A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications, 72(6), 2238–2242. 10.1016/j.na.2009.10.023.
IV. Branciari, A. (2000). A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publicationes Mathematicae Debrecen, 57, 31–37. 10.5486/PMD.2000.2133
V. Cherichi, M., & Samet, B. (2012). Fixed point theorems on ordered gauge spaces with applications to nonlinear integral equations. Fixed Point Theory and Applications, 2012, Article 13. 10.1186/1687-1812-2012-13
VI. Fernandez, J., Malviya, N., Savić, A., Paunović, M., & Mitrović, Z. D. (2022). The extended cone b-metric-like spaces over Banach algebra and some applications. Mathematics, 10(1), 149. 10.3390/math10010149.
VII. Fulga, A., Afshari, H., & Shojaat, H. (2021). Common fixed point theorems on quasi-cone metric space over a divisible Banach algebra. Advances in Difference Equations, 2021, Article 306. 10.1186/s13662-021-03464-z.
VIII. Gupta, V., Shatanawi, W., & Mani, N. (2016). Fixed point theorems for (Ψ,β)-Geraghty contraction type maps in ordered metric spaces and some applications to integral and ordinary differential equations. Journal of Fixed Point Theory and Applications, 19, 1251-1267. 10.1007/s11784-016-0303-2.
IX. Huang, H., & Xu, S. (2013). Fixed point theorems of contractive mappings in cone b-metric spaces and applications. Fixed Point Theory and Applications, 2013, Article 112. 10.1186/1687-18122013112.
X. Huang, L. G., & Zhang, X. (2007). Cone metric spaces and fixed-point theorems of contractive mappings. Journal of Mathematical Analysis and Applications, 332(2), 1468-1476.
XI. Jachymski, J. (2008). The contraction principle for mappings on a metric space with a graph. Proceedings of the American Mathematical Society, 136(4), 1359-1373. http://www.jstor.org/stable/20535302
XII. Jleli, M., & Samet, B. (2014). A new generalization of the Banach contraction principle. Journal of Inequalities and Applications, 2014, Article 38.
XIII. Karapınar, E., Fulga, A., & Roldán López de Hierro, A. F. (2021). Fixed point theory in the setting of ( α,β, ψ,ϕ )-interpolative contractions. Advances in Difference Equations, 2021, Article 339. 10.1186/s13662-021-03491-w.
XIV. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (Eds.). (2006). Theory and applications of fractional differential equations (Vol. 204, pp. 1-523). Elsevier. 10.1016/S0304-0208(06)80001-0
XV. Khojasteh, F., Shukla, S., & Radenovic, S. (2015). A new approach to the study of fixed point theory for simulation functions. Filomat, 26(6), 1189-1194. 10.2298/FIL1506189K
XVI. Kirk, W. A. (2003). Fixed points of asymptotic contractions. Journal of Mathematical Analysis and Applications. Vol. 277 (2), 15 January 2003, Pages 645-650. 10.1016/S0022-247X(02)00612-1
XVII. Li, X., Hussain, A., Adeel, M., & Savas, E. (2019). Fixed point theorems for Z_ ∀-contraction and applications to nonlinear integral equations. IEEE Access, 7, 120023-120032. 10.1109/ACCESS.2019.2933693
XVIII. Liu, X., Chang, S., Xiao, Y., & Zhao, L. (2016). Existence of fixed points for Θ-type contraction and Θ-type Suzuki contraction in complete metric spaces. Fixed Point Theory and Applications, 2016, Article 8. 10.1186/s13663-016-0496-5
XIX. Liu, X. L., Ansari, A. H., Chandok, S., & Radenovic, S. (2018). On some results in metric spaces using auxiliary simulation functions via new functions. Journal of Computational Analysis and Applications, 24(6), 1103-1114.
XX. Long, H. V., Son, N. T. K., & Rodríguez-López, R. (2017). Some generalizations of fixed point theorems in partially ordered metric spaces and applications to partial differential equations with uncertainty. Vietnam Journal of Mathematics, 46, 531-555. 10.1007/s10013-017-0254-y.
XXI. Nieto, J. J., & López, R. R. (2005). Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order, 22(3), 223-239. 10.1007/s11083-005-9018-5.
XXII. Radenovic, S., Vetro, F., & Vujakovic, J. (2017). An alternative and easy approach to fixed point results via simulation functions. Demonstratio Mathematica, 50(1), 224-231.
XXIII. Rao, N. S., Aloqaily, A., & Mlaiki, N. (2024). Result – n fixed points in b-metric space by altering distance functions. Heliyon, 10(7), e33962. 1016/j.heliyon.2024.e33962.
XXIV. Rashwan, R. A., Hammad, H. A., Gamal, M., Omran, S., & De la Sen, M. (2024). Fixed point methodologies for ψ-contraction mappings in cone metric spaces over Banach algebra with supportive applications. International Journal of Analysis and Applications, 22, 120. 10.28924/2291-8639-22-2024120
XXV. Rezapour, S., & Haghi, R. H. (2008). Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings.” Journal of Mathematical Analysis and Applications, 345(2), 719-724. 10.1016/j.jmaa.2008.04.049