Common Fixed Point Theorems for Nonlinear Contractive Mappings in Dislocated Metric Spaces

Abstract

In 1986, Matthews generalized Banach contraction mapping theorem in dislocated metric space that is a wider space than metric space. In this paper, we established common fixed point theorems for a class of contractive mappings. Our results extend the corresponding ones of other authors in dislocated metric spaces.

1. Introduction

Fixed point theory is an important branch of nonlinear analysis and can be used to many discipline branches, such as control theory, convex optimization, differential inclusion, and economics. Banach proved a celebrated fixed point theorem for contraction mappings in complete metric space which is one of the pivotal results of analysis. Dass and Gupta [1] generalized Banach contraction mapping in metric space. Also Rhoades [2] established a partial ordering for various definitions of contractive mapping. The concept of metric spaces, as an environmental space in fixed point theory, has been generalized in several directions. Some of such generalizations are dislocated spaces, quasimetric spaces, dislocated quasimetric spaces, and generalized quasimetric spaces. The concept of dislocated spaces is treated differently by different authors. Matthews [3] generalized Banach contraction mapping theorem in dislocated metric space. Hitzler [4] presented variants of Banach contraction principle for various modified forms of a metric space including dislocated metric space and applied them to semantic analysis of logic programs. In this context, Hitzler and Seda [5] raised some related questions on the topological aspects of dislocated metrics. In 2005, Zeyada et al. [6] generalized a fixed point theorem in dislocated quasimetric spaces. In 2008, Aage and Salunke [7] proved some results on fixed points in dislocated quasimetric space. Recently, Isufati [8] proved fixed point theorem for contractive type condition with rational expression in dislocated quasimetric space. In this paper, we study the mapping refereed by Xia Dafeng and obtained fixed point theorems in dislocated metric space. For fixed point theorems, see [9, 10]. The following definition is introduced by Xia et al. [11].

Definition 1 (see [11]). Let 𝑅₊=[0,∞). Let 𝐺₁∶𝑅²₊→𝑅₊, 𝐺₂∶𝑅³₊→𝑅₊, satisfying the following:(1)if 𝑤≤𝐺₁(𝑢,𝑣), then there exists 𝑐∈(0,1), such that 𝑤≤𝑐max{𝑢,𝑣};(2)if 𝑤≤𝐺₂(𝑢,𝑣,𝑟), then there exists 𝑐∈(0,1), such that 𝑤≤𝑐max{𝑢,𝑣,𝑟}.

Theorem 2 (see [11]). Let (𝑋,𝑑) be complete metric spaces, let 𝑓,𝑔∶𝑋→𝑋 be continuous mappings, and for all 𝑥,𝑦∈𝑋, such that 𝑑𝑓(𝑥),𝑔𝑦≤𝐺₁𝑑𝑥,𝑓(𝑥),𝑑𝑦,𝑔𝑦 (1) or 𝑑𝑓(𝑥),𝑔𝑦≤𝐺₂𝑑𝑥,𝑦,𝑑𝑥,𝑓(𝑥),𝑑𝑦,𝑔𝑦 (2) then 𝑓,𝑔 have a unique common fixed point.

2. Preliminaries

Definition 3 (see [6]). Let 𝑋 be a nonempty set and let 𝑑∶𝑋×𝑋→[0,∞) be a function called a distance function. If for all 𝑥,𝑦,𝑧∈𝑋,(1)nonnegativity: 𝑑(𝑥,𝑦)≥0;(2)faithful: 𝑑(𝑥,𝑦)=0⇔𝑥=𝑦;(3)the triangle inequality: 𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑥),So, here 𝑑 is a quasimetric on 𝑋, and (𝑋,𝑑) is called a quasimetric space.

Definition 4 (see [6]). Let 𝑋 be a nonempty set and let 𝑑∶𝑋×𝑋→[0,∞) be a function called a distance function. If for all 𝑥,𝑦,𝑧∈𝑋,(1)nonnegativity: 𝑑(𝑥,𝑦)≥0;(2)indistancy implies equality: 𝑑(𝑥,𝑦)=0=𝑑(𝑦,𝑥) implies 𝑥=𝑦;(3)the triangle inequality: 𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑥),so, here 𝑑 is called a dislocated quasimetric or 𝑑_𝑞-metric on 𝑋, and (𝑋,𝑑) is called a dislocated quasimetric space.

Definition 5 (see [6]). Let 𝑋 be a nonempty set and let 𝑑∶𝑋×𝑋→[0,∞) be a function called a distance function. If for 𝑥,𝑦,𝑧∈𝑋, (1)nonnegativity: 𝑑(𝑥,𝑦)≥0;(2)indistancy implies equality: 𝑑(𝑥,𝑦)=0=𝑑(𝑦,𝑥) implies 𝑥=𝑦;(3)symmetry: 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥);(4)the triangle inequality: 𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑥),so, here 𝑑 is called a dislocated metric or 𝑑-metric on 𝑋 and the pair (𝑋,𝑑) is called a dislocated metric space.

Definition 6 (see [6]). A sequence {𝑥_𝑛} in 𝑑_𝑞-metric space (dislocated quasimetric space) (𝑋,𝑑) is called a Cauchy sequence, if for a given 𝜖>0, there exists 𝑛₀∈𝑁 such that 𝑑(𝑥_𝑚,𝑥_𝑛)<𝜖 or 𝑑(𝑥_𝑛,𝑥_𝑚)<𝜖; that is, min{𝑑(𝑥_𝑚,𝑥_𝑛),𝑑(𝑥_𝑛,𝑥_𝑚)}<𝜖 for all 𝑚,𝑛≥𝑛₀.

Definition 7 (see [6]). A sequence {𝑥_𝑛} in 𝑑-metric spaces (𝑋,𝑑) is said to be 𝑑-converged to 𝑥∈𝑋 provided that lim𝑑𝑥_𝑛,𝑥=lim𝑑𝑥,𝑥_𝑛=0. (3) In this case, 𝑥 is called a 𝑑-limit of {𝑥_𝑛} and we write 𝑥_𝑛→𝑥.

Definition 8 (see [6]). A 𝑑-metric space (𝑋,𝑑) is called 𝑑-complete if every 𝑑-Cauchy sequence in 𝑋 converges with respect to 𝑥 in 𝑋.

Lemma 9. Every converging sequence in a 𝑑-metric space is a Cauchy sequence.

Proof. Let {𝑥_𝑛} be a sequence which converges to some 𝑥, and let 𝜖>0 be arbitrarily given. Then there exists 𝑛₀∈𝑁 with 𝑑(𝑥_𝑛,𝑥)<𝜖/2 for all 𝑛≥𝑛₀. For 𝑚,𝑛≥𝑛₀, then we obtain that 𝑑(𝑥_𝑚,𝑥)+𝑑(𝑥,𝑥_𝑛)<𝜖/2+𝜖/2=𝜖. Hence {𝑥_𝑛} is a Cauchy sequence.

Lemma 10. Limits in dislocated metric spaces are unique.

Proof. Let 𝑥 and 𝑦 be limits of the sequence {𝑥_𝑛}. Then 𝑑(𝑥_𝑛,𝑥)→0 and 𝑑(𝑥_𝑛,𝑦)→0 as 𝑛→∞. By the triangle inequality of Definition 5, we conclude that 𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑥_𝑛)+𝑑(𝑥_𝑛,𝑦)→0 as 𝑛→∞. Hence 𝑑(𝑥,𝑦)=0 and using the properties (2) of Definition 5, we conclude that 𝑥=𝑦.

Lemma 11. Limits in dislocated quasimetric spaces are unique.

Proof. Let 𝑥 and 𝑦 be limits of the sequence {𝑥_𝑛}. Then 𝑥_𝑛→𝑥 and 𝑥_𝑛→𝑦 as 𝑛→∞. By the triangle inequality, it follows that 𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑥_𝑛)+𝑑(𝑥_𝑛,𝑦). As 𝑛→∞, we have 𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑥)+𝑑(𝑦,𝑦). Similarly, 𝑑(𝑦,𝑥)≤𝑑(𝑥,𝑥)+𝑑(𝑦,𝑦). Hence |𝑑(𝑥,𝑦)−𝑑(𝑦,𝑥)|≤0; that is, 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥). Also 𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑥_𝑛)+𝑑(𝑥_𝑛,𝑦)→0 as 𝑛→∞. That is, 𝑑(𝑥,𝑦)=0, and 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥)=0. By the property (2) of Definition 4, we conclude that 𝑥=𝑦.

Example 12. Let 𝑋=𝑅⁺. Define 𝑑∶𝑋×𝑋→𝑅⁺ by 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Then the pair (𝑋,𝑑) is a dislocated metric space. We define an arbitrary sequence {𝑥_𝑛} in 𝑋; if 𝑘>𝑁, there exists an positive integer 𝑁 that satisfies |𝑥_𝑘−𝑎|<𝜖/2. Then, for any 𝑚,𝑛>𝑁, we have 𝑑(𝑥_𝑛,𝑥_𝑚)=|𝑥_𝑛−𝑥_𝑚|≤|𝑥_𝑛−𝑎|+|𝑥_𝑚−𝑎|<𝜖/2+𝜖/2=𝜖. Thus, {𝑥_𝑛} is a Cauchy sequence in 𝑋. Also as 𝑛→∞, then {𝑥_𝑛}→∞∈𝑋. Hence, every Cauchy sequence in 𝑋 is convergent with respect to 𝑑. Thus, (𝑋,𝑑) is a complete dislocated metric space.

3. Main Results

In this section, now we establish that common fixed points for mapping satisfying contractive condition are proved in the frame of dislocated metric spaces.

Definition 13 (see [9]). There exist 𝜙(𝑡) that satisfy the condition 𝜙^′, if one lets 𝜙∶[0,+∞]→[0,∞) be nondecreasing and non-negative, then lim𝜙_𝑛(𝑡)=0, for a given 𝑡>0.

Lemma 14 (see [9]). If 𝜙 satisfy the condition 𝜙^′, then 𝜙(𝑡)<𝑡, for a given 𝑡>0.

Lemma 15 (see [11]). Let 𝐹∶𝑅³₊→𝑅₊, and satisfy the condition 𝜙^′; for all 𝑢,𝑣≥0, if 𝑢≤𝐹(𝑣,𝑣,𝑢) or 𝑢≤𝐹(𝑣,𝑢,𝑣) or 𝑢≤𝐹(𝑢,𝑣,𝑣), then 𝑢≤𝜙(𝑣).

Theorem 16. Let (𝑋,𝑑) be a complete dislocated metric space, and let 𝑓,𝑔∶𝑋→𝑋 be self-mapping, if(1)either 𝑓 or 𝑔 is continuous;(2)there exists 𝐹 satisfying the condition 𝜙^′, for all 𝑥,𝑦∈𝑋, such that 𝑑𝑓(𝑥),𝑔𝑦≤𝐹𝑑𝑥,𝑦,𝑑𝑥,𝑓(𝑥),𝑑𝑦,𝑔𝑦, (4)then 𝑓,𝑔 have unique common fixed points.

Proof. Let 𝑔 be continuous, 𝑥₀ arbitrary in 𝑋, 𝑥_𝑛, 𝑦_𝑛 the sequence of 𝑋, and 𝑥_𝑛=𝑓𝑔^𝑛𝑥₀=𝑓𝑔𝑥_𝑛−1,𝑦_𝑛=𝑔𝑓𝑔^𝑛−1𝑥₀,𝑛=1,2,… (5) Obviously, 𝑦_𝑛=𝑔𝑥_𝑛−1,𝑓𝑦_𝑛=𝑥_𝑛,𝑔𝑓𝑦_𝑛=𝑔𝑥_𝑛=𝑦_𝑛+1,𝑛=1,2,…. (6) By the given condition, we have 𝑑𝑥_𝑛+1,𝑦_𝑛+1=𝑑𝑓𝑔𝑥_𝑛,𝑔𝑥_𝑛≤𝐹𝑑𝑔𝑥_𝑛,𝑥_𝑛,𝑑𝑔𝑥_𝑛,𝑓𝑔𝑥_𝑛,𝑑𝑥_𝑛,𝑔𝑥_𝑛. (7) By Lemma 15, we have 𝑑𝑥_𝑛+1,𝑦_𝑛+1≤𝜙𝑑𝑥_𝑛,𝑔𝑥_𝑛. (8) Also 𝑑𝑥_𝑛,𝑔𝑥_𝑛=𝑑𝑓𝑦_𝑛,𝑔𝑥_𝑛=𝑑𝑦_𝑛+1,𝑥_𝑛≤𝐹𝑑𝑦_𝑛,𝑥_𝑛,𝑑𝑦_𝑛,𝑓𝑦_𝑛,𝑑𝑥_𝑛,𝑔𝑥_𝑛=𝐹𝑑𝑦_𝑛,𝑓𝑦_𝑛,𝑑𝑦_𝑛,𝑓𝑦_𝑛,𝑑𝑥_𝑛,𝑔𝑥_𝑛. (9) Therefore 𝑑𝑥_𝑛,𝑔𝑥_𝑛=𝑑𝑦_𝑛+1,𝑥_𝑛≤𝜙𝑑𝑦_𝑛,𝑓𝑦_𝑛=𝜙𝑑𝑦_𝑛,𝑥_𝑛. (10) By Lemma 14, we have 𝑑𝑥_𝑛+1,𝑦_𝑛+1≤𝜙²𝑑𝑥^𝑛,𝑦_𝑛. (11) Hence, by induction, for all 𝑛∈𝑁, we obtain 𝑑𝑥_𝑛+1,𝑦_𝑛+1≤𝜙^2𝑛𝑑𝑥₁,𝑦₁=𝜙^2𝑛𝑑𝑥₁,𝑔𝑥₀. (12) Similarly 𝑑𝑦_𝑛+1,𝑥_𝑛≤𝜙^2𝑛−1𝑑𝑥₁,𝑦₁=𝜙^2𝑛−1𝑑𝑥₁,𝑔𝑥₀. (13) If 𝑛≥2, we have 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝑑𝑥_𝑛+1,𝑦_𝑛+1+𝑑𝑦_𝑛+1,𝑥_𝑛≤𝜙^2𝑛𝑑𝑥₁,𝑔𝑥₀+𝜙^2𝑛−1𝑑𝑥₁,𝑔𝑥₀≤2𝜙^2𝑛−1𝑑𝑥₁,𝑔𝑥₀. (14) Note that by the condition 𝜙^′ we know, for 𝑛,𝑚∈𝑁 such that 𝑚>𝑛, we have 𝑑𝑥_𝑛,𝑥_𝑛+𝑚≤𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑑𝑥_{𝑛+𝑚−1},𝑥_𝑛+𝑚≤𝜙^2𝑛−1𝑑𝑥₁,𝑔𝑥₀+2𝜙^{2(𝑛+1)−1}𝑑𝑥₁,𝑔𝑥₀+⋯+2𝜙^{2(𝑛+𝑚−1)−1}𝑑𝑥₁,𝑔𝑥₀≤^{2(𝑛+𝑚−1)−1}_{𝑖=2𝑛−1}𝜙^𝑖𝑑𝑥₁,𝑔𝑥₀≤^∞_{𝑖=2𝑛−1}𝜙^𝑖𝑑𝑥₁,𝑔𝑥₀→0. (15)
Hence, 𝑑(𝑥_𝑛,𝑥_𝑚)→0 as 𝑚,𝑛→∞. This forces that {𝑥_𝑛} is a Cauchy sequence in 𝑋. But 𝑋 is a completely dislocated metric space; hence, {𝑥_𝑛} is 𝑑-converges. Call the 𝑑-limit 𝑥_∗∈𝑋. Then, 𝑥_𝑛→𝑥_∗ as 𝑛→∞. By the continuity of 𝑔, as 𝑛→∞, 𝑦_𝑛=𝑦_∗=𝑔(𝑥_∗). So as 𝑛→∞, 𝑑(𝑥_𝑛+1,𝑦_𝑛+1)≤𝑑(𝑥_∗,𝑦_∗)≤0, 𝑥_∗ is the fixed point of 𝑔.
By the given condition, we have 𝑑𝑓𝑥_∗,𝑔𝑥_∗=𝑑𝑓𝑥_∗,𝑥_∗≤𝐹𝑑𝑥_∗,𝑥_∗,𝑑𝑥_∗,𝑓𝑥_∗,𝑑𝑥_∗,𝑔𝑥_∗=𝐹0,𝑑𝑥_∗,𝑓𝑥_∗,0. (16)
Hence, 𝑑(𝑓(𝑥_∗),𝑥_∗)≤𝜙(0)=0⇒𝑓(𝑥_∗)=𝑥_∗, so 𝑥_∗ is a common fixed point of 𝑓,𝑔.
Uniqueness. Let 𝑦_∗ be another common fixed point of 𝑓,𝑔. Then by the given condition, we have 𝑑𝑥_∗,𝑦_∗=𝑑𝑓𝑥_∗,𝑔𝑦_∗≤𝐹𝑑𝑥_∗,𝑦_∗,𝑑𝑦_∗,𝑓𝑦_∗,𝑑𝑥_∗,𝑔𝑥_∗=𝐹𝑑𝑥_∗,𝑦_∗,0,0. (17) Since 𝑑(𝑥_∗,𝑦_∗)≤𝜙(0)=0⇒𝑥_∗=𝑦_∗, 𝑥_∗ is the unique fixed point of 𝑓; similarly, we prove that 𝑥_∗ is also the unique fixed point of 𝑔. Thus the fixed point of 𝑓,𝑔 is unique, and we prove the theorem.

Theorem 17. Let (𝑋,𝑑) be a complete dislocated metric space; and let 𝑓,𝑔∶𝑋→𝑋 be continuous mapping, if(1)there exists 𝐹 satisfying the condition 𝜙^′, for all 𝑥,𝑦∈𝑋, if 𝑥≠𝑦, such that 𝑑𝑓(𝑥),𝑔𝑦≤𝐹𝑑𝑥,𝑦,𝑑𝑥,𝑓(𝑥),𝑑𝑦,𝑔𝑦, (18)(2)there exists 𝑥₀∈𝑋 such that {(𝑓𝑔)^𝑛(𝑥₀)} have a condensation point, then 𝑓,𝑔 have a unique common fixed point.

Proof. Let 𝑥_𝑛,𝑦_𝑛 be the sequence of 𝑋, and for all 𝑛, 𝑥_𝑛≠𝑦_𝑛, 𝑥_𝑛=𝑓𝑔^𝑛𝑥₀=𝑓𝑔𝑥_𝑛−1,𝑦_𝑛=𝑔𝑓𝑔^𝑛−1𝑥₀,𝑛=1,2,…. (19) Obviously, 𝑦_𝑛=𝑔𝑥_𝑛−1,𝑓𝑦_𝑛=𝑥_𝑛,𝑔𝑓𝑦_𝑛=𝑔𝑥_𝑛=𝑦_𝑛+1,𝑛=1,2,…. (20)
Suppose that 𝑥_∗ is the condensation point of {𝑥_𝑛}; there exists the subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖=𝑥_∗. Since 𝑔 is continuous, lim𝑔(𝑥_𝑛𝑖)=𝑔(𝑥_∗)=𝑦_∗.
Consider 𝑑𝑓𝑦_∗,𝑔𝑥_∗=𝑑𝑓𝑦_∗,𝑦_∗≤𝐹𝑑𝑦_∗,𝑥_∗,𝑑𝑦_∗,𝑓𝑦_∗,𝑑𝑥_∗,𝑔𝑥_∗=𝐹𝑑𝑦_∗,𝑥_∗,𝑑𝑦_∗,𝑓𝑦_∗,𝑑𝑥_∗,𝑦_∗,𝑑𝑓𝑦_∗,𝑦_∗≤𝜙𝑑𝑥_∗,𝑦_∗⟺𝑑𝑓𝑦_∗,𝑦_∗<𝑑𝑥_∗,𝑦_∗. (21)
Also consider 𝑑𝑥_𝑛+1,𝑦_𝑛+1=𝑑𝑓𝑔𝑥_𝑛,𝑔𝑥_𝑛≤𝐹𝑑𝑔𝑥_𝑛,𝑥_𝑛,𝑑𝑔𝑥_𝑛,𝑓𝑔𝑥_𝑛,𝑑𝑥_𝑛,𝑔𝑥_𝑛≤𝜙𝑑𝑥_𝑛,𝑔𝑥_𝑛,𝑑𝑥_𝑛,𝑔𝑥_𝑛=𝑑𝑓𝑦_𝑛,𝑔𝑥_𝑛=𝑑𝑦_𝑛+1,𝑥_𝑛≤𝐹𝑑𝑦_𝑛,𝑥_𝑛,𝑑𝑦_𝑛,𝑓𝑦_𝑛,𝑑𝑥_𝑛,𝑔𝑥_𝑛=𝐹𝑑𝑦_𝑛,𝑓𝑦_𝑛,𝑑𝑦_𝑛,𝑓𝑦_𝑛,𝑑𝑥_𝑛,𝑔𝑥_𝑛≤𝜙𝑑𝑦_𝑛,𝑓𝑦_𝑛=𝜙𝑑𝑦_𝑛,𝑥_𝑛. (22) Thus 𝑑𝑥_𝑛+1,𝑦_𝑛+1≤𝜙²𝑑𝑥_𝑛,𝑦_𝑛<𝑑𝑥_𝑛,𝑦_𝑛. (23)
Hence, we know that {𝑑(𝑥_𝑛+1,𝑦_𝑛+1)} is decreasing. Let lim𝑑(𝑥_𝑛+1,𝑦_𝑛+1)=𝜖 and lim𝑑(𝑦_𝑛𝑖+1,𝑥_𝑛𝑖)=𝑑(𝑦_∗,𝑥_∗)≤lim𝑑(𝑦_𝑛𝑖,𝑥_𝑛𝑖)=𝜖. Since {𝑥_𝑛𝑖} is the subsequence of {𝑥_𝑛}, we have 𝑑𝑓𝑦_∗,𝑦_∗=lim𝑑𝑓𝑦_𝑛𝑖+1,𝑦_𝑛𝑖+1=lim𝑑𝑥_𝑛𝑖,𝑦_𝑛𝑖=𝜖. (24)
Hence, we conclude that 𝑑(𝑥_∗,𝑦_∗)≤𝑑(𝑓(𝑦_∗),𝑦_∗), a contradiction. So 𝑥_∗=𝑦_∗, 𝑦_∗ is the fixed point of 𝑔. Similarly, 𝑦_∗ is the fixed point of 𝑓.
Uniqueness. Let 𝑦^′ be another common fixed point of 𝑓,𝑔. Then by the given condition, we have 𝑑𝑦_∗,𝑦^′=𝑑𝑓𝑦_∗,𝑔𝑦^′≤𝐹𝑑𝑦_∗,𝑦^′,𝑑𝑦^′,𝑓𝑦^′,𝑑𝑦_∗,𝑔𝑦_∗=𝐹𝑑𝑦_∗,𝑦^′,0,0. (25)
Since 𝑑(𝑦_∗,𝑦^′)≤𝜙(0)=0⇒𝑦_∗=𝑦^′, 𝑦_∗ is the unique fixed point of 𝑓. Similarly, we prove that 𝑦_∗ is also the unique fixed point of 𝑔. Thus the fixed point of 𝑓,𝑔 is unique.

Acknowledgments

This research was supported by NSFC Grants No: 11071279.