Monotone Relations in Hadamard Spaces

TL;DR

This paper introduces the concept of $\

Contribution

It defines the $\

Findings

Characterizes flat Hadamard spaces via the $\

paper_type":"theoretical"}}}# Answer {

contribution

Abstract

In this paper, the notion of $\mathcal{W}$-property for subsets of $X\times X^{\lozenge}$ is introduced and investigated, where $X$ is an Hadamard space and $X^{\lozenge}$ is its linear dual space. It is shown that an Hadamard space $X$ is flat if and only if $X\times X^{\lozenge}$ has $\mathcal{W}$-property. Moreover, the notion of monotone relation from an Hadamard space to its linear dual space is introduced. Finally, a characterization result for monotone relations with $\mathcal{W}$-property (and hence in flat Hadamard spaces) is proved.

Equations

$$\displaystyle\scalebox{1.5}{$\oplus$}^{n}_{i=1}\lambda_{i}v_{i}:=(1-\lambda_{n})\bigg{(}\frac{\lambda_{1}}{1-\lambda_{n}}v_{1}\oplus\frac{\lambda_{2}}{1-\lambda_{n}}v_{2}\oplus\cdots\oplus\frac{\lambda_{n-1}}{1-\lambda_{n}}v_{n}\bigg{)}\oplus\lambda_{n}v_{n}.$$

$$d(z,(1-\lambda )x\oplus \lambda y{)}^2\le (1-\lambda )d(z,x{)}^2+\lambda d(z,y{)}^2-\lambda (1-\lambda )d(x,y{)}^2\text{for all}x,y,z\in X,\lambda \in [0,1].$$

$$⟨\cdot ,\cdot ⟩$$

$$⟨\overrightarrow{ab},\overrightarrow{cd}⟩$$

$$⟨\overrightarrow{ab},\overrightarrow{cd}⟩={\phi }_{\overrightarrow{cd}}(b)-{\phi }_{\overrightarrow{cd}}(a)={\phi }_{\overrightarrow{ab}}(d)-{\phi }_{\overrightarrow{ab}}(c).$$

$$⟨\overrightarrow{ab},\overrightarrow{cd}⟩\le d(a,b)d(c,d)\text{for all}a,b,c,d\in X.$$

$$\Psi$$

$$(t,a,b)\mapsto \Psi (t,a,b)x=t⟨\overrightarrow{ab},\overrightarrow{ax}⟩;a,b,x\in X,t\in \mathbb{R},$$

$$L(\Psi (t,a,b))=|t|d(a,b),\text{for all}a,b\in X,\text{and all}t\in \mathbb{R},$$

$$L(\varphi)=\sup\bigg{\{}\frac{\varphi(x)-\varphi(y)}{d(x,y)}:x,y\in X,x\neq y\bigg{\}}.$$

$$D((t,a,b),(s,c,d))=L(\Psi (t,a,b)-\Psi (s,c,d));a,b,c,d\in X,t,s\in \mathbb{R}.$$

$$[t\overrightarrow{ab}]=\{s\overrightarrow{cd}:s\in \mathbb{R},c,d\in X,D((t,a,b),(s,c,d))=0\}.$$

$$⟨x^{\ast },\overrightarrow{xy}⟩=t⟨\overrightarrow{ab},\overrightarrow{xy}⟩,\text{where}{x}^{\ast }=[t\overrightarrow{ab}]\in X^{\ast }\text{and}\overrightarrow{xy}\in X^2.$$

$$\Big{\langle}\sum_{i=1}^{n}\alpha_{i}x_{i}^{*},\overrightarrow{xy}\Big{\rangle}:=\sum_{i=1}^{n}\alpha_{i}\langle x_{i}^{*},\overrightarrow{xy}\rangle,~{}\alpha_{i}\in\mathbb{R},\,x_{i}^{*}\in X^{*},\,n\in\mathbb{N},\,x,y\in X.$$

$$X^{\scalebox{0.55}{$\lozenge$}}=\bigg{\{}\sum_{i=1}^{n}\alpha_{i}x_{i}^{*}:\alpha_{i}\in\mathbb{R},\,x_{i}^{*}\in X^{*},\,n\in\mathbb{N}\bigg{\}}.$$

$$\displaystyle\|x^{\scalebox{0.5}{$\lozenge$}}\|_{{\scalebox{0.5}{$\lozenge$}}}:=\sup\Bigg{\{}\frac{\big{|}\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{ab}\rangle-\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{cd}\rangle\big{|}}{d(a,b)+d(c,d)}:a,b,c,d\in X,(a,c)\neq(b,d)\Bigg{\}},$$

$$\mathrm{Dom}(M):=\big{\{}x\in X:\exists\,x^{\scalebox{0.5}{$\lozenge$}}\in X^{\scalebox{0.55}{$\lozenge$}}~{}\text{such that}~{}(x,x^{\scalebox{0.5}{$\lozenge$}})\in M\big{\}},$$

$$\mathrm{Range}(M):=\big{\{}x^{\scalebox{0.5}{$\lozenge$}}\in X^{\scalebox{0.55}{$\lozenge$}}:\exists\,x\in X~{}\text{such that}~{}(x,x^{\scalebox{0.5}{$\lozenge$}})\in M\big{\}}.$$

$$\big{\langle}x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{p((1-\lambda)x_{1}\oplus\lambda x_{2})}\big{\rangle}\leq(1-\lambda)\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{px_{1}}\rangle+\lambda\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{px_{2}}\rangle,~{}\forall\lambda\in[0,1],\,\forall x^{\scalebox{0.5}{$\lozenge$}}\in\mathrm{Range}(M),\,\forall x_{1},x_{2}\in\mathrm{Dom}(M).$$

$$\displaystyle\big{\langle}x^{\scalebox{0.55}{$\lozenge$}},\overrightarrow{q(\oplus_{i=1}^{n}\lambda_{i}x_{i})}\big{\rangle}\leq\sum_{i=1}^{n}\lambda_{i}\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qx_{i}}\rangle,~{}\text{for all}~{}x^{\scalebox{0.5}{$\lozenge$}}\in\mathrm{Range}(M),\{x_{i}\}_{i=1}^{n}\subseteq\mathrm{Dom}(M),\{\lambda_{i}\}_{i=1}^{n}\subseteq[0,1].$$

$$\displaystyle\big{\langle}x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{q((1-\lambda)x_{1}\oplus\lambda x_{2})}\big{\rangle}$$

$$\displaystyle=\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qp}\rangle+\big{\langle}x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{p((1-\lambda)x_{1}\oplus\lambda x_{2})}\big{\rangle}$$

$$\displaystyle\leq(1-\lambda)(\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qp}\rangle+\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{px_{1}}\rangle)+\lambda(\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qp}\rangle+\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{px_{2}}\rangle)$$

$$\displaystyle=(1-\lambda)\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qp}+\overrightarrow{px_{1}}\rangle+\lambda\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qp}+\overrightarrow{px_{2}}\rangle$$

$$\displaystyle=(1-\lambda)\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qx_{1}}\rangle+\lambda\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qx_{2}}\rangle,$$

$$\displaystyle\big{\langle}x^{\scalebox{0.55}{$\lozenge$}},\overrightarrow{q(\oplus_{i=1}^{n}\lambda_{i}x_{i})}\big{\rangle}$$

$$\displaystyle\leq(1-\lambda_{n})\Big{\langle}x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{q\Big{(}\frac{\lambda_{1}}{1-\lambda_{n}}x_{1}\oplus\frac{\lambda_{2}}{1-\lambda_{n}}x_{2}\oplus\cdots\oplus\frac{\lambda_{n-1}}{1-\lambda_{n}}x_{n-1}}\Big{)}\Big{\rangle}+\lambda_{n}\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qx_{n}}\rangle$$

$$\displaystyle\leq(1-\lambda_{n})\sum_{i=1}^{n}\frac{\lambda_{i}}{1-\lambda_{n}}\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qx_{i}}\rangle+\lambda_{n}\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qx_{n}}\rangle$$

$$\displaystyle=\sum_{i=1}^{n-1}\lambda_{i}\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qx_{i}}\rangle+\lambda_{n}\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qx_{n}}\rangle$$

$$\displaystyle=\sum_{i=1}^{n}\lambda_{i}\langle x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{qx_{i}}\rangle.$$

$$d(z,(1-\lambda )x\oplus \lambda y{)}^2=(1-\lambda )d(z,x{)}^2+\lambda d(z,y{)}^2-\lambda (1-\lambda )d(x,y{)}^2,\text{for all}z\in X.$$

$${\phi }_{\overrightarrow{pz}}((1-\lambda )x\oplus \lambda y)=(1-\lambda ){\phi }_{\overrightarrow{pz}}(x)+\lambda {\phi }_{\overrightarrow{pz}}(y),\forall x,y\in X,\forall \lambda \in [0,1].$$

$$\displaystyle\big{\langle}x^{\scalebox{0.5}{$\lozenge$}},\overrightarrow{p\big{(}(1-\lambda)x\oplus\lambda y\big{)}}\big{\rangle}$$

$$\displaystyle=\sum^{n}_{i=1}\alpha_{i}t_{i}\big{(}\big{\langle}\overrightarrow{a_{i}b_{i}},\overrightarrow{px}\big{\rangle}+\big{\langle}\overrightarrow{a_{i}b_{i}},\overrightarrow{x\big{(}(1-\lambda)x\oplus\lambda y\big{)}}\big{\rangle}\big{)}$$

$$\displaystyle=\sum^{n}_{i=1}\alpha_{i}t_{i}\big{(}\big{\langle}\overrightarrow{a_{i}b_{i}},\overrightarrow{px}\big{\rangle}+\lambda\big{\langle}\overrightarrow{a_{i}b_{i}},\overrightarrow{xy}\big{\rangle}\big{)}$$

$$\displaystyle=\sum^{n}_{i=1}\alpha_{i}t_{i}\big{(}\big{\langle}\overrightarrow{a_{i}b_{i}},\overrightarrow{px}\big{\rangle}+\lambda\big{\langle}\overrightarrow{a_{i}b_{i}},\overrightarrow{py}-\overrightarrow{px}\big{\rangle}\big{)}$$

$$\displaystyle=\sum^{n}_{i=1}\alpha_{i}t_{i}\big{(}(1-\lambda)\big{\langle}\overrightarrow{a_{i}b_{i}},\overrightarrow{px}\big{\rangle}+\lambda\big{\langle}\overrightarrow{a_{i}b_{i}},\overrightarrow{py}\big{\rangle}\big{)}$$

Full text

1 Introduction and Preliminaries

Let $(X,d)$ be a metric space. We say that a mapping $c:[0,1]\rightarrow X$ is a geodesic path from $x\in X$ to $y\in X$ if $c(0)=x$, $c(1)=y$ and $d(c(t),c(s))=|t-s|d(x,y),$ for each $t,s\in[0,1]$. The image of $c$ is said to be a geodesic segment joining $x$ and $y$. A metric space $(X,d)$ is called a geodesic space if there is a geodesic path between every two points of $X$. Also, a geodesic space $X$ is called uniquely geodesic space if for each $x,y\in X$ there exists a unique geodesic path from $x$ to $y$. From now on, in a uniquely geodesic space, we denote the set $c([0,1])$ by $[x,y]$ and for each $z\in[x,y]$, we write $z=(1-t)x\oplus ty$, where $t\in[0,1]$. In this case, we say that $z$ is a convex combination of $x$ and $y$. Hence, $[x,y]=\{(1-t)x\oplus ty:t\in[0,1]\}$. More details can be found in Bacak2014 ; BridsonHaefliger .

Definition 1.1

(DhompongsaKaewkhaoPanyanak2012, , Definition 2.2)* *Let $(X,d)$ be a geodesic space, $v_{1},v_{2},v_{3},\ldots,v_{n}$ be $n$ points in $X$ and $\{\lambda_{1},\lambda_{2},\lambda_{3},\ldots,\lambda_{n}\}\subseteq(0,1)$ be such that $\sum^{n}_{i=1}\lambda_{i}=1$. We define convex combination of $\{v_{1},v_{2},v_{3},\ldots,v_{n}\}$ inductively as following:

[TABLE]

Note that for every $x\in X$, we have d\big{(}x,\scalebox{1.5}{\oplus}^{n}_{i=1}\lambda_{i}v_{i}\big{)}\leq\sum_{i=1}^{n}\lambda_{i}d(x,v_{i}).

According to (Bacak2014, , Definition 1.2.1), a geodesic space $(X,d)$ is a $\mathrm{CAT}(0)$* space*, if the following condition, so-called CN-inequality, holds:

[TABLE]

One can show that (for instance see (Bacak2014, , Theorem 1.3.3)) $\mathrm{CAT}(0)$ spaces are uniquely geodesic spaces. An Hadamard space is a complete $\mathrm{CAT}(0)$ space.

Let $X$ be an Hadamard space. For each $x,y\in X$, the ordered pair $(x,y)$ is called a bound vector and is denoted by $\overrightarrow{xy}$. Indeed, $X^{2}=\{\overrightarrow{xy}:x,y\in X\}$. For each $x\in X$, we apply $\mathbf{0}_{x}:=\overrightarrow{xx}$ as zero bound vector at $x$ and $-\overrightarrow{xy}$ as the bound vector $\overrightarrow{yx}$. The bound vectors $\overrightarrow{xy}$ and $\overrightarrow{uz}$ are called admissible if $y=u$. Therefore the sum of two admissible bound vectors $\overrightarrow{xy}$ and $\overrightarrow{yz}$ is defined by $\overrightarrow{xy}+\overrightarrow{yz}=\overrightarrow{xz}$. Ahmadi Kakavandi and Amini in KakavandiAmini have introduced the dual space of an Hadamard space, by using the concept of quasilinearization of abstract metric spaces presented by Berg and Nikolaev in BergNikolaev . The quasilinearization map is defined as following:

[TABLE]

Let $x,\,y\in X$, we define the mapping $\varphi_{\overrightarrow{xy}}:X\rightarrow\mathbb{R}$ by $\varphi_{\overrightarrow{xy}}(z)=\frac{1}{2}(d(x,z)^{2}-d(y,z)^{2})$; for each $z\in X$. We will see that $\varphi_{\overrightarrow{xy}}$ possess attractive properties that simplify some calculations. We observe that (3) can be rewritten as following:

[TABLE]

The metric space $(X,d)$ satisfies the Cauchy-Schwarz inequality if

[TABLE]

This inequality characterizes $\mathrm{CAT}(0)$ spaces. Indeed, it follows from (BergNikolaev, , Corollary 3) that a geodesic space $(X,d)$ is a $\mathrm{CAT}(0)$ space if and only if it satisfies in the Cauchy-Schwarz inequality. For an Hadamard space $(X,d)$, consider the mapping

[TABLE]

where $C(X,\mathbb{R})$ denotes the space of all continuous real-valued functions on $\mathbb{R}\times X^{2}$. It follows from Cauchy-Schwarz inequality that $\Psi(t,a,b)$ is a Lipschitz function with Lipschitz semi-norm

[TABLE]

where the Lipschitz semi-norm for any function $\varphi:(X,d)\rightarrow\mathbb{R}$ is defined by

[TABLE]

A pseudometric $D$ on $\mathbb{R}\times X^{2}$ induced by the Lipschitz semi-norm (4), is defined by

[TABLE]

For an Hadamard space $(X,d)$, the pseudometric space $(\mathbb{R}\times X^{2},D)$ can be considered as a subspace of the pseudometric space of all real-valued Lipschitz functions Lip$(X,\mathbb{R})$. Note that, in view of (KakavandiAmini, , Lemma 2.1), $D((t,a,b),(s,c,d))=0~{}\text{if and only if}~{}t\langle\overrightarrow{ab},\overrightarrow{xy}\rangle=s\langle\overrightarrow{cd},\overrightarrow{xy}\rangle~{}\text{for all}~{}x,y\in X$. Thus, $D$ induces an equivalence relation on $\mathbb{R}\times X^{2}$, where the equivalence class of $(t,a,b)\in\mathbb{R}\times X^{2}$ is

[TABLE]

The dual space of an Hadamard space $(X,d)$, denoted by $X^{*}$, is the set of all equivalence classes $[t\overrightarrow{ab}]$ where $(t,a,b)\in\mathbb{R}\times X^{2}$, with the metric $D([t\overrightarrow{ab}],[s\overrightarrow{cd}]):=D((t,a,b),(s,c,d))$. Clearly, the definition of equivalence classes implies that $[\overrightarrow{aa}]=[\overrightarrow{bb}]$ for all $a,b\in X$. The zero element of $X^{*}$ is $\mathbf{0}:=[t\overrightarrow{aa}]$, where $a\in X$ and $t\in\mathbb{R}$ are arbitrary. It is easy to see that the evaluation $\langle\mathbf{0},\cdot\rangle$ vanishes for any bound vectors in $X^{2}$. Note that in general $X^{*}$ acts on $X^{2}$ by

[TABLE]

The following notation will be used throughout this paper.

[TABLE]

For an Hadamard space $(X,d)$, Chaipunya and Kumam in (ChaipunyaKumam, ), defined the linear dual space of $X$ by

[TABLE]

Therefore, X^{\scalebox{0.55}{\lozenge}}={\mathrm{span}}\,X^{*}. It is easy to see that X^{\scalebox{0.55}{\lozenge}} is a normed space with the norm \|x^{\scalebox{0.5}{\lozenge}}\|_{{\scalebox{0.5}{\lozenge}}}=L(x^{\scalebox{0.5}{\lozenge}}) for all x^{\scalebox{0.5}{\lozenge}}\in X^{\scalebox{0.55}{\lozenge}}. Indeed:

Lemma 1.2

(ZamaniRaeisi, , Proposition 3.5)* Let $X$ be an Hadamard space with linear dual space X^{\scalebox{0.55}{\lozenge}}. Then*

[TABLE]

is a norm on X^{\scalebox{0.55}{\lozenge}}. In particular, \|[t\overrightarrow{ab}]\|_{{\scalebox{0.5}{\lozenge}}}=|t|d(a,b).

2 Flat Hadamard Spaces and $\mathcal{W}$-property

Let $M$ be a relation from $X$ to X^{\scalebox{0.55}{\lozenge}}; i.e., M\subseteq X\times X^{\scalebox{0.55}{\lozenge}}. The domain and range of $M$ are defined, respectively, by

[TABLE]

and

[TABLE]

Definition 2.1

Let $X$ be an Hadamard space with linear dual space X^{\scalebox{0.55}{\lozenge}}. We say that M\subseteq X\times X^{\scalebox{0.55}{\lozenge}} satisfies* $\mathcal{W}$-property *if there exists $p\in X$ such that the following holds: **

[TABLE]

Proposition 2.2

Let $X$ be an Hadamard space with linear dual space X^{\scalebox{0.55}{\lozenge}} and let M\subseteq X\times X^{\scalebox{0.55}{\lozenge}}. Then the following statements are equivalent:

(i)

M\subseteq X\times X^{\scalebox{0.55}{\lozenge}}* satisfies the $\mathcal{W}$-property for some $p\in X$.*

(ii)

M\subseteq X\times X^{\scalebox{0.55}{\lozenge}}* satisfies the $\mathcal{W}$-property for any $q\in X$.*

(iii)

For any $q\in X$,

[TABLE]

(iv)

For some $p\in X$, $(\mathcal{W}_{n}(p))$ holds.

Proof 2.3**.**

(i) $\Rightarrow$ (ii):

Let $q\in X$ be any arbitrary element of $X$, $\lambda\in[0,1]$,  x^{\scalebox{0.5}{\lozenge}}\in\mathrm{Range}(M), and $x_{1},x_{2}\in\mathrm{Dom}(M)$. Then

[TABLE]

as required.

(ii) $\Rightarrow$ (iii):

We proceed by induction on $n$. By Definition 2.1 the claim is true for $n=2$. Now assume that ($\mathcal{W}_{n-1}(q)$) is true. In view of equation (1),

[TABLE]

(iii) $\Rightarrow$ (iv):

Clear.

(iv) $\Rightarrow$ (i):

Take $n=2$ in $(\mathcal{W}_{n}(p))$.

We are done.

Remark 2.4**.**

It should be noticed that Proposition 2.2 implies that $\mathcal{W}$-property is independent of the choice of the element $p\in X$.

Definition 2.5**.**

(MovahediBehmardiSoleimani-Damaneh, , Definition 3.1) An Hadamard space $(X,d)$ is said to be flat if equality holds in the CN-inequality, i.e., for each $x,y\in X$ and $\lambda\in[0,1]$, the following holds:

[TABLE]

Proposition 2.6**.**

Let $X$ be an Hadamard space. The following statements are equivalent:

(i)

$X$* is a flat Hadamard space.*

(ii)

$\langle\overrightarrow{x((1-\lambda)x\oplus\lambda y)},\overrightarrow{ab}\rangle=\lambda\langle\overrightarrow{xy},\overrightarrow{ab}\rangle$, for all $a,b,x,y\in X$ and all $\lambda\in[0,1]$.

(iii)

X\times X^{\scalebox{0.55}{\lozenge}}* has $\mathcal{W}$-property.*

(iv)

Any subset of X\times X^{\scalebox{0.55}{\lozenge}} has $\mathcal{W}$-property.

(v)

For each $p,z\in X$, the mapping $\varphi_{\overrightarrow{pz}}$ is convex.

(vi)

For each $p,z\in X$, the mapping $\varphi_{\overrightarrow{pz}}$ is affine, in the sense that:

[TABLE]

Proof 2.7**.**

(i) $\Leftrightarrow$ (ii):

(MovahediBehmardiSoleimani-Damaneh*, *, Theorem 3.2)**.

(ii)$\Rightarrow$ (iii):

Let $x,y\in X$, $\lambda\in[0,1]$ and (x,x^{\scalebox{0.5}{\lozenge}})\in X\times X^{\scalebox{0.55}{\lozenge}}. Then x^{\scalebox{0.5}{\lozenge}}=\sum^{n}_{i=1}\alpha_{i}[t_{i}\overrightarrow{a_{i}b_{i}}]\in X^{\scalebox{0.55}{\lozenge}}, and hence by using (ii) we get:

[TABLE]

Therefore X\times X^{\scalebox{0.55}{\lozenge}} has $\mathcal{W}$-property.

(iii) $\Leftrightarrow$ (iv):

Straightforward.

(iv) $\Rightarrow$ (v):

Let $x,y\in X$ and $\lambda\in[0,1]$, then

[TABLE]

Therefore, $\varphi_{\overrightarrow{pz}}$ is convex.

(v) $\Rightarrow$ (vi):

It is easy.

(vi) $\Rightarrow$ (iii):

Let $x,y,p\in X$, $\lambda\in[0,1]$ and x^{\scalebox{0.5}{\lozenge}}=\sum^{n}_{i=1}\alpha_{i}[t_{i}\overrightarrow{p_{i}z_{i}}]\in X^{\scalebox{0.55}{\lozenge}} be given. Then

[TABLE]

i.e., X\times X^{\scalebox{0.55}{\lozenge}} has $\mathcal{W}$-property.

(iii)$\Rightarrow$ (ii):

For $a,b,x,y\in X$ and $\lambda\in[0,1]$, we have:

[TABLE]

where x^{\scalebox{0.5}{\lozenge}}=[\overrightarrow{ab}]\in X^{\scalebox{0.55}{\lozenge}}. Since X\times X^{\scalebox{0.55}{\lozenge}} has $\mathcal{W}$-property, one can deduce that:

[TABLE]

Hence, by interchanging the role of $a$ and $b$ in (5), we obtain:

[TABLE]

Finally, (5) and (6) yield:

[TABLE]

We are done.

The next example shows that there exists a relation M\subseteq X\times X^{\scalebox{0.55}{\lozenge}} in the non-flat Hadamard spaces which doesn’t have the $\mathcal{W}$-property.

Example 2.8**.**

Consider the following equivalence relation on $\mathbb{N}\times[0,1]$:

[TABLE]

Set $X:=\frac{\mathbb{N}\times[0,1]}{\sim}$ and let $d:X\times X\rightarrow\mathbb{R}$ be defined by

[TABLE]

The geodesic joining $x=[(n,t)]$ to $y=[(m,s)]$ is defined as follows:

[TABLE]

whenever $x\neq y$ and vacuously $(1-\lambda)x\oplus\lambda x:=x$. It is known that (see (Kakavandi, , Example 4.7)) $(X,d)$ is an $\mathbb{R}$-tree space. It follows from (Bacak2014, , Example 1.2.10), that any $\mathbb{R}$-tree space is an Hadamard space. Let $x=[(2,\frac{1}{2})]$, $y=[(1,\frac{1}{2})]$, $a=[(3,\frac{1}{3})]$, $b=[(2,\frac{1}{2})]$ and $\lambda=\frac{1}{5}$. Then $\frac{4}{5}x\oplus\frac{1}{5}y=[(2,\frac{3}{10})]$ and

[TABLE]

Now, Proposition 2.6(ii) implies that $(X,d)$ is not a flat Hadamard space. For each $n\in\mathbb{N}$, set $x_{n}:=[(n,\frac{1}{2})]$ and $y_{n}:=[(n,\frac{1}{n})]$. Now, we define

[TABLE]

Take $p=[(1,1)]\in X$, $[\overrightarrow{y_{5}y_{4}}]\in\mathrm{Range}(M)$ and $\lambda=\frac{1}{3}$. Clearly, $\tilde{x}:=(1-\lambda)x_{1}\oplus\lambda x_{3}=[(1,\frac{1}{6})]$ and \big{\langle}[\overrightarrow{y_{5}y_{4}}],\overrightarrow{p\tilde{x}}\big{\rangle}=\frac{1}{24}, while,

[TABLE]

Therefore, $M$ doesn’t have the $\mathcal{W}$-property.

3 Monotone Relations

Ahmadi Kakavandi and Amini KakavandiAmini introduced the notion of monotone operators in Hadamard spaces. In KhatibzadehRanjbar2017 , Khatibzadeh and Ranjbar, investigated some properties of monotone operators and their resolvents and also proximal point algorithm in Hadamard spaces. Chaipunya and Kumam ChaipunyaKumam studied the general proximal point method for finding a zero point of a maximal monotone set-valued vector field defined on Hadamard spaces. They proved the relation between the maximality and Minty’s surjectivity condition. Zamani Eskandani and Raeisi ZamaniRaeisi , by using products of finitely many resolvents of monotone operators, proposed an iterative algorithm for finding a common zero of a finite family of monotone operators and a common fixed point of an infinitely countable family of non-expansive mappings in Hadamard spaces. In this section, we will characterize the notation of monotone relations in Hadamard spaces based on characterization of monotone sets in Banach spaces CoodeySimons1996 ; Simons1998 ; Zalinescu .

Definition 3.1**.**

Let $X$ be an Hadamard space with linear dual space X^{\scalebox{0.55}{\lozenge}}. The set M\subseteq X\times X^{\scalebox{0.55}{\lozenge}} is called monotone if \langle x^{\scalebox{0.5}{\lozenge}}-y^{\scalebox{0.5}{\lozenge}},\overrightarrow{yx}\rangle\geq 0, for all (x,x^{\scalebox{0.5}{\lozenge}}),(y,y^{\scalebox{0.5}{\lozenge}}) in $M$.

Example 3.2**.**

Let $x_{n}$, $y_{n}$ and $M$ be the same as in Example 2.8. Let (u,u^{\scalebox{0.5}{\lozenge}}),(v,v^{\scalebox{0.5}{\lozenge}})\in M. There exists $m,n\in\mathbb{N}$ such that $u=x_{n}$, u^{\scalebox{0.5}{\lozenge}}:=[\overrightarrow{y_{n+1}y_{n}}], $v=x_{m}$ and v^{\scalebox{0.5}{\lozenge}}:=[\overrightarrow{y_{m+1}y_{m}}]. Then

[TABLE]

Therefore, \langle u^{\scalebox{0.5}{\lozenge}}-v^{\scalebox{0.5}{\lozenge}},\overrightarrow{vu}\rangle\geq 0 which shows that, $M$ is a monotone relation.

In the sequel, we need the following notations. Let $X$ be an Hadamard space and $Y\subseteq X$. Put

[TABLE]

where $\mathrm{supp}\,{\eta}=\{y\in Y:\eta(y)\neq 0\}$. Clearly, for each $\emptyset\neq A\subset Y$, $\varsigma_{A}=\{\eta\in\varsigma_{Y}:\mathrm{supp}\,\eta\subseteq A\}$. It is obvious that $\varsigma_{A}$ is a convex subset of $\mathbb{R}^{Y}$. Moreover, if $\emptyset\neq A\subseteq B$, then $\varsigma_{A}\subseteq\varsigma_{B}$. Suppose $u\in Y$ be fixed. Define $\delta_{u}\in\varsigma_{Y}$ by

[TABLE]

Let M\subseteq X\times X^{\scalebox{0.55}{\lozenge}} and $\eta\in\varsigma_{A}$. Then $\mathrm{supp}\eta=\{\lambda_{1},\ldots,\lambda_{n}\}$ where \lambda_{i}=\eta(x_{i},x_{i}^{\scalebox{0.5}{\lozenge}}),~{}\text{for each}~{}1\leq i\leq n. Let $p\in X$ be fixed. Define \alpha:\varsigma_{X\times X^{{\scalebox{0.55}{\lozenge}}}}\rightarrow X $($resp. \beta:\varsigma_{X\times X^{\scalebox{0.55}{\lozenge}}}\rightarrow X^{\scalebox{0.55}{\lozenge}} and \theta_{p}:\varsigma_{X\times X^{{\scalebox{0.55}{\lozenge}}}}\rightarrow\mathbb{R}) by

[TABLE]

Proposition 3.3**.**

Let $X$ be an Hadamard space, M\subseteq X\times X^{\scalebox{0.55}{\lozenge}} and $p\in X$. Set

[TABLE]

Then $\Theta_{p,M}=\Theta_{q,M}$ for any $q\in X$.

Proof 3.4**.**

It is enough to show that $\Theta_{p,M}\subseteq\Theta_{q,M}$. Let $\eta\in\Theta_{p,M}$ be such that $\mathrm{supp}\eta=\{\lambda_{1},\ldots,\lambda_{n}\}$ where \lambda_{i}=\eta(x_{i},x_{i}^{\scalebox{0.5}{\lozenge}}),~{}\text{for each}~{}1\leq i\leq n. Then

[TABLE]

Therefore, $\eta\in\Theta_{q,M}$, i.e., $\Theta_{p,M}\subseteq\Theta_{q,M}$.

According to Proposition 3.3, for each M\subseteq X\times X^{\scalebox{0.55}{\lozenge}}, the set $\Theta_{p,M}$ is independent of the choice of the element $p\in X$ and hence we denote the set $\Theta_{p,M}$ by $\Theta_{M}$.

Theorem 3.5**.**

Let $X$ be an Hadamard space and M\subseteq X\times X^{\scalebox{0.55}{\lozenge}} satisfies the $\mathcal{W}$-property. Then $M$ is a monotone set if and only if $\Theta_{M}=\varsigma_{M}$.

Proof 3.6**.**

Let $M$ be a monotone set. In view of (7), it is enough to show that $\varsigma_{M}\subseteq\Theta_{M}$. Let $\eta\in\varsigma_{M}$ be such that $\mathrm{supp}\eta=\{\lambda_{1},\ldots,\lambda_{n}\}$ where \lambda_{i}=\eta(x_{i},x_{i}^{\scalebox{0.5}{\lozenge}}),~{}\text{for each}~{}1\leq i\leq n. By using Proposition 2.2, we obtain:

[TABLE]

Then $\varsigma_{M}\subseteq\Theta_{M}$ and hence $\varsigma_{M}=\Theta_{M}$. For the converse, let (x,x^{{\scalebox{0.5}{\lozenge}}}),(y,y^{{\scalebox{0.5}{\lozenge}}})\in M and set \eta:=\frac{1}{2}\delta_{(x,x^{{\scalebox{0.5}{\lozenge}}})}+\frac{1}{2}\delta_{(y,y^{{\scalebox{0.5}{\lozenge}}})}\in\varsigma_{M}. By using $\mathcal{W}$-property, we get:

[TABLE]

Therefore, $M$ is monotone.

Corollary 3.7**.**

Let $X$ be a flat Hadamard space and M\subseteq X\times X^{\scalebox{0.55}{\lozenge}}. Then $M$ is a monotone set if and only if $\Theta_{M}=\varsigma_{M}$.

Proof 3.8**.**

Since $X$ is flat, Proposition 2.6 implies that M\subseteq X\times X^{\scalebox{0.55}{\lozenge}} satisfies the $\mathcal{W}$-property. Then the conclusion follows immediately from Theorem 3.5.

A fundamental result concerning monotone operators is the extension theorem of Debrunner-Flor (for a proof see (BurachikIusem2008, , Theorem 4.3.1) or (Zeidler1986, , Proposition 2.17)). In the sequel, we prove a type of this result for monotone relations from an Hadamard space to its linear dual space. First, we recall some notions and results.

Definition 3.9**.**

(KakavandiAmini, , Definition 2.4) Let $\{x_{n}\}$ be a sequence in an Hadamard space $X$. The sequence $\{x_{n}\}$ is said to be weakly convergent to $x\in X$, denoted by $x_{n}\overset{w}{\longrightarrow}x$, if $\lim_{n\rightarrow\infty}\langle\overrightarrow{xx_{n}},\overrightarrow{xy}\rangle=0$, for all $y\in X$.

One can easily see that convergence in the metric implies weak convergence.

Lemma 3.10**.**

(ZamaniRaeisi, , Proposition 3.6)* Let $\{x_{n}\}$ be a bounded sequence in an Hadamard space $(X,d)$ with linear dual space X^{\scalebox{0.55}{\lozenge}} and let \{x^{\scalebox{0.5}{\lozenge}}_{n}\} be a sequence in X^{{\scalebox{0.55}{\lozenge}}}. If $\{x_{n}\}$ is weakly convergent to $x$ and x^{{\scalebox{0.5}{\lozenge}}}_{n}\xrightarrow{\|\cdot\|_{\scalebox{0.5}{\lozenge}}}x^{\scalebox{0.5}{\lozenge}}, then \langle x^{\scalebox{0.5}{\lozenge}}_{n},\overrightarrow{x_{n}z}\rangle\rightarrow\langle x^{\scalebox{0.5}{\lozenge}},\overrightarrow{xz}\rangle, for all $z\in X$.*

Theorem 3.11**.**

Let $X$ be an Hadamard space and M\subseteq X\times X^{\scalebox{0.55}{\lozenge}} be a monotone relation satisfies the $\mathcal{W}$-property. Let C\subseteq X^{\scalebox{0.55}{\lozenge}} be a compact and convex set, and $\varphi:C\rightarrow X$ be a continuous function. Then there exists z^{\scalebox{0.5}{\lozenge}}\in C such that \{(\varphi(z^{\scalebox{0.5}{\lozenge}}),z^{\scalebox{0.5}{\lozenge}})\}\cup M is monotone.

Proof 3.12**.**

Let $x\in X$, u^{\scalebox{0.5}{\lozenge}},v^{\scalebox{0.5}{\lozenge}}\in X^{\scalebox{0.55}{\lozenge}} be arbitrary and fixed element. Consider the function $\tau:C\rightarrow\mathbb{R}$ defined by

[TABLE]

Let \{x_{n}^{\scalebox{0.5}{\lozenge}}\}\subseteq C be such that x^{\scalebox{0.5}{\lozenge}}_{n}\xrightarrow{\|\cdot\|_{\scalebox{0.5}{\lozenge}}}x^{\scalebox{0.5}{\lozenge}}, for some x^{\scalebox{0.5}{\lozenge}}\in C. By Lemma 3.10,

[TABLE]

Thus \tau(x_{n}^{\scalebox{0.5}{\lozenge}})\rightarrow\tau(x^{\scalebox{0.5}{\lozenge}}). Hence $\tau$ is continuous. For every (y,y^{\scalebox{0.5}{\lozenge}})\in M, set

[TABLE]

Continuity of $\tau$ implies that U(y,y^{\scalebox{0.5}{\lozenge}}) is an open subset of $C$. Suppose that the conclusion fails. Then for each u^{\scalebox{0.5}{\lozenge}}\in C there exists (y,y^{\scalebox{0.5}{\lozenge}})\in M such that u^{\scalebox{0.5}{\lozenge}}\in U(y,y^{\scalebox{0.5}{\lozenge}}). This means that the family of open sets \{U(y,y^{\scalebox{0.5}{\lozenge}})\}_{(y,y^{\scalebox{0.5}{\lozenge}})\in M} is an open cover of $C$. Using the compactness of $C$, we obtain that C=\bigcup_{i=1}^{n}U(y_{i},y_{i}^{\scalebox{0.5}{\lozenge}}). In addition, (Zeidler1986, , Page 756) implies that there exists a partition of unity associated with this finite subcover. Hence, there are continuous functions \psi_{i}:X^{\scalebox{0.55}{\lozenge}}\rightarrow\mathbb{R}~{}(1\leq i\leq n) satisfying

1. (i)

\sum_{i=1}^{n}\psi_{i}(x^{\scalebox{0.5}{\lozenge}})=1, for all x^{\scalebox{0.5}{\lozenge}}\in C. 2. (ii)

\psi_{i}(x^{\scalebox{0.5}{\lozenge}})\geq 0, for all x^{\scalebox{0.5}{\lozenge}}\in C and all $i\in\{1,\ldots,n\}$. 3. (iii)

\{x^{\scalebox{0.5}{\lozenge}}\in C:\psi_{i}(x^{\scalebox{0.5}{\lozenge}})>0\}\subseteq U_{i}:=U(y_{i},y_{i}^{\scalebox{0.5}{\lozenge}})* for all $i\in\{1,\ldots,n\}$.*

Set K:=\mathrm{co}(\{y_{1}^{\scalebox{0.5}{\lozenge}},\ldots,y_{n}^{\scalebox{0.5}{\lozenge}}\})\subseteq C and define

[TABLE]

Let \{u_{m}^{\scalebox{0.5}{\lozenge}}\}\subseteq K be such that u_{m}^{\scalebox{0.5}{\lozenge}}\rightarrow u^{\scalebox{0.5}{\lozenge}},

[TABLE]

By continuity of $\psi_{i}~{}(1\leq i\leq n)$, letting $m\rightarrow+\infty$, then \psi_{i}(u_{m}^{\scalebox{0.5}{\lozenge}})\rightarrow\psi_{i}(u^{\scalebox{0.5}{\lozenge}}) and this implies that \iota(u_{m}^{\scalebox{0.5}{\lozenge}})\rightarrow\iota(u^{\scalebox{0.5}{\lozenge}}) and so $\iota$ is continuous. One can identify $K$ with a finite-dimensional convex and compact set. By using Brouwer fixed point theorem (Zeidler1986, , Proposition 2.6), there exists w^{\scalebox{0.5}{\lozenge}}\in K such that \iota(w^{\scalebox{0.5}{\lozenge}})=w^{\scalebox{0.5}{\lozenge}}. Moreover, by using Proposition 2.2 we get:

[TABLE]

Set a_{ij}=\langle y_{i}^{\scalebox{0.5}{\lozenge}}-w^{\scalebox{0.5}{\lozenge}},\overrightarrow{\varphi(w^{\scalebox{0.5}{\lozenge}})y_{j}}\rangle. It follows from monotonicity of $M$ that

[TABLE]

i.e.,

[TABLE]

Applying (8) and (9), we obtain:

[TABLE]

Set I(w^{\scalebox{0.5}{\lozenge}}):=\big{\{}i\in\{1,\ldots,n\}:w^{\scalebox{0.5}{\lozenge}}\in U_{i}\big{\}}. Applying property (iii) of the partition of unity in (11) we get:

[TABLE]

By using property (iii) of the partition of unity and the definition of $U_{i}$, one deduce that all terms in the right-hand side of (12) are nonpositive. So all of \psi_{i}(w^{\scalebox{0.5}{\lozenge}})’s must be vanish, which contradicts with (i).

Corollary 3.13**.**

Let $X$ be a flat Hadamard space and M\subseteq X\times X^{\scalebox{0.55}{\lozenge}} be a monotone set. Let C\subseteq X^{\scalebox{0.55}{\lozenge}} be a compact and convex set, and $\varphi:C\rightarrow X$ be a continuous function. Then there exists z^{\scalebox{0.5}{\lozenge}}\in C such that \{(\varphi(z^{\scalebox{0.5}{\lozenge}}),z^{\scalebox{0.5}{\lozenge}})\}\cup M is monotone.

Proof 3.14**.**

Since $X$ is flat, it follows from Proposition 2.6 that M\subseteq X\times X^{\scalebox{0.55}{\lozenge}} has $\mathcal{W}$-property. The inclusion follows from Theorem 3.11.