Relational generalized iterated function systems
In this paper, we introduce a wider class of generalized iterated function systems, called relational generalized iterated function systems. More precisely, the classical contraction condition for functions defined on product spaces is weakened by means of an equivalence relation. In particular, if...
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| Vydáno v: | Chaos, solitons and fractals Ročník 182; s. 114823 |
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Elsevier Ltd
01.05.2024
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| ISSN: | 0960-0779, 1873-2887 |
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| Abstract | In this paper, we introduce a wider class of generalized iterated function systems, called relational generalized iterated function systems. More precisely, the classical contraction condition for functions defined on product spaces is weakened by means of an equivalence relation. In particular, if we consider the total equivalence relation, we recover the classical generalized iterated function systems. Our main result states that each compact subset of the underlying metric space generates, via a sequence of iterates, a fixed point of the associated fractal operator, called an attractor of the system. We also establish a structure result for the attractors and a theorem concerning the continuous dependence of the attractor on the associated compact set. Ultimately, we provide some examples which illustrate our main results.
•We introduce the class of relational generalized iterated function systems.•We weaken the classical contraction condition using an equivalence relation.•We prove that each compact set generates an attractor of the system.•We establish a structure result for the attractors.•We prove a theorem concerning the continuous dependence of the attractors. |
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| AbstractList | In this paper, we introduce a wider class of generalized iterated function systems, called relational generalized iterated function systems. More precisely, the classical contraction condition for functions defined on product spaces is weakened by means of an equivalence relation. In particular, if we consider the total equivalence relation, we recover the classical generalized iterated function systems. Our main result states that each compact subset of the underlying metric space generates, via a sequence of iterates, a fixed point of the associated fractal operator, called an attractor of the system. We also establish a structure result for the attractors and a theorem concerning the continuous dependence of the attractor on the associated compact set. Ultimately, we provide some examples which illustrate our main results.
•We introduce the class of relational generalized iterated function systems.•We weaken the classical contraction condition using an equivalence relation.•We prove that each compact set generates an attractor of the system.•We establish a structure result for the attractors.•We prove a theorem concerning the continuous dependence of the attractors. |
| ArticleNumber | 114823 |
| Author | Miculescu, Radu Abraham, Izabella Mihail, Alexandru |
| Author_xml | – sequence: 1 givenname: Izabella surname: Abraham fullname: Abraham, Izabella email: izabella.abraham@student.unitbv.ro organization: Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Iuliu Maniu Street, nr. 50, 500091, Braşov, Romania – sequence: 2 givenname: Radu orcidid: 0000-0002-5516-6193 surname: Miculescu fullname: Miculescu, Radu email: radu.miculescu@unitbv.ro organization: Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Iuliu Maniu Street, nr. 50, 500091, Braşov, Romania – sequence: 3 givenname: Alexandru surname: Mihail fullname: Mihail, Alexandru email: alex_m@fmi.unibuc.ro organization: Faculty of Mathematics and Computer Science, University of Bucharest, Academiei Street, nr. 14, 010014, Bucharest, Romania |
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| Cites_doi | 10.3390/math11132826 10.1007/s12346-020-00420-2 10.1007/s00009-020-01585-5 10.1016/j.jmaa.2014.08.029 10.1007/s11784-015-0235-2 10.1016/j.fss.2017.05.003 10.1016/j.chaos.2014.12.005 10.1016/j.jmaa.2019.123740 10.1007/s11075-016-0104-0 10.1016/0022-247X(85)90225-2 10.24193/fpt-ro.2017.2.55 10.1007/s00009-013-0300-2 10.1016/j.cnsns.2020.105423 10.1016/j.chaos.2020.110404 10.1512/iumj.1981.30.30055 10.1017/S0004972712000500 10.1007/s11075-019-00730-w 10.1007/s10440-013-9841-4 |
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| Keywords | 37C70 Relational generalized iterated function systems Weakly Picard operators 37B10 Equivalence relations 28A80 Attractors |
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| References | Oliveira, Strobin (b13) 2018; 331 Hiriart-Urruty (b25) 1985; 111 Mihail (b5) 2008; 53 R., Mihail (b3) 2008; 1 Strobin, Swaczyna (b7) 2016; 17 Miculescu, Mihail, Urziceanu (b15) 2020; 141 Jaros, Maślanka, Strobin (b20) 2016; 73 Strobin (b12) 2020; 19 Miculescu, Mihail (b23) 2017; 18 Şerban (b24) 2002; 3 Miculescu, Mihail, Urziceanu (b21) 2020; 83 Strobin, Swaczyna (b6) 2013; 87 Secelean (b10) 2014; 11 Maślanka (b9) 2020; 484 Miculescu (b11) 2014; 130 R., Mihail (b4) 2010; 1 Secelean (b22) 2002 Strobin (b8) 2015; 422 Leśniak (b26) 2003; 53 da Cunha, Oliveira, Strobin (b19) 2020; 91 Hutchinson (b1) 1981; 30 Secelean (b14) 2015; 17 Dumitru, Ioana, Sfetcu, Strobin (b16) 2015; 71 García (b18) 2020; 17 Barnsley (b2) 1993 Abraham, Miculescu (b17) 2023; 11 Jaros (10.1016/j.chaos.2024.114823_b20) 2016; 73 Maślanka (10.1016/j.chaos.2024.114823_b9) 2020; 484 Miculescu (10.1016/j.chaos.2024.114823_b21) 2020; 83 Hutchinson (10.1016/j.chaos.2024.114823_b1) 1981; 30 Mihail (10.1016/j.chaos.2024.114823_b5) 2008; 53 Miculescu (10.1016/j.chaos.2024.114823_b15) 2020; 141 Strobin (10.1016/j.chaos.2024.114823_b12) 2020; 19 Oliveira (10.1016/j.chaos.2024.114823_b13) 2018; 331 Secelean (10.1016/j.chaos.2024.114823_b14) 2015; 17 Strobin (10.1016/j.chaos.2024.114823_b8) 2015; 422 Secelean (10.1016/j.chaos.2024.114823_b22) 2002 Abraham (10.1016/j.chaos.2024.114823_b17) 2023; 11 Miculescu (10.1016/j.chaos.2024.114823_b23) 2017; 18 Barnsley (10.1016/j.chaos.2024.114823_b2) 1993 Miculescu (10.1016/j.chaos.2024.114823_b11) 2014; 130 García (10.1016/j.chaos.2024.114823_b18) 2020; 17 Dumitru (10.1016/j.chaos.2024.114823_b16) 2015; 71 Hiriart-Urruty (10.1016/j.chaos.2024.114823_b25) 1985; 111 R. (10.1016/j.chaos.2024.114823_b4) 2010; 1 da Cunha (10.1016/j.chaos.2024.114823_b19) 2020; 91 Strobin (10.1016/j.chaos.2024.114823_b6) 2013; 87 Strobin (10.1016/j.chaos.2024.114823_b7) 2016; 17 R. (10.1016/j.chaos.2024.114823_b3) 2008; 1 Leśniak (10.1016/j.chaos.2024.114823_b26) 2003; 53 Secelean (10.1016/j.chaos.2024.114823_b10) 2014; 11 Şerban (10.1016/j.chaos.2024.114823_b24) 2002; 3 |
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