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Keywords:
Best approximation; demiclosed mapping; fixed point; $I$-nonexpansive mapping; $q$-normed space
Best approximation; demiclosed mapping; fixed point; $I$-nonexpansive mapping; $q$-normed space
Summary:
Some existence results on best approximation are proved without starshaped subset and affine mapping in the set up of $q$-normed space. First, we consider the closed subset and then weakly compact subsets for said purpose. Our results improve the result of Mukherjee and Som (Mukherjee, R. N., Som, T., A note on an application of a fixed point theorem in approximation theory, Indian J. Pure Appl. Math. 16(3) (1985), 243–244.) and Jungck and Sessa (Jungck, G., Sessa, S., Fixed point theorems in best approximation theory, Math. Japonica 42(2) (1995), 249–252.) and some known results (Dotson,W. G., Jr., On fixed point of nonexpansive mappings in nonconvex sets, Proc. Amer. Math. Soc. 38(1) (1973), 155–156.), (Latif, A., A result on best approximation in p-normed spaces, Arch. Math. (Brno) 37 (2001), 71–75.), (Nashine,H. K., Common fixed point for best approximation for semi-convex structure, Bull. Kerala Math. Assoc. (communicated).) are obtained as consequence. To achieve our goal, we have introduced a property known as “Property(A)”.
References:
[1] Brosowski B.: Fixpunktsatze in der Approximationstheorie. Mathematica (Cluj) 11 (1969), 165–220. MR 0277979
[2] Carbone A.: Some results on invariant approximation. Internat. J. Math. Math. Soc. 17(3) (1994), 483–488. MR 1277733 | Zbl 0813.47067
[3] Dotson W. G.: Fixed point theorems for nonexpasive mappings on starshaped subsets of Banach space. J. London Math. Soc. 4(2) (1972), 408–410. MR 0296778
[4] Dotson W. G., Jr.: On fixed point of nonexpansive mappings in nonconvex sets. Proc. Amer. Math. Soc. 38(1) (1973), 155–156. MR 0313894
[5] Hicks T. L., Humpheries M. D.: A note on fixed point theorems. J. Approx. Theory 34 (1982), 221–225. MR 0654288
[6] Jungck G.: An iff fixed point criterion. Math. Mag. 49(1) (1976), 32–34. MR 0433425 | Zbl 0314.54054
[7] Jungck G., Sessa S.: Fixed point theorems in best approximation theory. Math. Japonica 42(2) (1995), 249–252. MR 1356383 | Zbl 0834.54026
[8] Köthe G.: Topological vector spaces I. Springer-Verlag, Berlin 1969. MR 0248498
[9] Latif A.: A result on best approximation in p-normed spaces. Arch. Math. (Brno) 37 (2001), 71–75. MR 1822766 | Zbl 1068.41055
[10] Meinardus G.: Invarianze bei linearen Approximationen. Arch. Rational Mech. Anal. 14 (1963), 301–303. MR 0156143
[11] Mukherjee R. N., Som T.: A note on an application of a fixed point theorem in approximation theory. Indian J. Pure Appl. Math. 16(3) (1985), 243–244. MR 0785288 | Zbl 0606.41048
[12] Nashine H. K.: Common fixed point for best approximation for semi-convex structure. Bull. Kerala Math. Assoc. (communicated).
[13] Park S.: Fixed points of f-contractive maps. Rocky Mountain J. Math. 8(4) (1978), 743–750. MR 0513947 | Zbl 0398.54030
[14] Sahab S. A., Khan M. S., Sessa S.: A result in best approximation theory. J. Approx. Theory 55 (1988), 349–351. MR 0968941 | Zbl 0676.41031
[15] Singh S. P.: An application of a fixed point theorem to approximation theory. J. Approx. Theory 25 (1979), 89–90. MR 0526280 | Zbl 0399.41032
[16] Singh S. P.: Application of fixed point theorems to approximation theory. in: V. Lakshmikantam (Ed.), Applied Nonlinear Analysis, Academic Press, New York 1979. MR 0537550
[17] Singh S. P.: Some results on best approximation in locally convex spaces. J. Approx. Theory 28 (1980), 329–332. MR 0589988 | Zbl 0444.41018
[18] Singh S. P., Watson B., Srivastava P.,: Fixed point theory and best approximation: The KKM-map principle. Vol. 424, Kluwer Academic Publishers 1997. MR 1483076 | Zbl 0901.47039
[19] Smoluk A.: Invariant approximations. Mat. Stos. 17 (1981), 17–22 [in Polish]. MR 0658256
[20] Subrahmanyam P. V.: An application of a fixed point theorem to best approximations. J. Approx. Theory 20 (1977), 165–172. MR 0445195
[21] Opial Z.: Weak convergence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967), 531–537. MR 0211301
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