Abstract
In this paper, we introduce the notion of generalized G-β-ψ contractive mappings which is inspired by the concept of α-ψ contractive mappings. We showed the existence and uniqueness of a fixed point for such mappings in the setting of complete G-metric spaces. The main results of this paper extend, generalize and improve some well-known results on the topic in the literature. We state some examples to illustrate our results. We consider also some applications to show the validity of our results.
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1 Introduction and preliminaries
In nonlinear functional analysis, the importance of fixed point theory has been increasing rapidly as an interesting research field. One of the most important reasons for this development is the potential of application of fixed point theory not only in various branches of applied and pure mathematics, but also in many other disciplines such as chemistry, biology, physics, economics, computer science, engineering etc. We also emphasize the crucial role of celebrated results of Banach [1], known as a Banach contraction mapping principle or a Banach fixed point theorem, in the growth of this theory. In 1922, Banach proved that every contraction in a complete metric space has a unique fixed point. After this remarkable paper, a number of authors have extended/generalized/improved the Banach contraction mapping principle in various ways in different abstract spaces (see, e.g., [2–22]). One of the interesting and recent results in this direction was given by Samet et al. [23]. They defined the notion of α-ψ contractive mappings and proved that including the Banach fixed point theorems, some well-known fixed point results turn into corollaries of their results. Another interesting result was given in 2004 by Mustafa and Sims [24] by introducing the notion of a G-metric space as a generalization of the concept of a metric space. The authors characterized the Banach fixed point theorem in the context of a G-metric space. After this result, many authors have paid attention to this space and proved the existence and uniqueness of a fixed point in the context of a G-metric space (see, e.g., [11, 17–20, 24–48]). In this paper, we combine these two notions by introducing a G-β-ψ contractive mapping which is a characterization α-ψ contractive mappings in the context of G-metric spaces. Our main results generalize, extend and improve the existence results on the topic in the literature.
Let Ψ be a family of functions satisfying the following conditions:
-
(i)
ψ is nondecreasing;
-
(ii)
there exist and and a convergent series of nonnegative terms such that
for and any , where .
These functions are known in the literature as Bianchini-Grandolfi gauge functions in some sources (see, e.g., [21, 22, 49]) and as -comparison functions in some other sources (see, e.g., [50]).
Lemma 1 (See [50])
If , then the following hold:
-
(i)
converges to 0 as for all ;
-
(ii)
for any ;
-
(iii)
ψ is continuous at 0;
-
(iv)
the series converges for any .
Very recently, Karapınar and Samet [32] introduced the following concepts.
Definition 2 Let be a metric space and be a given mapping. We say that T is a generalized α-ψ contractive mapping if there exist two functions and such that
for all , where
Clearly, since ψ is nondecreasing, every α-ψ contractive mapping, presented in [23], is a generalized α-ψ contractive mapping.
Definition 3 Let and . We say that T is α-admissible if for all , we have
Various examples of such mappings are presented in [23]. The main results in [32] are the following fixed point theorems.
Theorem 4 Let be a complete metric space and be a generalized α-ψ contractive mapping. Suppose that
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then there exists such that .
Theorem 5 Let be a complete metric space and be a generalized α-ψ contractive mapping. Suppose that
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as , then for all n.
Then there exists such that .
Theorem 6 Adding to the hypotheses of Theorem 4 (resp. Theorem 5) the condition: For all , there exists such that and , we obtain the uniqueness of the fixed point of T.
Mustafa and Sims [24] introduced the concept of G-metric spaces as follows.
Definition 7 [24]
Let X be a nonempty set and be a function satisfying the following properties:
(G1) if ;
(G2) for all with ;
(G3) for all with ;
(G4) (symmetry in all three variables);
(G5) for all (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair is called a G-metric space.
Every G-metric on X defines a metric on X by
Example 8 Let be a metric space. The function , defined as
or
for all , is a G-metric on X.
Definition 9 [24]
Let be a G-metric space, and let be a sequence of points of X. We say that is G-convergent to if
that is, for any , there exists such that for all . We call x the limit of the sequence and write or .
Proposition 10 [24]
Let be a G-metric space. The following are equivalent:
-
(1)
is G-convergent to x;
-
(2)
as ;
-
(3)
as ;
-
(4)
as .
Definition 11 [24]
Let be a G-metric space. A sequence is called a G-Cauchy sequence if for any , there is such that for all , that is, as .
Proposition 12 [24]
Let be a G-metric space. Then the following are equivalent:
-
(1)
the sequence is G-Cauchy;
-
(2)
for any , there exists such that for all .
Definition 13 [24]
A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
Lemma 14 [24]
Let be a G-metric space. Then, for any , it follows that
-
(i)
if , then ;
-
(ii)
;
-
(iii)
;
-
(iv)
;
-
(v)
;
-
(vi)
.
Definition 15 (See [24])
Let be a G-metric space. A mapping is said to be G-continuous if is G-convergent to , where is any G-convergent sequence converging to x.
In [36], Mustafa characterized the well-known Banach contraction principle mapping in the context of G-metric spaces in the following way.
Theorem 16 (See [36])
Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Theorem 17 (See [36])
Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Remark 18 The condition (1) implies the condition (2). The converse is true only if . For details, see [36].
From [24, 36], each G-metric G on X generates a topology on X whose base is a family of open G-balls , where for all and . Moreover,
Proposition 19 Let be a G-metric space and A be a nonempty subset of X. Then A is G-closed if for any G-convergent sequence in A with limit x, one has .
2 Main results
We introduce the concept of generalized α-ψ contractive mappings as follows.
Definition 20 Let be a G-metric space and be a given mapping. We say that T is a generalized G-β-ψ contractive mapping of type I if there exist two functions and such that for all , we have
where
Definition 21 Let be a G-metric space and be a given mapping. We say that T is a generalized G-β-ψ contractive mapping of type II if there exist two functions and such that for all , we have
where
Remark 22 Clearly, any contractive mapping, that is, a mapping satisfying (1), is a generalized G-β-ψ contractive mapping of type I with for all and , . Analogously, a mapping satisfying (2) is a generalized G-β-ψ contractive mapping of type II with for all and , where .
Definition 23 Let and . We say that T is β-admissible if for all , we have
Example 24 Let and . Define by and
Then T is β-admissible.
Our first result is the following.
Theorem 25 Let be a complete G-metric space. Suppose that is a generalized G-β-ψ contractive mapping of type I and satisfies the following conditions:
(i) a T is β-admissible;
(ii) a there exists such that ;
(iii) b T is G-continuous.
Then there exists such that .
Proof Let be such that (such a point exists from the condition (ii) a ). Define the sequence in X by for all . If for some , then is a fixed point of T. So, we can assume that for all n. Since T is β-admissible, we have
Inductively, we have
From (3) and (5), it follows that for all , we have
On the other hand, we have
Thus, we have
We consider the following two cases:
Case 1: If for some n, then
which is a contradiction.
Case 2: If , then
for all . Since ψ is nondecreasing, by induction, we get
Using (G5) and (6), we have
Since and , by Lemma 1, we get that
Thus, we have
By Proposition 12, this implies that is a G-Cauchy sequence in the G-metric space . Since is complete, there exists such that is G-convergent to u. Since T is G-continuous, it follows that is G-convergent to Tu. By the uniqueness of the limit, we get , that is, u is a fixed point of T. □
Definition 26 (See [51])
Let be a G-metric space and be a given mapping. We say that T is a G-β-ψ contractive mapping of type I if there exist two functions and such that for all , we have
by following the lines of the proof of Theorem 25.
Corollary 27 Let be a complete G-metric space. Suppose that is a G-β-ψ contractive mapping of type I and satisfies the following conditions:
(i) a T is β-admissible;
(ii) a there exists such that ;
(iii) b T is G-continuous.
Then there exists such that .
Example 28 Let be endowed with the G-metric
Define by for all . We define in the following way:
One can easily show that
Then T is a G-β-ψ contractive mapping of type I with for all . Take such that . By the definition of T, this implies that . Then we have , and so T is β-admissible. All the conditions of Corollary 27 are satisfied. Here, 0 is the fixed point of T. Notice also that the Banach contraction mapping principle is not applicable. Indeed, for all . Then we have for all .
It is clear that Theorem 16 is not applicable.
The following result can be easily concluded from Theorem 25.
Corollary 29 Let be a complete G-metric space. Suppose that is a generalized G-β-ψ contractive mapping of type II and satisfies the following conditions:
(i) a T is β-admissible;
(ii) a there exists such that ;
(iii) b T is G-continuous.
Then there exists such that .
The next theorem does not require the continuity of T.
Theorem 30 Let be a complete G-metric space. Suppose that is a generalized G-β-ψ contractive mapping of type I such that ψ is continuous and satisfies the following conditions:
(i) b T is β-admissible;
(ii) b there exists such that ;
(iii) b if is a sequence in X such that for all n and is a G-convergent to , then for all n.
Then there exists such that .
Proof Following the proof of Theorem 25, we know that the sequence defined by for all , is a G-Cauchy sequence in the complete G-metric space , that is, G-convergent to . From (5) and (iii) b , we have
Using (8), we have
where
Letting in the following inequality:
it follows that
which is a contradiction. Thus, we obtain , that is, by Lemma 14, . □
The following corollary can be easily derived from Theorem 30.
Corollary 31 Let be a complete G-metric space. Suppose that is a generalized G-β-ψ contractive mapping of type II such that ψ is continuous and satisfies the following conditions:
(i) b T is β-admissible;
(ii) b there exists such that ;
(iii) b if is a sequence in X such that for all n and is a G-convergent to , then for all n.
Then there exists such that .
With the following example, we will show that the hypotheses in Theorems 25 and 30 do not guarantee uniqueness.
Example 32 Let be endowed with the following G-metric:
for all . Obviously, is a complete metric space. The mapping is trivially continuous and satisfies, for any ,
for all , where
Thus T is a generalized G-β-ψ contractive mapping. On the other hand, for all , we have
which yields that
Hence T is β-admissible. Moreover, for all , we have . So, the assumptions of Theorem 25 are satisfied. Note that the assumptions of Theorem 30 are also satisfied, indeed, if is a sequence in X that converges to some point with for all n, then from the definition of β, we have for all n, which implies that for all n. However, in this case, T has two fixed points in X.
Let X be a set and T be a self-mapping on X. The set of all fixed points of T will be denoted by .
Theorem 33 Adding the following condition to the hypotheses of Theorem 25 (resp. Theorem 30, Corollary 29, Corollary 31), we obtain the uniqueness of the fixed point of T.
-
(iv)
For , for all .
Proof Let be two fixed points of T. By (iv), we derive
Notice that since u and v are fixed points of T. Consequently, we have
where
Thus, we get that
which is a contradiction. Therefore, , i.e., the fixed point of T is unique. □
Corollary 34 Let be a complete G-metric space and let be a given mapping. Suppose that there exists a continuous function such that
for all . Then T has a unique fixed point.
Corollary 35 Let be a complete G-metric space and let be a given mapping. Suppose that there exists a function such that
for all . Then T has a unique fixed point.
Corollary 36 Let be a complete G-metric space and let be a given mapping. Suppose that there exists such that
for all . Then T has a unique fixed point.
Corollary 37 Let be a complete G-metric space and let be a given mapping. Suppose that there exist nonnegative real numbers a, b, c, d and e with such that
for all . Then T has a unique fixed point.
Corollary 38 (See [40])
Let be a complete G-metric space and let be a given mapping. Suppose that there exists such that
for all . Then T has a unique fixed point.
3 Consequences
3.1 Fixed point theorems on metric spaces endowed with a partial order
Definition 39 Let be a partially ordered set and be a given mapping. We say that T is nondecreasing with respect to ⪯ if
Definition 40 Let be a partially ordered set. A sequence is said to be nondecreasing with respect to ⪯ if
Definition 41 Let be a partially ordered set and G be a G-metric on X. We say that is G-regular if for every nondecreasing sequence such that as , for all n.
Theorem 42 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is G-continuous or is G-regular and ψ is continuous.
Then there exists such that . Moreover, if for , for all , one has the uniqueness of the fixed point.
Proof Define the mapping by
From (9), for all , we have
It follows that T is a generalized G-β-ψ contractive mapping of type II. From the condition (i), we have
By the definition of β and since T is a nondecreasing mapping with respect to ⪯, we have
Thus T is β-admissible. Moreover, if T is G-continuous, by Theorem 25, T has a fixed point.
Now, suppose that is G-regular. Let be a sequence in X such that for all n and is G-convergent to . By Definition 41, for all n, which gives us for all k. Thus, all the hypotheses of Theorem 30 are satisfied and there exists such that . To prove the uniqueness, since , we have, for all . By the definition of β, we get that for all . Therefore, the hypothesis (iv) of Theorem 33 is satisfied and we deduce the uniqueness of the fixed point. □
Corollary 43 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is G-continuous or is G-regular.
Then there exists such that . Moreover, if for , for all , one has the uniqueness of the fixed point.
Corollary 44 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is G-continuous or is G-regular.
Then there exists such that . Moreover, if for , for all , one has the uniqueness of the fixed point.
Corollary 45 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exist nonnegative real numbers a, b, c and d with such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is G-continuous or is G-regular.
Then there exists such that . Moreover, if for , for all , one has the uniqueness of the fixed point.
Corollary 46 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a constant such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is G-continuous or is G-regular.
Then there exists such that . Moreover, if for , for all , one has the uniqueness of the fixed point.
3.2 Cyclic contraction
Now, we will prove our results for cyclic contractive mappings in a G-metric space.
Let A, B be a nonempty G-closed subset of a complete G-metric space . Suppose also that and is a given self-mapping satisfying
If there exists a continuous function such that
then T has a unique fixed point , that is, .
Proof Notice that is a complete G-metric space since A, B is a closed subset of a complete G-metric space . We define in the following way:
Due to the definition of β and the assumption (12), we have
Hence, T is a generalized G-β-ψ contractive mapping.
Let be such that . If then by the assumption (11), , which yields that . If , we get again by analogy. Thus, in any case, we have , that is, T is β-admissible. Notice also that for any , we have , which yields .
Take a sequence in X such that for all n and as . Regarding the definition of β, we derive that
By assumption, A, B and hence is a G-closed set. Hence, we get that , which implies that . We conclude, by the definition of β, that for all n.
Now, all hypotheses of Theorem 30 are satisfied and we conclude that T has a fixed point. Next, we show the uniqueness of a fixed point u of T. Since and , we get for all . Thus, the condition (iv) of Theorem 33 is satisfied. □
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Alghamdi, MA, Karapınar, E: G-β-ψ contractive type mappings and related fixed point theorems. Preprint
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The research of the first author was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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Alghamdi, M.A., Karapınar, E. G-β-ψ contractive-type mappings and related fixed point theorems. J Inequal Appl 2013, 70 (2013). https://doi.org/10.1186/1029-242X-2013-70
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DOI: https://doi.org/10.1186/1029-242X-2013-70