1. Introduction
The concept of fixed points has been extensively studied for single-valued mappings, leading to classical results such as Banach’s Fixed Point Theorem and Brouwer’s Fixed Point Theorem. However, real-world problems often involve scenarios where one state can lead to multiple possible subsequent states, necessitating the use of multivalued mappings. Examples include differential inclusions, control systems, and economic models where uncertainty or multiple choices are inherent. In recent years, various classes of multivalued mappings have been introduced to generalize and extend fixed point results. Notably, demicontractive and hemicontractive mappings have been studied due to their relevance in iterative methods and convergence analysis. However, the introduction of k-strictly asymptotically demicontractive and hemicontractive mappings presents new challenges and opportunities for further generalizations.
In a space
, a distance function
d is defined as symmetric if it fulfills the condition
for every pair of points
u and
v within
. A space that has such a symmetric distance function is known as a symmetric space. It is widely recognized that Hilbert spaces and metric spaces inherently have this symmetry. However, pseudo-metric spaces do not always exhibit this symmetric property. Several authors have proved fixed point results in symmetric spaces, for instance, see [
1].
Let
£ be a metric space. Fixed point theory is interested in the solution of the equation of the type
where ᘐ is considered as a nonlinear mapping on
£. Any point
for which (
1) holds is known as an invariant point of ᘐ, and a set consisting of all such points is represented with
. Fixed point theory is no doubt an indispensable tool in the arsenal of modern analysis. Currently, there is still an intensive research in this direction, as it has several utilities in establishing the existence and uniqueness of solutions of diverse mathematically oriented models such as solutions of optimisation problems, ordinary differential equations, and variational analysis. Research has revealed that these models represent different phenomena arising in several fields such as neutron transport theory, steady state temperature distributions, economic theory, optimal control of systems, chemical equations, and the flow of fluids.
In recent times, fixed point theory for multivalued mappings has received unwavering attention from well-known mathematicians such as Nadler [
2], Kakutani [
3], Nash [
4,
5], Gaenakoplos [
6], Downing and Kirk [
7], etc.
Following the results of Nadler [
2], who initiated fixed point for multivalued mappings, subsequent interesting works have been published by various researchers: for example, Markin [
8], Hu et al. [
9], Bunyawat and Suntain [
10], Isogugu [
11], etc.
In continuation, Agwu and Igbokwe [
12] introduced a new class of type-one asymptotically demicontractive multivalued mappings, which guarantees strong convergence without the condition that the sum of the finite family of the iteration parameters is unity. Moreover, Agwu and Igbokwe [
13] have obtained convergence results for finding common solutions for minimization problems and fixed point problems (FPPs) for asymptotically quasi-nonexpansive multivalued mappings (AQNMMs).
It is an established fact that the process involved in approximating fixed points of nonlinear single-valued mappings are relatively easier than that of multivalued mappings. Nevertheless, several researchers in this direction have deeply investigated fixed point results involving multivalued mappings due to their usefulness in the field of game theory and market economy (see, e.g., [
4,
5]), non-smooth differential equations (see, for instance, [
14]), control theory, convex optimization, variational inequalities, and differential inclusions (see [
2,
8,
15,
16,
17,
18] for further study).
Let
be a normed space and
. Then,
K is known as proximinal if we can find a point
that satisfies the identity
for each
. Let
be a (uniformly convex) Banach space and
. If
is either close and convex or weakly compact, then
is proximinal.
Throughout this paper, we denote the family of nonempty bounded closed subsets of
by
, the family of nonempty compact subsets of
K by
, the family of nonempty subsets of
by
, and the family of nonempty bounded proximinal subsets of
K by
. The Hausdorff distance induced by the metric
on
for any
is defined as follows:
where
is the distance of the point
℘ from the subset
B. Throughout the remainder of this section of the paper,
implies
.
Let be a multivalued mapping on , where denotes the domain of ᘐ (the range of ᘐ will be represented with ). Then, is said to be an invariant point of the multivalued mapping ᘐ if The set of invariant points of ᘐ is represented with . A point is known as a strict invariant point of ᘐ if .
Definition 1. ᘐ is known according to the following:
- (1)
uniformly ν-Lipschitzian if we can find for which the inequalityholds. If ν is replaced with in (4), where with , then ᘐ becomes a multivalued asymptotically nonexpansive mapping . - (2)
type-one [11] if, given and , we can find for which the inequalityholds, where - (3)
η-strictly asymptotically pseudocontractive (η-SAPC) if we can find a sequence with and a constant such that, given and we can find for which the inequalities andhold. If in (6), then ᘐ becomes asymptotically pseudocontractive, whereas ᘐ is 0-SAPC (or ASM) if in (6). - (4)
β-strictly asymptotically pseudononspreading (β-SAPN) if we can find a sequence with and a constant such that given and we can find for which the inequalities andhold. Note that for in (7), ᘐ is called asymptotically pseudononspreading. ᘐ is called 0-SAPN if in (7). - (5)
asymptotically demicontrctive (ADM) if ᘐ admits a nonempty fixed point set and Definition 1 hold; that is, ᘐ is asymptotically demicontractive (see, [12]) if and, and , we can find a sequence with and for which the inequalityholds.
Remark 1. If in Definition 1, then and reduce to . Therefore, the class of multivalued ADMs includes the classes of multivalued η-SAPC mappings and multivalued β-SAPN mappings.
At this point, we must mention that assuming the fixed point of a particular operator during the approximation process does not guarantee sound judgment in application. In this regard, Lim [
17] demonstrated a remarkable conclusion by establishing the existence of a fixed point for a multivalued nonexpansive mapping. Subsequently, several researchers have employed different fixed point algorithms to estimate the fixed point of multivalued nonexpansive mappings (MNN) using the Hausdorff metric (pick out [
11,
13,
18,
19] and the references therein for more clarification).
In the midst of this, Sastry and Babu [
20] initiated the notion of a Mann and Ishikawa approximation sequence as follows:
Let
. The Mann iteration (MI) sequence developed from an arbitrary
be given by
where
is such that
and
is a real sequence in
with
.
The sequence of iterates due to Ishikawa (SII), developed from an arbitrary
, is defined as
where
,
are such that
, and
are real sequences in
for which
are guaranteed. Using (
9) and (
10), the results obtained in [
20] were generalized by Panyanak [
21].
The following lemma, which was later used to modify the sequence of iterates defined by (
9) and (
10) (see [
22]), was due to Nadler [
2].
Lemma 1. Let and . If , then there exists such that In addition to the modification of (
9) and (
10) using (
11), Song and Wang [
22] observed that generating a Mann and Ishikawa sequence of iterates in [
20] is in some sense connected to the awareness of the invariant point. They presented their iteration scheme as follows:
Let
, and
with
. Choose
. Let
and choose
such that
and
Choose
such that
and
Choose
such that
and
Continuing in this manner, we obtain by induction that
where
are such that
, and
and, satisfying
. Using (
12), the authors in [
22] proved the following theorem:
Theorem 1 ([
22])
. Let be a Banach space and be compact and convex. Assume is an MNN such that and for all Then the iteration technique given by (12) guarantees a strong convergent point of . In [
23], the following observation was made: If
is any multivalued mapping where
is a normed space, then
given by
has the property that
for each
℘ and for all
. Using this idea, the restriction
for all
imposed on
by Song and Wang [
22] was removed.
In [
15], Khan and Yildrim initiated the following scheme for multivalued nonexpansive mappings:
where
, and
. Further, by means of a lemma introduced in [
24], the notion of removing the restriction imposed on
(i.e.,
) initiated in [
23], and the method of direct formulation of a Cauchy sequence established in [
25], they proved the following theorem:
Theorem 2 (Theorem 1, [
25])
. Let , satisfying Opial condition and admitting is closed and convex, as described above. Let be such that and is nonexpansive. Let be as described by (13). Let be demiclosed at zero. Then, converges weakly to an element of . In [
26], Isiogugu demonstrated a remarkable observation that gave birth to a new class of multivalued mappings known as nonexpansive-type (MNT), strictly pseudocontractive-type (MKSPT), and pseudocontractive-type (MPT). These classes of mappings stem from the fact that there are many multivalued mappings in which neither ᘐ nor
are nonexpansive. Precisely, the following definitions were given:
Definition 2. Let be as described above. A mapping is known as MKSPT in the sense of Browder and Petryshyn [
27]
if, for and , there exist and satisfying such that Note that ᘐ becomes MPT if
and MNT if
in (
14). It is not hard to see from Definition 2 that
The relationship illustrated by (
15) was further buttressed with examples (see, for instance, Example 1 and Example 2) in [
26].
Considering the invaluable nature of multivalued nonlinear mappings in a practical sense, it becomes pertinent to ask the following question:
Question. Is it possible to obtain the class of multivalued asymptotically k-strictly demicontractive mappings for which the associated fails to be asymptotically nonexpansive?
Motivated and inspired by the above results, in this study, we first present novel classes of multivalued asymptotically
k-strictly demicontractive-type mappings (MASDTMs) and multivalued asymptotically hemicontractive-type mappings (MAHTMs), which includes the class of multivalued asymptotically nonexpansive-type mappings (MANTMs). Also, we show that the class of MASDT mappings and the class of mappings studied in [
26] are independent (see our examples at the end of this paper).
In addition, if is weakly closed ( denoting a Hilbert space) and is a MASDTM, then is demiclosed at zero, where I represents the identity mapping on K. Finally, we present weak and strong convergence results for the aforementioned classes of mappings without any compactness assumption and without the application of Condition on the domain of the mappings via Mann and Ishikawa sequence of iterates. Our main results extend, improve, and generalize several results on single-valued and multivalued mappings in the current literature.
The organization of this paper is as follows: In
Section 2, we provide some necessary preliminaries that will be helpful in proving our main results. In
Section 3, strong and weak convergence of the new classes of mappings are obtained in the framework of
. Some corollaries, as direct consequences of our main results, are provided.
3. Results
Definition 6. Let £ be a normed space and be a multivalued mapping. Then, ᘐ is called a k-strictly asymptotically demicontractive-type mapping (kSADTM) in the sense of Isogugu et al. [31] if and, for all , we can find and a sequence with such thatIf , then ᘐ is called an asymptotically hemicontractive-type multivalued mapping (AHTM). Remark 3. From Definition 6, one can see that every multivalued asymptotically quasi-nonexpansive-type mapping (MAQNTM) is a kSADTM. The examples below show that the class of MAQNTMs is properly included into the class of kSADTMs and that the class of kSADTMs is properly included into the class of asymptotically hemicontrctive-type mappings (AHTMs).
Example 2. Let be endowed with the usual metric. We define the mapping byObserve that . For n odd , we obtainNow,Additionally, sinceIt follows from (23) and (24) that Therefore, ᘐ is a kSADTM with and . Observe that as . Consequently, ᘐ is an AHTM.
Example 3. Let be endowed with the usual metric. We define the mapping byThen, for n odd , we obtainThen, and for all . Additionally, observe that ; hence, it is not asymptotically quasi-nonexpansive. Indeed, For , we haveandTherefore, Therefore, ᘐ is a kSADTM with and . Note that ᘐ, not being an MAQNTM, demonstrates the conclusion that the class of MAQNTMs is included in the class of multivalued kSADTMs. The example that follows demonstrates the fact that the class of multivalued AHTMs possessed the class of multivalued kSADTMs.
Example 4. Let be endowed with the usual metric and define the mapping byThen, for n odd , we obtain Observe that . Now, for any , we haveSinceit follows from the last inequality thatTherefore, ᘐ is an AHTM with but not a kSADTM. Proposition 1. Let be a real Hilbert space, be compact and convex, and be an asymptotically k-strictly demicontractive-type mapping such that . Then, is closed, and is closed and convex.
Proof. Let such that converges to . It suffices to show that .
Since
it follows from the definition of the multivalued asymptotically demicontractiveness of ᘐ that
Thus, taking limits as
, we obtain that
Therefore, .
(a) The proof that is closed follows an identical technique as the one described by (b) above. It is left for us to demonstrate the convexity of . Let and . Then, and . Now,
Since
and
, it follows from the demicontractiveness of ᘐ and (
26) that
Thus, taking limits as
, we obtain that
Therefore,
. Additionally, if
, then we have
Consequently,
. □
Next, we prove the following important property of asymptotically demicontractive-type mappings.
Proposition 2. Let be as in Proposition 1, be compact and convex, and be an asymptotically k-strictly demicontractive-type mapping. Then, ᘐ is uniformly -Lipschitz at a fixed point; i.e., for all , there exists such that Proof. Since ᘐ is an asymptotically demidocontractive-type mapping, it follows that there exists
and a real sequence
with
such that for any
, we have
From the last inequality, we obtain
It then follows that
Therefore,
□
Proposition 3. Let be as in Proposition 1 and be weakly closed. Let be a multivalued mapping. Suppose is a kSADM. Then is demiclosed at zero.
Proof. Let
and
with
, such that
. We want to show that
(i.e.,
for a few
). Since
, boundedness is assured. Let
be arbitrary for all
. By Definition 6, for each
n, we can find
, which guarantees the inequalities
and
for all
. Now, since
, it follows that
For each
, define
by
Then, from Lemma 4, we obtain
Thus, for any
,
Consequently,
Further, observe that
It then follows from (
29) and (
30) that
Hence,
and
.
By Proposition 2,
is uniformly
L-Lipschitzian. Using this fact, we obtain the following estimates:
Since
, it follows from (
31) that
and so
as required. □
Theorem 3. Let be as in Proposition 1 and be closed and convex. Let ᘐ be as described in Proposition 2 such that with , , and for all . Suppose is demiclosed at zero. Then, the Mann-type iteration sequence generated from an arbitrary byconverges weakly to , where and , satisfying the conditions: . Proof. From
which is valid
, the following estimates ensued:
It follows from (
32) and Lemma 2 that
exists and hence
is bounded. Additionally, since
and
from condition
, it follows that
Again, since
K is closed and
with
bounded, we can find
such that
as
. In addition,
Using the fact that
is demiclosed at zero, we obtain
. Since
satisfies the Opial condition (see [
32]), we obtain that
converges weakly to
. □
Corollary 1. Let , K, and ᘐ be as described in Theorem 3. Suppose that is a multivalued kSADTM such that with , , and for all . Suppose is weakly demiclosed at zero. Then, the Mann-type iteration sequence generated from an arbitrary byconverges weakly to , where and , satisfying the conditions: . Proof. The proof of the above corollary immediately follows from Lemma 3, Proposition 3, and Theorem 3. □
Remark 4. Clearly, Examples 2 and 4 validate the requirement “given any pair and with , there exists with validating the conditions of Definition 6”. In addition, if ᘐ is a multivalued mapping such that is an asymptotically hemicontractive-type mapping, then, given any pair and with the corresponding validating the requirements of Definition 6, it is the case that and .
Dwelling on Remark 4, we present the following convergence results for the new class of multivalued asymptotically hemicontractive-type mappings (MAHTMs) based on these requirements: (i) given any pair and with , there exists with validating the requirements of Definition 6; (ii) for all , which leads to the case for an arbitrary multivalued mapping ᘐ for which is an MAHTM devoid of the requirements of as a corollary.
Theorem 4. Let be as described in Theorem 3. Suppose that is an L-Lipschitzian asymptotically hemicontractive-type multivalued mapping such that with , and for all . Then, the Ishikawa-type iteration sequence generated from an arbitrary byconverges strongly to , where and , satisfying the condition of Lemma 3, and and , satisfying the following conditions: . Moreover,
From (
34) and (
35), we obtain
Since
it follows from (
36) that
It then follows from the fact that
and Lemma 2 that
exists. Consequently,
is bounded, and so are the sequences
and
.
Next, we claim that the
is Cauchy. To prove our claim, observe in (
38) that
Now, setting
and using the fact that
, we obtain from (
39) that
which for
yields
Let
. Then, given any
, it follows from the fact that
that we can find a positive number
and a point
such that
Hence, for
, we obtain that
Thus,
is a Cauchy sequence in
K and converges to some
because
K is closed. Suppose
. Then, it is left for us to show that
. But, given any
, we can find a positive integer
for which
Similarly, we can find
for which
From the above estimates, we get
Hence,
. Since
exists, we have that
converges strongly to
. □
Corollary 2. Let and K be as described in Theorem 3. Let be a multivalued mapping such that . Suppose is an L-Lipschitzian and asymptotically hemicontractive-type mapping. Then, the Ishikawa-type iteration sequence generated by an arbitrary defined byconverges strongly to . Proof. By applying Lemma 3 and Theorem 4, the proof is completed. □
Remark 5. We remark that the results obtained in the paper can be applied in the following way:
- 1.
It is known that the class of asymptotically k-strictly demicontractive-type mappings contains the classes of asymptotically quasi- nonexpansive-type and quasi- nonexpansive-type mappings. Thus, the results established in the paper also hold for these classes of mappings so long as the indicated conditions are fulfilled.
- 2.
Since every asymptotically k-strictly pseudocontractive-type multivalued mapping ᘐ with and , , is an asymptotically k-strictly demicontractive-type multivalued mapping, our results can be applied for this class of mappings and hence for asymptotically nonexpansive multivalued mappings provided that the specified conditions are fulfilled.
- 3.
Since every asymptotically pseudocontractive-type multivalued mapping ᘐ with and , , is a asymptotically hemicontractive-type multivalued mapping, our results can be applied for this class of mappings provided that the specified conditions are fulfilled.