On Best Approximations for Set-Valued Mappings in $\mathcal{G}$-convex Spaces

Abstract

In this paper we obtain a best approximations theorem for multi-valued mappings in $\mathcal{G}$-convex spaces. As applications, we derive results on the best approximations in hyperconvex and normed spaces. The obtain results generalize many existing results in the literature.

1. Introduction and Preliminaries

S. Park and H. Kim [1] introduced the notion of generalized convex space or $\mathcal{G}$-convex space. In $\mathcal{G}$-convex space, many results were obtained in nonlinear analysis, see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. The aim of this paper is to obtain the best approximation theorem in $\mathcal{G}$-convex space. Our result generalize theorems of A. Amini-Harandi and A. P. Farajzadeh [3] (Theorem 2.1), W. A. Kirk, B. Sims and G. X. Z. Yuan [15] (Theorem 3.5), M. A. Khamsi [16], (Theorem 6), S. Park [18] (Theorem 5). Also we obtain that almost quasi-convex and almost affine conditions is unnecessary in results of J. B. Prolla [21] and A. Carbone [7].

A multifunction $\mathsf{\Phi }:X\to Y$ is a map such that $\mathsf{\Phi }\left(x\right)\subseteq Y$ for all $x\in X$. Let $S\subset X$, then $\mathsf{\Phi }\left(S\right)=\cup \left\{\mathsf{\Phi }\right(s):s\in S\}.$ Let $T\subset Y$, denote

$${\mathsf{\Phi }}^{-}\left(T\right)=\{s\in X:\mathsf{\Phi }\left(s\right)\cap T\ne \varnothing \} \mathrm{and} {\mathsf{\Phi }}^{+}\left(T\right)=\{s\in X:\mathsf{\Phi }\left(s\right)\subset T\}$$

as lower and upper inverses of T with respect to $\mathsf{\Phi }$ respectively. A multifunction $\mathsf{\Phi }:X\to Y$ is upper (lower) semi-continuous on X if for every open $U\subset Y$, the set ${\mathsf{\Phi }}^{+}\left(U\right)\left({\mathsf{\Phi }}^{-}\left(U\right)\right)$ is open. A multifunction $\mathsf{\Phi }$ is continuous if it is upper and lower semi-continuous. A multifunction $\mathsf{\Phi }$ with compact values is continuous if $\mathsf{\Phi }$ is a continuous multifunction in the Hausdorff distance.

Denote $Int\left(S\right),Bd\left(S\right)$ and $⟨S⟩$, the interior, boundary and the set of all nonempty finite subsets of S respectively.

Let $r\in {\mathbb{R}}^{+}\cup \left\{0\right\}$ and $\varnothing \ne S\subset X$, we denote the $r-$parallel set of S by

$$S+r=\bigcup \left\{B\right(s,r):s\in S\},$$

where $B(s,r)=\{t\in X:d(s,t)\le r\}.$

For nonempty subsets S and T of X, we define

$$d(S,T)=inf\left\{d\right(s,t):s\in S,t\in T\}.$$

We call a set K is metrically convex if for any $x,y\in K$ and positive numbers $p_i$ and $p_j$ such that $d(x,y)\le p_i+p_j$, there exists $z\in K$ such that $z\in B(x,p_i)\cap B(y,p_j).$

Denote ${\Delta }_n$, the standard $n-$simplex having vertices $e_1,e_2,\dots ,e_{n+1},$ where $e_i$ is the ith unit vector in ${\mathbb{R}}^{n+1}.$ A $\mathcal{G}$-convex space $(X,D;\mathsf{\Omega })$ consists of a topological space X, a nonempty set D and a multifunction $\mathsf{\Omega }:⟨D⟩\to X$ such that for each $S\in ⟨D⟩$ with the cardinality $\left|S\right|=n+1$, there exists a continuous function ${\phi }_S:{\Delta }_n\to \mathsf{\Omega }\left(S\right),$ such that each $J\in ⟨S⟩$ implies ${\phi }_S\left({\Delta }_J\right)\subset \mathsf{\Omega }\left(J\right),$ where ${\Delta }_J$ denotes the faces of ${\Delta }_n$ corresponding to $J\in ⟨S⟩.$ We write $\mathsf{\Omega }\left(S\right)={\mathsf{\Omega }}_S$ for each $S\in ⟨D⟩$. Note that S may or may not be a subset of ${\mathsf{\Omega }}_S$. For $(X,D;\mathsf{\Omega })$ a subset K of X is called $\mathsf{\Omega }-$convex if for each $S\in ⟨D⟩,S\subset K$ implies ${\mathsf{\Omega }}_S\subset K.$ If $D=X$, then $(X,D;\mathsf{\Omega })$ announced as $(X,\mathsf{\Omega }).$ For any $K\subset X$, the $\mathsf{\Omega }-$convex hull of K is denoted and defined by

$$c{o}_{\mathsf{\Omega }}K:=\bigcup \{{\mathsf{\Omega }}_S:S\in ⟨K⟩\}.$$

A multifunction $\mathsf{\Phi }:K\to X$ is a KKM map if ${\mathsf{\Omega }}_S\subset \mathsf{\Phi }\left(S\right)$ for each $S\in ⟨K⟩,$ where K is a $\mathsf{\Omega }-$convex subset of X, see for example [25]. A multifunction $\mathsf{\Phi }:K\to X$ is called generalized KKM map if for each $S\in ⟨K⟩,$ there exists a function $\varrho :S\to X$ such that ${\mathsf{\Omega }}_{\varrho \left(T\right)}\subset \mathsf{\Phi }\left(T\right)$ for each $T\in ⟨S⟩.$

H. Kim and S. Park in [14] (Theorem 3), obtained an extension of KKM theorem of Ky Fan, see [12] (Lemma 1) and [13] (Theorem 4).

Theorem 1.

Let $(X,D;\mathsf{\Omega })$ be a $\mathcal{G}$-convex space, S a nonempty set and $\mathsf{\Phi }:S\to X$ a multifunction with closed (resp. open) values. If Φ is a generalized KKM map, then the class of its values has the finite intersection property (More precisely, for each $T\in ⟨S⟩,$ there exists and $T^{\prime }\in ⟨D⟩$ such that ${\mathsf{\Omega }}_{T^{\prime }}\cap {\bigcap }_{t\in T}\mathsf{\Phi }\left(t\right)\ne \varnothing .$

In this paper we use the following Corollary of Theorem 1.

Theorem 2.

Let $(X,\mathsf{\Omega })$ be a $\mathcal{G}$-convex space, S a nonempty set and $\mathsf{\Phi }:S\to X$ a generalized KKM map with closed values. If there exists a nonempty compact subset L of X such that ${\bigcap }_{t\in T}\mathsf{\Phi }\left(t\right)\subset L$ for some $T\in ⟨S⟩$ then ${\bigcap }_{s\in S}\mathsf{\Phi }\left(s\right)\ne \varnothing .$

2. Main Results

In this section, by using Theorem 2, we prove a new best approximation theorem in $\mathcal{G}$-convex spaces.

Theorem 3.

Let $\mathsf{\Phi }:S\to X$ be a continuous multi map with compact values such that

$$\mathsf{\Phi }\left(x\right)+ris\mathsf{\Omega }-convex for all x\in S,r\ge 0$$

(1)

and $g:S\to S$ is a continuous onto map, where $(X,\mathsf{\Omega })$ a $\mathcal{G}$-convex space with metric d and S a nonempty Ω-convex subset of X. If there exists a nonempty compact subset K of X such that

$$\bigcap _{y\in M}\{x\in S:d(g\left(x\right),\mathsf{\Phi }\left(x\right))\le d(g\left(y\right),\mathsf{\Phi }\left(x\right))\}\subset K for some M\in ⟨S⟩,$$

(2)

then there exists ${\upsilon }_0\in S$ such that

$$d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right))=\underset{x\in S}{inf}d(x,\mathsf{\Phi }\left({\upsilon }_0\right)).$$

If S is metrically convex and $g\left({\upsilon }_0\right)\notin \mathsf{\Phi }\left({\upsilon }_0\right)$, then ${\upsilon }_0\in Bd\left(S\right).$

Proof. 

Define the multimaps $H,T:S\to S$ by

$$H\left(x\right)=\left\{y\in S:d\left(g\right(y),\mathsf{\Phi }(y\left)\right)\le d\left(g\right(x),\mathsf{\Phi }(y\left)\right)\right\},$$

$$T\left(x\right)=(g\circ H)\left(x\right).$$

We have that $T\left(x\right)$ is nonempty for each $x\in S,$ because $g\left(x\right)\in T\left(x\right)$ for each $x\in S.$ We prove that T is generalized KKM map. Suppose that there exists $\{x_1,\dots ,x_n\}\in ⟨K⟩$ such that $g^{-1}\left({\mathsf{\Omega }}_{\{g\left(x_1\right),\dots ,g\left(x_n\right)\}}\right)$ is not a subset of $H\left(\{x_1,\dots ,x_n\}\right).$ Then there exists $z\in g^{-1}\left({\mathsf{\Omega }}_{\{g\left(x_1\right),\dots ,g\left(x_n\right)\}}\right)$ such that $z\notin H\left(x_k\right)$ for every $k\in \{1,\dots ,n\}.$ So, we have

$$d(g\left(z\right),\mathsf{\Phi }\left(z\right))>d(g\left(x_k\right),\mathsf{\Phi }\left(z\right)) \mathrm{for} \mathrm{every} k\in \{1,\dots ,n\}.$$

Let

$$r=\underset{1\le k\le n}{max}\left\{d(g\left(x_k\right),\mathsf{\Phi }\left(z\right))\right\},$$

we have

$$g\left(x_k\right)\in \mathsf{\Phi }\left(z\right)+r \mathrm{for} \mathrm{all} k\in \{1,\dots ,n\}.$$

This implies that

$${\mathsf{\Omega }}_{\{g\left(x_1\right),\dots ,g\left(x_n\right)\}}\subset c{o}_{\mathsf{\Omega }}(\mathsf{\Phi }\left(z\right)+r).$$

Since

$$z\in g^{-1}\left({\mathsf{\Omega }}_{\{g\left(x_1\right),\dots ,g\left(x_n\right)\}}\right),$$

we have

$$g\left(z\right)\in {\mathsf{\Omega }}_{\{g\left(x_1\right),\dots ,g\left(x_n\right)\}},$$

so,

$$g\left(z\right)\in c{o}_{\mathsf{\Omega }}(\mathsf{\Phi }\left(z\right)+r).$$

From condition (1) we obtain

$$g\left(z\right)\in \mathsf{\Phi }\left(z\right)+r.$$

So, exists $u\in \mathsf{\Phi }\left(z\right)$ such that

$$d\left(g\right(z),u)\le r,$$

that is why

$$d\left(g\right(z),\mathsf{\Phi }(z\left)\right)\le d\left(g\right(z),u)\le r<d\left(g\right(z),\mathsf{\Phi }(z\left)\right).$$

This is a contradiction. Therefore, for each $D\in ⟨S⟩$ we have

$$g^{-1}\left({\mathsf{\Omega }}_{g\left(D\right)}\right)\subseteq H\left(D\right).$$

Since g is onto map we have that

$${\mathsf{\Omega }}_{g\left(D\right)}\subseteq T\left(D\right) \mathrm{for} \mathrm{each} D\in ⟨S⟩.$$

This implies that T is a generalized KKM map. Since maps $\mathsf{\Phi }$ and g are continuous we get that $T\left(x\right)$ is closed for each $x\in S$. Hence, by condition (2) and Theorem 2, there exists ${\upsilon }_0\in S$ such that

$$d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right))=\underset{x\in S}{inf}d(x,\mathsf{\Phi }\left({\upsilon }_0\right)).$$

If S is metrically convex and $g\left({\upsilon }_0\right)\notin \mathsf{\Phi }\left({\upsilon }_0\right)$ then ${\upsilon }_0\in Bd\left(S\right).$ Namely, if ${\upsilon }_0\in Int\left(S\right)$, then there exists $\gamma >0$ such that

$$B_S({\upsilon }_0,\gamma )=\{x\in S:d({\upsilon }_0,x)<\gamma \}\subset S$$

and

$$\gamma <d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right))\le d(x,\mathsf{\Phi }\left({\upsilon }_0\right)) \mathrm{for} \mathrm{all} x\in B_S(g\left({\upsilon }_0\right),\gamma ).$$

Let $u_0\in \mathsf{\Phi }\left({\upsilon }_0\right)$ such that $d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right))=d(g\left({\upsilon }_0\right),u_0).$ Then, if S is metrically convex, we obtain

$$B_S(g\left({\upsilon }_0\right),\gamma )\cap B_S(u_0,d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right))-\gamma )\ne \varnothing .$$

Since

$$B_S(u_0,d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right)-\gamma )\subset \mathsf{\Phi }\left({\upsilon }_0\right)+d(\mathsf{\Phi }\left({\upsilon }_0\right),g\left({\upsilon }_0\right))-\gamma ,$$

we have

$$B_S(g\left({\upsilon }_0\right),\gamma )\cap (\mathsf{\Phi }\left({\upsilon }_0\right)+d(\mathsf{\Phi }\left({\upsilon }_0\right),g\left({\upsilon }_0\right))-\gamma )\ne \varnothing .$$

Let $z\in S$ such that

$$z\in B_S(g\left({\upsilon }_0\right),\gamma )\cap (\mathsf{\Phi }\left({\upsilon }_0\right)+d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right)-\gamma ),$$

we obtain

$$d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right))\le d(z,\mathsf{\Phi }\left({\upsilon }_0\right))\le d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right))-\gamma <d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right)),$$

a contradiction. Therefore, ${\upsilon }_0\in Bd\left(S\right).$  □

Next results follows from Theorem 3.

Corollary 1.

Let $\mathsf{\Phi }:S\to X$ be a continuous multi map with compact values such that condition (1) is satisfied and $g:S\to S$ is a continuous onto map, where $(X,\mathsf{\Omega })$ a $\mathcal{G}$-convex space with metric d and S a nonempty Ω-convex set contained in compact subset of X. Then there exists ${\upsilon }_0\in K$ such that

$$d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right))=\underset{x\in S}{inf}d(x,\mathsf{\Phi }\left({\upsilon }_0\right)).$$

If K is metrically convex and $g\left({\upsilon }_0\right)\notin \mathsf{\Phi }\left({\upsilon }_0\right)$, then ${\upsilon }_0\in Bd\left(S\right).$

Corollary 2.

Let the metric space $(X,\mathsf{\Omega })$ be a $\mathcal{G}$-convex space with metric d, S a nonempty $\mathsf{\Omega }-$convex set contained in compact subset of X, $\mathsf{\Phi }:S\to X$ is a continuous multimap with compact values such that condition (1) is satisfied. Then there exists ${\upsilon }_0\in K$ such that

$$d({\upsilon }_0,\mathsf{\Phi }\left({\upsilon }_0\right))=\underset{x\in S}{inf}d(x,\mathsf{\Phi }\left({\upsilon }_0\right)).$$

If K is metrically convex and ${\upsilon }_0\notin \mathsf{\Phi }\left({\upsilon }_0\right)$ then ${\upsilon }_0\in Bd\left(S\right).$

3. Some Applications

As some applications of our results, we give the versions of Fan’ best approximation theorem in hyperconvex and normed spaces.

Recall that a metric space $(X,d)$ is called a hyperconvex metric space if for any class of elements $x_i$ of X and any class $p_i\in {\mathbb{R}}^{+}\cup \left\{0\right\}$ with $d(x_i,x_j)\le p_i+p_j,$ we have

$$\bigcap _i\mathcal{B}(x_i,p_i)\ne \varnothing .$$

Let $\mathcal{U}$ be a nonempty bounded subset of a hyperconvex metric space X, denote

$$co\mathcal{U}=\bigcap \{\mathcal{V}:\mathcal{V} \mathrm{is} \mathrm{closed} \mathrm{ball} \mathrm{in} X \mathrm{containing} \mathcal{U}\}.$$

Denote $\mathcal{U}\left(X\right)=\{\mathcal{U}\subset X:\mathcal{U}=co\mathcal{U}\},$ the elements of this set are known as admissible subset of X. Moreover, any hyperconvex metric space $(X,d)$ is an $\mathcal{G}$-convex space $(X,\mathsf{\Omega }),$ with ${\mathsf{\Omega }}_{\mathcal{U}}=co\mathcal{U}$ for each $\mathcal{U}\in ⟨X⟩.$ The r-parallel set of an admissible subset of a hyperconvex metric space is also an admissible set, see R. Espínola and M. A. Khamsi [11] (Lemma 4. 10). In this case the condition (1) is satisfied.

Following Corollary 1, we obtained best approximation result for hyperconvex metric spaces due to A. Amini-Harandi and A. P. Farajzadeh [3] (Theorem 2.1).

Corollary 3.

Let $(X,d)$ be hyperconvex metric space and S be a compact admissible subset of X. Suppose that $\mathsf{\Phi }:S\to \mathcal{U}\left(X\right)$ continuous multimmap with compact values and $g:S\to S$ is a continuous onto map. Then there exists ${\upsilon }_0\in K$ such that

$$d(g\left({\upsilon }_0\right),\mathsf{\Phi }\left({\upsilon }_0\right))=\underset{x\in S}{inf}d(g\left(x\right),\mathsf{\Phi }\left({\upsilon }_0\right)).$$

Moreover, if $g\left({\upsilon }_0\right)\notin \mathsf{\Phi }\left({\upsilon }_0\right)$ then ${\upsilon }_0\in Bd\left(S\right).$

In view of Corollary 2, the result of G. X. Z. Yuan, [25] (Theorem 2. 11. 16) and for single-valued maps, the result of M. A Khamsi, [16] (Lemma) are obtain as follows:

Corollary 4.

Let $\mathsf{\Phi }:K\to X$ be a continuous multimap on a nonempty admissible compact set K to hyperconvex metric space X. Then there exists an element ${\upsilon }_0$ in K such that

$$d({\upsilon }_0,\mathsf{\Phi }\left({\upsilon }_0\right))=\underset{x\in K}{inf}d(x,\mathsf{\Phi }\left({\upsilon }_0\right)).$$

Corollary 5.

Let $\varphi :K\to X$ be a continuous map on a nonempty admissible compact set K to hyperconvex metric space X. Then there exists an element ${\upsilon }_0$ in K such that

$$d({\upsilon }_0,\varphi \left({\upsilon }_0\right))=\underset{x\in K}{inf}d(x,\varphi \left({\upsilon }_0\right)).$$

If X is a normed linear space, then condition (1) in Theorem 3 is satisfied. So, from Theorem 3 we obtain the next result for normed linear spaces.

Theorem 4.

Let X be a normed linear space, S a nonempty convex set contained in compact subset of X, $\mathsf{\Phi }:S\to X$ is a continuous multimap with convex compact values and $g:S\to S$ is a continuous onto map. Then there exists ${\upsilon }_0\in S$ such that

$$\left|\right|g\left({\upsilon }_0\right)-\mathsf{\Phi }\left({\upsilon }_0\right)\left|\right|=\underset{x\in S}{inf}d(x,\mathsf{\Phi }\left({\upsilon }_0\right)).$$

J. B. Prolla [21] and A. Carbone [7] obtained a form of Theorem 4 using almost affine and almost quasi-convex maps in normed vector spaces.

Definition 1.

Let S a nonempty convex subset of a normed space X. A map $g:S\to X$ is

(i) 

almost affine if for all $x,y\in S$ and $u\in S$

$$\left|\right|g(\lambda x+(1-\lambda \left)y\right)-u\left|\right|\le \lambda \left|\right|g\left(x\right)-u\left|\right|+(1-\lambda )\left|\right|g\left(y\right)-u\left|\right|,$$

for each λ with $0<\lambda <1.$

(ii) 

almost quasi-convex if for all $u\in S$ and $r>0,$ the set

$$\{x\in K:|\left|g\right(x)-u||<r\} is convex.$$

Note that the mapping to be an almost quasi-convex is unnecessary in Theorem 4.

Corollary 6.

Let X be a normed linear space, S a nonempty convex compact subset of X, $\varphi :S\to X$ is a continuous map and $g:S\to S$ is a continuous, almost affine, onto map. Then there exists ${\upsilon }_0\in S$ such that

$$\left|\right|g\left({\upsilon }_0\right)-\varphi \left({\upsilon }_0\right)\left|\right|=\underset{x\in S}{inf}d(x,\varphi \left({\upsilon }_0\right)).$$

Corollary 7.

Let X be a normed linear space, S a nonempty convex compact subset of X, $\varphi :S\to X$ is a continuous map and $g:S\to S$ is a continuous, almost quasi-convex, onto map. Then there exists ${\upsilon }_0\in S$ such that

$$\left|\right|g\left({\upsilon }_0\right)-\varphi \left({\upsilon }_0\right)\left|\right|=\underset{x\in S}{inf}d(x,\varphi \left({\upsilon }_0\right)).$$

Example 1.

Let $X=\mathbb{R},S=[0,1]$ and define maps $\varphi :S\to S$ and $g:S\to S$ by

$$\varphi \left(x\right)=x,g\left(x\right)=4x-4x^2.$$

Then map g is not almost quasi-convex and results of J. B. Prolla [21] and A. Carbone [7] are not applicable. Note that the maps ϕ and g satisfy all hypotheses of Theorem 4 and ${\upsilon }_0\in \{0,\frac{3}{4}\}.$