# Fixed point theory for cyclic (*ϕ* - *ψ*)-contractions

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## Abstract

In this article, the concept of cyclic (*ϕ* - *ψ*)-contraction and a fixed point theorem for this type of mappings in the context of complete metric spaces have been presented. The results of this study extend some fixed point theorems in literature.

**2000 Mathematics Subject Classification:** 47H10;46T99 54H25.

## Keywords

cyclic (*ϕ*-

*ψ*)-contraction fixed point theory.

## 1. Introduction and preliminaries

One of the most important results used in nonlinear analysis is the well-known Banach's contraction principle. Generalization of the above principle has been a heavily investigated branch research. Particularly, in [1] the authors introduced the following definition.

**Definition 1**.

*Let X be a nonempty set, m a positive integer, and T: X*→

*X a mapping*. $X={\cup}_{i=1}^{m}{A}_{i}$

*is said to be a cyclic representation of X with respect to T if*

- (i)
*A*_{ i },*i*= 1, 2, ...,*m are nonempty sets*. - (ii)
*T*(*A*_{1}) ⊂*A*_{2}, ...,*T*(*A*_{m-1}) ⊂*A*_{ m },*T*(*Am*) ⊂*A*_{1}.

Recently, fixed point theorems for operators *T* defined on a complete metric space *X* with a cyclic representation of *X* with respect to *T* have appeared in the literature (see, e.g., [2, 3, 4, 5]). Now, we present the main result of [5]. Previously, we need the following definition.

**Definition 2**.

*Let*(

*X*,

*d*)

*be a metric space, m a positive integer A*

_{1},

*A*

_{2}, ...,

*A*

_{ m }

*nonempty closed subsets of X and*$X={\cup}_{i=1}^{m}{A}_{i}$.

*An operator T: X*→

*X is said to be a cyclic weak ϕ-contraction if*

- (i)
$X={\cup}_{i=1}^{m}{A}_{i}$

*is a cyclic representation of X with respect to T*. - (ii)
*d*(*Tx*,*Ty*) ≤*d*(*x*,*y*) -*ϕ*(*d*(*x*,*y*)),*for any X*∈*A*_{ i }*, y*∈*A*_{i+1},*i*= 1, 2, ...,*m, where A*_{m+1}=*A*_{1}*and ϕ:*[0, ∞) → [0, ∞)*is a nondecreasing and continuous function satisfying ϕ*(*t*) > 0*for t*∈ (0, ∞)*and ϕ*(0) = 0

**Remark 3**. *For convenience, we denote by* **F***the class of functions ϕ:* [0, ∞) → [0, ∞) *nondecreasing and continuous satisfying ϕ*(*t*) > 0 *for t* ∈ (0, ∞) *and ϕ*(0) = 0.

The main result of [5] is the following.

**Theorem 4**. [[5], Theorem 6] *Let* (*X*, *d*) *be a complete metric space, m a positive integer, A*_{1}, *A*_{2}, ..., *A*_{ m } *nonempty closed subsets of X and*$X={\cup}_{i=1}^{m}{A}_{i}$. *Let T: X* → *X be a cyclic weak ϕ-contraction with ϕ* ∈ **F***. Then T has a unique fixed point* $z\in {\cap}_{i=1}^{m}{A}_{i}$.

The main purpose of this article is to present a generalization of Theorem 4.

## 2. Main results

First, we present the following definition.

**Definition 5**.

*Let*(

*X*,

*d*)

*be a metric space, m a positive integer, A*

_{1},

*A*

_{2}, ...,

*A*

_{ m }

*nonempty subsets of X and*$X={\cup}_{i=1}^{m}{A}_{i}$.

*An operator T: X*→

*X is a cyclic weak*(

*ϕ*-

*ψ*)-

*contraction if*

- (i)
$X={\cup}_{i=1}^{m}{A}_{i}$

*is a cyclic representation of X with respect to T*, - (ii)
*ϕ*(*d*(*Tx*,*Ty*)) ≤*ϕ*(*d*(*x*,*y*)) -*ψ*(*d*(*x*,*y*)),*for any X*∈*A*_{ i }*, y*∈*A*_{i+1},*i*= 1, 2, ...,*m, where A*_{m+1}=*A*_{1}*and ϕ*,*ψ*∈.**F**

Our main result is the following.

**Theorem 6**. *Let* (*X*, *d*) *be a complete metric space, m a positive integer, A*_{1}, *A*_{2}, ..., *A*_{ m } *nonempty subsets of X and*$X={\cup}_{i=1}^{m}{A}_{i}$. *Let T: X* → *X be a cyclic* (*ϕ* - *ψ*)-*contraction. Then T has a unique fixed point*$z\in {\cap}_{i=1}^{m}{A}_{i}$.

*Proof*. Take

*x*

_{0}∈

*X*and consider the sequence given by

*n*

_{0}∈ ℕ such that ${x}_{{n}_{0}+1}={x}_{{n}_{0}}$ then, since ${x}_{{n}_{0}+1}=T{x}_{{n}_{0}}={x}_{{n}_{0}}$, the part of existence of the fixed point is proved. Suppose that

*x*

_{n+1}≠

*x*

_{ n }for any

*n*= 0, 1, 2, .... Then, since $X={\cup}_{i=1}^{m}{A}_{i}$, for any

*n*> 0 there exists

*i*

_{ n }∈ {1, 2, ...,

*m*} such that ${x}_{n-1}\in {A}_{{i}_{n}}$ and ${x}_{n}\in {A}_{{i}_{n+1}}$. Since

*T*is a cyclic (

*ϕ*-

*ψ*)-contraction, we have

*ϕ*is nondecreasing we obtain

*d*(

*x*

_{ n },

*x*

_{n+1})} is a nondecreasing sequence of nonnegative real numbers. Consequently, there exists

*γ*≥ 0 such that $\underset{n\to \infty}{lim}d\left({x}_{n},{x}_{n+1}\right)=\gamma $. Taking

*n*→ ∞ in (2.1) and using the continuity of

*ϕ*and

*ψ*, we have

*ψ*(

*γ*) = 0. Since

*ψ*∈

**F**,

*γ*= 0, that is,

In the sequel, we will prove that {*x*_{ n } } is a Cauchy sequence.

First, we prove the following claim.

**Claim**: For every *ε* > 0 there exists *n* ∈ ℕ such that if *p*, *q* ≥ *n* with *p* - *q* ≡ 1(*m*) then *d*(*x*_{ p } , *x*_{ q } ) < *ε*.

*ε*> 0 such that for any

*n*∈ ℕ we can find

*p*

_{ n }

*> q*

_{ n }≥

*n*with

*p*

_{ n }-

*q*

_{ n }≡ 1(

*m*) satisfying

*n*> 2

*m*. Then, corresponding to

*q*

_{ n }≥

*n*use can choose

*p*

_{ n }in such a way that it is the smallest integer with

*p*

_{ n }

*> q*

_{ n }satisfying

*p*

_{ n }-

*q*

_{ n }≡ 1(

*m*) and $d\left({x}_{{q}_{n}},{x}_{{p}_{n}}\right)\ge \epsilon $. Therefore, $d\left({x}_{{q}_{n}},{x}_{{p}_{n-m}}\right)\le \epsilon $. Using the triangular inequality

*n*→ ∞ in the last inequality and taking into account that lim

_{n→∞}

*d*(

*x*

_{ n },

*x*

_{n+1}) = 0, we obtain

*n*→ ∞ in (2.4) and taking into account that $\underset{n\to \infty}{lim}d\left({x}_{n},{x}_{n+1}\right)=0$ and (2.4), we get

*A*

_{ i }and

*A*

_{i+1}for certain 1 ≤

*i*≤

*m*, using the fact that

*T*is a cyclic (

*ϕ*-

*ψ*)-contraction, we have

*ϕ*and

*ψ*, letting

*n*→ ∞ in the last inequality, we obtain

and consequently, *ψ*(*ε*) = 0. Since *ψ* ∈ **F**, then *ε* = 0 which is contradiction.

Therefore, our claim is proved.

*X*,

*d*) is a Cauchy sequence. Fix

*ε*> 0. By the claim, we find

*n*

_{0}∈ ℕ such that if

*p*,

*q*≥

*n*

_{0}with

*p*-

*q*≡ 1(

*m*)

*n*

_{1}∈ ℕ such that

for any *n* ≥ *n*_{1}.

*r*,

*s*≥ max{

*n*

_{0},

*n*

_{1}} and

*s > r*. Then there exists

*k*∈ {1, 2, ...,

*m*} such that

*s*-

*r*≡

*k*(

*m*). Therefore,

*s*-

*r*+

*j*≡ 1(

*m*) for

*j*=

*m*-

*k*+ 1. So, we have

*d*(

*x*

_{ r },

*x*

_{ s }) ≤

*d*(

*x*

_{ r },

*x*

_{s+j})+

*d*(

*x*

_{s+j},

*x*

_{s+j-1})+ ⋯ +

*d*(

*x*

_{s+1},

*x*

_{ s }). By (2.7) and (2.8) and from the last inequality, we get

This proves that (*x*_{ n } ) is a Cauchy sequence. Since *X* is a complete metric space, there exists *x* ∈ *X* such that lim_{n→∞}*x*_{ n } = *x*. In what follows, we prove that *x* is a fixed point of *T*. In fact, since $\underset{n\to \infty}{lim}{x}_{n}=x$ and, as $X={\cup}_{i=1}^{m}{A}_{i}$ is a cyclic representation of *X* with respect to *T*, the sequence (*x*_{ n } ) has infinite terms in each *A*_{ i } for *i* ∈ {∈ 1, 2, ..., *m*}.

*x*∈

*A*

_{ i },

*Tx*∈

*A*

_{i+1}and we take a subsequence ${x}_{{n}_{k}}$ of (

*x*

_{ n }) with ${x}_{{n}_{k}}\in {A}_{i-1}$ (the existence of this subsequence is guaranteed by the above-mentioned comment). Using the contractive condition, we can obtain

*ϕ*and

*ψ*belong to

**F**, letting

*k*→ ∞ in the last inequality, we have

or, equivalently, *ϕ*(*d*(*x*, *Tx*)) = 0. Since *ϕ* ∈ **F**, then *d*(*x*, *Tx*) = 0 and, therefore, *x* is a fixed point of *T*.

*y*,

*z*∈

*X*with

*y*and

*z*fixed points of

*T*. The cyclic character of

*T*and the fact that

*y*,

*z*∈

*X*are fixed points of

*T*, imply that $y,z\in {\cap}_{i=1}^{m}{A}_{i}$. Using the contractive condition we obtain

Since *ψ* ∈ **F**, *d*(*y*, *z*) = 0 and, consequently, *y* = *z*. This finishes the proof.

In the sequel, we will show that Theorem 6 extends some recent results.

If in Theorem 6 we take as *ϕ* the identity mapping on [0, ∞) (which we denote by *Id*_{[0, ∞)}), we obtain the following corollary.

**Corollary 7**. *Let* (*X*, *d*) *be a complete metric space m a positive integer, A*_{1}, *A*_{2}, ..., *A*_{ m } *nonempty subsets of X and*$X={\cup}_{i=1}^{m}{A}_{i}$. *Let T: X* → *X be a cyclic* (*Id*_{[0, ∞)} - *ψ*) *contraction. Then T has a unique fixed point*$z\in {\cap}_{i=1}^{m}{A}_{i}$.

Corollary 7 is a generalization of the main result of [5] (see [[5], Theorem 6]) because we do not impose that the sets *A*_{ i } are closed.

If in Theorem 6 we consider *ϕ* = *Id*_{[0, ∞)} and *ψ* = (1 - *k*)*Id*_{[0, ∞)} for *k* ∈ [0, 1) (obviously, *ϕ*, *ψ* ∈ **F**), we have the following corollary.

**Corollary 8**. *Let* (*X*, *d*) *be a complete metric space m a positive integer, A*_{1}, *A*_{2}, ..., *A*_{ m } *nonempty subsets of X and*$X={\cup}_{i=1}^{m}{A}_{i}$. *Let T: X* → *X be a cyclic* (*Id*_{[0, ∞)} - (1 - *k*)*Id*_{[0, ∞)}) *contraction, where k* ∈ [0, 1). *Then T has a unique fixed point*$z\in {\cap}_{i=1}^{m}{A}_{i}$.

Corollary 8 is Theorem 1.3 of [1].

The following corollary gives us a fixed point theorem with a contractive condition of integral type for cyclic contractions.

**Corollary 9**.

*Let*(

*X*,

*d*)

*be a complete metric space, m a positive integer, A*

_{1},

*A*

_{2}, ...,

*A*

_{ m }

*nonempty closed subsets of X and*$X={\cup}_{i=1}^{m}{A}_{i}$.

*Let T: X*→

*X be an operator such that*

- (i)
$X={\cup}_{i=1}^{m}{A}_{i}$

*is a cyclic representation of X with respect to T*. - (ii)
*There exists k*∈ [0, 1)*such that*${\int}_{0}^{d\left(Tx,Ty\right)}\rho \left(t\right)dt\le k{\int}_{0}^{d\left(x,y\right)}\rho \left(t\right)dt$

*for any X* ∈ *A*_{ i }*, y* ∈ *A*_{i+1}, *i* = 1, 2, ..., *m where A*_{m+1}= *A*_{1}, *and ρ:* [0, ∞) → [0, ∞) *is a Lebesgue-integrable mapping satisfying* ${\int}_{0}^{\epsilon}\rho \left(t\right)dt$ *for ε*> 0.

*Then T has unique fixed point*$z\in {\cap}_{i=1}^{m}{A}_{i}$.

*Proof*. It is easily proved that the function *ϕ:* [0, ∞) → [0, ∞) given by $\phi \left(t\right)={\int}_{0}^{t}p\left(s\right)ds$ satisfies that *ϕ* ∈ **F**. Therefore, Corollary 9 is obtained from Theorem 6, taking as *ϕ* the above-defined function and as *ψ* the function *ψ*(*t*) = (1 - *k*)*ϕ*(*t*).

If in Corollary 9, we take *A*_{ i } = *X* for *i* = 1, 2, ..., *m* we obtain the following result.

**Corollary 10**.

*Let*(

*X*,

*d*)

*be a complete metric space and T: X*→

*X a mapping such that for x, y*∈

*X*,

*where ρ:* [0, ∞) → [0, ∞) *is a Lebesgue-integrable mapping satisfying*${\int}_{0}^{\epsilon}\rho \left(t\right)dt$*for ε* > 0 *and the constant k* ∈ [0, 1). *Then T has a unique fixed point*.

Notice that this is the main result of [6]. If in Theorem 6 we put *A*_{ i } = *X* for *i* = 1, 2, ..., *m* we have the result.

**Corollary 11**.

*Let*(

*X*,

*d*)

*be a complete metric space and T: X*→

*X an operator such that for x, y*∈

*X*,

*where ϕ, ψ* ∈ **F***. Then T has a unique fixed point*.

This result appears in [7].

## 3. Example and remark

In this section, we present an example which illustrates our results. Throughout the article, we let ℕ* = ℕ\{0}.

**Example 12**.

*Consider*$X=\left\{\frac{1}{n}:n\in {\mathbb{N}}^{*}\right\}\cup \left\{0\right\}$

*with the metric induced by the usual distance in*ℝ,

*i.e., d*(

*x*,

*y*) = |

*x*-

*y*|.

*Since X is a closed subset of*ℝ,

*it is a complete metric space. We consider the following subsets of X:*

*Obviously, X*=

*A*

_{1}∪

*A*

_{2}.

*Let T: X*→

*X be the mapping defined by*

*It is easily seen that X*=

*A*

_{1}∪

*A*

_{2}

*is a cyclic representation of X with respect to T. Now we consider the function ρ:*[0, ∞) → [0, ∞)

*defined by*

*It is easily proved that*${\int}_{0}^{t}\rho \left(s\right)ds={t}^{1\u2215t}$*for t* ≤ 1.

*In what follows, we prove that T satisfies condition (ii) of Corollary 9*.

*In fact, notice that the function ρ*(

*t*)

*is a Lebesgue-integrable mapping satisfying*${\int}_{0}^{\epsilon}\rho \left(t\right)dt>0$

*for ε*> 0.

*We take m*,

*n*∈ ℕ*

*with m*≥

*n and we will prove*

*Since*${\int}_{0}^{t}\rho \left(s\right)ds={t}^{1\u2215t}$

*for t*≤ 1

*and, as diam*(

*X*) ≤ 1,

*the last inequality can be written as*

*or equivalently*,

*or equivalently*,

*or equivalently*,

*In order to prove that this last inequality is true, notice that*

*and, therefore*,

*On the other hand, from*

*we obtain*

*and, thus*,

*Since*$\frac{n+m+1}{m-n}\ge 1$,

*Finally*, (3.2) *and* (3.3) *give us* (3.1).

*n*∈ ℕ* and

*y*= 0. In this case, condition (

*ii*) of Corollary 9 for $k=\frac{1}{2}$ has the form

Consequently, since assumptions of Corollary 9 are satisfied, this corollary gives us the existence of a unique fixed point (which is obviously *x* = 0).

This example appears in [6].

Now, we connect our results with the ones appearing in [3]. Previously, we need the following definition.

**Definition 13**. *A function φ:* [0, ∞) → [0, ∞) *is a (c)-comparison function if* $\sum _{k=0}^{\infty}{\phi}^{k}\left(t\right)$ *converges for any t*∈ [0, ∞). *The main result of*[3]*is the following*.

**Theorem 14**.

*Let*(

*X*,

*d*)

*be a complete metric space, m a positive integer, A*

_{1},

*A*

_{2}, ...,

*A*

_{ m }

*nonempty subsets of X*, $X={\cup}_{i=1}^{m}{A}_{i}$

*and φ:*[0, ∞) → [0, ∞)

*a (c)-comparison function. Let T: X*→

*X be an operator and we assume that*

- (i)
$X={\cup}_{i=1}^{m}{A}_{i}$

*is a cyclic representation of X with respect to T*. - (ii)
*d*(*Tx*,*Ty*) ≤*φ*(*d*(*x*,*y*)),*for any X*∈*A*_{ i }*and y*∈*A*_{i+1},*where A*_{m+1}=*A*.

*Then T has a unique fixed point*$z\in {\cap}_{i=1}^{m}{A}_{i}$.

for any *x* ∈ *A*_{ i } , *y* ∈ *A*_{i+1}, where *A*_{m+1}= *A*_{1}, and *ϕ*, *φ* ∈ **F**.

*ϕ*=

*Id*

_{[0, ∞)}and $\phi \left(t\right)=\frac{{t}^{2}}{1+t}$, it is easily seen that

*ϕ*,

*φ*∈

**F**. On the other hand,

Moreover, for every *t* ∈ (0, ∞), $\sum _{k=0}^{\infty}{\left(\varphi -\phi \right)}^{\left(k\right)}\left(t\right)$ diverges. Therefore, *ϕ* - *φ* is not a (c)-comparison function. Consequently, our Theorem 6 can be applied to cases which cannot treated by Theorem 14.

## Notes

### Acknowledgements

KS was partially supported by the "Ministerio de Education y Ciencia", Project MTM 2007/65706.

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