Abstract and Applied Analysis

Volume 2014, Article ID 325840, 5 pages

http://dx.doi.org/10.1155/2014/325840

## Two New Types of Fixed Point Theorems in Complete Metric Spaces

^{1}Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran^{2}Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan^{3}Department of Mathematics and Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania

Received 12 May 2014; Accepted 13 June 2014; Published 26 June 2014

Academic Editor: Abdul Latif

Copyright © 2014 Farshid Khojasteh 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

We introduce two new types of fixed point theorems in the collection of multivalued and single-valued mappings in complete metric spaces.

#### 1. Introduction

Let be a mapping on a complete (or compact) metric space . We do not assume richer structure such as convex metric spaces and Banach spaces. There are thousands of theorems which assure the existence of a fixed point of . We can categorize these theorems into the following four types.(T1)Leader type [1]: has a unique fixed point and converges to the fixed point for all . Such a mapping is called a* Picard operator* in [2].(T2)Unnamed type: has a unique fixed point and does not necessarily converge to the fixed point.(T3)Subrahmanyam type [3]: may have more than one fixed point and converges to a fixed point for all . Such a mapping is called a* weakly Picard operator* in [3, 4].(T4)Caristi type [5, 6]: may have more than one fixed point and does not necessarily converge to a fixed point.

We know that most of the theorems such as Banach’s [7], Ćirić’s [8], Kannan’s [9], Kirk’s [10], Matkowski’s [11], Meir and Keeler’s [12], and Suzuki’s [13, 14] belong to . Also, very recently, Suzuki [15] characterized . Subrahmanyam’s theorem [3] belongs to , and Caristi’s theorem [5, 6] and its generalizations [15–17] belong to . On the other hand, as far as the authors do know, there are no theorems belonging to ; see Kirk’s survey [18]. Also, recently many interesting fixed point theorems are proved in the framework of ordered metric spaces; see [18–35] and others.

In this paper, motivated by the above, we introduce two new types of fixed point theorems in the collection of multivalued and single-valued mappings and will prove them, which belong to .

Let be a metric space, and let denote the class of all nonempty, closed, and bounded subsets of . Let be a multivalued mapping on . A point is called a fixed point of if . Set .

A famous theorem on multivalued mappings is due to Nadler [36], which extended the Banach contraction principle to multivalued mappings. Many authors have studied the existence and uniqueness of strict fixed points for multivalued mappings in metric spaces; see, for example, [37–44] and references therein.

Let be the Hausdorff metric on induced by ; that is, Denote and , where .

#### 2. Main Results

The following is the first our main results.

Theorem 1. *Let be a complete metric space and let be a mapping from into itself. Suppose that satisfies the following condition:
**
for all . Then*(a)* has at least one fixed point ;*(b)* converges to a fixed point, for all ;*(c)*if are two distinct fixed points of , then .*

*Proof. *Let be arbitrary and choose a sequence such that . We have
Given
we have

Observe that is nonincreasing, with positive terms. So and . It follows that
Thus, it is verified that

Now for all we have
Suppose that . Since
. It means that
as . In other words, is a Cauchy sequence and so converges to .

We claim that is a fixed point.

Note that
On taking limit on both sides of (11), we have . Thus, .

If there exist two distinct fixed points , then

Therefore, and we find the desired results.

*In the following, two examples of such type of mappings, which satisfy (2), are given.*

*Example 2. *Let and let be defined by
is a complete metric space. Let be defined by
and we have
and also
Therefore, satisfies all the conditions of Theorem 1. Also, has two distinct fixed points and .

*Example 3. *Let be endowed with Euclidean metric and let be defined by
Then we claim that satisfies all the conditions of Theorem 1.

If and , we have
Thus,
Similar argument holds for the other conditions.

*Remark 4. *Note that in (2) the ratio
might be greater or less than 1 and has not introduced an upper bound. Note that if, for every , , then we have
It means that
and thus Theorem 1 is a special case of Banach contraction principle. Therefore, when is a complete metric space such that, for all , , Theorem 1 is valuable because (20) might be greater than 1. Example 2 shows this note precisely.

*The following is the second in our main results.*

*Theorem 5. Let be a complete metric space and let be a multivalued mapping from into . Let satisfy the following:
for all . Then has a fixed point .*

*Proof. *Let and . For each one can choose such that
For each we can choose such that
Specifically if
then
Therefore,
It can easily be seen that
Thus, it is easily verified that
Now for all we have
Suppose that . Since
. It means that
as . In other words, is a Cauchy sequence and so converges to . We claim that is a fixed point. Consider
On taking limit on both sides of (31) we have . It means that .

*Remark 6. *Note that Theorem 5 is a generalization of Theorem 1 because by taking and applying Theorem 5 for we obtain Theorem 1.

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*References*

- S. Leader, “Equivalent Cauchy sequences and contractive fixed points in metric spaces,”
*Studia Mathematica*, vol. 76, no. 1, pp. 63–67, 1983. View at Google Scholar · View at MathSciNet - I. A. Rus, “Picard operators and applications,”
*Scientiae Mathematicae*, vol. 58, no. 1, pp. 191–219, 2003. View at Google Scholar · View at MathSciNet - P. V. Subrahmanyam, “Remarks on some fixed point theorems related to Banach's contraction principle,”
*Journal of Mathematical and Physical Sciences*, vol. 8, pp. 445–457, 1974. View at Google Scholar - I. A. Rus, A. S. Muresan, and V. Muresan, “Weakly Picard operators on a set with two metrics,”
*Fixed Point Theory*, vol. 6, no. 2, pp. 323–331, 2005. View at Google Scholar · View at MathSciNet - J. Caristi, “Fixed point theorems for mappings satisfying inwardness conditions,”
*Transactions of the American Mathematical Society*, vol. 215, pp. 241–251, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Caristi and W. A. Kirk, “Geometric fixed point theory and inwardness conditions,” in
*The Geometry of Metric and Linear Spaces*, vol. 490 of*Lecture Notes in Mathematics*, pp. 74–83, Springer, Berlin, Germany, 1975. View at Publisher · View at Google Scholar · View at MathSciNet - S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integrales,”
*Fundamenta Mathematicae*, vol. 3, pp. 133–181, 1922. View at Google Scholar - L. B. Ćirić, “A generalization of Banach's contraction principle,”
*Proceedings of the American Mathematical Society*, vol. 45, pp. 267–273, 1974. View at Google Scholar · View at MathSciNet - R. Kannan, “Some results on fixed points—II,”
*The American Mathematical Monthly*, vol. 76, no. 4, pp. 405–408, 1969. View at Publisher · View at Google Scholar · View at MathSciNet - W. A. Kirk, “Fixed points of asymptotic contractions,”
*Journal of Mathematical Analysis and Applications*, vol. 277, no. 2, pp. 645–650, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Matkowski, “Integrable solutions of functional equations,”
*Dissertationes Mathematicae*, vol. 127, pp. 1–68, 1975. View at Google Scholar - A. Meir and E. Keeler, “A theorem on contraction mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 28, pp. 326–329, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - T. Suzuki, “Generalized distance and existence theorems in complete metric spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 253, no. 2, pp. 440–458, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - T. Suzuki, “Several fixed point theorems concerning $\tau $-distance,”
*Fixed Point Theory and Applications*, vol. 2004, no. 3, pp. 195–209, 2004. View at Publisher · View at Google Scholar · View at Scopus - T. Suzuki, “A sufficient and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point,”
*Proceedings of the American Mathematical Society*, vol. 136, no. 11, pp. 4089–4093, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. S. Bae, “Fixed point theorems for weakly contractive multivalued maps,”
*Journal of Mathematical Analysis and Applications*, vol. 284, no. 2, pp. 690–697, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. S. Bae, E. W. Cho, and S. H. Yeom, “A generalization of the Caristi-Kirk fixed point theorem and its applications to mapping theorems,”
*Journal of the Korean Mathematical Society*, vol. 31, no. 1, pp. 29–48, 1994. View at Google Scholar · View at MathSciNet - W. A. Kirk, “Contraction mappings and extensions,” in
*Handbook of Metric Fixed Point Theory*, W. A. Kirk and B. Sims, Eds., pp. 1–34, Kluwer Academic, Dordrecht, The Netherlands, 2001. View at Google Scholar · View at MathSciNet - H. Aydi, W. Shatanawi, M. Postolache, Z. Mustafa, and N. Tahat, “Theorems for Boyd-Wong-type contractions in ordered metric spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 359054, 14 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus - S. Chandok and M. Postolache, “Fixed point theorem for weakly Chatterjea-type cyclic contractions,”
*Fixed Point Theory and Applications*, vol. 2013, no. 28, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. Chandok, Z. Mustafa, and M. Postolache, “Coupled common fixed point results for mixed
*g*-monotone mapps in partially ordered*G*-metric spaces,”*University Politehnica of Bucharest Scientific Bulletin A*, vol. 75, no. 4, pp. 13–26, 2013. View at Google Scholar · View at MathSciNet - Y. Chen, “Stability of positive fixed points of nonlinear operators,”
*Positivity*, vol. 6, no. 1, pp. 47–57, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. S. Choudhury, N. Metiya, and M. Postolache, “A generalized weak contraction principle with applications to coupled coincidence point problems,”
*Fixed Point Theory and Applications*, vol. 2013, article 152, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - R. H. Haghi, M. Postolache, and S. Rezapour, “On T-stability of the Picard iteration for generalized $\phi $-contraction mappings,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 658971, 7 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - D. Ilić and V. Rakočević, “Common fixed points for maps on cone metric space,”
*Journal of Mathematical Analysis and Applications*, vol. 341, no. 2, pp. 876–882, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - J. Jachymski, “The contraction principle for mappings on a metric space with a graph,”
*Proceedings of the American Mathematical Society*, vol. 136, no. 4, pp. 1359–1373, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - V. Lakshmikantham and L. Ćirić, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 12, pp. 4341–4349, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - J. J. Nieto, R. L. Pouso, and R. Rodriguez-Lopez, “Fixed point theorems in ordered abstract spaces,”
*Proceedings of the American Mathematical Society*, vol. 135, no. 8, pp. 2505–2517, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. J. Nieto and R. Rodriguez-Lopez, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,”
*Order*, vol. 22, no. 3, pp. 223–239, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,”
*Acta Mathematica Sinica*, vol. 23, no. 12, pp. 2205–2212, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. O'Regan and A. Petruşel, “Fixed point theorems for generalized contractions in ordered metric spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 341, no. 2, pp. 1241–1252, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - H. K. Pathak and N. Shahzad, “Fixed points for generalized contractions and applications to control theory,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 68, no. 8, pp. 2181–2193, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W. Shatanawi and M. Postolache, “Common fixed point theorems for dominating and weak annihilator mappings in ordered metric spaces,”
*Fixed Point Theory and Applications*, vol. 2013, article 271, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - W. Shatanawi and M. Postolache, “Common fixed point results for mappings under nonlinear contraction of cyclic form in ordered metric spaces,”
*Fixed Point Theory and Applications*, vol. 2013, no. 60, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Wu, “New fixed point theorems and applications of mixed monotone operator,”
*Journal of Mathematical Analysis and Applications*, vol. 341, no. 2, pp. 883–893, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Nadler, “Multi-valued contraction mappings,”
*Pacific Journal of Mathematics*, vol. 30, pp. 475–488, 1969. View at Publisher · View at Google Scholar · View at MathSciNet - A. Amini-Harandi, “Endpoints of set-valued contractions in metric spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 1, pp. 132–134, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Amini-Harandi, “Fixed point theory for set-valued quasi-contraction maps in metric spaces,”
*Applied Mathematics Letters*, vol. 24, no. 11, pp. 1791–1794, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - L. B. Ciric and J. S. Ume, “Multi-valued non-self-mappings on convex metric spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 60, no. 6, pp. 1053–1063, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Fakhar, “Endpoints of set-valued asymptotic contractions in metric spaces,”
*Applied Mathematics Letters*, vol. 24, no. 4, pp. 428–431, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - N. Hussain, A. Amini-Harandi, and Y. J. Cho, “Approximate endpoints for set-valued contractions in metric spaces,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 614867, 13 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Kadelburg and S. Radenović, “Some results on set-valued contractions in abstract metric spaces,”
*Computers & Mathematics with Applications*, vol. 62, no. 1, pp. 342–350, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Moradi and F. Khojasteh, “Endpoints of multi-valued generalized weak contraction mappings,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 6, pp. 2170–2174, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Khojasteh and Rakočević V., “Some new common fixed point results for generalized contractive multi-valued non-self-mappings,”
*Applied Mathematics Letters*, vol. 25, no. 3, pp. 287–293, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus

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