Geodesic
Best approximation for nonconvex set in $q$-normed space
Archivum mathematicum, Tome 42 (2006) no. 1, pp. 51-58.

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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)”.
Classification : 41A50, 41A65, 46B20, 47H10, 54H25
Keywords: Best approximation; demiclosed mapping; fixed point; $I$-nonexpansive mapping; $q$-normed space 
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Nashine, Hemant Kumar. Best approximation for nonconvex set in $q$-normed space. Archivum mathematicum, Tome 42 (2006) no. 1, pp. 51-58. http://geodesic.mathdoc.fr/item/ARM_2006__42_1_a5/