Generalized hybrid mappings on $CAT(\kappa )$ spaces

Abstract

In this paper, we obtain the demiclosed principle, fixed point theorems, and Δ-convergence theorems for the class of generalized hybrid mappings on $CAT(\kappa )$ spaces with $\kappa >0$. Our results extend the results of Lin et al. (Fixed Point Theory Appl. 2011:49, 2011) and many others.

1 Introduction

For a real number κ, a $CAT(\kappa )$ space is a geodesic metric space whose geodesic triangle is thinner than the corresponding comparison triangle in a model space with curvature κ. The precise definition is given below. The letters C, A, and T stand for Cartan, Alexandrov, and Toponogov, who have made important contributions to the understanding of curvature via inequalities for the distance function.

Fixed point theory in $CAT(\kappa )$ spaces was first studied by Kirk [1, 2]. His works were followed by a series of new works by many authors, mainly focusing on $CAT(0)$ spaces (see e.g., [3–18]). Since any $CAT(\kappa )$ space is a $CAT({\kappa }^{\prime })$ space for ${\kappa }^{\prime }\ge \kappa$, all results for $CAT(0)$ spaces immediately apply to any $CAT(\kappa )$ space with $\kappa \le 0$. However, there are only a few articles that contain fixed point results in the setting of $CAT(\kappa )$ spaces with $\kappa >0$.

The concept of generalized hybrid mappings was introduced in Hilbert spaces by Kocourek et al. [19]. Later on, Lin et al. [10] defined a generalized hybrid mapping, which is more general than that of Kocourek et al. [19], in a $CAT(0)$ space setting. This class of mappings properly contains the class of nonspreading mappings and the class of hybrid mappings; see [10] for more details. In [10], the authors also obtained the demiclosed principle, fixed point theorems as well as Δ-convergence theorems for generalized hybrid mappings in $CAT(0)$ spaces. In this paper, we extend the results of Lin et al. [10] to the general setting of $CAT(\kappa )$ spaces with $\kappa >0$.

2 Preliminaries

Let $(X,\rho )$ be a metric space. A geodesic path joining $x\in X$ to $y\in X$ (or, more briefly, a geodesic from x to y) is a map c from a closed interval $[0,l]\subset \mathbb{R}$ to X such that $c(0)=x$, $c(l)=y$, and $\rho (c(t),c(t^{\prime }))=|t-t^{\prime }|$ for all $t,t^{\prime }\in [0,l]$. In particular, c is an isometry and $\rho (x,y)=l$. The image $c([0,l])$ of c is called a geodesic segment joining x and y. When it is unique this geodesic segment is denoted by $[x,y]$. This means that $z\in [x,y]$ if and only if there exists $\alpha \in [0,1]$ such that

$$\rho (x,z)=(1-\alpha )\rho (x,y)\text{and}\rho (y,z)=\alpha \rho (x,y).$$

In this case, we write $z=\alpha x\oplus (1-\alpha )y$. For $D\in (0,+\mathrm{\infty }]$, the space X is called a D-geodesic space if every two points of X with their distance smaller than D are joined by a geodesic segment. An ∞-geodesic space is simply called a geodesic space. The space X is said to be uniquely geodesic (D-uniquely geodesic) if there is exactly one geodesic segment joining x and y for each $x,y\in X$ (for $x,y\in X$ with $\rho (x,y)<D$). A subset C of X is said to be convex if C includes every geodesic segment joining any two of its points. The set C is said to be bounded if

$$diam(C):=sup\{\rho (x,y):x,y\in C\}<\mathrm{\infty }.$$

Now we present the model spaces $M_{\kappa }^n$, for more details on these spaces the reader is referred to [20]. Let $n\in \mathbb{N}$. We denote by ${\mathbb{E}}^n$ the metric space ${\mathbb{R}}^n$ endowed with the usual Euclidean distance. We denote by $(\cdot |\cdot )$ the Euclidean scalar product in ${\mathbb{R}}^n$, that is,

$$(x|y)=x_1{y}_1+\cdots +x_n{y}_n,\text{where }x=(x_1,\dots ,x_n),y=(y_1,\dots ,y_n).$$

Let ${\mathbb{S}}^n$ denote the n-dimensional sphere defined by

$${\mathbb{S}}^n=\{x=(x_1,\dots ,x_{n+1})\in {\mathbb{R}}^{n+1}:(x|x)=1\},$$

with metric $d_{{\mathbb{S}}^n}(x,y)=arccos(x|y)$, $x,y\in {\mathbb{S}}^n$.

Let ${\mathbb{E}}^{n,1}$ denote the vector space ${\mathbb{R}}^{n+1}$ endowed with the symmetric bilinear form which associates to vectors $u=(u_1,\dots ,u_{n+1})$ and $v=(v_1,\dots ,v_{n+1})$ the real number $〈u|v〉$ is defined by

$$〈u|v〉=-u_{n+1}{v}_{n+1}+\sum _{i=1}^n{u}_i{v}_i.$$

Let ${\mathbb{H}}^n$ denote the hyperbolic n-space defined by

$${\mathbb{H}}^n=\{u=(u_1,\dots ,u_{n+1})\in {\mathbb{E}}^{n,1}:〈u|u〉=-1,u_{n+1}>0\},$$

with metric $d_{{\mathbb{H}}^n}$ such that

$$cosh{d}_{{\mathbb{H}}^n}(x,y)=-〈x|y〉,x,y\in {\mathbb{H}}^n.$$

Definition 2.1 Given $\kappa \in \mathbb{R}$, we denote by $M_{\kappa }^n$ the following metric spaces:

• (i) if $\kappa =0$ then $M_0^n$ is the Euclidean space ${\mathbb{E}}^n$;

• (ii) if $\kappa >0$ then $M_{\kappa }^n$ is obtained from the spherical space ${\mathbb{S}}^n$ by multiplying the distance function by the constant $1/\sqrt{\kappa }$;

• (iii) if $\kappa <0$ then $M_{\kappa }^n$ is obtained from the hyperbolic space ${\mathbb{H}}^n$ by multiplying the distance function by the constant $1/\sqrt{-\kappa }$.

A geodesic triangle $\mathrm{△}(x,y,z)$ in a geodesic space $(X,\rho )$ consists of three points x, y, z in X (the vertices of △) and three geodesic segments between each pair of vertices (the edges of △). A comparison triangle for a geodesic triangle $\mathrm{△}(x,y,z)$ in $(X,\rho )$ is a triangle $\overline{\mathrm{△}}(\overline{x},\overline{y},\overline{z})$ in $M_{\kappa }^2$ such that

$$\rho (x,y)=d_{M_{\kappa }^2}(\overline{x},\overline{y}),\rho (y,z)=d_{M_{\kappa }^2}(\overline{y},\overline{z})\text{and}\rho (z,x)=d_{M_{\kappa }^2}(\overline{z},\overline{x}).$$

If $\kappa \le 0$ then such a comparison triangle always exists in $M_{\kappa }^2$. If $\kappa >0$ then such a triangle exists whenever $\rho (x,y)+\rho (y,z)+\rho (z,x)<2D_{\kappa }$, where $D_{\kappa }=\pi /\sqrt{\kappa }$. A point $\overline{p}\in [\overline{x},\overline{y}]$ is called a comparison point for $p\in [x,y]$ if $\rho (x,p)=d_{M_{\kappa }^2}(\overline{x},\overline{p})$.

A geodesic triangle $\mathrm{△}(x,y,z)$ in X is said to satisfy the $CAT(\kappa )$ inequality if for any $p,q\in \mathrm{△}(x,y,z)$ and for their comparison points $\overline{p},\overline{q}\in \overline{\mathrm{△}}(\overline{x},\overline{y},\overline{z})$, one has

$$\rho (p,q)\le d_{M_{\kappa }^2}(\overline{p},\overline{q}).$$

Definition 2.2 If $\kappa \le 0$, then X is called a $CAT(\kappa )$ space if X is a geodesic space such that all of its geodesic triangles satisfy the $CAT(\kappa )$ inequality.

If $\kappa >0$, then X is called a $CAT(\kappa )$ space if X is $D_{\kappa }$-geodesic and any geodesic triangle $\mathrm{△}(x,y,z)$ in X with $\rho (x,y)+\rho (y,z)+\rho (z,x)<2D_{\kappa }$ satisfies the $CAT(\kappa )$ inequality.

Now, we recall the concepts of comparison angle and upper (Alexandrov) angle (cf. [8]).

Definition 2.3 Let p, q, and r be three points in a geodesic space. The interior angle of $\overline{\mathrm{△}}(\overline{p},\overline{q},\overline{r})\subseteq {\mathbb{E}}^2$ at $\overline{p}$ is called the comparison angle between q and r at p and will be denoted by ${\overline{\mathrm{\angle }}}_p(q,r)$.

Definition 2.4 Let X be a geodesic space and let $c:[0,a]\to X$ and $c^{\prime }:[0,a^{\prime }]\to X$ be two geodesic paths with $c(0)=c^{\prime }(0)$. Given $t\in (0,a]$ and $t^{\prime }\in (0,a^{\prime }]$, we consider the comparison triangle $\overline{\mathrm{△}}(\overline{c(0)},\overline{c(t)},\overline{c^{\prime }(t^{\prime })})$ and the comparison angle ${\overline{\mathrm{\angle }}}_{c(0)}(c(t),c^{\prime }(t^{\prime }))$ in ${\mathbb{E}}^2$. The (Alexandrov) angle or the upper angle between the geodesic paths c and $c^{\prime }$ is the number $\mathrm{\angle }(c,c^{\prime })$ defined by

$$\mathrm{\angle }(c,c^{\prime }):=\underset{t,t^{\prime }\to 0^{+}}{lim\hspace{0.17em}sup}{\overline{\mathrm{\angle }}}_{c(0)}(c(t),c^{\prime }\left(t^{\prime }\right)).$$

The angle between the geodesic segments $[p,x]$ and $[p,y]$ will be denoted by ${\mathrm{\angle }}_p(x,y)$. Notice that the Alexandrov angle coincides with the spherical angle on ${\mathbb{S}}^n$ and the hyperbolic angle on ${\mathbb{H}}^n$.

In a $CAT(0)$ space $(X,\rho )$, if $x,y,z\in X$ then the $CAT(0)$ inequality implies

$$(\mathrm{CN}){\rho }^2(x,\frac{1}{2}y\oplus \frac{1}{2}z)\le \frac{1}{2}{\rho }^2(x,y)+\frac{1}{2}{\rho }^2(x,z)-\frac{1}{4}{\rho }^2(y,z).$$

This is the (CN) inequality of Bruhat and Tits [21]. This inequality is extended by Dhompongsa and Panyanak [22] to

$$\left({\mathrm{CN}}^{\ast }\right){\rho }^2(x,(1-\alpha )y\oplus \alpha z)\le (1-\alpha ){\rho }^2(x,y)+\alpha {\rho }^2(x,z)-\alpha (1-\alpha ){\rho }^2(y,z)$$

for all $\alpha \in [0,1]$ and $x,y,z\in X$. In fact, if X is a geodesic space then the following statements are equivalent:

• (i) X is a $CAT(0)$ space;

• (ii) X satisfies (CN);

• (iii) X satisfies (CN^∗).

Let $R\in (0,2]$. Recall that a geodesic space $(X,\rho )$ is said to be R-convex for R (see [23]) if for any three points $x,y,z\in X$, we have

$${\rho }^2(x,(1-\alpha )y\oplus \alpha z)\le (1-\alpha ){\rho }^2(x,y)+\alpha {\rho }^2(x,z)-\frac{R}{2}\alpha (1-\alpha ){\rho }^2(y,z).$$

(1)

It follows from (CN^∗) that a geodesic space $(X,\rho )$ is a $CAT(0)$ space if and only if $(X,\rho )$ is R-convex for $R=2$. The following lemma is a consequence of Proposition 3.1 in [23].

Lemma 2.5 Let $\kappa >0$ and $(X,\rho )$ be a $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Then $(X,\rho )$ is R-convex for $R=(\pi -2\epsilon )tan(\epsilon )$.

We now collect some elementary facts about $CAT(\kappa )$ spaces. Most of them are proved in the setting of $CAT(1)$ spaces. For completeness, we state the results in $CAT(\kappa )$ with $\kappa >0$.

Lemma 2.6 ([[8], Proposition 3.5])

Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let $x\in X$ and C be a nonempty closed convex subset of X. Then

• (i) the metric projection $P_C(x)$ of x onto C is a singleton;

• (ii) if $x\notin C$ and $y\in C$ with $y\ne P_C(x)$, then ${\mathrm{\angle }}_{P_C(x)}(x,y)\ge \pi /2$;

• (iii) for each $y\in C$, $\rho (P_C(x),P_C(y))\le \rho (x,y)$.

Let $\{x_n\}$ be a bounded sequence in a $CAT(\kappa )$ space $(X,\rho )$. For $x\in X$, we set

$$r(x,\{x_n\})=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\rho (x,x_n).$$

The asymptotic radius $r(\{x_n\})$ of $\{x_n\}$ is given by

$$r(\{x_n\})=inf\{r(x,\{x_n\}):x\in X\},$$

and the asymptotic center $A(\{x_n\})$ of $\{x_n\}$ is the set

$$A(\{x_n\})=\{x\in X:r(x,\{x_n\})=r(\{x_n\})\}.$$

It is well known from Proposition 4.1 of [8] that in a $CAT(\kappa )$ space with diameter smaller than $\frac{\pi }{2\sqrt{\kappa }}$, $A(\{x_n\})$ consists of exactly one point. We now give the concept of Δ-convergence and collect some of its basic properties.

Definition 2.7 ([6, 24])

A sequence $\{x_n\}$ in X is said to Δ-converge to $x\in X$ if x is the unique asymptotic center of $\{u_n\}$ for every subsequence $\{u_n\}$ of $\{x_n\}$. In this case we write $\mathrm{\Delta }\text{-}{lim}_n{x}_n=x$ and call x the Δ-limit of $\{x_n\}$.

Lemma 2.8 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Then the following statements hold:

• (i) [[8], Corollary  4.4] every sequence in X has a Δ-convergence subsequence;

• (ii) [[8], Proposition  4.5] if $\{x_n\}\subseteq X$ and $\mathrm{\Delta }\text{-}{lim}_n{x}_n=x$, then $x\in {\bigcap }_{k=1}^{\mathrm{\infty }}\overline{conv}\{x_k,x_{k+1},\dots \}$, where $\overline{conv}(A)=\bigcap \{B:B\supseteq A\mathit{\text{ and }}B\mathit{\text{ is closed and convex}}\}$.

By the uniqueness of asymptotic centers, we can obtain the following lemma (cf. [[22], Lemma 2.8]).

Lemma 2.9 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. If $\{x_n\}$ is a sequence in X with $A(\{x_n\})=\{x\}$ and $\{u_n\}$ is a subsequence of $\{x_n\}$ with $A(\{u_n\})=\{u\}$ and the sequence $\{\rho (x_n,u)\}$ converges, then $x=u$.

Definition 2.10 Let C be a nonempty subset of a $CAT(\kappa )$ space $(X,\rho )$. A mapping $T:C\to X$ is called a generalized hybrid mapping [10] if there exist functions $a_1,a_2,a_3,k_1,k_2:C\to [0,1)$ such that

• (P1) ${\rho }^2(T(x),T(y))$ ≤ $a_1(x){\rho }^2(x,y)$ + $a_2(x){\rho }^2(T(x),y)$ + $a_3(x){\rho }^2(T(y),x)$ + $k_1(x){\rho }^2(T(x),x)$ + $k_2(x){\rho }^2(T(y),y)$ for all $x,y\in C$;

• (P2) $a_1(x)+a_2(x)+a_3(x)\le 1$ for all $x,y\in C$;

• (P3) $2k_1(x)<1-a_2(x)$ and $k_2(x)<1-a_3(x)$ for all $x\in C$.

A point $x\in C$ is called a fixed point of T if $x=T(x)$. We denote the set of all fixed points of T with $F(T)$.

3 Main results

3.1 Demiclosed principle

Theorem 3.1 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let C be a nonempty closed convex subset of X, and $T:C\to X$ be a generalized hybrid mapping with $\frac{2k_1(x)}{1-a_2(x)}<\frac{R}{2}$ for all $x\in C$ where $R=(\pi -2\epsilon )tan(\epsilon )$. Let $\{x_n\}$ be a sequence in C with $\mathrm{\Delta }\text{-}{lim}_n{x}_n=z$ and ${lim}_n\rho (x_n,T(x_n))=0$. Then $z\in C$ and $z=T(z)$.

Proof Since $\mathrm{\Delta }\text{-}{lim}_n{x}_n=z$, by Lemma 2.8, $z\in C$. Since T is a generalized hybrid mapping,

$$\begin{array}{rcl}{\rho }^2(T(x_n),T(z))& \le & a_1(z){\rho }^2(z,x_n)+a_2(z){\rho }^2(T(z),x_n)+a_3(z){\rho }^2(T(x_n),z)\\ +k_1(z){\rho }^2(T(z),z)+k_2(z){\rho }^2(T(x_n),x_n)\\ \le & a_1(z){\rho }^2(z,x_n)+a_2(z){[\rho (T(z),T(x_n))+\rho (T(x_n),x_n)]}^2\\ +a_3(z){[\rho (T(x_n),x_n)+\rho (x_n,z)]}^2+k_1(z){\rho }^2(T(z),z)\\ +k_2(z){\rho }^2(T(x_n),x_n),\end{array}$$

yielding

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(T(x_n),T(z))\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(z,x_n)+\frac{k_1(z)}{1-a_2(z)}{\rho }^2(z,T(z)).$$

This implies that

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,T(z))& \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{[\rho (x_n,T(x_n))+\rho (T(x_n),T(z))]}^2\\ \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(T(x_n),T(z))\\ \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(z,x_n)+\frac{k_1(z)}{1-a_2(z)}{\rho }^2(z,T(z)).\end{array}$$

(2)

On the other hand, by Lemma 2.5 we have

$${\rho }^2(x_n,\frac{1}{2}z\oplus \frac{1}{2}T(z))\le \frac{1}{2}{\rho }^2(x_n,z)+\frac{1}{2}{\rho }^2(x_n,T(z))-\frac{R}{8}{\rho }^2(z,T(z)).$$

(3)

By (2) and (3), we get

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,\frac{1}{2}z\oplus \frac{1}{2}T(z))& \le & \frac{1}{2}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,z)+\frac{1}{2}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,T(z))\\ -\frac{R}{8}{\rho }^2(z,T(z))\\ \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,z)+\frac{k_1(z)}{2(1-a_2(z))}{\rho }^2(z,T(z))\\ -\frac{R}{8}{\rho }^2(z,T(z)).\end{array}$$

Thus

$$(\frac{R}{8}-\frac{k_1(z)}{2(1-a_2(z))}){\rho }^2(z,T(z))\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,z)-\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,\frac{1}{2}z\oplus \frac{1}{2}T(z))\le 0.$$

Since $\frac{2k_1(z)}{1-a_2(z)}<\frac{R}{2}$, we get $\frac{k_1(z)}{2(1-a_2(z))}<\frac{R}{8}$ and so ${\rho }^2(z,T(z))=0$. Hence $z=T(z)$. □

The following corollary shows that how we derive a result for $CAT(0)$ spaces from Theorem 3.1.

Corollary 3.2 Let $(X,\rho )$ be a complete $CAT(0)$ space, C be a nonempty bounded closed convex subset of X, and $T:C\to C$ be a generalized hybrid mapping. Let $\{x_n\}$ be a sequence in C with $\mathrm{\Delta }\text{-}{lim}_n{x}_n=z$ and ${lim}_n\rho (x_n,T(x_n))=0$. Then $z\in C$ and $z=T(z)$.

Proof It is well known that every convex subset of a $CAT(0)$ space, equipped with the induced metric, is a $CAT(0)$ space (cf. [20]). Then $(C,\rho )$ is a $CAT(0)$ space and hence it is a $CAT(\kappa )$ space for all $\kappa >0$. Notice also that C is R-convex for $R=2$. Since C is bounded, we can choose $\epsilon \in (0,\pi /2)$ and $\kappa >0$ so that $diam(C)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$. The conclusion follows from Theorem 3.1. □

3.2 Fixed point theorems

Theorem 3.3 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let C be a nonempty closed convex subset of X, and $T:C\to C$ be a generalized hybrid mapping with $k_1(x)=k_2(x)=0$ for all $x\in C$. Then T has a fixed point.

Proof Fix $x\in C$ and define $x_n:=T^n(x)$ for $n\in \mathbb{N}$. Suppose that $A(\{x_n\})=\{z\}$. Then by Lemma 2.8, $z\in C$. Since T is generalized hybrid and $k_1(z)=k_2(z)=0$,

$${\rho }^2(x_n,T(z))\le a_1(z){\rho }^2(z,x_{n-1})+a_2(z){\rho }^2(T(z),x_{n-1})+a_3(z){\rho }^2(x_n,z).$$

Taking the limit superior on both sides, we get

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,T(z))& \le & a_1(z)\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(z,x_{n-1})+a_2(z)\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(T(z),x_{n-1})\\ +a_3(z)\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,z)\\ \le & (a_1(z)+a_3(z))\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,z)+a_2(z)\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\rho }^2(x_n,T(z)).\end{array}$$

This implies by (P2) that ${lim\hspace{0.17em}sup}_n{\rho }^2(x_n,T(z))\le {lim\hspace{0.17em}sup}_n{\rho }^2(x_n,z)$. But, since $A(\{x_n\})=\{z\}$, it must be the case that $z=T(z)$ and the proof is complete. □

As a consequence of Theorem 3.3, we obtain:

Corollary 3.4 Let $(X,\rho )$ be a complete $CAT(0)$ space, C be a nonempty bounded closed convex subset of X, and $T:C\to C$ be a generalized hybrid mapping with $k_1(x)=k_2(x)=0$ for all $x\in C$. Then T has a fixed point.

3.3 Δ-Convergence theorems

We begin this section by proving a crucial lemma.

Lemma 3.5 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let C be a nonempty closed convex subset of X, and $T:C\to X$ be a generalized hybrid mapping with $\frac{2k_1(x)}{1-a_2(x)}<\frac{R}{2}$ for all $x\in C$ where $R=(\pi -2\epsilon )tan(\epsilon )$. Suppose $\{x_n\}$ is a sequence in C such that ${lim}_n\rho (x_n,T{x}_n)=0$ and $\{\rho (x_n,v)\}$ converges for all $v\in F(T)$, then ${\omega }_w(x_n)\subseteq F(T)$. Here ${\omega }_w(x_n):=\bigcup A(\{u_n\})$ where the union is taken over all subsequences $\{u_n\}$ of $\{x_n\}$. Moreover, ${\omega }_w(x_n)$ consists of exactly one point.

Proof Let $u\in {\omega }_w(x_n)$, then there exists a subsequence $\{u_n\}$ of $\{x_n\}$ such that $A(\{u_n\})=\{u\}$. By Lemma 2.8, there exists a subsequence $\{v_n\}$ of $\{u_n\}$ such that $\mathrm{\Delta }\text{-}{lim}_n{v}_n=v\in C$. By Theorem 3.1, $v\in F(T)$. By Lemma 2.9, $u=v$. This shows that ${\omega }_w(x_n)\subseteq F(T)$. Next, we show that ${\omega }_w(x_n)$ consists of exactly one point. Let $\{u_n\}$ be a subsequence of $\{x_n\}$ with $A(\{u_n\})=\{u\}$ and let $A(\{x_n\})=\{x\}$. Since $u\in {\omega }_w(x_n)\subseteq F(T)$, $\{\rho (x_n,u)\}$ converges. Again, by Lemma 2.9, $x=u$. This completes the proof. □

Theorem 3.6 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let C be a nonempty closed convex subset of X, and $T:C\to X$ be a generalized hybrid mapping with $F(T)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\}$ be a sequence in $[0,1]$ and define a sequence $\{x_n\}$ in C by

$$\{\begin{array}{c}\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrary}},\\ x_{n+1}:=P_C((1-{\alpha }_n)x_n\oplus {\alpha }_{n}T(x_n)),n\in \mathbb{N}.\end{array}\hfill \end{array}$$

Let $R=(\pi -2\epsilon )tan(\epsilon )$ and suppose that

• (i) $\frac{2k_1(x)}{1-a_2(x)}<\frac{R}{2}$ for all $x\in C$,

• (ii) ${lim\hspace{0.17em}inf}_n{\alpha }_n[\frac{(1-{\alpha }_n)R}{2}-\frac{k_2(z)}{1-a_3(z)}]>0$ for all $z\in F(T)$.

Then $\{x_n\}$ Δ-converges to an element of $F(T)$.

Proof Let $z\in F(T)$. Since T is generalized hybrid,

$${\rho }^2(T(x),z)\le {\rho }^2(z,x)+\frac{k_2(z)}{1-a_3(z)}{\rho }^2(T(x),x)\text{for all }x\in C.$$

By Lemmas 2.5 and 2.6, we have

$$\begin{array}{rcl}{\rho }^2(x_{n+1},z)& =& {\rho }^2(P_C((1-{\alpha }_n)x_n\oplus {\alpha }_{n}T(x_n)),z)\\ \le & {\rho }^2((1-{\alpha }_n)x_n\oplus {\alpha }_{n}T(x_n),z)\\ \le & (1-{\alpha }_n){\rho }^2(x_n,z)+{\alpha }_n{\rho }^2(T(x_n),z)-\frac{R}{2}{\alpha }_n(1-{\alpha }_n){\rho }^2(x_n,T(x_n))\\ \le & {\rho }^2(x_n,z)+{\alpha }_n[\frac{k_2(z)}{1-a_3(z)}-\frac{R(1-{\alpha }_n)}{2}]{\rho }^2(x_n,T(x_n)).\end{array}$$

(4)

By (ii), there exist $\delta >0$ and $N\in \mathbb{N}$ such that

$${\alpha }_n[\frac{(1-{\alpha }_n)R}{2}-\frac{k_2(z)}{1-a_3(z)}]\ge \delta >0\text{for all }n\ge N.$$

Without loss of generality, we may assume that

$${\alpha }_n[\frac{(1-{\alpha }_n)R}{2}-\frac{k_2(z)}{1-a_3(z)}]>0\text{for all }n\in \mathbb{N}.$$

(5)

It follows from (4) and (5) that $\{\rho (x_n,z)\}$ is a nonincreasing sequence and hence ${lim}_n\rho (x_n,z)$ exists. Again, by (4), we have

$$\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n[\frac{(1-{\alpha }_n)R}{2}-\frac{k_2(z)}{1-a_3(z)}]{\rho }^2(x_n,T(x_n))=0.$$

This implies by (ii) that ${lim}_n\rho (x_n,T(x_n))=0$. By Lemma 3.5, ${\omega }_w(x_n)$ consists of exactly one point and is contained in $F(T)$. This shows that $\{x_n\}$ Δ-converges to an element of $F(T)$. □

Theorem 3.7 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let C be a nonempty closed convex subset of X, and $T:C\to X$ be a generalized hybrid mapping with $F(T)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be sequences in $[0,1]$ and define a sequence $\{x_n\}$ in C by

$$\{\begin{array}{c}\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrary}},\\ x_{n+1}:=P_C((1-{\alpha }_n)T(x_n)\oplus {\alpha }_{n}T(y_n)),\\ y_n:=P_C((1-{\beta }_n)x_n\oplus {\beta }_{n}T(x_n)).\end{array}\hfill \end{array}$$

Assume that

• (i) $k_2(z)=0$ for all $z\in F(T)$,

• (ii) ${lim\hspace{0.17em}inf}_n{\alpha }_n>0$ and ${lim\hspace{0.17em}inf}_n{\beta }_n(1-{\beta }_n)>0$.

Then $\{x_n\}$ Δ-converges to an element of $F(T)$.

Proof Fix $z\in F(T)$. By (i), we have $\rho (T(x),z)\le \rho (x,z)$ for all $x\in C$. Let $R=(\pi -2\epsilon )tan(\epsilon )$. By Lemmas 2.5 and 2.6, we have

$$\begin{array}{rcl}{\rho }^2(y_n,z)& =& {\rho }^2(P_C((1-{\beta }_n)x_n\oplus {\beta }_{n}T(x_n)),z)\\ \le & {\rho }^2((1-{\beta }_n)x_n\oplus {\beta }_{n}T(x_n),z)\\ \le & (1-{\beta }_n){\rho }^2(x_n,z)+{\beta }_n{\rho }^2(T(x_n),z)-\frac{R}{2}{\beta }_n(1-{\beta }_n){\rho }^2(x_n,T(x_n))\\ \le & {\rho }^2(x_n,z)-\frac{R}{2}{\beta }_n(1-{\beta }_n){\rho }^2(x_n,T(x_n))\\ \le & {\rho }^2(x_n,z).\end{array}$$

(6)

This implies that

$$\begin{array}{rcl}{\rho }^2(x_{n+1},z)& =& {\rho }^2(P_C((1-{\alpha }_n)T(x_n)\oplus {\alpha }_{n}T(y_n)),z)\\ \le & {\rho }^2((1-{\alpha }_n)T(x_n)\oplus {\alpha }_{n}T(y_n),z)\\ \le & (1-{\alpha }_n){\rho }^2(T(x_n),z)+{\alpha }_n{\rho }^2(T(y_n),z)-\frac{R}{2}{\alpha }_n(1-{\alpha }_n){\rho }^2(T(x_n),T(y_n))\\ \le & (1-{\alpha }_n){\rho }^2(x_n,z)+{\alpha }_n{\rho }^2(y_n,z)-\frac{R}{2}{\alpha }_n(1-{\alpha }_n){\rho }^2(T(x_n),T(y_n))\\ \le & {\rho }^2(x_n,z)-\frac{R}{2}{\alpha }_n(1-{\alpha }_n){\rho }^2(T(x_n),T(y_n))\\ \le & {\rho }^2(x_n,z).\end{array}$$

Hence ${lim}_n\rho (x_n,z)$ exists and

$$0\le \frac{R}{2}{\alpha }_n(1-{\alpha }_n){\rho }^2(T(x_n),T(y_n))\le {\rho }^2(x_n,z)-{\rho }^2(x_{n+1},z)+{\alpha }_n[{\rho }^2(y_n,z)-{\rho }^2(x_n,z)].$$

So,

$${\alpha }_n[{\rho }^2(x_n,z)-{\rho }^2(y_n,z)]\le {\rho }^2(x_n,z)-{\rho }^2(x_{n+1},z).$$

Since ${lim\hspace{0.17em}inf}_n{\alpha }_n>0$, ${lim\hspace{0.17em}sup}_n[{\rho }^2(x_n,z)-{\rho }^2(y_n,z)]=0$. By (6), we have

$$\frac{R}{2}{\beta }_n(1-{\beta }_n){\rho }^2(x_n,T(x_n))\le {\rho }^2(x_n,z)-{\rho }^2(y_n,z).$$

This implies by (ii) that ${lim}_n\rho (x_n,T(x_n))=0$. By Lemma 3.5, ${\omega }_w(x_n)$ consists of exactly one point and is contained in $F(T)$. This shows that $\{x_n\}$ Δ-converges to an element of $F(T)$. □

The following lemma is also needed (cf. [[10], Lemma 4.2]).

Lemma 3.8 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let $\{x_n\}$ and $\{y_n\}$ be sequences in X with ${lim}_n\rho (x_n,y_n)=0$. If $\mathrm{\Delta }\text{-}{lim}_n{x}_n=x$ and $\mathrm{\Delta }\text{-}{lim}_n{y}_n=y$, then $x=y$.

Theorem 3.9 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let C be a nonempty closed convex subset of X, and $T,S:C\to X$ be a two generalized hybrid mappings with $F(T)\cap F(S)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be a sequence in $[0,1]$ and define a sequence $\{x_n\}$ in C by

$$\{\begin{array}{c}\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrary}},\\ x_{n+1}:=P_C((1-{\alpha }_n)x_n\oplus {\alpha }_{n}T(y_n)),\\ y_n:=P_C((1-{\beta }_n)x_n\oplus {\beta }_{n}S(x_n)).\end{array}\hfill \end{array}$$

Let $R=(\pi -2\epsilon )tan(\epsilon )$ and suppose that

• (i) ${lim\hspace{0.17em}inf}_n{\alpha }_n(1-{\alpha }_n)>0$,

• (ii) $k_2^T(z)=0$ and ${lim\hspace{0.17em}inf}_n{\beta }_n[\frac{(1-{\beta }_n)R}{2}-\frac{k_2^S(z)}{1-a_3^S(z)}]>0$ for all $z\in F(T)\cap F(S)$.

Then $\{x_n\}$ Δ-converges to a common fixed point of S and T.

Proof Let $z\in F(T)\cap F(S)$. Since $k_2^T(z)=0$, $\rho (T(x),z)\le \rho (x,z)$ for all $x\in C$. By Lemmas 2.5 and 2.6, we have

$$\begin{array}{rcl}{\rho }^2(y_n,z)& =& {\rho }^2(P_C((1-{\beta }_n)x_n\oplus {\beta }_{n}S(x_n)),z)\\ \le & {\rho }^2((1-{\beta }_n)x_n\oplus {\beta }_{n}S(x_n),z)\\ \le & (1-{\beta }_n){\rho }^2(x_n,z)+{\beta }_n{\rho }^2(S(x_n),z)-\frac{R}{2}{\beta }_n(1-{\beta }_n){\rho }^2(x_n,S(x_n))\\ \le & (1-{\beta }_n){\rho }^2(x_n,z)+{\beta }_n[{\rho }^2(x_n,z)+\frac{k_2^S(z)}{1-a_3^S(z)}{\rho }^2(S(x_n),x_n)]\\ -\frac{R}{2}{\beta }_n(1-{\beta }_n){\rho }^2(x_n,S(x_n))\\ \le & {\rho }^2(x_n,z)-{\beta }_n[\frac{(1-{\beta }_n)R}{2}-\frac{k_2^S(z)}{1-a_3^S(z)}]{\rho }^2(S(x_n),x_n).\end{array}$$

(7)

By (ii), there exist $\delta >0$ and $N\in \mathbb{N}$ such that

$${\beta }_n[\frac{(1-{\beta }_n)R}{2}-\frac{k_2^S(z)}{1-a_3^S(z)}]\ge \delta >0\text{for all }n\ge N.$$

Without loss of generality, we may assume that

$${\beta }_n[\frac{(1-{\beta }_n)R}{2}-\frac{k_2^S(z)}{1-a_3^S(z)}]>0\text{for all }n\in \mathbb{N}.$$

By (7), $\rho (y_n,z)\le \rho (x_n,z)$. Thus

$$\begin{array}{rcl}{\rho }^2(x_{n+1},z)& =& {\rho }^2(P_C((1-{\alpha }_n)x_n\oplus {\alpha }_{n}T(y_n)),z)\\ \le & {\rho }^2((1-{\alpha }_n)x_n\oplus {\alpha }_{n}T(y_n),z)\\ \le & (1-{\alpha }_n){\rho }^2(x_n,z)+{\alpha }_n{\rho }^2(T(y_n),z)-\frac{R}{2}{\alpha }_n(1-{\alpha }_n){\rho }^2(x_n,T(y_n))\\ \le & (1-{\alpha }_n){\rho }^2(x_n,z)+{\alpha }_n{\rho }^2(y_n,z)-\frac{R}{2}{\alpha }_n(1-{\alpha }_n){\rho }^2(x_n,T(y_n))\\ \le & {\rho }^2(x_n,z)-\frac{R}{2}{\alpha }_n(1-{\alpha }_n){\rho }^2(x_n,T(y_n))\\ \le & {\rho }^2(x_n,z).\end{array}$$

(8)

Hence ${lim}_n\rho (x_n,z)$ exists and

$$\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n(1-{\alpha }_n){\rho }^2(x_n,T(y_n))=0.$$

By (i), ${lim}_n{\rho }^2(x_n,T(y_n))=0$. It follows from (8) that

$$0\le \frac{R}{2}{\alpha }_n(1-{\alpha }_n){\rho }^2(x_n,T(y_n))\le {\rho }^2(x_n,z)-{\rho }^2(x_{n+1},z)+{\alpha }_n[{\rho }^2(y_n,z)-{\rho }^2(x_n,z)].$$

Thus

$${\alpha }_n(1-{\alpha }_n)[{\rho }^2(x_n,z)-{\rho }^2(y_n,z)]\le {\rho }^2(x_n,z)-{\rho }^2(x_{n+1},z).$$

Again, by (i), ${lim\hspace{0.17em}sup}_n[{\rho }^2(x_n,z)-{\rho }^2(y_n,z)]=0$. By (7), we have

$${\beta }_n[\frac{(1-{\beta }_n)R}{2}-\frac{k_2^S(z)}{1-a_3^S(z)}]{\rho }^2(x_n,S(x_n))\le {\rho }^2(x_n,z)-{\rho }^2(y_n,z).$$

This implies by (ii) that ${lim}_n\rho (x_n,S(x_n))=0$. Hence,

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\rho (y_n,x_n)& =& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\rho (P_C((1-{\beta }_n)x_n\oplus {\beta }_{n}S(x_n)),P_C(x_n))\\ \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\rho ((1-{\beta }_n)x_n\oplus {\beta }_{n}S(x_n),x_n)\\ =& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\beta }_n\rho (S(x_n),x_n)\\ =& 0.\end{array}$$

So, ${lim}_n\rho (y_n,T(y_n))=0$. By Lemma 3.5, there exist $u,v\in C$ such that ${\omega }_w(x_n)=\{u\}\subseteq F(S)$ and ${\omega }_w(y_n)=\{v\}\subseteq F(T)$. This means that $\mathrm{\Delta }\text{-}{lim}_n{x}_n=u$ and $\mathrm{\Delta }\text{-}{lim}_n{y}_n=v$. Hence, by Lemma 3.8, $u=v$ and the proof is complete. □