Cyclic admissible contraction and applications to functional equations in dynamic programming
 Huseyin Isik^{1},
 Bessem Samet^{2}Email author and
 Calogero Vetro^{3}
https://doi.org/10.1186/s1366301504106
© Isik et al. 2015
Received: 24 April 2015
Accepted: 21 August 2015
Published: 15 September 2015
Abstract
In this paper, we introduce the notion of Tcyclic \(( \alpha ,\beta ) \)contraction and give some common fixed point results for this type of contractions. The presented theorems extend, generalize, and improve many existing results in the literature. Several examples and applications to functional equations arising in dynamic programming are also given in order to illustrate the effectiveness of the obtained results.
Keywords
common fixed point point of coincidence Tcyclic \((\alpha,\beta)\)admissible mapping weakly compatible mappings functional equationsMSC
47H10 54H25 65Q201 Introduction and preliminaries
Fixed point theorems play a crucial role to constructing methods for solving problems in applied mathematics and the majority of other sciences. Thus, a large number of mathematicians have focused on this interesting topic. The Banach contraction mapping principle [1] is one of the pivotal results in fixed point theory. It is widely considered as the source of metric fixed point theory. Also its significance lies in its vast applicability in a number of branches of mathematics.
A new category of contractive fixed point problems was addressed by Khan et al. [2]. In this study they introduced the notion of altering distance function which is a control function that alters distance between two points in a metric space. This function and its extensions have been used in several problems of fixed point theory, some of which are noted in [3–6].
Definition 1
([2])
 (i)
φ is nondecreasing and continuous,
 (ii)
\(\varphi ( t ) =0\) if and only if \(t=0\).
Recently, Samet et al. [7] presented the notions of αψcontractive and αadmissible mappings. The results obtained by Samet et al. [7] extended and generalized many existing fixed point results in the literature, in particular the Banach contraction principle. After that, several authors considered the generalizations of this new approach (see [8–13]). Very recently, Alizadeh et al. [12] offered the concept of a cyclic \(( \alpha,\beta ) \)admissible mapping and proved some new fixed point results which generalize and modify some recent results in the literature.
Definition 2
([12])
 (i)
\(\alpha ( x ) \geq1\) for some \(x\in X\) implies \(\beta ( fx ) \geq1\);
 (ii)
\(\beta ( x ) \geq1\) for some \(x\in X\) implies \(\alpha ( fx ) \geq1\).
The purpose of this paper is to formulate the above definition in terms of two mappings so that we can prove existence and uniqueness of common fixed points for these mappings on a complete metric space. Our results improve and extend the results of [7, 12, 14] and many others. Several examples and interesting consequences of our theorems are also given. As a consequence of the presented results, we discuss the existence and uniqueness of the common bounded solution of a functional equation arising in dynamic programming.
Definition 3
([15])
Let X be a nonempty set and \(f,T:X\rightarrow X\). The pair \((f,T)\) is said to be weakly compatible if f and T commute at their coincidence points (i.e. \(fTx=Tfx\) whenever \(fx=Tx\)). A point \(y\in X\) is called a point of coincidence of f and T if there exists a point \(x\in X\) such that \(y=fx=Tx\).
 (\(\psi_{1}\)):

ψ is nondecreasing in each coordinate and continuous;
 (\(\psi_{2}\)):

\(\psi ( t,t,t,t ) \leq t\), \(\psi ( t,0,0,t ) \leq t\) and \(\psi ( 0,0,t,\frac{t}{2} ) \leq t\) for all \(t>0\);
 (\(\psi_{3}\)):

\(\psi ( t_{1},t_{2},t_{3},t_{4} ) =0\) if and only if \(t_{1}=t_{2}=t_{3}=t_{4}=0\).
2 Main results
Before proceeding with our results, let us give the following definitions which will be used efficiently in the proof of main results.
Definition 4
 (i)
\(\alpha ( Tx ) \geq1\) for some \(x\in X\) implies \(\beta ( fx ) \geq1\);
 (ii)
\(\beta ( Tx ) \geq1\) for some \(x\in X\) implies \(\alpha ( fx ) \geq1\).
Example 1
Let \(f,T:\mathbb{R} \rightarrow \mathbb{R} \) be defined by \(fx=x\) and \(Tx=x\). Suppose that \(\alpha,\beta:\mathbb{R} \rightarrow \mathbb{R} ^{+}\) are given by \(\alpha ( x ) =e^{x}\) for all \(x\in \mathbb{R} \) and \(\beta ( y ) =e^{y}\) for all \(y\in \mathbb{R} \). Then f is a Tcyclic \(( \alpha,\beta ) \) admissible mapping. Indeed, if \(\alpha ( Tx ) =e^{x}\geq1\), then \(x\geq0\) which implies \(fx\geq0\) and so \(\beta ( fx ) =e^{fx}\geq1\). Also, if \(\beta ( Ty ) =e^{y}\geq1\), then \(y\leq0\), which implies \(fy\leq0\) and so \(\alpha ( fy ) =e^{fy}\geq1\).
Definition 5
Theorem 1
 (i)
there exists \(x_{0}\in X\) such that \(\alpha ( Tx_{0} ) \geq1\) and \(\beta ( Tx_{0} ) \geq1\);
 (ii)
if \(\{ x_{n} \} \) is a sequence in X such that \(x_{n}\rightarrow x\) and \(\beta ( x_{n} ) \geq1\) for all n, then \(\beta ( x ) \geq1\);
 (iii)
\(\alpha ( Tu ) \geq1\) and \(\beta ( Tv ) \geq1\) whenever \(fu=Tu\) and \(fv=Tv\).
Proof
Example 2
Let \(X=\mathbb{R} \) be endowed with the usual metric \(d ( x,y ) =\vert xy\vert \) for all \(x,y\in X\). Also, let \(\varphi ( t ) =t\) and \(\eta ( t ) =\frac{2}{3}t\) for all \(t \geq0\), and \(\psi ( t_{1},t_{2},t_{3},t_{4} ) =\max \{ t_{1},t_{2},t_{3},t_{4} \} \) for all \(t_{1},t_{2},t_{3},t_{4}\geq 0\).
Let \(x\in X\) such that \(\alpha ( Tx ) \geq1\) so that \(Tx\in [ \frac{1}{3},0 ] \) and hence \(x\in [ 1,0 ] \). By the definitions of f and β, we have \(fx\in [ \frac {1}{9},0 ] \) and so \(\beta ( fx ) \geq1\).
Similarly, one can show that if \(\beta ( Tx ) \geq1\) then \(\alpha ( fx ) \geq 1\). Thus, f is a Tcyclic \(( \alpha,\beta ) \)admissible mapping. Moreover, the conditions \(\alpha ( Tx_{0} ) \geq 1\) and \(\beta ( Tx_{0} ) \geq1\) are satisfied with \(x_{0}=\frac {1}{9}\).
Now, let \(\{ x_{n} \} \) be a sequence in X such that \(\beta (x_{n} ) \geq1\) for all \(n\in \mathbb{N} \) and \(x_{n}\rightarrow x\) as \(n\rightarrow\infty\). Then, by the definition of β, we have \(x_{n}\in [ \frac{1}{9},0 ] \) for all \(n\in \mathbb{N} \) and so \(x\in [ \frac{1}{9},0 ] \), that is, \(\beta ( x ) \geq1\).
Corollary 1
 (i)
there exists \(x_{0}\in X\) such that \(\alpha ( Tx_{0} ) \geq1\) and \(\beta ( Tx_{0} ) \geq1\);
 (ii)
if \(\{ x_{n} \} \) is a sequence in X such that \(x_{n}\rightarrow x\) and \(\beta ( x_{n} ) \geq1\) for all n, then \(\beta ( x ) \geq1\);
 (iii)
\(\alpha ( Tu ) \geq1\) and \(\beta ( Tv ) \geq1\) whenever \(fu=Tu\) and \(fv=Tv\).
Proof
This implies that the inequality (2.1) holds. Therefore, the proof follows from Theorem 1. □
If we choose \(T=I_{X}\) in Theorem 1, we have the following corollary.
Corollary 2
 (i)
there exists \(x_{0}\in X\) such that \(\alpha ( x_{0} ) \geq1\) and \(\beta ( x_{0} ) \geq1\);
 (ii)
if \(\{ x_{n} \} \) is a sequence in X such that \(x_{n}\rightarrow x\) and \(\beta ( x_{n} ) \geq1\) for all n, then \(\beta ( x ) \geq1\);
 (iii)
\(\alpha ( u ) \geq1\) and \(\beta ( v ) \geq1\) whenever \(fu=u\) and \(fv=v\).
If we take \(\eta ( t ) =\varphi ( t ) \eta ^{1} ( t ) \) in Corollary 2, we have the following corollary.
Corollary 3
 (i)
there exists \(x_{0}\in X\) such that \(\alpha ( x_{0} ) \geq1\) and \(\beta ( x_{0} ) \geq1\);
 (ii)
if \(\{ x_{n} \} \) is a sequence in X such that \(x_{n}\rightarrow x\) and \(\beta ( x_{n} ) \geq1\) for all n, then \(\beta ( x ) \geq1\);
 (iii)
\(\alpha ( u ) \geq1\) and \(\beta ( v ) \geq1\) whenever \(fu=u\) and \(fv=v\).
If we take \(\varphi ( t ) =t\) in Corollary 3, we have the following corollary.
Corollary 4
 (i)
there exists \(x_{0}\in X\) such that \(\alpha ( x_{0} ) \geq1\) and \(\beta ( x_{0} ) \geq1\);
 (ii)
if \(\{ x_{n} \} \) is a sequence in X such that \(x_{n}\rightarrow x\) and \(\beta ( x_{n} ) \geq1\) for all n, then \(\beta ( x ) \geq1\);
 (iii)
\(\alpha ( u ) \geq1\) and \(\beta ( v ) \geq1\) whenever \(fu=u\) and \(fv=v\).
 (\(\phi_{1}\)):

ϕ is nondecreasing in each coordinate and continuous;
 (\(\phi_{2}\)):

\(\phi ( t,t,t,t ) \leq t\), \(\phi ( t,\frac{t}{2},t,0 ) \leq t\) and \(\phi ( 0,\frac{t}{2},0,t ) \leq t\) for all \(t>0\);
 (\(\phi_{3}\)):

\(\phi ( t_{1},t_{2},t_{3},t_{4} ) =0\) if and only if \(t_{1}=t_{2}=t_{3}=t_{4}=0\).
Definition 6
Theorem 2
 (i)
there exists \(x_{0}\in X\) such that \(\alpha ( Tx_{0} ) \geq1\) and \(\beta ( Tx_{0} ) \geq1\);
 (ii)
if \(\{ x_{n} \} \) is a sequence in X such that \(x_{n}\rightarrow x\) and \(\beta ( x_{n} ) \geq1\) for all n, then \(\beta ( x ) \geq1\);
 (iii)
\(\alpha ( Tu ) \geq1\) and \(\beta ( Tv ) \geq1\) whenever \(fu=Tu\) and \(fv=Tv\).
Proof
Since TX is closed, by (2.32), \(w\in TX\). Therefore, there exists \(v\in X\) such that \(Tv=w\). As \(y_{n}\rightarrow w\) and \(\beta ( y_{n} ) =\beta ( Tx_{n+1} ) \geq1\) for all \(n\in \mathbb{N} \), by (ii), \(\beta ( w ) =\beta ( Tv ) \geq1\). Thus, \(\alpha ( Tx_{n} ) \beta ( Tv ) \geq1\) for all \(n\in \mathbb{N} \).
Corollary 5
 (i)
there exists \(x_{0}\in X\) such that \(\alpha ( Tx_{0} ) \geq1\) and \(\beta ( Tx_{0} ) \geq1\);
 (ii)
if \(\{ x_{n} \} \) is a sequence in X such that \(x_{n}\rightarrow x\) and \(\beta ( x_{n} ) \geq1\) for all n, then \(\beta ( x ) \geq1\);
 (iii)
\(\alpha ( Tu ) \geq1\) and \(\beta ( Tv ) \geq1\) whenever \(fu=Tu\) and \(fv=Tv\).
Then f and T have a unique point of coincidence in X. Moreover, if f and T are weakly compatible, then f and T have a unique common fixed point.
If we take \(T=I_{X}\) and \(\eta ( t ) =\varphi ( t ) \eta^{1} ( t ) \) in Theorem 2, we have the following corollary.
Corollary 6
 (i)
there exists \(x_{0}\in X\) such that \(\alpha ( x_{0} ) \geq1\) and \(\beta ( x_{0} ) \geq1\);
 (ii)
if \(\{ x_{n} \} \) is a sequence in X such that \(x_{n}\rightarrow x\) and \(\beta ( x_{n} ) \geq1\) for all n, then \(\beta ( x ) \geq1\);
 (iii)
\(\alpha ( u ) \geq1\) and \(\beta ( v ) \geq1\) whenever \(fu=u\) and \(fv=v\).
Then f has a unique fixed point.
If we take \(\varphi ( t ) =t\) in Corollary 6, we have the following corollary.
Corollary 7
 (i)
there exists \(x_{0}\in X\) such that \(\alpha ( x_{0} ) \geq1\) and \(\beta ( x_{0} ) \geq1\);
 (ii)
if \(\{ x_{n} \} \) is a sequence in X such that \(x_{n}\rightarrow x\) and \(\beta ( x_{n} ) \geq1\) for all n, then \(\beta ( x ) \geq1\);
 (iii)
\(\alpha ( u ) \geq1\) and \(\beta ( v ) \geq1\) whenever \(fu=u\) and \(fv=v\).
Then f has a unique fixed point.
3 Cyclic results
The mappings \(f, T:A\cup B\rightarrow A\cup B\) are called cyclic if \(fA\subseteq TB\) and \(fB\subseteq TA\), where A, B are nonempty subsets of a metric space \((X,d)\). Moreover, f and T are called cyclic contraction if there exists \(k\in(0,1)\) such that \(d(fx,fy)\leq kd(Tx,Ty)\) for all \(x\in A\) and \(y\in B\). For more results see [16–19].
In this section we give some fixed point results involving cyclic mappings which can be regarded as consequences of the theorems presented in the previous section.
Theorem 3
 (i)
If T is one to one then there exists \(z\in A\cap B\) such that \(fz=Tz\).
 (ii)
If f and T are weakly compatible, then f and T have a unique common fixed point \(z\in A\cap B\).
Proof
There exists an \(x_{0}\in A\cap B\), as \(A\cap B\) is nonempty. This implies that \(Tx_{0}\in TA\) and \(Tx_{0}\in TB\) and so \(\alpha ( Tx_{0} ) \geq1\) and \(\beta ( Tx_{0} ) \geq1\).
Let \(\{ x_{n} \} \) be a sequence in X such that \(\beta ( x_{n} ) \geq1\) for all \(n\in \mathbb{N} \) and \(x_{n}\rightarrow x\) as \(n\rightarrow\infty\). Then \(x_{n}\in TB\) for all \(n\in \mathbb{N} \) and so \(x\in TB\). This implies that \(\beta ( x ) \geq1\).
Then the conditions (i) and (ii) of Theorem 1 hold. So there exist \(u,z\in A\cup B\) such that \(u=fz=Tz\). On the other hand, since T is one to one, there exist \(z_{1}\in A\), \(z_{2}\in B\) such that \(Tz_{1}=Tz_{2}=u\) implies \(z_{1}=z_{2}=z\). Therefore, \(u=Tz\) for \(z\in A\cap B\). If f and T are weakly compatible, following the proof of Theorem 1, we have \(u=fu=Tu\). The uniqueness of the common fixed point follows from (3.1). □
Similarly, we can prove the following theorem.
Theorem 4
 (i)
If T is one to one then there exists \(z\in A\cap B\) such that \(fz=Tz\).
 (ii)
If f and T are weakly compatible, then f and T have a unique common fixed point \(z\in A\cap B\).
4 Application to functional equations
The existence and uniqueness of solutions of functional equations and system of functional equations arising in dynamic programming have been studied by using different fixed point theorems (see [20–22]).
Let \(B(W)\) denote the space of all bounded realvalued functions defined on the set W. Meanwhile, \(B(W)\) endowed with the sup metric \(d(h,k)=\sup_{x\in W}hxkx\) for all \(h,k\in B(W)\) is a complete metric space.
 (A1)For any \(h\in B(W)\), there exists \(k\in B(W)\) such that$$ fh(x)=Tk(x),\quad x\in W. $$
 (A2)There exists \(h\in B(W)\) such that$$ fh(x)=Th(x)\quad\mbox{implies}\quad Tfh(x)=fTh(x),\quad x\in W. $$
 (A3)
\(p,q:W\times D\rightarrow \mathbb{R} \) and \(G,K:W\times D\times\ \mathbb{R} \rightarrow \mathbb{R} \) are bounded.
 (A4)
\(\xi ( Th ) \geq0\) for some \(h\in B(W)\) implies \(\xi ( fh ) \geq0\).
 (A5)\(\vert G ( x,y,h ( x ) ) G ( x,y,k ( x ) ) \vert \leq\ln ( 1+M ( h,k ) ) \) where \(h,k\in B(W)\), \(\xi ( Th ) \geq 0\), and \(\xi ( Tk ) \geq0\), \(( x,y ) \in W\times D\), \(t\in W\) and$$\begin{aligned} M ( h,k ) =&\max\biggl\{ d \bigl( Th ( t ) ,Tk ( t ) \bigr) ,d \bigl( Th ( t ) ,fh ( t ) \bigr) ,d \bigl( Tk ( t ) ,fk ( t ) \bigr) ,\\ &{}\frac{1}{2} \bigl[ d \bigl( Th ( t ) ,fk ( t ) \bigr) +d \bigl( Tk ( t ) ,fh ( t ) \bigr) \bigr] \biggr\} . \end{aligned}$$
 (A6)
If \(\{ h_{n} \} \) is a sequence in \(B(W)\) such that \(\xi ( h_{n} ) \geq0\) for all \(n\in \mathbb{N} \cup \{ 0 \} \) and \(h_{n}\rightarrow h^{\ast}\) as \(n\rightarrow \infty\), then \(\xi ( h^{\ast} ) \geq0\).
 (A7)
There exists \(h_{0}\in B(W)\) such that \(\xi ( Th_{0} ) \geq0\).
Theorem 5
Assume that conditions (A1)(A7) are satisfied and \(T(B(W))\) is a closed and bounded subspace of \(B(W)\). Then the functional equations (4.1) and (4.2) have a unique common bounded solution in W.
Proof
One easily shows that all the hypotheses of Corollary 1 are satisfied. Therefore f and T have a unique common fixed point, that is, the functional equations (4.1) and (4.2) have a unique bounded common solution. □
 (A5′):

\(\vert G ( x,y,h ( x ) ) G ( x,y,k ( x ) ) \vert \leq\ln ( 1+N ( h,k ) ) \) where \(h,k\in B(W)\), \(\xi ( Th ) \geq0\), and \(\xi ( Tk ) \geq0\); and \(( x,y ) \in W\times D\), \(t\in W\) and$$\begin{aligned} N ( h,k ) =&\max\biggl\{ d \bigl( Th ( t ) ,Tk ( t ) \bigr) ,\frac{1}{2}d \bigl( Th ( t ) ,fk ( t ) \bigr) ,d \bigl( Tk ( t ) ,fh ( t ) \bigr) ,\\ &{}\frac{ [ 1+d ( Th ( t ) ,fh ( t ) ) ] d ( Tk ( t ) ,fk ( t ) ) }{1+d ( Th ( t ) ,Tk ( t ) ) }\biggr\} . \end{aligned}$$
Declarations
Acknowledgements
The second author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this Prolific Research Group Project No. PRG143610.
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
References
 Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133181 (1922) MATHGoogle Scholar
 Khan, MS, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30(1), 19 (1984) MATHMathSciNetView ArticleGoogle Scholar
 Alber, YI, GuerreDelabriere, S: Principles of weakly contractive maps in Hilbert spaces. Oper. Theory, Adv. Appl. 98, 722 (1997) MathSciNetGoogle Scholar
 Rhoades, BE: Some theorems on weakly contractive maps. Nonlinear Anal. 47(4), 26832693 (2001) MATHMathSciNetView ArticleGoogle Scholar
 Dutta, PN, Choudhury, BS: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl. 2008, 406368 (2008) MathSciNetView ArticleGoogle Scholar
 Yan, F, Su, Y, Feng, Q: A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory Appl. 2012, 152 (2012) MathSciNetView ArticleGoogle Scholar
 Samet, B, Vetro, C, Vetro, P: Fixed point theorems for αψcontractive type mappings. Nonlinear Anal. 75, 21542165 (2012) MATHMathSciNetView ArticleGoogle Scholar
 La Rosa, V, Vetro, P: Common fixed points for αψφcontractions in generalized metric spaces. Nonlinear Anal., Model. Control 19(1), 4354 (2014) MATHMathSciNetGoogle Scholar
 Kumam, P, Vetro, C, Vetro, F: Fixed points for weak αψcontractions in partial metric spaces. Abstr. Appl. Anal. 2013, 986028 (2013) MathSciNetGoogle Scholar
 Cho, SH, Bae, JS: Fixed points of weak αcontraction type maps. Fixed Point Theory Appl. 2014, 175 (2014) MathSciNetView ArticleGoogle Scholar
 Salimi, P, Vetro, C, Vetro, P: Fixed point theorems for twisted \((\alpha ,\beta )\)ψcontractive type mappings and applications. Filomat 27(4), 605615 (2013) MathSciNetView ArticleGoogle Scholar
 Alizadeh, S, Moradlou, F, Salimi, P: Some fixed point results for \((\alpha ,\beta )\)\((\psi ,\varphi )\)contractive mappings. Filomat 28(3), 635647 (2014) MathSciNetView ArticleGoogle Scholar
 Moradlou, F, Salimi, P, Vetro, P: Fixed point results for r\((\eta ,\xi ,\psi )\)contractive mappings of type (I), (II) and (III). Filomat 27(2), 403410 (2013) MathSciNetView ArticleGoogle Scholar
 Su, Y: Contraction mapping principle with generalized altering distance function in ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory Appl. 2014, 227 (2014) View ArticleGoogle Scholar
 Jungck, G, Rhoades, BE: Fixed points for set valued functions without continuity. Indian J. Pure Appl. Math. 29, 227238 (1998) MATHMathSciNetGoogle Scholar
 Kirk, WA, Srinavasan, PS, Veeramani, P: Fixed points for mapping satisfying cyclical contractive conditions. Fixed Point Theory 4, 7989 (2003) MATHMathSciNetGoogle Scholar
 Pacurar, M, Rus, IA: Fixed point theory for cyclic ϕcontractions. Nonlinear Anal. 72, 11811187 (2010) MATHMathSciNetView ArticleGoogle Scholar
 Agarwal, RP, Alghamdi, MA, Shahzad, N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012, 40 (2012) MathSciNetView ArticleGoogle Scholar
 Karapınar, E, Shobkolaei, N, Sedghi, S, Vaezpour, SM: A common fixed point theorem for cyclic operators on partial metric spaces. Filomat 26(2), 407414 (2012) MATHMathSciNetView ArticleGoogle Scholar
 Abbas, M, Ali, B, Vetro, C: A Suzuki type fixed point theorem for a generalized multivalued mapping on partial Hausdorff metric spaces. Topol. Appl. 160(3), 553563 (2013) MATHMathSciNetView ArticleGoogle Scholar
 Imdad, M, Chauhan, S, Kadelburg, Z, Vetro, C: Fixed point theorems for nonself mappings in symmetric spaces under phiweak contractive conditions and an application to functional equations in dynamic programming. Appl. Math. Comput. 227, 469479 (2014) MathSciNetView ArticleGoogle Scholar
 Aydi, H, Abbas, M, Vetro, C: Common fixed points for multivalued generalized contractions on partial metric spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 108(2), 483501 (2014) MATHMathSciNetView ArticleGoogle Scholar
 Bellman, R, Lee, ES: Functional equations in dynamic programming. Aequ. Math. 17, 118 (1978) MATHMathSciNetView ArticleGoogle Scholar
 Bhakta, TC, Mitra, S: Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl. 98, 348362 (1984) MATHMathSciNetView ArticleGoogle Scholar