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Fixed point results involving a finite family of enriched strictly pseudocontractive and pseudononspreading mappings
Journal of Inequalities and Applications volume 2024, Article number: 58 (2024)
Abstract
In this study, we introduce a method for finding common fixed points of a finite family of \((\eta _{i}, k_{i})\)-enriched strictly pseudocontractive (ESPC) maps and \((\eta _{i}, \beta _{i})\)-enriched strictly pseudononspreading (ESPN) maps in the setting of real Hilbert spaces. Further, we prove the strong convergence theorem of the proposed method under mild conditions on the control parameters. Our main results are also applied in proving strong convergence theorems for \(\eta _{i}\)-enriched nonexpansive, strongly inverse monotone, and strictly pseudononspreading maps. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature.
1 Introduction
The process of finding solutions to real-life problems has continually defied human intellect. To develop an approach for solving nonlinear problems, fixed point algorithmic technique has emerged as one of the indispensable tools. Consequently, it has drawn the attention of well-established mathematicians all over the world (see, for instance, [1–9] and the references therein), considering the vast applications of results obtained through this means in diverse fields, from pure mathematics to engineering to applied mathematics.
In this paper, we assume that \(\mathcal{H}\), with inner product \(\langle \cdot ,\cdot \rangle \) and induced norm \(\Vert \cdot \Vert \), is a real Hilbert space, and \(\emptyset \neq \Omega \subset \mathcal{H}\) is closed and convex; \(\mathcal{N}\) and \(\mathcal{R}\) will represent the set of all positive integers and the set of real numbers, respectively. If \(\Im :\Omega \longrightarrow \Omega \) is a nonlinear map, then \(F(\Im )=\{\wp \in \Omega :\Im \wp =\wp \}\) will denote the set of fixed point of ℑ.
Definition 1.1
Recall that the map \(\Im :\Omega \longrightarrow \Omega \)
-
1.
is known as nonexpansive if
$$ \Vert \Im \psi -\Im \wp \Vert \leq \Vert \psi -\wp \Vert ,\quad \forall \psi ,\phi \in \Omega ; $$(1.1) -
2.
is called quasi-nonexpansive if \(F(\Im )\neq \emptyset \) and \(\forall (\psi ,\vartheta )\in \Omega \times F(\Im )\), we have
$$ \Vert \Im \psi -\vartheta \Vert \leq \Vert \psi -\vartheta \Vert ; $$(1.2) -
3.
is called nonspreading [10] if \(\forall \psi ,\wp \in \Omega \), we have
$$ 2 \Vert \Im \psi -\Im \wp \Vert ^{2}\leq \Vert \Im \psi -\wp \Vert ^{2} + \Vert \Im \wp -\psi \Vert ^{2}; $$(1.3) -
4.
is called β-strictly pseudononspreading [11] if there exists \(\beta \in [0,1)\) such that \(\forall \psi ,\wp \in \Omega \),
$$ \Vert \Im \psi -\Im \wp \Vert ^{2}\leq \Vert \psi -\wp \Vert ^{2}+ \beta \Vert \psi -\Im \psi -(\wp -\Im \wp ) +2\langle \wp -\Im \wp , \psi -\Im \rangle . $$(1.4)
It is not difficult to show that (1.3) is equivalent to
Remark 1.1
It is easy to see from Definition 1.1\([(3) \text{and} (4)]\) that
-
(a)
if (1.3) holds and \(F(\Im )\neq \emptyset \), then (1.2) surfaces immediately for all \(\vartheta \in F(\Im )\);
-
(b)
if (1.3) holds, then (1.4) holds with \(\beta =0\). However, the opposite is not true, as shown in the following example.
Example 1.1
Let \(\Im :\mathcal{R}\longrightarrow \mathcal{R}\) be defined by
with the usual norm. Then, ℑ satisfies (1.4), but not (1.3). Thus, the class of maps satisfying (1.4) is more general than that of (1.3).
In 2011, Osilike and Isiogugu [11] initiated the concept of β-strictly pseudononspreading (SPN) maps and established weak convergence result of Bailion-type similar to that obtained in [10] and [12]. In addition, using the notion of mean convergence, they obtained strong convergence results similar to the those established in [10] and thus resolved an open problem posed by Kurokawa and Takahashi [10] for the case where the map ℑ is averaged.
A map \(\Im :\mathcal{K}\longrightarrow \mathcal{H}\) is called ϑ-inverse strongly monotone if there exists a positive number ϑ such that
Finding fixed points of nonexpansive, nonspreading, strictly pseudononspreading, and strictly pseudocontractive maps is an important topic in fixed point theory, and they have far-reaching applications in applied areas such as signal processing [13], split feasibility [14], and convex feasibility problems [15]. Subsequently, as an important generation of the above-mentioned maps, the notion of enriched nonlinear maps was first introduced by Berinde [16] (see also [17] and [5]) in the setup of a real Hilbert space. This concept was later extended to the more general Banach space by Saleem [18].
Definition 1.2
A map \(\Im :\Omega \longrightarrow \Omega \) is called \(\Psi _{\Im}\)-enriched Lipschitizian (or \((\eta ,\Psi _{\Im})\)-enriched Lipshitizian) (see [18]) if for all \(\psi ,\phi \in \Omega \), there exist \(\eta \in [0,+\infty )\) and a continuous nondecreasing function \(\Psi _{\Im}:R^{+}\longrightarrow R^{+}\), with \(\Psi _{\Im}(0)=0\), such that
The following special cases emanating from inequality (1.7) are worth mentioning:
-
(i)
if \(\eta =0\), inequality (1.7) reduces to a class of maps known as \(\Psi _{\Im}\)-Lipschitizian;
-
(ii)
if \(\eta =0\) and \(\Psi (t)=Lt\), for \(L>0\), then (1.7) reduces to a class of maps called L-Lipschitizian with L as the Lipschitz constant. In a more special case where \(\eta =0\), \(\Psi _{\eta}(t)=Lt\) and \(L=1\), \(\Psi _{\Im}\)-enriched Lipschitizian map immediately reduces to the class of nonexpansive maps on Ω;
-
(iii)
if \(\Psi _{\Im}(s)=1\), then inequality (1.7) becomes
$$ \bigl\Vert \eta (\psi -\phi )+\Im \psi -\Im \phi \bigr\Vert \leq (\eta +1) \Vert \psi -\phi \Vert , $$(1.8)which known as an η-enriched nonexpansive map. This class of maps was first studied by Berinde [5, 17] as a generalization of a well-known class of maps called nonexpansive.
Note that if \(\Psi _{\Im}\) is not necessarily nondecreasing and satisfies the condition
then we have the class of η-enriched contraction maps.
Definition 1.3
A map ℑ is known as \((\eta ,k)\)-ESPC (see [18]) if for all \(\psi ,\phi \in \Omega \), there exist \(\eta \in [0,+\infty )\) and \(j(\psi -\phi )\in J(\psi -\phi )\) such that
where \(k=\frac {1}{2}(1-\lambda )\) for some \(\lambda \in [0,1)\).
In the setup of a real Hilbert space, inequality (1.9) is equivalent to the following:
where \(\lambda =1-2k\).
In [18], Saleem et al. established that if Ω is a bounded close and convex subset of a real Banach space and \(\Im :\Omega \longrightarrow \Omega \) is a finite family of \((\eta ,k)\)-ESPC maps, then ℑ has a fixed point in Ω.
In 2009, Takahashi and Shimoji [19] initiated the concept of W-map developed from \(\Im _{1}, \Im _{2}, \ldots , \Im _{t}\) and \(\alpha _{1}, \alpha _{2},\ldots , \alpha _{t}\) in the following way:
Definition 1.4
Let \(\mathcal{E}\) be a Banach space and \(\mathcal{C}\subset \mathcal{E}\) be convex. Let \(N\in \mathcal{N}\), \(\{\Im _{i}\}_{i=1}^{N}:\mathcal{C}\longrightarrow \mathcal{C}\) and \(\alpha _{1}, \alpha _{2},\ldots , \alpha _{N}\) be real numbers such that \(0\leq \alpha _{k}\leq 1\) for very \(k=1, 2, \ldots , N\). Then, the map W is defined as follows:
This is known as W-map developed from \(\Im _{1}, \Im _{2},\ldots , \Im _{N}\) and \(\alpha _{1}, \alpha _{2},\ldots , \alpha _{N}\).
The results obtained using W-map in [19, 20] were generalized in [20, 21] through the instrument of K-map, while, in [22, 23], the notion of S-map was studied and applied in generalizing the main results of [20, 21].
More recently, Ke and Ma [24] introduced and studied the following nonlinear maps:
Definition 1.5
Let \(\mathcal{E}\) be a real Banach space and \(\emptyset \neq \mathcal{K}\subset \mathcal{E}\). Let \(\{\Im _{i}\}_{i=1}^{N}\) be a finite family of maps of \(\mathcal{K}\) into itself. For \(i=1,2,\ldots , \mathcal{N}\), let \(\tau _{i}=(\alpha _{i},\beta _{i},\gamma _{i},\delta _{i})\), where \(\alpha _{i},\beta _{i},\gamma _{i},\delta _{i}\in [0,1]\) and \(\alpha _{i}+\beta _{i}+\gamma _{i}+\delta _{i}=1\). The map D is defined as follows:
This is known as a G-map developed from \(\Im _{1}, \Im _{2},\ldots , \Im _{N}\) and \(\tau _{1}, \tau _{2},\ldots , \tau _{N}\).
Using this map, they obtained the following main results as a generalization of the results in [22, 23].
Lemma 1.1
[24] Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. Let \(\{\Im _{i}\}_{i=1}^{N}:\mathcal{K}\longrightarrow \mathcal{K}\) be \(k_{i}\)-strictly pseudocontractive (SPC) maps with \(\bigcap_{i=1}^{N}F(\Im _{i})\neq \emptyset \), and let \(\tau _{i}=(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i})\), where \(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i}\in [0,1]\) with \(\alpha _{i}+\beta _{i}+\gamma _{i}+\delta _{i}=1\). Let D be the G-map developed from the sequences \(\{\Im _{i}\}_{i=1}^{N}\) and \(\{\tau _{i}\}_{i=1}^{N}\). If the following conditions are satisfied:
-
(a)
\(\alpha _{1}\leq \beta _{1}<1-k_{1}\) and \((k_{1}+\beta _{1})\alpha _{1}<\beta _{1}(1-\alpha _{1}-\beta _{1})\);
-
(b)
\(\beta _{i}\geq k_{i}\), \(k_{i}<\gamma _{i}<1\) and \(k_{i}\alpha _{i}\leq \beta _{i}\gamma _{i}-\beta _{i}k_{i}\), for \(i=1,2,\ldots ,N\),
then \(F(G)=\bigcap_{i=1}^{N}F(\Im _{i})\), and D is nonexpansive.
Theorem 1.2
[24] Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. Let \(\{\Im _{i}\}_{i=1}^{N}:\mathcal{K}\longrightarrow \mathcal{K}\) be \(k_{i}\)-SPC maps and \(S:\mathcal{K}\longrightarrow \mathcal{K}\) be a β-SPN map for \(\beta \in [0,1)\). Let \(\tau _{i}=(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i})\), for \(i=1,2,\ldots , N\), where \(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i}\in [0,1]\) with \(\alpha _{i}+\beta _{i}+\gamma _{i}+\delta _{i}=1\) such that
-
(a)
\(\alpha _{1}\leq \beta _{1}<1-k_{1}\) and \((k_{1}+\beta _{1})\alpha _{1}<\beta _{1}(1-\alpha _{1}-\beta _{1})\);
-
(b)
\(\beta _{i}\geq k_{i}\), \(k_{i}<\gamma _{i}<1\) and \(k_{i}\alpha _{i}\leq \beta _{i}\gamma _{i}-\beta _{i}k_{i}\), for \(i=1,2,\ldots ,N\).
Let D be the G-map generated by the sequences \(\{\Im _{\omega ,i}\}_{i=1}^{N}\) and \(\{\tau _{i}\}_{i=1}^{N}\), where \(\Im _{\omega ,i}=(1-\omega )I+\omega \Im _{i}\). Suppose \(\mathcal{F}=F(S)\cap \bigcap_{i=1}^{N}F(\Im _{i})\neq \emptyset \). Let \(\{\wp _{n}\}\) be a sequence developed from arbitrary \(u, \wp _{0}\in \mathcal{K}\) by
where \(\{\pi _{n}\}, \{\sigma _{n}\}, \{a_{n}\}, \{b_{n}\}, \{c_{n}\} \subset [0,1]\) and \(\{\lambda _{n}\} \subset (0,1-\beta )\) satisfying the conditions:
-
(i)
\(a_{n}+b_{n}+c_{n}=1\);
-
(ii)
\(\lim_{n\rightarrow \infty}a_{n}=0\) and \(\sum_{n=0}^{\infty}a_{n}=\infty \);
-
(iii)
\(\liminf_{n\rightarrow \infty}b_{n}>0\) and \(\liminf_{n\rightarrow \infty}c_{n}>0\);
-
(iv)
\(\sum_{n=0}^{\infty}\lambda _{n}<\infty \);
-
(v)
\(\sum_{n=0}^{\infty}\vert \lambda _{n+1}-\lambda _{n}\vert <\infty \), \(\sum_{n=0}^{\infty}\vert \pi _{n+1}-\pi _{n}\vert <\infty \), \(\sum_{n=0}^{ \infty}\vert \sigma _{n+1}-\sigma _{n}\vert <\infty \), \(\sum_{n=0}^{ \infty}\vert a_{n+1}-a_{n}\vert <\infty \), \(\sum_{n=0}^{\infty}\vert b_{n+1}-b_{n} \vert <\infty \), \(\sum_{n=0}^{\infty}\vert c_{n+1}-c_{n}\vert <\infty \).
Then, \(\{\wp _{n}\}\) converges strongly to \(\bar{\wp}=P_{\mathcal{F}}u\).
Considering the results of Ke and Ma [24] and other results in the reviewed works, the following question arises:
Question 1.1
Could there be a nonlinear map that contains the G-map for which we would obtain the results in [24] as special cases?
Ke and Ma [24] considered the G-map and proved Lemma 1.1 and Theorem 1.2 as their main results in [24]. The results they extended and generalized were consistent with those from [22, 23]. In this paper, we first introduce a new class of nonlinear maps called η-enriched D-maps and give some nontrivial examples to demonstrate its existence. Further, we modify the iterative method studied in [24] and after that give an affirmative answer to Question 1.1.
2 Preliminaries
Further, in the course of establishing our main results, the following well-known results should help us. Let \(\mathcal{H}\) be a real Hilbert space, and let \(\{\psi _{n}\}\subset \mathcal{H}\). We shall represent weak convergence of \(\{\psi _{n}\}\) to \(\psi \in \mathcal{H}\) by \(\psi _{n}\rightharpoonup \psi \) and the strong convergence of \(\{\psi _{n}\}\) to \(\psi \in \mathcal{H}\) by \(\psi _{n}\rightarrow \psi \) as \(n\rightarrow \infty \), respectively.
Lemma 2.1
([11, 24]) Let \(\mathcal{H}\) be a real Hilbert space. Then, the following results are valid:
-
(i)
$$ \Vert \wp +\hbar \Vert ^{2}= \Vert \wp \Vert ^{2}+ 2 \langle \wp , \hbar \rangle + \Vert \hbar \Vert ^{2},\quad \forall \hbar ,\wp \in \mathcal{H}; $$
-
(ii)
$$ \Vert \hbar +\wp \Vert ^{2}\leq \Vert \hbar \Vert ^{2} +2\langle \wp , \hbar +\wp \rangle , \quad \forall \hbar ,\wp \in \mathcal{H}; $$
-
(iii)
$$ \Biggl\Vert \sum_{i=0}^{m}\beta _{i}\wp _{i} \Biggr\Vert ^{2}= \sum _{i=0}^{m} \beta _{i} \Vert \wp _{i} \Vert ^{2}- \sum_{i=0}^{m} \beta _{i}\beta _{j} \Vert \wp _{i}-\wp _{j} \Vert ^{2}; $$
-
(iv)
if \(\{\psi _{n}\}\) is a sequence in \(\mathcal{H}\) such that \(\psi _{n}\rightharpoonup \wp \in \mathcal{H}\), then
$$ \limsup_{n\rightarrow \infty} \Vert \psi _{n}-\hbar \Vert ^{2}= \limsup_{n\rightarrow \infty} \Vert \psi _{n}-\wp \Vert ^{2}+ \Vert \wp -\hbar \Vert ^{2}. $$
Definition 2.1
Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. The nearest point projection \(P_{\mathcal{K}}:\mathcal{H}\longrightarrow \mathcal{K}\) defined from \(\mathcal{H}\) onto \(\mathcal{K}\) is a operator that assigns to each \(\psi \in \mathcal{H}\) its nearest point represented with \(P_{\mathcal{K}}\psi \) in Ω. Thus, \(P_{\mathcal{K}}\) is the unique point in Ω such that
Lemma 2.2
[24] Let \(\mathcal{H}\) be a real Hilbert space, \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex and \(P_{\mathcal{K}}:\mathcal{H}\longrightarrow \mathcal{K}\) be a metric projection. Then,
-
(i)
$$ \Vert P_{\mathcal{K}}\wp -P_{\mathcal{K}}\hbar \Vert \leq \langle \wp - \hbar ,P_{\mathcal{K}}\wp -P_{\mathcal{K}}\hbar \rangle , \quad \forall \wp , \hbar \in \mathcal{H}; $$
-
(ii)
\(P_{\mathcal{K}}\) is a nonexpansive map, that is, \(\Vert P_{\mathcal{K}}\wp -P_{\mathcal{K}}\hbar \Vert \leq \Vert \wp - \hbar \Vert \);
-
(iii)
$$ \langle \wp -P_{\mathcal{K}}\wp ,\hbar -P_{\mathcal{K}} \wp \rangle \leq 0, \quad \forall \wp ,\hbar \in \mathcal{K}. $$
Lemma 2.3
([25]) Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. Let \(P_{\mathcal{K}}:\mathcal{H}\longrightarrow \Omega \) be the metric projection of \(\mathcal{H}\) onto \(\mathcal{K}\). Let \(\{\psi _{n}\}\) be a sequence in \(\mathcal{K}\) and
Then, \(\{P_{\Omega}\psi _{n}\}\) converges strongly.
Lemma 2.4
Let \(\mathcal{H}\) be a real Hilbert space, \(\emptyset \neq \mathcal{K} \subset \) \(\mathcal{H}\) be closed and convex and \(\Im :\mathcal{K}\longrightarrow \mathcal{K}\) be a β-ESPN map such that \(F(\Im )\neq \emptyset \). Let \(\Im _{\xi}=\xi I+(1-\xi )\Im \), \(\xi \in [\beta ,1)\). Then, the following conclusions hold:
-
1.
\(F(\Im )=F(\Im _{\xi})\);
-
2.
\(I-\Im _{\xi}\) is demiclosed at zero;
-
3.
\(\Vert \Im \psi -\Im \psi \Vert ^{2}\leq \Vert \psi -\phi \Vert ^{2}+ \frac {2}{1-\xi}\langle \psi -\Im \psi ,\phi -\Im \phi \rangle \);
-
4.
\(\Im _{\xi}\) is quasi-nonexpansive map.
Lemma 2.5
([26, 27]) Let \(\{\nu _{n}\}\) be a sequence of nonnegative real numbers satisfying
where \(\{\pi _{n}\}\) and \(\{\mu _{n}\}\) are real sequences such that
-
(i)
\(\{\pi _{n}\}\subset [0,1]\) and \(\sum_{n=1}^{\infty}\pi _{n}=\infty \);
-
(ii)
\(\limsup_{n\rightarrow \infty} \frac {\mu _{n}}{\pi _{n}}\leq 0\) or \(\sum_{n=0}^{\infty}\vert \mu _{n}\vert <\infty \).
Then, \(\lim_{n\rightarrow \infty}\nu _{n}=0\).
Lemma 2.6
([26, 27]) Let \(\{\nu _{n}\}\subset [0,+\infty )\) be satisfying
where \(\{\pi _{n}\}\) and \(\{\mu _{n}\}\) are real sequences such that
-
(i)
\(\{\pi _{n}\}\subset [0,1]\) and \(\sum_{n=1}^{\infty}\pi _{n}=\infty \);
-
(ii)
\(\limsup_{n\rightarrow \infty}\mu _{n}\leq 0\) or \(\sum_{n=0}^{\infty}\vert \mu _{n}\vert <\infty \).
Then, \(\lim_{n\rightarrow \infty}\nu _{n}=0\).
Let \(\mathcal{X}\) be a real Banach space. A map \(\Im :\mathcal{D}(\Im )\longrightarrow \mathcal{R}(\Im )\), with domain \(D(\Im )\) and range \(R(\Im )\) in \(\mathcal{X}\), is called demiclosed at a point ϑ (see, for instance, [28]) if whenever \(\{\psi _{n}\}\) is a sequence in \(D(\Im )\) such that \(\psi _{n}\rightharpoonup \psi \in D(\Im )\) and \(\{\Im \psi _{n}\}\) converges strongly to ℘, then \(\Im \psi =\wp \).
Lemma 2.7
[22] Let \(\mathcal{H}\) be a real Hilbert space, \(\emptyset \neq \mathcal{K} \subset \) \(\mathcal{H}\) be closed and convex and \(\Im :\mathcal{K}\longrightarrow \mathcal{H}\) be a nonexpansive map. Then, the map \(I-\Im \) is demiclosed at zero.
Lemma 2.8
(Opial property [29]) Let \(\mathcal{H}\) be a real Hilbert space. Suppose \(\wp _{n}\rightharpoonup \omega \), then
3 Results and discussion
Further, we state the following definition.
Definition 3.1
Let \(\mathcal{H}\) be a real Hilbert space. A map ℑ with domain \(\mathcal{D}(\Im )\) and range \(\mathcal{R}(\Im )\) in \(\mathcal{H}\) is known as \((\eta ,\beta )\)-ESPN in the sense of Browder and Petryshyn [30] if there exist \(\eta \in [0,\infty )\) and \(\beta \in [0,1)\) such that for all \((\psi ,\phi )\in \mathcal{D}(\Im )\),
Let \(\omega =\frac {1}{\eta +1}\), then it is clear that \(\omega \in (0,1]\). In this case, inequality (3.1) becomes
which, on simplification, yields
Inequality (3.2) is equivalently written as
where \(\Im _{\omega}=(1-\omega )I+\omega \Im \), \(\lambda =\frac {1}{2}(1- \beta )\), and I denotes the identity map Ω. Here, it is not difficult to see from (3.2) that the average operator \(\Im _{\omega}\) is β-SPN.
The following example shows that the class of \((\eta ,\beta )\)-ESPN maps is larger than the class of β-SPN maps.
Example 3.1
Let \(\Im :[-2,2]\longrightarrow [-2,2]\) be defined by
Then, we have
Thus, for \(\eta =\frac{5}{3}\), \(\beta =\frac{1}{4}\) and \(\Phi (\psi ,\phi )=(\eta +1)^{2}\vert \psi -\phi \vert ^{2}+ \frac {1}{4}\vert \psi -\Im \psi -(\phi -\Im \phi )\vert ^{2}+ 2 \langle \psi -\Im \psi ,\phi -\Im \phi \rangle \), we get
Hence, ℑ is \((\frac{5}{3},\frac{1}{4})\)-ESPN map, but ℑ is not β-SPN since for \(\psi =\frac {3}{2}\) and \(\phi =-\frac {3}{2}\), we obtain
and
Therefore,
for \(\beta =\frac {1}{4}\).
The examples below demonstrate the conclusion that the class of \((\eta ,\lambda )\)-ESPC maps and the class of maps studied in this paper are independent.
Example 3.2
Let \(\Im :\mathcal{R}\longrightarrow \mathcal{R}\) be defined, for each \(\psi \in \mathcal{R}\), by
where \(\mathcal{R}\) denotes the reals with the usual norm. Then, for all \(\psi ,\phi \in (-\infty ,2]\) and for all \(\beta \in [0,1)\), ℑ is \((\eta ,\beta )\)-ESPN map with \(\eta =0\) (see [11] for details). However, ℑ is not \((\eta ,\lambda )\)-ESPC map since every \((\eta ,\lambda )\)-ESPC map satisfies the Lipschitz condition (see, Proposition 3.3 below).
Example 3.3
Let \(\Im :\mathcal{R}\longrightarrow \mathcal{R}\) be defined, for each \(\psi \in \mathcal{R}\), by
where \(\mathcal{R}\) denotes the reals with the usual norm. It is shown in [11] that ℑ is \((\eta ,\lambda )\)-ESPC map with \(\eta =0\). Nevertheless, it is not difficult to see that ℑ is not \((\eta ,\beta )\)-ESPN map. Indeed, for \(\eta =0\), if \(\psi =\frac {1}{2}\) and \(\phi =-\frac {1}{2}\), then
for all \(\beta \in [0,1)\).
Proposition 3.1
Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. Let \(\Im :\mathcal{K}\longrightarrow \mathcal{K}\) be an \((\eta ,\beta )\)-ESPN map with \(F(\Im )\neq \emptyset \). Then,
Proof
Since ℑ is an \((\eta ,\beta )\)-ESPN, we get
from which
Set \(C=\Vert \eta (\wp _{n}-\wp ^{\star})+\Im \wp _{n}-\wp ^{\star}\Vert \) and \(D=\Vert \wp _{n}-\wp ^{\star}\Vert \) in (3.5) so that
The last inequality implies that
and this completes the proof. □
Proposition 3.2
Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. Let \(\Im :\mathcal{K}\longrightarrow \mathcal{K}\) be an \((\eta ,\beta )\)-ESPN map with \(F(\Im )\neq \emptyset \). Then, \(F(\Im )=VI(\mathcal{K},(I-\Im ))\).
Proof
It is not difficult to see that \(F(\Im )\subseteq VI(\mathcal{K},(I-\Im ))\). Let \(A=I-\Im \), \(\wp \in VI(\mathcal{K},(I-\Im ))\) and \(\wp ^{\star}\in F(\Im )\). Since \(\wp \in VI(\mathcal{K},A)\), it follows that
Since ℑ is an \((\eta ,\beta )\)-ESPN map with \(F(\Im )\neq \emptyset \), it follows that
from which
Consequently, \(\wp \in F(\Im )\) and \(VI(\mathcal{K},A)\subseteq F(\Im )\). Hence, \(VI(\mathcal{K},A)=F(\Im )\). □
Remark 3.1
From Lemma 2.2 and (3.7), we obtain
Proposition 3.3
([31]) Let \(\mathcal{E}\) be a normed space and \(\Im :D(\Im )\subseteq \mathcal{E}\longrightarrow \mathcal{E}\) be an \((\eta ,k)\)-SPC map. Then, ℑ is an L-Lipschitizian map.
Proof
Since ℑ is an \((\eta ,k)\)-SPC map, \(\exists k\in [0,1)\) such that \(\forall \psi ,\phi \in d(\Im )\),
From the above inequality, we obtain
Therefore,
with \(L=\frac {(\eta +1)(1+\sqrt{k})}{1-\sqrt{k}}\). □
Proposition 3.4
([31]) Let \(\mathcal{H}\) be a real Hilbert space, \(\emptyset \neq \Omega \subset \) \(\mathcal{H}\) and \(\Im :\Omega \longrightarrow \Omega \) be an \((\eta ,\beta )\)-ESPN map. Then, \((I-\Im )\) is demiclosed at 0.
Proof
Let \(\{\psi _{n}\}\) be a sequence in Ω, which converges weakly to ϑ and \(\{\psi _{n}-\Im \psi _{n}\}\) converges strongly to 0. We want to show that \(\vartheta \in F(\Im )\). Now, since \(\{\psi _{n}\}\) converges weakly, it is bounded.
For each \(\psi \in \mathcal{H}\), define \(f:\mathcal{H}\longrightarrow [0,\infty )\) by
Then, using Lemma 2.1(iii), we get
Consequently,
and
Observe that
Then, (3.8) and (3.9) give that
so that \(\vartheta \in F(\Im )\) as required. □
Proposition 3.5
([31]) Let \(\mathcal{H}\) be a real Hilbert space, \(\emptyset \neq \Omega \subset \) \(\mathcal{H}\) and \(\Im :\Omega \longrightarrow \Omega \) be an \((\eta ,\beta )\)-ESPN map. Then, \(F(\Im )\) is closed and convex.
Proof
Let \(\{\psi _{n}\}\) be a sequence in Ω, which converges to ψ. We want to show that \(\psi \in F(\Im )\). Since
which follows that \(\psi =\Im \psi \). Hence, \(\psi \in F(\Im )\).
Next, let \(\vartheta _{1},\vartheta _{2}\in F(\Im )\). We prove that \(\lambda \vartheta _{1}+(1-\lambda )\vartheta _{2}\in F(\Im )\). Set \(\wp =\lambda \vartheta _{1}+(1-\lambda )\vartheta _{2}\). Then, \(\vartheta _{1}-\wp =(1-\lambda )(\vartheta _{1}-\vartheta _{2})\) and \(\vartheta _{2}-\wp =\lambda (\vartheta _{2}-\vartheta _{1})\). Since
it follows that \((1-\beta )\Vert \wp -\Im \wp \Vert \leq 0\). Therefore, \(\wp =\Im \wp \) and \(\wp \in F(\Im )\) as required. □
Definition 3.2
Let \(\mathcal{E}\) be a real Banach space and \(\emptyset \neq \mathcal{K}\subset \mathcal{E}\). Let \(\{\Im _{i}\}_{i=1}^{N}\) be a finite family of maps of \(\mathcal{K}\) into itself. For \(i=1,2,\ldots , \mathcal{N}\), let \(\tau _{i}=(\alpha _{i},\beta _{i},\gamma _{i},\delta _{i})\), where \(\alpha _{i},\beta _{i},\gamma _{i},\delta _{i}\in [0,1]\) with \(\alpha _{i}+\beta _{i}+\gamma _{i}+\delta _{i}=1\). The map \(D:\mathcal{K}\longrightarrow \mathcal{K}\) is defined as follows:
where \(\Im _{\omega ,i}=(1-\omega )I+\omega \Im _{i}\), \(i=1, 2, \ldots , N\). This is known as an η-enriched D-map developed from \(\Im _{\omega ,1}, \Im _{\omega ,2},\ldots , \Im _{\omega ,N}\) and \(\tau _{1}, \tau _{2},\ldots , \tau _{N}\).
Remark 3.2
If \(\eta =0\), then \(\omega =1\) and η-enriched D-map become G-map; if \(\eta =0\) and for every \(i=1,2,\ldots ,N\), \(\alpha _{i}=0\), then η-enriched D-map becomes S-map; if \(\eta =0\) and for every \(i=1,2,\ldots ,N\), \(\alpha _{i}=0\) and \(\gamma _{i}=0\), then η-enriched D-map becomes W-map, and if \(\eta =0\) and for every \(i=1,2,\ldots ,N\), \(\alpha _{i}=0\) and \(\delta _{i}=0\), then η-enriched D-map becomes K-map.
Lemma 3.6
Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. Let \(\{\Im _{i}\}_{i=1}^{N}:\mathcal{K}\longrightarrow \mathcal{K}\) be \((\eta _{i},\beta _{i})\)-ESPN maps with \(\bigcap_{i=1}^{N}F(\Im _{i})\neq \emptyset \), and let \(\tau _{i}=(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i})\), where \(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i}\in [0,1]\) and \(\alpha _{i}+\beta _{i}+\gamma _{i}+\delta _{i}=1\). Let D be an η-enriched D-map developed from the sequences \(\{\Im _{i}\}_{i=1}^{N}\) and \(\{\tau _{i}\}_{i=1}^{N}\). If the following conditions are satisfied:
-
(a)
\(\alpha _{1}\leq \beta _{1}<1-k_{1}\) and \(\beta _{1}\gamma _{1}<\beta _{1}(1-\alpha _{1}-\beta _{1})\);
-
(b)
\(\beta _{i}\geq k_{i}\), \(k_{i}<\gamma _{i}<1\) and \(\eta _{i}\leq \sqrt{ \frac {\beta _{i}\gamma _{i}-(\alpha _{i}+\beta _{i})k_{i}}{\alpha _{i}}}\), for \(i=1,2,\ldots ,N\).
Then, \(F(D)=\bigcap_{i=1}^{N}F(\Im _{i})\) and D is nonexpansive.
Proof
It is not difficult to see that \(\bigcap_{i=1}^{N}F(\Im _{i})\subseteq F(D)\). So, it suffices for us to show that \(F(D)\subseteq \bigcap_{i=1}^{N}F(\Im _{i})\). Let \(\wp _{0}\in F(D)\) and \(\wp ^{\star}\in \bigcap_{i=1}^{N}F(\Im _{i})\) so that
Since
and
it follows from (3.10) that
If we set
then by conditions [(a) and (b)], we obtain
Using (3.12) and the fact that \(\delta _{k}<1\) for \(k=1,2,\ldots ,N\), it follows that
Consequently, \(\Im _{1}\wp _{0}=\wp _{0}\), that is, \(\wp _{0}\in F(\Im _{1})= F(\Im _{\omega ,1})\). Using the definition \(G_{1}\) and the fact that \(\Im _{\omega ,1}=(1-\omega )I+\omega \Im _{1}\), we obtain
Again, set
then by (3.12), (3.16), and the fact that \(\delta _{k}<1\) for \(k=3,4,\ldots ,N\), we obtain
Using condition [(a) and (b)], it follows that
Hence, \(\Im _{2}\wp _{0}=\wp _{0}\); that is, \(\wp _{0}\in F(\Im _{2})= F(\Im _{\omega ,2})\). By the definition \(G_{2}\), we obtain
Continuing in this manner, we obtain
Consequently,
Next, we show that D is nonexpansive. Indeed, for any \(\wp ,\hbar \in \mathcal{K}\), we have
□
Remark 3.3
Since \(F(D)=\bigcap_{i}^{N}F(\Im _{i})\neq \emptyset \), it follows that the map D is quasi-nonexpansive, that is,
Example 3.4
Let \(\Im _{1},\Im _{2}:\mathcal{R}\longrightarrow \mathcal{R}\) be defined as follows:
and
Then, \(F(\Im _{1})=(-\infty ,0]\) and \(F(\Im _{2})=[0,+\infty )\). Consequently, \(F(\Im _{1})\cap F(\Im _{2})=\{0\}\). Also, it is shown in [24] that \(\Im _{1}\) is a \((0,k_{1})\)-ESPC map (with \(k_{1}=\frac{1}{5}\)) and \(\Im _{2}\) is a \((0,k_{2})\)-ESPC map (with \(k_{2}=\frac{1}{3}\)). Further, if we set \(\tau _{1}= (\frac {1}{5}, \frac {1}{5}, \frac {2}{5}, \frac {1}{5} )\), which satisfies condition (a) of Lemma 3.1, then it follows from
that
Since \(\Im _{1}\) is \((0,k_{1})\)-ESPC, it follows that \(\eta _{1}=0\) so that \(\omega =\frac {1}{\eta _{1}+1}=1\). Thus,
and
Hence,
Again, if we set \(\tau _{2}= (\frac {1}{7}, \frac {1}{3}, \frac {1}{2}, \frac {1}{42} )\), which satisfies condition (b) of Lemma 3.1, then it follows from
that
Since \(\Im _{2}\) is \((0,k_{2})\)-ESPC, it follows that \(\eta _{2}=0\) so that \(\omega =\frac {1}{\eta _{2}+1}=1\). Thus,
and
Hence,
Using the above information, it is not difficult to see that \(F(D)=\{0\}=F(\Im _{1})\cap F(\Im _{2})\). It has also been demonstrated in [24] that D is nonexpansive.
Theorem 3.7
Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. Let \(\{\Im _{i}\}_{i=1}^{N}:\mathcal{K}\longrightarrow \mathcal{K}\) be \((\eta _{i},k_{i})\)-ESPC maps and \(S:\mathcal{K}\longrightarrow \mathcal{K}\) be an \((\eta ,\beta )\)-ESPN map for \(\beta \in [0,1)\). Let \(\tau _{i}=(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i})\), for \(i=1,2,\ldots , N\), where \(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i}\in [0,1]\) with \(\alpha _{i}+\beta _{i}+\gamma _{i}+\delta _{i}=1\) such that
-
(a)
\(\alpha _{i}\leq \beta _{i}<1-k_{i}\); \(\beta _{i}\gamma _{i}<\beta _{i}(1-\alpha _{i}-\beta _{i})\) for \(i=1,2,\ldots ,N\);
-
(b)
\(\beta _{i}\geq k_{i}\), \(k_{i}<\gamma _{i}<1\) and \(\eta _{i}\leq \sqrt{ \frac {\beta _{i}\gamma _{i}-(\alpha _{i} +\beta _{i})k_{i}}{\alpha _{i}}}\), for \(i=1,2,\ldots ,N\).
Let D be the D-map generated by the sequences \(\{\Im _{\omega ,i}\}_{i=1}^{N}\) and \(\{\tau _{i}\}_{i=1}^{N}\), where \(\Im _{\omega ,i}=(1-\omega )I+\omega \Im _{i}\). Suppose \(\mathcal{F}=F(S)\cap \bigcap_{i=1}^{N}F(\Im _{i})\neq \emptyset \). Let \(\{\wp _{n}\}\) be a sequence as defined in (1.11) with the conditions \((i)-(v) \). Then, \(\{\wp _{n}\}_{n=0}^{\infty}\) converges strongly to \(\bar{\wp}=P_{\mathcal{F}}u\).
Proof
First, we show that \((I-\lambda _{n}A)\wp _{n}\), where \(A=I-S\), is nonexpansive. Now, since S is \((\eta ,\beta )\)-ESPN map, it follows from Definition 3.1 that
or equivalently
so that
which, when \(A=I-S\), yields
Also, since
it follows from (3.23) that
Thus, if S is an \((\eta ,\beta )\)-ESPN, then \(A=I-S\) is \(\frac{1-\beta}{2}\)-inverse strongly monotone map. From (3.24), we obtain, for any \(\hbar =\bar{\wp ^{\star}}\in F(S)\), that
Next, we set
and then show that Q is bounded. To do this, let \(\wp ^{\star}\in \mathcal{F}(\Im )\) be arbitrarily chosen. Then, using (1.11), we get
Using Lemma 2.2(ii) and (3.25), we also obtain from (1.11) that
and
Thus, (3.26), (3.27), and (3.28) imply that
It is now easy to see using (3.29) and inductional hypothesis that
which consequently implies that \(\{\wp _{n}\}\) is bounded. In addition, \(\{\hbar _{n}\}\), \(\{w_{n}\}\), and \(\{Dw_{n}\}\) are also bounded. Now, from Remark 3.1, we have \(\wp ^{\star}\in F(P_{\mathcal{K}}(I-\lambda (I-\Im )))\), and from the nonexpansiveness of \(P_{\mathcal{K}}\), we obtain
Thus, using the boundedness of \(\{\wp _{n}\}\) and \(\{\hbar _{n}\}\), it will not be difficult to see that \(\{P_{\mathcal{K}}(I-\lambda (I-\Im ))\wp _{n}\}\) and \(\{P_{\mathcal{K}}(I-\lambda (I-\Im ))\hbar _{n}\}\) are bounded. From Proposition 3.1, we also obtain that \(\{(I-\Im )\wp _{n}-(I-\Im )\wp _{n-1}\}\) and \(\{(I-\Im )\hbar _{n}-(I-\Im )\hbar _{n-1}\}\) are bounded. Hence, Q is bounded.
Next, we show that \(\lim_{n\rightarrow \infty}\Vert \wp _{n+1}-\wp _{n}\Vert =0\). Using (1.11), we obtain the following estimates
and
Using (3.32) and (3.33) in (3.31), we infer
where
Since \(\sum_{n=0}^{\infty}\ell _{n}<\infty \) (by conditions [(iv) and (v)]), it follows from Lemma 2.5 and (3.34) that
Next, we show that \(\lim_{n\rightarrow \infty}\Vert Dw_{n}-w_{n}\Vert =0\) and \(\lim_{n\rightarrow \infty}\Vert \wp _{n}-w_{n}\Vert =0\). Since, from (1.11), (3.32), (3.33) and Lemma 2.1,
and
it follows that
Using conditions [(ii) and (iii)], the boundedness of \(\Vert u-\wp ^{\star}\Vert \) and \(\{\wp _{n}\}\) and the fact that \(\lim_{n\rightarrow \infty}\Vert \wp _{n+1}-\wp _{n}\Vert =0\), we obtain from (3.36) that
Also, since
it follows that
Next, let \(\bar{\wp}=P_{\mathcal{F}}u\). Then, we show that \(\limsup_{n\rightarrow \infty}\langle u-\bar{\wp},\wp _{n}-\bar{\wp} \rangle \leq 0\). Consider a subsequence \(\{\wp _{n_{k}}\}\) of \(\{\wp _{n}\}\) such that
From the boundedness of \(\{\wp _{n}\}\), we can find a subsequence \(\{\wp _{n_{k}}\}\) that converges weakly to a point of \(\mathcal{K}\). Without loss of generality, we may consider that \(\wp _{n_{k}}\rightharpoonup \wp ^{\star}\). Hence, from (3.38), we get \(w_{n_{k}}\rightharpoonup \wp ^{\star}\). Using Lemma 2.7 and (3.37), we also obtain \(D\wp ^{\star}=\wp ^{\star}\); or equivalently, \(\wp ^{\star}\in F(D)\). Since \(w_{n_{k}}\rightharpoonup \wp ^{\star}\), it follows that \(\wp ^{\star}\in F(S)\). Now, suppose otherwise and consider \(\wp ^{\star}\in F(S)\). Then,
and by Lemma 2.8, we obtain
which is a contradiction. Consequently,
From (3.40) in combination with the property of metric projection, we get
Last, we shall prove that \(\wp _{n}\rightarrow \wp ^{\star}\) as \(n\rightarrow \infty \). Now, from (1.11)
it follows that
It is not difficult to see that \(\sum_{n=0}^{\infty}\frac{2a_{n}}{1+a_{n}}= \infty \). Using this fact, together with (3.41), (3.42) and Lemma 2.6, we get that \(\wp _{n}\rightarrow \bar{\wp}\) as \(n\rightarrow \infty \). The proof is completed. □
4 Application
The following theorems can easily be obtained from Theorem 3.7.
Theorem 4.1
Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. Let \(\{\Im _{i}\}_{i=1}^{N}:\mathcal{K}\longrightarrow \mathcal{K}\) be \(\eta _{i}\)-enriched nonexpansive maps and \(S:\mathcal{K}\longrightarrow \mathcal{K}\) be an \((\eta ,\beta )\)-ESPN map for \(\beta \in [0,1)\). Let \(\tau _{i}=(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i})\), for \(i=1,2,\ldots , N\), where \(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i}\in [0,1]\) with \(\alpha _{i}+\beta _{i}+\gamma _{i}+\delta _{i}=1\) and the conditions (a) and (b) in Theorem 3.7.
Let D be the D-map generated by the sequences \(\{\Im _{\omega ,i}\}_{i=1}^{N}\) and \(\{\tau _{i}\}_{i=1}^{N}\), where \(\Im _{\omega ,i}=(1-\omega )I+\omega \Im _{i}\). Suppose \(\mathcal{F}=F(S)\cap \bigcap_{i=1}^{N}F(\Im _{i})\neq \emptyset \). Let \(\{\wp _{n}\}\) be a sequence as defined in (1.11) with the conditions (i)–(v) . Then, \(\{\wp _{n}\}_{n=0}^{\infty}\) converges strongly to \(\bar{\wp}=P_{\mathcal{F}}u\).
Lemma 4.2
[24] Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. Let \(\Im :\mathcal{K}\longrightarrow \mathcal{H}\) be ϑ-inverse strongly monotone map. Then, for all \(\wp ,\hbar \in \mathcal{K}\) and \(\nu >0\),
Thus, if \(0<\nu \leq 2\vartheta \), then \(I-\nu \Im \) is a nonexpansive map.
Using (4.1) and Lemma 4.2, the following result emerges as an immediate consequence of Theorem 4.1.
Theorem 4.3
Let \(\mathcal{H}\) be a real Hilbert space and \(\emptyset \neq \mathcal{K}\subset \mathcal{H}\) be closed and convex. Let \(\{\mathcal{B}_{i}\}_{i=1}^{N}:\mathcal{K}\longrightarrow \mathcal{H}\) be an \(\vartheta _{i}\)-inverse strongly monotone maps and let \(S:\mathcal{K}\longrightarrow \mathcal{K}\) be an \((\eta ,\beta )\)-ESPN map for \(\beta \in [0,1)\). Let \(\{\Im _{i}\}_{i=1}^{N}:\mathcal{K}\longrightarrow \mathcal{K}\) be defined \(\Im _{i}\wp =P_{\mathcal{K}}(I-\nu _{i}\mathcal{B}_{i})\wp \) for every \(\wp \in \mathcal{K}\) and \(\nu _{i}\in (0,2\vartheta )\), and let \(\tau _{i}=(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i})\), for \(i=1,2,\ldots , N\), where \(\alpha _{i}, \beta _{i},\gamma _{i},\delta _{i}\in [0,1]\) with \(\alpha _{i}+\beta _{i}+\gamma _{i}+\delta _{i}=1\) and the conditions (a) and (b) in Theorem 3.7.
Let D be the D-map generated by the sequences \(\{\Im _{\omega ,i}\}_{i=1}^{N}\) and \(\{\tau _{i}\}_{i=1}^{N}\), where \(\Im _{\omega ,i}=(1-\omega )I+\omega \Im _{i}\). Suppose \(\mathcal{F}=F(S)\cap \bigcap_{i=1}^{N}F(\Im _{i})\neq \emptyset \). Let \(\{\wp _{n}\}\) be a sequence as defined in (1.11) with the conditions (i)–(v) . Then, \(\{\wp _{n}\}\) converges strongly to \(\bar{\wp}=P_{\mathcal{F}}u\).
5 Conclusion
In this paper, a method for finding common fixed points of a finite family of \((\eta _{i}, k_{i})\)-ESPC maps and \((\eta _{i}, \beta _{i})\)-ESPN maps have been introduced in the setup of a real Hilbert space. Further, strong convergence theorems of the proposed method under mild conditions on the control parameters have been established. The main results have been applied in proving strong convergence theorems for \(\eta _{i}\)-enriched nonexpansive, strongly inverse monotone, and strictly pseudononspreading maps. Some nontrivial examples have also been constructed to demonstrate the effectiveness of the proposed method.
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I.K.A. made conceptualization, methodology and writing draft preparation. H.I. performed the formal analysis, writing-review and editing. D.I.I. made investigation, review and validation. All authors read and approved the final version.
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Agwu, I.K., Işık, H. & Igbokwe, D.I. Fixed point results involving a finite family of enriched strictly pseudocontractive and pseudononspreading mappings. J Inequal Appl 2024, 58 (2024). https://doi.org/10.1186/s13660-024-03120-6
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DOI: https://doi.org/10.1186/s13660-024-03120-6