# Nonlinear cyclic weak contractions in *G*-metric spaces and applications to boundary value problems

- Hemant Kumar Nashine
^{1}, - Zoran Golubović
^{2}and - Zoran Kadelburg
^{3}Email author

**2012**:227

https://doi.org/10.1186/1687-1812-2012-227

© Nashine et al.; licensee Springer. 2012

**Received: **20 August 2012

**Accepted: **25 November 2012

**Published: **17 December 2012

## Abstract

We present fixed point theorems for nonlinear cyclic mappings under a generalized weakly contractive condition in *G*-metric spaces. We furnish examples to demonstrate the usage of the results and produce an application to second-order periodic boundary value problems for ODEs.

**MSC:**47H10, 34B15.

### Keywords

cyclic contraction fixed point*G*-metric space weakly contractive condition altering distance function boundary value problem

## 1 Introduction

Nonlinear analysis is a remarkable confluence of topology, analysis and applied mathematics. The fixed point theory is one of the most rapidly growing topics of nonlinear functional analysis. It is a vast and inter-disciplinary subject whose study belongs to several mathematical domains such as classical analysis, differential and integral equations, functional analysis, operator theory, topology and algebraic topology *etc.* Most important nonlinear problems of applied mathematics reduce to finding solutions of nonlinear functional equations (*e.g.*, nonlinear integral equations, boundary value problems for nonlinear ordinary or partial differential equations, the existence of periodic solutions of nonlinear partial differential equations). They can be formulated in terms of finding the fixed points of a given nonlinear mapping on an infinite dimensional function space into itself.

the conclusion is that *T* has a fixed point, in fact, exactly one of them.

The simplicity of its proof and the possibility of attaining the fixed point by using successive approximations let this theorem become a very useful tool in analysis and its applications. There is a great number of generalizations of the Banach contraction principle in the literature (see, *e.g.*, [2] and the references cited therein).

It is important to note that the inequality (1.1) implies the continuity of *T*. A natural question is whether we can find contractive conditions which will imply the existence of a fixed point in a complete metric space but will not imply continuity. The question was answered by Kirk *et al.* [3] and turned the area of investigation of fixed points by introducing cyclic representations and cyclic contractions, which can be stated as follows.

**Definition 1** [3]

*p*be a positive integer, ${\mathcal{A}}_{1},{\mathcal{A}}_{2},\dots ,{\mathcal{A}}_{p}$ be subsets of

*X*, $Y={\bigcup}_{i=1}^{p}{\mathcal{A}}_{i}$ and $T:Y\to Y$. Then

*Y*is said to be a cyclic representation of

*Y*with respect to

*T*if

- (i)
${\mathcal{A}}_{i}$, $i=1,2,\dots ,p$ are nonempty closed sets, and

- (ii)
$T({\mathcal{A}}_{1})\subseteq {\mathcal{A}}_{2}$, …, $T({\mathcal{A}}_{p-1})\subseteq {\mathcal{A}}_{p}$, $T({\mathcal{A}}_{p})\subseteq {\mathcal{A}}_{1}$.

Moreover, *T* is called a cyclic contraction if there exists $k\in (0,1)$ such that $d(Tx,Ty)\le kd(x,y)$ for all $x\in {\mathcal{A}}_{i}$ and $y\in {\mathcal{A}}_{i+1}$, with ${\mathcal{A}}_{p+1}={\mathcal{A}}_{1}$.

Notice that although a contraction is continuous, a cyclic contraction need not to be. This is one of the important gains of this notion. Kirk *et al.* obtained, among others, cyclic versions of the Banach contraction principle [1], of the Boyd and Wong fixed point theorem [4] and of the Caristi fixed point theorem [5]. Following the paper [3], a number of fixed point theorems on cyclic representation with respect to a self-mapping have appeared (see, *e.g.*, [6–13]).

On the other hand, in 2006, Mustafa and Sims [14, 15] introduced the notion of generalized metric spaces or simply *G*-metric spaces. Several authors studied these spaces a lot and obtained several fixed and common fixed point theorems (see, *e.g.*, [16–26]).

Khan *et al.* [27] introduced the concept of an altering distance function.

**Definition 2** [27]

- (i)
*ψ*is continuous and non-decreasing, - (ii)
$\psi (t)=0$ if and only if $t=0$.

They proved the following theorem.

**Theorem 1** [27]

*Let*$(X,d)$

*be a complete metric space*,

*let*

*φ*

*be an altering distance function*,

*and let*$T:X\to X$

*be a self*-

*mapping which satisfies the following inequality*:

*for all* $x,y\in X$ *and for some* $0<c<1$. *Then* *T* *has a unique fixed point*.

Putting $\phi (t)=t$ in the previous theorem, (1.2) reduces to (1.1).

Rhoades [28] extended Banach’s principle by introducing weakly contractive mappings in complete metric spaces.

**Definition 3** [28]

where *φ* is an altering distance function.

He proved the following result.

**Theorem 2** [28]

*Let* $(X,d)$ *be a complete metric space*. *If* $T:X\to X$ *is a weakly contractive mapping*, *then* *T* *has a unique fixed point*.

If one takes $\phi (t)=(1-k)t$, where $0<k<1$, then (1.3) reduces to (1.1).

Dutta and Choudhury obtained in [29] the following generalization of Theorems 1 and 2.

**Theorem 3** [29]

*Let*$(X,d)$

*be a complete metric space and let*$T:X\to X$

*satisfy*

*for all* $x,y\in X$, *where* *ψ* *and* *φ* *are altering distance functions*. *Then* *T* *has a unique fixed point*.

Weak inequalities of the above type have been used to establish fixed point results in a number of subsequent works, some of which are noted in [2, 30–35] and the references cited therein.

In the present paper, we introduce nonlinear cyclic contraction mappings under a generalized weakly contraction condition in *G*-metric spaces. It is followed by the proof of existence and uniqueness of fixed points for such mappings. The obtained result generalizes and improves many existing theorems in the literature. Some examples are given in support of our results. In conclusion, we apply accomplished fixed point results for generalized cyclic contraction type mappings to the study of existence and uniqueness of solutions for a class of second-order periodic boundary value problems for ODEs.

## 2 Preliminaries

For more details on the following definitions and results, we refer the reader to [15].

**Definition 4** Let *X* be a nonempty set and let $G:X\times X\times X\to {\mathbb{R}}^{+}$ be a function satisfying the following properties:

(G1) $G(x,y,z)=0$ if $x=y=z$;

(G2) $0<G(x,x,y)$ for all $x,y\in X$ with $x\ne y$;

(G3) $G(x,x,y)\le G(x,y,z)$ for all $x,y,z\in X$ with $z\ne y$;

(G4) $G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots $ (symmetry in all three variables);

(G5) $G(x,y,z)\le G(x,a,a)+G(a,y,z)$ for all $x,y,z,a\in X$ (rectangle inequality).

Then the function *G* is called a *G*-metric on *X* and the pair $(X,G)$ is called a *G*-metric space.

holds for all $x,y,z\in X$.

**Definition 5**Let $(X,G)$ be a

*G*-metric space and let $({x}_{n})$ be a sequence of points in

*X*.

- 1.
A point $x\in X$ is said to be the limit of the sequence $({x}_{n})$ if ${lim}_{n,m\to \mathrm{\infty}}G(x,{x}_{n},{x}_{m})=0$, and one says that the sequence $({x}_{n})$ is

*G*-convergent to*x*. - 2.
The sequence $({x}_{n})$ is said to be a

*G*-Cauchy sequence if for every $\epsilon >0$, there is a positive integer*N*such that $G({x}_{n},{x}_{m},{x}_{l})<\epsilon $ for all $n,m,l\ge N$; that is, if $G({x}_{n},{x}_{m},{x}_{l})\to 0$ as $n,m,l\to \mathrm{\infty}$. - 3.
$(X,G)$ is said to be

*G*-complete (or a complete*G*-metric space) if every*G*-Cauchy sequence in $(X,G)$ is*G*-convergent in*X*.

Thus, if ${x}_{n}\to x$ in a *G*-metric space $(X,G)$, then for any $\epsilon >0$, there exists a positive integer *N* such that $G(x,{x}_{n},{x}_{m})<\epsilon $ for all $n,m\ge N$. It was shown in [15] that the *G*-metric induces a Hausdorff topology and that the convergence, as described in the above definition, is relative to this topology. The topology being Hausdorff, a sequence can converge to at most one point.

**Lemma 1**

*Let*$(X,G)$

*be a*

*G*-

*metric space*, $({x}_{n})$

*be a sequence in*

*X*

*and*$x\in X$.

*Then the following are equivalent*:

- (1)
$({x}_{n})$

*is**G*-*convergent to**x*. - (2)
$G({x}_{n},{x}_{n},x)\to 0$

*as*$n\to \mathrm{\infty}$. - (3)
$G({x}_{n},x,x)\to 0$

*as*$n\to \mathrm{\infty}$.

**Lemma 2**

*If*$(X,G)$

*is a*

*G*-

*metric space*,

*then the following are equivalent*:

- (1)
*The sequence*$({x}_{n})$*is**G*-*Cauchy*. - (2)
*For every*$\epsilon >0$,*there exists a positive integer**N**such that*$G({x}_{n},{x}_{m},{x}_{m})<\epsilon $*for all*$n,m\ge N$.

**Lemma 3** *Let* $(X,G)$, $({X}^{\prime},{G}^{\prime})$ *be two* *G*-*metric spaces*. *Then a function* $f:X\to {X}^{\prime}$ *is* *G*-*continuous at a point* $x\in X$ *if and only if it is* *G*-*sequentially continuous at* *x*, *that is*, *if* $(f{x}_{n})$ *is* ${G}^{\prime}$-*convergent to* *fx* *whenever* $({x}_{n})$ *is* *G*-*convergent to* *x*.

**Definition 6**A

*G*-metric space $(X,G)$ is said to be symmetric if

holds for arbitrary $x,y\in X$. If this is not the case, the space is called asymmetric.

*G*-metric on the set

*X*, a standard metric can be associated by

*G*is symmetric, then obviously ${d}_{G}(x,y)=2G(x,y,y)$, but in the case of an asymmetric

*G*-metric, only

holds for all $x,y\in X$.

The following are some easy examples of *G*-metric spaces.

**Example 1**(1) Let $(X,d)$ be an ordinary metric space. Define ${G}_{s}$ by

*G*-metric space.

- (2)Let $X=\{a,b\}$. Define$G(a,a,a)=G(b,b,b)=0,\phantom{\rule{2em}{0ex}}G(a,a,b)=1,\phantom{\rule{2em}{0ex}}G(a,b,b)=2,$

and extend *G* to $X\times X\times X$ by using the symmetry in the variables. Then it is clear that $(X,G)$ is an asymmetric *G*-metric space.

## 3 Main results

First, we define the notion of a generalized weakly cyclic contraction in a *G*-metric space.

**Definition 7**Let $(X,G)$ be a

*G*-metric space. Let

*p*be a positive integer, ${\mathcal{A}}_{1},{\mathcal{A}}_{2},\dots ,{\mathcal{A}}_{p}$ be nonempty subsets of

*X*and $Y={\bigcup}_{i=1}^{p}{\mathcal{A}}_{i}$. An operator $T:Y\to Y$ is a generalized weakly cyclic contraction if

- (I)
$Y={\bigcup}_{i=1}^{p}{\mathcal{A}}_{i}$ is a cyclic representation of

*Y*with respect to*T*; - (II)for any $(x,y,z)\in {\mathcal{A}}_{i}\times {\mathcal{A}}_{i+1}\times {\mathcal{A}}_{i+1}$, $i=1,2,\dots ,p$ (with ${\mathcal{A}}_{p+1}={\mathcal{A}}_{1}$),$\psi (G(Tx,Ty,Tz))\le \psi (\mathrm{\Theta}(x,y,z))-\phi (\theta (x,y,z)),$(3.1)

$\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is an altering distance function and $\phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is a continuous function with $\phi (t)=0$ if and only if $t=0$.

Our main result is the following.

**Theorem 4** *Let* $(X,G)$ *be a complete* *G*-*metric space*, $p\in \mathbb{N}$, ${\mathcal{A}}_{1},{\mathcal{A}}_{2},\dots ,{\mathcal{A}}_{p}$ *be nonempty closed subsets of* *X* *and* $Y={\bigcup}_{i=1}^{p}{\mathcal{A}}_{i}$. *Suppose* $T:Y\to Y$ *is a generalized weakly cyclic contraction*. *Then* *T* *has a unique fixed point*. *Moreover*, *the fixed point of* *T* *belongs to* ${\bigcap}_{i=1}^{p}{\mathcal{A}}_{i}$.

*Proof*Let ${x}_{0}\in {\mathcal{A}}_{1}$ (such a point exists since ${\mathcal{A}}_{1}\ne \mathrm{\varnothing}$). Define the sequence $({x}_{n})$ in

*X*by

*n*. From the condition (I), we observe that for all

*n*, there exists $i=i(n)\in \{1,2,\dots ,p\}$ such that $({x}_{n},{x}_{n+1},{x}_{n+1})\in {\mathcal{A}}_{i}\times {\mathcal{A}}_{i+1}\times {\mathcal{A}}_{i+1}$. Putting $x={x}_{n}$ and $y={x}_{n+1}=z$ in the condition (II), we have

*φ*, we have $G({x}_{{n}_{0}+1},{x}_{{n}_{0}+2},{x}_{{n}_{0}+2})=0$, which contradicts the condition (3.4). Therefore, (3.7) is true and so the sequence $(G({x}_{n},{x}_{n+1},{x}_{n+1}))$ is non-increasing and bounded below. Thus, there exists $\rho \ge 0$ such that

*ψ*and

*φ*, we obtain

*G*-Cauchy sequence in

*X*. Suppose the contrary, that is, $({x}_{n})$ is not

*G*-Cauchy. Then there exists $\epsilon >0$ for which we can find two subsequences $({x}_{m(k)})$ and $({x}_{n(k)})$ of $({x}_{n})$ such that $n(k)$ is the smallest index for which

*k*, there exists $j(k)\in \{1,\dots ,p\}$ such that $n(k)-m(k)+j(k)\equiv 1[p]$. Then ${x}_{m(k)-j(k)}$ (for

*k*large enough, $m(k)>j(k)$) and ${x}_{n(k)}$ lie in different adjacently labeled sets ${\mathcal{A}}_{i}$ and ${\mathcal{A}}_{i+1}$ for certain $i\in \{1,\dots ,p\}$. Using (2.1) and (G5), we have

*G*-Cauchy sequence in $(X,G)$. Since $(X,G)$ is

*G*-complete, there exists ${x}^{\ast}\in X$ such that

From condition (I), and since ${x}_{0}\in {\mathcal{A}}_{1}$, we have ${({x}_{np})}_{n\ge 0}\subseteq {\mathcal{A}}_{1}$. Since ${\mathcal{A}}_{1}$ is closed, from (3.15) we get that ${x}^{\ast}\in {\mathcal{A}}_{1}$. Again, from the condition (I) we have ${({x}_{np+1})}_{n\ge 0}\subseteq {\mathcal{A}}_{2}$. Since ${\mathcal{A}}_{2}$ is closed, from (3.15) we get that ${x}^{\ast}\in {\mathcal{A}}_{2}$. Continuing this process, we obtain (3.16).

*T*. Indeed, from (3.16), since for all

*n*there exists $i(n)\in \{1,2,\dots ,p\}$ such that ${x}_{n}\in {\mathcal{A}}_{i(n)}$, applying (II) with $x={x}_{n}$ and $y=z={x}^{\ast}$, we obtain

*G*is continuous in its variables, we obtain

This implies that $\phi (G({x}^{\ast},T{x}^{\ast},T{x}^{\ast}))=0$ and hence ${x}^{\ast}=T{x}^{\ast}$. Thus, ${x}^{\ast}$ is a fixed point of *T*.

*T*. Assume that ${y}^{\ast}$ is another fixed point of

*T*, that is, $T{y}^{\ast}={y}^{\ast}$. By the condition (I), this implies that ${y}^{\ast}\in {\bigcap}_{i=1}^{p}{\mathcal{A}}_{i}$. Then we can apply (II) for $x={x}^{\ast}$ and $y=z={y}^{\ast}$. We obtain

*T*, we have from (3.20)-(3.22)

Hence, $G({x}^{\ast},{y}^{\ast},{y}^{\ast})=0$, that is, ${x}^{\ast}={y}^{\ast}$. Thus, we have proved the uniqueness of the fixed point. □

**Remark 1**Following the proof of Theorem 4, we can derive a similar conclusion if $\mathrm{\Theta}(x,y,z)$ and $\theta (x,y,z)$ are replaced, respectively, by

**Corollary 1**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space*, $p\in \mathbb{N}$, ${\mathcal{A}}_{1},{\mathcal{A}}_{2},\dots ,{\mathcal{A}}_{p}$

*be nonempty closed subsets of*

*X*, $Y={\bigcup}_{i=1}^{p}{\mathcal{A}}_{i}$

*and*$T:Y\to Y$

*such that*

- (I)
$Y={\bigcup}_{i=1}^{p}{\mathcal{A}}_{i}$

*is a cyclic representation of**Y**with respect to**T*;

*for any*$(x,y,z)\in {\mathcal{A}}_{i}\times {\mathcal{A}}_{i+1}\times {\mathcal{A}}_{i+1}$, $i=1,2,\dots ,p$ (

*with*${\mathcal{A}}_{p+1}={\mathcal{A}}_{1}$),

*where* $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ *is an altering distance function and* $\phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ *is a continuous function with* $\phi (t)=0$ *if and only if* $t=0$.

*Then* *T* *has a unique fixed point*. *Moreover*, *the fixed point of* *T* *belongs to* ${\bigcap}_{i=1}^{p}{\mathcal{A}}_{i}$.

## 4 Examples

In this section, we furnish some examples to demonstrate the validity of the hypotheses of Theorem 4.

**Example 2** (inspired by [31])

with symmetry in all variables. Note that *G* is asymmetric since $G(x,x,y)\ne G(x,y,y)$ whenever $x\ne y$. Let ${\mathcal{A}}_{1}=\{a,b\}$ and ${\mathcal{A}}_{2}=\{a,c\}$ and consider the mapping $T:X\to X$ given by $Ta=Tb=a$ and $Tc=b$. Obviously, ${\mathcal{A}}_{1}\cup {\mathcal{A}}_{2}=X$ is a cyclic representation of *X* with respect to *T*.

Hence, all the conditions of Theorem 4 are fulfilled and it follows that *T* has a unique fixed point $a\in {\mathcal{A}}_{1}\cap {\mathcal{A}}_{2}$.

**Example 3**Let $X={\mathbb{R}}^{3}$ and let $G:{X}^{3}\to {\mathbb{R}}^{+}$ be given as

*G*-metric space. Consider the following closed subsets of

*X*:

It is clear that $Y={\mathcal{A}}_{1}\cup {\mathcal{A}}_{2}\cup {\mathcal{A}}_{3}$ is a cyclic representation of *Y* with respect to *T*. We will check that *T* satisfies the contraction condition (II).

*e.g.*, $(x,y,z)\in {\mathcal{A}}_{1}\times {\mathcal{A}}_{2}\times {\mathcal{A}}_{2}$ (the other two cases are treated analogously), and let $x=(a,0,0)$, $y=(0,b,0)$, $z=(0,c,0)$. Without loss of generality, we can assume that $b\ge c$. Then $Tx=(0,\frac{a}{2},0)$, $Ty=(0,0,\frac{b}{4})$, $Tz=(0,0,\frac{c}{4})$ and

Thus, the conditions of Theorem 4 are fulfilled and *T* has a unique fixed point $(0,0,0)\in {\bigcap}_{i=1}^{3}{\mathcal{A}}_{i}$.

## 5 An application to boundary value problems

In this section, we present another example where Theorem 4 and its corollaries can be applied. The example is inspired by [36].

*X*with the

*G*-metric

for $u,v,w\in X$. Then $(X,G)$ is a complete *G*-metric space.

Clearly, ${u}^{\ast}$ is a solution of (5.2) if and only if ${u}^{\ast}$ is a fixed point of *T*.

*T*under the following conditions.

- (A)$K(s,\cdot )$ is a non-increasing function for any fixed $s\in [0,1]$, that is,$x,y\in {\mathbb{R}}^{+},x\ge y\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}K(s,x)\le K(s,y).$
- (B)
$|K(s,x)-K(s,y)|\le |x-y|$ for all $s\in [0,1]$ and $x,y\in {\mathbb{R}}^{+}$.

- (C)There exist $\alpha ,\beta \in X$ such that $\alpha (t)\le \beta (t)$ for $t\in [0,1]$ and that$T\alpha (t)\le \beta (t)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}T\beta (t)\ge \alpha (t)\phantom{\rule{1em}{0ex}}\text{for}t\in [0,1].$

**Theorem 5** *Under the conditions* (A)-(C), *equation* (5.2) *has a unique solution* ${u}^{\ast}\in X$ *and it belongs to* $\mathcal{C}=\{u\in X:\alpha (t)\le u(t)\le \beta (t),t\in [0,1]\}$.

*Proof*In order to prove the existence of a (unique) fixed point of

*T*, we construct closed subsets ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ of

*X*as follows:

for all $t\in [0,1]$. Then we have $Tu\in {\mathcal{A}}_{2}$. Similarly, the other inclusion is proved. Hence, $Y={\mathcal{A}}_{1}\cup {\mathcal{A}}_{2}$ is a cyclic representation of *Y* with respect to *T*.

for $\psi (t)=t$ and $\phi (t)=\frac{7}{8}t$.

Using the same technique, we can show that the above inequality also holds if we take $(u,v,w)\in {\mathcal{A}}_{2}\times {\mathcal{A}}_{1}\times {\mathcal{A}}_{1}$. Thus, *T* satisfies the contractive condition of Corollary 1.

Consequently, by Corollary 1, *T* has a unique fixed point ${u}^{\ast}\in {\mathcal{A}}_{1}\cap {\mathcal{A}}_{2}$, that is, ${u}^{\ast}\in \mathcal{C}$ is the unique solution to (5.2). □

## Declarations

### Acknowledgements

The authors thank the referees for their valuable comments that helped them to correct the first version of the manuscript. The second and third authors are thankful to the Ministry of Science and Technological Development of Serbia.

## Authors’ Affiliations

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