An extension of Brosowski-Meinardus theorem on invariant approximation.

*(English)*Zbl 0856.41023Let \(E\) be a vector space over a field \(K\) (\(K= \mathbb{R}\) or \(\mathbb{C}\)). For \(0< p\leq 1\), a real-valued function \(|\cdot|_p\) on \(E\) is called a \(p\)-norm if

(i) \(|x|_p\geq 0\) and \(|x|_p= 0\) iff \(x= 0\),

(ii) \(|\alpha x|_p= |\alpha|^p |x|_p\),

(iii) \(|x+ y|_p\leq |x|_p+ |y|_p\) for \(x, y\in E\) and \(\alpha\in K\).

(\(E, |\cdot|_p\)) is called a \(p\)-normed linear space. A subset \(C\) of a vertex space \(E\) is called starshaped if there exists at least one point \(z\in C\) such that \(tz+ (1- t) x\in C\) for all \(x\in C\) and \(0< t< 1\); in this case \(z\) is called a starcentre of \(C\).

If \(C\) is a closed subset of a metric space \((E, d)\), then a mapping \(T: C\to C\) is called a compact mapping if, for every bounded subset \(A\) of \(C\), \(\overline{T(A)}\) is compact in \(C\). If \(T: E\to E\) is a mapping with \(T(C)\subseteq C\), then \(C\) is called a \(T\)-invariant subset of \(E\).

By extending a result of W. G. Dotson jun. [J. Lond. Math. Soc., II. Ser. 4, 408-410 (1972; Zbl 0229.47047), Theorem 1] for non-expansive mappings on starshaped sets, a Brosowski-Meinardus [B. Brosowski, Mathematica Cluj. 11(34), 195-220 (1969; Zbl 0207.45502) and Cs. Meinardus, Arch. Rat. Mech. Anal. 14, 301-303 (1963; Zbl 0012.30801)] type theorem on invariant approximation is proved in the setting of \(p\)-normed linear spaces in this paper. Main results (Theorems 2 and 4):

(1) Let \((E, |\cdot|_p)\) be a \(p\)-normed linear space and \(C\) a closed and starshaped subset of \(E\). If \(T: C\to C\) is a non-expansive mapping with \(\overline{T(C)}\) compact, then \(T\) has a fixed point in \(C\).

(2) Let \((E, |\cdot|_p)\) a \(p\)-normed linear space, \(T: E\to E\) a non-expansive mapping with a fixed point \(u\in E\), and \(C\) a closed \(T\)-invariant subset of \(E\) with \(T|C\) compact. If \(P_C(u)= \{y\in C: d(u, y)= d(u, C)\}\) – the set of best approximations to \(u\) in \(C\), is starshaped, then there exists an element in \(P_C(u)\) which is also a fixed point of \(T\).

(i) \(|x|_p\geq 0\) and \(|x|_p= 0\) iff \(x= 0\),

(ii) \(|\alpha x|_p= |\alpha|^p |x|_p\),

(iii) \(|x+ y|_p\leq |x|_p+ |y|_p\) for \(x, y\in E\) and \(\alpha\in K\).

(\(E, |\cdot|_p\)) is called a \(p\)-normed linear space. A subset \(C\) of a vertex space \(E\) is called starshaped if there exists at least one point \(z\in C\) such that \(tz+ (1- t) x\in C\) for all \(x\in C\) and \(0< t< 1\); in this case \(z\) is called a starcentre of \(C\).

If \(C\) is a closed subset of a metric space \((E, d)\), then a mapping \(T: C\to C\) is called a compact mapping if, for every bounded subset \(A\) of \(C\), \(\overline{T(A)}\) is compact in \(C\). If \(T: E\to E\) is a mapping with \(T(C)\subseteq C\), then \(C\) is called a \(T\)-invariant subset of \(E\).

By extending a result of W. G. Dotson jun. [J. Lond. Math. Soc., II. Ser. 4, 408-410 (1972; Zbl 0229.47047), Theorem 1] for non-expansive mappings on starshaped sets, a Brosowski-Meinardus [B. Brosowski, Mathematica Cluj. 11(34), 195-220 (1969; Zbl 0207.45502) and Cs. Meinardus, Arch. Rat. Mech. Anal. 14, 301-303 (1963; Zbl 0012.30801)] type theorem on invariant approximation is proved in the setting of \(p\)-normed linear spaces in this paper. Main results (Theorems 2 and 4):

(1) Let \((E, |\cdot|_p)\) be a \(p\)-normed linear space and \(C\) a closed and starshaped subset of \(E\). If \(T: C\to C\) is a non-expansive mapping with \(\overline{T(C)}\) compact, then \(T\) has a fixed point in \(C\).

(2) Let \((E, |\cdot|_p)\) a \(p\)-normed linear space, \(T: E\to E\) a non-expansive mapping with a fixed point \(u\in E\), and \(C\) a closed \(T\)-invariant subset of \(E\) with \(T|C\) compact. If \(P_C(u)= \{y\in C: d(u, y)= d(u, C)\}\) – the set of best approximations to \(u\) in \(C\), is starshaped, then there exists an element in \(P_C(u)\) which is also a fixed point of \(T\).

Reviewer: T.D.Narang (Amritsar)

##### MSC:

41A50 | Best approximation, Chebyshev systems |