Fixed point theory plays a fundamental role in nonlinear analysis and has significant applications in differential equations, integral equations, optimization, and applied mathematics. Inspired by developments in generalized metric structures, we introduced and studied the Geraghty-type contractive conditions in the setting of cone $ \theta $-type multiplicative metric spaces defined on ordered Banach algebras. By extending the classical notions of $ \theta $-metric and multiplicative metric spaces, we constructed a framework based on solid multiplicative cones and established their essential topological and convergence properties. Within this generalized structure, we formulated new fixed point theorems for self-mappings satisfying cone-valued $ \theta $-type multiplicative Geraghty contractions, which weakened the traditional Lipschitz condition while preserving the existence and uniqueness of fixed points in complete spaces. The proposed results significantly extended and unified several well-known contraction principles, including those of Banach, Kannan, Chatterjea, and standard Geraghty contractions, under a broader cone-valued multiplicative setting. Furthermore, we developed extensions to higher-dimensional frameworks to enhance applicability in complex nonlinear systems. Illustrative examples are presented to substantiate the theoretical findings and to demonstrate the effectiveness of the introduced approach in generalized analytical environments.
Citation: Pravin Singh, Shivani Singh, Sani Salisu, Virath Singh. Fixed point theorems for nonlinear contractive mappings in cone-valued $ \theta $-type multiplicative metric spaces[J]. AIMS Mathematics, 2026, 11(6): 17794-17819. doi: 10.3934/math.2026725
Fixed point theory plays a fundamental role in nonlinear analysis and has significant applications in differential equations, integral equations, optimization, and applied mathematics. Inspired by developments in generalized metric structures, we introduced and studied the Geraghty-type contractive conditions in the setting of cone $ \theta $-type multiplicative metric spaces defined on ordered Banach algebras. By extending the classical notions of $ \theta $-metric and multiplicative metric spaces, we constructed a framework based on solid multiplicative cones and established their essential topological and convergence properties. Within this generalized structure, we formulated new fixed point theorems for self-mappings satisfying cone-valued $ \theta $-type multiplicative Geraghty contractions, which weakened the traditional Lipschitz condition while preserving the existence and uniqueness of fixed points in complete spaces. The proposed results significantly extended and unified several well-known contraction principles, including those of Banach, Kannan, Chatterjea, and standard Geraghty contractions, under a broader cone-valued multiplicative setting. Furthermore, we developed extensions to higher-dimensional frameworks to enhance applicability in complex nonlinear systems. Illustrative examples are presented to substantiate the theoretical findings and to demonstrate the effectiveness of the introduced approach in generalized analytical environments.
| [1] | S. Banach, Sur les operations dans les ensembles abstraits et leurs application aux equations integrals, Fund. Math., 3 (1922), 133–181. |
| [2] |
F. Khojasteh, E. Karapinar, S. Radenović, $\theta$-metric space: A generalization, Math. Probl. Eng., 2013 (2013), 504609. https://doi.org/10.1155/2013/504609 doi: 10.1155/2013/504609
|
| [3] | M. Grossman, R. Katz, Non-Newtonian calculus, Pigeon Cove, MA: Lee Press, 1972. |
| [4] |
A. E. Bashirov, E. M. Kurpinar, A. Özyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36–48. https://doi.org/10.1016/j.jmaa.2007.03.081 doi: 10.1016/j.jmaa.2007.03.081
|
| [5] | M. Özavsar, Fixed points of multiplicative contraction mappings on multiplicative metric spaces, J. Eng. Technol. Appl. Sci., 2 (2017), 65–79. |
| [6] |
M. Abbas, M. De La Sen, T. Nazir, Common fixed points of generalized rational co-cyclic mappings in multiplicative metric spaces, Discrete Dyn. Nat. Soc., 2015 (2015), 532725. http://doi.org/10.1155/2015/532725. doi: 10.1155/2015/532725
|
| [7] |
X. He, M. Song, D. Chen, Common fixed points for weak commutative mappings on a multiplicative metric space, Fixed Point Theory Appl., 2014 (2014), 48. http://doi.org/10.1186/1687-1812-2014-48 doi: 10.1186/1687-1812-2014-48
|
| [8] |
O. Yamaod, W. Sintunavarat, Some fixed point results for generalized contraction mappings with cyclic $(\alpha, \beta)$-admissible mapping in multiplicative metric spaces, J. Inequal. Appl., 2014 (2014), 488. http://doi.org/10.1186/1029-242X-2014-488. doi: 10.1186/1029-242X-2014-488
|
| [9] |
F. Gu, Y. J. Cho, Common fixed point results for four maps satisfying phicontractive condition in multiplicative metric spaces, Fixed Point Theory Appl., 2015 (2015), 165. https://doi.org/10.1186/s13663-015-0412-4 doi: 10.1186/s13663-015-0412-4
|
| [10] |
C. Mongkolkeha, W. Sintunavarat, Best proximity points for multiplicative proximal contraction mapping on multiplicative metric spaces, J. Nonlinear Sci. Appl., 8 (2015), 1134–1140. https://doi.org/10.22436/jnsa.008.06.22 doi: 10.22436/jnsa.008.06.22
|
| [11] | B. E Rome, M. Sarwar, Characterization of multiplicative metric completeness, Int. J. Anal. Appl., 10 (2016), 90–94. |
| [12] |
P. Singh, S. Singh, V. Singh, Fixed point results in a multiplicative $b_2$-metric space, Nonlinear Funct. Anal. Appl., 30 (2025), 361–372. https://doi.org/10.22771/nfaa.2025.30.02.04 doi: 10.22771/nfaa.2025.30.02.04
|
| [13] |
E. Kaplan, N. Tas, S. Haque, N. Mlaiki, Multiplicative modular metric spaces and some fixed point results, J. Inequal. Appl., 2025 (2025), 54. https://doi.org/10.1186/s13660-025-03303-9 doi: 10.1186/s13660-025-03303-9
|
| [14] |
S. Saejung, On multiplicative modular metric spaces of Kaplan et al.: corrections and clarifications, Fixed Point Theory Algorithms Sci. Eng., 2025 (2025), 1–11. https://doi.org/10.1186/s13663-025-00797-2 doi: 10.1186/s13663-025-00797-2
|
| [15] |
S. Babak, A. Shloof, N. Senu, D. Baleanu, A rational power function-based approach for solving rational fractional differential equations, Int. J. Optim. Contro. Theor. Appl., 16 (2026), 370–382. https://doi.org/10.36922/IJOCTA025320139 doi: 10.36922/IJOCTA025320139
|
| [16] |
H. Liu, S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl., 2013 (2013), 320. https://doi.org/10.1186/1687-1812-2013-320 doi: 10.1186/1687-1812-2013-320
|
| [17] | F. Riesz, S. B. Nagy, Functional analysis, New York: Dover, 1990. |
| [18] | M. Luczak, Holomorphic functional calculus in A-pseudoconvex algebras, Comment. Math., 29 (2005), 161–169. |
| [19] | R. Kannan, Some results on fixed points-Ⅱ, Amer. Math. Mon., 76 (1969), 405–408. |
| [20] | S. K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727–730. |
Pravin Singh, Shivani Singh, Sani Salisu, Virath Singh. Fixed point theorems for nonlinear contractive mappings in cone-valued $ \theta $-type multiplicative metric spaces[J]. AIMS Mathematics, 2026, 11(6): 17794-17819. doi: 10.3934/math.2026725
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