Common Fixed Points for Multivalued Mappings in Complex Valued Metric Spaces with Applications

Abstract

In this paper, we introduce the notion of multivalued contractive mappings in complex valued metric space and prove common fixed point theorems for two multivalued contractive mappings in complex valued metric spaces without using the notion of continuity. Our results improve and extend the results of Azam et al. (2011).

1. Introduction

The Banach fixed point theorem was used to establish the existence of a unique solution for a nonlinear integral equation [1]. Moreover, this theorem plays an important role in several branches of mathematics. For instance, it has been used to show the existence of solutions of nonlinear Volterra integral equations and nonlinear integrodifferential equations in the Banach spaces and to show the convergence of algorithms in computational mathematics. Because of its importance and usefulness for mathematical theory, it has become a very popular tool of mathematical analysis in many directions. Nadler [2] introduced the concept of multivalued contraction mappings and obtained the fixed points results for multivalued mappings. Huang and Zhang [3] introduced the notion of cone metric space which is a generalization of metric spaces. They extended Banach contraction principle to cone metric spaces. Since then, Arshad et al. [4], Azam and Arshad [5], Latif and Shaddad [6], Karapınar [7], and many others obtained fixed point theorems in cone metric spaces (see [8]).

The fixed point results regarding rational contractive conditions cannot be extended in cone metric spaces. Azam et al. [9] introduced the concept of complex valued metric spaces and obtained sufficient conditions for the existence of common fixed points of a pair of mappings satisfying contractive type condition involving rational inequalities. In the same way, Rouzkard and Imdad [10] established some common fixed point theorems satisfying certain rational expressions in complex valued metric spaces which generalize, unify, and complement the results of Azam et al. [9]. Recently, Sintunavarat and Kumam [11] obtained common fixed point results by replacing constant of contractive condition with control functions. For more details in the subject, we refer to [12–19].

The aim of this paper is to extend the results of Azam et al. [9] to multivalued mappings in complex valued metric spaces.

2. Preliminaries

Let ℂ be the set of complex numbers and 𝑧₁,𝑧₂∈ℂ. Define a partial order ≾ on ℂ as follows: 𝑧₁≾𝑧₂iffRe𝑧₁⩽Re𝑧₂,Im𝑧₁⩽Im𝑧₂. (1) It follows that 𝑧₁≾𝑧₂ (2) if one of the following conditions is satisfied: (i)Re𝑧₁=Re𝑧₂,Im𝑧₁<Im𝑧₂,(ii)Re𝑧₁<Re𝑧₂,Im𝑧₁=Im𝑧₂,(iii)Re𝑧₁<Re𝑧₂,Im𝑧₁<Im𝑧₂,(iv)Re𝑧₁=Re𝑧₂,Im𝑧₁=Im𝑧₂. (3) In particular, we will write 𝑧₁⋨𝑧₂ if𝑧₁≠𝑧₂ and one of (i), (ii), and (iii) is satisfied and we will write 𝑧₁≺𝑧₂ if only (iii) is satisfied. Note that 0≾𝑧₁⋨𝑧₂⟹𝑧₁<𝑧₂,𝑧₁⪯𝑧₂,𝑧₂≺𝑧₃⟹𝑧₁≺𝑧₃. (4)

Definition 1. Let 𝑋 be a nonempty set. Suppose that a mapping 𝑑∶𝑋×𝑋→ℂ satisfies(1)0≾𝑑(𝑥,𝑦), for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦;(2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(3)𝑑(𝑥,𝑦)≾𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦), for all 𝑥,𝑦,𝑧∈𝑋.

Then 𝑑 is called a complex valued metric on 𝑋, and (𝑋,𝑑) is called a complex valued metric space. A point 𝑥∈𝑋 is called interior point of a set 𝐴⊆𝑋 whenever there exists 0≺𝑟∈ℂ such that 𝐵(𝑥,𝑟)=𝑦∈𝑋∶𝑑𝑥,𝑦≺𝑟⊆𝐴. (5)

A point 𝑥∈𝑋 is called a limit point of 𝐴 whenever for every 0≺𝑟∈ℂ, 𝐵(𝑥,𝑟)(𝐴⧵{𝑥})≠𝜙. (6)

𝐴 is called open whenever each element of 𝐴 is an interior point of 𝐴. Moreover, a subset 𝐵⊆𝑋 is called closed whenever each limit point of 𝐵 belongs to 𝐵. The family 𝐹={𝐵(𝑥,𝑟)∶𝑥∈𝑋,0≺𝑟} (7) is a subbasis for a Hausdorff topology 𝜏 on 𝑋.

Let {𝑥_𝑛} be a sequence in 𝑋 and 𝑥∈𝑋. If for every 𝑐∈ℂ with 0≺𝑐 there is 𝑛₀∈ℕ such that for all 𝑛>𝑛₀, 𝑑(𝑥_𝑛,𝑥)≺𝑐, then {𝑥_𝑛} is said to be convergent, {𝑥_𝑛} converges to 𝑥, and 𝑥 is thelimit point of {𝑥_𝑛}. We denote this by lim_𝑛→∞𝑥_𝑛=𝑥, or 𝑥_𝑛→𝑥, as 𝑛→∞. If for every 𝑐∈ℂ with 0≺𝑐 there is 𝑛₀∈ℕ such that for all 𝑛>𝑛₀, 𝑑(𝑥_𝑛,𝑥_𝑛+𝑚)≺𝑐, then {𝑥_𝑛} is called a Cauchy sequence in (𝑋,𝑑). If every Cauchy sequence is convergent in (𝑋,𝑑), then (𝑋,𝑑) is called a complete complex valued metric space. We require the following lemmas.

Lemma 2 (see [9]). Let (𝑋,𝑑) be a complex valued metric space and let {𝑥_𝑛} be a sequence in 𝑋. Then {𝑥_𝑛} converges to 𝑥 if and only if |𝑑(𝑥_𝑛,𝑥)|→0 as 𝑛→∞.

Lemma 3 (see [9]). Let (𝑋,𝑑)be a complex valued metric space and let {𝑥_𝑛} be a sequence in 𝑋. Then {𝑥_𝑛} is a Cauchy sequence if and only if|𝑑(𝑥_𝑛,𝑥_𝑛+𝑚)|→0 as 𝑛→∞.

3. Main Results

Let (𝑋,𝑑) be a complex valued metric space.

We denote by 𝑁(𝑋) (resp., 𝐶𝐵(𝑋)) the set of nonempty (resp., closed and bounded) subsets of a complex valued metric space. Now, we denote 𝑠(𝑧₁)={𝑧₂∈ℂ∶𝑧₁⪯𝑧₂} for 𝑧₁∈ℂ, and 𝑠(𝑎,𝐵)=∪_𝑏∈𝐵𝑠(𝑑(𝑎,𝑏))=∪_𝑏∈𝐵{𝑧∈ℂ∶𝑑(𝑎,𝑏)⪯𝑧} for 𝑎∈𝑋 and 𝐵∈𝐶(𝑋). For 𝐴,𝐵∈𝐶(𝑋), we denote 𝑠(𝐴,𝐵)=_𝑎∈𝐴𝑠(𝑎,𝐵)_𝑏∈𝐵𝑠(𝑏,𝐴). (8)

Remark 4. Let (𝑋,𝑑) be a complex valued metric space. If ℂ=𝑅, 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.

Let (𝑋,𝑑) be a complex valued metric space and let 𝐶𝐵(𝑋) be a collection of nonempty closed subsets of 𝑋. Let 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued map. For 𝑥∈𝑋 and 𝐴∈𝐶𝐵(𝑋), define 𝑊_𝑥(𝐴)={𝑑(𝑥,𝑎)∶𝑎∈𝐴}. (9) Thus, for 𝑥,𝑦∈𝑋𝑊_𝑥𝑇𝑦=𝑑(𝑥,𝑢)∶𝑢∈𝑇𝑦. (10)

Definition 5. Let (𝑋,𝑑) be acomplex valued metric space. A nonempty subset 𝐴 of 𝑋 is called bounded from below if there exists some 𝑧∈ℂ, such that 𝑧⪯𝑎 for all 𝑎∈𝐴.

Definition 6. Let (𝑋,𝑑) be acomplex valued metric space. A multivalued mapping 𝐹∶𝑋→2^ℂ is called bounded from below if for 𝑥∈𝑋 there exists 𝑧_𝑥∈ℂ such that 𝑧_𝑥⪯𝑢,∀𝑢∈𝐹𝑥. (11)

Definition 7. Let (𝑋,𝑑) be a complex valued metric space. The multivalued mapping 𝑇∶𝑋→𝐶𝐵(𝑋) is said to have lower bound property (l.b. property) on (𝑋,𝑑) if for any 𝑥∈𝑋, the multivalued mapping 𝐹_𝑥∶𝑋→2^ℂ defined by 𝐹_𝑥𝑇𝑦=𝑊_𝑥𝑇𝑦 (12) is bounded from below. That is, for 𝑥,𝑦∈𝑋 there exists an element 𝑙_𝑥(𝑇𝑦)∈ℂ such that 𝑙_𝑥𝑇𝑦⪯𝑢 (13) for all 𝑢∈𝑊_𝑥(𝑇𝑦), where 𝑙_𝑥(𝑇𝑦) is called lower bound of 𝑇 associated with (𝑥,𝑦).

Definition 8. Let (𝑋,𝑑) be a complex valued metric space. The multivalued mapping 𝑇∶𝑋→𝐶𝐵(𝑋) is said to have the greatest lower bound property (g.l.b. property) on (𝑋,𝑑) if the greatest lower bound of 𝑊_𝑥(𝑇𝑦) exists in ℂ for all 𝑥,𝑦∈𝑋. We denote 𝑑(𝑥,𝑇𝑦) by the g.l.b of 𝑊_𝑥(𝑇𝑦). That is, 𝑑𝑥,𝑇𝑦=inf𝑑(𝑥,𝑢)∶𝑢∈𝑇𝑦. (14)

Theorem 9. Let (𝑋,𝑑) be a complete complex valued metric space and let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mappings with g.l.b. property such that +()𝑑𝑑𝜆𝑑𝑥,𝑦𝜇𝑑𝑥,𝑆𝑥𝑦,𝑇𝑦+𝛾𝑑𝑦,𝑆𝑥𝑥,𝑇𝑦1+𝑑𝑥,𝑦∈𝑠𝑆𝑥,𝑇𝑦 (15) for all 𝑥,𝑦∈𝑋 and 𝜆, 𝜇, and 𝛾 are nonnegative real numbers with 𝜆+𝜇+𝛾<1. Then 𝑆,𝑇 have a common fixed point.

Proof. Let 𝑥₀ be an arbitrary point in 𝑋 and 𝑥₁∈𝑆𝑥₀. From (15), we have 𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₀,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁+𝛾𝑑𝑥₁,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑆𝑥₀,𝑇𝑥₁. (16) This implies that 𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₀,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁+𝛾𝑑𝑥₁,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈_{𝑥∈𝑆𝑥0}𝑠𝑥,𝑇𝑥₁,𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₀,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁+𝛾𝑑𝑥₁,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑥,𝑇𝑥₁∀𝑥∈𝑆𝑥₀. (17) Since 𝑥₁∈𝑆𝑥₀, we have 𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₀,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁+𝛾𝑑𝑥₁,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑥₁,𝑇𝑥₁,𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₀,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁+𝛾𝑑𝑥₁,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑥₁,𝑇𝑥₁=_{𝑥∈𝑇𝑥1}𝑠𝑑𝑥₁,𝑥. (18) So there exists some 𝑥₂∈𝑇𝑥₁ such that 𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₀,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁+𝛾𝑑𝑥₁,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑑𝑥₁,𝑥₂. (19) Therefore, 𝑑𝑥₁,𝑥₂⪯𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₀,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁+𝛾𝑑𝑥₁,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁. (20) By using the greatest lower bound property (g.l.b property) of 𝑆 and 𝑇, we get 𝑑𝑥₁,𝑥₂⪯𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₀,𝑥₁𝑑𝑥₁,𝑥₂+𝛾𝑑𝑥₁,𝑥₁𝑑𝑥₀,𝑥₂1+𝑑𝑥₀,𝑥₁=𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₀,𝑥₁𝑑𝑥₁,𝑥₂1+𝑑𝑥₀,𝑥₁, (21) which implies that 𝑑𝑥₁,𝑥₂≤𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₀,𝑥₁𝑑𝑥₁,𝑥₂1+𝑑𝑥₀,𝑥₁=𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₁,𝑥₂𝑑𝑥₀,𝑥₁1+𝑑𝑥₀,𝑥₁,𝑑𝑥₁,𝑥₂≤𝜆𝑑𝑥₀,𝑥₁+𝜇𝑑𝑥₁,𝑥₂,1−𝜇𝑑𝑥₁,𝑥₂≤𝜆𝑑𝑥₀,𝑥₁,𝑑𝑥₁,𝑥₂≤𝜆𝑑𝑥1−𝜇⁰,𝑥₁=ℎ𝑑𝑥₀,𝑥₁𝜆,whereℎ=1−𝜇<1. (22) Inductively, we can construct a sequence {𝑥_𝑛} in 𝑋 such that for 𝑛=0,1,2,…𝑑𝑥_𝑛,𝑥_𝑛+1≤ℎ^𝑛𝑑𝑥₀,𝑥₁ (23) with ℎ=𝜆/(1−𝜇)<1, for 𝑥_2𝑛+1∈𝑆𝑥_2𝑛 and 𝑥_2𝑛+2∈𝑇𝑥_2𝑛+1.
Now for 𝑚>𝑛, we get 𝑑𝑥_𝑛,𝑥_𝑚≤𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑑𝑥_𝑚−1,𝑥_𝑚≤ℎ^𝑛+ℎ^𝑛+1+⋯+ℎ^𝑚−1𝑑𝑥₀,𝑥₁≤ℎ^𝑛𝑑𝑥1−ℎ⁰,𝑥₁. (24) Therefore, 𝑑𝑥_𝑛,𝑥_𝑚≤ℎ^𝑛𝑑𝑥1−ℎ⁰,𝑥₁⟶0as𝑚,𝑛⟶∞. (25) This implies that {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑢∈𝑋 such that 𝑥_𝑛→𝑢 as 𝑛→∞. We now show that 𝑢∈𝑇𝑢 and 𝑢∈𝑆𝑢. From (15), we have 𝜆𝑑𝑥_2𝑘+𝑥,𝑢𝜇𝑑_2𝑘,𝑆𝑥_2𝑘𝑑(𝑢,𝑇𝑢)+𝛾𝑑𝑢,𝑆𝑥_2𝑘𝑑𝑥_2𝑘,𝑇𝑢1+𝑑𝑥_2𝑘,𝑢∈𝑠𝑆𝑥_2𝑘,𝑇𝑢. (26) This implies that 𝜆𝑑𝑥_2𝑘+𝑥,𝑢𝜇𝑑_2𝑘,𝑆𝑥_2𝑘𝑑(𝑢,𝑇𝑢)+𝛾𝑑𝑢,𝑆𝑥_2𝑘𝑑𝑥_2𝑘,𝑇𝑢1+𝑑𝑥_2𝑘∈,𝑢_{𝑥∈𝑆𝑥2𝑘}𝑠(),𝑥𝑥,𝑇𝑢𝜆𝑑_2𝑘+𝑥,𝑢𝜇𝑑_2𝑘,𝑆𝑥_2𝑘𝑑(𝑢,𝑇𝑢)+𝛾𝑑𝑢,𝑆𝑥_2𝑘𝑑𝑥_2𝑘,𝑇𝑢1+𝑑𝑥_2𝑘(),𝑢∈𝑠𝑥,𝑇𝑢∀𝑥∈𝑆𝑥_2𝑘. (27) Since 𝑥_2𝑘+1∈𝑆𝑥_2𝑘, we have 𝜆𝑑𝑥_2𝑘+𝑥,𝑢𝜇𝑑_2𝑘,𝑆𝑥_2𝑘𝑑(𝑢,𝑇𝑢)+𝛾𝑑𝑢,𝑆𝑥_2𝑘𝑑𝑥_2𝑘,𝑇𝑢1+𝑑𝑥_2𝑘𝑥,𝑢∈𝑠_2𝑘+1,𝑇𝑢. (28) By definition, 𝜆𝑑𝑥_2𝑘+𝑥,𝑢𝜇𝑑_2𝑘,𝑆𝑥_2𝑘𝑑(𝑢,𝑇𝑢)+𝛾𝑑𝑢,𝑆𝑥_2𝑘𝑑𝑥_2𝑘,𝑇𝑢1+𝑑𝑥_2𝑘𝑥,𝑢∈𝑠_2𝑘+1,𝑇𝑢=_{𝑢′∈𝑇𝑢}𝑠𝑑𝑥_2𝑘+1,𝑢^′. (29) There exists some 𝑢_𝑘∈𝑇𝑢 such that 𝜆𝑑𝑥_2𝑘+𝑥,𝑢𝜇𝑑_2𝑘,𝑆𝑥_2𝑘𝑑(𝑢,𝑇𝑢)+𝛾𝑑𝑢,𝑆𝑥_2𝑘𝑑𝑥_2𝑘,𝑇𝑢1+𝑑𝑥_2𝑘𝑑𝑥,𝑢∈𝑠_2𝑘+1,𝑢_𝑘; (30) that is, 𝑑𝑥_2𝑘+1,𝑢_𝑘⪯𝜆𝑑𝑥_2𝑘+𝑥,𝑢𝜇𝑑_2𝑘,𝑆𝑥_2𝑘𝑑(𝑢,𝑇𝑢)+𝛾𝑑𝑢,𝑆𝑥_2𝑘𝑑𝑥_2𝑘,𝑇𝑢1+𝑑𝑥_2𝑘.,𝑢 (31) By using the greatest lower bound property (g.l.b. property) of 𝑆 and 𝑇, we have 𝑑𝑥_2𝑘+1,𝑢_𝑘⪯𝜆𝑑𝑥_2𝑘,𝑢+𝜇𝑑𝑥_2𝑘,𝑥_2𝑘+1𝑑𝑢,𝑢_𝑘+𝛾𝑑𝑢,𝑥_2𝑘+1𝑑𝑥_2𝑘,𝑢_𝑘1+𝑑𝑥_2𝑘,𝑢⁻¹. (32) Since 𝑑𝑢,𝑢_𝑘⪯𝑑𝑢,𝑥_2𝑘+1+𝑑𝑥_2𝑘+1,𝑢_𝑘, (33) using (32), we get 𝑑𝑢,𝑢_𝑘⪯𝑑𝑢,𝑥_2𝑘+1+𝜆𝑑𝑢,𝑥_2𝑘+1+𝜇𝑑𝑥_2𝑘,𝑥_2𝑘+1𝑑𝑢,𝑢_𝑘+𝛾𝑑𝑢,𝑥_2𝑘+1𝑑𝑥_2𝑘,𝑢_𝑘1+𝑑𝑥_2𝑘,𝑑,𝑢𝑢,𝑢_𝑘≤𝑑𝑢,𝑥_2𝑘+1+𝜆𝑑𝑢,𝑥_2𝑘+1+𝜇𝑑𝑥_2𝑘,𝑥_2𝑘+1𝑑𝑢,𝑢_𝑘+𝛾𝑑𝑢,𝑥_2𝑘+1𝑑𝑥_2𝑘,𝑢_𝑘×1+𝑑𝑥_2𝑘,𝑢⁻¹. (34) Taking the limit as 𝑘→∞, we get |𝑑(𝑢,𝑢_𝑘)|→0 as 𝑘→∞. By  [9, lemma  2], we have 𝑢_𝑘→𝑢 as 𝑘→∞. Since 𝑇𝑢 is closed, 𝑢∈𝑇𝑢. Similarly, it follows that 𝑢∈𝑆𝑢. Thus, 𝑆 and 𝑇 have a common fixed point.

By setting 𝛾=0 in Theorem 9, we get the following corollary.

Corollary 10. Let (𝑋,𝑑) be a complete complex valued metric space and let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mappings with g.l.b property such that𝑑()𝑑𝜆𝑑𝑥,𝑦+𝜇𝑥,𝑆𝑥𝑦,𝑇𝑦1+𝑑𝑥,𝑦∈𝑠𝑆𝑥,𝑇𝑦 (35) for all 𝑥,𝑦∈𝑋 and 𝜆,𝜇 are nonnegative real numbers with 𝜆+𝜇<1. Then 𝑆,𝑇 have a common fixed point.

By setting 𝑆=𝑇 in Theorem 9, we get the following Corollary.

Corollary 11. Let (𝑋,𝑑) be a complete complex valued metric space and let 𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mapping with g.l.b property such that+()𝑑𝑑𝜆𝑑𝑥,𝑦𝜇𝑑𝑥,𝑇𝑥𝑦,𝑇𝑦+𝛾𝑑𝑦,𝑇𝑥𝑥,𝑇𝑦1+𝑑𝑥,𝑦∈𝑠𝑇𝑥,𝑇𝑦 (36) for all 𝑥,𝑦∈𝑋 and 𝜆, 𝜇, and 𝛾 are nonnegative real numbers with 𝜆+𝜇+𝛾<1. Then 𝑇 has a fixed point in 𝑋.

By Remark 4, we have the following corollaries from Theorem 9.

Corollary 12. Let (𝑋,𝑑) be a complete metric space and let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mappings such that 𝐻+()𝑑𝑑𝑆𝑥,𝑇𝑦≤𝜆𝑑𝑥,𝑦𝜇𝑑𝑥,𝑆𝑥𝑦,𝑇𝑦+𝛾𝑑𝑦,𝑆𝑥𝑥,𝑇𝑦1+𝑑𝑥,𝑦 (37) for all 𝑥,𝑦∈𝑋 and 𝜆, 𝜇, and 𝛾 are nonnegative real numbers with 𝜆+𝜇+𝛾<1. Then 𝑆,𝑇 have a common fixed point.

Remark 13. By equating 𝜆, 𝜇, 𝛾 to 0 in all possible combinations, one can derive a host of corollaries which include the Banach fixed point theorem for multivalued mappings in complete metric space.

Example 14. Let 𝑋=[0,1]. Define 𝑑∶𝑋×𝑋→ℂ by 𝑑𝑥,𝑦=𝑥−𝑦𝑒^𝑖𝜃,𝜃=tan⁻¹𝑦𝑥. (38) Then (𝑋,𝑑) is a complex valued metric space. Consider the mappings 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) defined by 1𝑆𝑥=0,5𝑥1,𝑇𝑥=0,𝑥.10 (39) The contractive condition of main theorem is trivial for the case when 𝑥=𝑦=0. Suppose without any loss of generality that all 𝑥,𝑦 are nonzero and 𝑥<𝑦. Then 𝑑𝑥,𝑦=𝑦−𝑥𝑒^𝑖𝜃,𝑑()=𝑥𝑥,𝑆𝑥𝑥−5𝑒^𝑖𝜃,𝑑=𝑦𝑦,𝑇𝑦𝑦−𝑒10^𝑖𝜃,𝑑=𝑥𝑦,𝑆𝑥𝑦−5𝑒^𝑖𝜃,𝑑=𝑦𝑥,𝑇𝑦𝑥−𝑒10^𝑖𝜃,𝑠𝑥𝑆𝑥,𝑇𝑦=𝑠5−𝑦𝑒10^𝑖𝜃. (40) Consider, 𝜆𝑑+𝜇|𝑑(𝑑𝑑𝑑𝑥,𝑦𝑥,𝑆𝑥)|𝑦,𝑇𝑦+𝛾𝑦,𝑆𝑥𝑥,𝑇𝑦+𝜇|((1+𝑑𝑥,𝑦=𝜆𝑦−𝑥𝑥−𝑥/5)|𝑦−𝑦/10+𝛾𝑦−𝑥/5)𝑥−𝑦/10+𝜇|((1+𝑑𝑥,𝑦=𝜆𝑦−𝑥𝑥−𝑥/5)|𝑦−𝑦/10+𝛾𝑦−𝑥/5)𝑥−𝑦/10.1+𝑑𝑥,𝑦 (41) Clearly, for any value of 𝜇 and 𝛾 and 𝜆=1/5, we have 𝑥5−𝑦≤1105+𝜇|((𝑦−𝑥𝑥−𝑥/5)|𝑦−𝑦/10+𝛾𝑦−𝑥/5)𝑥−𝑦/10.1+𝑑𝑥,𝑦 (42) Thus, +()𝑑𝑑𝜆𝑑𝑥,𝑦𝜇𝑑𝑥,𝑆𝑥𝑦,𝑇𝑦+𝛾𝑑𝑦,𝑆𝑥𝑥,𝑇𝑦.1+𝑑𝑥,𝑦∈𝑠𝑆𝑥,𝑇𝑦 (43) Hence, all the conditions of our main theorem are satisfied and 0 is a common fixed point of 𝑆 and 𝑇.

In the following results, we considered the Kannan type contractive condition involving rational expressions.

Theorem 15. Let (𝑋,𝑑) be a complete complex valued metric space and let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mappings with g.l.b property such that ()𝑑()𝑑𝑎𝑑𝑥,𝑆𝑥+𝑏𝑑𝑦,𝑇𝑦+𝑐𝑥,𝑆𝑥𝑦,𝑇𝑦1+𝑑𝑥,𝑦∈𝑠𝑆𝑥,𝑇𝑦 (44) for all 𝑥,𝑦∈𝑋 and 𝑎+𝑏+𝑐<1. Then 𝑆 and 𝑇 have a common fixed point.

Proof. Let 𝑥₀∈𝑋 and 𝑥₁∈𝑆𝑥₀. From (44), we get 𝑎𝑑𝑥₀,𝑆𝑥₀+𝑏𝑑𝑥₁,𝑇𝑥₁𝑑𝑥+𝑐⁰,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑆𝑥₀,𝑇𝑥₁. (45) This implies that 𝑎𝑑𝑥₀,𝑆𝑥₀+𝑏𝑑𝑥₁,𝑇𝑥₁𝑑𝑥+𝑐⁰,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈_{𝑥∈𝑆𝑥0}𝑠𝑥,𝑇𝑥₁,𝑎𝑑𝑥₀,𝑆𝑥₀+𝑏𝑑𝑥₁,𝑇𝑥₁𝑑𝑥+𝑐⁰,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑥,𝑇𝑥₁∀𝑥∈𝑆𝑥₀. (46) Since 𝑥₁∈𝑆𝑥₀, we have 𝑎𝑑𝑥₀,𝑆𝑥₀+𝑏𝑑𝑥₁,𝑇𝑥₁𝑑𝑥+𝑐⁰,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑥₁,𝑇𝑥₁,𝑎𝑑𝑥₀,𝑆𝑥₀+𝑏𝑑𝑥₁,𝑇𝑥₁𝑑𝑥+𝑐⁰,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑥₁,𝑇𝑥₁=_{𝑥∈𝑇𝑥1}𝑠𝑑𝑥₁,𝑥. (47) So there exists some 𝑥₂∈𝑇𝑥₁, such that 𝑎𝑑𝑥₀,𝑆𝑥₀+𝑏𝑑𝑥₁,𝑇𝑥₁𝑑𝑥+𝑐⁰,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑑𝑥₁,𝑥₂. (48) That is, 𝑑𝑥₁,𝑥₂⪯𝑎𝑑𝑥₀,𝑆𝑥₀+𝑏𝑑𝑥₁,𝑇𝑥₁𝑑𝑥+𝑐⁰,𝑆𝑥₀𝑑𝑥₁,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁. (49) By using the greatest lower bound property (g.l.b property) of 𝑆 and 𝑇, we get 𝑑𝑥₁,𝑥₂⪯𝑎𝑑𝑥₀,𝑥₁+𝑏𝑑𝑥₁,𝑥₂𝑑𝑥+𝑐⁰,𝑥₁𝑑𝑥₁,𝑥₂1+𝑑𝑥₀,𝑥₁ (50) which implies that 𝑑𝑥₁,𝑥₂≤𝑎𝑑𝑥₀,𝑥₁+𝑏𝑑𝑥₁,𝑥₂𝑑𝑥+𝑐⁰,𝑥₁𝑑𝑥₁,𝑥₂1+𝑑𝑥₀,𝑥₁. (51) Since |𝑑(𝑥₀,𝑥₁)|<|1+𝑑(𝑥₀,𝑥₁)|, we have 𝑑𝑥₁,𝑥₂≤𝑎𝑑𝑥₀,𝑥₁+𝑏𝑑𝑥₁,𝑥₂+𝑐𝑑𝑥₁,𝑥₂,(1−𝑏−𝑐)𝑑𝑥₁,𝑥₂≤𝑎𝑑𝑥₀,𝑥₁𝑑𝑥₁,𝑥₂≤𝑎()𝑑𝑥1−𝑏−𝑐⁰,𝑥₁. (52) Thus, 𝑑𝑥₁,𝑥₂≤𝑎()𝑑𝑥1−𝑏−𝑐⁰,𝑥₁=𝑙𝑑𝑥₀,𝑥₁𝑎,where𝑙=(1−𝑏−𝑐)<1. (53) Inductively, we can construct a sequence {𝑥_𝑛} in 𝑋 such that for 𝑛=0,1,2,…𝑑𝑥_𝑛,𝑥_𝑛+1≤𝑙^𝑛𝑑𝑥₀,𝑥₁ (54) with 𝑙=𝑎/(1−𝑏−𝑐)<1, for 𝑥_2𝑛+1∈𝑆𝑥_2𝑛 and 𝑥_2𝑛+2∈𝑇𝑥_2𝑛+1. Now for 𝑚>𝑛, we get 𝑑𝑥_𝑛,𝑥_𝑚≤𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑑𝑥_𝑚−1,𝑥_𝑚≤𝑙^𝑛+𝑙^𝑛+1+⋯+𝑙^𝑚−1𝑑𝑥₀,𝑥₁≤𝑙^𝑛𝑑𝑥1−𝑙⁰,𝑥₁, (55) and so𝑑𝑥_𝑛,𝑥_𝑚≤𝑙^𝑛𝑑𝑥1−𝑙⁰,𝑥₁⟶0as𝑚,𝑛⟶∞. (56) This implies that {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝜐∈𝑋 such that 𝑥_𝑛→𝜐 as 𝑛→∞. We now show that 𝜐∈𝑇𝜐 and 𝜐∈𝑆𝜐. So from (44), we get 𝑎𝑑𝑥_2𝑛,𝑆𝑥_2𝑛()𝑑𝑥+𝑏𝑑𝜐,𝑇𝜐+𝑐^2𝑛,𝑆𝑥_2𝑛𝑑(𝜐,𝑇𝜐)1+𝑑𝑥_2𝑛,𝜐∈𝑠𝑆𝑥_2𝑛,𝑇𝜐. (57) This implies that 𝑎𝑑𝑥_2𝑛,𝑆𝑥_2𝑛()𝑑𝑥+𝑏𝑑𝜐,𝑇𝜐+𝑐^2𝑛,𝑆𝑥_2𝑛𝑑(𝜐,𝑇𝜐)1+𝑑𝑥_2𝑛∈,𝜐_{𝑥∈𝑆𝑥2𝑛}𝑠(),𝑥𝑥,𝑇𝜐𝑎𝑑_2𝑛,𝑆𝑥_2𝑛()𝑑𝑥+𝑏𝑑𝜐,𝑇𝜐+𝑐^2𝑛,𝑆𝑥_2𝑛𝑑(𝜐,𝑇𝜐)1+𝑑𝑥_2𝑛(),𝜐∈𝑠𝑥,𝑇𝜐∀𝑥∈𝑆𝑥_2𝑛. (58) Since 𝑥_2𝑛+1∈𝑆𝑥_2𝑛, we have 𝑎𝑑𝑥_2𝑛,𝑆𝑥_2𝑛()𝑑𝑥+𝑏𝑑𝜐,𝑇𝜐+𝑐^2𝑛,𝑆𝑥_2𝑛𝑑(𝜐,𝑇𝜐)1+𝑑𝑥_2𝑛𝑥,𝜐∈𝑠_2𝑛+1,𝑇𝜐. (59) By definition, 𝑎𝑑𝑥_2𝑛,𝑆𝑥_2𝑛()𝑑𝑥+𝑏𝑑𝜐,𝑇𝜐+𝑐^2𝑛,𝑆𝑥_2𝑛𝑑(𝜐,𝑇𝜐)1+𝑑𝑥_2𝑛𝑥,𝜐∈𝑠_2𝑛+1,𝑇𝜐=_{𝜐′∈𝑇𝜐}𝑠𝑑𝑥_2𝑛+1,𝜐^′. (60) There exists some 𝜐_𝑛∈𝑇𝜐 such that 𝑎𝑑𝑥_2𝑛,𝑆𝑥_2𝑛()𝑑𝑥+𝑏𝑑𝜐,𝑇𝜐+𝑐^2𝑛,𝑆𝑥_2𝑛𝑑(𝜐,𝑇𝜐)1+𝑑𝑥_2𝑛𝑑𝑥,𝜐∈𝑠_2𝑛+1,𝜐_𝑛; (61) that is, 𝑑𝑥_2𝑛+1,𝜐_𝑛⪯𝑎𝑑𝑥_2𝑛,𝑆𝑥_2𝑛()𝑑𝑥+𝑏𝑑𝜐,𝑇𝜐+𝑐^2𝑛,𝑆𝑥_2𝑛𝑑(𝜐,𝑇𝜐)1+𝑑𝑥_2𝑛.,𝜐 (62) By using the greatest lower bound property (g.l.b property) of 𝑆 and 𝑇, we get 𝑑𝑥_2𝑛+1,𝜐_𝑛⪯𝑎𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑏𝑑𝜐,𝜐_𝑛𝑑𝑥+𝑐^2𝑛,𝑥_2𝑛+1𝑑𝜐,𝜐_𝑛1+𝑑𝑥_2𝑛.,𝜐 (63) Now by using the triangular inequality, we get 𝑑𝜐,𝜐_𝑛⪯𝑑𝜐,𝑥_2𝑛+1+𝑑𝑥_2𝑛+1,𝜐_𝑛⪯𝑑𝜐,𝑥_2𝑛+1+𝑎𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑏𝑑𝜐,𝜐_𝑛𝑑𝑥+𝑐^2𝑛,𝑥_2𝑛+1𝑑𝜐,𝜐_𝑛1+𝑑𝑥_2𝑛.,𝜐 (64) That is, (1−𝑏)𝑑𝜐,𝜐_𝑛≤𝑑𝜐,𝑥_2𝑛+1+𝑎𝑑𝑥_2𝑛,𝑥_2𝑛+1𝑑𝑥+𝑐^2𝑛,𝑥_2𝑛+1𝑑𝜐,𝜐_𝑛1+𝑑𝑥_2𝑛.,𝜐 (65) It follows that 𝑑𝜐,𝜐_𝑛≤1()𝑑1−𝑏𝜐,𝑥_2𝑛+1+𝑎()𝑑𝑥1−𝑏^2𝑛,𝑥_2𝑛+1+𝑐()𝑑𝑥1−𝑏^2𝑛,𝑥_2𝑛+1𝑑𝜐,𝜐_𝑛1+𝑑𝑥_2𝑛.,𝜐 (66) By letting 𝑛→∞ in the above inequality, we get 𝑑𝜐,𝜐_𝑛⟶0as𝑛⟶∞. (67)
By  [9, Lemma  2], we have 𝜐_𝑛→𝜐 as 𝑛→∞. Since 𝑇𝜐 is closed, 𝜐∈𝑇𝜐. Similarly, it follows that 𝜐∈𝑆𝜐. Thus, 𝑆 and 𝑇 have a common fixed point.

Corollary 16. Let (𝑋,𝑑) be a complete complex valued metric space and let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mappings with g.l.b property such that 𝛼𝑑(𝑥,𝑆𝑥)+𝑑𝑦,𝑇𝑦∈𝑠𝑆𝑥,𝑇𝑦 (68) for all 𝑥,𝑦∈𝑋 and 0≤𝛼<1/2. Then 𝑆 and 𝑇 have a common fixed point.

By setting 𝑆=𝑇 in Theorem 15, we get the following corollary.

Corollary 17. Let (𝑋,𝑑) be a complete complex valued metric space and let 𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mappings with g.l.b property such that ()𝑑()𝑑𝑎𝑑𝑥,𝑇𝑥+𝑏𝑑𝑦,𝑇𝑦+𝑐𝑥,𝑇𝑥𝑦,𝑇𝑦1+𝑑𝑥,𝑦∈𝑠𝑇𝑥,𝑇𝑦 (69) for all 𝑥,𝑦∈𝑋 and 𝑎+𝑏+𝑐<1. Then 𝑇 has a fixed point.

By Remark 4 we have the following corollaries.

Corollary 18. Let (𝑋,𝑑) be a complete metric space and let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mappings such that 𝐻()𝑑()𝑑𝑆𝑥,𝑇𝑦≤𝑎𝑑𝑥,𝑆𝑥+𝑏𝑑𝑦,𝑇𝑦+𝑐𝑥,𝑆𝑥𝑦,𝑇𝑦1+𝑑𝑥,𝑦 (70) for all 𝑥,𝑦∈𝑋 and 𝑎, 𝑏, and 𝑐 are nonnegative real numbers with 𝑎+𝑏+𝑐<1. Then 𝑆,𝑇 have a common fixed point.

Remark 19. By writing 𝑎, 𝑏, 𝑐 in all possible combinations, one can derive a host of corollaries which include the Kannan fixed point theorem for multivalued mappings in complete metric space.

In the following results, we considered the Chatterjea type locally contractive condition involving rational expressions.

Theorem 20. Let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mappings with the g.l.b property on complete complex valued metric space (𝑋,𝑑), 𝑥₀∈𝑋, and 0≺𝑟∈ℂ. If 𝑆 and 𝑇 satisfy 𝑑𝑑𝑎𝑑𝑦,𝑆𝑥+𝑏𝑑𝑥,𝑇𝑦+𝑐𝑦,𝑆𝑥𝑥,𝑇𝑦1+𝑑𝑥,𝑦∈𝑠𝑆𝑥,𝑇𝑦 (71) for all 𝑥,𝑦∈𝐵(𝑥₀,𝑟) and (1−𝑘)𝑟∈𝑠𝑥₀,𝑆𝑥₀, (72) where 𝑎,𝑏, and 𝑐 are nonnegative real numbers with 𝑘=𝑏/(1−𝑏)<1, then 𝑆 and 𝑇 have a common fixed point in 𝐵(𝑥₀,𝑟).

Proof. Let 𝑥₀ be an arbitrary point in 𝑋. From (72), one can easily prove that 𝑑𝑥₀,𝑥₁≤(1−𝑘)|𝑟|. (73) Thus, we have 𝑥₁∈𝐵(𝑥₀,𝑟). From (71), we get 𝑎𝑑𝑥₁,𝑆𝑥₀+𝑏𝑑𝑥₀,𝑇𝑥₁𝑑𝑥+𝑐¹,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑆𝑥₀,𝑇𝑥₁. (74) This implies that 𝑎𝑑𝑥₁,𝑆𝑥₀+𝑏𝑑𝑥₀,𝑇𝑥₁𝑑𝑥+𝑐¹,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑥₁,𝑇𝑥₁. (75) By the definition, we can take 𝑥₂∈𝑇𝑥₁ such that 𝑎𝑑𝑥₁,𝑆𝑥₀+𝑏𝑑𝑥₀,𝑇𝑥₁𝑑𝑥+𝑐¹,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁∈𝑠𝑑𝑥₁,𝑥₂. (76) By the definition, we get 𝑑𝑥₁,𝑥₂⪯𝑎𝑑𝑥₁,𝑆𝑥₀+𝑏𝑑𝑥₀,𝑇𝑥₁𝑑𝑥+𝑐¹,𝑆𝑥₀𝑑𝑥₀,𝑇𝑥₁1+𝑑𝑥₀,𝑥₁. (77) By using the g.l.b property of 𝑆 and 𝑇, we get 𝑑𝑥₁,𝑥₂⪯𝑎𝑑𝑥₁,𝑥₁+𝑏𝑑𝑥₀,𝑥₂𝑑𝑥+𝑐¹,𝑥₁𝑑𝑥₀,𝑥₂1+𝑑𝑥₀,𝑥₁. (78) Hence, we have 𝑑𝑥₁,𝑥₂⪯𝑏𝑑𝑥₀,𝑥₂ (79) which implies that 𝑑𝑥₁,𝑥₂≤𝑘𝑑𝑥₀,𝑥₁, (80) where 𝑘=𝑏/(1−𝑏)<1. From (73), we get 𝑑𝑥₁,𝑥₂≤𝑘(1−𝑘)|𝑟|. (81) Note that 𝑑𝑥₀,𝑥₂≤𝑑𝑥₀,𝑥₁+𝑑𝑥₁,𝑥₂≤(𝑟|(𝑟|=(𝑟|≤1−𝑘)|+𝑘1−𝑘)|1−𝑘)(1+𝑘)|1−𝑘²|𝑟|. (82) Thus, we have 𝑥₂∈𝐵(𝑥₀,𝑟). From (71), we get 𝑎𝑑𝑥₂,𝑇𝑥₁+𝑏𝑑𝑥₁,𝑆𝑥₂+𝑐𝑑𝑥₂,𝑇𝑥₁𝑑𝑥₁,𝑆𝑥₂1+𝑑𝑥₁,𝑥₂∈𝑠𝑇𝑥₁,𝑆𝑥₂. (83) This implies that 𝑎𝑑𝑥₂,𝑇𝑥₁+𝑏𝑑𝑥₁,𝑆𝑥₂+𝑐𝑑𝑥₂,𝑇𝑥₁𝑑𝑥₁,𝑆𝑥₂1+𝑑𝑥₁,𝑥₂∈𝑠𝑥₂,𝑆𝑥₂. (84) By the definition, there exists 𝑥₃∈𝑆𝑥₂ such that 𝑎𝑑𝑥₂,𝑇𝑥₁+𝑏𝑑𝑥₁,𝑆𝑥₂+𝑐𝑑𝑥₂,𝑇𝑥₁𝑑𝑥₁,𝑆𝑥₂1+𝑑𝑥₁,𝑥₂∈𝑠𝑑𝑥₂,𝑥₃. (85) By the definition, we get 𝑑𝑥₂,𝑥₃⪯𝑎𝑑𝑥₂,𝑇𝑥₁+𝑏𝑑𝑥₁,𝑆𝑥₂+𝑐𝑑𝑥₂,𝑇𝑥₁𝑑𝑥₁,𝑆𝑥₂1+𝑑𝑥₁,𝑥₂. (86) By using the g.l.b property of 𝑆 and 𝑇, we get 𝑑𝑥₂,𝑥₃⪯𝑎𝑑𝑥₂,𝑥₂+𝑏𝑑𝑥₁,𝑥₃+𝑐𝑑𝑥₂,𝑥₂𝑑𝑥₁,𝑥₃1+𝑑𝑥₁,𝑥₂. (87) Hence, we have 𝑑𝑥₂,𝑥₃⪯𝑏𝑑𝑥₁,𝑥₃ (88) which implies that 𝑑𝑥₂,𝑥₃≤𝑘𝑑𝑥₁,𝑥₂, (89) where 𝑘=𝑏/(1−𝑏)<1. Consider that 𝑑𝑥₀,𝑥₃≤𝑑𝑥₀,𝑥₁+𝑑𝑥₁,𝑥₂+𝑑𝑥₂,𝑥₃≤(1−𝑘)|𝑟|+𝑘(1−𝑘)|𝑟|+𝑘²(𝑟|=1−𝑘)|1−𝑘³|𝑟|≤|𝑟|. (90) So 𝑥₃∈𝐵(𝑥₀,𝑟). Continuing in this way, we can construct a sequence {𝑥_𝑛} in 𝐵(𝑥₀,𝑟) such that for each 𝑛≥1, 𝑑𝑥_2𝑛,𝑥_2𝑛+1≤𝑘^2𝑛𝑑𝑥₀,𝑥₁,𝑑𝑥_2𝑛+1,𝑥_2𝑛+2≤𝑘^2𝑛+1𝑑𝑥₀,𝑥₁, (91) where 𝑘=𝑏/(1−𝑏)<1, 𝑥_2𝑛+1∈𝑆𝑥_2𝑛, and 𝑥_2𝑛+2∈𝑇𝑥_2𝑛+1. Now, inductively, we can construct a sequence {𝑥_𝑛} in 𝑋 such that for each 𝑛≥0, 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝑘^𝑛𝑑𝑥₀,𝑥₁. (92) Thus, as in the proof of Theorem 15, {𝑥_𝑛} is a Cauchy sequence in 𝐵(𝑥₀,𝑟). Since 𝑋 is complete and 𝐵(𝑥₀,𝑟) is a closed subspace of 𝑋, there exists 𝜐∈𝐵(𝑥₀,𝑟) such that 𝑥_𝑛→𝑢 as 𝑛→∞. Now, we show that 𝜐∈𝑇𝜐 and 𝜐∈𝑆𝜐. From (71), we get 𝑎𝑑𝜐,𝑆𝑥_2𝑛+𝑏𝑑𝑥_2𝑛𝑑,𝑇𝜐+𝑐𝜐,𝑆𝑥_2𝑛𝑑𝑥_2𝑛,𝑇𝜐1+𝑑𝑥_2𝑛,𝜐∈𝑠𝑆𝑥_2𝑛,𝑇𝜐. (93) This implies that 𝑎𝑑𝜐,𝑆𝑥_2𝑛+𝑏𝑑𝑥_2𝑛𝑑,𝑇𝜐+𝑐𝜐,𝑆𝑥_2𝑛𝑑𝑥_2𝑛,𝑇𝜐1+𝑑𝑥_2𝑛𝑥,𝜐∈𝑠_2𝑛+1,𝑇𝜐. (94) By the definition, there exists 𝜐_𝑛∈𝑇𝜐 such that 𝑎𝑑𝜐,𝑆𝑥_2𝑛+𝑏𝑑𝑥_2𝑛𝑑,𝑇𝜐+𝑐𝜐,𝑆𝑥_2𝑛𝑑𝑥_2𝑛,𝑇𝜐1+𝑑𝑥_2𝑛𝑑𝑥,𝜐∈𝑠_2𝑛+1,𝜐_𝑛. (95) By the definition, we get 𝑑𝑥_2𝑛+1,𝜐_𝑛⪯𝑎𝑑𝜐,𝑆𝑥_2𝑛+𝑏𝑑𝑥_2𝑛𝑑,𝑇𝜐+𝑐𝜐,𝑆𝑥_2𝑛𝑑𝑥_2𝑛,𝑇𝜐1+𝑑𝑥_2𝑛.,𝜐 (96) By using the g.l.b property of 𝑆 and 𝑇, we get 𝑑𝑥_2𝑛+1,𝜐_𝑛⪯𝑎𝑑𝜐,𝑥_2𝑛+1+𝑏𝑑𝑥_2𝑛,𝜐_𝑛𝑑+𝑐𝜐,𝑥_2𝑛+1𝑑𝑥_2𝑛,𝜐_𝑛1+𝑑𝑥_2𝑛.,𝜐 (97) Hence, we have 0≺𝑑𝑥_2𝑛+1,𝜐_𝑛⪯𝑎𝑑𝜐,𝑥_2𝑛+1+𝑏𝑑𝑥_2𝑛,𝜐_𝑛𝑑+𝑐𝜐,𝑥_2𝑛+1𝑑𝑥_2𝑛,𝜐_𝑛1+𝑑𝑥_2𝑛.,𝜐 (98) Now, by using the triangular inequality, we get 𝑑𝜐,𝜐_𝑛⪯𝑑𝜐,𝑥_2𝑛+1+𝑑𝑥_2𝑛+1,𝜐_𝑛⪯𝑑𝜐,𝑥_2𝑛+1+𝑎𝑑𝜐,𝑥_2𝑛+1+𝑏𝑑𝑥_2𝑛,𝜐_𝑛𝑑+𝑐𝜐,𝑥_2𝑛+1𝑑𝑥_2𝑛,𝜐_𝑛1+𝑑𝑥_2𝑛,,𝜐 (99) and so 𝑑𝜐,𝜐_𝑛≤𝑑𝜐,𝑥_2𝑛+1+𝑎𝑑𝜐,𝑥_2𝑛+1+𝑏𝑑𝑥_2𝑛,𝜐_𝑛𝑑+𝑐𝜐,𝑥_2𝑛+1𝑑𝑥_2𝑛,𝜐_𝑛1+𝑑𝑥_2𝑛.,𝜐 (100) By letting 𝑛→∞ in the above inequality, we get |𝑑(𝜐,𝜐_𝑛)|→0 as 𝑛→∞. By [9, Lemma  2.2], it follows that 𝜐_𝑛→𝜐 as 𝑛→∞. Since 𝑇𝜐 is closed, 𝜐∈𝑇𝜐. Similarly, it follows that 𝜐∈𝑆𝜐. Thus, 𝑆 and 𝑇 have a common fixed point in 𝑋. This completes the proof.

By taking 𝑐=0 in Theorem 20, we get the following Corollary.

Corollary 21. Let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mappings with the g.l.b property in complete complex valued metric space (𝑋,𝑑), 𝑥₀∈𝑋, and 0≺𝑟∈ℂ. If 𝑆 and 𝑇 satisfy 𝑎𝑑𝑦,𝑆𝑥+𝑏𝑑𝑥,𝑇𝑦∈𝑠𝑆𝑥,𝑇𝑦 (101) for all 𝑥,𝑦∈𝐵(𝑥₀,𝑟) and (1−𝑘)𝑟∈𝑠𝑥₀,𝑆𝑥₀, (102) where 𝑎 and 𝑏 are nonnegative real numbers with 𝑘=𝑏/(1−𝑏)<1, then 𝑆 and 𝑇 have a common fixed point in 𝐵(𝑥₀,𝑟).

By taking 𝑆=𝑇 in Theorem 20, we get the following.

Corollary 22. Let 𝑇∶𝑋→𝐶𝐵(𝑋) be multivalued mapping with the g.l.b property in complete complex valued metric space (𝑋,𝑑), 𝑥₀∈𝑋, and 0≺𝑟∈ℂ. If 𝑇 satisfies 𝑑𝑑𝑎𝑑𝑦,𝑇𝑥+𝑏𝑑𝑥,𝑇𝑦+𝑐𝑦,𝑇𝑥𝑥,𝑇𝑦1+𝑑𝑥,𝑦∈𝑠𝑇𝑥,𝑇𝑦 (103) for all 𝑥,𝑦∈𝐵(𝑥₀,𝑟) and (1−𝑘)𝑟∈𝑠𝑥₀,𝑇𝑥₀, (104) where 𝑎,𝑏, and 𝑐 are nonnegative real numbers with 𝑘=𝑏/(1−𝑏)<1, then 𝑇 has a fixed point in 𝐵(𝑥₀,𝑟).

4. Applications

As an application of the main result (Corollary 22), we prove the following homotopy result.

Theorem 23. Let (𝑋,𝑑) be a complete complex valued metric space and let 𝑈 be an open subset of 𝑋. Let 𝐹∶[0,1]×𝑈→𝐶𝐵(𝑋) be multivalued mapping with the g.l.b property. Suppose that there exist 𝜁^∘∈𝑋 and 0≺𝑟∈ℂ such that(a)𝜁∉[𝐹(𝑡,𝜁)] for all 𝜁∈𝜕𝑈 and 𝑡∈[0,1];(b)𝐹(𝑡,⋅)∶𝑈→𝐶𝐵(𝑋) is a multivalued mapping satisfying ́()́𝜁́𝑑́()𝑑́𝜁́𝑎𝑑𝜁,𝐹𝑡,𝜁+𝑏𝑑𝜁,𝐹𝑡,+𝑐𝜁,𝐹𝑡,𝜁𝜁,𝐹𝑡,𝜁́𝐹()́𝜁́,()1+𝑑𝜁,∈𝑠𝑡,𝜁,𝐹𝑡,1−𝑘𝑟∈𝑠𝜁^∘,𝐹𝑡^∘,𝜁^∘, (105) where 𝑘=𝑏/(1−𝑏)<1;(c)there exists a continuous increasing function 𝜑∶(0,1]→𝑃∪{0} such that 𝜑(𝑠)−𝜑(𝑡)∈𝑠𝐹(𝑠,𝜁),𝐹𝑡,𝜁́,𝜑(𝑠)∈𝜑(𝑡) (106) for all 𝑠,𝑡∈[0,1] and 𝜁∈𝑈, where 𝑃={𝑧∈ℂ∶0≺𝑧}. Then 𝐹(0,⋅) has a fixed point if and only if 𝐹(1,⋅) has a fixed point.

Proof. Suppose that 𝐹(0,⋅) has a fixed point 𝑧; so 𝑧∈𝐹(0,𝑧). From (a), 𝑧∈𝑈. Define the following set: 𝒬∶=(𝑡,𝜁)∈[0,1]×𝑈∶𝜁∈𝐹(𝑡,𝜁). (107)
Clearly, 𝒬≠∅. We define the partial ordering ≾ in 𝒬 as follows: ()≾𝜁́𝑑𝜁́⪯2𝑡,𝜁𝑠,⟺𝑡≤𝑠,𝜁,𝜑(𝑠)(𝑡).1−𝑘−𝜑 (108)
Let ℳ be a totally ordered subset of 𝒬 and 𝑡^∘=sup{𝑡∶(𝑡,𝜁)∈ℳ}. Consider a sequence {(𝑡_𝑛,𝜁_𝑛)} in ℳ such that (𝑡_𝑛,𝜁_𝑛)≾(𝑡_𝑛+1,𝜁_𝑛+1) and 𝑡_𝑛→𝑡^∘ as 𝑛→∞. Then, for any 𝑛≥1 with 𝑚>𝑛, we have 𝑑𝜁_𝑚,𝜁_𝑛⪯2𝜑𝑡1−𝑘^𝑚−𝜑𝑡_𝑛⟶0 (109) as 𝑛,𝑚→∞, which implies that {𝜁_𝑛} is a Cauchy sequence. Since (𝑋,𝑑) is a complete complex valued metric space, there exists 𝜁^∘∈𝑋 such that 𝜁_𝑛→𝜁^∘. From (a), choose 𝑛₀∈ℕ such that for all 𝑛≥𝑛₀, 𝑎𝑑𝜁^∘,𝐹𝑡_𝑛,𝜁_𝑛+𝑏𝑑𝜁_𝑛,𝐹𝑡^∘,𝜁^∘𝑑+𝑐^∘𝜁,𝐹𝑡_𝑛,𝜁_𝑛𝑑𝜁_𝑛,𝐹𝑡^∘,𝜁^∘1+𝑑𝜁_𝑛,𝜁^∘∈𝑠𝐹𝑡_𝑛,𝜁_𝑛,𝐹𝑡^∘,𝜁^∘,𝑎𝑑𝜁^∘,𝐹𝑡_𝑛,𝜁_𝑛+𝑏𝑑𝜁_𝑛,𝐹𝑡^∘,𝜁^∘+𝑐𝑑𝜁^∘,𝐹𝑡_𝑛,𝜁_𝑛𝑑𝜁_𝑛,𝐹𝑡^∘,𝜁^∘1+𝑑𝜁_𝑛,𝜁^∘∈𝑠𝜁_𝑛,𝐹𝑡^∘,𝜁^∘, (110) since 𝜁_𝑛∈𝐹(𝑡_𝑛,𝜁_𝑛). So there exists 𝜁_𝑘∈𝐹(𝑡^∘,𝜁^∘) such that 𝑑𝜁_𝑛,𝜁_𝑘⪯𝑎𝑑𝜁^∘,𝐹𝑡_𝑛,𝜁_𝑛+𝑏𝑑𝜁_𝑛,𝐹𝑡^∘,𝜁^∘+𝑐𝑑𝜁^∘,𝐹𝑡_𝑛,𝜁_𝑛𝑑𝜁_𝑛,𝐹𝑡^∘,𝜁^∘1+𝑑𝜁_𝑛,𝜁^∘. (111) By using the g.l.b property of 𝐹, we get 𝑑𝜁_𝑛,𝜁_𝑘⪯𝑎𝑑𝜁^∘,𝜁_𝑛+𝑏𝑑𝜁_𝑛,𝜁_𝑘+𝑐𝑑𝜁^∘,𝜁_𝑛𝑑𝜁_𝑛,𝜁_𝑘1+𝑑𝜁_𝑛,𝜁^∘, (112) which implies that 𝑑𝜁_𝑛,𝜁_𝑘≤𝑎𝑑𝜁^∘,𝜁_𝑛+𝑏𝑑𝜁_𝑛,𝜁_𝑘𝑑+𝑐^∘𝜁,𝜁_𝑛1+𝑑𝜁_𝑛,𝜁^∘𝑑𝜁_𝑛,𝜁_𝑘. (113) Since |1+𝑑(𝜁_𝑛,𝜁^∘)|>|𝑑(𝜁_𝑛,𝜁^∘)|, so we have 𝑑𝜁_𝑛,𝜁_𝑘≤𝑎𝑑𝜁^∘,𝜁_𝑛+𝑏𝑑𝜁_𝑛,𝜁_𝑘+𝑐𝑑𝜁_𝑛,𝜁_𝑘,𝑑𝜁_𝑛,𝜁_𝑘≤𝑎𝑑𝜁1−𝑏−𝑐^𝑛,𝜁^∘. (114) Note that 𝑑𝜁^∘,𝜁_𝑘≤𝑑𝜁^∘,𝜁_𝑛+𝑑𝜁_𝑛,𝜁_𝑘≤𝑑𝜁^∘,𝜁_𝑛+𝑎𝑑𝜁1−𝑏−𝑐^𝑛,𝜁^∘⟶0 (115) for all 𝑛≥𝑛₀. Thus 𝜁_𝑘→𝜁^∘∈𝐹(𝑡^∘,𝜁^∘) and hence 𝜁^∘∈𝑈, which implies that (𝑡^∘,𝜁^∘)∈𝒬. Thus, (𝑡,𝜁)≾(𝑡^∘,𝜁^∘) for all (𝑡,𝜁)∈ℳ; that is, (𝑡^∘,𝜁^∘) is an upper bound of ℳ. Hence, by Zorn's Lemma, 𝒬 has the maximal element (𝑡^∘,𝜁^∘).
Now, we claim that 𝑡^∘=1. Suppose that 𝑡^∘≤1. Choose 0≺𝑟∈ℂ and 𝑡^∘≤𝑡 such that 𝐵(𝜁^∘,𝑟)⊂𝑈, where 𝑟=(2/(1−𝑘))(𝜑(𝑡)−𝜑(𝑡^∘)). Using (c), we have 𝜑(𝑡)−𝜑𝑡^∘∈𝑠𝐹(𝑡,𝜁),𝐹𝑡^∘,𝜁^∘,𝜑(𝑡)−𝜑𝑡^∘∈𝑠𝜁^∘,𝐹(𝑡,𝜁) (116) for all 𝜁^∘∈𝐹(𝑡^∘,𝜁^∘). So there exists 𝜁∈𝐹(𝑡,𝜁) such that 𝜑(𝑡)−𝜑(𝑡^∘)∈𝑠(𝑑(𝜁^∘,𝜁)) and so 𝑑𝜁,𝜁^∘⪯𝜑(𝑡)−𝜑𝑡^∘⪯(1−𝑘)𝑟2≺(1−𝑘)𝑟, (117) which implies that 𝑑𝜁,𝜁^∘≤(1−𝑘)|𝑟|. (118) Also, by using (b), we conclude that the mapping 𝐹(𝑡,⋅)∶𝐵(𝜁^∘,𝑟)→𝐶𝐵(𝑋) satisfies all the assumptions of Corollary 22 for all 𝑡∈[0,1]. Hence, for all 𝑡∈[0,1], there exists 𝜁∈𝐵(𝜁^∘,𝑟) such that 𝜁∈𝐹(𝑡,𝜁). Thus, (𝜁,𝑡)∈𝒬.
Consider 𝑑𝜁,𝜁^∘2≺𝑟=𝜑(𝑡)1−𝑘−𝜑𝑡^∘, (119) which implies that (𝑡^∘,𝜁^∘)≾(𝑡,𝜁), which is a contradiction. Thus, we have 𝑡^∘=1. Therefore, 𝐹(⋅,1) has a fixed point.
Conversely, if 𝐹(1,⋅) has a fixed point, then, in the same way, we can prove that 𝐹(0,⋅) has a fixed point. This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the Faculty of Science, Naresuan University, Phitsanulok, Thailand, for the financial support.