Periodic Points and Fixed Points for the Weaker (𝜙,𝜑)-Contractive Mappings in Complete Generalized Metric Spaces

Abstract

We introduce the notion of weaker (𝜙,𝜑)-contractive mapping in complete metric spaces and prove the periodic points and fixed points for this type of contraction. Our results generalize or improve many recent fixed point theorems in the literature.

1. Introduction and Preliminaries

Let (𝑋,𝑑) be a metric space, 𝐷 a subset of 𝑋 and 𝑓∶𝐷→𝑋 a map. We say 𝑓 is contractive if there exists 𝛼∈[0,1) such that, for all 𝑥,𝑦∈𝐷, 𝑑(𝑓𝑥,𝑓𝑦)≤𝛼⋅𝑑(𝑥,𝑦).(1.1) The well-known Banach’s fixed point theorem asserts that if 𝐷=𝑋, 𝑓 is contractive and (𝑋,𝑑) is complete, then 𝑓 has a unique fixed point in 𝑋. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. In 1969, Boyd and Wong [2] introduced the notion of Φ-contraction. A mapping 𝑓∶𝑋→𝑋 on a metric space is called Φ-contraction if there exists an upper semicontinuous function Φ∶[0,∞)→[0,∞) such that𝑑(𝑓𝑥,𝑓𝑦)≤Φ(𝑑(𝑥,𝑦))∀𝑥,𝑦∈𝑋.(1.2)

In 2000, Branciari [3] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involving three terms instead of two. Later, many authors worked on this interesting space (e.g., [4–9]).

Let (𝑋,𝑑) be a generalized metric space. For 𝛾>0 and 𝑥∈𝑋, we define 𝐵_𝛾(𝑥)∶={𝑦∈𝑋∣𝑑(𝑥,𝑦)<𝛾}.(1.3) Branciari [3] also claimed that {𝐵_𝛾(𝑥)∶𝛾>0,𝑥∈𝑋} is a basis for a topology on 𝑋, 𝑑 is continuous in each of the coordinates and a generalized metric space is a Hausdorff space. We recall some definitions of a generalized metric space, as follows.

Definition 1.1 (see [3]). Let 𝑋 be a nonempty set and 𝑑∶𝑋×𝑋→[0,∞) a mapping such that for all 𝑥,𝑦∈𝑋 and for all distinct point 𝑢,𝑣∈𝑋 each of them different from 𝑥 and 𝑦, one has(i)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦;(ii)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥);(iii)𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑢)+𝑑(𝑢,𝑣)+𝑑(𝑣,𝑦) (rectangular inequality). Then (𝑋,𝑑) is called a generalized metric space (or shortly g.m.s).

Definition 1.2 (see [3]). Let (𝑋,𝑑) be a g.m.s, {𝑥_𝑛} 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 by 𝑥_𝑛→𝑥 as 𝑛→∞.

Definition 1.3 (see [3]). Let (𝑋,𝑑) be a g.m.s, {𝑥_𝑛} a sequence in 𝑋, and 𝑥∈𝑋. We say that {𝑥_𝑛} is g.m.s Cauchy sequence if and only if for each 𝜀>0, there exists 𝑛₀∈ℕ such that 𝑑(𝑥_𝑚,𝑥_𝑛)<𝜀 for all 𝑛>𝑚>𝑛₀.

Definition 1.4 (see [3]). Let (𝑋,𝑑) be a g.m.s. Then 𝑋 is called complete g.m.s if every g.m.s Cauchy sequence is g.m.s convergent in 𝑋.

In this paper, we also recall the notion of Meir-Keeler function (see [10]). A function 𝜙∶[0,∞)→[0,∞) is said to be a Meir-Keeler function if for each 𝜂>0, there exists 𝛿>0 such that for 𝑡∈[0,∞) with 𝜂≤𝑡<𝜂+𝛿, we have 𝜙(𝑡)<𝜂. Generalization of the above function has been a heavily investigated branch research. Particularly, in [11, 12], the authors proved the existence and uniqueness of fixed points for various Meir-Keeler-type contractive functions. We now introduce the notion of weaker Meir-Keeler function 𝜙∶[0,∞)→[0,∞), as follows.

Definition 1.5. We call 𝜙∶[0,∞)→[0,∞) a weaker Meir-Keeler function if for each 𝜂>0, there exists 𝛿>0 such that for 𝑡∈[0,∞) with 𝜂≤𝑡<𝜂+𝛿, there exists 𝑛₀∈ℕ such that 𝜙^𝑛0(𝑡)<𝜂.

2. Main Results

In the paper, we denote by Φ the class of functions 𝜙∶[0,∞)→[0,∞) satisfying the following conditions:

(𝜙₁) 𝜙∶[0,∞)→[0,∞) is a weaker Meir-Keeler function;

(𝜙₂) 𝜙(𝑡)>0 for 𝑡>0, 𝜙(0)=0;

(𝜙₃) for all 𝑡∈(0,∞), {𝜙^𝑛(𝑡)}_𝑛∈ℕ is decreasing;

(𝜙₄) if lim_𝑛→∞𝑡_𝑛=𝛾, then lim_𝑛→∞𝜙(𝑡_𝑛)≤𝛾.

And we denote by Θ the class of functions 𝜑∶[0,∞)→[0,∞) satisfying the following conditions:

(𝜑₁) 𝜑 is continuous;

(𝜑₂) 𝜑(𝑡)>0 for 𝑡>0 and 𝜑(0)=0.

Our main result is the following.

Theorem 2.1. Let (𝑋,𝑑) be a Hausdorff and complete g.m.s, and let 𝑓∶𝑋→𝑋 be a function satisfying 𝑑(𝑓𝑥,𝑓𝑦)≤𝜙(𝑑(𝑥,𝑦))−𝜑(𝑑(𝑥,𝑦))(2.1) for all 𝑥,𝑦∈𝑋 and 𝜙∈Φ, 𝜑∈Θ. Then 𝑓 has a periodic point 𝜇 in 𝑋, that is, there exists 𝜇∈𝑋 such that 𝜇=𝑓^𝑝𝜇 for some 𝑝∈ℕ.

Proof. Given 𝑥₀, define a sequence {𝑥_𝑛} in 𝑋 by 𝑥_𝑛+1=𝑓𝑥_𝑛for𝑛∈ℕ∪{0}.(2.2)Step 1. We will prove that lliimm_{𝑛→∞𝑛→∞}𝑑𝑥_𝑛,𝑥_𝑛+1=0,𝑑𝑥_𝑛,𝑥_𝑛+2=0.(2.3) Using inequality (2.1), we have that for each 𝑛∈ℕ∪{0}𝑑𝑥_𝑛,𝑥_𝑛+1=≤𝑑𝜙𝑓𝑑𝑥𝑥_𝑛−1,_𝑛−1𝑓𝑥_𝑛≤𝜙𝑑𝑥_𝑛−1,𝑥_𝑛−𝜑𝑑𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,(2.4) and so 𝑑𝑥_𝑛,𝑥_𝑛+1≤≤𝜙𝜙𝑑𝜙𝑥𝑑𝑥_𝑛−1,𝑥_𝑛𝑛−2≤≤𝜙⋯_𝑛,𝑥_𝑛−1=𝜙²𝑑𝑥_𝑛−2,𝑥_𝑛−1𝑑𝑥₀,𝑥₁.(2.5) Since {𝜙^𝑛(𝑑(𝑥₀,𝑥₁))}_𝑛∈ℕ is decreasing, it must converge to some 𝜂≥0. We claim that 𝜂=0. On the contrary, assume that 𝜂>0. Then by the definition of weaker Meir-Keeler function 𝜙, there exists 𝛿>0 such that for 𝑥₀,𝑥₁∈𝑋 with 𝜂≤𝑑(𝑥₀,𝑥₁)<𝛿+𝜂, there exists 𝑛₀∈ℕ such that 𝜙^𝑛0(𝑑(𝑥₀,𝑥₁))<𝜂. Since lim_𝑛→∞𝜙^𝑛(𝑑(𝑥₀,𝑥₁))=𝜂, there exists 𝑝₀∈ℕ such that 𝜂≤𝜙^𝑝(𝑑(𝑥₀,𝑥₁))<𝛿+𝜂, for all 𝑝≥𝑝₀. Thus, we conclude that 𝜙^𝑝0+𝑛0(𝑑(𝑥₀,𝑥₁))<𝜂. So we get a contradiction. Therefore, lim_𝑛→∞𝜙^𝑛(𝑑(𝑥₀,𝑥₁))=0, that is, lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0.(2.6) Using inequality (2.1), we also have that for each 𝑛∈ℕ𝑑𝑥_𝑛,𝑥_𝑛+2≤=𝜙𝑑𝑓𝑑𝑥𝑥_𝑛−1,_𝑛−1𝑓𝑥_𝑛+1≤𝜙𝑑𝑥_𝑛−1,𝑥_𝑛+1−𝜑𝑑𝑥_𝑛−1,𝑥_𝑛+1,𝑥_𝑛+1,(2.7) and so 𝑑𝑥_𝑛,𝑥_𝑛+2≤≤𝜙𝜙𝑑𝜙𝑥𝑑𝑥_𝑛−1,𝑥_{𝑛+1𝑛−2}≤≤𝜙⋯_𝑛,𝑥_𝑛=𝜙²𝑑𝑥_𝑛−2,𝑥_𝑛𝑑𝑥₀,𝑥₂.(2.8) Since {𝜙^𝑛(𝑑(𝑥₀,𝑥₂))}_𝑛∈ℕ is decreasing, by the same proof process, we also conclude lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+2=0.(2.9)
Next, we claim that {𝑥_𝑛} is g.m.s Cauchy. We claim that the following result holds.Step 2. Claim that for every 𝜀>0, there exists 𝑛∈ℕ such that if 𝑝,𝑞≥𝑛 then 𝑑(𝑥_𝑝,𝑥_𝑞)<𝜀.
Suppose the above statement is false. Then there exists 𝜖>0 such that for any 𝑛∈ℕ, there are 𝑝_𝑛,𝑞_𝑛∈ℕ with 𝑝_𝑛>𝑞_𝑛≥𝑛 satisfying 𝑑𝑥_𝑞𝑛,𝑥_𝑝𝑛≥𝜖.(2.10) Further, corresponding to 𝑞_𝑛≥𝑛, we can choose 𝑝_𝑛 in such a way that it the smallest integer with 𝑝_𝑛>𝑞_𝑛≥𝑛 and 𝑑(𝑥_𝑞𝑛,𝑥_𝑝𝑛)≥𝜖. Therefore, 𝑑(𝑥_𝑞𝑛,𝑥_𝑝𝑛−1)<𝜖. By the rectangular inequality and (2.3), we have 𝜖≤≤𝑑𝑑𝑥𝑥_{𝑝𝑛𝑝𝑛},𝑥_𝑞𝑛<𝑑𝑥_𝑝𝑛,𝑥_𝑝𝑛−2+𝑑𝑥_𝑝𝑛−2,𝑥_𝑝𝑛−1+𝑑𝑥_𝑝𝑛−1,𝑥_𝑞𝑛,𝑥_𝑝𝑛−2+𝑑𝑥_𝑝𝑛−2,𝑥_𝑝𝑛−1+𝜖.(2.11) Let 𝑛→∞. Then we get lim_𝑛→∞𝑑𝑥_𝑝𝑛,𝑥_𝑞𝑛=𝜖.(2.12) On the other hand, we have 𝑑𝑑𝑥𝑥_𝑝𝑛,𝑥_{𝑞𝑛𝑝𝑛−1}≤𝑑𝑥_𝑝𝑛,𝑥_𝑝𝑛−1+𝑑𝑥_𝑝𝑛−1,𝑥_𝑞𝑛−1+𝑑𝑥_𝑞𝑛−1,𝑥_𝑞𝑛,,𝑥_𝑞𝑛−1≤𝑑𝑥_𝑝𝑛−1,𝑥_𝑝𝑛+𝑑𝑥_𝑝𝑛,𝑥_𝑞𝑛+𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛−1.(2.13) Let 𝑛→∞. Then we get lim_𝑛→∞𝑑𝑥_𝑝𝑛−1,𝑥_𝑞𝑛−1=𝜖.(2.14) Using inequality (2.1), we have 𝑑𝑥_𝑝𝑛,𝑥_𝑞𝑛=≤𝑑𝜙𝑓𝑑𝑥𝑥_𝑝𝑛−1,_𝑝𝑛−1𝑓𝑥_𝑞𝑛−1,𝑥_𝑞𝑛−1−𝜑𝑑𝑥_𝑝𝑛−1,𝑥_𝑞𝑛−1.(2.15) Letting 𝑛→∞, using the definitions of the functions 𝜙 and 𝜑, we have 𝜖≤𝜖−𝜑(𝜖),(2.16) which implies that 𝜑(𝜖)=0. By the definition of the function 𝜑, we have 𝜖=0. So we get a contradiction. Therefore {𝑥_𝑛} is g.m.s Cauchy.Step 3. We claim that 𝑓 has a periodic point in 𝑋.
Suppose, on contrary, 𝑓 has no periodic point. Then {𝑥_𝑛} is a sequence of distinct points, that is, 𝑥_𝑝≠𝑥_𝑞 for all 𝑝,𝑞∈ℕ with 𝑝≠𝑞. By Step 2, since 𝑋 is complete g.m.s, there exists 𝜈∈𝑋 such that 𝑥_𝑛→𝜈. Using inequality (2.1), we have 𝑑𝑓𝑥_𝑛,𝑓𝜈≤𝜙𝑑𝑥_𝑛,𝜈−𝜑𝑑𝑥_𝑛,𝜈.(2.17) Letting 𝑛→∞, we have 𝑑𝑓𝑥_𝑛,𝑓𝜈⟶0,as𝑛⟶∞,(2.18) that is, 𝑥_𝑛+1=𝑓𝑥_𝑛⟶𝑓𝜈,as𝑛⟶∞.(2.19) As (𝑋,𝑑) is Hausdorff, we have 𝜈=𝑓𝜈, a contradiction with our assumption that 𝑓 has no periodic point. Therefore, there exists 𝜈∈𝑋 such that 𝜈=𝑓^𝑝(𝜈) for some 𝑝∈ℕ. So 𝑓 has a periodic point in 𝑋.

Following Theorem 2.1, it is easy to get the below fixed point result.

Theorem 2.2. Let (𝑋,𝑑) be a Hausdorff and complete g.m.s, and let 𝑓∶𝑋→𝑋 be a function satisfying 𝑑(𝑓𝑥,𝑓𝑦)≤𝜙(𝑑(𝑥,𝑦))−𝜑(𝑑(𝑥,𝑦))(2.20) for all 𝑥,𝑦∈𝑋, where 𝜙∈Φ with 0<𝜙(𝑡)<𝑡 for all 𝑡>0, and 𝜑∈Θ. Then 𝑓 has a unique fixed point in 𝑋.

Proof. From Theorem 2.1, we conclude that 𝑓 has a periodic point 𝜈∈𝑋, that is, there exists 𝜈∈𝑋 such that 𝜈=𝑓^𝑝(𝜈) for some 𝑝∈ℕ. If 𝑝=1, then we complete the proof, that is, 𝜈 is a fixed point of 𝑓. If 𝑝>1, then we will show that 𝜇=𝑓^𝑝−1𝜈 is a fixed point of 𝑓. Suppose that it is not the case, that is, 𝑓^𝑝−1𝜈≠𝑓^𝑝𝜈. Then Using inequality (2.1), we have 𝑑(𝜈,𝑓𝜈)=≤𝜙𝑑𝑓𝑑^𝑝𝜈𝑓,𝑓^𝑝+1_𝑝−1𝜈<𝜙𝑑𝑓_𝑝−1𝜈,𝑓^𝑝𝜈−𝜑𝑑𝑓^𝑝−1𝜈,𝑓^𝑝𝜈≤𝑑𝑓_𝑝−1𝜈,𝑓^𝑝𝜈𝜈,𝑓^𝑝𝜈.(2.21) Using inequality (2.1), we also have 𝑑𝑓^𝑝−1𝜈,𝑓^𝑝𝜈≤≤𝜙𝜙𝑑𝑑𝑓𝑓^𝑝−2_𝑝−2𝜈,𝑓^𝑝−1𝜈−𝜑𝑑𝑓^𝑝−2𝜈,𝑓^𝑝−1𝜈≤𝑑𝑓_𝑝−2𝜈,𝑓^𝑝−1𝜈𝜈,𝑓^𝑝−1𝜈.(2.22) Continuing this process, we conclude that 𝑑(𝜈,𝑓𝜈)<𝑑𝑓^𝑝−1𝜈,𝑓^𝑝𝜈≤𝑑𝑓^𝑝−2𝜈,𝑓^𝑝−1𝜈≤⋯≤𝑑(𝜈,𝑓𝜈),(2.23) which implies a contradiction. Thus, 𝜇=𝑓^𝑝−1𝜈 is a fixed point of 𝑓.
Finally, to prove the uniqueness of the fixed point, suppose 𝜇,𝜈 are fixed points of 𝑓. Then, 𝑑(𝜇,𝜈)=𝑑(𝑓𝜇,𝑓𝜈)≤𝜙(𝑑(𝜇,𝜈))−𝜑(𝑑(𝜇,𝜈)),(2.24) which implies that 𝑑(𝜇,𝜈)=0, that is, 𝜇=𝜈. So we complete the proof.

Acknowledgment

This research was supported by the National Science Council of the Republic of China.