On rough convergence in amenable semigroups and some properties

Abstract

The authors of the present paper, firstly, investigated relations between the notions of rough convergence and classical convergence, and studied on some properties of the rough convergence notion which the set of rough limit points and rough cluster points of a sequence of functions defined on amenable semigroups. Then, they examined the dependence of r-limit LIM^rf of a fixed function f ∈ G on varying parameter r.

Keywords

Rough convergence rough limit point rough cluster point Folner sequence amenable semigroups

1 Introduction and background

Day [3] studied the notions of amenable semigroups (or briefly ASG). Then, some authors [5 , 9] studied the notions of summability in ASG. Douglas [4] extended the notion of arithmetic mean to ASG and obtained a characterization for almost convergence in ASG. In [11], Nuray and Rhoades presented the notions of convergence and statistical convergence in ASG. Recently, some authors studied the new notions in ASG (see, [6 , 15–17]).

Phu [12], firstly, introduced the notion of rough convergence in finite-dimensional normed spaces. In [12], he investigated some properties of LIM^rx such as boundedness, closedness and convexity, and also he defined the notion of rough Cauchy sequence. Then, Phu [13, 14] studied the notions of rough convergence and some important properties of this notion. Also, some authors [1, 2] investigated the rough convergence types in some normed spaces. Furthermore, Dündar et al. [7] introduced rough convergent functions defined on amenable semigroups.

The purpose of this study, firstly, is to reveal the relations between the notions of rough convergence and classical convergence, and to study some properties of the rough convergence notion which the set of rough limit points and rough cluster points of a sequence of functions defined on ASG. Next, we aim to examine the dependence of r-limit LIM^rf of a fixed function f ∈ G on varying parameter r.

First of all, let’s remind the basic definitions and notions that we will use in our study such as amenable semigroups, rough convergence, rough Cauchy sequence, etc. (see, [3–5 , 8–14]).

Let G be a discrete countable amenable semigroup (or briefly DCASG) with identity in which both left and right cancelation laws hold, and w (G) denotes the space of all real-valued functions on G.

If G is a countable amenable group, then there exists a sequence {S_n} of finite subsets of G such that

$G = ⋃_{n = 1}^{\infty} S_{n}$ ,

S_n ⊂ S_n+1 (n = 1, 2, …),

$lim_{n \to \infty} \frac{| S_{n} g \cap S_{n} |}{| S_{n} |} = 1, lim_{n \to \infty} \frac{| {gS}_{n} \cap S_{n} |}{| S_{n} |} = 1,$ for all g ∈ G.

If a sequence of finite subsets of G satisfy (i) - (iii), then it is called a Folner sequence (or briefly FS) of G.

A familiar FS giving rise to the classical Cesàro method of summability is the sequence $\begin{matrix} S_{n} = {0, 1, 2, . . ., n - 1} . \end{matrix}$

Let G be a DCASG with identity in which both left and right cancelation laws hold. For any FS {S_n} of G, a function f ∈ w (G) is said to be convergent to $t \in ℝ$ if for every ɛ > 0 there exists a $k_{ɛ} \in ℕ$ such that $\begin{matrix} | f (g) - t | < ɛ, \end{matrix}$ for all m > k_ɛ and g ∈ G \ S_m.

Let a real number r ≥ 0 and $ℝ^{n}$ (the real n-dimensional space) with the norm ∥ .∥, and a sequence $x = (x_{n}) \subset ℝ^{n}$ .

A sequence (x_n) is said to be r-convergent to ξ, denoted by $x_{n} \overset{r}{⟶} ξ,$ provided that $\begin{matrix} \forall ɛ > 0 \exists n_{ɛ} \in ℕ : n \geq n_{ɛ} \Rightarrow ∥ x_{n} - ξ ∥ < r + ɛ . \end{matrix}$

The rough limit set of the sequence x = (x_n) is denoted by $\begin{matrix} {LIM}^{r} x = {ξ \in ℝ^{n} : x_{n} \overset{r}{⟶} ξ} . \end{matrix}$

A sequence x = (x_n) is said to be r-convergent if LIM^rx≠ ∅. In this instance, r is called the convergence degree of the sequence x. For r = 0, we get the ordinary convergence.

For any FS {S_n} of G, a function f ∈ w (G) is said to be rough convergent (r-convergent) to $t \in ℝ$ if $\forall ɛ > 0 \exists k_{ɛ} \in ℕ : m \geq k_{ɛ} \Rightarrow | f (g) - t | < r + ɛ,$ (1) for all g ∈ G \ S_m, or equivalently,

$lim sup | f (g) - t | \leq r,$ (2) for all g ∈ G \ S_m. In this instance, it is denoted by $r - lim f (g) = t or f (g) \overset{r}{\to} t .$

If (1) holds, then t is an r-limit point of the function f ∈ w (G), which is usually no longer unique (for r > 0). Hence, we have to think the so-called rough limit set (r-limit set) of the function f ∈ w (G) defined by $\begin{matrix} {LIM}^{r} f = {t \in ℝ : f (g) \overset{r}{\to} t} . \end{matrix}$

For any FS {S_n} of G, a function f ∈ w (G) is said to be r-convergent if LIM^rf ≠ ∅ . In this instance, r is called the convergence degree of the function f ∈ w (G).

1.1 Lemma

[[7], Theorem 2.2] For any FS {S_n} of G, a function f ∈ w (G) is bounded if and only if there exists an r ≥ 0 such that LIM^rf ≠ ∅ .

1.2 Lemma

[[7], Theorem 2.3] For a function f ∈ w (G) and all r ≥ 0, the rough limit set LIM^rf is closed.

1.3 Lemma

[[7], Theorem 2.4] Let a function f ∈ w (G). If s₀ ∈ LIM^{r
₀}fands₁ ∈ LIM^{r
₁}f then, $\begin{matrix} s_{α} : = (1 - α) s_{0} + α s_{1} \in {LIM}^{(1 - α) r_{0} + α r_{1}} f, \end{matrix}$ for α ∈ [0, 1].

2 Main results

2.1 Theorem

Let r₁ ≥ 0 and r₂ > 0. For any FS {S_n} of G, a function f ∈ w (G) is (r₁ + r₂)-convergent to t if and only if there exists a function h ∈ w (G)

$h (g) \to r_{1} ⟶ t and | f (g) - h (g) | \leq r_{2}$ (3) for all g ∈ G \ S_m.

Proof. Assume that (3) is true. Then, for every ɛ > 0 there exists a k_ɛ such that for all m ≥ k_ɛ $\begin{matrix} | h (g) - t | < r_{1} + ɛ, \end{matrix}$ for all g ∈ G \ S_m. Since $\begin{matrix} | f (g) - h (g) | \leq r_{2}, \end{matrix}$ then for all m ≥ k_ɛ, we have $\begin{matrix} | f (g) - t | & \leq & | f (g) - h (g) | + | h (g) - t | \\ < & r_{1} + r_{2} + ɛ, \end{matrix}$ for all g ∈ G \ S_m. Hence, f ∈ w (G) is (r₁ + r₂)-convergent to t .

Now, for any FS {S_n} of G, we assume that f ∈ w (G) is (r₁ + r₂)-convergent to t . Then, for every ɛ > 0 there exists a k_ɛ such that for all m ≥ k_ɛ, we define the h (g) as following: $\begin{matrix} h (g) : = {\begin{matrix} t & , & if | f (g) - t | \leq r_{2} \\ [0, 3 cm] f (g) + r_{2} \frac{t - f (g)}{| t - f (g) |} & , & if | f (g) - t | > r_{2}, \end{matrix} \end{matrix}$ for all g ∈ G \ S_m. Hence, we have $\begin{matrix} | h (g) - t | \leq {\begin{matrix} 0 & , & if | f (g) - t | \leq r_{2} \\ [0, 3 cm] | f (g) - t | - r_{2} & , & if | f (g) - t | > r_{2} \end{matrix} \end{matrix}$ and so, $\begin{matrix} | f (g) - h (g) | \leq r_{2}, \end{matrix}$ for all g ∈ G \ S_m. By (2), since t ∈ LIM^r₁+r₂f, we get $\begin{matrix} lim sup | f (g) - t | \leq r_{1} + r_{2}, \end{matrix}$ and so, $\begin{matrix} lim sup | h (g) - t | \leq r_{1}, \end{matrix}$ for all g ∈ G \ S_m. Hence, f (g) → r₁ ⟶ t. □

Particularly, let r₁ = 0 and r₂ = r > 0. Then, the second result says that for any FS {S_n} of G, if a function f ∈ w (G) is r-convergent to t then there exists a function h ∈ w (G) such that $\begin{matrix} h (g) \to t and | f (g) - h (g) | \leq r, \end{matrix}$ for all g ∈ G \ S_m.

Requirement means that for any FS {S_n} of G, f ∈ w (G) is an approximation of a convergent function h (g) → t with r as an upper bound of approximation error then it is still r-convergent to t. This is what has already been said in the introduction. Conversely, for any FS {S_n} of G, a function f ∈ w (G) is r-convergent to t, then there exists a function h ∈ w (G) near f ∈ w (G) (that is, $\begin{matrix} | f (g) - h (g) | \leq r, \end{matrix}$ for all g ∈ G \ S_m) which converges to t.

2.2 Theorem

For any FS {S_n} of G, a function f ∈ w (G) is r-convergent to t if and only if LIM^rf = ${\bar{B}}_{r}$ (t), where ${\bar{B}}_{r} (t) : = {s : | s - t | \leq r} .$

Proof. It goes on to show that ${LIM}^{r} f = {\bar{B}}_{r} (t)$ implies f (g) → t, for any FS {S_n} of G. Contrary, assume that f ∈ w (G) has a cluster point t′ different from t. Hence, the point $\begin{matrix} \bar{t} : = t + \frac{r}{| t - t^{'} |} (t - t^{'}) \end{matrix}$ satisfies

$| \bar{t} - t^{'} | = r + | t - t^{'} | > r .$ (4) Since t′ is a cluster point, by definition, inequality (4) implies that $\begin{matrix} \bar{t} \notin {LIM}^{r} f, \end{matrix}$ which contradicts $\begin{matrix} | \bar{t} - t^{'} | = r and {LIM}^{r} f = {\bar{B}}_{r} (t) . \end{matrix}$ Therefore, t is only cluster point of f ∈ w (G) as a bounded sequence (by Lemma 1.1). As a result, f ∈ w (G) converges to t for any FS {S_n} of G. □

2.3 Theorem

For any FS {S_n} of G, let f ∈ w (G) be a function.

If ℓ is a cluster point of a function f ∈ w (G), then

${LIM}^{r} f \subseteq {\bar{B}}_{r} (ℓ) .$ (5)

Let $C$ be the set of all cluster points of a function f ∈ w (G). Then,

${LIM}^{r} f = ⋂_{ℓ \in C} {\bar{B}}_{r} (ℓ) = {t \in ℝ : C \subseteq {\bar{B}}_{r} (t)} .$ (6)

Proof. (i) For an arbitrary cluster point ℓ of f ∈ w (G), we have

$| t - ℓ | \leq r,$ (7) for all t ∈ LIM^rf . Otherwise, since ℓ is a cluster point of f ∈ w (G), there are infinite f ∈ w (G) for any FS {S_n} of G such that for every ɛ > 0, there exists a k_ɛ such that for all m ≥ k_ɛ, $\begin{matrix} | t - f (g) | > r + ɛ, \end{matrix}$ for all g ∈ G \ S_m, where $\begin{matrix} ɛ : = (| t - ℓ | - r) / 2 > 0, \end{matrix}$ which contradicts (1). Therefore, we have $\begin{matrix} {LIM}^{r} f \subseteq {\bar{B}}_{r} (ℓ) . \end{matrix}$

(ii) From (i), for any FS {S_n} of G, it is clearly

${LIM}^{r} f \subseteq ⋂_{ℓ \in C} {\bar{B}}_{r} (ℓ) .$ (8) If we let $s \in ⋂_{ℓ \in C} {\bar{B}}_{r} (ℓ),$ then for all $ℓ \in C$ $\begin{matrix} | s - ℓ | \leq r, \end{matrix}$ which is equivalent to $C \subseteq {\bar{B}}_{r} (s),$ that is, we have

$⋂_{ℓ \in C} {\bar{B}}_{r} (ℓ) \subseteq {t : C \subseteq {\bar{B}}_{r} (t)} .$ (9)

Conversely, if we let s ∉ LIM^rf then, by definition, there exists an ɛ > 0 such that for any FS {S_n} of G, there exists infinite f ∈ w (G) such that $\begin{matrix} | f (g) - s | \geq r + ɛ, \end{matrix}$ which implies the existence of a cluster point ℓ of f ∈ w (G) (with |s - ℓ | ≥ r + ɛ), that is, $\begin{matrix} C ⊊ {\bar{B}}_{r} (s) and s \notin {t : C \subseteq {\bar{B}}_{r} (t)} . \end{matrix}$ Therefore, if $\begin{matrix} s \in {LIM}^{r} f \end{matrix}$ then $\begin{matrix} s \in {t : C \subseteq {\bar{B}}_{r} (t)}, \end{matrix}$ that is, we have

${t : C \subseteq {\bar{B}}_{r} (t)} \subseteq {LIM}^{r} f .$ (10) Hence, from (8)-(10) we have $\begin{matrix} {LIM}^{r} f = ⋂_{ℓ \in C} {\bar{B}}_{r} (ℓ) = {t : C \subseteq {\bar{B}}_{r} (t)} . \end{matrix}$ □

By definition, Limsup f is the set of cluster points of f ∈ w (G) for any FS {S_n} of G. Hence, by (7) we get $\begin{matrix} Lim sup f \subset {\bar{B}}_{r} (t), \end{matrix}$ for all t ∈ LIM^rf and by (6), for any FS {S_n} of G $\begin{matrix} {LIM}^{r} f = ⋂_{ℓ \in Lim sup f} {\bar{B}}_{r} (s) . \end{matrix}$

2.4 Theorem

For any FS {S_n} of G, $\begin{matrix} {LIM}^{r} f = Lim inf {\bar{B}}_{r} (f) . \end{matrix}$

Proof. Firstly for any FS {S_n} of G, let s ∈ LIM^rf . Then, for every ɛ > 0 there exists a k_ɛ such that for all m ≥ k_ɛ, we define h (g) as following: $\begin{matrix} h (g) = {\begin{matrix} f (g) + \frac{r}{| s - f (g) |} (s - f (g)) & , & if | s - f (g) | > r \\ [0, 3 cm] s & , & otherwise, \end{matrix} \end{matrix}$ for all g ∈ G \ S_m. For any FS {S_n} of G, since $\begin{matrix} | h (g) - s | & = & | \frac{r}{| s - f (g) |} - 1 | \cdot | s - f (g) | \\ = & | | s - f (g) | - r |, \end{matrix}$ then we have $\begin{matrix} | h (g) - s | = {\begin{matrix} | s - f (g) | - r & , & if | s - f (g) | > r \\ [0, 3 cm] 0 & , & otherwise . \end{matrix} \end{matrix}$ Hence, s ∈ LIM^rf yields that h ∈ w (G) tends to s for any FS {S_n} of G, as n → ∞ . But |f (g) - h (g) | ≤ r, that is, $h \in {\bar{B}}_{r} (f) .$ As a results, $\begin{matrix} lim d (s, {\bar{B}}_{r} (f)) = 0, \end{matrix}$ and so by definition, $s \in Lim inf {\bar{B}}_{r} (f) .$ Hence, $\begin{matrix} {LIM}^{r} f \subset Lim inf {\bar{B}}_{r} (f) . \end{matrix}$

Now, let $s \in Lim inf {\bar{B}}_{r} (f)$ . By definition, there exists a sequence h ∈ w (G) satisfying h (g) → s and $h \in {\bar{B}}_{r} (f)$ for any FS {S_n} of G, i.e., $\begin{matrix} | f (g) - h (g) | \leq r . \end{matrix}$ Therefore, by Theorem 2.1 we get s ∈ LIM^rf . Hence, $\begin{matrix} Lim inf {\bar{B}}_{r} (f) \subset {LIM}^{r} f \end{matrix}$ and as a results, we have $\begin{matrix} {LIM}^{r} f = Lim inf {\bar{B}}_{r} (f) . \end{matrix}$ □

The previous theorems deals with the r-limit properties determined for a constant degree of roughness r. Let us now investigate the dependence of LIM^rf of a fixed sequence f ∈ w (G) for any FS {S_n} of G on a varying parameter r.

It follows from definition

${LIM}^{r_{1}} f \subseteq {LIM}^{r_{2}} f, if r_{1} < r_{2} .$ (11)

2.5 Theorem

For any FS {S_n} of G, suppose that r ≥ 0 and ρ > 0 . Then,

LIM^rf + ${\bar{B}}_{ρ}$ (0) ⊆ LIM^r+ρ f.

${\bar{B}}_{ρ} (s) \subseteq {LIM}^{r} f$ implies s ∈ LIM^r-ρf.

Proof. (i) Let s ∈ LIM^rf for any FS {S_n} of G and $s^{'} \in {\bar{B}}_{ρ} (0) .$ By definition for all ɛ > 0, there exists an k_ɛ such that for all g ∈ G ∖ S_m $\begin{matrix} | f (g) - s | < r + ɛ, (if m \geq k_{ɛ}) \end{matrix}$ which implies by |s′| < ρ that $\begin{matrix} | f (g) - s - s^{'} | \leq r + ρ + ɛ, (if m \geq k_{ɛ}) . \end{matrix}$ Therefore, s + s′ ∈ LIM^r+ρf .

(ii) Let ℓ be an arbitrary cluster point of f ∈ w (G) for any FS {S_n} of G. If $\begin{matrix} | s - ℓ | > r - ρ, \end{matrix}$ then the point $\begin{matrix} t : = s + \frac{ρ}{| s - ℓ |} (s - ℓ) \end{matrix}$ satisfies $\begin{matrix} | t - ℓ | = ρ + | s - ℓ | > ρ + (r - ρ) = r . \end{matrix}$ By (5), this yields t ∉ LIM^rf, a contradiction to |t - s| = ρ and ${\bar{B}}_{ρ} (s) \subseteq {LIM}^{r} f .$ Hence, for all cluster points $ℓ \in C,$ |s - ℓ | ≤ r - ρ. As a results, it follows from (6) $\begin{matrix} s \in ⋂_{ℓ \in C} {\bar{B}}_{r - ρ} (ℓ) = {LIM}^{r - ρ} f . \end{matrix}$ □

Now, define $\begin{matrix} \bar{r} : = inf {r > 0 : {LIM}^{r} f \neq \emptyset} . \end{matrix}$ By the monotonicity given in (11), we get

${LIM}^{r} f {\begin{matrix} = \emptyset, & if r < \bar{r} \\ [0.3 cm] \neq \emptyset, & if r > \bar{r} . \end{matrix}$ (12) Furthermore, by Theorem 2.5, for all $r > \bar{r}$ and $ρ \in (0, r - \bar{r})$ , LIM^rf always contains some ball with radius ρ. For $r > \bar{r}$ , this is at least $\begin{matrix} {int (LIM}^{r} f) \neq \emptyset . \end{matrix}$ For this reason, ${int (LIM}^{r} f) = \emptyset implies r \leq \bar{r} and {LIM}^{r^{'}} f = \emptyset,$ (13) for r′ ∈ [0, r).

2.6 Theorem

$r = \bar{r}$ if and only if

${LIM}^{r} f \neq \emptyset and {int (LIM}^{r} f) = \emptyset,$ (14) for any FS {S_n} of G.

Proof. Assume that $r = \bar{r} .$ We must show (14). It follows from Theorem 2.7 proved later that $\begin{matrix} {LIM}^{r} f = ⋂_{r^{'} > \bar{r}} {LIM}^{r^{'}} f . \end{matrix}$ For any FS {S_n} of G, by (12) LIM^r′f is non-empty and closed by Lemma 1.2, for $r^{'} > \bar{r}$ . Also, by (11), we get $\begin{matrix} ⋂_{r^{'} > \bar{r}} {LIM}^{r^{'}} f = ⋂_{\bar{r} < r^{'} \leq \bar{r} + 1} {LIM}^{r^{'}} f \end{matrix}$ and LIM^r′f, $r^{'} \in (\bar{r}, \bar{r} + 1]$ is a family of non-empty closed subsets in the compact set ${LIM}_{2}^{\bar{r} + 1} f$ having the finite intersection property. Hence, their intersection is non-empty and so for any FS {S_n} of G, $\begin{matrix} {LIM}^{\bar{r}} f \neq \emptyset . \end{matrix}$

If int (LIM^rf)≠ ∅, it contains some ball ${\bar{B}}_{ρ} (s)$ with ρ > 0 and by Theorem 2.5 we get LIM^r-ρf ≠ ∅ , that is, $r > \bar{r} .$ Therefore, $r = \bar{r}$ yields int (LIM^rf) =∅ for any FS {S_n} of G.

Suppose (14) holds. For any FS {S_n} of G, since LIM^rf≠ ∅, we get $r \geq \bar{r} .$ Moreover, by (13), $r \leq \bar{r}$ follows from int (LIM^rf) = ∅ . As a results, $r = \bar{r} .$ □

2.7 Theorem

For any FS {S_n} of G, $\begin{matrix} cl (⋃_{0 \leq r^{'} < r} {LIM}^{r^{'}} f) \subseteq {LIM}^{r} f = ⋂_{r^{'} > r} {LIM}^{r^{'}} f . \end{matrix}$ If $r \neq \bar{r}$ , then $\begin{matrix} cl (⋃_{0 \leq r^{'} < r} {LIM}^{r^{'}} f) = {LIM}^{r} f . \end{matrix}$

Proof. The closedness of r-limit (by Lemma 1.2) and by ((11)), we get $\begin{matrix} cl (⋃_{0 \leq r^{'} < r} {LIM}^{r^{'}} f) \subseteq {LIM}^{r} f \subseteq ⋂_{r^{'} > r} {LIM}^{r^{'}} f . \end{matrix}$ Now, let an arbitrary s ∉ LIM^rf . By definition for all ɛ > 0 there exists an k_ɛ such that for all m ≥ k_ɛ $\begin{matrix} | f (g) - s | \geq r + ɛ, \end{matrix}$ for all g ∈ G ∖ S_m. From here, for r′ < r + ɛ that ɛ′ : = r + ɛ - r′ > 0 and for all m ≥ k_ɛ and all g ∈ G ∖ S_m $\begin{matrix} | f (g) - s | \geq r^{'} + ɛ^{'} . \end{matrix}$ Hence, for r′ < r + ɛ, s ∉ LIM^r′f which implies $\begin{matrix} s \notin ⋂_{r^{'} > r} {LIM}^{r^{'}} f . \end{matrix}$ Thus, $\begin{matrix} {LIM}^{r} f = ⋂_{r^{'} > r} {LIM}^{r^{'}} f . \end{matrix}$ and so, for $r < \bar{r}$ we get clearly $\begin{matrix} cl (⋃_{0 \leq r^{'} < r} {LIM}^{r^{'}} f) = {LIM}^{r} f = \emptyset . \end{matrix}$ Choose $r = r_{1} > \bar{r}$ and $r_{0} = (\bar{r} + r_{1}) / 2 .$ Because of $r_{0} > \bar{r}$ , we can let a s₀ ∈ LIM^{r
₀}f ≠ ∅ . Choose an arbitrary s₁ ∈ LIM^{r
₁}f . By Lemma 1.3, for α ∈ [0, 1], we get $\begin{matrix} s_{α} = (1 - α) s_{0} + α s_{1} \in {LIM}^{(1 - α) r_{0} + α r_{1}} f . \end{matrix}$ As a results, for α ∈ [0, 1), we get $\begin{matrix} s_{α} \in ⋃_{0 \leq r^{'} < r} {LIM}^{r^{'}} f . \end{matrix}$ For α → 1, since $\begin{matrix} ∥ s_{α} - s_{1}, z ∥ = (1 - α) ∥ s_{0} - s_{1}, z ∥ \to 0, \end{matrix}$ then $\begin{matrix} s_{1} \in cl (⋃_{0 \leq r^{'} < r} {LIM}^{r^{'}} f) . \end{matrix}$ Thus, for $r > \bar{r}$ $\begin{matrix} cl (⋃_{0 \leq r^{'} < r} {LIM}^{r^{'}} f) = {LIM}^{r} f \end{matrix}$ holds true, too. □

References

Arslan

and Dündar

, Rough convergence in 2-normed spaces, Bull Math Anal Appl 10(3) (2018), 1–9.

Aytar

, Rough statistical convergence, Numer Funct Anal Optimiz 29(3-4) (2008), 291–303.

Day

, Amenable semigroups, Illinois J Math 1 (1957), 509–544.

Douglass

S.A.

, On a concept of summability in amenable semigroups, Math Scand 28 (1968), 96–102.

Douglass

S.A.

, Summing sequences for amenable semigroups, Michigan Math J 20 (1973), 169–179.

Dündar

, Nuray

and Ulusu

I

-convergent functions defined on amenable semigroups, (in review).

Dündar

, Ulusu

and Nuray

, Rough convergent functions defined on amenable semigroups, (in review).

Mah

P.F.

, Summability in amenable semigroups, Trans Amer Math Soc 156 (1971), 391–403.

Mah

P.F.

, Matrix summability in amenable semigroups, Proc Amer Math Soc 36 (1972), 414–420.

10.

Namioka

, Følner’s conditions for amenable semigroups, Math Scand 15 (1964), 18–28.

11.

Nuray

and Rhoades

B.E.

, Some kinds of convergence defined by Folner sequences, Analysis 31(4) (2011), 381–390.

12.

Phu

H.X.

, Rough convergence in normed linear spaces, Numer Funct Anal Optimiz 22 (2001), 199–222.

13.

Phu

H.X.

, Rough continuity of linear operators, Numer Funct Anal Optimiz 23 (2002), 139–146.

14.

Phu

H.X.

, Rough convergence in infinite dimensional normed spaces, Numer Funct Anal Optimiz 24 (2003), 285–301.

15.

Ulusu

, Dündar

and Nuray

, Some generalized convergence types using ideals in amenable semigroups, Bull Math Anal Appl 11(1) (2019), 28–35.

16.

Ulusu

, Dündar

and Aydýn

, Asymptotically

I

-statistical equivalent functions defined on amenable semigroups, Sigma J Engineering and Natural Sci 37(4) (2019), 1367–1373.

17.

Ulusu

, Nuray

and Dündar

I

-limit and

I

-cluster points for functions defined on amenable semigroups, Fundamental J Math Appl 4(1) (2021), 45–48.