Fixed points for cyclic orbital generalized contractions on complete metric spaces

• Autorzy: Erdal Karapınar , Salvador Romaguera , Kenan Taş
• Języki publikacji: EN

Abstrakty

EN

We prove a fixed point theorem for cyclic orbital generalized contractions on complete metric spaces from which we deduce, among other results, generalized cyclic versions of the celebrated Boyd and Wong fixed point theorem, and Matkowski fixed point theorem. This is done by adapting to the cyclic framework a condition of Meir-Keeler type discussed in [Jachymski J., Equivalent conditions and the Meir-Keeler type theorems, J. Math. Anal. Appl., 1995, 194(1), 293–303]. Our results generalize some theorems of Kirk, Srinavasan and Veeramani, and of Karpagam and Agrawal.

Słowa kluczowe

EN

Fixed point; Cyclic generalized contraction; Complete metric space

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 3
• Strony: 552-560

Daty

wydano

2013-03-01

online

2012-12-22

Twórcy

autor

Erdal Karapınar

• Department of Mathematics, Atılım University, 06836, İncek, Ankara, Turkey,

autor

Salvador Romaguera

• Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camí de Vera s/n, 46022, Valencia, Spain,

autor

Kenan Taş

• Department of Mathematics and Computer Science, Çankaya University, 06530, Yuzuncuyil, Ankara, Turkey,

Bibliografia

• [1] Al-Thagafi M.A., Shahzad N., Convergence and existence results for best proximity points, Nonlinear Anal., 2009, 70(10), 3665–3671 http://dx.doi.org/10.1016/j.na.2008.07.022[WoS][Crossref]
• [2] Anuradha J., Veeramani P., Proximal pointwise contraction, Topology Appl., 2009, 156(18), 2942–2948 http://dx.doi.org/10.1016/j.topol.2009.01.017[Crossref]
• [3] Boyd D.W., Wong J.S.W., On nonlinear contractions, Proc. Amer. Math. Soc., 1969, 20, 458–464 http://dx.doi.org/10.1090/S0002-9939-1969-0239559-9[Crossref]
• [4] Derafshpour M., Rezapour Sh., Shahzad N., Best proximity point of cyclic φ-contractions in ordered metric spaces, Topol. Methods Nonlinear Anal., 2011, 37(1), 193–202
• [5] Di Bari C., Suzuki T., Vetro C., Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal., 2008, 69(11), 3790–3794 http://dx.doi.org/10.1016/j.na.2007.10.014[WoS][Crossref]
• [6] Eldred A.A., Veeramani P., Existence and convergence of best proximity points, J. Math. Anal. Appl., 2006, 323(2), 1001–1006 http://dx.doi.org/10.1016/j.jmaa.2005.10.081[Crossref]
• [7] Jachymski J., Equivalent conditions and the Meir-Keeler type theorems, J. Math. Anal. Appl., 1995, 194(1), 293–303 http://dx.doi.org/10.1006/jmaa.1995.1299[Crossref]
• [8] Jachymski J.R., Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc., 1997, 125(8), 2327–2335 http://dx.doi.org/10.1090/S0002-9939-97-03853-7[Crossref]
• [9] Karapınar E., Fixed point theory for cyclic weak ϕ-contraction, Appl. Math. Lett., 2011, 24(6), 822–825 http://dx.doi.org/10.1016/j.aml.2010.12.016[WoS][Crossref]
• [10] Karpagam S., Agrawal S., Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps, Nonlinear Anal., 2011, 74(4), 1040–1046 http://dx.doi.org/10.1016/j.na.2010.07.026[Crossref]
• [11] Kirk W.A., Srinavasan P.S., Veeramani P., Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 2003, 4(1), 79–89
• [12] Kosuru G.S.R., Veeramani P., Cyclic contractions and best proximity pair theorems, preprint available at http://arxiv.org/abs/1012.1434
• [13] Matkowski J., Integrable Solutions of Functional Equations, Dissertationes Math. (Rozprawy Mat.), 127, Polish Academy of Sciences, Warsaw, 1975
• [14] Meir A., Keeler E., A theorem on contraction mappings, J. Math. Anal. Appl., 1969, 28, 326–329 http://dx.doi.org/10.1016/0022-247X(69)90031-6[Crossref]
• [15] Păcurar M., Rus I.A., Fixed point theory for cyclic φ-contractions, Nonlinear Anal., 2010, 72(3–4), 1181–1187 http://dx.doi.org/10.1016/j.na.2009.08.002[Crossref]
• [16] Petruşel G., Cyclic representations and periodic points, Studia Univ. Babeş-Bolyai Math., 2005, 50(3), 107–112
• [17] Rus I.A., Cyclic representations and fixed points, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity, 2005, 3, 171–178

Identyfikatory

DOI

10.2478/s11533-012-0145-0