#### Abstract

In this article, the concept of sequential -metric spaces has been introduced as a generalization of usual metric spaces, -metric spaces, -metric spaces, and mainly -metric spaces. Some topological properties of such spaces have been discussed here. By considering this notion, we prove fixed-point theorems for some classes of contractive mappings over such spaces. Examples have been given in order to examine the validity of the underlying space and in support of our fixed-point theorems. Moreover, our fixed-point theorem is applied to obtain solution of a system of linear algebraic equations.

#### 1. Introduction and Preliminaries

Two distance-controlled functions have been used extensively by the researchers working on fixed-point theory for obtaining fixed points of mappings such as contractive or expansive mappings in nature. Also, the polygonal inequality involved in a metric-like structure plays vital role for defining the topology on such space. But nowadays, after the introduction of -metric space, the latest fashion is to define a metric-type space which does not involve any type of polygonal inequality (see [1–3]). There is an immense literature in fixed-point theory and applications. For instance, in [4], a class of generalized -weak contraction has been introduced, and some fixed-point theorems in the framework of partially ordered metric spaces have been proved. The authors also applied their results to a first-order ordinary differential equation.

Now, we remember some efforts on -metric spaces.

In [5], Asif and Nazam noticed that the existence of fixed points of -contractions, in an -metric space, can be ensured with restricted conditions on the Wardowski function . They obtained some fixed-point results for both single and set-valued Reich-type -contractions in -metric spaces. To show the usability of our results, we present two examples. Also, an application to functional equations is presented.

In [6], Jahangir et al. investigated some properties of -metric spaces. They presented a simple proof to show that the natural topology induced by an -metric is metrizable. They also presented a method to construct -metric spaces from bounded metric spaces. They also showed that -metrics are not necessarily jointly continuous functions. They showed that the Nadler fixed-point theorem and, therefore, the Banach contraction principle in the framework of -metric spaces, the Schauder fixed-point theorem in -normed spaces, and also some related -metric fixed-point results can be reduced to their original metric versions.

We now give some definitions of generalized metric-type spaces which are relevant to our research work.

*Definition 1 (-metric space) [7, 8]. *Let be a nonempty set and be a real number satisfying A function is a metric on provided that
(1) if and only if (2) for all (3) for all .The space is called a -metric space.

Let be a nonempty set and be a mapping. For any , let us define the set

*Definition 2 (-metric space) [9]. *Let be a mapping such that
(1) implies (2)for every , we have (3)if and , then , for some .The pair is called a generalized metric space, usually known as -metric space (JSMS).

*Definition 3 (-metric space) [10]. *Let be a nonempty set. A mapping is said to be an -metric on , if for all , satisfies
(1) if and only if (2)For every with , and for every with , we have
where is an increasing function such that for all 0-convergent sequence and .The pair is called an -metric space.

Motivated from the previous definitions and based on these ideas, we now define a new generalized metric-type space in our next section.

#### 2. Sequential -Metric Spaces

In this section, we introduce the concept of sequential -metric space. To develop such a notion, first we define , where is a given mapping.

*Definition 4. *Let be a nonempty set. A mapping is said to be a sequential -metric if for all

() implies

()

() , for all , where is an increasing function with iff , for all 0-convergent sequence and .

The triplet is called a sequential -metric space (SFMS). A SFMS indicated simply as .

*Example 5. *Let and the metric be defined by
Also, let .

For , . Let . If all but finitely many terms of are , then we have nothing to prove. So, suppose that only have finitely many 1’s. Without loss of generality, we can exclude such 1’s, and then, we get . Therefore, for all .

Hence, is a sequential -metric on for for all and .

Note that taking , , and , we see that . So, is not a usual metric. Now, if are sufficiently large and , again, the left-hand side in triangular inequality in a -metric space is greater than the right-hand side. So, is not also a -metric.

To show that is not an -metric space, it is sufficient to take , are sufficiently large and . So

Proposition 6. *Any -metric space is a SFMS.*

*Proof. *Since is an -metric space, then for every , for every with and for every with , we have
Therefore, it follows that
Thus, for any , if we take , then we see that
where is defined by , , and for all So
and therefore, also satisfies condition (). Hence, is a sequential -metric on for the mapping and

Proposition 7. *Any -metric space is a SFMS.*

*Proof. *Since is a -metric, then there exists such that for all Thus
for all . So, also satisfies the third condition of Definition 4 for the function for all and , , and Hence, is a sequential -metric on .

*Remark 8. *(i)A metric space, *-*metric space [7, 8], metric-like space [11], and modular metric space with the Fatou property [12] are -metric spaces. Therefore, these spaces are also SFMS(ii)Any metric space [13] is an -metric space and therefore is also a SFMS. There exist -metric spaces which are not -metric spaces (see [10]); therefore, our SFMS is a stronger concept than the concept of -metric space.The following is an example of a SFMS which is not an -metric space as well as not a -metric space.

*Example 9. *Let and be defined by , , for all , and for all with Then for all , Let . If all but finitely many terms of are , then we are done. So, let only have finitely many 1’s. Without loss of generality, we can ignore such 1’s, and then, we get . Therefore, for all . So, is a SFMS for for all , , and . If it is an -metric space, then there exists such that
Taking , we see that , a contradiction. Hence, is not an -metric. In a similar way, we can show that is not a -metric on

*Definition 10. *Let be a SFMS. Also let be a sequence in and (i)if , is called convergent and converges to (ii)if is called Cauchy(iii)if any Cauchy sequence in is convergent, is called complete.

*Definition 11. *Let and be two SFMS. If for any there exists such that for any , whenever , then is called continuous at a point . is said to be continuous on if is continuous at each point of .

Proposition 12. *Let be a SFMS and be a convergent sequence converging to some ; then, .*

*Proof. *If possible, let . Then
since , a contradiction. Hence, the result.

Proposition 13. *Let be a SFMS. If converges to some , then .*

*Proof. *From the condition () of Definition 4, we have
implying that .

Proposition 14. *Let be a Cauchy sequence in a SFMS . If has a convergent subsequence which converges to , then also converges to .*

*Proof. *From condition () of Definition 4, we have
Taking , we get as . Therefore, as , that is, as .

*Remark 15. *Here is an example of -metric space which is given by Senapati et al. [14]. Let and be defined by
(i)In this space, we see that the sequence converges to , but it is not a Cauchy sequence. Since any -metric space is a SFMS also, therefore, in a SFMS, a convergent sequence is not necessarily Cauchy(ii)Also, in a SFMS, if and are two sequences convergent to and , respectively, then may not be convergent to . For this, let us consider two sequences and , and then, both the sequences converge to but .

Proposition 16. *In a SFMS , if a self mapping is continuous at , then implies that .*

*Proof. *Let be given. Since is continuous at , then for any , there exists such that , , implies

Let . Since converges to ; for , there exists such that for all . Therefore, for any , , and thus, as , that is, .

Let be a SFMS with supporting function and . Define
for all and .

*Remark 17. *The family
forms a topology on .

*Definition 18. *If there exists an open set such that in a SFMS , then is said to be closed.

Proposition 19. *Let be a SFMS and be closed. Let such that as . Then, .*

Proposition 20. *Let be a complete SFMS and be closed. Then, the subspace is also complete.*

*Definition 21. *In a SFMS , for , we define

Theorem 22. *Let be a complete SFMS and be a decreasing sequence of nonempty closed subsets of such that as . Then, .*

#### 3. Some Fixed-Point Theorems

Theorem 23 (Banach-type fixed-point theorem). *Let be a complete SFMS and be a mapping which satisfies the following conditions:
*(i)* for all and for some *(ii)*there exists such that .**Then, has at least one fixed-point in . Moreover, if and are two fixed points of in with , then .*

*Proof. *Define for every . Since , then for all Now
for all and

Therefore, for all , from which it follows that Now, for any , we have
So, is Cauchy in , and so, there exists some such that as Thus, as . From Proposition 12, it follows that and is a fixed point of .

Now, if and are two fixed points of in with , then we have which gives implying that .

*Example 24. *Let endowed with the distance function:
Then, is a SFMS for and Now, let us define as follows:
Then, has all contractive conditions of Theorem 23 for and also satisfies all other additional conditions. Here, has a unique fixed-point in .

Theorem 25 (Reich-type fixed-point theorem). *Let be a complete SFMS and satisfy
*(i)* for all and for with *(ii)*there exists such that .**Then, the Picard iterating sequence , for all converges to some . If and for all then is a fixed point of . Moreover, if is a fixed point of in such that and , then .*

*Proof. *Let us define for every Since , then for all . Now
Proceeding in a similar way as in Theorem 23, it follows that , that is, is Cauchy in , and so, there exists some such that as Therefore, we get
which implies that .

Thus, using condition of Definition 4, we have
Now, if , then by the assumed condition of the theorem, we see that , a contradiction. Thus, gives , and is a fixed point of .

Now, if is a fixed point of in with and , then we have , as , implying that , that is, .

Corollary 26. *Let be a complete SFMS and satisfies
*(i)* for all and for *(ii)*there exists such that .**Then the Picard iterating sequence , for all converges to some . If and for all , then is a fixed point of . Moreover, if is a fixed point of in such that and , then .*

*Proof. *If we take and , then this corollary follows from our Theorem 25.

*Example 27. *Let endowed with the distance function:
Then is a SFMS for and . Now, let be defined as
Then, satisfies the contractive condition of Corollary 26 for . Here, all other additional conditions are also satisfied. We see that has a unique fixed-point in

Theorem 28 (Chatterjea-type fixed-point theorem). *Let be a complete SFMS and be a mapping satisfying
*(i)* for all and for some *(ii)*there exists such that .**Then the Picard iterating sequence , for all converges to some . If , then is a fixed point of . Also, if is a fixed point of in such that , then .*

*Proof. *By similar argument as in Theorem 23, is a Cauchy sequence in , and by the completeness of , it converges to an element say .

Now, for all , , which implies that , and therefore, . Thus, we have
which gives , that is, , and is a fixed point of .

If is a fixed point of in with , then we have . Consequently, , that is, .

#### 4. An Application to the System of Linear Algebraic Equations

An application of Theorem 23 for solving a system of linear algebraic equations has been presented in this section.

Consider the following system of linear algebraic equations with unknowns: where for all . We can write the system of linear equations in matrix notation as , where , , , and . To find a solution of the system of linear Equations (30), we have to find a fixed point of the mapping defined by , where , that is, with if and for all .

Now, we define by

Then, is a sequential -metric for and

Theorem 29. *If
then the system of linear Equations (30) has a unique solution in .*

*Proof. *To find a unique solution of (30), we show that the mapping defined by for all , where , that is, with if and for all , satisfies the contractive condition of Theorem 23. Now, for any and in , we have
Since is complete, therefore, due to Theorem 23, has a unique fixed point, that is, the system of linear Equations (30) has a unique solution in .

We now give a numerical example in respect of Theorem 23.

*Example 30. *Let us consider the following system of linear algebraic equations in three variables:
Then, the system of linear algebraic Equations (34) has a unique solution.

*Solution. *Let be the SFMS endowed with the metric defined by
We can write the above system of linear algebraic Equations (34) as
Here, , , , , , , , , , , , and

Thus,