Common coupled fixed point results for operators without mixed monotone type properties and application to nonlinear integral equations
 Nabil Machrafi^{1}Email authorView ORCID ID profile
DOI: 10.1186/s136630170595y
© The Author(s) 2017
Received: 6 October 2016
Accepted: 18 January 2017
Published: 31 January 2017
Abstract
We establish some new common coupled fixed point theorems for a pair of operators not assumed to satisfy mixed monotone type properties in the ordered Banach space setting. For that purpose, the notions of weakly inflationary and weakly deflationary operators are introduced in the twodimensional setting, and existence and uniqueness common coupled fixed point theorems for such operators are established under certain condensing and contractive conditions involving the twodimensional setting. As an application, we study the existence and uniqueness of nonnegative solutions for nonlinear integral equations.
Keywords
ordered Banach space coupled fixed point mixed monotone property inflationary mappingMSC
46T99 47H10 54H251 Introduction and preliminaries
The notion of a coupled fixed point was introduced and studied by Opoitsev [1] and investigated later by GuoLakshmikantham [2]. Among investigations of coupled fixed point results for nonlinear operators in the ordered Banach space setting, there are more results on the existence of coupled fixed points than on the existence of common coupled fixed points of a pair of operators, and most of these results were established for mixed monotone operators; see for instance [2–8] and the references therein. The notion of a mixed monotone operator was introduced in [2] as follows.
Definition 1.1
In some recent work, the authors have followed a line of research consisting in replacing the mixed monotonicity of operators by other properties, since the mixed monotonicity is not often easy to check; see [9] for a property based on the comparability of elements in ordered metric spaces, and [3, 10] for the following alternative mixed monotone property.
Definition 1.2
Hence, the mixed gmonotone property of an operator \(A:\Omega\times \Omega\rightarrow\Omega\) extends its mixed monotone property (which is the mixed Idmonotone property where Id is the identity mapping on Ω).
In this paper, we continue on this path by investigating the existence of common coupled fixed points of a pair of operators \(A,B:\Omega\times \Omega \rightarrow\Omega\), where mixed monotone type properties of the operators are not assumed, while the pair of operators is assumed to satisfy a new twodimensional order type property, extending a wellknown onedimensional equivalent property (Definition 2.3 and the remarks following), and guaranteeing the use of monotone iterative technique. To prove the existence of a common coupled fixed point for such pair of operators, it is assumed for operators to satisfy a useful condensing and contractive conditions (conditions (\(C_{1}\)) and (\(C_{2}\)) hereafter) involving the twodimensional setting. The consideration of the latter condensing and contractive conditions is motivated by the fact that these conditions are satisfied in particular when the operators satisfy the standard condensing and contractive conditions as defined in the literature; see Definition 1.5 and the remarks following Lemma 2.1.
The first main result extends the wellknown results in the literature on a fixed point theorem for monotone operators and coupled fixed point theorem for mixed monotone operators; see [4], Theorem 2.1.1 and Theorem 2.1.7. We look, also, at the equivalent of our main result when the Banach space is endowed with its weak topology. Finally, we illustrate the applicability of our results by studying the existence and uniqueness of nonnegative solution for a twodimensional nonlinear integral equations.
Throughout this paper X will be a real Banach space, \(B_{r}\) will denote the closed ball in X centered at 0 with radius \(r>0\). In particular, we use the notation \(B_{X}:=B_{1}\). For a subset \(\Omega\subset X\), \(\pi _{\Omega}:\Omega\times\Omega\rightarrow\Omega\) will denote the first projection mapping, i.e. \(\pi_{\Omega} ( x,y ) =x\) (\(x,y\in\Omega \)). We will mean in the sequel by the term ‘operator’ between two Banach spaces a mapping which is (nonlinear in general) continuous and bounded (i.e. takes bounded sets to bounded sets).
The following two lemmas will be useful in the proofs of our results.
Lemma 1.3
[11], Lemmas 2.3 and 2.22
 (1)
X is Hausdorff and the order intervals of X are closed;
 (2)
if the cone \(X^{+}\) is normal, then every order interval is bounded.
Lemma 1.4
Let X be an ordered Banach space with a normal cone \(X^{+}\). Then a monotone sequence \(( u_{n} ) \subset X\) is convergent if and only if it has a weakly convergent subsequence.
Proof
 (1)
\(\phi(U\cup V)=\max(\phi(U),\phi(V))\) (the set additivity);
 (2)
\(\phi(U+V)\leq\phi(U)+\phi(V)\) (the algebraic subadditivity);
 (3)
\(\phi(\lambda U)=\vert \lambda \vert \phi(U)\) (the homogeneity);
 (4)
\(\phi(U)\leq\phi ( V ) \) if \(U\subset V\) (the monotonicity);
 (5)
\(\phi( [ 0,1 ] \cdot U)=\phi(U)\) (the absorption invariance);
 (6)
\(\phi(\overline{\operatorname{co}}U)=\phi(U)\) (the convex closure invariance).
Definition 1.5
[15], p.21
Remark 1.6
Clearly every nonlinear \(( \phi,\psi ) \)setcontraction \(T:\Omega\rightarrow Y\) is \(( \phi,\psi ) \)condensing. Also, it is easy to see that every φnonlinear contraction \(T:\Omega\rightarrow Y\) is a nonlinear \(( \alpha _{X},\alpha_{Y} ) \)setcontraction, where \(\alpha_{X}\), \(\alpha_{Y}\) denote, respectively, the Kuratowski measures of noncompactness in X, Y.
When \(Y=X\), an operator \(T:\Omega\rightarrow\Omega\) is said to be nondecreasing (resp. nonincreasing) if for all \(x,y\in\Omega\), \(x\leq y\) implies \(Tx\leq Ty\) (resp. \(Tx\geq Ty\)).

A point \((x_{\ast},y_{\ast})\in\Omega\times\Omega\) is said to be a coupled fixed point of A if\(x_{\ast}\in\Omega\) is called a fixed point of A if \(A(x_{\ast },x_{\ast})=x_{\ast}\). Evidently, if \((x_{\ast},y_{\ast})\) is a coupled fixed point of A, then \((y_{\ast},x_{\ast})\) is also a coupled fixed point of A. Also, \((x_{\ast},x_{\ast})\) is a coupled fixed point of A whenever \(x_{\ast}\) is a fixed point of A.$$ A(x_{\ast},y_{\ast})=x_{\ast}\quad \text{and} \quad A(y_{\ast},x_{\ast })=y_{\ast}. $$

A point \(( x,y ) \in\Omega\times\Omega\) is said to be a lower (resp. upper) coupled fixed point of A ifand$$ x\leq A(x,y)\quad \bigl(\mbox{resp. }x\geq A(x,y)\bigr) $$$$ A(y,x)\leq y \quad \bigl(\mbox{resp. }A(y,x)\geq y\bigr). $$
Clearly, if \(( x,y ) \) is a lower (resp. upper) coupled fixed point of A then \(( y,x ) \) is an upper (resp. lower) coupled fixed point of A.
2 Main results
In the sequel, we consider the product space \(X\times X\) equipped with \(\Vert ( x,y ) \Vert _{\infty}=\max \{ \Vert x\Vert , \Vert y\Vert \} \) and the Kuratowski measure of noncompactness and the De Blasi measure of weak noncompactness \(\alpha ^{\times}\) and \(\omega^{\times}\), respectively, and \(\gamma=\alpha\) or ω. The facts in the following lemma are obtained in a simple way and therefore their proofs are omitted.
Lemma 2.1
 (1)
\(\operatorname {dia}( D_{1}\times D_{2} ) =\max \{ \operatorname {dia}( D_{1} ) ,\operatorname {dia}( D_{2} ) \} \).
 (2)
\(\gamma^{\times} ( D_{1}\times D_{2} ) =\max \{ \gamma ( D_{1} ) ,\gamma ( D_{2} ) \} \).
Example 2.2
 (1)Let \(0< k<1\). Define the (continuous and bounded) operator \(A:B_{X}\times B_{X}\rightarrow B_{X}\) by$$ A ( x,y ) =\frac{k}{2} \bigl( \Vert x\Vert \cdot x+\Vert y\Vert \cdot y \bigr) . $$Then A satisfies the αcondensing condition (\(C_{1}\)). Indeed, let \(G:B_{X}\times B_{X}\rightarrow B_{X}\) and \(T:B_{X}\rightarrow B_{X}\) be such thatSince \(G ( \cdot,y ) \) is contractive with contraction constant k for every \(y\in B_{X}\), and \(G ( x,\cdot ) \) is compact for every \(x\in B_{X}\) (because the range of \(G ( x,\cdot ) \) lies in a finite dimensional subspace of X, for every \(x\in B_{X}\)), T is αcondensing (see [12], p.198). Now, let \(U,V\subset B_{X}\) be such that \(\alpha ( U ) >0\) or \(\alpha ( V ) >0\). Then, from \(A ( U\times V ) =\frac{1}{2} ( T ( U ) +T ( V ) ) \), we see that$$\begin{aligned}& G ( x,y ) =k\cdot \Vert y\Vert \cdot x, \\& T ( x ) =G ( x,x ) . \end{aligned}$$which proves that A satisfies the αcondensing condition (\(C_{1}\)).$$\begin{aligned} \alpha\bigl( A ( U\times V ) \bigr) \leq&\frac{1}{2} \bigl( \alpha \bigl( T ( U ) \bigr) +\alpha\bigl( T ( V ) \bigr) \bigr) \\ < &\frac{1}{2} \bigl( \alpha( U ) +\alpha( V ) \bigr) \\ \leq&\max\bigl( \alpha( U ) ,\alpha( V ) \bigr) , \end{aligned}$$
 (2)More generally, if \(T,S:\Omega\rightarrow\Omega\) are two operators such that both T and S are γcondensing, then the operator \(A:\Omega\times\Omega\rightarrow\Omega\) defined bysatisfies the γcondensing condition (\(C_{1}\)).$$ A ( x,y ) =\frac{1}{2} \bigl( T ( x ) +S ( y ) \bigr) $$
 (3)
In (1), if \(B=k^{\prime}\cdot A\) (\(k^{\prime}\geq 0 \)), then it follows that the operator B satisfies the \(k^{\prime }\)αcontraction condition (\(C_{2}\)).
Recall from [16], p.263 that an operator \(T:\Omega \rightarrow\Omega\) on a partially ordered set is said to be inflationary (or progressing, see [17], p.34) if \(Tx\geq x\) for every \(x\in\Omega\). An example of such operator is the operator that associates to every element of a vector lattice its positive part. We introduce the following similar twodimensional concepts.
Definition 2.3
 (1)
An operator \(A:\Omega\times\Omega\rightarrow\Omega\) is said to be inflationary, if A is inflationary with respect to its first argument, that is, \(A ( x,y ) \geq x\) for every \(x,y\in\Omega\);
 (2)a pair of operators \(A,B:\Omega\times\Omega\rightarrow\Omega \) is said to be weakly inflationary if A is inflationary on \(T_{B} ( \Omega \times\Omega ) \) and B is inflationary on \(T_{A} ( \Omega \times\Omega ) \), that is,and$$ A \bigl( B ( x,y ) ,B ( y,x ) \bigr) \geq B ( x,y ) $$for all \(x,y\in\Omega\). If the preceding inequalities are satisfied only on \(T_{B} ( D ) \) and \(T_{A} ( D ) \) respectively, where D is a subset of \(\Omega\times\Omega\), then the pair A, B is said to be weakly inflationary on D;$$ B \bigl( A ( x,y ) ,A ( y,x ) \bigr) \geq A ( x,y ) $$
 (3)an operator \(A:\Omega\times\Omega\rightarrow\Omega\) is said to be weakly inflationary, if the pair A, A is weakly inflationary, that is,for all \(x,y\in\Omega\).$$ A \bigl( A ( x,y ) ,A ( y,x ) \bigr) \geq A ( x,y ) $$
Clearly, if A and B are both inflationary (resp. deflationary), then the pair A, B is weakly inflationary (resp. weakly deflationary). The converse does not hold in general.
Example 2.4
 (1)Since \(h^{\prime} ( x ) =2e^{2x}1<0\) for all \(x\in \mathbb{R}\), it follows thatfor all \(x\in [ 0,1 ] \). This shows that \(f ( x,x ) \geq x \) for all \(x\in [ 0,1 ] \), i.e., f is inflationary on Δ. Since g is decreasing,$$ h ( x ) \geq h ( 1 ) =e^{2}>0 $$for all \(x\in [ 0,1 ] \). This shows that f is weakly deflationary on Δ. Therefore, the restriction of f on Δ is a weakly deflationary mapping which is not deflationary.$$ f \bigl( f ( x,x ) ,f ( x,x ) \bigr) \leq f ( x,x ) $$(2.1)
 (2)On the other hand, it follows from (2.1) that f is deflationary on \(\Delta^{\prime}\), and since g is decreasing, thenfor all \(y\in f ( \Delta ) \). This shows that the restriction of f on \(\Delta^{\prime}\) is a weakly inflationary mapping which is not inflationary.$$ f \bigl( f ( y,y ) ,f ( y,y ) \bigr) \geq f ( y,y ) $$
 (3)
The function f does not satisfy the mixed monotone property since f is decreasing in x.
 (4)
From (1) and (2), it follows that the pair f, g (i.e. f, \(g\circ\pi_{\mathbb{R}}\)) is weakly deflationary on Δ (resp. weakly inflationary on \(\Delta^{\prime}\)), but f is not mixed gmonotone. Indeed, if \(x,y_{1},y_{2}\in\mathbb{R}\) such that \(g ( y_{1} ) \leq g ( y_{2} ) \), since g is decreasing and one to one from \(\mathbb{R}\) to \(\mathbb{(}1,+\infty)\), then \(y_{1}\geq y_{2}\) and hence \(f ( x,y_{1} ) \leq f ( x,y_{2} ) \) since f is decreasing in y. This shows that f is not gnonincreasing in y.
Theorem 2.5
 (1)
A satisfies the αcondensing condition (\(C_{1}\));
 (2)
B satisfies the \(1\alpha\)contraction condition (\(C_{2}\));
 (3)
the pair A, B is weakly inflationary (resp. weakly deflationary).
From the above theorem, we derive the following two corollaries, which extend the wellknown results in the literature; see Remark 2.8 hereafter.
Corollary 2.6
 (1)
T and S commute, that is, \(TSx=STx\) for each \(x\in\Omega\);
 (2)
T and S have at least one common lower (resp. upper) fixed point \(u_{0}\in\Omega\) (resp. \(v_{0}\in\Omega\));
 (3)
T is αcondensing;
 (4)
S is a \(1\alpha\)contraction.
Proof
Now, applying Theorem 2.5, there exists a common coupled fixed point \(( u_{\ast},v_{\ast} ) \in\Gamma\times\Gamma\) of A and B, that is, \(u_{\ast}\) is a common fixed point of T and S. From (2.2) we see that \(u_{\ast}=\lim u_{n}\) and (2.4) holds true. Finally, to prove the minimality of \(u_{\ast}\), let \(u\in[ u_{0})\cap\Omega\) be such that \(Tu=Su=u\). Since T is nondecreasing, it follows from \(u_{0}\leq u\) that \(Tu_{0}\leq Tu\), that is, \(u_{1}\leq u\). Again, since S is nondecreasing, \(Su_{1}\leq Su\), that is, \(u_{2}\leq u\). Proceeding inductively, we get \(u_{n}\leq u\) for each \(n=0,1,2,\ldots\) . Now, taking the limit \(n\rightarrow\infty\), we obtain \(u_{\ast}\leq u\) as desired.
For the existence of a maximal common fixed point \(u^{\ast}\) of T and S, consider the subset \(\Gamma^{\prime}= \{ v\in(v_{0}]\cap\Omega :Tv\leq v\text{ and }Sv\leq v \} \), and by the same preceding arguments, such common fixed point exists with \(u^{\ast}=\lim v_{n}\) and (2.5) holds true. □
It is easy to see that if \(X^{+}\) is a normal cone in X then \(( X\times X ) ^{+}\) is also a normal cone in \(X\times X\).
Corollary 2.7
 (1)A and B satisfy the following commutation property:for all \(x,y\in\Omega\);$$ A \bigl( B ( x,y ) ,B ( y,x ) \bigr) =B \bigl( A ( x,y ) ,A ( y,x ) \bigr) $$
 (2)
A and B have at least one common lower (resp. upper) coupled fixed point \(( u_{0},v_{0} ) \) with \(u_{0}\leq v_{0}\) (resp. \(u_{0}\geq v_{0}\));
 (3)
A is \(( \alpha^{\times},\alpha ) \)condensing;
 (4)
B is a \(1 ( \alpha^{\times},\alpha ) \)contraction.
Proof
Now, applying Corollary 2.6, the operators \(T_{A}\) and \(T_{B}\) have a minimal common fixed point \(( u_{\ast},v_{\ast} ) \in\Omega _{0}\times\Omega_{0}\), that is, \(( u_{\ast},v_{\ast} ) \) is a minimal common coupled fixed point of A and B, and (2.2) and (2.3) follow from (2.4) and (2.5) applied for \(T_{A}\) and \(T_{B}\).
Note that in the case \(( u_{0},v_{0} ) \) is a common upper coupled fixed point of A and B, \(( v_{0},u_{0} ) \) is a common lower coupled fixed point of A and B, and the required conclusions follow from the preceding case. □
Remark 2.8
 (1)
If we take \(S=T\) in Corollary 2.6 and \(A=B\) in Corollary 2.7 then we obtain the wellknown results [4], Theorem 2.1.1 and Theorem 2.1.7.
 (2)
In Corollary 2.7, the operator A is supposed to be only \(( \alpha^{\times},\alpha ) \)condensing, which is an hypothesis weaker than the complete continuity supposed in [4], Theorem 2.1.7.
Remark 2.9
 (1)
Weakly continuous operators \(A:\Omega \times\Omega\rightarrow\Omega\) satisfy the condition (\(C_{3}\)). However, the converse is false in general. Indeed, if \(\mathcal {N}_{\varphi }:L_{1} [ 0,1 ] \rightarrow L_{1} [ 0,1 ] \) is the Nemytskii operator generated by a Caratheodory function \(\varphi: [ 0,1 ] \times\mathbb{R\rightarrow R}\), then \(\mathcal{N}_{\varphi}\circ \pi_{L_{1} [ 0,1 ] }\) satisfies the condition (\(C_{3}\)) (see Lemma 4.2 hereafter for \(n=1\)), but \(\mathcal{N}_{\varphi}\) is weakly continuous if and only if φ is linear (see [20], Theorem 2.6). Furthermore, it is clear that the condition (\(C_{4}\)) is in particular satisfied by weakly compact operators (i.e., operators that map bounded sets to a relatively weakly compact ones);
 (2)
clearly, if A satisfies the condition (\(C_{4}\)), then A satisfies the condition (\(C_{3}\)) and the ωcondensing condition (\(C_{1}\)).
Theorem 2.10
 (1)
A satisfies the condition (\(C_{3}\));
 (2)
A satisfies the ωcondensing condition (\(C_{1}\));
 (3)
B satisfies the \(1\omega\)contraction condition (\(C_{2}\));
 (4)
the pair A, B is weakly inflationary (resp. weakly deflationary).
Corollary 2.11
 (1)
A satisfies the condition (\(C_{4}\));
 (2)
B satisfies the \(1\omega\)contraction condition (\(C_{2}\));
 (3)
the pair A, B is weakly inflationary (resp. weakly deflationary).
Lemma 2.12
Proof
Therefore, from the condition (\(C_{3}\)), we see that \(\omega ( A ( U\times V ) ) \leq\varphi ( r ) \). Now, since φ is continuous, letting \(r\rightarrow\max ( \omega ( U ) ,\omega ( V ) ) \), we get the required conclusion. □
Corollary 2.13
 (1)The operators A and B have at least one common fixed point \(u_{\ast}\in\Omega\), and the common fixed point iteration is given by$$ u_{2n+1}=A ( u_{2n},u_{2n} ) \quad \textit{and} \quad u_{2n+2}=B ( u_{2n+1},u_{2n+1} ) \quad ( n=0,1,2,\ldots ) . $$
 (2)
If for the operator A, we replace the condensing condition with “A is a φnonlinear contraction”, then A and B have a unique common fixed point \(u_{\ast}\) and \(( u_{\ast},u_{\ast} ) \) is their unique common coupled fixed point.
 (3)If A satisfies in addition the following weak φnonlinear contraction condition:for all \(x,y\in\Omega\), then all coupled fixed points of A (and hence all common coupled fixed points of A and B) are in the form \(( u_{\ast},u_{\ast} ) \), \(u_{\ast}\) is a fixed point of A (common fixed point of A and B).$$ \bigl\Vert A ( x,y ) A ( y,x ) \bigr\Vert \leq\varphi\bigl( \Vert xy \Vert \bigr) $$
3 Proofs of the main theorems
Proof of Theorem 2.5
Similarly, we have \(\alpha ( V_{2} ) <\max \{ \alpha ( V_{2} ) ,\alpha ( U_{2} ) \} \) which is a contradiction. Therefore, \(U_{1}\) and \(V_{1}\) are a relatively compact sets. Since A is continuous, \(A ( U_{1}\times V_{1} ) \) and \(A ( V_{1}\times U_{1} ) \) are relatively compact, and hence so are \(U_{2}\subset A ( U_{1}\times V_{1} ) \) and \(V_{2}=A ( \Delta ^{\ast} ) \subset A ( V_{1}\times U_{1} ) \). Thus, we conclude that U and V are a relatively compact sets. It follows from Lemma 1.4 that \(\lim u_{n}=u_{\ast}\) and \(\lim v_{n}=v_{\ast } \) for some \(( u_{\ast},v_{\ast} ) \in\Omega\times\Omega\). Now, since A and B are continuous then by (2.2) and (2.3) we get \(A ( u_{\ast},v_{\ast} ) =B ( u_{\ast },v_{\ast} ) =u_{\ast}\) and \(A ( v_{\ast},u_{\ast} ) =B ( v_{\ast},u_{\ast} ) =v_{\ast}\), that is, \(( u_{\ast },v_{\ast} ) \) is a common coupled fixed point of A and B.
If A, B is weakly deflationary, then in this case the sequences of common coupled fixed point iteration are nonincreasing, and the desired conclusions are obtained by similar arguments. □
Proof of Theorem 2.10
Let U and V be the sets defined as in the proof of Theorem 2.5. Using the condition (\(C_{3}\)), it can be shown by a similar arguments of the proof of Theorem 2.5 that U and V are relatively weakly compact. Since the sequences \(( u_{n} ) \) and \(( v_{n} ) \) are nondecreasing, it follows from Lemma 1.4 that \(\lim u_{n}=u_{\ast}\) and \(\lim v_{n}=v_{\ast}\) for some \(( u_{\ast},v_{\ast} ) \in\Omega\times\Omega\). Now, since A and B are continuous, by (2.2) and (2.3) we get \(A ( u_{\ast},v_{\ast} ) =B ( u_{\ast},v_{\ast} ) =u_{\ast}\) and \(A ( v_{\ast},u_{\ast} ) =B ( v_{\ast },u_{\ast} ) =v_{\ast}\), that is, \(( u_{\ast},v_{\ast} ) \) is a common coupled fixed point of A and B. □
4 Application to nonlinear integral equations
An element \(u\in X\) is called a solution of (4.1) if \(( u,u ) \) is a solution of (4.1).

the function \(t\rightarrow\varphi ( t,u ) \) is measurable on I for any \(u\in\mathbb{R}^{n}\);

the function \(u\rightarrow\varphi ( t,u ) \) is continuous on \(\mathbb{R}^{n}\) for almost all \(t\in I\).
Lemma 4.1
[21], p.154
If a Caratheodory function \(\varphi:I\times \mathbb{R}^{n}\rightarrow\mathbb{R}\) satisfies the separated domination property, then the operator \(\mathcal{N}_{\varphi}:X^{n}\rightarrow X\) is continuous and bounded.
Lemma 4.2
Proof
So, since \(\omega ( \{ u_{0} \} ) =0\), under the supremum on all subsets M and all \(( u_{1},\ldots,u_{n} ) \in W_{1}\times\cdots\times W_{n}\) respectively, and letting \(\varepsilon \rightarrow0\), the desired conclusion follows from equation (1.2). □
 \(( A_{1} ) \) :

f is a Caratheodory function satisfying the following positivity property:for almost all \(t\in I\) and for all \(x,y\in\mathbb{R}^{+}\).$$ f \bigl( t, ( x,y ) \bigr) >0 $$
 \(( A_{2} ) \) :

f satisfies the separated domination property, that is, there exist a constant \(k>0\) and a positive function \(w_{0}\in X\) such thatfor almost all \(t\in I\) and for all \(u\in\mathbb{R}^{2}\).$$ \bigl\vert f ( t,u ) \bigr\vert \leq w_{0} ( t ) +k\vert u \vert $$(4.3)
 \(( A_{3} ) \) :

The kernel mapping K is positive, i.e. \(K ( t,x ) >0\) for all \(t,x\in I\), and satisfies the following upper estimate:for almost all \(t,x\in I\), and for some positive functions \(\psi ,\varphi \in L_{\infty} ( I ) \) such that$$ K ( t,x ) \leq\psi( t ) \varphi( x ) $$where ρ is some (arbitrary) positive number.$$ \delta=\Vert \psi \Vert _{\infty}\cdot \Vert \varphi \Vert _{\infty}\leq\frac{\rho}{1+2\rho k}, $$(4.4)
 \(( A_{4} ) \) :

g is a Caratheodory function such that there exists a nondecreasing mapping \(\phi:\mathbb{R}^{+}\rightarrow\mathbb {R}^{+}\) with \(\phi ( r ) \leq r\), \(r>0\), such that, for almost all \(t\in I\) and for all \(x\geq0\), we havewhere \(\mu_{0}=\rho \Vert w_{0}\Vert \).$$ \frac{\mu_{0}}{\Vert \psi \Vert _{\infty}}\psi( t ) \leq g ( t,x ) \leq\phi( x ), $$(4.5)
 \(( A_{5} ) \) :

g is nonexpansive with respect to the second variable, that is, for almost all \(t\in I\) and every \(x,y\in\mathbb {R}\), we have$$ \bigl\vert g ( t,x ) g ( t,y ) \bigr\vert \leq \vert xy\vert . $$
Theorem 4.3
Proof
 (a)
B satisfies the \(1\omega\)contraction condition (\(C_{2}\));
 (b)
A satisfies the condition (\(C_{4}\));
 (c)
the pair A, B is weakly deflationary.
Finally, applying Corollary 2.11, we obtain all conclusions of Theorem 4.3. □
Corollary 4.4
Proof
Remark 4.5
 (1)If we assume g to be nondecreasing with respect to the second variable, i.e.,for almost all \(t\in I\) and for all \(x,y\in\mathbb{R}\), then assumption \(( A_{4} ) \) may be reduced to$$ x\leq y\quad \Rightarrow \quad g ( t,x ) \leq g ( t,y ) $$and$$ \frac{\mu_{0}}{\Vert \psi \Vert _{\infty}}\psi( t ) \leq g ( t,0 ) $$(4.10)for almost all \(t\in I\) and for all \(x>0\).$$ 0\leq g ( t,x ) \leq x $$
 (2)If the function \(\mathcal{N}_{g} ( \theta ) \in L_{\infty } ( I ) \), then it is easy to see that the upper estimate of K in assumption \(( A_{3} ) \) and assumption (4.10) are, respectively, guaranteed by the following inequalities:and$$ K ( t,x ) \leq g ( t,0 ) \leq\frac{\rho}{1+2\rho k} $$for almost all \(t,x\in I\) (here \(\psi=\mathcal{N}_{g} ( \theta ) \) and φ is the constant function equal to one).$$ w_{0} ( t ) \leq\frac{1}{\rho}g ( t,0 ) $$
 (3)If f is nonexpansive with respect to the second variable, then from (4.9) we have \(w_{0}=\mathcal{N}_{f} ( \theta,\theta ) \) and \(k=1\), and hence the preceding inequalities becomeand$$ K ( t,x ) \leq g ( t,0 ) \leq\frac{\rho}{1+2\rho} $$for almost all \(t,x\in I\).$$ f \bigl( t, ( 0,0 ) \bigr) \leq\frac{1}{\rho}g ( t,0 ) $$
Declarations
Acknowledgements
The author would like to thank the anonymous reviewers for their comments and suggestions to improve the quality of the paper.
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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