On Fixed Points of 𝛼-𝜓-Contractive Multivalued Mappings in Cone Metric Spaces

Abstract

We define the notion of 𝛼^∗-𝜓-contractive mappings for cone metric space and obtain fixed points of multivalued mappings in connection with Hausdorff distance function for closed bounded subsets of cone metric spaces. We obtain some recent results of the literature as corollaries of our main theorem. Moreover, a nontrivial example of 𝛼^∗-𝜓-contractive mapping satisfying all conditions of our main result has been constructed.

1. Introduction

Banach contraction principle is widely recognized as the source of metric fixed point theory. Also, this principle plays an important role in several branches of mathematics. For instance, it has been used to study the existence of solutions for nonlinear equations, systems of linear equations, and linear integral equations and to prove the convergence of algorithms in computational mathematics. Because of its importance for mathematical theory, Banach contraction principle has been extended in many directions.

In 2007, Huang and zhang [1] introduced cone metric space with normal cone, as a generalization of metric space. Rezapour and Hamlbarani [2] presented the results of [1] for the case of cone metric space without normality in cone. Many authors work out on it (see [3, 4]). Cho and Bae [5] introduced the Hausdorff distance function on cone metric spaces and generalized the result of [6] for multivalued mappings.

In 2012, Samet et al. [7] introduced the concept of 𝛼-𝜓-contractive type mappings. Their results generalized some ordered fixed point results (see [7]). In [8], Karapinar et al. introduced the notion of a 𝐺^𝑚-Meir-Keeler contractive mapping and established some fixed point theorems for the G𝑚-Meir-Keeler contractive mapping in the setting of G-metric spaces. For more details in fixed point theory related to our paper, we refer to the reader [9–19]. Asl et al. [20] introduced the notion of 𝛼^∗-𝜓-contractive mappings and improved the concept of 𝛼-𝜓-contractive mappings along with some fixed point theorems in metric space. Consequently, Ali et al. [21], Mohammadi et al. [22] and Salimi et al. [23] studied the concept of 𝛼^∗-𝜓-contractive mappings for proving fixed point results by using generalized contractive conditions in complete metric spaces.

In this paper, we first define the notion of 𝛼^∗-𝜓-contractive mappings for cone metric spaces and then we use it to study fixed point theorems for multivalued mappings satisfying 𝛼^∗-𝜓-contractive conditions in a complete cone metric space without the assumption of normality. We also furnish a nontrivial example to support our main result.

2. Preliminaries

In the following, we always suppose that 𝔼 is a real Banach space, 𝑃 is a cone in 𝔼 with nonempty interior, and ⪯ is the partial ordering with respect to 𝑃. By 𝜃, we denote the zero element of 𝔼. A subset 𝑃 is called a cone if and only if(i)𝑃is closed, nonempty, and 𝑃={𝜃};(ii)𝑎,𝑏∈ℝ,𝑎,𝑏≥0,𝑥,𝑦∈𝑃⇒𝑎𝑥+𝑏𝑦∈𝑃;(iii)𝑃∩(−𝑃)={𝜃}.

For a given cone 𝑃⊆𝔼, we 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 stand for 𝑦−𝑥∈int𝑃, where int𝑃 denotes the interior of 𝑃.

Definition 1 (see [1]). Let 𝑋 be a nonempty set. A function 𝑑∶𝑋×𝑋→𝔼 is said to be a 𝑐𝑜𝑛𝑒 metric, if the following conditions hold:(𝐶1)𝜃⪯𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦;(𝐶2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(𝐶3)𝑑(𝑥,𝑧)⪯𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋.
The pair (𝑋,𝑑) is then called a 𝑐𝑜𝑛𝑒 metric space.

Lemma 2 (see [1]). Let (𝑋,𝑑) be a 𝑐𝑜𝑛𝑒 metric space, 𝑥∈𝑋, and let {𝑥_𝑛}be a sequence in 𝑋. Then (i){𝑥_𝑛} converges to 𝑥 whenever for every 𝑐∈𝔼 with 𝜃≪𝑐 there is a natural number 𝑛₀ such that 𝑑(𝑥_𝑛,𝑥)≪𝑐, for all 𝑛≥𝑛₀. We denote this by lim_𝑛→∞𝑥_𝑛=𝑥;(ii){𝑥_𝑛} is a Cauchy sequence whenever for every 𝑐∈𝔼 with 𝜃≪𝑐 there is a natural number 𝑛₀ such that 𝑑(𝑥_𝑛,𝑥_𝑚)≪𝑐, for all 𝑛,𝑚≥𝑛₀;(iii)(𝑋,𝑑) is complete cone metric if every Cauchy sequence in 𝑋 is convergent.

Remark 3 (see [3]). The results concerning fixed points and other results, in case of cone spaces with nonnormal cones, cannot be provided by reducing to metric spaces, because in this case neither of the conditions of the lemmas 1–4, in [1] hold. Further, the vector cone metric is not continuous in the general case; that is, from 𝑥_𝑛→𝑥, 𝑦_𝑛→𝑦 it need not follow that 𝑑(𝑥_𝑛,𝑦_𝑛)→𝑑(𝑥,𝑦).

Let (𝑋,𝑑) be a cone metric space. The following properties of cone metric spaces have been noticed [3].(𝑃𝑇1) If 𝑢⪯𝑣 and 𝑣≪𝑤, then 𝑢≪𝑤.(𝑃𝑇2) If 𝑢≪𝑣 and 𝑣⪯𝑤, then 𝑢≪𝑤.(𝑃𝑇3) If 𝑢≪𝑣 and 𝑣≪𝑤, then 𝑢≪𝑤.(𝑃𝑇4) If 𝜃⪯𝑢≪𝑐 for each 𝑐∈int𝑃, then 𝑢=𝜃.(𝑃𝑇5) If 𝑎⪯𝑏+𝑐, for each 𝑐∈int𝑃, then 𝑎⪯𝑏.(𝑃𝑇6){𝑎_𝑛} be a sequence in 𝔼. If 𝑐∈int𝑃 and 𝜃⪯𝑎_𝑛→𝜃 (as 𝑛→∞), then there exists 𝑛₀∈𝑁 such that for all 𝑛≥𝑛₀, we have 𝑎_𝑛≪𝑐.

With some modifications, we have the following definition from [24].

Definition 4. Let Ψ be a family of nondecreasing functions, 𝜓∶𝑃→𝑃 such that(i)𝜓(𝜃)=𝜃 and 𝜃<𝜓(𝑡)<𝑡 for 𝑡∈𝑃⧵{𝜃},(ii)𝑡∈Int𝑃 implies 𝑡−𝜓(𝑡)∈Int𝑃,(iii)lim_𝑛→+∞𝜓^𝑛(𝑡)=𝜃 for every 𝑡∈𝑃⧵{𝜃}.

3. Main Result

For a cone metric space (𝑋,𝑑), denote (see [5]) 𝑁(𝑋)=𝐴∶𝐴isnonemptysubsetof𝑋𝐶𝐵(𝑋)=𝐴∶𝐴isnonempty,closedandboundedsubsetof𝑋},𝑠𝑝=𝑞∈𝔼∶𝑝⪯𝑞for𝑠()=𝑞∈𝔼,𝑎,𝐵_𝑏∈𝐵𝑠(𝑑(𝑎,𝑏))=_𝑏∈𝐵{𝑥∈𝔼∶𝑑(𝑎,𝑏)⪯𝑥}for𝑎∈𝑋and𝐵∈𝐶𝐵(𝑋). (1)

For 𝐴,𝐵∈𝐶𝐵(𝑋) we denote 𝑠(𝐴,𝐵)=_𝑎∈𝐴𝑠(𝑎,𝐵)∩_𝑏∈𝐵𝑠(𝑏,𝐴). (2)

Lemma 5. Let (𝑋,𝑑) be a cone metric space, and let 𝑃 be a cone in Banach space 𝔼.(1)Let 𝑝,𝑞∈𝔼. If 𝑝⪯𝑞, 𝑠(𝑞)⊂𝑠(𝑝).(2)Let 𝑥∈𝑋 and 𝐴∈𝑁(𝑋). If 𝜃∈𝑠(𝑥,𝐴), then 𝑥∈𝐴.(3)Let 𝑞∈𝑃 and let 𝐴,𝐵∈𝐶𝐵(𝑋) and 𝑎∈𝐴. If 𝑞∈𝑠(𝐴,𝐵), then 𝑞∈𝑠(𝑎,𝐵) for all 𝑎∈𝐴 or 𝑞∈𝑠(𝐴,𝑏) for all 𝑏∈𝐵.(4)Let 𝑞∈𝑃 and let 𝜆≥0, then 𝜆𝑠(𝑞)⊆𝑠(𝜆𝑞).

Remark 6. Let (𝑋,𝑑) be a 𝑐𝑜𝑛𝑒 metric space. If 𝔼=𝑅 and 𝑃=[0,+∞), then (𝑋,𝑑) is a metric space. Moreover, for 𝐴,𝐵∈𝐶𝐵(𝑋), 𝐻(𝐴,𝐵)=inf𝑠(𝐴,𝐵) is the Hausdorff distance induced by 𝑑.

Definition 7. Let (𝑋,𝑑) be a complete cone metric space with cone 𝑃,𝜓∈Ψ, 𝛼∶𝑋×𝑋→[0,+∞), and 𝐹∶𝑋→𝐶𝐵(𝑋) is known as 𝛼^∗-𝜓-contractive multivalued mapping whenever 𝜓𝑑𝑥,𝑦∈𝛼^∗𝐹𝑥,𝐹𝑦𝑠𝐹𝑥,𝐹𝑦 (3) for all 𝑥,𝑦∈𝑋, where 𝛼^∗(𝐴,𝐵)=inf{𝛼(𝑎,𝑏)∶𝑎∈𝐴,𝑏∈𝐵}. Also, we say that 𝐹 is 𝛼^∗-admissible whenever 𝛼(𝑥,𝑦)≥1 implies 𝛼^∗(𝐹𝑥,𝐹𝑦)≥1.

Note that an 𝛼^∗-𝜓-contractive multivalued mappings for cone metric space is generalized 𝛼^∗-𝜓-contractive. When 𝜓∈Ψ is a strictly increasing mapping, 𝛼^∗-𝜓-contractive is called strictly generalized 𝛼^∗-𝜓-contractive.

Theorem 8. Let (𝑋,𝑑) be a complete cone metric space with cone 𝑃,𝛼∶𝑋×𝑋→[0,+∞) be a function, 𝜓∈Ψ be a strictly increasing map and 𝐹∶𝑋→𝐶𝐵(𝑋), and 𝐹 be 𝛼^∗-admissible and 𝛼^∗-𝜓-contractive multivalued mapping on 𝑋. Suppose that there exist 𝑥₀∈𝑋,𝑥₁∈𝐹𝑥₀ such that 𝛼(𝑥₀,𝑥₁)≥1. Assume that if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑢 as 𝑛→+∞ then 𝛼(𝑥_𝑛,𝑢)≥1 for all 𝑛. Then, there exists a point 𝑥^∗ in 𝑋 such that 𝑥^∗∈𝐹𝑥^∗.

Proof. We may suppose that 𝑥₀≠𝑥₁. Then 𝛼^∗(𝐹𝑥₀,𝐹𝑥₁)≥1 and 𝜓𝑑𝑥₀,𝑥₁∈𝛼^∗𝐹𝑥₀,𝐹𝑥₁𝑠𝐹𝑥₀,𝐹𝑥₁. (4) By Lemma 5(3), we have 𝜓𝑑𝑥₀,𝑥₁∈𝛼^∗𝐹𝑥₀,𝐹𝑥₁𝑠𝑥₁,𝐹𝑥₁. (5) By definition, we can take 𝑥₂∈𝐹𝑥₁ such that 𝜓𝑑𝑥₀,𝑥₁∈𝛼^∗𝐹𝑥₀,𝐹𝑥₁𝑠𝑑𝑥₁,𝑥₂. (6) By Lemma 5(4), we have 𝜓𝑑𝑥₀,𝑥₁∈𝛼^∗𝐹𝑥₀,𝐹𝑥₁𝑠𝑑𝑥₁,𝑥₂⊆𝑠𝛼^∗𝐹𝑥₀,𝐹𝑥₁𝑑𝑥₁,𝑥₂. (7) So, 𝛼^∗𝐹𝑥₀,𝐹𝑥₁𝑑𝑥₁,𝑥₂⪯𝜓𝑑𝑥₀,𝑥₁. (8) Hence, 0≺𝑑𝑥₁,𝑥₂⪯𝛼^∗𝐹𝑥₀,𝐹𝑥₁𝑑𝑥₁,𝑥₂⪯𝜓𝑑𝑥₀,𝑥₁ (9)𝑥₁≠𝑥₂ and 𝛼(𝑥₁,𝑥₂)≥1. Thus 𝛼^∗(𝐹𝑥₁,𝐹𝑥₂)≥1 and 𝑑(𝑥₁,𝑥₂)⪯𝜓(𝑑(𝑥₀,𝑥₁)). If 𝑥₂∈𝐹𝑥₂, then 𝑥₂ is a fixed point of 𝐹. Assume that 𝑥₂∉𝐹𝑥₂. Then 𝜓𝑑𝑥₁,𝑥₂∈𝛼^∗𝐹𝑥₁,𝐹𝑥₂𝑠𝐹𝑥₁,𝐹𝑥₂. (10) By Lemma 5(3), we have 𝜓𝑑𝑥₁,𝑥₂∈𝛼^∗𝐹𝑥₁,𝐹𝑥₂𝑠𝑥₂,𝐹𝑥₂. (11) By definition, we can take 𝑥₃∈𝐹𝑥₂ such that 𝜓𝑑𝑥₁,𝑥₂∈𝛼^∗𝐹𝑥₁,𝐹𝑥₂𝑠𝑑𝑥₂,𝑥₃. (12) By Lemma 5(4), we have 𝜓𝑑𝑥₁,𝑥₂∈𝛼^∗𝐹𝑥₁,𝐹𝑥₂𝑠𝑑𝑥₂,𝑥₃⊆𝑠𝛼^∗𝐹𝑥₁,𝐹𝑥₂𝑑𝑥₂,𝑥₃. (13) So, 𝛼^∗𝐹𝑥₁,𝐹𝑥₂𝑑𝑥₂,𝑥₃⪯𝜓𝑑𝑥₁,𝑥₂. (14) Hence, 0≺𝑑𝑥₂,𝑥₃⪯𝛼^∗𝐹𝑥₁,𝐹𝑥₂𝑑𝑥₂,𝑥₃⪯𝜓𝑑𝑥₁,𝑥₂. (15) It is clear that 𝑥₂≠𝑥₃ and 𝛼(𝑥₂,𝑥₃)≥1. Thus, 𝛼^∗(𝐹𝑥₂,𝐹𝑥₃)≥1 and 𝑑(𝑥₂,𝑥₃)<𝜓²(𝜓(𝑑(𝑥₀,𝑥₁))).
If 𝑥₃∈𝐹𝑥₃, then 𝑥₃ is a fixed point of 𝐹. Assume that 𝑥₃∉𝐹𝑥₃: 𝜓𝑑𝑥₂,𝑥₃∈𝛼^∗𝐹𝑥₂,𝐹𝑥₃𝑠𝐹𝑥₂,𝐹𝑥₃. (16) By Lemma 5(3), we have 𝜓𝑑𝑥₂,𝑥₃∈𝛼^∗𝐹𝑥₂,𝐹𝑥₃𝑠𝑥₃,𝐹𝑥₃. (17) By definition, we can take 𝑥₄∈𝐹𝑥₃ such that 𝜓𝑑𝑥₂,𝑥₃∈𝛼^∗𝐹𝑥₂,𝐹𝑥₃𝑠𝑑𝑥₃,𝑥₄. (18) By Lemma 5(4), we have 𝜓𝑑𝑥₂,𝑥₃∈𝛼^∗𝐹𝑥₂,𝐹𝑥₃𝑠𝑑𝑥₃,𝑥₄⊆𝑠𝛼^∗𝐹𝑥₂,𝐹𝑥₃𝑑𝑥₃,𝑥₄. (19) So 𝛼^∗𝐹𝑥₂,𝐹𝑥₃𝑑𝑥₃,𝑥₄⪯𝜓𝑑𝑥₂,𝑥₃. (20) Hence 0≺𝑑𝑥₃,𝑥₄⪯𝛼^∗𝐹𝑥₂,𝐹𝑥₃𝑑𝑥₃,𝑥₄⪯𝜓𝑑𝑥₂,𝑥₃. (21) It is clear that 𝑥₃≠𝑥₄ and 𝛼(𝑥₃,𝑥₄)≥1. Thus, 𝛼^∗(𝐹𝑥₃,𝐹𝑥₄)≥1 and 𝑑(𝑥₃,𝑥₄)<𝜓³(𝜓(𝑑(𝑥₀,𝑥₁))). By continuing this process, we obtain a sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛∈𝑇𝑥_𝑛−1, 𝑥_𝑛≠𝑥_𝑛−1, 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 and 𝑑𝑥_𝑛,𝑥_𝑛+1⪯𝜓𝑑𝑥_𝑛−1,𝑥_𝑛⪯⋯⪯𝜓^𝑛𝑑𝑥₀,𝑥₁ (22) for all 𝑛.

Fix 𝜃≪𝑐. We choose a positive real number 𝛿 such that (𝑐−𝜓(𝑐))+𝐼(𝜃,𝛿)⊂Int𝑃, where 𝐼(𝜃,𝛿)={𝑦∈𝐸∶‖𝑦‖<𝛿}. By (iii) of Definition 4, there exists a natural number 𝑁 such that 𝜓^𝑚(𝑑(𝑥₀,𝑥₁))∈𝐼(𝜃,𝛿), for all 𝑚≥𝑁. Then 𝜓^𝑚𝑑𝑥₀,𝑥₁≪𝑐−𝜓(𝑐), (23) for all 𝑚≥𝑁. Consequently, 𝑑(𝑥_𝑚,𝑥_𝑚+1)≪𝑐−𝜓(𝑐), for all 𝑚≥𝑁. Fix 𝑚≥𝑁. Now we prove 𝑑𝑥_𝑚,𝑥_𝑛+1≪𝑐 (24)

for all 𝑛≥𝑚. Note that (24) holds when 𝑛=𝑚. Assume that (24) holds for some 𝑛≥𝑚. Then, we have 𝑑𝑥_𝑚,𝑥_𝑛+1≪𝑐. (25)

Now by (22), we have 𝑑𝑥_𝑚,𝑥_𝑛+2≤𝑑𝑥_𝑚,𝑥_𝑚+1+𝑑𝑥_𝑚+1,𝑥_𝑛+2≪𝑐−𝜓(𝑐)+𝜓𝑑𝑥_𝑚,𝑥_𝑛+1≪𝑐−𝜓(𝑐)+𝜓(𝑐)=𝑐. (26) Therefore, (24) holds when 𝑛=𝑛+1. By induction, we deduce that (24) holds for all 𝑛≥𝑚. This is sufficient to conclude that {𝑥_𝑛} is a Cauchy sequence. Choose 𝑢∈𝑋 such that 𝑥_𝑛→𝑢. Since 𝛼(𝑥_𝑛,𝑢)≥1 for all 𝑛 and 𝐹 is 𝛼^∗-admissible, so 𝛼^∗(𝐹𝑥_𝑛,𝐹𝑢)≥1 for all 𝑛. From (3), we have 𝜓𝑑𝑥_𝑛,𝑢∈𝛼^∗𝐹𝑥_𝑛,𝐹𝑢𝑠𝐹𝑥_𝑛,𝐹𝑢 (27)

for all 𝑛∈ℕ. By Lemma 5(3), we have 𝜓𝑑𝑥_𝑛,𝑢∈𝛼^∗𝐹𝑥_𝑛,𝐹𝑢𝑠𝑥_𝑛+1,𝐹𝑢. (28)

By definition, we can take 𝑣_𝑛∈𝐹𝑢 such that 𝜓𝑑𝑥_𝑛,𝑢∈𝛼^∗𝐹𝑥_𝑛,𝐹𝑢𝑠𝑑𝑥_𝑛+1,𝑣_𝑛. (29)

By Lemma 5(4), we have 𝜓𝑑𝑥_𝑛,𝑢∈𝛼^∗𝐹𝑥_𝑛,𝐹𝑢𝑠𝑑𝑥_𝑛+1,𝑣_𝑛⊆𝑠𝛼^∗𝐹𝑥_𝑛,𝐹𝑢𝑑𝑥_𝑛+1,𝑣_𝑛. (30)

So 𝛼^∗𝐹𝑥_𝑛,𝐹𝑢𝑑𝑥_𝑛+1,𝑣_𝑛⪯𝜓𝑑𝑥_𝑛,𝑢. (31)

Hence 0≺𝑑𝑥_𝑛+1,𝑣_𝑛⪯𝛼^∗𝐹𝑥_𝑛,𝐹𝑢𝑑𝑥_𝑛+1,𝑣_𝑛⪯𝜓𝑑𝑥_𝑛𝑥,𝑢⪯𝑑_𝑛,𝑢. (32)

Moreover, for a given 𝑐∈Int𝑃, we have 𝑑𝑢,𝑣_𝑛⪯𝑑𝑢,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑣_𝑛⪯𝑑𝑢,𝑥_𝑛+1+𝛼^∗𝐹𝑥_𝑛,𝐹𝑢𝑑𝑥_𝑛+1,𝑣_𝑛⪯𝑑𝑢,𝑥_𝑛+1+𝜓𝑑𝑥_𝑛,𝑢⪯𝑑𝑢,𝑥_𝑛+1+𝑑𝑥_𝑛≪𝑐,𝑢2+𝑐2=𝑐,for𝑛≥𝑘=𝑘(𝑐). (33) Hence, according to Lemma 2(i), we have lim_𝑛→∞𝑣_𝑛=𝑢. Since 𝐹𝑢 is closed, 𝑢∈𝐹𝑢.

Theorem 9. Let (𝑋,𝑑) be a complete cone metric space with cone 𝑃,𝛼∶𝑋×𝑋→[0,+∞) be a function, and 𝐹∶𝑋→𝐶𝐵(𝑋) be 𝛼^∗-admissible. If there exists a constant 𝑘∈[0,1) such that 𝑘𝑑𝑥,𝑦∈𝛼^∗𝐹𝑥,𝐹𝑦𝑠𝐹𝑥,𝐹𝑦 (34) for all 𝑥,𝑦∈𝑋. Suppose that there exist 𝑥₀∈𝑋,𝑥₁∈𝐹𝑥₀ such that 𝛼(𝑥₀,𝑥₁)≥1. Assume that if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑢 as 𝑛→+∞; then 𝛼(𝑥_𝑛,𝑢)≥1 for all 𝑛. Then, there exists a point 𝑥^∗ in 𝑋 such that 𝑥^∗∈𝐹𝑥^∗.

Proof. Take 𝜓(𝑡)=𝑘𝑡 in Theorem 8.

Theorem 10. Let (𝑋,𝑑) be a complete cone metric space with cone 𝑃,𝜓∈Ψ be a strictly increasing map, and 𝐹∶𝑋→𝐶𝐵(𝑋) be multivalued mapping such that 𝜓𝑑𝑥,𝑦∈𝑠𝐹𝑥,𝐹𝑦 (35) for all 𝑥,𝑦∈𝑋. Then, there exists a point 𝑥^∗ in 𝑋 such that 𝑥^∗∈𝐹𝑥^∗.

Proof. Take 𝛼^∗(𝐹𝑥,𝐹𝑦)=1 in the Theorem 8.

Corollary 11. Let (𝑋,𝑑) be a complete cone metric space with cone 𝑃 and let 𝐹∶𝑋→𝐶𝐵(𝑋) be a multivalued mapping. If there exists a constant 𝑘∈[0,1) such that 𝑘𝑑𝑥,𝑦∈𝑠𝐹𝑥,𝐹𝑦 (36) for all 𝑥,𝑦∈𝑋, then, there exists a point 𝑥^∗ in 𝑋 such that 𝑥^∗∈𝐹𝑥^∗.

Proof. Take 𝜓(𝑡)=𝑘𝑡 and 𝛼^∗(𝐹𝑥,𝐹𝑦)=1 in the Theorem 8.

Corollary 12 (see [20]). Let (𝑋,𝑑) be a complete metric space,𝛼∶𝑋×𝑋→[0,+∞) be a function, 𝜓∈Ψ be a strictly increasing map, and 𝐹∶𝑋→𝐶𝐵(𝑋) be 𝛼^∗-admissible such that 𝛼^∗𝐹𝑥,𝐹𝑦𝐻𝐹𝑥,𝐹𝑦≤𝜓𝑑𝑥,𝑦 (37) for all 𝑥,𝑦∈𝑋. Suppose that there exist 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝐹𝑥₀)≥1. Assume that if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑢 as 𝑛→+∞ then 𝛼(𝑥_𝑛,𝑢)≥1 for all 𝑛. Then, there exists a point 𝑥^∗ in 𝑋 such that 𝑥^∗∈𝐹𝑥^∗.

By Remark 6, we have the following corollaries.

Corollary 13 (see [20]). Let (𝑋,𝑑) be a complete metric space,𝜓∈Ψ be a strictly increasing map, and 𝐹∶𝑋→𝐶𝐵(𝑋) be a multivalued mapping such that 𝐻𝐹𝑥,𝐹𝑦≤𝜓𝑑𝑥,𝑦 (38) for all 𝑥,𝑦∈𝑋. Then, there exists a point 𝑥^∗ in 𝑋 such that 𝑥^∗∈𝐹𝑥^∗.

Proof. Take 𝛼^∗(𝐹𝑥,𝐹𝑦)=1 in the Corollary 12.

Corollary 14 (see [20]). Let (𝑋,𝑑) be a complete metric space, 𝛼∶𝑋×𝑋→[0,+∞) be a function, and 𝐹∶𝑋→𝐶𝐵(𝑋) be 𝛼^∗-admissible. If there exists a constant 𝑘∈[0,1) such that 𝛼^∗𝐹𝑥,𝐹𝑦𝐻𝐹𝑥,𝐹𝑦≤𝑘𝑑𝑥,𝑦 (39) for all 𝑥,𝑦∈𝑋. Suppose that there exist 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝐹𝑥₀)≥1. Assume that if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑢 as 𝑛→+∞ then 𝛼(𝑥_𝑛,𝑢)≥1 for all 𝑛. Then, there exists a point 𝑥^∗ in 𝑋 such that 𝑥^∗∈𝐹𝑥^∗.

Proof. Take 𝜓(𝑡)=𝑘𝑡 in the Corollary 13.

Corollary 15 (see [25]). Let (𝑋,𝑑) be a complete metric space and let 𝐹∶𝑋→𝐶𝐵(𝑋) be a multivalued mapping. If there exists a constant 𝑘∈[0,1) such that 𝐻𝐹𝑥,𝐹𝑦≤𝑘𝑑𝑥,𝑦 (40) for all 𝑥,𝑦∈𝑋. Then, there exists a point 𝑥^∗ in 𝑋 such that 𝑥^∗∈𝐹𝑥^∗.

Proof. Take 𝛼^∗(𝐹𝑥,𝐹𝑦)=1 in the Corollary 14.

Example 16. Let 𝑋=[0,1], 𝐸=𝐶¹[0,1], 𝑃={𝜃⪯𝑥(𝑡)∶𝑡∈𝑋}, where 𝜃(𝑡)=0 for all 𝑡∈𝑋. Define 𝑑∶𝑋×𝑋→𝐸 by 𝑑𝑥,𝑦=𝑥−𝑦𝑒^𝑡 (41) and 𝐹∶𝑋→2^𝑋 by 𝐹𝑥=[0,(𝑥/12)] for all 𝑥∈𝑋 and 𝛼=1𝑥,𝑦𝑥−𝑦if1𝑥≠𝑦if𝑥=𝑦. (42) Then, 𝛼(𝑥,𝑦)≥1⇒𝛼^∗(𝐹𝑥,𝐹𝑦)=inf{𝛼(𝑥,𝑦)∶𝑥∈𝐹𝑥,𝑦∈𝐹𝑦}≥1. Then, clearly 𝐹 is 𝛼^∗-admissible. Now for 𝑥,𝑦 and 𝑥≤𝑦, we get 𝜓𝑑=1𝑥,𝑦5𝑥−𝑦𝑒^𝑡⪰1𝑒12𝑥−𝑦^𝑡=𝛼^∗𝐹𝑥,𝐹𝑦𝑠𝐹𝑥,𝐹𝑦 (43) which implies that 𝜓𝑑𝑥,𝑦∈𝛼^∗𝐹𝑥,𝐹𝑦𝑠𝐹𝑥,𝐹𝑦. (44) So 𝐹 is 𝛼^∗-𝜓-contractive multivalued mapping on 𝑋 where 𝜓(𝑡)=𝑡/5. Thus, all the conditions of main result are satisfied to obtain the fixed point of 𝐹.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

The authors thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper. The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.