Some Fixed Point Theorem for Expansive Type Mapping in Dislocated Metric Space

Abstract

The purpose of this paper is to present some fixed point theorem in dislocated quasimetric space for expansive type mappings.

1. Introduction and Preliminaries

It is well known that Banach Contraction mappings principle is one of the pivotal results of analysis. Generalizations of this principle have been obtained in several directions. Dass and Gupta [1] generalized Banach’s Contraction principle in metric space. Also Rhoades [2] established a partial ordering for various definitions of contractive mappings. In 2005, Zeyada et al. [3] established a fixed point theorem in dislocated quasimetric spaces. In 2008, Aage and Salunke [4] proved some results on fixed point in dislocated quasimetric spaces. Recently, Isufati [5], proved fixed point theorem for contractive type condition with rational expression in dislocated quasimetric spaces. The following definitions will be needed in the sequel.

Definition 1.1 (see [3]). Let 𝑋 be a nonempty set, and let 𝑑∶𝑋×𝑋→[0,∞) be a function, called a distance function. One needs the following conditions:(M1)𝑑(𝑥,𝑥)=0, (M2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥)=0, then 𝑥=𝑦,(M3)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥), (M4)𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦), (M4)^’𝑑(𝑥,𝑦)≤max{𝑑(𝑥,𝑧),𝑑(𝑧,𝑦)}, for all 𝑥,𝑦,𝑧∈𝑋.
If 𝑑 satisfies conditions (M1)–(M4), then it is called a metric on 𝑋. If 𝑑 satisfies conditions (M1), (M2), and (M4), it is called a quasimetric on 𝑋. If it satisfies conditions (M2)–(M4) ((M2) and (M4)), it is called a dislocated metric (or simply 𝑑-metric) (a dislocated quasimetric (or simply 𝑑𝑞-metric)) on 𝑋, respectively. If a metric 𝑑 satisfies the strong triangle inequality (M4)^’, then it is called an ultrametric.

Definition 1.2 (see [3]). A sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑑𝑞-metric space (dislocated quasimetric space) (𝑋,𝑑) is called a Cauchy sequence if, for 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)}<𝜖for all𝑚,𝑛≥𝑛₀.

Definition 1.3 (see [3]). A sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑑𝑞-metric space [𝑑-metric space] is said to be 𝑑-converge to 𝑥∈𝑋 provided that lim_𝑛→∞𝑑𝑥_𝑛,𝑥=lim_𝑛→∞𝑑𝑥,𝑥_𝑛=0.(1.1) In this case, 𝑥 is called a 𝑑𝑞-limit [𝑑-limit] of {𝑥_𝑛} and we write 𝑥_𝑛→𝑥.

Definition 1.4 (see [3]). A 𝑑𝑞-metric space (𝑋,𝑑) is called complete if every Cauchy sequence in it is a dq-convergent.

Example 1.5. Let𝑋=[0,1]. Define 𝑑∶𝑋×𝑋→ℝ⁺⁰ by 𝑑(𝑥,𝑦)=max{𝑥,𝑦}. Then the pair (𝑋,𝑑) is a dislocated metric space. We define an arbitrary sequence {𝑥_𝑛}in 𝑋 by 𝑥_𝑛=2/(2^𝑛+1), 𝑛∈𝑁∪{0}. Let 𝜀=sup_𝑛∈ℕ{2/(2^𝑛+1)}, then, for 𝑛,𝑚∈𝑁 and 𝑛>𝑚, we have𝑑(𝑥_𝑛,𝑥_𝑚)=𝑑(2/(2^𝑛+1),2/(2^𝑚+1))=2/(2^𝑚+1)≤𝜀. Thus, {𝑥_𝑛} is a Cauchy sequence in 𝑋. Also as 𝑛→∞, then 𝑥_𝑛→0∈𝑋. Hence, every Cauchy sequence in 𝑋 is convergent with respect to 𝑑. Thus, (𝑋,𝑑) is a complete dislocated metric space.

2. Main Results

In this paper, we prove some fixed point theorem for continuous mapping satisfying expansion condition in complete 𝑑𝑞-metric space.

Theorem 2.1. Let (𝑋,𝑑) be a complete dislocated metric space and 𝑇 a continuous mapping satisfying the following condition: 𝑑(𝑇𝑥,𝑇𝑦)+𝛼max{𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥)}≥𝛽⋅𝑑(𝑥,𝑇𝑥)[1+𝑑(𝑦,𝑇𝑦)]1+𝑑(𝑥,𝑦)+𝛾⋅𝑑(𝑥,𝑦)(2.1) for all 𝑥,𝑦∈𝑋, 𝑥≠𝑦, where 𝛼,𝛽,𝛾≥0 are real constants and𝛽+𝛾>1+2𝛼, 𝛾>1+𝛼. Then, 𝑇 has a fixed point in 𝑋.

Proof. Choose 𝑥₀∈𝑋 be arbitrary, to define the iterative sequence {𝑥_𝑛}_𝑛∈ℕ as follows and 𝑇𝑥_𝑛=𝑥_𝑛−1 for 𝑛=1,2,3,…. Then, using (2.1), we obtain 𝑑𝑇𝑥≥𝛽_𝑛+1,⋅𝑇𝑑𝑥𝑥_𝑛+2_𝑛+1+𝛼max𝑑𝑥_𝑛+1,𝑇𝑥_𝑛+2,𝑑𝑥_𝑛+2,𝑇𝑥_𝑛+1,𝑇𝑥_𝑛+11+𝑑𝑥_𝑛+2,𝑇𝑥_𝑛+21+𝑑𝑥_𝑛+1,𝑥_𝑛+2+𝛾𝑑𝑥^𝑛+1⟹𝑑𝑥_𝑛,𝑥_𝑛+2≥𝛽⋅𝑑,𝑥𝑥_𝑛+1_𝑛+1+𝛼max𝑑𝑥_𝑛+1,𝑥_𝑛+1,𝑑𝑥_𝑛+2,𝑥_𝑛,𝑥_𝑛1+𝑑𝑥_𝑛+2,𝑥_𝑛+11+𝑑𝑥_𝑛+1,𝑥_𝑛+2+𝛾⋅𝑑𝑥^𝑛+1⟹𝑑𝑥_𝑛,𝑥_𝑛+2≥𝛽⋅𝑑,𝑥𝑥_𝑛+1_𝑛+1+𝛼max𝑑𝑥_𝑛+1,𝑥_𝑛+1,𝑑𝑥_𝑛+2,𝑥_𝑛,𝑥_𝑛1+𝑑𝑥_𝑛+1,𝑥_𝑛+21+𝑑𝑥_𝑛+1,𝑥_𝑛+2+𝛾⋅𝑑𝑥^𝑛+1⟹𝑑𝑥_𝑛,𝑥_𝑛+2⟹𝑑𝑥_𝑛,𝑥_𝑛+1+𝛼𝑑𝑥_𝑛,𝑥_𝑛+2≥𝛽⋅𝑑𝑥_𝑛,𝑥_𝑛+1+𝛾⋅𝑑𝑥_𝑛+1,𝑥_𝑛+2⟹(1+𝛼,𝑥−𝛽_𝑛+1)𝑑+𝛼𝑥𝑑_𝑛𝑥_𝑛,𝑥_𝑛+1+𝛼𝑑𝑥_𝑛+1,𝑥_𝑛+2≥𝛽⋅𝑑𝑥_𝑛,𝑥_𝑛+1+𝛾⋅𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+1≥(𝛾−𝛼)𝑑𝑥_𝑛+1,𝑥_𝑛+2.(2.2) The last inequality gives 𝑑𝑥_𝑛+1,𝑥_𝑛+2≤1+𝛼−𝛽𝛾−𝛼𝑑𝑥^𝑛=𝛿𝑑𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,(2.3) where 𝛿=(1+𝛼−𝛽)/(𝛾−𝛼)<1. Hence, by induction, we obtain 𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝛿^𝑛+1𝑑𝑥₀,𝑥₁.(2.4) Note that, for 𝑚,𝑛∈ℕ such that 𝑚>𝑛, we have 𝑑𝑥_𝑚,𝑥_𝑛≤≤𝑑𝛿𝑥_𝑚𝑚−1,𝑥_𝑚−1+𝑑𝑥_𝑚−1,𝑥_𝑚−2+⋯+𝑑𝑥_𝑛+1,𝑥_𝑛=𝛿_𝑛+𝛿^𝑚−2+⋯+𝛿^𝑛𝑑𝑥₀,𝑥₁≤𝛿_𝑛∞1+𝛿+⋯+𝛿^{𝑚−𝑛−1}𝑑𝑥₀,𝑥₁_𝑟=0=𝛿_𝑛𝛿^𝑟𝑑𝑥₀,𝑥₁1−𝛿𝑑𝑥⁰,𝑥₁.(2.5) Since 0≤𝛿<1, then as𝑛→∞, 𝛿^𝑛(1−𝛿)⁻¹→0. Hence, 𝑑(𝑥_𝑚,𝑥_𝑛)→0 as𝑚,𝑛→∞. This forces that {𝑥_𝑛}_𝑛∈ℕis a Cauchy sequence in 𝑋. But 𝑋 is a complete dislocated metric space; hence, {𝑥_𝑛}_𝑛∈ℕ is 𝑑-converges. Call the 𝑑-limit𝑥^∗∈𝑋. Then, 𝑥_𝑛→𝑥^∗ as𝑛→∞. By continuity of 𝑇 we have, 𝑇𝑥^∗=𝑇𝑑−lim_𝑛→∞𝑥_𝑛=𝑑−lim_𝑛→∞𝑇𝑥_𝑛=𝑑−lim_𝑛→∞𝑥_𝑛−1=𝑥^∗(2.6) that is, 𝑇𝑥^∗=𝑥^∗; thus, 𝑇 has a fixed point in𝑋.

Uniqueness
Let 𝑦^∗be another fixed point of 𝑇 in𝑋, then 𝑇𝑦^∗=𝑦^∗and 𝑇𝑥^∗=𝑥^∗. Now, 𝑑𝑇𝑥^∗,𝑇𝑦^∗+𝛼max𝑑𝑥^∗,𝑇𝑦^∗,𝑑𝑦^∗,𝑇𝑥^∗≥𝛽⋅𝑑𝑥^∗,𝑇𝑥^∗1+𝑑𝑦^∗,𝑇𝑦^∗1+𝑑(𝑥^∗,𝑦^∗)+𝛾⋅𝑑𝑥^∗,𝑦^∗.(2.7) This implies that thatis𝑑𝑑𝑥𝑥^∗_∗,𝑦^∗+𝛼𝑑𝑥^∗,𝑦^∗≥𝛾⋅𝑑𝑥^∗,𝑦^∗⟹𝑑𝑥_∗,𝑦^∗≥(𝛾−𝛼)𝑑𝑥^∗,𝑦^∗,𝑦^∗≤1𝛾−𝛼𝑑𝑥^∗,𝑦^∗.(2.8) This is true only when𝑑(𝑥^∗,𝑦^∗)=0. Similarly,𝑑(𝑦^∗,𝑥^∗)=0. Hence, 𝑑(𝑥^∗,𝑦^∗)=𝑑(𝑦^∗,𝑥^∗)=0and so 𝑥^∗=𝑦^∗. Hence, 𝑇 has a unique fixed point in𝑋.

Next we prove Theorem 2.1 for surjective mapping.

Theorem 2.2. Let (𝑋,𝑑) be a complete dislocated metric space and 𝑇 a surjective mapping satisfying the condition (2.1) for all𝑦∈𝑋, ≠𝑦, where 𝛼,𝛽,𝛾≥0 are real constants and 𝛽+𝛾>1+2𝛼, 𝛾>1+𝛼. Then, 𝑇 has a fixed point in 𝑋.

Proof. Choose 𝑥₀∈𝑋 to be arbitrary, and define the iterative sequence {𝑥_𝑛}_𝑛∈ℕ as follows: 𝑇𝑥_𝑛=𝑥_𝑛−1 for 𝑛=1,2,3,…. Then, using (2.1), we obtain, sequence {𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence in 𝑋. But 𝑋 is a complete dislocated metric space; hence {𝑥_𝑛}_𝑛∈ℕ is 𝑑-converges. Call the 𝑑-limit𝑥^∗∈𝑋. Then, 𝑥_𝑛→𝑥^∗ as𝑛→∞. Existence of Fixed Point
Since 𝑇 is a Surjective map, so there exists a point 𝑦 in𝑋, such that𝑥=𝑇𝑦. Consider 𝑑𝑥_𝑛,𝑥=≥𝑑−𝛼𝑇𝑥max_𝑛+1,𝑇𝑑𝑦𝑥_𝑛+1,𝑇𝑦,𝑑𝑦,𝑇𝑥_𝑛+1+𝛽⋅𝑑𝑥^𝑛+1,𝑇𝑥_𝑛+1[1+𝑑(𝑦,𝑇𝑦)]+𝛾⋅𝑑𝑥_𝑛+11+𝑑𝑥_𝑛+1,𝑦,𝑦.(2.9) Taking 𝑛→∞, we get 𝑑(𝑥,𝑥)≥−𝛼max{𝑑(𝑥,𝑥),𝑑(𝑦,𝑥)}+𝛽⋅𝑑(𝑥,𝑥)[1+𝑑(𝑦,𝑥)]⟹⟹⟹0𝑑(𝛾(≥𝑥−,−𝑦𝛼))𝛼=𝑑𝑑(0(𝑥𝑥,a,𝑦s𝑦))𝛾≤+>0𝛾𝛼𝑑.(𝑦,𝑥)1+𝑑(𝑥,𝑦)+𝛾⋅𝑑(𝑥,𝑦)(2.10) Similarly, 𝑑(𝑦,𝑥)=0. Hence, 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥)=0⇒𝑥=𝑦 and so 𝑇𝑥=𝑥, that is, 𝑥 is a fixed point of𝑇.Uniqueness
Let 𝑦^∗be another fixed point of 𝑇 in𝑋, then 𝑇𝑦^∗=𝑦^∗and 𝑇𝑥^∗=𝑥^∗. Now, 𝑑𝑇𝑥^∗,𝑇𝑦^∗+𝛼max𝑑𝑥^∗,𝑇𝑦^∗,𝑑𝑦^∗,𝑇𝑥^∗≥𝛽⋅𝑑𝑥^∗,𝑇𝑥^∗1+𝑑𝑦^∗,𝑇𝑦^∗1+𝑑(𝑥^∗,𝑦^∗)+𝛾⋅𝑑𝑥^∗,𝑦^∗(2..11) This implies that thatis,𝑑𝑑𝑥𝑥^∗_∗,𝑦^∗+𝛼𝑑𝑥^∗,𝑦^∗≥𝛾⋅𝑑𝑥^∗,𝑦^∗⟹𝑑𝑥_∗,𝑦^∗≥(𝛾−𝛼)𝑑𝑥^∗,𝑦^∗,𝑦^∗≤1𝛾−𝛼𝑑𝑥^∗,𝑦^∗.(2.12) This is true only when𝑑(𝑥^∗,𝑦^∗)=0. Similarly, 𝑑(𝑦^∗,𝑥^∗)=0. Hence, 𝑑(𝑥^∗,𝑦^∗)=𝑑(𝑦^∗,𝑥^∗)=0and so 𝑥^∗=𝑦^∗. Hence 𝑇 has a unique fixed point in𝑋. The proof is completed.