We apply a fixed point result for multivalued contractions on complete metric spaces endowed with a graph to graph-directed iterated function systems. More precisely, we construct a suitable metric space endowed with a graph and a suitable -contraction such that its fixed points permit us to obtain more information on the attractor of a graph-directed iterated function system.
References
J. E. Hutchinson, “Fractals and self-similarity,” Indiana University Mathematics Journal, vol. 30, no. 5, pp. 713–747, 1981. MR625600 0598.28011 10.1512/iumj.1981.30.30055 J. E. Hutchinson, “Fractals and self-similarity,” Indiana University Mathematics Journal, vol. 30, no. 5, pp. 713–747, 1981. MR625600 0598.28011 10.1512/iumj.1981.30.30055
M. F. Barnsley, Fractals Everywhere, Academic Press, Boston, Mass, USA, 1988. MR977274 0691.58001 M. F. Barnsley, Fractals Everywhere, Academic Press, Boston, Mass, USA, 1988. MR977274 0691.58001
A. C. Ran and M. C. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004. MR2053350 10.1090/S0002-9939-03-07220-4 1060.47056 A. C. Ran and M. C. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004. MR2053350 10.1090/S0002-9939-03-07220-4 1060.47056
J. J. Nieto, R. L. Pouso, and R. Rodríguez-López, “Fixed point theorems in ordered abstract spaces,” Proceedings of the American Mathematical Society, vol. 135, pp. 2505–2517, 2007. 1126.47045 10.1090/S0002-9939-07-08729-1 J. J. Nieto, R. L. Pouso, and R. Rodríguez-López, “Fixed point theorems in ordered abstract spaces,” Proceedings of the American Mathematical Society, vol. 135, pp. 2505–2517, 2007. 1126.47045 10.1090/S0002-9939-07-08729-1
J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, pp. 223–239, 2005. MR2212687 10.1007/s11083-005-9018-5 1095.47013 J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, pp. 223–239, 2005. MR2212687 10.1007/s11083-005-9018-5 1095.47013
J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, English Series, vol. 23, no. 12, pp. 2205–2212, 2007. 1140.47045 10.1007/s10114-005-0769-0 J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, English Series, vol. 23, no. 12, pp. 2205–2212, 2007. 1140.47045 10.1007/s10114-005-0769-0
A. Petruşel and I. A. Rus, “Fixed point theorems in ordered $L$-spaces,” Proceedings of the American Mathematical Society, vol. 134, no. 2, pp. 411–418, 2006. MR2176009 1086.47026 A. Petruşel and I. A. Rus, “Fixed point theorems in ordered $L$-spaces,” Proceedings of the American Mathematical Society, vol. 134, no. 2, pp. 411–418, 2006. MR2176009 1086.47026
J. Jachymski, “The contraction principle for mappings on a metric space with a graph,” Proceedings of the American Mathematical Society, vol. 136, no. 4, pp. 1359–1373, 2008. MR2367109 10.1090/S0002-9939-07-09110-1 1139.47040 J. Jachymski, “The contraction principle for mappings on a metric space with a graph,” Proceedings of the American Mathematical Society, vol. 136, no. 4, pp. 1359–1373, 2008. MR2367109 10.1090/S0002-9939-07-09110-1 1139.47040
G. Gwóźdź-Łukawska and J. Jachymski, “IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem,” Journal of Mathematical Analysis and Applications, vol. 356, no. 2, pp. 453–463, 2009. MR2524281 10.1016/j.jmaa.2009.03.023 1171.28002 G. Gwóźdź-Łukawska and J. Jachymski, “IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem,” Journal of Mathematical Analysis and Applications, vol. 356, no. 2, pp. 453–463, 2009. MR2524281 10.1016/j.jmaa.2009.03.023 1171.28002
T. Dinevari and M. Frigon, “Fixed point results for multivalued contractions on a metric space with a graph,” Journal of Mathematical Analysis and Applications, vol. 405, no. 2, pp. 507–517, 2013. MR3061029 10.1016/j.jmaa.2013.04.014 1306.47062 T. Dinevari and M. Frigon, “Fixed point results for multivalued contractions on a metric space with a graph,” Journal of Mathematical Analysis and Applications, vol. 405, no. 2, pp. 507–517, 2013. MR3061029 10.1016/j.jmaa.2013.04.014 1306.47062
A. Nicolae, D. O'Regan, and A. Petruşel, “Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph,” Georgian Mathematical Journal, vol. 18, no. 2, pp. 307–327, 2011. 1227.54053 MR2805983 A. Nicolae, D. O'Regan, and A. Petruşel, “Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph,” Georgian Mathematical Journal, vol. 18, no. 2, pp. 307–327, 2011. 1227.54053 MR2805983
R. D. Mauldin and S. C. Williams, “Hausdorff dimension in graph directed constructions,” Transactions of the American Mathematical Society, vol. 309, no. 2, pp. 811–829, 1988. MR961615 0706.28007 10.1090/S0002-9947-1988-0961615-4 R. D. Mauldin and S. C. Williams, “Hausdorff dimension in graph directed constructions,” Transactions of the American Mathematical Society, vol. 309, no. 2, pp. 811–829, 1988. MR961615 0706.28007 10.1090/S0002-9947-1988-0961615-4
M. Das, “Contraction ratios for graph-directed iterated constructions,” Proceedings of the American Mathematical Society, vol. 134, no. 2, pp. 435–442, 2006. MR2176012 1125.28001 10.1090/S0002-9939-05-08146-3 M. Das, “Contraction ratios for graph-directed iterated constructions,” Proceedings of the American Mathematical Society, vol. 134, no. 2, pp. 435–442, 2006. MR2176012 1125.28001 10.1090/S0002-9939-05-08146-3
M. Das and S.-M. Ngai, “Graph-directed iterated function systems with overlaps,” Indiana University Mathematics Journal, vol. 53, no. 1, pp. 109–134, 2004. MR2048186 1065.28003 10.1512/iumj.2004.53.2342 M. Das and S.-M. Ngai, “Graph-directed iterated function systems with overlaps,” Indiana University Mathematics Journal, vol. 53, no. 1, pp. 109–134, 2004. MR2048186 1065.28003 10.1512/iumj.2004.53.2342
G. A. Edgar and J. Golds, “A fractal dimension estimate for a graph-directed iterated function system of non-similarities,” Indiana University Mathematics Journal, vol. 48, no. 2, pp. 429–447, 1999. \endinput MR1722803 10.1512/iumj.1999.48.1641 0944.28008 G. A. Edgar and J. Golds, “A fractal dimension estimate for a graph-directed iterated function system of non-similarities,” Indiana University Mathematics Journal, vol. 48, no. 2, pp. 429–447, 1999. \endinput MR1722803 10.1512/iumj.1999.48.1641 0944.28008
