Approximating common endpoints of multivalued generalized nonexpansive mappings in hyperbolic 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,If 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,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, impliesThey 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; Definition 1.1 For a multivalued mapping G: X → 2X, we say m ∈ X is (i) a fixed point of G if m ∈ Gm, (ii) an endpoint of G if . By End(G) and F(G), we mean the set of endpoints and fixed points of G, respectively. Definition 1.2 [33] Suppose 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 {wn} (i.e. ). 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 → 2X is:
- (1)
nonexpansive if
- (2)
quasi-nonexpansive if F(R) ≠ ∅ and
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
Contrary to Proposition 4 in [24], a multivalued mapping satisfying condition (E) need not be quasi-nonexpansive. Example 1 Let be endowed with the usual metric. Define G: U → CB(U) byIt is simple to ascertain that and G satisfies condition (E). However, for and we get and . 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) satisfiesthen, 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 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
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 . Define a metric and the mapping G byThen G satisfies condition (E′), and . For and and any α < 1, we haveHence, 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:where w1 ∈ U, yn ∈ Swn with and vn ∈ Szn with .
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].
Section snippets
Preliminaries
We present some basic definitions and useful technical lemmas:
Existence of endpoint
Lemma 3.1 Let U be a nonempty subset of a metric space (X, d). For a multivalued α-nonexpansive mapping G: U → CB(U), we have Proof For any m, w ∈ X, Lemma 2.6 implies thatBy the definition of the mapping G, we haveUsingin (3.2), we have
Convergence analysis
By Theorems 3.3–3.4, we can easily deduce the following technical result. Lemma 4.1 Let U be a nonempty closed and convex subset of X and G: U → K(U) be one of the following: Multivalued α-nonexpansive mapping; Multivalued mapping satisfying condition (E′).
If a sequence {mn} in X △-converges to m and then m ∈ End(G).
Inspired by the works of Iqbal et al. [20] and Panyanak [34], we come up with a new modified algorithm to approximate common endpoint of two multivalued mappings G,R: U → K(U
Numerical experiment
Example 6 Let be endowed with the usual metric. Define R: U → K(U) and G: U → K(U) as follows :
Note that and U is compact. Moreover, it is easy to show that R and G are multivalued nonexpansive mappings.
We note that for any m ∈ [0, 1] and w ∈ Rm, the equation implies that . Moreover, for any m ∈ [0, 1], define w ∈ Gm as follows :
Now,
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