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

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), let𝐻()=𝐴,𝐵maxsup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑𝑦,𝐴,ifthemaximumexists;∞,otherwise. (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 𝐻𝑇𝑥,𝑇𝑦≤𝑟𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋, (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 𝐻𝑇𝑥,𝑇𝑦≤𝛽𝑑𝑥,𝑦𝑑𝑥,𝑦, (3) where 𝑥,𝑦∈𝑋 and 𝛽 is a function from (0,∞) into [0,1) such that limsup_𝑟→𝑡+𝛽(𝑟)<1,𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑡∈(0,∞). (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 𝐻𝑇𝑥,𝑇𝑦≤𝛽𝑑𝑥,𝑦𝑑𝑥,𝑦, (5) where 𝑥,𝑦∈𝑋 and 𝛽 is a function from [0,∞) into [0,1) such that limsup_𝑟→𝑡+𝛽(𝑟)<1,𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑡∈[0,∞). (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 𝑥,𝑦∈𝑋,𝛼𝑥,𝑦≥𝜂𝑥,𝑦⟹𝛼_∗𝐺𝑥,𝐺𝑦≥𝜂_∗𝐺𝑥,𝐺𝑦, (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 𝑥,𝑦∈𝑋,𝛼𝑥,𝑦≥𝜂𝑥,𝑦⟹𝛼(𝑢,𝑣)≥𝜂(𝑢,𝑣),∀𝑢∈𝐺𝑥,𝑣∈𝐺𝑦. (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 𝛼𝑥,𝑦=1,if1𝑥,𝑦∈𝑛∶𝑛∈ℕ∪{0}0,otherwise, (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 𝑑𝑑,𝑥,𝑦∈𝑋,𝛼𝑥,𝑦≥𝜂𝑥,𝑦⟹𝐻𝐺𝑥,𝐺𝑦≤𝛽𝑥,𝑦𝑥,𝑦 (10) where 𝛽∶[0,∞)→[0,1) satisfying limsup_𝑟→𝑡+𝛽(𝑟)<1 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 {𝑥_𝑛} is a sequence in 𝑋 with 𝑥_𝑛→𝑥 as 𝑛→∞ and 𝛼(𝑥_𝑛−1,𝑥_𝑛)≥𝜂(𝑥_𝑛−1,𝑥_𝑛) for each 𝑛∈ℕ, then one has 𝛼(𝑥_𝑛−1,𝑥)≥𝜂(𝑥_𝑛−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 𝑑𝑥₁,𝐺𝑥₁≤𝐻𝐺𝑥₀,𝐺𝑥₁≤𝛽𝑑𝑥₀,𝑥₁𝑑𝑥₀,𝑥₁. (11) There exists 𝑥₂∈𝐺𝑥₁ such that 𝑑𝑥₁,𝑥₂≤𝐻𝐺𝑥₀,𝐺𝑥₁+1−𝛽𝑑𝑥₀,𝑥₁2𝑑𝑥₀,𝑥₁≤𝛽𝑑𝑥₀,𝑥₁𝑑𝑥₀,𝑥₁+1−𝛽𝑑𝑥₀,𝑥₁2𝑑𝑥₀,𝑥₁. (12) Consider 𝛾(𝑡)=(𝛽(𝑡)+1)/2 for all 𝑡∈[0,∞). Then limsup_𝑟→𝑡+𝛾(𝑟)<1 for all 𝑡∈[0,∞). From (12), we have 𝑑𝑥₁,𝑥₂≤𝛾𝑑𝑥₀,𝑥₁𝑑𝑥₀,𝑥₁. (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 𝑑𝑥₂,𝐺𝑥₂≤𝐻𝐺𝑥₁,𝐺𝑥₂≤𝛽𝑑𝑥₁,𝑥₂𝑑𝑥₁,𝑥₂. (14) There exists 𝑥₃∈𝐺𝑥₂ such that 𝑑𝑥₂,𝑥₃≤𝐻𝐺𝑥₁,𝐺𝑥₂+1−𝛽𝑑𝑥₁,𝑥₂2𝑑𝑥₁,𝑥₂≤𝛽𝑑𝑥₁,𝑥₂𝑑𝑥₁,𝑥₂+1−𝛽𝑑𝑥₁,𝑥₂2𝑑𝑥₁,𝑥₂=𝛾𝑑𝑥₁,𝑥₂𝑑𝑥₁,𝑥₂. (15) Continuing the same process, we get a sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛∈𝐺𝑥_𝑛−1, 𝑥_𝑛≠𝑥_𝑛−1, 𝛼(𝑥_𝑛,𝑥_𝑛−1)≥𝜂(𝑥_𝑛,𝑥_𝑛−1), and 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝛾𝑑𝑥_𝑛−1,𝑥_𝑛𝑑𝑥_𝑛−1,𝑥_𝑛foreach𝑛∈ℕ. (16) It follows from 𝛾(𝑡)<1 for all 𝑡∈[0,∞) that {𝑑(𝑥_𝑛,𝑥_𝑛+1)} is a strictly decreasing sequence in ℝ. Hence it converges to some nonnegative real number 𝜐. Since limsup_𝑟→𝜐+𝛾(𝑟)<1 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(𝑥_𝑛,𝑥_𝑛+1)≤𝜐+𝜖 for all 𝑛∈ℕ with 𝑛≥𝑤. Then 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝛾𝑑𝑥_𝑛−1,𝑥_𝑛𝑑𝑥_𝑛−1,𝑥_𝑛≤𝑠𝑑𝑥_𝑛−1,𝑥_𝑛, (17) for each 𝑛≥𝑤. Also, we have ^∞_𝑛=1𝑑𝑥_𝑛−1,𝑥_𝑛≤^𝑤_𝑛=1𝑑𝑥_𝑛−1,𝑥_𝑛+^∞_𝑛=1𝑠^𝑛𝑑𝑥_𝑤−1,𝑥_𝑤<∞. (18) Hence {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, then there exists 𝑥^∗∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝑥^∗. If we suppose that 𝐺 is continuous, then 𝑑𝑥^∗,𝐺𝑥^∗=lim_𝑛→∞𝑑𝑥_𝑛,𝐺𝑥^∗≤lim_𝑛→∞𝐻𝐺𝑥_𝑛−1,𝐺𝑥^∗=0. (19) On the other hand, since 𝛼𝑥_𝑛−1,𝑥_𝑛≥𝜂𝑥_𝑛−1,𝑥_𝑛, (20) for each 𝑛∈ℕ and 𝑥_𝑛→𝑥^∗ as 𝑛→∞, then we have 𝛼𝑥_𝑛−1,𝑥^∗≥𝜂𝑥_𝑛−1,𝑥^∗, (21) for each 𝑛∈ℕ. Then from (10), we have 𝑑𝑥^∗,𝐺𝑥^∗=lim_𝑛→∞𝑑𝑥_𝑛,𝐺𝑥^∗≤lim_𝑛→∞𝐻𝐺𝑥_𝑛−1,𝐺𝑥^∗≤lim_𝑛→∞𝛽𝑑𝑥_𝑛,𝑥^∗𝑑𝑥_𝑛,𝑥^∗≤lim_𝑛→∞𝑑𝑥_𝑛,𝑥^∗=0. (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 [)𝐺𝑥=𝑥,∞if𝑥𝑥>10,2if0≤𝑥≤1−∞,𝑥²if𝑥<0, (23)𝛼∶𝑋×𝑋→[0,∞) by 𝛼=1𝑥,𝑦if[)0𝑥,𝑦∈0,∞otherwise, (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 𝐻𝐺𝑥,𝐺𝑦=𝑥−𝑦2=𝛽𝑑𝑥,𝑦𝑑𝑥,𝑦. (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 {𝑥_𝑛} in 𝑋 with 𝑥_𝑛→𝑥 as 𝑛→∞ and 𝛼(𝑥_𝑛−1,𝑥_𝑛)≥𝜂(𝑥_𝑛−1,𝑥_𝑛) for each 𝑛∈ℕ, we have 𝛼(𝑥_𝑛−1,𝑥)≥𝜂(𝑥_𝑛−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 𝑑𝑑,𝑥,𝑦∈𝑋,𝛼𝑥,𝑦≥𝜂𝑥,𝑦⟹𝐻𝐺𝑥,𝐺𝑦≤𝛽𝑥,𝑦𝑥,𝑦 (26) where 𝛽∶[0,∞)→[0,1) satisfying limsup_𝑟→𝑡+𝛽(𝑟)<1 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 {𝑥_𝑛} is a sequence in 𝑋 with 𝑥_𝑛→𝑥 as 𝑛→∞ and 𝛼(𝑥_𝑛−1,𝑥_𝑛)≥𝜂(𝑥_𝑛−1,𝑥_𝑛) for each 𝑛∈ℕ, then we have 𝛼(𝑥_𝑛−1,𝑥)≥𝜂(𝑥_𝑛−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).