#### Abstract

The existence of the endpoints and approximate endpoints are studied in a general setting for the operators satisfying various contractive conditions. Some recent results are also derived as special cases.

#### 1. Introduction

Among several generalizations of celebrated Banach fixed point theorem, one interesting extension is Nadler’s [1] fixed point theorem for multivalued contraction. He exactly proved that a multivalued contraction has a fixed point in a complete metric space. Subsequently, it received great attention in applicable mathematics and was extended and generalized on various settings. Indeed, these extensions and generalizations have been influenced by the applications of the multivalued fixed point theory in mathematical economics, game theory, differential inclusions, interval arithmetic, Hammerstein equations, convex optimization, duality theory in optimization, variational inequalities and control theory, nonlinear evolution equations and nonlinear semigroups, quasivariational inequalities, and elasticity and plasticity theory (see, for instance, [2–8] and several references thereof). The results related to existence of endpoints or strict fixed-points were first given by Rus [9] in 2003. Thereafter, a number of authors established interesting results concerning existence and uniqueness of endpoints for multivalued contractions in different settings; see, for example, [10–15]. The main purpose of this paper is to establish some existence and uniqueness results for endpoints using different multivalued contractions. Our results include some recent results.

#### 2. Preliminaries

Let be a multivalued mapping. An element is said to be an endpoint of if . We say that a multivalued mapping has the approximate endpoint property (AEPP) if (also see [3, 10]). Throughout the paper, let be a -metric space, and let denote the family of all nonempty subsets of and the family of all nonempty closed subsets of . For any , the Hausdorff metric is defined as where is the distance from the point to the set .

Let be a single-valued mapping and a multivalued contraction. We say that the mappings and have an AEPP provided .

A point is called an endpoint of and if . For each , let be the set of all approximate endpoints of the mappings and .

*Example 1. *Let with Euclidean norm. Assume that and defined by
where is a positive constant. Clearly, and .

*Definition 2 (see [16]). *Let be a nonempty set and a given real number. A function (set of nonnegative real numbers) is said to be a -metric iff for all the following conditions are satisfied:(i) ,(ii) ,(iii) .A pair is called a -metric space.

The class of -metric spaces is effectively larger than that of metric spaces, since a -metric space is a metric space when in the above condition (iii). The following example shows that a -metric on need not be a metric on (see also [16, page 264]).

*Example 3 (see [17]). *Let and

, for all and , . Then,
and if , the ordinary triangle inequality does not hold.

*Definition 4 (see [16]). *Let be a -metric space. Then, a sequence in is called(a)*convergent* if and only if there exists such that as . In this case, one writes ,(b)*Cauchy* if and only if as .

*Remark 5 (see [16]). *In a -metric space , the following assertions hold.(i)A convergent sequence has a unique limit.(ii)Each convergent sequence is Cauchy.(iii)In general, a -metric is not continuous.

*Definition 6 (see [16]). *Let be a -metric space. If is a nonempty subset of , then the closure of is the set of limits of all convergent sequences of points in , i.e.,

*Definition 7 (see [16]). *Let be a -metric space. Then, a subset is called(a)*closed* if and only if for each sequence in which converges to an element , one has ,(b)*compact* if and only if for every sequence of elements of , there exists a subsequence that converges to an element of ,(c)*bounded* if and only if .

*Definition 8 (see [16]). *The -metric space is complete iff every Cauchy sequence in converges.

*Definition 9. *Let and be two Hausdorff topological spaces and , a multivalued mapping with nonempty values. Then, is said to be(i)upper semicontinuous (u.s.c.) if, for each closed set , is closed in ;(ii)lower semicontinuous (l.s.c.) if, for each open set , is open in ;(iii)continuous if it is both u.s.c. and l.s.c.;(iv)closed if its graph is closed;(v)compact if closure of is a compact subset of .

*Definition 10 (see [5]). *Let be a single-valued mapping and a multivalued mapping. Then, is called(i)a multivalued -contraction if there exists a number (ii)a multivalued -Kannan contraction if there exists a number (iii)a multivalued -Chatterjea contraction if there exists a number (iv)a multivalued -quasi-contraction if
for some and all in ,(v)a multivalued -weak or almost contraction if there exist and (vi)a multivalued generalized -almost contraction if there exists a function satisfying for every such that

*Remark 11. *A multivalued mapping is called a multivalued -Zamfirescu operator if it satisfies at least one of the conditions (i), (ii), and (iii).

We have used following Cantor’s intersection theorem in our results.

Theorem 12. *Let be a compact space, and let be a nested chain of nonempty closed subsets of . Then, .*

#### 3. Main Results

Lemma 13. *Let be a -metric space with the -metric as a continuous functional. Let be a single-valued mapping such that for all , where is a constant. If satisfies with , then
*

*Proof. *For any , we have
So,
Since , we have

Lemma 14. *Let be a -metric space with the -metric as a continuous functional. Let be a continuous single-valued mapping. If is a lower semicontinuous multivalued mapping. Then, for each , is closed.*

*Proof. *The proof follows from Lemma 16 of Hussain et al. [11].

Theorem 15. *Let be a complete -metric space with the -metric as a continuous functional. Let be a continuous single-valued mapping such that , where is a constant. Let be a lower semicontinuous map satisfying . Then, and have a unique endpoint if and only if and have the AEPP.*

*Proof. *It is clear that if and have an endpoint, then and have the AEPP.

Then,
Also, we have for each , . By Lemma 14, is closed for each . Since and satisfy AEPP, then for each . Now, we show that . To show this, let . Then, from Lemma 13,
and so . It follows from the Cantor intersection theorem that
Thus, is the unique endpoint of and .

Lemma 16. *Let be a -metric space with the -metric as a continuous functional. Let be a single-valued mapping such that for all , where is a constant. If satisfies , then
*

*Proof. *For any , we have
So,
Since , we have

Theorem 17. *Let be a complete -metric space with the -metric as a continuous functional. Let be a continuous single-valued mapping such that , where is a constant. Let be a lower semicontinuous map satisfying . Then, and have a unique endpoint if and only if and have the AEPP.*

*Proof. *It is clear that if and have an endpoint, then and have AEPP. Then,
Also, we have for each , . By Lemma 14, is closed for each . Since and satisfy AEPP, then for each . Now, we show that . To show this, let . Then, from Lemma 16,
and so . It follows from the Cantor intersection theorem that
Thus, is the unique endpoint of and .

Lemma 18. *Let be a -metric space with the -metric as a continuous functional. Let be a single-valued mapping such that for all , where is a constant. If satisfies with , then
*

*Proof. *For any , we have
So,
Since , we have

Theorem 19. *Let be a complete -metric space with the -metric as a continuous functional. Let be a continuous single-valued mapping such that , where is a constant. Let be a lower semicontinuous map satisfying . Then, and have a unique endpoint if and only if and have the AEPP.*

*Proof. *It is clear that if and have an endpoint, then and have the AEPP.

Then,
Also, we have for each , . By Lemma 14, is closed for each . Since and satisfy AEPP, then for each . Now, we show that . To show this, let . Then, from Lemma 18,
and so . It follows from the Cantor intersection theorem that
Thus, is the unique endpoint of and .

Lemma 20. *Let be a -metric space with the -metric as a continuous functional. Let be a single-valued mapping such that for all , where is a constant. If satisfies with , , then
*

*Proof. *Suppose that satisfies , then we have
So, we have
If satisfies , then we have
Let . Then,
Thus, for any , we have
So,
Since , we have

Theorem 21.

*Proof. *It is clear that if and have an endpoint, then and have the AEPP. Then,
Further, we have for each , . By Lemma 14, is closed for each . Since and satisfy AEPP, then for each . Now, we show that . To show this, let . Then, from Lemma 20,
and so . It follows from the Cantor intersection theorem that
Thus, is the unique endpoint of and .

Lemma 22. *Let be a -metric space with the -metric as a continuous functional. Let be a single-valued mapping such that for all , where is a constant. If satisfies with , then
*

*Proof. *For any , we have
So,
Since , we have

Theorem 23.

*Proof. *It is clear that if and have an endpoint, then and have the AEPP.

Then,
Also, it is clear that, for each , . By the above Lemma 14, is closed for each . Since and satisfy AEPP, then for each . Now, we show that . To show this, let . Then, from Lemma 22,
and so . It follows from the Cantor intersection theorem that
Thus, is the unique endpoint of and .

Lemma 24. *Let be a -metric space and a single-valued mapping such that for all , where is a constant. If satisfies with , then
*

*Proof. *For any , we have
So,
Since , we have

Theorem 25.

*Proof. *It is clear that if and have an endpoint, then and have the AEPP. Then,
Also, we have for each , . By Lemma 14, is closed for each . Since and satisfy AEPP, then for each . Now, we show that . To show this, let . Then, from Lemma 24,
and so . It follows from the Cantor intersection theorem that
Thus, is the unique endpoint of and .

On putting in the above Theorem 25, we obtain the following result of [11].

Corollary 26 (see [11]). *Let be a complete metric space. Let be a continuous single-valued mapping such that , where is a constant. Let be a lower semicontinuous map satisfying . Then, and have a unique endpoint if and only if and have the AEPP.*

If is the identity mapping on and , then the above result reduces to the following results:

Corollary 27 (see [11, Corollary 3.5]). *Let be a metric space, and let satisfy with . Then, for each ,
**
where .*

Corollary 28 (see [11, Corollary 3.6]). *Let be a complete metric space. Let be a lower semicontinuous map satisfying with . Then, has a unique endpoint if and only if has the AEPP.*

If , in almost contraction, then we have following result in metric space.

Corollary 29 ([10, Corollary 2.2]). *Let be a complete metric space. Let satisfy . Then, has a unique endpoint if and only if has the AEPP.*

#### Acknowledgments

The authors would like to sincerely thank the referees and the editors for their valuable comments to improve the paper in its present form.