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Common fixed points in best approximation for Banach operator pairs with Ćirić type \(I\)-contractions. (English) Zbl 1134.47039
A pair \((T,I)\) of two self maps of a metric space \((X,d)\) is called a Banach operator pair if the set \(F(I)\) of fixed points of \(I\) is \(T\)-invariant, i.e., \(T(F(I))\subseteq F(I)\). A mapping \(T:(X,\|\cdot\|)\to (X,\|\cdot\|\}\) is said to satisfy Ćirić type contractive condition if
\[ \|Tx-Ty\|\leq a\max\{\|x-y\|, c(\|x-Ty\|+ \|y-Tx\|)\}+ b\max\{\|x-Tx\|, \|y-Ty\|\}, \] where \(a\in (0,1)\), \(a+b=1\), \(0\leq c<n\), and \(n= \min\{\frac{2+q}{5+q}, \frac{2-q}{4}, \frac {4}{q+a}\}\). In this paper, some common fixed point theorems, similar to those of Lj. B. Ćirić [Publ. Inst. Math., Nouv. Sér. 49(63), 174–178 (1991; Zbl 0753.54023); Arch. Math., Brno 29, No. 3–4, 145–152 (1993; Zbl 0810.47051); Czech. Math. J. 50, No. 3, 449–458 (2000; Zbl 1079.47509)], B. Fisher and S. Sessa [Int. J. Math. Math. Sci. 9, 23–28 (1986; Zbl 0597.47036)], G. Jungck [Int. J. Math. Math. Sci. 13, No. 3, 497–500 (1990); Zbl 0705.54034)], and R. N. Mukherjee and V. Verma [Math. Jap. 33, No. 5, 745–749 (1988; Zbl 0655.47047)], are proved for a Banach operator pair. As applications, common fixed point and approximation results for Banach operator pair satisfying Ćirić type contractive conditions are obtained without the assumption of linearity or affinity of either \(T\) or \(I\). The results proved in the present paper unify and generalize various known results to a more general class of noncommuting mappings.

MSC:
47H10 Fixed-point theorems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A50 Best approximation, Chebyshev systems
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[1] Al-Thagafi, M.A., Common fixed points and best approximation, J. approx. theory, 85, 318-323, (1996) · Zbl 0858.41022
[2] Berinde, V., A common fixed point theorem for quasi-contractive type mappings, Ann. univ. sci. Budapest., 46, 81-90, (2003)
[3] J. Chen, Z. Li, Common fixed points for Banach operator pairs in best approximation, J. Math. Anal. Appl. (2007), doi:10.1016/j.jmaa.2007.01.064, in press · Zbl 1128.47050
[4] Ćirić, Lj.B., A generalization of Banach’s contraction principle, Proc. amer. math. soc., 45, 267-273, (1974) · Zbl 0291.54056
[5] Ćirić, Lj.B., On a common fixed point theorem of a Gregus type, Publ. inst. math. (beograd) (N.S.), 49, 174-178, (1991) · Zbl 0753.54023
[6] Ćirić, Lj.B., On diviccaro, Fisher and sessa open questions, Arch. math. (Brno), 29, 145-152, (1993) · Zbl 0810.47051
[7] Ćirić, Lj.B., On a generalization of Gregus fixed point theorem, Czechoslovak math. J., 50, 449-458, (2000) · Zbl 1079.47509
[8] Ćirić, Lj.B., Contractive-type non-self mappings on metric spaces of hyperbolic type, J. math. anal. appl., 317, 28-42, (2006) · Zbl 1089.54019
[9] Fisher, B.; Sessa, S., On a fixed point theorem of Gregus, Internat. J. math. math. sci., 9, 23-28, (1986) · Zbl 0597.47036
[10] Gregus, M., A fixed point theorem in Banach space, Boll. unione mat. ital. sez. A mat. soc. cult. (8), 517, 193-198, (1980) · Zbl 0538.47035
[11] Habiniak, L., Fixed point theorems and invariant approximation, J. approx. theory, 56, 241-244, (1989) · Zbl 0673.41037
[12] N. Hussain, V. Berinde, Common fixed point and invariant approximation results in certain metrizable topological vector spaces, Fixed Point Theory Appl., vol. 2006, Article ID 23582, pp. 1-13 · Zbl 1103.47046
[13] Hussain, N.; Jungck, G., Common fixed point and invariant approximation results for noncommuting generalized \((f, g)\)-nonexpansive maps, J. math. anal. appl., 321, 851-861, (2006) · Zbl 1106.47048
[14] Hussain, N.; Khan, A.R., Common fixed point results in best approximation theory, Appl. math. lett., 16, 575-580, (2003) · Zbl 1063.47055
[15] N. Hussain, B.E. Rhoades, \(C_q\)-commuting maps and invariant approximations, Fixed Point Theory Appl., vol. 2006, Article ID 24543, pp. 1-9 · Zbl 1131.54027
[16] N. Hussain, B.E. Rhoades, G. Jungck, Common fixed point and invariant approximation results for Gregus type I-contractions, Numer. Funct. Anal. Optim., in press · Zbl 1140.47043
[17] Jungck, G., Commuting mappings and fixed points, Amer. math. monthly, 83, 261-263, (1976) · Zbl 0321.54025
[18] Jungck, G., Common fixed points for commuting and compatible maps on compacta, Proc. amer. math. soc., 103, 977-983, (1988) · Zbl 0661.54043
[19] Jungck, G., On a fixed point theorem of Fisher and sessa, Internat. J. math. math. sci., 13, 497-500, (1990) · Zbl 0705.54034
[20] Jungck, G.; Hussain, N., Compatible maps and invariant approximations, J. math. anal. appl., 325, 1003-1012, (2007) · Zbl 1110.54024
[21] Jungck, G.; Sessa, S., Fixed point theorems in best approximation theory, Math. japon., 42, 249-252, (1995) · Zbl 0834.54026
[22] Khan, A.R.; Hussain, N., Iterative approximation of fixed points of nonexpansive maps, Sci. math. jpn., 54, 503-511, (2001) · Zbl 0997.41021
[23] Khan, A.R.; Hussain, N.; Thaheem, A.B., Applications of fixed point theorems to invariant approximation, Approx. theory appl., 16, 48-55, (2000) · Zbl 0995.41018
[24] Khan, A.R.; Latif, A.; Bano, A.; Hussain, N., Some results on common fixed points and best approximation, Tamkang J. math., 36, 33-38, (2005) · Zbl 1091.47509
[25] Khan, L.A.; Khan, A.R., An extention of brosowski – meinardus theorem on invariant approximations, Approx. theory appl., 11, 1-5, (1995) · Zbl 0856.41023
[26] Kothe, G., Topological vector spaces I, (1969), Springer-Verlag Berlin
[27] Meinardus, G., Invarianze bei linearen approximationen, Arch. ration. mech. anal., 14, 301-303, (1963) · Zbl 0122.30801
[28] Mukherjee, R.N.; Verma, V., A note on fixed point theorem of Gregus, Math. japon., 33, 745-749, (1988) · Zbl 0655.47047
[29] J.O. Olaleru, H. Akewe, An extension of Gregus fixed point theorem, Fixed Point Theory Appl., vol. 2007, Article ID 78628, pp. 1-8 · Zbl 1158.47043
[30] D. O’Regan, N. Hussain, Generalized I-contractions and pointwise R-subweakly commuting maps, Acta Math. Sinica 23 (8) (2007) · Zbl 1139.47041
[31] H.K. Pathak, N. Shahzad, Fixed points for generalized contractions and applications to control theory, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.01.041, in press
[32] Rhoades, B.E., Some applications of contractive type mappings, Math. sem. notes Kobe univ., 5, 2, 137-139, (1977) · Zbl 0362.54044
[33] Sahab, S.A.; Khan, M.S.; Sessa, S., A result in best approximation theory, J. approx. theory, 55, 349-351, (1988) · Zbl 0676.41031
[34] Schaffer, H.H., Topological vector spaces, (1999), Springer-Verlag
[35] Shahzad, N., Remarks on invariant approximations, Int. J. math. game theory algebra, 13, 157-159, (2003) · Zbl 1077.41022
[36] Singh, S.P., An application of fixed point theorem to approximation theory, J. approx. theory, 25, 89-90, (1979) · Zbl 0399.41032
[37] Singh, S.P.; Watson, B.; Srivastava, P., Fixed point theory and best approximation: the KKM-map principle, (1997), Kluwer Academic Publishers Dordrecht · Zbl 0901.47039
[38] Smoluk, A., Invariant approximations, Mat. stosow., 17, 17-22, (1981) · Zbl 0539.41038
[39] Subrahmanyam, P.V., An application of a fixed point theorem to best approximation, J. approx. theory, 20, 165-172, (1977) · Zbl 0349.41013
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