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Nonlinear cyclic weak contractions in G-metric spaces and applications to boundary value problems
Fixed Point Theory and Applications volume 2012, Article number: 227 (2012)
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.
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.
In 1922, Banach’s contraction mapping principle appeared  and it was known for its simple and elegant proof by using the Picard iteration. If is a complete metric space and is a contraction, that is, there exists 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.,  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.  and turned the area of investigation of fixed points by introducing cyclic representations and cyclic contractions, which can be stated as follows.
Definition 1 
Let be a metric space. Let p be a positive integer, be subsets of X, and . Then Y is said to be a cyclic representation of Y with respect to T if
, are nonempty closed sets, and
, …, , .
Moreover, T is called a cyclic contraction if there exists such that for all and , with .
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 , of the Boyd and Wong fixed point theorem  and of the Caristi fixed point theorem . Following the paper , 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.  introduced the concept of an altering distance function.
Definition 2 
A function is called an altering distance function if the following properties are satisfied:
ψ is continuous and non-decreasing,
if and only if .
They proved the following theorem.
Theorem 1 
Let be a complete metric space, let φ be an altering distance function, and let be a self-mapping which satisfies the following inequality:
for all and for some . Then T has a unique fixed point.
Putting in the previous theorem, (1.2) reduces to (1.1).
Rhoades  extended Banach’s principle by introducing weakly contractive mappings in complete metric spaces.
Definition 3 
Let be a metric space. A mapping is called weakly contractive if
where φ is an altering distance function.
He proved the following result.
Theorem 2 
Let be a complete metric space. If is a weakly contractive mapping, then T has a unique fixed point.
If one takes , where , then (1.3) reduces to (1.1).
Dutta and Choudhury obtained in  the following generalization of Theorems 1 and 2.
Theorem 3 
Let be a complete metric space and let satisfy
for all , where ψ and φ are altering distance functions. Then T has a unique fixed point.
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.
For more details on the following definitions and results, we refer the reader to .
Definition 4 Let X be a nonempty set and let be a function satisfying the following properties:
(G1) if ;
(G2) for all with ;
(G3) for all with ;
(G4) (symmetry in all three variables);
(G5) for all (rectangle inequality).
Then the function G is called a G-metric on X and the pair is called a G-metric space.
Note that it can be easily deduced from (G4) and (G5) that
holds for all .
Definition 5 Let be a G-metric space and let be a sequence of points in X.
A point is said to be the limit of the sequence if , and one says that the sequence is G-convergent to x.
The sequence is said to be a G-Cauchy sequence if for every , there is a positive integer N such that for all ; that is, if as .
is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in is G-convergent in X.
Thus, if in a G-metric space , then for any , there exists a positive integer N such that for all . It was shown in  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 be a G-metric space, be a sequence in X and . Then the following are equivalent:
is G-convergent to x.
Lemma 2 If is a G-metric space, then the following are equivalent:
The sequence is G-Cauchy.
For every , there exists a positive integer N such that for all .
Lemma 3 Let , be two G-metric spaces. Then a function is G-continuous at a point if and only if it is G-sequentially continuous at x, that is, if is -convergent to fx whenever is G-convergent to x.
Definition 6 A G-metric space is said to be symmetric if
holds for arbitrary . If this is not the case, the space is called asymmetric.
To every G-metric on the set X, a standard metric can be associated by
If G is symmetric, then obviously , but in the case of an asymmetric G-metric, only
holds for all .
The following are some easy examples of G-metric spaces.
Example 1 (1) Let be an ordinary metric space. Define by
for all . Then it is clear that is a symmetric G-metric space.
Let . Define
and extend G to by using the symmetry in the variables. Then it is clear that 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 be a G-metric space. Let p be a positive integer, be nonempty subsets of X and . An operator is a generalized weakly cyclic contraction if
is a cyclic representation of Y with respect to T;
for any , (with ),(3.1)
is an altering distance function and is a continuous function with if and only if .
Our main result is the following.
Theorem 4 Let be a complete G-metric space, , be nonempty closed subsets of X and . Suppose is a generalized weakly cyclic contraction. Then T has a unique fixed point. Moreover, the fixed point of T belongs to .
Proof Let (such a point exists since ). Define the sequence in X by
Without loss of the generality, we can assume that
We shall prove that
By the assumption, for all n. From the condition (I), we observe that for all n, there exists such that . Putting and in the condition (II), we have
By (G5) we have
From (3.1) we have
We claim that
for all . Suppose this is not true, that is, there exists an such that . By the inequality (3.6), for these elements, we have
This implies that and by the property of φ, we have , which contradicts the condition (3.4). Therefore, (3.7) is true and so the sequence is non-increasing and bounded below. Thus, there exists such that
Now suppose that . Taking in (3.6), then using (3.8) and the continuity of ψ and φ, we obtain
Therefore, and hence . Thus,
Next, we show that is a G-Cauchy sequence in X. Suppose the contrary, that is, is not G-Cauchy. Then there exists for which we can find two subsequences and of such that is the smallest index for which
This means that
From (3.9), (3.10) and (G5), we get
Passing to the limit as and using (3.5), we get
On the other hand, for all k, there exists such that . Then (for k large enough, ) and lie in different adjacently labeled sets and for certain . Using (2.1) and (G5), we have
as (from (3.5)), which, by (3.11), implies that
Using (3.5), we have
Again, using (2.1), we get
Passing to the limit as in the above inequality, and using (3.13) and (3.12), we get
Similarly, we have
Passing to the limit as and using (3.5) and (3.12), we obtain
Similarly, we have
Now, from the definitions of and and from the obtained limits, we have
Putting , , in (II), we obtain
Passing to the limit as , utilizing (3.14) and the obtained limits, we get
which is a contradiction if . We have proved that is a G-Cauchy sequence in . Since is G-complete, there exists such that
We shall prove that
From condition (I), and since , we have . Since is closed, from (3.15) we get that . Again, from the condition (I) we have . Since is closed, from (3.15) we get that . Continuing this process, we obtain (3.16).
Now, we shall prove that is a fixed point of T. Indeed, from (3.16), since for all n there exists such that , applying (II) with and , we obtain
Passing to the limit as 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 and hence . Thus, is a fixed point of T.
Finally, we prove that is the unique fixed point of T. Assume that is another fixed point of T, that is, . By the condition (I), this implies that . Then we can apply (II) for and . We obtain
Since and are fixed points of T, we have from (3.20)-(3.22)
Hence, , that is, . 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 and are replaced, respectively, by
and the condition (3.1) by
Corollary 1 Let be a complete G-metric space, , be nonempty closed subsets of X, and such that
is a cyclic representation of Y with respect to T;
(II′) for any , (with ),
where is an altering distance function and is a continuous function with if and only if .
Then T has a unique fixed point. Moreover, the fixed point of T belongs to .
In this section, we furnish some examples to demonstrate the validity of the hypotheses of Theorem 4.
Example 2 (inspired by )
Define a G-metric G on the set by
with symmetry in all variables. Note that G is asymmetric since whenever . Let and and consider the mapping given by and . Obviously, is a cyclic representation of X with respect to T.
Take and and denote , . The following table shows that the contractive condition (3.1) is fulfilled whenever or .
Hence, all the conditions of Theorem 4 are fulfilled and it follows that T has a unique fixed point .
Example 3 Let and let be given as
where . It is easy to see that is a complete G-metric space. Consider the following closed subsets of X:
and the mapping given by
It is clear that is a cyclic representation of Y with respect to T. We will check that T satisfies the contraction condition (II).
Take (which is an altering distance function) and a continuous function such that . Let, e.g., (the other two cases are treated analogously), and let , , . Without loss of generality, we can assume that . Then , , and
Thus, the conditions of Theorem 4 are fulfilled and T has a unique fixed point .
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 .
We study the existence of a solution for the following two-point boundary value problem for a second-order differential equation:
where is a continuous function. This problem is equivalent to the integral equation
where is the Green function
Denote by the set of non-negative continuous real functions on . We endow X with the G-metric
for . Then is a complete G-metric space.
Consider the self-map defined by
Clearly, is a solution of (5.2) if and only if is a fixed point of T.
We will prove the existence and uniqueness of the fixed point of T under the following conditions.
is a non-increasing function for any fixed , that is,
for all and .
There exist such that for and that
Theorem 5 Under the conditions (A)-(C), equation (5.2) has a unique solution and it belongs to .
Proof In order to prove the existence of a (unique) fixed point of T, we construct closed subsets and of X as follows:
We shall prove that
Let , that is,
Since for all , we deduce from (A) and (C) that
for all . Then we have . Similarly, the other inclusion is proved. Hence, is a cyclic representation of Y with respect to T.
Finally, we will show that, for each and , one has
for and .
To this end, let and . Therefore, by (B) we deduce that for each ,
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 . Thus, T satisfies the contractive condition of Corollary 1.
Consequently, by Corollary 1, T has a unique fixed point , that is, 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.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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Nashine, H.K., Golubović, Z. & Kadelburg, Z. Nonlinear cyclic weak contractions in G-metric spaces and applications to boundary value problems. Fixed Point Theory Appl 2012, 227 (2012). https://doi.org/10.1186/1687-1812-2012-227
- cyclic contraction
- fixed point
- G-metric space
- weakly contractive condition
- altering distance function
- boundary value problem