- Open Access
Nonlinear cyclic weak contractions in G-metric spaces and applications to boundary value problems
© Nashine et al.; licensee Springer. 2012
Received: 20 August 2012
Accepted: 25 November 2012
Published: 17 December 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.
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 
, 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 
ψ is continuous and non-decreasing,
if and only if .
They proved the following theorem.
Theorem 1 
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 
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 
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.
holds for all .
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.
is G-convergent to x.
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.
holds for arbitrary . If this is not the case, the space is called asymmetric.
holds for all .
The following are some easy examples of G-metric spaces.
- (2)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.
is a cyclic representation of Y with respect to T;
- (II)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 .
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).
This implies that and hence . Thus, is a fixed point of T.
Hence, , that is, . Thus, we have proved the uniqueness of the fixed point. □
is a cyclic representation of Y with respect to T;
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 )
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.
Hence, all the conditions of Theorem 4 are fulfilled and it follows that T has a unique fixed point .
It is clear that is a cyclic representation of Y with respect to T. We will check that T satisfies the contraction condition (II).
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 .
for . Then is a complete G-metric space.
Clearly, is a solution of (5.2) if and only if is a fixed point of T.
- (A)is a non-increasing function for any fixed , that is,
for all and .
- (C)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 .
for all . Then we have . Similarly, the other inclusion is proved. Hence, is a cyclic representation of Y with respect to T.
for and .
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). □
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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