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Nonlinear cyclic weak contractions in Gmetric spaces and applications to boundary value problems
Fixed Point Theory and Applications volume 2012, Article number: 227 (2012)
Abstract
We present fixed point theorems for nonlinear cyclic mappings under a generalized weakly contractive condition in Gmetric spaces. We furnish examples to demonstrate the usage of the results and produce an application to secondorder periodic boundary value problems for ODEs.
MSC:47H10, 34B15.
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 interdisciplinary 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.
In 1922, Banach’s contraction mapping principle appeared [1] and it was known for its simple and elegant proof by using the Picard iteration. If (X,d) is a complete metric space and T:X\to X is a contraction, that is, there exists k\in (0,1) such that
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]
Let (X,d) be a metric space. Let 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}}_{p1})\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 selfmapping 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 Gmetric 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]
A function \psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) is called an altering distance function if the following properties are satisfied:

(i)
ψ is continuous and nondecreasing,

(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 selfmapping 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]
Let (X,d) be a metric space. A mapping T:X\to X is called weakly contractive if
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)=(1k)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 Gmetric 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 secondorder 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 Gmetric on X and the pair (X,G) is called a Gmetric space.
Note that it can be easily deduced from (G4) and (G5) that
holds for all x,y,z\in X.
Definition 5 Let (X,G) be a Gmetric 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 Gconvergent to x.

2.
The sequence ({x}_{n}) is said to be a GCauchy 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 Gcomplete (or a complete Gmetric space) if every GCauchy sequence in (X,G) is Gconvergent in X.
Thus, if {x}_{n}\to x in a Gmetric 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 Gmetric 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 Gmetric space, ({x}_{n}) be a sequence in X and x\in X. Then the following are equivalent:

(1)
({x}_{n}) is Gconvergent 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 Gmetric space, then the following are equivalent:

(1)
The sequence ({x}_{n}) is GCauchy.

(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 Gmetric spaces. Then a function f:X\to {X}^{\prime} is Gcontinuous at a point x\in X if and only if it is Gsequentially continuous at x, that is, if (f{x}_{n}) is {G}^{\prime}convergent to fx whenever ({x}_{n}) is Gconvergent to x.
Definition 6 A Gmetric 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.
To every Gmetric on the set X, a standard metric can be associated by
If G is symmetric, then obviously {d}_{G}(x,y)=2G(x,y,y), but in the case of an asymmetric Gmetric, only
holds for all x,y\in X.
The following are some easy examples of Gmetric spaces.
Example 1 (1) Let (X,d) be an ordinary metric space. Define {G}_{s} by
for all x,y,z\in X. Then it is clear that (X,{G}_{s}) is a symmetric Gmetric 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 Gmetric space.
3 Main results
First, we define the notion of a generalized weakly cyclic contraction in a Gmetric space.
Definition 7 Let (X,G) be a Gmetric 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)
where
and
\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 Gmetric 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
Without loss of the generality, we can assume that
We shall prove that
By the assumption, G({x}_{n},{x}_{n+1},{x}_{n+1})>0 for all 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
By (G5) we have
Thus,
From (3.1) we have
We claim that
for all n\ge 0. Suppose this is not true, that is, there exists an {n}_{0}\ge 0 such that G({x}_{{n}_{0}+1},{x}_{{n}_{0}+2},{x}_{{n}_{0}+2})>G({x}_{{n}_{0}},{x}_{{n}_{0}+1},{x}_{{n}_{0}+1}). By the inequality (3.6), for these elements, we have
This implies that \phi (G({x}_{{n}_{0}+1},{x}_{{n}_{0}+2},{x}_{{n}_{0}+2}))=0 and by the property of 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 nonincreasing and bounded below. Thus, there exists \rho \ge 0 such that
Now suppose that \rho >0. Taking n\to \mathrm{\infty} in (3.6), then using (3.8) and the continuity of ψ and φ, we obtain
Therefore, \phi (\rho )=0 and hence \rho =0. Thus,
Next, we show that ({x}_{n}) is a GCauchy sequence in X. Suppose the contrary, that is, ({x}_{n}) is not GCauchy. 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
This means that
From (3.9), (3.10) and (G5), we get
Passing to the limit as k\to \mathrm{\infty} and using (3.5), we get
On the other hand, for all 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
as k\to \mathrm{\infty} (from (3.5)), which, by (3.11), implies that
Using (3.5), we have
and
Again, using (2.1), we get
Passing to the limit as k\to \mathrm{\infty} in the above inequality, and using (3.13) and (3.12), we get
Similarly, we have
Passing to the limit as k\to \mathrm{\infty} and using (3.5) and (3.12), we obtain
Similarly, we have
Now, from the definitions of \mathrm{\Theta}(x,y,z) and \theta (x,y,z) and from the obtained limits, we have
Putting x={x}_{n(k)+1}, y={x}_{m(k)j(k)+1}, z={x}_{m(k)j(k)+1} in (II), we obtain
Passing to the limit as k\to \mathrm{\infty}, utilizing (3.14) and the obtained limits, we get
which is a contradiction if \epsilon >0. We have proved that ({x}_{n}) is a GCauchy sequence in (X,G). Since (X,G) is Gcomplete, there exists {x}^{\ast}\in X such that
We shall prove 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).
Now, we shall prove that {x}^{\ast} is a fixed point of 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
where
and
Passing to the limit as n\to \mathrm{\infty} in the inequality (3.17) and using (3.18), (3.19) and the fact that 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.
Finally, we prove that {x}^{\ast} is the unique fixed point of 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
where
and
Since {x}^{\ast} and {y}^{\ast} are fixed points of 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
and
and the condition (3.1) by
Corollary 1 Let (X,G) be a complete Gmetric 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;
(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}),
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])
Define a Gmetric G on the set X=\{a,b,c\} by
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.
Take \psi (t)={t}^{2} and \phi (t)=t and denote L=\psi (G(Tx,Ty,Tz)), R=\psi (\mathrm{\Theta}(x,y,z))\phi (\theta (x,y,z)). The following table shows that the contractive condition (3.1) is fulfilled whenever (x,y,z)\in {\mathcal{A}}_{1}\times {\mathcal{A}}_{2}\times {\mathcal{A}}_{2} or (x,y,z)\in {\mathcal{A}}_{2}\times {\mathcal{A}}_{1}\times {\mathcal{A}}_{1}.
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
where ({x}_{4},{y}_{4},{z}_{4})=({x}_{1},{y}_{1},{z}_{1}). It is easy to see that (X,G) is a complete Gmetric space. Consider the following closed subsets of X:
and the mapping T:Y\to Y given by
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).
Take \psi (t)={t}^{2} (which is an altering distance function) and a continuous function \phi (t)=\frac{3{t}^{2}}{4} such that \phi (0)=0. Let, 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
Hence,
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].
We study the existence of a solution for the following twopoint boundary value problem for a secondorder differential equation:
where K:[0,1]\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+} is a continuous function. This problem is equivalent to the integral equation
where \mathcal{G}(t,s) is the Green function
Denote by X=C([0,1],{\mathbb{R}}^{+}) the set of nonnegative continuous real functions on [0,1]. We endow X with the Gmetric
for u,v,w\in X. Then (X,G) is a complete Gmetric space.
Consider the selfmap T:X\to X defined by
Clearly, {u}^{\ast} is a solution of (5.2) if and only if {u}^{\ast} is a fixed point of T.
We will prove the existence and uniqueness of the fixed point of T under the following conditions.

(A)
K(s,\cdot ) is a nonincreasing 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 xy 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:
and
We shall prove that
Let u\in {\mathcal{A}}_{1}, that is,
Since \mathcal{G}(t,s)\ge 0 for all t,s\in [0,1], we deduce from (A) and (C) that
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.
Finally, we will show that, for each u\in {\mathcal{A}}_{1} and v,w\in {\mathcal{A}}_{2}, one has
for \psi (t)=t and \phi (t)=\frac{7}{8}t.
To this end, let u\in {\mathcal{A}}_{1} and (v,w)\in {\mathcal{A}}_{2}\times {\mathcal{A}}_{2}. Therefore, by (B) we deduce that for each t\in [0,1],
It is easy to verify that
With these facts, the inequality (5.3) gives us
and we have
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). □
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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.
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Nashine, H.K., Golubović, Z. & Kadelburg, Z. Nonlinear cyclic weak contractions in Gmetric spaces and applications to boundary value problems. Fixed Point Theory Appl 2012, 227 (2012). https://doi.org/10.1186/168718122012227
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DOI: https://doi.org/10.1186/168718122012227
Keywords
 cyclic contraction
 fixed point
 Gmetric space
 weakly contractive condition
 altering distance function
 boundary value problem