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A minimax inequality and its applications to fixed point theorems in CAT(0) spaces
 S Shabanian^{1} and
 SM Vaezpour^{1}Email author
https://doi.org/10.1186/16871812201161
© Shabanian and Vaezpour; licensee Springer. 2011
 Received: 25 December 2010
 Accepted: 4 October 2011
 Published: 4 October 2011
Abstract
In this paper, a CAT(0) version of famous Fan's minimax inequality is established and as its application, we obtain some fixed point theorems and best approximation theorems in CAT(0) spaces.
2000 Mathematics Subject Classification: 47H10.
Keywords
 CAT(0) space
 minimax inequality
 fixed point
1 Introduction
A metric space is a CAT(0) space if it is geodesically connected and if every geodesic triangle in this space is at least as thin as its comparison triangle in Euclidean plane. CAT(0) spaces play fundamental role in various areas of mathematics [1]. Moreover, there are applications in biology and computer science as well [2, 3].
Fixed point theory in a CAT(0) space was first studied by Kirk [4]. Since then, the fixed point theory for single valued and multivalued mappings in CAT(0) spaces has been developed [5–8].
The famous KnasterKuratowskiMazurkiewicz theorem (in short, KKM theorem) and its generalization have a fundamental importance in modern nonlinear analysis [9, 10]. Recently, Niculescu and Roventa established the KKM mapping principle for CAT(0) spaces [11].
In this paper, a minimax inequality in CAT(0) spaces is established and as its application, some fixed point and best approximation theorems in CAT(0) spaces are proved.
2 Preliminaries
Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ Y (briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊆ ℝ to X such that c(0) = x, c(l) = y and d(c(t), c(t')) = t  t' for all t, t' ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image of c is called a geodesic segment joining x and y. When it is unique, this geodesic is denoted by [x, y].
The metric space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y of X is said to be convex if Y includes every geodesic segment joining any of two its points.
for i, j ∈ {1, 2, 3}.
A geodesic space is called a CAT(0) space if all geodesic triangles satisfy the CAT(0) inequality:
We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.
Lemma 2.1[1]Every CAT(0) space (X, d) is uniquely geodesic, and the balls in (X, d) are convex.
Lemma 2.2[12]Let (X, d) be a CAT(0) space. Then,
1. for each x, y ∈ X such that x ≠ y then d(x, z) + d(z, y) = d(x, y) if and only if z ∈ [x, y],
2. for each x, y ∈ X and t ∈ [0, 1], there exists a unique point z ∈ [x, y] such that d(x, z) = td(x, y) and d(y, z) = (1  t)d(x, y).
and let int(B), ∂B and $\mathcal{F}\left(B\right)$ denote the interior, boundary and the set of all nonempty finite subsets of B.
where F_{0} = F and for n ≥ 1, the set F_{ n } consists of all points in E which lie on geodesics which start and end in F_{n1}. The convex hull of a finite subset is not necessarily closed, but in any locally convex Hausdorff space, if K_{1},..., K_{ n } are compact convex subsets, then the convex hull of their union is compact too [13].
for every nonempty finite set $F\in \mathcal{F}\left(C\right)$.
We show that ϕ is a KKM map. By contradiction, suppose y ∈ co({x_{1},..., x_{ n } }) and y ∉ ∪_{ i } ϕ(x_{ i } ). Therefore, there exist a_{1},..., a_{ n } ∈ ℝ with ${\sum}_{i=1}^{n}{a}_{i}=1$ such that $f\left(y\right)\le {\sum}_{i=1}^{n}{a}_{i}f\left({x}_{i}\right)$. Since f (x_{ i } ) < f (y) for each i = 1,..., n, so we have a contradiction. Thus, ϕ is a KKM map.
Definition 2.2 We say that a CAT(0) space X has the convex hull finite property if the closed convex hull of every nonempty finite family of points of X has the fixed point property.
Example 2.2[14] In a locally compact CAT(0) space, the closed convex hull of each finite family of points has the fixed point property. So, every locally compact CAT(0) space has the convex hull finite property.
The following important result is established in [11].
3 Main results
The following theorem is a direct application of KKM mapping principle.
which implies that y ∈ B(F(y), d(y, F(y))). Clearly, this gets a contradiction.
and the proof is complete. □
Then, F has a fixed point.
Since d(y_{0}, F(y_{0})) ≤ d(x, F(y_{0})), we clearly get a contradiction and this completes the proof. □
Definition 3.1 Let E be a CAT(0) space, and C be a convex subset of E. A function f : C → ℝ is said to be metrically quasiconcave (resp., metrically quasiconvex) if for each λ ∈ ℝ, the set {x ∈ C : f(x) > λ} (resp., {x → C : f(x) < λ}) is convex.
and g(x) = x. It is easy to see that f is metrically quasiconcave and is not quasiconvex, and g is metrically quasiconvex and not metrically quasiconcave.
Lemma 3.1 Let C be a convex subset of a CAT(0) space X, and the function f : C × C → ℝ satisfies the following conditions.
1. for each x ∈ C, the function f(·, x) : C → ℝ is metrically quasiconcave (resp., metrically quasiconvex),
2. there exists γ ∈ ℝ such that f(x, x) ≤ γ (resp., f(x, x) ≥ ) for each x ∈ C.
is a KKM mapping.
where λ > λ_{0}> γ. For each i, we have x_{ i } ∈ B. According to hypothesis 1, B is convex and hence co(A) ⊆ B. So, x_{0} ∈ B, and we have f (x_{0}, x_{0}) > λ_{0}> γ which is a contradiction by (2). Thus, G is a KKM mapping. □
Definition 3.2 Let X, Y be CAT(0) spaces. A map F : X → 2 ^{ Y } is said to be

upper semicontinuous if for each closed set B ⊆ Y , F^{}(B) is closed in X.

lower semicontinuous if for each open set B ⊆ Y , F^{}(B) is open in X.
It is well known that if F(x) is compact for each x ∈ X, then F is upper semicontinuous if and only if for each x ∈ X and ε > 0, there exist δ > 0 such that for each x' ∈ B(x, δ), we have F(x') ⊆ B(F(x), ε).
The following is a CAT(0) version of the Fan's minimax inequality [15].
Theorem 3.3 Suppose C is a compact and convex subset of a complete CAT(0) space E with convex hull finite property and f : C×C → ℝ satisfies the following,
1. for each x ∈ C, the function f(x,·) : C → ℝ is lower semicontinuous (resp., upper semicontinuous),
2. for each y ∈ C, the function f(·, y) : C → ℝ is metrically quasiconcave (resp., metrically quasiconvex),
3. there exists γ ∈ ℝ such that f(x, x) ≤ γ (resp., f(x, x) ≥ γ) for each x ∈ C.
Therefore, there exists a ${y}_{0}\in {\bigcap}_{x\in C}G\left(x\right)$. Thus, f (x, y_{0}) ≤ λ for every x ∈ C.
This completes the proof. □
Definition 3.3 Let X be a CAT(0) space and D ⊆ X. The map G : D → 2 ^{ X } is called quasiconvex if the set G^{}(C) is convex for each convex subset C of Y.
For each y ∈ X, since y ∈ H(y), so H(y) ≠ ∅.
and this proves our claim.
This completes the proof. □
Corollary 3.1 Suppose X is a compact subset of a complete CAT(0) space E with convex hull finite property and G : X → 2 ^{ X } is an onto, quasiconvex and upper semicontinuous map with nonempty compact convex values and f : X → X is a continuous single valued map. Then, there exists x_{0} ∈ X such that f(x_{0}) ∈ G(x_{0}).
Moreover, if x_{0} ∉ G(x_{0}) then x_{0} ∈ ∂X.
which is a contradiction by (2). Therefore, x_{0} ∈ ∂X. □
Corollary 3.4 Suppose X is a compact subset of a complete CAT(0) space E with convex hull finite property and G : X → 2 ^{ X } is an upper semicontinuous map with nonempty compact convex values. If G(x) ∩ X = ∅ for all x ∈ ∂X, then G has a fixed point.
for all x ∈ X. Since x_{0} ∈ ∂X, we have Gx_{0} ∩ X ≠ ∅. This is a contradiction by (3). □
If in Theorem 3.4, G is single valued, then it reduces to the following analog of Fan's best approximation to singlevalued mappings in CAT(0) spaces.
for all x ∈ X.
The following is an analog of Fan's fixed point theorem in CAT(0) spaces [16].
contains at least one point of X, then G has a fixed point.
for all x ∈ X.
We claim that x_{0} is a fixed point of T. On the contrary, assume that x_{0} ≠ Gx_{0}. Then, by assumptions, there exists z ∈ X such that z ∈ (x_{0}, Gx_{0}].
which by (4) it is a contradiction. □
Declarations
Authors’ Affiliations
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