Abstract
The purpose of this paper is to prove some existence theorems for fixed point problem by using a generalization of metric distance, namely u-distance. Consequently, some special cases are discussed and an interesting example is also provided. Presented results are generalizations of the important results due to Ume (Fixed Point Theory Appl 2010(397150), 21 pp, 2010) and Suzuki and Takahashi (Topol Methods Nonlinear Anal 8, 371-382, 1996).
2010 Mathematics Subject Classification: 47H09, 47H10.
Similar content being viewed by others
1. Introduction and preliminaries
Let (X, d) be a metric space. A mapping T: X → X is said to be contraction if there exists r ∈ [0, 1) such that
In 1922, Banach [1] proved that if (X, d) is a complete metric space and the mapping T satisfies (1.1), then T has a unique fixed point, that is T(u) = u for some u ∈ X. Such a result is well known and called the Banach contraction mapping principle. Following the Banach contraction principle, Nadler Jr. [2] established the fixed point result for multi-valued contraction maps, which in turn is a generalization of the Banach contraction principle. Since then, there are several extensions and generalizations of these two important principles, see [3, 4] and [5–11] for examples.
In 1996, Kada et al. [4] introduced the concept of w-distance on a metric space (X, d). By using such a w-distance concept, they improved some important theorems such as Caristi's fixed point theorem, Ekeland's variational principle and the nonconvex minimization theorem. Recently, Suzuki [7] introduced the concept of generalization metric distance, which is called τ-distance. By using concepts of τ-distance, he proved some results on fixed point problems and also showed that the class of w-distance is properly contained in the class of τ-distance. Most recently, Ume [11] introduced another concept of distance as the following.
Definition 1.1. [11]. Let (X, d) be a metric space. Then, a function p: X × X → [0, ∞) is called u-distance on X if there exists a function θ: X × X × [0, ∞) × [0, ∞) → [0, ∞) such that
(u 1) p(x, z) ≤ p(x, y) + p(y, z) for all x, y, z ∈ X;
(u 2) θ(x, y, 0, 0) = 0 and θ(x, y, s, t) ≥ min{s, t} for all x, y ∈ X and s, t ∈ [0, ∞), and for any x ∈ X and for every ε > 0, there exists δ > 0 such that |s - s0| < δ, |t - t0| < δ, s, s0, t, t0 ∈ [0, ∞) and y ∈ X imply
(u 3)
imply
(u 4)
imply
or
imply
(u 5)
imply
or
imply
We give the following remark and example, which can be found in [11].
Remark 1.2. Suppose that θ: X × X × [0, ∞) × [0, ∞) → [0, ∞) is a mapping satisfying (u 2) ~ (u 5). Then, there exists a mapping η: X × X × [0, ∞) × [0, ∞) → [0, ∞) such that η is nondecreasing in its third and fourth variable, respectively, satisfying (u 2)η ~ (u 5)η, where (u 2)η ~ (u 5)η stand for substituting η for θ in (u 2) ~ (u 5), respectively.
Example 1.3. Let p be a τ-distance on metric space (X, d), then p is also a u-distance on X. On the other hand, let (X, || · ||) be a normed space then a function p: X × X → [0, ∞) defined by p(x, y) = ||x|| for every x, y ∈ X is a u-distance on X but not a τ-distance. These imply that the class of τ-distance is properly contained in the class of u-distance.
In this paper, we will prove some fixed point theorems in metric spaces by using such a u-distance concept. Consequently, as shown by Example 1.3, our results generalize many of the existing results presented in metric spaces. Indeed, it provides more choices of tool implements to check whether a fixed point of considered mapping exists.
Our main results are concerned with the following class of mappings.
Definition 1.4. Let (X, d) be a metric space and 2 X be a set of all nonempty subset of X. A multi-valued mapping T: X → 2 X is called p-contractive if there exist a u-distance p on X and r ∈ [0, 1) such that for any x1, x2 ∈ X and y1 ∈ T(x1) there is y2 ∈ T(x2) such that
Remark 1.5. Definition 1.4 was introduced and its fixed point theorems were proved in [9], but by using the concept of w-distance. Note that in the case p = d, the mapping T is called a contraction.
We now recall some basic concepts and well-known results.
Definition 1.6. [11]. Let (X, d) be a metric space and p be a u-distance on X. Then, a sequence {x n } of X is called p-Cauchy sequence if there exists a sequence {z n } of X such that
or
where θ: X × X × [0, ∞) × [0, ∞) → [0, ∞) is a function satisfying (u 2) ~ (u 5) for a u-distance p.
Lemma 1.7. [11]. Let (X, d) be a metric space and let p be a u-distance on X. Suppose that a sequence {x n } of X satisfies
or
Then, {x n } is a p-Cauchy sequence.
Lemma 1.8. [11]. Let (X, d) be a metric space and let p be a u-distance on X. If {x n } is a p-Cauchy sequence, then {x n } is a Cauchy sequence.
Lemma 1.9. [11]. Let (X, d) be a metric space and let p be a u-distance on X. Let x, y ∈ X. If there exists z ∈ X such that p(z, x) = 0 and p(z, y) = 0, then x = y.
Definition 1.10. Let (X, d) be a metric space and T: X → 2 X be a mapping. For any fixed x0 ∈ X, a sequence {x n } = {x0, x1, x2, ...} ⊂ X such that xn+1∈ T (x n ) is called an orbit of x0 with respect to mapping T. We will denote by the set of all orbital sequences of x0 with respect to mapping T.
2. Main results
From now on, in view of Remark 1.2, if θ: X × X × [0, ∞) × [0, ∞) → [0, ∞) is a mapping satisfying (u 2) ~ (u 5) for the considered u-distance, we will always understand that θ is a nondecreasing function in its third and fourth variables.
Inspired by an idea presented by Suzuki [8], we have an important tool for proving our main result.
Lemma 2.1. Let (X, d) be a metric space and let p be a u-distance on X. If {x n } is a p-Cauchy sequence and {y n } is a sequence satisfying
then {y n } is also a p-Cauchy sequence and.
Proof. Let θ: X × X × [0, ∞) × [0, ∞) → [0, ∞) satisfying (u 2) ~ (u 5) for a u-distance p. Since {x n } is a p-Cauchy sequence, there exists a sequence {z n } of X such that
Now, let {y n } be a sequence satisfying . Let us put
and
We note that {β n } is a nonincreasing sequence and converge to 0. Thus, from (u 2) we can define a strictly increasing function f from ℕ into itself such that
for all n ∈ ℕ. Using such a function f, we now define a function g: ℕ → ℕ by
Then, we can see that
• g(n) ≤ f (g(n)) ≤ n for all n ∈ ℕ with g(n) ≥ 2.
•.
•.
Now we consider
This means {y n } is a p-Cauchy sequence. Furthermore, since
we have , by (u 5). This completes the proof. □
Now we present our main results, which are related to p-contractive mapping.
Lemma 2.2. Let (X, d) be a metric space and let T: X → 2 X be a p-contractive mapping. Then, for each u0 ∈ X, there exists an orbitsuch that
Consequently, {u n } is a p-Cauchy sequence.
Proof. Let u0 ∈ X be arbitrary and u1 ∈ T(u0) be chosen. Then, by T as a p-contractive mapping, there exists u2 ∈ T(u1) such that
For this u2 ∈ T(u1), again by T as a p-contractive, we can find u3 ∈ T(u2) such that
Continuing this process, we obtain a sequence {u n } in X such that un+1∈ T(u n ) and
Notice that we have
for each n ∈ ℕ. This gives,
where n, m ∈ ℕ with m ≥ n. Consequently,
This proves the first part of this lemma. Furthermore, the second part is followed from (2.6) and Lemma 1.7. □
Lemma 2.3. Let (X, d) be a complete metric space and T: X → 2 X be a p-contractive mapping. Then, there exist a sequence {w n } and v0in × such that {w n } ⊂ T(v0) and {w n } converges to v0.
Proof. Let θ: X × X × [0, ∞) × [0, ∞) → [0, ∞) be a mapping satisfying (u 2) ~ (u 5) for this u-distance p.
Let u0 ∈ X be chosen. By Lemma 2.2, we know that there exists such that {u n } is a p-Cauchy sequence. Moreover, it satisfies
where n, m ∈ ℕ with m ≥ n. Since {u n } is a p-Cauchy sequence in a metric complete space (X, d), it is a convergent sequence, say limn→∞u n = v0, for some v0 ∈ X. Consequently, by (u3) and (2.7), we have
For this v0 ∈ X, by using the p-contractiveness of mapping T, we can find a sequence {w n } in T(v0) such that
It follows that
for any n ∈ ℕ.
Now we show that {w n } converges to v0. In fact, since θ is a nondecreasing function in its third and fourth variables, then (2.8) and (2.9) imply
and
Hence, by (u 5), we conclude that limn→∞d(v0, w n ) = 0. This means that {w n } converges to v0, and the proof is completed. □
For a metric space (X, d), we will denote by Cl(X) the set of all closed subsets of X. In view of proving Lemma 2.3, we can obtain a fixed point theorem in the general metric space setting.
Theorem 2.4. Let (X, d) be a metric space and T: X → 2 X be a p-contractive mapping. If there exist u0, v0 ∈ X andsuch that
-
(i)
;
-
(ii)
.
Then, F(T) ≠ Ø. Furthermore, v0 ∈ F (T).
Next, we provide some fixed point theorems for p-contractive mapping in a complete metric space. □
Theorem 2.5. Let (X, d) be a complete metric space and T: X → Cl(X) be a p-contractive mapping. Then, there exists v0 ∈ X such that v0 ∈ T (v0) and p(v0, v0) = 0.
Proof. Let u0 ∈ X be chosen. From Lemma 2.2 and Theorem 2.4, we know that there exist a p-Cauchy sequence and v0 ∈ F(T) such that {u n } converges to v0,
and
We now show that p(v0, v0) = 0. Observe that, since T is a p-contractive mapping and v0 ∈ T (v0), we can find v1 ∈ T (v0) such that
In fact, by using this process, we can obtain a sequence {v n } in X such that vn+1∈ T (v n ) and
It follows that
By using (2.11) and (2.12), we have
Consequently, by using this one together with (2.10), we get
Thus, since {u n } is a p-Cauchy sequence, we know from (2.13) and Lemma 2.1 that {v n } is a p-Cauchy sequence and . Thus, since (X, d) is a complete metric space, there exists x0 ∈ X such that . Consequently, by using (u 3), we obtain
This implies
On the other hand, since , and , we know that x0 = v0. Hence, from (2.14), we conclude that p(v0, v0) = 0. This completes the proof. □
Remark 2.6. Theorem 2.5 extends a result presented by Suzuki and Takahashi [9], from the concept of w-distance to the concept of u-distance.
By using Theorem 2.5, we can obtain the following result.
Corollary 2.7. Let (X, d) be a complete metric space, and let T: X → X be a p-contractive mapping. Then, T has a unique fixed point v0 ∈ X. Further, such v0satisfies p(v0, v0) = 0.
Proof. It follows from Theorem 2.5 that there exists v0 ∈ X such that T(v0) = v0 and p(v0, v0) = 0. Now if y0 = T(y0), we see that
where r ∈ [0, 1) satisfies the condition of p-contractive mapping T. Consequently, since r ∈ [0, 1), we have p(v0, y0) = 0. Hence, by p(v0, v0) = 0 and Lemma 1.9, we conclude that v0 = y0. This completes the proof. □
Obviously, our Corollary 2.7 is a generalization of Banach contraction principle. Now we provide an interesting example.
Example 2.8. Let a ∈ (1, ∞) be a fixed real number. Let c and d be two positive real numbers such that and , respectively. Let X = [0, a] and d: X × X → [0, ∞) be a usual metric. Let us consider a mapping T: X → X, which is defined by
Observe that for each , we have
since c > 1. This fact implies that the Banach contraction principle cannot be used for guaranteeing the existence of fixed point of our considered mapping T.
On the other hand, define now a function p: X × X → [0, ∞) by
It follows that p is a u-distance, see [11].
Let us choose . Notice that, by the choice of d, we have . We will show that T satisfies all hypotheses of our Corollary 2.7, with respect to this real number r and u-distance p. To do this, we consider the following cases:
Case 1: If . We have
Case 2: If . We have
By using above facts, we can show that all assumptions of Corollary 2.7 are satisfied. In fact, we can check that F(T) = {0}.
Remark 2.9. Example 2.8 shows that Corollary 2.7 is a genuine generalization of the Banach contraction principle.
References
Banach S: Surles opérations dans les ensembles abstraits et leurs applications aux êquations intégrales. Fund Math 1922, 3: 133–181.
Nadler SB Jr: Multi-valued contraction mappings. Pac J Math 1969,30(2):475–488.
Cho YJ, Hirunworakit S, Petrot N: Set-valued fixed points theorems for generalized contractive mappings without the Hausdorff metric. Appl Math Lett 2011, 24: 1959–1967. 10.1016/j.aml.2011.05.030
Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math Japonica 1996,44(2):381–391.
Sintunavart W, Kumam P: Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition. Appl Math Lett 2009, 22: 1877–1881. 10.1016/j.aml.2009.07.015
Sintunavart W, Kumam P: Weak condition for generalized multi-valued ( f ; α ; β )-weak contraction mappings. Appl Math Lett 2011, 24: 460–465. 10.1016/j.aml.2010.10.042
Suzuki T: Generalized distance and existence theorems in complete metric spaces. J Math Anal Appl 2001,253(2):440–458. 10.1006/jmaa.2000.7151
Suzuki T: Subrahmanyam's fixed point theorem. Nonlinear Anal 2009, 71: 1678–1683. 10.1016/j.na.2009.01.004
Suzuki T, Takahashi W: Fixed point theorems and characterizations of metric completeness. Topol Methods Nonlinear Anal 1996, 8: 371–382.
Suwannawit J, Petrot N: Common fixed point theorem for hybrid generalized multivalued. Thai J Math 2011,9(2):417–427.
Ume J-S: Existence theorems for generalized distance on complete metric spaces. Fixed Point Theory Appl 2010,2010(397150):21.
Acknowledgements
The authors would like to thank the anonymous referees for a careful reading of the manuscript and helpful suggestions. Narin Petrot was supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Both authors contributed equally in this paper. They read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Hirunworakit, S., Petrot, N. Some fixed point theorems for contractive multi-valued mappings induced by generalized distance in metric spaces. Fixed Point Theory Appl 2011, 78 (2011). https://doi.org/10.1186/1687-1812-2011-78
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2011-78