In this paper, we define the set of all real numbers as , the set of all non-negative real numbers as , the set of all non-negative integers by .

In 2012, Samet et al. [1] proposed the concept of α-admissible mapping, which plays a very important role in fixed point theory. In 2014, on the basis of α-admissible mapping, the concepts of α-orbit admissible and triangular α-orbit admissible mapping were raised by Popescu [2] .

Next, we will recall a new metric space. In 2013, Alghamdi et al. [3] introduced b-metric-like space for the first time. It is also known as a dislocated metric space by Karapınar [4] . In 2015, Chen et al. [5] established several novel theorems concerning fixed points and common fixed points within the context of b-metric spaces, and gave some examples and applications to prove the accuracy and practicability of their conclusions. In 2016, Gholamian et al. [6] introduced generalized Meir-Keeler compression in b-metric-like spaces, and proved the existence of fixed points by function. In 2017, Zoto et al. [7] introduced α-admissible mapping to prove the uniqueness of fixed point in b-metric-like spaces. In 2018, Gholizadeh et al. [8] proved the results of the best proximity point in b-metric-like spaces, and provided an example to prove the accuracy of the result. In the same year, Dakun et al. [9] introduced the concept of -contractions, and studied the common fixed point theorems for such contractions in b-metric-like spaces. In 2021, Javed et al. [10] extended b-metric-like-space to fuzzy b-metric-like-space, proved the uniqueness of fixed point, and applies it to integral equation, which shows the practicability of the results in this paper.

After the above mappings and related metric spaces, we continue to understand a contraction, that is, Geraghty type hybrid contractions. In 2020, Alzaid et al. [11] studied Geraghty type hybrid contractions and got some fixed point results in a b-metric space.

Subsequently, Karapinar et al. [12] introduced admissible hybrid Geraghty contractions by combining hybrid Geraghty contractions with α-admissible mapping, and obtained some fixed point results of such contractions in a complete metric space.

In 1996, a class of asymmetric structures were introduced firstly in terms of w-distance by Kada et al. [13] in metric spaces. In 2023, Hu et al. [14] generalized the w-distance and obtained the φ-fixed point result for nonlinear compression. Inspired by it, we also prove some fixed point theorems by w-distance. Next, we introduce some related lemmas about w-distance.

Inspired by the aforementioned research outcomes and the latest advancements in fixed point theory within b-metric-like spaces, we decide to expand the research results of Karapinar et al. [12] to the fixed disc of . And inspired by [14] , employing a w-distance, we introduce a new type contraction in b-metric-like spaces, called β-ζ-contraction, and some new fixed point theorems are obtained by weakening their conditions. On the one hand, our results improve the research results of Alzaid et al. [11] and expand the research results of Karapinar et al. [12] . On the other hand, it enriches the fixed point theory results under b-metric-like spaces.

The geometric characteristics of non-unique fixed points have been extensively examined from multiple perspectives. For instance, the fixed-disc problem, fixed-circle problem and so on. Özgür and Taş [15] introduced the concept of a fixed circle, and the fixed circle problem in metric space is proposed as a new direction for the promotion of fixed point theory. Later, Hussain et al. [16] presented some fixed disc results for improving contractions in F-metric space.

In 2019, the concept of α--admissible was raised by Aydi et al. [17] in rectangular metric spaces. Now we give a corresponding definition in b-metric-like spaces.

Definition 1. [2] Let be a function. We say that a mapping is α-orbital admissible if , for all .

Definition 2. [2] Let be a function. If meets the following conditions:

(i) , for all ;

(ii) , for all ,

then is called a triangular α-orbital admissible mapping.

Based on a certain understanding, below, we give the related concepts of b-metric-like-space.

Definition 3. [3] Let be a non-empty set and be a given real number. is a b-metric-like, if it meets the following conditions:

;

, for all ;

, for all ,

then is said to be a b-metric-like space.

Remark* [3] (i) If is a b-metric-like, then the self-distance might not be zero for some ;

(ii) Every b-metric is a b-metric-like, but the converse is not true in general. For example, the readers can refer to [3] .

Example 1. Let . Defined by:

Theorem 2. [11] Let be a complete b-metric space. If there exists such that satisfying the following conditions:

(i) , for all ;

(ii) is continuous or or , and

(iv) , for all ,

and

Definition 8. [15] Suppose that is a b-metric-like space and a nonempty set , is the set of all the fixed point of mapping and is a mapping. Define by

In this part, we get a new fixed circle result through the new contractions.

Now, we present the definitions of fixed circle and fixed disc in b-metric-like spaces.

Definition 10. Suppose that is a b-metric-like space with a nonempty set , is a point in and is a function. The mapping is α--admissible if

Case 1. .

By (2), it follows that

This is a contradiction. So, in all cases, we have , for all . The proof is completed. □

Corollary 1. Suppose that is a b-metric-like space with a nonempty set and is a function. If mapping is α--admissible and satisfies

where , with .

Definition 12. be a complete b-metric like space with a w-distance , . is called an admissible β-ζ-contraction if there exists a function such that

(4)

and

and

(6)

Case 1. There exists such that .

If , then . Therefore, by (c) of Definition 7, we have the limit of , and taking the limits at both sides of the inequality, we get , so that . When , it is a constant sequence, which is or . Then

(9)

where

(10)

By (9), we have . Then

Taking the limits on both sides of (13), and combining it with (12), we can get

(14)

and

(15)

Then, and

(16)

Let , in (4), where and by (6), it follows that

(17)

Then, we have

(18)

where

(19)

(20)

Taking the upper limits in (19), we have

(21)

Now, we take the upper limits on the both sides of (18), that is

This is a contradiction. Therefore, is a Cauchy sequence.

Step 3. We prove the existence of fixed points.

Since is complete, there exists such that . And by (iv), we have

(24)

The proof is completed. □

Theorem 8. When the condition (iv) is removed in Theorem 7, it can also be proved that has a fixed point .

Proof. Since , for each , there exists such that for all . As and is lower semicontinuous, we get

(25)

Taking the limits on both sides of (25), we obtain

(26)

Next, we will prove that . It can be divided into the following two cases.

Case 1. .

Suppose that there exists such that , and , by (c), we can conclude that , it says that . If for all ,

(27)

that is , where

(28)

Then, we can conclude that

It is obvious that is a complete b-metric-like space with . Let , and . Now we show that is an admissible hybrid Geraghty contraction. Indeed,

(i) if , , then (4) holds clearly.

(ii) if with or with ,

(29)

When , then

(iii''') is regular with respect to and .

We provide sufficient conditions for the existence of fixed points of in Theorem 7 (rep., Theorem 9, Theorem 10), but it can’t guarantee the uniqueness of fixed point by Examples 2. Now in order to assure the uniqueness of fixed point, consider the following condition:

(iv) for all , where is the set of fixed points of .

(v) , where .

Theorem 11. Adding (iv) and (v) to the conditions of Theorem 7 (rep., Theorem 9, Theorem 10), we can obtain the uniqueness of the fixed point of .

Proof. We prove the uniqueness of fixed points.

For , let and . Let in Theorem 7 and then , this is a contraction, so , take the same method, we have , then . Furthermore,

where is the set of all continuous real-valued functions defined on and . It is easy to see that is a complete b-metric-like space with . Let be a mapping defined by

(33)

Suppose that the following conditions hold:

(i) there exist a function and with , such that

We can attain that

This research was funded by the Natural Science Foundation of Sichuan Province (Grant No. 2023NSFSC1299), the Scientific Research and Innovation Team Program of Sichuan University of Science and Engineering (SUSE652B002), the Innovation Fund of Postgraduate, Sichuan University of Science and Engineering (Grant No. Y2023336).