The aim of this paper is to define a Berinde type ($ \rho $, $ \mu $)-$ \vartheta $ contraction and establish some fixed point results for self mappings in the setting of complete metric spaces. We derive new fixed point results, which extend and improve some results in the literature. We also supply a non trivial example to support the obtained results. Finally, we investigate the existence of solutions for the nonlinear fractional differential equation.
Citation: Saleh Abdullah Al-Mezel, Jamshaid Ahmad. Fixed point results with applications to nonlinear fractional differential equations[J]. AIMS Mathematics, 2023, 8(8): 19743-19756. doi: 10.3934/math.20231006
The aim of this paper is to define a Berinde type ($ \rho $, $ \mu $)-$ \vartheta $ contraction and establish some fixed point results for self mappings in the setting of complete metric spaces. We derive new fixed point results, which extend and improve some results in the literature. We also supply a non trivial example to support the obtained results. Finally, we investigate the existence of solutions for the nonlinear fractional differential equation.
| [1] |
S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fund. Math., 3 (1922), 133–181. http://doi.org/10.4064/FM-3-1-133-181 doi: 10.4064/FM-3-1-133-181
|
| [2] | V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Analysis Forum, 9 (2004), 43–53. |
| [3] | V. Berinde, General constructive fixed point theorem for Ćirić-type almost contractions in metric spaces, Carpathian J. Math., 24 (2008), 10–19. |
| [4] |
B. Samet, C. Vetro, P. Vetro, Fixed point theorem for $\alpha$-$\psi$ contractive type mappings, Nonlinear Anal. Thero., 75 (2012), 2154–2165. https://doi.org/10.1016/j.na.2011.10.014 doi: 10.1016/j.na.2011.10.014
|
| [5] |
P. Salimi, C. Vetro, P. Vetro, Fixed point theorems for twisted ($\alpha $, $\beta $)-$\psi$-contractive type mappings and applications, Filomat, 27 (2013), 605–615. https://doi.org/10.2298/FIL1304605S doi: 10.2298/FIL1304605S
|
| [6] |
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
|
| [7] |
J. Ahmad, A. E. Al-Mazrooei, Y. J. Cho, Y. O. Yang, Fixed point results for generalized $\Theta$-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
|
| [8] |
M. Abbas, H. Iqbal, A. Petrusel, Fixed points for multivalued Suzuki type $(\Theta, \mathfrak{R})$-contraction mapping with applications, J. Funct. Space., 2019 (2019), 9565804. https://doi.org/10.1155/2019/9565804 doi: 10.1155/2019/9565804
|
| [9] |
J. Ahmad, A. S. Al-Rawashdeh, Common fixed points of set mappings endowed with directed graph, Tbilisi Math. J., 11 (2018), 107–123. https://doi.org/10.32513/tbilisi/1538532030 doi: 10.32513/tbilisi/1538532030
|
| [10] | J. Ahmad, A. E. Al-Mazrooei, Common fixed point theorems for multivalued mappings on metric spaces with a directed graph, Bull. Math. Anal. Appl., 10 (2018), 26–37. |
| [11] |
J. Ahmad, A. E. Al-Mazrooei, I. Altun, Generalized $\Theta$ -contractive fuzzy mappings, J. Intell. Fuzzy Syst., 35 (2018), 1935–1942. https://doi.org/10.3233/JIFS-171515 doi: 10.3233/JIFS-171515
|
| [12] |
J. Ahmad, D. Lateef, Fixed point theorems for rational type ($\alpha, \Theta $)-contractions in controlled metric spaces, J. Nonlinear Sci. Appl., 13 (2020), 163–170. https://doi.org/10.22436/jnsa.013.03.05 doi: 10.22436/jnsa.013.03.05
|
| [13] | A. E. Al-Mazrooei, J. Ahmad, Multivalued fixed point theorems of generalized (K, J)-contractions with applications, J. Math. Anal., 11 (2020), 113–122. |
| [14] | A. Al-Rawashdeh, J. Ahmad, Common fixed point theorems for JS-contractions, Bull. Math. Anal. Appl., 8 (2016), 12–22. |
| [15] | Z. Aslam, J. Ahmad, N. Sultana, New common fixed point theorems for cyclic compatible contractions, J. Math. Anal., 8 (2017), 1–12. |
| [16] |
N. Hussain, A. E. Al-Mazrooei, J. Ahmad, Fixed point results for generalized ($\alpha $-$\eta$)-$\Theta$ contractions with applications, J. Nonlinear Sci. Appl., 10 (2017), 4197–4208. https://doi.org/10.22436/JNSA.010.08.15 doi: 10.22436/JNSA.010.08.15
|
| [17] |
Z. Ma, J. Ahmad, A. E. Al-Mazrooei, D. Lateef, Fixed point results for rational orbitally ($\Theta$, $\delta_{b}$)-contractions with an application, J. Funct. Space., 2021 (2021), 9946125. https://doi.org/10.1155/2021/9946125 doi: 10.1155/2021/9946125
|
| [18] |
D. Baleanu, S. Rezapour, H. Mohammadi, Some existence results on nonlinear fractional differential equations, Phil. Trans. R. Soc. A, 371 (2013), 20120144. http://doi.org/10.1098/rsta.2012.0144 doi: 10.1098/rsta.2012.0144
|
| [19] |
H. H. Al-Sulami, J. Ahmad, N. Hussain, A. Latif, Relation theoretic ($\Theta$, $R$) contraction results with applications to nonlinear matrix equations, Symmetry, 10 (2018), 767. http://doi.org/10.3390/sym10120767 doi: 10.3390/sym10120767
|
| [20] | W. Onsod, T. Saleewong, J. Ahmad, A. E. Al-Mazrooei, P. Kumam, Fixed points of a $\Theta$-contraction on metric spaces with a graph, Commun. Nonlinear Anal., 2 (2016), 139–149. |
| [21] |
J. Zhou, S. Zhang, Y. He, Existence and stability of solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 498 (2021), 124921. https://doi.org/10.1016/j.jmaa.2020.124921 doi: 10.1016/j.jmaa.2020.124921
|
| [22] |
J. Zhou, S. Zhang, Y. He, Existence and stability of solution for nonlinear differential equations with $\psi$-Hilfer fractional derivative, Appl. Math. Lett., 121 (2021), 107457. https://doi.org/10.1016/j.aml.2021.107457 doi: 10.1016/j.aml.2021.107457
|
Saleh Abdullah Al-Mezel, Jamshaid Ahmad. Fixed point results with applications to nonlinear fractional differential equations[J]. AIMS Mathematics, 2023, 8(8): 19743-19756. doi: 10.3934/math.20231006
DownLoad: