Best approximation for nonconvex set in $q$-normed space
Nashine, Hemant Kumar
Archivum Mathematicum, Tome 042 (2006), p. 51-58 / Harvested from Czech Digital Mathematics Library

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)”.

Publié le : 2006-01-01
Classification:  41A50,  41A65,  46B20,  47H10,  54H25
@article{107981,
     author = {Hemant Kumar Nashine},
     title = {Best approximation for nonconvex set in $q$-normed space},
     journal = {Archivum Mathematicum},
     volume = {042},
     year = {2006},
     pages = {51-58},
     zbl = {1164.41347},
     mrnumber = {2227112},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107981}
}
Nashine, Hemant Kumar. Best approximation for nonconvex set in $q$-normed space. Archivum Mathematicum, Tome 042 (2006) pp. 51-58. http://gdmltest.u-ga.fr/item/107981/

Brosowski B. Fixpunktsatze in der Approximationstheorie, Mathematica (Cluj) 11 (1969), 165–220. (1969) | MR 0277979

Carbone A. Some results on invariant approximation, Internat. J. Math. Math. Soc. 17(3) (1994), 483–488. (1994) | MR 1277733 | Zbl 0813.47067

Dotson W. G. Fixed point theorems for nonexpasive mappings on starshaped subsets of Banach space, J. London Math. Soc. 4(2) (1972), 408–410. (1972) | MR 0296778

Dotson W. G., Jr. On fixed point of nonexpansive mappings in nonconvex sets, Proc. Amer. Math. Soc. 38(1) (1973), 155–156. (1973) | MR 0313894

Hicks T. L.; Humpheries M. D. A note on fixed point theorems, J. Approx. Theory 34 (1982), 221–225. (1982) | MR 0654288

Jungck G. An iff fixed point criterion, Math. Mag. 49(1) (1976), 32–34. (1976) | MR 0433425 | Zbl 0314.54054

Jungck G.; Sessa S. Fixed point theorems in best approximation theory, Math. Japonica 42(2) (1995), 249–252. (1995) | MR 1356383 | Zbl 0834.54026

Köthe G. Topological vector spaces I, Springer-Verlag, Berlin 1969. (1969) | MR 0248498

Latif A. A result on best approximation in p-normed spaces, Arch. Math. (Brno) 37 (2001), 71–75. | MR 1822766 | Zbl 1068.41055

Meinardus G. Invarianze bei linearen Approximationen, Arch. Rational Mech. Anal. 14 (1963), 301–303. (1963) | MR 0156143

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. (1985) | MR 0785288 | Zbl 0606.41048

Nashine H. K. Common fixed point for best approximation for semi-convex structure, Bull. Kerala Math. Assoc. (communicated).

Park S. Fixed points of f-contractive maps, Rocky Mountain J. Math. 8(4) (1978), 743–750. (1978) | MR 0513947 | Zbl 0398.54030

Sahab S. A.; Khan M. S.; Sessa S. A result in best approximation theory, J. Approx. Theory 55 (1988), 349–351. (1988) | MR 0968941 | Zbl 0676.41031

Singh S. P. An application of a fixed point theorem to approximation theory, J. Approx. Theory 25 (1979), 89–90. (1979) | MR 0526280 | Zbl 0399.41032

Singh S. P. Application of fixed point theorems to approximation theory, in: V. Lakshmikantam (Ed.), Applied Nonlinear Analysis, Academic Press, New York 1979. (1979) | MR 0537550

Singh S. P. Some results on best approximation in locally convex spaces, J. Approx. Theory 28 (1980), 329–332. (1980) | MR 0589988 | Zbl 0444.41018

Singh S. P.; Watson B.; Srivastava P., Fixed point theory and best approximation: The KKM-map principle, Vol. 424, Kluwer Academic Publishers 1997. (1997) | MR 1483076 | Zbl 0901.47039

Smoluk A. Invariant approximations, Mat. Stos. 17 (1981), 17–22 [in Polish]. (1981) | MR 0658256

Subrahmanyam P. V. An application of a fixed point theorem to best approximations, J. Approx. Theory 20 (1977), 165–172. (1977) | MR 0445195

Opial Z. Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 531–537. (1967) | MR 0211301