Common fixed point and approximation results for noncommuting maps on locally convex spaces. (English) Zbl 1168.47044

Summary: Common fixed point results for some new classes of nonlinear noncommuting maps on a locally convex space are proved. As applications, related invariant approximation results are obtained. Our work includes improvements and extension of several recent developments of the existing literature on common fixed points. We also provide illustrative examples to demonstrate the generality of our results over the known ones.


47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46A03 General theory of locally convex spaces
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[1] Köthe G: Topological Vector Spaces. I, Die Grundlehren der mathematischen Wissenschaften. Volume 159. Springer, New York, NY, USA; 1969:xv+456.
[2] Cheng LX, Zhou Y, Zhang F: Danes’ drop theorem in locally convex spaces.Proceedings of the American Mathematical Society 1996,124(12):3699-3702. 10.1090/S0002-9939-96-03404-1 · Zbl 0863.46003
[3] Tarafdar E: Some fixed-point theorems on locally convex linear topological spaces.Bulletin of the Australian Mathematical Society 1975,13(2):241-254. 10.1017/S0004972700024436 · Zbl 0318.47032
[4] Sessa S: On a weak commutativity condition of mappings in fixed point considerations.Publications de l’Institut Mathématique 1982, 32(46): 149-153. · Zbl 0523.54030
[5] Jungck G: Common fixed points for commuting and compatible maps on compacta.Proceedings of the American Mathematical Society 1988,103(3):977-983. 10.1090/S0002-9939-1988-0947693-2 · Zbl 0661.54043
[6] Hussain N, Khan AR: Common fixed-point results in best approximation theory.Applied Mathematics Letters 2003,16(4):575-580. 10.1016/S0893-9659(03)00039-9 · Zbl 1063.47055
[7] Hussain N, Jungck G: Common fixed point and invariant approximation results for noncommuting generalized -nonexpansive maps.Journal of Mathematical Analysis and Applications 2006,321(2):851-861. 10.1016/j.jmaa.2005.08.045 · Zbl 1106.47048
[8] Hussain, N.; Rhoades, BE, [InlineEquation not available: see fulltext.]-commuting maps and invariant approximations, No. 2006, 9 (2006)
[9] Jungck G, Hussain N: Compatible maps and invariant approximations.Journal of Mathematical Analysis and Applications 2007,325(2):1003-1012. 10.1016/j.jmaa.2006.02.058 · Zbl 1110.54024
[10] Meinardus G: Invarianz bei linearen Approximationen.Archive for Rational Mechanics and Analysis 1963,14(1):301-303. · Zbl 0122.30801
[11] Singh SP: An application of a fixed-point theorem to approximation theory.Journal of Approximation Theory 1979,25(1):89-90. 10.1016/0021-9045(79)90036-4 · Zbl 0399.41032
[12] Sahab SA, Khan MS, Sessa S: A result in best approximation theory.Journal of Approximation Theory 1988,55(3):349-351. 10.1016/0021-9045(88)90101-3 · Zbl 0676.41031
[13] Jungck G, Sessa S: Fixed point theorems in best approximation theory.Mathematica Japonica 1995,42(2):249-252. · Zbl 0834.54026
[14] Al-Thagafi MA: Common fixed points and best approximation.Journal of Approximation Theory 1996,85(3):318-323. 10.1006/jath.1996.0045 · Zbl 0858.41022
[15] Pant RP: Common fixed points of noncommuting mappings.Journal of Mathematical Analysis and Applications 1994,188(2):436-440. 10.1006/jmaa.1994.1437 · Zbl 0830.54031
[16] Pathak HK, Cho YJ, Kang SM: Remarks on -weakly commuting mappings and common fixed point theorems.Bulletin of the Korean Mathematical Society 1997,34(2):247-257. · Zbl 0878.54032
[17] Khamsi MA, Kirk WA: An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics. Wiley-Interscience, New York, NY, USA; 2001:x+302. · Zbl 1318.47001
[18] Singh S, Watson B, Srivastava P: Fixed Point Theory and Best Approximation: The KKM-Map Principle, Mathematics and Its Applications. Volume 424. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:x+220. · Zbl 0901.47039
[19] Khan AR, Akbar F: Best simultaneous approximations, asymptotically nonexpansive mappings and variational inequalities in Banach spaces.Journal of Mathematical Analysis and Applications 2009,354(2):469-477. 10.1016/j.jmaa.2009.01.007 · Zbl 1179.47046
[20] Khan AR, Akbar F: Common fixed points from best simultaneous approximations.Taiwanese Journal of Mathematics 2009.,13(4): · Zbl 1182.41034
[21] Pathak HK, Hussain N: Common fixed points for Banach operator pairs with applications.Nonlinear Analysis: Theory, Methods & Applications 2008,69(9):2788-2802. 10.1016/j.na.2007.08.051 · Zbl 1170.47036
[22] Cain GL Jr., Nashed MZ: Fixed points and stability for a sum of two operators in locally convex spaces.Pacific Journal of Mathematics 1971, 39: 581-592. · Zbl 0229.47044
[23] Chen J, Li Z: Common fixed-points for Banach operator pairs in best approximation.Journal of Mathematical Analysis and Applications 2007,336(2):1466-1475. 10.1016/j.jmaa.2007.01.064 · Zbl 1128.47050
[24] Hussain N: Common fixed points in best approximation for Banach operator pairs with Ćirić type -contractions.Journal of Mathematical Analysis and Applications 2008,338(2):1351-1363. 10.1016/j.jmaa.2007.06.008 · Zbl 1134.47039
[25] Hussain, N.; Berinde, V., Common fixed point and invariant approximation results in certain metrizable topological vector spaces, No. 2006, 13 (2006) · Zbl 1103.47046
[26] Sahney BN, Singh KL, Whitfield JHM: Best approximations in locally convex spaces.Journal of Approximation Theory 1983,38(2):182-187. 10.1016/0021-9045(83)90125-9 · Zbl 0525.41027
[27] Singh SP: Some results on best approximation in locally convex spaces.Journal of Approximation Theory 1980,28(4):329-332. 10.1016/0021-9045(80)90067-2 · Zbl 0444.41018
[28] Taylor WW: Fixed-point theorems for nonexpansive mappings in linear topological spaces.Journal of Mathematical Analysis and Applications 1972,40(1):164-173. 10.1016/0022-247X(72)90040-6 · Zbl 0207.45501
[29] Khan AR, Akbar F, Sultana N: Random coincidence points of subcompatible multivalued maps with applications.Carpathian Journal of Mathematics 2008,24(2):63-71. · Zbl 1174.41027
[30] Singh SL, Tomar A: Weaker forms of commuting maps and existence of fixed points.Journal of the Korea Society of Mathematical Education. Series B 2003,10(3):145-161. · Zbl 1203.54048
[31] Fabian M, Habala P, Hájek P, Montesinos Santalucía V, Pelant J, Zizler V: Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8. Springer, New York, NY, USA; 2001:x+451. · Zbl 0981.46001
[32] Ćirić, LB; Husain, N.; Akbar, F.; Ume, JS, Common fixed points for Banach operator pairs from the set of best approximations, No. 16 (2009) · Zbl 1204.47065
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