Journal of Applied Mathematics

1. Introduction and Preliminaries

In 2007, Huang and Zhang [1] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space, and they showed some fixed point theorems of contractive-type mappings on cone metric spaces. The category of cone metric spaces is larger than metric spaces. Subsequently, many authors studied this subject and many results on fixed point theory are proved (see, e.g., [2–15]). Recently, Du [16] introduced the concept of -cone metric and -cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [1]. Later, in the papers [16–21], the authors tried to generalize this approach by using cones in topological vector spaces instead of Banach spaces. However, it should be noted that an old result shows that if the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space. Thus, proper generalizations when passing from norm-valued cone metric spaces to -valued cone metric spaces can be obtained only in the case of nonnormal cones (for details, see [19]). Further, Radenović et al. [22] introduced the concept of set-valued contraction of Nadler type in the setting of -cone spaces and proved a fixed point theorem in the setting of tvs-cone spaces with respect to a solid cone.

Definition 1.1 (see [22]). Let be a -cone metric space with a solid cone , and let be a collection of nonempty subsets of . A map is called a --cone metric with respect to if for any the following conditions hold:,,, one of the following is satisfied:(i),(ii).

Theorem 1.2 (see [22]). Let be a -cone complete metric space with a solid cone and let be a collection of nonempty closed subsets of , , and let be a --cone metric with respect to . If the mapping the condition that exists a such that for all holds then has a fixed point in .

We recall some definitions and results of the -cone metric spaces that introduced in [19, 23], which will be needed in the sequel.

Let be be a real Hausdorff topological vector space ( for short) with the zero vector . A nonempty subset of is called a convex cone if and for . A convex cone is said to be pointed (or proper) if ; is normal (or saturated) if has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone , we can define a partial ordering with respect to by if and only if ; will stand for and , while will stand for , where denotes the interior of . The cone is said to be solid if it has a nonempty interior.

In the sequel, will be a locally convex Hausdorff with its zero vector , a proper, closed, and convex pointed cone in with and a partial ordering with respect to .

Definition 1.3 (see [16, 18, 19]). Let be a nonempty set and an ordered . A vector-valued function is said to be a -cone metric, if the following conditions hold:,,,. Then the pair is called a -cone metric space.

Definition 1.4 (see [16, 18, 19]). Let be a -cone metric space, , and a sequence in .(1)-cone converges to whenever for every with , there exists such that for all . We denote this by cone-;(2) is a -cone Cauchy sequence whenever for every with , there exists such that for all ;(3) is -cone complete if every -cone Cauchy sequence in is -cone convergent in .

Remark 1.5. Clearly, a cone metric space in the sense of Huang and Zhang [1] is a special case of -cone metric spaces when is a -cone metric space with respect to a normal cone .

Remark 1.6 (see [19, 22, 23]). Let be a -cone metric space with a solid cone . The following properties are often used, particularly in the case when the underlying cone is nonnormal.(p1) If and , then ,(p2) If and , then ,(p3) If and , then ,(p4) If for each , then ,(p5) If for each , then ,(p6) If is with a cone , and if where and , then ,(p7) If , , and in locally convex , then there exists such that for all .

Metric spaces are playing an important role in mathematics and the applied sciences. To overcome fundamental laws in Dhage’s theory of generalized metric spaces [24]. In 2003, Mustafa and Sims [25] introduced a more appropriate and robust notion of a generalized metric space as follows.

Definition 1.7 (see [25]). Let be a nonempty set, and let be a function satisfying the following axioms:(G1),(G2),(G3),(G4) (symmetric in all three variables),(G5). Then the function is called a generalized metric, or, more specifically a -metric on , and the pair is called a -metric space.

By using the notions of generalized metrics and -cone metrics, we introduce the below notion of -generalized-cone metrics.

Definition 1.8. Let be a nonempty set and an ordered , and let be a function satisfying the following axioms:(G1) if and only if ,(G2),(G3),(G4) (symmetric in all three variables),(G5). Then the function is called a -generalized-cone metric, or, more specifically, a --cone metric on , and the pair is called a --cone metric space.

Definition 1.9. Let be a --cone metric space, , and a sequence in .(1)--cone converges to whenever, for every with , there exists such that for all . Here is called the limit of the sequence and is denoted by -cone-;(2) is a --cone Cauchy sequence whenever, for every with , there exists such that for all ;(3) is --cone complete if every --cone Cauchy sequence in is --cone convergent in .

Proposition 1.10. Let be a --cone metric space, , and a sequence in . The following are equivalent:(i)--cone converges to ,(ii) as ,(iii) as ,(iv) as .

In this paper, we also introduce the below concept of the ---cone-type function.

Definition 1.11. One calls a ---cone-type function if the function satisfies the following condition for all and ; for all .
In this paeper, for a --cone metric space and for the family of subsets of , we introduce a new notion of the --cone metric with respect to , and we get a fixed result for the ---cone-type function in a complete --cone metric space . Our results generalize some recent results in the literature.

2. Main Results

Let be a locally convex Hausdorff with its zero vector , a proper, closed and convex pointed cone in with , and a partial ordering with respect to . We introduce the below notion of the --cone metric with respect to --cone metric .

Definition 2.1. Let be a --cone metric space with a solid cone , and let be a collection of nonempty subsets of . A map is called a --cone metric with respect to if for any the following conditions hold:(),()(symmetry in all variables),,(),() one of the following is satisfied:(i),(ii),(iii).

We will prove that a --cone metric satisfies the conditions of .

Lemma 2.2. Let be a --cone metric space with a solid cone , and let be a collection of nonempty subsets of , . If is a --cone metric with respect to , then pair is a --cone metric space.

Proof. Let be a sequence such that for all and --. Take any and , . From , for each , there exists such that Therefore, for each . By the closedness of , we conclude that .
Assume that . From H₅, we obtain for any . So .
Let . Assume that satisfy condition . Then, for each , there exists such that for all and . From , there exists a sequence satisfying for every . Obviously, for any and any and , we have Now for each , there exists , such that . Consequently, we obtain that for each Therefore, In the case when or holds, we use the analogous method.

In the sequel, we denote by the class of functions satisfying the following conditions: is a ---cone-type-function; is subadditive, that is, for all .

Our main result is the following.

Theorem 2.3. Let be a --cone complete metric space with a solid cone , let be a collection of nonempty closed subsets of , , and let be a --cone metric with respect to . If the mapping satisfies the condition that exists a such that for all holds then has a fixed point in .

Proof. Let us choose arbitrarily, and let be a sequence such that and . Let us choose arbitrarily and . If , then , and we are done. Assume that . Taking into account (2.5) and , there exists such that Taking into account (2.5), (2.6), and and since , there exists such that We continue in this manner. In general, for , , is chosen such that and Since is arbitrary, letting and by the definition of the ---cone-type function, we obtain that
Next, we let , and we claim that the following result holds: for each ,  there is such that for all m, , We will prove (2.10) by contradiction. Suppose that (2.10) is false. Then there exists some such that for all , there are with satisfying(i) is even and is odd,(ii), and(iii) is the smallest even number such that conditions (i), (ii) hold.Since , by (ii), we have that and It follows from ; let us choose arbitrarily such that Taking into account (2.5), (2.11), and (2.12), we have that Since is arbitrarily, letting and by letting , we have a contradiction. So is a --cone Cauchy sequence. Since is a --cone complete metric space, is --cone convergent in and --. Thus, for every and sufficiently large , we have that Since for , , by we obtain that for all there exists such that Since , then for sufficiently large , we obtain that which implies --. Since is closed, we obtain that .

For the case , , then and it is easy to get the following corollary.

Corollary 2.4. Let be a --cone complete metric space with a solid cone , let be a collection of nonempty closed subsets of , , and let be a --cone metric with respect to . If the mapping satisfies the condition that exists such that for all holds then has a fixed point in .

Remark 2.5. Following Corollary 2.4, it is easy to get Theorem 1.2. So our results generalize some recent results in the literature (e.g., [22]).

Acknowledgment

The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.