Definition 1. Let $(X,\varrho)$ be a metric or a semimetric space (symmetric or asymmetric), $\varepsilon>0$, $M\subset X$. We say that $\varphi\colon X\to M$ is an additive (multiplicative) $\varepsilon$-selection if, for all $x\in X$,

Definition 2. We say that $\nu\colon X\times X\to \mathbb{R}_+$ is an asymmetric semimetric on $X$ if:

1) $\nu(x,x)= 0$ for all $x\in X$;

2) $\nu(x,z)\leqslant \nu(x,y)+ \nu(y,z)$ for all $x,y,z\in X$.

In this case, we say that $\mathcal{X}=(X,\nu)$ is an asymmetric semimetric space. The function $\sigma(x,y):=\max\{\nu(x,y), \nu(y,x)\}$ is known as the symmetrization semimetric. A space $\mathcal{X}$ is said to be complete if it is complete with respect to the semimetric $\sigma$.

Definition 3. A subset $A$ of a (symmetric or asymmetric) semimetric space $(X,\nu) $ is called infinitely connected if, for all $n\in \mathbb{N}$, any continuous mapping $\varphi\colon \partial B\to A$ of the boundary $ \partial B$ of the unit ball $B\subset \mathbb{R}^n$ has a continuous extension $\widetilde{\varphi}\colon B\to A$ to the whole of the unit ball $B$. A set $M\subset X$ is called $\mathring{B}$-infinitely connected ($B$-infinitely connected) if its intersection with any open (closed) ball is either infinitely infinitely connected or empty. A set $M\subset X $ is called $\mathring{B}$-contractible ($B$-contractible) if its intersection with any open (closed) ball is contractible or empty.

Definition 4. Let $(X,q)$ be a semimetric space. A mapping $\chi\colon X\to \overline{\mathbb{R}}$ is lower semicontinuous on $X$ if $\varliminf_{n\to\infty}\chi(x_n)\geqslant \chi(x)$ whenever $x\in X$ and $\{x_n\}\subset X$, $ q(x,x_n)\to 0$ as $n\to\infty$.

Definition 5. Let $(X,q)$ and $(Y,\nu)$ be semimetric spaces, $M\subset Y$. A mapping $F\colon X\to 2^M$ is called lower stable if $F(x)\neq \varnothing$ for all $x\in X$, and if, for any $x_0\in X$ and any $\varepsilon>0$, there exists $\delta>0$ such that $\varrho(y,F(x))-\varrho(y,F(x_0))\leqslant\varepsilon$ for all $y\in Y$ and $x\in X$ with $q(x,x_0)\leqslant\delta$.

Remark 1. Let $A\subset X$ be a closed subset of a semimetric space $(X,q)$, and let a mapping $F\colon A\to 2^M$, where $M\subset Y$ and $(A,q)$, $(Y,\nu) $ are semimetric spaces, be lower stable. Then the mapping $F$ defined by $F(x)=M$ for all $x\in X\setminus A$ is lower stable on the entire $X$.

Lemma 1. Let $S\subset X$ be a non-degenerate simplex of dimension $n$, and $Q$ be the union of some proper faces of $S$. Next, let $\varphi\colon Q\to M$ be a continuous mapping such that $\varphi(x)\in F(x,t_0)$, for some $t_0\in (0,\vartheta)$, and

For each point $x\in S$, there exists a number $r=r(x)\in (0,\delta)$ such that, for any $y\in O_{2r}(x)$,

$1^\circ$. For each zero-dimensional face (a vertex) $s$ of any simplex $S_i$, there exists an index $j=j_s$ for which the number $r(x_j)$ is maximal among those for which the face $s$ lies in the neighbourhood $O_{r(x_j)}(x_j)$. If $s\notin Q$, then with the point $s$ we associate some point $y := \varphi(s)\in F(x_j,t_0)\subset F(x_j,t'_0)$. We have

$2^\circ$. Assume that the mapping $\varphi$ extends continuously to the set $T_m$ defined as the union of the faces $\{\Delta_{ik}\}$ of dimension $m$ of simplexes in the family $\{S_i\}$. Note that $\varphi(x)\in F(x,t_{m+1})$ for all $x\in T_m$.

Moreover, for each face $\Delta_{ik}$, there exists an index $j=j_{ik}=j_{\Delta_{ik}}$ such that $\Delta_{ik}\subset O_{r(x_j)}(x_j)$ and $\varphi(\Delta_{ik})\subset F(x_j,t'_{m})$. We take an arbitrary $(m+1)$-dimensional face $\Delta\not\subset Q$ of some simplex in the family $\{S_i\}$. The function $\varphi\colon T_m\to M$ is defined on its relative boundary $\partial\Delta$. With each face $P$ of the boundary $\partial\Delta$ of dimension $k\leqslant m$ we associate the greatest of the numbers $r(x_l)$, $(l=l_P)$, for which this face lies in the neighbourhood $O_{r(x_l)}(x_l)$. These neighbourhoods will also be associated with a face $P$. Let $j=j_{\Delta}$ be an index for which the number $r(x_j)$ is maximal among the numbers $r(x_k)$ for which the face $\Delta$ lies in $O_{r(x_k)}(x_k)$. For each face $\Delta_{ik}$ in the boundary $\partial\Delta$, the values of the function $\varphi$ on this face lie in $F(x_{j_{ik}},t'_m)\subset F(x_{j_{ik}},t_{m+1})$ $({j_{ik}}=j_{\Delta_{ik}})$. Taking into account that the neighbourhood $O_{r(x_{j_{ik}})}(x_{j_{ik}})$ has the maximal radius among those neighbourhoods that contain the face $\Delta_{ik}$, we get the inequality $r(x_{j_{ik}})\geqslant r(x_j)$. Since $x_j\in O_{2r(x_{j_{ik}})}(x_{j_{ik}})$, this implies that

Since distinct $(m+ 1)$-dimensional faces intersect only in proper faces, the mapping $\varphi$ is well defined on the set $T_{m+1}$ that is the union of all faces of $(m+ 1)$-dimensional simplexes in the family $\{S_i\}$. Note that $\varphi \in C(T_{m+1})$.

$3^\circ$. By induction, on $T_n = S$ we can construct a continuous mapping $\varphi\colon T_n\to M$ such that $\varphi(x)\in F(x,t)$ $(x\in S)$. This shows that, for all $y\in S $,

Remark 2. The proof of Lemma 1 shows that if $y\in S_i$ does not lie initially in $Q$, then $\varphi(y)\in F(x_j,t)$ for some index $j$ for which $y\in O_{r(x_j)}(x_j)$, and

Lemma 2. Let $(X,\|\,{\cdot}\,\|)$ and $(Y,\|\,{\cdot}\,\|) $ be seminormed linear spaces, $M\subset Y$, $\{x_\alpha\}_\alpha$ be a family of vectors in $X$, $\Sigma=\{S_\beta\}_\beta$ be a family of simplexes whose vertices form a finite subfamily of $\{x_\alpha\}_\alpha$ under the condition that any two distinct simplexes $S_\beta$ either have no common points, or intersect in some proper faces of them, or one of them is a proper face of the other. Next, let $T$ be the union of some simplexes in $\Sigma$ and let $F$ be the set-valued mapping defined above. Then there exists a mapping $\varphi\colon T\to M$ such that $\varrho(\varphi(x),F(x,0))<\chi(x)$ for all $x\in T$, the mapping $\varphi$ being continuous on each simplex in $T$.

$1^\circ$. By Lemma 1, there exists a continuous extension of the mapping $\varphi$ to each one-dimensional simplex $S\in \Sigma$ such that $\varrho(\varphi(y),F(y,0))<\chi(y)$ for all $y\in S$.

$2^\circ$. Assume that the mapping $\varphi\colon T_m\to M$ is constructed on the union $T_m$ of all $m$-dimensional simplexes in $\Sigma$ and that $\varphi$ is continuous on each such simplex. In addition, for any $(m+1)$-dimensional simplex $S\in \Sigma$, the continuous mapping $\varphi$ satisfies $\varrho(\varphi(y),F(y,0))<\chi(y)$ for all $y\in \partial S$ on the relative boundary $\partial S$ of $S$. By Lemma 1, there also exists a continuous extension $\varphi\colon S\to M$ such that $\varrho(\varphi(y),F(y,0))<\chi(y)$ for all $y\in S$. This mapping is well defined on the union of all $(m+1)$-dimensional simplexes in $\Sigma$, because distinct $(m+1)$-dimensional simplexes intersect only in proper faces (if the intersection is non-empty). This mapping is continuous on any such simplex.

$3^\circ$. So, we have constructed a mapping $\varphi\colon T\to M$ such that

Remark 3. Since any finite-dimensional space $X$ can be represented as a set $T$ from Lemma 2, there exists a function $\varphi\in C(X,M)$ such that $\varrho(\varphi(x),F(x,0))<\chi(x))$ for all $x\in X$.

Remark 4. Let $\Sigma$ be a family of simplexes from Lemma 2. Since the proof of this lemma and the construction of the function $\varphi$ depend on Lemma 1, it follows by Remark 2 that, for each simplex $S_i$ from the family $\Sigma$ (after a refinement), we have $\varphi(y)\in F(x_j,t)$ for some index $j$ such that $y\in O_{r(x_j)}(x_j)$, and

Lemma 3. Let $(X,\|\,{\cdot}\,\|)$ and $(Y,\|\,{\cdot}\,\|) $ be normed linear spaces, $M\subset Y$, and $F$ be a set-valued mapping from Lemma 1. Then there exists a mapping $\widehat{\varphi}\in C(X, M)$ such that $\varrho (\widehat{\varphi}(x),F(x,0))<\chi(x) $ for all $x\in X$.

For each point $x\in X $, there exists a largest number $\delta=\delta(x)\in (0,1]$ such that $\inf_{O_{3\delta}(x)}\widehat{\chi}\geqslant 4\delta$. Note that if $O_{\delta(y)}(y)\cap O_{\delta(z)}(z)\neq\varnothing$, and, for definiteness $\delta(y)\geqslant \delta(z)$, then $z\in O_{2\delta(y)}(y)$. Hence $\delta(z)\geqslant \delta(y)/3$, for otherwise we would have $O_{3\delta(y)}(y)\supset O_{\delta'}(z)\supset O_{3\delta(z)}(z)$ $(\delta'>3\delta(z))$, which would imply $\widehat{\chi}\geqslant 4\delta(y)\geqslant 12 \delta(z)$ on $O_{\delta'}(z)\supset O_{3\delta(z)}(z)$, contradicting the maximality of $\delta(z)$.

Let $r=r(x)\in (0,\delta(x)/3)$ be such that $\sup_{u\in Y}(\varrho(u,F(y,0))-\varrho(u,F(x,t)))<\delta=\delta(x)$ for any $y\in O_{2r}(x)$. Note that if $y\in O_{2r(x)}(x)$ ($O_{2r(x)}(x)\subset O_{\delta(x)}(x)$), then $r(x)<\delta(x)/3\leqslant\delta (y)$.

Let $\{U_\alpha\}_\alpha$ be an open locally finite subcovering of the open covering $\{O_{r(x)}(x)\}$ For each $\alpha$, there exists a point $y_\alpha$ such that $U_\alpha\subset O_{r(y_\alpha)}(y_\alpha)$. Let $\{g_\alpha\}_\alpha$ be a partition of unity corresponding to this covering, that is, $g_\alpha\equiv 0$ outside $ U_\alpha$; $\sum_\alpha g_\alpha\equiv 1$ on $X$, $g_\alpha\geqslant 0$ for all $\alpha$. We take a family of vectors $\{x_\alpha\}$ such that $x_\alpha\in U_\alpha$ for all $\alpha$ and any finite subfamily of vectors is linearly independent. Consider the continuous mapping $q(t)=\sum_\alpha x_\alpha g_\alpha(t)$. For each point $y\in X$, the index set $\mathcal{A}=\{\alpha\mid y\in \overline{U_\alpha} \}=\{\alpha_1,\dots,\alpha_N\}$ is finite, and there exists a neighbourhood $O(y)$ such that $\{\alpha\mid U_\alpha\cap O(y)\neq\varnothing\}=\mathcal{A} =\{\alpha_1,\dots,\alpha_N\}$. Let an index $i$ be such that $\max_{j=1,\dots,N}r(y_{\alpha_j})=r(y_{\alpha_i}) := r_y=r$. Then $U_{\alpha_j}\subset O_{2r}(y_{\alpha_i})$ for all $j=1,\dots,N$. Therefore, the simplex $S_y$ with vertices at the points $\{x_{\alpha_1},\dots,x_{\alpha_N}\}$ is contained in $O_{2r}(y_{\alpha_i})$. All distinct simplexes of the form $S_y$ $(y\in X)$ are either disjoint, or intersect in some proper faces of them, or one of them is a proper face of another one. By Lemma 2, there exists a mapping $\varphi\colon T\to M$ on the set $T=\bigcup_yS_y$ which is continuous on each $S_y$ and such that $\varrho (\varphi(u),F(u,0))<\widehat{\chi}(u) $ for all $u\in T$.

For the continuous function $q(u)=\sum_\alpha x_\alpha g_\alpha(u)\in S_y\subset O_{2r}(y_{\alpha_i})$, we have, for all $u\in O(y)$,

We set $\widehat{\varphi}(y)=\varphi(q(y))$. Then $\widehat{\varphi}\in C(X,M)$ and $\varrho(\widehat{\varphi}(y),F(y,0))\leqslant 2\delta(y)\,{<}\,\chi(y)$ for all $y\in X$. Lemma is proved.

Remark 5. The conclusion of Lemma 3 remains true if in it the space $(X,\|\,{\cdot}\,\|)$ is replaced by a metric linear space, and then this space is replaced by an arbitrary metric space $(D,\nu)$ via an isometrical embedding $\tau\colon D\to X$ into the space of continuous functions $X:= C_b(D,\mathbb{R})$ with the metric $\upsilon(f,g):=\sup_D|f(\,{\cdot}\,)-g(\,{\cdot}\,)|$, where $\tau(x):=\nu(x,{\cdot}\,)\in C_b(D,\mathbb{R})$. Next, the mapping $F$ extends to the entire space $C_b(D,\mathbb{R})$ by setting $F(x,t):=M$ for all points $x\notin \tau(X)$ and $t\in \mathbb{R}$. So, in Lemma 3, we can assume that $X$ is an arbitrary metric space.

Theorem 1. Let $(X,\upsilon)$ be a metric space, $(Y,\|\,{\cdot}\,\|)$ be a complete seminormed linear space, and let a set-valued mapping $\Phi\colon X\to 2^Y$ be lower stable and have closed $\mathring{B} $-infinitely connected values in $ (Y,\|\,{\cdot}\,\|) $. Then there exists a mapping $\varphi\in C(X, Y)$ such that $ \varphi(x)\in \Phi(x)$ for all $x\in X$.

2. Assume that we have already constructed a mapping $\psi_n\in C(X, Y)$ such that $ \varrho(\psi_n(x),\Phi(x))<1/2^n$. We set $\chi(x)=2^{-n-1}-(1/2)\varrho(\psi_n(x),\Phi(x))$ and $\delta(x)=(1/2)\varrho(\psi_n(x),\Phi(x))$. Consider the set-valued mappings $H(x,t)=O_{\delta(x)+t}(\Phi(x))$ and $G(x,t)=O_{2^{-n-1}+t}(\psi_n(x))$ ($x\in X$). For all $t\geqslant 0$, the sets $F(x,t)=H (x,t)\cap G (x,t)$ are non-empty and infinitely connected. In addition, it is easily checked that the mapping $F$ satisfies the conditions formulated before Lemma 1, and, therefore, for it the conclusions Lemma 3 and Remark 5 are true. Hence there exists a mapping $\psi_{n+1}\in C(X, Y)$ such that $ \varrho(\psi_{n+1}(x),F(x,0))<\chi(x)$ ($x\in X$). It follows that $\varrho(\psi_{n+1}(x),\Phi(x))<\delta(x)+\chi(x)=2^{-n-1}$ and $\|\psi_{n+1}(x)-\psi_n(x)\|<2^{-n}$.

3. The sequence $\{\psi_n(x)\}$ is a Cauchy sequence uniformly on $(Y,\|\,{\cdot}\,\|)$, and hence, converges to some continuous mapping $\varphi\colon X\to Y$. In addition, $\varrho(\psi_{n+1}(x),\Phi(x))<2^{-n-1}\to 0$ as $n\to\infty$, and, therefore, $\varrho(\varphi(x),\Phi(x))=0$, that is, $\varphi(x)\in \Phi(x)$ for all $x\in X$.

Theorem 1 is proved.

Corollary 1. Let $ (X,\|\,{\cdot}\,\|) $ be a seminormed linear space, let $\mathcal{K}\subset X$ be a compact $\mathring{B}$-infinitely connected set, and let $\Psi\colon \mathcal{K}\to 2^\mathcal{K}$ be a lower semicontinuous mapping with $\mathring{B}$-infinitely connected compact values. Then there exists a fixed point $x_0\in \mathcal{K}$ such that $ x_0\in \Psi(x_0)$.

By Schauder’s fixed point theorem, there exists a fixed point $x_0\in \mathcal{K}$ for the restriction of the continuous mapping $\varphi$ to the convex hull of the compact set $\mathcal{K}$. Hence $x_0=\varphi(x_0)\in \Phi(x_0)=\Psi(x_0)$, proving the corollary.

Definition 6. A set $X$ will be called a semilinear space (or a cone) over $\mathbb{R}$ if the operation of addition of vectors and multiplication by non-negative scalars are defined on $X$ and if, for all $x,y,z\in X$ and $\alpha,\beta\in \mathbb{R}_+$:

1) $x+y=y+x$;

2) $x+(y+z)=(x+y)+z$;

3) there exists a unique zero element $\theta\in X$ such that $x+\theta=x$;

4) $\alpha(x+y)=\alpha x+\alpha y$;

5) $(\alpha+\beta)x=\alpha x+\beta y$;

6) $\alpha(\beta x)=(\alpha\beta)x$;

7) $0\cdot x=\theta$; $1\cdot x=x$.

When no confusion will ensue, we denote the zero element $\theta$ by $0$.

Definition 7. A pair $\mathcal{X}=(X,\varrho)$, where $X$ is a semilinear space, will be called an asymmetric semimetric semilinear space if $\varrho$ is an asymmetric semimetric on $X$, and if the operations of addition of elements and multiplication by non-negative scalars are continuous; that is, if $\{\alpha_n\},\{\beta_n\}\subset \mathbb{R}_+$ and $\{x_n\},\{y_n\}\subset X$ converge, respectively, to $\alpha,\beta\in \mathbb{R}_+$ and $x,y\in X$, then the sequence $\sigma(\alpha_n x_n+\beta_n y_n,\alpha x +\beta y)$ converges to $0$, where $\sigma$ is the symmetrization semimetric.

A set $M\subset X$ is called convex if $[a,b]\subset M$ for all $a,b\in M$.

An asymmetric semimetric $\varrho$ is called generalized convex (respectively, convex) on a convex set $M\subset X$ if $\varrho(z,\alpha x+(1-\alpha)y)\leqslant \max\{\varrho(z,x),\varrho(z,y)\}$ (respectively, $\varrho(z,\alpha x+(1-\alpha)y)\leqslant \alpha\varrho(z,x)+(1-\alpha)\varrho(z,y)\bigr)$ for all $\alpha\in[0,1]$, $z\in X$ and $x,y \in M$. In particular, if $\varrho$ is convex, then it is generalized convex.

Remark 6. If an asymmetric semimetric is generalized convex on a $M$, then, for any interval interval $[x,y]\subset M$,

Remark 7. If an asymmetric semimetric is generalized convex on a convex set $M$, then $M$ has contractible intersections with open balls, and, therefore, $M$ is $\mathring{B}_\varrho$-infinitely connected (that is, the intersection of $M$ with any open ball is infinitely connected). Hence by Theorem 5 in [12], the convex set $M$ admits a continuous $\varepsilon$-selection for any $\varepsilon>0$.

Theorem A. Let $(X,\varrho)$ be a semimetric semilinear space with generalized convex semimetric $\varrho$ satisfying $\varrho(x,y)=\varrho(y,x)$ and $\varrho(x,(1-\alpha)x+\alpha y)= \alpha \varrho(x,y) $ for all $x,y\in X$ and $\alpha\in [0,1]$ (in particular, this condition is met for the space $\mathbf{L}_h$; see § 4 below). Also let $M $ be either $\mathring{B}_\varrho$-infinitely connected in $X$, or admit a continuous additive $\varepsilon$-selection for any $\varepsilon>0$. Then any $r$-neighbourhood $(r>0)$ of $M$ is $\mathring{B}_\varrho$-infinitely connected.

Theorem 2. Let $(X,\upsilon)$ be a metric space, $(Y,\varrho)$ be a complete semimetric semilinear space with generalized convex semimetric $\varrho$ satisfying $\varrho(x,y)=\varrho(y,x)$ and $\varrho(x,(1-\alpha)x+\alpha y)= \alpha \varrho(x,y) $ for all $x,y\in X$ and $\alpha\in [0,1]$. Next, let a set-valued mapping $\Phi\colon X\to 2^Y$ be lower stable and have closed $\mathring{B}$-infinitely connected values in $(Y,\varrho)$. Then there exists a mapping $\varphi\in C(X, Y)$ such that $ \varphi(x)\in \Phi(x)$ for all $x\in X$.

Remark 8. In Remark 1 of [11] it was noted that in an asymmetric normed linear space $(Y,\|\,{\cdot}\,|)$ in which the asymmetric norm is equivalent to the symmetrization norm $\|\,{\cdot}\,\|_{\mathrm{sym}}$ (see the definition below), any (closed) $r$-neighbourhood of a $\mathring{B}$-infinitely connected set $M\subset Y$ is $\mathring{B}$-infinitely connected in $Y$. Using this fact, one can easily carry over Theorem 2 to the case where $Y$ is a complete asymmetric normed space in which the asymmetric norm is equivalent to the symmetrization norm.

Corollary 2. Let either $(Y,\|\,{\cdot}\,|)$ be a complete asymmetric linear space in which the asymmetric norm is equivalent to the symmetrization norm or $(Y,\varrho)$ be a complete semimetric semilinear space in which the semimetric $\varrho$ is generalized convex and satisfies $\varrho(x,y)=\varrho(y,x)$ and $\varrho(x,(1-\alpha)x+\alpha y)= \alpha \varrho(x,y)$ for all $x,y\in X$ and $\alpha\in [0,1]$. Next, let $M\subset Y$ and the metric projection $P_M$ (respectively, $P_M^-)$ have closed $\mathring{B}$-infinitely connected values in $Y$. Then there exists a mapping $\varphi\in C(X, M)$ such that $ \varphi(x)\in P_M(x)$ (respectively, $ \varphi(x)\in P_M^-(x)$) for all $x\in X$.

Corollary 3. Let either $(Y,\|\,{\cdot}\,|)$ be a complete asymmetric linear space in which the asymmetric norm is equivalent to the symmetrization norm or $(Y,\varrho)$ be a complete semimetric semilinear space in which the semimetric $\varrho$ is generalized convex and satisfies $\varrho(x,y)=\varrho(y,x)$ and $\varrho(x,(1-\alpha)x+\alpha y)= \alpha \varrho(x,y)$ for all $x,y\in X$ and $\alpha\in [0,1]$. Next, let $M\subset Y$ admit a set-valued selection $\Psi\colon X\to 2^M$ of the metric projection $P_M$ (respectively, $P_M^-)$ and $\Psi$ be lower stable and have closed $\mathring{B}$-infinitely connected values in $Y$. Then there exists a mapping $\varphi\in C(X, M)$ such that $ \varphi(x)\in \Psi(x)\subset P_M(x)$ (respectively, $\varphi(x)\in \Psi(x)\subset P_M^-(x)$) for all $x\in X$.

Remark 9. Under the conditions of Corollary 2 (or Corollary 3), there exists a continuous selection of the metric projection operator, and hence, if a “closed” right ball $B(x,r):=\{y\in Y\mid \varrho(x,y)\leqslant r\}$ or a left ball $B^-(x,r):=\{y\in Y\mid \varrho(y,x)\leqslant r\}$ has non-empty intersection with $M$, then the corresponding continuous selection $\varphi$ maps the cone $K:=\{z\in [x,y]\mid y\in M\cap B(x,r)\}\subset M\cap B(x,r)$ (or, respectively, the cone $K^-:=\{z\in [x,y]\mid y\in M\cap B^-(x,r)\}\subset M\cap B^-(x,r)$) into itself and is identical on $M\cap B(x,r)$ (respectively, $M\cap B^-(x,r)$). As a result, in both these cases, the sets $ P_M(x)$ ($P_M^-(x)$) and $M\cap B(x,r)$ ($M\cap B^-(x,r)$) are retracts of the corresponding cone, and, therefore, are contractible in itself. Hence, the intersection of $M$ with any closed right (left) ball is contractible.

Definition 8. Let $\varnothing \ne M\subset X$. We say that $x\in X\setminus M$ is a solar point for $M$ if there exists a point $y\in P_Mx\ne \varnothing$ (called a luminosity point) such that $y\in P_M((1-\lambda)y+\lambda x)$ for all $\lambda\geqslant 0$ (geometrically, this means that there is a “solar” ray emanating from $y$ and passing through $x$ such that $y$ is a nearest point in $M$ for any point on the ray).

We say that $x\in X\setminus M$ is a strict solar point if $P_Mx\ne\varnothing$ and each $y\in P_Mx$ is a luminosity point for $x$. A set $M$ is a sun (a strict sun) if any point from $ X\setminus M$ is a solar (strict solar) point for $M$.

Theorem B. Let $(X,\|\,{\cdot}\,|)$ be a finite-dimensional asymmetric space, $M\subset X$, and let the metric projection $P_M$ onto $M$ be lower semicontinuous. Then there exists a continuous mapping $\varphi\colon X\to M$ such that $\varphi(x)\in P_Mx$ for all $x\in X$.

Theorem 3. Let $(X,\|\,{\cdot}\,|)$ be a finite-dimensional asymmetric space and let $M\subset X$ be a set with lower semicontinuous metric projection. Then there exists a continuous mapping $\varphi\colon X\to M$ such that $\varphi(x)\in P_Mx$ for all $x\in X$. Moreover, $M$ is $B$-contractible, and if, in addition, $X$ is polyhedral, then $M$ is a strict sun.

Definition 9. A path $p\colon [0,1]\to X$ in an asymmetric linear space $X$ is said to be monotone if $x^*(p(\tau))$ is a monotone function for any norm-one extreme functional $x^*$. A set $M\subset X$ is called monotone path-connected if any two points from $M$ can be connected by a monotone path whose trace lies in $M$.

A subset $M$ of a normed linear space $X$ is called stably monotone path-connected if there exists a continuous mapping $p\colon M\times M\times[0,1]\to M$ such that, for all $x,y\in M$, $p(x,y,{\cdot}\,)$ is a monotone path connecting $x$ and $y$.

Theorem 4. Let $(Y,\|\,{\cdot}\,|)$ be a complete asymmetric linear space in which the asymmetric norm is equivalent to the symmetrization norm $\|\,{\cdot}\,\|_{\mathrm{sym}}$ and let $M\subset Y$ be either a stably monotone path-connected set or an approximatively compact (see Definition 11 below) monotone path-connected set. Then the following conditions are equivalent:

1) the set $M$ admits a lower stable (respectively, lower semicontinuous) set-valued selection $\Psi\colon X\to 2^M$ of the metric projection $P_M$ (respectively, $P_M^-$);

2) the set $M$ admits a continuous selection $\varphi\colon X\to M$ of the metric projection $P_M$ (respectively, $P_M^-$).

Example. In the space of all continuous functions on a compact set $Q$, consider the asymmetric norm $\|f|_{\psi_+,\psi_-}:=\max_{x\in Q}\{f_+/\psi_+,f_-/\psi_-\}$, where $f_+=\max\{f,0\}$ and $f_-=\max\{-f,0\}$ for $f\in C(Q,\mathbb{R})$, and $\psi_+$ and $\psi_-$ are fixed functions satisfying $0<c<\psi_+,\psi_-<C$ for some positive constants $c$ and $C$. The asymmetric ball $B(0,R)$ consists of all functions $f$ lying between $R\psi_+$ and $-R\psi_-$, that is, $-R\psi_-(x) \leqslant f(x)\leqslant R\psi_+(x)$ for all $x\in Q$. Hence the ball $B(g,R)$ consists of all functions $f$ such that $f-g$ lies between $R\psi_+$ and $-R\psi_-$. Let $C_{\psi_+,\psi_-}(Q)$ be the space of continuous functions on a compact set $Q$ with asymmetric norm $\|\,{\cdot}\,|_{\psi_+,\psi_-}$. This norm is equivalent to the symmetrization norm, which, in turn, is equivalent to the classical uniform norm on $C(Q)$. A path $p(\,{\cdot}\,)\colon [0,1]\to C_{\psi_+,\psi_-}(Q) $ will be said to be monotone if $(p(\tau))[x]$ is a monotone function of $\tau$ for all $x\in Q$; a set $M\subset C_{\psi_+,\psi_-}(Q) $ will be said to be monotone path-connected if any two points $M$ can be connected by a monotone path whose trace lies in $M$. In this setting, Theorem 4 can be applied to the space $Y=C_{\psi_+,\psi_-}(Q) $ and to its approximatively compact subset $M$.

Theorem 5. Let $M$ be a boundedly weakly compact approximatively compact sun in $C(Q)$. Then the following conditions are equivalent:

1) $M$ admits a set-valued lower semicontinuous selection $\Psi\colon X\to 2^M$ of the metric projection operator $P_M$;

2) $M$ admits a continuous selection $\varphi\colon X\to M$ of the metric projection $P_M$.

Theorem C. Let $(X,\|\,{\cdot}\,|) $ be an asymmetric seminormed linear space, $M\subset X$ be a $\mathring{B}$-infinitely connected (in particular, convex) set. Then, for any lower semicontinuous (with respect to the symmetrization norm) function $\psi\colon X\to\overline{\mathbb{R}}$ such that $ \varrho(x,M)<\psi(x)$ ($x\in X$), there exists a mapping $\varphi\in C(X^{\mathrm{sym}},M)$ such that $ \|\varphi(x)-x|<\psi(x)$ ($x\in X$). Similarly, for a non-empty open set $D\subset X$ and a lower semicontinuous (with respect to the symmetrization norm) function $\psi\colon D\to\overline{\mathbb{R}}$ such that $\varrho(x,M)<\psi(x)$ ($x\in D$), there exists a mapping $\varphi\in C(D^{\mathrm{sym}},M)$ such that $ \|\varphi(x)-x|<\psi(x)$ ($x\in D$).

Theorem 6. Let $(X,\|\,{\cdot}\,|) $ be an asymmetric seminormed linear space, let $M\subset X$ be a convex set, and let $K\subset X$ be a non-empty Hausdorff compact set on which the metric function $\varrho(\,{\cdot}\,,M)$ is continuous. Then, for any $\varepsilon>0$, there exists a continuous function $\varphi\in C(K,M)$ such that

By Theorems 1 and 2, there exists a continuous mapping $g\colon K\to T$ such that $g(x)\in \Phi(x)$ (that is, $\|g(x)-x|<\varepsilon/4$) for $x\in K)$. By Theorem C, for the finite-dimensional subspace $T^0$ spanned by $T$, there exists a continuous function $\vartheta\colon T^0\to M$ such that $\|\vartheta(x)-x|<\varrho(x,M)+\varepsilon/2$ ($x\in T^0$). Hence $\varphi=\vartheta\circ g\colon K\to M$ is a continuous mapping, and $\|\varphi(x)-x|=\|\vartheta\circ g(x)-x|\leqslant \|\vartheta\circ g(x)- g(x)|+\|g(x)-x|<\varrho(g(x),M)+\varepsilon/2+\varepsilon/4<\varrho(x,M)+\varepsilon$. Theorem 6 is proved.

Theorem 7. Let $(X,\|\,{\cdot}\,|) $ be an asymmetric seminormed linear space, let $M\subset X$ be a convex set, and let $K\subset X$ be a non-empty Hausdorff compact set (in the left topology) on which the metric function $\varrho^-(\,{\cdot}\,,M)$ is continuous in the left topology. Then, for any $\varepsilon>0$, there exists a left-left continuous function (both the image and the pre-image are equipped with the left topology) $\varphi\colon K\to M$ such that

Remark 10. For a paracompact asymmetric normed space $X$, Theorems 6 and 7 are also true if the compact set $K$ is replaced by the paracompact set $X$ (with respect to the right or left topology). In the proof of this result, we may avoid using Theorem C by constructing an appropriate locally finite covering and a corresponding partition of unity. Since any separable asymmetric normed space $X$ with closed unit ball is a Lindelöf space, $X$ is paracompact. Therefore, in this case, the set $K$ from Theorems 6 and 7 can be replaced by the entire space $X$.

Definition 10. Let $X=(X,\|\,{\cdot}\,|)$ be a space with asymmetric seminorm. A set $M\subset X$ is a left- (right)-approximatively) stable at a point $x\in X$ if, for any $\{x_n\}\subset X$, any $\{y_n\}\subset M$, and any null sequence $\{\varepsilon_n\}\subset \mathbb{R}_+$ such that $\|x_n-x|\to 0$ as $n\to\infty$, the condition

Definition 11. Let $ M$ be a non-empty subset of an asymmetric seminormed space $(X,\|\,{\cdot}\,|)$. We say that $x\in X$ is a point of left (right) approximative compactness for $M$ (written $x \in \mathrm{AC}^-(M)$ ($x \in \mathrm{AC}(M)$) if, for any minimizing sequence $\{y_n\}\subset M$ (that is, $\|x-y_n|\to \varrho^-(x,M)$ (respectively, $\|y_n- x|\to \varrho(x,M)$ as $n\to\infty$)), there exists a subsequence $\{y_{n_k}\}$ such that $\|y-y_{n_k}|\to 0$ ($\|y_{n_k}-y|\to 0$) as $k\to\infty$ for some point $y \in M$. If $\mathrm{AC}^-(M)=X$ ($\mathrm{AC}(M)=X$), we say that $M$ is left- (right-)approximatively compact.

Theorem D. Let $X=(X,\|\,{\cdot}\,|)$ be an asymmetric seminormed space in which the unit ball $B(0,1)$ is closed and let $M\subset X$ be left-approximatively stable at $x\in X$. Then $x\in X$ is a point of left-approximative compactness for $M$, and the function $\varrho^-(\,{\cdot}\,,M)$ is continuous at $x$.

Theorem 8. Let $X=(X,\|\,{\cdot}\,|)$ be an asymmetric seminormed space in which the unit ball $B(0,1)$ is closed and let $M\subset X$ be a left-approximatively stable convex set. Then, for any $\varepsilon>0$, there exists a continuous function $\varphi\in C(X,M)$ such that

Definition 12. Let $\mathcal{X}=(X,\|\,{\cdot}\,\|)$ be a seminormed linear space. Consider the family $\mathcal{M}=\mathcal{M}(X)$ of all bounded subsets of $X$. Given two sets $M_1,M_2\in \mathcal{M}$, the linear combination $\alpha M_1+\beta M_2$ is defined by $\{z=\alpha x_1+\beta x_2\mid x_j\in M_j,\ j=1,2\}$. On the set $\mathcal{M}$, we consider the symmetric semimetric $h(\,{\cdot}\,,{\cdot}\,)$ defined as the Hausdorff distance; we also consider the asymmetric semimetric $d(\,{\cdot}\,,{\cdot}\,)$ defined as the directed Hausdorff distance. The spaces $\mathbf{M}_h=(\mathcal{M}(X),h)$ and $ \mathbf{M}_d=(\mathcal{M}(X),d)$ are asymmetric semimetric spaces. Note that the semimetric $h$ is the symmetrization of the metric $d$. On the space $\mathcal{M}=\mathcal{M}(X)$, we also consider the semilinear subspace of all convex bounded subsets $\mathcal{L}=\mathcal{L}(X)$. When equipped with the semimetric $d$ (respectively, $h$), this space defines the asymmetric semimetric semilinear space $\mathbf{L}_d=\mathbf{L}_d(X)$ (respectively, $\mathbf{L}_h=\mathbf{L}_h(X)$).

Remark 11. Let $\mathcal{X}=(X,\|\,{\cdot}\,\|)$ be a seminormed linear space. Then the semimetrics of the spaces $\mathbf{L}_d$ and $ \mathbf{L}_h$ are generalized convex. In addition, if $\mathcal{X}$ is complete, then so are $ \mathbf{M}_h$ and $ \mathbf{L}_h$. We also note that $h(A,B)=h(A+C,B+C)$ and $d(A,B)=d(A+C,B+C)$, $h(\alpha A,\alpha B)=\alpha h(A,B)$, and $d(\alpha A,\alpha B)=\alpha d(A,B)$ for all $A,B,C\in \mathcal{L}$ and $\alpha\geqslant 0$.

Corollary 4. Let $\mathcal{X}=(X,\|\,{\cdot}\,\|)$ be a seminormed linear space. Then

Theorem 9. Let $X$ be a real reflexive normed space. Then each non-empty bounded subset of the space $\mathbf{L}_h=\mathbf{L}_h(X)$ admits a Chebyshev centre.