Existential examination of the coupled fixed point in generalized b-metric spaces and an application

Abstract

In this work, we build up some coupled fixed point results in b-metric spaces and employ the same to study the stability for a coupled system of functional equations.

Keywords

Coupled fixed point Hausdorff distance stability of a set-valued functional equation

1 Introduction and preliminaries

The fixed point results in metric spaces is a significant piece of nonlinear mathematical analysis. During the 1920s, Banach proposed and established the well known Banach Contraction Principle. From that point forward, many researchers suggest new ideas for generalized metric spaces and new works for the existence of fixed points. For example, in 1958, Luxemburg [12] gave the following concept for one of generalization of metric spaces

Definition 1.1. [12] Let X be a set. A distance mapping d : X × X → [0, ∞] is said to be a generalized metric on X, for all a, b, u ∈ X the following conditions are hold:

d (a, b) =0 if and only if a = b;

d (a, b) = d (b, a) ;

d (a, u)≤ d (a, b) + d (b, u) ;

every d-Cauchy sequence in X is d-convergent, i.e. $lim_{n, m \to \infty} d (x_{n}, x_{m}) = 0$ for a sequence x_n ∈ X, n = 1, 2, … implies the existence of an element x ∈ X with $lim_{n \to \infty} d (x, x_{n}) = 0$ , (x is unique by (1) and (3)).

The concept of a b-metric space has been presented in the references [6, 7]. Very newly, this concept was generalized in [1] to a generalized b-metric space (GbMS) in the next way.

Definition 1.2. [1] Let X be a nonempty set and b ≥ 1 be a given real number. A function M_b on X × X such that 0≤ M_b (a, b) ≤ ∞ for all a, b ∈ X is called a b-generalized metric distance if it satisfy the following properties

M_b (a, b) ≥0 if and only if a = b,

M_b (a, b) = M_b (b, a) (symmetry),

M_b (a, c) ≤ b [M_b (a, b) + M_b (b, c)] , (triangle inequality).

For all a, b, c ∈ X .

On the other hand, there are many important fixed point applications in metric areas. One of these important applications is concerned with using the results of fixed points to study the stability of Cauchy Jenson functional equations (CJFE). It worth to mention that the stability of functional equations due to Cauchy was initiated by Ulam [19] in 1940 which states that the solution of an equation varying marginally from a given solution, should of need be near the solution for the given equation.

In 2008, Park and An [17] studeid the stability of CJFE by a fixed point technique. In 2009, Gao et al. [9] gave a new generalized version of CJFE as follows:

Let S be an abelian group and n-divisible, where n ∈ N, the set of all natural numbers, and X be a normed space with the norm ∥, ∥ _X. For any function f : S → X, the equation $\begin{matrix} nf (\frac{x + y}{n} + z) = f (x) + f (y) + nf (z), \end{matrix}$ for each x, y, z ∈ S and n ∈ N is said to be the generalized CJFE. In special case, when n = 2, the equation is called the Cauchy Jensen equation.

In this paper we establish a new contribution of fixed point technique in b-metric spaces approach to study the stability for the following coupled system of the generalized set-valued CJFE: $\begin{matrix} nf (\frac{x + u}{n} + z, \frac{y + v}{n} + w) = \\ f (x, y) \oplus f (u, v) \oplus nf (z, w), \\ ng (\frac{x + u}{n} + z, \frac{y + v}{n} + w) = \\ g (x, y) \oplus g (u, v) \oplus ng (z, w), \end{matrix}$ (1) for all $x, y, z, u, v, w \in X, n \in ℕ$ and f, g : G → X .

Next, we recall some preliminaries that will be used in the main results of this paper Let Y be a Banach space. We defined the following:

2^Y = the set of power sets of Y;

C_b (Y) = all bounded and closed subsets of Y;

C_c (Y) = all convex and closed subsets of Y;

C_cb (Y) = all convex closed and bounded subsets of Y.

Definition 1.3. [9] On 2^Y, they consider the addition and the scalar multiplication as follows: $\begin{matrix} C + C^{'} = {x + x^{'} | x \in C, x^{'} \in C^{'}} and \end{matrix}$

$\begin{matrix} λ C = {λ x | x \in C}, \end{matrix}$ where C, C′ ∈ 2^Y and λ ∈ R, the set of all real numbers. Also, we define the following: $\begin{matrix} C \oplus C^{'} = \bar{C + C^{'}} . \end{matrix}$ Then $\begin{matrix} λ C + λ C^{'} = λ (C + C^{'}) and \end{matrix}$ $\begin{matrix} (λ + μ) C \subseteq λ C + μ C . \end{matrix}$ Also, when C is convex, we obtain $\begin{matrix} (λ + μ) C = λ C + μ C . \end{matrix}$ for all λ, μ ∈ R. For any set C ∈ 2^Y, the distance function d (. , C) and the support function s (. , C) are defined by $\begin{matrix} d (x, C) & = & inf_{x \in Y} {∥ x - y ∥ | y \in C}, \\ d (x^{*}, C) & = & sup_{x^{*} \in Y^{*}} {〈 x - y 〉 | x \in C} . \end{matrix}$ In the following, we give some definitions related to coupled fixed points which will be important in our results and for more information about this subject see [18 , 24–28]

Definition 1.4. [11] Let (X, ⪯) be a partially ordered space and let F : X × X → X. The function F is said to have the mixed monotone property if F (a, b) is non-decreasing monotone in a and is non-increasing monotone in b, that is, for each a, b ∈ X, $\begin{matrix} a_{1}, a_{2} \in X, a_{1} ⪯ a_{2} \Rightarrow F (a_{1}, b) ⪯ F (a_{2}, b), \end{matrix}$ and $\begin{matrix} b_{1}, b_{2} \in Y, b_{1} ⪯ b_{2} \Rightarrow F (a, b_{1}) ⪯ F (a, b_{2}) . \end{matrix}$

Definition 1.5. [11] A pair (a, b) ∈ X × X is called a coupled fixed point of the function F : X × X → X if a = F (a, b) and b = F (b, a).

2 On b-generalized metric spaces

Theorem 2.1. Suppose that (X, M_b) is a complete b-G.M.S and the function F : X × X → X be a continuous mapping having the mixed monotone property on X. Assume that there exists a β ∈ [0, 1) such that for w, t, u, v ∈ X, the following property holds: $M_{b} (F (w, t), F (u, v)) \leq \frac{β}{2} [M_{b} (w, u) + M_{b} (t, v)]$ (2) for all w ≤ u, t ≥ v and M_b (w, t)< ∞. If there exist w₀, t₀ ∈ X such that w₀ ≤ F (w₀, t₀) and t₀ ≥ F (t₀, w₀). Then the following alternative holds: either

for all n ≥ 0, we have $\begin{matrix} M_{b} (F^{n} (w, t), F^{n + 1} (w, t)) = \\ M_{b} (F^{n} (t, w), F^{n + 1} (t, w)) = \infty, & or \end{matrix}$

F has a coupled fixed point in X, that is, there exist w, t ∈ X such that w = F (w, t) and t = F (t, w).

Proof. By the given assumptions, there exists (w₀, t₀) ∈ X × X such that w₀ ≤ F (u₀, v₀) and t₀ ≥ F (t₀, w₀). Then, we can define (w₁, t₁) ∈ X × X such that w₁ = F (w₀, t₀) and t₁ = F (t₀, w₀), then w₀ ≤ F (w₀, t₀) = w₁ and t₀ ≥ F (t₀, w₀) = t₁. Also there exists (w₂, t₂) ∈ X × X such that w₂ = F (w₁, t₁) and t₂ = F (t₁, w₁). Since F has the mixed monotone property, we have, $\begin{matrix} w_{1} & = & F (w_{0}, t_{0}) \\ \leq & F (w_{1}, t_{1}) = FF (w_{0}, t_{0}) \\ = & F (F (w_{0}, t_{0}), F (w_{0}, t_{0})) = w_{2}, \\ t_{1} & = & F (t_{0}, w_{0}) \\ \leq & F (t_{1}, w_{1}) = FF (t_{0}, w_{0}) \\ = & F (F (t_{0}, w_{0}), F (w_{0}, t_{0})) = t_{2} . \end{matrix}$ Continuing in this way, we construct two sequences {w_n} and {t_n} in X such that $\begin{matrix} w_{n + 1} & = & F^{n + 1} (w_{0}, t_{0}) = {FF}^{n} (w_{0}, t_{0}) \\ = & F (F^{n} (w_{0}, t_{0}), F^{n} (t_{0}, w_{0})) \\ t_{n + 1} & = & F^{n + 1} (t_{0}, w_{0}) = {FF}^{n} (t_{0}, w_{0}) \\ = & F (F^{n} (t_{0}, w_{0}), F^{n} (w_{0}, t_{0})) \end{matrix}$ for all n = 0, 1, 2, . . ..

There are two mutually exclusive possibilities: either

for every integer i = 0, 1, 2 . . . . , one has $\begin{matrix} M_{b} (F^{n} (w_{0}, t_{0}), F^{n + 1} (w_{0}, t_{0})) = \infty and \\ M_{b} (F^{n} (t_{0}, w_{0}), F^{n + 1} (t_{0}, w_{0})) = \infty . \end{matrix}$ Which is exactly the alternative (i) of the conclusion of the theorem, or else

some integer i = 0, 1, 2 . . . . , one has $\begin{matrix} M_{b} (F^{n} (w_{0}, t_{0}), F^{n + 1} (w_{0}, t_{0})) & < & \infty and \\ M_{b} (F^{n} (t_{0}, w_{0}), F^{n + 1} (t_{0}, w_{0})) & < & \infty . \end{matrix}$

Now, we need to show that (b) implies alternative (ii) of the conclusion of the theorem.

If case (b) holds, let N = N (w₀, t₀) denote a particular one. For definiteness, one could choose the smallest of all integer n ≥ 0, such that

$\begin{matrix} M_{b} (F^{n} (w_{0}, t_{0}), F^{n + 1} (w_{0}, t_{0})) & < & \infty and \\ M_{b} (F^{n} (t_{0}, w_{0}), F^{n + 1} (t_{0}, w_{0})) & < & \infty . \end{matrix}$ Then, by (2), since M_b (F^N (w₀, t₀) , F^N+1 (w₀, t₀))< ∞ and M_b (F^N (t₀, w₀) , F^N+1 (t₀, w₀)) < ∞ , we get

$\begin{matrix} M_{b} (F^{N + 1} (w_{0}, t_{0}), F^{N + 2} (w_{0}, t_{0})) \\ = & M_{b} ({FF}^{N} (w_{0}, t_{0}), {FF}^{N + 1} (w_{0}, t_{0})) \\ = & M_{b} (F (F^{N} (w_{0}, t_{0}), F^{N} (t_{0}, w_{0})), \\ (F^{N + 1} (w_{0}, t_{0}), F^{N + 1} (t_{0}, w_{0})))) \\ \leq & \frac{β}{2} [M_{b} (F^{N} (w_{0}, t_{0}), F^{N + 1} (w_{0}, t_{0})) + \\ M_{b} (F^{N} (t_{0}, w_{0}), F^{N + 1} (t_{0}, w_{0})))] \\ < & \infty . \end{matrix}$ Also,

$\begin{matrix} M_{b} (F^{N + 1} (t_{0}, w_{0}), F^{N + 2} (t_{0}, w_{0})) \leq \\ \frac{β}{2} [M_{b} (F^{N} (t_{0}, w_{0}), F^{N + 1} (t_{0}, w_{0})) + \\ M_{b} (F^{N} (w_{0}, t_{0}), F^{N + 1} (w_{0}, t_{0})))] \\ < \infty . \end{matrix}$ However at this point, the triangle property (3) in Definition 1.1 infers that, at whatever point n > N, one has for each L = 1, 2, …, that

$\begin{matrix} M_{b} (F^{n} (w_{0}, t_{0}), F^{n + 1} (w_{0}, t_{0})) \leq \\ b Σ_{i = 1}^{L} M_{b} (F^{n + i - 1} (w_{0}, t_{0}), F^{n + i} (w_{0}, t_{0})) \\ \leq b Σ_{i = 1}^{L} (\frac{β^{n + 1 - i - N}}{2^{n + 1 - i - N}}) \\ M_{b} (F^{N} (w_{0}, t_{0}), F^{N + 1} (w_{0}, t_{0})) \\ \leq b {(\frac{β}{2})}^{n - N} \frac{1 - (\frac{β}{2})^{L}}{1 - \frac{β}{2}} \\ M_{b} (F^{N} (w_{0}, t_{0}), F^{N + 1} (w_{0}, t_{0})) . \end{matrix}$ Since 0 < L < 1, then the sequence ${F^{n} (w_{0}, t_{0})}_{n = 0}^{\infty}$ and similarly the sequence ${F^{n} (t_{0}, w_{0})}_{n = 0}^{\infty}$ are M_b-Cauchy sequences and by (4) in Definition 1.1 they are M_b-convergent. In other words, there exist w, t ∈ X such that $\begin{matrix} lim_{n \to \infty} M_{b} (F^{n} (w_{0}, t_{0}), w) = 0 & and \\ lim_{n \to \infty} M_{b} (F^{n} (t_{0}, w_{0}), t) = 0 . \end{matrix}$ At last, we guarantee F (w, t) = w and F (t, w) = t, since F is continuous at (w, t) then we have $\begin{matrix} F (w, t) & = & lim_{n \to \infty} F (F^{n} (w_{0}, t_{0}), F^{n} (t_{0}, w_{0})) \\ = & lim_{n \to \infty} F^{n + 1} (w_{0}, t_{0}) = w, \end{matrix}$ and $\begin{matrix} F (t, w) & = & lim_{n \to \infty} F (F^{n} (t_{0}, w_{0}), F^{n} (w_{0}, t_{0})) \\ = & lim_{n \to \infty} F^{n + 1} (t_{0}, w_{0}) = t . \end{matrix}$

Remark 2.1. Let f : X → X be a mapping from X into itself. If we put f (w) = F (w, t) and f (t) = F (t, w) in Theorem 2.1, then one can deduce the following theorem.

Theorem 2.2. Suppose that (X, M_b) is a partially ordered complete b-generalized metric space and the function f : X → X be a continuous strictly contractive mapping, that is, there exists a number β < 1 such that $M_{b} (fw, ft) \leq β M_{b} (w, t), \forall w \geq t .$ (3) If there exists w₀ ∈ X with w₀ ≤ f (w₀) , then f has a fixed point.

Remark 2.2. We note that the contractive condition in [8, Theorem 1.6] is slightly stronger than the condition (3) of Theorem 2.2.

3 An application

Let X be a real vector space and Y be a subset of X .

Definition 3.1. Let f, g : X × X → C_cb (Y) be two set-valued mappings.

The coupled generalized Cauchy-Jensen set-valued functional equation is defined by $\begin{matrix} nf (\frac{x + u}{n} + z, \frac{y + v}{n} + w) = \\ f (x, y) \oplus f (u, v) \oplus nf (z, w), \\ ng (\frac{x + u}{n} + z, \frac{y + v}{n} + w) = \\ g (x, y) \oplus g (u, v) \oplus ng (z, w), \end{matrix}$ (4) for all x, y, z, u, v, w ∈ X and $n \in ℕ, n \geq 2$

Every solution of the generalized Cauchy-Jensen set-valued functional equation is called a Cauchy-Jensen set-valued mapping.

Definition 3.2. For any set C ∈ 2^Y, the distance function d (. , C) and the support function s (. , C) are defined by $\begin{matrix} M_{b} (x, C) & = & inf_{x \in Y} {∥ x - y ∥ | y \in C} \\ M_{b} (x^{*}, C) & = & sup_{x^{*} \in Y^{*}} {〈 x - y 〉 | x \in C} \end{matrix}$ For all sets C, C′ ∈ C_b (Y), the Hausdorff b-distance between C and C′ is defined by $\begin{matrix} h_{b} (C, C^{'}) = \\ {(inf {λ > 0 | C \subseteq C^{'} + λ B_{Y}, C^{'} \subseteq C + λ B_{Y}})}^{s}, \end{matrix}$ where b = 2^s-1, B_Y is the closed unit ball in Y.

Proposition 3.1. For any C, C′, K, K′ ∈ C_cb (Y) and λ > 0, the following properties hold:

h_b (C ⊕ C′, K ⊕ K′) ≤ h_b (C ⊕ K) + (C′ ⊕ K′);

h_b (λC, λK) = λ^sh_b (C, K).

Theorem 3.1. Let f, g be two set-valued mappings defined on X × X into (C_cb (Y) , ⊕ , h_b) such that there exists a function

ψ : X × X × X → [0, ∞) satisfying

$\begin{matrix} h_{b} (nf (\frac{x + u}{n} + z, \frac{y + v}{n} + w), \\ f (x, y) \oplus f (u, v) \oplus nf (z, w))) \leq \\ ψ (x, u, z) + ψ (y, v, w) \end{matrix}$ (5) for all x, y, z, u, v, w ∈ X. If there exists L < 1 such that

$ψ (x, y, z) \leq \frac{L}{2} n ψ (nx, ny, nz),$ (6)

$ψ (u, v, w) \leq \frac{L}{2} n ψ (nu, nv, nw)$ (7) for all x, y, z, u, v, w ∈ X and n ∈ N, then there exists unique generalized Cauchy-Jensen set-valued mappings F, G : X × X → (C_cb (Y) , ⊕ , h_b) such that

$\begin{matrix} h_{b} (f (x, y), F (x, y)) \leq \\ \frac{1}{(1 - L) k} [ψ (x, x, x) + ψ (y, y, y)], \\ h_{b} (g (y, x), G (y, x)) \leq \\ \frac{1}{(1 - L) k} [ψ (x, x, x) + ψ (y, y, y)] . \end{matrix}$ (8) for all $x, y \in X, k = 1 + \frac{2}{n} .$

Proof. First, we consider the set S = {g : X × X → C_cb (Y) | g (0, 0) =0} and introduce the b-G.M.S on X as follows:

$M_{b} (g (x, y), f (x, y)) =$ (9) $inf {M \in [0, \infty) | h_{b} (g (x, y), f (x, y)) \leq M η},$ where η = ψ (x, x, x) + ψ (y, y, y) and inf ψ =+ ∞. Then (S, M_b) is a complete b-G.M.S (see [[10], Theorem (2.4)]). Now, we consider a linear mapping T : S × S → S such that

$T (f (x, y), g (x, y)) = \frac{1}{k} f (kx, ky),$ (10) and

$T (g (x, y), f (x, y)) = \frac{1}{k} g (kx, ky)$ (11) for all x, y ∈ X and $k = 1 + \frac{2}{n} .$

Next, we show that T is a strictly contractive mapping with Lipschitz constant L. Let g, f ∈ S with M_b (f (x, y) , u (x, y)) = S and $M_{b} (g (x, y), v (x, y)) = S^{'}$ for some $S, S^{'} \in R_{+}$ . It follows from (9) that

$\begin{array}{l} h_{b} (f (x, y), u (x, y)) \leq S [ψ (x, x, x) + ψ (y, y, y), \\ h_{b} (g (x, y), v (x, y)) \leq S^{'} [ψ (x, x, x) + ψ (y, y, y), \end{array}$ (12) for all x, y ∈ X . From Proposition 3.1, (6), (7) and (12) we obtain that $\begin{matrix} h_{b} (T (f, g), T (u, v)) \\ = & h_{b} (\frac{1}{k} f (kx, ky), \frac{1}{k} u (kx, ky)) \end{matrix}$ (13) $\begin{matrix} = & (\frac{1}{k})^{s} h_{b} (f (kx, ky), u (kx, ky)) \\ \leq & \frac{L}{2} (S + S') [ψ (x, x, x) + ψ (y, y, y)]}, \end{matrix}$ for all x ∈ X.

Hence, $M_{b} (T (f, g), T (u, v)) \leq \frac{L}{2} (S + S')$ , that is, $M_{b} (T (f, g), T (u, v)) \leq \frac{L}{2} [M_{b} (f, u) + M_{b} (g, v)]$ . Therefore, we suppose that x = u = z, y = v = w and in (5) since f (x, y) is convex, we have $\begin{matrix} h_{b} (nf (kx, ky), (n + 2) f (x, y)) \\ \leq ψ (x, x, x) + ψ (y, y, y), \end{matrix}$ for all x, y ∈ X. Then, we have $\begin{matrix} (n + 2)^{s} h_{b} (\frac{1}{k} f (kx, ky), f (x, y)) \\ \leq ψ (x, x, x) + ψ (y, y, y), \end{matrix}$ for all x, y ∈ X . Thus, by proposition 3.1, we have $\begin{matrix} h_{b} (T (f, g), f) \leq \frac{1}{(n + 2)^{s}} [ψ (x, x, x) + ψ (y, y, y)], \end{matrix}$ for all x, y ∈ X .

Similarly, one can deduce that $\begin{matrix} h_{b} (T (g, f), g) \leq \frac{1}{(n + 2)^{s}} [ψ (x, x, x) + ψ (y, y, y)], \end{matrix}$ and so $\begin{matrix} M_{b} (T (f, g), f) \leq \frac{1}{(n + 2)^{s}} < \infty, \end{matrix}$ and $\begin{matrix} M_{b} (T (g, f), g) \leq \frac{1}{(n + 2)^{s}} < \infty . \end{matrix}$ for all x, y ∈ X.

By Theorem 2.1, there exist two mappings F, G : X × X → (C_cb (Y) , h_b) such that the following conditions hold;

(F, G) is a coupled fixed point of T, that is, F (x, y) = T (F (x, y) , G (x, y)) and

G (x, y) = T (G (x, y) , F (x, y)) , for all x, y ∈ X. Then we have $\begin{matrix} F (x, y) = T (F (x, y), G (x, y)) = \frac{1}{k} F (kx, ky) \\ \Rightarrow F (kx, ky) = kF (x, y), \end{matrix}$ and $\begin{matrix} G (x, y) = T (G (x, y), F (x, y)) = \frac{1}{k} G (kx, ky) \\ \Rightarrow G (kx, ky) = kG (x, y) . \end{matrix}$

The sequences {T_n (f, g)} and {T_n (g, f)} converge to F, G respectively. This implies the following equality: $\begin{matrix} F (x, y) = lim_{n \to \infty} \frac{1}{k^{n}} f (k^{n} x, k^{n} y) \forall x, y \in X, \end{matrix}$ and $\begin{matrix} G (x, y) = lim_{n \to \infty} \frac{1}{k^{n}} g (k^{n} x, k^{n} y) \forall x, y \in X . \end{matrix}$ (c) We obtain that $M_{b} (f, F) \leq \frac{1}{1 - L} M_{b} (f, T$ (f, g)) and $M_{b} (g, G) \leq \frac{1}{1 - L} M_{b} (g, T (g, f))$ which implies to the following inequality: $\begin{matrix} M_{b} (f, F) \leq \frac{1}{(1 - L) (n + 2)^{s}} and \\ M_{b} (g, G) \leq \frac{1}{(1 - L) (n + 2)^{s}} . \end{matrix}$ Thus the inequalities (8) hold.

It follows from (5) and (6) that $\begin{matrix} h_{b} (nF (\frac{x + u}{n} + z, \frac{y + v}{n} + w), \\ (x, u) \oplus (y, v) \oplus n (z, w)) \\ = & lim_{n \to \infty} \frac{1}{k^{n}} \\ h_{b} (nf (\frac{k^{n} x + k^{n} u}{n} + k^{n} z, \frac{k^{n} y + k^{n} v}{n} + k^{n} w), \\ f (k^{n} x, k^{n} y) \oplus f (k^{n} u, k^{n} v) \oplus f (k^{n} z, k^{n} w)) \\ \leq & lim_{n \to \infty} \frac{1}{k^{n}} ψ (k^{n} x, k^{n} u, k^{n} z) + ψ (k^{n} y, k^{n} v, k^{n} w) \\ = 0, \end{matrix}$ for all x, y, z, u, v, w ∈ X. Thus, we have $\begin{matrix} h_{b} (nF (\frac{x + u}{n} + z, \frac{y + v}{n} + w), \\ F (x, y) \oplus F (u, v) \oplus nF (z, w)) = 0 . \end{matrix}$ So, we have $\begin{matrix} nF (\frac{x + u}{n} + z, \frac{y + v}{n} + w) \\ = & F (x, u) \oplus F (y, v) \oplus nF (z, w), \end{matrix}$ and similarly, one can get that $\begin{matrix} nG (\frac{x + u}{n} + z, \frac{y + v}{n} + w) \\ = & G (x, u) \oplus G (y, v) \oplus nG (z, w), \end{matrix}$ for all x, y, z, u, v, w ∈ X.

Remark. It will be interesting to find more applications to our current paper in another fields see [2–5 , 21]

Open problem. Can our results in this paper be obtained of n-tupled fixed point for nonexpansive mappings in generalized b-metric spaces with normal structure as in A. H. soliman et al [24 –28]?.

Availability of supporting data

No data were used to support this study.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

The author read and approved the final manuscript.

Funding

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research General Project under grant number R.G.P.1-162-40.

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