# zbMATH — the first resource for mathematics

An extension of Brosowski-Meinardus theorem on invariant approximation. (English) Zbl 0856.41023
Let $$E$$ 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 $$T$$.

##### MSC:
 41A50 Best approximation, Chebyshev systems