# zbMATH — the first resource for mathematics

A result in best approximation theory. (English) Zbl 0676.41031
Let T, I be two commuting operators on a subset c of a Banach space X, and let F(T) (respectively F(I)) be the set of fixed points of T (respectively I). For $$\bar x\in F(T)\cap F(I)$$, D denotes the set of best approximation elements of $$\bar x$$ in C. The authors prove that $$D\cap F(T)\cap F(I)$$ is non-empty if C, F and I satisfies certain conditions. If the operator I is the identity on X, one obtains a result of S. P. Singh [J. Approximation Theory 25, 89-90 (1979; Zbl 0399.41032)].
Reviewer: I.Şerb

##### MSC:
 41A50 Best approximation, Chebyshev systems 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 47H10 Fixed-point theorems
##### Keywords:
set of best approximation elements
Full Text:
##### References:
 [1] Brosowski, B, Fixpunktsatze in der approximations-theorie, Mathematica (cluj), 11, 195-220, (1969) · Zbl 0207.45502 [2] Hicks, T.L; Humphries, M.D, A note on fixed point theorems, J. approx. theory, 34, 221-225, (1982) · Zbl 0483.47039 [3] Jungck, G, An iff fixed point criterion, Math. mag., 49, No. 1, 32-34, (1976) · Zbl 0314.54054 [4] Park, S, Fixed points of f-contractive maps, Rocky mountain J. math., 8, No. 4, 743-750, (1978) · Zbl 0398.54030 [5] Singh, S.P, An application of a fixed point theorem to approximation theory, J. approx. theory, 25, 89-90, (1979) · Zbl 0399.41032 [6] Singh, S.P, Applications of fixed point theorems in approximation theory, (), 389-397
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.