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


41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
47H10 Fixed-point theorems


Zbl 0399.41032
Full Text: DOI


[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, (Lakshmikantham, V., Applied Nonlinear Analysis (1979), Academic Press: Academic Press New York), 389-397
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