Abstract and Applied Analysis

Volume 2014 (2014), Article ID 657858, 7 pages

http://dx.doi.org/10.1155/2014/657858

Research Article

ON Modified -Contractive Mappings

Marwan Amin Kutbi,¹ Muhammad Arshad,² and Aftab Hussain²

¹Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

²Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan

Received 6 May 2014; Accepted 6 July 2014; Published 21 July 2014

Academic Editor: Abdul Latif

Copyright © 2014 Marwan Amin Kutbi et al. 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.

Hussain et al. (2013) established new fixed point results in complete metric space. In this paper, we prove fixed point results of α-admissible mappings with respect to η, for modified contractive condition in complete metric space. An example is given to show the validity of our work. Our results generalize/improve several recent and classical results existing in the literature.

1. Preliminaries and Scope

The study of fixed point problems in nonlinear analysis has emerged as a powerful and very important tool in the last 60 years. Particularly, the technique of fixed point theory has been applicable to many diverse fields of sciences such as engineering, chemistry, biology, physics, and game theory. Over the years, fixed point theory has been generlized in many directions by several mathematicians (see [1–36]).

In 1973, Geraghty [12] studied different contractive conditions and established some useful fixed point theorems.

In 2012, Samet et al. [33] introduced a concept of -contractive type mappings and established various fixed point theorems for mappings in complete metric spaces. Afterwards Karapinar and Samet [10] refined the notions and obtained various fixed point results. Hussain et al. [17] extended the concept of -admissible mappings and obtained useful fixed point theorems. Subsequently, Abdeljawad [4] introduced pairs of -admissible mappings satisfying new sufficient contractive conditions different from those in [17, 33] and proved fixed point and common fixed point theorems. Lately, Salimi et al. [32] modified the concept of -contractive mappings and established fixed point results.

We define the family of nondecreasing functions such that , and for each where is the th term of .

Lemma 1 (see [32]). If , then for all .

Definition 2 (see [33]). Let be a metric space and let be a given mapping. We say that is an -contractive mapping if there exist two functions and such that

for all .

Definition 3 (see [33]). Let and . One says that is -admissible if , .

Example 4. Consider . Define and by , for all and

Then is -admissible.

Definition 5 (see [32]). Let and let be two functions. One says that is -admissible mapping with respect to if , . Note that if one takes , then this definition reduces to definition [33]. Also if we take , then one says that is an -subadmissible mapping.

2. Main Results

In this section, we prove fixed point theorems for -admissible mappings with respect to , satisfying modified ()-contractive condition in complete metric space.

Theorem 6. Let be a complete metric space and let is -admissible mappings with respect to . Assume that there exists a function such that, for any bounded sequence of positive reals, implies such that

for all where ; then suppose that one of the following holds:

(i)is continuous;(ii)if is a sequence in such that for all and as , then

If there exists such that , then has a unique fixed point.

Proof. Let and define

We will assume that for each . Otherwise, there exists an such that . Then and is a fixed point of . Since and is -admissible mapping with respect to , we have

By continuing in this way, we have

for all . From (7), we have

Thus applying the inequality (3), with and , we obtain

which implies that

We suppose that

Then we prove that . It is clear that is a decreasing sequence. Therefore, there exists some positive number such that . Now we will prove that . From (10), we have

Now by taking limit , we have

By using property of function, we have . Thus

Now we prove that sequence is Cauchy sequence. Suppose on contrary that is not a Cauchy sequence. Then there exists and sequences and such that, for all positive integers , we have ,

By the triangle inequality, we have

for all . Now taking limit as in (16) and using (14), we have

Again using triangle inequality, we have

Taking limit as and using (14) and (17), we obtain

By using (3), (17), and (19), we have

which implies that

Therefore, we have

Now taking limit as in (22), we get

Hence , which is a contradiction. Hence is a Cauchy sequence. Since is complete so there exists such that . Now we prove that . Suppose (i) holds; that is, is continuous, so we get

Thus . Now we suppose that (ii) holds. Since

for all . By the hypotheses of (ii), we have

Using the triangle inequality and (3), we have

which implies that

Letting then we have . Thus . Let there exists to be another fixed point of , s.t ;

which implies that

By the property of function, , implies ; then we have . Hence has a unique fixed point.

If in Theorem 6,   we get the following corollary.

Corollary 7 (see [17]). Let be a complete metric space and let be -admissible mapping. Assume that there exists a function such that, for any bounded sequence of positive reals, implies such that

for all , where . Suppose that either

(i) is continuous, or(ii)if is a sequence in such that for all and as , then

If there exists such that ; then has a fixed point.

If in Theorem 6, we get the following corollary.

Corollary 8. Let be a complete metric space and let be -subadmissible mapping. Assume that there exists a function such that, for any bounded sequence of positive reals, implies such that

for all where ; then suppose that one of the following holds:

(i) is continuous;(ii)if is a sequence in such that for all and as , then

If there exists such that , then has a fixed point.

Example 9. Let with usual metric for all and , and for all be defined by

We prove that Corollary 7 can be applied to . Let ; clearly and , then of -admissible mapping , and , , and imply that

If , then we have

Let and ; then

Theorem 10. Let be a complete metric space and let be -admissible mappings with respect to . Assume that there exists a function such that, for any bounded sequence of positive reals, implies such that

for all ; then suppose that one of the following holds:

(i) is continuous;(ii)if is a sequence in such that for all and as , then

If there exists such that , then has a fixed point.

Proof. Let and define

We will assume that for each . Otherwise, there exists an such that . Then and is a fixed point of . Since and is -admissible mapping with respect to , we have

By continuing in this way, we have

for all . From (43), we have

Thus applying the inequality (39), with and , we obtain

which implies that

We suppose that

Then we prove that . It is clear that is a decreasing sequence. Therefore, there exists some positive number such that . Now we will prove that . From (47), we have

Now by taking limit , we have

By using property of function, we have . Thus

Now we prove that sequence is Cauchy sequence. Suppose on contrary that is not a Cauchy sequence. Then there exists and sequences and such that, for all positive integers , we have ,

By the triangle inequality, we have

for all . Now taking limit as in (52) and using (50), we have

Again using triangle inequality, we have

Taking limit as and using (50) and (53), we obtain

By using (39), (53), and (55), we have

which implies that

Therfore, we have

Now taking limit as in (58), we get

Hence , which is a contradiction. Hence is a Cauchy sequence. Since is complete so there exists such that . Now we prove that . Suppose (i) holds; that is, is continuous, so we get

Thus . Now we suppose that (ii) holds. Since

for all . By the hypotheses of (ii), we have

Using the triangle inequality and (39), we have

which implies that

Letting , we have . Thus . Let there exists to be another fixed point of , s.t ;

implies

By the property of function, implies ; then we have . Hence has a unique fixed point.

If in Theorem 10,   we get the following corollary.

Corollary 11 (see [17]). Let be a complete metric space and let be -admissible mapping. Assume that there exists a function such that, for any bounded sequence of positive reals, implies such that

for all . Suppose that either

(i) is continuous, or(ii)if is a sequence in such that for all and as , then

If there exists such that , then has a fixed point. Our results are more general than those in [17, 32, 33] and improve several results existing in the literature.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors contributed equally and significantly to writing this paper. All authors read and approved the final paper.

Acknowledgments

Marwan Amin Kutbi gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The authors thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper.

References

M. Abbas and B. E. Rhoades, “Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type,” Fixed Point Theory and Applications, vol. 2007, Article ID 54101, 9 pages, 2007.

M. Abbas and B. E. Rhoades, “Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition,” Mathematical Communications, vol. 13, no. 2, pp. 295–301, 2008.

M. Abbas and A. R. Khan, “Common fixed points of generalized contractive hybrid pairs in symmetric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 869407, 11 pages, 2009.

T. Abdeljawad, “Meir-Keeler -contractive fixed and common fixed point theorems,” Fixed Point Theory and Applications, vol. 2013, article 19, 2013.

A. Aliouche, “A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 796–802, 2006.

M. Arshad, “Some fixed point results for -contractive multi-valued mapping in partial metric spaces,” Journal of Advanced Research in Applied Mathematics. In press

M. Arshad, A. Azam, and P. Vetro, “Some common fixed point results in cone metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 493965, 11 pages, 2009.

S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.

A. Branciari, “A fixed point theorem for mappings satisfying a general contractive condition of integral type,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 9, pp. 531–536, 2002.

E. Karapinar and B. Samet, “Generalized contractive type mappings and related fixed point theorems with applications,” Abstract and Applied Analysis, vol. 2012, Article ID 793486, 17 pages, 2012.

U. C. Gairola and A. S. Rawat, “A fixed point theorem for integral type inequality,” International Journal of Mathematical Analysis, vol. 2, no. 13–16, pp. 709–712, 2008.

M. A. Geraghty, “On contractive mappings,” Proceedings of the American Mathematical Society, vol. 40, pp. 604–608, 1973.

F. Gu and H. Ye, “Common fixed point theorems of Altman integral type mappings in -metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 630457, 13 pages, 2012.

V. Gupta and N. Mani, “A common fixed point theorem for two weakly compatible mappings satisfying a new contractive condition of integral type,” Mathematical Theory and Modeling, vol. 1, no. 1, 2011.

R. H. Haghi, S. Rezapour, and N. Shahzad, “Some fixed point generalizations are not real generalizations,” Nonlinear Analysis, vol. 74, no. 5, pp. 1799–1803, 2011.

N. Hussain, M. Arshad, and A. Shoaib, “Shoaib and Fahimuddin, Common fixed point results for --contractions on a metric space endowed with graph,” Journal of Inequalities and Applications, vol. 2014, article 136, 2014.

N. Hussain, E. Karapınar, P. Salimi, and F. Akbar, “α-admissible mappings and related fixed point theorems,” Journal of Inequalities and Applications, vol. 2013, article 114, 11 pages, 2013.

N. Hussain and M. Abbas, “Common fixed point results for two new classes of hybrid pairs in symmetric spaces,” Applied Mathematics and Computation, vol. 218, no. 2, pp. 542–547, 2011.

N. Hussain and Y. J. Cho, “Weak contractions, common fixed points, and invariant approximations,” Journal of Inequalities and Applications, vol. 2009, Article ID 390634, 10 pages, 2009.

G. Jungck and B. E. Rhoades, “Fixed points for set valued functions without continuity,” Indian Journal of Pure and Applied Mathematics, vol. 29, no. 3, pp. 227–238, 1998.

G. Jungck and N. Hussain, “Compatible maps and invariant approximations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1003–1012, 2007.

G. Jungck and B. E. Rhoades, “Fixed point theorems for occasionally weakly compatible mappings,” Fixed Point Theory, vol. 7, no. 2, pp. 287–296, 2006.

G. Jungck and B. E. Rhoades, “Erratum: “Fixed point theorems for occasionally weakly compatible mappings” [Fixed Point Theory, vol. 7 (2006), no. 2, 287–296],” Fixed Point Theory, vol. 9, no. 1, pp. 383–384, 2008.

R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp. 71–76, 1968.

S. Moradi and M. Omid, “A fixed point theorem for integral type inequality depending on another function,” Research Journal of Applied Sciences, Engineering and Technology, vol. 2, no. 3, pp. 239–2442, 2010.

S. B. Nadler, “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.

D. B. Ojha, M. K. Mishra, and U. Katoch, “A common fixed point theorem satisfying integral type for occasionally weakly compatible maps,” Research Journal of Applied Sciences, Engineering and Technology, vol. 2, no. 3, pp. 239–244, 2010.

H. K. Pathak, R. Tiwari, and M. S. Khan, “A common fixed point theorem satisfying integral type implicit relations,” Applied Mathematics E-Notes, vol. 7, pp. 222–228, 2007.

B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977.

B. E. Rhoades, “Two fixed-point theorems for mappings satisfying a general contractive condition of integral type,” International Journal of Mathematics and Mathematical Sciences, no. 63, pp. 4007–4013, 2003.

P. K. Shrivastava, N. P. S. Bawa, and S. K. Nigam, “Fixed point theorems for hybrid contractions,” Varahmihir Journal of Mathematical Sciences, vol. 2, no. 2, pp. 275–281, 2002.

P. Salimi, A. Latif, and N. Hussain, “Modified α-ψ-contractive mappings with applications,” Fixed Point Theory and Applications, vol. 2013, 19 pages, 2013.

B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for -contractive type mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2154–2165, 2012.

P. Vijayaraju, B. E. Rhoades, and R. Mohanraj, “A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type,” International Journal of Mathematics and Mathematical Sciences, no. 15, pp. 2359–2364, 2005.

Y. Li and F. Gu, “Common fixed point theorem of altman integral type mappings,” Journal of Nonlinear Science and Its Applications, vol. 2, no. 4, pp. 214–218, 2009.

X. Zhang, “Common fixed point theorems for some new generalized contractive type mappings,” Journal of Mathematical Analysis and Applications, vol. 333, no. 2, pp. 780–786, 2007.