On Multivalued Contractions in Cone Metric Spaces without Normality

Abstract

Wardowski (2011) in this paper for a normal cone metric space (𝑋,𝑑) and for the family 𝒜 of subsets of 𝑋 established a new cone metric 𝐻∶𝒜×𝒜→𝐸 and obtained fixed point of set-valued contraction of Nadler type. Further, it is noticed in the work of Jankovic et al., 2011 that the ?fixed-point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is nonnormal. In the present paper we improve Wardowski's result by proving the same without the assumption of normality on cones.

1. Introduction and Preliminaries

Huang and Zhang [1] generalized the notion of metric space by replacing the set of real numbers by ordered Banach space and defined cone metric space and extended Banach type fixed-point theorems for contractive type mappings. Subsequently, some other authors (e.g., see [2–15] and references therein) studied properties of cone metric spaces and fixed points results of mappings satisfying contractive type condition in cone metric spaces. Recently, Choa et al. [9], Kadelburg and Radenovic [16], Klim and Wardowski [17], Latif and Shaddad [18], Radenovic and Kadelburg [19], Rezapour and Haghi [20], and Wardowski [14, 21] obtained fixed points of set-valued mappings in normal cone metric spaces. On the other hand, it is shown in [11] that most of the fixed points results of mappings satisfying contractive type condition in cone metric spaces with a normal cone can be reduced to the corresponding results from metric space theory. The fixed-point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is nonnormal, because the results concerning fixed points and other results in the case of cone metric spaces with nonnormal solid cones cannot be proved by reducing to metric spaces. In this paper, we prove the result of Wardowski [14] without the assumption of normality of cones. We need the following definitions and results, consistent with [1, 11, 14].

Let 𝐸 be a Banach Space and 𝑃 a subset of 𝐸. Then, 𝑃 is called a cone whenever(i)𝑃is closed, nonempty, and 𝑃≠{𝜃},(ii)𝑎𝑥+𝑏𝑦∈𝑃 for all 𝑥,𝑦∈𝑃 and nonnegative real numbers 𝑎,𝑏,(iii)𝑃∩(−𝑃)={𝜃}.

Each cone 𝑃 induces a partial ordering ≼ on 𝐸 by 𝑥≼𝑦 if and only if 𝑦−𝑥∈𝑃. So 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. The cone 𝑃 is called normal if there is a number 𝐾>0 such that, for all 𝑥,𝑦∈𝐸, 𝜃≼𝑥≼𝑦⟹‖𝑥‖≤𝐾𝑦. (1) The least positive number 𝐾satisfying (1) is called the normal constant of 𝑃.

Definition 1. Let 𝑋 be a nonempty set. Suppose the mapping 𝑑∶𝑋×𝑋→𝐸 satisfies(d₁)𝜃≼𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=𝜃 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 cone metric space.

Let (𝑋,𝑑) be a cone metric space, 𝑥∈𝑋 and {𝑥_𝑛}_𝑛≥1 a sequence in 𝑋. Then, {𝑥_𝑛}_𝑛≥1 converges to 𝑥 whenever for every 𝑐∈𝐸 with 𝜃≪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. {𝑥_𝑛}_𝑛≥1 is a Cauchy sequence whenever for every 𝑐∈𝐸 with 𝜃≪𝑐 there is a natural number 𝑁 such that 𝑑(𝑥_𝑛,𝑥_𝑚)≪𝑐 for all 𝑛,𝑚≥𝑁.(𝑋,𝑑) is called a complete cone metric space if every Cauchy sequence in 𝑋 is convergent.

A set 𝐴⊂(𝑋,𝑑) is called closed if, for any sequence {𝑥_𝑛}⊂𝐴convergent to 𝑥, we have 𝑥∈𝐴. Denote by 𝑁(𝑋) the collection of all nonempty subsets of 𝑋 and by 𝐶(𝑋) a collection of all nonempty closed subsets of 𝑋. Denote by Fix𝑇 a set of all fixed points of a mapping 𝑇. In the present paper, we assume that 𝐸 is a real Banach space, 𝑃 is a cone in 𝐸 with nonempty interior (such cones are called solid), and ≼ is a partial ordering with respect to 𝑃. In accordance with [14, Definition 3.1 and Lemma 3.1], we minutely modify the idea of𝐻-cone metric to make it more comparable with a standard metric.

Definition 2. Let (𝑋,𝑑) be a cone metric space and 𝒜be a collection of nonempty subsets of 𝑋. A map 𝐻∶𝒜×𝒜→𝐸is called an 𝐻-cone metric on 𝒜 induced by 𝑑 if the following conditions hold: (H₁)𝜃≼𝐻(𝐴,𝐵)for all 𝐴,𝐵∈𝒜and𝐻(𝐴,𝐵)=𝜃if and only if 𝐴=𝐵,(H₂)𝐻(𝐴,𝐵)=𝐻(𝐵,𝐴)for all𝐴,𝐵∈𝒜,(H₃)𝐻(𝐴,𝐵)≼𝐻(𝐴,𝐶)+𝐻(𝐶,𝐵) for all𝐴,𝐵,𝐶∈𝒜,(H₄) If 𝐴,𝐵∈𝒜,𝜃≺𝜀∈𝐸with𝐻(𝐴,𝐵)≺𝜀, then for each𝑎∈𝐴 there exists𝑏∈𝐵 such that𝑑(𝑎,𝑏)≺𝜀.

Examples can be seen in [14, examples 3.1 and 3.2].

2. Main Result

Theorem 3. Let (𝑋,𝑑)be a complete cone metric space. Let 𝒜 be a nonempty collection of nonempty closed subsets of 𝑋, and let 𝐻∶𝒜×𝒜→𝐸 be an 𝐻-cone metric induced by 𝑑. If for a map 𝑇∶𝑋→𝒜there exists 𝜆∈(0,1) such that for all 𝑥,𝑦∈𝑋𝐻𝑇𝑥,𝑇𝑦≼𝜆𝑑𝑥,𝑦, (2) then Fix𝑇≠∅.

Proof. Let 𝑥₀ be an arbitrary but fixed element of 𝑋 and 𝑥₁∈𝑇𝑥₀. If 𝑥₀=𝑥₁, then 𝑥₀∈Fix𝑇, and if 𝑥₀≠𝑥₁, using the fact that 𝐻𝑇𝑥₀,𝑇𝑥₁≼𝜆𝑑𝑥₀,𝑥₁≺√𝜆𝑑𝑥₀,𝑥₁, (3) we may choose 𝑥₂∈𝑋 such that 𝑥₂∈𝑇𝑥₁and 𝑑𝑥₁,𝑥₂≺√𝜆𝑑𝑥₀,𝑥₁. (4) Similarly, in case 𝑥₁≠𝑥₂, we may choose 𝑥₃∈𝑋 such that 𝑥₃∈𝑇𝑥₂and 𝑑𝑥₂,𝑥₃≺√𝜆𝑑𝑥₁,𝑥₂≺√𝜆²𝑑𝑥₀,𝑥₁. (5) We can continue this process to find a sequence {𝑥_𝑛} of points of 𝑋such that 𝑥_𝑛+1∈𝑇𝑥_𝑛𝑑𝑥,𝑛=0,1,2,…,_𝑛,𝑥_𝑛+1≺√𝜆𝑑𝑥_𝑛−1,𝑥_𝑛≺√𝜆²𝑑𝑥_𝑛−2,𝑥_𝑛−1≺⋯≺√𝜆^𝑛𝑑𝑥₀,𝑥₁. (6) Now for any 𝑚>𝑛, 𝑑𝑥_𝑚,𝑥_𝑛≼𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑑𝑥_𝑚−1,𝑥_𝑚≼𝜆^𝑛/2+𝜆^(𝑛+1)/2+⋯+𝜆^(𝑚−1)/2𝑑𝑥₀,𝑥₁≼𝜆^𝑛/21−𝜆^1/2𝑑𝑥₀,𝑥₁. (7) Let 𝜃≪𝑐be given. Choose a symmetric neighborhood 𝑉 of 𝜃 such that 𝑐+𝑉⊆int𝑃. Also, choose a natural number 𝑁₁ such that [𝜆^𝑛/2/(1−𝜆^1/2)]𝑑(𝑥₀,𝑥₁)∈𝑉, for all 𝑛≥𝑁₁. Then, (𝜆^𝑛/2/(1−𝜆^1/2))𝑑(𝑥₁,𝑥₀)≪𝑐, for all 𝑛≥𝑁₁. Thus, 𝑑𝑥_𝑚,𝑥_𝑛≼𝜆^𝑛/21−𝜆^1/2𝑑𝑥₀,𝑥₁≪𝑐, (8) for all 𝑚>𝑛. Therefore, {𝑥_𝑛}_𝑛≥1 is a Cauchy sequence. Since 𝑋 is complete, there exists 𝑢∈𝑋 such that 𝑥_𝑛→𝑢. Since 𝐻𝑇𝑥_𝑛,𝑇𝑢≼𝜆𝑑𝑥_𝑛,𝑢≺√𝜆𝑑𝑥_𝑛,𝑢, (9) for each 𝑛,𝑥_𝑛+1∈𝑇𝑥_𝑛, we have 𝑦_𝑛∈𝑇𝑢 such that 𝑑(𝑥_𝑛+1,𝑦_𝑛)≺√𝜆𝑑(𝑥_𝑛,𝑢).Now, choose a natural number 𝑁₂such that 𝑑𝑥_𝑛≪𝑐,𝑢2∀𝑛≥𝑁₂. (10) Then for all 𝑛≥𝑁₂, 𝑑𝑢,𝑦_𝑛≼𝑑𝑢,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑦_𝑛≼𝑑𝑢,𝑥_𝑛+1+√𝜆𝑑𝑥_𝑛,𝑢≼𝑑𝑢,𝑥_𝑛+1+𝑑𝑥_𝑛≪𝑐,𝑢2+𝑐2=𝑐. (11) It follows that 𝑦_𝑛→𝑢, and it implies that 𝑢∈𝑇𝑢.

Example 4. Suppose 𝑋=[0,1],𝐸=𝐶²_𝑅[0,1]with the norm ‖𝑥‖=‖𝑥‖_∞+‖𝑥^/‖_∞, 𝑃={𝑥∈𝐸∶𝑥≥0}, 𝑥(𝑡)=𝑡, and 𝑦(𝑡)=𝑡^2𝐾. Then,0≤𝑥≤𝑦,??‖𝑥‖=2, and ‖𝑦‖=1+2𝐾. For all 𝐾≥1, since 𝐾‖𝑥‖<‖𝑦‖. Therefore, 𝑃 is non-normal. Define 𝑑∶𝑋×𝑋→𝐸 as follows: 𝑑𝑥,𝑦(𝑡)=𝑥−𝑦𝑒^𝑡. (12) Let 𝒜 be a family of subsets of 𝑋 of the form 𝒜={[0,𝑥]∶𝑥∈𝑋}∪{{𝑥}∶𝑥∈𝑋}, and define 𝐻∶𝒜×𝒜→𝐸 as follows: 𝐻()=𝑒𝐴,𝐵𝑥−𝑦^𝑡[],𝑒for𝐴=0,𝑥,𝐵=0,𝑦𝑥−𝑦^𝑡{𝑥}𝑦,𝑒for𝐴=,𝐵=Max𝑦,𝑥−𝑦^𝑡[]𝑦,𝑒for𝐴=0,𝑥,𝐵=Max𝑥,𝑥−𝑦^𝑡for𝐴={𝑥},𝐵=0,𝑦. (13) It is easy to observe that 𝐻satisfies (H₁)–(H₄) of Definition 2. Define 𝑇∶𝑋→𝒜as {0}1𝑇𝑥=,for𝑥∈0,2,10,21𝑥−2²1,for𝑥∈2,1. (14) Note that 𝑇 satisfies the conditions of Theorem 3 with 𝜆=1/2 and 0∈Fix𝑇.