1. Introduction
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.
2. Preliminaries
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:
It is obvious that 
is b-metric-like space with 
. However it is not a b-metric space, indeed, 
.
Definition 4. [3] Let 
be a b-metric-like space.
(i) A sequence 
in 
is said to be convergent to 
if
;
(ii) A sequence in 
is said to be a Cauchy sequence if 
exists and be finite;
(iii) 
is said to be complete if every Cauchy sequence in 
converges to some point in 
such that 
.
Proposition 1. [3] Let 
be a b-metric-like space. If is a sequence in 
such that 
, then has a unique limit 
.
Definition 5. [5] Let 
be a b-metric-like space and 
be a mapping. We say that 
is continuous at , if 
implies that 
, for each sequence in 
.
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
where 
, and 
, with 
, then 
has a unique fixed point and converges to some point 
in 
, where is produced by 
, for all 
.
Definition 6. [12] Let 
be a function. If for every sequence in 
such that 
, for all 
and 
, implies that 
, for all 
, we say that 
is regular with respect to 
.
Let 
satisfying that 
implies that 
.
Theorem 3. [12] Let 
be a complete metric space and 
be a mapping. If 
satisfies the following conditions:
(i) 
is triangular α-orbital admissible;
(ii) there exists 
such that 
;
(iii) one of the conditions satisfies:
(iiia) 
is continuous,
(iiib) 
is continuous and 
for all 
with 
,
(iiic) 
is regular with respect to 
;
(iv) 
, for all 
,
and
where 
, 
with 
, then 
has a fixed point in 
.
Definition 7. [13] Assume that 
be a metric space. If the following conditions are met:
(a) 
for all 
;
(b) 
is lower semi-continuous for any 
;
(c) there exists 
such that 
and 
imply
for any 
, then a function 
is called a w-distance on 
.
Lemma 4. [13] 
is a metric space equipped with a w-distance 
, 
is a sequence in 
, and it satisfies that for any 
, there exists 
such that 
(or 
) for 
. Then 
is a Cauchy sequence.
Lemma 5. [13] Suppose that 
is a metric space, and 
is a w-distance on 
and 
are three sequences in 
, 
.
(i) If 
and 
, then 
In particular, if
and 
, then 
;
(ii) If 
and , then 
converges to 
;
(iii) If 
and 
, then 
converges to 0.
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
(1) A circle 
in 
is said to be a fixed circle of 
if and only if 
.
(2) A disc 
in 
is said to be a fixed disc of 
if and only if 
.
Definition 9. [16] A simulation function is a mapping 
satisfying the following conditions:
(i) 
for all 
;
(ii) if 
are sequences in 
such that 
, then
3. Main Results
In the section, we present the main results by the following definitions.
Let 
satisfying 
implies that 
. And if 
, then 
, where 
.
3.1. Fixed Circle Result
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
Inspired by the above results, in virtue of α--admissible mapping, we obtain new fixed circle and fixed disc results in b-metric-like spaces.
Theorem 6. Suppose that 
is a b-metric-like space with a nonempty set 
and 
is a function. If the mapping 
is α-
-admissible and satisfies
(1)
and
where 
, 
with 
. In addition, 
and 
, for all 
, then 
is a fixed circle of 
.
Proof. Suppose that there exists such that 
. And from (1), , , for all and 
is α-
-admissible, we obtain
(2)
where
Case 1. 
.
By (2), it follows that
Both sides of the above inequality are simultaneously raised to the 
power, and we get
Taking account of the definition of 
, we have
or
This is contradictory with 
.
Case 2. 
.
It deduces from (2) 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
and
where , with 
. In addition, 
and 
, for all 
, then 
is fixed disc of 
.
Proof. The result follows by a similar discussion method as Theorem 6. The proof is completed. □
3.2. Fixed Point Results
In this part, we get some new fixed point results through some new contractions.
Definition 11. Let 
be a b-metric-like space. A mapping 
is called an α-β-admissible contraction if there exists a function 
such that
(3)
and
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
where , with .
Theorem 7. Let 
be a complete b-metric-like space with a w-distance 
, 
and 
be an admissible β-ζ-contraction. If the following conditions hold:
(i) 
is triangular α-orbital admissible;
(ii) there exists 
such that 
;
(iii) 
is continuous;
(iv) 
for any 
,
then 
has a unique fixed point 
.
Proof. Step 1. we will get 
, 
, for all 
,
, 
.
From (ii), there exists 
such that 
. Define a sequence 
by 
, for all 
. By (i) and induction, it follows easily that
(5)
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
Let 
in (4), we have
(7)
It follows that
where
(8)
that is 
, it is a contraction. So 
. Continuing this process, we can get 
. Similarly, we can prove 
. Case 2. 
.
Let 
in (4), then we have
that is
(9)
where
(10)
By (9), we have 
. Then
Then, we can get 
by definition of 
, and so
. By the same way, we can get 
.
To sum up, we have
Step 2. we suffice to prove that 
.
Suppose on the contrary that there exist 
and two sequences
such that
(11)
where 
. Then
(12)
By the triangle inequality, we have
(13)
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
By (16) and (21), thus it can be seen that
and then
By definition of 
, we can get
(22)
Indeed, taking in (20), (22) hold. Taking the limits in (19), by (22), we can get
(23)
By 
, we have
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
Then, 
, that is 
. By (i) of Lemma 5, we have
Case 2. 
.
This is easy to prove the above results. The proof is completed. □
Example 2 Let 
be endowed with the usual metric 
, and let 
for 
. 
is a mapping defined as follows:
Take 
and 
as follows:
and
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 
,
(iii) if 
, 
or 
, 
, then 
, that is (4) holds.
In all cases, (4) is satisfied. Thus, 
is an β-ζ contraction and it satisfies all conditions of Theorem 7. So 
has a fixed point 
such that 
.
Theorem 9 Let 
be a complete b-metric-like space with a w-distance 
, and 
be an admissible hybrid Geraghty contraction. If the following conditions hold:
(i) 
is triangular α-orbital admissible;
(ii) there exists 
such that 
;
(iii') 
is continuous and 
for any 
;
then 
has a fixed point in 
.
Proof. Following the proof of Theorem 7, we attain 
is a Cauchy sequence and there exists 
such that (24) holds. By (iii'), it deduces easily that
If 
. Let 
in (4), by (iii'), it follows that
(29)
When 
, then
This is a contradiction. So 
, that is, 
has a fixed point.
When 
, the same result can be proved. The proof is completed. □
Theorem 10. Let 
be a complete b-metric-like space with a w-distance 
, and 
be an admissible hybrid Geraghty contraction. If the following conditions hold:
(i) 
is triangular α-orbital admissible;
(ii) there exists 
such that 
;
(iii'') 
is regular,
then 
has a fixed point.
Proof. Similar to Theorem 7, we can also get 
. The proof is completed.
□
Remark* We can get the same result, if (iii'') replaced by the following condition:
(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,
The proof is completed. □
3.3. Application to Differential Equations
Consider the two-point boundary value problem of the second-order differential equation:
(30)
where 
is continuous. The boundary value problems (30) is equivalent to the following integral equation:
(31)
The Green function associated to (31) is defined by
Let 
be a function defined by
(32)
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
for all 
, 
, with 
, where
(ii) there exists 
such that 
for all 
;
(iii) for all 
, if 
is a sequence in 
such that
and 
, for all 
, then 
, for all 
.
Now we prove that existence of a solution of the above mentioned second-order differential equation.
Theorem 12. Under conditions (i)-(iv), (30) has a solution in 
.
Proof. It is well known that the solution of (30) is equivalent to the fixed point of 
in (33). Assume that 
such that 
, for all 
, and let 
. By (i), we get that
Let 
and
We can attain that
Thus, 
is an β-ζ-contraction. It is obvious that there exists 
such that 
by (ii), (iii) and the definition of 
. In addition, by (iv), if is a sequence in 
such that 
and 
for all 
, then 
, for all 
. 
satisfies the all conditions of Theorem 10, so (30) has a solution on 
. □
4. Conclusion
In this article, we firstly obtain some fixed circle result of α-
-admissible mapping in b-metric-like spaces. Secondly, with the help of w-distance 
, the new fixed point results are obtained. Finally, we give some examples to show the validity of our main results and apply our main results to the second-order differential equation. In addition, on the one hand, fixed point theory is an important component of nonlinear functional analysis theory, it is applied to multiple field, especially in the problem of the existence and uniqueness of solutions to various types of equations, for example, nonlinear functional differential equations, nonlinear Volterra integral equations. On the other hand, the fixed points of a neural network were determined by the fixed points of the employed activation function. So, in the future, we can use the fixed disc and β-ζ-contraction results obtained in this paper to judge whether some activation functions have fixed points.
Acknowledgements
The authors thank the anonymous reviewers for their excellent comments, suggestions, and ideas that helped improve this article.
Funding
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).