This paper aims to present the new concept of generalized $ \alpha $-$ \psi $-Geraghty contraction type for both single-valued and multi-valued mappings in a strong $ b $-metric space. Our approach extends the concepts introduced by Karapınar for single-valued mappings and by Afshari et al. for multi-valued mappings. Several results on the fixed point theory of this contraction type are provided both globally (in the entire space) and locally (in the closed ball). Our findings expand and enhance several fixed point theorems, including the famous Geraghty's theorem. Examples and an application of integral equations are included to illustrate these results.
Citation: Saud M. Alsulami, Thanaa A. Alarfaj. Generalized $ \alpha $-$ \psi $-Geraghty contraction type fixed point theorems in strong $ b $-metric spaces[J]. AIMS Mathematics, 2025, 10(9): 21104-21125. doi: 10.3934/math.2025943
This paper aims to present the new concept of generalized $ \alpha $-$ \psi $-Geraghty contraction type for both single-valued and multi-valued mappings in a strong $ b $-metric space. Our approach extends the concepts introduced by Karapınar for single-valued mappings and by Afshari et al. for multi-valued mappings. Several results on the fixed point theory of this contraction type are provided both globally (in the entire space) and locally (in the closed ball). Our findings expand and enhance several fixed point theorems, including the famous Geraghty's theorem. Examples and an application of integral equations are included to illustrate these results.
| [1] | H. Afshari, M. Atapour, H. Aydi, Generalized $\alpha$-$\psi$-Geraghty multi-valued mappings on $b$-metric spaces endowed with a graph, TWMS J. Appl. Eng. Math., 7 (2017), 248–260. |
| [2] |
H. Afshari, H. Aydi, E. Karapinar, Existence of fixed points of set-valued mappings in $b$-metric spaces, E. Asian Math. J., 32 (2016), 319–332. https://doi.org/10.7858/eamj.2016.024 doi: 10.7858/eamj.2016.024
|
| [3] |
H. Afshari, H. Aydi, E. Karapınar, On generalized $\alpha$-$\psi$-Geraghty contractions on $b$-metric spaces, Georgian Math. J., 27 (2020), 9–21. https://doi.org/10.1515/gmj-2017-0063 doi: 10.1515/gmj-2017-0063
|
| [4] |
T. A. Alarfaj, S. M. Alsulami, On a version of Dontchev and Hager's inverse mapping theorem, Axioms, 13 (2024), 445. https://doi.org/10.3390/axioms13070445 doi: 10.3390/axioms13070445
|
| [5] |
M. U. Ali, T. Kamran, On ($\alpha$*, $\psi$)-contractive multi-valued mappings, Fixed Point Theory A., 2013 (2013), 1–7. https://doi.org/10.1186/1687-1812-2013-137 doi: 10.1186/1687-1812-2013-137
|
| [6] |
I. Altun, K. Sadarangani, Generalized Geraghty type mappings on partial metric spaces and fixed point results, Arab. J. Math., 2 (2013), 247–253. https://doi.org/10.1007/s40065-013-0073-2 doi: 10.1007/s40065-013-0073-2
|
| [7] |
T. V. An, N. V. Dung, Answers to Kirk-Shahzad's questions on strong $b$-metric spaces, Taiwan. J. Math., 20 (2016), 1175–1184. https://doi.org/10.11650/tjm.20.2016.6359 doi: 10.11650/tjm.20.2016.6359
|
| [8] |
M. Arab, M. S. Abdo, N. Alghamdi, M. Awadalla, Analyzing coupled delayed fractional systems: Theoretical insights and numerical approaches, Mathematics, 13 (2025), 1113. https://doi.org/10.3390/math13071113 doi: 10.3390/math13071113
|
| [9] |
M. Asadi, E. Karapınar, A. Kumar, $\alpha$-$\psi$-Geraghty contractions on generalized metric spaces, J. Inequal. Appl., 2014 (2014), 1–21. https://doi.org/10.1186/1029-242X-2014-423 doi: 10.1186/1029-242X-2014-423
|
| [10] |
V. Berinde, A. Petrusel, I. A. Ru, Remarks on the terminology of the mappings in fixed point iterative methods in metric spaces, Fixed Point Theor., 24 (2023), 525–540. http://dx.doi.org/10.24193/fpt-ro.2023.2.05 doi: 10.24193/fpt-ro.2023.2.05
|
| [11] | S. Czerwik, Nonlinear set-valued contraction mappings in $b$-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263–276. |
| [12] |
N. V. Dung, V. T. L. Hang, On relaxations of contraction constants and Caristi's theorem in $b$-metric spaces, J. Fixed Point Theory A., 18 (2016), 267–284. https://doi.org/10.1007/s11784-015-0273-9 doi: 10.1007/s11784-015-0273-9
|
| [13] |
H. E. Mamouni, K. Hattaf, N. Yousfi, Dynamics of an epidemic model for COVID-19 with Hattaf fractal-fractional operator and study of existence of solutions by means of fixed point theory, J. Math. Comput. S., 36 (2025), 371–385. https://doi.org/10.22436/jmcs.036.04.02 doi: 10.22436/jmcs.036.04.02
|
| [14] | V. Enescu, H. Sahbi, Learning conditionally untangled latent spaces using fixed point iteration, In: The British Machine Vision Conference (BMVC), Glasgow, United Kingdom, 2024. |
| [15] |
Z. Feinstein, B. Rudloff, Technical note–characterizing and computing the set of Nash equilibria via vector optimization, Oper. Res., 72 (2023), 2082–2096. https://doi.org/10.1287/opre.2023.2457 doi: 10.1287/opre.2023.2457
|
| [16] | R. George, Z. D. Mitrović, Result on best approximations in strong $b$-normed space, J. Nonlinear Convex A., 26 (2025), 1457–1462. |
| [17] | M. A. Geraghty, On contractive mappings, P. Am. Math. Soc., 40 (1973), 604–608. |
| [18] |
N. Hussain, S. M. Alsulami, H. Alamri, Solving fractional differential equations via fixed points of chatterjea maps, CMES-Comp. Model. Eng., 135 (2023), 2617–2648. https://doi.org/10.32604/cmes.2023.023143 doi: 10.32604/cmes.2023.023143
|
| [19] |
E. Karapınar, $\alpha$-$\psi$-Geraghty contraction type mappings and some related fixed point results, Filomat, 28 (2014), 37–48. https://doi.org/10.2298/FIL1401037K doi: 10.2298/FIL1401037K
|
| [20] |
E. Karapınar, A discussion on "$\alpha$-$\psi$-Geraghty contraction type mappings", Filomat, 28 (2014), 761–766. https://doi.org/10.2298/FIL1404761K doi: 10.2298/FIL1404761K
|
| [21] |
E. Karapinar, P. Kumam, P. Salimi, On alpha-psi-Meir-Keeler contractive mappings, Fixed Point Theory A., 2013 (2013), 94. https://doi.org/10.1186/1687-1812-2013-94 doi: 10.1186/1687-1812-2013-94
|
| [22] | W. Kirk, N. Shahzad, Fixed point theory in distance spaces, Springer Cham, 2014. https://doi.org/10.1007/978-3-319-10927-5 |
| [23] |
B. Mohammadi, S. Rezapour, N. Shahzad, Some results on fixed points of $\alpha$-$\psi$-Ćirić generalized multifunctions, Fixed Point Theory A., 2013 (2013), 1–10. https://doi.org/10.1186/1687-1812-2013-24 doi: 10.1186/1687-1812-2013-24
|
| [24] |
R. Pant, R. Panicker, Geraghty and Ćirić type fixed point theorems in $b$-metric spaces, J. Nonlinear Sci. Appl., 9 (2016), 5741–5755. https://doi.org/10.22436/jnsa.009.11.03 doi: 10.22436/jnsa.009.11.03
|
| [25] |
O. Popescu, Some new fixed point theorems for $\alpha$-Geraghty contraction type maps in metric spaces, Fixed Point Theory A., 2014 (2014), 1–12. https://doi.org/10.1186/1687-1812-2014-190 doi: 10.1186/1687-1812-2014-190
|
| [26] |
B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $\alpha$-$\psi$-contractive type mappings, Nonlinear Anal.-Theor., 75 (2012), 2154–2165. https://doi.org/10.1016/j.na.2011.10.014 doi: 10.1016/j.na.2011.10.014
|
| [27] |
M. Younis, H. Ahmad, M. Öztürk, 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
|
| [28] |
M. Younis, M. Öztürk, Some novel proximal point results and applications, Univ. J. Math. Appl., 8 (2025), 8–20. https://doi.org/10.32323/ujma.1597874 doi: 10.32323/ujma.1597874
|
Saud M. Alsulami, Thanaa A. Alarfaj. Generalized $ \alpha $-$ \psi $-Geraghty contraction type fixed point theorems in strong $ b $-metric spaces[J]. AIMS Mathematics, 2025, 10(9): 21104-21125. doi: 10.3934/math.2025943
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