Certain Fixed Point Results On 𝔄-Metric Space Using Banach Orbital Contraction and Asymptotic Regularity

Kushal Roy; Debashis Dey; Mantu Saha

doi:10.1515/ms-2023-0036

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Certain Fixed Point Results On 𝔄-Metric Space Using Banach Orbital Contraction and Asymptotic Regularity

Kushal Roy , Debashis Dey and Mantu Saha

Published/Copyright: March 30, 2023

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From the journal Mathematica Slovaca Volume 73 Issue 2

ABSTRACT

In this paper, we investigate the existence of φ-fixed point for Banach orbital contraction over 𝔄-metric space. Also a fixed point result has been established via asymptotic regularity property over such generalized metric space. Our fixed point theorems have also been applied to the fixed circle problem. Moreover, we give some new solutions to the open problem raised by Özgür and Taş on the geometric properties of φ-fixed points of self-mappings and the existence and uniqueness of φ-fixed circles and φ-fixed discs for various classes of self-mappings.

MSC 2010: Primary 47H10; Secondary 54H25

Keywords: Fixed point; φ-fixed point; 𝔘-metric space; Banach orbital (F; φ 𝔘)-contraction; asymptotic regularity; fixed circle and φ-fixed circle problem

1. Introduction and preliminaries

A generalized metric type space namely 𝔘-metric space has recently been introduced by Roy et al. [21] The 𝔘-metric space [21] generalizes several metric structures like S-metric space by Shedghi et al. [23], A-metric space by Abbas et al. [1], A_b-metric space by Ughade et al. [26], extended b-metric space by Kamran et al. [11] and parametric metric space by Hussain et al. [8] with an unified approach. We extend our investigation on fixed point results for Banach orbital contraction over such generalized metric type space.

Definition 1

([21]). Let X be a nonempty set and let 𝔘: Xⁿ × (0,+∞) → [0,+∞) be a mapping. Then the function 𝔘 is called an 𝔘-metric if it satisfies the following conditions:

𝔘 (x₁, x₂, …, x_n; θ) = 0 if and only if x₁ = x₂ = ⋯ = x_n for any θ ∈ (0,+∞),
U(x1,x2,…,xn;θ)≤α(x1,x2,…,xn;θ)∑i=1nU(xi,xi,…,(xi)n−1,a;θ) for all x₁, x₂, …, x_n, a ∈ X and θ > 0, where α: Xⁿ × (0, + ∞) → [1, + ∞) is a given function.

The 𝔘-metric is called symmetric, if, for any x₁, x₂ ∈ X, 𝔘(x₁, x₁, …, x₁, x₂; θ) = 𝔘(x₂, x₂, …, x₂, x₁; θ) for all θ ∈ (0,+∞).

Example 1

([21]). Let X be the set of all real valued continuous functions with domain (0, + ∞) and 𝔘: Xⁿ × (0, + ∞) → [0, + ∞) be defined by U(f1,f2,…,fn;θ)=∑i=1n−1∑j=i+1n|fi(θ)−fj(θ)|2 for all f₁, f₂, …, f_n ∈ X and θ > 0. Then 𝔘 is an 𝔘-metric on X with α(f₁, f₂, …, f_n; θ) = 2 for all f₁, f₂, …, f_n ∈ X and θ ∈ (0, + ∞).

Example 2

([21]). Let X = C⁰ (0, + ∞), the space of all real valued continuous mappings with domain (0, + ∞) and let us define

U(f1,f2,…,fn;θ)=∑i=1n−1∑i<j|fi(θ)−fj(θ)|2+(∑i=1n−1∑i<j|fi(θ)−fj(θ)|2)2

for all f₁, f₂, …, f_n ∈ X and for all θ ∈ (0,∞). Then 𝔘 is an 𝔘-metric space with α(f1,f2,…,fn;θ)=n(1+∑i=1n−1∑i<j|fi(θ)−fj(θ)|2) for all f₁, f₂, …, f_n ∈ X and for all θ ∈ (0, + ∞).

Example 3

([21]). Let us consider the space X = C⁰(0, + ∞) and define the mapping

U(f1,f2,…,fn;θ)={ (1+1∑i=1n|fi(θ)|)∑i=1n−1∑i<j|fi(θ)−fj(θ)|2if any one of fi(θ)is non zero,0if f1(θ)=f2(θ)=…=fn(θ)=0,

for all f₁, f₂, …, f_n ∈ X and for all θ ∈ (0, + ∞). Then one can verify that 𝔘 is an 𝔘-metric space with

α(f1,f2,…,fn;θ)={n(1+1∑i=1n|fi(θ)|)if any one of fi(θ)is non zero,nif f1(θ)=f2(θ)=…=fn(θ)=0,

for all f₁, f₂, …, f_n ∈ X and for all θ ∈ (0,∞).

The notion of 𝔘-metric [21] space generalizes several of known metric type spaces, for example:

For n = 2 and for each x₁, x₂ ∈ X, if 𝔘 (x₁, x₂; θ) is a constant function with α (x₁, x₂; θ) = 1 for all x₁, x₂ ∈ X, θ∈ (0,∞), then a symmetric 𝔘-metric is the usual metric.
For n = 2 and for each x₁, x₂ ∈ X, if 𝔘(x₁, x₂; θ) is a constant function with α(x₁, x₂;θ) = s > 1 for all x₁, x₂ ∈ X, θ ∈ (0,∞), then a symmetric 𝔘-metric is a b-metric [4].
For n = 2, if α(x₁, x₂; θ) = 1 for all x₁, x₂ ∈ X, θ ∈ (0,∞), then a symmetric 𝔘-metric is a parametric metric space [9].
For n = 2, if α(x₁,x₂;θ) = s > 1 for all x₁, x₂ ∈ X, θ ∈ (0,∞), then a symmetric 𝔘-metric is a parametric b-metric.
For n = 2 and for each x₁, x₂ ∈ X, if 𝔘(x₁, x₂;θ) is a constant function with α(x₁, x₂; θ) is independent of θ for all x₁, x₂ ∈ X, then a symmetric 𝔘-metric is an extended b-metric [11].
For n = 3 and for each x₁, x₂, x₃ ∈ X, if 𝔘(x₁, x₂, x₃; θ) is a constant function with (x₁, x₂, x₃; θ) = 1 for all x₁, x₂, x₃ ∈ X, θ ∈ (0,∞), then an 𝔘-metric is an S-metric [23].
For n = 3 and for each x₁, x₂, x₃ ∈ X, if 𝔘 (x₁, x₂, x₃; θ) is a constant function with α(x₁, x₂, x₃; θ) = s > 1 for all x₁, x₂, x₃ ∈ X, θ ∈ (0,∞), then an 𝔘-metric is an S_b-metric [24].
For n = 3, if α(x₁, x₂, x₃; θ) = 1 for all x₁, x₂, x₃ ∈ X, θ ∈ (0,∞), then an 𝔘-metric is a parametric S-metric [25].
For n = 3, if α(x₁, x₂, x₃; θ) = s > 1 for all x₁, x₂, x₃ ∈ X, θ ∈ (0,∞), then an 𝔘-metric is a parametric S_b-metric.
For n = 3 and for each x₁, x₂, x₃ ∈ X, if 𝔘(x₁, x₂, x₃; θ) is a constant function with α(x₁, x₂, x₃; θ) independent of θ for all x₁, x₂, x₃ ∈ X, then an 𝔘-metric is an extended S_b-metric [12].
If for each x₁, x₂,…,x_n ∈ X, 𝔘 (x₁, x₂, …,x_n; θ) is a constant function with α(x₁, x₂,…,x_n; θ) = 1 for all x₁, x₂,…,x_n ∈ X, θ ∈ (0,∞), then an 𝔘-metric is an A-metric. [1].
If for each x₁, x₂,…,x_n ∈ X, 𝔘 (x₁, x₂,…,x_n; θ) is a constant function with α(x₁, x₂,…,x_n; θ) = s> 1 for all x₁, x₂,…,x_n ∈ X and θ ∈ (0,∞), then an 𝔘 -metric is an A_b-metric [26].
If α(x₁, x₂,…,x_n; θ) = 1 for all x₁, x₂,…,x_n ∈ X, θ ∈ (0,∞), then an 𝔘-metric is a parametric A-metric [20].

Definition 2.

Let (X, 𝔘) be an 𝔘-metric space. A sequence {x_k}⊂ X is said to be

convergent to an element x ∈ X if for any ε > 0, there exists N ∈ ℕ such that for any k ≥ N, we have 𝔘 (x_k, x_k, …,(x_k)_{n − 1}, x;θ) < ε for all θ > 0 that is 𝔘 (x_k, x_k,…,(x_k)_{n − 1}, x;θ) → 0 as k→ ∞ for all θ ∈ (0,∞);
Cauchy sequence if for any ε > 0, there exists M ∈ ℕ such that for any k,m≥ M, we have 𝔘 (x_k, x_k, …,(x_k)_{n − 1}, x_m; θ) < ε for all θ >0 that is 𝔘 (x_k,x_k,…,(x_k)_{n − 1},x_m; θ)→0 as k,m → ∞ for all θ ∈ (0,∞);
X is called complete if every Cauchy sequence in X is convergent.

Definition 3.

Let (X, 𝔘₁) and (Y, 𝔘₂) be two 𝔘-metric spaces and T: X → Y a mapping. Then T is said to be continuous at x₀ ∈ X if for any ε > 0 there exists δ > 0 such that 𝔘₂(Tx,Tx, …,(Tx)_{n − 1},Tx₀) < ε whenever 𝔘₁(x,x,…,(x)_n−1, x0) < δ for all θ > 0.

Proposition 1.1

Let (X, 𝔘) be an 𝔘-metric space. Then

U(x,x,…,(x)n−1,y;θ)≤α(x,x,…,(x)n−1,y;θ)U(y,y,…,(y)n−1,x;θ)

for all x,y ∈ X and for all θ > 0.

Lemma 1.1.

Let (X,𝔘) be an 𝔘-metric space which is either symmetric or there exists some s > 1 such that α(x₁, x₂,…,x_n; θ) ≤ s for all x₁,…,x_n ∈ X and θ ∈ (0,∞), then any convergent sequence in X has a unique limit.

In 2014, Jleli et al.[10] introduced the concept of φ-fixed point which generalizes the concept of usual fixed point of a mapping. Let us define the following sets for a self mapping T and a mapping φ: X→ [0,∞) on a nonempty set X,

FT={x∈X:Tx=x}andZφ={x∈X:φ(x)=0}.

Definition 4.

An element u ∈ X is called a φ-fixed point of the mapping T if u ∈ F_T ∩ Z_φ.

Jleli et al. have proved some φ-fixed point theorems for self mappings on a metric space with the help of some special type functions.

Let ℱ be the set of all functions F:[0,∞)³ → [0,∞) satisfying the following condition:

max{a,b} ≤ F(a,b,c) for all a,b,c ∈ [0,∞);
F(0,0,0) = 0;
F is continuous.

Some examples of functions belonged to the collection ℱ are given below.

F(a,b,c) = max{a,b}+c, (ii) F (a,b,c) = a + b + f(c), where f:[0,∞) → [0,∞) is a continuous function such that f(0) = 0.

The following theorem was proved in [10].

Theorem 1.2.

Let (X,d) be a complete metric space, φ:X → [0,∞) a lower semi-continuous mapping and F ∈ ℱ. A mapping T:X→X satisfies

F(d(Tx,Ty),φ(Tx),φ(Ty))≤tF(d(x,y),φ(x),φ(y)),t∈(0,1)

for all x,y ∈ X. Then F_T ⊂ Z_φ.

The concept of asymptotic regularity of a mapping is an useful tool for researchers working in the area of fixed point theory. Several researchers have used this concept to find fixed point of different types of mappings in metric spaces (see[6,19,22]).

Definition 5

([3,5]). In a metric space (X, d), a map T : X → X is said to be asymptotically regular at some point x ∈ X if limn→∞d(Tnx,Tn+1x)=0.

If T is asymptotically regular at all x ∈ X, then T is said to be asymptotically regular.

The asymptotic regularity like concept has been instrumental in applying fixed point theory to the different branches of non-linear analysis. One such application is well-posedness of fixed point problem. The definition of well-posedness of fixed point problem over metric spaces is as follows:

Definition 6

([2]) Let (X, d) be a metric space and F: (X, d) → (X, d) be a mapping. The fixed point problem of F is said to be well-posed if (i) F has a unique fixed point z ∈ X, (ii) for any sequence {x_n} in X with d(x_n, F(x_n)) → 0 as n → ∞ we have d(z, x_n) → 0 as n→ ∞.

Recently, Dey et al. [7] have established different conditions for well-posedness of a fixed point problem for single and multivalued operators, where these operators are either deterministic or random.

2. φ-fixed point theorems using Banach orbital (F,φ, 𝔘)-contraction

Definition 7.

Let (X, 𝔘) be an 𝔘-metric space, φ: X → [0,∞) be a given function and F ∈ ℱ. We say that the mapping T: X → X is a Banach orbital (F,φ, 𝔘)-contraction, if it satisfies the following condition:

1 F(U(T2x,…,(T2x)n−1,Tx;θ),φ(T2x),φ(Tx))≤qF(U(Tx,…,(Tx)n−1,x;θ),φ(Tx),φ(x))

for some q ∈ (0,1), for all x ∈ X and for all θ > 0.

Definition 8.

Let (X, 𝔘) be an 𝔘 -metric space. A self mapping T on X is said to be 𝔘-orbitally continuous if limi→∞U(Tkix,…,(Tkix)n−1,z)=0 implies

limi→∞U(Tki+1x,…,(Tki+1x)n−1,Tz)=0,x,z∈X.

Theorem 2.1

Let (X, 𝔘) be a complete symmetric 𝔘-metric space, φ: X → [0,∞) be a given function and F ∈ 𝓕. Suppose that the following conditions hold:

φ is lower semi-continuous;
T:X → X is a Banach orbital (F,φ,𝔘)-contraction with Lipschitz constant q;
limk,m→∞α(Tkx,…,(Tkx)n−1,Tmx;θ)<1q for all x ∈ X;
T is 𝔘-orbitally continuous.

Then T has a φ-fixed point in X and F_T ⊂ Z_φ.

Proof.

First we show that F_T⊂ Zφ. Suppose ζ ∈ X is a fixed point of T. Now,

2 F(U(T2ζ,…,(T2ζ)n−1,Tζ;θ),φ(T2ζ),φ(Tζ))≤qF(U(Tζ,…,(Tζ)n−1,ζ;θ),φ(Tζ),φ(ζ)),

which implies F(0,φ(ζ),φ(ζ)) ≤ qF(0,φ(ζ),φ(ζ)), that is F(0,φ (ζ) ,φ(ζ)) = 0. Therefore by the property of F we see that max{0,φ(ζ)} ≤ F(0, φ(ζ),φ(ζ)) = 0. Thus φ(ζ) = 0 and so ζ is a φ-fixed point of T. Hence any fixed point of T is also a φ-fixed point of T.

Now we prove the existence of φ-fixed point of T. Let x₀ ∈X be arbitrary. Let x_m = T^mx₀ for all m ∈ ℕ. Then for all m ∈ ℕ, we have

3 F(U(xm+1,...,(xm+1)n−1,xm;θ),φ(xm+1),φ(xm)) =F(U(Tm+1x,...,(Tm+1x)n−1,Tmx;θ),φ(Tm+1x),φ(Tmx)) ≤qF(U(Tmx,...,(Tmx)n−1,Tm−1x;θ),φ(Tmx),φ(Tm−1x)) =F(U(xm,...,(xm)n−1,xm−1;θ),φ(xm),φ(xm−1)).

Therefore from (3) we get

4 F(𝔘(xm+1,...,(xm+1)n−1,xm;θ),φ(xm+1),φ(xm))≤qmF(𝔘(x1,...,(x1)n−1,x0;θ,φ(x1),φ(x0))

for all m ∈ ℕ. Thus by the first property of F, it follows that for all m ≥ 1,

5 max{𝔘(xm+1,...,(xm+1)n−1,xm;θ),φ(xm+1)}≤qmF(𝔘(x1,...,(x1)n−1,x0;θ),φ(x1),φ(x0)).

Now for any 1 ≤ k < m, we have

6 U(xk,...,(xk)n−1,xm;θ)≤α(xk,...,(xk)n−1,xm;θ)[(n−1)U(xk,...,(xk)n−1,xk+1;θ)+U(xm,...,(xm)n−1,xk+1;θ)]=α(xk,...,(xk)n−1,xm;θ)[(n−1)U(xk,...,(xk)n−1,xk+1;θ)+U(xk+1,...,(xk+1)n−1,xm;θ)]≤(n−1)α(xk,...,(xk)n−1,xm;θ)U(xk+1,...,(xk+1)n−1,xk;θ)+ α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xk;θ)[(n−1)U(xk+1,...,(xk+1)n−1,xk+2;θ)+ U(xm,...,(xm)n−1,xk+2;θ)]=(n−1)α(xk,...,(xk)n−1,xm;θ)U(xk+1,...,(xk+1)n−1,xk;θ)+ (n−1)α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xm;θ)U(xk+2,...,(xk+2)n−1,xm;θ)+ α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xm;θ)U(xk+2,...,(xk+2)n−1,xm;θ)...

≤(n−1) [α(xk,...,(xk)n−1,xm;θ)U(xk+1,...,(xk+1)n−1,xk;θ)+ α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xm;θ)U(xk+2,...,(xk+2)n−1,xk+1;θ)+...+α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xm;θ)...... α(xm−2,...,(xm−2)n−1,xm;θ)U(xm−1,...,(xm−1)n−1,xm−2;θ)]+ α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xm;θ)...... α(xm−2,...,(xm−2)n−1,xm;θ)U(xm−1,...,(xm−1)n−1,xm;θ)

≤(n−1)[α(xk,...,(xk)n−1,xm;θ)U(xk+1,...,(xk+1)n−1,xk;θ)+ α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xm;θ)U(xk+2,...,(xk+2)n−1,xk+1;θ)+ ...+α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xm;θ)...... α(xm−2,...,(xm−2)n−1,xm;θ)U(xm−1,...,(xm−1)n−1,xm−2;θ)]+ α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xm;θ) ...... α(xm−2,...,(xm−2)n−1,xm;θ)U(xm−1,...,(xm−1)n−1,xm;θ)U(xm,...,(xm)n−1,xm−1;θ)]

≤(n−1)[α(xk,...,(xk)n−1,xm;θ)qk+α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xm;θ)qk+1 +...+α(xk,...,(xk)n−1,xm;θ)α(xk+1,...,(xk+1)n−1,xm;θ)... ... α(xm−2,...,(xm−2)n−1,xm;θ)α(xm−1,...,(xm−1)n−1,xm;θ)qm−1]F(U(x1,...,(x1)n−1,x0;θ),φ(x1),φ(x0)).

Let us denote Pj(m)(x0,θ)=(n−1)qj∏i=1jα(xi,…,(xi)n−1,xm;θ) for all j ∈ ℕ and for all θ > 0. Then from (6), we get

7 U(xk,…,(xk)n−1,xm;θ)≤(∑j=km−1Pj(m)(x0,θ))F(U(x1,…,(x1)n−1,x0;θ),φ(x1),φ(x0)),

Where 1 ≤ k < m. Now since limk,m→∞α(Tkx0,…,(Tkx0)n−1,Tmx0;θ)<1q , by ratio test we see that the series limm→∞∑r=1∞Pr(m)(x0,θ) is convergent and thus limk→∞∑j=km−1Pj(m)(x0,θ)=0. So (7) implies that x_m is a Cauchy sequence in X and by the completeness of X we get some z ∈ X such that x_m → z as m → ∞. Since T is 𝔘-orbitally continuous, by Lemma (1.1) it follows that T_z = z. Also (5) gives limi→∞φ(xi+1)=0. Since φ is lower semi-continuous it follows that φ(z) = 0. Therefore z is a φ-fixed point of T in X.

Example 4.

Let us consider the complete 𝔘-metric space X given in Example 1 with n = 3. Also let E = {f ∈ X : |f(θ)| ≤ 1 for all θ > 0}. Define T : X → X by Tf(θ)={11+eθ if f∈E,f(θ)1+|f(θ)| if f∉E.. Then T is a Banach orbital (F, φ, 𝔘)-contraction with Lipschitz constant 25, for the function φ(f) = 0 for all f ∈ X and for F(a, b, c) = a+b+c for all a, b, c ≥ 0. Then all the conditions of Theorem (2.1) are satisfied and f₀ ∈ X is a fixed point of T, where f0(θ)=11+eθ for all θ ∈ (0, ∞).

Now, some corollaries follow from Theorem (2.1).

Corollary 2.1.1.

Let(X, 𝔘) be a complete symmetric 𝔘-metric space and T : X → X be a mapping satisfying

8 U(Tx,Tx,…,(Tx)n−1,Ty;θ)≤rU(x,x,…,(x)n−1,y;θ)

for all x, y ∈ X and for all θ > 0, where r ∈ (0,1) is such that

limk,m→∞α(Tkx,…,(Tkx)n−1,Tmx;θ)<1rfor allx∈X.

Then T has a fixed point in X.

Proof.

Let limi→∞U(Tkix,…,(Tkix)n−1,z)=0, where x, z∈ X and {k_i} ⊂ ℕ. Then

U(Tki+1x,…,(Tki+1x)n−1,Tz)≤rU(Tkix,…,(Tkix)n−1,z)→0asi→∞.

Thus limi→∞Tki+1x=Tz. Hence T is 𝔘-orbitally continuous in X. Also for F(a, b, c) = a + b + c for all a, b, c∈ [0, ∞) and for the function φ such that φ(x) = 0 for all x∈ X we see that T is a Banach orbital (F, φ, 𝔘)-contraction with Lipschitz constant r ∈ (0,1). Therefore T has a φ-fixed point that is a fixed point in X.

Corollary 2.1.2.

Let (X, 𝔘) be a complete symmetric 𝔘-metric space and T: X → X be a 𝔘-orbitally continuous mapping satisfying

9 U(T2x,T2x,…,(T2x)n−1,Tx;θ)≤sU(Tx,Tx,…,(Tx)n−1,x;θ)

for all x ∈ X and for all θ > 0, where s ∈ (0, 1) is such that limk,m→∞α(Tkx,…,(Tkx)n−1,Tmx;θ)<1s for all x ∈ X. Then T has a fixed point in X.

Proof.

For F(a, b, c) = a + b + c for all a, b, c ∈ [0, ∞) and for the function φ such that φ(x) = 0 for all x ∈ X, we see that T is a Banach orbital (F, φ, 𝔘)-contraction with Lipschitz constant s ∈ (0, 1). Therefore T has a φ-fixed point that is a fixed point in X.

Corollary 2.1.3.

Let (X, 𝔘) be a complete symmetric 𝔘 -metric space and T : X → X be a 𝔘 -orbitally continuous mapping. If T satisfies

𝔘 (Tx, Tx,...,(Tx)_n–1, Ty; θ) ≤ t[𝔘(x, x,...,(x)_n–1, Tx; θ) + 𝔘 (y, y, ...,(y)_n–1, Ty; θ)] for all x, y ∈ X and for all θ > 0, where t∈(0,12)) is such that
limk,m→∞α(Tkx,…,(Tkx)n−1,Tmx;θ)<1−ttfor allx∈X,
or 𝔘(Tx, Tx,...,(Tx)_n–1, Ty; θ) ≤ emax{𝔘(x,x,...,(x)_n–1, Tx; θ), 𝔘(y,y,–,(y)_n–1, Ty; θ)} for all x, y ∈ X and for all θ > 0, where e ∈ (0,1) is such that
limk,m→∞α(Tkx,…,(Tkx)n−1,Tmx;θ)<1e for all x∈X,
or 𝔘(Tx, Tx, ...,(Tx)_n–1, Ty; θ) ≤ α𝔘(x, x,...,(x)_n–1, y; θ) + β𝔘 (x, x,...,(x)_n–1, Tx; θ) + γ 𝔘(y, y,...,(y)_n–1,Ty;θ)] for all x, y ∈ X and for all θ > 0, where α, β, γ ∈(0,1) with α + β + γ < 1 is such that
limk,m→∞α(Tkx,…,(Tkx)n−1,Tmx;θ)<1−γα+β for all x∈X,
or 𝔘 (Tx, Tx,...,(Tx)_n–1, Ty; θ)≤ μ𝔘 (x, x,...,(x)_n–1, y; θ) + ν𝔘 (y, y,...,(y)_n–1, Tx; θ) for all x, y ∈ X and for all θ > 0, where μ ∈ (0,1) and ν ≥ 0 is such that
limk,m→∞α(Tkx,…,(Tkx)n−1,Tmx;θ)<1μ for all x∈X.

Then T has a fixed point in X.

3. Fixed points via asymptotic regularity

Definition 9.

Let (X, 𝔘) be an 𝔘-metric space and T: X → X be a self mapping. Then T is said to be 𝔘-asymptotically regular at a point x ∈ X, if for all θ ∈ (0,∞)

10 U(Tmx,…,(Tmx)n−1,Tm+1x;θ)→0 as m→∞.

Definition 10.

Let (X, 𝔘) be an 𝔘 -metric space and T: X → X be a given mapping. A point u is said to be an orbital limit of T, if there exists a sequence {a_i} ⊂ ℕ such that limi→∞Taix0=u for some x₀ ∈ X.

Let H_i: [0,∞) → [0,∞) be functions such that H_i(0) = 0 and H_i is continuous at 0, i = 1,2.

Theorem 3.1.

Let (X, 𝔘) be a complete symmetric 𝔘-metric space and T : X → X be a mapping satisfying the following condition:

11 U(Tx,Tx,…,(Tx)n−1,Ty;θ) ≤e1(x,y;θ)H1(min{U(x,x,…,(x)n−1,Tx;θ),U(y,y,…,(y)n−1,Ty;θ)}) +e2(x,y;θ)H2(U(x,x,…,(x)n−1,Tx;θ)U(y,y,…,(y)n−1,Ty;θ)) +e3(x,y;θ)(U(x,x,…,(x)n−1,Tx;θ)+U(y,y,…,(y)n−1,Ty;θ))

for all x, y ∈ X and for all θ ∈ (0,∞), where e₁(x, y; θ), e₂ (x, y; θ) ≤ δ for some δ > 0 and e3(x,y;θ)≤δ<1λ, with λ=supz∈S,θ>0α(z,z,…,(z)n−1,Tz;θ)<∞, S is the collection of all orbital limits of T. If T is 𝔘-asymptotically regular at some point x₀ in X, then T has a unique fixed point in X.

Proof.

Let x_k = T^k_x₀ for all k ≥ 1. Then

12 U(xm,xm,…,(xm)n−1,xk;θ) =U(Tmx0,Tmx0,…,(Tmx0)n−1,Tkx0;θ)≤e1(Tm−1x0,Tk−1x0;θ)H1(min{U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ),U(Tk−1x0,Tk−1x0,…,(Tk−1x0)n−1,Tkx0;θ)})+ e2(Tm−1x0,Tk−1x0;θ)H2(U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ) ×U(Tk−1x0,Tk−1x0,…,(Tk−1x0)n−1,Tkx0;θ))+ e3(Tm−1x0,Tk−1x0;θ)(U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ)+ U(Tk−1x0,Tk−1x0,…,(Tk−1x0)n−1,Tkx0;θ))≤ΔH1(min{U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ),U(Tk−1x0,Tk−1x0,…,(Tk−1x0)n−1,Tkx0;θ)})+ ΔH2(U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ) ×U(Tk−1x0,Tk−1x0,…,(Tk−1x0)n−1,Tkx0;θ))+ δ(U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ)+ U(Tk−1x0,Tk−1x0,…,(Tk−1x0)n−1,Tkx0;θ))→0 as m,k→∞.

Therefore {x_p}_p∈ℕ is a Cauchy sequence in X. Since X is complete, there exists z ∈ X such that x_p → z as p → ∞. Now for any m ∈ ℕ,

13 U(xm,xm,…,(xm)n−1,Tz;θ) =U(Tmx0,Tmx0,…,(Tmx0)n−1,Tz;θ) ≤e1(Tm−1x0,z;θ)H1(min{U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ),U(z,z,…,(z)n−1,Tz;θ)}) +e2(Tm−1x0,z;θ)H2(U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ) ×U(z,z,…,(z)n−1,Tz;θ)) +e3(Tm−1x0,z;θ)(U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ) +U(z,z,…,(z)n−1,Tz;θ))

≤ΔH1(min{U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ),U(z,z,…,(z)n−1,Tz;θ)})+ΔH2(U(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ) ×U(z,z,…,(z)n−1,Tz;θ))+δU(Tm−1x0,Tm−1x0,…,(Tm−1x0)n−1,Tmx0;θ)

+δα(z,z,…,(z)n−1,Tz;θ)[∑(n−1)timesU(z,z,…,(z)n−1,Tmx0;θ)+U(Tz,Tz,…,(Tz)n−1,Tmx0;θ)].

Thus [1 – λδ] 𝔘(x_m, x_m,...,(x_m)_n–1, Tz; θ) → 0 as n → ∞, i.e., x_m → Tz as m → ∞. Hence by Lemma 1.1, it follows that Tz = z and T has a fixed point in X. The uniqueness of fixed point follows immediately. □

4. An application to the fixed-circle problem

In this section, we investigate new solutions to the fixed-circle problem raised by Ӧzgür and Taȿ (see [13, 17, 18]) in the context of 𝔘-metric spaces.

First, we introduce the notions of a circle and a fixed-circle in an 𝔘-metric space as following:

Definition 11.

Let (X, 𝔘) be an 𝔘-metric space. We denote a circle in X with center at x₀ ∈ X and radius l > 0 by Cx0,lU={x∈X:U(x,x,…,(x)n−1,x0;θ)=l. This circle is said to be a fixed circle of a mapping T : X → X if T x = x for every x ∈ C x∈Cx0,lU..

Now we present some existence theorems for a fixed circle of a self mapping T and give some related corollaries to show that our results extend and generalize several results proved earlier.

Theorem 4.1.

Let (X, 𝔘) be an 𝔘 -metric space and Cx0,lU,l be a circle on X. Let us define the mapping

14 FX×(0,∞)→[0,∞),F(x;θ)=U(x,x,…,(x)n−1,x0;θ)

for all x ∈ X and θ > 0. If there exists a self mapping T : X → X satisfying

(CI1) 𝔘(x,x,…,(x)n−1,Tx;θ)α(x,x,…,(x)n−1,Tx;θ)≤[F(x;θ)+𝔘(Tx;θ)−2l] and

(CI2) 𝔘 (Tx, Tx, …, (Tx)n−1,x₀;θ) ≤ l for all x∈Cx0,lU, and for any θ > 0, then the circle Cx0,lU, is a fixed circle of T.

Proof.

Let us take an arbitrary element x∈Cx0,lU. We show that Tx = x. Using the condition (CI1) for all θ > 0, we obtain

15 U(x,x,...,(x)n−1,Tx;θ)≤α(x,x,...,(x)n−1,Tx;θ)[ℱ(x;θ)+ℱ(Tx;θ)−2l]=α(x,x,...,(x)n−1,Tx;θ)[U(x,x,...,(x)n−1,x0;θ)+U(Tx,Tx,...,(Tx)n−1,x0;θ)−2l]=α(x,x,...,(x)n−1,Tx;θ)[U(Tx,Tx,...,(Tx)n−1,x0;θ)−l]

From the condition (CI2), we have two cases. If 𝔘 (Tx, Tx, …, (Tx)_n−1,x₀;θ) < l for some θ ∈ (0,∞), then [15] leads us to a contradiction. Therefore, 𝔘(Tx, Tx,…,(Tx)_n−1, x0;θ) = l for all θ > 0 and we have 𝔘(x, x,…,(x)_n−1,Tx; θ) ≤ α(x, x,…,(x)n−1,Tx;θ)[l−l] implying that 𝔘(x, x,…,(x)_n−1,Tx;θ) = 0 for all θ ∈ (0,∞), that is Tx = x. Hence the circle Cx0,lU, is a fixed circle of T.

Example 5

Let us take the 𝔘-metric space given in Example [1] with n = 3. Also let T : X → X be given by

Tf(θ)=f(θ) if |f(θ)|=1 for all θ>0,f(θ)+12exp⁡(θ) if |f(θ)|≠1 for some θ>0.

Then T satisfies all the conditions of Theorem [4.1] with respect to the circle C0,2U, where 0 is the zero function. Here we see that C0,2U is a fixed circle of T.

Corollary 4.1.1

(Theorem 2.2 [18]). Let (X,d) be a metric space and C_x0,t be any circle on X. Define the mapping ϑ : X → [0,∞) as ϑ(x) = d(x,x₀) for all x ∈ X. If there exists a self-mapping T : X → X satisfying

d(x,Tx) ≤ ϑ(x) + ϑ(Tx) − 2t and
d(Tx,x₀) ≤ t for all x ∈ C_x0,t,

then the circle C_x0,t is a fixed circle of T.

Corollary 4.1.2.

Let (X, S) be an S-metric space and x∈Cx0,tS be any circle on X. Define the mapping : X → [0,∞) as ς(x) = S(x, x, x₀) for all x ∈ X. If there exists a self-mapping T : X → X satisfying

S(x,x,Tx) ≤ ς(x)+ς(Tx)−2t and
S(Tx,Tx,x0) ≤ t for all x ∈ Cx0,tS

then the circle Cx0,tS is a fixed circle of T.

Theorem 4.2.

Let (X, 𝔘) be an 𝔘 -metric space and Cx0,lU be a circle on X. Let us take the mapping defined in Theorem [4.1.] If there exists a self mapping T : X→ X satisfying

(CII1) U(x,x,…,(x)n−1,Tx;θ)α(x,x,…,(x)n−1,Tx;θ)≤[F(x;θ)−F(Tx;θ)] and

(CII2) κU(x,x,…,(x)n−1,Tx;θ)α(x,x,…,(x)n−1,Tx;θ)+U(Tx,Tx,…,(Tx)n−1,x0;θ)≥l

for all x ∈ x∈Cx0,lU for any θ > 0 and for some κ ∈ [0,1), then the circle Cx0,lU is a fixed circle of T. □

Proof.

Consider an arbitrary element x ∈ x∈Cx0,lUWe show that x is a fixed point of T. Using conditions (CII1) and (CII2), we see that

16 U(x,x,...,(x)n−1,Tx;θ)≤α(x,x,...,(x)n−1,Tx;θ)[ℱ(x;θ)−ℱ(Tx;θ)]=α(x,x,...,(x)n−1,Tx;θ)[U(x,x,...,(x)n−1,x0;θ)−U(Tx,Tx,...,(Tx)n−1,x0;θ)]=α(x,x,...,(x)n−1,Tx;θ)[l−U(Tx,Tx,...,(Tx)n−1,x0;θ)]=α(x,x,...,(x)n−1,Tx;θ)[kU(x,x,...,(x)n−1,Tx0;θ)α(x,x,...,(x)n−1,Tx0;θ)+U(Tx,Tx,...,(Tx)n−1,x0;θ)−U(Tx,Tx,...,(Tx)n−1,x0;θ)]≤kU(x,x,...,(x)n−1,Tx0;θ)for allθ∈(0,∞),

which implies that 𝔘(x, x,…, (x)_n−1, Tx;θ) = 0 for all θ ∈ (0,∞), that is Tx = x. Thus the circle Cx0,lU is a fixed circle of T.

Example 6

Consider the 𝔘-metric space given in Example 1 and take n=4. Let T : X → X be given by

Tf(θ)=f(θ) if |f(θ)|=13 for all θ>0,sin⁡(θ)+{f(θ)}3 if |f(θ)|≠13 for some θ>0.

Then T satisfies all the conditions of Theorem 4.2 with respect to the circle C0,1U 0 is the null function. Here, we see that C0,1U is a fixed circle of T.

Corollary 4.2.1

([18 : Theorem 2.3]). Let (X, d) be a metric space and Cx0,t be any circle on X. Define the mapping ϑ : X→[0,∞) as ϑ(x) = d(x,x0) for all x ∈ X. If there exists a self-mapping T : X→X satisfying

d(x, Tx) ≤ ϑ(x) − ϑ(Tx) and
wd(x, Tx) + d(Tx, x0) ≥ t for all x ∈ Cx0,t and for some w ∈ [0,1),

then the circle Cx0,t is a fixed circle of T.

Corollary 4.2.2

([13: Theorem 3.2]). Let (X, S) be an S-metric space and Cx0,tS be any circle on X. Define the mapping ς : X → [0,∞) as ς(x) = S(x,x,x0) for all x ∈ X. If there exists a self-mapping T : X → X satisfying

S(x, x, Tx) ≤ ς(x) − ς(Tx) and
wS(x, x, Tx) + S(Tx, Tx, x0) ≥ t for all x ∈ x∈Cx0,tS and for some w ∈ [0,1),

then the circle Cx0,tS is a fixed circle of T.

Theorem 4.3.

Let (X, 𝔘) be an 𝔘 -metric space and Cx0,lU be a circle on X. Let us define the mapping ϒ_l : [0,∞) → ℝ by

Υl(t)={t−l if t>0,0 if t=0.

A self mapping T: X → X satisfies the following conditions:

(CIII1) 𝔘 (Tx, Tx,…,(Tx)_n−1, x₀;θ) = l for each x ∈ x∈Cx0,lU and for all θ > 0,

(CIII2) for each x,y(x≠y)∈Cx0,lU there exists some θ ∈ (0,∞) such that 𝔘 (Tx, Tx,…, (Tx)_n−1, Ty; θ) > l and

(CIII3) 𝔘 (Tx, Tx,…,(Tx)_n−1,T_y;θ) + ϒ_l(𝔘(x, x,…,(x)_n−1, Tx;θ))

≤ 𝔘(x,x,…,(x)_n−1,y;θ) for all x, y ∈ x,y∈Cx0,lU and for any θ > 0.

Then the circle Cx0,lU is a fixed circle of T.

Proof.

Let x ∈ x∈Cx0,lU be taken as arbitrary. If possible let x ≠ Tx. By using conditions (CIII1) and (CIII2), we get (Tx, Tx,…,(Tx)_n−1,T²x;θ1) > l for some θ₁ > 0. Also from the condition (CIII3) it follows that

17 U(Tx,Tx,…,(Tx)n−1,T2x;θ1)≤U(x,x,…,(x)n−1,Tx;θ1)−Υl(U(x,x,…,(x)n−1,Tx;θ1))=U(x,x,…,(x)n−1,Tx;θ1)−U(x,x,…,(x)n−1,Tx;θ1)+l=l, a contradiction.

Therefore x = Tx and since x is arbitrary, Cx0,lU is a fixed circle of T.

Corollary 4.3.1

([17:Theorem 3]). Let (X, d) be a metric space and C_x_0,t be any circle on X. Let us define the mapping γ_t : [0,∞)→ ℝ by Γt(j)=j−t if j>00 if j=0.. If there exists a self-mapping T : X→X satisfying

d(Tx,x0) = t for each x ∈ C_x_0,t
d(Tx, Ty) > t for each x,y ∈ C_x_0,t with x≠y and
d(Tx, Ty) ≤ d(x,y) − γ_t(d(x, Tx)) for each x,y ∈ C_x_0,t,

then the circle C_x_0,t is a fixed circle of T.

In the following, a uniqueness theorem for fixed circle of a self mapping T is given.

Theorem 4.4.

Let (X, 𝔘) be an 𝔘-metric space and T : X → X be a mapping satisfying the conditions (CI1) and (CI2) given in Theorem 4.1 (or, (CII1) and (CII2) given in Theorem 4.2 or (CIII1), (CIII2) and (CIII3) given in Theorem 4.3). If for any θ>0, T satisfies the contractive condition

18 U(Tx,Tx,…,(Tx)n−1,Ty;θ)≤rU(x,x,…,(x)n−1,y;θ)

for all x ∈ x∈Cx0,lU and y ∈ X ∖y∈X∖Cx0,lU, where Cx0,lU is a fixed circle of T and r ∈ [0,1), then T has a unique fixed circle in X.

Proof.

Since T satisfies the conditions (CI1) and (CI2) ((or, (CII1) and (CII2) or (CIII1), (CIII2) and (CIII3)), it is guaranteed that T has atleast one fixed circle in X. Let us assume that T has two different fixed circles Cx0,r0U and Cx1,r1U in X. Then either Cx0,r0U∖Cx1,r1U≠∅ or Cx1,r1U∖Cx0,r0U≠∅ Without loss of generality, let us take Cx1,r1U∖Cx0,r0U≠∅. Then we can choose some u ∈ u∈Cx0,r0U and v∈Cx1,r1U∖Cx0,r0U and by applying contraction condition (18) we have

19 U(u,u,…,(u)n−1,v;θ)=U(Tu,Tu,…,(Tu)n−1,Tv;θ)≤rU(u,u,…,(u)n−1,v;θ),

from which, it follows that 𝔘(u, u,…,(u)_n−1,v;θ) = 0 for all θ > 0, i.e., u = v leading to a contradiction. Thus Cx0,r0U=Cx1,r1U and it is the unique fixed circle of T. □

5. Solution to an open problem concerning to the geometric properties of non-unique φ-fixed points

Recently, N. Özgür and N. Taş proposed an open problem in their paper [15] regarding the geometric properties of φ-fixed points of self-mappings and the existence and uniqueness of φ-fixed circles and φ-fixed discs for various classes of self-mappings over metric spaces. In the paper [14 ,16], Özgür et al. found some solutions of this open problem in the setting of metric spaces.

We now give some new solutions to this open problem with the help of new contractive type self mappings over 𝔘 -metric spaces.

Definition 12.

Let (X, 𝔘) be an 𝔘-metric space, T : X → X be a mapping and φ : X → [0,∞) be a given function.

A circle Cx0,lU={x∈X:U(x,x,…,(x)n−1,x0;θ)=l, for all θ > 0}, x₀ ∈ X; l > 0, is said to be a φ-fixed circle of T if Cx0,lU⊂Fix⁡(T)∩Zφ.
A disc Dx0,lU={x∈X:U(x,x,…,(x)n−1,x0;θ)≤lfor all θ > 0}, x₀ ∈ X ; l > 0, is called a φ-fixed disc of T if Dx0,lU⊂Fix⁡(T)∩Zφ .

In this section, we give several φ-fixed circle (resp. φ-fixed disc) results using various geometric conditions and techniques on symmetric 𝔘 -metric spaces.

First we define ρ = inf{𝔘(x,x,...(x)_n–1,Tx;θ): x ∈ X with x ≠ Tx and for all θ > 0}.

Definition 13.

Let (X, 𝔘) be an 𝔘 -metric space, T : X → X be a mapping and φ : X → [0,∞) be a function. If there exists some x₀ ∈ X such that for all x ≠ T_x

20 F(U(x,x,…,(x)n−1,Tx;θ),φ(Tx),φ(x))≤hF(U(x,x,…,(x)n−1,x0;θ),φ(x),φ(x0)),

for any θ > 0 and some h ∈ (0,1), then T is called an F-type φ_x0-contraction.

Let ℱ’⊂ ℱ be taken as ℱ’ ={F ∈ ℱ : F(a,b,0) ≤ s_max{a, b} for s ≥ 1 chosen to be least and for all a, b ≥ 0}. s is called the coefficient of ℱ in ℱ’.

Some examples of functions belonged to the collection ℱ’ are given below.

F(a, b, c) = max{a, b, c},
F(a, b, c) = max{a, b} + c,
F(a,b,c)=a2+b2+c2,
F(a,b,c)=a2+b2+c,
F(a, b, c) = a + b + c,

where a, b, c ≥ 0

Theorem 5.1.

Let (X, 𝔘) be an 𝔘 -metric space, the number ρ is previously defined, φ : X → [0,∞) be a given function and T : X → X be an F-type φ_x₀-contraction with the point x₀ ∈ X, F ∈ ℱ’ be of coefficient s and h∈(0,1s). If x₀ ∈ Zφ and φ(x) ≤ 𝔘 (x, x,…,(x)_n−1,T_x;θ) for all x ∈ x∈Cx0,ρU, for all θ > 0, then the circle Cx0,ρU is a φ-fixed circle of T.

Proof.

We first prove that x₀ is a fixed point of T. If possible let x₀ ≠ Tx0. Then from (20) we have

21 F(U(x0,x0,…,(x0)n−1,Tx0;θ),φ(Tx0),φ(x0))≤hF(U(x0,x0,…,(x0)n−1,x0;θ),φ(x0),φ(x0))=0 for all θ>0,

implying that

max{U(x0,x0,…,(x0)n−1,Tx0;θ),φ(Tx0)}≤F(U(x0,x0,…,(x0)n−1,Tx0;θ),φ(Tx0),φ(x0))=0,

a contradiction. Hence x₀ ∈ Fix(T) ∩ Zφ. Now we have to consider two cases.

Case-I: If ρ=0, then clearly Cx0,ρU={x0}⊂Fix⁡(T)∩Zφ.

Case-II: If ρ > 0, then for any x ∈ x∈Cx0,ρU with x ≠ Tx,

22 F(U(x,x,…,(x)n−1,Tx;θ),φ(Tx),φ(x))≤hF(U(x,x,…,(x)n−1,x0;θ),φ(x),φ(x0)) =hF(U(x,x,…,(x)n−1,x0;θ),φ(x),0)=hF(ρ,φ(x),0)≤hsmax{ρ,φ(x)}.

If max{ρ, φ(x)} = ρ, then due to the property of F it follows that

0<ρ≤U(x,x,…,(x)n−1,Tx;θ)≤hsρ<ρ,

a contradiction. Also if max{ρ, φ(x)} = φ(x), then from the property of F and by the assumed condition,

0<ρ≤φ(x)≤U(x,x,…,(x)n−1,Tx;θ)≤hsφ(x),

leads to a contradiction. Therefore, it should be Tx = x. Thus x ∈ Fix(T) for all x ∈ x∈Cx0,ρU . Also the assumed condition gives φ(x) = 0 for all x∈Cx0,ρU and Cx0,ρU ⊂ Fix(T) ∩ Zφ. Hence in any case Cx0,ρU is a φ-fixed circle of T.

Example 7.

Let us take the 𝔘-metric space given in Example 1 with n = 3. Define T : X → X by Tf=f if f=g32f−12g if f≠g, where g : (0, ∞) → ℝ is given by g(θ)=eθcos⁡(θ)+θ2sin−1⁡θθ+1 for all θ > 0. Also let φ : X → [0, ∞) be defined by φ(g) = 0 and φ(f)=213infθ>0|f(θ)−g(θ)|2 for all f ≠ g. Then T is an F-type φ_x₀-contraction for the function F ∈ 𝓕′ given by F(a,b,c)=a2+b2+c for all a, b, c ≥ 0. Then the coefficient of F is 2. Here T satisfies all the conditions of Theorem 5.1 for h=12. Here we see that ρ = 0 and Cg,0U is a φ-fixed circle of T.

Corollary 5.1.1

([14: Theorem 2.1]). Let (X, d) be an usual metric space, ρ = inf{d(x,Tx : x ∈ X, x ≠ Tx}, φ : X → [0,∞) be a given function and T : X → X be a type 1 φ_x₀-contraction with the point x₀ ∈ X. If x₀ ∈ Z_φ and φ(x) ≤ d(x, Tx) for all x ∈ C_x₀,ρ = {x ∈ X : d(x, x₀) = ρ}, then the circle C_x₀,ρ is a φ-fixed circle of T.

Proof.

We know that any metric space is an 𝔘-metric space also. If we take F ∈ 𝓕′ defined by F(a, b, c) = max{a, b, c} for all a, b, c ∈ [0,∞), then an F-type φ_x₀-contraction reduces to type 1 φ_x₀-contraction and the corollary follows from our Theorem 5.1.

Corollary 5.1.2

([14: Theorem 2.5]). Let (X, d) be an usual metric space, ρ = inf{d(x, Tx) : x ∈ X, x ≠ Tx}, φ : X → [0,∞) be a given function and T : X → X be a type 2 φ_x₀-contraction with the point x₀ ∈ X. If x₀ ∈ Z_φ and φ(x) ≤ d(x, Tx) for all x ∈ C_x₀,ρ = {x ∈ X : d(x, x₀) = ρ}, then the circle C_x₀,ρ is a φ-fixed circle of T.

Proof.

We know that any metric space is an 𝔘-metric space also. If we take F ∈ 𝓕′ defined by F(a, b, c) = max{a, b}+c for all a, b, c ∈ [0,∞), then an F-type φ_x₀-contraction reduces to type 2 φ_x₀-contraction and the corollary follows from our Theorem 5.1.

Our Theorem partially generalizes Theorem 2.9 of [14].

Corollary 5.1.3

([14]). Let (X, d) be an usual metric space, ρ = inf{d(x, Tx) : x ∈ X, x ≠ Tx}, φ : X → [0,∞) be a given function and T : X → X be a type 3 φ_x₀-contraction with the point x₀ ∈ X and h∈(0,12). If x₀ ∈ Z_φ and φ(x) ≤ d(x, Tx) for all x ∈ C_x₀,ρ ={x ∈ X : d(x, x₀) = ρ}, then the circle C_x₀,ρ is a φ-fixed circle of T.

Proof.

We know that any metric space is an 𝔘-metric space also. If we take F ∈ 𝓕′ defined by F(a, b, c) = a + b + c for all a, b, c ∈ [0,∞), then an F-type φ_x₀-contraction reduces to type 3 φ_x₀-contraction and the corollary follows from our Theorem 5.1. □

Theorem 5.2.

Let (X, 𝔘) be an 𝔘-metric space, the number ρ is previously defined, φ : X → [0,∞) be a given function and T : X → X be an F-type φ_x₀-contraction with the point x₀ ∈ X, F ∈ 𝓕′ be of coefficient s and h∈(0,1s). If x₀ ∈ Z_φ and φ(x) ≤ 𝔘(x, x,…,(x)_n−1,Tx; θ) for all x∈Dx0,ρU, for all θ > 0, then the disc Dx0,ρU is a φ-fixed disc of T.

Proof.

The proof is similar to that of Theorem 5.1. □

Corollary 5.2.1

([14: Theorem 2.2]) Let (X, d) be an usual metric space, ρ = inf{d(x, Tx) : x ∈ X, x ≠ Tx}, φ : X → [0,∞) be a given function and T : X → X be a type 1 φ_x₀-contraction with the point x₀ ∈ X. If x₀ ∈ Z_φ and φ(x) ≤ d(x, Tx) for all x ∈ D_x₀,ρ = {x ∈ X : d(x, x₀) ≤ ρ}, then the disc D_x₀,ρ is a φ-fixed disc of T.

Proof.

Clearly any metric space is an 𝔘-metric space. If we consider F ∈ 𝓕′ defined by F(a, b, c) = max{a, b, c} for all a, b, c ∈ [0,∞), then an F-type φ_x₀-contraction reduces to type 1 φ_x₀-contraction and the corollary follows from our Theorem 5.2.

Corollary 5.2.2

([14: Theorem 2.6]). Let (X, d) be an usual metric space, ρ = inf{d(x, Tx) : x ∈ X, x ≠ Tx}, φ : X → [0,∞) be a given function and T : X → X be a type 1 φ_x₀-contraction with the point x₀ ∈ X. If x₀ ∈ Z_φ and φ(x) ≤ d(x, Tx) for all x ∈ D_x₀,ρ = {x ∈ X : d(x, x₀) ≤ ρ}, then the disc D_x₀,ρ is a φ-fixed disc of T.

Proof.

Clearly any metric space is an 𝔘-metric space. If we consider F ∈ 𝓕′ defined by F(a, b, c) = max{a, b}+c for all a, b, c ∈ [0,∞), then an F-type φ_x₀-contraction reduces to type 2 φ_x₀-contraction and the corollary follows from our Theorem 5.2.

Our Theorem partially generalizes Theorem 2.10 of [14]

Corollary 5.2.3

([14]). Let (X, d) be an usual metric space, ρ = inf{d(x, Tx) : x ∈ X, x ≠ Tx}, φ : X → [0,∞) be a given function and T : X → X be a type 1 φ_x₀-contraction with the point x₀ ∈ X and h∈(0,12). If x₀ ∈ Z_φ and φ(x) ≤ d(x, Tx) for all x ∈ D_x₀,ρ = {x ∈ X : d(x, x₀) ≤ ρ}, then the disc D_x₀,ρ is a φ-fixed disc of T.

Proof.

Clearly any metric space is an 𝔘 -metric space. If we consider F ∈ 𝓕′ defined by F(a, b, c) = a + b + c for all a, b, c ∈ [0,∞), then an F-type φ_x₀-contraction reduces to type 3 φ_x₀-contraction and the corollary follows from our Theorem 5.2. □

(Communicated by Marcus Waurick )

Acknowledgement.

Kushal Roy acknowledges financial support awarded by the Council of Scientific and Industrial Research, New Delhi, India, through research fellowship for carrying out research work leading to the preparation of this manuscript.

The authors also acknowledge the valuable observation and suggestions of the anonymous referees for improvement of the paper.

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Received: 2021-11-21

Accepted: 2022-04-23

Published Online: 2023-03-30

Published in Print: 2023-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ms-2023-0036

Keywords for this article

Fixed point; φ-fixed point; 𝔘-metric space; Banach orbital (F; φ 𝔘)-contraction; asymptotic regularity; fixed circle and φ-fixed circle problem

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