#### Abstract

The purpose of this paper is to prove end-point theorems for multivalued mappings satisfying comparatively a more general contractive condition in ordered complete metric spaces. Afterwards, we extend the results of previous sections and prove common end-point results for a pair of -weakly isotone increasing multivalued mappings in the underlying spaces. Finally, we present common end point for a pair of -weakly isotone increasing multivalued mappings satisfying weakly contractive condition.

#### 1. Introduction and Preliminaries

Fixed-point theory for multivalued mappings was originally initiated by Von Neumann in the study of game theory. Fixed-point theorem for multivalued mappings is quite useful in control theory and has been frequently used in solving the problem of economics and game theory.

The theory of multivalued nonexpansive mappings is comparatively complicated as compare to the corresponding theory of single-valued nonexpansive mappings. It is therefore natural to expect that the theory of noncontinuous nonself-multivalued mappings would be much more complicated.

The study of fixed-points for multivalued contraction mappings was equally an active topic as single-valued mappings. The development of geometric fixed-point theory for multivalued was initiated with the work of Nadler Jr. [1] in the year 1969. He used the concept of Hausdorff metric to establish the multivalued contraction principle containing the Banach contraction principle as a special case, as following.

Theorem 1.1. *Let be a complete metric space and is a mapping from into such that for all ,
**
where . Then has a fixed-point. *

Since then, this discipline has been further developed, and many profound concepts and results have been established; for example, the work of Border [2], Ä†iriÄ‡ [3], Corley [4], Itoh and Takahashi [5], Mizoguchi and Takahashi [6], PetruĹźel and Luca [7], Rhoades [8], Tarafdar and Yuan [9], and references cited therein.

Let be a metric space. We denote the class of nonempty and bounded subsets of by . For , , functions , and are defined as follows: If , then we write and . Also in addition, if , then and . Obviously, .

For all , the definition of yields the following:

A point is called a fixed-point of a multivalued mapping if . If there exists a point such that , then is called an end-point of [10].

*Definition 1.2. *Let be a nonempty set. Then is called an ordered metric space if and only if:(i) is a metric space,(ii) is a partially ordered set.

*Definition 1.3. *Let be a partial ordered set. Then are called comparable if or holds.

*Definition 1.4 (see [11]). *Let and be two nonempty subsets of a partially ordered set . The relation between and is denoted and defined as follows:

*Definition 1.5 (see [12]). *A function is called an altering distance function if the following properties are satisfied: (i) is monotone increasing and continuous,(ii) if and only if .

On the other hand, fixed-point theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [13, Theoremâ€‰â€‰2.1] who presented its applications to matrix equations. Subsequently, Nieto and RodrĂguez-LĂłpez [14] extended the result of Ran and Reurings for nondecreasing mappings and applied it to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Thereafter, several authors obtained many fixed-point theorems in ordered metric spaces. For detail see [14â€“28] and references cited therein. Beg and Butt [11, 17, 29] worked on set-valued mappings and proved common fixed-point for mapping satisfying implicit relation in partially ordered metric space. Recently, Choudhury and Metiya [30] proved fixed-point theorems for multivalued mappings in the framework of a partially ordered metric space.

The results of this paper are divided in three sections. In the first section we establish the existence of end-points for a multivalued mapping under a more general contractive condition in partially ordered metric spaces. The consequences of the main theorem are also given. The second section is devoted for common end-point results for a pair of weakly isotone increasing multivalued mappings. In the third section, we present common end-point results for a pair of weakly isotone increasing multivalued mappings satisfying weakly contractive condition.

#### 2. End-Point Theorems for a Multivalued Mapping

In this section, we prove end-point theorems for a multivalued mapping in ordered complete metric space.

Theorem 2.1. *Let be an ordered complete metric space. Let be such that the following conditions are satisfied: *(i)*there exists such that , *(ii)*for , implies ,*(iii)*for all comparable , where , and is an altering distance function and
**
If the property
**
holds, then has a end-point. *

*Proof. *By the assumption (i), there exists such that . By the assumption (ii), . Then there exists such that . Continuing this process we construct a monotone increasing sequence in such that , for all . Thus we have
If for some , then the proof is finished. So assume for all .

Using the monotone property of and the condition (iii), we have for all ,
Since max, it follows that
Suppose that , for some positive integer .

Then from (2.6), we have
it implies that , or that , contradicting our assumption that , for each .

Therefore, , for all and is a monotone decreasing sequence of nonnegative real numbers. Hence there exists an such that
Taking the limit as in (2.6) and using the continuity of , we have , which is a contradiction unless . Hence
Next we show that is a Cauchy sequence. If otherwise, there exists an for which we can find two sequences of positive integers and such that for all positive integers , and .

Assuming that is the smallest such positive integer, we get ,
Now,
that is,
Taking the limit as in the above inequality and using (2.9), we have
Again,
Taking the limit as in the above inequalities and using (2.9) and (2.13), we have
Again,
Letting in the above inequalities and using (2.9) and (2.13), we have
Similarly, we have that
For each positive integer , and are comparable. Then using the monotone property of and the condition (iii), we have
Letting in above inequality, using (2.9), (2.13), (2.15), (2.17), and (2.18) and the continuity of , we have
which is a contradiction by virtue of a property of .

Hence is a Cauchy sequence. From the completeness of , there exists a such that
By the assumption (2.3), , for all .

Then by the monotone property of and the condition (iii), we have
Taking the limit as in the above inequality, using (2.9) and (2.21) and the continuity of , we have
which implies that , or that . Moreover, is a end-point of .

Taking an identity function in Theorem 2.1, we have the following result.

Corollary 2.2. *Let be an ordered complete metric space. Let be such that the following conditions are satisfied: *(i)*there exists such that , *(ii)*for , implies , *(iii)*for all comparable , where , and
**
If the property
**
holds, then has a end-point. *

The following corollary is a special case of Theorem 2.1 when is a single-valued mapping.

Corollary 2.3. *Let be an ordered complete metric space. Let be such that the following conditions are satisfied: *(i)*there exists such that , *(ii)*for , implies , *(iii)*for all comparable , where , and is an altering distance function and
**
If the property (2.3) holds, then has a fixed-point. *

In the following theorem we replace condition (2.3) of the above corollary by requiring to be continuous.

Theorem 2.4. *Let be an ordered complete metric space. Let be a continuous mapping such that the following conditions are satisfied: *(i)*there exists such that , *(ii)*for , implies ,*(iii)*for all comparable , where , and is an altering distance function and
**
Then has a end-point. *

*Proof. *If we assume as a multivalued mapping in which is a singleton set for every . Then we consider the same sequence as in the proof of Theorem 2.1. Follows the line of proof of Theorem 2.1, we have that is a Cauchy sequence and

Then, the continuity of implies that
and this proves that is a end-point of .

#### 3. Common End-Point Theorems for a Pair of Multivalued Mappings

In this section, we prove common end-point theorems for a pair of -weakly isotone increasing multivalued mappings.

To complete the result, we need notion of -weakly isotone increasing for multivalued mappings given by Vetro [31, Definitionâ€‰â€‰4.2].

*Definition 3.1. *Let be a partially ordered set and be two maps. The mapping is said to be -weakly isotone increasing if for all any , and .

Note that, in particular, for single-valued mappings , mapping is said to be -weakly isotone increasing if [31, Definitionâ€‰â€‰2.2] if for each we have .

Theorem 3.2. *Let be an ordered complete metric space. Let be such that
**
for all comparable , where , and is an altering distance function and
**
Also suppose that is -weakly isotone increasing and there exists an such that . If the property
**
holds, then and have a common end-point. *

*Proof. *Define a sequence and prove that the limit point of that sequence is a unique common end-point for and . For a given and nonnegative integer let
If or for some , then the proof is finished. So assume for all .

Since , can be chosen so that . Since is -weakly isotone increasing, it is ; in particular, can be chosen so that . Now, (since ); in particular, can be chosen so that .

Continuing this process we construct a monotone increasing sequence in such that
If or for some , then the proof is finished. So assume for all .

Suppose that is an odd number. Substituting and in (3.1) and using properties of function , we have for all ,
Since max, it follows that
Suppose that , for some positive integer .

Then from (3.7), we have
it implies that , or that , contradicting our assumption that , for each and so we have
In the similar fashion, we can also show inequalities (3.9) when is an even number. Therefore, the sequence is a monotone decreasing sequence of nonnegative real numbers. Hence there exists an such that
Taking the limit as in (3.7) and using the continuity of , we have , which is a contradiction unless . Hence
Next we show that is a Cauchy sequence. If otherwise, there exists an for which we can find two sequences of positive integers and such that for all positive integers , and .

Assuming that is the smallest such positive integer, we get ,
Now,
that is,
Taking the limit as in the above inequality and using (3.11), we have
Again,
Taking the limit as in the above inequalities and using (3.11) and (3.15), we have
Again,
Letting in the above inequalities and using (2.9) and (3.15), we have
Similarly, we have that
For each positive integer , and are comparable. Then using the monotone property of and the condition (3.1), we have
Letting in above inequality, using (3.11), (3.15), (3.17), (3.19), and (3.20) and using the continuity of , we have
which is a contradiction by virtue of a property of .

Hence is a Cauchy sequence. From the completeness of , there exists a such that
By the assumption (3.3), , for all .

Then by the monotone property of and the condition (3.1), we have
Taking the limit as in the above inequality, using (3.11) and (3.23) and the continuity of , we have
it implies that , or that . Similarly . Moreover, is a common end-point of and .

Putting in Theorem 3.2, we immediately obtain the following result.

Corollary 3.3. *Let be an ordered complete metric space. Let be such that
**
for all comparable , where , and is an altering distance function and
**
Also suppose that for all and there is such that . If the property
**
holds, then has a end-point. *

In Theorem 3.2, if are single valued mappings, then we have the following result.

Theorem 3.4. *Let be an ordered complete metric space. Let be such that
**
for all comparable , where , and is an altering distance function and
**
Also suppose that and are weakly isotone increasing. If
**
or
**
or
**
holds, then and have a common end-point. *

*Proof. *If we assume and as a multivalued mapping in which and are a singleton set for every . Then we consider the same sequence as in the proof of Theorem 3.4. Follows the line of proof of Theorem 3.4, we have that is a Cauchy sequence and
Then, if is continuous, we have
and this proves that is a end-point of and so is a end-point of . Similarly, if is continuous, we have the result. Thus it is immediate to conclude that and have a common end-point.

#### 4. Common End-Point Theorems for a Pair of Multivalued Mappings Satisfying Weakly Contractive Condition

In this section, we prove common end-point theorems for a pair of weakly isotone increasing multivalued mappings under weakly contractive condition.

To complete the result, we need notion of weakly contractive condition given by Rhoades [32].

*Definition 4.1 (Weakly Contractive Mapping). *Let be a metric space. A mapping is called weakly contractive if and only if
where is an altering distance function.

Theorem 4.2. *Let be an ordered complete metric space. Let be such that
**
for all comparable , where are an altering distance functions.**Also suppose that is -weakly isotone increasing and there exists an such that . If the property
**
holds, then and have a common end-point. *

*Proof. *Define a sequence and prove that the limit point of that sequence is a unique common end-point for and . For a given and nonnegative integer let
Since , can be chosen so that . Since is -weakly isotone increasing, it is ; in particular, can be chosen so that . Now, (since ); in particular, can be chosen so that .

Continuing this process, we conclude that can be an increasing sequence in :
If there exists a positive integer such that , then is a common end-point of and . Hence we will assume that , for all .

Suppose that is an odd number. Substituting and in (2.6) and using properties of functions and , we have for all ,
Since , it follows that
Suppose that , for some positive integer . Then from (4.7), we have
that is, , which implies that , or that , contradicting our assumption that . So we have
In the similar fashion, we can also show inequalities (4.9) when is an even number. Therefore, for all and is a monotone decreasing sequence of nonnegative real numbers. Hence there exists an such that
In view of the above facts, from (4.7) we have for all ,
Taking the limit as in the above inequality, using (4.10) and the continuities of and , we have
which is a contradiction unless . Hence
Next we show that is a Cauchy sequence. If is not a Cauchy sequence, then using an argument similar to that given in Theorem 3.2, we can find two sequences of positive integers and for which
For each positive integer , and are comparable. Then using the monotone property of and (4.2), we have
Letting in the above inequality, using (4.14) and the continuities of and , we have
which is a contradiction by virtue of a property of . Hence is a Cauchy sequence. From the completeness of , there exists a such that
By the condition (4.3), , for all . Then by the monotone property of and (4.2), we have
Taking the limit as in the above inequality, using (4.13), (4.17) and the continuities of and , we have
which implies that
which is a contradiction unless , or that ; that is, is a end-point of . Similarly . Moreover, is a common end-point of and .

Similar corollaries can be derived from Theorem 4.2.

#### Acknowledgment

The present version of the paper owes much to the precise and kind remarks of the learned referees.