1. Introduction

Fixed point theory has got wide applications in different branches of mathematics. Since the work of ^[3] known as the Banach Contraction Principle, many mathematicians have extended and generalized the results in ^[3]. Some of the classical generalizations of ^[3] are presented in ^[14]. The concept of a b-metric space as a generalization of a metric space is introduced in ^[2] and a contraction mapping theorem is proved there. Since then results about fixed points, variational principles and applications were obtained in b-metric spaces. We will cite just a few recent results in these directions ^{[1, 5, 7, 8, 9, 10, 11, 12, 13, 16]}.

We recall some definitions and properties for b-metric spaces ^{[12, 13, 16]}.

Definition 1.1. Let be a non-empty set, . A functional is called a b-metric if it satisfies the following conditions:

for all and iff ;

for all ;

for all .

The ordered pair is called a b-metric space (with constant s).

Any metric space is a b-metric space with .

An example of b-metric is the functional . It is easy to see that in this case .

Other classical example of b-metric space is endowed with the b-metric function for . It is easy to see that in this case and for we get the metric space of the real numbers with a metric .

Definition 1.2. Let be a b-metric space.

A sequence is called b-convergent if there exists , such that for any there exists such that the inequality holds true for all ;

A sequence is called b-Cauchy sequence if for any there exists such that the inequality holds true for all ;

The b-metric space is called complete b-metric space if any Cauchy sequence is convergent;

A subset is called b-bounded if ;

If the set is b-bounded then the number is called its b-diameter and is denoted with .

A subset is called b-closed if for any convergent sequence the convergence implies .

A b-metric function is called continuous if for any and any there exists such that there holds the inequality , provided that It is easy to observe that if is continuous and is b-convergent to then

Every b-convergent sequence in b-metric space is a b-Cauchy sequence. If a sequence is a b-convergent in b-metric space then its limit is unique. In general a b-metric function is not continuous ^{[5, 10]}.

As far as we will consider only b-metrics we will omit the letter b in the above definitions.

Definition 1.3. (^[14]) Let be a metric space. A map is a Hardy Rogers map is there exist nonnegative constants , satisfying such that for each the inequality

holds for all .

As pointed in ^[15] from the symmetry of the function it follows that and . Therefore if is a Hardy-Rogers contraction then there exist , such that and there holds the inequality

Generalizations of Hardy Rogers map in b-metric space are investigated in ^{[8, 13]}.

If and in the above inequality we get a generalization of Chatterjea’s map ^[6] in b-metric space.

Definition 1.4. Let be a b-metric space. A map is called Chatterjea’s map if there exists such that the inequality

holds for all .

We will denote for the rest of the article , where is the constant from the definition of Chatterjea’s map. From it follows that .

2. Fixed Points for Chatterjea’s Maps in b-Metric Spaces

Theorem 2.1. Let be a complete b-metric space, be a continuous function, be a Chatterjea’s map, such that the inequality holds for any . Then

(i) there exists a unique fixed point say of ;

(ii) for any the sequence converges to , where , ;

(iii) there holds the a priori error estimate

(2.1)

Lemma 2.2. Let be a b-metric space and let be a Chatterjea’s map. Then for any there holds the inequality

(2.2)

for any .

Proof. Let us denote and . We consider the sequence

(2.3)

We will prove inequality (2.2) by induction on the sequence (2.3). Let us denote by the sum of the indices of the sequence in (2.3).

Let , i.e. and . Then .

Let, i.e. and . Then

Let inequality (2.2) holds for .

We will prove that (2.2) holds true for . Let . There are two cases: If then we consider , if then we consider .

Case I) There are two subcases: and . Let first . Then

Let now . Then

Case II)

Proof. of Theorem 2.1 (i) Let be arbitrary.

Let us put . From Lemma 2.2 we have that the inequality

holds for every . Consequently the sequence is a Cauchy sequence. From the assumption that is complete b-metric space it follows that the sequence is b-convergent. Therefore it follows that there exists . Let us fix . After taking a limit on from the assumption that the b-metric is continuous and using that is Chatterjea’s map we get the inequality

and therefore i.e. is a fixed point for . Let suppose that there are two fixed points . Then from the inequality

and the assumption that it follows that .

(ii) The proof follows from (i), because any sequence is convergent to the fixed point of , which is unique.

(iii) Let be arbitrary. From Lemma 2.2 we have the inequality

holds for every and every . From (ii) it follows that the sequence converges to the unique fixed point. Therefore using the continuity of and Lemma 2.2 we get

As far as any metric space is a b-metric space, then Theorem 2.1 holds true for arbitrary metric space. If is a complete metric space and be Chatterjea’s map then the a priori error estimate is well known ^[4]

(2.4)

If we assume that then we will get from Theorem 2.1 the a priori estimate

(2.5)

Let us mention that in this case the a priori estimate (2.5) is better, than (2.4).

Let , be the smallest number, that satisfies (2.5) and be the smallest number, that satisfies (2.4). Then

If gets close to then gets closer to 1 and therefore gets closer to infinity.

We would like to point out that if the space is a metric space than using the triangle inequality we can obtain (2.5) from (2.1).

Example 2.3. Let us consider the b-metric space for . Let be two arbitrary positive real numbers. Let us define the map , by (Figure 1), which is a variation of the classical examples from ^[14]. It is well known that is Chatterjea’s map and is not Chatterjea’s map in the metric space ^[14]. It is easy to observe that the Picard iteration sequence converges to the fixed point for any initial point .

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Figure 1

If or , then satisfies the condition in Definition 1.4 for any , because . If and , then we get and . Using the inequality

we get that there holds

(2.6)

for any . Therefore if then is not a Chatterjea’s map in . For any arbitrary we can choose , such that . Consequently for any map we can endow with a suitable -metric so that to satisfy the condition in Definition 1.4 in .

Let us consider the particular case and . If we choose in this case , provided that we have considered the b-metric space , , then , because in . Consequently does not satisfy the conditions in (^[16] Theorem 3) for any in and thus Theorem 2.1 extends (^[12] Theorem 3) in the case when .

In the particular case we get that , provided that is chosen so that inequality (2.6) to hold in and therefore (^[12] Theorem 3) could not be applied.

When applying fixed point theorems for approximating of a solution of the equation we usually find an initial starting point , which belongs to a neighborhood of the solution , such that and is bounded and closed. Thus the next Corollary can be applied in a wide class of problems.

Corollary 2.3. Let be a complete b-metric space, be a continuous function, be a b-bounded and b-closed set, be Chatterjea’s map. Then

there exists a unique fixed point say of ;

for any the sequence converges to , where , ;

there holds the a priori error estimate .

Acknowledgement

We would like to thank the anonymous reviewer for the valuable suggestions that have improved the article.

The second author is partially suppored by Plovdiv University “Paisii Hilendarski” NPD Project NI 15 – FMI – 004.

References

[1]	Allahyari, R., Arab, R., Haghighi A.S., “A generalization on weak contractions in partially ordered b-metric spaces and its application to quadratic integral equations”, Journal of Inequalities and Applications, 2014 (355), 2014.
	In article
[2]	Bakhtin, I.A., “The contraction mapping principle in almost metric spaces”, Funct. Anal., Gos. Ped. Inst., Unianowsk, 30, 26-37, 1989.
	In article
[3]	Banach, S., “Sur les opérations dans les ensembles abstraits et leur application aux équations integrals”, Fund. Math., 3, 133-181, 1922.
	In article
[4]	Berinde, V., Iterative approximation on fixed points, Springer, Berlin, 2007.
	In article
[5]	Bota, M., Molnár, A., Varga, C., “On Ekeland’s variationals principle in b-metric spaces”, Fixed point theory, 12 (2), 21-28, 2011.
	In article
[6]	Chatterjea, S., “Fixed point theorems”, C. R. Acad. Bulgare Sci. 25, 727-730, 1972.
	In article
[7]	Czrezvik, S., “Contraction mappings in b-metric spaces”, Acta Mathematica et Informatica Universitatis Ostravlensis, 1 (1), 5-11, 1993.
	In article
[8]	Dubey, A.K., Shukla, R., Dubey, R.P., “Some fixed point results in b-metric spaces”, Asian journal of mathematics and applications, 2014, Article ID ama0147, 6 pages, 2014.
	In article
[9]	George, R., Fisher, B., “Some generalized results of fixed points in cone b-metric spaces”, Mathematica Moravica, 17 (2), 39-50, 2013.
	In article
[10]	Farkas, C., Molnár, A.E., Nagy, S., “A generalized variational principle in b-metric spaces”, Le Matematiche, LXIX, 205-221, 2014.
	In article
[11]	Isufati, A., “Rational Contractions in b-Metric Spaces”, Journal of Advances in Mathematics, 5 (3), 803-811, Jan, 2014.
	In article
[12]	Mehmet K., Hükmi K., “On Some Well Known Fixed Point Theorems in b-Metric Spaces”, Turkish Journal of Analysis and Number Theory, 1 (1), 13-16, 2013.
	In article
[13]	Mishra, P.K., Sachdeva, S., Banerjee, S.K., “Some fixed point theorems in b-metric space”,Turkish Journal of Analysus and Number Theory, 2 (1), 19-22, 2014.
	In article
[14]	Rhoades, B.E., “A comparison of various definitions of contractive mappings”, Transactions of the American Mathematical Society, 226, 257-290, 1977.
	In article
[15]	Rhoades, B.E., Sessa, S., Khan, M.S., Khan, M.D., “Some fixed point theorems for Hardy-Rogers type mappings, Internat. J. Math. & Math. Sci., 7(1), 75-87, 1984.
	In article
[16]	Sintunavarat, W., Plubtieng, S., Katchang, P., “Fixed point result and applications on b-metric space endowed with an arbitrary binary relation”, Fixed Point Theory Appl., Article ID 296, 2013.
	In article