A Note on the Fixed Point of Generalized Eventual Cyclic Gross Contractive Mappings

Abstract

Based on the work of Agarwal et al., in press, we introduce a concept of generalized eventual cyclic gross contractive mapping in metric spaces, which is a generalization of the eventual cyclic gross contractions. Furthermore, we analyze the existence of a fixed point for this type of contractive mapping, and present a new and general fixed point theorem.

1. Introduction

The following classical Banach contraction principle is well-known.

Theorem 1. If 𝑇 is a self-mapping in a complete metric space 𝑋 satisfying the following condition: 𝑑𝑇𝑥,𝑇𝑦<𝜆𝑑𝑥,𝑦,𝜆∈[0,1), (1) then 𝑇 has a unique fixed point.

Based on this principle and the idea of its proof, many researchers have presented numerous extensions of the principle and new types of fixed point theorems (cf, e.g., [1–3]).

In [3], Karpagam and Agrawal proved the following unique fixed point for cyclic mappings.

Theorem 2 (see [3]). Let 𝐴 and 𝐵 be two nonempty closed subsets of a complete metric space (𝑋,𝑑). Suppose 𝜙∶𝐴∪𝐵→𝐴∪𝐵 is a cyclic mapping such that for some 𝑥∈𝐴, there exists a 𝑘_𝑥∈(0,1) such that 𝑑𝜙^2𝑛𝑥,𝜙𝑦≤𝑘_𝑥𝑑𝜙^2𝑛−1𝑥,𝑦,𝑛∈𝑁,𝑦∈𝐴. (2) Then, 𝐴∩𝐵 is nonempty and 𝜙 has a unique fixed point in 𝐴∩𝐵.

Very recently, Agarwal et al. [1] defined a new type of cyclic mappings, “eventual cyclic gross contraction” in complete metric spaces, and established a fixed point theorem for the eventual cyclic gross contraction.

Theorem 3 (see [1]). Let 𝐴 and 𝐵 be two nonempty closed subsets of a complete metric space (𝑋,𝑑), and let 𝑇∶𝐴∪𝐵→𝐴∪𝐵 be an eventual cyclic gross contraction. Then 𝐴∩𝐵 is nonempty and 𝑇 has a unique fixed point in 𝐴∩𝐵.

Inspired by the work in [1], in this paper, we introduce a concept of generalized eventual cyclic gross contractive mapping in metric spaces, which is a generalization of the eventual cyclic gross contractions. Furthermore, we obtain a fixed point theorem for this type of contractive mappings, which extends Theorem 3.

2. Generalized Eventual Cyclic Gross Contractive Mappings

Let 𝑅⁺ be the set of all nonnegative real numbers and 𝑁 the set of all natural numbers. Let (𝑋,𝑑) be a metric space, and let 𝐴 and 𝐵 be nonempty subsets of 𝑋.

Now, we introduce the concept of generalized eventual cyclic gross contractive mappings.

Definition 4. A mapping 𝑇∶𝐴∪𝐵→𝐴∪𝐵 is called a generalized eventual cyclic gross contractive mapping if the following are satisfied: (1)𝑇(𝐴)⊂𝐵,𝑇(𝐵)⊂𝐴,  (2)for some 𝑥∈𝐴, 𝑓𝑑𝑇^2𝑛𝑇𝑥,𝑇𝑦≤𝑓𝜆𝑑^2𝑛−1𝑥,𝑇𝑦+(1−𝜆)𝑑𝑇^2𝑛𝑑𝑇𝑥,𝑦−𝑔^2𝑛−1𝑥,𝑇𝑦,𝑑𝑇^2𝑛𝑥,𝑦,𝑛≥𝑛₀∈𝑁,𝑦∈𝐴, (3)where 𝜆∈[0,1), 𝑓∶𝑅⁺→𝑅⁺ is a monotone increasing and continuous function, 𝑔∶𝑅⁺×𝑅⁺→𝑅⁺ is a lower semicontinuous mapping such that 𝑔(𝑎,𝑏)=0 if and only if 𝑎=𝑏=0, and 𝑛₀ is sufficiently large.

Lemma 5. Let 𝑇 be a generalized eventual cyclic gross contractive mapping and 𝜆∈(1/2,1]. Then {𝑇^𝑛𝑥} is a Cauchy sequence for every 𝑥∈𝐴∪𝐵.

Proof. For every 𝑥∈𝐴∪𝐵, let 𝑛≥𝑛₀,𝑦=𝑇^2𝑛−2𝑥. (4) Then, (3) and the monotone increasing property of 𝑓 imply that 𝑑𝑇^2𝑛𝑥,𝑇^2𝑛−1𝑥≤(1−𝜆)𝑑𝑇^2𝑛𝑥,𝑇^2𝑛−2𝑥≤(1−𝜆)𝑑𝑇^2𝑛𝑥,𝑇^2𝑛−1𝑥+(1−𝜆)𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛−2𝑥, (5) since 𝑓𝑑𝑇^2𝑛𝑑𝑇𝑥,𝑇𝑦=𝑓^2𝑛𝑥,𝑇^2𝑛−1𝑥≤𝑓𝜆𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛−1𝑥+(1−𝜆)𝑑𝑇^2𝑛𝑥,𝑇^2𝑛−2𝑥−𝑔𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛−1𝑥,𝑑𝑇^2𝑛𝑥,𝑇^2𝑛−2𝑥=𝑓(1−𝜆)𝑑𝑇^2𝑛𝑥,𝑇^2𝑛−2𝑥−𝑔0,𝑑𝑇^2𝑛𝑥,𝑇^2𝑛−2𝑥≤𝑓(1−𝜆)𝑑𝑇^2𝑛𝑥,𝑇^2𝑛−2𝑥. (6) Hence, 𝑑𝑇^2𝑛𝑥,𝑇^2𝑛−1𝑥≤1−𝜆𝜆𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛−2𝑥. (7) Thus, 𝑑𝑇^𝑛𝑥,𝑇^𝑛+1𝑥≤1−𝜆𝜆𝑑𝑇^𝑛−1𝑥,𝑇^𝑛𝑥. (8) Since 𝜆∈(1/2,1], we deduce that lim_𝑛→∞𝑑(𝑇^𝑛𝑥,𝑇^𝑛+1𝑥) exists. Set lim_𝑛→∞𝑑𝑇^𝑛𝑥,𝑇^𝑛+1𝑥=𝑙≥0. (9) If 𝑙>0, then (5) implies that 𝑙≤(1−𝜆)lim_𝑛→∞𝑑𝑇^2𝑛𝑥,𝑇^2𝑛−2𝑥≤2(1−𝜆)𝑙. (10) So, we have 11−𝜆≥2. (11) This is impossible since 𝜆≤1/2. Therefore, lim_𝑛→∞𝑑𝑇^𝑛𝑥,𝑇^𝑛+1𝑥=0. (12) This means that, for any 𝜀>0, there exists a natural number 𝑁₁ such that for any natural number 𝑛≥𝑁₁, 𝑑𝑇^𝑛𝑥,𝑇^𝑛+1𝑥<𝜀. (13) Moreover, for any natural number 𝑚>𝑛>0, we have 𝑑𝑇^𝑛+𝑁1𝑥,𝑇^𝑚+𝑁1𝑥≤^𝑚−1_𝑖=𝑛𝑑𝑇^𝑁1+𝑖𝑥,𝑇^𝑁1+𝑖+1𝑥≤^𝑚−1_𝑖=𝑛1−𝜆𝜆^𝑖𝑑𝑇^𝑁1𝑥,𝑇^𝑁1+1𝑥<((1−𝜆)/𝜆)^𝑛1−((1−𝜆)/𝜆)𝜀. (14) This implies that 𝑑𝑇^𝑛+𝑁1𝑥,𝑇^𝑚+𝑁1𝑥⟶0 (15) as 𝑛→∞, since 1−𝜆𝜆∈[0,1). (16) Therefore, {𝑇^𝑛𝑥} is a Cauchy sequence.

Lemma 6. Let 𝑇 be a generalized eventual cyclic gross contractive mapping and 𝜆∈[0,1/2]. Then {𝑇^𝑛𝑥} is a Cauchy sequence for every 𝑥∈𝐴∪𝐵.

Proof. For every 𝑥∈A∪𝐵, let 𝑛≥𝑛₀,𝑦=𝑇^2𝑛𝑥. (17) Then, 𝑑𝑇^2𝑛𝑥,𝑇^2𝑛+1𝑥≤𝜆𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛+1𝑥≤𝜆𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛𝑥+𝜆𝑑𝑇^2𝑛𝑥,𝑇^2𝑛+1𝑥, (18) since 𝑓𝑑𝑇^2𝑛𝑑𝑇𝑥,𝑇𝑦=𝑓^2𝑛𝑥,𝑇^2𝑛+1𝑥≤𝑓𝜆𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛+1𝑥+(1−𝜆)𝑑𝑇^2𝑛𝑥,𝑇^2𝑛𝑥−𝑔𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛+1𝑥,𝑑𝑇^2𝑛𝑥,𝑇^2𝑛𝑥=𝑓𝜆𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛+1𝑥−𝑔𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛𝑥𝑇,0≤𝑓𝜆𝑑^2𝑛−1𝑥,𝑇^2𝑛+1𝑥. (19) So, 𝑑𝑇^2𝑛𝑥,𝑇^2𝑛+1𝑥≤𝜆𝑑𝑇1−𝜆^2𝑛−1𝑥,𝑇^2𝑛𝑥. (20) Hence, 𝑑𝑇^𝑛𝑥,𝑇^𝑛+1𝑥≤𝜆𝑑𝑇1−𝜆^𝑛−1𝑥,𝑇^𝑛𝑥. (21) Since 𝜆∈[0,1/2], we see that for all 𝑛≥𝑛₀, the nonnegative sequence {𝑑(𝑇^𝑛𝑥,𝑇^𝑛+1𝑥)} is decreasing. Let lim_𝑛→∞𝑑𝑇^𝑛𝑥,𝑇^𝑛+1𝑥=𝑙≥0. (22) Then we obtain 𝑙≤𝜆lim_𝑛→∞𝑑𝑇^2𝑛−1𝑥,𝑇^2𝑛+1𝑥≤2𝜆𝑙. (23) This implies that 1𝜆≥2 (24) if 𝑙≠0. Clearly, 𝜆>1/2 is impossible. Moreover, 𝜆=1/2 is also impossible by the proof of [1, Theorem 2.4].
Now, we show that {𝑇^𝑛𝑥} is a Cauchy sequence. It follows from lim_𝑛→∞𝑑𝑇^𝑛𝑥,𝑇^𝑛+1𝑥=0 (25) that for any 𝜀>0, there exists a natural number 𝑁₀ such that for any natural number 𝑛≥𝑁₀, 𝑑𝑇^𝑛𝑥,𝑇^𝑛+1𝑥<𝜀. (26) For 𝜆∈[0,1/2) and any natural number 𝑚>𝑛>0, we have 𝑑𝑇^𝑛+𝑁0𝑥,𝑇^𝑚+𝑁0𝑥≤^𝑚−1_𝑖=𝑛𝑑𝑇^𝑁0+𝑖𝑥,𝑇^𝑁0+𝑖+1𝑥≤^𝑚−1_𝑖=𝑛𝜆1−𝜆^𝑖𝑑𝑇^𝑁0𝑥,𝑇^𝑁0+1𝑥<(𝜆/(1−𝜆))^𝑛1−(𝜆/(1−𝜆))𝜀. (27) So 𝑑𝑇^𝑛+𝑁0𝑥,𝑇^𝑚+𝑁0𝑥⟶0 (28) as 𝑛→∞, since 𝜆∈[).1−𝜆0,1 (29) Therefore, {𝑇^𝑛𝑥} is a Cauchy sequence.
For 𝜆=1/2, by the proof of [1, Theorem 2.4], we know that {𝑇^𝑛𝑥} also is a Cauchy sequence.

The following is our central theorem about cycle mapping in complete metric spaces.

Theorem 7. Assume that (𝑋,𝑑) is a complete metric space, 𝐴 and 𝐵 are closed, and 𝑇 is a generalized eventual cyclic gross contractive mapping. Then, 𝐴∩𝐵 is nonempty and 𝑇 has a unique fixed point in 𝐴∩𝐵.

Proof. By the virtue of Lemmas 5 and 6, we know that for every 𝑥∈𝐴∪𝐵, {𝑇^𝑛𝑥} is a Cauchy sequence. Since (𝑋,𝑑) is a complete metric space, 𝐴 and 𝐵 are closed, there exists some 𝑥₀∈𝐴∪𝐵 such that lim_𝑛→∞𝑇^𝑛𝑥=𝑥₀. (30) Therefore, 𝑥₀=lim_𝑛→∞𝑇^2𝑛+1𝑥𝑥∈𝐴,₀=lim_𝑛→∞𝑇^2𝑛𝑥∈𝐵. (31) So, 𝑥₀∈𝐴∩𝐵; that is, 𝐴∩𝐵≠∅.
On the other hand, we obtain 𝑓𝑑𝑇𝑥₀,𝑥₀≤𝑓𝜆𝑑𝑥₀,𝑥₀+(1−𝜆)𝑑𝑇𝑥₀,𝑥₀−𝑔𝑑𝑥₀,𝑥₀,𝑑𝑇𝑥₀,𝑥₀=𝑓(1−𝜆)𝑑𝑥₀,𝑇𝑥₀−𝑔0,𝑑𝑇𝑥₀,𝑥₀, (32) in view of 𝑓𝑑𝑇𝑥₀,𝑇^2𝑛𝑥≤𝑓𝜆𝑑𝑥₀,𝑇^2𝑛𝑥+(1−𝜆)𝑑𝑇𝑥₀,𝑇^2𝑛−1𝑥−𝑔𝑑𝑥₀,𝑇^2𝑛𝑥,𝑑𝑇𝑥₀,𝑇^2𝑛−1𝑥. (33)
If 𝑑(𝑇𝑥₀,𝑥₀)=0, then 𝑥₀ is a fixed point of 𝑇.
Otherwise, if 𝑑(𝑇𝑥₀,𝑥₀)>0, then we have 𝑔0,𝑑𝑇𝑥₀,𝑥₀>0. (34) Equation (32) implies that 𝑓𝑑𝑇𝑥₀,𝑥₀<𝑓(1−𝜆)𝑑𝑥₀,𝑇𝑥₀. (35) Hence, 𝑑𝑇𝑥₀,𝑥₀<(1−𝜆)𝑑𝑥₀,𝑇𝑥₀. (36) This is impossible since 1−𝜆∈[0,1].
According to previous discussions, it is concluded that 𝑑𝑇𝑥₀,𝑥₀=0, (37) and therefore, 𝑥₀ is a fixed point of 𝑇.
If there is 𝑦₀∈𝐴∩𝐵 such that 𝑇𝑦₀=𝑦₀, then 𝑑(𝑥₀,𝑦₀)=0; that is, the fixed point of 𝑇 is unique, since 𝑔𝑑𝑥₀,𝑦₀,𝑑𝑥₀,𝑦₀≤0, (38) in view of 𝑓𝑑𝑥₀,𝑦₀=𝑓𝑑𝑇𝑥₀,𝑇𝑦₀≤𝑓𝜆𝑑𝑥₀,𝑇𝑦₀+(1−𝜆)𝑑𝑇𝑥₀,𝑦₀−𝑔𝑑𝑥₀,𝑇𝑦₀,𝑑𝑇𝑥₀,𝑦₀≤𝑓𝑑𝑥₀,𝑦₀−𝑔𝑑𝑥₀,𝑦₀,𝑑𝑥₀,𝑦₀. (39) The proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the NSF of China (11171210).