1. Introduction
In 1964, Edelstein [
1] proposed the existence of a fixed point (FP) in a non-expansive mapping
T with a non-empty
T-closure. The basic concept of asymptotically non-expansive mappings was first introduced and thoroughly explained by Goebel and Kirk [
2]. After this, many authors proved various FP results by using a class of asymptotically non-expansive mappings. Some of these contributions are listed here:
Nanjaras and Panyanak [
3] established the principle of demiclosedness for single-valued asymptotically non-expansive mappings in CAT(0) spaces.
Alber et al. [
4] initiated the idea of total asymptotically non-expansive mappings and approximated the FP for these mappings.
Strong and weak convergence for asymptotically non-expansive mappings have been established in hyperbolic spaces; for example, see [
5,
6].
In 1969, Nadler [
7] presented an FP result for multi-valued contractions. This article brought a revolution in the area of FP theory, as well as applications in multiple disciplines. (For more details, readers are referred to [
8,
9].) Khan et al. [
10] evaluated the common FPs for the two multi-valued non-expansive mappings in hyperbolic spaces by using a unit-step implicit algorithm. Zhang et al. [
11] proved the strong convergence result for multi-valued total Bregmann quasi-asymptotically non-expansive mappings. In [
12], Khamsi and Khan generalized the results of [
2] by introducing the class of multi-valued asymptotically non-expansive mappings.
In 2008, Jachymski [
13] presented an innovative generalization of the Banach contraction principle by merging the notions of FP theory with graph theory. Furthermore, Beg et al. [
14] utilized the idea of Jachymski toward the general class of multi-valued contractions. In 2015, Alfuraidan and Khamsi [
15] proved an existence result for the newly introduced structure of a monotone, which increased the
G-non-expansive mappings in the setting of hyperbolic metric spaces. In [
16], Panyanak and Suantai provided an extension of Wangkeeree and Preechasilp’s result [
17] by utilizing multi-valued non-expansive mappings. In [
18], Anakkamatee and Tongnoi extended Browder’s convergence result for the collection of
-non-expansive mappings in
spaces. In [
19], Chifu et al. applied an FP theorem for an appropriate operator on the Cartesian product of a
b-metric space in the presence of a graph. Afterward, numerous FP results for generalized metric spaces have been equipped with graphs and have flourished (see, e.g., [
20,
21,
22,
23,
24]).
In this article, inspired by the abovementioned developments, some strong convergence theorems for the class of asymptotically
-non-expansive mappings in the setting of uniformly convex hyperbolic metric space are presented. These results will provide the generalizations of the consequences of Goebel and Kirk [
2], Khamsi and Khan [
12], and many others.
2. Preliminaries
The theory of multi-valued mappings is a compelling fusion of analysis, topology, and geometry. It has been receiving an degree of important attention by researchers working in a variety of fields in the mathematical sciences. All mappings that are single-valued in traditional analysis are inherently multi-valued, whereas many problems in applied mathematics are multi-valued in nature. For example, the problems of stability and control theory can be solved with the aid of FP methods for multi-valued mappings. The inverse of a single-valued map is the first naturally occurring instance of a set-valued map. The importance of multi-valued mappings can be judged by a beginner when they look at the inverse of basic trigonometric functions (for example, , , etc., in a given domain of 0 to are multi-valued mappings).
In this article, we consider a useful metric known as the Hausdorff–Pompeiu distance function on the collection of non-empty bounded and closed subsets of a metric space to generalize some FP findings in a traditional single-valued F.P theory.
Consider two non-empty sets X and Y. Suppose we have a function T that maps elements from X to a collection of subsets in Y. For any x in X, is a set contained in Y, and this is called the image of x under T. If a point x of X is an element of , it is referred to as an FP of T. In the following, some examples of multi-valued mappings have been provided with regard to the existence and uniqueness of their FPs.
Let us start with an illustration of the usual problems involving multi-valued mappings. For the two sets
X and
Y, a multi-valued mapping is a set valued function from
X to
and the power set of
Y. Consider a function
, such that
. Then,
T is a multi-valued mapping with
Here, it should be noted that
and
, that is
T, have FPs.
Now, for
and
, suppose
is a multi-valued map defined as
Then,
F has a unique FP
, that is,
.
Suppose that
and
is defined as
Then,
has infinitely many FPs
for all
.
Now, if
and the multi-valued map
are defined as
then the whole domain of
H form the set of FPs.
The graphs of the functions defined by (
1)–(
3) are depicted below as
Figure 1,
Figure 2 and
Figure 3, respectively. It is evident that these graphs exhibit multi-valued behavior and possess FPs.
Consider a metric space , and let A be any non-empty subset of X. We symbolize a couple of collections of subsets as follows:
: all of the non-empty subsets of A;
: all of the non-empty, convex, and compact subsets of A;
: all of the non-empty, closed subsets of A;
: all of the non-empty, closed, and bounded subsets of A;
: all of the non-empty, closed, and convex subsets of A;
: all of the non-empty, closed, convex, and bounded subsets of A.
Definition 1. Let A self map g on A is named a contraction if a constant exists such that we have the following:If the above inequality is accurate for , then the map g is called non-expansive. An FP of g is an element x in A, for which The generalized multi-valued Hausdorff distance
is given as follows:
where
and
.
In 1988, Assad [
25] initiated the notions of an
-general orbit and an
-starred general orbit. These concepts were further generalized by Rus [
26] in terms of a generalized orbit. Afterward, many authors utilized this idea in subsequent directions (see, for example, [
8,
27,
28]).
Definition 2. Let and be a multi-valued mapping. For , a generalized orbit of x is the sequence that is generated from by for any nonnegative integer n. Evidently, the generalized orbits generated from x may differ in values for a given .
Remark 1. It was observed that, for a single-valued mapping T, the generalized orbit coincides with the conventional definition of an orbit.
The class of asymptotically non-expansive mappings has been playing a vital part in the advancement of FP theory due to it being a generalized version of non-expansive mappings ([
6,
9], and many others). In 2017, Khamsi and Khan [
12] extended the idea of asymptotically non-expansive mappings for multi-valued cases. The authors proposed the solutions for some problems that are related to these mappings in the context of [
8]. The following definition and theorems have been taken from [
12]:
Definition 3. We say that a mapping is multi-valued asymptotically non-expansive if there exists a sequence of positive real numbers such that . In addition, for any generalized orbit of x and for any , there exists a generalized orbit of y such thatwhere . In simpler terms, this means that the mapping T does not increase the distances between points in X as they are iterated along their generalized orbits, and the rate at which distances increase is controlled by the sequence .
Convexity is an important concept in mathematics and optimization theory that characterizes the curved form of certain geometric shapes or functions. A set or function is said to be convex if every point on a line segment joining two points in the set or on the graph of the function lies within the set or above the graph. The concept of convexity finds broad applications in different areas, including economics, optimization, and physics. For example, convex optimization problems arise in many engineering and financial applications, and convex functions are used to model the behavior of various physical systems.
For the introduction of convex structure in metric spaces, Menger [
29] considered the concept of metric segments as a vital component. The element
in the metric segment
was defined in terms of
where some
are unique. A metric space along with these groups of segments is understood as a convex metric space. If the subsequent axiom holds
for all
, and
, then the space is termed a hyperbolic metric space [
30].
Definition 4. In a hyperbolic metric space , the modulus of uniform convexity is defined as per the following:which applies for any and . The space is understood as uniformly convex provided that , whenever and .
Throughout this article, our underlying space is supposed to be a complete uniformly convex hyperbolic metric space, which is abbreviated as CUCHMS.
Theorem 1 ([
12]).
For CUCHMS X, the following assertions hold.- 1.
X has the property , i.e., any decreasing sequence of non-empty, convex, bounded, and closed sets that have a non-empty intersection.
- 2.
If then any type function attains a minimal point u in Z that is unique, thereby satisfying Furthermore, any minimizing sequence in Z is convergent, that is, .
- 3.
Let and . Suppose and are any two arbitrary sequences in X satisfying then .
Definition 5. Consider a multi-valued mapping and a sequence in C. Then, T is called H-continuous if whenever converges to x in C, we havefor any sequence , where belongs to the set , for all . Remark 2. - 1.
In the case of a compact valued operator T, H-continuity coincides with the lower and upper semi-continuity.
- 2.
An asymptotically non-expansive map always fulfills the criterion of H-continuity.
Theorem 2. Let Then, an asymptotically non-expansive map attains an FP.
According to [
13], the following concepts are defined with respect to CUCHMS.
Let symbolize the diagonal of the Cartesian product . Suppose that characterizes a directed graph (whereby represents vertices, and represents edges), which includes all the loops when assuming that does not have any parallel edges, and where the symbol designates the undirected graph associated with .
Definition 6. A self map on X is known as the Banach -contraction if it fulfills the following axioms
- 1.
The edges of under are preserved, that is, for all elements in X, such that - 2.
The corresponding weights of edges of under decrease in a subsequent manner, that is, an element exists by satisfying
3. Convergence Results for Multi-Valued -Asymptotically Non-Expansive Mappings
In this section, we define the notion of multi-valued
-asymptotically non-expansive mappings by combining the concept of asymptotically non-expansive mappings with a graph. We also list two main conditions (namely
and
), which will be utilized further. In 2017, an extension of Goebel and Kirk’s FP theorem for multi-valued asymptotically non-expansive mappings has been proposed by Khamsi and Khan [
12]. Inspired by this work, we are also extending this classical result for the class of multi-valued
-asymptotically non-expansive mappings in the setting of CUCHMS.
Definition 7 (Multi-valued -Asymptotically Non-expansive Mapping). Let represent a directed graph on X. Then, a mapping is said to be a multi-valued -asymptotically non-expansive mapping if the following conditions hold:
- 1.
There exists with ;
- 2.
preserves the edges, that is, where is an element of and belongs to
- 3.
Let q and r be any two elements of X. Then, for any generalized orbit of q, there exists a generalized orbit of r such that and
Condition ). Let represent a directed graph on X. Let and be a generalized orbit of q in X. Then, the type functionattains a minimum point z in Z, which is unique, that is, for any convergent minimizing sequence in Z, whereCondition (). Let represent a directed graph on X and . Let and be the generalized orbit of q. Then, for , we have - (i)
,
- (ii)
.
Theorem 3. Let be the directed graph on X, and let , such that . Let , and, for any let be a generalized orbit of q that satisfies Condition (), such that . If is an H-continuous -asymptotically non-expansive mapping, then has an FP.
Proof. Suppose that
and
is a generalized orbit of
The boundedness of
Z ensures the boundedness of
. Consider a type function
produced by
, that is,
. By Condition
,
has a unique minimum point
in
Z. Let
. Then, by Condition
, we have
. Since
is
-asymptotically non-expansive, one has
This ensures that
is for all
. Since
, we achieve that
is a minimizing sequence for
as well. Again, by utilizing Condition
, we obtain that
is convergent to
. Then, the
H-continuity of
and
for any
implies the following:
Since
is closed and
is convergent toward
, we determine that
z exists in
, thus indicating that
is the FP of
. □
Before stating the next result, we will form a sequence with the help of generalized orbits and Condition . This formation will be utilized in the upcoming result.
Let , be a -asymptotically non-expansive mapping with and . Suppose is a generalized orbit of .
Set , then, by Condition , we have
- (i)
;
- (ii)
.
Assume that
is the generalized orbit of
. Since
is a
-asymptotically non-expansive mapping, we therefore have
Set
. Then, again by Condition
, we obtain
- (i)
- (ii)
and also
By repeating the above steps, we create a sequence
in
Z and respectively
for any
as the generalized orbit of
thereby satisfying
and
Theorem 4. Let be the directed graph on X, and let . Let be a -asymptotically non-expansive mapping. Assume that is an FP of , thereby satisfying . Assume that
- (i)
is the Lipschitz sequence associated with and
- (ii)
the series is convergent.
Suppose , , is a generalized orbit of and , and let this be the sequence generated by Equation (4), such that for each . Then,thus implying , that is, that will be an approximated FP sequence of . Proof. In view of
we have
Using Equation (
4) and the definition of CUCHMS, we obtain
From the above inequality, we obtain
for any
and
, which indicates the diameter. As we have
we can write
for any
By letting
s approach infinity, one obtains
Now, by letting
t approach infinity, and by using the given assumption, we have
thus implying the convergence of the sequence
Assume that
.
If
, then it follows from Inequality (
5) that
Then,
By using Theorem 1,
Now, consider the case for
By repeating the above steps, we have
With the selection of our generalized orbits, we now assert that
Evidently, we now have
for any
, which ultimately implies
□