The Krasnoselskii–Mann Method for Approximation of Coincidence Points of Set-Valued Mappings

Abstract

In the present paper, we use the Krasnoselskii–Mann method in order to obtain approximate coincidence points of set-valued mappings in metric spaces with a hyperbolic structure.

1. Introduction

The fixed point theory of nonlinear operators has been a rapidly growing area of research. The starting point of this theory is Banach’s classical theorem [1] concerning the existence of a unique fixed point for a strict contraction. The main goals of this theory are to show the existence of a fixed point for a given nonlinear mapping and to construct an iterative process which generates an approximate fixed point and, in some cases, converges to a fixed point of the mapping [2,3,4,5,6,7,8,9,10]. Fixed point theory contains the study of various classes of nonlinear common fixed point problems which have applications in engineering, medical and the natural sciences [9,10,11,12,13,14,15,16,17,18]. In this paper, we study the Krasnoselskii–Mann method, which is of great importance and interest [19,20,21,22,23,24,25], in order to obtain approximate coincidence points of set-valued mappings in metric spaces with a hyperbolic structure. It should be mentioned that the study of coincidence points of nonlinear mappings is an important topic in fixed point theory [26,27,28,29,30,31]. Krasnoselskii–Mann iterations for a construction of approximate fixed points of a (quasi)-nonexpansive operator in a Hilbert space were independently introduced by Mann in 1953 [23] and by Krasnoselskii in 1955 [22]. Their weak convergence is well known in the literature. Krasnoselskii–Mann iterations are important for numerical optimization and variational analysis, where many problems can be reduced to finding fixed points of appropriate operators. In the current paper, our goal is to show that most of iterates obtained by the Krasnoselskii–Mann method are approximate coincidence points of set-valued mappings in metric spaces with a hyperbolic structure.

Assume that $(X,d)$ is a metric space. For each $x\in X$ and each $r>0$ set

$$B(x,r)=\{y\in X:d(x,y)\le r\}.$$

For each $x\in X$ and each set $E\subset X$ put

$$d(x,E)=inf\left\{d\right(x,y):y\in E\}.$$

Assume that $d(x,\varnothing )=\infty$, $x\in X$. For each mapping $P:X\to X$ set $P^0\left(x\right)=x$ for each $x\in X$ and for each integer $i\ge 0$ define $P^{i+1}=P\circ P^i$. Denote the cardinality of a set E with Card$\left(E\right)$. We assume that the sum over an empty set is zero and that the infimum of an empty set is equal to infinity.

A W-space is a structure $(X,d,W)$ where $(X,d)$ is a metric space and $W:X\times X\times [0,1]\to X$. We think of $W(x,y,\lambda )$ as a convex combination of the points $x,y\in X$ with the coefficients $1-\lambda ,\lambda$ and use the notation

$$W(x,y,\lambda ):=(1-\lambda )x\oplus \lambda y,x,y\in X,\lambda \in [0,1].$$

A W-space $(X,d,W)$ is called a W-hyperbolic space [32] if for all $x,y,w,z\in X$ and all $\lambda ,\tilde{\lambda }\in [0,1]$ the following properties hold:

(W1) $d(z,(1-\lambda )x\oplus \lambda y)\le (1-\lambda )d(z,x)+\lambda d(z,y);$

(W2) $d((1-\lambda )x\oplus \lambda y,(1-\tilde{\lambda })x\oplus \tilde{\lambda }y)\le \mid \lambda -\tilde{\lambda }\mid d(x,y);$

(W3) $(1-\lambda )x\oplus \lambda y=\lambda y\oplus (1-\lambda )x;$

(W4) $d\left(\right(1-\lambda )x\oplus \lambda z,(1-\lambda )y\oplus \lambda w)\le (1-\lambda )d(x,y)+\lambda d(z,w).$

It is obvious that convex subsets of normed spaces are W-hyperbolic spaces. Other examples of W-hyperbolic spaces are hyperbolic spaces [7], Busemann spaces [2], and $CAT\left(0\right)$ spaces [33].

In the following, we assume that $(X,d,W)$ is a W-hyperbolic space, which is denoted by X for simplicity.

A nonempty set $C\subset X$ is convex if for each $x,y\in C$ and each $\lambda \in [0,1]$,

$$(1-\lambda )x\oplus \lambda y\in C.$$

For each $x\in X$, each nonempty set $E\subset X$ and each $\lambda \in [0,1]$, set

$$\lambda x\oplus (1-\lambda )E=\{\lambda x\oplus (1-\lambda )y:y\in E\}.$$

The following result is well-known in the literature [34,35].

Proposition 1. 

Let $x,y,w,z\in X$ and let $\lambda ,\tilde{\lambda }\in [0,1]$. Then,

$$d(x,(1-\lambda )x\oplus \lambda y)\le \lambda d(x,y),d(y,(1-\lambda )x\oplus \lambda y)\le (1-\lambda )d(x,y)$$

and

$$d((1-\lambda )x\oplus \lambda z,(1-\tilde{\lambda })y\oplus \tilde{\lambda }w)$$

$$\le (1-\lambda )d(x,y)+\lambda d(z,w)+\mid \lambda -\tilde{\lambda }\mid d(y,w).$$

Let $X=(X,d,W)$ be a W-hyperbolic space. We assume that the W-hyperbolic space X has a structure $(X,\eta )$, where $\eta :(0,\infty )\times (0,2]\to (0,1]$ is a so-called monotone modulus of uniform convexity (see [2,36,37,38]) such that the following assumptions hold.

(B1) For each $r>0$, each $ϵ\in (0,2]$ and each $x,y,a\in X$, if

$$d(x,a),d(y,a)\le r \mathrm{and} d(x,y)\ge ϵr,$$

then

$$d(2^{-1}x\oplus 2^{-1}y,a)\le (1-\eta (r,ϵ))r.$$

(B2) For each $ϵ\in (0,2]$ and each numbers $s,r$ satisfying $0<r\le s$,

$$\eta (s,ϵ)\le \eta (r,ϵ).$$

It turns out that this class of spaces is an appropriate setting for obtaining quantitative results on the asymptotic behavior of the Mann iteration for nonexpansive mappings, as well as of the Picard iteration for firmly nonexpansive mappings [2]. It contains uniformly convex normed spaces and CAT(0) space.

In this chapter, we use the following lemma. For its proof, see Lemma 2.1 (iv) of [38].

Lemma 1. 

Let $r>0$, $ϵ\in (0,2]$, $x,y,a\in X$,

$$d(x,a)\le r,d(y,a)\le r,d(x,y)\ge ϵr.$$

Then, for each $\lambda \in [0,1]$ and each $s\ge r$,

$$d\left(\right(1-\lambda )x\oplus \lambda y,a)\le (1-2\lambda (1-\lambda \left)\eta \right(s,ϵ\left)\right)r.$$

In the following, we assume that $X=(X,d,W)$ is a W-hyperbolic space equipped with the structure $(X,\eta )$, where $\eta :(0,\infty )\times (0,2]\to (0,1]$ is the modulus of uniform convexity. We also assume that (B1) and (B2) hold, C is a nonempty convex set and $\theta \in C$. For each pair of nonempty sets $A,B\subset X$ put

$$H(A,B)=max\{sup\{d(x,B):x\in A\},sup\{d(y,A):y\in B\left\}\right\}.$$

Clearly, for each triplet of nonempty sets $A,B,D\subset X$,

$$H(A,B)=H(B,A),$$

$$H(A,D)\le H(A,B)+H(B,D).$$

2. The Coincidence Point Problem

Assume that $S:C\to C$ satisfies

$$S\left(C\right)=C,$$

(1)

$T:C\to 2^C\setminus \{\varnothing \}$, $T\left(x\right)$ is bounded for each $x\in X$, $p\in C$,

$$T\left(p\right)=\left\{S\right(p\left)\right\},$$

(2)

and that for each $x\in C$,

$$H\left(T\right(x),T(p\left)\right)\le d\left(S\right(x),S(p\left)\right).$$

(3)

Set

$$F=\{x\in C:T(x)=\{S\left(x\right)\left\}\right\}$$

and for each $ϵ>0$,

$$F_{ϵ}=\{x\in C:T\left(x\right)\subset B(S\left(x\right),ϵ)\}.$$

(4)

A point belonging to the set F is a solution of our coincidence point problem while a point that belongs to the set $F_{ϵ}$ is its $ϵ$-approximate coincidence point. In this paper, we use the Krasnoselskii–Mann method in order to generate $ϵ$-approximate coincidence points and generalize analogous results obtained in [39] in the case when for each $x\in X$,

$$T\left(x\right)=\{T_i\left(x\right):ß=1,\dots ,m\},$$

where m is a natural number, for all $i=1,\dots ,m$, $T_i:C\to C$,

$$T_i\left(p\right)=S\left(p\right)$$

and

$$d(T_i\left(x\right),T_i\left(p\right))\le d(S\left(x\right),S\left(p\right)),x\in C.$$

3. The Basic Lemma

The following lemma is an important ingredient in our study.

Lemma 2. 

Assume that $M_0>0$, $\alpha ,\gamma \in (0,1)$, $u\in C$,

$$S\left(p\right)\in B(\theta ,M_0),$$

(5)

$$S\left(u\right)\in B(\theta ,M_0),$$

(6)

$$H\left(\right\{S\left(u\right)\},T(u\left)\right)\ge \gamma ,$$

(7)

$$\xi \in T\left(u\right),$$

(8)

$$d(\xi ,S\left(u\right))\ge 2^{-1}H(\left\{S\left(u\right)\right\},T\left(u\right)),$$

(9)

$$v=(1-\alpha )S\left(u\right)\oplus \alpha \xi$$

(10)

and $w\in C$ satisfies

$$S\left(w\right)=v.$$

(11)

Then,

$$d(v,S\left(p\right))\le 2M_0$$

and

$$d(S\left(w\right),S\left(p\right))=d(v,S\left(p\right))\le d(S\left(u\right),S\left(p\right))-2^{-1}\gamma \alpha (1-\alpha )\eta (2M_0,\gamma {\left(4M_0\right)}^{-1}).$$

Proof. 

Equations (2), (3), (5), (6), (8), (10), and properties (W1), (W3) imply that

$$d(v,S(p\left)\right)=d\left(\right(1-\alpha \left)S\right(u)\oplus \alpha \xi ,S(p\left)\right)$$

$$\le (1-\alpha )d\left(S\right(u),S(p\left)\right)+\alpha d(\xi ,S(p\left)\right)$$

$$\le (1-\alpha )d\left(S\right(u),S(p\left)\right)+\alpha H\left(T\right(u),T(p\left)\right)$$

$$\le d(S\left(u\right),S\left(p\right))\le 2M_0.$$

(12)

From (2), (3), and (7),

$$\gamma \le H\left(\right\{S\left(u\right)\},T(u\left)\right)\le H\left(T\right(u),T(p\left)\right)+d\left(S\right(p),S(u\left)\right)\le 2d\left(S\right(u),S(p\left)\right),$$

$$d(S\left(p\right),S\left(u\right))\ge 2^{-1}\gamma .$$

(13)

It follows from (3), (5), and (6) that

$$H(T\left(u\right),T\left(p\right))\le d(S\left(u\right),S\left(p\right))\le 2M_0.$$

(14)

We show that

$$d\left(S\right(u),S(p\left)\right)\ge \gamma /4.$$

(15)

Assume the contrary. Then,

$$d\left(S\right(u),S(p\left)\right)<\gamma /4$$

(16)

and in view of (3) and (16),

$$H\left(T\right(u),T(p\left)\right)<\gamma /4.$$

From (2), (3), and (16),

$$H\left(\right\{S\left(u\right)\},T(u\left)\right)\le H\left(T\right(u),T(p\left)\right)+d\left(S\right(p),S(u\left)\right)\le 2d\left(S\right(p),S(u\left)\right)\le \gamma /2.$$

This contradicts (7). The contradiction we have reached proves (15). From (7), (9), and (14),

$$d(S\left(u\right),\xi )\ge 2^{-1}H(\left\{S\left(u\right)\right\},T\left(u\right))\ge 2^{-1}\gamma \ge \gamma (d(S\left(u\right),S\left(p\right)){\left(4M_0\right)}^{-1}.$$

(17)

In view of (3) and (8),

$$d(\xi ,S(p\left)\right)\le H\left(T\right(u),T(p\left)\right)\le d\left(S\right(u),S(p\left)\right).$$

(18)

Equations (10), (14), (15), (17), (18), and Lemma 1 applied with

$$x=S\left(u\right),y=\xi ,a=S\left(p\right),r=d\left(S\right(u),S(p\left)\right),$$

$$s=2M_0,ϵ=\gamma {\left(4M_0\right)}^{-1},\lambda =\alpha$$

imply that

$$d(v,S(p\left)\right)=d\left(\right(1-\alpha \left)S\right(u)\oplus \alpha \xi ,S(p\left)\right)$$

$$\le (1-2\alpha (1-\alpha )\eta (2M_0,\gamma {\left(4M_0\right)}^{-1}))d(S\left(u\right),S\left(p\right))$$

$$\le d(S\left(u\right),S\left(p\right))-2^{-1}\alpha (1-\alpha )\eta (2M_0,\gamma {\left(4M_0\right)}^{-1})\gamma .$$

Lemma 2 is proven. □

4. The First Main Result

The following theorem shows that our algorithm generates approximate solutions of the coincidence point problem.

Theorem 1. 

Assume that $M>0$,

$$S\left(p\right)\in B(\theta ,M),$$

(19)

${\left\{{\alpha }_i\right\}}_{i=0}^{\infty }\subset (0,1),$${\left\{{\gamma }_i\right\}}_{i=0}^{\infty }\subset (0,1),$

$$\underset{i\to \infty }{lim}{\gamma }_i=0,$$

(20)

${\left\{x_i\right\}}_{i=0}^{\infty }\subset C$,

$$S\left(x_0\right)\in B(\theta ,M),$$

(21)

for each integer $n\ge 0$,

$$y_n\in T\left(x_n\right),$$

(22)

$$d(S\left(x_n\right),y_n)\ge H(\left\{S\left(x_n\right)\right\},T\left(x_n\right))-{\gamma }_n,$$

(23)

$$S\left(x_{n+1}\right)=(1-{\alpha }_n)S\left(x_n\right)\oplus {\alpha }_n{y}_n,$$

(24)

$ϵ\in (0,1)$. Then,

$$d(S\left(x_n\right),S\left(p\right))\le 2M,n=0,1,\dots$$

and the following assertion holds.

1. Let $n_0$ and Q be natural numbers such that

$${\gamma }_n<ϵ/2 for each integer n\ge n_0,$$

(25)

$$\sum _{n=n_0}^{n_0+Q-1}{\alpha }_n(1-{\alpha }_n)$$

$$>8M{ϵ}^{-1}{\eta }^{-1}(6M,ϵ{\left(6M\right)}^{-1}).$$

(26)

Then, there exists an integer $n\in \{n_0,\dots ,n_0+Q-1\}$ such that

$$x_n\in F_{ϵ}.$$

2. Assume that $\Lambda \in (0,2^{-1})$,

$${\left\{{\alpha }_i\right\}}_{i=0}^{\infty }\subset (\Lambda ,1-\Lambda )$$

(27)

and $n_0$ is a natural number such that (25) holds. Then,

$$Card\left(\{n\in \{0,1,\dots \}:x_n\notin F_{ϵ}\}\right)$$

$$\le 4M{ϵ}^{-1}{\Lambda }^{-2}{\eta }^{-1}(6M,ϵ{\left(6M\right)}^{-1})+n_0.$$

Proof. 

In view of (19) and (21),

$$d(S\left(x_0\right),S\left(p\right))\le d(S\left(x_0\right),\theta )+d(\theta ,S\left(p\right))\le 2M.$$

Properties (W1) and (W3) and (2), (3), (22), and (24) imply that for each integer $n\ge 0$,

$$d(S\left(x_{n+1}\right),S\left(p\right))=d((1-{\alpha }_n)S\left(x_n\right)\oplus {\alpha }_n{y}_n,S\left(p\right))$$

$$\le (1-{\alpha }_n)d(S\left(x_n\right),S\left(p\right))+{\alpha }_{n}d(y_n,S\left(p\right))$$

$$\le (1-{\alpha }_n)d(S\left(x_n\right),S\left(p\right))+{\alpha }_{n}H(T\left(x_n\right),T\left(p\right))$$

$$\le (1-{\alpha }_n)d(S\left(x_n\right),S\left(p\right))+{\alpha }_{n}d(S\left(x_n\right),S\left(p\right))$$

$$=d(S\left(x_n\right),S\left(p\right)).$$

Thus, for each integer $n\ge 0$,

$$d(S\left(x_{n+1}\right),S\left(p\right))\le d(S(x_n,S\left(p\right))\le 2M$$

(28)

and in view of (19),

$$d(S\left(x_n\right),\theta )\le 3M.$$

(29)

Assume that a nonnegative integer $n\in \{0,1,\dots ,\}$ and that

$${\gamma }_n\le ϵ/2,x_n\ne F_{ϵ}.$$

(30)

From (4) and (30),

$$T\left(x_n\right)\setminus B(S\left(x_n\right),ϵ)\ne \varnothing ,H(\left\{S\left(x_n\right)\right\},T\left(x_n\right))>ϵ.$$

(31)

It follows from (23), (30), and (31) that

$$d(S\left(x_n\right),y_n)\ge H(\left\{S\left(x_n\right)\right\},T\left(x_n\right))-{\gamma }_n$$

$$\ge H(\left\{S\left(x_n\right)\right\},T\left(x_n\right))-ϵ/2\ge 2^{-1}H(\left\{S\left(x_n\right)\right\},T\left(x_n\right)).$$

(32)

From (19), (24), (28), (31), (32), and Lemma 2 applied with $u=x_n$, $\alpha ={\alpha }_n$, $M_0=3M$, $\gamma =ϵ$, $\xi =y_n$, and $v=(1-{\alpha }_n)S\left(x_n\right)\oplus {\alpha }_n{y}_n$ we have

$$d(S\left(x_{n+1}\right),S\left(p\right))=d((1-{\alpha }_n)S\left(x_n\right)\oplus {\alpha }_n{y}_n,S\left(p\right))$$

$$=d(v,S\left(p\right))\le d(S\left(x_n\right),S\left(p\right))-2^{-1}ϵ{\alpha }_n(1-{\alpha }_n)\eta (6M,ϵ{\left(6M\right)}^{-1}).$$

(33)

Thus, the following property holds:

(P1) if $n\ge 0$ is an integer, ${\gamma }_n\le ϵ/2$ and $x_n\ne F_{ϵ}$, then (33) is true.

Let us prove Assertion 1. Assume that it does not hold. Then, for each $n\in \{n_0,\dots ,n_0+Q-1\}$,

$$x_n\notin F_{ϵ},$$

and property (P1) and (25) imply that (33) holds. From (28) and (33),

$$2M\ge d(S\left(x_{n_0}\right),S\left(p\right))$$

$$\ge d(S\left(x_{n_0}\right),S\left(p\right))-d(S\left(x_{n_0+Q}\right),S\left(p\right))$$

$$=\sum _{n=n_0}^{n_0+Q-1}(d(S\left(x_n\right),S\left(p\right))-d(S\left(x_{n+1}\right),S\left(p\right)))$$

$$\ge 2^{-1}ϵ\left(\sum _{n=n_0}^{n_0+Q-1}{\alpha }_n(1-{\alpha }_n)\right)\eta (6M,ϵ{\left(6M\right)}^{-1}),$$

$$\sum _{n=n_0}^{n_0+Q-1}{\alpha }_n(1-{\alpha }_n)\le 4M{ϵ}^{-1}{\eta }^{-1}(6M,ϵ{\left(6M\right)}^{-1}).$$

The relation above contradicts (26). The contradiction we have reached proves Assertion 1.

Let us prove Assertion 2. Recall that

$${\gamma }_n<ϵ/2 \mathrm{for} \mathrm{each} \mathrm{integer} n\ge n_0.$$

(34)

Set

$$E=\{n\in \{n_0,n_0+1,\dots ,\}:x_n\notin F_{ϵ}\}.$$

(35)

Let $Q>n_0$ be an integer,

$$E_Q=E\cap \{0,\dots ,Q\}.$$

(36)

Property (P1) and Equations (28) and (33)–(36) imply that

$$2M\ge d(S\left(x_{n_0}\right),S\left(p\right))\ge d(S\left(x_{n_0}\right),S\left(p\right))-d(S\left(x_{Q+1}\right),S\left(p\right))$$

$$=\sum _{n=n_0}^Q(d(S\left(x_n\right),S\left(p\right))-d(S\left(x_{n+1}\right),S\left(p\right)))$$

$$\ge \sum \{d(S\left(x_n\right),S\left(p\right))-d(S\left(x_{n+1}\right),S\left(p\right)):n\in E_Q\}$$

$$\ge 2^{-1}ϵ\eta (6M,ϵ{\left(6M\right)}^{-1})ϵ\sum \{{\alpha }_n(1-{\alpha }_n):n\in E_Q\},$$

$$\ge 2^{-1}{\Lambda }^2ϵ\eta (6M,ϵ{\left(6M\right)}^{-1})\mathrm{Card}\left(E_Q\right),$$

$$\mathrm{Card}\left(E_Q\right)\le 4M{\Lambda }^{-2}{ϵ}^{-1}{\eta }^{-1}(6M,ϵ{\left(6M\right)}^{-1}).$$

Since Q is an arbitrary natural number larger that $n_0$ the relation above imply that

$$\mathrm{Card}\left(E\right)\le 4M{\Lambda }^{-2}{ϵ}^{-1}{\eta }^{-1}(6M,ϵ{\left(6M\right)}^{-1}).$$

Together with (35) this implies that

$$\mathrm{Card}\left(\{n\in \{0,1,\dots \}:x_n\notin F_{ϵ}\}\right)$$

$$\le n_0+4M{ϵ}^{-1}{\Lambda }^{-2}{\eta }^{-1}(6M,ϵ{\left(6M\right)}^{-1}).$$

Assertion 2 and Theorem 1 are proven. □

5. The Second Main Result

Assume that $S:C\to C$ satisfies

$$S\left(C\right)=C,$$

(37)

$T:C\to 2^C\setminus \{\varnothing \}$, $p\in C$,

$$S\left(p\right)\in T\left(p\right),$$

(38)

and that for each $x\in C$,

$$d\left(S\right(p),T(x\left)\right)\le d\left(S\right(x),S(p\left)\right).$$

(39)

Theorem 2. 

Assume that $M>0$,

$$S\left(p\right)\in B(\theta ,M),$$

(40)

${\left\{{\alpha }_i\right\}}_{i=0}^{\infty }\subset (0,1),$${\left\{{ϵ}_i\right\}}_{i=0}^{\infty }\subset (0,1),$

$$\sum _{n=0}^{\infty }{ϵ}_n<\infty ,$$

(41)

${\left\{x_i\right\}}_{i=0}^{\infty }\subset C$,

$$S\left(x_0\right)\in B(\theta ,M),$$

(42)

for each integer $n\ge 0$,

$$y_n\in T\left(x_n\right),$$

(43)

$$d(S\left(p\right),y_n)\le d(S\left(p\right),T\left(x_n\right))+{ϵ}_n,$$

(44)

$$S\left(x_{n+1}\right)=(1-{\alpha }_n)S\left(x_n\right)\oplus {\alpha }_n{y}_n,$$

(45)

$ϵ\in (0,1)$. Then,

$$d(S\left(x_n\right),S\left(p\right))\le 2M+\sum _{i=0}^{\infty }{ϵ}_i,n=0,1,\dots$$

and the following assertion holds.

1. Let $n_0\ge$ be an integer and Q be a natural number such that

$$\sum _{n=n_0}^{n_0+Q-1}{\alpha }_n(1-{\alpha }_n)$$

$$>(2M+3\sum _{i=0}^{\infty }{ϵ}_i){ϵ}^{-1}{\eta }^{-1}(2M+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1}).$$

(46)

Then, there exists an integer $n\in \{n_0,\dots ,n_0+Q-1\}$ such that

$$B(S\left(x_n\right),ϵ)\cap T\left(x_n\right)\ne \varnothing .$$

2. Assume that $\Lambda \in (0,2^{-1})$,

$${\left\{{\alpha }_i\right\}}_{i=0}^{\infty }\subset (\Lambda ,1-\Lambda ).$$

(47)

Then,

$$Card\left(\{n\in \{0,1,\dots \}:B(S\left(x_n\right),ϵ)\cap T\left(x_n\right)\ne \varnothing \}\right)$$

$$\le (2M+2\sum _{i=0}^{\infty }{ϵ}_i){ϵ}^{-1}{\Lambda }^{-2}{\eta }^{-1}(2M+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1}).$$

Proof. 

Let $n\ge 0$ be an integer. Properties (W1) and (W3) and (39), (44), and (45) imply that

$$d(S\left(x_{n+1}\right),S\left(p\right))=d((1-{\alpha }_n)S\left(x_n\right)\oplus {\alpha }_n{y}_n,S\left(p\right))$$

$$\le (1-{\alpha }_n)d(S\left(x_n\right),S\left(p\right))+{\alpha }_{n}d(y_n,S\left(p\right))$$

$$\le (1-{\alpha }_n)d(S\left(x_n\right),S\left(p\right))+{\alpha }_n{ϵ}_n+d(S\left(p\right),T\left(x_n\right))$$

$$\le (1-{\alpha }_n)d(S\left(x_n\right),S\left(p\right))+{\alpha }_{n}d(S\left(x_n\right),S\left(p\right))+{ϵ}_n$$

$$\le d(S\left(x_n\right),S\left(p\right))+{ϵ}_n.$$

(48)

In view of (40), (44), and (48), for each integer $n\ge 0$,

$$d(S\left(x_n\right),S\left(p\right))\le d(S(x_0,S\left(p\right))+\sum _{i=0}^{\infty }{ϵ}_i\le 2M+\sum _{i=0}^{\infty }{ϵ}_i.$$

(49)

Set

$${ϵ}_0=ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1}.$$

* Assume that $n\in \{0,1,\dots ,\}$. From (39), (44), and (49),

$$d(S\left(p\right),y_n)\le d(S\left(p\right),T\left(x_n\right))+{ϵ}_n\le d(S\left(x_n\right),S\left(p\right))+{ϵ}_n$$

$$\le 2M+2\sum _{i=0}^{\infty }{ϵ}_i.$$

(50)

Assume that

$$d(S\left(x_n\right),y_n)>ϵ.$$

(51)

From (39), (44), and (51),

$$ϵ<d(S\left(x_n\right),y_n)\le d(S\left(x_n\right),S\left(p\right))+d(S\left(p\right),y_n)$$

$$\le d(S\left(x_n\right),S\left(p\right))+ϵ+d(S\left(p\right),T\left(x_n\right))$$

$$\le d(S\left(x_n\right),S\left(p\right))+{ϵ}_n+d(S\left(x_n\right),S\left(p\right))$$

$$\le {ϵ}_n+2d(S\left(p\right),S\left(x_n\right)),$$

$$d(S\left(p\right),S\left(x_n\right))\ge 2^{-1}ϵ-{ϵ}_n/2.$$

(52)

In view of (49) and (51),

$$d(y_n,S\left(x_n\right))>ϵ=ϵ{(d(S\left(p\right),S\left(x_n\right))+{ϵ}_n)}^{-1}(d(S\left(p\right),S\left(x_n\right))+{ϵ}_n)$$

$$\ge ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1}(d(S\left(p\right),S\left(x_n\right))+{ϵ}_n).$$

(53)

From (45), (49), (50), (52), (53), relation ${ϵ}_0=ϵ{(2M+{\sum }_{i=0}^{\infty }{ϵ}_i)}^{-1}$ and Lemma 1 applied with $x=S\left(x_n\right)$, $y=y_n$, $a=S\left(p\right)$,

$$r=d(S\left(p\right),S\left(x_n\right))+{ϵ}_n,$$

$\lambda ={\alpha }_n$, $ϵ={ϵ}_0$,

$$S=2M+2\sum _{i=0}^{\infty }{ϵ}_i$$

we have

$$d(S\left(x_{n+1}\right),S\left(p\right))=d((1-{\alpha }_n)S\left(x_n\right)\oplus {\alpha }_n{y}_n,S\left(p\right))$$

$$\le (1-2{\alpha }_n(1-{\alpha }_n)\eta (2M+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ(2M$$

$$+2\sum _{i=0}^{\infty }{ϵ}_i{)}^{-1})(d(S\left(x_n\right),S\left(p\right))+{ϵ}_n)$$

$$\le d(S\left(x_n\right),S\left(p\right))-ϵ{\alpha }_n(1-{\alpha }_n)\eta (2M+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1})+2{ϵ}_n$$

and

$$d(S\left(x_{n+1}\right),S\left(p\right))$$

$$\le d(S\left(x_n\right),S\left(p\right))-ϵ{\alpha }_n(1-{\alpha }_n)\eta (2M+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1})+2{ϵ}_n.$$

(54)

Thus, the following property holds:

(P2) if $n\ge 0$ is an integer, $d(S\left(x_n\right),y_n)>ϵ$, then (54) is true.

Let us prove Assertion 1. Assume that it does not hold. Then, for each $n\in \{n_0,\dots ,n_0+Q-1\}$,

$$B(S\left(x_n\right),ϵ)\cap T\left(x_n\right)=\varnothing$$

and in view of (43),

$$d(S\left(x_n\right),y_n)>ϵ.$$

Property (P2), (49), and the relation above imply that for each $n\in \{n_0,\dots ,n_0+Q-1\}$, Equation (54) holds. From (54),

$$2M+\sum _{i=0}^{\infty }{ϵ}_i\ge d(S\left(x_{n_0}\right),S\left(p\right))$$

$$\ge d(S\left(x_{n_0}\right),S\left(p\right))-d(S\left(x_{n_0+Q}\right),S\left(p\right))$$

$$=\sum _{n=n_0}^{n_0+Q-1}(d(S\left(x_n\right),S\left(p\right))-d(S\left(x_{n+1}\right),S\left(p\right)))$$

$$\ge \left(\sum _{n=n_0}^{n_0+Q-1}{\alpha }_n(1-{\alpha }_n)\right)ϵ\eta \left(2M+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1}\right)-2\sum _{n=n_0}^{n_0+Q-1}{ϵ}_n,$$

$$\sum _{i=n_0}^{n_0+Q-1}{\alpha }_n(1-{\alpha }_n)\le (2M+3\sum _{i=0}^{\infty }{ϵ}_i){ϵ}^{-1}{\eta }^{-1}\left(2M+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1}\right).$$

The relation above contradicts (46). The contradiction we have reached proves Assertion 1.

Let us prove Assertion 2. Set

$$E=\{n\in \{0,1,\dots ,\}:B(S\left(x_n\right),ϵ)\cap T\left(x_n\right)=\varnothing \}.$$

(55)

Let Q be a natural number. Set

$$E_Q=E\cap \{0,\dots ,Q-1\}.$$

(56)

Property (P2) and Equations (43), (55), and (56) imply that for each integer $n\in E_Q$,

$$d(S\left(x_n\right),y_n)>ϵ,$$

and (54) holds. From (40), (42), (47), (48), and (54) holding for each $n\in E_Q$,

$$2M\ge d(S\left(x_0\right),S\left(p\right))\ge d(S\left(x_0\right),S\left(p\right))-d(S\left(x_Q\right),S\left(p\right))$$

$$=\sum _{n=0}^{Q-1}(d(S\left(x_n\right),S\left(p\right))-d(S\left(x_{n+1}\right),S\left(p\right)))$$

$$\ge \sum \{d(S\left(x_n\right),S\left(p\right))-d(S\left(x_{n+1}\right),S\left(p\right)):n\in E_Q\}$$

$$+\sum \{d(S\left(x_n\right),S\left(p\right))-d(S\left(x_{n+1}\right),S\left(p\right)):n\in \{0,\dots ,Q-1\}\setminus E_Q\}$$

$$\ge \sum \{{\alpha }_n(1-{\alpha }_n)\eta (2M+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1})ϵ-2ϵ:n\in E_Q\},$$

$$-\sum \{{ϵ}_n:n\in \{0,\dots ,Q-1\}\setminus E_Q\}$$

$$\ge {\Lambda }^2ϵ\eta (2M+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1})\mathrm{Card}\left(E_Q\right)$$

$$-2\sum _{n=0}^{\infty }{ϵ}_n,$$

$$\mathrm{Card}\left(E_Q\right)\le (2M+2\sum _{i=0}^{\infty }{ϵ}_i){\Lambda }^{-2}{ϵ}^{-1}{\eta }^{-1}(2M$$

$$+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1}).$$

Since Q is an arbitrary natural number the relation above imply that

$$\mathrm{Card}\left(E\right)\le (2M+2\sum _{i=0}^{\infty }{ϵ}_i){\Lambda }^{-2}{ϵ}^{-1}{\eta }^{-1}(2M$$

$$+2\sum _{i=0}^{\infty }{ϵ}_i,ϵ{(2M+2\sum _{i=0}^{\infty }{ϵ}_i)}^{-1}).$$

Assertion 2 and Theorem 2 are proved. □

6. Conclusions

In this paper, we use the Krasnoselskii–Mann method in order to generate approximate coincidence points of set-valued mappings in metric spaces with a hyperbolic structure. We generalize analogous results obtained in [39] in the case where the set-valued mapping is a union of a finite family of single-valued mappings.