Abstract
In this paper, we introduce G-β-ψ-contractive mappings which are generalizations of α-ψ-contractive mappings in the context of G-metric spaces. Additionally, we prove existence and uniqueness of fixed points of such contractive mappings. Our results generalize, extend and improve the existing results in the literature. We state some examples to illustrate our results.
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1 Introduction and preliminaries
In the last few decades, fixed point theory has been one of the most interesting research fields in nonlinear functional analysis. In addition to many branches of applied and pure mathematics, fixed point theory results have wide application areas in many disciplines such as economics, computer science, engineering etc. The most remarkable results in this direction were given by Banach [1] in 1922. He proved that each contraction in a complete metric space has a unique fixed point. Due to application potential of the theory, many authors have directed their attention to this field and have generalized the Banach fixed point theorem in various ways (see, e.g., [1–50]). Very recently, Samet et al. [38] introduced the notion of α-ψ-contractive mappings and proved the related fixed point theorems. The authors [38] showed that Banach fixed point theorems and some other theorems in the literature became direct consequences of their results. On the other hand, in 2004, Mustafa and Sims [24] defined the notion of a G-metric space and characterized the Banach fixed point theorem in the context of a G-metric space. Following these results, many authors have discussed fixed point theorems in the framework of G-metric spaces; see, e.g., [9, 10, 14–16, 18–20, 23–29, 40–43, 50]. In this paper, we combine these two notions by introducing G-β-ψ-contractive mappings, which are a characterization of α-ψ-contractive mappings in the context of G-metric spaces. Our main results generalize, extend and improve the existing results on the topic in the literature.
Throughout this paper, ℕ denotes the set of nonnegative integers, and denotes the set of nonnegative reals.
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 .
These functions are known in the literature as -comparison functions.
Lemma 1 (See [5])
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 .
Remark 2 In some sources, -comparison functions are called Bianchini-Grandolfi gauge functions (see, e.g., [34–36]).
Very recently, Samet et al. [38] introduced the following concepts.
Definition 3 Let be a metric space and let be a given mapping. We say that T is an α-ψ-contractive mapping if there exist two functions and such that
for all .
Clearly, any contractive mapping, that is, a mapping satisfying Banach contraction, is an α-ψ-contractive mapping with for all and , for all and some .
Definition 4 Let and . We say that T is α-admissible if for all , we have
Various examples of such mappings are presented in [38]. The main results in [38] are the following fixed point theorems.
Theorem 5 Let be a complete metric space and be an 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 6 Let be a complete metric space and be an 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 7 Adding to the hypotheses of Theorem 5 (resp. Theorem 6) the condition: For all , there exists such that and , we obtain the uniqueness of a fixed point of T.
Mustafa and Sims [24] introduced the concept of G-metric spaces as follows.
Definition 8 ([24])
Let X be a non-empty 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 9 Let be a metric space. The function , defined as either
or
for all , is a G-metric on X.
Definition 10 ([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 11 ([24])
Let be a G-metric space. The following are equivalent:
-
(1)
as ;
-
(2)
as ;
-
(3)
as .
Definition 12 ([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 13 ([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 14 ([24])
A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
Lemma 15 ([24])
Let be a G-metric space. Then, for any , it follows that:
-
(i)
if , then ;
-
(ii)
;
-
(iii)
;
-
(iv)
;
-
(v)
;
-
(vi)
.
Definition 16 (See [24])
Let be a G-metric space. A mapping is said to be G-continuous if is G-convergent to , where is a G-convergent sequence converging to x.
In [23], Mustafa characterized the well-known Banach contraction principle mapping in the context of G-metric spaces in the following ways.
Theorem 17 (See [23])
Let be a complete G-metric space and let be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Theorem 18 (See [23])
Let be a complete G-metric space and let be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Remark 19 The condition (1) implies the condition (2). The converse is true only if . For details, see [23].
From [23, 24], each G-metric G on X generates a topology on X whose base is a family of open G-balls , where for all and . A nonempty set A in the G-metric space is G-closed if . Moreover,
Proposition 20 Let be a G-metric space and let A be a nonempty subset of X. The set 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 21 Let be a G-metric space and let 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
Definition 22 Let be a G-metric space and let be a given mapping. We say that T is a G-β-ψ-contractive mapping of type II if there exist two functions and such that for all , we have
Definition 23 Let be a G-metric space and let be a given mapping. We say that T is a G-β-ψ-contractive mapping of type A if there exist two functions and such that for all , we have
Remark 24 Clearly, any contractive mapping, that is, a mapping satisfying (1), is a G-β-ψ-contractive mapping of type I with for all and , . Analogously, a mapping satisfying (2), is a G-β-ψ-contractive mapping of type II with for all and , .
Definition 25 Let and . We say that T is β-admissible if for all , we have
Example 26 Let and . Define by
Then T is β-admissible.
Our first result is the following.
Theorem 27 Let be a complete G-metric space. Suppose that is a G-β-ψ-contractive mapping of type A and satisfies the following conditions:
-
(i)
T is β-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is G-continuous.
Then there exists such that .
Proof Let such that (such a point exists from the condition (ii)). 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 (5) and (6), it follows that for all , we have
Since ψ is nondecreasing, by induction, we have
Using (G5) and (7), we have
Since and , by Lemma 1, we get
Thus, we have
By Proposition 13, 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. □
The next theorem does not require continuity.
Theorem 28 Let be a complete G-metric space. Suppose that is a G-β-ψ-contractive mapping of type A and satisfies the following conditions:
-
(i)
T is β-admissible;
-
(ii)
there exists such that ;
-
(iii)
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 29, 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 (6) and (iii), we have
Using the basic properties of G-metric together with (5) and (8), we have
Letting , using Proposition 11 and since ψ is continuous at , it follows that
By Lemma 15, we obtain . □
The following theorem can be derived easily from Theorems 27 and 28.
Theorem 29 Let be a complete G-metric space. Suppose that is a G-β-ψ-contractive mapping of type A and satisfies the following conditions:
-
(i)
T is β-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is G-continuous.
Then there exists such that .
Theorem 30 Let be a complete G-metric space. Suppose that is a G-β-ψ-contractive mapping of type II and satisfies the following conditions:
-
(i)
T is β-admissible;
-
(ii)
there exists such that ;
-
(iii)
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 .
Remark 31 We notice that some fixed point theorems in the context of G-metric space can be derived from usual fixed point results via certain substitutions (see, e.g., [21, 39]). On the other hand, our main result cannot be obtained via a substitution technique because the expressions in our statements do not allow one to achieve a metric by writing a simple substitution.
With the following example, we show that the hypotheses in Theorems 27-30 do not guarantee uniqueness.
Example 32 Let be the G-metric space, where
for all . Consider the self-mapping given by
Notice that Theorem 18 in [23], a characterization of the Banach fixed point theorem, cannot be applied in this case because .
Define as
Let for . Then we conclude that T is a G-β-ψ-contractive mapping. In fact, for all , we have
On the other hand, there exists such that . Indeed, for , we have .
Notice also that T is continuous. To show that T satisfies all the hypotheses of Theorem 29, it is sufficient to observe that T is β-admissible. For this purpose, let such that , which is equivalent to saying that . Due to the definitions of β and T, we have
Hence, . As a result, all the conditions of Theorem 29 are satisfied. Note that Theorem 29 guarantees the existence of a fixed point but not the uniqueness. In this example, 0 and are two fixed points of T.
In the following example, T is not continuous.
Example 33 Let X, G and β be defined as in Example 32. Let be a map given by
Let for . Then we conclude that T is a G-β-ψ-contractive mapping. In fact, for all , we have
Furthermore, there exists such that . For , we have .
Let be a sequence such that for all and as . By the definition of β, we have for all . Then we see that . Thus, .
To show that T satisfies all of the hypotheses of Theorem 30, it is sufficient to observe that T is β-admissible. For this purpose, let such that . It is equivalent to saying that . Due to the definitions of β and T, we have
Hence, .
As a result, all the conditions of Theorem 30 are satisfied. Note that Theorem 30 guarantees the existence of a fixed point but not uniqueness. In this example, 0 and are two fixed points of T.
Theorem 34 Adding the following condition to the hypotheses of Theorem 27 (resp. Theorem 29-Theorem 30) we obtain the uniqueness of a fixed point of T.
-
(iv)
For all , there exists such that and .
Proof Let be two fixed points of T. By (iv), there exists such that
Since T is β-admissible, we get by induction that
From (9) and (5), we have
Thus, we get by induction that
By (G4), we get
Letting , and since , we have
This implies that is G-convergent to u. Similarly, we get is G-convergent to . By the uniqueness of the limit, we get , that is, the fixed point of T is unique. □
3 Consequences
3.1 Cyclic contraction
Now, we prove our results for cyclic contractive mappings in a G-metric space.
Theorem 35 Let A, B be a non-empty G-closed subset of a complete G-metric space, let , and let be a given self-mapping satisfying
If there exists a function such that
then T has a unique fixed point , that is, .
Proof Notice that is a complete G-metric space because A, B are closed subsets of a complete G-metric space . We define in the following way:
Due to the definition of β and assumption (11), we have
Hence, T is a G-β-ψ-contractive mapping.
Let such that . If , then by assumption (10), , 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 that .
Take a sequence in X such that for all n and as . Regarding the definition of β, we derive that
By the assumption, A, B and are closed sets. Hence we get that , which implies that . We conclude, by the definition of β, that for all n.
Now all the hypotheses of Theorem 30 are satisfied, and we conclude that T has a fixed point. Next, we show the uniqueness of a fixed point z of T. Suppose that , where . Since , we have and . Thus the condition (iv) of Theorem 34 is satisfied. □
3.2 Coupled fixed point theorems
For the rest of the paper, we suppose that all G-metric spaces are symmetric, that is, for all .
In 1987, Guo and Lakshmikantham [8] introduced the notion of a coupled fixed point. The concept of a coupled fixed point was reconsidered by Gnana-Bhaskar and Lakshmikantham [7] in 2006. In this paper, they proved the existence and uniqueness of a coupled fixed point of an operator on a partially ordered metric space under a condition called the mixed monotone property.
Definition 36 ([7])
Let be a partially ordered set and . The mapping F is said to have the mixed monotone property if is monotone non-decreasing in x and monotone non-increasing in y, that is, for any ,
and
Definition 37 ([7])
An element is called a coupled fixed point of the mapping if
Lemma 38 (See [38])
Let be a given mapping. Define the mapping by for all . Then is a fixed point of if and only if is a coupled fixed point of F.
Definition 39 Let be a G-metric space. A mapping is said to be continuous if for any two G-convergent sequences and converging to x and y, respectively, is G-convergent to .
Theorem 40 Let be a complete G-metric space and let be a given mapping. Suppose there exist and a function such that
for all . Suppose also that
-
(a)
for all , we have
-
(b)
there exists such that
-
(c)
F is continuous.
Then F has a coupled fixed point, that is, there exists such that and .
Proof Let be a complete G-metric space with and
for all . By using (14) and (G4), we get
and
Combining (15) and (16), we have
for all , where is defined by
and is given by
It follows that T is a G-continuous and G-γ-ψ-contractive mapping of type II.
Suppose that for . Then, by the condition (a), we have . Therefore, T is γ-admissible.
From the condition (b), there exists such that
Since all the hypotheses of Theorem 29 are satisfied, it follows that T has a fixed point, and by Lemma 38, F has a coupled fixed point. □
Theorem 41 Let be a complete G-metric space and let be a given mapping. Suppose there exist and a function such that
for all . Suppose also that
-
(a)
for all , we have
-
(b)
there exists such that
-
(c)
if and are sequences in X such that
and
and are G-convergent to x and y, respectively, then
and
for all n.
Then F has a coupled fixed point, that is, there exists such that and .
Proof Let be a sequence in Y such that
and is G-convergent to . From the condition (c), we get
This implies that all the hypotheses of Theorem 30 are satisfied. It follows that T has a fixed point, and by Lemma 38, the mapping F has a coupled fixed point. □
Theorem 42 Adding the following condition to the hypotheses of Theorem 40 (resp. Theorem 41), we obtain the uniqueness of a coupled fixed point of F.
-
(d)
For all , there exists such that
and
Proof With the condition (d), T and γ satisfy the condition (iv) of Theorem 34. From Theorem 34 and Lemma 38, the result follows. □
3.3 Choudhury and Maity’s coupled fixed point results in a G-metric space
Definition 43 Let be a partially ordered set, and let be a G-metric space. A partially ordered G-metric space, , is called ordered complete if for each convergent sequence , the following conditions hold:
(OC1) if is a non-increasing sequence in X such that , then ;
(OC2) if is a non-decreasing sequence in X such that , then .
Choudhury and Maity [10] proved the following coupled fixed point theorems on ordered G-metric spaces.
Theorem 44 Let be a partially ordered set and let G be a G-metric on X such that is a complete G-metric space. Let be a G-continuous mapping having the mixed monotone property on X. Suppose that there exists a such that
for all with and , where either or . If there exist such that and , then F has a coupled fixed point, that is, there exists such that and .
Proof Let . Suppose that such that
where
From (17), for all , we have
and
It follows that T is a G-γ-ψ-contractive mapping of type II with , . Let such that
By the definition of γ, we get and . This implies that
since F has the mixed monotone property. Thus,
By the assumption, there exist such that and . By the definition of γ, it implies that
where and . From Theorem 40, F has a coupled fixed point. □
Theorem 45 If, instead of G-continuity of F in the theorem above, we assume that X is ordered complete, then F has a coupled fixed point.
Proof It is sufficient to prove that the condition (c) of Theorem 41 is satisfied under the setting of (18). For this purpose, we take two sequences and in X such that and as . Assume that and . Due to the definition of β, the sequences and are nonincreasing and nondecreasing, respectively. Regarding (i) and (ii), we derive that
which yields that
Then, the assumption (c) of Theorem 41 holds. Hence, F has a coupled fixed point. □
Remark 46 Notice that analogs of all of the theorems proved in Section 2 can be derived by replacing type I and type II with type A.
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Acknowledgements
The authors express their gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions. The authors also thanks to Professor İlker Savas Yüce for his help for improving the presentation of the paper. The first author 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 in G-metric spaces. Fixed Point Theory Appl 2013, 123 (2013). https://doi.org/10.1186/1687-1812-2013-123
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DOI: https://doi.org/10.1186/1687-1812-2013-123