Research Article | Open Access

W. Sintunavarat, P. Kumam, "Gregus-Type Common Fixed Point Theorems for Tangential Multivalued Mappings of Integral Type in Metric Spaces", *International Journal of Mathematics and Mathematical Sciences*, vol. 2011, Article ID 923458, 12 pages, 2011. https://doi.org/10.1155/2011/923458

# Gregus-Type Common Fixed Point Theorems for Tangential Multivalued Mappings of Integral Type in Metric Spaces

**Academic Editor:**Frank Werner

#### Abstract

The concept of tangential for single-valued mappings is extended to multivalued mappings and used to prove the existence of a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type. Consequently, several known fixed point results generalized and improved the corresponding recent result of Pathak and Shahzad (2009) and many authors.

#### 1. Introduction

The first important result on fixed points for contractive-type mappings was the well-known Banach contraction principle, published for the first time in 1922 in [1] (see also [2]). Banach contraction principle has been extended in many different directions, see [3–5], and so forth. Many authors in [3, 5–12] established fixed point theorems involving more general contractive conditions. In 1969, Nadler [13] combines the ideas of set-valued mapping and Lipschitz mapping and prove some fixed point theorems about multivalued contraction mappings. Afterward, the study of fixed points for multivalued contractions using the Hausdorff metric was initiated by Markin [14]. Later, an interesting and rich fixed point theory for such maps was developed (see [15–18]). The theory of multivalued maps has applications in optimization problems, control theory, differential equations, and economics.

Sessa [19] introduced the concept of weakly commuting maps. Jungck [20] defined the notion of compatible maps in order to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible but the converse is not true. This concept was further improved by Jungck and Rhoades [21] with the notion of weakly compatible mappings. In 2002, Aamri and Moutawakil [22] defined property (E.A). This concept was frequently used to prove existence theorems in common fixed point theory. Three years later, Liu et al. [23] introduced common property (E.A). The class of (E.A) maps contains the class of noncompatible maps. Branciari [3] studied contractive conditions of integral type, giving an integral version of the Banach contraction principle, that could be extended to more general contractive conditions. Recently, Pathak and Shahzad [24] introduced the new concept of weak tangent point and tangential property for single-valued mappings and established common fixed point theorems. Very recently, Vetro [25] obtained an interesting theorem for mappings satisfying a contractive condition of integral type which is a generalization of Branciari [3, Theorem 2].

The aim of this paper is to define a tangential property for multivalued mappings which generalize the concept of tangential property for single-valued mappings of Pathak and Shahzad [24] and prove a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type.

#### 2. Preliminary

Throughout this paper, denotes a metric space. We denote by , the class of all nonempty bounded closed subsets of . The Hausdorff metric induced by on is given by
for every , where is the distance from to . Let and . A point is a *fixed point* of (resp. ) if (resp. ). The set of all fixed points of (resp. ) is denoted by (resp. ). A point is a *coincidence point* of and if . The set of all coincidence points of and is denoted by . A point is a *common fixed point* of and if . The set of all common fixed points of and is denoted by .

*Definition 2.1. *The maps and are said to be commuting if , for all .

*Definition 2.2 (see [19]). *The maps and are said to be weakly commuting if , for all .

*Definition 2.3 (see [20]). *The maps and are said to be compatible if whenever is a sequence in such that , for some .

*Definition 2.4 (see [26]). *The maps and are said to be weakly compatible , for all .

*Definition 2.5 (see [22]). *Let and . The pair satisfies property (E.A) if there exist the sequence in such that

See example of property (E.A) in Kamran [27, 28] and Sintunavarat and Kumam [11].

*Definition 2.6 (see [23]). *Let . The pair and satisfy a common property (E.A) if there exist sequences and in such that

*Remark 2.7. *If , , and in (2.3), then we get the definition of property (E.A).

*Definition 2.8 (see [24]). *Let . A point is said to be a weak tangent point to if there exist sequences and in such that

*Remark 2.9. *If in (2.4), we get the definition of property (E.A).

*Definition 2.10 (see [24]). *Let . The pair is called tangential with respect to the pair if there exist sequences and in such that

#### 3. Main Results

In this section, we first introduce the notion of tangential property for two single-valued and two multivalued mappings. Throughout this section, denotes the set of nonnegative real numbers.

*Definition 3.1. *Let and . The pair is called tangential with respect to the pair if
whenever sequences and in such that
for some .

*Example 3.2. *Let be a metric space with usual metric . Let and be mappings defined by , , , and , for all . Clearly, there exists two sequences and such that
whenever
So, the pair is tangential with respect to the pair .

*Definition 3.3. *Let and . The mapping is called tangential with respect to the mapping if
whenever sequences and in such that
for some .

*Example 3.4. *Let be a metric space with usual metric . Let and be mappings defined by
Clearly, there exist two sequences and such that
whenever
So, the mapping is tangential with respect to the mapping .

Now, we state and prove our main result.

Theorem 3.5. *Let and satisfy
**
for all for which the right-hand side of (3.10) is positive, where , and is a Lebesgue integrable mapping which is a summable nonnegative and such that
**
for each . If the following conditions (a)–(d) hold: *(a)*there exists a point which is a weak tangent point to , *(b)* is tangential with respect to , *(c)*, , and for and , *(d)*the pairs and are weakly compatible. **Then, , , , and have a common fixed point in .*

* Proof. *Since , for some . It follows from a point which is a weak tangent point to that there exist sequences and in such that
Because the pair is tangential with respect to the pair , we get
for some . Since and (3.12) and (3.13) are true, we have

We claim that . If not, then condition (3.10) implies
Letting , we get
Since
which is a contradiction, then .

Again, we claim that . If not, then condition (3.10) implies
Letting , we get
Since
which is a contradiction, then .

Now, we conclude and . It follows from , that , , and . Hence, , and .

Since the pair is weakly compatible, . Thus . Similarly, we can prove that . Consequently, . Therefore the maps , and have a common fixed point.

If in Theorem 3.5, we get the following corollary.

Corollary 3.6. *Let and satisfy
**
for all for which the right-hand side of (3.21) is positive, where and is a Lebesgue integrable mapping which is a summable nonnegative and such that
**
for each . If the following conditions (a)–(d) hold: *(a)*there exists a point which is a weak tangent point to , *(b)* is tangential with respect to , *(c)*, and for and , *(d)*the pairs and are weakly compatible. ** Then, , , , and have a common fixed point in .*

If , , and in Theorem 3.5, we get the following corollary.

Corollary 3.7. *Let and satisfy
**
for all for which the right-hand side of (3.23) is positive, where and is a Lebesgue integrable mapping which is a summable nonnegative and such that
**
for each . If the following conditions (a)–(d) hold: *(a)*there exists sequence in such that , *(b)* is tangential with respect to , *(c)* for , *(d)*the pair is weakly compatible. **Then, and have a common fixed point in .*

If in Theorem 3.5, we get the following corollary.

Corollary 3.8. *Let and satisfy
**
for all for which the right-hand side of (3.25) is positive, where and . If the following conditions (a)–(d) holds: *(a)*there exists a point which is a weak tangent point to , *(b)* is tangential with respect to , *(c)*, and for and , *(d)*the pairs and are weakly compatible. **Then, , , , and have a common fixed point in .*

If and in Theorem 3.5, we get the following corollary.

Corollary 3.9. *Let and satisfy
**
for all for which the right-hand side of (3.26) is positive, where . If the following conditions (a)–(d) hold: *(a)*there exists a point which is a weak tangent point to , *(b)* is tangential with respect to , *(c)*, and for and , *(d)*the pairs and are weakly compatible. **Then, , , , and have a common fixed point in .*

If , , and in Theorem 3.5, we get the following corollary.

Corollary 3.10. *Let and satisfy
**
for all for which the right-hand side of (3.27) is positive, where . If the following conditions (a)–(d) holds: *(a)*there exists sequence in such that , *(b)* is tangential with respect to , *(c)* for , *(d)*the pairs is weakly compatible. **Then, and have a common fixed point in .*

#### Acknowledgments

W. Sintunavarat would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the Faculty of Science, KMUTT for financial support during the preparation of this manuscript for Ph.D. Program at KMUTT. Moreover, the authors also would like to thank the National Research University Project of Thailand's Office of the Higher Education Commission for financial support (under the CSEC project no. 54000267). Finally, the authors would like to thank Professor Frank Werner for your help and encouragement. Special thanking are also due to the reviewers, who have made a number of valuable comments and suggestions which have improved the manuscript greatly.

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#### Copyright

Copyright © 2011 W. Sintunavarat and P. Kumam. 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.