Abstract
In this paper, first we introduce the notion of a -Meir-Keeler contractive mapping and establish some fixed point theorems for the -Meir-Keeler contractive mapping in the setting of G-metric spaces. Further, we introduce the notion of a -Meir-Keeler contractive mapping in the setting of G-cone metric spaces and obtain a fixed point result. Later, we introduce the notion of a -Meir-Keeler contractive mapping and prove some fixed point theorems for this class of mappings in the setting of G-metric spaces.
MSC:46N40, 47H10, 54H25, 46T99.
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1 Introduction
In nonlinear functional analysis, the study of fixed points of given mappings satisfying certain contractive conditions in various abstract spaces has been at the center of vigorous research activity in the last decades. The Banach contraction mapping principle is one of the initial and crucial results in this direction: In a complete metric space each contraction has a unique fixed point. Following this celebrated result, many authors have devoted their attention to generalizing, extending and improving this theory. For this purpose, the authors consider to extend some well-known results to different abstract spaces such as symmetric spaces, quasi-metric spaces, fuzzy metric, partial metric spaces, probabilistic metric spaces and a G-metric space (see, e.g., [1–9]). Several authors have reported interesting (common) fixed point results for various classes of functions in the setting of such abstract spaces (see, e.g., [6, 7, 10–32]).
In this paper, we consider especially a G-metric space and cone metric spaces which are introduced by Mustafa-Sims [9] and Huang-Zhang [3], respectively. Roughly speaking, a G-metric assigns a real number to every triplet of an arbitrary set. On the other hand, a cone metric space is obtained by replacing the set of real numbers by an ordered Banach space. Very recently, a number of papers on these concepts have appeared [9, 33–48].
One of the remarkable notions in fixed point theory is Meir-Keeler contractions [49] which have been studied by many authors (see, e.g., [50–56]). In this paper, first we introduce the notion of a -Meir-Keeler contractive mapping and establish some fixed point theorems for the -Meir-Keeler contractive mapping in the setting of G-metric spaces. In Section 4, we introduce the notion of a -Meir-Keeler contractive mapping in the setting of cone G-metric spaces and establish a fixed point result. Later, we introduce the notion of a -Meir-Keeler contractive mapping and prove some fixed point theorems for this class of mappings in the setting of G-metric spaces.
2 Preliminaries
We present now the necessary definitions and results in G-metric spaces which will be useful; for more details, we refer to [9, 57]. In the sequel, ℝ, and ℕ denote the set of real numbers, the set of nonnegative real numbers and the set of positive integers, respectively.
Definition 1 Let X be a nonempty set. A function is called a G-metric if the following conditions are satisfied:
-
(G1)
if , then ;
-
(G2)
for any with ;
-
(G3)
for any points , with ;
-
(G4)
, symmetry in all three variables;
-
(G5)
for any .
Then the pair is called a G-metric space.
Definition 2 Let be a G-metric space, and let be a sequence of points of X. A point is said to be the limit of the sequence if , and we say that the sequence is G-convergent to x and denote it by .
We have the following useful results.
Proposition 3 (see [44])
Let be a G-metric space. Then the following are equivalent:
-
(1)
is G-convergent to x;
-
(2)
;
-
(3)
.
Definition 4 ([44])
Let be a G-metric space, the sequence is called G-Cauchy if for every , there is such that for all , that is, as .
Proposition 5 ([44])
Let be a G-metric space. Then the following are equivalent:
-
(1)
the sequence is G-Cauchy;
-
(2)
for every , there is such that for all .
Definition 6 ([44])
A G-metric space is called G-complete if every G-Cauchy sequence in is G-convergent in .
Proposition 7 (see [44])
Let be a G-metric space. Then, for any , it follows that
-
(i)
if , then ;
-
(ii)
;
-
(iii)
;
-
(iv)
;
-
(v)
;
-
(vi)
.
Proposition 8 (see [44])
Let be a G-metric space. Then the function is jointly continuous in all three of its variables.
Now, we introduce the following notion of a -Meir-Keeler contractive mapping.
Definition 9 Let be a G-metric space. Suppose that is a self-mapping satisfying the following condition:
For each , there exists such that for all and for all , we have
Then f is called a -Meir-Keeler contractive mapping.
Remark 10 If is a -Meir-Keeler contractive mapping on a G-metric space X, then
holds for all and for all when . On the other hand, if , by Proposition 7, , and so . Hence, for all and for all , we have
3 Fixed point result for -Meir-Keeler contractive mappings
Now, we are ready to state and prove our main result.
Theorem 11 Let be a G-complete G-metric space and let f be a -Meir-Keeler contractive mapping on X. Then f has a unique fixed point.
Proof Define the sequence in X as follows:
Suppose that there exists such that . Since , then is the fixed point of f. Hence, we assume that for all , and so
By Remark 10 with , we get
for all . Define . Then is a strictly decreasing sequence in and so it is convergent, say, to . Now, we show that s must be equal to 0. Suppose, to the contrary, that . Clearly, we have
Assume . Then by hypothesis, there exists a convenient such that (2.1) holds. On the other hand, by the definition of ε, there exists such that
Now, by condition (2.1) with and (3.4), we get
which contradicts (3.3). Hence , that is, .
We will show that is a G-Cauchy sequence. For all , by the hypothesis, there exists a suitable such that (2.1) holds. Without loss of generality, we assume . Since , there exists such that
We assert that for any fixed , the condition
holds. To prove it, we use the method of induction. By Remark 10 and (3.6), assertion (3.7) is satisfied for . Suppose that (3.7) is satisfied for for some . Now, for , using (3.6), we obtain
If , then by (2.1) we get
and hence (3.7) is satisfied.
If , then and hence . This implies
and (3.7) is satisfied.
If , by Remark 10, we obtain
Consequently, (3.7) is satisfied for and hence
Now, if , by (3.9) and Proposition 7, we have
Hence, for all , the following holds:
Thus is a G-Cauchy sequence. Since is G-complete, there exists such that is G-convergent to z. Now, by Remark 10 with , we have
By taking the limit as in the above inequality and using the continuity of the function G, we get
and hence, , that is, z is a fixed point of f. To prove the uniqueness, we assume that is another fixed point of f such that . Then . Now, by Remark 10, we get
which is a contradiction and hence . □
Example 12 Let and
be a G-metric on X. Define by . Then . Assume that . Then
and
Let, . Then, for any , condition (2.1) holds. Similarly, condition (2.1) holds when . That is, f is a -Meir-Keeler contractive mapping. The condition of Theorem 11 holds, and so f has a unique fixed point.
4 Fixed point for -Meir-Keeler contractive mappings
In this section we introduce a notion of a -Meir-Keeler contractive mapping and establish some results of a fixed point for such class of mappings.
Denote with Ψ the family of nondecreasing functions continuous in such that
-
if and only if ;
-
.
Samet, Vetro and Vetro [19] introduced the following concept.
Definition 13 Let and . We say that f is an α-admissible mapping if
Now, we apply this concept in the following definition.
Definition 14 Let be a G-metric space and . Suppose that is an α-admissible mapping satisfying the following condition:
For each , there exists such that
for all . Then f is called a -Meir-Keeler contractive mapping.
Remark 15 Let f be a -Meir-Keeler contractive mapping. Then
for all when . Also, if , then , which implies , i.e.,
for all .
Theorem 16 Let be a G-complete G-metric space. Suppose that f is a continuous -Meir-Keeler contractive mapping and that there exists such that . Then f has a fixed point.
Proof Let and define the sequence by for all . Since f is an α-admissible mapping and , we deduce that . By continuing this process, we get for all . If for some , then obviously f has a fixed point. Hence, we suppose that
for all . By (G2), we have
for all . Now, define . By Remark 15, we deduce that for all ,
which implies
Hence, the sequence is decreasing in and so it is convergent to . We will show that . Suppose, to the contrary, that . Hence, we have
Let . Then by hypothesis, there exists a such that (4.10) holds. On the other hand, by the definition of ε, there exists such that
Now, by (4.10) we have
which is a contradiction. Hence , that is, . Now, by the continuity of ψ in , we have
For given , by the hypothesis, there exists a such that (4.10) holds. Without loss of generality, we assume . Since , then there exists such that
We will prove that for any fixed ,
holds. Note that (4.6), by (4.5), holds for . Suppose condition (4.10) is satisfied for some . For , by (G5) and (4.5), we get
If , then by (4.10) we get
and hence (4.6) holds.
If , by Remark 15, we get
Consequently, (4.6) holds for . Hence, for all and , which means
Then, for all , by (4.8) and Proposition 7, we have
That is, for all , the following condition holds:
Consequently, . By the continuity of ψ in , we get . Hence is a G-Cauchy sequence. Since is G-complete, there exists such that
Also, by the continuity of f, we have
and hence
that is, . □
Theorem 17 Let be a G-complete G-metric space and let f be a -Meir-Keeler contractive mapping. If the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
if is a sequence in X such that for all n and as , then ,
then f has a fixed point.
Proof Let such that . Define the sequence in X by for all . Following the proof of Theorem 16, we say that for all and that there exists such that as . Hence, from (ii) . By Remark 15, we have
By taking limit as , in the above inequality, we get , that is, . Hence . □
Theorem 18 Assume that all the hypotheses of Theorem 16 (and 17) hold. Adding the following conditions:
-
(iii)
for all ,
we obtain the uniqueness of the fixed point of f.
Proof Suppose that z and are two fixed points of f such that . Then . Now, by Remark 15, we have
which is a contradiction. Hence, . □
If in Theorems 17 and 18 we take and where , then we have the following corollary.
Corollary 19 Let be a G-complete G-metric space. Suppose that is a mapping satisfying the following condition:
For each , there exists such that
for all where . Then f has a unique fixed point.
5 Fixed point in G-cone metric spaces
In this section we recall the notion of a cone G-metric [36], we introduce the notion of a -Meir-Keeler contractive mapping and establish the result of a fixed point for such class of mappings.
Definition 20 ([3])
Let E be a real Banach space with θ as the zero element and with the norm . A subset P of E is called a cone if and only if the following conditions are satisfied:
-
(i)
P is closed, nonempty and ;
-
(ii)
and implies ;
-
(iii)
and implies .
Let be a cone, we define a partial ordering ⪯ on E with respect to P by if and only if ; we write whenever and , while will stand for (the interior of P). The cone is called normal if there is a positive real number K such that for all , . The least positive number satisfying the above inequality is called the normal constant of P. If , then the cone P is called monotone.
Definition 21 Let be a real Banach space with a monotone solid cone P. A mapping satisfying the following conditions:
-
(F1)
if , then ;
-
(F2)
for any with ;
-
(F3)
for any points , with ;
-
(F4)
, symmetry in all three variables;
-
(F5)
for any
is a cone G-metric on X and is a cone G-metric space.
Let be a real Banach space with a monotone solid cone P. Then
Proposition 23 ([8])
Let be a real Banach space with a monotone solid cone P. If is a G-cone metric on X, then the function defined by is a G-metric on X and a G-metric space.
Definition 24 Let be a real Banach space with a monotone solid cone P and be a cone G-metric space. Suppose that is a self-mapping satisfying the following condition:
For each , there exists such that for all and for all ,
Then f is called a -Meir-Keeler contractive mapping.
Theorem 25 Let be a real Banach space with a monotone solid cone P and be a G-complete G-cone metric space and f be a -Meir-Keeler contractive mapping on X. Then f has a unique fixed point.
Proof For a given , let , where . This implies
for given . Indeed, if , then
and so by Lemma 22, we get , which is a contradiction. Therefore (5.2) holds.
Now suppose that . This implies
Indeed if
then
and so , which is a contradiction. This implies that (5.3) holds.
Now, by (5.4), (5.2) and (5.3), we have
Again, by Lemma 22, we get
Thus f is a -Meir-Keeler contractive mapping, and by Theorem 11, f has a unique fixed point. □
Similarly, we have the following corollary.
Corollary 26 Let be a real Banach space with a monotone solid cone P and be a G-complete G-cone metric space and f be a mapping such that for each , there exists such that
for all , where . Then f has a unique fixed point.
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Acknowledgements
This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR, KAU for financial support. The authors would like to thank the editor and the referees for their suggestions to improve the presentation of the paper. The 3rd author is thankful for support of Islamic Azad University, Astara, during this research.
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Hussain, N., Karapınar, E., Salimi, P. et al. Fixed point results for -Meir-Keeler contractive and -Meir-Keeler contractive mappings. Fixed Point Theory Appl 2013, 34 (2013). https://doi.org/10.1186/1687-1812-2013-34
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DOI: https://doi.org/10.1186/1687-1812-2013-34