Contributions to the fixed point theory of diagonal operators
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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 directionMSC
47H10 54H25 47J25 65J15 47H091 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, 5, 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, 8, 9, 10, 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, 13, 14, 15, 16, 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, 20, 21, 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, 26, 27, 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, 32, 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, 42, 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, 20, 21, 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 \(x\in Y\).
Notes
Acknowledgements
The authors are thankful to the anonymous reviewer(s) for his (her, their) useful remarks and constructive suggestions.
References
 1.Guo, D, Lakshmikantham, V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11, 623632 (1987) MathSciNetCrossRefMATHGoogle Scholar
 2.Amann, H: Order Structures and Fixed Points. SAFA, vol. 2, pp. 151, University of Calabria (1977) Google Scholar
 3.Bota, MF, Petruşel, A, Petruşel, G, Samet, B: Coupled fixed point theorems for singlevalued operators in bmetric spaces. Fixed Point Theory Appl. 2015, 231 (2015) CrossRefMATHGoogle Scholar
 4.Fréchet, M: Les espaces abstraits. GauthierVillars, Paris (1928) MATHGoogle Scholar
 5.Rus, IA: Picard operators and applications. Sci. Math. Jpn. 58, 191219 (2003) MathSciNetMATHGoogle Scholar
 6.Rus, IA: Some nonlinear functional differential and integral equations, via weakly Picard operator theory: a survey. Carpath. J. Math. 26(2), 230258 (2010) MathSciNetMATHGoogle Scholar
 7.Berinde, V, Păcurar, M, Rus, IA: From a Dieudonné theorem concerning the Cauchy problem to an open problem in the theory of weakly Picard operators. Carpath. J. Math. 30, 283292 (2014) MATHGoogle Scholar
 8.Rus, IA: Weakly Picard mappings. Comment. Math. Univ. Carol. 34(4), 769773 (1993) MathSciNetMATHGoogle Scholar
 9.Rus, IA, Petruşel, A, Şerban, MA: Weakly Picard operators: equivalent definitions, applications and open problems. Fixed Point Theory 7(1), 322 (2006) MathSciNetMATHGoogle Scholar
 10.Rus, IA: Generalized Contractions and Applications. Transilvania Press, ClujNapoca (2001) MATHGoogle Scholar
 11.Rus, IA: A Müller’s example to Cauchy problem: an operatorial point of view. Creative Math. Inform. 21(2), 215220 (2012) MathSciNetMATHGoogle Scholar
 12.Nussbaum, RD: The fixed point index and fixed point theorems for ksetcontractions. Ph.D. Dissertation, University of Chicago (1969) Google Scholar
 13.Petryshyn, WV: Fixed point theorems for various classes of 1setcontractive and 1ballcontractive mappings in Banach spaces. Trans. Am. Math. Soc. 182, 323352 (1973) MathSciNetMATHGoogle Scholar
 14.Rus, IA: Fixed Point Structure Theory. Cluj University Press, ClujNapoca (2006) MATHGoogle Scholar
 15.Andres, J, Górniewicz, L: Topological Fixed Point Principles for Boundary Value Problems. Kluwer Academic, Dordrecht (2003) CrossRefMATHGoogle Scholar
 16.Browder, FE, Nussbaum, RD: The topological degree for noncompact nonlinear mappings in Banach spaces. Bull. Am. Math. Soc. 74, 671676 (1968) MathSciNetCrossRefMATHGoogle Scholar
 17.Granas, A, Dugundji, J: Fixed Point Theory. Springer, Berlin (2003) CrossRefMATHGoogle Scholar
 18.Brown, RF, Furi, M, Górniewicz, L, Jiang, B (eds.): Handbook of Topological Fixed Point Theory. Springer, Berlin (2005) MATHGoogle Scholar
 19.Ortega, J, Rheinboldt, WC: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1960) MATHGoogle Scholar
 20.Rus, IA: An abstract point of view in the nonlinear difference equations. In: Conference on Analysis, Functional Equations, Approximation and Convexity, in Honour of Professor Elena Popoviciu, ClujNapoca, pp. 272276 (1999) Google Scholar
 21.Şerban, MA: Fixed Point Theory for Operators on Cartesian Product Spaces. Cluj University Press, ClujNapoca (2002) (in Romanian) MATHGoogle Scholar
 22.Rus, IA: An iterative method for the solution of the equation \(x=f(x,\ldots ,x)\). Rev. Anal. Numér. Théor. Approx. 10(1), 95100 (1981) MathSciNetMATHGoogle Scholar
 23.Krasnoselskii, MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 10(1), 123127 (1955) MathSciNetGoogle Scholar
 24.Browder, FE: Fixed point theorems for nonlinear semicontractive mappings in Banach spaces. Arch. Ration. Mech. Anal. 21(4), 259269 (1966) MathSciNetCrossRefMATHGoogle Scholar
 25.Browder, FE: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. Am. Math. Soc. 74, 660665 (1968) MathSciNetCrossRefMATHGoogle Scholar
 26.Kirk, WA: On nonlinear mappings of strongly semicontractive type. J. Math. Anal. Appl. 27, 409412 (1969) MathSciNetCrossRefMATHGoogle Scholar
 27.Webb, JRL: Fixed point theorems for nonlinear semicontractive operators in Banach spaces. J. Lond. Math. Soc. 1, 683688 (1969) CrossRefMATHGoogle Scholar
 28.Soardi, P: Punti uniti di semicontrazioni. Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 108, 5157 (1974) MathSciNetMATHGoogle Scholar
 29.Ziebur, AD: Uniqueness and the convergence of successive approximations. Proc. Am. Math. Soc. 13, 899903 (1962) MathSciNetCrossRefMATHGoogle Scholar
 30.Ziebur, AD: Uniqueness and the convergence of successive approximations. II. Proc. Am. Math. Soc. 16, 335340 (1965) MathSciNetCrossRefMATHGoogle Scholar
 31.Opoitsev, VI: Heterogenic and combinedconcave operators. Sib. Math. J. 16, 781792 (1975) (in Russian) Google Scholar
 32.Opoitsev, VI: A generalization of the theory of monotone and concave operators. Trans. Mosc. Math. Soc. 36, 243280 (1979) MathSciNetGoogle Scholar
 33.Diamond, P, Opoitsev, VI: Stability of nonlinear difference inclusions. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 8, 353371 (2001) MathSciNetMATHGoogle Scholar
 34.Anichini, G: Nonlinear problems for systems of differential equations. Nonlinear Anal. 1(6), 691699 (1977) MathSciNetCrossRefMATHGoogle Scholar
 35.Anichini, G, Conti, G: How to make use of the solution set to solve boundary value problems. Prog. Nonlinear Differ. Equ. Appl. 40, 1525 (2000) MathSciNetMATHGoogle Scholar
 36.Deimling, K, Lakshmikantham, V: Quasisolutions and their role in the qualitative theory of differential equations. Nonlinear Anal. 4, 657663 (1980) MathSciNetCrossRefMATHGoogle Scholar
 37.Opoitsev, VI, Khurodze, TA: Nonlinear Operators in Spaces with a Cone. Tbilis Gos. University, Tbilisi (1984) (in Russian), 271 pp. Google Scholar
 38.Cronin, J: Fixed Points and Topological Degree in Nonlinear Analysis. Amer. Math. Soc. Math. Survey, vol. 11 (1964) MATHGoogle Scholar
 39.Petruşel, A, Petruşel, G, Samet, B, Yao, JC: Coupled fixed point theorems for symmetric contractions in bmetric spaces with applications to operator equation systems. Fixed Point Theory 17(2), 457476 (2016) MATHGoogle Scholar
 40.Kirk, WA, Shahzad, N: Fixed Point Theory in Distance Spaces. Springer, Heidelberg (2014) CrossRefMATHGoogle Scholar
 41.Petruşel, A, Petruşel, G, Samet, B, Yao, JC: A study of the coupled fixed point problem for operators satisfying a maxsymmetric condition in bmetric spaces with applications to a boundary value problem. Miskolc Math. Notes 17(1), 501516 (2016) Google Scholar
 42.Radenović, S: BhaskarLakshmikantham type results for monotone mappings in partially ordered metric spaces. Int. J. Nonlinear Anal. Appl. 5(2), 3749 (2014) MATHGoogle Scholar
 43.Radenović, S: Coupled fixed point theorems for monotone mappings in partially ordered metric spaces. Kragujev. J. Math. 38(2), 249257 (2014) MathSciNetCrossRefGoogle Scholar
 44.Bainov, D, Hristova, S: Monotoneiterative techniques of V. Lakshmikantham for a boundary value problem for systems of integrodifferential equations. Math. J. Toyama Univ. 18, 169178 (1995) MathSciNetMATHGoogle Scholar
 45.Berinde, V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 74, 73477355 (2011) MathSciNetCrossRefMATHGoogle Scholar
 46.Berinde, V: Coupled fixed point theorems for φcontractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 75(6), 31283228 (2012) MathSciNetCrossRefGoogle Scholar
 47.Chen, YZ: Existence theorems of coupled fixed points. J. Math. Anal. Appl. 154, 142150 (1991) MathSciNetCrossRefMATHGoogle Scholar
 48.Gnana Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 13791393 (2006) MathSciNetCrossRefMATHGoogle Scholar
 49.Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988) MATHGoogle Scholar
 50.Guo, D, Cho, YJ, Zhu, J: Partial Ordering Methods in Nonlinear Problems. Nova Science Publishers, Hauppauge (2004) MATHGoogle Scholar
 51.Krasnoselskii, MA, Zabreiko, PP: Geometrical Methods of Nonlinear Analysis. Springer, Berlin (1984) CrossRefGoogle Scholar
 52.Lakshmikantham, V, Ćirić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70, 43414349 (2009) MathSciNetCrossRefMATHGoogle Scholar
 53.Lishan, L: Iterative method for solutions and coupled quasisolutions of nonlinear Fredholm integral equations in ordered Banach spaces. Indian J. Pure Appl. Math. 27(10), 959972 (1996) MathSciNetMATHGoogle Scholar
 54.Xiao, JZ, Zhu, XH, Shen, ZM: Common coupled fixed point results for hybrid nonlinear contractions in metric spaces. Fixed Point Theory 14, 235250 (2013) MathSciNetCrossRefMATHGoogle Scholar
 55.Popa, D, Lungu, N: On an exponential inequality. Demonstr. Math. 38(3), 667674 (2005) MathSciNetMATHGoogle Scholar
 56.Soleimani Rad, G, Shukla, S, Rahimi, H: Some relations between ntuple fixed point and fixed point results. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 109(2), 471481 (2015) MathSciNetCrossRefMATHGoogle Scholar
 57.Rus, IA: Remarks on a LaSalle conjecture on global asymptotic stability. Fixed Point Theory 17(1), 159172 (2016) MathSciNetMATHGoogle Scholar
 58.Presić, SB: Sur une classe d’inequations aux differences finies et sur la convergence de certaines suites. Publ. Inst. Math. 5(19), 7578 (1965) MathSciNetMATHGoogle Scholar
 59.Djebali, S, Gorniewicz, L, Ouahab, A: Solution Sets for Differential Equations and Inclusions. de Gruyter, Berlin (2012) MATHGoogle Scholar
 60.Heikkilá, S, Lakshmikantham, V: Monotone Iterative Techniques for Discountinuous Nonlinear Differential Equations. Dekker, New York (1991) Google Scholar
 61.Rus, MD: Fixed point theorems for generalized contractions in partially ordered metric spaces, with semimonotone metric. Nonlinear Anal. 74, 18041813 (2011) MathSciNetCrossRefMATHGoogle Scholar
 62.Rus, MD: The method of monotone iterations for mixed monotone operators in partially ordered sets and orderattractive fixed points. Fixed Point Theory 15, 579594 (2014) MathSciNetMATHGoogle Scholar
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