Common Fixed Points for Maps on Topological Vector Space Valued Cone Metric Spaces

Abstract

We introduced a notion of topological vector space valued cone metric space and obtained some common fixed point results. Our results generalize some recent results in the literature.

1. Introduction

Huang and Zhang [1] generalized the notion of metric space by replacing the set of real numbers by ordered Banach space, deffined a cone metric space, and established some fixed point theorems for contractive type mappings in a normal cone metric space. Subsequently, several other authors [2–5] studied the existence of common fixed point of mappings satisfying a contractive type condition in normal cone metric spaces. Afterwards, Rezapour and Hamlbarani [6] studied fixed point theorems of contractive type mappings by omitting the assumption of normality in cone metric spaces (see also [7–14]). In this paper we obtain common fixed points for a pair of self-mappings satisfying a generalized contractive type condition without the assumption of normality in a class of topological vector space valued cone metric spaces which is bigger than that introduced by Huang and Zhang [1].

Let (𝐸,𝜏) be always a topological vector space and 𝑃 a subset of 𝐸. Then, 𝑃 is called a cone whenever

(i)𝑃 is closed, nonempty and 𝑃≠{0},(ii)𝑎𝑥+𝑏𝑦∈𝑃 for all 𝑥,𝑦∈𝑃 and nonnegative real numbers 𝑎,𝑏,(iii)𝑃∩(−𝑃)={0}.

For a given cone 𝑃⊆𝐸, we can define a partial ordering ≤ with respect to 𝑃 by 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝑃.𝑥<𝑦 will stand for 𝑥≤𝑦 and 𝑥≠𝑦, while 𝑥≪𝑦 will stand for 𝑦−𝑥∈int𝑃, where int𝑃 denotes the interior of 𝑃.

Definition 1.1. Let 𝑋 be a nonempty set. Suppose that the mapping 𝑑∶𝑋×𝑋→𝐸 satisfies(𝑑₁)0≤𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦,(𝑑₂)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋,(𝑑₃)𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋.Then 𝑑 is called a cone metric on 𝑋 and (𝑋,𝑑) is called a topological vector space valued cone metric space.

Note that Huang and Zhang [1] notion of cone metric space is a special case of our notion of topological vector space valued cone metric space.

Example 1.2. Let 𝑋=[0,1],andlet𝐸 be the set of all real valued functions on 𝑋 which also have continuous derivatives on𝑋,then 𝐸is a vector space over ℝ under the following operations: (𝑓+𝑔)(𝑡)=𝑓(𝑡)+𝑔(𝑡),(𝛼𝑓)(𝑡)=𝛼𝑓(𝑡), (1.1) for all 𝑓,𝑔∈𝐸,𝛼∈ℝ.Let 𝜏be the strongest vector (locally convex) topology on 𝐸,then (𝑋,𝜏) is a topological vector space which is not normable and is not even metrizable (see [15]). Define 𝑑∶𝑋×𝑋→𝐸 as follows: (𝑑(𝑥,𝑦))(𝑡)=|𝑥−𝑦|𝑒^𝑡,𝑃={𝑥∈𝐸∶𝑥(𝑡)⩾0∀𝑡∈𝑋}. (1.2) Then (𝑋,𝑑)is a topological vector space valued cone metric space.

Example 1.2 shows that this category of cone metric spaces is larger than that considered in [1–8] .

Definition 1.3. Let (𝑋,𝑑) be a topological vector space valued cone metric space, and let 𝑥∈𝑋 and {𝑥_𝑛}_𝑛≥1 be a sequence in 𝑋. Then
(i)??{𝑥_𝑛}_𝑛≥1 converges to 𝑥 whenever for every 𝑐∈𝐸 with 0≪𝑐 there is a natural number 𝑁 such that 𝑑(𝑥_𝑛,𝑥)≪𝑐 for all 𝑛≥𝑁. We denote this by lim_𝑛→∞𝑥_𝑛=𝑥 or 𝑥_𝑛→𝑥.
(ii)??{𝑥_𝑛}_𝑛≥1 is a Cauchy sequence whenever for every 𝑐∈𝐸 with 0≪𝑐 there is a natural number 𝑁 such that 𝑑(𝑥_𝑛,𝑥_𝑚)≪𝑐 for all 𝑛,𝑚≥𝑁.
(iii)??(𝑋,𝑑) is a complete topological vector space valued cone metric space if every Cauchy sequence is convergent.

2. Fixed Point

In this section, we shall give some results which generalize [6, Theorems 2.3, 2.6, 2.7, and 2.8] (and so [1, Theorems 1, 3, and 4]).

Theorem 2.1. Let (𝑋,𝑑) be a complete topological vector space valued cone metric space and let the self-mappings 𝑆,𝑇∶𝑋→𝑋 satisfy 𝑑(𝑆𝑥,𝑇𝑦)≤𝑘𝑑(𝑥,𝑦)+𝑙(𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑆𝑥)), (2.1) for all 𝑥,𝑦∈𝑋, where 𝑘,𝑙∈[0,1) with 𝑘+2𝑙<1. Then 𝑆 and 𝑇 have a unique common fixed point.

Proof. For 𝑥₀∈𝑋 and 𝑛≥0, define 𝑥_2𝑛+1=𝑆𝑥_2𝑛 and 𝑥_2𝑛+2=𝑇𝑥_2𝑛+1. Then, 𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)=𝑑(𝑆𝑥_2𝑛,𝑇𝑥_2𝑛+1)⩽𝑘𝑑(𝑥_2𝑛,𝑥_2𝑛+1)+𝑙[𝑑(𝑥_2𝑛,𝑇𝑥_2𝑛+1)+𝑑(𝑥_2𝑛+1,𝑆𝑥_2𝑛)]=𝑘𝑑(𝑥_2𝑛,𝑥_2𝑛+1)+𝑙[𝑑(𝑥_2𝑛,𝑇𝑥_2𝑛+1)]⩽𝑘𝑑(𝑥_2𝑛,𝑥_2𝑛+1)+𝑙[𝑑(𝑥_2𝑛,𝑥_2𝑛+1)+𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)]=[𝑘+𝑙]𝑑(𝑥_2𝑛,𝑥_2𝑛+1)+𝑙𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2). (2.2) It implies that 𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)⩽[(𝑘+𝑙)/(1−𝑙)]𝑑(𝑥_2𝑛,𝑥_2𝑛+1). Similarly, 𝑑(𝑥_2𝑛+2,𝑥_2𝑛+3)=𝑑(𝑆𝑥_2𝑛+2,𝑇𝑥_2𝑛+1)⩽𝑘𝑑(𝑥_2𝑛+2,𝑥_2𝑛+1)+𝑙[𝑑(𝑥_2𝑛+2,𝑇𝑥_2𝑛+1)+𝑑(𝑥_2𝑛+1,𝑆𝑥_2𝑛+2)]⩽𝑘𝑑(𝑥_2𝑛+2,𝑥_2𝑛+1)+𝑙[𝑑(𝑥_2𝑛+2,𝑥_2𝑛+3)+𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)]=[𝑘+𝑙]𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)+𝑙𝑑(𝑥_2𝑛+2,𝑥_2𝑛+3). (2.3) Hence, 𝑑(𝑥_2𝑛+2,𝑥_2𝑛+3)≤[(𝑘+𝑙)/(1−𝑙)]𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2). Thus, 𝑑(𝑥_𝑛,𝑥_𝑛+1)⩽𝜆^𝑛𝑑(𝑥₀,𝑥₁), (2.4) for all 𝑛≥0, where 𝜆=((𝑘+𝑙)/(1−𝑙))<1. Now, for𝑛>𝑚 we have 𝑑(𝑥_𝑛,𝑥_𝑚)⩽𝑑(𝑥_𝑛,𝑥_𝑛−1)+𝑑(𝑥_𝑛−1,𝑥_𝑛−2)+⋯+𝑑(𝑥_𝑚+1,𝑥_𝑚)⩽(𝜆^𝑛−1+𝜆^𝑛−2+⋯+𝜆^𝑚)𝑑(𝑥₀,𝑥₁)⩽𝜆^𝑚1−𝜆𝑑(𝑥₀,𝑥₁). (2.5) Let 0≪𝑐. Take a symmetric neighborhood 𝑉 of 0 such that 𝑐+𝑉⊆int𝑃. Also, choose a natural number 𝑁₁ such that (𝜆^𝑚/(1−𝜆))𝑑(𝑥₁,𝑥₀)∈𝑉, for all 𝑚≥𝑁₁. Then, (𝜆^𝑚/(1−𝜆))𝑑(𝑥₁,𝑥₀)≪𝑐, for all 𝑚≥𝑁₁. Thus, 𝑑(𝑥_𝑛,𝑥_𝑚𝜆)≤^𝑚1−𝜆𝑑(𝑥₁,𝑥₀)≪𝑐, (2.6) for all 𝑛>𝑚. Therefore, {𝑥_𝑛}_𝑛≥1 is a Cauchy sequence in (𝑋,𝑑). Since 𝑋 is complete, there exists 𝑢∈𝑋 such that 𝑥_𝑛→𝑢. Choose a natural number 𝑁₂ such that 𝑑(𝑥_𝑛,𝑢)≪[𝑐(1−𝑙)/2(1+𝑙)] for all 𝑛⩾𝑁₂. Thus, 𝑑(𝑢,𝑇𝑢)⩽𝑑(𝑢,𝑥_2𝑛+1)+𝑑(𝑥_2𝑛+1,𝑇𝑢)=𝑑(𝑢,𝑥_2𝑛+1)+𝑑(𝑆𝑥_2𝑛,𝑇𝑢)⩽𝑑(𝑢,𝑥_2𝑛+1)+𝑘𝑑(𝑢,𝑥_2𝑛)+𝑙[𝑑(𝑢,𝑆𝑥_2𝑛)+𝑑(𝑥_2𝑛,𝑇𝑢)]⩽𝑑(𝑢,𝑥_2𝑛+1)+𝑘𝑑(𝑢,𝑥_2𝑛)+𝑙[𝑑(𝑢,𝑥_2𝑛+1)+𝑑(𝑥_2𝑛,𝑢)+𝑑(𝑢,𝑇𝑢)]=(1+𝑙)𝑑(𝑢,𝑥_2𝑛+1)+(𝑘+𝑙)𝑑(𝑢,𝑥_2𝑛)+𝑙𝑑(𝑢,𝑇𝑢). (2.7) So, 𝑑(𝑢,𝑇𝑢)⩽[1+𝑙1−𝑙]𝑑(𝑢,𝑥_2𝑛+1)+[𝑘+𝑙1−𝑙]𝑑(𝑢,𝑥_2𝑛)⩽[1+𝑙1−𝑙]𝑑(𝑢,𝑥_2𝑛+1)+[1+𝑙1−𝑙]𝑑(𝑢,𝑥_2𝑛)=𝑐2+𝑐2=𝑐, (2.8) for all 𝑛≥𝑁₂. Therefore, 𝑑(𝑢,𝑇𝑢)≪𝑐/𝑖 for all 𝑖⩾1. Hence, (𝑐/𝑖)−𝑑(𝑢,𝑇𝑢)∈𝑃 for all𝑖⩾1. Since 𝑃 is closed, −𝑑(𝑢,𝑇𝑢)∈𝑃 and so 𝑑(𝑢,𝑇𝑢)=0. Hence, 𝑢 is a fixed point of 𝑇. Similarly, we can show that 𝑢=𝑆𝑢. Now, we show that 𝑆 and 𝑇 have a unique fixed point. For this, assume that there exists another point 𝑢^∗ in 𝑋 such that 𝑢^∗=𝑇𝑢^∗=𝑆𝑢^∗. Then, 𝑑(𝑢,𝑢^∗)=𝑑(𝑆𝑢,𝑇𝑢^∗)⩽𝑘𝑑(𝑢,𝑢^∗)+𝑙[𝑑(𝑢,𝑇𝑢^∗)+𝑑(𝑢^∗,𝑆𝑢)]⩽𝑘𝑑(𝑢,𝑢^∗)+𝑙[𝑑(𝑢,𝑢^∗)+𝑑(𝑢^∗,𝑢)]⩽(𝑘+2𝑙)𝑑(𝑢,𝑢^∗). (2.9) Since 𝑘+2𝑙<1,𝑑(𝑢,𝑢^∗)=0 and so 𝑢=𝑢^∗.

The following corollary generalizes [6, Theorems 2.3, 2.7, and 2.8] (and so [1, Theorems 1 and 4]).

Corollary 2.2. Let (𝑋,𝑑) be a complete topological vector space valued cone metric space and let the self-mapping 𝑇∶𝑋→𝑋 satisfy 𝑑(𝑇𝑥,𝑇𝑦)⩽𝑎𝑑(𝑥,𝑦)+𝑏𝑑(𝑥,𝑇𝑦)+𝑐𝑑(𝑦,𝑇𝑥) for all 𝑥,𝑦∈𝑋, where 𝑎,𝑏,𝑐∈[0,1) with 𝑎+𝑏+𝑐<1. Then 𝑇 has a unique fixed point.

Proof. The symmetric property of 𝑑 and the above inequality imply that 𝑑(𝑇𝑥,𝑇𝑦)⩽𝑎𝑑(𝑥,𝑦)+𝑏+𝑐2[𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)]. (2.10) By substituting 𝑆=𝑇𝑎=𝑘 and (𝑏+𝑐)/2=𝑙 in Theorem 2.1, we obtain the required result.

Theorem 2.3. Let (𝑋,𝑑) be a complete topological vector space valued cone metric space and let the self-mappings 𝑆,𝑇∶𝑋→𝑋 satisfy 𝑑(𝑆𝑥,𝑇𝑦)≤𝑘𝑑(𝑥,𝑦)+𝑙(𝑑(𝑥,𝑆𝑥)+𝑑(𝑦,𝑇𝑦)), (2.11) for all 𝑥,𝑦∈𝑋, where 𝑘,𝑙∈[0,1) with 𝑘+2𝑙<1. Then 𝑆 and 𝑇 have a unique common fixed point.

Proof. For 𝑥₀∈𝑋 and 𝑛≥0, define 𝑥_2𝑛+1=𝑆𝑥_2𝑛 and 𝑥_2𝑛+2=𝑇𝑥_2𝑛+1. Then, 𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)=𝑑(𝑆𝑥_2𝑛,𝑇𝑥_2𝑛+1)⩽𝑘𝑑(𝑥_2𝑛,𝑥_2𝑛+1)+𝑙[𝑑(𝑥_2𝑛,𝑆𝑥_2𝑛)+𝑑(𝑥_2𝑛+1,𝑇𝑥_2𝑛+1)]=𝑘𝑑(𝑥_2𝑛,𝑥_2𝑛+1)+𝑙[𝑑(𝑥_2𝑛,𝑥_2𝑛+1)+𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)]=[𝑘+𝑙]𝑑(𝑥_2𝑛,𝑥_2𝑛+1)+𝑙𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2). (2.12) It implies that 𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)⩽[(𝑘+𝑙)/(1−𝑙)]𝑑(𝑥_2𝑛,𝑥_2𝑛+1). Similarly, 𝑑(𝑥_2𝑛+2,𝑥_2𝑛+3)=𝑑(𝑆𝑥_2𝑛+2,𝑇𝑥_2𝑛+1)⩽𝑘𝑑(𝑥_2𝑛+2,𝑥_2𝑛+1)+𝑙[𝑑(𝑥_2𝑛+2,𝑆𝑥_2𝑛+2)+𝑑(𝑥_2𝑛+1,𝑇𝑥_2𝑛+1)]=𝑘𝑑(𝑥_2𝑛+2,𝑥_2𝑛+1)+𝑙[𝑑(𝑥_2𝑛+2,𝑥_2𝑛+3)+𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)]=[𝑘+𝑙]𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)+𝑙𝑑(𝑥_2𝑛+2,𝑥_2𝑛+3). (2.13) Hence, 𝑑(𝑥_2𝑛+2,𝑥_2𝑛+3)≤[(𝑘+𝑙)/(1−𝑙)]𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2). Thus, 𝑑(𝑥_𝑛,𝑥_𝑛+1)⩽𝜆^𝑛𝑑(𝑥₀,𝑥₁), (2.14) for all 𝑛≥0, where 𝜆=((𝑘+𝑙)/(1−𝑙))<1. Now, for𝑛>𝑚 we have 𝑑(𝑥_𝑛,𝑥_𝑚)⩽𝑑(𝑥_𝑛,𝑥_𝑛−1)+𝑑(𝑥_𝑛−1,𝑥_𝑛−2)+⋯+𝑑(𝑥_𝑚+1,𝑥_𝑚)⩽(𝜆^𝑛−1+𝜆^𝑛−2+⋯+𝜆^𝑚)𝑑(𝑥₀,𝑥₁)⩽𝜆^𝑚1−𝜆𝑑(𝑥₀,𝑥₁). (2.15) Let 0≪𝑐. Take a symmetric neighborhood 𝑉 of 0 such that 𝑐+𝑉⊆int𝑃. Also, choose a natural number 𝑁₁ such that (𝜆^𝑚/(1−𝜆))𝑑(𝑥₁,𝑥₀)∈𝑉, for all 𝑚≥𝑁₁. Then, (𝜆^𝑚/(1−𝜆))𝑑(𝑥₁,𝑥₀)≪𝑐, for all 𝑚≥𝑁₁. Thus, 𝑑(𝑥_𝑛,𝑥_𝑚𝜆)≤^𝑚1−𝜆𝑑(𝑥₁,𝑥₀)≪𝑐, (2.16) for all 𝑛>𝑚. Therefore, {𝑥_𝑛}_𝑛≥1 is a Cauchy sequence in (𝑋,𝑑). Since 𝑋 is complete, there exists 𝑢∈𝑋 such that 𝑥_𝑛→𝑢. Choose a natural number 𝑁₂ such that 𝑑(𝑥_𝑛,𝑢)≪[𝑐(1−𝑙)/2(1+𝑙)] for all 𝑛⩾𝑁₂. Thus, 𝑑(𝑢,𝑇𝑢)⩽𝑑(𝑢,𝑥_2𝑛+1)+𝑑(𝑥_2𝑛+1,𝑇𝑢)=𝑑(𝑢,𝑥_2𝑛+1)+𝑑(𝑆𝑥_2𝑛,𝑇𝑢)⩽𝑑(𝑢,𝑥_2𝑛+1)+𝑘𝑑(𝑢,𝑥_2𝑛)+𝑙[𝑑(𝑢,𝑇𝑢)+𝑑(𝑥_2𝑛,𝑆𝑥_2𝑛)]⩽𝑑(𝑢,𝑥_2𝑛+1)+𝑘𝑑(𝑢,𝑥_2𝑛)+𝑙[𝑑(𝑢,𝑥_2𝑛+1)+𝑑(𝑥_2𝑛,𝑢)+𝑑(𝑢,𝑇𝑢)]=(1+𝑙)𝑑(𝑢,𝑥_2𝑛+1)+(𝑘+𝑙)𝑑(𝑢,𝑥_2𝑛)+𝑙𝑑(𝑢,𝑇𝑢). (2.17) So, 𝑑(𝑢,𝑇𝑢)⩽[1+𝑙1−𝑙]𝑑(𝑢,𝑥_2𝑛+1)+[𝑘+𝑙1−𝑙]𝑑(𝑢,𝑥_2𝑛)⩽[1+𝑙1−𝑙]𝑑(𝑢,𝑥_2𝑛+1)+[1+𝑙1−𝑙]𝑑(𝑢,𝑥_2𝑛)≪𝑐2+𝑐2=𝑐, (2.18) for all 𝑛≥𝑁₂. Therefore, 𝑑(𝑢,𝑇𝑢)≪𝑐/𝑖 for all 𝑖⩾1. Hence, (𝑐/𝑖)−𝑑(𝑢,𝑇𝑢)∈𝑃 for all 𝑖⩾1. Since 𝑃 is closed, −𝑑(𝑢,𝑇𝑢)∈𝑃 and so 𝑑(𝑢,𝑇𝑢)=0. Hence, 𝑢 is a fixed point of 𝑇. Similarly, we can show that 𝑢=𝑆𝑢. Now, we show that 𝑆 and 𝑇 have a unique fixed point. For this, assume that there exists another point 𝑢^∗ in 𝑋 such that 𝑢^∗=𝑇𝑢^∗=𝑆𝑢^∗. Then, 𝑑(𝑢,𝑢^∗)=𝑑(𝑆𝑢,𝑇𝑢^∗)⩽𝑘𝑑(𝑢,𝑢^∗)+𝑙[𝑑(𝑢,𝑢^∗)+𝑑(𝑢^∗,𝑢)]=𝑘𝑑(𝑢,𝑢^∗). (2.19) Since 𝑘<1,??𝑑(𝑢,𝑢^∗)=0 and so 𝑢=𝑢^∗.

The following corollary generalizes [6, Theorem 2.6] (and so [1, Theorem 3]).

Corollary 2.4. Let (𝑋,𝑑) be a complete topological vector space valued cone metric space and let the self-mapping 𝑇∶𝑋→𝑋 satisfy 𝑑(𝑇𝑥,𝑇𝑦)⩽𝑎𝑑(𝑥,𝑦)+𝑏𝑑(𝑥,𝑇𝑥)+𝑐𝑑(𝑦,𝑇𝑦) for all 𝑥,𝑦∈𝑋, where 𝑎,𝑏,𝑐∈[0,1) with 𝑎+𝑏+𝑐<1. Then 𝑇 has a unique fixed point.

Proof is similar to the proof of Corollary 2.2.

Example 2.5. Let (𝑋,𝑑)be a topological vector space valued cone metric space of Example 1.2. Define 𝑆,𝑇∶𝑋→𝑋as follows: 𝑡𝑆(𝑡)=𝑇(𝑡)=31if𝑥≠1,6if𝑥=1. (2.20) Then, |𝑆𝑥−𝑇𝑦|𝑒^𝑡≤𝑘|𝑥−𝑦|𝑒^𝑡+𝑙[|𝑥−𝑆𝑥|𝑒^𝑡+|𝑦−𝑇𝑦|𝑒^𝑡], (2.21) if 𝑘=1/6,𝑙=5/18.Hence all conditions of Theorem 2.3 are satisfied.

Acknowledgments

The present version of the paper owes much to the precise and kind remarks of the learned referees.