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Contributions to the fixed point theory of diagonal operators
Fixed Point Theory and Applications volume 2016, Article number: 95 (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.
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.
Diagonal operators
Let X be a nonempty set and \(V:X\times X\to X\) be an operator. By definition, the operator \(U_{V}:X\to X\), defined by
is called the diagonal operator corresponding to the operator V.
We also consider the following operators generated by an operator \(V:X\times X\to X\):
(1) The operator \(C_{V}:X\times X\to X\times X\) defined by
By definition an element
is called a coupled fixed point of V (see [1]; see also [2, 3]). We remark that
(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 \(x\in F_{U_{V}}\).
(3) The operator \(T_{V}:X\to\mathcal{P}(X)\) is defined by
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 \(U_{V}\).
4. Iterations of the operator \(D_{V}\) and the difference equation
5. Fixed point results for the operator \(T_{V}\).
6. Applications.
7. Research directions.
References.
LSpaces [4–6]
Following Fréchet [4], we present now the concept of Lspace.
Definition 1.1
Let X be a nonempty set. Let
Let \(c(X)\) be a subset of \(s(X)\) and \(\operatorname{Lim}:c(X)\rightarrow X\) be an operator. By definition, the triple \((X,c(X),\operatorname{Lim})\) is called an Lspace (denoted by \((X,\rightarrow)\)) if the following conditions are satisfied:

(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
$$\operatorname{Lim}(x_{n_{i}})_{i\in\mathbb{N}}=x. $$
By definition, an element of \(c(X)\) is said to be a convergent sequence and \(\operatorname{Lim}(x_{n})_{n\in\mathbb{N}}\) is the limit of this sequence. If \(\operatorname{Lim}(x_{n})_{n\in\mathbb{N}}=x\), then we will write
Remark 1.1
An Lspace is any set endowed with a structure implying a notion of convergence for sequences. As examples of Lspaces we mention the following:

(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.
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).
If \(f:X\to X\) is WPO, then we define the operator \(f^{\infty}:X\to X\) by
Now let \((X,d)\) be a metric space. By definition, a WPO \(f:X\to X\) is called ψWPO if \(\psi:\mathbb{R}_{+}\to\mathbb{R}_{+}\) is increasing, continuous in 0 with \(\psi(0)=0\), and
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
Let X be a Banach space, \(Y\in P_{b,cl,cv}(X)\) and \(f:Y\to Y\) be an operator. We suppose that:

(i)
f is continuous;

(ii)
there exists \(l\in[0,1[\) such that:
$$\alpha_{K} \bigl(f(A) \bigr)\le l\cdot\alpha_{K}(A), \quad\textit{for all } A\in P(Y). $$
Then:

(a)
\(F_{f}\ne\emptyset\);

(b)
\(F_{f}\) is a compact subset of Y.
Sadovskii’s theorem
Let X be a Banach space, \(Y\in P_{b,cl,cv}(X)\) and \(f:Y\to Y\) be an operator. We suppose that:

(i)
f is continuous;

(ii)
\(\alpha_{H}(f(A))<\alpha_{H}(A)\), for all \(A\in P(Y)\) with \(\alpha_{H}(A)\ne0\).
Then:

(a)
\(F_{f}\ne\emptyset\);

(b)
\(F_{f}\) is a compact subset of Y.
Fixed point structures [14]
Let \(\mathcal{C}\) be a class of structured sets (ordered sets, topological spaces, metric spaces, Banach spaces, Hilbert spaces, …). Let we denote by Set^{∗} the class of all nonempty sets. For \(X\in \operatorname{Set}^{*}\) we shall use the notations:
Now we consider the following multivalued operators:

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

\(M:D_{M}\subset P(\mathcal{C})\times P(\mathcal{C})\multimap \mathbb{M}(P(\mathcal{C}),P(\mathcal{C})), (A,B)\multimap M(A,B)\subset\mathbb{M}(A,B)\).
By a fixed point structure on \(X\in\mathcal{C}\), we understand a triple \((X,S(X),M)\) with the following properties:

(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
Let \(\mathcal{C}\) be the class of complete lattices. If \((X,\preceq)\) is a complete lattice,
and
then, by Tarski’s fixed point theorem, we see that \((X,S(X),M)\) is a f.p.s.
(2) The f.p.s. of contractions
Let \(\mathcal{C}\) be the class of complete metric spaces. For a complete metric space \((X,d)\), we consider
and
Then the Banach contraction principle implies that \((X,S(X),M)\) is a f.p.s.
(3) The f.p.s. of Schauder
Let \(\mathcal{C}\) be the class of Banach spaces. For a Banach space X, if we consider
and
then, by Schauder’s theorem, we see that \((X,S(X),M)\) is a f.p.s.
A similar notion of fixed point structure can be defined for multivalued operators (see [14], pp.139142).
A triple \((X,S(X),M^{0})\) is a multivalued fixed point structure (m.f.p.s.) if the following properties hold:

(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)
\(Y\in S(X), T\in M^{0}(Y)\Rightarrow F_{T}\ne\emptyset\).
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 \(F_{T}\neq\emptyset\).
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])
Let \((X,\to)\) be an Lspace and \(V:X\times X\to X\) be an operator. We consider the following difference equation:
Let us suppose that \((x_{n})_{n\in\mathbb{N}}\) is a solution of this equation with the property that
If the function V is continuous, then we have \(x^{*}=V(x^{*},x^{*})\). Thus, \(x^{*}\) is a fixed point of the diagonal operator corresponding to V.
Example 2.2
(Krasnoselskii (1955) [23])
Let X be a Banach space, \(Y\in P_{b,cl,cv}(X)\) and \(f,g:Y\to Y\) be two operators. We suppose that:

(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.
If we consider the operator \(V:Y\times Y\to Y\), defined by
then \(f+g:Y\to Y\), \(x\mapsto f(x)+g(x)\) is the diagonal operator corresponding to V.
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 \(U(x):=V(x,x)\).
Moreover, Browder introduced the following notions:

(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])
Let \(b\in\mathbb{R}^{m}\) and \(h,k\in\mathbb{R}_{+}^{*}\). One consider a set \(\Omega:=\prod_{i=1}^{m} [b_{i}k,b_{i}+k]\) and a function \(f\in C([0,h]\times\Omega,\mathbb{R}^{m})\). Let us consider the Cauchy problem
Then Ziebur introduced a function \(F\in C([0,h]\times\Omega\times \Omega,\mathbb{R}^{m})\) with the following property:

(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.
The following Cauchy problem was also considered:
and it is proved that if the Cauchy problem (C_{2}) has a unique solution then the problem (C_{1}) has a unique solution too and the Picard sequence converges to that solution.
Example 2.5
(Amann (1973, 1977), Opoitsev (1975; [2, 31–33]))
In [2] the author presents the following result ‘concerning socalled intervined’:
Let X be a chain complete ordered set possessing a least and a greatest element. Let \(g:X\times X\to X\) be a mapping such that:

(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 \(x\in X\).
Then there exist two points \(\overline {x},\widehat {x}\in X\) such that \(\overline {x}\preceq \widehat {x}\), \(g(\overline {x},\widehat {x})=\overline {x}\), and \(g(\widehat {x},\overline {x})=\widehat {x}\). Moreover, if \(f(x):=g(x,x)\) for all \(x\in X\), then
Remark 2.1
For other examples on this topic see [1, 33–37], etc.
Example 2.6
(Quasilinear differential equations; see [17, 34, 35, 38], etc.)
Diagonal operators also appears by the linearization of a quasilinear differential equations. For example, let us consider the Cauchy problem:
where \(A\in C([a,b],\mathbb{R}^{m\times m})\) and \(f\in C([a,b]\times \mathbb{R}^{m},\mathbb{R}^{m})\).
Then, for each \(u\in C([a,b],\mathbb{R}^{m})\) with \(u(a)=x_{0}\) we consider the linearized Cauchy problem
Let \(S\subset C^{1}([a,b],\mathbb{R}^{m})\) be the solution set of the problem (2.1) and let \(T(u)\) the solution set of the problem (2.2). Then \(S=F_{T}\).
Iterations of \(C_{V}\) and \(U_{V}\)
The following result is the starting point for this section.
Lemma 3.1
Let \((X,\to)\) be an Lspace and \(V:X\times X\to X\) be an operator. We suppose that the operator \(C_{V}\) is WPO. Then we have:

(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
(a) + (b). We remark that \(C_{V}(x,x)=(U_{V}(x),U_{V}(x))\). From this we have
and
(c). Follows from the definition of PO and from (a) + (b).
Now we consider instead of the Lspace \((X,\to)\) a metric space \((X,d)\).
We will consider, on \(X\times X\), the following metrics:
and
□
In the case of metric spaces we have the following result.
Theorem 3.1
Let \(( X,d ) \) be a metric space and \(V:X\times X\rightarrow X\) be an operator. Then:

(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
(a). From
it follows that
so
(b). If \(C_{V}\) is \(\psiWPO\) with respect to the metric \(d_{2}\) then
which means that
so we get the conclusion.
(c). If \(C_{V}\) is \(\psiWPO\) with respect to the metric \(d_{\infty}\) then
so
which proves that \(U_{V}\) is a \(\psiWPO\). □
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])
Let \((X,d)\) be a complete bmetric space with constant \(s\ge1\). Let \(V:X\times X\to X\) be an operator. Assume that there exists \(k\in(0,1)\) such that, for all \((x,y),(u,v)\in X\times X\), we have
Then there exists a unique solution \((x^{*},y^{*})\in X\times X\) of the following coupled fixed point problem :
and, for any initial point \((x_{0},y_{0})\in X\times X\), the sequences \((x_{n})_{n\in\mathbb{N}}, (y_{n})_{n\in\mathbb{N}}\) defined, for \(n\in\mathbb{N}\), by
converge to \(x^{*}\) and, respectively, to \(y^{*}\) as \(n\to\infty\).
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
For the sake of completeness we present here the sketch of the proof. We introduce on \(Z:=X\times X\) the functional \(\tilde{d}:Z\times Z\to \mathbb{R}_{+}\) defined by
Notice that, as before, d̃ is a bmetric on Z with the same constant \(s\ge1\) and, if the space \((X,d)\) is complete, then \((Z,\tilde{d})\) is complete too.
We consider now the operator \(F:Z\to Z\) given by
It is easy to prove now that F is a contraction in \((Z,\tilde{d})\) with constant \(k\in(0,1)\), i.e.,
Thus, we can apply for F the bmetric space version of the contraction principle given by Czerwik (see, for example, Theorem 12.2, p.115 in [40]) and we get the conclusion. □
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
Let \((X,d)\) be a complete metric space and \(V:X\times X\to X\) be an operator. Assume that there exists \(k\in(0,1)\) such that, for all \((x,y) \in X\times X\), we have
Then there exists at least one solution \((x^{*},y^{*})\in X\times X\) of the coupled fixed point problem (3.4) and, for any initial point \((x_{0},y_{0})\in X\times X\), the sequences \((x_{n})_{n\in\mathbb{N}}, (y_{n})_{n\in\mathbb{N}}\) defined by (3.5) converge to \(x^{*}\) and, respectively, to \(y^{*}\) as \(n\to\infty\).
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
We consider again on \(Z:=X\times X\) the functional \(\tilde{d}:Z\times Z\to\mathbb{R}_{+}\) defined by
As before, \((Z,\tilde{d})\) is a complete metric space.
We consider now the operator \(F:Z\to Z\) given by
It is easy to prove now that F is a graphic contraction in \((Z,\tilde {d})\) with constant \(k\in(0,1)\), i.e.,
Thus, the conclusion follows by the graphic contraction principle; see, for example, [8] or [10]. □
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:
(i) the fixed point equation
is called wellposed if \(F_{A}=\{x_{A}^{\ast}\}\) and for any \(x_{n}\in X\), \(n\in\mathbb{N}\) a sequence in X such that
we have
(ii) The operator A has the Ostrowski property (or the operator A has the limit shadowing property) if \(F_{A}= \{ x^{\ast } \} \) and for any \(x_{n}\in X\), \(n\in\mathbb{N}\) a sequence in X such that
we have
(iii) The fixed point equation
is generalized UlamHyers stable if there exists an increasing function \(\theta:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\), continuous in 0 with \(\theta ( 0 ) =0\), and for each \(\varepsilon>0\) and for each solution \(y^{\ast}\) of the inequality
there exists a solution \(x^{\ast}\) of the fixed point equation with
By the above notions, we have the following result.
Theorem 3.4
Let \(( X,d ) \) be a metric space and \(V:X\times X\rightarrow X\) be an operator. Then:

(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
(a). If \(F_{C_{V}}= \{ ( x^{\ast},x^{\ast } ) \} \) then \(F_{U_{V}}= \{ x^{\ast} \} \). Let \(( x_{n} ) _{n\in\mathbb{N}}\subset X\) such that \(d ( x_{n},U_{V} ( x_{n} ) ) \rightarrow0\) as \(n\rightarrow +\infty\). Then
where \(d_{\ast}\) is one of the metrics on \(X\times X\) defined by (3.1)(3.3). Since the fixed point equation for \(C_{V}\) is wellposed
therefore
so the fixed point equation for \(U_{V}\) is wellposed.
(b). Let \(( x_{n} ) _{n\in\mathbb {N}}\subset X\) such that \(d ( x_{n},U_{V} ( x_{n+1} ) ) \rightarrow0\) as \(n\rightarrow+\infty\). Then
where \(d_{\ast}\) is one of the metrics on \(X\times X\) defined by (3.1)(3.3). Since the operator \(C_{V}\) has the Ostrowski property
therefore
so the operator \(U_{V}\) has the Ostrowski property.
(c). If \(( x^{\ast},x^{\ast} ) \in F_{C_{V}}\) then \(x^{\ast}\in F_{U_{V}}\). Let \(\varepsilon>0\) and \(y^{\ast}\in X\) be a solution of the inequality
then
From the UlamHyers stability of the fixed point equation for \(C_{V}\) we see that there exists \(( x^{\ast},x^{\ast} ) \in F_{C_{V}}\) such that
so
where \(\theta_{1} ( t ) =\frac{1}{2}\theta ( 2t ) \) which proves that the fixed point equations for \(U_{V}\) is generalized UlamHyers stable.
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 \(\theta _{\infty } ( t ) =\theta ( t ) \). □
Remark 3.2
For other considerations on the operator \(C_{V}\) see [39, 44–56], etc.
Iterations of the operator \(D_{V}\) and the difference equation \(x_{n+2}=V(x_{n},x_{n+1})\), \(n\in\mathbb{N}\), \(x_{0},x_{1}\in X\)
Let X be a nonempty set and \(V:X\times X\to X\). Let \((x_{n},y_{n}):=D_{V}^{n}(x_{0},y_{0})\), \((x_{0},y_{0})\in X\times X\). We remark that \(( x_{n} ) _{n\in\mathbb{N}}\) is a solution of the difference equation
Moreover, we have the following.
Lemma 4.1
Let \((X,\to)\) be an Lspace and \(V:X\times X\to X\) a continuous operator. Then the following statements are equivalent:

(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
$$x_{n+2}=V(x_{n},x_{n+1}),\quad n\in\mathbb{N}. $$
Proof
(i) ⇔ (ii). See Lemma 2.2 in [57]. (i) ⇔ (iii). See [20, 21]. □
Theorem 4.1
Let \(( X,d ) \) be a complete metric space, \(V:X\times X\rightarrow X\) and \(\varphi:\mathbb{R}_{+}^{2}\rightarrow\mathbb{R}_{+}\). We suppose that:

(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 \(x_{0},x_{1},x_{2}\in X\).
Then

(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
Let \(x_{0},x_{1}\in X\) and \(( x_{n} ) _{n\in\mathbb{N}}\) be defined by the difference equation (4.1). We have
By induction we get
thus
so \(( x_{n} ) _{n\in\mathbb{N}}\) is fundamental, therefore there exists \(x^{\ast}\in X\) such that \(x_{n}\rightarrow x^{\ast}\) as \(n\rightarrow+\infty\).
By the continuity assumption on φ in 0 we have
thus \(x^{\ast}=V ( x^{\ast},x^{\ast} ) \), which means that \(x^{\ast }\in F_{U_{V}}\).
If there exist \(x^{\ast},y^{\ast}\in F_{U_{V}}\) then
but \(\phi ( r ) < r\) for all \(r\in\mathbb{R}_{+}^{\ast}\), so \(d ( x^{\ast},y^{\ast} ) =0\). □
Theorem 4.2
Let \((X,d)\) be a complete metric space and \(V:X\times X\to X\). We suppose that there exist \(l_{1},l_{2}\in\mathbb{R}_{+}\), \(l_{1}+l_{2}<1\) such that
Then

(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
$$x_{n+2}=V(x_{n},x_{n+1}),\quad n\in\mathbb{N}, $$
then \(x_{n}\to x^{*}\) as \(x\to\infty\).
Proof
(a). Let us consider the complete metric space \(( X\times X,d_{\infty} ) \) where \(d_{\infty}\) is defined by (3.3). We have
and
Thus
so \(D_{V}^{2}\) is an \(( l_{1}+l_{2} ) \)contraction.
From Lemma 4.1 we get (b) and (c). □
For related results concerning the difference equation (4.1) see [19–22, 57, 58], etc.
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
Let X be a nonempty set, \((X,S_{1}(X), M_{1})\) be a f.p.s. on X and \((X,S_{2}(X),M_{2}^{0})\) be a m.f.p.s. on X. Let \(Y\in S_{1}(X)\cap S_{2}(X)\) and \(V:Y\times Y\to Y\). We suppose that:

(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
Let X be a Banach space and \(Y\in P_{cp,cv}(X)\). Let \(V:Y\times Y\to Y\) be an operator such that:

(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
Let X be a uniformly convex Banach space and \(Y\in P_{cp,cv}(X)\). Let \(V:Y\times Y\to Y\) be an operator such that:

(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
Let \((X,\Vert \cdot \Vert ) \) be a Banach space and \(Y\in P_{cl,cv}(X)\). Let \(V:Y\times Y\to Y\) be an operator such that:

(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
Since, for every \(x\in Y\), the operator \(V(\cdot,x):Y\to Y\) is a contraction, for each \(x\in Y\) there exists a unique \(u^{*}=u^{*}(x)\in Y\) such that \(V(u^{*},x)=u^{*}\). Thus, the operator \(T_{V}:Y\to Y\) given by \(T_{V}(x):=u^{*}(x)\) is a self singlevalued operator on Y. Notice that
Moreover, \(T_{V}\) is continuous since
As a consequence, we get the continuity of the singlevalued opertor \(T_{V}\):
By (iii), \(T_{V}(Y)\) is relatively compact. Thus, by Schauder’s fixed point theorem, there exists \(x^{*}\in Y\) such that \(x^{*}=T_{V}(x^{*})\). As a consequence, \(x^{*}=u^{*}=V(u^{*},x^{*})=V(x^{*},x^{*})\). □
Theorem 5.4
Let \((X,\Vert \cdot \Vert ) \) be a Banach space and \(Y\in P_{cp,cv}(X)\). Let \(V:Y\times Y\to Y\) be an operator such that:

(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
Let \((X,\Vert \cdot \Vert ) \) be a Banach space and \(Y\in P_{cp,cv}(X)\). Let \(V:Y\times Y\to Y\) be an operator such that:

(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. □
Remark 5.1
For the case of the EilenbergMontgomery fixed point theorem see [15, 34, 35], etc.
Remark 5.2
For the fixed point theory of multivalued operators see [14, 15, 17, 18], etc.
Applications
Fredholm type integral equations
Let \(\Omega\subset\mathbb{R}^{m}\) be a bounded domain and \(C(\overline {\Omega})\) be the Banach space with
We consider the integral equation, in \(C(\overline {\Omega})\),
where \(K\in C(\overline {\Omega}, \overline {\Omega}\times\mathbb{R}^{2})\).
Let us consider now the operator \(V:C(\overline {\Omega})\times C(\overline {\Omega})\to C(\overline {\Omega})\) defined by
By Theorem 4.1 we have the following.
Theorem 6.1
We suppose that:

(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 \(t,s\in \overline {\Omega}\), \(u,v,w\in\mathbb{R}\).
Then:

(a)
Equation (6.1) has a unique solution, \(x^{*}\in C(\overline {\Omega})\).

(b)
The sequence \((x_{n})_{n\in\mathbb{N}}\), defined by
$$x_{n+2}(t)= \int_{\Omega}K \bigl(t,s,x_{n}(s),x_{n+1}(s) \bigr)\,ds, \quad t\in \overline {\Omega}, $$converges to \(x^{*}\) for all \(x_{0},x_{1}\in C(\overline {\Omega})\).
A periodic boundary value problem
We will consider now a periodic boundary value problem of the following type:
where \(f:[a,b]\times\mathbb{R}^{2}\to\mathbb{R}\) is a given continuous function.
This problem is equivalent to a Fredholm type integral equation of the following form:
where \(G:[a,b]\times[a,b]\to\mathbb{R}_{+}\) is the corresponding Green function.
Let us define now the operator \(V:C[a,b]\times C[a,b]\to C[a,b]\) defined by
By Theorem 4.1 we have the following.
Theorem 6.2
We suppose that:

(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 \(t,s\in[a,b]\), \(u,v,w\in\mathbb{R}\).
Then:

(a)
The boundary value problem (6.2) has a unique solution, \(x^{*}\).

(b)
The sequence \((x_{n})_{n\in\mathbb{N}}\), defined by
$$x_{n+2}(t)= \int_{a}^{b} f \bigl(s,x_{n}(s),x_{n+1}(s) \bigr)\,ds,\quad t\in[a,b] $$converges to \(x^{*}\) for all \(x_{0},x_{1}\in C[a,b]\).
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.
Research directions
Mixed monotone operators
Let \((X,\le)\) be an ordered set and \(U:X\to X\). Under these conditions there exists \(V:X\times X\to X\) such that:

(i)
\(V(\cdot,x)\) is increasing;

(ii)
\(V(x,\cdot)\) is decreasing;

(iii)
\(U=U_{V}\).
Difference equations for diagonal operators
Let \((X,\to)\) be an Lspace and \(U:X\to X\) an operator. Under which conditions is U a diagonal operator with respect to some \(V:X\times X\to X\) such that each solution \((x_{n})_{n\in\mathbb{N}}\) of the difference equation
converges to a fixed point of U?
References: [19, 20, 22, 57, 61, 62], etc.
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
Let X be a Banach space, \((X,P_{b,cl,cv}(X),M)\) the fixed point structure of Darbo. Let \(Y\in P_{b,cl,cv}(X)\) and \(V:Y\times Y\to Y\). We suppose that:

(i)
V is continuous;

(ii)
\(V(\cdot,y)\) is a lcontraction, for all \(y\in Y\);

(iii)
\(V(x,\cdot)\) is compact, for all \(x\in Y\).
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Petruşel, A., Rus, I.A. & Şerban, M. Contributions to the fixed point theory of diagonal operators. Fixed Point Theory Appl 2016, 95 (2016). https://doi.org/10.1186/s1366301605891
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MSC
 47H10
 54H25
 47J25
 65J15
 47H09
Keywords
 diagonal operator
 fixed point
 fixed point structure
 coupled fixed point
 weakly Picard operator
 difference equation
 multivalued operator
 differential equation
 integral equation
 research direction