Leray-Schauder-type fixed point theorems in Banach algebras and application to quadratic integral equations
- Abdelmjid Khchine^{1},
- Lahcen Maniar^{1} and
- Mohamed-Aziz Taoudi^{2}Email author
DOI: 10.1186/s13663-016-0579-3
© Khchine et al. 2016
Received: 20 February 2016
Accepted: 18 August 2016
Published: 1 September 2016
Abstract
In this paper, we present new fixed point theorems in Banach algebras relative to the weak topology. Our fixed point results are obtained under Leray-Schauder-type boundary conditions. These results improve and complement a number of earlier works. As an application, we establish some existence results for a broad class of quadratic integral equations.
Keywords
fixed point theorem Banach algebra quadratic integral equation measure of weak noncompactness weakly condensingMSC
47H10 45G101 Introduction
The need for a fixed point theory in Banach algebras arose out of the study of quadratic integral equations. Those equations have received increasing attention during recent years due to their applications in diverse fields of science and engineering, for example, the theory of radiative transfer [1], statistical mechanics [2], biomathematics [3], signal theory [4], kinetic theory of gases [5], transport theory [6], and so on.
In recent years, significant advances have been made in the development of fixed point theory in Banach algebras using the norm topology and applications to quadratic integral equations. We quote the contributions by Leggett [7], Majorana and Marano [8], Banas and Lecko [9], Dhage [10–14], and many others (see, e.g., [15–18] and the references therein).
However, only a few papers have been up to now devoted to the existence of fixed points for mappings acting on Banach algebras relative to the weak topology [19–21].
One of the most important results for solving (1.1) using the norm topology is the following result due to Dhage [12]. This result can be seen as an analogue of Krasnosel’skii’s fixed point theorem in Banach algebras.
Theorem 1.1
- (i)
B is completely continuous,
- (ii)
A and C are Lipschitzian mappings with constants \(k_{A}\) and \(k_{C}\), respectively,
- (iii)
the equality \(x=AxBy+Cx\) with \(y\in\Omega\) implies \(x\in \Omega\).
Very recently, Banas and Taoudi [19] proved the following analogue of Theorem 1.1 for the weak topology.
Theorem 1.2
- (i)
\(B(\Omega)\) is relatively weakly compact,
- (ii)
A and C are Lipschitzian mappings with constants \(k_{A}\) and \(k_{C}\), respectively,
- (iii)
the equality \(x=AxBy+Cx\) with \(y\in\Omega\) implies \(x\in \Omega\).
Recall that a Banach algebra X is said to have property \((\mathcal{P})\) if for every two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in X such that \(x_{n}\rightharpoonup x\) and \(y_{n}\rightharpoonup y \) for some \(x,y\in X\), we have \(x_{n}y_{n}\rightharpoonup xy\).
In some applications, it is difficult and sometimes impossible to construct a set Ω so that (1.2) holds. This entails the importance of considering other ‘boundary conditions,’ namely Leray-Schauder-type conditions [14], Furi-Pera-type conditions [18], and many others.
The aim of this paper is to establish new fixed point theorems in Banach algebras relative to the weak topology under Leray-Schauder-type boundary conditions. Then, we replace the weak continuity of the involved operators by the concepts of ww-compactness and ws-compactness and interchange the weak compactness by some alternative conditions formulated using the axiomatic measure of weak noncompactness. Our results improve and encompass several earlier related works. As an application, we prove the existence of a continuous solution for a broad class of quadratic integral equations.
This paper is arranged as follows. In Section 1, we fix the notation and present some key tools that will be used to prove our main results. In Section 2, we prove some fixed point results in Banach algebras under boundary conditions of Leray-Schauder type. Finally, in the last section, we use some materials from previous sections to develop an existence theory for a broad class of quadratic integral equations. For the remainder of this section, we gather some notations and preliminary facts. Let X be a Banach algebra with norm \(\Vert \cdot \Vert \), and let \(X^{*}\) denote the topological dual of X. We use the symbol \(B(x, r)\) to denote the closed ball centered at x with radius r. We write \(B_{r}\) to denote \(B(0, r)\). Further, we denote by \(B(X)\) the family of all nonempty and bounded subsets of X. The symbol \(\mathcal{K}^{w}(X)\) stands for the family of all weakly compact subsets of X. For any bounded subset Ω of X, we put \(\Vert \Omega \Vert =\sup \{ \Vert x \Vert : x \in\Omega\}\).
In the sequel, we will adopt the following definition of the axiomatic measure of weak noncompactness [22].
Definition 1.3
- (i)
The family \(\ker\psi= \{M\in B(X): \psi(M)=0\}\) is nonempty, and kerψ is contained in the set of relatively weakly compact subsets of X,
- (ii)
\(M_{1} \subset M_{2} \Longrightarrow\psi(M_{1}) \leq\psi(M_{2})\),
- (iii)
\(\psi(co (M))= \psi(M)\), where \(co(M)\) is the convex hull of M,
- (iv)
\(\psi(\lambda M_{1}+(1-\lambda)M_{2}) \leq\lambda\psi (M_{1})+ (1- \lambda) \psi(M_{2})\) for \(\lambda\in[0,1]\),
- (v)
if \((M_{n})_{n \geq1} \) is a sequence of nonempty, weakly closed subsets of X with \(M_{1}\) bounded and \(M_{1} \supseteq M_{2} \supseteq\cdots \supseteq M_{n} \supseteq\cdots\) such that \(\lim_{n \to+\infty} \psi(M_{n})= 0\), then the set \(M_{\infty}:=\bigcap_{n=1}^{\infty} M_{n}\) is nonempty.
We summarize some useful properties of w in the following proposition. For a proof, we refer the reader to [23].
Proposition 1.4
- (a)
\(w(M_{1})=0\) if and only if \(M_{1}\) is relatively weakly compact,
- (b)
\(w(M_{1}+M_{2})\leq w(M_{1})+w(M_{2})\),
- (c)
\(w(M_{1}\cup M_{2})=\max(w(M_{1}),w(M_{2}))\).
Definition 1.5
- (i)
T is weakly sequentially continuous if it maps weakly convergent sequences into weakly convergent sequences,
- (ii)
T is ψ-condensing if T is bounded (i.e., it takes bounded sets into bounded sets) and \(\psi(T (M )) < \psi(M )\) for any bounded subset M of Ω with \(\psi(M ) > 0\),
- (iii)
T is weakly compact if \(T (\Omega)\) is relatively weakly compact,
- (iv)
T is ws- compact if it is continuous and for any weakly convergent sequence \((x_{n} )\) in Ω, the sequence \((T x_{n} )\) has a strongly convergent subsequence in X,
- (v)
T is ww- compact if it is continuous and for any weakly convergent sequence \((x_{n})\) in Ω, the sequence \((T x_{n} )\) has a weakly convergent subsequence in X.
Notice that the concepts of ww-compact and ws-compact mappings arise naturally in the study of integral and partial differential equations (see [24–32]). It is worth noting that ww-compact and ws-compact mappings are not necessarily weakly sequentially continuous [25].
The following result is crucial for our purpose.
Theorem 1.6
([33], Theorem 1.1)
- (i)Let H be a bounded subset of \(C([0,T], X)\). Thenwhere \(H(t)=\{ x(t); x\in H\}\);$$\sup_{t\in[0,T ]} w\bigl(H(t)\bigr) \leq w(H), $$
- (ii)Let \(H \subset C([0, T ], X)\) be bounded and equicontinuous. Thenwhere \(H[0, T ] = \bigcup_{t\in[0,T ]} H(t)\).$$w(H) = \sup_{t\in[0,T ]} w\bigl(H(t)\bigr) = w\bigl(H[0, T ]\bigr), $$
For later use, we recall the mean value theorem for the Pettis integral; see [34], Theorem 3.
Lemma 1.7
The following lemma is well known.
Lemma 1.8
([35], Lemma 2.8)
Let X be a Banach space, and \(F : X \longrightarrow X\) be a Lipschitzian mapping with constant k. If, moreover, F is ww-compact, then for any bounded subset M of X, we have \(w(F(M))\leq k w(M)\).
Definition 1.9
[19]
Let X be a Banach algebra. We say that X is a WC-Banach algebra if the product \(K\cdot K'\) of arbitrary weakly compact subsets K and \(K'\) of X is weakly compact.
Lemma 1.10
[19]
Let M and \(M^{\prime}\) be bounded subsets of a WC-Banach algebra X. Then \(w(M\cdot M^{\prime}) \leq \Vert M^{\prime} \Vert w(M) + \Vert M \Vert w(M^{\prime})+w(M)w(M^{\prime})\).
In particular, if \(M^{\prime}\) is relatively weakly compact, then \(w(M\cdot M^{\prime}) \leq \Vert M^{\prime} \Vert w(M)\).
Remark 1.11
It is worth noticing that every Banach algebra with property (\(\mathcal{P}\)) is a WC-Banach algebra. Also, every commutative Banach algebra with the Dunford-Pettis property has property (\(\mathcal{P}\)) (see [19]). In particular, \(L^{1}(\mu), L^{\infty}(\mu)\), and \(C(K)\) have property (\(\mathcal{P}\)). Recall that a Banach space X is said to have the Dunford-Pettis property (DPP) if for each Banach space Y, every weakly compact linear operator \(T \colon X\to Y\) maps weakly convergent sequences into strongly convergent sequences.
2 Leray-Schauder-type fixed point theorems in Banach algebras relative to the weak topology
In this section, we prove some fixed point theorems in Banach algebras relative to the weak topology. Our results are formulated using some Leray-Schauder-type boundary conditions. Before proceeding with the main results, we give a key lemma, which we will employ several times in the sequel.
Lemma 2.1
- (i)
\(\tau_{A,C}(M)\) is bounded.
- (ii)
\(\tau_{A,C}\) is continuous.
- (iii)
If X is a WC-Banach algebra and if A and C are ww-compact, then \(\tau_{A,C}\) is ww-compact.
- (iv)
If X is a Banach algebra with property \((\mathcal{P})\) and if A and C are weakly sequentially continuous, then \(\tau_{A,C}\) is weakly sequentially continuous on M.
Proof
Now, we establish our first fixed point theorem under Leray-Schauder boundary conditions.
Theorem 2.2
- (i)
\(B(\overline{U^{w}})\) is bounded with bound \(Q = \Vert B(\overline{U^{w}}) \Vert \),
- (ii)
A and C are Lipschitzian mappings with constants \(\alpha_{A}\) and \(\alpha_{C}\) such that \(Q \alpha_{A} + \alpha_{C} < 1\),
- (iii)
\(T:= \tau_{A,C}B \colon\overline{U^{w}}\to X \) is ψ-condensing.
- (A1)
there exists \(u\in\overline{U^{w}} \) such that \(Au Bu + Cu = u\), or
- (A2)there exists \(u \in\partial_{\Omega} U\) such that$$\lambda A \biggl(\frac{u}{\lambda}\biggr) B u + \lambda C \biggl( \frac{u}{\lambda}\biggr) = u $$
Proof
Remark 2.3
The statement of Theorem 2.2 remains true if we replace ‘A and C Lipschitzian’ with ‘A and C \(\mathcal{D}\)-Lipschitzian,’ provided that \(\tau_{A,C}B(\overline{U^{w}})\) is bounded. We emphasize that Theorem 2.2 improves [20], Theorem 3.1. In contrast to [20], the operator A in Theorem 2.2 needs not be regular.
Now, we present some important corollaries of Theorem 2.2, which extend and encompass several well-known results in the literature. Firstly, note that hypothesis (iii) in Theorem 2.2 may be replaced with (iii′): \(T( \overline{U^{w}})= \tau_{A,C}B (\overline{U^{w}}) \) is relatively weakly compact or (iii″): T maps bounded sets into relatively weakly compact sets. From an application point of view, it is convenient to require the weak compactness on B instead of \(\tau_{A,C}B\). We may therefore state the following result.
Corollary 2.4
- (i)
\(B (\overline{U^{w}})\) is relatively weakly compact,
- (ii)
A and C are Lipschitzian mappings with constants \(\alpha_{A}\) and \(\alpha_{C}\) such that \(Q \alpha_{A} + \alpha_{C} < 1\), where \(Q = \Vert B(\overline{U^{w}}) \Vert \).
- (A1)
there exists \(u\in\overline{U^{w}} \) such that \(Au Bu + Cu = u\), or
- (A2)
there exist \(u \in\partial_{\Omega} U\) and \(\lambda \in (0, 1)\) such that \(\lambda A (\frac{u}{\lambda}) B u + \lambda C (\frac{u}{\lambda}) = u\).
Proof
Remark 2.5
- (i)
Corollary 2.4 is an analogue of [14], Theorem 2.4, for the weak topology and extends [20], Theorem 3.4.
- (ii)
In Corollary 2.4, condition (i) may be replaced with (i′): ‘B maps bounded sets into relatively weakly compact sets, and \(B(\overline {U^{w}})\) is bounded.’ In this case, if we take \(A=1_{X}\), then we obtain the Krasnosel’skii-Leray-Schauder-type fixed point theorem for the weak topology [39], Theorem 2.1.
- (iii)
If we assume that \(\overline{U^{w}}\) is a weakly compact subset of Ω, then hypothesis (iii) in Theorem 2.2 is redundant.
If we take \(C=0\) in Theorem 2.2, then we get the following result.
Corollary 2.6
- (i)
\(B(\overline{U^{w}})\) is bounded with bound \(Q = \Vert B(\overline{U^{w}}) \Vert \),
- (ii)
A is Lipschitzian mapping with constant \(\alpha_{A}\) such that \(Q \alpha_{A} < 1\),
- (iii)
\(T:= \tau_{A,0}B \colon\overline{U^{w}}\to X \) is a ψ-condensing mapping.
- (A1)
there exists \(u\in\overline{U^{w}} \) such that \(Au Bu = u\), or
- (A2)
there exist \(u \in\partial_{\Omega} U\) and \(\lambda \in (0, 1)\) such that \(\lambda A (\frac{u}{\lambda}) B u = u\).
We describe now an important special case where \(C\equiv z_{0}\in X\). We will obtain a sharpening of [20], Theorem 3.2, and a weak Leray-Schauder version of [8], Proposition 2.
Corollary 2.7
- (i)
A maps bounded sets into bounded sets and there exists a constant \(\lambda\geq 0\) such that \(\psi(A (M)) \leq \lambda \psi(M)\) for every bounded subset M of X, and
- (ii)
B maps bounded sets into relatively weakly compact sets, \(B(\overline{U^{w}})\) is bounded, and \(\lambda\cdot\sup_{z \in\overline{U^{w}}} \Vert B(z) \Vert < 1\).
- (A1)
there exists \(z\in\overline{U^{w}} \) such that \(z_{0} + Az Bz = z\), or
- (A2)there exist \(z \in\partial_{\Omega} U\) and \(\lambda \in (0, 1)\) such that$$\lambda z_{0} + \lambda A \biggl(\frac{z}{\lambda}\biggr) B z = z. $$
Proof
The reasoning in the proof of Corollary 2.4 yields that \(T=\tau_{A,C} B\) is ψ-condensing, where \(C\equiv z_{0}\). The result follows from Theorem 2.2. □
Remark 2.8
If we take \(Ax=x\) (A is not regular), in Corollary 2.7, we obtain a weak Leray-Schauder version of the Leggett result [7], Theorem 2.
Corollary 2.9
Let X be a Banach algebra satisfying property \((\mathcal{P})\), and let ψ be a nonsingular measure of weak noncompactness on X. Let Ω be a nonempty closed and convex subset of X, and \(U \subset\Omega\) be a weakly open set (with respect to the weak topology of Ω) such that \(0 \in U\). Let \(A, C: X \rightarrow X\) and \(B : \overline{U^{w}} \rightarrow X\) be weakly sequentially continuous operators satisfying conditions (i), (ii), (iii) of Theorem 2.2.
Proof
Remark 2.10
In many practical situations, the weak (sequential) continuity is not easy to be checked or even not satisfied. So, we will interchange this condition by the ws-compactness. Our approach combines the advantages of the strong topology (i.e., the involved mappings will be continuous) with the advantages of the weak topology (i.e., the maps will be weakly compact). This enlarges considerably the applicability of our fixed point theorems. In analogy to what we have done in beginning of Section 2, we will try to develop a parallel fixed point theory for ws-compact operators under Leray-Schauder-type boundary conditions.
Theorem 2.11
- (i)
B is ws-compact, and the set \(B(\overline{U})\) is bounded with bound \(Q = \Vert B(\overline{U}) \Vert \),
- (ii)A and C are Lipschitzian mappings with constants \(\alpha_{A}\) and \(\alpha_{C}\) such that$$Q \alpha_{A} + \alpha_{C} < 1, $$
- (iii)
\(T:= \tau_{A,C}B \colon\overline{U}\to X \) is ψ-condensing.
- (A1)
there exists \(u\in\overline{U} \) such that \(Au Bu + Cu = u\), or
- (A2)
there exist \(u \in\partial_{\Omega} U\) and \(\lambda \in (0, 1)\) such that \(\lambda A (\frac{u}{\lambda}) B u + \lambda C (\frac{u}{\lambda}) = u\).
Proof
Remark 2.12
We should mention that property \((\mathcal{P})\) is essential in the proof of Theorem 2.2 as far as the weak sequential continuity is concerned. This is not the case in Theorem 2.11 since the involved mappings are continuous (for the norm topology) and the algebra multiplication is always continuous.
As a consequence of Theorem 2.11, we get the following corollaries and related results.
Corollary 2.13
- (i)
B is ws-compact, and the set \(B(\overline{U})\) is bounded with bound \(Q = \Vert B(\overline{U}) \Vert \),
- (ii)A and C are Lipschitzian mappings with constants \(\alpha _{A}\) and \(\alpha_{C}\) such that$$Q \alpha_{A} + \alpha_{C} < 1, $$
- (iii)
\(T:= \tau_{A,C}B \colon\overline{U}\to X \) is weakly compact.
- (A1)
there exists \(u\in\overline{U} \) such that \(Au Bu + Cu = u\), or
- (A2)
there exist \(u \in\partial_{\Omega} U\) and \(\lambda \in (0, 1)\) such that \(\lambda A (\frac{u}{\lambda}) B u + \lambda C (\frac{u}{\lambda}) = u\).
Remark 2.14
In Corollary 2.13, hypothesis (iii) may be replaced with (iii′): ‘T maps bounded sets into relatively weakly compact sets.’
Corollary 2.15
- (i)
B is ws-compact, B maps bounded sets into relatively weakly compact sets, and \(B(\overline{U})\) is bounded with bound Q,
- (ii)A and C are ww-compact Lipschitzian mappings with constants \(\alpha_{A}\) and \(\alpha_{C}\) such that$$Q \alpha_{A} + \alpha_{C} < 1. $$
- (A1)
there exists \(u\in\overline{U} \) such that \(Au Bu + Cu = u\), or
- (A2)
there exist \(u \in\partial_{\Omega} U\) and \(\lambda \in (0, 1)\) such that \(\lambda A (\frac{u}{\lambda}) B u + \lambda C (\frac{u}{\lambda}) = u\).
Proof
According to Corollary 2.13, we have only to show that T maps bounded sets into relatively weakly compact sets. To see this, let M a bounded subset of U̅. From our hypotheses we know that \(B(M)\) is relatively weakly compact. Notice \(\tau_{A,C}\) is ww-compact implies that \(T(M)=\tau_{A,C} (B(M))\) is relatively weakly compact. This achieves the proof. □
Remark 2.16
Considering the particular case \(C\equiv0\), we get the following result.
Corollary 2.17
- (i)
B is ws-compact, and \(B(\overline{U})\) is bounded with bound \(Q = \Vert B(\overline{U}) \Vert \),
- (ii)
A is a Lipschitzian mapping with constant \(\alpha_{A}\) such that \(Q \alpha_{A} < 1\),
- (iii)
\(T:= \tau_{A,0}B \colon\overline{U}\to X \) is ψ-condensing.
- (A1)
there exists \(u\in\overline{U} \) such that \(Au Bu = u\), or
- (A2)
there exist \(u \in\partial_{\Omega} U\) and \(\lambda \in (0, 1)\) such that \(\lambda A (\frac{u}{\lambda}) B u = u\).
Remark 2.18
The results of Section 2 remain true if we replace A and C ‘Lipschitzian’ with ‘\(\mathcal{D}\)-Lipschitzian,’ provided that \(\tau_{A,C}B(\overline{U^{w}})\) is bounded.
3 Application
- (H1)
- (i)
The functions \(\mu, \sigma, \eta: J \longrightarrow J\) are continuous,
- (ii)
the function \(q: J \longrightarrow X\) is continuous. Let \(Q_{1}= \sup_{t\in J} \Vert q(t) \Vert \),
- (i)
- (H2)
- (i)
for all \(t\in[0,1]\), \(k(t,\cdot) : X \longrightarrow X\) is weakly sequentially continuous,
- (ii)
for each \(x \in X\), \(k(\cdot, x) : J \longrightarrow X\) is continuous,
- (iii)
there is a continuous function \(\delta\colon J \longrightarrow\mathbb{R}^{+}\) with bound \(\Delta= \Vert \delta \Vert _{\infty} \) such that \(\Vert k(t,x) - k(t,y) \Vert \leq\delta(t) \Vert x-y \Vert \) for all \(x,y\in X\) and \(t\in[0,1]\),
- (i)
- (H3)the operator \(A : E \longrightarrow E\) satisfies
- (i)there exists a continuous function \(\gamma\colon J \longrightarrow \mathbb{R}^{+}\) with bound \(\Gamma= \Vert \gamma \Vert _{\infty} \) such that, for all \(x, y \in E\) and \(t \in [0, 1]\),$$ \bigl\Vert Ax(t) - Ay(t) \bigr\Vert \leq\gamma(t) \bigl\Vert x (t)-y(t) \bigr\Vert , $$
- (ii)
A is weakly sequentially continuous on E,
- (iii)
A is weakly compact, and \(M_{0} = \sup_{x \in E} \Vert Ax \Vert \),
- (i)
- (H4)
- (i)
for each continuous \(x:[0,1] \longrightarrow X\), the function \(s\mapsto g(s, x(s))\) is weakly measurable on \([0,1]\), and for almost every \(t \in [0,1]\), the map \(x \mapsto g(t, x)\) is weakly sequentially continuous,
- (ii)there are a function \(\phi \in L^{1}( [0,1], \mathbb{R}^{+})\) and a continuous nondecreasing function \(\theta: \mathbb{R}^{+} \longrightarrow\mathbb{R}^{+}\) such thatand all x in X,$$\bigl\Vert g(s,x) \bigr\Vert \leq \phi(s) \theta\bigl(\Vert x \Vert \bigr) \quad\mbox{for a.e. }s \in[0,1] $$
- (iii)there is a constant \(\tau \geq0\) such that \(\tau \frac {M_{0}}{1-k_{C}} <1\) andfor any bounded subset V of X,$$ w\bigl(g\bigl([0, 1] \times V\bigr)\bigr) \leq\tau w(V) $$
- (i)
- (H5)there is \(r > 0\) such that \(Q \Gamma+ \Delta< 1\), where$$Q= Q_{1}+ \theta(r) \int_{0}^{1}\phi(s)\,ds. $$
Now, we are ready to state the main result of this section.
Theorem 3.1
Let X be a Banach algebra satisfying condition \((\mathcal{P})\) and suppose that assumptions (H1)-(H5) hold.
Then equation (3.1) has at least one solution in E.
To prove Theorem 3.1, we need the following result.
Lemma 3.2
([41], p.36)
Let K be a compact Hausdorff space, and X be a Banach space. Let \((f_{n})_{n}\) be a bounded sequence in \(C(K,X)\), and \(f \in C(K,X)\). Then \((f_{n})_{n}\) is weakly convergent to f if and only if \((f_{n}(t))_{n}\) is weakly convergent to \(f(t)\) for each \(t \in K\).
Proof
We show that the operators \(A, B\), and C satisfy all conditions of Corollary 2.9. This will be achieved in five steps.
Remark 3.3
Our results are more general than those obtained in [20]. In [20], X is assumed to be a reflexive Banach algebra, and A is assumed to be quasi-regular. In our considerations, it might be, from technical viewpoint, relatively simple to require X to be reflexive and to use standard arguments based on the weak version of the Arzelà-Ascoli theorem to obtain existence results under weaker assumptions. Unfortunately, the only known examples of Banach algebras with property \((\mathcal{P})\) are Banach algebras with the Dunford-Pettis property (see [19]). Nevertheless, the Dunford-Pettis property is thought of as the opposite of reflexivity because it is known that a reflexive Banach algebra with the Dunford-Pettis property is necessarily finite-dimensional. For this reason, we may ask whether some infinite-dimensional reflexive Banach algebra could have property \((\mathcal{P})\).
Declarations
Acknowledgements
The authors express their gratitude to the anonymous referees for their very valuable remarks and improvements.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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