Iterated Function Systems Consisting of Generalized Convex Contractions in the Framework of Complete Strong b-metric Spaces

Georgescu, Flavian

doi:10.1515/awutm-2017-0018

Abstract

The concept of generalized convex contraction was introduced and studied by V. Istrăţescu and the notion of b-metric space was introduced by I. A. Bakhtin and S. Czerwik. In this paper we combine these two elements by studying iterated function systems consisting of generalized convex contractions on the framework of b-metric spaces. More precisely we prove the existence and uniqueness of the attractor of such a system providing in this way a generalization of Istrăţescu’s convex contractions fixed point theorem in the setting of complete strong b-metric spaces.

References

[1] Aghajani A., Abbas M., and Roshan J.R., Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca, 64, (2014), 941-960.10.2478/s12175-014-0250-6
Search in Google Scholar Back to article
[2] Alghamdia M.A., Alnafeia S.H., Radenović S., and Shahzad N., Fixed point theorems for convex contraction mappings on cone metric spaces, Math. Comput. Modelling, 54, (2011), 2020–2026.10.1016/j.mcm.2011.05.010
Search in Google Scholar Back to article
[3] An T.V., Tuyen L.Q., and Dung N.V., Stone-type theorem on b-metric spaces and applications, Topology Appl., 185/186, (2015), 50-64.10.1016/j.topol.2015.02.005
Search in Google Scholar Back to article
[4] András Sz., Fiber Picard operators and convex contractions, Fixed Point Theory, 4, (2003), 121-129.
Search in Google Scholar Back to article
[5] András Sz., Ecuaţii integrale Fredholm-Volterra, Editura Didactică şi Pedagogică, Bucureşti, 2005.
Search in Google Scholar Back to article
[6] Andres J. and Rypka M., Multivalued fractals and hyperfractals, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22, (2012), DOI 10.1142/S02181127412500095.10.1142/S02181127412500095
Open DOI Search in Google Scholar Back to article
[7] Aydi H., Bota M.F., Karapinar E., and Mitrović S., A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl., (2012), 2012–88.10.1186/1687-1812-2012-88
Open DOI Search in Google Scholar Back to article
[8] Bakhtin I.A., The contraction mapping principle in quasimetric spaces, Funct. Anal., Unianowsk Gos. Ped. Inst., 30, (1989), 26–37.
Search in Google Scholar Back to article
[9] Berinde V., Generalized contractions in quasimetric spaces, Seminar on Fixed Point Theory, (1993), 3–9.
Search in Google Scholar Back to article
[10] Boriceanu M., Bota M., and Petruşel A., Multivalued fractals in b-metric spaces, Cent. Eur. J. Math., 8, (2010), 367–377.10.2478/s11533-010-0009-4
Search in Google Scholar Back to article
[11] Boriceanu M., Petruşel A., and Rus A.I., Fixed point theorems for some multivalued generalized contraction in b-metric spaces, Int. J. Math. Stat., 6, (2010), 65–76.
Search in Google Scholar Back to article
[12] Bota M., Molnár A., and Varga C., On Ekeland’s variational principle in b-metric spaces, Fixed Point Theory, 12, (2011), 21–28.
Search in Google Scholar Back to article
[13] Chifu C. and Petruşel A., Multivalued fractals and generalized multivalued contractions, Chaos Solitons Fractals, 36, (2008), 203–210.10.1016/j.chaos.2006.06.027
Open DOI Search in Google Scholar Back to article
[14] Chifu C. and Petruşel G., Fixed points for multivalued contractions in b-metric spaces with applications to fractals, Taiwanese J. Math., 18, (2014), 1365–1375.10.11650/tjm.18.2014.4137
Search in Google Scholar Back to article
[15] Czerwik S., Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1, (1993), 5–11.
Search in Google Scholar Back to article
[16] Czerwik S., Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46, (1998), 263–276.
Search in Google Scholar Back to article
[17] Dubey A.K., Shukla R., and Dubey R.P., Some fixed point results in b-metric spaces, Asian Journal of Mathematics and Applications, (2014), Article ID ama014710.14445/22315373/IJMTT-V9P510
Search in Google Scholar Back to article
[18] Dumitru D., Generalized iterated function systems containing Meir-Keeler functions, An. Univ. Bucur., Mat., 58, (2009), 109–121.
Search in Google Scholar Back to article
[19] Istrăţescu V., Some fixed point theorems for convex contraction mappings and convex nonexpansive mappings (I), Libertas Math., 1, (1981), 151–164.10.1007/BF01761490
Search in Google Scholar Back to article
[20] V. Istrăţescu, Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters - I, Annali di Mat. Pura Appl., 130, (1982), 89–104.10.1007/BF01761490
Search in Google Scholar Back to article
[21] V. Istrăţescu, Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters, II, Annali di Mat. Pura Appl., 134, (1983), 327–362.10.1007/BF01773511
Search in Google Scholar Back to article
[22] Georgescu F., IFSs consisting of generalized convex contractions, An. Ştiinţ. Univ. “Ovidius” Constanţa, Ser. Mat., in print
Search in Google Scholar Back to article
[23] Ghorbanian V., Rezapour S., and N. Shahzad, Some ordered fixed point results and the property, (P), Comput. Math. Appl., 63, (2012), 1361–1368.10.1016/j.camwa.2011.12.071
Open DOI Search in Google Scholar Back to article
[24] Gwóźdź-Lukowska G. and J. Jachymski, IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem, J. Math. Anal. Appl., 356, (2009), 453–463.10.1016/j.jmaa.2009.03.023
Search in Google Scholar Back to article
[25] N. Hussain, M.A. Kutbi, Khaleghizadeh S., and P. Salimi, Discussions on recent results for α–Ψ-contractive mappings, Abstr. Appl. Anal., 2014, (2014), Article ID 456482, 13 pages.10.1155/2014/456482
Search in Google Scholar Back to article
[26] Hutchinson J.E., Fractals and self similarity, Indiana Univ. Math. J., 30, (1981), 713–747.10.1512/iumj.1981.30.30055
Search in Google Scholar Back to article
[27] Khamsi M.A. and N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal., 73, (2010), 3123–3129.10.1016/j.na.2010.06.084
Search in Google Scholar Back to article
[28] Kir M. and Kizitune H., On some well known fixed point theorems in b-metric spaces, Turkish Journal of Analysis and Number Theory, 1, (2013), 13–16.10.12691/tjant-1-1-4
Search in Google Scholar Back to article
[29] Kirk W.A. and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer Heidelberg, 2014.10.1007/978-3-319-10927-5
Search in Google Scholar Back to article
[30] Klimek M. and Kosek M., Generalized iterated function systems, multifunctions and Cantor sets, Ann. Polon. Math., 96, (2009), 25–41.10.4064/ap96-1-2
Open DOI Search in Google Scholar Back to article
[31] Kutbi M.A., Karapinar E., Ahmad J., and Azam A., Some fixed point results for multi-valued mappings in b-metric spaces, J. Inequal. Appl, 2014, (2014), article ID 126.10.1186/1029-242X-2014-126
Open DOI Search in Google Scholar Back to article
[32] Latif A., Sintunavarat W., and Ninsri A., Approximate fixed point theorems for partial generalized convex contraction mappings in α-complete metric spaces, Taiwanese J. Math., 19, (2015), 315–333.10.11650/tjm.19.2015.4746
Search in Google Scholar Back to article
[33] Leśniak K., Infinite iterated function systems: a multivalued approach, Bull. Pol. Acad. Sci. Math., 52, (2004), 1–8.10.4064/ba52-1-1
Open DOI Search in Google Scholar Back to article
[34] Máté L., The Hutchinson-Barnsley theory for certain noncontraction mappings, Period. Math. Hungar., 27, (1993), 21–33.10.1007/BF01877158
Search in Google Scholar Back to article
[35] Miandaragh M.A., Postolache M., and Rezapour S., Approximate fixed points of generalized convex contractions, Fixed Point Theory Appl., 2013, (2013), article 255.10.1186/1687-1812-2013-255
Open DOI Search in Google Scholar Back to article
[36] Mihail A. and Miculescu R., Applications of fixed point theorems in the theory of generalized IFS, Fixed Point Theory Appl., (2008), Art. ID 312876, 11 pp.10.1155/2008/312876
Open DOI Search in Google Scholar Back to article
[37] Generalized IFSs on noncompact spaces A. Mihail and R. Miculescu, Fixed Point Theory Appl., (2010), Art. ID 584215, 15 pp.10.1155/2010/584215
Search in Google Scholar Back to article
[38] Miculescu R. and Mihail A., Reich-type iterated function systems, J. Fixed Point Theory Appl., 18, (2016), 285–296.10.1007/s11784-015-0264-x
Open DOI Search in Google Scholar Back to article
[39] Miculescu R. and Mihail A., A generalization of Istrăţescu’s fixed point theorem for convex contractions, in print, Fixed Point Theory (available at arXiv:1512.05490).
Search in Google Scholar Back to article
[40] Miculescu R. and Mihail A., Caristi-Kirk type and Boyd&Wong–Browder-Matkowski-Rus type fixed point results in b-metric spaces, in print, Filomat (available at arXiv:1512.03968)
Search in Google Scholar Back to article
[41] Mishra P.K., Sachdeva S., and Banerjee S.K., Some fixed point theorems in b-metric space, Turkish Journal of Analysis and Number Theory, 2, (2014), 19–22.10.12691/tjant-2-1-5
Search in Google Scholar Back to article
[42] Mohanta S.K., Some fixed point theorems using wt-distance in b-metric spaces, Fasc. Math., 54, (2015), 125–140.10.1515/fascmath-2015-0008
Search in Google Scholar Back to article
[43] Mureşan V. and Mureşan A., On the theory of fixed point theorems for convex contraction mappings, Carpathian J. Math., 31, (2015), 365–371.
Search in Google Scholar Back to article
[44] Nashine H.N. and Kadelburg Z., Cyclic generalized ϕ-contractions in b-metric spaces and an application to integral equations, Filomat, 28, (2014), 2047–2057.10.2298/FIL1410047N
Search in Google Scholar Back to article
[45] Nazir T., Silvestrov S., and Qi Xiaomin, Fractals of generalized F-Hutchinson operator in b-metric spaces, J. Oper., 2016, (2016), article ID 5250394.10.1155/2016/5250394
Search in Google Scholar Back to article
[46] Olatinwo M.O., A fixed point theorem for multi-valued weakly Picard operators in b-metric spaces, Demonstratio Math., 42, (2009), 599–606.10.1515/dema-2013-0178
Search in Google Scholar Back to article
[47] Păcurar M., Sequences of almost contractions and fixed points in b-metric spaces, An. Univ. Vest Timiş. Ser. Mat.-Inform., 48, (2010), 125–137.
Search in Google Scholar Back to article
[48] Petruşel A., Iterated function system of locally contractive operators, Rev. Anal. Numér. Théor. Approx., 33, (2004), 215–219.10.33993/jnaat332-779
Search in Google Scholar Back to article
[49] Roshan J.R., Hussain N., Sedghi S., and Shobkolaei N., Suzuki-type fixed point results in b-metric spaces, Math. Sci. (Springer), 9, (2015), 153–160.10.1007/s40096-015-0162-9
Search in Google Scholar Back to article
[50] Roshan J.R., Parvaneh V., and Altun I., Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations, Appl. Math. Comput., 226, (2014), 725–737.10.1016/j.amc.2013.10.043
Search in Google Scholar Back to article
[51] M. Sarwar and Rahman M.U., Fixed point theorems for Ciric’s and generalized contractions in b-metric spaces, International Journal of Analysis and Applications, 7, (2015), 70–78.
Search in Google Scholar Back to article
[52] N.A.. Secelean, Iterated function systems consisting of F-contractions, Fixed Point Theory Appl., (2013), 2013–277.10.1186/1687-1812-2013-277
Open DOI Search in Google Scholar Back to article
[53] Secelean N.A., Generalized iterated function systems on the space l^∞(X), J. Math. Anal. Appl., 410, (2014), 847–458.10.1016/j.jmaa.2013.09.007
Search in Google Scholar Back to article
[54] Secelean N.A., Countable Iterated Function Systems, Lambert Academic Publishing, 201310.1186/1687-1812-2013-277
Search in Google Scholar Back to article
[55] Shukla S., Partial b-metric spaces and fixed point theorems, Mediterr. J. Math., 11, (2014), 703–711.10.1007/s00009-013-0327-4
Search in Google Scholar Back to article
[56] Singh S.L., Czerwik S., Król K., and Singh A., Coincidences and fixed points of hybrid contractions, Tamsui Oxf. J. Math. Sci., 24, (2008), 401–416.
Search in Google Scholar Back to article
[57] Strobin F. and J. Swaczyna, On a certain generalization of the iterated function systems, Bull. Australian Math. Soc., 87, (2013), 37–54.10.1017/S0004972712000500
Search in Google Scholar Back to article
[58] Strobin F., Attractors of generalized IFSs that are not attractors of IFSs, J. Math. Anal. Appl., 422, (2015), 99–108.10.1016/j.jmaa.2014.08.029
Search in Google Scholar Back to article
[59] Yingtaweesittikul H., Suzuki type fixed point for generalized multi-valued mappings in b-metric spaces, Fixed Point Theory and Applications, (2013), 215.10.1186/1687-1812-2013-215
Search in Google Scholar Back to article

Iterated Function Systems Consisting of Generalized Convex Contractions in the Framework of Complete Strong b-metric Spaces

Abstract

Paradigm

My account