On Statistical Convergence Of Order ${\alpha}$ In Partial Metric Spaces

One of the two notions that constitute the concern of our study is the partial metric space, which is a generalization of the usual metric spaces as introduced by Matthews[13]. The main difference between the partial metric and the standard metric is that the distance of an arbitrary point does not need to be equal to zero. The partial metric in which was first used in computer science to be later applied to many other fields.

The second notion is statistical convergence, which is defined as a generalization of sequential convergence. In the first edition of Zygmund’s monograph, published in Warsaw, the definition of statistical convergence (as almost convergence) was given by Zygmund [19]. Steinhaus [18] and Fast [7] and later Schoenberg [17] introduced the concept of statistical convergence, independently.This subject has been studied by many mathematicians, (for example, see Altın et al. [1] , Bilalov and Nazarova [2], Kayan et al. [10], Küçükaslan et al. [11]). The concept of statistical convergence has been used in many areas of mathematics, such as Number theory, Probabilistic normed spaces, Ergodic theory, Fourier analysis, Measure theory, Trigonometric series, and others. Therefore, it is a very active and intensively studied subject in many contexts.

Before proceeding with results, for the convenience of the reader, let us recall the definitions and terminology that this work involves.

The definition of the partial metric space is as it follows:

Let $X$ be a non-empty set and $\rho:X$ $\times X\rightarrow\mathbb{R}$ be a function such that for all $x,y,z\in X,$

$i)$ $0\leq\rho(x,x)\leq\rho(x,y),$

$ii)$ If $\rho(x,x)=\rho(x,y)=\rho(y,y)$ then $x=y,$

$iii)$ $\rho(x,y)=\rho(y,x),$

$iv)$ $\rho(x,z)\leq\rho(x,y)+$ $\rho(y,z)-\rho(y,y).$

Then $\rho$ is called a partial metric and the pair $(X,\rho)$ is called a partial metric space (see Matthews [13]).

We will now give the concepts of convergence and bounded in the partial metric space.

Let $(X,\rho)$ be a partial metric space and $(x_{n})$ be a sequence in $X.$ Then

$i)$ $(x_{n})$ is bounded if there exists $a$ real number $M>0$ such that $\rho(x_{n},x_{m})\leq M$ for all $n,m\in\mathbb{N}$,

$ii)$ $\left(x_{n}\right)$ is called convergent to $x$ in $(X,\rho),$ written as $\lim\limits_{n\rightarrow\infty}x_{n}=x,$ if

The statistical convergence depends on density of subsets of $\mathbb{N}$. The natural density of $K\subset\mathbb{N}$ which is the main tool for this convergence is defined by

where $\left|\left\{k\leq n:k\in K\right\}\right|$ denotes the number of elements of $K$ in which does not exceed $n$ (see Fridy [8]). Clearly, any finite subset of $\mathbb{N}$ has zero natural density and $\delta\left(K^{c}\right)=1-\delta\left(K\right)$. For a detailed description of the density of subsets of $\mathbb{N}$, reference can be made to Niven and Zuckerman [15].

A sequence $\left(x_{k}\right)$ of complex numbers is said to be statistically convergent to a number $L$ if for every positive number $\varepsilon,$ $\delta\left(\left\{k\in\mathbb{N}\mathbf{:}\text{ }\left|x_{k}-L\right|\geq\varepsilon\right\}\right)$ has natural density zero. The number $L$ is called statistical limit of $\left(x_{k}\right)$ and is written as $S-\lim x_{k}=L$ or $x_{k}\rightarrow L\left(S\right)$ and the set of all statistically convergent sequences is denoted by $S.$

Leindler [12] introduced $(V,\lambda)$-summability by the help of sequence $\lambda=\left(\lambda_{n}\right)$ as in the following: Let $\lambda=\left(\lambda_{n}\right)$ be a non-decreasing sequence of positive numbers tending to $\infty$ with $\lambda_{n+1}\leq\lambda_{n}+$ $1$, $\lambda_{1}=1$. The generalized de la Vallee-Poussin mean is defined by

where $I_{n}=[n-\lambda_{n}+1,n]$. Accordingly, a sequence $\left(x_{k}\right)$ of numbers is said to be $(V,\lambda)$-summable to a number $L$ if $t_{n}\left(x\right)\rightarrow L$ as $n\rightarrow\infty$.

and

denote the sets of sequences $\left(x_{k}\right)$ which are strongly Cesáro summable and strongly $(V,\lambda)$-summable to $L$. It is noted that for $\lambda_{n}=n$, $\left(V,\lambda\right)$-summability reduces to $\left(C,1\right)$-summability.

Mursaleen [14] introduced the $\lambda$-density of $K\subset\mathbb{N}$ as defined by

and $\lambda$-statistical convergence as it follows:

A sequence $\left(x_{k}\right)$ of numbers is said to be $\lambda$-statistically convergent to a number $L$ provided that for every $\varepsilon>0$,

In this case, the number $L$ is called $\lambda$-statistical limit of the sequence $\left(x_{k}\right)$.

The statistical convergence with degree $0<\beta<1$ was introduced by Gadjiev and Orhan [9]. The statistical convergence of order $\alpha$ and strong $p$-Cesàro summability of order $\alpha$ were later studied by Çolak [4]. Also Çolak and Bektaş [6] introduced $\lambda$-statistical convergence of order $\alpha$, as in the following:

Let the sequence $\lambda=\left(\lambda_{n}\right)$ of real numbers be defined as above and $0<\alpha\leq 1$. The sequence $x=\left(x_{k}\right)\in w$ is said to be $\lambda$-statistically convergent of order $\alpha$ if there is a complex number $L$ such that

where $I_{n}=[n-\lambda_{n}+1,$ $n]$ and $\lambda_{n}^{\alpha}$ is the coordinates of $\alpha$ th power of the sequence $\lambda$, that is, $\lambda^{\alpha}=\left(\lambda_{n}^{\alpha}\right)=\left(\lambda_{1}^{\alpha},\lambda_{2}^{\alpha},...,\lambda_{n}^{\alpha},...\right)$.

Some new sequence spaces for $\lambda$ and $\mu$ different sequences of $\Lambda$ class were later defined by Çolak [5] and some inclusion theorems were examined.

In this study, we denote the class of all decreasing sequence of positive real numbers tending to $\infty,$ such that $\lambda_{n+1}\leq\lambda_{n}+1$, $\lambda_{1}=1$ by $\Lambda$. Also, unless stated otherwise, by "for all $n\in\mathbb{N}_{n_{0}}$" we mean "for all $n\in\mathbb{N}$ except finite numbers of positive integers", where $\mathbb{N}_{n_{0}}=\left\{n_{0},n_{0}+1,n_{0}+2,...\right\}$ for some $n_{0}\in\mathbb{N}=\{1,2,3,...\}$.

The concept of statistical convergence in partial metric spaces was given by Nuray [16] as it follows:

Let $\left(x_{k}\right)$ be a sequence in partial metric space $(X,\rho)$. The sequence $\left(x_{k}\right)$ is said to be $\rho-$ statistically convergent to $x$ if there exists a point $L\in X$ such that

for every $\varepsilon>0$. In addition, Nuray [16] also examined the relationship between the concept of statistical convergence in partial metric spaces and strong Cesàro summability.

In this section, we introduce the notions of statistical convergence of order $\alpha$ and strong $p-$ Cesàro summability of order $\alpha$ in partial metric spaces. Also, some relations between statistical convergence of order $\alpha$ and strongly $p-$ Cesàro summable sequences of order $\alpha$ are given.

Throughout this paper, $S_{\rho}^{\alpha}(X)$ will denote the class of sequences in partial metric space $(X,\rho)$ which are $\rho-$ statistically convergent of order $\alpha$.

For the sake of simplicity, it will be considered that the sequence $\left(x_{k}\right)$ and the element $x$ we use in the proofs are chosen from the partial metric space $(X,\rho)$, although we do not emphasize it every time.

That the inclusion may be strict can be seen by the following example.

Taking $\beta=1$, we get the following result from the last inequality above.

From Theorem 2.2 we have the following results in which the proofs are easy.

In the following theorem, we examine the relationship between statistical convergence and Cesàro convergence in partial metric spaces.

If we take $\theta=\alpha$ in Theorem 2.7, we obtain the following result:

Hence, from Corollary 2.8, we have the necessary part of Theorem 4.4 in Nuray [16] in case $\alpha=1$ as if a sequence is strongly $p-$ Cesaro summable to $x\in X,$ then it is statistically convergent to $x\in X$.

In this section, we introduce the notion of $\lambda-$statistical convergence of order $\alpha$ and $[V,\lambda]-$summability of order $\alpha$ in partial metric spaces. In this setting, some inclusion results related these concepts are also included in this section.

The following example states that the inclusion in the previous theorem may be strict.

For $\alpha\in\left(0,1\right]$ the inclusion $S_{\rho}^{\alpha}(\lambda,X)\subseteq S_{\rho}^{\alpha}(X)$ clearly hold. The following theorem gives a case where the reverse of this inclusion is also holds.

By choosing the value of $\beta$, as a result of Theorem 3.6, we can give the following two results.

Now, we introduce $[V_{\rho}^{\alpha},\lambda]-$summability of order $\alpha$ for the partial metric spaces.

Again, by choosing the value of $\beta$, the following two results follows directly from the last theorem.