Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 408086, 17 pages
doi:10.1155/2010/408086
Research Article

Common Fixed Point Theorem for Non-Self Mappings Satisfying Generalized Ćirić Type Contraction Condition in Cone Metric Space

R. Sumitra,1 V. Rhymend Uthariaraj,2 R. Hemavathy,3 and P. Vijayaraju4

1Department of Mathematics, SMK Fomra Institute of Technology, Chennai-603 103, Tamilnadu, India
2Ramanujan Computing Centre, Anna University Chennai, Chennai-600 025, Tamilnadu, India
3Department of Mathematics, Easwari Engineering College, Ramapuram, Chennai-600 089, Tamilnadu, India
4Department of Mathematics, Anna University Chennai, Chennai-600 025, Tamilnadu, India

Received 27 November 2009; Revised 21 February 2010; Accepted 9 March 2010

Academic Editor: Naseer Shahzad

Copyright © 2010 R. Sumitra 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.

Abstract

We prove common fixed point theorem for coincidentally commuting nonself mappings satisfying generalized contraction condition of Ćirić type in cone metric space. Our results generalize and extend all the recent results related to non-self mappings in the setting of cone metric space.

1. Introduction

Recently, Huang and Zhang [1] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. The category of cone metric spaces is larger than metric spaces and there are different types of cones. Subsequently, many authors like Abbas and Jungck [2], Abbas and Rhoades [3], Ilić and Rakočević [4], Raja andVaezpour[5] have generalized the results of Huang and Zhang [1] and studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Janković et al. [6], Jungck et al. [7], Kadelburg et al. [8, 9], Radenović and Rhoades [10], Rezapour and Hamlbarani [11] studied the existence of common fixed points of a pair of self and nonself mappings satisfying a contractive type condition in the situation in which the cone does not need be normal.

The study of fixed point theorems for nonself mappings in metrically convex metric spaces was initiated by Assad and Kirk [12]. Utilizing the induction method of Assad and Kirk [12], many authors like Assad [13], Ćirić [14], Hadžić [15], Hadžić and Gajić [16], Imdad and Kumar [17], Rhoades [18, 19] have obtained common fixed point in metrically convex spaces. Recently, Ćirić and Ume [20] defined a wide class of multivalued nonself mappings which satisfy a generalized contraction condition and proved a fixed point theorem which generalize the results of Itoh [21] and Khan [22].

Very recently, Radenović and Rhoades [10] extended the fixed point theorem of Imdad and Kumar [17] for a pair of nonself mappings to nonnormal cone metric spaces. Janković et al. [6] proved new common fixed point results for a pair of nonself mappings defined on a closed subset of metrically convex cone metric space which is not necessarily normal by adapting Assad-Kirk’s method.

The aim of this paper is to prove common fixed point theorems for coincidentally commuting nonself mappings satisfying a generalized contraction condition of Ćirić type in the setting of cone metric spaces. Our results generalize mainly results of Ćirić and Ume [20] and all the recent results related to nonself mappings in the setting of cone metric space.

2. Definitions and Preliminaries

We recall some basic definitions and preliminaries that will be needed in the sequel.

Definition 2.1 (see [1]). Let 𝐸 be a real Banach space. A subset 𝑃 of 𝐸 is called a Cone if and only if(1) 𝑃 is nonempty, closed and 𝑃 ≠ { 0 } ;(2) 𝛼 , 𝛽 ∈ 𝑅 , 𝛼 , 𝛽 ≥ 0 and 𝑥 , 𝑦 ∈ 𝑃 ⇒ 𝛼 𝑥 + 𝛽 𝑦 ∈ 𝑃 ;(3) 𝑃 ∩ ( − 𝑃 ) = { 0 } .For a given cone 𝑃 ⊆ 𝐸 , a partial ordering is defined as ≤ on 𝐸 with respect to 𝑃 by 𝑥 ≤ 𝑦 , if and only if 𝑦 − 𝑥 ∈ 𝑃 . It is denoted as 𝑥 < 𝑦 to indicate that 𝑥 ≤ 𝑦 but 𝑥 ≠ 𝑦 , while 𝑥 ≪ 𝑦 will stand for 𝑦 − 𝑥 ∈ i n t 𝑃 , where i n t 𝑃 denotes the interior of 𝑃 .
The cone 𝑃 ⊂ 𝐸 is called normal, if there is a number 𝐾 > 0 such that for all 𝑥 , 𝑦 ∈ 𝐸 , 0 ≤ 𝑥 ≤ 𝑦 implies
‖ 𝑥 ‖ ≤ 𝐾 ‖ 𝑦 ‖ . ( 2 . 1 ) The least positive number 𝐾 satisfying (2.1) is called the normal constant of 𝑃 . It is clear that 𝐾 ≥ 1 . There are nonnormal cones also.

The definition of a cone metric space given by Huang and Zhang [1] is as follows.

Definition 2.2 (see [1]). Let 𝑋 be a nonempty set. Suppose that 𝐸 is a real Banach space, 𝑃 is a cone with i n t 𝑃 ≠ ∅ and ≤ is a partial ordering with respect to 𝑃 .
If the mapping 𝑑 ∶ 𝑋 × 𝑋 → 𝐸 satisfies the following:
(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 ( 𝑦 , 𝑧 ) for all 𝑥 , 𝑦 , 𝑧 ∈ 𝑋 ;then 𝑑 is called a cone metric on 𝑋 and ( 𝑋 , 𝑑 ) is called a cone metric space.

Example 2.3 (see [1]). Let 𝐸 = ℝ 2 , 𝑃 = { ( 𝑥 , 𝑦 ) ∈ 𝐸 ∶ 𝑥 , 𝑦 ≥ 0 } ⊂ ℝ 2 , 𝑋 = ℝ , and 𝑑 ∶ 𝑋 × 𝑋 → 𝐸 such that 𝑑 ( 𝑥 , 𝑦 ) = ( | 𝑥 − 𝑦 | , 𝛼 | 𝑥 − 𝑦 | ) , where 𝛼 ≥ 0 is a constant. Then ( 𝑋 , 𝑑 ) is a cone metric space.

Definition 2.4 (see [1]). Let ( 𝑋 , 𝑑 ) be a cone metric space and { 𝑥 𝑛 } a sequence in 𝑋 . Then, one has the following.(1) { 𝑥 𝑛 } converges to 𝑥 ∈ 𝑋 , if for every 𝑐 ∈ 𝐸 with 0 ≪ 𝑐 , there is 𝑛 0 ∈ 𝑁 such that for all 𝑛 ≥ 𝑛 0 , 𝑑 𝑥 𝑛 , 𝑥 ≪ 𝑐 . ( 2 . 2 ) It is denoted by l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑥 or 𝑥 𝑛 → 𝑥 , ( 𝑛 → ∞ ) . (2)If for any 𝑐 ∈ 𝐸 , there is a number 𝑛 0 ∈ 𝑁 such that for all 𝑚 , 𝑛 ≥ 𝑛 0 𝑑 𝑥 𝑛 , 𝑥 𝑚 ≪ 𝑐 , ( 2 . 3 ) then { 𝑥 𝑛 } is called a Cauchy sequence in 𝑋 .(3) ( 𝑋 , 𝑑 ) is a complete cone metric space, if every Cauchy sequence in 𝑋 is convergent.(4)A self mapping 𝑇 ∶ 𝑋 → 𝑋 is said to be continuous at a point 𝑥 ∈ 𝑋 , if l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑥 implies that l i m 𝑛 → ∞ 𝑇 𝑥 𝑛 = 𝑇 𝑥 for every { 𝑥 𝑛 } in 𝑋 .

The following two lemmas of Huang and Zhang [1] will be required in the sequel.

Lemma 2.5 (see [1]). Let ( 𝑋 , 𝑑 ) be a cone metric space and 𝑃 a normal cone with normal constant 𝐾 . A sequence { 𝑥 𝑛 } in 𝑋 converges to 𝑥 if and only if 𝑑 ( 𝑥 𝑛 , 𝑥 ) → 0 as 𝑛 → ∞ .

Lemma 2.6 (see [1]). Let ( 𝑋 , 𝑑 ) be a cone metric space and 𝑃 a normal cone with normal constant 𝐾 . A sequence { 𝑥 𝑛 } in 𝑋 is a Cauchy sequence if and only if 𝑑 ( 𝑥 𝑛 , 𝑥 𝑚 ) → 0 as 𝑛 , 𝑚 → ∞ .

The following Corollary of Rezapour [23] will be needed in the sequel.

Corollary 2.7 (see [23]). Let 𝑎 , 𝑏 , 𝑐 , 𝑢 ∈ 𝐸 , the real Banach space. (i)If 𝑎 ≤ 𝑏 and 𝑏 ≪ 𝑐 , then 𝑎 ≪ 𝑐 .(ii)If 𝑎 ≪ 𝑏 and 𝑏 ≪ 𝑐 , then 𝑎 ≪ 𝑐 .(iii)If 0 ≤ 𝑢 ≪ 𝑐 for each 𝑐 ∈ i n t 𝑃 , then 𝑢 = 0 .

The following remarks of Radenović and Rhoades [10] will be needed in the sequel.

Remark 2.8 (see [10]). If 𝑐 ∈ i n t 𝑃 , 0 ≤ 𝑎 𝑛 and 𝑎 𝑛 → 0 , then there exists 𝑛 0 such that for all 𝑛 > 𝑛 0 , it follows that 𝑎 𝑛 ≪ 𝑐 .

Remark 2.9 (see [10]). If 0 ≤ 𝑑 ( 𝑥 𝑛 , 𝑥 ) ≤ 𝑏 𝑛 and 𝑏 𝑛 → 0 , then 𝑑 ( 𝑥 𝑛 , 𝑥 ) ≪ 𝑐 , where { 𝑥 𝑛 } is a sequence and 𝑥 is a given point in 𝑋 .

Remark 2.10 (see [10]). If 0 ≤ 𝑎 𝑛 ≤ 𝑏 𝑛 and 𝑎 𝑛 → 𝑎 , 𝑏 𝑛 → 𝑏 , then 𝑎 ≤ 𝑏 for each cone 𝑃 .

Remark 2.11 (see [10]). If 𝐸 is a real Banach space with a cone 𝑃 and if 𝑎 ≤ 𝜆 𝑎 , where 𝑎 ∈ 𝑃 and 0 < 𝜆 < 1 , then 𝑎 = 0 .

3. Main Results

In the following, we suppose that 𝐸 is a Banach space, 𝑃 is a cone in 𝐸 with i n t 𝑃 ≠ ∅ , and ≤ is partial ordering with respect to 𝑃 .

Theorem 3.1. Let ( 𝑋 , 𝑑 ) be a complete cone metric space and 𝑀 a nonempty closed subset of 𝑋 such that for each 𝑥 ∈ 𝑀 and 𝑦 ∉ 𝑀 there exists a point 𝑧 ∈ 𝜕 𝑀 such that 𝑑 ( 𝑥 , 𝑧 ) + 𝑑 ( 𝑧 , 𝑦 ) = 𝑑 ( 𝑥 , 𝑦 ) . ( 3 . 1 ) Suppose that 𝑓 , 𝑇 ∶ 𝑀 → 𝑋 are two nonself mappings satisfying for all 𝑥 , 𝑦 ∈ 𝑀 with 𝑥 ≠ 𝑦 , 𝑑 ( 𝑇 𝑥 , 𝑇 𝑦 ) ≤ 𝛼 𝑑 ( 𝑓 𝑥 , 𝑓 𝑦 ) + 𝛽 𝑢 + 𝛾 𝑣 , w h e r e 𝑢 ∈ { 𝑑 ( 𝑓 𝑥 , 𝑇 𝑥 ) , 𝑑 ( 𝑓 𝑦 , 𝑇 𝑦 ) } , 𝑣 ∈ { 𝑑 ( 𝑓 𝑥 , 𝑇 𝑥 ) + 𝑑 ( 𝑓 𝑦 , 𝑇 𝑦 ) , 𝑑 ( 𝑓 𝑥 , 𝑇 𝑦 ) + 𝑑 ( 𝑓 𝑦 , 𝑇 𝑥 ) } , ( 3 . 2 ) and 𝛼 , 𝛽 , 𝛾 are nonnegative real numbers such that 𝛼 + 2 𝛽 + 3 𝛾 + 𝛼 𝛾 < 1 . ( 3 . 3 ) Also assume that (i) 𝜕 𝑀 ⊆ 𝑓 𝑀 , 𝑇 𝑀 ∩ 𝑀 ⊂ 𝑓 𝑀 ;(ii) 𝑓 𝑥 ∈ 𝜕 𝑀 ⇒ 𝑇 𝑥 ∈ 𝑀 ;(iii) 𝑓 𝑀 is closed in 𝑋 ;Then there exists a coincidence point of 𝑓 and 𝑇 in 𝑀 . Moreover, if 𝑇 and 𝑓 are coincidentally commuting, then 𝑇 and 𝑓 have a unique common fixed point in 𝑀 .

Proof. Two sequences { 𝑥 𝑛 } and { 𝑦 𝑛 } are constructed in the following way. Let 𝑥 ∈ 𝜕 𝑀 . As 𝜕 𝑀 ⊆ 𝑓 𝑀 , by (i) there exists a point 𝑥 0 ∈ 𝑀 such that 𝑥 = 𝑓 𝑥 0 ∈ 𝜕 𝑀 . Since 𝑓 𝑥 ∈ 𝜕 𝑀 ⇒ 𝑇 𝑥 ⊂ 𝑀 , from (ii) it follows that 𝑓 𝑥 0 ∈ 𝜕 𝑀 ⇒ 𝑇 𝑥 0 ∈ 𝑀 ⇒ 𝑇 𝑥 0 ∈ 𝑀 ∩ 𝑇 𝑀 ⊂ 𝑓 𝑀 . Let 𝑥 1 ∈ 𝑀 be such that 𝑦 1 = 𝑓 𝑥 1 = 𝑇 𝑥 0 ∈ 𝑀 . Since 𝑦 1 = 𝑇 𝑥 0 , there exists 𝑦 2 = 𝑇 𝑥 1 such that 𝑑 ( 𝑦 1 , 𝑦 2 ) = 𝑑 ( 𝑇 𝑥 0 , 𝑇 𝑥 1 ) .
If 𝑦 2 ∈ 𝑀 , then 𝑦 2 ∈ 𝑀 ∩ 𝑇 𝑀 ⊆ 𝑓 𝑀 which implies that there exists a point 𝑥 2 ∈ 𝑀 such that 𝑦 2 = 𝑓 𝑥 2 . Otherwise, if 𝑦 2 ∉ 𝑀 , then there exists a point 𝑢 ∈ 𝜕 𝑀 such that
𝑑 𝑓 𝑥 1 , 𝑢 + 𝑑 𝑢 , 𝑦 2 = 𝑑 𝑓 𝑥 1 , 𝑦 2 . ( 3 . 4 ) Since 𝑢 ∈ 𝜕 𝑀 ⊆ 𝑓 𝑀 , there exists a point 𝑥 2 ∈ 𝑀 such that 𝑢 = 𝑓 𝑥 2 and thus 𝑑 𝑓 𝑥 1 , 𝑓 𝑥 2 + 𝑑 𝑓 𝑥 2 , 𝑦 2 = 𝑑 𝑓 𝑥 1 , 𝑦 2 . ( 3 . 5 ) Assume that 𝑦 3 = 𝑇 𝑥 2 .
Thus repeating the arguments, two sequences { 𝑥 𝑛 } ⊆ 𝑀 and { 𝑦 𝑛 } ⊆ 𝑇 𝑀 ⊂ 𝑋 are obtained such that
(i) 𝑦 𝑛 + 1 = 𝑇 𝑥 𝑛 ;(ii) 𝑦 𝑛 ∈ 𝑀 ⇒ 𝑦 𝑛 = 𝑓 𝑥 𝑛 ;(iii) 𝑦 𝑛 ≠ 𝑓 𝑥 𝑛 whenever 𝑦 𝑛 ∉ 𝑀 , then there exists 𝑓 𝑥 𝑛 ∈ 𝜕 𝑀 such that 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 = 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 . ( 3 . 6 ) Next, we claim that { 𝑓 𝑥 𝑛 } is a Cauchy sequence in 𝑓 𝑀 . The following are derived. Let us denote 𝑃 = 𝑓 𝑥 𝑖 ∈ 𝑓 𝑥 𝑛 ∶ 𝑓 𝑥 𝑖 = 𝑦 𝑖 , 𝑄 = 𝑓 𝑥 𝑖 ∈ 𝑓 𝑥 𝑛 ∶ 𝑓 𝑥 𝑖 ≠ 𝑦 𝑖 . ( 3 . 7 ) Obviously, two consecutive terms cannot lie in 𝑄 . Note that, if 𝑓 𝑥 𝑛 ∈ 𝑄 , then 𝑓 𝑥 𝑛 − 1 and 𝑓 𝑥 𝑛 + 1 belong to 𝑃 . Now, three cases are distinguished.Case 1. If 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ∈ 𝑃 , then 𝑦 𝑛 = 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 = 𝑓 𝑥 𝑛 + 1 . Now from (3.2), 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑦 = 𝑑 𝑛 , 𝑦 𝑛 + 1 = 𝑑 𝑇 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 ≤ 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑢 + 𝛾 𝑣 , ( 3 . 8 ) where 𝑑 𝑢 ∈ 𝑓 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 − 1 , 𝑑 𝑓 𝑥 𝑛 , 𝑇 𝑥 𝑛 = 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 , 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 , 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑣 ∈ 𝑓 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 − 1 + 𝑑 𝑓 𝑥 𝑛 , 𝑇 𝑥 𝑛 , 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑇 𝑥 𝑛 − 1 = 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 1 . ( 3 . 9 ) Thus 𝑑 𝑢 ∈ 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 , 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑣 ∈ 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 1 . ( 3 . 1 0 )
Now foursubcasesarise.
Subcase 1.1. If 𝑢 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 ) and 𝑣 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 ) + 𝑑 ( 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ) , then (3.8) becomes 𝑑 𝑦 𝑛 , 𝑦 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 + 𝛽 + 𝛾 𝑑 1 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 . ( 3 . 1 1 ) Subcase 1.2. If 𝑢 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 ) and 𝑣 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 1 ) , then (3.8) becomes 𝑑 𝑦 𝑛 , 𝑦 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 + 𝛽 + 𝛾 𝑑 1 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 . ( 3 . 1 2 ) Subcase 1.3. If 𝑢 = 𝑑 ( 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ) and 𝑣 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 ) + 𝑑 ( 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ) , then (3.8) becomes 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 + 𝛾 𝑑 1 − 𝛽 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 . ( 3 . 1 3 ) Subcase 1.4. If 𝑢 = 𝑑 ( 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ) and 𝑣 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 1 ) , then (3.8) becomes 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 1 , ≤ 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 + 𝛾 𝑑 1 − 𝛽 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 . ( 3 . 1 4 ) Combining allSubcases1.1, 1.2, 1.3, and 1.4, it follows that 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ ℎ 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 , ( 3 . 1 5 ) where ℎ = m a x { ( 𝛼 + 𝛽 + 𝛾 ) / ( 1 − 𝛾 ) , ( 𝛼 + 𝛾 ) / ( 1 − 𝛽 − 𝛾 ) } = ( 𝛼 + 𝛽 + 𝛾 ) / ( 1 − 𝛾 ) . Hence 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑦 = 𝑑 𝑛 , 𝑦 𝑛 + 1 ≤ 𝛼 + 𝛽 + 𝛾 𝑑 1 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 . ( 3 . 1 6 ) Case 2. If 𝑓 𝑥 𝑛 ∈ 𝑃 , 𝑓 𝑥 𝑛 + 1 ∈ 𝑄 , then 𝑦 𝑛 = 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 ≠ 𝑓 𝑥 𝑛 + 1 . Now, 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 + 𝑑 𝑓 𝑥 𝑛 + 1 , 𝑦 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 𝑦 b y ( 3 . 6 ) = 𝑑 𝑛 , 𝑦 𝑛 + 1 = 𝑑 𝑇 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 . ( 3 . 1 7 ) Proceeding as in Case 1, 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑦 ≤ 𝑑 𝑛 , 𝑦 𝑛 + 1 = 𝑑 𝑇 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 ≤ 𝛼 + 𝛽 + 𝛾 𝑑 1 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 . ( 3 . 1 8 ) Case 3. If 𝑓 𝑥 𝑛 ∈ 𝑄 , 𝑓 𝑥 𝑛 + 1 ∈ 𝑃 , then 𝑓 𝑥 𝑛 − 1 ∈ 𝑃 , 𝑦 𝑛 ≠ 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 = 𝑓 𝑥 𝑛 + 1 , 𝑦 𝑛 − 1 = 𝑓 𝑥 𝑛 − 1 and 𝑦 𝑛 = 𝑇 𝑥 𝑛 − 1 . Now, 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 ≤ 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 𝑦 + 𝑑 𝑛 , 𝑦 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 𝑑 𝑇 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 = 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑢 + 𝛾 𝑣 . ( 3 . 1 9 ) Thus 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑢 + 𝛾 𝑣 , ( 3 . 2 0 ) where 𝑑 𝑢 ∈ 𝑓 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 − 1 , 𝑑 𝑓 𝑥 𝑛 , 𝑇 𝑥 𝑛 = 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 , 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 , 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑣 = 𝑓 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 − 1 + 𝑑 𝑓 𝑥 𝑛 , 𝑇 𝑥 𝑛 , 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑇 𝑥 𝑛 − 1 = 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 1 + 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 . ( 3 . 2 1 ) Thus 𝑑 𝑢 ∈ 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 , 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 , 𝑑 𝑣 ∈ 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 , 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 1 + 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 . ( 3 . 2 2 )
Again foursubcasesarise.
Subcase 3.1. If 𝑢 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 ) , 𝑣 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 ) + 𝑑 ( 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ) , then (3.20) becomes 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 ≤ 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 − 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + ( 𝛽 + 𝛾 ) 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 𝛾 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ b y ( 3 . 6 ) 1 + 𝛽 + 𝛾 𝑑 1 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 𝛼 − 1 𝑑 1 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 = 1 + 𝛽 + 𝛾 + 𝛼 − 1 𝑑 1 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 . ( 3 . 2 3 ) Thus 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 + 𝛽 + 𝛾 𝑑 1 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 , ( 3 . 2 4 ) where using Case 2, 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 𝑦 = 𝑑 𝑛 − 1 , 𝑦 𝑛 = 𝑑 𝑇 𝑥 𝑛 − 2 , 𝑇 𝑥 𝑛 − 1 ≤ 𝛼 + 𝛽 + 𝛾 𝑑 1 − 𝛾 𝑓 𝑥 𝑛 − 2 , 𝑓 𝑥 𝑛 − 1 . ( 3 . 2 5 ) Then (3.24) becomes 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 + 𝛽 + 𝛾 1 − 𝛾 2 𝑑 𝑓 𝑥 𝑛 − 2 , 𝑓 𝑥 𝑛 − 1 . ( 3 . 2 6 ) Subcase 3.2. If 𝑢 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 ) , 𝑣 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 1 ) + 𝑑 ( 𝑓 𝑥 𝑛 , 𝑦 𝑛 ) , then (3.20) becomes 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 ≤ 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 1 + 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 ≤ 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 + 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 ≤ 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 b y ( 3 . 6 ) . ( 3 . 2 7 ) Proceeding as in Subcase 3.1, it follows that 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 + 𝛽 + 𝛾 1 − 𝛾 2 𝑑 𝑓 𝑥 𝑛 − 2 , 𝑓 𝑥 𝑛 − 1 . ( 3 . 2 8 ) Subcase 3.3. If 𝑢 = 𝑑 ( 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ) , 𝑣 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 ) + 𝑑 ( 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ) , then (3.20) becomes 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 ≤ 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 − 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 + 𝛾 𝑑 1 − 𝛽 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 . ( 3 . 2 9 ) Thus 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 + 𝛾 𝑑 1 − 𝛽 − 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 , ( 3 . 3 0 ) where using Case 2, 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 𝑦 = 𝑑 𝑛 − 1 , 𝑦 𝑛 = 𝑑 𝑇 𝑥 𝑛 − 2 , 𝑇 𝑥 𝑛 − 1 ≤ 𝛼 + 𝛽 + 𝛾 𝑑 1 − 𝛾 𝑓 𝑥 𝑛 − 2 , 𝑓 𝑥 𝑛 − 1 . ( 3 . 3 1 ) Then (3.30) becomes 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ≤ 𝛼 + 𝛽 + 𝛾 , 1 − 𝛾 𝛼 + 𝛾 𝑑 1 − 𝛽 − 𝛾 𝑓 𝑥 𝑛 − 2 , 𝑓 𝑥 𝑛 − 1 . ( 3 . 3 2 ) Subcase 3.4. If 𝑢 = 𝑑 ( 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 ) , 𝑣 = 𝑑 ( 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 1 ) + 𝑑 ( 𝑓 𝑥 𝑛 , 𝑦 𝑛 ) , then (3.20) becomes 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 = 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 1 ≤ 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 1 + 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 ≤ 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 − 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 + 𝑑 𝑓 𝑥 𝑛 , 𝑦 𝑛 = 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 − 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛼 𝑑 𝑓 𝑥 𝑛 − 1 , 𝑓 𝑥 𝑛 + 𝛽 𝑑 𝑓 𝑥 𝑛 , 𝑓 𝑥 𝑛 + 1 𝑑 + 𝛾 𝑓 𝑥 𝑛 − 1 , 𝑦 𝑛 + 𝑑 𝑓 𝑥 𝑛