In this paper, we aim to obtain some new common attractors with the assistance of finite families of generalized contractive mappings, that belong to the special class of mappings defined on a partial metric space. Consequently, a variety of results for iterated function systems satisfying a different set of generalized contractive conditions are acquired. We present some examples to reinforce the results proved herein. These results generalize, unify and extend a variety of results that exist in current literature.
Citation: Melusi Khumalo, Talat Nazir, Vuledzani Makhoshi. Generalized iterated function system for common attractors in partial metric spaces[J]. AIMS Mathematics, 2022, 7(7): 13074-13103. doi: 10.3934/math.2022723
In this paper, we aim to obtain some new common attractors with the assistance of finite families of generalized contractive mappings, that belong to the special class of mappings defined on a partial metric space. Consequently, a variety of results for iterated function systems satisfying a different set of generalized contractive conditions are acquired. We present some examples to reinforce the results proved herein. These results generalize, unify and extend a variety of results that exist in current literature.
| [1] |
E. Ameer, H. Aydi, M. Arshad, H. Alsamir, M. S. Noorani, Hybrid multivalued type contraction mappings in $\alpha _{K}$-complete partial $b$-metric spaces and applications, Symmetry, 11 (2019), 86. https://doi.org/10.3390/sym11010086 doi: 10.3390/sym11010086
|
| [2] |
H. Aydi, A. Felhi, E. Karapinar, S. Sahmim, A Nadler-type fixed point theorem in dislocated spaces and applications, Miskolc Math. Notes, 19 (2018), 111–124. https://doi.org/10.18514/MMN.2018.1652 doi: 10.18514/MMN.2018.1652
|
| [3] |
H. Aydi, M. Abbas, C. Vetro, Partial Hausdorff metric and Nadler's fixed point theorem on partial metric spaces, Topol. Appl., 159 (2012), 3234–3242. https://doi.org/10.1016/j.topol.2012.06.012 doi: 10.1016/j.topol.2012.06.012
|
| [4] | I. Altun, H. Simsek, Some fixed point theorems on dualistic partial metric spaces, J. Adv. Math. Stud., 1 (2008), 1–8. |
| [5] |
M. Abbas, B. Ali, Fixed point of Suzuki-Zamfirescu hybrid contractions in partial metric spaces via partial Hausdorff metric, Fixed Point Theory Appl., 2013 (2013), 21. https://doi.org/10.1186/1687-1812-2013-21 doi: 10.1186/1687-1812-2013-21
|
| [6] | M. F. Barnsley, H. Rising, Fractals everywhere, Morgan Kaufmann, 1993. |
| [7] |
M. Barnsley, A. Vince, Developments in fractal geometry, Bull. Math. Sci., 3 (2013), 299–348. https://doi.org/10.1007/s13373-013-0041-3 doi: 10.1007/s13373-013-0041-3
|
| [8] | S. Banach, Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math., 3 (1922), 133–181. |
| [9] | P. Debnath, N. Konwar, S. Radenovic, Metric fixed point theory: Applications in science, engineering and behavioural sciences, Springer, 2021. |
| [10] |
K. Goyal, B. Prasad, Generalized iterated function systems in multi-valued mapping, AIP Conf. Proc., 2316 (2021), 040001. https://doi.org/10.1063/5.0036921 doi: 10.1063/5.0036921
|
| [11] |
H. A. Hammad, M. De 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. https://doi.org/10.1186/s13660-020-02477-8 doi: 10.1186/s13660-020-02477-8
|
| [12] |
H. A. Hammad, P. Agarwal, L. G. J. Guirao, Applications to boundary value problems and homotopy theory via tripled fixed point techniques in partially metric spaces, Mathematics, 9 (2021), 1–22. https://doi.org/10.3390/math9162012 doi: 10.3390/math9162012
|
| [13] | J. Hutchinson, Fractals and self-similarity, Indiana U. Math. J., 30 (1981), 713–747. |
| [14] |
K. Javed, H. Aydi, F. Uddin, M. Arshad, On orthogonal partial $b$-metric spaces with an application, J. Math., 2021 (2021), 6692063. https://doi.org/10.1155/2021/6692063 doi: 10.1155/2021/6692063
|
| [15] |
E. Karapinar, R. Agarwal, H. Aydi, Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces, Mathematics, 6 (2018), 256. https://doi.org/10.3390/math6110256 doi: 10.3390/math6110256
|
| [16] |
M. A. Kutbi, A. Latif, T. Nazir, Generalized rational contractions in semi metric spaces via iterated function system, RACSAM, 114 (2020), 187. https://doi.org/10.1007/s13398-020-00915-2 doi: 10.1007/s13398-020-00915-2
|
| [17] |
G. Lin, X. Cheng, Y. Zhang, A parametric level set based collage method for an inverse problem in elliptic partial differential equations, J. Comput. Appl. Math., 340 (2018), 101–121. https://doi.org/10.1016/j.cam.2018.02.008 doi: 10.1016/j.cam.2018.02.008
|
| [18] |
S. G. Matthews, Partial metric topology, Ann. N. Y. Acad. Sci., 728 (1994), 183–197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x doi: 10.1111/j.1749-6632.1994.tb44144.x
|
| [19] |
S. B. Nadler, Multivalued contraction mappings, Pacific J. Math., 30 (1969), 475–488. https://doi.org/10.2140/pjm.1969.30.475 doi: 10.2140/pjm.1969.30.475
|
| [20] |
T. Nazir, S. Silverstrov, M. Abbas, Fractals of generalized $F$-Hutchinson operator, Waves Wavelets Fractals Adv. Anal., 2 (2016), 29–40. https://doi.org/10.1515/wwfaa-2016-0004 doi: 10.1515/wwfaa-2016-0004
|
| [21] |
N. A. Secelean, Generalized countable iterated function systems, Filomat, 25 (2011), 21–36. https://doi.org/10.2298/FIL1101021S doi: 10.2298/FIL1101021S
|
| [22] |
Y. Zhang, B. Hofmann, Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems, Inverse Probl. Imag., 15 (2021), 229–256. https://doi.org/10.3934/ipi.2020062 doi: 10.3934/ipi.2020062
|
| [23] | V. Todorcevic, Harmonic quasiconformal mappings and hyperbolic type metrics, Springer, 2019. |
Melusi Khumalo, Talat Nazir, Vuledzani Makhoshi. Generalized iterated function system for common attractors in partial metric spaces[J]. AIMS Mathematics, 2022, 7(7): 13074-13103. doi: 10.3934/math.2022723
DownLoad: