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Journal of Mathematics
Volume 2013 (2013), Article ID 967045, 5 pages
http://dx.doi.org/10.1155/2013/967045
Research Article

Weak Contraction Condition Involving Cubic Terms of under the Fixed Point Consideration

Department of Pure and Applied Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur, Chhattisgrah 495009, India

Received 31 January 2013; Accepted 14 April 2013

Academic Editor: Krassimir T. Atanassov

Copyright © 2013 Penumurthy Parvateesam Murthy and K. N. V. V. Vara Prasad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A fixed point theorem is presented for single-valued map with using generalized -weak contractive condition involving various combinations of on a complete metric space. Our result is an extension as well as a generalization of Alber and Guerre-Delabriere (1997) in particular. It also generalizes the results of Rhoades (2001), Choudhury and Dutta, (2000), and Dutta and Choudhury, (2008).

1. Introduction

Let be a metric space. A map is a contraction if, for each , there exists a constant such that .

A map is a -weak contraction if, for each , there exists a function , for all and such that .

In [1] Alber and Guerre-Delabriere introduced the concept of weak contraction in Hilbert spaces. Rhoades [2] has shown that the result which Alber and Guerre-Delabriere had proved in [2] is also valid in complete metric spaces.

In this paper we introduced generalized -weak contractive condition involving various combinations of . Our result is an extension as well as a generalization of Alber and Guerre-Delabriere [1] and Rhoades [2] in particular. It also generalizes the results of [3, 4].

Now, We state the result of Rhoades as follows.

Theorem 1 (see [2, Theorem 2]). Let be a complete metric space, and let be a -weak contraction on which is a continuous and nondecreasing function with for all and then has a unique fixed point in .
If one takes , where , then weak contraction reduces to contraction mapping.
In this paper, a new type of inequality is introduced with cubic terms involving called a “generalized -weak contractive condition with cubic terms involving ”.
Let be a metric space and a self-map of satisfying the following condition: where is a real number is and is a continuous function with and for each .

2. Main Result

Lemma 2. Let be a self map of a metric space satisfying (1). For any sequence in defined by , . Then the sequence is Cauchy in .

Proof. Let be an arbitrary point. Constructing the sequence follows If for some , then trivially has a fixed point. We assume , for all . We write .
First, we prove that is a nonincreasing sequence and converges to .
Case 1. If is even, taking and in (1), we get where
By using (3), we get where Now consider ; then we have where ,  ,.
By triangular inequality and using property of , we get Then
If , then (8) reduces to , a contradiction. Thus implies that .
In a similar way, if is odd, we can obtain .
It follows that the sequence is decreasing.
Let , for some .
Suppose ; then from inequality (1), we have where By using (3), we get where
Using triangular inequality and property of and taking limits , we get Then , since is positive; then by the property of , we get . We conclude that
Now we show that is a Cauchy sequence. Suppose, we assume that is not a Cauchy sequence; then there is for which we can find two sequences of positive integers and such that for all positive integers ,
Now, Letting , we get Now, from the triangular inequality, we have
Taking limits as and using (16) and (19), we have
Again from the triangular inequality, we have
Taking limits as and using (16) and (19), we have Again by using triangular inequality, we have Taking limit in the above inequality and using (16) and (19), we have
Again putting and in (1), we get where Using (3), then we obtain where Letting and using (16)–(25), we get a contradiction. Thus is a Cauchy in .

Theorem 3. Let be a self-map of a complete metric space satisfying (1). Then has a unique fixed point in .

Proof. From Lemma 2 the sequence is a Cauchy in . Since is a complete metric space, then there exists a point such that
Now we prove that is a fixed point of .
Taking and in (1), we have where
Using (31) and (3), we get
Hence .
Then has a fixed point in .
To prove the uniqueness of the fixed point, we assume that and are two fixed points of . Taking and in (1), we easily get wich implies that . Therefore has a unique fixed point in .

Corollary 4. Let be a self-map of a complete metric space satisfying the condition where for all and is a continuous function with and for each . Then has a unique fixed point in .

Proof. in Theorem 3; we have the result.

Now we give an example to support our result.

Example 5. Let , and let be the usual metric on . Let be defined by and . And define by . For any value of and , then it is easy to verify that inequality (1) holds. Hence Theorem 3 holds well.

Acknowledgment

Penumurthy Parvateesam Murthy is thankful to the University Grants Commission, New Delhi, India, for financial assistance through Major Reserch Project File no. 42-32/2013 (SR).

References

  1. Y. I. Alber and S. Guerre-Delabriere, “Principle of weakly contractive maps in Hilbert spaces,” in New Results in Operator Theory and its Applications, I. Gohberg and Y. Lybich, Eds., vol. 98 of Operator Theory: Advances and Applications, pp. 7–22, Birkhäuser, Basel, Switzerland, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. B. E. Rhoades, “Some theorems on weakly contractive maps,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 47, no. 4, pp. 2683–2693, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  3. B. S. Choudhury and P. N. Dutta, “A unified fixed point result in metric spaces involving a two variable function,” Filomat, no. 14, pp. 43–48, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. P. N. Dutta and B. S. Choudhury, “A generalisation of contraction principle in metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 406368, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet