Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 854965, 12 pages http://dx.doi.org/10.1155/2013/854965
Research Article Common Fixed Points for Multivalued Mappings in Complex Valued Metric Spaces with Applications Jamshaid Ahmad,1 Chakkrid Klin-Eam,2 and Akbar Azam1 1 2
Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad 44000, Pakistan Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Correspondence should be addressed to Chakkrid Klin-Eam;
[email protected] Received 31 July 2013; Revised 21 November 2013; Accepted 24 November 2013 Academic Editor: Irena Rach˚unkov´a Copyright © 2013 Jamshaid Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 ≾ 𝑧2
iff Re (𝑧1 ) ⩽ Re (𝑧2 ) ,
Im (𝑧1 ) ⩽ Im (𝑧2 ) .
(1)
It follows that 𝑧1 ≾ 𝑧2
(2)
2
Abstract and Applied Analysis
3. Main Results
if one of the following conditions is satisfied: (i)
Re (𝑧1 ) = Re (𝑧2 ) ,
Im (𝑧1 ) < Im (𝑧2 ) ,
(ii)
Re (𝑧1 ) < Re (𝑧2 ) ,
Im (𝑧1 ) = Im (𝑧2 ) ,
(iii)
Re (𝑧1 ) < Re (𝑧2 ) ,
Im (𝑧1 ) < Im (𝑧2 ) ,
(iv)
Re (𝑧1 ) = Re (𝑧2 ) ,
Im (𝑧1 ) = Im (𝑧2 ) .
(3)
In particular, we will write 𝑧1 ⋨𝑧2 if 𝑧1 ≠ 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 , (4) 𝑧1 ⪯ 𝑧2 , 𝑧2 ≺ 𝑧3 ⇒ 𝑧1 ≺ 𝑧3 . 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; 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 (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}
𝑠 (𝐴, 𝐵) = ( ⋂ 𝑠 (𝑎, 𝐵)) ⋂ (⋂ 𝑠 (𝑏, 𝐴)) . 𝑎∈𝐴
(7)
is a subbasis for a Hausdorff topology 𝜏 on 𝑋. Let {𝑥𝑛 } 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(𝑥𝑛 , 𝑥) ≺ 𝑐, then {𝑥𝑛 } is said to be convergent, {𝑥𝑛 } converges to 𝑥, and 𝑥 is the limit point of {𝑥𝑛 }. We denote this by lim𝑛 → ∞ 𝑥𝑛 = 𝑥, or 𝑥𝑛 → 𝑥, as 𝑛 → ∞. If for every 𝑐 ∈ C with 0 ≺ 𝑐 there is 𝑛0 ∈ N such that for all 𝑛 > 𝑛0 , 𝑑(𝑥𝑛 , 𝑥𝑛+𝑚 ) ≺ 𝑐, then {𝑥𝑛 } 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 {𝑥𝑛 } be a sequence in 𝑋. Then {𝑥𝑛 } converges to 𝑥 if and only if |𝑑(𝑥𝑛 , 𝑥)| → 0 as 𝑛 → ∞. Lemma 3 (see [9]). Let (𝑋, 𝑑) be a complex valued metric space and let {𝑥𝑛 } be a sequence in 𝑋. Then {𝑥𝑛 } is a Cauchy sequence if and only if |𝑑(𝑥𝑛 , 𝑥𝑛+𝑚 )| → 0 as 𝑛 → ∞.
(8)
𝑏∈𝐵
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) = inf 𝑠(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 𝑊𝑥 (𝐴) = {𝑑 (𝑥, 𝑎) : 𝑎 ∈ 𝐴} .
(3) 𝑑(𝑥, 𝑦) ≾ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦), for all 𝑥, 𝑦, 𝑧 ∈ 𝑋.
𝐵 (𝑥, 𝑟) = {𝑦 ∈ 𝑋 : 𝑑 (𝑥, 𝑦) ≺ 𝑟} ⊆ 𝐴.
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∈𝐵 𝑠(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
(9)
Thus, for 𝑥, 𝑦 ∈ 𝑋 𝑊𝑥 (𝑇𝑦) = {𝑑 (𝑥, 𝑢) : 𝑢 ∈ 𝑇𝑦} .
(10)
Definition 5. Let (𝑋, 𝑑) be a complex 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 a complex valued metric space. A multivalued mapping 𝐹 : 𝑋 → 2C is called bounded from below if for 𝑥 ∈ 𝑋 there exists 𝑧𝑥 ∈ C such that 𝑧𝑥 ⪯ 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 𝐹𝑥 : 𝑋 → 2C defined by 𝐹𝑥 (𝑇𝑦) = 𝑊𝑥 (𝑇𝑦)
(12)
is bounded from below. That is, for 𝑥, 𝑦 ∈ 𝑋 there exists an element 𝑙𝑥 (𝑇𝑦) ∈ C such that 𝑙𝑥 (𝑇𝑦) ⪯ 𝑢
(13)
for all 𝑢 ∈ 𝑊𝑥 (𝑇𝑦), where 𝑙𝑥 (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 𝑊𝑥 (𝑇𝑦) exists in C for all 𝑥, 𝑦 ∈ 𝑋. We denote 𝑑(𝑥, 𝑇𝑦) by the g.l.b of 𝑊𝑥 (𝑇𝑦). That is, 𝑑 (𝑥, 𝑇𝑦) = inf {𝑑 (𝑥, 𝑢) : 𝑢 ∈ 𝑇𝑦} .
(14)
Abstract and Applied Analysis
3
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 𝜆𝑑 (𝑥, 𝑦) +
𝜇𝑑 (𝑥, 𝑆𝑥) 𝑑 (𝑦, 𝑇𝑦) + 𝛾𝑑 (𝑦, 𝑆𝑥) 𝑑 (𝑥, 𝑇𝑦) 1 + 𝑑 (𝑥, 𝑦)
So there exists some 𝑥2 ∈ 𝑇𝑥1 such that 𝜆𝑑 (𝑥0 , 𝑥1 ) +
𝜇𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝛾𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) 1 + 𝑑 (𝑥0 , 𝑥1 )
∈ 𝑠 (𝑑 (𝑥1 , 𝑥2 )) .
∈ 𝑠 (𝑆𝑥, 𝑇𝑦)
(19)
(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 𝜆𝑑 (𝑥0 , 𝑥1 ) +
𝜇𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝛾𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) 1 + 𝑑 (𝑥0 , 𝑥1 )
∈ 𝑠 (𝑆𝑥0 , 𝑇𝑥1 ) . (16)
𝑑 (𝑥1 , 𝑥2 ) ⪯ 𝜆𝑑 (𝑥0 , 𝑥1 ) +
𝜇𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝛾𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) . 1 + 𝑑 (𝑥0 , 𝑥1 ) (20)
By using the greatest lower bound property (g.l.b property) of 𝑆 and 𝑇, we get 𝑑 (𝑥1 , 𝑥2 ) ⪯ 𝜆𝑑 (𝑥0 , 𝑥1 )
This implies that 𝜆𝑑 (𝑥0 , 𝑥1 ) +
Therefore,
+
𝜇𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝛾𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) 1 + 𝑑 (𝑥0 , 𝑥1 )
𝜇𝑑 (𝑥0 , 𝑥1 ) 𝑑 (𝑥1 , 𝑥2 ) + 𝛾𝑑 (𝑥1 , 𝑥1 ) 𝑑 (𝑥0 , 𝑥2 ) 1 + 𝑑 (𝑥0 , 𝑥1 )
= 𝜆𝑑 (𝑥0 , 𝑥1 ) +
𝜇𝑑 (𝑥0 , 𝑥1 ) 𝑑 (𝑥1 , 𝑥2 ) , 1 + 𝑑 (𝑥0 , 𝑥1 )
∈ ( ⋂ 𝑠 (𝑥, 𝑇𝑥1 )) , 𝑥∈𝑆𝑥0
which implies that
𝜆𝑑 (𝑥0 , 𝑥1 ) 𝜇𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝛾𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) + 1 + 𝑑 (𝑥0 , 𝑥1 ) ∈ 𝑠 (𝑥, 𝑇𝑥1 )
∀𝑥 ∈ 𝑆𝑥0 . (17)
Since 𝑥1 ∈ 𝑆𝑥0 , we have 𝜆𝑑 (𝑥0 , 𝑥1 ) 𝜇𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝛾𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) + 1 + 𝑑 (𝑥0 , 𝑥1 ) ∈ 𝑠 (𝑥1 , 𝑇𝑥1 ) ,
𝜇 𝑑 (𝑥0 , 𝑥1 ) 𝑑 (𝑥1 , 𝑥2 ) 𝑑 (𝑥1 , 𝑥2 ) ≤ 𝜆 𝑑 (𝑥0 , 𝑥1 ) + 1 + 𝑑 (𝑥0 , 𝑥1 ) 𝑑 (𝑥0 , 𝑥1 ) = 𝜆 𝑑 (𝑥0 , 𝑥1 ) + 𝜇 𝑑 (𝑥1 , 𝑥2 ) , 1 + 𝑑 (𝑥0 , 𝑥1 ) 𝑑 (𝑥1 , 𝑥2 ) ≤ 𝜆 𝑑 (𝑥0 , 𝑥1 ) + 𝜇 𝑑 (𝑥1 , 𝑥2 ) , (1 − 𝜇) 𝑑 (𝑥1 , 𝑥2 ) ≤ 𝜆 𝑑 (𝑥0 , 𝑥1 ) , 𝜆 𝑑 (𝑥1 , 𝑥2 ) ≤ 𝑑 (𝑥0 , 𝑥1 ) (1 − 𝜇) = ℎ 𝑑 (𝑥0 , 𝑥1 ) ,
where ℎ =
𝜆𝑑 (𝑥0 , 𝑥1 ) +
(21)
𝜇𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝛾𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) 1 + 𝑑 (𝑥0 , 𝑥1 )
∈ 𝑠 (𝑥1 , 𝑇𝑥1 ) = ⋃ 𝑠 (𝑑 (𝑥1 , 𝑥)) . 𝑥∈𝑇𝑥1
(18)
𝜆 < 1. (1 − 𝜇) (22)
Inductively, we can construct a sequence {𝑥𝑛 } in 𝑋 such that for 𝑛 = 0, 1, 2, . . . 𝑛 𝑑 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ ℎ 𝑑 (𝑥0 , 𝑥1 )
(23)
with ℎ = 𝜆/(1 − 𝜇) < 1, for 𝑥2𝑛+1 ∈ 𝑆𝑥2𝑛 and 𝑥2𝑛+2 ∈ 𝑇𝑥2𝑛+1 .
4
Abstract and Applied Analysis Now for 𝑚 > 𝑛, we get 𝑑 (𝑥𝑛 , 𝑥𝑚 ) ≤ 𝑑 (𝑥𝑛 , 𝑥𝑛+1 ) + 𝑑 (𝑥𝑛+1 , 𝑥𝑛+2 ) + ⋅ ⋅ ⋅ + 𝑑 (𝑥𝑚−1 , 𝑥𝑚 ) ≤ [ℎ𝑛 + ℎ𝑛+1 + ⋅ ⋅ ⋅ + ℎ𝑚−1 ] 𝑑 (𝑥0 , 𝑥1 ) ≤ [
By definition, 𝜆𝑑 (𝑥2𝑘 , 𝑢) 𝜇𝑑 (𝑥2𝑘 , 𝑆𝑥2𝑘 ) 𝑑 (𝑢, 𝑇𝑢) + 𝛾𝑑 (𝑢, 𝑆𝑥2𝑘 ) 𝑑 (𝑥2𝑘 , 𝑇𝑢) 1 + 𝑑 (𝑥2𝑘 , 𝑢)
+ (24)
𝑛
ℎ ] 𝑑 (𝑥0 , 𝑥1 ) . 1−ℎ
∈ 𝑠 (𝑥2𝑘+1 , 𝑇𝑢) = ⋃ 𝑠 (𝑑 (𝑥2𝑘+1 , 𝑢 )) . 𝑢 ∈𝑇𝑢
(29) There exists some 𝑢𝑘 ∈ 𝑇𝑢 such that
Therefore,
𝜆𝑑 (𝑥2𝑘 , 𝑢)
𝑛
ℎ 𝑑 (𝑥𝑛 , 𝑥𝑚 ) ≤ 𝑑 (𝑥0 , 𝑥1 ) → 0 as 𝑚, 𝑛 → ∞. 1−ℎ (25) This implies that {𝑥𝑛 } is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑢 ∈ 𝑋 such that 𝑥𝑛 → 𝑢 as 𝑛 → ∞. We now show that 𝑢 ∈ 𝑇𝑢 and 𝑢 ∈ 𝑆𝑢. From (15), we have
𝜇𝑑 (𝑥2𝑘 , 𝑆𝑥2𝑘 ) 𝑑 (𝑢, 𝑇𝑢) + 𝛾𝑑 (𝑢, 𝑆𝑥2𝑘 ) 𝑑 (𝑥2𝑘 , 𝑇𝑢) 1 + 𝑑 (𝑥2𝑘 , 𝑢)
+
∈ 𝑠 (𝑑 (𝑥2𝑘+1 , 𝑢𝑘 )) ; (30) that is, 𝑑 (𝑥2𝑘+1 , 𝑢𝑘 )
𝜆𝑑 (𝑥2𝑘 , 𝑢) 𝜇𝑑 (𝑥2𝑘 , 𝑆𝑥2𝑘 ) 𝑑 (𝑢, 𝑇𝑢) + 𝛾𝑑 (𝑢, 𝑆𝑥2𝑘 ) 𝑑 (𝑥2𝑘 , 𝑇𝑢) + 1 + 𝑑 (𝑥2𝑘 , 𝑢)
⪯ 𝜆𝑑 (𝑥2𝑘 , 𝑢) +
∈ 𝑠 (𝑆𝑥2𝑘 , 𝑇𝑢) . (26)
𝜇𝑑 (𝑥2𝑘 , 𝑆𝑥2𝑘 ) 𝑑 (𝑢, 𝑇𝑢) + 𝛾𝑑 (𝑢, 𝑆𝑥2𝑘 ) 𝑑 (𝑥2𝑘 , 𝑇𝑢) . 1 + 𝑑 (𝑥2𝑘 , 𝑢) (31)
By using the greatest lower bound property (g.l.b. property) of 𝑆 and 𝑇, we have
This implies that
𝑑 (𝑥2𝑘+1 , 𝑢𝑘 )
𝜆𝑑 (𝑥2𝑘 , 𝑢)
⪯ 𝜆𝑑 (𝑥2𝑘 , 𝑢) + (𝜇𝑑 (𝑥2𝑘 , 𝑥2𝑘+1 ) 𝑑 (𝑢, 𝑢𝑘 )
𝜇𝑑 (𝑥2𝑘 , 𝑆𝑥2𝑘 ) 𝑑 (𝑢, 𝑇𝑢) + 𝛾𝑑 (𝑢, 𝑆𝑥2𝑘 ) 𝑑 (𝑥2𝑘 , 𝑇𝑢) + 1 + 𝑑 (𝑥2𝑘 , 𝑢)
(32) −1
+ 𝛾𝑑 (𝑢, 𝑥2𝑘+1 ) 𝑑 (𝑥2𝑘 , 𝑢𝑘 )) (1 + 𝑑 (𝑥2𝑘 , 𝑢)) . Since
∈ ( ⋂ 𝑠 (𝑥, 𝑇𝑢)) ,
𝑑 (𝑢, 𝑢𝑘 ) ⪯ 𝑑 (𝑢, 𝑥2𝑘+1 ) + 𝑑 (𝑥2𝑘+1 , 𝑢𝑘 ) ,
𝑥∈𝑆𝑥2𝑘
using (32), we get
𝜆𝑑 (𝑥2𝑘 , 𝑢) +
𝜇𝑑 (𝑥2𝑘 , 𝑆𝑥2𝑘 ) 𝑑 (𝑢, 𝑇𝑢) + 𝛾𝑑 (𝑢, 𝑆𝑥2𝑘 ) 𝑑 (𝑥2𝑘 , 𝑇𝑢) 1 + 𝑑 (𝑥2𝑘 , 𝑢)
∈ 𝑠 (𝑥, 𝑇𝑢)
∀𝑥 ∈ 𝑆𝑥2𝑘 . (27)
Since 𝑥2𝑘+1 ∈ 𝑆𝑥2𝑘 , we have 𝜆𝑑 (𝑥2𝑘 , 𝑢) +
(33)
𝜇𝑑 (𝑥2𝑘 , 𝑆𝑥2𝑘 ) 𝑑 (𝑢, 𝑇𝑢) + 𝛾𝑑 (𝑢, 𝑆𝑥2𝑘 ) 𝑑 (𝑥2𝑘 , 𝑇𝑢) 1 + 𝑑 (𝑥2𝑘 , 𝑢)
∈ 𝑠 (𝑥2𝑘+1 , 𝑇𝑢) . (28)
𝑑 (𝑢, 𝑢𝑘 ) ⪯ 𝑑 (𝑢, 𝑥2𝑘+1 ) + 𝜆𝑑 (𝑢, 𝑥2𝑘+1 ) 𝜇𝑑 (𝑥2𝑘 , 𝑥2𝑘+1 ) 𝑑 (𝑢, 𝑢𝑘 ) + 𝛾𝑑 (𝑢, 𝑥2𝑘+1 ) 𝑑 (𝑥2𝑘 , 𝑢𝑘 ) , 1 + 𝑑 (𝑥2𝑘 , 𝑢) 𝑑 (𝑢, 𝑢𝑘 ) ≤ 𝑑 (𝑢, 𝑥2𝑘+1 ) + 𝜆 𝑑 (𝑢, 𝑥2𝑘+1 ) +(𝜇 𝑑 (𝑥2𝑘 , 𝑥2𝑘+1 ) 𝑑 (𝑢, 𝑢𝑘 ) + 𝛾 𝑑 (𝑢, 𝑥2𝑘+1 ) 𝑑 (𝑥2𝑘 , 𝑢𝑘 )) −1 × 1 + 𝑑 (𝑥2𝑘 , 𝑢) . (34) +
Abstract and Applied Analysis
5
Taking the limit as 𝑘 → ∞, we get |𝑑(𝑢, 𝑢𝑘 )| → 0 as 𝑘 → ∞. By [9, lemma 2], we have 𝑢𝑘 → 𝑢 as 𝑘 → ∞. Since 𝑇𝑢 is closed, 𝑢 ∈ 𝑇𝑢. Similarly, it follows that 𝑢 ∈ 𝑆𝑢. Thus, 𝑆 and 𝑇 have a common fixed point.
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 𝑑 (𝑥, 𝑦) = 𝑦 − 𝑥 𝑒𝑖𝜃 ,
By setting 𝛾 = 0 in Theorem 9, we get the following corollary.
𝑥 𝑑 (𝑥, 𝑆𝑥) = 𝑥 − 𝑒𝑖𝜃 , 5 𝑦 𝑑 (𝑦, 𝑇𝑦) = 𝑦 − 𝑒𝑖𝜃 , 10 𝑥 𝑑 (𝑦, 𝑆𝑥) = 𝑦 − 𝑒𝑖𝜃 , 5 𝑦 𝑑 (𝑥, 𝑇𝑦) = 𝑥 − 𝑒𝑖𝜃 , 10 𝑥 𝑦 𝑖𝜃 𝑠 (𝑆𝑥, 𝑇𝑦) = 𝑠 ( − 𝑒 ) . 5 10
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 that 𝜆𝑑 (𝑥, 𝑦) + 𝜇
𝑑 (𝑥, 𝑆𝑥) 𝑑 (𝑦, 𝑇𝑦) ∈ 𝑠 (𝑆𝑥, 𝑇𝑦) 1 + 𝑑 (𝑥, 𝑦)
(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 𝜆𝑑 (𝑥, 𝑦) +
𝜇𝑑 (𝑥, 𝑇𝑥) 𝑑 (𝑦, 𝑇𝑦) + 𝛾𝑑 (𝑦, 𝑇𝑥) 𝑑 (𝑥, 𝑇𝑦) 1 + 𝑑 (𝑥, 𝑦)
∈ 𝑠 (𝑇𝑥, 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.
Consider, 𝜇 |𝑑 (𝑥, 𝑆𝑥)| 𝑑 (𝑦, 𝑇𝑦) + 𝛾 𝑑 (𝑦, 𝑆𝑥) 𝑑 (𝑥, 𝑇𝑦) 𝜆 𝑑 (𝑥, 𝑦) + 1 + 𝑑 (𝑥, 𝑦) = 𝜆 𝑦 − 𝑥 𝜇 |𝑥 − (𝑥/5)| 𝑦 − (𝑦/10) + 𝛾 𝑦 − (𝑥/5) 𝑥 − (𝑦/10) + 1 + 𝑑 (𝑥, 𝑦) = 𝜆 𝑦 − 𝑥 𝜇 |𝑥 − (𝑥/5)| 𝑦 − (𝑦/10) + 𝛾 𝑦 − (𝑥/5) 𝑥 − (𝑦/10) + . 1 + 𝑑 (𝑥, 𝑦)
Corollary 12. Let (𝑋, 𝑑) be a complete metric space and let 𝑆, 𝑇 : 𝑋 → 𝐶𝐵(𝑋) be multivalued mappings such that
𝜇𝑑 (𝑥, 𝑆𝑥) 𝑑 (𝑦, 𝑇𝑦) + 𝛾𝑑 (𝑦, 𝑆𝑥) 𝑑 (𝑥, 𝑇𝑦) 1 + 𝑑 (𝑥, 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 𝑦 𝑑 (𝑥, 𝑦) = 𝑥 − 𝑦 𝑒𝑖𝜃 , 𝜃 = tan−1 . (38) 𝑥 Then (𝑋, 𝑑) is a complex valued metric space. Consider the mappings 𝑆, 𝑇 : 𝑋 → 𝐶𝐵(𝑋) defined by 1 𝑆𝑥 = [0, 𝑥] , 5
(41) Clearly, for any value of 𝜇 and 𝛾 and 𝜆 = 1/5, we have
𝐻 (𝑆𝑥, 𝑇𝑦) ≤ 𝜆𝑑 (𝑥, 𝑦) +
(40)
𝑇𝑥 = [0,
1 𝑥] . 10
(39)
𝑥 𝑦 − 5 10 ≤
1 𝑦 − 𝑥 5 +
𝜇 |𝑥 − (𝑥/5)| 𝑦 − (𝑦/10) + 𝛾 𝑦 − (𝑥/5) 𝑥 − (𝑦/10) . 1 + 𝑑 (𝑥, 𝑦) (42)
Thus, 𝜆𝑑 (𝑥, 𝑦) +
𝜇𝑑 (𝑥, 𝑆𝑥) 𝑑 (𝑦, 𝑇𝑦) + 𝛾𝑑 (𝑦, 𝑆𝑥) 𝑑 (𝑥, 𝑇𝑦) 1 + 𝑑 (𝑥, 𝑦) (43)
∈ 𝑠 (𝑆𝑥, 𝑇𝑦) . Hence, all the conditions of our main theorem are satisfied and 0 is a common fixed point of 𝑆 and 𝑇.
6
Abstract and Applied Analysis
In the following results, we considered the Kannan type contractive condition involving rational expressions.
By using the greatest lower bound property (g.l.b property) of 𝑆 and 𝑇, we get
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 (𝑥1 , 𝑥2 ) ⪯ 𝑎𝑑 (𝑥0 , 𝑥1 ) + 𝑏𝑑 (𝑥1 , 𝑥2 ) + 𝑐
𝑎𝑑 (𝑥, 𝑆𝑥) + 𝑏𝑑 (𝑦, 𝑇𝑦) + 𝑐
𝑑 (𝑥, 𝑆𝑥) 𝑑 (𝑦, 𝑇𝑦) ∈ 𝑠 (𝑆𝑥, 𝑇𝑦) 1 + 𝑑 (𝑥, 𝑦) (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 𝑎𝑑 (𝑥0 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝑐
𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) 1 + 𝑑 (𝑥0 , 𝑥1 ) (45)
∈ 𝑠 (𝑆𝑥0 , 𝑇𝑥1 ) . This implies that 𝑎𝑑 (𝑥0 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝑐
𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) 1 + 𝑑 (𝑥0 , 𝑥1 )
which implies that 𝑑 (𝑥1 , 𝑥2 ) ≤ 𝑎 𝑑 (𝑥0 , 𝑥1 ) + 𝑏 𝑑 (𝑥1 , 𝑥2 ) 𝑑 (𝑥 , 𝑥 ) 𝑑 (𝑥1 , 𝑥2 ) + 𝑐 0 1 . 1 + 𝑑 (𝑥0 , 𝑥1 )
Since |𝑑(𝑥0 , 𝑥1 )| < |1 + 𝑑(𝑥0 , 𝑥1 )|, we have 𝑑 (𝑥1 , 𝑥2 ) ≤ 𝑎 𝑑 (𝑥0 , 𝑥1 ) + 𝑏 𝑑 (𝑥1 , 𝑥2 ) + 𝑐 𝑑 (𝑥1 , 𝑥2 ) , (1 − 𝑏 − 𝑐) 𝑑 (𝑥1 , 𝑥2 ) ≤ 𝑎 𝑑 (𝑥0 , 𝑥1 ) 𝑎 𝑑 (𝑥1 , 𝑥2 ) ≤ 𝑑 (𝑥0 , 𝑥1 ) . (1 − 𝑏 − 𝑐) (52) Thus,
𝑥∈𝑆𝑥0
𝑎𝑑 (𝑥0 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝑐
(46) Since 𝑥1 ∈ 𝑆𝑥0 , we have 𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) 1 + 𝑑 (𝑥0 , 𝑥1 )
∈ 𝑠 (𝑥1 , 𝑇𝑥1 ) , 𝑎𝑑 (𝑥0 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝑐
= 𝑙 𝑑 (𝑥0 , 𝑥1 ) ,
𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) 1 + 𝑑 (𝑥0 , 𝑥1 )
∀𝑥 ∈ 𝑆𝑥0 .
𝑎𝑑 (𝑥0 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝑐
𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) 1 + 𝑑 (𝑥0 , 𝑥1 )
∈ 𝑠 (𝑥1 , 𝑇𝑥1 ) = ⋃ 𝑠 (𝑑 (𝑥1 , 𝑥)) .
𝑎 < 1. (1 − 𝑏 − 𝑐)
Inductively, we can construct a sequence {𝑥𝑛 } in 𝑋 such that for 𝑛 = 0, 1, 2, . . . 𝑛 (54) 𝑑 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ 𝑙 𝑑 (𝑥0 , 𝑥1 ) with 𝑙 = 𝑎/(1−𝑏−𝑐) < 1, for 𝑥2𝑛+1 ∈ 𝑆𝑥2𝑛 and 𝑥2𝑛+2 ∈ 𝑇𝑥2𝑛+1 . Now for 𝑚 > 𝑛, we get 𝑑 (𝑥𝑛 , 𝑥𝑚 ) ≤ 𝑑 (𝑥𝑛 , 𝑥𝑛+1 ) + 𝑑 (𝑥𝑛+1 , 𝑥𝑛+2 ) + ⋅ ⋅ ⋅ + 𝑑 (𝑥𝑚−1 , 𝑥𝑚 ) (55) ≤ [𝑙𝑛 + 𝑙𝑛+1 + ⋅ ⋅ ⋅ + 𝑙𝑚−1 ] 𝑑 (𝑥0 , 𝑥1 ) ≤ [
(47)
𝑙𝑛 ] 𝑑 (𝑥0 , 𝑥1 ) , 1−𝑙
and so
So there exists some 𝑥2 ∈ 𝑇𝑥1 , such that 𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) 𝑎𝑑 (𝑥0 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥1 , 𝑇𝑥1 ) + 𝑐 1 + 𝑑 (𝑥0 , 𝑥1 ) ∈ 𝑠 (𝑑 (𝑥1 , 𝑥2 )) . (48) That is, 𝑑 (𝑥1 , 𝑥2 ) ⪯ 𝑎𝑑 (𝑥0 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥1 , 𝑇𝑥1 ) 𝑑 (𝑥0 , 𝑆𝑥0 ) 𝑑 (𝑥1 , 𝑇𝑥1 ) . 1 + 𝑑 (𝑥0 , 𝑥1 )
where 𝑙 =
(53)
𝑥∈𝑇𝑥1
+𝑐
(51)
𝑎 𝑑 (𝑥1 , 𝑥2 ) ≤ 𝑑 (𝑥0 , 𝑥1 ) (1 − 𝑏 − 𝑐)
∈ ( ⋂ 𝑠 (𝑥, 𝑇𝑥1 )) ,
∈ 𝑠 (𝑥, 𝑇𝑥1 )
𝑑 (𝑥0 , 𝑥1 ) 𝑑 (𝑥1 , 𝑥2 ) 1 + 𝑑 (𝑥0 , 𝑥1 ) (50)
(49)
𝑙𝑛 𝑑 (𝑥𝑛 , 𝑥𝑚 ) ≤ 𝑑 (𝑥0 , 𝑥1 ) → 0 as 𝑚, 𝑛 → ∞. 1−𝑙 (56) This implies that {𝑥𝑛 } is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝜐 ∈ 𝑋 such that 𝑥𝑛 → 𝜐 as 𝑛 → ∞. We now show that 𝜐 ∈ 𝑇𝜐 and 𝜐 ∈ 𝑆𝜐. So from (44), we get 𝑎𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝜐, 𝑇𝜐) + 𝑐 ∈ 𝑠 (𝑆𝑥2𝑛 , 𝑇𝜐) .
𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) 𝑑 (𝜐, 𝑇𝜐) 1 + 𝑑 (𝑥2𝑛 , 𝜐)
(57)
Abstract and Applied Analysis
7
This implies that
That is,
𝑎𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝜐, 𝑇𝜐) + 𝑐
(1 − 𝑏) 𝑑 (𝜐, 𝜐𝑛 ) ≤ 𝑑 (𝜐, 𝑥2𝑛+1 ) + 𝑎 𝑑 (𝑥2𝑛 , 𝑥2𝑛+1 )
𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) 𝑑 (𝜐, 𝑇𝜐) 1 + 𝑑 (𝑥2𝑛 , 𝜐)
𝑑 (𝑥 , 𝑥 2𝑛 2𝑛+1 ) 𝑑 (𝜐, 𝜐𝑛 ) . + 𝑐 1 + 𝑑 (𝑥2𝑛 , 𝜐)
∈ ( ⋂ 𝑠 (𝑥, 𝑇𝜐)) ,
It follows that
𝑥∈𝑆𝑥2𝑛
𝑎𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝜐, 𝑇𝜐) + 𝑐 ∈ 𝑠 (𝑥, 𝑇𝜐)
𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) 𝑑 (𝜐, 𝑇𝜐) 1 + 𝑑 (𝑥2𝑛 , 𝜐)
∀𝑥 ∈ 𝑆𝑥2𝑛 . (58)
Since 𝑥2𝑛+1 ∈ 𝑆𝑥2𝑛 , we have
1 𝑎 𝑑 (𝜐, 𝜐𝑛 ) ≤ 𝑑 (𝜐, 𝑥2𝑛+1 ) + 𝑑 (𝑥2𝑛 , 𝑥2𝑛+1 ) (1 − 𝑏) (1 − 𝑏) 𝑐 𝑑 (𝑥2𝑛 , 𝑥2𝑛+1 ) 𝑑 (𝜐, 𝜐𝑛 ) . + 1 + 𝑑 (𝑥2𝑛 , 𝜐) (1 − 𝑏) (66) By letting 𝑛 → ∞ in the above inequality, we get 𝑑 (𝜐, 𝜐𝑛 ) → 0 as 𝑛 → ∞.
𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) 𝑑 (𝜐, 𝑇𝜐) 𝑎𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝜐, 𝑇𝜐) + 𝑐 1 + 𝑑 (𝑥2𝑛 , 𝜐) ∈ 𝑠 (𝑥2𝑛+1 , 𝑇𝜐) . (59) By definition,
By [9, Lemma 2], we have 𝜐𝑛 → 𝜐 as 𝑛 → ∞. Since 𝑇𝜐 is closed, 𝜐 ∈ 𝑇𝜐. Similarly, it follows that 𝜐 ∈ 𝑆𝜐. Thus, 𝑆 and 𝑇 have a common fixed point.
𝛼 (𝑑 (𝑥, 𝑆𝑥) + 𝑑 (𝑦, 𝑇𝑦)) ∈ 𝑠 (𝑆𝑥, 𝑇𝑦)
∈ 𝑠 (𝑥2𝑛+1 , 𝑇𝜐) = ⋃ 𝑠 (𝑑 (𝑥2𝑛+1 , 𝜐 )) . 𝜐 ∈𝑇𝜐
(60) There exists some 𝜐𝑛 ∈ 𝑇𝜐 such that
∈ 𝑠 (𝑑 (𝑥2𝑛+1 , 𝜐𝑛 )) ; (61) that is,
for all 𝑥, 𝑦 ∈ 𝑋 and 0 ≤ 𝛼 < 1/2. Then 𝑆 and 𝑇 have a common fixed point.
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 𝑎𝑑 (𝑥, 𝑇𝑥) + 𝑏𝑑 (𝑦, 𝑇𝑦) + 𝑐
𝑑 (𝑥2𝑛+1 , 𝜐𝑛 ) ⪯ 𝑎𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝜐, 𝑇𝜐) (62)
By using the greatest lower bound property (g.l.b property) of 𝑆 and 𝑇, we get 𝑑 (𝑥2𝑛+1 , 𝜐𝑛 ) ⪯ 𝑎𝑑 (𝑥2𝑛 , 𝑥2𝑛+1 ) + 𝑏𝑑 (𝜐, 𝜐𝑛 ) 𝑑 (𝑥2𝑛 , 𝑥2𝑛+1 ) 𝑑 (𝜐, 𝜐𝑛 ) . +𝑐 1 + 𝑑 (𝑥2𝑛 , 𝜐)
(63)
Now by using the triangular inequality, we get
+ 𝑏𝑑 (𝜐, 𝜐𝑛 ) + 𝑐
𝑑 (𝑥2𝑛 , 𝑥2𝑛+1 ) 𝑑 (𝜐, 𝜐𝑛 ) . 1 + 𝑑 (𝑥2𝑛 , 𝜐)
𝑑 (𝑥, 𝑇𝑥) 𝑑 (𝑦, 𝑇𝑦) ∈ 𝑠 (𝑇𝑥, 𝑇𝑦) 1 + 𝑑 (𝑥, 𝑦) (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 (𝑆𝑥, 𝑇𝑦) ≤ 𝑎𝑑 (𝑥, 𝑆𝑥) + 𝑏𝑑 (𝑦, 𝑇𝑦) + 𝑐
𝑑 (𝑥, 𝑆𝑥) 𝑑 (𝑦, 𝑇𝑦) 1 + 𝑑 (𝑥, 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.
𝑑 (𝜐, 𝜐𝑛 ) ⪯ 𝑑 (𝜐, 𝑥2𝑛+1 ) + 𝑑 (𝑥2𝑛+1 , 𝜐𝑛 ) ⪯ 𝑑 (𝜐, 𝑥2𝑛+1 ) + 𝑎𝑑 (𝑥2𝑛 , 𝑥2𝑛+1 )
(68)
By setting 𝑆 = 𝑇 in Theorem 15, we get the following corollary.
𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) 𝑑 (𝜐, 𝑇𝜐) 𝑎𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝜐, 𝑇𝜐) + 𝑐 1 + 𝑑 (𝑥2𝑛 , 𝜐)
𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) 𝑑 (𝜐, 𝑇𝜐) . 1 + 𝑑 (𝑥2𝑛 , 𝜐)
(67)
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
𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) 𝑑 (𝜐, 𝑇𝜐) 𝑎𝑑 (𝑥2𝑛 , 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝜐, 𝑇𝜐) + 𝑐 1 + 𝑑 (𝑥2𝑛 , 𝜐)
+𝑐
(65)
(64)
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.
8
Abstract and Applied Analysis
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 𝑎𝑑 (𝑦, 𝑆𝑥) + 𝑏𝑑 (𝑥, 𝑇𝑦) + 𝑐 [
𝑑 (𝑦, 𝑆𝑥) 𝑑 (𝑥, 𝑇𝑦) ] ∈ 𝑠 (𝑆𝑥, 𝑇𝑦) 1 + 𝑑 (𝑥, 𝑦) (71)
Hence, we have 𝑑 (𝑥1 , 𝑥2 ) ⪯ 𝑏𝑑 (𝑥0 , 𝑥2 )
(79)
𝑑 (𝑥1 , 𝑥2 ) ≤ 𝑘 𝑑 (𝑥0 , 𝑥1 ) ,
(80)
which implies that
where 𝑘 = 𝑏/(1 − 𝑏) < 1. From (73), we get 𝑑 (𝑥1 , 𝑥2 ) ≤ 𝑘 (1 − 𝑘) |𝑟| .
(81)
Note that
for all 𝑥, 𝑦 ∈ 𝐵(𝑥0 , 𝑟) and (1 − 𝑘) 𝑟 ∈ 𝑠 (𝑥0 , 𝑆𝑥0 ) ,
(72)
𝑑 (𝑥0 , 𝑥2 ) ≤ 𝑑 (𝑥0 , 𝑥1 ) + 𝑑 (𝑥1 , 𝑥2 ) ≤ (1 − 𝑘) |𝑟| + 𝑘 (1 − 𝑘) |𝑟|
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 𝑑 (𝑥0 , 𝑥1 ) ≤ (1 − 𝑘) |𝑟| .
(73)
𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) ] 1 + 𝑑 (𝑥0 , 𝑥1 )
∈ 𝑠 (𝑆𝑥0 , 𝑇𝑥1 ) .
(82)
≤ (1 − 𝑘2 ) |𝑟| . Thus, we have 𝑥2 ∈ 𝐵(𝑥0 , 𝑟). From (71), we get 𝑎𝑑 (𝑥2 , 𝑇𝑥1 ) + 𝑏𝑑 (𝑥1 , 𝑆𝑥2 ) + [𝑐
Thus, we have 𝑥1 ∈ 𝐵(𝑥0 , 𝑟). From (71), we get 𝑎𝑑 (𝑥1 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥0 , 𝑇𝑥1 ) + 𝑐 [
= (1 − 𝑘) (1 + 𝑘) |𝑟|
𝑑 (𝑥2 , 𝑇𝑥1 ) 𝑑 (𝑥1 , 𝑆𝑥2 ) ] 1 + 𝑑 (𝑥1 , 𝑥2 )
∈ 𝑠 (𝑇𝑥1 , 𝑆𝑥2 ) . (83) This implies that
(74) 𝑎𝑑 (𝑥2 , 𝑇𝑥1 ) + 𝑏𝑑 (𝑥1 , 𝑆𝑥2 ) + [𝑐
This implies that 𝑎𝑑 (𝑥1 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥0 , 𝑇𝑥1 ) + 𝑐 [
𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) ] 1 + 𝑑 (𝑥0 , 𝑥1 )
𝑑 (𝑥2 , 𝑇𝑥1 ) 𝑑 (𝑥1 , 𝑆𝑥2 ) ] 1 + 𝑑 (𝑥1 , 𝑥2 )
∈ 𝑠 (𝑥2 , 𝑆𝑥2 ) . (84) By the definition, there exists 𝑥3 ∈ 𝑆𝑥2 such that
∈ 𝑠 (𝑥1 , 𝑇𝑥1 ) . (75) By the definition, we can take 𝑥2 ∈ 𝑇𝑥1 such that 𝑎𝑑 (𝑥1 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥0 , 𝑇𝑥1 ) + 𝑐 [
𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) ] 1 + 𝑑 (𝑥0 , 𝑥1 )
𝑎𝑑 (𝑥2 , 𝑇𝑥1 ) + 𝑏𝑑 (𝑥1 , 𝑆𝑥2 ) + [𝑐
𝑑 (𝑥2 , 𝑇𝑥1 ) 𝑑 (𝑥1 , 𝑆𝑥2 ) ] 1 + 𝑑 (𝑥1 , 𝑥2 )
∈ 𝑠 (𝑑 (𝑥2 , 𝑥3 )) . (85) By the definition, we get
∈ 𝑠 (𝑑 (𝑥1 , 𝑥2 )) . (76)
𝑑 (𝑥2 , 𝑥3 ) ⪯ 𝑎𝑑 (𝑥2 , 𝑇𝑥1 ) + 𝑏𝑑 (𝑥1 , 𝑆𝑥2 )
By the definition, we get
+ [𝑐
𝑑 (𝑥1 , 𝑥2 ) ⪯ 𝑎𝑑 (𝑥1 , 𝑆𝑥0 ) + 𝑏𝑑 (𝑥0 , 𝑇𝑥1 ) + 𝑐[
𝑑 (𝑥1 , 𝑆𝑥0 ) 𝑑 (𝑥0 , 𝑇𝑥1 ) ]. 1 + 𝑑 (𝑥0 , 𝑥1 )
(77)
𝑑 (𝑥2 , 𝑥3 ) ⪯ 𝑎𝑑 (𝑥2 , 𝑥2 ) + 𝑏𝑑 (𝑥1 , 𝑥3 ) + [𝑐
𝑑 (𝑥1 , 𝑥2 ) ⪯ 𝑎𝑑 (𝑥1 , 𝑥1 ) + 𝑏𝑑 (𝑥0 , 𝑥2 ) 𝑑 (𝑥1 , 𝑥1 ) 𝑑 (𝑥0 , 𝑥2 ) ]. 1 + 𝑑 (𝑥0 , 𝑥1 )
(86)
By using the g.l.b property of 𝑆 and 𝑇, we get
By using the g.l.b property of 𝑆 and 𝑇, we get
+ 𝑐[
𝑑 (𝑥2 , 𝑇𝑥1 ) 𝑑 (𝑥1 , 𝑆𝑥2 ) ]. 1 + 𝑑 (𝑥1 , 𝑥2 )
(78)
𝑑 (𝑥2 , 𝑥2 ) 𝑑 (𝑥1 , 𝑥3 ) ]. 1 + 𝑑 (𝑥1 , 𝑥2 )
(87)
Hence, we have 𝑑 (𝑥2 , 𝑥3 ) ⪯ 𝑏𝑑 (𝑥1 , 𝑥3 )
(88)
Abstract and Applied Analysis
9 By using the g.l.b property of 𝑆 and 𝑇, we get
which implies that 𝑑 (𝑥2 , 𝑥3 ) ≤ 𝑘 𝑑 (𝑥1 , 𝑥2 ) ,
(89)
𝑑 (𝑥2𝑛+1 , 𝜐𝑛 ) ⪯ 𝑎𝑑 (𝜐, 𝑥2𝑛+1 ) + 𝑏𝑑 (𝑥2𝑛 , 𝜐𝑛 )
where 𝑘 = 𝑏/(1 − 𝑏) < 1. Consider that
+ 𝑐[
𝑑 (𝑥0 , 𝑥3 ) ≤ 𝑑 (𝑥0 , 𝑥1 ) + 𝑑 (𝑥1 , 𝑥2 ) + 𝑑 (𝑥2 , 𝑥3 )
0 ≺ 𝑑 (𝑥2𝑛+1 , 𝜐𝑛 ) ⪯ 𝑎𝑑 (𝜐, 𝑥2𝑛+1 ) + 𝑏𝑑 (𝑥2𝑛 , 𝜐𝑛 )
= (1 − 𝑘3 ) |𝑟| ≤ |𝑟| . (90) So 𝑥3 ∈ 𝐵(𝑥0 , 𝑟). Continuing in this way, we can construct a sequence {𝑥𝑛 } in 𝐵(𝑥0 , 𝑟) such that for each 𝑛 ≥ 1, (91)
(92)
Thus, as in the proof of Theorem 15, {𝑥𝑛 } 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 𝑥𝑛 → 𝑢 as 𝑛 → ∞. Now, we show that 𝜐 ∈ 𝑇𝜐 and 𝜐 ∈ 𝑆𝜐. From (71), we get 𝑎𝑑 (𝜐, 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝑥2𝑛 , 𝑇𝜐) + 𝑐 [
𝑑 (𝜐, 𝑆𝑥2𝑛 ) 𝑑 (𝑥2𝑛 , 𝑇𝜐) ] 1 + 𝑑 (𝑥2𝑛 , 𝜐)
∈ 𝑠 (𝑆𝑥2𝑛 , 𝑇𝜐) . (93) This implies that 𝑎𝑑 (𝜐, 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝑥2𝑛 , 𝑇𝜐) + 𝑐 [
𝑑 (𝜐, 𝑆𝑥2𝑛 ) 𝑑 (𝑥2𝑛 , 𝑇𝜐) ] 1 + 𝑑 (𝑥2𝑛 , 𝜐)
∈ 𝑠 (𝑥2𝑛+1 , 𝑇𝜐) . (94) By the definition, there exists 𝜐𝑛 ∈ 𝑇𝜐 such that 𝑎𝑑 (𝜐, 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝑥2𝑛 , 𝑇𝜐) + 𝑐 [
𝑑 (𝜐, 𝑥2𝑛+1 ) 𝑑 (𝑥2𝑛 , 𝜐𝑛 ) ]. 1 + 𝑑 (𝑥2𝑛 , 𝜐)
(98)
Now, by using the triangular inequality, we get
⪯ 𝑑 (𝜐, 𝑥2𝑛+1 ) + 𝑎𝑑 (𝜐, 𝑥2𝑛+1 ) + 𝑏𝑑 (𝑥2𝑛 , 𝜐𝑛 ) + 𝑐[
(99)
𝑑 (𝜐, 𝑥2𝑛+1 ) 𝑑 (𝑥2𝑛 , 𝜐𝑛 ) ], 1 + 𝑑 (𝑥2𝑛 , 𝜐)
and so 𝑑 (𝜐, 𝜐𝑛 ) ≤ 𝑑 (𝜐, 𝑥2𝑛+1 ) + 𝑎 𝑑 (𝜐, 𝑥2𝑛+1 ) 𝑑 (𝜐, 𝑥2𝑛+1 ) 𝑑 (𝑥2𝑛 , 𝜐𝑛 ) ] . + 𝑏 𝑑 (𝑥2𝑛 , 𝜐𝑛 ) + 𝑐 [ 1 + 𝑑 (𝑥2𝑛 , 𝜐) (100) By letting 𝑛 → ∞ in the above inequality, we get |𝑑(𝜐, 𝜐𝑛 )| → 0 as 𝑛 → ∞. By [9, Lemma 2.2], it follows that 𝜐𝑛 → 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 𝑎𝑑 (𝑦, 𝑆𝑥) + 𝑏𝑑 (𝑥, 𝑇𝑦) ∈ 𝑠 (𝑆𝑥, 𝑇𝑦)
(101)
for all 𝑥, 𝑦 ∈ 𝐵(𝑥0 , 𝑟) and (1 − 𝑘) 𝑟 ∈ 𝑠 (𝑥0 , 𝑆𝑥0 ) ,
𝑑 (𝜐, 𝑆𝑥2𝑛 ) 𝑑 (𝑥2𝑛 , 𝑇𝜐) ] 1 + 𝑑 (𝑥2𝑛 , 𝜐)
(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.
∈ 𝑠 (𝑑 (𝑥2𝑛+1 , 𝜐𝑛 )) . (95) By the definition, we get 𝑑 (𝑥2𝑛+1 , 𝜐𝑛 ) ⪯ 𝑎𝑑 (𝜐, 𝑆𝑥2𝑛 ) + 𝑏𝑑 (𝑥2𝑛 , 𝑇𝜐) 𝑑 (𝜐, 𝑆𝑥2𝑛 ) 𝑑 (𝑥2𝑛 , 𝑇𝜐) ]. + 𝑐[ 1 + 𝑑 (𝑥2𝑛 , 𝜐)
+ 𝑐[
𝑑 (𝜐, 𝜐𝑛 ) ⪯ 𝑑 (𝜐, 𝑥2𝑛+1 ) + 𝑑 (𝑥2𝑛+1 , 𝜐𝑛 )
where 𝑘 = 𝑏/(1 − 𝑏) < 1, 𝑥2𝑛+1 ∈ 𝑆𝑥2𝑛 , and 𝑥2𝑛+2 ∈ 𝑇𝑥2𝑛+1 . Now, inductively, we can construct a sequence {𝑥𝑛 } in 𝑋 such that for each 𝑛 ≥ 0, 𝑛 𝑑 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ 𝑘 𝑑 (𝑥0 , 𝑥1 ) .
(97)
Hence, we have
≤ (1 − 𝑘) |𝑟| + 𝑘 (1 − 𝑘) |𝑟| + 𝑘2 (1 − 𝑘) |𝑟|
2𝑛 𝑑 (𝑥2𝑛 , 𝑥2𝑛+1 ) ≤ 𝑘 𝑑 (𝑥0 , 𝑥1 ) , 2𝑛+1 𝑑 (𝑥2𝑛+1 , 𝑥2𝑛+2 ) ≤ 𝑘 𝑑 (𝑥0 , 𝑥1 ) ,
𝑑 (𝜐, 𝑥2𝑛+1 ) 𝑑 (𝑥2𝑛 , 𝜐𝑛 ) ]. 1 + 𝑑 (𝑥2𝑛 , 𝜐)
(96)
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 𝑎𝑑 (𝑦, 𝑇𝑥)+𝑏𝑑 (𝑥, 𝑇𝑦)+𝑐 [
𝑑 (𝑦, 𝑇𝑥) 𝑑 (𝑥, 𝑇𝑦) ] ∈ 𝑠 (𝑇𝑥, 𝑇𝑦) 1 + 𝑑 (𝑥, 𝑦) (103)
10
Abstract and Applied Analysis ∘
for all 𝑥, 𝑦 ∈ 𝐵(𝑥0 , 𝑟) and
(𝑡𝑛 , 𝜁𝑛 ) ≾ (𝑡𝑛+1 , 𝜁𝑛+1 ) and 𝑡𝑛 → 𝑡 as 𝑛 → ∞. Then, for any 𝑛 ≥ 1 with 𝑚 > 𝑛, we have
(1 − 𝑘) 𝑟 ∈ 𝑠 (𝑥0 , 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
𝑑 (𝜁𝑚 , 𝜁𝑛 ) ⪯
2 (𝜑 (𝑡𝑚 ) − 𝜑 (𝑡𝑛 )) → 0 1−𝑘
as 𝑛, 𝑚 → ∞, which implies that {𝜁𝑛 } is a Cauchy sequence. Since (𝑋, 𝑑) is a complete complex valued metric space, there ∘ ∘ exists 𝜁 ∈ 𝑋 such that 𝜁𝑛 → 𝜁. From (a), choose 𝑛0 ∈ N such that for all 𝑛 ≥ 𝑛0 , ∘
∘
+𝑐
∘
∘
1 + 𝑑 (𝜁𝑛 , 𝜁) ∘
(b) 𝐹(𝑡, ⋅) : 𝑈 → 𝐶𝐵(𝑋) is a multivalued mapping satisfying
∘
+
́ 𝑑 (𝜁,́ 𝐹 (𝑡, 𝜁)) 𝑑 (𝜁, 𝐹 (𝑡,́ 𝜁)) 1 + 𝑑 (𝜁, 𝜁)́
(105)
́ , ∈ 𝑠 (𝐹 (𝑡, 𝜁) , 𝐹 (𝑡,́ 𝜁))
∘
∘
𝑐𝑑 (𝜁, 𝐹 (𝑡𝑛 , 𝜁𝑛 )) 𝑑 (𝜁𝑛 , 𝐹 (𝑡, 𝜁)) ∘
1 + 𝑑 (𝜁𝑛 , 𝜁) ∘
∘
∈ 𝑠 (𝜁𝑛 , 𝐹 (𝑡, 𝜁)) , ∘
∘
since 𝜁𝑛 ∈ 𝐹(𝑡𝑛 , 𝜁𝑛 ). So there exists 𝜁𝑘 ∈ 𝐹(𝑡, 𝜁) such that
∘
(1 − 𝑘) 𝑟 ∈ 𝑠 (𝜁, 𝐹 (𝑡, 𝜁)) ,
∘
∘
∘
𝑑 (𝜁𝑛 , 𝜁𝑘 ) ⪯ 𝑎𝑑 (𝜁, 𝐹 (𝑡𝑛 , 𝜁𝑛 )) + 𝑏𝑑 (𝜁𝑛 , 𝐹 (𝑡, 𝜁))
where 𝑘 = 𝑏/(1 − 𝑏) < 1;
∘
(c) there exists a continuous increasing function 𝜑 (0, 1] → 𝑃 ∪ {0} such that 𝜑 (𝑠) ∈ 𝜑 (𝑡)
:
(106)
for all 𝑠, 𝑡 ∈ [0, 1] and 𝜁 ∈ 𝑈, where 𝑃 = {𝑧 ∈ C : 0 ≺ 𝑧}.
+
∘
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 ≠ 0. We define the partial ordering ≾ in Q as follows:
∘
1 + 𝑑 (𝜁𝑛 , 𝜁)
(111) .
By using the g.l.b property of 𝐹, we get ∘
∘
𝑑 (𝜁𝑛 , 𝜁𝑘 ) ⪯ 𝑎𝑑 (𝜁, 𝜁𝑛 ) + 𝑏𝑑 (𝜁𝑛 , 𝜁𝑘 ) +
𝑐𝑑 (𝜁, 𝜁𝑛 ) 𝑑 (𝜁𝑛 , 𝜁𝑘 ) ∘
1 + 𝑑 (𝜁𝑛 , 𝜁)
,
(112) which implies that ∘ 𝑑 (𝜁𝑛 , 𝜁𝑘 ) ≤ 𝑎 𝑑 (𝜁, 𝜁𝑛 ) + 𝑏 𝑑 (𝜁𝑛 , 𝜁𝑘 ) ∘ 𝑑 (𝜁, 𝜁𝑛 ) 𝑑 (𝜁𝑛 , 𝜁𝑘 ) . + 𝑐 ∘ 1 + 𝑑 (𝜁 , 𝜁) 𝑛 ∘
(𝑡, 𝜁) ≾ (𝑠, 𝜁)́ ⇐⇒ 𝑡 ≤ 𝑠,
∘
𝑐𝑑 (𝜁, 𝐹 (𝑡𝑛 , 𝜁𝑛 )) 𝑑 (𝜁𝑛 , 𝐹 (𝑡, 𝜁))
Then 𝐹(0, ⋅) has a fixed point if and only if 𝐹(1, ⋅) has a fixed point.
𝑑 (𝜁, 𝜁)́ ⪯
(110)
∘
∘
𝑎𝑑 (𝜁, 𝐹 (𝑡𝑛 , 𝜁𝑛 )) + 𝑏𝑑 (𝜁𝑛 , 𝐹 (𝑡, 𝜁)) ∘
́ 𝑎𝑑 (𝜁,́ 𝐹 (𝑡, 𝜁)) + 𝑏𝑑 (𝜁, 𝐹 (𝑡,́ 𝜁))
́ , 𝜑 (𝑠) − 𝜑 (𝑡) ∈ 𝑠 (𝐹 (𝑠, 𝜁) , 𝐹 (𝑡, 𝜁))
∘
∈ 𝑠 (𝐹 (𝑡𝑛 , 𝜁𝑛 ) , 𝐹 (𝑡, 𝜁)) ,
(a) 𝜁 ∉ [𝐹(𝑡, 𝜁)] for all 𝜁 ∈ 𝜕𝑈 and 𝑡 ∈ [0, 1];
∘
∘
𝑑 (𝜁, 𝐹 (𝑡𝑛 , 𝜁𝑛 )) 𝑑 (𝜁𝑛 , 𝐹 (𝑡, 𝜁))
that there exist 𝜁 ∈ 𝑋 and 0 ≺ 𝑟 ∈ C such that
∘
∘
∘
𝑎𝑑 (𝜁, 𝐹 (𝑡𝑛 , 𝜁𝑛 )) + 𝑏𝑑 (𝜁𝑛 , 𝐹 (𝑡, 𝜁))
∘
+𝑐
(109)
(113)
∘
Since |1 + 𝑑(𝜁𝑛 , 𝜁)| > |𝑑(𝜁𝑛 , 𝜁)|, so we have (108)
2 (𝜑 (𝑠) − 𝜑 (𝑡)) . 1−𝑘 ∘
Let M be a totally ordered subset of Q and 𝑡 = sup{𝑡 : (𝑡, 𝜁) ∈ M}. Consider a sequence {(𝑡𝑛 , 𝜁𝑛 )} in M such that
∘ 𝑑 (𝜁𝑛 , 𝜁𝑘 ) ≤ 𝑎 𝑑 (𝜁, 𝜁𝑛 ) + 𝑏 𝑑 (𝜁𝑛 , 𝜁𝑘 ) + 𝑐 𝑑 (𝜁𝑛 , 𝜁𝑘 ) , ∘ 𝑎 𝑑 (𝜁𝑛 , 𝜁) . 𝑑 (𝜁𝑛 , 𝜁𝑘 ) ≤ 1 − 𝑏 − 𝑐
(114)
Abstract and Applied Analysis
11
References
Note that ∘ ∘ 𝑑 (𝜁, 𝜁𝑘 ) ≤ 𝑑 (𝜁, 𝜁𝑛 ) + 𝑑 (𝜁𝑛 , 𝜁𝑘 ) ∘ ∘ 𝑎 ≤ 𝑑 (𝜁, 𝜁𝑛 ) + 𝑑 (𝜁𝑛 , 𝜁) → 0 1−𝑏−𝑐 ∘
∘
∘
(115)
∘
for all 𝑛 ≥ 𝑛0 . Thus 𝜁𝑘 → 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) − 𝜑 (𝑡) ∈ 𝑠 (𝐹 (𝑡, 𝜁) , 𝐹 (𝑡, 𝜁)) , ∘
∘
(116)
𝜑 (𝑡) − 𝜑 (𝑡) ∈ 𝑠 (𝜁, 𝐹 (𝑡, 𝜁)) ∘
∘
∘
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 ∘
∘
𝑑 (𝜁, 𝜁) ⪯ 𝜑 (𝑡) − 𝜑 (𝑡) ⪯
(1 − 𝑘) 𝑟 ≺ (1 − 𝑘) 𝑟, 2
(117)
which implies that ∘ 𝑑 (𝜁, 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 ∘
𝑑 (𝜁, 𝜁) ≺ 𝑟 = ∘
∘ 2 (𝜑 (𝑡) − 𝜑 (𝑡)) , 1−𝑘
(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.
[1] S. Banach, “Sur les op´erations dans les ensembles abstraits et leurs applications aux e´quations int´egrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922. [2] S. B. Nadler, “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969. [3] L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007. [4] M. Arshad, A. Azam, and P. Vetro, “Some common fixed point results in cone metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 493965, 11 pages, 2009. [5] A. Azam and M. Arshad, “Common fixed points of generalized contractive maps in cone metric spaces,” Iranian Mathematical Society, vol. 35, no. 2, pp. 255–264, 2009. [6] A. Latif and F. Y. Shaddad, “Fixed point results for multivalued maps in cone metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 941371, 2010. [7] E. Karapınar, “Some nonunique fixed point theorems of Ciric type on cone metric spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 123094, 14 pages, 2010. [8] E. Karapinar, P. Kumam, and W. Sintunavarat, “Coupled fixed point theorems in cone metric spaces with a 𝑐-distance and applications,” Fixed Point Theory and Applications, vol. 2012, article 194, 2012. [9] A. Azam, B. Fisher, and M. Khan, “Common fixed point theorems in complex valued metric spaces,” Numerical Functional Analysis and Optimization, vol. 32, no. 3, pp. 243–253, 2011. [10] F. Rouzkard and M. Imdad, “Some common fixed point theorems on complex valued metric spaces,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1866–1874, 2012. [11] W. Sintunavarat and P. Kumam, “Generalized common fixed point theorems in complex valued metric spaces and applications,” Journal of Inequalities and Applications, vol. 2012, article 84, 2012. [12] M. Abbas, M. Arshad, and A. Azam, “Fixed points of asymptotically regular mappings in complex-valued metric spaces,” Georgian Mathematical Journal, vol. 20, no. 2, pp. 213–221, 2013. [13] M. Abbas, B. Fisher, and T. Nazir, “Well-Posedness and periodic point property of mappings satisfying a rational inequality in an ordered complex valued metric space,” Numerical Functional Analysis and Optimization, vol. 243, 2011. [14] M. Arshad and J. Ahmad, “On multivalued contractions in cone metric spaces with out normality,” The Scientific World Journal, vol. 2013, Article ID 481601, 3 pages, 2013. [15] C. Klin-eam and C. Suanoom, “Some common fixed-point theorems for generalized-contractive-type mappings on complexvalued metric spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 604215, 6 pages, 2013. [16] M. A. Kutbi, J. Ahmad, and A. Azam, “On fixed points of 𝛼 -𝜓-contractive multivalued mappings in cone metric spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 313782, 6 pages, 2013. [17] M. A. Kutbi, J. Ahmad, N. Hussain, and M. Arshad, “Common fixed point results for mappings with rational expressions,” Abstract and Applied Analysis, vol. 2013, Article ID 549518, 11 pages, 2013. [18] W. Sintunavarat, Y. J. Cho, and P. Kumam, “Urysohn integral equations approach by common fixed points in complex-valued
12 metric spaces,” Advances in Difference Equations, vol. 2013, article 49, 2013. [19] K. Sitthikul and S. Saejung, “Some fixed point theorems in complex valued metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 189, 2012.
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