This paper develops a relaxed fixed-point framework for nonlinear operators acting on suprametric spaces. A new class of control functions $ \Theta^{R} $ is introduced, allowing strictly increasing but possibly discontinuous behaviors that go beyond the classical $ \vartheta $-contraction structure. Within this setting, several relaxed $ \vartheta _{\mathcal{R}} $-type contractive conditions are formulated through max-based suprametric functionals and on complete suprametric spaces. These conditions guarantee the existence and uniqueness of fixed-points under a suitable jump requirement on the control function. The theory is supported by explicit examples showing how discontinuities and nonlinear growth patterns influence convergence. Finally, two differential models, namely a second-order particle motion problem and a fourth-order beam equation, are used to demonstrate that their associated integral operators admit unique solutions within the proposed relaxed suprametric framework.
Citation: Abdurrahman Büyükkaya, Ekber Girgin, Haroon Ahmad, Mudasir Younis, Mahpeyker Öztürk. Relaxed contractions in suprametric spaces: A unified framework with applications to nonlinear differential models[J]. AIMS Mathematics, 2026, 11(4): 9008-9040. doi: 10.3934/math.2026372
This paper develops a relaxed fixed-point framework for nonlinear operators acting on suprametric spaces. A new class of control functions $ \Theta^{R} $ is introduced, allowing strictly increasing but possibly discontinuous behaviors that go beyond the classical $ \vartheta $-contraction structure. Within this setting, several relaxed $ \vartheta _{\mathcal{R}} $-type contractive conditions are formulated through max-based suprametric functionals and on complete suprametric spaces. These conditions guarantee the existence and uniqueness of fixed-points under a suitable jump requirement on the control function. The theory is supported by explicit examples showing how discontinuities and nonlinear growth patterns influence convergence. Finally, two differential models, namely a second-order particle motion problem and a fourth-order beam equation, are used to demonstrate that their associated integral operators admit unique solutions within the proposed relaxed suprametric framework.
| [1] | S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math., 3 (1922), 133–181. |
| [2] |
M. Berzig, First results in suprametric spaces with applications, Mediterr. J. Math., 19 (2022), 226. https://doi.org/10.1007/s00009-022-02148-6 doi: 10.1007/s00009-022-02148-6
|
| [3] |
M. Berzig, Fixed point results in generalized suprametric spaces, Topol. Algebra Appl., 11 (2023), 20230105. https://doi.org/10.1515/taa-2023-0105 doi: 10.1515/taa-2023-0105
|
| [4] |
M. Berzig, Nonlinear contraction in $b$-suprametric spaces, J. Anal., 32 (2024), 2401–2414. https://doi.org/10.1007/s41478-024-00732-5 doi: 10.1007/s41478-024-00732-5
|
| [5] |
S. K. Panda, R. P. Agarwal, E. Karapınar, Extended suprametric spaces and Stone-type theorem, AIMS Math., 8 (2023), 23183–23199. https://doi.org/10.3934/math.20231179 doi: 10.3934/math.20231179
|
| [6] |
M. Younis, H. Ahmad, M. Ozturk, D. Singh, A novel approach to the convergence analysis of chaotic dynamics in fractional order Chua's attractor model employing fixed points, Alex. Eng. J., 110 (2025), 363–375. https://doi.org/10.1016/j.aej.2024.10.001 doi: 10.1016/j.aej.2024.10.001
|
| [7] |
M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), 38. https://doi.org/10.1186/1029-242X-2014-38 doi: 10.1186/1029-242X-2014-38
|
| [8] | A. Branciari, A fixed point theorem of Banach–Caccioppoli type on generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31–37. |
| [9] |
M. Jleli, E. Karapınar, B. Samet, Further generalizations of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), 439. https://doi.org/10.1186/1029-242X-2014-439 doi: 10.1186/1029-242X-2014-439
|
| [10] |
J. Ahmad, A. E. Al-Mazrooeia, Y. J. Cho, Y. Yang, Fixed point results for generalized $\vartheta$-contractions, J. Nonlinear Sci. Appl., 10 (2017), 2350–2358. https://doi.org/10.22436/jnsa.010.05.07 doi: 10.22436/jnsa.010.05.07
|
| [11] |
X. Liu, S. Chang, Y. Xiao, L. C. Zao, Existence of fixed points for $\vartheta$-type contraction and $\vartheta$-type Suzuki contraction in complete metric spaces, Fixed Point Theory Appl., 2016 (2016), 8. https://doi.org/10.1186/s13663-016-0496-5 doi: 10.1186/s13663-016-0496-5
|
| [12] |
N. Hussain, V. Parvaneh, B. Samet, C. Vetro, Some fixed point theorems for generalized contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2015 (2015), 185. https://doi.org/10.1186/s13663-015-0433-z doi: 10.1186/s13663-015-0433-z
|
| [13] |
M. Cvetković, E. Karapınar, A. Petruşel, Fixed point theorems for basic $\vartheta$-contraction and applications, Carpathian J. Math., 40 (2024), 789–804. https://doi.org/10.37193/CJM.2024.03.16 doi: 10.37193/CJM.2024.03.16
|
| [14] |
H. Ahmad, F. U. Din, M. Younis, L. Guran, Analyzing the chaotic dynamics of a fractional-order Dadras–Momeni system using relaxed contractions, Fractal Fract., 8 (2024), 699. https://doi.org/10.3390/fractalfract8120699 doi: 10.3390/fractalfract8120699
|
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Abdurrahman Büyükkaya, Ekber Girgin, Haroon Ahmad, Mudasir Younis, Mahpeyker Öztürk. Relaxed contractions in suprametric spaces: A unified framework with applications to nonlinear differential models[J]. AIMS Mathematics, 2026, 11(4): 9008-9040. doi: 10.3934/math.2026372
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