Existence of Picard operator and iterated function system

[EN] In this paper, we define weak θm− contraction mappings and give a new class of Picard operators for such class of mappings on a complete metric space. Also, we obtain some new results on the existence and uniqueness of attractor for a weak θm− iterated multifunction system. Moreover, we introduce (α, β, θm)− contractions using cyclic (α, β)− admissible mappings and obtain some results for such class of mappings without the continuity of the operator. We also provide an illustrative example to support the concepts and results proved herein.The authors are thankful to the learned referee for valuable suggestions. The second author is also thankful to AISTDF, DST for the research grant vide project No. CRD/2018/000017.Garg, M.; Chandok, S. (2020). Existence of Picard operator and iterated function system. Applied General Topology. 21(1):57-70. https://doi.org/10.4995/agt.2020.11992OJS5770211S. Alizadeh, F. Moradlou and P. Salimi, Some fixed point results for (α, β) − (ψ, φ)- contractive mappings, Filomat 28 (2014), 635-647. https://doi.org/10.2298/FIL1403635AM. F. Barnsley, Fractals Everywhere, Revised with the Assistance of and with a Foreword by Hawley Rising, III. Academic Press Professional, Boston (1993).R. M. T. Bianchini, Su un problema di S. Reich riguardante la teoria dei punti fissi, Boll. Un. Mat. Ital. 5 (1972), 103-108.E. L. Fuster, A. Petrusel and J. C. Yao, Iterated function system and well-posedness, Chaos Sol. Fract. 41 (2009), 1561-1568. https://doi.org/10.1016/j.chaos.2008.06.019R. H. Haghi, Sh. Rezapour and N. Shahzad, Some fixed point generalizations are not real generalizations, Nonlinear Anal. 74 (2011), 1799-1803. https://doi.org/10.1016/j.na.2010.10.052N. Hussain, V. Parvaneh, B. Samet and C. Vetro, Some fixed point theorems for generalized contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2015, 185 (2015). https://doi.org/10.1186/s13663-015-0433-zJ. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30, no. 5 (1981), 713-747. https://doi.org/10.1512/iumj.1981.30.30055M. Imdad, W. M. Alfaqih and I. A. Khan, Weak θ−contractions and some fixed point results with applications to fractal theory, Adv. Diff. Eq. 439 (2018). https://doi.org/10.1186/s13662-018-1900-8M. Jleli and B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl. 38 (2014). https://doi.org/10.1186/1029-242X-2014-38M. Radenovic and S. Chandok, Simulation type functions and coincidence points, Filomat, 32, no. 1 (2018), 141-147. https://doi.org/10.2298/FIL1801141RB. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. American Math. Soc. 226 (1977), 257-290. https://doi.org/10.1090/S0002-9947-1977-0433430-4I. A. Rus, Picard operators and applications, Sci. Math. Jpn. 58, no. 1 (2003), 191-219.I. A. Rus, A. Petrusel and G. Petrusel, Fixed Point Theory, Cluj University Press, Cluj-Napoca, 2008.N. A. Secelean, Countable Iterated Function Systems, LAP LAMBERT Academic Publishing (2013). https://doi.org/10.1186/1687-1812-2013-277N. A. Secelean, Iterated function systems consisting of F-contractions, Fixed Point Theory Appl. 2013, 277 (2013). https://doi.org/10.1186/1687-1812-2013-277V. M. Sehgal, On fixed and periodic points for a class of mappings, J. London Math. Soc. 5 (1972), 571-576. https://doi.org/10.1112/jlms/s2-5.3.571S.-A. Urziceanu, Alternative charaterizations of AGIFSs having attactors, Fixed Point Theory 20 (2019), 729-740. https://doi.org/10.24193/fpt-ro.2019.2.4

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