1. Introduction
Fixed point theory has been develo** extensively, since the ground breaking contraction principle [
1], due to the fact that a significant number of problems may be reduced to the determination of such a point associated to adequate operators. From that point onward, scientists focused on generalizations of this result. One of the directions of approach has been related to the class of map**s involved in the fixed point problem. Kannan [
2] changed the inequality of Banach so that the map**s in view need not to be continuous anymore—idea followed by Chatterjea [
3]. Hardy and Rogers [
4] referred to a linear contractive condition, which generalizes the ones previously mentioned. Ćirić [
5] introduced the contractive type map**s and related them to the orbitally completness. Berinde [
6] replaced the contractive term from the Banach contraction by a sum which involves it and an additional quantity. Another interesting generalization in this direction is due to Geraghty [
7], who employed appropriate functions in order to define contractive operators with the fixed point property. Ansari et al. [
8] combined further various types of map**s by the use of the operators of class C. Suzuki [
9] introduced a new generalized type of contractive condition, and provided conditions related to the domain of definition of the map**s involved, ensuring the existence of their fixed points. These results were even more generalized by García-Falset et al. [
10] to the operators with condition (E). Another fruitful idea of develo** fixed point results belonged to Wardowski [
11], who introduced implicit contractive conditions in this regard.
Another direction of extensions refers to the underlying space used to prove fixed point results. Generalizations had in view of drop** one or more conditions from the metric definition, or enlarge the triangle inequality. One extension in this direction is the
b-metric, whose early developments in fixed point theory were presented by Berinde and Păcurar [
12]. Shatanawi et al. [
13] used comparison functions to prove common fixed point results on
b-metric spaces. Amini-Hanandi [
14] used this context to develop a theory for quasicontractive operators. Dung and Hang [
15] studied features of the Caristi theorem in this background. The same setting was also used by Kamran et al. [
16] to develop Feng and Liu type F-contractions, or by Ali et al. [
17] to solve integral equations. Real world applications in this context were given by McConnell et al. [
18]. Interesting surveys on
b-metrics can be found in Berinde and Păcurar [
12] or Karapınar [
19], also some interesting generalizations can be found in Samreen et al. [
20]. The lack of continuity of
b-metrics imposed the introduction of strong
b-metric spaces by Kirk and Shahzad [
21], which allowed for obtaining stronger results. A useful comparison of these metric spaces with other generalized metrics was made by Cobzaş and Czerwik [
22].
The paper is organized as follows.
Section 2 presents some preliminary aspects used in the sequel.
Section 3 contains theorems on the existence of fixed points for weakly contractive contractions.
Section 4 presents some uniqueness results on the fixed points of operators introduced in the previous section.
2. Preliminary Issues
Bakhtin [
23] and Czerwik [
24,
25] relaxed the triangle inequality from the metric spaces, moving the focus towards the next definition.
Definition 1. Let S be a nonempty set and . A function is called b-metric if the following hypotheses are fulfilled.
(i) if and only if ;
(ii) δ is symmetric;
(iii)
if x, y, , thenThe pair is b-metric space.
The problem of this generalization is the lack of continuity of the distance function, as proved by Husain et al. [
26]. Still, in such spaces the uniqueness of limits of convergent sequences is satisfied. A method to overcome this difficulty is given by the next class of metric spaces.
Definition 2 ([
21]).
Let S be a nonempty set and . A function is called strong b-metric if the following hypotheses are fulfilled.(i) if and only if ;
(ii) if x, , ;
(iii)
if x, y, , thenThe pair is a strong b-metric space.
The class of metric spaces is strictly included into this one. In this regard, we give the next example inspired by Doan [
27].
Example 1. Let , and But all the properties of strong b-metrics are satisfied.
Note that, throughout the paper, designates a strong b-metric space with parameter .
For the convergence, Cauchy property and completeness, we have the following definitions.
Definition 3. Let be a sequence and . Then:
(i) The sequence is called convergent to x if ;
(ii) is called Cauchy sequence in S if ;
(iii) The strong b-metric space satisfies the completness condition if every Cauchy sequence in S is convergent.
Proposition 1 ([
21]).
Let . Then(i) The limit of a convergent sequence in a strong b-metric space is unique.
(ii) The strong b-metric is a continuous function.
Note that the last property means that strong b-metrics are continuous functions, while b-metrics are not. This fact allows us to obtain improved results for the case of strong b-metrics.
Another property of this class of metrics is the next.
Proposition 2 ([
21]).
Let be a sequence in a strong b-metric space and supposeThen, is a Cauchy sequence.
The contractive inequalities we will work with have at the base, the Geraghty map**s.
Definition 4 ([
7]).
A function , which satisfies the conditionis called the Geraghty function. We denote by the set of Geraghty functions.
As examples of Geraghty functions, we refer to the following ones.
Example 2. (1) , satisfies the conditions of Geraghty.
(2) The same conclusion is for ,
(3) The function , is a Geraghty operator.
(4) The function , is a Geraghty operator.
3. Existence Results
We are now ready to formulate our main results, using Geraghty functions and map**s inspired by works of Samet [
28], and Shatanawi and Pitea [
29].
Theorem 1. Let be a map**. Suppose that the following conditions are satisfied:
is a Geraghty function,
For all x, , the next relation is truewhere is continuous, such that , , , x, . Then, G has a fixed point and , converges to .
Proof. Define a sequence by , .
If
for some
, then obviously
G has a fixed point. Hence, we suppose that
,
. Then, for all
, we obtain
So is decreasing, let us say to .
If
, for
, we obtain
By the condition satisfied by Geraghty functions, it follows that goes to zero, hence .
Suppose
does not satisfy the Cauchy sequence condition. Let
. From [
30], there exists a subsequence
, with
,
,
bounded, with
From the generalized triangle inequality, we have
Taking lim sup after
, we have
and again, assuming that
, we obtain
, contradiction.
So is a Cauchy sequence in S, hence it is convergent, let us say, to .
Now, we have to prove that and from the uniqueness of the limit in strong b-metric spaces, it will result that .
From the hypotheses, we obtain
Taking , and using if necessary lim sup, we have and from here we obtain the conclusion. □
Taking particular cases for the Geraghty functions or for the continuous function from the definition of the generalized contraction, we obtain the known results in literature.
Corollary 1. Let be a map**. Suppose that the following condition is satisfied, for any x, :where is continuous, such that , , , x, . Then, G has a fixed point and , , converges to .
Corollary 2. Let be a map**. Suppose that the following condition is satisfied, for any x, ,where is continuous, such that , , , x, . Then, G has a fixed point and , , converges to .
Corollary 3. Let be a map**. Suppose that the following condition is satisfied, for any x, :where is continuous, such that , . Then G has a fixed point and , converges to .
Considering now , Theorem 1 gives the next consequence.
Corollary 4. Let be a map**. Suppose that the following conditions are satisfied:
is a Geraghty function,
For all y, , the next relation is true Then, G has a fixed point and , , converges to .
From all these results we can obtain outcomes regarding the classic metric spaces, which are strong b-metric spaces.
We continue with another fixed point existence result. In the following a and b are numbers from the interval , so that .
Theorem 2. Let be an operator. Suppose that the following conditions are satisfied:
β is a Geraghty function ,
, where is continuous, such that , , .
Then, G has a fixed point and converges to .
Proof. Define a sequence by .
If for some , then obviously G has a fixed point.
Hence, we suppose that .
Since .
Thus, the sequence has positive terms and is decreasing. Therefore, there exists such that . We will demonstrate that .
Suppose, to the contrary, that .
Since we have already proved that
This implies that . Since , then . This is a contradiction. So .
Next, we shall show that is a Cauchy sequence.
Let m, , m, .
From the generalized triangle inequality, we have
Taking lim after
, we have
Thus, is Cauchy sequence in S, hence it is convergent, let us say, to .
Now, we have to prove that and from the uniqueness of the limit in strong b-metric spaces, it will result that .
Taking
and using the continuity of the strong
b-metric, we obtain
Therefore and we obtain the conclusion. □
Taking in Theorem 2, we obtain a result in the direction of Kannan.
Corollary 5. Let be an operator. Suppose that the following conditions are satisfied:
β is a Geraghty function ,
, where is continuous, such that , , .
Then, G has a fixed point and converges to .
Considering now particular choices for and in Corollary 5, the next corollaries arise.
Corollary 6. Let be a map**. Suppose that the following condition is satisfied, for any x, :where , continuous, such that , , , x, . Then, G has a fixed point and , , converges to .
Corollary 7. Let be a map**. Suppose that the following condition is satisfied, for any x, :where , continuous, such that , , , x, . Then, G has a fixed point and , , converges to .
Corollary 8. Let be a map**. Suppose that the following condition is satisfied, for any x, where , continuous, such that , , , x, . Then, G has a fixed point and , converges to .
Corollary 9. Let be a map**. Suppose that the following conditions are satisfied:
is a Geraghty function,
For all , the next relation is true Then, G has a fixed point and , , converges to .
We provide now an example of a map**, which fulfills the conditions in Corollary 4.
Example 3. Let , andwhich is a strong b-metric, but not a classic metric. Consider , , , , and , and , .
The map** G satisfies the conditions in Corollary 4, and its fixed point is zero.