Abstract and Applied Analysis
Volume 2013, Issue 1 418798
Research Article
Open Access

Mizoguchi-Takahashi’s Fixed Point Theorem with α, η Functions

Muhammad Usman Ali, 

Muhammad Usman Ali

Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12, Islamabad, Pakistan nust.edu.pk

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Tayyab Kamran, 

Tayyab Kamran

Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan qau.edu.pk

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Wutiphol Sintunavarat, 

Corresponding Author

Wutiphol Sintunavarat

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand tu.ac.th

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Phayap Katchang, 

Corresponding Author

Phayap Katchang

Division of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand rmutl.ac.th

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First published: 26 December 2013
https://doi.org/10.1155/2013/418798
Citations: 10
Academic Editor: Hassen Aydi

Abstract

We introduce the notion of generalized α*-admissible mappings. By using this notion, we prove a fixed point theorem. Our result generalizes Mizoguchi-Takahashi’s fixed point theorem. We also provide some examples to show the generality of our work.

1. Introduction and Preliminaries

Let (X, d) be a metric space. For each x ∈ X and A⊆X, d(x, A)∶ = {d(x, y) : y ∈ A}. We denote by K(X) the class of all nonempty compact subset of X, by CB(X) the class of all nonempty closed and bounded subsets of X, and by CL(X) the class of all nonempty closed subsets of X. For every A, B ∈ CL(X), let
(1)
Such a map H is called generalized Hausdorff metric induced by d.

Nadler [1] extended the Banach contraction principle to multivalued mappings as follows.

Theorem 1 (see [1].)Let (X, d) be a complete metric space and T is a mapping from X into CB(X) such that

(2)
where r ∈ [0,1). Then T has a fixed point.

Reich [2] extended the above result in the following way.

Theorem 2 (see [2].)Let (X, d) be a complete metric space and T : X → K(X) is a mapping satisfying

(3)
where x, y ∈ X and β is a function from (0, ∞) into [0,1) such that
(4)
Then T has a fixed point.

Reich [2] raised the question: whether the range of T, K(X) can be replaced by CB(X) or CL(X). Mizoguchi and Takahashi [3] gave a positive answer to the conjecture of Reich [2], when the inequality holds also for t = 0; in particular they proved the following.

Theorem 3 (see [3].)Let (X, d) be a complete metric space and T : X → CB(X) is a mapping satisfying

(5)
where x, y ∈ X and β is a function from [0, ∞) into [0,1) such that
(6)
Then T has a fixed point.

The other proofs of Theorem 3 have been given by Daffer and Kaneko [4] and Chang [5]. Eldred et al. [6] claimed that Theorem 3 is equivalent to Theorem 1. Suzuki produced an example [7, page 753] to disprove their claim and showed that Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s fixed point theorem. Reader can find some more results related to Mizoguchi-Takahashi’s fixed point theorem in [8–14].

Samet et al. [15] introduced the notion of α-ψ-contractive and α-admissible self-mappings and proved some fixed point results for such mappings in complete metric spaces. Karapinar and Samet [16] generalized these notions and obtained some fixed point results. Asl et al. [17] extended these notions to multifunctions by introducing the notions of α*-ψ-contractive and α*-admissible mappings and proved some fixed point results. Some results in this direction are also given by the authors [18, 19]. Ali and Kamran [20] further generalized the notion of α*-ψ-contractive mappings and obtained some fixed point theorems for multivalued mappings.

Recently, Salimi et al. [21] modified the notions of α-ψ-contractive and α-admissible self-mappings by introducing another function η and established some fixed point theorems for such mappings in complete metric spaces. Hussain et al. [22] extended the result of Asl et al. and introduced the following definition.

Definition 4 (see [22].)Let G : X → CL(X) be a mapping on a metric space (X, d). Let α, η : X × X → [0, ∞) be two functions, where η is bounded. We say that G is an α*-admissible mapping with respect to η if we have

(7)
where α*(Gx, Gy) = inf {α(a, b) : a ∈ Gx, b ∈ Gy} and η*(Gx, Gy) = sup {η(a, b) : a ∈ Gx, b ∈ Gy}. In case when α(x, y) = 1 for all x, y ∈ X, then G is η*-subadmissible mapping. In case when η(x, y) = 1 for all x, y ∈ X, then G is α*-admissible.

Definition 5 (see [17].)Let (X, d) be a metric space and let α : X × X → [0, ∞) be a mapping. A mapping G : X → CL(X) is α*-admissible if α(x, y) ≥ 1⇒α*(Gx, Gy) ≥ 1, where α*(Gx, Gy) = inf {α(a, b) : a ∈ Gx, b ∈ Gy}.

In this paper, we generalize Definition 4 and provide some examples to show generality of such concept. We also establish a fixed point theorem which generalizes Mizoguchi-Takahashi’s fixed point theorem. Some illustrative examples to claim that our results properly generalize some results in the literature are presented. Furthermore, at the end of this paper, we give an open problem for further investigation.

2. Main Results

We begin this section by generalizing Definition 4.

Definition 6. Let G : X → CL(X) be a mapping on a metric space (X, d). Let α, η : X × X → [0, ∞) be two functions. We say that G is generalized α*-admissible mapping with respect to η if we have

(8)
When α(x, y) = 1 for all x, y ∈ X, then G is a generalized η*-subadmissible mapping. When η(x, y) = 1 for all x, y ∈ X, then G is α*-admissible.

Remark 7. Note that inequality (8) is weaker than (7). Moreover, η involved in inequality (8) is not necessarily bounded.

Example 8. Let X = {1/n : n ∈ ℕ}∪{0}∪{n + 1 : n ∈ ℕ} be endowed with the usual metric d. Define G : X → CL(X) by Gx = {0, x2} for all x, y ∈ X, α : X × X → [0, ∞) by

(9)
and η : X × X → [0, ∞) by η(x, y) = x + y for each x, y ∈ X. Then for x, y ∈ X with α(x, y) ≥ η(x, y), we have α(u, v) ≥ η(u, v) for all u ∈ Gx and v ∈ Gy. Therefore G is generalized α*-admissible mapping with respect to η but it is not α*-admissible mapping with respect to η.

Theorem 9. Let (X, d) be a complete metric space and let G : X → CL(X) be a generalized α*-admissible mapping with respect to η such that

(10)
where β : [0, ∞)→[0,1) satisfying for all t ∈ [0, ∞). Assume that the following conditions hold:
  • (i)

    there exist x0 ∈ X and x1 ∈ Gx0 such that α(x0, x1) ≥ η(x0, x1);

  • (ii)

    either

    • (1)

      G is continuous

    •  

      or

    • (2)

      if {xn} is a sequence in X with xn → x as n → ∞ and α(xn−1, xn) ≥ η(xn−1, xn) for each n ∈ ℕ, then one has α(xn−1, x) ≥ η(xn−1, x) for each n ∈ ℕ.

Then G has a fixed point.

Proof. By hypothesis, there exist x0 ∈ X and x1 ∈ Gx0 such that α(x0, x1) ≥ η(x0, x1). If x0 = x1, then we have nothing to prove. Let x0 ≠ x1. Then from (10), we have

(11)
There exists x2 ∈ Gx1 such that
(12)
Consider γ(t) = (β(t) + 1)/2 for all t ∈ [0, ∞). Then for all t ∈ [0, ∞). From (12), we have
(13)
Since G is an α*-admissible mapping with respect to η, then α(x1, x2) ≥ η(x1, x2). If x1 = x2, then we have nothing to prove. Let x1 ≠ x2. Then from (10), we have
(14)
There exists x3 ∈ Gx2 such that
(15)
Continuing the same process, we get a sequence {xn} in X such that xn ∈ Gxn−1, xn ≠ xn−1, α(xn, xn−1) ≥ η(xn, xn−1), and
(16)
It follows from γ(t) < 1 for all t ∈ [0, ∞) that {d(xn, xn+1)} is a strictly decreasing sequence in ℝ. Hence it converges to some nonnegative real number υ. Since and γ(υ) < 1, there exists s ∈ [0,1) and ϵ > 0 such that γ(t) ≤ s for all t ∈ [υ, υ + ϵ]. We can find w ∈ ℕ such that υ ≤ d(xn, xn+1) ≤ υ + ϵ for all n ∈ ℕ with n ≥ w. Then
(17)
for each n ≥ w. Also, we have
(18)
Hence {xn} is a Cauchy sequence in X. Since X is complete, then there exists x* ∈ X such that lim n→∞xn = x*. If we suppose that G is continuous, then
(19)
On the other hand, since
(20)
for each n ∈ ℕ and xn → x* as n → ∞, then we have
(21)
for each n ∈ ℕ. Then from (10), we have
(22)
Therefore x* ∈ Gx*. This completes the proof.

The following example shows that Theorem 9 properly generalizes Theorem 3, in Section 1.

Example 10. Let X = ℝ be endowed with the usual metric d. Define G : X → CL(X) by

(23)
α : X × X → [0, ∞) by
(24)
and η : X × X → [0, ∞) by η(x, y) = |x | +|y| for each x, y ∈ X. Take β(t) = 1/2 for all t ≥ 0. Then for x, y ∈ X with α(x, y) ≥ η(x, y), we have
(25)
Also, G is generalized α*-admissible mapping with respect to η. For x0 = 1 and x1 = 0 ∈ Gx0 we have α(x0, x1) ≥ η(x0, x1). Further, for any sequence {xn} in X with xn → x as n → ∞ and α(xn−1, xn) ≥ η(xn−1, xn) for each n ∈ ℕ, we have α(xn−1, x) ≥ η(xn−1, x) for each n ∈ ℕ. Therefore, all conditions of Theorem 9 are satisfied and G has infinitely many fixed points.

Corollary 11. Let (X, d) be a complete metric space and let G : X → CL(X) be an α*-admissible mapping with respect to η such that

(26)
where β : [0, ∞)→[0,1) satisfying for all t ∈ [0, ∞). Assume that the following conditions hold:
  • (i)

    there exist x0 ∈ X and x1 ∈ Gx0 such that α(x0, x1) ≥ η(x0, x1);

  • (ii)

    either

    • (1)

      G is continuous

    •  

      or

    • (2)

      if {xn} is a sequence in X with xn → x as n → ∞ and α(xn−1, xn) ≥ η(xn−1, xn) for each n ∈ ℕ, then we have α(xn−1, x) ≥ η(xn−1, x) for each n ∈ ℕ.

Then G has a fixed point.

Proof. We can prove this result by using Theorem 9 and the fact that inequality (8) is weaker than (7).

Acknowledgments

This research was supported by the Commission on Higher Education, the Thailand Research Fund, and the Rajamangala University of Technology Lanna Tak (Grant no. MRG5580233).

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