The purpose of this article is to obtain common fixed point results in $ b $-metric spaces for generalized rational contractions involving control functions of two variables. We provide an example to show the originality of our main result. As outcomes of our results, we derive certain fixed and common fixed point results for rational contractions presuming control functions of one variable and constants. As an application, we investigate the solution of an integral equation.
Citation: Badriah Alamri, Jamshaid Ahmad. Fixed point results in $ b $-metric spaces with applications to integral equations[J]. AIMS Mathematics, 2023, 8(4): 9443-9460. doi: 10.3934/math.2023476
The purpose of this article is to obtain common fixed point results in $ b $-metric spaces for generalized rational contractions involving control functions of two variables. We provide an example to show the originality of our main result. As outcomes of our results, we derive certain fixed and common fixed point results for rational contractions presuming control functions of one variable and constants. As an application, we investigate the solution of an integral equation.
| [1] |
M. Frechet, Sur quelques points du calcul fonctionnel, Rendiconti del circolo matematico di Palermo, Rend. Circ. Matem. Palermo, 22 (1906), 1–72. http://dx.doi.org/10.1007/BF03018603 doi: 10.1007/BF03018603
|
| [2] | S. Banach, Sur les operations dans les ensembles abstracts ET leur applications aux equations integrals, Fund. Math., 3 (1922), 133–181. |
| [3] | R. Kannan, Some results on fixed points-Ⅱ, The American Mathematical Monthly, 76 (1969), 405–408. |
| [4] | S. Chatterjea, Fixed point theorems, C. R. Acad. Bulgara Sci., 25 (1972), 727–730. |
| [5] | B. Fisher, Mappings satisfying a rational inequality, Bull. Math. de la Soc. Sci. Math. de la R. S. de Roumanie, 24 (1980), 247–251. |
| [6] | I. Bakhtin, The contraction mapping principle in almost metric spaces, Functional Analysis, 30 (1989), 26–37. |
| [7] | S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1 (1993), 5–11. |
| [8] |
R. Koleva, B. Zlatanov, On fixed points for Chatterjea's maps in $b$-metric spaces, Turkish Journal of Analysis and Number Theory, 4 (2016), 31–34. http://dx.doi.org/10.12691/tjant-4-2-1 doi: 10.12691/tjant-4-2-1
|
| [9] |
M. Abbas, I. Chema, A. Razani, Existence of common fixed point for $b$-metric rational type contraction, Filomat, 30 (2016), 1413–1429. http://dx.doi.org/10.2298/FIL1606413A doi: 10.2298/FIL1606413A
|
| [10] |
H. Hammad, M. la Sen, A solution of Fredholm integral equation by using the cyclic $\eta _{s}^{q}$ -rational contractive mappings technique in $b$-metric-like spaces, Symmetry, 11 (2019), 1184. http://dx.doi.org/10.3390/sym11091184 doi: 10.3390/sym11091184
|
| [11] |
H. Hammad, M. la Sen, Generalized contractive mappings and related results in $b$-metric like spaces with an application, Symmetry, 11 (2019), 667. http://dx.doi.org/10.3390/sym11050667 doi: 10.3390/sym11050667
|
| [12] |
E. Ameer, H. Aydi, M. Arshad, M. la Sen, Hybrid Ćirić type graphic ($Y$-$\Lambda)$-contraction mappings with applications to electric circuit and fractional differential equations, Symmetry, 12 (2020), 467. http://dx.doi.org/10.3390/sym12030467 doi: 10.3390/sym12030467
|
| [13] |
M. Seddik, H. Taieb, Some fixed point theorems of rational type contraction in $b$-metric spaces, Moroccan Journal of Pure and Applied Analysis, 7 (2021), 350–363. http://dx.doi.org/10.2478/mjpaa-2021-0023 doi: 10.2478/mjpaa-2021-0023
|
| [14] |
H. Huang, G. Deng, S. Radenović, Fixed point theorems for $ C$-class functions in $b$-metric spaces and applications, J. Nonlinear Sci. Appl., 10 (2017), 5853–5868. http://dx.doi.org/10.22436/jnsa.010.11.23 doi: 10.22436/jnsa.010.11.23
|
| [15] |
H. Huang, Y. Singh, M. Khan, S. Radenović, Rational type contractions in extended $b$-metric spaces, Symmetry, 13 (2021), 614. http://dx.doi.org/10.3390/sym13040614 doi: 10.3390/sym13040614
|
| [16] |
A. Shoaib, A. Asif, M. Arshad, E. Ameer, Generalized dynamic process for generalized multivalued $F$-contraction of Hardy Rogers type in $ b$-metric spaces, Turkish Journal of Analysis and Number Theory, 6 (2018), 43–48. http://dx.doi.org/10.12691/tjant-6-2-2 doi: 10.12691/tjant-6-2-2
|
| [17] |
H. Aydi, M. Bota, E. Karapınar, S. Mitrović, A fixed point theorem for set-valued quasi-contractions in $b$-metric spaces, Fixed Point Theory Appl., 2012 (2012), 88. http://dx.doi.org/10.1186/1687-1812-2012-88 doi: 10.1186/1687-1812-2012-88
|
| [18] | H. Aydi, M. Bota, E. Karapinar, S. Moradi, A common fixed point for weak $\varphi $-contractions on $b$-metric spaces, Fixed Point Theory, 13 (2012), 337–346. |
| [19] | H. Aydi, A. Felhi, S. Sahmim, Common fixed points via implicit contractions on $b$-metric-like spaces, J. Nonlinear Sci. Appl., 10 (2017), 1524–1537. |
| [20] |
H. Afshari, H. Aydi, E. Karapınar, On generalized $\alpha $- $\psi $-Geraghty contractions on $b$-metric spaces, Georgian Math. J., 27 (2020), 9–21. http://dx.doi.org/10.1515/gmj-2017-0063 doi: 10.1515/gmj-2017-0063
|
| [21] |
Z. Ma, J. Ahmad, A. Al-Mazrooei, D. Lateef, Fixed point results for rational orbitally ($\Theta, \delta _{b}$)-contractions with an application, J. Funct. Space., 2021 (2021), 9946125. http://dx.doi.org/10.1155/2021/9946125 doi: 10.1155/2021/9946125
|
| [22] |
K. Haji, K. Tola, M. Mamud, Fixed point results for generalized rational type $\alpha $-admissible contractive mappings in the setting of partially ordered $b$-metric spaces, BMC Res. Notes, 15 (2022), 242. http://dx.doi.org/10.1186/s13104-022-06122-z doi: 10.1186/s13104-022-06122-z
|
| [23] |
T. Stephen, Y. Rohen, M. Singh, K. Devi, Some rational $F$-contractions in $b$-metric spaces, Nonlinear Functional Analysis and Applications, 27 (2022), 309–322. http://dx.doi.org/10.22771/nfaa.2022.27.02.07 doi: 10.22771/nfaa.2022.27.02.07
|
| [24] |
H. Huang, G. Deng, S. Radenović, Fixed point theorems in $ b$-metric spaces with applications to differential equations, J. Fixed Point Theory Appl., 20 (2018), 52. http://dx.doi.org/10.1007/s11784-018-0491-z doi: 10.1007/s11784-018-0491-z
|
| [25] | S. Aleksic, Z. Mitrovic, S. Radenovic, On some recent fixed point results for single and multivalued mappings in $b$-metric spaces, Fasc. Math., 61 (2018), 5–16. |
| [26] |
H. Hammad, M. la Sen, A technique of tripled coincidence points for solving a system of nonlinear integral equations in POCML spaces, J. Inequal. Appl., 2020 (2020), 211. http://dx.doi.org/10.1186/s13660-020-02477-8 doi: 10.1186/s13660-020-02477-8
|
| [27] |
H. Hammad, M. la Sen, H. Aydi, Generalized dynamic process for an extended multi-valued $F$-contraction in metric-like spaces with applications, Alex. Eng. J., 59 (2020), 3817–3825. http://dx.doi.org/10.1016/j.aej.2020.06.037 doi: 10.1016/j.aej.2020.06.037
|
| [28] |
H. Hammad, M. la Sen, H. Aydi, Analytical solution for differential and nonlinear integral equations via $F_{\omega _{\varrho }}$ -suzuki contractions in modified $\omega _{\varrho }$-metric-like spaces, J. Funct. Space., 2021 (2021), 6128586. http://dx.doi.org/10.1155/2021/6128586 doi: 10.1155/2021/6128586
|
| [29] |
P. Debnath, M. la Sen, Set-valued interpolative Hardy-Rogers and set-valued Reich-Rus-Ćirić-type contractions in $ b$-metric spaces, Mathematics, 7 (2019), 849. http://dx.doi.org/10.3390/math7090849 doi: 10.3390/math7090849
|
| [30] |
N. Alamgir, Q. Kiran, H. Aydi, Y. Gaba, Fuzzy fixed point results of generalized almost $F$-contractions in controlled metric spaces, Adv. Differ. Equ., 2021 (2021), 476. http://dx.doi.org/10.1186/s13662-021-03598-0 doi: 10.1186/s13662-021-03598-0
|
Badriah Alamri, Jamshaid Ahmad. Fixed point results in $ b $-metric spaces with applications to integral equations[J]. AIMS Mathematics, 2023, 8(4): 9443-9460. doi: 10.3934/math.2023476
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