Common Fixed Point Results for Six Mappings via Integral ...

Hindawi Publishing Corporation International Journal of Analysis Volume 2016, Article ID 7480469, 13 pages http://dx.doi.org/10.1155/2016/7480469

Research Article Common Fixed Point Results for Six Mappings via Integral Contractions with Applications Mian Bahadur Zada,1 Muhammad Sarwar,1 and Nayyar Mehmood2 1

Department of Mathematics, University of Malakand, Chakdara, Pakistan Department of Mathematics and Statistics, International Islamic University, Sector H-10, Islamabad, Pakistan

2

Correspondence should be addressed to Mian Bahadur Zada; [email protected] and Muhammad Sarwar; [email protected] Received 3 July 2016; Accepted 20 September 2016 Academic Editor: Remi Léandre Copyright © 2016 Mian Bahadur Zada 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. Common fixed point theorems for six self-mappings under integral type inequality satisfying (E.A) and (CLR) properties in the context of complex valued metric space (not necessarily complete) are established. The derived results are new even for ordinary metric spaces. We prove existence result for optimal unique solution of the system of functional equations used in dynamical programming with complex domain.

1. Introduction and Preliminaries Metric fixed point theory is the most impressive and active branch of modern mathematics that has vast applications in applied functional and numerical analysis. Banach contraction principle [1] is one of the best known results in this theory. This principle can be considered as the launch of metric fixed point theory that guarantees the existence and uniqueness of fixed points of mappings. In the following years, various efforts have been done to further generalize Banach contraction principle in different direction for a single map. The exploration of common fixed point theory is an active field of research activity since 1976. The work of Jungck [2] is considered as major achievement in the field of common fixed point theory. Jungck presented the notion of commuting maps to introduce the common fixed point results for two self-maps on complete metric space. To improve common fixed point theorems, researchers began to utilize weaker conditions than commuting mappings such as weakly commuting maps, compatible mappings, compatible mappings of type (A), compatible mappings of type (B), compatible mappings of type (P), and compatible mappings of type

(C). In the study of common fixed point results of weakly compatible mappings we often require the assumption of the continuity of mappings or the completeness of underlying space. As a consequence a natural question arises as to whether there exist common fixed point theorems, which do not enforce such conditions. Regarding this Aamri and El Moutawakil [3] relaxed these conditions by introducing the notion of (E.A) property and it was marked that (E.A) property does not require the condition of continuity of mappings and completeness of the underlying space. However, (E.A) property tolerates the condition of closeness of the range subspaces of the involved mappings. In 2011, the new notion of Common Limit in the range property (shortly (CLR) property) was given by Sintunavarat and Kumam [4] that does not enforce the above-mentioned conditions. Moreover, the significance of (CLR) property reveals that closeness of range subspaces is not essential. Using these two important notions many fixed point theorems were established [3–6]. One of the most pleasant generalizations of Banach principle is the Branciari [7] fixed point theorem for a single mapping satisfying an integral type inequality. After that, serval researchers ([8–11], etc.) generalize the result of Branciari in ordinary metric spaces.

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International Journal of Analysis

On the other hand Azam et al. [12] studied complex valued metric space and proved common fixed point theorems for two self-mappings satisfying a rational type inequality. Manro et al. [13] generalized the theorem of Branciari [7] for two self-maps under contractive condition of integral type satisfying property (E.A) and (CLR) property in the setting of complex valued metric spaces. Bahadur Zada et al. [6] generalized the results of [13] for four self-maps in the context of complex valued metric spaces. The aim of this paper is to prove common fixed point theorems for six self-maps, satisfying integral type contractive condition using property (E.A) and (CLR) property in complex valued metric spaces, which extends and generalizes many results of the existing literature. Throughout the paper C+ = {𝑧 ∈ C : 𝑧 ≿ (0, 0)}, opt stand for sup or inf. 𝑍 and 𝑌 are Banach spaces, Ω ⊆ 𝑍 is the state space, 𝐷 ⊆ 𝑌 is the decision space, Φ = {𝜙 : 𝜙 : [0, ∞[→ [0, ∞[ is a Lebesgue integrable mapping which is summable on each compact subset of [0, ∞[, nonnegative 𝜀 and nondecreasing such that, for each 𝜀 > 0, ∫0 𝜙(𝑡)𝑑𝑡 > 0}, ∗ 𝑛 and Φ = {𝜑 : R → C is a complex valued Lebesgue integrable mapping, which is summable and nonvanishing on𝜀 each measurable subset of R𝑛 , such that, for each 𝜀 ≻ 0, ∫0 𝜑(𝑡)𝑑𝑡 ≻ 0}. Definition 1 (see [12]). Let C be the set of complex numbers and 𝑧, 𝑤 ∈ C. Define a partial order ≾ on C as follows: 𝑧≾𝑤

iff Re (𝑧) ≤ Re (𝑤) , Im (𝑧) ≤ Im (𝑤) ,

𝑧≺𝑤

iff Re (𝑧) < Re (𝑤) , Im (𝑧) < Im (𝑤) .

(1)

Note that

Then (C, 𝑑) is a complex valued metric space. Definition 5 (see [12]). Let {𝑧𝑛 } be a sequence in complex valued metric (𝑋, 𝑑) and 𝑧 ∈ 𝑋. Then 𝑧 is called the limit of {𝑧𝑛 } if for every 𝑤 ∈ C, with 0 ≺ 𝑤, there is 𝑛0 ∈ 𝑁 such that 𝑑(𝑧𝑛 , 𝑧) ≺ 𝑤 for all 𝑛 > 𝑛0 and one writes lim𝑛→∞ 𝑧𝑛 = 𝑧. Lemma 6 (see [12]). Any sequence {𝑧𝑛 } in complex valued metric space (𝑋, 𝑑) converges to 𝑧 if and only if |𝑑(𝑧𝑛 , 𝑧)| → 0 as 𝑛 → ∞. Definition 7 (see [4]). Let 𝑋 be a nonempty set and 𝐾, 𝐿 : 𝑋 → 𝑋 be two self-maps. Then (i) 𝑧 ∈ 𝑋 is called a fixed point of 𝐿 if 𝐿𝑧 = 𝑧; (ii) 𝑧 ∈ 𝑋 is called a coincidence point of 𝐾 and 𝐿 if 𝐾𝑧 = 𝐿𝑧; (iii) 𝑧 ∈ 𝑋 is called a common fixed point of 𝐾 and 𝐿 if 𝐾𝑧 = 𝐿𝑧 = 𝑧. Jungck [2] initiated the concept of commuting maps in the following way. Definition 8. Two self-maps 𝐾 and 𝐿 of nonempty set 𝑋 are commuting if 𝐿𝐾𝑧 = 𝐾𝐿𝑧, for all 𝑧 ∈ 𝑋. Jungck [15] initiated the concept of weakly compatible maps in ordinary metric spaces while Bhatt et al. [16] refined this notion in the complex valued metric space in the following way.

(i) 𝑘1 , 𝑘2 ∈ 𝑅 and 𝑘1 ≤ 𝑘2 ⇒ 𝑘1 𝑧 ≾ 𝑘2 𝑧 for all 𝑧 ∈ C; (ii) 0 ≾ 𝑧 ≾ 𝑤 ⇒ |𝑧| < |𝑤| for all 𝑧, 𝑤 ∈ C; (iii) 𝑧 ≾ 𝑤 and 𝑤 ≺ 𝑤∗ ⇒ 𝑧 ≺ 𝑤∗ for all 𝑧, 𝑤, 𝑤∗ ∈ C.

Definition 9. Two self-maps 𝐾 and 𝐿 on complex valued metric space 𝑋 are weakly compatible if there exists point 𝑧 ∈ 𝑋 such that 𝐾𝐿𝑧 = 𝐿𝐾𝑧 whenever 𝐾𝑧 = 𝐿𝑧.

Definition 2 (see [14]). The “max” function for the partial order relation “≾” is defined by the following:

Aamri and El Moutawakil [3] initiated the concept of (E.A) property in ordinary metric spaces while Verma and Pathak [14] defined this concept in complex valued metric space as follows.

(1) max{𝑤1 , 𝑤2 } = 𝑤2 ⇔ 𝑤1 ≾ 𝑤2 . (2) If 𝑤1 ≾ max{𝑤2 , 𝑤3 }, then 𝑤1 ≾ 𝑤2 or 𝑤1 ≾ 𝑤3 . (3) max{𝑤1 , 𝑤2 } = 𝑤2 ⇔ 𝑤1 ≾ 𝑤2 or |𝑤1 | ≤ |𝑤2 |. Definition 3 (see [12]). Let 𝑋 be a nonempty set and 𝑑 : 𝑋 × 𝑋 → C be the mapping satisfying the following axioms: (1) 0 ≾ 𝑑(𝑧1 , 𝑧2 ), for all 𝑧1 , 𝑧2 ∈ 𝑋 and 𝑑(𝑧1 , 𝑧2 ) = 0 if and only if 𝑧1 = 𝑧2 . (2) 𝑑(𝑧1 , 𝑧2 ) = 𝑑(𝑧2 , 𝑧1 ), for all 𝑧1 , 𝑧2 ∈ 𝑋. (3) 𝑑(𝑧1 , 𝑧2 ) ≾ 𝑑(𝑧1 , 𝑧3 ) + 𝑑(𝑧3 , 𝑧2 ), for all 𝑧1 , 𝑧2 , 𝑧3 ∈ 𝑋. Then pair (𝑋, 𝑑) is called a complex valued metric space. Example 4. Let 𝑧1 , 𝑧2 ∈ C and define the mapping 𝑑 : C × C → C by {0 𝑑 (𝑧1 , 𝑧2 ) = { 𝜄̇ {

if 𝑧1 = 𝑧2 , if 𝑧1 ≠ 𝑧2 .

Definition 10. Two self-maps 𝐾 and 𝐿 on a complex valued metric space 𝑋 satisfy property (E.A) if there exists sequence {𝑧𝑛 } in 𝑋 such that lim 𝐿𝑧𝑛 = lim 𝐾𝑧𝑛 = 𝑧 for some 𝑧 ∈ 𝑋.

𝑟→∞

𝑟→∞

(3)

Sintunavarat and Kumam [4] introduced the notion of (CLR) property in ordinary metric spaces, in a similar mode. Verma and Pathak [14] defined this notion in a complex valued metric space in the following way. Definition 11. Two self-maps 𝐾 and 𝐿 on a complex valued metric space 𝑋 satisfy (CLR𝐾 ) if there exists sequence {𝑧𝑛 } in 𝑋 such that

(2) lim 𝐿𝑧𝑛 = lim 𝐾𝑧𝑛 = 𝐾𝑧 for some 𝑧 ∈ 𝑋.

𝑛→∞

𝑟→∞

(4)


3

Remark 12 (see [6]). Let 𝜑 ∈ Φ∗ , such that Re(𝜑), Im(𝜑) ∈ Φ and {𝑧𝑛 } is a sequence in C+ converges to 𝑧, and then 𝑧 𝑧 lim𝑛→∞ ∫0 𝑛 𝜑(𝑠)𝑑𝑠 = ∫0 𝜑(𝑠)𝑑𝑠.

Since 𝐾(𝑋) ⊆ 𝑀𝑆(𝑋), there exists {𝑤𝑛 } in 𝑋 such that 𝐾𝑧𝑛 = 𝑀𝑆𝑤𝑛 and thus, from (7), we get lim 𝐾𝑧𝑛 = lim 𝑁𝑅𝑧𝑛 = lim 𝑀𝑆𝑤𝑛 = 𝑧.

𝑛→∞

∗

Lemma 13 (see [6]). Let 𝜑 ∈ Φ , such that Re(𝜑), Im(𝜑) ∈ Φ 𝑧 and {𝑧𝑛 } is a sequence in C+ , and then lim𝑛→∞ ∫0 𝑛 𝜑(𝑠)𝑑𝑠 = 0 if and only if 𝑧𝑛 → (0, 0), as 𝑛 → ∞.

∫

Let Ψ be the class of all functions 𝜓 : C+ → C+ that satisfy the following properties:

𝑑(𝐾𝑧𝑛 ,𝐿𝑤𝑛 )

0

Now, we present our first result. Theorem 14. Let (𝑋, 𝑑) be a complex valued metric space and 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, 𝑆 : 𝑋 → 𝑋 be six self-mappings satisfying the following conditions: (1) One of pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) satisfies property (𝐸.𝐴) such that 𝐾(𝑋) ⊆ 𝑀𝑆(𝑋) and 𝐿(𝑋) ⊆ 𝑁𝑅(𝑋).

𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

Δ 1 (𝑧𝑛 , 𝑤𝑛 ) = 𝑑 (𝑀𝑆𝑤𝑛 , 𝐿𝑤𝑛 )

1 + 𝑑 (𝑁𝑅𝑧𝑛 , 𝐾𝑧𝑛 ) ; 1 + 𝑑 (𝑁𝑅𝑧𝑛 , 𝑀𝑆𝑤𝑛 )

Δ 2 (𝑧𝑛 , 𝑤𝑛 ) = 𝑑 (𝑁𝑅𝑧𝑛 , 𝐾𝑧𝑛 )

1 + 𝑑 (𝑀𝑆𝑤𝑛 , 𝐿𝑤𝑛 ) ; 1 + 𝑑 (𝑁𝑅𝑧𝑛 , 𝑀𝑆𝑤𝑛 )

Δ 3 (𝑧𝑛 , 𝑤𝑛 ) = max {𝑑 (𝑁𝑅𝑧𝑛 , 𝑀𝑆𝑤𝑛 ) , 𝑑 (𝑁𝑅𝑧𝑛 , 𝐾𝑧𝑛 ) ,

(2) ∀𝑧1 , 𝑧2 ∈ 𝑋.

(10)

𝑑 (𝑀𝑆𝑤𝑛 , 𝐿𝑤𝑛 ) , 1 [𝑑 (𝐾𝑧𝑛 , 𝑀𝑆𝑤𝑛 ) + 𝑑 (𝐿𝑤𝑛 , 𝑁𝑅𝑧𝑛 )]} . 2

𝜑 (𝑡) 𝑑𝑡 Δ 𝑗 (𝑧1 ,𝑧2 )

≾ 𝜓 (max {∫

0

(5) 𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

Δ 1 (𝑧1 , 𝑧2 ) = 𝑑 (𝑀𝑆𝑧2 , 𝐿𝑧2 ) Δ 2 (𝑧1 , 𝑧2 ) = 𝑑 (𝑁𝑅𝑧1 , 𝐾𝑧1 )

Taking upper limit as 𝑛 → ∞ in (9), we have Δ 1 (𝑧𝑛 , 𝑤𝑛 ) 󳨀→ 𝑑 (𝑧, 𝑤) ,

where 𝜓 ∈ Ψ, 𝜑 ∈ Φ∗ and

Δ 2 (𝑧𝑛 , 𝑤𝑛 ) 󳨀→ 0,

1 + 𝑑 (𝑁𝑅𝑧1 , 𝐾𝑧1 ) ; 1 + 𝑑 (𝑁𝑅𝑧1 , 𝑀𝑆𝑧2 )

Δ 3 (𝑧𝑛 , 𝑤𝑛 ) 󳨀→ 𝑑 (𝑧, 𝑤) , ∫

1 + 𝑑 (𝑀𝑆𝑧2 , 𝐿𝑧2 ) ; 1 + 𝑑 (𝑁𝑅𝑧1 , 𝑀𝑆𝑧2 )

Δ 3 (𝑧1 , 𝑧2 ) = max {𝑑 (𝑁𝑅𝑧1 , 𝑀𝑆𝑧2 ) , 𝑑 (𝑁𝑅𝑧1 , 𝐾𝑧1 ) ,

𝑑(𝑧,𝑤)

𝑑(𝐾𝑧𝑛 ,𝐿𝑤𝑛 )

𝜑 (𝑡) 𝑑𝑡 = lim sup ∫ 𝑛→∞

0

0

Δ 𝑗 (𝑧𝑛 ,𝑤𝑛 )

(6)

≾ lim sup 𝜓 (max {∫ 𝑛→∞

0 Δ 𝑗 (𝑧𝑛 ,𝑤𝑛 )

𝑑 (𝑀𝑆𝑧2 , 𝐿𝑧2 ) ,

≾ 𝜓 (lim sup max {∫

1 [𝑑 (𝐾𝑧1 , 𝑀𝑆𝑧2 ) + 𝑑 (𝐿𝑧2 , 𝑁𝑅𝑧1 )]} . 2

= 𝜓 (max {∫

𝑛→∞

Proof. Let pair (𝐾, 𝑁𝑅) satisfy (E.A) property, so there exists sequence {𝑧𝑛 } in 𝑋 such that lim 𝐾𝑧𝑛 = lim 𝑁𝑅𝑧𝑛 = 𝑧 for some 𝑧 ∈ 𝑋. 𝑛→∞

0

𝑑(𝑧,𝑤)

0

If one of 𝑀𝑆(𝑋) and 𝑁𝑅(𝑋) is closed subspace of 𝑋 such that pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) are weakly compatible, then each pair of pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) has a coincidence point in 𝑋. Moreover, if (𝐾, 𝑆), (𝐿, 𝑅), (𝑀𝑆, 𝑅), and (𝑁𝑅, 𝑆) are commuting pairs, then 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆 have a unique common fixed point in 𝑋.

𝑛→∞

(9)

where

(3) 𝜓(0) = 0 and 𝜓(𝑧) ≺ 𝑧 for every 𝑧 ≻ 0.

0

Δ 𝑗 (𝑧𝑛 ,𝑤𝑛 )

0

(2) 𝜓 is upper semicontinuous on C+ .

(8)

𝜑 (𝑡) 𝑑𝑡

≾ 𝜓 (max {∫

(1) 𝜓 is nondecreasing on C+ .

𝑑(𝐾𝑧1 ,𝐿𝑧2 )

𝑛→∞

We assert that lim𝑛→∞ 𝐿𝑤𝑛 = 𝑧. If lim𝑛→∞ 𝐿𝑤𝑛 = 𝑤 ≠ 𝑧, then, upon putting 𝑧1 = 𝑧𝑛 and 𝑧2 = 𝑤𝑛 in condition (2) of Theorem 14, we have

2. Main Results

∫

𝑛→∞

(7)

= 𝜓 (∫

𝑑(𝑧,𝑤)

0


𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3})

(11) 𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) 𝑑(𝑧,𝑤)

𝜑 (𝑡) 𝑑𝑡, 0, ∫

0

𝜑 (𝑡) 𝑑𝑡) ≺ ∫

𝑑(𝑧,𝑤)

0

𝜑 (𝑡) 𝑑𝑡})

𝜑 (𝑡) 𝑑𝑡 󳨐⇒

󵄨󵄨 𝑑(𝑧,𝑤) 󵄨󵄨 󵄨󵄨 𝑑(𝑧,𝑤) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨∫ 󵄨󵄨 < 󵄨󵄨∫ 󵄨󵄨 , 𝜑 𝑑𝑡 𝜑 𝑑𝑡 (𝑡) (𝑡) 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 󵄨 󵄨 󵄨 󵄨

which contradict with our assumption; thus 𝑧 = 𝑤 and lim𝑛→∞ 𝐿𝑤𝑛 = 𝑧. Therefore (8) becomes lim 𝐾𝑧𝑛 = lim 𝑁𝑅𝑧𝑛 = lim 𝐿𝑤𝑛 = lim 𝑀𝑆𝑤𝑛 = 𝑧. (12) 𝑛→∞ 𝑛→∞ 𝑛→∞

𝑛→∞

4


Also, since 𝑀𝑆(𝑋) is closed subspace of 𝑋, there exists 𝑢 ∈ 𝑋 such that 𝑀𝑆𝑢 = 𝑧 and, using (12), we get lim 𝐾𝑧𝑛 = lim 𝑁𝑅𝑧𝑛 = lim 𝐿𝑤𝑛 = lim 𝑀𝑆𝑤𝑛 = 𝑧

𝑛→∞

𝑛→∞

𝑛→∞

𝑛→∞

= 𝑀𝑆𝑢.

𝐿𝑢 = 𝑀𝑆𝑢 = 𝑁𝑅V = 𝑧.

𝑑(𝐾𝑧𝑛 ,𝐿𝑢)

0

We show that 𝐾V = 𝑁𝑅V. Let on contrary 𝐾V ≠ 𝑁𝑅V; then, using condition (2) of Theorem 14 with 𝑧1 = V and 𝑧2 = 𝑢, we have ∫

𝜑 (𝑡) 𝑑𝑡 Δ 𝑗 (𝑧𝑛 ,𝑢)

0

𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

Δ 1 (𝑧𝑛 , 𝑢) = 𝑑 (𝑀𝑆𝑢, 𝐿𝑢)

1 + 𝑑 (𝑀𝑆𝑢, 𝐿𝑢) ; 1 + 𝑑 (𝑁𝑅𝑧𝑛 , 𝑀𝑆𝑢)

0

1 + 𝑑 (𝑁𝑅V, 𝐾V) = 0; 1 + 𝑑 (𝑁𝑅V, 𝑀𝑆𝑢)

Δ 2 (V, 𝑢) = 𝑑 (𝑁𝑅V, 𝐾V)

1 + 𝑑 (𝑀𝑆𝑢, 𝐿𝑢) = 𝑑 (𝑧, 1 + 𝑑 (𝑁𝑅V, 𝑀𝑆𝑢)

𝑑 (𝑀𝑆𝑢, 𝐿𝑢) ,

Therefore, ∫

Δ 2 (𝑧𝑛 , 𝑢) 󳨀→ 0, Δ 3 (𝑧𝑛 , 𝑢) 󳨀→ 𝑑 (𝑧, 𝐿𝑢) , 𝜑 (𝑡) 𝑑𝑡 = lim sup ∫ 𝑛→∞

0


0

0

Δ 𝑗 (𝑧𝑛 ,𝑢)

0

= 𝜓 (max {∫ 𝑑(𝑧,𝐿𝑢)




(16) 𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3})

0

𝑑(𝑧,𝐿𝑢)

0

𝜑 (𝑡) 𝑑𝑡, ∫

𝑑(𝑧,𝐾V)

0

𝑑(𝑧,𝐾V)

𝜑 (𝑡) 𝑑𝑡) ≺ ∫

0

𝜑 (𝑡) 𝑑𝑡}) (21)

𝜑 (𝑡) 𝑑𝑡,

which is a contradiction to our assumption that 𝐾V ≠ 𝑁𝑅V. Thus 𝐾V = 𝑁𝑅V and hence, from (18), we get 𝐾V = 𝐿𝑢 = 𝑀𝑆𝑢 = 𝑁𝑅V = 𝑧.


(22)

Now, using the weak compatibility of pairs (𝐾, 𝑁𝑅), (𝐿, 𝑀𝑆), and (22), we have

𝜑 (𝑡) 𝑑𝑡 󳨐⇒

󵄨󵄨 𝑑(𝑧,𝐿𝑢) 󵄨󵄨 󵄨󵄨 𝑑(𝑧,𝐿𝑢) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨∫ 󵄨󵄨 < 󵄨󵄨∫ 󵄨󵄨 , 𝜑 𝑑𝑡 𝜑 𝑑𝑡 (𝑡) (𝑡) 󵄨󵄨 󵄨󵄨󵄨 0 󵄨󵄨󵄨 󵄨󵄨󵄨 0 󵄨

which is a contradiction. Thus, 𝐿𝑢 = 𝑧 and hence 𝐿𝑢 = 𝑀𝑆𝑢 = 𝑧.

𝑑(𝑧,𝐾V)

0

𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3})

𝑑(𝑧,𝐿𝑢)

𝑑(𝑧,𝐾V)

0

≾ 𝜓 (∫

𝜑 (𝑡) 𝑑𝑡, 0, ∫

𝜑 (𝑡) 𝑑𝑡) ≺ ∫


≾ 𝜓 (max {0, ∫

0

≾ lim sup 𝜓 (max {∫

= 𝜓 (∫

𝑑(𝐾V,𝑧)

0

𝑑(𝑧,𝐿𝑢)

1 [𝑑 (𝐾V, 𝑀𝑆𝑢) + 𝑑 (𝐿𝑢, 𝑁𝑅V)]} 2

= 𝑑 (𝑧, 𝐾V) .

Δ 1 (𝑧𝑛 , 𝑢) 󳨀→ 𝑑 (𝑧, 𝐿𝑢) ,

𝑛→∞

(20)

Δ 3 (V, 𝑢) = max {𝑑 (𝑁𝑅V, 𝑀𝑆𝑢) , 𝑑 (𝑁𝑅V, 𝐾V) ,

Taking upper limit as 𝑛 → ∞ in (14), we have

𝑛→∞

𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

Δ 1 (V, 𝑢) = 𝑑 (𝑀𝑆𝑢, 𝐿𝑢)

1 𝑑 (𝑀𝑆𝑢, 𝐿𝑢) , [𝑑 (𝐾𝑧𝑛 , 𝑀𝑆𝑢) + 𝑑 (𝐿𝑢, 𝑁𝑅𝑧𝑛 )]} . 2

0

(19)

𝐾V) ; (15)

Δ 3 (𝑧𝑛 , 𝑢) = max {𝑑 (𝑁𝑅𝑧𝑛 , 𝑀𝑆𝑢) , 𝑑 (𝑁𝑅𝑧𝑛 , 𝐾𝑧𝑛 ) ,

∫

Δ 𝑗 (V,𝑢)

where

1 + 𝑑 (𝑁𝑅𝑧𝑛 , 𝐾𝑧𝑛 ) ; 1 + 𝑑 (𝑁𝑅𝑧𝑛 , 𝑀𝑆𝑢)

Δ 2 (𝑧𝑛 , 𝑢) = 𝑑 (𝑁𝑅𝑧𝑛 , 𝐾𝑧𝑛 )


≾ 𝜓 (max {∫

(14)

where

𝑑(𝑧,𝐿𝑢)

𝑑(𝐾V,𝐿𝑢)

0

≾ 𝜓 (max {∫

(18)

(13)

Now, we claim that 𝐿𝑢 = 𝑀𝑆𝑢. To support the claim, let 𝐿𝑢 ≠ 𝑀𝑆𝑢. Then, using condition (2) of Theorem 14 with 𝑧1 = 𝑧𝑛 and 𝑧2 = 𝑢, one can get ∫

Since 𝐿(𝑋) ⊆ 𝑁𝑅(𝑋), there exists V ∈ 𝑋 such that 𝐿𝑢 = 𝑁𝑅V and it follows from (17) that

(17)

𝐾V = 𝑁𝑅V 󳨐⇒ 𝑁𝑅𝐾V = 𝐾𝑁𝑅V 󳨐⇒ 𝐾𝑧 = 𝑁𝑅𝑧,

(23)

𝐿𝑢 = 𝑀𝑆𝑢 󳨐⇒ 𝑀𝑆𝐿𝑢 = 𝐿𝑀𝑆𝑢 󳨐⇒ 𝐿𝑧 = 𝑀𝑆𝑧.

(24)

Hence 𝑧 is the coincident point of each pair (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆). Next, we have to show that 𝑧 is the common fixed point of 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆. For this, we claim that 𝐾𝑧 = 𝑧.


5

If 𝐾𝑧 ≠ 𝑧, then upon putting 𝑧1 = 𝑧, 𝑧2 = 𝑢 in condition (2) of Theorem 14 and using (22) and (23) we have ∫

𝑑(𝐾𝑧,𝐿𝑢)

0

Further, assume the 𝑆𝑧 ≠ 𝑧. Then upon putting 𝑧1 = 𝑆𝑧, 𝑧2 = 𝑧 in condition (2) of Theorem 14 and using (29) and (31), we have

𝜑 (𝑡) 𝑑𝑡 (25)

Δ 𝑗 (𝑧,𝑢)

𝑑(𝐾𝑆𝑧,𝐿𝑧)

∫

0

𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

≾ 𝜓 (max {∫

0

≾ 𝜓 (max {∫

Δ 1 (𝑧, 𝑢) = 𝑑 (𝑀𝑆𝑢, 𝐿𝑢)

1 + 𝑑 (𝑁𝑅𝑧, 𝐾𝑧) = 0; 1 + 𝑑 (𝑁𝑅𝑧, 𝑀𝑆𝑢)

Δ 2 (𝑧, 𝑢) = 𝑑 (𝑁𝑅𝑧, 𝐾𝑧)

1 + 𝑑 (𝑀𝑆𝑢, 𝐿𝑢) = 0; 1 + 𝑑 (𝑁𝑅𝑧, 𝑀𝑆𝑢)

Δ 3 (𝑧, 𝑢) = max {𝑑 (𝑁𝑅𝑧, 𝑀𝑆𝑢) , 𝑑 (𝑁𝑅𝑧, 𝐾𝑧) , 𝑑 (𝑀𝑆𝑢, 𝐿𝑢) ,

Δ 1 (𝑆𝑧, 𝑧) = 𝑑 (𝑀𝑆𝑧, 𝐿𝑧) (26)

1 [𝑑 (𝐾𝑧, 𝑀𝑆𝑢) + 𝑑 (𝐿𝑢, 𝑁𝑅𝑧)]} 2

𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

1 + 𝑑 (𝑁𝑅𝑆𝑧, 𝐾𝑆𝑧) = 0; 1 + 𝑑 (𝑁𝑅𝑆𝑧, 𝑀𝑆𝑧)

Δ 2 (𝑆𝑧, 𝑧) = 𝑑 (𝑁𝑅𝑆𝑧, 𝐾𝑆𝑧)

1 + 𝑑 (𝑀𝑆𝑧, 𝐿𝑧) = 0; 1 + 𝑑 (𝑁𝑅𝑆𝑧, 𝑀𝑆𝑧)

Δ 3 (𝑆𝑧, 𝑧) = max {𝑑 (𝑁𝑅𝑆𝑧, 𝑀𝑆𝑧) , 𝑑 (𝑁𝑅𝑆𝑧, 𝐾𝑆𝑧) , 1 𝑑 (𝑀𝑆𝑧, 𝐿𝑧) , [𝑑 (𝐾𝑆𝑧, 𝑀𝑆𝑧) + 𝑑 (𝐿𝑧, 𝑁𝑅𝑆𝑧)]} 2

Therefore, 𝑑(𝐾𝑧,𝑧)

(33)

where

= 𝑑 (𝐾𝑧, 𝑧) .

0

Δ 𝑗 (𝑆𝑧,𝑧)

0

where

∫


𝜑 (𝑡) 𝑑𝑡 ≾ 𝜓 (max {0, 0, ∫

𝑑(𝐾𝑧,𝑧)

0

≾ 𝜓 (∫

𝑑(𝐾𝑧,𝑧)

0

≺∫

𝑑(𝐾𝑧,𝑧)

0

= max {𝑑 (𝑆𝑧, 𝑧) , 𝑑 (𝑆𝑧, 𝑆𝑧) , 𝑑 (𝑧, 𝑧) ,


𝜑 (𝑡) 𝑑𝑡)

(34)

(27)

1 [𝑑 (𝑆𝑧, 𝑧) + 𝑑 (𝑧, 𝑆𝑧)]} = 𝑑 (𝑆𝑧, 𝑧) . 2 Therefore,


which is impossible. Thus 𝐾𝑧 = 𝑧 and hence, in view of (23), we get 𝐾𝑧 = 𝑁𝑅𝑧 = 𝑧.

(28)

Similarly, we can show that 𝐿𝑧 = 𝑀𝑆𝑧 = 𝑧.

(29)

Hence, from (28) and (29), we get 𝐾𝑧 = 𝐿𝑧 = 𝑀𝑆𝑧 = 𝑁𝑅𝑧 = 𝑧.

∫

𝑑(𝑆𝑧,𝑧)

0

𝜑 (𝑡) 𝑑𝑡 ≾ 𝜓 (max {0, 0, ∫

(𝑆𝑧,𝑧)

0

≺∫

(𝑆𝑧,𝑧)

0



which is a contradiction; thus 𝑆𝑧 = 𝑧. Also 𝑀𝑧 = 𝑧 as 𝑀𝑆𝑧 = 𝑧, so from (30) it follows that 𝐾𝑧 = 𝐿𝑧 = 𝑀𝑧 = 𝑆𝑧 = 𝑁𝑅𝑧 = 𝑧.

(30)

(35)

(36)

Now, by commuting conditions of pairs (𝐾, 𝑆) and (𝑁𝑅, 𝑆) and using (28) and (30), we have 𝐾(𝑆𝑧) = 𝑆(𝐾𝑧) = 𝑆𝑧 and 𝑁𝑅(𝑆𝑧) = 𝑆(𝑁𝑅𝑧) = 𝑆𝑧; from here it follows that

Similarly, using condition (2) of Theorem 14 with 𝑧1 = 𝑧 and 𝑧2 = 𝑅𝑧 and taking (28) and (32), one can easily obtain that 𝑅𝑧 = 𝑧. Also 𝑁𝑧 = 𝑧 as 𝑁𝑅𝑧 = 𝑧. Hence, from (36), we get

𝐾 (𝑆𝑧) = 𝑁𝑅 (𝑆𝑧) = 𝑆𝑧.

𝐾𝑧 = 𝐿𝑧 = 𝑀𝑧 = 𝑁𝑧 = 𝑅𝑧 = 𝑆𝑧 = 𝑧.

(31)

Also, by commuting conditions of pairs (𝐿, 𝑅) and (𝑀𝑆, 𝑅) and taking (29) and (30), we have 𝐿(𝑅𝑧) = 𝑅(𝐿𝑧) = 𝑅𝑧 and 𝑀𝑆(𝑅𝑧) = 𝑅(𝑀𝑆𝑧) = 𝑅𝑧; from here it follows that 𝐿 (𝑅𝑧) = 𝑀𝑆 (𝑅𝑧) = 𝑅𝑧.

(32)

(37)

That is 𝑧 is a common fixed point of 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆 in 𝑋. Similarly, if (𝐿, 𝑀𝑆) satisfies property (E.A) and 𝑁𝑅(𝑋) is closed subspace of 𝑋, then we can prove that 𝑧 is a common

6


fixed point of 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆 in 𝑋 in the same arguments as above. Uniqueness. For the uniqueness of common fixed point, let 𝑧∗ ≠ 𝑧 be another fixed point of 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆. Then, using condition (2) of Theorem 14, we have ∗

∫

𝑑(𝑧,𝑧 )

0

∗

𝑑(𝐾𝑧,𝐿𝑧 )

𝜑 (𝑡) 𝑑𝑡 = ∫

0

≾ 𝜓 (max {∫

Δ 𝑗 (𝑧,𝑧∗ )

0

Δ 1 (𝑧1 , 𝑧2 ) = 𝑑 (𝑀𝑆𝑧2 , 𝐾𝑧2 )

1 + 𝑑 (𝑁𝑅𝑧1 , 𝐾𝑧1 ) ; 1 + 𝑑 (𝑁𝑅𝑧1 , 𝑀𝑆𝑧2 )

Δ 2 (𝑧1 , 𝑧2 ) = 𝑑 (𝑁𝑅𝑧1 , 𝐾𝑧1 )

1 + 𝑑 (𝑀𝑆𝑧2 , 𝐾𝑧2 ) ; 1 + 𝑑 (𝑁𝑅𝑧1 , 𝑀𝑆𝑧2 )

𝜑 (𝑡) 𝑑𝑡 (38)

𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,


(42)

𝑑 (𝑀𝑆𝑧2 , 𝐾𝑧2 ) , 1 [𝑑 (𝐾𝑧1 , 𝑀𝑆𝑧2 ) + 𝑑 (𝐾𝑧2 , 𝑁𝑅𝑧1 )]} . 2

where Δ 1 (𝑧, 𝑧∗ ) = 𝑑 (𝑀𝑆𝑧∗ , 𝐿𝑧∗ ) Δ 2 (𝑧, 𝑧∗ ) = 𝑑 (𝑁𝑅𝑧, 𝐾𝑧)

1 + 𝑑 (𝑁𝑅𝑧, 𝐾𝑧) = 0; 1 + 𝑑 (𝑁𝑅𝑧, 𝑀𝑆𝑧∗ )

1 + 𝑑 (𝑀𝑆𝑧∗ , 𝐿𝑧∗ ) = 0; 1 + 𝑑 (𝑁𝑅𝑧, 𝑀𝑆𝑧∗ )

Δ 3 (𝑧, 𝑧∗ ) = max {𝑑 (𝑁𝑅𝑧, 𝑀𝑆𝑧∗ ) , 𝑑 (𝑁𝑅𝑧, 𝐾𝑧) , 𝑑 (𝑀𝑆𝑧∗ , 𝐿𝑧∗ ) ,

(39)

If one of 𝑀𝑆(𝑋) and 𝑁𝑅(𝑋) is closed subspace of 𝑋 such that pairs (𝐾, 𝑁𝑅) and (𝐾, 𝑀𝑆) are weakly compatible, then each pair of pairs (𝐾, 𝑁𝑅) and (𝐾, 𝑀𝑆) has a coincidence point in 𝑋. Moreover, if (𝐾, 𝑆), (𝐾, 𝑅), (𝑀𝑆, 𝑅), and (𝑁𝑅, 𝑆) are commuting pairs, then 𝐾, 𝑀, 𝑁, 𝑅, and 𝑆 have a unique common fixed point in 𝑋. Corollary 16. Let (𝑋, 𝑑) be a complex valued metric space and 𝐾, 𝐿, 𝑅, 𝑆 : 𝑋 → 𝑋 be four self-mappings satisfying the following conditions:

𝑑 (𝐾𝑧, 𝑀𝑆𝑧∗ ) + 𝑑 (𝐿𝑧∗ , 𝑁𝑅𝑧) } 2

(1) One of the pairs (𝐾, 𝑆) and (𝐿, 𝑅) satisfies property (𝐸.𝐴) such that 𝐾(𝑋) ⊆ 𝑅(𝑋) and 𝐿(𝑋) ⊆ 𝑆(𝑋).

= 𝑑 (𝑧, 𝑧∗ ) .

(2) ∀𝑧1 , 𝑧2 ∈ 𝑋.

Thus, ∫

where 𝜓 ∈ Ψ, 𝜑 ∈ Φ∗ and

𝑑(𝑧,𝑧∗ )

0

𝜑 (𝑡) 𝑑𝑡 ≾ 𝜓 (max {0, 0, ∫

𝑑(𝑧,𝑧∗ )

0


∗

𝑑(𝑧,𝑧 )

≺∫

0

(40)


Corollary 15. Let (𝑋, 𝑑) be a complex valued metric space and 𝐾, 𝑀, 𝑁, 𝑅, 𝑆 : 𝑋 → 𝑋 be five self-mappings satisfying the following conditions: (1) One of pairs (𝐾, 𝑁𝑅) and (𝐾, 𝑀𝑆) satisfies property (𝐸.𝐴) such that 𝐾(𝑋) ⊆ 𝑀𝑆(𝑋) and 𝐾(𝑋) ⊆ 𝑁𝑅(𝑋). (2) ∀𝑧1 , 𝑧2 ∈ 𝑋.

0

0


𝜑 (𝑡) 𝑑𝑡 Δ 𝑗 (𝑧1 ,𝑧2 )

0

(41) 𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

(43) 𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

where 𝜓 ∈ Ψ, 𝜑 ∈ Φ∗ and Δ 1 (𝑧1 , 𝑧2 ) = 𝑑 (𝑅𝑧2 , 𝐿𝑧2 )

1 + 𝑑 (𝑆𝑧1 , 𝐾𝑧1 ) ; 1 + 𝑑 (𝑆𝑧1 , 𝑅𝑧2 )

Δ 2 (𝑧1 , 𝑧2 ) = 𝑑 (𝑆𝑧1 , 𝐾𝑧1 )

1 + 𝑑 (𝑅𝑧2 , 𝐿𝑧2 ) ; 1 + 𝑑 (𝑆𝑧1 , 𝑅𝑧2 )

(44)

Δ 3 (𝑧1 , 𝑧2 ) = max {𝑑 (𝑆𝑧1 , 𝑅𝑧2 ) , 𝑑 (𝑆𝑧1 , 𝐾𝑧1 ) , 𝑑 (𝑅𝑧2 , 𝐿𝑧2 ) ,

≾ 𝜓 (max {∫

Δ 𝑗 (𝑧1 ,𝑧2 )

0

Now we present some corollaries; their proofs are easily followed from Theorem 14, so we omit the proofs.

𝑑(𝐾𝑧1 ,𝐾𝑧2 )


≾ 𝜓 (max {∫

which is a contradiction; hence 𝑧 is a unique common fixed point of 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆 in 𝑋.

∫

∫

1 [𝑑 (𝐾𝑧1 , 𝑅𝑧2 ) + 𝑑 (𝐿𝑧2 , 𝑆𝑧1 )]} . 2

If one of 𝑅(𝑋) and 𝑆(𝑋) is closed subspace of 𝑋, then pairs (𝐾, 𝑆) and (𝐿, 𝑅) have a coincidence point in 𝑋. Moreover, if (𝐾, 𝑆) and (𝐿, 𝑅) are weakly compatible, then 𝐾, 𝐿, 𝑅, and 𝑆 have a unique common fixed point in 𝑋.


7

Corollary 17. Let (𝑋, 𝑑) be a complex valued metric space and 𝐾, 𝐿, 𝑅 : 𝑋 → 𝑋 be three self-mappings satisfying the following conditions: (1) One of the pairs (𝐾, 𝑅) and (𝐿, 𝑅) satisfies property (𝐸.𝐴) such that 𝐾(𝑋) ⊆ 𝑅(𝑋) and 𝐿(𝑋) ⊆ 𝑅(𝑋). (2) ∀𝑧1 , 𝑧2 ∈ 𝑋. ∫


0


≾ 𝜓 (max {∫

Δ 𝑗 (𝑧1 ,𝑧2 )

0

(45) 𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

Δ 2 (𝑧1 , 𝑧2 ) = 𝑑 (𝑅𝑧1 , 𝐾𝑧1 )

1 + 𝑑 (𝑅𝑧1 , 𝐾𝑧1 ) ; 1 + 𝑑 (𝑅𝑧1 , 𝑅𝑧2 ) 1 + 𝑑 (𝑅𝑧2 , 𝐿𝑧2 ) ; 1 + 𝑑 (𝑅𝑧1 , 𝑅𝑧2 )

∫

(46)

(1) Pair (𝐾, 𝐿) satisfies property (𝐸.𝐴). (2) ∀𝑧1 , 𝑧2 ∈ 𝑋.

0


≾ 𝜓 (max {∫

Δ 𝑗 (𝑧1 ,𝑧2 )

0

(47) 𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

where 𝜓 ∈ Ψ, 𝜑 ∈ Φ∗ and Δ 1 (𝑧1 , 𝑧2 ) = 𝑑 (𝐾𝑧2 , 𝐿𝑧2 )

1 + 𝑑 (𝐿𝑧1 , 𝐾𝑧1 ) ; 1 + 𝑑 (𝐿𝑧1 , 𝐾𝑧2 )

Δ 2 (𝑧1 , 𝑧2 ) = 𝑑 (𝐿𝑧1 , 𝐾𝑧1 )

1 + 𝑑 (𝐾𝑧2 , 𝐿𝑧2 ) ; 1 + 𝑑 (𝐿𝑧1 , 𝐾𝑧2 )

Δ 3 (𝑧1 , 𝑧2 ) = max {𝑑 (𝐿𝑧1 , 𝐾𝑧2 ) , 𝑑 (𝐿𝑧1 , 𝐾𝑧1 ) , 𝑑 (𝐾𝑧2 , 𝐿𝑧2 ) ,

1 [𝑑 (𝐾𝑧1 , 𝐾𝑧2 ) + 𝑑 (𝐿𝑧2 , 𝐿𝑧1 )]} . 2

𝜑 (𝑡) 𝑑𝑡 ≾ 𝜓 (∫

Δ 3 (𝑧1 ,𝑧2 )

0

𝜑 (𝑡) 𝑑𝑡) ,

(49)

where 𝜓 ∈ Ψ, 𝜑 ∈ Φ∗ and Δ 3 (𝑧1 , 𝑧2 ) = max {𝑑 (𝑁𝑅𝑧1 , 𝑀𝑆𝑧2 ) , 𝑑 (𝑁𝑅𝑧1 , 𝐾𝑧1 ) ,

1 [𝑑 (𝐾𝑧1 , 𝑅𝑧2 ) + 𝑑 (𝐿𝑧2 , 𝑅𝑧1 )]} . 2

Corollary 18. Let (𝑋, 𝑑) be a complex valued metric space and 𝐾, 𝐿 : 𝑋 → 𝑋 be two self-mappings satisfying the following conditions:

∫


0

If 𝑅(𝑋) is closed subspace of 𝑋, then pairs (𝐾, 𝑅) and (𝐿, 𝑅) have a coincidence point in 𝑋. Moreover, if (𝐾, 𝑅) and (𝐿, 𝑅) are weakly compatible, then 𝐾, 𝐿, and 𝑅 have a unique common fixed point in 𝑋.


Theorem 19. Let (𝑋, 𝑑) be a complex valued metric space and 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, 𝑆 : 𝑋 → 𝑋 be six self-mappings satisfying the following conditions:

(2) ∀𝑧1 , 𝑧2 ∈ 𝑋.

Δ 3 (𝑧1 , 𝑧2 ) = max {𝑑 (𝑅𝑧1 , 𝑅𝑧2 ) , 𝑑 (𝑅𝑧1 , 𝐾𝑧1 ) , 𝑑 (𝑅𝑧2 , 𝐿𝑧2 ) ,

Similar to the arguments of Theorem 14, we conclude the following result and omit their proof.

(1) One of pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) satisfies property (𝐸.𝐴) such that 𝐾(𝑋) ⊆ 𝑀𝑆(𝑋) and 𝐿(𝑋) ⊆ 𝑁𝑅(𝑋).

where 𝜓 ∈ Ψ, 𝜑 ∈ Φ∗ and Δ 1 (𝑧1 , 𝑧2 ) = 𝑑 (𝑅𝑧2 , 𝐿𝑧2 )

If 𝐾(𝑋) is closed subspace of 𝑋, then pair (𝐾, 𝐿) has a coincidence point in 𝑋. Moreover, if (𝐾, 𝐿) is weakly compatible, then mappings 𝐾 and 𝐿 have a unique common fixed point in 𝑋.

𝑑 (𝑀𝑆𝑧2 , 𝐿𝑧2 ) ,

(50)

1 [𝑑 (𝐾𝑧1 , 𝑀𝑆𝑧2 ) + 𝑑 (𝐿𝑧2 , 𝑁𝑅𝑧1 )]} . 2

If one of 𝑀𝑆(𝑋) and 𝑁𝑅(𝑋) is closed subspace of 𝑋 such that pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) are weakly compatible, then each pair of pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) has a coincidence point in 𝑋. Moreover, if (𝐾, 𝑆), (𝐿, 𝑅), (𝑀𝑆, 𝑅), and (𝑁𝑅, 𝑆) are commuting pairs, then 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆 have a unique common fixed point in 𝑋. Theorem 20. Let (𝑋, 𝑑) be a complex valued metric space and 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, 𝑆 : 𝑋 → 𝑋 be six self-mappings satisfying condition (2) of Theorem 14 and either pair (𝐾, 𝑁𝑅) satisfies (𝐶𝐿𝑅𝐾 ) property or pair (𝐿, 𝑀𝑆) satisfies (𝐶𝐿𝑅𝐿 ) property such that 𝐾(𝑋) ⊆ 𝑀𝑆(𝑋) and 𝐿(𝑋) ⊆ 𝑁𝑅(𝑋). If pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) are weakly compatible, then each pair of pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) has a coincidence point in 𝑋. Moreover, if (𝐾, 𝑆), (𝐿, 𝑅), (𝑀𝑆, 𝑅), and (𝑁𝑅, 𝑆) are commuting pairs, then 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆 have a unique common fixed point in 𝑋. Proof. Suppose that pair (𝐾, 𝑁𝑅) satisfies (CLR𝐾 ) property, then there exists sequence {𝑧𝑛 } in 𝑋 such that

(48)

lim 𝐾𝑧𝑛 = lim 𝑁𝑅𝑧𝑛 = 𝐾𝑡 for some 𝑡 ∈ 𝑋.

𝑛→∞

𝑛→∞

(51)

Since 𝐾(𝑋) ⊆ 𝑀𝑆(𝑋), there exists 𝑢 ∈ 𝑋 such that 𝐾𝑡 = 𝑀𝑆𝑢. We claim that 𝐿𝑢 = 𝑀𝑆𝑢. To support the claim, let 𝐿𝑢 ≠ 𝑀𝑆𝑢. Then on using condition (2) of Theorem 14, with setting

8


𝑧1 = 𝑧𝑛 and 𝑧2 = 𝑢, we have ∫


0

Now, we assert that 𝐾V = 𝑁𝑅V. Let on contrary 𝐾V ≠ 𝑁𝑅V; then setting 𝑧1 = V and 𝑧2 = 𝑢, in condition (2) of Theorem 14, we get

𝜑 (𝑡) 𝑑𝑡 Δ 𝑗 (𝑧𝑛 ,𝑢)

≾ 𝜓 (max {∫

0

(52) 𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

∫

𝑑(𝐾V,𝐿𝑢)

0

≾ 𝜓 (max {∫

where

Δ 𝑗 (V,𝑢)

0

Δ 1 (𝑧𝑛 , 𝑢) = 𝑑 (𝑀𝑆𝑢, 𝐿𝑢)

1 + 𝑑 (𝑁𝑅𝑧𝑛 , 𝐾𝑧𝑛 ) ; 1 + 𝑑 (𝑁𝑅𝑧𝑛 , 𝑀𝑆𝑢)

1 + 𝑑 (𝑀𝑆𝑢, 𝐿𝑢) ; Δ 2 (𝑧𝑛 , 𝑢) = 𝑑 (𝑁𝑅𝑧𝑛 , 𝐾𝑧𝑛 ) 1 + 𝑑 (𝑁𝑅𝑧𝑛 , 𝑀𝑆𝑢)

(53)

Δ 1 (V, 𝑢) = 𝑑 (𝑀𝑆𝑢, 𝐿𝑢)

1 + 𝑑 (𝑁𝑅V, 𝐾V) ; 1 + 𝑑 (𝑁𝑅V, 𝑀𝑆𝑢)

Δ 2 (V, 𝑢) = 𝑑 (𝑁𝑅V, 𝐾V)

1 + 𝑑 (𝑀𝑆𝑢, 𝐿𝑢) ; 1 + 𝑑 (𝑁𝑅V, 𝑀𝑆𝑢)


Taking upper limit as 𝑛 → ∞ in (52) and using (51), we get

Δ 2 (𝑧𝑛 , 𝑢) 󳨀→ 0,

∫

Δ 3 (𝑧𝑛 , 𝑢) 󳨀→ 𝑑 (𝐿𝑢, 𝐾𝑡) ,

≾ lim sup 𝜓 (max {∫ 𝑛→∞

𝑛→∞

𝑑(𝐾𝑡,𝐿𝑢)

0

𝑑(𝐿𝑢,𝐾𝑡)

0

0



0

= 𝜓 (max {∫ ∫


0


0

𝜑 (𝑡) 𝑑𝑡}) ≾ 𝜓 (∫

𝑑(𝐾𝑡,𝐾V)


0


≺∫

0

𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) (54)


𝜑 (𝑡) 𝑑𝑡 ≾ 𝜓 (max {0, ∫


0

𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3})

0

≺∫

1 [𝑑 (𝐾V, 𝑀𝑆𝑢) + 𝑑 (𝐿𝑢, 𝑁𝑅V)]} . 2

0

∫



𝜑 (𝑡) 𝑑𝑡) (59)

𝜑 (𝑡) 𝑑𝑡 󳨐⇒

󵄨󵄨 󵄨󵄨 𝑑(𝐾V,𝐾𝑡) 󵄨󵄨󵄨 󵄨󵄨󵄨 𝑑(𝐾𝑡,𝐾V) 󵄨󵄨 󵄨󵄨 󵄨󵄨 < 󵄨󵄨∫ 󵄨󵄨 , 󵄨󵄨∫ 𝜑 𝑑𝑡 𝜑 𝑑𝑡 (𝑡) (𝑡) 󵄨 󵄨 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 0 󵄨 󵄨 󵄨 󵄨 which is impossible. Thus 𝐾V = 𝐾𝑡 and hence

𝜑 (𝑡) 𝑑𝑡, 0,

𝜑 (𝑡) 𝑑𝑡}) = 𝜓 (∫


𝑑(𝐾V,𝐾𝑡)

0

𝜑 (𝑡) 𝑑𝑡 = lim sup ∫ 𝑛→∞

0

(58)

Using (56), we have

Δ 1 (𝑧𝑛 , 𝑢) 󳨀→ 𝑑 (𝐾𝑡, 𝐿𝑢) ,

𝑑(𝐾𝑡,𝐿𝑢)

𝜑 (𝑡) 𝑑𝑡 : 1 ≤ 𝑗 ≤ 3}) ,

Δ 3 (V, 𝑢) = max {𝑑 (𝑁𝑅V, 𝑀𝑆𝑢) , 𝑑 (𝑁𝑅V, 𝐾V) ,

1 [𝑑 (𝐾𝑧𝑛 , 𝑀𝑆𝑢) + 𝑑 (𝐿𝑢, 𝑁𝑅𝑧𝑛 )]} . 2


(57)

where

Δ 3 (𝑧𝑛 , 𝑢) = max {𝑑 (𝑁𝑅𝑧𝑛 , 𝑀𝑆𝑢) , 𝑑 (𝑁𝑅𝑧𝑛 , 𝐾𝑧𝑛 ) ,

∫


𝐾V = 𝑁𝑅V = 𝐾𝑡.

𝜑 (𝑡) 𝑑𝑡)

Therefore, from (56) and (60), we get

𝜑 (𝑡) 𝑑𝑡 󳨐⇒

𝐾V = 𝐿𝑢 = 𝑀𝑆𝑢 = 𝑁𝑅V = 𝐾𝑡 = 𝑧 (say) .

󵄨󵄨 󵄨󵄨 𝑑(𝐾𝑡,𝐿𝑢) 󵄨󵄨 󵄨󵄨 𝑑(𝐾𝑡,𝐿𝑢) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 < 󵄨󵄨∫ 󵄨󵄨 , 󵄨󵄨∫ 𝜑 𝑑𝑡 𝜑 𝑑𝑡 (𝑡) (𝑡) 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 0 󵄨 󵄨 󵄨 󵄨

(55)

Also, since 𝐿(𝑋) ⊆ 𝑁𝑅(𝑋), there exists V ∈ 𝑋 such that 𝐿𝑢 = 𝑁𝑅V. Thus (55) becomes 𝐿𝑢 = 𝑀𝑆𝑢 = 𝑁𝑅V = 𝐾𝑡.

(61)

Finally, following the lines in the proof of Theorem 14 we can show that 𝑧 is the coincident point of pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) and is a unique common fixed point of the mappings 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆.

which is a contradiction. Thus 𝐿𝑢 = 𝐾𝑡 and hence 𝐿𝑢 = 𝑀𝑆𝑢 = 𝐾𝑡.

(60)

(56)

Similar to the arguments of Theorem 20, we conclude the following results and omit their proofs. Theorem 21. Let (𝑋, 𝑑) be a complex valued metric space and 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, 𝑆 : 𝑋 → 𝑋 be six self-mappings satisfying the following conditions:


9

(1) Either pair (𝐾, 𝑁𝑅) satisfies (𝐶𝐿𝑅𝐾 ) property or pair (𝐿, 𝑀𝑆) satisfies (𝐶𝐿𝑅𝐿 ) property such that 𝐾(𝑋) ⊆ 𝑀𝑆(𝑋) and 𝐿(𝑋) ⊆ 𝑁𝑅(𝑋). (2) ∀𝑧1 , 𝑧2 ∈ 𝑋. ∫


0

𝜑 (𝑡) 𝑑𝑡 ≾ 𝜓 (∫

Δ 3 (𝑧1 ,𝑧2 )

0

𝜑 (𝑡) 𝑑𝑡) ,

(62)

Similarly to Theorem 14 one can derive variant of corollaries from Theorems 19, 20, and 21. Remark 23. The conclusions of Theorems 14, 19, 20, and 21 are still valid if we replace Δ 3 with Δ∗3 , where Δ∗3 (𝑧1 , 𝑧2 ) = max {𝑑 (𝑁𝑅𝑥, 𝑀𝑆𝑦) , 𝑑 (𝑁𝑅𝑥, 𝐾𝑡) , 𝑑 (𝑀𝑆𝑦, 𝐿𝑦) , 𝑑 (𝐾𝑡, 𝑀𝑆𝑦) , 𝑑 (𝐿𝑦, 𝑁𝑅𝑥)} .

Remark 24. Theorems 14 and 20 and Corollary 15 extends Theorem 2.1 of [11] in complex valued metric space. Corollary 16 generalizes the results of [8–11] in complex valued metric space. Moreover, the real valued metric space version of our main results generalizes the results of [8–11].

where 𝜓 ∈ Ψ, 𝜑 ∈ Φ∗ and Δ 3 (𝑧1 , 𝑧2 ) = max {𝑑 (𝑁𝑅𝑧1 , 𝑀𝑆𝑧2 ) , 𝑑 (𝑁𝑅𝑧1 , 𝐾𝑧1 ) , 𝑑 (𝑀𝑆𝑧2 , 𝐿𝑧2 ) ,

(63)

1 [𝑑 (𝐾𝑧1 , 𝑀𝑆𝑧2 ) + 𝑑 (𝐿𝑧2 , 𝑁𝑅𝑧1 )]} . 2

To support Theorem 21, we present the following example. Example 25. Let 𝑋 = {𝑧 = 𝑥 + 𝜄𝑦̇ : 𝑥, 𝑦 ∈ [0, 1)} be a complex valued metric space with metric 𝑑 : 𝑋 × 𝑋 → C defined by

If pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) are weakly compatible, then each pair of pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) has a coincidence point in 𝑋. Moreover, if (𝐾, 𝑆), (𝐿, 𝑅), (𝑀𝑆, 𝑅), and (𝑁𝑅, 𝑆) are commuting pairs, then 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆 have a unique common fixed point in 𝑋. Corollary 22. Let (𝑋, 𝑑) be a metric space and 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, 𝑆 : 𝑋 → 𝑋 be six self-mappings satisfying the following conditions:

(2) ∀𝑧1 , 𝑧2 ∈ 𝑋. 𝑑(𝐾𝑧1 ,𝐿𝑧2 )

0

𝜑 (𝑡) 𝑑𝑡 ≤ 𝛼 ∫

Δ 3 (𝑧1 ,𝑧2 )

0


(64)

𝑧 𝑧 𝑀𝑆𝑧 = 𝑀 ( ) = , 6 12 𝑧 𝑧 𝑁𝑅𝑧 = 𝑁 ( ) = . 3 12

lim 𝐾𝑧𝑛 = lim 𝐾 (


𝑛→∞

1 𝜄̇ + ) = 0, 𝑛+1 𝑛+1

lim 𝑁𝑅𝑧𝑛 = lim 𝑁𝑅 (

𝑛→∞

(65)

If pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) are weakly compatible, then each pair (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) has a coincidence point in 𝑋. Moreover, if (𝐾, 𝑆), (𝐿, 𝑅), (𝑀𝑆, 𝑅), and (𝑁𝑅, 𝑆) are commuting pairs, then 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆 have a unique common fixed point in 𝑋.

(68)

Also, we define 𝜑 : R2 → C by 𝜑(𝑥, 𝑦) = 2 + 0𝜄 ̇ and 𝜓 : C+ → C+ by 𝜓(𝑧) = 𝑧/2. Clearly 𝐾(𝑋) = {0} ⊆ 𝑀𝑆(𝑋) = {𝑧 = 𝑥 + 𝜄𝑦̇ : 𝑥, 𝑦 ∈ [0, 1/12)} and 𝐿(𝑋) ⊆ 𝑁𝑅(𝑋). Now, we construct sequence 𝑧𝑛 = 𝑥𝑛 + 𝜄𝑦̇ 𝑛 = 1/(𝑛 + 1) + ̇ + 1) in 𝑋 such that 𝜄/(𝑛

𝑛→∞

1 [𝑑 (𝐾𝑧1 , 𝑀𝑆𝑧2 ) + 𝑑 (𝐿𝑧2 , 𝑁𝑅𝑧1 )]} . 2

(67)

Define self-maps 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆 on 𝑋 by 𝐾𝑧 = 0, 𝐿𝑧 = 0, 𝑀𝑧 = 𝑧/2, 𝑁𝑧 = 𝑧/4, 𝑅𝑧 = 𝑧/3, and 𝑆𝑧 = 𝑧/6. Then,

where 0 ≤ 𝛼 < 1, 𝜑 ∈ Φ and

𝑑 (𝑀𝑆𝑧2 , 𝐿𝑧2 ) ,

𝜋 for a given 𝜃 ∈ [0, ] . 2

󵄨 󵄨 𝑑 (𝑧1 , 𝑧2 ) = 󵄨󵄨󵄨𝑧1 − 𝑧2 󵄨󵄨󵄨 𝑒𝑖𝜃

(1) Either pair (𝐾, 𝑁𝑅) satisfies (𝐶𝐿𝑅𝐾 ) property or pair (𝐿, 𝑀𝑆) satisfies (𝐶𝐿𝑅𝐿 ) property such that 𝐾(𝑋) ⊆ 𝑀𝑆(𝑋) and 𝐿(𝑋) ⊆ 𝑁𝑅(𝑋).

∫

(66)

𝑛→∞

= lim

1

𝑛→∞ 12

(

1 𝜄̇ + ) 𝑛+1 𝑛+1

(69)

1 𝜄̇ + ) = 0. 𝑛+1 𝑛+1

that is, there exists sequence {𝑧𝑛 } in 𝑋 such that lim 𝐾𝑧𝑛 = lim 𝑁𝑅𝑧𝑛 = 0 = 𝐾𝑧 for 𝑧 = 0 + 0𝜄 ̇ ∈ 𝑋. (70) 𝑛→∞

𝑛→∞

Hence (𝐾, 𝑁𝑅) satisfies (CLR𝐾 ) property.

10

International Journal of Analysis Hence from Theorem 21, 0 is a unique common fixed point of 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, and 𝑆.

Next, check the following condition ∫


0

𝜑 (𝑡) 𝑑𝑡 ≾ 𝜓 (∫

Δ(𝑧1 ,𝑧2 )

0

𝜑 (𝑡) 𝑑𝑡) (71)

= 𝜓 ( 2𝑡|Δ(𝑧1 ,𝑧2 ) ) = Δ (𝑧1 , 𝑧2 ) , where Δ (𝑧1 , 𝑧2 ) = max {𝑑 (𝑁𝑅𝑧1 , 𝑀𝑆𝑧2 ) , 𝑑 (𝑁𝑅𝑧1 , 𝐾𝑧1 ) , 𝑑 (𝑀𝑆𝑧2 , 𝐿𝑧2 ) ,

(72) 1 󵄨󵄨 𝑧 [𝑑 (𝐾𝑧1 , 𝑀𝑆𝑧2 ) + 𝑑 (𝐿𝑧2 , 𝑁𝑅𝑧1 )]} = max {󵄨󵄨󵄨󵄨 1 2 󵄨 12 𝑧2 󵄨󵄨󵄨 𝑖𝜃 󵄨󵄨󵄨 𝑧1 󵄨󵄨󵄨 𝑖𝜃 󵄨󵄨󵄨 𝑧2 󵄨󵄨󵄨 𝑖𝜃 1 󵄨󵄨󵄨 𝑧1 󵄨󵄨󵄨 𝑖𝜃 󵄨󵄨󵄨 𝑧1 󵄨󵄨󵄨 𝑖𝜃 − 󵄨󵄨󵄨 𝑒 , 󵄨󵄨󵄨 󵄨󵄨󵄨 𝑒 , 󵄨󵄨󵄨 󵄨󵄨󵄨 𝑒 , {󵄨󵄨󵄨 󵄨󵄨󵄨 𝑒 + 󵄨󵄨󵄨 󵄨󵄨󵄨 𝑒 }} . 12 󵄨 2 󵄨 36 󵄨 󵄨 12 󵄨 󵄨 12 󵄨 󵄨 12 󵄨 Since

3. Applications Many researchers study the applications of common fixed point theorems in complex valued metric spaces; see for instance [17, 18] and the references therein. On the other hand, Liu et al. [19] and Sarwar et al. [20] study the existence and uniqueness of common solution for the system of functional equations arising in dynamic programming with real domain. We apply Corollary 22 for the existence and uniqueness of a common solution for the following system of functional equations arising in dynamic programming with complex domain (see [21]). 𝑝1 (𝑧) = opt {𝑢 (𝑧, 𝑤) + Θ1 (𝑧, 𝑤, 𝑝1 (𝜏1 (𝑧, 𝑤)))} 𝑤∈𝐷

∀𝑧 ∈ Ω, 𝑝2 (𝑧) = opt {𝑢 (𝑧, 𝑤) + Θ2 (𝑧, 𝑤, 𝑝2 (𝜏2 (𝑧, 𝑤)))}

󵄨󵄨 𝑧 󵄨󵄨 𝑧 󵄨󵄨 󵄨󵄨 𝑧 󵄨󵄨 𝑧 󵄨󵄨 0 ≾ max {󵄨󵄨󵄨󵄨 1 − 2 󵄨󵄨󵄨󵄨 𝑒𝑖𝜃 , 󵄨󵄨󵄨󵄨 1 󵄨󵄨󵄨󵄨 𝑒𝑖𝜃 , 󵄨󵄨󵄨󵄨 2 󵄨󵄨󵄨󵄨 󵄨 12 12 󵄨 󵄨 12 󵄨 󵄨 12 󵄨 󵄨 󵄨 󵄨 󵄨 1 󵄨𝑧 󵄨 󵄨𝑧 󵄨 ⋅ 𝑒𝑖𝜃 , {󵄨󵄨󵄨󵄨 1 󵄨󵄨󵄨󵄨 𝑒𝑖𝜃 + 󵄨󵄨󵄨󵄨 1 󵄨󵄨󵄨󵄨 𝑒𝑖𝜃 }} , 2 󵄨 36 󵄨 󵄨 12 󵄨

𝑤∈𝐷

∀𝑧 ∈ Ω, (73)

𝑝3 (𝑧) = opt {V (𝑧, 𝑤) + Θ3 (𝑧, 𝑤, 𝑝3 (𝜏3 (𝑧, 𝑤)))} 𝑤∈𝐷

∀𝑧 ∈ Ω,

therefore

𝑝4 (𝑧) = opt {V (𝑧, 𝑤) + Θ4 (𝑧, 𝑤, 𝑝4 (𝜏4 (𝑧, 𝑤)))}

(77)

𝑤∈𝐷

∫


0

∀𝑧 ∈ Ω,

𝜑 (𝑡) 𝑑𝑡 ≾ max {𝑑 (𝑁𝑅𝑧1 , 𝑀𝑆𝑧2 ) ,

𝑑 (𝑁𝑅𝑧1 , 𝐾𝑧1 ) , 𝑑 (𝑀𝑆𝑧2 , 𝐿𝑧2 ) ,

(74)

1 [𝑑 (𝐾𝑧1 , 𝑀𝑆𝑧2 ) + 𝑑 (𝐿𝑧2 , 𝑁𝑅𝑧1 )]} . 2


0

𝜑 (𝑡) 𝑑𝑡 ≾ 𝜓 (∫

Δ(𝑧1 ,𝑧2 )

0

𝜑 (𝑡) 𝑑𝑡) ,


(75)

where Δ (𝑧1 , 𝑧2 ) = max {𝑑 (𝑁𝑅𝑧1 , 𝑀𝑆𝑧2 ) , 𝑑 (𝑁𝑅𝑧1 , 𝐾𝑧1 ) , 𝑑 (𝑀𝑆𝑧2 , 𝐿𝑧2 ) ,

𝑤∈𝐷

∀𝑧 ∈ Ω,

Thus, from (71), (73), and (74) and by using the value of 𝜓, we have ∫


𝑤∈𝐷

∀𝑧 ∈ Ω, where 𝑧 and 𝑤 signify the state and decision vectors, respectively, 𝑝𝑖 (𝑧) denotes the optimal return functions with initial state 𝑧, 𝜏𝑖 : Ω × 𝐷 → Ω, Θ𝑖 : Ω × 𝐷 × C → R ∀𝑖 ∈ {1, 2, 3, 4, 5, 6}, and 𝑢, V : Ω × 𝐷 → C. Let 𝐶(Ω) be the space of all continuous real valued functions on possibly complex domain Ω with metric 𝑑 (ℎ, 𝑘) = sup |ℎ (𝑧) − 𝑘 (𝑧)|

(76)

1 [𝑑 (𝐾𝑧1 , 𝑀𝑆𝑧2 ) + 𝑑 (𝐿𝑧2 , 𝑁𝑅𝑧1 )]} . 2 Also pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) are weakly compatible and (𝐾, 𝑆), (𝐿, 𝑅), (𝑀𝑆, 𝑅), and (𝑁𝑅, 𝑆) are commuting pairs.

𝑧∈Ω

∀ℎ, 𝑘 ∈ 𝐶 (Ω) .

(78)

We prove the following result. Theorem 26. Let 𝑢, V and Θ𝑖 : Ω × 𝐷 × C → R, 𝑖 = 1, 2, . . . , 6, be bounded functions and let 𝐾, 𝐿, 𝑀, 𝑁, 𝑅, 𝑆 : 𝐶(Ω) → 𝐶(Ω) be six operators defined as


11

𝐾ℎ1 (𝑧) = opt {𝑢 (𝑧, 𝑤) + Θ1 (𝑧, 𝑤, ℎ1 (𝜏1 (𝑧, 𝑤)))}

Proof. Notice that the system of functional equations (77) has a unique bounded solution if and only if the system of operators (79) have a unique common fixed point. Now since 𝑢, V, and Θ𝑖 are bounded, there exists positive number 𝜆 such that 󵄨 󵄨 sup {|𝑢 (𝑧, 𝑤)| , |V (𝑧, 𝑤)| , 󵄨󵄨󵄨Θ𝑖 (𝑧, 𝑤, 𝑤∗ )󵄨󵄨󵄨 : (𝑧, 𝑤, 𝑤∗ ) (83) ∈ Ω × 𝐷 × C, 𝑖 = 1, 2, . . . , 6} ≤ 𝜆.

𝑤∈𝐷

∀𝑧 ∈ Ω, 𝐿ℎ2 (𝑧) = opt {𝑢 (𝑧, 𝑤) + Θ2 (𝑧, 𝑤, ℎ2 (𝜏2 (𝑧, 𝑤)))} 𝑤∈𝐷

∀𝑧 ∈ Ω, 𝑀ℎ3 (𝑧) = opt {V (𝑧, 𝑤) + Θ3 (𝑧, 𝑤, ℎ3 (𝜏3 (𝑧, 𝑤)))} 𝑤∈𝐷

∀𝑧 ∈ Ω, 𝑁ℎ4 (𝑧) = opt {V (𝑧, 𝑤) + Θ4 (𝑧, 𝑤, ℎ4 (𝜏4 (𝑧, 𝑤)))}

(79)

Now, by using properties of the theory of integration and definition of 𝜙, we conclude that, for each positive number 𝜆, there exists positive 𝛿(𝜆), such that

𝑤∈𝐷

∫ 𝜙 (𝑠) 𝑑𝑠 ≤ 𝜆 Γ

∀𝑧 ∈ Ω,

for all Γ ⊆ [0, 2𝜆] with 𝑚(Γ) ≤ 𝛿(𝜆), where 𝑚(Γ) is the Lebesgue measure of Γ. Now, we consider two possible cases.

𝑅ℎ5 (𝑧) = opt {V (𝑧, 𝑤) + Θ5 (𝑧, 𝑤, ℎ5 (𝜏5 (𝑧, 𝑤)))} 𝑤∈𝐷

∀𝑧 ∈ Ω,

Case 1. Suppose that opt𝑤∈𝐷 = sup𝑤∈𝐷. Let 𝑧 ∈ Ω and ℎ1 , ℎ2 ∈ 𝐶(Ω); then for 𝛿(𝜆) > 0 there exist 𝑤1 , 𝑤2 ∈ 𝐷 such that

𝑆ℎ6 (𝑧) = opt {V (𝑧, 𝑤) + Θ6 (𝑧, 𝑤, ℎ6 (𝜏6 (𝑧, 𝑤)))} 𝑤∈𝐷

𝐾ℎ1 (𝑧) < 𝑢 (𝑧, 𝑤1 ) + Θ1 (𝑧, 𝑤1 , ℎ1 (𝜏1 (𝑧, 𝑤1 )))

∀𝑧 ∈ Ω, for all ℎ𝑖 ∈ 𝐶(Ω) and 𝑧 ∈ Ω. Assume that the following conditions hold: (i) There exist {ℎ𝑛 } ∈ 𝐶(Ω) such that lim𝑛→∞ 𝐾ℎ𝑛 = lim𝑛→∞ 𝑁𝑅ℎ𝑛 = 𝐾ℎ∗ , for some ℎ∗ ∈ 𝐶(Ω). (ii) 𝐾(𝐶(Ω)) ⊆ 𝑀𝑆(𝐶(Ω)) such that pairs (𝐾, 𝑁𝑅) and (𝐿, 𝑀𝑆) are weakly compatible. (iii) Pairs (𝐾, 𝑆), (𝐿, 𝑅), (𝑀𝑆, 𝑅), and (𝑁𝑅, 𝑆) are commuting. (iv) For ℎ1 , ℎ2 ∈ 𝐶(Ω). |Θ1 (𝑧,𝑤,ℎ1 (𝜏(𝑧,𝑤)))−Θ2 (𝑧,𝑤,ℎ2 (𝜏(𝑧,𝑤)))|

∫

0

≤ 𝛼∫

Δ 3 (ℎ1 ,ℎ2 )

0

(84)

+ 𝛿 (𝜆) , 𝐿ℎ2 (𝑧) < 𝑢 (𝑧, 𝑤2 ) + Θ2 (𝑧, 𝑤2 , ℎ2 (𝜏2 (𝑧, 𝑤2 ))) + 𝛿 (𝜆) ,

(87)

𝐿ℎ2 (𝑧) ≥ 𝑢 (𝑧, 𝑤1 ) + Θ2 (𝑧, 𝑤1 , ℎ2 (𝜏2 (𝑧, 𝑤1 ))) .

(88)

From inequalities (85) and (88) it follows that 𝐾ℎ1 (𝑧) − 𝐿ℎ2 (𝑧) < Θ1 (𝑧, 𝑤1 , ℎ1 (𝜏1 (𝑧, 𝑤1 ))) 󵄨 ≤ 󵄨󵄨󵄨Θ1 (𝑧, 𝑤1 , ℎ1 (𝜏1 (𝑧, 𝑤1 )))

(80) 𝜑 (𝑡) 𝑑𝑡,

(86)

𝐾ℎ1 (𝑧) ≥ 𝑢 (𝑧, 𝑤2 ) + Θ1 (𝑧, 𝑤2 , ℎ1 (𝜏1 (𝑧, 𝑤2 ))) ,

− Θ2 ((𝑧, 𝑤1 , ℎ2 (𝜏2 (𝑧, 𝑤1 )))) + 𝛿 (𝜆) 𝜑 (𝑡) 𝑑𝑡

(85)

(89)

󵄨 − Θ2 ((𝑧, 𝑤1 , ℎ2 (𝜏2 (𝑧, 𝑤1 ))))󵄨󵄨󵄨 + 𝛿 (𝜆) which gives 󵄨 𝐾ℎ1 (𝑧) − 𝐿ℎ2 (𝑧) < max {󵄨󵄨󵄨Θ1 (𝑧, 𝑤1 , ℎ1 (𝜏1 (𝑧, 𝑤1 )))

where

󵄨 − Θ2 ((𝑧, 𝑤1 , ℎ2 (𝜏2 (𝑧, 𝑤1 ))))󵄨󵄨󵄨 + 𝛿 (𝜆) ,

󵄨 󵄨 󵄨 󵄨 Δ 3 (ℎ1 , ℎ2 ) = max {󵄨󵄨󵄨𝑁𝑅ℎ1 − 𝑀𝑆ℎ2 󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑁𝑅ℎ1 − 𝐾ℎ1 󵄨󵄨󵄨 , 󵄨󵄨 󵄨 1 󵄨 󵄨 󵄨 󵄨 󵄨󵄨𝑀𝑆ℎ2 − 𝐿ℎ2 󵄨󵄨󵄨 , {󵄨󵄨󵄨𝐾ℎ1 − 𝑀𝑆ℎ2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐿ℎ2 − 𝑁𝑅ℎ1 󵄨󵄨󵄨}} , 2

(81)

𝜀

0

(90)

󵄨 − Θ2 ((𝑧, 𝑤2 , ℎ2 (𝜏2 (𝑧, 𝑤2 ))))󵄨󵄨󵄨 + 𝛿 (𝜆)} .

where ℎ1 ∈ 𝐶(Ω), 0 ≤ 𝛼 < 1, and 𝜙 : R+ → R+ is a nonnegative summable Lebesgue integrable function such that ∫ 𝜙 (𝑠) 𝑑𝑠 > 0

󵄨󵄨 󵄨󵄨Θ1 (𝑧, 𝑤2 , ℎ1 (𝜏1 (𝑧, 𝑤2 )))

(82)

for each 𝜀 > 0. Then the system of functional equations (77) has a unique bounded solution.

Similarly, using inequalities (86) and (87) we obtain 󵄨 𝐿ℎ2 (𝑧) − 𝐾ℎ1 (𝑧) < max {󵄨󵄨󵄨Θ1 (𝑧, 𝑤1 , ℎ1 (𝜏1 (𝑧, 𝑤1 ))) 󵄨 − Θ2 ((𝑧, 𝑤1 , ℎ2 (𝜏2 (𝑧, 𝑤1 ))))󵄨󵄨󵄨 + 𝛿 (𝜆) ,

󵄨󵄨 󵄨󵄨Θ1 (𝑧, 𝑤2 , ℎ1 (𝜏1 (𝑧, 𝑤2 )))

󵄨 − Θ2 ((𝑧, 𝑤2 , ℎ2 (𝜏2 (𝑧, 𝑤2 ))))󵄨󵄨󵄨 + 𝛿 (𝜆)} .

(91)

12


Therefore from (90) and (91) we get

∫

0

󵄨 󵄨 󵄨󵄨 󵄨󵄨𝐾ℎ1 (𝑧) − 𝐿ℎ2 (𝑧)󵄨󵄨󵄨 < max {󵄨󵄨󵄨Θ1 (𝑧, 𝑤1 , ℎ1 (𝜏1 (𝑧, 𝑤1 ))) 󵄨 − Θ2 ((𝑧, 𝑤1 , ℎ2 (𝜏2 (𝑧, 𝑤1 ))))󵄨󵄨󵄨 + 𝛿 (𝜆) , 󵄨󵄨 󵄨󵄨Θ1 (𝑧, 𝑤2 , ℎ1 (𝜏1 (𝑧, 𝑤2 ))) 󵄨 − Θ2 ((𝑧, 𝑤2 , ℎ2 (𝜏2 (𝑧, 𝑤2 ))))󵄨󵄨󵄨 + 𝛿 (𝜆)} < max {𝐴

(92)

𝐵+𝛿(𝜆)

where 𝐴 = |Θ1 (𝑧, 𝑤1 , ℎ1 (𝜏1 (𝑧, 𝑤1 ))) − Θ2 ((𝑧, 𝑤1 , ℎ2 (𝜏2 (𝑧, 𝑤1 ))))| and 𝐵 = |Θ1 (𝑧, 𝑤2 , ℎ1 (𝜏1 (𝑧, 𝑤2 ))) − Θ2 ((𝑧, 𝑤2 , ℎ2 (𝜏2 (𝑧, 𝑤2 ))))|. Case 2. Suppose that opt𝑤∈𝐷 = inf 𝑤∈𝐷. By following the procedure in Case 1, one can check that (92) holds. Now, from (3.10), we have

0

max{𝐴+𝛿(𝜆),𝐵+𝛿(𝜆)}

𝜙 (𝑡) 𝑑𝑡, ∫

𝐵+𝛿(𝜆)

0

𝐴

𝐴+𝛿(𝜆)

0

𝐴

= max {∫ 𝜑 (𝑡) 𝑑𝑡 + ∫

𝐵

𝜙 (𝑡) 𝑑𝑡 < 𝛼 ∫

𝐴+𝛿(𝜆)

0

+∫

|𝐾ℎ1 (𝑧)−𝐿ℎ2 (𝑧)|

𝜙 (𝑡) 𝑑𝑡 < ∫

𝐵

0

𝐴

(93)

𝜑 (𝑡) 𝑑𝑡} 𝐴

𝐵

0

0

= max {∫ 𝜑 (𝑡) 𝑑𝑡, ∫ 𝜑 (𝑡) 𝑑𝑡} + max {∫

𝐴+𝛿(𝜆)

𝐴

𝜑 (𝑡) 𝑑𝑡, ∫

𝐵+𝛿(𝜆)

𝐵

𝜑 (𝑡) 𝑑𝑡} .

And, by condition (iv) of Theorem 26, we get

0

+ max {∫

𝜙 (𝑡) 𝑑𝑡}

𝜑 (𝑡) 𝑑𝑡, ∫ 𝜑 (𝑡) 𝑑𝑡

max{|𝑁𝑅ℎ1 −𝑀𝑆ℎ2 |,|𝑁𝑅ℎ1 −𝐾ℎ1 |,|𝑀𝑆ℎ2 −𝐿ℎ2 |,(1/2){|𝐾ℎ1 −𝑀𝑆ℎ2 |+|𝐿ℎ2 −𝑁𝑅ℎ1 |}}

𝐴+𝛿(𝜆)

𝜙 (𝑡) 𝑑𝑡

0

= max {∫

+ 𝛿 (𝜆) , 𝐵 + 𝛿 (𝜆)} ,

∫

|𝐾ℎ1 (𝑧)−𝐿ℎ2 (𝑧)|

𝜑 (𝑡) 𝑑𝑡, ∫

𝐵+𝛿(𝜆)

𝐵

𝜙 (𝑡) 𝑑𝑡 (94)

𝜑 (𝑡) 𝑑𝑡} ,

and using (84) we get ∫

|𝐾ℎ1 (𝑧)−𝐿ℎ2 (𝑧)|

0

𝜙 (𝑡) 𝑑𝑡 < 𝛼 ∫

max{|𝑁𝑅ℎ1 −𝑀𝑆ℎ2 |,|𝑁𝑅ℎ1 −𝐾ℎ1 |,|𝑀𝑆ℎ2 −𝐿ℎ2 |,(1/2){|𝐾ℎ1 −𝑀𝑆ℎ2 |+|𝐿ℎ2 −𝑁𝑅ℎ1 |}}

0

Since above inequality is true for each 𝑧 ∈ Ω and 𝜆 > 0 is taken arbitrarily, we deduce that 𝑑(𝐾ℎ1 ,𝐿ℎ2 )

∫

0

𝜙 (𝑡) 𝑑𝑡 ≤ 𝛼 ∫

Δ 3 (ℎ1 ,ℎ2 )

0

𝜙 (𝑡) 𝑑𝑡,

(95)

Authors’ Contributions All authors read and approved the final version.

(96)

References

where Δ 3 (ℎ1 , ℎ2 ) = max {𝑑 (𝑁𝑅ℎ1 , 𝑀𝑆ℎ2 ) , 𝑑 (𝑁𝑅ℎ1 , 𝐾ℎ1 ) , 𝑑 (𝑀𝑆ℎ2 , 𝐿ℎ2 ) ,

𝜙 (𝑡) 𝑑𝑡 + 𝜆.

(97)

1 {𝑑 (𝐾ℎ1 , 𝑀𝑆ℎ2 ) + 𝑑 (𝐿ℎ2 , 𝑁𝑅ℎ1 )}} . 2

Also, from condition (i) of Theorem 26 pair (𝐾, 𝑁𝑅) satisfies (CLR) property. Thus all hypothesis of Corollary 22 are satisfied. Consequently operators (79) have a unique common fixed point, that is, system (77) of functional equations has a unique bounded solution.

Competing Interests The authors declare that they have no competing interests regarding this manuscript.

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