 Research
 Open Access
Contributions to the fixed point theory of diagonal operators
 Adrian Petruşel^{1, 2}Email author,
 Ioan A Rus^{1} and
 MarcelAdrian Şerban^{1}
https://doi.org/10.1186/s1366301605891
© Petruşel et al. 2016
 Received: 12 August 2016
 Accepted: 22 September 2016
 Published: 3 October 2016
Abstract
In this paper, we introduce the notion of diagonal operator, we present the historical roots of diagonal operators and we give some fixed point theorems for this class of operators. Our approaches are based on the weakly Picard operator technique, difference equation techniques, and some fixed point theorems for multivalued operators. Some applications to differential and integral equations are given. We also present some research directions.
Keywords
 diagonal operator
 fixed point
 fixed point structure
 coupled fixed point
 weakly Picard operator
 difference equation
 multivalued operator
 differential equation
 integral equation
 research direction
MSC
 47H10
 54H25
 47J25
 65J15
 47H09
1 Introduction and preliminary notions and results
In this section we will present some useful notions and results concerning diagonal operators, coupled fixed point operators, and iterations of some operators generated by the above concepts.
1.1 Diagonal operators
We also consider the following operators generated by an operator \(V:X\times X\to X\):
(2) The operator \(D_{V}:X\times X\to X\times X\) defined by \(D_{V}(x,y):=(y,V(x,y))\).
We have \((x,y)\in F_{D_{V}}\Leftrightarrow x=y\) and \(x\in F_{U_{V}}\).
It is clear that \(F_{U_{V}}=F_{T_{V}}\).
The aim of this paper is to present some historical roots of the diagonal operators, to study the fixed points of this class of operators, and to give some applications. Some new research directions are also presented.
More precisely, the plan of the paper is the following:
1. Introduction and preliminary notions and results.
2. Historical roots of the diagonal operators.
3. Iterations of the operators \(C_{V}\) and \(U_{V}\).
5. Fixed point results for the operator \(T_{V}\).
6. Applications.
7. Research directions.
References.
1.2 LSpaces [4–6]
Following Fréchet [4], we present now the concept of Lspace.
Definition 1.1
 (i)
if \(x_{n}=x\), for all \(n\in\mathbb{N}\), then \((x_{n})_{n\in \mathbb{N}}\in c(X)\) and \(\operatorname{Lim}(x_{n})_{n\in\mathbb{N}}=x\);
 (ii)if \((x_{n})_{n\in\mathbb{N}}\in c(X)\) and \(\operatorname{Lim}(x_{n})_{n\in \mathbb{N}}=x\), then, for all subsequences \((x_{n_{i}})_{i\in\mathbb{N}}\) of \((x_{n})_{n\in\mathbb{N}}\), we have \((x_{n_{i}})_{i\in\mathbb{N}}\in c(X)\) and$$\operatorname{Lim}(x_{n_{i}})_{i\in\mathbb{N}}=x. $$
Remark 1.1
 (1)
If \((X,\tau)\) is a Hausdorff topological space, then \((X,\overset{\tau}{\longrightarrow})\) is an Lspace.
 (2)
If \((X,d)\) is a metric space then \((X,\overset{d}{\longrightarrow})\) is an Lspace.
 (3)
If \((X,\Vert \cdot \Vert )\) is a normed space then \((X,\overset{\Vert \cdot \Vert }{\longrightarrow})\) and \((X,\rightharpoonup)\) are Lspaces.
1.3 Weakly Picard operators [5, 7–11], etc.
Let \((X,\to)\) be an Lspace. By definition, \(f:X\to X\) is said to be a weakly Picard operator (WPO) if the sequence \((f^{n}(x))_{n\in\mathbb{N}}\) of successive approximations converges for all \(x\in X\) and the limit (which may depend on x) is a fixed point of f. If f is weakly Picard operator and \(F_{f}=\{x^{*}\}\), then, by definition, f is called a Picard operator (PO).
1.4 Measures of noncompactness [12–17], etc.
Let X be a Banach space. We will denote by \(P_{b}(X)\) the family of all nonempty bounded subsets of S.
We will use the symbol \(\alpha_{K}:P_{b}(X)\to\mathbb{R}_{+}\), for the Kuratowski measure of noncompactness on X, while \(\alpha_{H}:P_{b}(x)\to\mathbb{R}_{+}\) will denote the Hausdorff measure of noncompactness on X. The following results are well known.
Darbo’s theorem
 (i)
f is continuous;
 (ii)there exists \(l\in[0,1[\) such that:$$\alpha_{K} \bigl(f(A) \bigr)\le l\cdot\alpha_{K}(A), \quad\textit{for all } A\in P(Y). $$
 (a)
\(F_{f}\ne\emptyset\);
 (b)
\(F_{f}\) is a compact subset of Y.
Sadovskii’s theorem
 (i)
f is continuous;
 (ii)
\(\alpha_{H}(f(A))<\alpha_{H}(A)\), for all \(A\in P(Y)\) with \(\alpha_{H}(A)\ne0\).
 (a)
\(F_{f}\ne\emptyset\);
 (b)
\(F_{f}\) is a compact subset of Y.
1.5 Fixed point structures [14]

\(S:\mathcal{C}\multimap \operatorname{Set}^{*}, X\multimap S(X)\subset P(X)\),

\(M:D_{M}\subset P(\mathcal{C})\times P(\mathcal{C})\multimap \mathbb{M}(P(\mathcal{C}),P(\mathcal{C})), (A,B)\multimap M(A,B)\subset\mathbb{M}(A,B)\).
 (i)
\(A\in S(X)\Rightarrow (A,A)\in D_{M}\);
 (ii)
\(A\in S(X), f\in M(U)\Rightarrow F_{f}\ne\emptyset\).
The following examples illustrate this notion.
(1) The fixed point structure (f.p.s.) of Tarski
(2) The f.p.s. of contractions
(3) The f.p.s. of Schauder
A similar notion of fixed point structure can be defined for multivalued operators (see [14], pp.139142).
 (i)
\(S(X)\subset P(X)\) and \(S(X)\neq\emptyset\);
 (ii)
\(M^{0}: P(X)\multimap \bigcup_{Y\in P(X)}M^{0}(Y), Y\multimap M^{0}(Y)\subset\mathbb{M}(Y)\), where \(\mathbb{M}(Y)\) is the set of all self multivalued operators on Y;
 (iii)
\(Y\in S(X), T\in M^{0}(Y)\Rightarrow F_{T}\ne\emptyset\).
1.6 Acyclic topological spaces [15, 18]
Let X be a compact metric space and \(H_{q}(x)\) be the qdimensional Čech homology on \(\mathbb{Q}\) of X. By definition, X is called acyclic if \(H_{q}(X)=0\) for \(q\ge1\) and \(H_{q}(X)\approx\mathbb{Q}\).
The following result is a particular case of the EilenbergMontgomery theorem (see [15, 17, 18]).
Theorem 1.1
Let Y be a compact convex subset of a Banach space E and \(T:Y\to P(Y)\) be an upper semicontinuous multivalued operator with acyclic values. Then \(F_{T}\neq\emptyset\).
2 Historical roots of the diagonal operators
There are some roots of the diagonal operators as the following examples reveal.
Example 2.1
(Difference equations [19–22])
Example 2.2
(Krasnoselskii (1955) [23])
 (i)
f is a contraction;
 (ii)
g is complete continuous;
 (iii)
\(f(x)+g(y)\in Y, \mbox{ for all } x,y\in Y\).
Under these conditions, Krasnoselskii proved that the operator \(f+g:Y\to Y\) has at least a fixed point.
Example 2.3
(Browder (1966; [24]; see also [12, 13, 16, 25–28], etc.)
Let X be a Banach space, \(Y\in P_{b,op}(X)\) and \(V:X\times X\to X\) be a continuous operator. Then Browder considered the operator \(U:\overline {Y}\to X\) defined by \(U(x):=V(x,x)\).
 (1)
U is strictly semicontractive if, for each fixed x in X, \(V(\cdot,x)\) is Lipschitzian with constant \(l<1\) and \(V(x,\cdot)\) is compact.
 (2)
U is weakly semicontractive if, for each x in X, the operator \(V(\cdot,x)\) is nonexpansive and \(V(x,\cdot)\) is compact.
Example 2.4
(Ziebur (1962 and 1965); [29, 30])
 (a)
\(F(t,x,x)=f(t,x)\), for all \(t\in[0,h], x\in\mathbb{R}^{m}\);
 (b)
\(F(t,\cdot,x)\) is increasing;
 (c)
\(F(t,x,\cdot)\) is decreasing.
Example 2.5
(Amann (1973, 1977), Opoitsev (1975; [2, 31–33]))
In [2] the author presents the following result ‘concerning socalled intervined’:
 (i)
\(g(\cdot,y):X\to X\) is increasing for every \(y\in X\);
 (ii)
\(g(x,\cdot):X\to X\) is decreasing for every \(x\in X\).
Example 2.6
(Quasilinear differential equations; see [17, 34, 35, 38], etc.)
3 Iterations of \(C_{V}\) and \(U_{V}\)
The following result is the starting point for this section.
Lemma 3.1
 (a)
\(U_{V}\) is WPO;
 (b)
\(C_{V}^{\infty}(x,x)=(U_{V}^{\infty}(x),U_{V}^{\infty}(x))\), for all \(x\in X\);
 (c)if \(C_{V}\) is a PO, then:
 (1)
\(F_{C_{V}}=\{(x^{*},x^{*})\}\);
 (2)
\(U_{V}\) is PO and \(F_{U_{V}}=\{x^{*}\}\).
 (1)
Proof
(c). Follows from the definition of PO and from (a) + (b).
Now we consider instead of the Lspace \((X,\to)\) a metric space \((X,d)\).
In the case of metric spaces we have the following result.
Theorem 3.1
 (a)If \(C_{V}\) is \(\psiWPO\) with respect to the metric \(d_{1}\), then \(U_{V}\) is a \(\thetaWPO\) where$$\theta ( r ) :=\frac{1}{2}\psi ( 2r ) , \quad r\in \mathbb{R}_{+}. $$
 (b)If \(C_{V}\) is \(\psiWPO\) with respect to the metric \(d_{2}\), then \(U_{V}\) is a \(\thetaWPO\) where$$\theta ( r ) :=\frac{1}{\sqrt{2}}\psi ( \sqrt {2}r ),\quad r\in \mathbb{R}_{+}. $$
 (c)
If \(C_{V}\) is \(\psiWPO\) with respect to the metric \(d_{\infty}\), then \(U_{V}\) is a \(\psiWPO\).
Proof
The following result is a coupled fixed point theorem in a complete bmetric space, which has as an additional conclusion the fact that the operator \(C_{V}\) is a Picard operator.
Theorem 3.2
([39])
In particular, the operator \(C_{V}:X\times X\to X\times X\) given by \(C_{V}(x,y):=(V(x,y),V(y,x))\) is a Picard operator.
Proof
Another result involves the coupled fixed point problem in a complete metric space under a contraction condition on the graphic of the operator. In this case, we will see that \(C_{V}\) is a weakly Picard operator. Let us also point out that we denote \(V^{2}(x,y):=V(V(x,y),V(y,x))\) and \(V^{2}(y,x):=V(V(y,x),V(x,y))\), while the graphic of an operator \(U:X\to X\) is denoted by \(\operatorname{Graph}(U):=\{(x,y)\in X\times X : y=U(x) \}\).
Theorem 3.3
In particular, \(C_{V}:X\times X\to X\times X\) given by \(C_{V}(x,y):=(V(x,y),V(y,x))\) is a weakly Picard operator.
Proof
Remark 3.1
It is worth to mention that the above results can easily be considered in the framework of an ordered metric space X, under contraction type conditions imposed for comparable elements (with respect to a partial order relation ⪯ on X); see for example [3, 39, 41–43], etc.
We will consider now some qualitative properties concerning the behavior of an operator \(A:X\rightarrow X\), where \((X,d)\) is a metric space. More precisely, we consider the following notions:
By the above notions, we have the following result.
Theorem 3.4
 (a)
If the fixed point equations for \(C_{V}\) is wellposed and \(F_{C_{V}}= \{ ( x^{\ast},x^{\ast} ) \} \), then the fixed point equations for \(U_{V}\) is wellposed.
 (b)
If the operator \(C_{V}\) has the Ostrowski property and \(F_{C_{V}}= \{ ( x^{\ast},x^{\ast} ) \} \), then the operator \(U_{V}\) has the Ostrowski property.
 (c)
If the fixed point equations for \(C_{V}\) is generalized UlamHyers stable and all the fixed points of \(C_{V}\) are of the form \(( x^{\ast},x^{\ast} ) \), then the fixed point equations for \(U_{V} \) is generalized UlamHyers stable.
Proof
If we replace the metric \(d_{1}\) on \(X\times X\) defined by (3.1) with the metric \(d_{2}\) or \(d_{\infty}\) on \(X\times X\) defined by (3.2), respectively, by (3.3), we get the same conclusion but instead of \(\theta_{1} ( t ) \) we have a different function, in the case of \(d_{2}\) we have \(\theta_{2} ( t ) =\frac{1}{\sqrt {2}}\theta ( \sqrt{2}t ) \), and in the case of \(d_{\infty}\) we have \(\theta _{\infty } ( t ) =\theta ( t ) \). □
4 Iterations of the operator \(D_{V}\) and the difference equation \(x_{n+2}=V(x_{n},x_{n+1})\), \(n\in\mathbb{N}\), \(x_{0},x_{1}\in X\)
Moreover, we have the following.
Lemma 4.1
 (i)
\(D_{V}\) is Picard with \(F_{D_{V}}=\{(x^{*},x^{*})\}\);
 (ii)
\(D_{V}^{2}\) is Picard operator with \(F_{D^{2}_{V}}=\{(x^{*},x^{*})\}\);
 (iii)\(x^{*}\) is globally asymptotically stable solution of the difference equation$$x_{n+2}=V(x_{n},x_{n+1}),\quad n\in\mathbb{N}. $$
Theorem 4.1
 (i)
φ is increasing;
 (ii)
\(\sum_{k=0}^{\infty}\phi^{k} ( r ) <+\infty\), where \(\phi ( r ) =\varphi ( r,r ) \), \(r\in \mathbb {R}_{+}\);
 (iii)
\(\varphi ( r,0 ) +\varphi ( 0,r ) \leq \phi ( r ) \), \(r\in\mathbb{R}_{+}\);
 (iv)
\(d ( V ( x_{0},x_{1} ) ,V ( x_{1},x_{2} ) ) \leq\varphi ( d ( x_{0},x_{1} ) ,d ( x_{1},x_{2} ) ) \), for all \(x_{0},x_{1},x_{2}\in X\).
 (a)
\(F_{U_{V}}= \{ x^{\ast} \} \).
 (b)
If \(( x_{n} ) _{n\in\mathbb{N}}\) is a solution of the difference equation (4.1) then \(x_{n}\rightarrow x^{\ast}\) as \(n\rightarrow+\infty\).
Proof
Theorem 4.2
 (a)
\(D_{V}^{2}\) is a \((l_{1}+l_{2})\)contraction;
 (b)
\(F_{D_{V}}=\{(x^{*},x^{*})\}\) and \(F_{U_{V}}=\{x^{*}\}\);
 (c)if \((x_{n})_{n\in\mathbb{N}}\) is a solution of the difference equation$$x_{n+2}=V(x_{n},x_{n+1}),\quad n\in\mathbb{N}, $$
Proof
From Lemma 4.1 we get (b) and (c). □
For related results concerning the difference equation (4.1) see [19–22, 57, 58], etc.
5 Fixed point results for the operator \(T_{V}\)
A possible approach for the study of the fixed points of the operator \(T_{V}\) is given by the following general result.
Lemma 5.1
 (i)
\(S_{1}(X)\cap S_{2}(X)\neq\emptyset\);
 (ii)
\(V(\cdot,x)\in M_{1}(Y)\), for each \(x\in Y\);
 (iii)
\(T_{V}\in M^{0}_{1}(Y)\).
Then \(F_{T_{V}}\neq\emptyset\) and \(F_{U_{V}}=F_{T_{V}}\).
Proof
Since \(Y\in S_{1}(X)\cap S_{2}(X)\) and using (ii) we obtain \(T_{V}(x)\neq \emptyset\), for each \(x\in Y\). Since \(Y\in S_{2}(X)\) and using (iii) we get \(F_{T_{V}}\neq\emptyset\). On the other hand, \(F_{U_{V}}=F_{T_{V}}\). □
In particular, we have the following consequences of the above approach.
Theorem 5.1
 (i)
\(V:Y\times Y\to Y\) is continuous;
 (ii)
the set \(\{u\in Y \mid u=V(u,x) \}\) is convex, for each \(x\in Y\).
Then \(F_{T_{V}}\neq\emptyset\), i.e., there exists \(x^{*}\in Y\) such that \(x^{*}=V(x^{*},x^{*})\).
Proof
Since \(V(\cdot,x):Y\to Y\) is continuous and \(Y\in P_{cp,cv}(X)\), by Schauder’s fixed point theorem, we get \(F_{V(\cdot,x)}\neq\emptyset\), for each \(x\in Y\). Moreover, by (ii), the set \(F_{V(\cdot,x)}\) is convex, for each \(x\in Y\). On the other hand, by the continuity of V, we see that the set \(\{ (x,u)\in Y\times Y \mid u=V(u,x) \}\) is closed in \(Y\times Y\). Thus, the multivalued operator \(T_{V}:Y\to P(Y)\) given by \(T_{V}(x):=F_{V(\cdot ,x)}\) has a closed graphic. Since the codomain Y is compact, \(T_{V}\) is upper semicontinuous on Y. Hence, we get \(T_{V}:Y\to P_{cp,cv}(Y)\) and it is upper semicontinuous. By BohnenblustKarlin’s fixed point theorem we get \(F_{T_{V}}\neq\emptyset\). □
Theorem 5.2
 (i)
\(V:Y\times Y\to Y\) is continuous;
 (ii)
\(V(\cdot,x):Y\to Y\) is nonexpansive, for each \(x\in Y\).
Then \(F_{T_{V}}\neq\emptyset\).
Proof
Since \(V(\cdot,x):Y\to Y\) is nonexpansive and \(Y\in P_{cp,cv}(X)\), by BrowderGhödeKirk’s fixed point theorem, we see that the set \(F_{V(\cdot,x)}\) is nonempty and convex, for each \(x\in Y\). On the other hand, by the continuity of V, we see that the set \(\{ (x,u)\in Y\times Y \mid u=V(u,x) \}\) is closed in \(Y\times Y\). Thus, the multivalued operator \(T_{V}:Y\to P(Y)\) given by \(T_{V}(x):=F_{V(\cdot ,x)}\) has a closed graphic. Since the codomain Y is compact, \(T_{V}\) is upper semicontinuous on Y. Hence, we get \(T_{V}:Y\to P_{cp,cv}(Y)\) and it is upper semicontinuous. Our conclusion follows by BohnenblustKarlin’s fixed point theorem. □
Theorem 5.3
 (i)there exists \(\alpha\in(0,1)\) such that, for each \(x\in X\), we have$$\bigl\Vert V(u,x)V(v,x)\bigr\Vert \le\alpha \Vert uv\Vert , \quad \textit{for all } u,v\in Y; $$
 (ii)
for each \(u\in Y\) the operator \(V(u,\cdot):Y\to Y\) is continuous;
 (iii)
for each \(u\in Y\) the set \(V(u,Y)\) is relatively compact.
Then \(F_{T_{V}}\neq\emptyset\).
Proof
Theorem 5.4
 (i)there exists \(\alpha\in(0,1)\) such that, for each \(x\in X\), we have$$\bigl\Vert V(u,x)V \bigl(V(u,x),x \bigr)\bigr\Vert \le\alpha\bigl\Vert uV(u,x)\bigr\Vert ,\quad \textit{for all } x,u \in Y; $$
 (ii)
\(V:Y\times Y\to Y\) is continuous;
 (iii)
the set \(\{u\in Y \mid u=V(u,x) \}\) is convex, for each \(x\in Y\).
Then \(F_{T_{V}}\neq\emptyset\).
Proof
Notice first that, for every \(x\in Y\), the operator \(V(\cdot,x):Y\to Y\) is a graphic contraction. Thus, for each \(x\in Y\), the set \(F_{V(\cdot,x)}\) is nonempty. Moreover, by the continuity of V, the set \(F_{V(\cdot,x)}\) is closed. Thus, the operator \(T_{V}:Y\to P(Y)\) given by \(T_{V}(x):=F_{V(\cdot,x)}\) is a multivalued operator with closed graph. Since Y is compact, we see that \(T_{V}\) is upper semicontinuous on Y with compact and (by (iii)) convex values. The conclusion follows by BohnenblustKarlin’s fixed point theorem. □
Another result of this type can be reached using the above mentioned particular variant of the EilenbergMontgomery theorem; see Theorem 1.1.
Theorem 5.5
 (i)
for each \(x\in X\) the operator \(V(\cdot,x)\) is nonexpansive and compact;
 (ii)
the operator \(V:Y\times Y\to Y\) is continuous.
Then \(F_{T_{V}}\neq\emptyset\).
Proof
Notice first that, by Theorem 1.63 in [59], the set \(F_{V(\cdot ,x)}\) is nonempty and acyclic, for each \(x\in Y\). On the other hand, by the continuity of V, we see that the set \(\{ (x,u)\in Y\times Y \mid u=V(u,x) \}\) is closed in \(Y\times Y\). Thus, the multivalued operator \(T_{V}:Y\to P(Y)\) given by \(T_{V}(x):=F_{V(\cdot ,x)}\) has a closed graphic. Since the codomain Y is compact, \(T_{V}\) is upper semicontinuous on Y. Hence, we see that \(T_{V}:Y\to P(Y)\) has acyclic values and it is upper semicontinuous. The conclusion follows by Theorem 1.1. □
6 Applications
6.1 Fredholm type integral equations
By Theorem 4.1 we have the following.
Theorem 6.1
 (i)
\(\varphi:\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}\) satisfies conditions (i)(iii) in Theorem 4.1;
 (ii)
\(\vert K(t,s,u,v)K(t,s,v,w)\vert \le\frac{1}{\operatorname{mes}(\Omega)}\varphi (\vert uv\vert ,\vert vw\vert )\) for all \(t,s\in \overline {\Omega}\), \(u,v,w\in\mathbb{R}\).
 (a)
Equation (6.1) has a unique solution, \(x^{*}\in C(\overline {\Omega})\).
 (b)The sequence \((x_{n})_{n\in\mathbb{N}}\), defined byconverges to \(x^{*}\) for all \(x_{0},x_{1}\in C(\overline {\Omega})\).$$x_{n+2}(t)= \int_{\Omega}K \bigl(t,s,x_{n}(s),x_{n+1}(s) \bigr)\,ds, \quad t\in \overline {\Omega}, $$
6.2 A periodic boundary value problem
By Theorem 4.1 we have the following.
Theorem 6.2
 (i)
\(\varphi:\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}\) satisfies conditions (i)(iii) in Theorem 4.1;
 (ii)
\(\vert f(s,u,v)f(s,v,w)\vert \le\frac{8}{(ba)^{2}}\cdot\varphi(\vert uv\vert ,\vert vw\vert )\) for all \(t,s\in[a,b]\), \(u,v,w\in\mathbb{R}\).
 (a)
The boundary value problem (6.2) has a unique solution, \(x^{*}\).
 (b)The sequence \((x_{n})_{n\in\mathbb{N}}\), defined byconverges to \(x^{*}\) for all \(x_{0},x_{1}\in C[a,b]\).$$x_{n+2}(t)= \int_{a}^{b} f \bigl(s,x_{n}(s),x_{n+1}(s) \bigr)\,ds,\quad t\in[a,b] $$
6.3 Other applications
Other applications of the abstract results given in this paper can be obtained for the case of functionaldifferential equations and functionalintegral equations (or inclusions) which appear in [6, 21, 36, 44, 53, 60], etc.
7 Research directions
7.1 Mixed monotone operators
 (i)
\(V(\cdot,x)\) is increasing;
 (ii)
\(V(x,\cdot)\) is decreasing;
 (iii)
\(U=U_{V}\).
7.2 Difference equations for diagonal operators
7.3 Fixed point structures approach to diagonal operators
Let \((X,S(X),M)\) be a fixed point structure on X, \(Y\in S(X)\) and \(V:Y\times Y\to Y\). Under which conditions on V do we have \(U_{V}\in M(Y)\)?
Commentaries
 (i)
V is continuous;
 (ii)
\(V(\cdot,y)\) is a lcontraction, for all \(y\in Y\);
 (iii)
\(V(x,\cdot)\) is compact, for all \(x\in Y\).
Declarations
Acknowledgements
The authors are thankful to the anonymous reviewer(s) for his (her, their) useful remarks and constructive suggestions.
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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