Approximating common endpoints of multivalued generalized nonexpansive mappings in hyperbolic spaces

Highlights

• This paper states some sufficient conditions for the existence of endpoints for some multivalued generalized nonexpansive mappings.

• Convergence analysis of a new modified algorithm for approximating common endpoints of two generalized nonexpansive mappings in uniformly convex hyperbolic spaces is studied.

• Some numerical examples are given to substantiate the common endpoint results.

Abstract

In this paper, we introduce a new modified iteration process for approximating common endpoint of a multivalued α-nonexpansive mapping and a multivalued mapping satisfying condition (E′) in uniformly convex hyperbolic spaces. As a by-product, we improve the main results of Panyanak (2018) with a new and faster algorithm for two multivalued mappings. Moreover, we give a numerical example to substantiate our results. Our work is new and holds simultaneously in uniformly convex Banach spaces as well as CAT(0) spaces.

Introduction

Let U be a nonempty subset of a metric space (X, d). R: U → U is said to be nonexpansive if for any m, w ∈ U,

$$d(Rm,Rw)\le d(m,w).$$

If $R\left(z\right)=z$ for z ∈ U, we say z is a fixed point of R. For years, several authors have considered extensions of nonexpansive mappings on both linear and non-linear domains([13], [16]).

In 2011, Aoyama and Kohsaka [4] initiated the study of α-nonexpansive mapping. For α < 1, R: U → X is α-nonexpansive mapping if for any m, w ∈ U,

$$d^2(Rm,Rw)\le \alpha d^2(Rm,w)+\alpha d^2(Rw,m)+(1-2\alpha )d^2(m,w).$$

Ariza et al. [5] showed that if α < 0, then R becomes the identity mapping. Among various generalizations of nonexpansive mapping, this mapping is pertinent because it extends several nonlinear mappings with applications to minimization problems, variational inequality, and zeros of maximal operators(See [9], [11]).

Recently, Pant and Shukla [32] introduced a mapping R: U → U as generalized α-nonexpansive mapping if for 0 ≤ α < 1 and any m, w ∈ U, $\frac{1}{2}d(m,Rw)\le d(m,w)$ implies

$$d(Rm,Rw)\le \alpha d(Rm,w)+\alpha d(Rm,w)+(1-2\alpha )d(m,w).$$

They showed by means of an illustrative example that it is independent of (1.1).

Extending valid single-valued fixed point results to the multivalued case is natural in view of their useful applications in applied sciences(See [25], [30]).

In what follows, we shall employ the following notations:

K(U) and CB(U) will stand for the family of compact and closed and bounded subsets of U, respectively;    

$$Thedistancefromm\in XtoU=d(m,U):=\inf \left\{d\right(m,w):w\in U\};$$

   

$$TheradiusofUrelativetom=\eta (m,U):=\sup \left\{d\right(m,w):w\in U\};$$

   

$$\mathit{The}\mathit{metr}\mathit{ic}\mathit{on}\mathit{CB}\left(X\right),H(U,Q):=\max \{\underset{m\in U}{sup}d(m,Q),\underset{q\in Q}{sup}d(q,U)\}.$$

Definition 1.1

For a multivalued mapping G: X → 2^X, we say m ∈ X is (i) a fixed point of G if m ∈ Gm, (ii) an endpoint of G if $Gm=\left\{m\right\}$.

By End(G) and F(G), we mean the set of endpoints and fixed points of G, respectively.

Definition 1.2 [33]

Suppose $\underset{w\in U}{inf}\eta (w,Gw)=0$ for a multivalued mapping G: U → CB(U). Then we say G has an approximate endpoint property. Equivalently, G has approximate endpoint property if it has an approximate endpoint sequence {w_n} (i.e. $\underset{n\to \infty }{lim}\eta (w_n,G{w}_n)=0$).

Remark 1.3

Note that End(G)⊆F(G). Moroever, the existence of fixed point of G does not guarantee the existence of its endpoint(See [33]). For a single-valued mapping, these two notions coincide.

A multivalued mapping R: X → 2^X is:

• nonexpansive if

  $$H(Rm,Rw)\le d(m,w)\forall m,w\in X;$$

• quasi-nonexpansive if F(R) ≠ ∅ and

  $$H(Rm,R\overline{w})\le d(m,w)\forall m\in X,\overline{w}\in F\left(R\right).$$

Motivated by the work of Garcia-Falset et al. [14], several authors have studied multivalued mappings satisfying condition (E)(See [1], [24]).

Definition 1.4 [1]

We say a multivalued mapping G: U → CB(U) satisfies condition (E) if for some μ ≥ 1, we have

$$d(m,Gw)\le \mu d(m,Gm)+d(m,w)\text{for}\text{all}m,w\in U.$$

Contrary to Proposition 4 in [24], a multivalued mapping satisfying condition (E) need not be quasi-nonexpansive.

Example 1

Let $U=[-1,1]$ be endowed with the usual metric. Define G: U → CB(U) by

$$Gw=\{\begin{array}{cc}[-w,1],\hfill & w\in [-1,0)\hfill \\ [-1,1-w],\hfill & w\in [0,1]\hfill \end{array}.$$

It is simple to ascertain that $F\left(G\right)=[0,\frac{1}{2}]$ and G satisfies condition (E). However, for $m=-1$ and $w=0,$ we get $H(Gm,G0)=2$ and $d(m,w)=1$. Therefore, G can not be quasi-nonexpansive.

Inspired by the contributions in [2], we consider a rich subclass of mappings satisfying condition (E).

Definition 1.5

If for some μ ≥ 1, G: U → CB(U) satisfies

$$H(Gm,Gw)\le \mu d(m,Gm)+d(m,w)\text{for}\text{all}m,w\in U,$$

then, we say G satisfies condition (E′).

Example 2 [20], [37]

A multivalued generalized α-nonexpansive mapping G: U → CB(U) satisfies condition (E′). In fact, for any m, w ∈ U, we have

$$H(Gm,Gw)\le 2\frac{1+\alpha }{1-\alpha }d(m,Gm)+d(m,w).$$

Definition 1.6 [17]

A mapping R: U → CB(U) is α-nonexpansive if for α ∈ [0, 1) and any m, w ∈ C, we have

$$H^2(Rm,Rw)\le \alpha d^2(Rm,w)+\alpha d^2(Rw,m)+(1-2\alpha )d^2(m,w).$$

Hajisharifi [17] showed that the class of multivalued mean nonexpansive mappings [8] forms a subclass of the class of α-nonexpansive mappings.

We present an example to show that the set of mappings satisfying condition (E′) and the class of α-nonexpansive mappings are different.

Example 3

Let $U=\{(4,0),(4,5),(5,4),(0,0),(2,0),(0,4)\}\subset {\mathbb{R}}^2$. Define a metric $d((w_1,m_1),(w_2,m_2))=|w_1-w_2|+|m_1-m_2|$ and the mapping G by

$$Gm=\{\begin{array}{cc}\left\{\right(0,0),(2,0\left)\right\},\hfill & m\in \left\{\right(2,0),(0,4),(4,0\left)\right\}\hfill \\ \left\{\right(4,0\left)\right\},\hfill & m=(4,5)\hfill \\ \left\{\right(0,4\left)\right\},\hfill & m=(5,4)\hfill \\ \left\{\right(0,0\left)\right\},\hfill & m=(0,0)\hfill \end{array}.$$

Then G satisfies condition (E′), $F\left(G\right)=\left\{\right(0,0),(2,0\left)\right\}$ and $End\left(G\right)=\left\{\right(0,0\left)\right\}$. For $m=(5,4)$ and $w=(4,5)$ and any α < 1, we have

$$H^2(Gm,Gw)=64>4+42\alpha =4(1-2\alpha )+25\alpha +25\alpha =(1-2\alpha )d^2(m,w)+\alpha d^2(m,Gw)+\alpha d^2(Gm,w).$$

Hence, G is not α-nonexpansive.

Many researchers have shown that fixed point and endpoint exist for different multivalued contractive-type mappings(See [3], [21]). Some applications of endpoint results to optimization problems can be found in the works of Lin and Du [31] and Corley [10]. Corley, for example, showed that maximization in relation to a cone is comparable to the case of finding endpoints of multivalued mappings. By setting sufficient conditions in various spaces, several authors ([8], [15], [18], [27], [33], [38]) have established existence of endpoint of nonexpansive mappings and their generalizations.

In 2018, Panyanak [34] introduced a modified Ishikawa iteration process for approximating endpoints of multivalued nonexpansive mapping S: U → K(U) in a uniformly convex Banach space. Particularly, he showed that under some given conditions, the following algorithm converges to an endpoint of a multivalued nonexpansive mapping:

$$\begin{array}{ccc}\hfill z_n& =& {\beta }_n{y}_n+(1-{\beta }_n)w_n\hfill \\ \hfill w_{n+1}& =& {\alpha }_n{v}_n+(1-{\alpha }_n)w_n,\hfill \end{array}$$

where w₁ ∈ U, y_n ∈ Sw_n with $\parallel w_n-y_n\parallel =\eta (w_n,S{w}_n)$ and v_n ∈ Sz_n with $\parallel z_n-v_n\parallel =\eta (z_n,S{z}_n)$.

This result naturally gives rise to the following question:

Question: Can common endpoints of two multivalued nonexpansive mappings be approximated with a faster iteration process ?

For a possible solution to the above question, we follow new results of Piri et al. [36] and Iqbal et al. [20] to introduce a new algorithm for approximating common endpoint in uniformly convex hyperbolic spaces. Consequently, we find an affirmative answer. In Section 3, we show that endpoint of multivalued α-nonexpansive mappings and mappings satisfying condition (E′) exist. In Section 4, we establish convergence theorems for the proposed algorithm. In Section 5, we provide an example to substantiate our results. Moreover, we show the efficiency of our algorithm in approximating common endpoints of two nonexpansive mappings by comparing its results with common fixed point results of Fukhar-ud-din et al. [12] and Chang et al. [7].