Convexity property and fixed-point theorems in C Ω -modular space

Abstract

The purpose of this study is to introduce a new concept of the modular space, which is C_Ω-modular space, and then some of the convex properties are discussed. We also study finding fixed-point in C_Ω-modular space.

Keywords

𝒢-convergence 𝒢-Cauchy sequence

1 Introduction

The notion of modular space was introduced in 1950 s by Nakano [12]. After a while, it was redefined by Musielak and Orlicz [7]. A function $G : R \to [0, \infty]$ , where $R \neq \emptyset$ is a real vector space, if the following hold:

$G (r) = 0 \Rightarrow r = 0;$

$G (σ r) = | σ | G (r)$ , if |σ| = 1, then $G (σ r) = G (r);$

$G (σ r + (1 - σ) v) ⩽ σ G (r) + (1 - σ) G (v)$ , where σ ∈ [0, 1] .

Then the function $G$ is called convex modular.

A modular $G$ defines a modular space corresponding to that, i.e. the modular space $R_{G}$ is given by

$R_{G} = {u \in R : G (σ u) \to 0 as σ \to 0} .$

After that, many authors studied geometric properties of modular space, see ([4, 7] and, [8]). However, fixed point theory in modular space, was presented by many researchers (see [1–3 , 9–13] and, [15]).

The purpose of this article paper is to present a new concept of the modular space which is C_Ω-modular space, and then some of the convex properties will be discussed. We also studied finding fixed-point in C_Ω-modular space.

2 Basic definitions and main results

Definition 2.1. If $R_{G}$ and $T_{Σ}$ are two convex modular spaces, the operator $Ω : R \to T$ is called convex modular operator if $\begin{matrix} Ω (G (σ r + (1 - σ) v)) \\ = σ Σ (Ω (u)) + (1 - σ) Σ (Ω (v)) . \end{matrix}$

Definition 2.2. A function $G : R \to [0, \infty]$ , where $R \neq \emptyset$ is a real vector space, is called C_Ω-modular if the following hold:

$G (r) = 0 \Leftrightarrow r = 0;$

$G (σ r) = | σ | G (r)$ , if |σ| = 1, then $G (σ r) = G (r);$

If Ω is convex-modular operator on $R$ and $Ω (G (σ u + (1 - σ) v)) ⩽ σ G (Ω (u)) + (1 - σ) G (Ω (u))$ , for all $u, v \in R$ and σ ∈ [0, 1].

A modular $G$ defines a modular space corresponding to that, i.e. the space $R_{G}^{Ω}$ is given by

$R_{G}^{Ω} = {u \in R : Ω (G (σ u)) \to 0 as σ \to 0} .$

Then $R_{G}^{Ω}$ is called C_Ω-modular space.

Theorem 2.3. The function $ω_{u} : R \to [0, 1]$ defined as follow $ω_{u} (w) = \inf {0 ⩽ μ ⩽ 1 : w \in G (μ u + (1 - μ) R), \forall u \in R}$ is convex.

Proof. We must prove $ω_{u} (w) ⩽ (1 - μ) ω_{u} (u) + μ ω_{u} (v)$ for all $u, v \in R$ and 0 ⩽ μ ⩽ 1 with $w = G (μ u + (1 - μ) v)$

Now, let 0 ⩽ τ, ζ ⩽ 1 on condition $u = G (τ u + (1 - τ) a)$ and $v = G (ζ u + (1 - ζ) b)$ where $a, b \in R$ .

Suppose that ξ = (1 - μ) τ + μζ, thus 0 ⩽ ξ ⩽ 1 then

$\begin{matrix} w = G (μ G (τ u + (1 - τ) a) + (1 - μ) G (ζ u + (1 - ζ) b)) \\ = G ([(1 - μ) τ + μ ζ] u + [((1 - μ) (1 - τ) + μ (1 - ζ) \\ + μ ζ) a + ((1 - μ) (1 - τ) + μ (1 - ζ) + (1 - μ) τ) b] \\ = G (ξ u + (1 - ξ) e) \end{matrix}$

where, $e = G (\frac{1}{ξ} μ ζ a + (\frac{1}{ξ} (1 - μ) τ) b) \in R$ .

From this, we get ω_u (w) ⩽ (1 - μ) ω_u (u) + μω_u (v).

Hence ω_u (w) is convex.

Definition 2.4. Let $R_{G}^{Ω}$ be a C_Ω-modular space, then

$R_{G}^{Ω}$ -sequence {η_k} satisfies $G$ -convergence property if and only if $\lim_{k \to \infty} G_{η_{k} - η} = 0 .$

$R_{G}^{Ω}$ -sequence {η_k} is $G$ -Cauchy sequence if and only if $\lim_{k, j \to \infty} G_{η_{k} - η_{j}} = 0$ .

$R_{G}^{Ω}$ is $G$ -complete if every $G$ -Cauchy sequence is $G$ -convergent.

Definition 2.5. The function $φ : R_{G}^{Ω} \to R_{G}^{Ω}$ is considered a $G$ -contraction function if $G_{φ η - φ ζ} ⩽ G_{η - ζ}$ for all $η, ζ \in R_{G}^{Ω}$ .

Definition 2.6. The function $φ : R_{G}^{Ω} \to R_{G}^{Ω}$ is considered a $[σ, G]$ -contraction function if $G_{φ η - φ ζ} ⩽ G_{[σ (φ η - φ ζ) + (1 - σ) (η - ζ)]}$ for all $η, ζ \in R_{G}^{Ω}$ and σ ∈ (0, 1).

Theorem 2.7. $[σ, G]$ -contraction function is $G$ -contraction function.

Proof. Let $φ : R_{G}^{Ω} \to R_{G}^{Ω}$ be a $[σ, G]$ -contraction function, then $\begin{matrix} G_{φ η - φ ζ} ⩽ G_{[σ (φ η - φ ζ) + (1 - σ) (η - ζ)]} \\ ⩽ σ G_{φ η - φ ζ} + (1 - σ) G_{η - ζ} \\ \Rightarrow (1 - σ) G_{φ η - φ ζ} ⩽ (1 - σ) G_{η - ζ} . \end{matrix}$

Hence $G_{φ η - φ ζ} ⩽ G_{η - ζ}$ for all $η, ζ \in R_{G}^{Ω}$ and σ ∈ (0, 1).

Theorem 2.8. Let $R_{G}^{Ω}$ be a $G$ -complete C_Ω-modular space and the function $φ : R_{G}^{Ω} \to R_{G}^{Ω}$ , satisfies the following property $\begin{matrix} G_{φ η - φ ζ} ⩽ & π_{1} (x) max {G_{η - ζ}, G_{φ η - η}, G_{φ ζ - ζ}, G_{η - φ ζ}} \\ + π_{2} (x) G_{φ η - ζ} \end{matrix}$ where $η, ζ \in R_{G}^{Ω}$ and π₁, π₂ : (0, ∞) → (0, 1), then φ has a unique fixed-point in $R_{G}^{Ω}$ .

Proof.

Let {η_k} be a sequence in $R_{G}^{Ω}$ and η_k+1 = φη_k, we get $\begin{matrix} G_{η_{k + 1} - η_{k}} = G_{φ η_{k} - φ η_{k - 1}} ⩽ π_{1} (x) max {G_{η_{k} - η_{k - 1}}, \\ G_{φ η_{k} - η_{k}}, G_{φ η_{k - 1} - η_{k - 1}}, G_{η_{k} - φ η_{k - 1}}} + π_{2} (x) G_{φ η_{k} - η_{k - 1}} \\ = π_{1} (x) max {G_{η_{k} - η_{k - 1}}, G_{η_{k + 1} - η_{k}}, G_{η_{k} - η_{k - 1}}, G_{η_{k} - η_{k}}} \\ + π_{2} (x) G_{η_{k + 1} - η_{k - 1}} \\ ⩽ π_{1} (x) max {G_{η_{k} - η_{k - 1}}, G_{η_{k + 1} - η_{k}}} + π_{2} (x) \\ [G_{η_{k + 1} - η_{k}} + G_{η_{k} - η_{k - 1}}] \\ = [π_{1} (x) + π_{2} (x)] max {G_{η_{k} - η_{k - 1}}, G_{η_{k + 1} - η_{k}}} \\ = [π_{1} (x) + π_{2} (x)] G_{η_{k} - η_{k - 1}} \\ ⩽ {[π_{1} (x) + π_{2} (x)]}^{2} G_{η_{k - 1} - η_{k - 2}} ⩽ \dots ⩽ \\ {[π_{1} (x) + π_{2} (x)]}^{k} G_{η_{k} - η_{k - 1}} \end{matrix}$ as k → 0, we deduce $\lim_{k \to \infty} G_{η_{k + 1} - η_{k}} = 0, (because π_{1} (x), π_{2} (x) < 1)$

We have to prove {η_k} be a $G$ -Cauchy sequence

$\begin{matrix} G_{η_{k} - η_{l}} = G_{φ η_{k - 1} - φ η_{l - 1}} ⩽ π_{1} (x) \max {G_{η_{k - 1} - η_{l - 1}}, \\ G_{φ η_{k - 1} - η_{k - 1}}, G_{φ η_{l - 1} - η_{l - 1}}, G_{η_{k - 1} - φ η_{l - 1}}} + π_{2} (x) G_{φ η_{k - 1} - η_{l - 1}} \\ = π_{1} (x) \max {G_{η_{k - 1} - η_{l - 1}}, G_{η_{k} - η_{k - 1}}, G_{η_{l} - η_{l - 1}}, G_{η_{k - 1} - η_{l}}} \\ + π_{2} (x) G_{η_{k} - η_{l - 1}} \\ ⩽ π_{1} (x) \max {G_{η_{k - 1} - η_{l}} + G_{η_{l} - η_{l - 1}}, G_{η_{k} - η_{k - 1}}, G_{η_{l} - η_{l - 1}}, \\ G_{η_{k - 1} - η_{l}}} + π_{2} (x) G_{η_{k} - η_{l - 1}} \\ = π_{1} (x) \max {G_{η_{k - 1} - η_{l}} + 0, 0, 0, G_{η_{k - 1} - η_{l}}} \\ + π_{2} (x) G_{η_{k} - η_{l - 1}} \\ = π_{1} (x) G_{η_{k - 1} - η_{l}} + π_{2} (x) G_{η_{k} - η_{l - 1}} \\ ⩽ π_{1} (x) [G_{η_{k - 1} - η_{k}} + G_{η_{k} - η_{l}}] + π_{2} (x) \\ [G_{η_{k} - η_{k - 1}} + G_{η_{k - 1} - η_{l - 1}}] . \end{matrix}$

This implies $G_{η_{k} - η_{l}} ⩽ \frac{π_{2} (x)}{1 - π_{1} (x)} G_{η_{k - 1} - η_{l - 1}}$ . As k, l→ ∞, we get $\lim_{k, l \to \infty} G_{η_{k} - η_{l}} = 0$ .

So that {η_k} is a $G$ -Cauchy sequence, as $R_{G}^{Ω}$ is $G$ -complete. This implies that there exists $η \in R_{G}^{Ω}$ , such that $\lim_{k \to \infty} G_{η_{k} - η} = 0$ .

Now, we show that φη = η. $\begin{matrix} G_{η_{k} - φ η} = G_{φ η_{k - 1} - φ η} ⩽ π_{1} (x) max {G_{η_{k - 1} - η}, \\ G_{φ η_{k - 1} - η_{k - 1}}, G_{φ η - η}, G_{η_{k} - φ η}} + π_{2} (x) G_{φ η_{k - 1} - η} \\ = π_{1} (x) max {G_{η_{k - 1} - η}, G_{η_{k} - η_{k - 1}}, G_{φ η - η}, G_{η_{k} - φ η}} \\ + π_{2} (x) G_{η_{k} - η .} \end{matrix}$

Taking $\lim_{k \to \infty}$ , we have $G_{η_{k} - φ η} ⩽ π_{1} (x) G_{φ η - η}$ .

Since 0 < π₃ (x) < 1, it implies that $G_{φ η - η} = 0 \Rightarrow φ η = η$ .

Suppose that ζ is another fixed-point in $R_{G}^{Ω}$ and ζ ≠ η. $\begin{matrix} G_{η - ζ} = G_{φ η - φ ζ} ⩽ π_{1} (x) max {G_{η - ζ}, G_{φ η - η}, \\ G_{ζ - φ ζ}, G_{η - φ ζ}} + π_{2} (x) G_{φ η - ζ} \\ = π_{1} (x) max {G_{η - ζ}, 0, 0, G_{η - ζ}} + π_{2} (x) G_{η - ζ} \\ ⩽ [π_{1} (x) + π_{2} (x)] G_{η - ζ} . \end{matrix}$

Therefore, $G_{η - ζ} = 0 \Rightarrow η = ζ$ . (Because 0 < π₁ (x) + π₂ (x) < 1).

Hence φ has a unique fix-point in $R_{G}^{Ω}$ .

Theorem 2.9. Let $R_{G}^{Ω}$ be a $G$ -complete C_Ω-modular space and the function $φ : R_{G}^{Ω} \to R_{G}^{Ω}$ , satisfy the following property $G_{φ η - φ ζ} ⩽ π_{1} (x) G_{η - ζ} + π_{2} (x) G_{φ η - η} + π_{3} (x) G_{φ ζ - ζ}$ where $η, ζ \in R_{G}^{Ω}$ and π₁, π₂, π₃ : (0, ∞) → (0, 1), then φ has a unique fixed-point in $R_{G}^{Ω}$ .

Proof. Let {η_k} be a sequence in $R_{G}^{Ω}$ and η_k+1 = φη_k, we get $\begin{matrix} G_{η_{k + 1} - η_{k}} = G_{φ η_{k} - φ η_{k - 1}} ⩽ π_{1} (x) G_{η_{k} - η_{k - 1}} \\ + π_{2} (x) G_{φ η_{k} - η_{k}} + π_{3} (x) G_{φ η_{k - 1} - η_{k - 1}} \\ = π_{1} (x) G_{η_{k} - η_{k - 1}} + π_{2} (x) G_{η_{k + 1} - η_{k}} + π_{3} (x) G_{η_{k} - η_{k - 1}} \\ (1 - π_{2} (x)) G_{η_{k + 1} - η_{k}} ⩽ (π_{1} (x) + π_{3} (x)) G_{η_{k} - η_{k - 1}} \\ G_{η_{k + 1} - η_{k}} ⩽ ϑ_{π (x)} G_{η_{k} - η_{k - 1}} ⩽ ϑ_{π (x)}^{2} G_{η_{k - 1} - η_{k - 2}} ⩽ \dots \\ ⩽ ϑ_{π (x)}^{k} G_{η_{1} - η_{0}} \end{matrix} .$ where $ϑ_{π (x)}^{k} = \frac{{(π_{1} (x) + π_{3} (x))}^{k}}{1 - π_{2} (x)} < 1$ .

As k → 0, we deduce $G_{η_{k + 1} - η_{k}} \to 0 .$

We will verify {η_k} be a $G$ -Cauchy sequence $\begin{matrix} G_{η_{k} - η_{l}} = G_{φ η_{k - 1} - φ η_{l - 1}} ⩽ π_{1} (x) G_{η_{k - 1} - η_{l - 1}} + π_{2} (x) \\ G_{φ η_{k - 1} - η_{k - 1}} + π_{3} (x) G_{φ η_{l - 1} - η_{l - 1}} \\ = π_{1} (x) G_{η_{k - 1} - η_{l - 1}} + π_{2} (x) G_{η_{k} - η_{k - 1}} + π_{3} (x) G_{η_{l} - η_{l - 1}} \\ ⩽ π_{1} (x) [G_{η_{k - 1} - η_{l}} + G_{η_{l} - η_{l - 1}}] + π_{2} (x) G_{η_{k} - η_{k - 1}} \\ + π_{3} (x) G_{η_{l} - η_{l - 1}} \\ = π_{1} (x) G_{η_{k - 1} - η_{l}} + π_{2} (x) G_{η_{k} - η_{k - 1}} \\ + [π_{1} (x) + π_{3} (x)] G_{η_{l} - η_{l - 1} .} \end{matrix}$

As k, l→ ∞, we get $\lim_{k, l \to \infty} G_{η_{k} - η_{l}} = 0$ .

So that {η_k} is a $G$ -Cauchy sequence, as $R_{G}^{Ω}$ is $G$ -complete, this indicate that there exists $η \in R_{G}^{Ω}$ , such that $\lim_{k \to \infty} G_{η_{k} - η} = 0$ .

Now, we show that φη = η. $\begin{matrix} G_{η_{k + 1} - φ η} = G_{φ η_{k} - φ η} ⩽ π_{1} (x) G_{η_{k} - η} \\ + π_{2} (x) G_{φ η_{k} - η_{k}} + π_{3} (x) G_{φ η - η} \\ = π_{1} (x) G_{η_{k} - η} + π_{2} (x) G_{η_{k + 1} - η_{k}} + π_{3} (x) G_{φ η - η} . \end{matrix}$

Taking $\lim_{k \to \infty}$ , we obtain $G_{η_{k + 1} - φ η} ⩽ π_{3} (x) G_{φ η - η}$ .

It implies that $G_{φ η - η} = 0 \Rightarrow φ η = η$ .

Suppose that ζ ≠ η such that ζ be another fixed-point in $R_{G}^{Ω}$ . $\begin{matrix} G_{η - ζ} = G_{φ η - φ ζ} ⩽ π_{1} (x) G_{η - ζ} + π_{2} (x) G_{φ η - η} \\ + G_{φ ζ - ζ} = π_{1} (x) G_{η - ζ} (1 - π_{1} (x)) G_{η - ζ} ⩽ 0 \\ \Rightarrow G_{η - ζ} = 0 . \end{matrix}$

Then ζ = η, hence φ has a unique fixed-point in $R_{G}^{Ω}$ .

Footnotes

Acknowledgments

The authors would like to thank the referees for their comments and suggestions to improve the quality of the manuscript.

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