Cyclic admissible contraction and applications to functional equations in dynamic programming

(1)

R E S E A R C H

Open Access

Cyclic admissible contraction and

applications to functional equations in

dynamic programming

Huseyin Isik

1

_{, Bessem Samet}

2*

_{and Calogero Vetro}

3

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

In this paper, we introduce the notion ofT-cyclic (α,

β)-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.

MSC: Primary 47H10; secondary 54H25; 65Q20

Keywords: common ﬁxed point; point of coincidence;T-cyclic (α,β)-admissible mapping; weakly compatible mappings; functional equations

1 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 mathe-maticians have focused on this interesting topic. The Banach contraction mapping prin-ciple [] 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 Khanet al.[]. 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 [–].

Deﬁnition ([]) A functionϕ: [,∞)→[,∞) is called an altering distance function if the following properties are satisﬁed:

(i) ϕis nondecreasing and continuous, (ii) ϕ(t) = if and only ift= .

Recently, Sametet al.[] presented the notions ofα-ψ-contractive andα-admissible mappings. The results obtained by Sametet al.[] extended and generalized many existing ﬁxed point results in the literature, in particular the Banach contraction principle. After

(2)

that, several authors considered the generalizations of this new approach (see [–]). Very recently, Alizadehet al.[] oﬀered the concept of a cyclic (α,β)-admissible mapping and proved some new ﬁxed point results which generalize and modify some recent results in the literature.

Deﬁnition ([]) Letf :X→Xandα,β:X→[,∞). We say thatf is a cyclic (α,β )-admissible mapping if

(i) α(x)≥for somex∈Ximpliesβ(fx)≥; (ii) β(x)≥for somex∈Ximpliesα(fx)≥.

The purpose of this paper is to formulate the above deﬁnition in terms of two map-pings 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 [, , ] 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 ([]) LetXbe a nonempty set andf,T:X→X. The pair (f,T) is said to be weakly compatible iff andTcommute at their coincidence points (i.e. fTx=Tfxwhenever

fx=Tx). A pointy∈Xis called a point of coincidence off andT if there exists a point

x∈Xsuch thaty=fx=Tx.

Following the direction in [], we denote bythe family of all functionsψ:R +→R+ such that:

(ψ) ψis nondecreasing in each coordinate and continuous;

(ψ) ψ(t,t,t,t)≤t,ψ(t, , ,t)≤tandψ(, ,t,_t)≤tfor allt> ;

(ψ) ψ(t,t,t,t) = if and only ift=t=t=t= .

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  Letf,T :X→Xandα,β:X→[,∞). We say thatf is aT-cyclic (α,β )-admissible mapping if

(i) α(Tx)≥for somex∈Ximpliesβ(fx)≥; (ii) β(Tx)≥for somex∈Ximpliesα(fx)≥.

Example  Letf,T:R→Rbe deﬁned byfx=xandTx= –x. Suppose thatα,β:R→R+ are given byα(x) =e–x_{for all}_x_∈_R_and_β₍_y_{) =}_ey_{for all}_y_∈_R_{. Then}_f _{is a}_T_{-cyclic (}_α_,_β₎ admissible mapping. Indeed, ifα(Tx) =ex_≥_{, then}_x_≥_{ which implies}_fx_≥_{ and so}

β(fx) =efx_≥_{. Also, if}_β₍_Ty_{) =}_e–y_≥_{, then}_y_≤_{, which implies}_fy_≤_{ and so}_α₍_fy_{) =}

e–fy_≥_.

Deﬁnition  Let (X,d) be a metric space and letf be aT-cyclic (α,β)-admissible map-ping. We say thatf is aT-cyclic (α,β)-contractive mapping if

(3)

for allx,y∈X, where

M(x,y) =ψ

d(Tx,Ty),d(Tx,fx),d(Ty,fy), 

d(Tx,fy) +d(Ty,fx),

ψ∈,ϕis an altering distance function andη: [,∞)→[,∞) is a nondecreasing func-tion and continuous from the right with the condifunc-tionϕ(t) >η(t) for allt> .

Theorem  Let(X,d)be a complete metric space and let f and T be self-mappings on X such that fX⊂TX.Let f be a T -cyclic(α,β)-contractive mapping.Assume that TX is a closed subset of X and the following conditions are satisﬁed:

(i) there existsx∈Xsuch thatα(Tx)≥andβ(Tx)≥;

(ii) if{xn}is a sequence inXsuch thatxn→xandβ(xn)≥for alln,thenβ(x)≥;

(iii) α(Tu)≥andβ(Tv)≥wheneverfu=Tuandfv=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.

Proof Letx∈Xsuch thatα(Tx)≥ andβ(Tx)≥. Deﬁne the sequences{xn}and{yn} inXby

yn=fxn=Txn+, n∈N∪ {}. (.)

If yn=yn+, then yn+ is a point of coincidence of f and T. Suppose that yn=yn+ for all n∈N. Since f is a T-cyclic (α,β)-admissible mapping and α(Tx) ≥  then

β(fx) =β(Tx)≥, which implies α(Tx) =α(fx)≥. By continuing this process, we get α(Txn)≥ and β(Txn+)≥ for all n∈N∪ {}. Similarly, since f is a T-cyclic (α,β)-admissible mapping andβ(Tx)≥, we haveβ(Txn)≥ andα(Txn+)≥ for all n∈N∪ {}, that is, α(Txn)≥ and β(Txn)≥ for all n∈N∪ {}. Equivalently,

α(Txn)β(Txn+)≥ for alln∈N∪ {}. Therefore by (.) and using (.), we get

ϕd(yn,yn+)

=ϕd(fxn,fxn+)

≤ηM(xn,xn+)

<ϕM(xn,xn+)

. (.)

Sinceϕis nondecreasing, we have

d(yn,yn+) <M(xn,xn+), (.)

where

M(xn,xn+)

=ψ

d(Txn,Txn+),d(Txn,fxn),d(Txn+,fxn+), 

d(Txn,fxn+) +d(Txn+,fxn)

=ψ

d(yn–,yn),d(yn–,yn),d(yn,yn+),  

d(yn–,yn+) +d(yn,yn)

≤ψ

d(yn–,yn),d(yn–,yn),d(yn,yn+), 

(4)

Thus, from (.), we obtain

d(yn,yn+) <M(xn,xn+)

≤ψ

d(yn–,yn),d(yn–,yn),d(yn,yn+),  

d(yn–,yn) +d(yn,yn+).

Ifd(yn–,yn)≤d(yn,yn+) for somen∈N, then

d(yn,yn+) <ψ

d(yn–,yn),d(yn–,yn),d(yn,yn+),  

d(yn–,yn) +d(yn,yn+)

≤ψd(yn,yn+),d(yn,yn+),d(yn,yn+),d(yn,yn+)

≤d(yn,yn+),

which is a contradiction, and henced(yn,yn+) <d(yn–,yn) for alln∈N. Therefore, the sequence{d(yn,yn+)}is decreasing and bounded below. Thus, there existsr≥ such that limn→∞d(yn,yn+) =r. Assumer> . Also, from (.), (.) and using the properties ofψ, we deduce

ϕd(yn,yn+)

≤ηM(xn,xn+)

≤η

ψ

d(yn–,yn),d(yn–,yn),d(yn,yn+), 

d(yn–,yn) +d(yn,yn+)

≤ηψd(yn–,yn),d(yn–,yn),d(yn–,yn),d(yn–,yn) ≤ηd(yn–,yn)

. (.)

Considering the properties ofϕandη, lettingn→ ∞in (.), we get

ϕ(r) = lim n→∞ϕ

d(yn,yn+)

≤ lim n→∞η

d(yn–,yn)=η(r) <ϕ(r),

which impliesr=  and so

lim

n→∞d(yn,yn+) = . (.)

Now, we prove that{yn}is a Cauchy sequence. Suppose, to the contrary, that{yn}is not a Cauchy sequence. Then there exists anε>  for which we can ﬁnd two subsequences {ymk}and{ynk}of{yn}such thatmkis the smallest index for whichmk>nk>kand

d(ymk,ynk)≥ε and d(ymk–,ynk) <ε. (.)

Using the triangular inequality and (.), we have

ε≤d(ynk,ymk)≤d(ynk,ymk–) +d(ymk–,ymk)

(5)

By takingk→ ∞in the above inequality and using (.), we obtain

lim

k→∞d(ynk,ymk) =ε. (.)

By using (.), (.), and the triangular inequality, we deduce

lim

k→∞d(ynk–,ymk) =ε, lim

k→∞d(ymk–,ynk) =ε, lim

k→∞d(ymk–,ynk–) =ε.

(.)

From (.), we get

ϕd(ynk,ymk)

=ϕd(fxnk,fxmk)

≤ηM(xnk,xmk)

, (.)

where

M(xnk,xmk) =ψ

d(Txnk,Txmk),d(Txnk,fxnk),d(Txmk,fxmk),

 

d(Txnk,fxmk) +d(Txmk,fxnk)

≤ψ

maxε,d(ynk–,ymk–) ,d(ynk–,ynk),d(ymk–,ymk),

max

ε, 

d(ynk–,ymk) +d(ymk–,ynk)

.

Now, from the properties ofϕ,ψ, andηand using (.), (.), (.), and the above inequality, ask→ ∞in (.), we have

ϕ(ε)≤ηψ(ε, , ,ε)≤η(ε) <ϕ(ε),

which implies thatε= , a contradiction withε> . Thus{yn}is a Cauchy sequence inX. From the completeness of (X,d), there existsz∈Xsuch that

lim

n→∞yn=z. (.)

From (.) and (.), we obtain

fxn→z and Txn+→z. (.)

Since TX is closed, by (.),z∈TX. Therefore, there exists u∈X such that Tu=z. As yn →z and β(yn) =β(Txn+)≥ for all n∈ N, by (ii), β(z) =β(Tu)≥. Thus,

(6)

Now, applying inequality (.), we get

ϕd(fxn,fu)≤ηM(xn,u), (.)

where

M(xn,u)

=ψ

d(Txn,Tu),d(Txn,fxn),d(Tu,fu), 

d(Txn,fu) +d(Tu,fxn)

≤ψ

d(Txn,Tu),d(Txn,fxn),d(Tu,fu), max

d(Tu,fu),d(Txn,fu) +d(Tu,fxn) .

Takingk→ ∞in the inequality (.) and using the properties ofϕ,ψ,η, and the above inequality we have

ϕd(z,fu)≤η

ψ

, ,d(z,fu), d(z,fu)

≤ηd(z,fu)<ϕd(z,fu),

which impliesd(z,fu) = , that is,z=fu. Thus, we deduce

z=fu=Tu, (.)

and sozis a point of coincidence forf andT. The uniqueness of the point of coincidence is a consequence of the conditions (.) and (iii), and so we omit the details.

By (.) and using the weak compatibility off andT, we obtain

fz=fTu=Tfu=Tz (.)

and sofz=Tz. Uniqueness of the point of coincidence impliesz=fz=Tz. Consequently,

zis a unique common ﬁxed point off andT.

Example  LetX=Rbe endowed with the usual metricd(x,y) =|x–y|for allx,y∈X. Also, let ϕ(t) =tandη(t) = 

tfor allt≥, andψ(t,t,t,t) =max{t,t,t,t} for all

t,t,t,t≥.

Now, deﬁne the self-mappingsf andTonXby

fx=

⎧ ⎨ ⎩

–x

 ifx∈[, ], x

 ifx∈R\[, ]

and Tx=

⎧ ⎨ ⎩

x

 ifx∈[–, ], x

 ifx∈R\[–, ].

Then it is clear thatfX⊂TX. Also, deﬁne the mappingsα,β:X→[,∞) by

α(x) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

ex ifx∈(–∞, –_),

e–x _if_x_∈_[– , ],

(7)

Figure 1 Plot ofTx(in blue) andfx(in red) on [–2, 2].

Figure 2 Zoom of the plot in Figure 1.In violet, we have{yn}={fxn}={Txn+1}at the starting pointx0

= –1/9.

and

β(x) =

⎧ ⎨ ⎩

e–x _if_x_∈_[– , ],  ifx∈R\[–_, ].

Letx∈Xsuch thatα(Tx)≥ so thatTx∈[–_, ] and hencex∈[–, ]. By the deﬁnitions off andβ, we havefx∈[–_, ] and soβ(fx)≥.

Similarly, one can show that if β(Tx)≥ thenα(fx)≥. Thus,f is aT-cyclic (α,β )-admissible mapping. Moreover, the conditions α(Tx)≥ andβ(Tx)≥ are satisﬁed withx= –_.

Now, let{xn}be a sequence inXsuch thatβ(xn)≥ for alln∈Nandxn→xasn→ ∞. Then, by the deﬁnition ofβ, we havexn∈[–_, ] for alln∈Nand sox∈[–_, ], that is,

β(x)≥.

Next, we prove thatf is aT-cyclic (α,β)-contractive mapping. Letα(Tx)β(Ty)≥, then

Tx∈[–_, ],Ty∈[–_, ] and sox∈[–, ],y∈[–_, ]. Thus, we get

ϕd(fx,fy)=|fx–fy|= |x–y|

≤ 

|x–y|= 

|Tx–Ty|

≤

M(x,y) =η

(8)

Obviously, assumption (iii) of Theorem  is satisfied. Consequently, all conditions of The-orem  hold, and hencefandThave a unique common fixed point. Here  is the common fixed point off andT; see Figures  and .

Corollary  Let(X,d)be a complete metric space and let f and T be self-mappings on X such that fX⊂TX.Let f be a T -cyclic(α,β)-admissible mapping such that

α(Tx)β(Ty)ϕd(fx,fy)≤ηM(x,y), (.)

for all x,y∈X,whereϕis an altering distance function andη: [,∞)→[,∞)is a nonde-creasing function and continuous from the right with the conditionϕ(t) >η(t)for all t> 

and

M(x,y) =max

d(Tx,Ty),d(Tx,fx),d(Ty,fy), 

d(Tx,fy) +d(Ty,fx).

Assume that TX is a closed subset of X and the following conditions are satisﬁed:

Proof Letα(Tx)β(Ty)≥ forx,y∈X. If we takeψ(t,t,t,t) =max{t,t,t,t}in Theo-rem , then from (.), we have

ϕd(fx,fy)≤ηM(x,y).

This implies that the inequality (.) holds. Therefore, the proof follows from

Theo-rem .

If we chooseT=IXin Theorem , we have the following corollary.

Corollary  Let (X,d) be a complete metric space and f :X→X be a cyclic (α,β) -admissible mapping such that

α(x)β(y)≥ ⇒ ϕd(fx,fy)≤ηMf(x,y)

, (.)

for all x,y∈X,whereψ∈,ϕis an altering distance function andη: [,∞)→[,∞)is a nondecreasing function and continuous from the right with the conditionϕ(t) >η(t)for all t> and

Mf(x,y) =ψ

d(x,y),d(x,fx),d(y,fy), 

d(x,fy) +d(y,fx).

Assume that the following conditions are satisﬁed:

(i) there existsx∈Xsuch thatα(x)≥andβ(x)≥;

(9)

(iii) α(u)≥andβ(v)≥wheneverfu=uandfv=v.

Then f has a unique ﬁxed point.

If we takeη(t) =ϕ(t) –η(t) in Corollary , we have the following corollary.

Corollary  Let (X,d) be a complete metric space and f :X→X be a cyclic (α,β) -admissible mapping such that

α(x)β(y)≥ ⇒ ϕd(fx,fy)≤ϕMf(x,y)

–ηMf(x,y)

, (.)

for all x,y∈X,whereψ∈,ϕis an altering distance function andη: [,∞)→[,∞)

is such thatϕ(t) –η(t)is nondecreasing andη(t)is continuous from the right,with the conditionϕ(t) >η(t)for all t> .

(ii) if{xn}is a sequence inXsuch thatxn→xandβ(xn)≥for alln,thenβ(x)≥;

If we takeϕ(t) =tin Corollary , we have the following corollary.

Corollary  Let (X,d) be a complete metric space and f :X→X be a cyclic (α,β) -admissible mapping such that

α(x)β(y)≥ ⇒ d(fx,fy)≤Mf(x,y) –η

Mf(x,y)

, (.)

for all x,y∈X,whereψ∈andη_{: [,}_∞₎_→_[,_∞₎_{is such that t}_–_η₍_t₎_{is nondecreasing}

andη₍_t₎_{is continuous from the right}_,_{with the condition}_η₍_t_{) > }_{for all t}_{> .}

We denote by the family of all functionsφ:R

+→R+such that

(φ) φis nondecreasing in each coordinate and continuous;

(φ) φ(t,t,t,t)≤t,φ(t,_t,t, )≤tandφ(,_t, ,t)≤tfor allt> ;

(φ) φ(t,t,t,t) = if and only ift=t=t=t= .

Deﬁnition  Let (X,d) be a metric space and letf be aT-cyclic (α,β)-admissible map-ping. We say thatf is aT-cyclic (α,β)-rational contractive mapping if

α(Tx)β(Ty)≥ ⇒ ϕd(fx,fy)≤ηN(x,y), (.)

for allx,y∈X, where

N(x,y) =φ

d(Tx,Ty),

d(Tx,fy),d(Ty,fx),

[ +d(Tx,fx)]d(Ty,fy)  +d(Tx,Ty)

(10)

φ∈ ,ϕis an altering distance function andη: [,∞)→[,∞) is a nondecreasing func-tion and continuous from the right with the condifunc-tionϕ(t) >η(t) for allt> .

Theorem  Let(X,d)be a complete metric space and let f and T be self-mappings on X such that fX⊂TX.Let f be a T -cyclic(α,β)-rational contractive mapping.Assume that TX is a closed subset of X and the following conditions are satisﬁed:

Proof Similar to the proof of Theorem , we deﬁne sequences{xn}and{yn}inXbyyn=

fxn=Txn+and note thatα(Txn)β(Txn+)≥ for alln∈N∪ {}. Also we assume thatyn=

yn–for alln∈N. Then by (.), we have

ϕd(yn,yn+)=ϕd(fxn,fxn+)≤ηN(xn,xn+)<ϕN(xn,xn+). (.)

Sinceϕis nondecreasing, we get

d(yn,yn+) <N(xn,xn+), (.)

where

N(xn,xn+) =φ

d(Txn,Txn+), 

d(Txn,fxn+),d(Txn+,fxn), [ +d(Txn,fxn)]d(Txn+,fxn+)

 +d(Txn,Txn+)

=φ

d(yn–,yn),

d(yn–,yn+),d(yn,yn),

[ +d(yn–,yn)]d(yn,yn+)  +d(yn–,yn)

≤φ

d(yn–,yn), 

d(yn–,yn) +d(yn,yn+), ,d(yn,yn+)

. (.)

Thus, from (.), we deduce

d(yn,yn+) <N(xn,xn+)

≤φ

d(yn–,yn), 

.

Ifd(yn–,yn)≤d(yn,yn+) for somen∈N, then

d(yn,yn+) <φ

d(yn–,yn),  

d(yn–,yn) +d(yn,yn+)

, ,d(yn,yn+)

≤φd(yn,yn+),d(yn,yn+),d(yn,yn+),d(yn,yn+)

(11)

which is a contradiction, and henced(yn,yn+) <d(yn–,yn) for alln∈N. Therefore, the sequence {d(yn,yn+)}is decreasing and bounded from below. Thus, there existsδ≥ such thatlim_n_→∞d(yn,yn+) =δ. Also, from (.), (.) and using the properties ofϕ andη, we obtain

ϕd(yn,yn+) ≤ηN(xn,xn+)

≤η

φ

d(yn–,yn), 

≤ηφd(yn–,yn),d(yn–,yn),d(yn–,yn),d(yn–,yn)

≤ηd(yn–,yn)<ϕd(yn–,yn). (.)

Consider the properties ofϕandη, lettingn→ ∞in (.), we get

ϕ(δ) = lim n→∞ϕ

d(yn,yn+)

≤ lim n→∞η

d(yn–,yn)=η(δ) <ϕ(δ),

which impliesδ=  and so

lim

n→∞d(yn,yn+) = . (.)

Now, we want to show that{yn}is a Cauchy sequence. Suppose, to the contrary, that {yn}is not a Cauchy sequence. Then there exists anε>  for which we can ﬁnd two sub-sequences{ynk}and{ymk}of{yn}such thatnkis the smallest index for whichnk>mk>k

and

d(ynk,ymk)≥ε and d(ynk–,ymk) <ε. (.)

Using the triangular inequality and (.), we have

ε≤d(ymk,ynk)≤d(ymk,ynk–) +d(ynk–,ynk)

< ε+d(ynk–,ynk).

By takingk→ ∞in the above inequality and using (.), we obtain

lim

k→∞d(ymk,ynk) =ε. (.)

By using (.), (.), and the triangular inequality, we deduce

lim

k→∞d(ynk–,ymk) =ε, lim

k→∞d(ymk–,ynk) =ε, lim

k→∞d(ymk–,ynk–) =ε.

(12)

From (.), we get

ϕd(ynk,ymk)

=ϕd(fxnk,fxmk)

≤ηN(xnk,xmk)

, (.)

where

N(xnk,xmk) =φ

d(Txnk,Txmk),



d(Txnk,fxmk),d(Txmk,fxnk),

[ +d(Txnk,fxnk)]d(Txmk,fxmk)

 +d(Txnk,Txmk)

=φ

d(ynk–,ymk–),



d(ynk–,ymk),d(ymk–,ynk),

[ +d(ynk–,ynk)]d(ymk–,ymk)

 +d(ynk–,ymk–)

≤maxε,N(xnk,xmk)

=φ

maxε,d(ynk–,ymk–) ,

 max

ε,d(ynk–,ymk) ,

maxε,d(ymk–,ynk) ,

[ +d(ynk–,ynk)]d(ymk–,ymk)

 +d(ynk–,ymk–)

.

Thereforelimk→∞max{ε,N(xnk,xmk)}=φ(ε, ε

,ε, )≤ε.

Now, from the properties ofϕandηand using (.), (.), (.), and the previous inequality, ask→ ∞in (.), we have

ϕ(ε) = lim n→∞ϕ

d(yn,yn+)

≤ lim n→∞η

maxε,N(xnk,xmk) ≤η(ε) <ϕ(ε),

which implies thatε= , a contradiction withε> . Thus{yn}is a Cauchy sequence inX. From the completeness of (X,d), there existsw∈Xsuch that

lim

n→∞yn=w (.)

and so by (.), we obtain

fxn→w and Txn+→w. (.)

SinceTXis closed, by (.),w∈TX. Therefore, there existsv∈Xsuch thatTv=w. As yn→ w and β(yn) =β(Txn+)≥  for all n ∈N, by (ii), β(w) =β(Tv)≥. Thus,

α(Txn)β(Tv)≥ for alln∈N.

Now, applying inequality (.), we get

ϕd(fxn,fv)≤ηN(xn,v), (.)

where

N(xn,v)≤φ

d(Txn,Tv), max

d(v,fv),d(Txn,fv) ,d(Tv,fxn),

d(Tv,fv)max

[ +d(Txn,fxn)]  +d(Txn,Tv) , 

(13)

Takingk→ ∞in the inequality (.), using the properties ofϕ,ηand the previous in-equality we have

ϕd(w,fv)≤η

φ

,

d(w,fv), ,d(w,fv)

≤ηd(w,fv)<ϕd(w,fv),

which impliesd(w,fv) = , that is,w=fv. Thus, we deduce

w=fv=Tv, (.)

and sowis a point of coincidence forf andT. The uniqueness of the point of coincidence is a consequence of the conditions (.) and (iii), and so we omit the details.

By (.) and using the weak compatibility off andT, we obtain

fw=fTv=Tfv=Tw. (.)

The uniqueness of the point of coincidence impliesw=fw=Tw. Consequently,wis the

unique common ﬁxed point off andT.

Corollary  Let(X,d)be a complete metric space and let f and T be self-mappings on X such that fX⊂TX.Let f be a T -cyclic(α,β)-admissible mapping such that

α(Tx)β(Ty)ϕd(fx,fy)≤ηN(x,y),

for all x,y∈X,whereϕis an altering distance function andη: [,∞)→[,∞)is a nonde-creasing function and continuous from the right with the conditionϕ(t) >η(t)for all t> 

and

N(x,y) =max

d(Tx,Ty),

d(Tx,fy),d(Ty,fx),

[ +d(Tx,fx)]d(Ty,fy)  +d(Tx,Ty)

.

Assume that TX is a closed subset of X and that the following conditions are satisﬁed:

If we takeT=IXandη(t) =ϕ(t) –η(t) in Theorem , we have the following corollary.

Corollary  Let (X,d)be a complete metric space and f :X→X be a cyclic (α,β) -admissible mapping such that

α(x)β(y)≥ ⇒ ϕd(fx,fy)≤ϕNf(x,y)

–ηNf(x,y)

,

for all x,y∈X,whereφ∈ ,ϕis an altering distance function andη: [,∞)→[,∞)

(14)

conditionϕ(t) >η₍_t₎_{for all t}_{> ,}_and

Nf(x,y) =φ

d(x,y),

d(x,fy),d(y,fx),

[ +d(x,fx)]d(y,fy)  +d(x,y)

.

If we takeϕ(t) =tin Corollary , we have the following corollary.

Corollary  Let (X,d) be a complete metric space and f :X→X be a cyclic (α,β) -admissible mapping such that

α(x)β(y)≥ ⇒ d(fx,fy)≤Nf(x,y) –η

Nf(x,y)

,

for all x,y∈X,whereη_{: [,}_∞₎_→_[,_∞₎_{is such that t}_–_η₍_t₎_{is nondecreasing and}_η₍_t₎

is continuous from the right,with the conditionη(t) > for all t> .

3 Cyclic results

The mappingsf,T:A∪B→A∪Bare called cyclic iffA⊆TBandfB⊆TA, whereA,Bare nonempty subsets of a metric space (X,d). Moreover,f andTare called cyclic contraction if there existsk∈(, ) such thatd(fx,fy)≤kd(Tx,Ty) for allx∈Aandy∈B. For more results see [–].

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  Let A and B be two closed subsets of complete metric space(X,d)such that

A∩B=∅and f,T:A∪B→A∪B be mappings such that fA⊆TB and fB⊆TA.Assume that T(A∪B)is a closed subset of X such that

ϕd(fx,fy)≤ηM(x,y), (.)

for all x∈A and y∈B,whereϕis an altering distance function andη: [,∞)→[,∞)is a nondecreasing function and continuous from the right with the conditionϕ(t) >η(t)for all t> .

(i) IfT is one to one then there existsz∈A∩Bsuch thatfz=Tz.

(ii) Iff andT are weakly compatible,thenf andT have a unique common ﬁxed point

(15)

Proof Deﬁneα,β:X→[,∞) by

α(x) =

, ifx∈TA,

, otherwise and β(x) =

, ifx∈TB, , otherwise.

Letα(Tx)β(Ty)≥. ThenTx∈TAandTy∈TB. SinceT is one to one, we havex∈Aand

y∈B. Hence, from (.) we obtain

ϕd(fx,fy)≤ηM(x,y).

Letα(Tx)≥ for somex∈X, soTx∈TAand thenx∈A. Hence,fx∈TBand soβ(fx)≥. Again, letβ(Tx)≥ for somex∈X. ThenTx∈TBand sox∈B. Hence,fx∈TAand then

α(fx)≥. Therefore,f is aT-cyclic (α,β)-admissible mapping.

There exists anx∈A∩B, asA∩Bis nonempty. This implies thatTx∈TAandTx∈

TBand soα(Tx)≥ andβ(Tx)≥.

Let{xn}be a sequence inXsuch thatβ(xn)≥ for alln∈Nandxn→xasn→ ∞. Then

xn∈TBfor alln∈Nand sox∈TB. This implies thatβ(x)≥.

Then the conditions (i) and (ii) of Theorem  hold. So there existu,z∈A∪Bsuch that

u=fz=Tz. On the other hand, sinceT is one to one, there existz∈A,z∈Bsuch that

Tz=Tz=uimpliesz=z=z. Therefore,u=Tzforz∈A∩B. Iff andT are weakly compatible, following the proof of Theorem , we haveu=fu=Tu. The uniqueness of the

common ﬁxed point follows from (.).

Similarly, we can prove the following theorem.

Theorem  Let A and B be two closed subsets of complete metric space(X,d)such that A∩B=∅and f,T:A∪B→A∪B be mappings such that fA⊆TB and fB⊆TA.Assume that T(A∪B)is a closed subset of X such that

ϕd(fx,fy)≤ηN(x,y), (.)

for all x∈A and y∈B,whereϕis an altering distance function andη: [,∞)→[,∞)is a nondecreasing function and continuous from the right with the conditionϕ(t) >η(t)for all t> .

(i) IfT is one to one then there existsz∈A∩Bsuch thatfz=Tz.

(ii) Iff andT are weakly compatible,thenf andT have a unique common ﬁxed point

z∈A∩B.

4 Application to functional equations

The existence and uniqueness of solutions of functional equations and system of func-tional equations arising in dynamic programming have been studied by using diﬀerent ﬁxed point theorems (see [–]).

Throughout this section, we assume thatUandVare Banach spaces,W⊆Uis a state space,D⊆Vis a decision space. Now, we apply our results in order to prove the existence and uniqueness of the common solution of the following functional equations:

P=sup y∈D

(16)

and

Q=sup y∈D

q(x,y) +Kx,y,Qτ(x,y) , x∈W, (.)

whereτ:W×D→W,p,q:W×D→RandG,K:W×D×R→R. It is well known that equations of the type (.) and (.) provide useful tools for mathematical optimization, computer programming, and in dynamic programming (see [, ]).

LetB(W) denote the space of all bounded real-valued functions deﬁned on the setW. Meanwhile, B(W) endowed with the sup metricd(h,k) =sup_x_∈_W|hx–kx|for allh,k∈ B(W) is a complete metric space.

We consider the operatorsf,T:B(W)→B(W) given by

fh(x) =sup y∈D

p(x,y) +Gx,y,hτ(x,y) , x∈W,h∈B(W),

Th(x) =sup y∈D

q(x,y) +Kx,y,hτ(x,y) , x∈W,h∈B(W).

Suppose that the following conditions hold:

(A) For anyh∈B(W), there existsk∈B(W)such that

fh(x) =Tk(x), x∈W.

(A) There existsh∈B(W)such that

fh(x) =Th(x) implies Tfh(x) =fTh(x), x∈W.

(A) p,q:W×D→RandG,K:W×D× R→Rare bounded. (A) ξ(Th)≥for someh∈B(W)impliesξ(fh)≥.

(A) |G(x,y,h(x)) –G(x,y,k(x))| ≤ln( +M(h,k))whereh,k∈B(W),ξ(Th)≥, and

ξ(Tk)≥,(x,y)∈W×D,t∈W and

M(h,k) =max

dTh(t),Tk(t),dTh(t),fh(t),dTk(t),fk(t),

 

dTh(t),fk(t)+dTk(t),fh(t).

(A) If{hn}is a sequence inB(W)such thatξ(hn)≥for alln∈N∪ {}andhn→h∗

asn→ ∞, thenξ(h∗)≥.

(A) There existsh∈B(W)such thatξ(Th)≥.

Theorem  Assume that conditions(A)-(A)are satisﬁed and T(B(W))is a closed and bounded subspace of B(W).Then the functional equations(.)and(.)have a unique common bounded solution in W.

Proof Letλbe an arbitrary positive number andx∈h,h∈B(W) such thatξ(Th)≥ andξ(Th)≥. Then there existy,y∈Dsuch that

fh(x) <p(x,y) +Gx,y,h

(17)

fh(x) <p(x,y) +Gx,y,h

τ(x,y)+λ, (.)

fh(x)≥p(x,y) +G

x,y,h

τ(x,y)

, (.)

fh(x)≥p(x,y) +Gx,y,h

τ(x,y). (.)

Next, by using (.) and (.), we have

fh(x) –fh(x) <Gx,y,h

τ(x,y)–Gx,y,h

τ(x,y)+λ

≤Gx,y,h

τ(x,y)–Gx,y,h

τ(x,y)+λ

≤ln +M(h,h)+λ. (.)

Analogously, by using (.) and (.), we obtain

fh(x) –fh(x) <ln

 +M(h,h)

+λ. (.)

Therefore, from (.) and (.), we deduce

fh(x) –fh(x)<ln +M(h,h)+λ

or, equivalently,

d(fh,fh)≤ln +M(h,h)+λ.

Sinceλ>  is arbitrary, we get

d(fh,fh)≤ln +M(h,h).

Now, deﬁneα,β:B(W)→[,∞) by

α(h) =β(h) =

, ifξ(h)≥ whereh∈B(W), , otherwise.

Also, deﬁneϕ,η: [,∞)→[,∞) byϕ(t) =tandη(t) =ln( +t). Therefore, using the last inequality, we have

α(Th)β(Th)ϕd(fh,fh)

≤ηM(h,h).

One easily shows that all the hypotheses of Corollary  are satisﬁed. Thereforef andT

have a unique common ﬁxed point, that is, the functional equations (.) and (.) have a

unique bounded common solution.

Now, we give the following condition to use in the next theorem:

(A) |G(x,y,h(x)) –G(x,y,k(x))| ≤ln( +N(h,k)) where h,k ∈B(W), ξ(Th)≥, and

(18)

N(h,k) =max

dTh(t),Tk(t), d

Th(t),fk(t),dTk(t),fh(t),

[ +d(Th(t),fh(t))]d(Tk(t),fk(t))  +d(Th(t),Tk(t))

.

Theorem  Let all the conditions of Theorembe fulﬁlled.If we write the condition(A)

instead of the condition(A)in Theorem,then functional equations(.)and(.)have a unique common bounded solution in W.

Proof Following similar arguments to the proof of Theorem , one can easily show that all the hypotheses of Corollary  hold. Then the functional equations (.) and (.) have

a unique bounded common solution.

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript. Author details

1_{Department of Mathematics, Gazi University, Ankara, Turkey.}2_{Department of Mathematics, College of Science, King}

Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia.3_{Department of Mathematics and Computer Sciences,}

University of Palermo, Via Archiraﬁ 34, Palermo, 90123, Italy. 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. PRG-1436-10.

Received: 24 April 2015 Accepted: 21 August 2015 References

1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math.3, 133-181 (1922)

2. Khan, MS, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30(1), 1-9 (1984)

3. Alber, YI, Guerre-Delabriere, S: Principles of weakly contractive maps in Hilbert spaces. Oper. Theory, Adv. Appl.98, 7-22 (1997)

4. Rhoades, BE: Some theorems on weakly contractive maps. Nonlinear Anal.47(4), 2683-2693 (2001)

5. Dutta, PN, Choudhury, BS: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl.2008, 406368 (2008)

6. 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)

7. Samet, B, Vetro, C, Vetro, P: Fixed point theorems forα-ψ-contractive type mappings. Nonlinear Anal.75, 2154-2165 (2012)

8. La Rosa, V, Vetro, P: Common ﬁxed points forα-ψ-ϕ-contractions in generalized metric spaces. Nonlinear Anal., Model. Control19(1), 43-54 (2014)

9. Kumam, P, Vetro, C, Vetro, F: Fixed points for weakα-ψ-contractions in partial metric spaces. Abstr. Appl. Anal.2013, 986028 (2013)

10. Cho, SH, Bae, JS: Fixed points of weakα-contraction type maps. Fixed Point Theory Appl.2014, 175 (2014) 11. Salimi, P, Vetro, C, Vetro, P: Fixed point theorems for twisted (α,β)-ψ-contractive type mappings and applications.

Filomat27(4), 605-615 (2013)

12. Alizadeh, S, Moradlou, F, Salimi, P: Some ﬁxed point results for (α,β)-(ψ,ϕ)-contractive mappings. Filomat28(3), 635-647 (2014)

13. Moradlou, F, Salimi, P, Vetro, P: Fixed point results forr-(η,ξ,ψ)-contractive mappings of type (I), (II) and (III). Filomat 27(2), 403-410 (2013)

14. 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)

15. Jungck, G, Rhoades, BE: Fixed points for set valued functions without continuity. Indian J. Pure Appl. Math.29, 227-238 (1998)

16. Kirk, WA, Srinavasan, PS, Veeramani, P: Fixed points for mapping satisfying cyclical contractive conditions. Fixed Point Theory4, 79-89 (2003)

17. Pacurar, M, Rus, IA: Fixed point theory for cyclicφ-contractions. Nonlinear Anal.72, 1181-1187 (2010)

18. Agarwal, RP, Alghamdi, MA, Shahzad, N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl.2012, 40 (2012)

(19)

20. 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), 553-563 (2013)

21. Imdad, M, Chauhan, S, Kadelburg, Z, Vetro, C: Fixed point theorems for non-self mappings in symmetric spaces under phi-weak contractive conditions and an application to functional equations in dynamic programming. Appl. Math. Comput.227, 469-479 (2014)

22. 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), 483-501 (2014)

23. Bellman, R, Lee, ES: Functional equations in dynamic programming. Aequ. Math.17, 1-18 (1978)