Abstract
Brianciari (‘A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces,’ Publ. Math. Debrecen 57 (2000) 31-37) initiated the notion of the generalized metric space as a generalization of a metric space in such a way that the triangle inequality is replaced by the ‘quadrilateral inequality,’ d(x,y) ≤ d(x,a) + d(a,b) + d(b,y) for all pairwise distinct points x,y,a, and b of X. In this paper, we establish a fixed point result for weak contractive mappings T : X → X in complete Hausdorff generalized metric spaces. The obtained result is an extension and a generalization of many existing results in the literature.
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Introduction
It is well known that the Banach contraction principle[1] is a very useful and classical tool in nonlinear analysis. Later, this principle has been generalized in many directions. For instance, a very interesting generalization of the concept of a metric space was obtained by Branciari[2] by replacing the triangle inequality of a metric space with a more general inequality. Thereafter, many authors initiated and studied many existing fixed point theorems in such spaces. For more details about the fixed point theory in generalized metric spaces, we refer the reader to[3–15].
In the sequel, the letters,, and will denote the set of real numbers, the set of nonnegative real numbers, and the set of nonnegative integer numbers, respectively. The following definitions will be needed in the sequel.
Definition 1.1
[2] Let X be a non-empty set and d : X × X → [0, + ∞)such that for all x,y ∈ X and for all distinct points u, v ∈ X, each of them different from x and y, one has the following:
Then, (X, d) is called a generalized metric space (or shortly g.m.s.).
Any metric space is a generalized metric space, but the converse is not true[2]. We confirm this by the following.
Example 1.2
Let X = A ∪ B, where and. Define the generalized metric d on X as follows:
It is clear that d does not satisfy the triangle inequality on A. Indeed,
Notice that (p3) holds, so d is a generalized metric.
Definition 1.3
[2] Let (X, d) be a g.m.s., {x n }be a sequence in X, and x ∈ X. We say that {x n } is g.m.s. convergent to x if and only if d(x n x) → 0 as n → + ∞. We denote this by x n → x.
Definition 1.4
[2] Let (X, d) be a g.m.s. and {x n } be a sequence in X. We say that {x n } is a g.m.s. Cauchy sequence if and only if for each ε > 0, there exists a natural number N such that d(x n x m ) < N for all n > m > N.
Definition 1.5
[2] Let (X, d) be a g.m.s. Then, (X, d) is called a complete g.m.s. if every g.m.s. Cauchy sequence is g.m.s convergent in X.
Recently, Miheţ established the following theorem, extending Kannan’s Theorem[16] to generalized metric spaces.
Theorem 1.6
Let (X,d) be a T-orbitally g.m.s. and T : X → X be a self-map. Assume that there existssuch that
for all x,y ∈ X. Then, T has a unique fixed point in X.
In complete metric spaces, an important fixed point theorem has been proved by Choudhury[17].
Theorem 1.7
Let (X, d) be a complete metric space and T : X → X be a self-map such that for all x,y ∈ X
where ϕ :[0,∞) × [0,∞) → [0,∞) is continuous, and ϕ(a,b) = 0 if and only if a = b = 0. Then, there exists a unique point u ∈ X such that u = Tu.
Several papers attempting to generalize fixed point theorems in metric spaces to g.m.s. are plagued by the use of some false properties given in[2] (see, for example,[3, 4, 6–8]). This was observed first by Samet[13, 18] and then by Sarma et al.[14] by assuming that the generalized metric space is Hausdorff. In this paper, we prove a fixed point result involving weak contractive mappings T : X → X in complete generalized metric spaces by assuming, in particular, that (X,d) is Hausdorff. As a corollary, we derive a Kannan-type[16, 19] fixed point result in such spaces. Also, we state some examples to illustrate our results.
Main results
Our main result is the following.
Theorem 2.1
Let (X, d) be a Hausdorff and complete generalized metric space. Suppose that T : X → X such that for all x,y ∈ X,
where ϕ :[0,∞) × [0,∞) → [0,∞) is continuous, and ϕ(a,b) = 0 if and only if a = b = 0. Then, there exists a unique point u ∈ X such that u = Tu.
Proof
Let x0 ∈ X be an arbitrary point. By induction, we easily construct a sequence {x n } such that
If for some, x n = xn + 1, the proof is completed. For the rest, assume that x n ≠ xn + 1 for all. □
Step 1. We claim that
Substituting x = x n and y = xn−1 in (2.1) and using the properties of ϕ, we obtain
which implies that
Therefore, the sequence {d(x n ,xn + 1)} is monotone non-increasing and bounded below. So, there exists r ≥ 0 such that
Letting n → ∞ in (2.4) and using the continuity of ϕ, we get, which implies that ϕ(r,r) = 0, so r = 0 by a property of ϕ. Thus, (2.3) is proved.
Step 2. We shall prove that
By (2.1), we have
By (2.3), we get that
so (2.5) is proved.
Step 3. We claim that T has a periodic point.
We argue by contradiction. Assume that T has no periodic point; then, {x n } is a sequence of distinct points, that is, x n ≠ x m for all m ≠ n. We will show that in this case, {x n } is a g.m.s. Cauchy. Suppose to the contrary. Then, there is a ε > 0 such that for an integer k, there exist integers m(k) > n(k) > k such that
For every integer k, let m(k) be the least positive integer exceeding n(k) satisfying (2.7) and such that
Now, using (2.7) and (2.8) and the rectangular inequality (because {x n } is a sequence of distinct points), we find that
Then, by (2.3) and (2.5), it follows that
Applying (2.1) with x = xm(k)−1 and y = xn(k)−1, we have
Letting k → ∞ in the above inequality and using (2.3) and (2.9), we obtain
It is a contradiction.
Hence, {x n } is a g.m.s. Cauchy. Since (X, d) is a complete g.m.s., there exists u ∈ X such that x n → u. Applying again (2.1) with x = x n and y = u, we obtain
which implies that
By (2.3), it follows that
Next, we shall find a contradiction of the fact that T has no periodic point in each of two following cases:
-
If for all n ≥ 2, x n ≠ u and x n ≠ Tu. Then, by rectangular inequality
and using (2.3), we get that
From (2.11) and (2.12),
which holds unless d(u,Tu) = 0, so Tu = u, that is, u is a fixed point of T, so u is a periodic point of T. It contradicts the fact that T has no periodic point.
-
If for some q ≥ 2, x q = u or x q = Tu. Since T has no periodic point, so obviously u ≠ x0. Indeed, if x q = u = x0, so Tqx0 = x0, i.e, x0 is a periodic point of T. On the other hand, if x q = Tu and x0 = u, so T x0 = Tu = x q = Tqx0 = Tq−1(T x0), i.e, T x0 is a periodic point of T.
For all n ≥ 1, we have
In the two precedent identities, the integer q ≥ 2 is fixed, so {xn + q} and {xn + q−1} are subsequences from {x n }, and since {x n } g.m.s. converges to u in (X,d) which is assumed to be Hausdorff, so the two subsequence g.m.s. converge to the same unique limit u, i.e,
Thus,
Again, since (X,d) is Hausdorff, then by (2.14),
On the other hand, since T has no periodic point, then it is easy that
Using (2.16) and a rectangular inequality, we may write
Letting n → ∞ in the above limit and proceeding as (2.3) (since the point x0is arbitrary and so the same for the point u) and using (2.15), we obtain
Similarly,
Now, by (2.1),
Letting n → ∞ in (2.19) and using (2.17) and (2.18), we get that
which holds unless d(u,Tu) = 0, so Tu = u; hence, u is a periodic point of T. It is a contradiction with the fact that T has no periodic point.Consequently, T admits a periodic point, that is, there exists u∈X such that u = Tpu for some p ≥ 1.
Step 4. Existence of a fixed point of T.
If p = 1, then u = Tu, that is, u is a fixed point of T. Suppose now that p > 1. We will prove that a = Tp−1u is a fixed point of T. Suppose that Tp−1u ≠ Tpu; then, d(Tp−1u,Tpu) > 0, and so ϕ(d(Tp−1u,Tpu),d(Tp−1u,Tpu)) > 0. Now, using the inequality (2.1), we obtain
which implies that
Again, by (2.1), we have
Again, this implies that
Continuing this process as in (2.20) and (2.21), we find that
which is a contradiction. We deduce that a = Tp−1u is a fixed point of T.
Step 5. Uniqueness of the fixed point of T.
Suppose that there are two points b,c ∈ X such that Tb = b and Tc = c. By (2.1), we obtain
so b = c. Thus, T has a unique fixed point. This completes the proof of Theorem 2.1.
Now, we state a corollary of Theorem 2.1 (a Kannan-type contraction[16, 19]) given in the following.
Corollary 2.2
Let (X, d) be a Hausdorff and complete generalized metric space. Suppose that T : X → X such that for all x,y ∈ X, there exists k ∈[0,1) and
Then, T has a unique fixed point.
Proof
It suffices to take in Theorem 2.1. □
Also, we have the following consequence from Theorem 2.1.
Corollary 2.3
Let (X, d) be a Hausdorff and complete generalized metric space. Suppose that T : X → X such that for all x,y ∈ X
where ψ :[0,∞)→[0,∞) is continuous, and ψ−1({0}) = {0}. Then, T has a unique fixed point.
Proof
We have only to show that satisfies the hypotheses of Theorem 2.1. □
Remark 2.4
-
(1)
Corollary 2.2 corresponds to the main result of Miheţ [11], except that we assumed, in addition, that the generalized metric space is Hausdorff.
-
(2)
Theorem 2.1 extends the results of Branciari [2], Azam and Arshad [4], and Sarma et al. [14].
We give some examples illustrating Theorem 2.1.
Example 2.5
Following[4, 12], let X = {1,2,3,4} and define as follows:
Then, (X, d) is a complete generalized metric space, but (X, d) is not a metric space because it lacks the triangular property:
Now, define a mapping T:X → X as follows:
Take for all a,b ≥ 0. It is easy that all hypotheses of Theorem 2.1 are satisfied, and u = 3 is the unique fixed of T.
On the other hand, Banach’s theorem[1] is not applicable (for the metric d0|x y) = |x − y| for all x,y ∈ X). Indeed, taking x = 2 and y = 4, we have
Also, Theorem 1.7 is not applicable by taking, for example, x = 3 and y = 4,
for all ϕ(given as our Theorem 2.1), and so in particular, it is the same for Chatterjea’s theorem[20].
Example 2.6
Let X=A∪B, where and. Define the generalized metric d on X as follows:
It is clear that d does not satisfy triangle inequality on A. Indeed,
Notice that (p3) holds, so d is a generalized metric.
Let T:X → X be defined as
Choose. Then, it easy to check that T satisfies the conditions of Theorem 2.1 and has a unique fixed point on X, i.e.,.
Note that, Banach’s theorem[1] is not applicable (it suffices to take and). Also, we couldn’t apply Theorem 1.7 by taking, for example, and.
References
Banach S: Su r les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math 1922, 3: 133–181.
Branciari A: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ. Math. Debrecen 2000, 57: 31–37.
Akram M, Siddiqui AA: A fixed-point theorem for A-contractions on a class of generalized metric spaces. Korean J. Math. Sciences 2003, 10(2):1–5.
Azam A, Arshad M: Kannan fixed point theorem on generalized metric spaces. J. Nonlinear Sci. Appl 2008, 1(1):45–48.
Chen CM, Chen CH: Periodic points for the weak contraction mappings in complete generalized metric spaces. Fixed Point Theory Appl 2012, 2012: 79. 10.1186/1687-1812-2012-79
Das P: A fixed point theorem on a class of generalized metric spaces. Korean J. Math. Sci 2002, 9: 29–33.
Das P: A fixed point theorem in a generalized metric space. Soochow J. Math 2007, 33(1):33–39.
Das P, Lahiri BK: Fixed point of a Ljubomir Ćirić’s quasi-contraction mapping in a generalized metric space. Publ. Math. Debrecen 2002, 61: 589–594.
Das P, Lahiri BK: Fixed point of contractive mappings in generalized metric spaces. Math. Slovaca 2009, 59(4):499–504. 10.2478/s12175-009-0143-2
Fora A, Bellour A, Al-Bsoul A: Some results in fixed point theory concerning generalized metric spaces. Matematicki Vesnik 2009, 61(3):203–208.
Miheţ D: On Kannan fixed point principle in generalized metric spaces. J. Nonlinear Sci. Appl 2009, 2(2):92–96.
Samet B: A fixed point theorem in a generalized metric space for mappings satisfying a contractive condition of integral type. Int. Journal Math. Anal 2009, 3(26):1265–1271.
Samet B: Discussion on “A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces” by A. Branciari. Publ. Math. Debrecen 2010, 76(4):493–494.
Sarma IR, Rao JM, Rao SS: Contractions over generalized metric spaces. J. Nonlinear Sci. Appl 2009, 2(3):180–182.
Shatanawi W, Al-Rawashdeh A, Aydi H, Nashine HK: On a fixed point for generalized contractions in generalized metric spaces. 2012. 10.1155/2012/246085
Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc 1968, 60: 71–76.
Choudhury Binayak S: Unique fixed point theorem for weakly C-contractive mappings. Kathmandu Univ J. Science Engineering and Technology 2009, 5(1):6–13.
Lakzian H, Samet B: Fixed point for ( ψ,ϕ )-weakly contractive mappings in generalized metric spaces. Appl. Math. Lett 2012, 25(5):902–906. 10.1016/j.aml.2011.10.047
Kannan R: Some results on fixed points-II. Amer. Math. Monthly 1969, 76: 405–408. 10.2307/2316437
Chatterjea SK: Fixed point theorems. C. R. Acad. Bulgare Science 1972, 25: 727–730.
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Aydi, H., Karapinar, E. & Lakzian, H. Fixed point results on a class of generalized metric spaces. Math Sci 6, 46 (2012). https://doi.org/10.1186/2251-7456-6-46
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DOI: https://doi.org/10.1186/2251-7456-6-46