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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 793486, 17 pages
http://dx.doi.org/10.1155/2012/793486
Research Article

Generalized 𝜢-𝝍 Contractive Type Mappings and Related Fixed Point Theorems with Applications

1Department of Mathematics, Atilim University, Δ°ncek, 06836 Ankara, Turkey
2Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia

Received 28 May 2012; Accepted 25 July 2012

Academic Editor: Jean MichelΒ Combes

Copyright Β© 2012 Erdal KarapΔ±nar and Bessem Samet. 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 establish fixed point theorems for a new class of contractive mappings. As consequences of our main results, we obtain fixed point theorems on metric spaces endowed with a partial order and fixed point theorems for cyclic contractive mappings. Various examples are presented to illustrate our obtained results.

1. Introduction and Preliminaries

Let Ξ¨ be the family of functions πœ“βˆΆ[0,∞)β†’[0,∞) satisfying the following conditions: (Ξ¨1)πœ“ is nondecreasing; (Ξ¨2)βˆ‘+βˆžπ‘›=1πœ“π‘›(𝑑)<∞ for all 𝑑>0, where πœ“π‘› is the 𝑛th iterate of πœ“. These functions are known in the literature as (c)-comparison functions. It is easily proved that if πœ“ is a (c)-comparison function, then πœ“(𝑑)<𝑑 for any 𝑑>0.

Very recently, Samet et al. [1] introduced the following concepts.

Definition 1.1. Let (𝑋,𝑑) be a metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a given mapping. We say that 𝑇 is an 𝛼-πœ“ contractive mapping if there exist two functions π›ΌβˆΆπ‘‹Γ—π‘‹β†’[0,∞) and πœ“βˆˆΞ¨ such that 𝛼(π‘₯,𝑦)𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ“(𝑑(π‘₯,𝑦)),βˆ€π‘₯,π‘¦βˆˆπ‘‹.(1.1)
Clearly, any contractive mapping, that is, a mapping satisfying Banach contraction, is an 𝛼-πœ“ contractive mapping with 𝛼(π‘₯,𝑦)=1 for all π‘₯,π‘¦βˆˆπ‘‹ and πœ“(𝑑)=π‘˜π‘‘, π‘˜βˆˆ(0,1).

Definition 1.2. Let π‘‡βˆΆπ‘‹β†’π‘‹ and π›ΌβˆΆπ‘‹Γ—π‘‹β†’[0,∞). We say that 𝑇 is 𝛼-admissible if for all π‘₯,π‘¦βˆˆπ‘‹, and we have 𝛼(π‘₯,𝑦)β‰₯1βŸΉπ›Ό(𝑇π‘₯,𝑇𝑦)β‰₯1.(1.2) Various examples of such mappings are presented in [1].
The main results in [1] are the following fixed point theorems.

Theorem 1.3. Let (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be an 𝛼-πœ“ contractive mapping. Suppose that(i)𝑇 is 𝛼 admissible; (ii)there exists π‘₯0βˆˆπ‘‹ such that 𝛼(π‘₯0,𝑇π‘₯0)β‰₯1; (iii)𝑇 is continuous. Then there exists π‘’βˆˆπ‘‹ such that 𝑇𝑒=𝑒.

Theorem 1.4. Let (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be an 𝛼-πœ“ contractive mapping. Suppose that (i)𝑇 is 𝛼 admissible; (ii)there exists π‘₯0βˆˆπ‘‹ such that 𝛼(π‘₯0,𝑇π‘₯0)β‰₯1; (iii)if {π‘₯𝑛} is a sequence in 𝑋 such that 𝛼(π‘₯𝑛,π‘₯𝑛+1)β‰₯1 for all 𝑛 and π‘₯𝑛→π‘₯βˆˆπ‘‹ as π‘›β†’βˆž, then 𝛼(π‘₯𝑛,π‘₯)β‰₯1 for all 𝑛. Then there exists π‘’βˆˆπ‘‹ such that 𝑇𝑒=𝑒.

Theorem 1.5. Adding to the hypotheses of Theorem 1.3 (resp., Theorem 1.4) the condition, for all π‘₯,π‘¦βˆˆπ‘‹, there exists π‘§βˆˆπ‘‹ such that 𝛼(π‘₯,𝑧)β‰₯1 and 𝛼(𝑦,𝑧)β‰₯1, and one obtains uniqueness of the fixed point.

In the present work, we introduce the concept of generalized 𝛼-πœ“ contractive type mappings, and we study the existence and uniqueness of fixed points for such mappings. Presented theorems in this paper extend and generalize the above results derived by Samet et al. in [1]. Moreover, from our fixed point theorems, we will deduce various fixed point results on metric spaces endowed with a partial order and fixed point results for cyclic contractive mappings.

2. Main Results

We introduce the concept of generalized 𝛼-πœ“ contractive type mappings as follows.

Definition 2.1. Let (𝑋,𝑑) be a metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a given mapping. We say that 𝑇 is a generalized 𝛼-πœ“ contractive mapping if there exist two functions π›ΌβˆΆπ‘‹Γ—π‘‹β†’[0,∞) and πœ“βˆˆΞ¨ such that for all π‘₯,π‘¦βˆˆπ‘‹, and we have 𝛼(π‘₯,𝑦)𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ“(𝑀(π‘₯,𝑦)),(2.1) where 𝑀(π‘₯,𝑦)=max{𝑑(π‘₯,𝑦),(𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦))/2,(𝑑(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯))/2}.

Remark 2.2. Clearly, since πœ“ is nondecreasing, every 𝛼-πœ“ contractive mapping is a generalized 𝛼-πœ“ contractive mapping.

Our first result is the following.

Theorem 2.3. Let (𝑋,𝑑) be a complete metric space. Suppose that π‘‡βˆΆπ‘‹β†’π‘‹ is a generalized 𝛼-πœ“ contractive mapping and satisfies the following conditions: (i)𝑇 is 𝛼 admissible; (ii)there exists π‘₯0βˆˆπ‘‹ such that 𝛼(π‘₯0,𝑇π‘₯0)β‰₯1; (iii)𝑇 is continuous. Then there exists π‘’βˆˆπ‘‹ such that 𝑇𝑒=𝑒.

Proof. Let π‘₯0βˆˆπ‘‹ such that 𝛼(π‘₯0,𝑇π‘₯0)β‰₯1 (such a point exists from condition (ii)). Define the sequence {π‘₯𝑛} in 𝑋 by π‘₯𝑛+1=𝑇π‘₯𝑛 for all 𝑛β‰₯0. If π‘₯𝑛0=π‘₯𝑛0+1 for some 𝑛0, then 𝑒=π‘₯𝑛0 is a fixed point of 𝑇. So, we can assume that π‘₯𝑛≠π‘₯𝑛+1 for all 𝑛. Since 𝑇 is 𝛼 admissible, we have 𝛼π‘₯0,π‘₯1ξ€Έξ€·π‘₯=𝛼0,𝑇π‘₯0ξ€Έξ€·β‰₯1βŸΉπ›Όπ‘‡π‘₯0,𝑇π‘₯1ξ€Έξ€·π‘₯=𝛼1,π‘₯2ξ€Έβ‰₯1.(2.2) Inductively, we have 𝛼π‘₯𝑛,π‘₯𝑛+1ξ€Έβ‰₯1,βˆ€π‘›=0,1,….(2.3) From (2.1) and (2.3), it follows that for all 𝑛β‰₯1, we have 𝑑π‘₯𝑛+1,π‘₯𝑛=𝑑𝑇π‘₯𝑛,𝑇π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯≀𝛼𝑛,π‘₯π‘›βˆ’1𝑑𝑇π‘₯𝑛,𝑇π‘₯π‘›βˆ’1𝑀π‘₯β‰€πœ“π‘›,π‘₯π‘›βˆ’1.ξ€Έξ€Έ(2.4) On the other hand, we have 𝑀π‘₯𝑛,π‘₯π‘›βˆ’1𝑑π‘₯=max𝑛,π‘₯π‘›βˆ’1ξ€Έ,𝑑π‘₯𝑛,𝑇π‘₯𝑛π‘₯+π‘‘π‘›βˆ’1,𝑇π‘₯π‘›βˆ’1ξ€Έ2,𝑑π‘₯𝑛,𝑇π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯+π‘‘π‘›βˆ’1,𝑇π‘₯𝑛2𝑑π‘₯=max𝑛,π‘₯π‘›βˆ’1ξ€Έ,𝑑π‘₯𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯+π‘‘π‘›βˆ’1,π‘₯𝑛2,𝑑π‘₯π‘›βˆ’1,π‘₯𝑛+1ξ€Έ2𝑑π‘₯≀max𝑛,π‘₯π‘›βˆ’1ξ€Έ,𝑑π‘₯𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯+π‘‘π‘›βˆ’1,π‘₯𝑛2𝑑π‘₯≀max𝑛,π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝑑𝑛,π‘₯𝑛+1ξ€Έξƒ°.(2.5) From (2.4) and taking in consideration that πœ“ is a nondecreasing function, we get that 𝑑π‘₯𝑛+1,π‘₯𝑛𝑑π‘₯β‰€πœ“max𝑛,π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝑑𝑛,π‘₯𝑛+1,ξ€Έξ€Ύξ€Έ(2.6) for all 𝑛β‰₯1. If for some 𝑛β‰₯1, we have 𝑑(π‘₯𝑛,π‘₯π‘›βˆ’1)≀𝑑(π‘₯𝑛,π‘₯𝑛+1), from (2.6), we obtain that 𝑑π‘₯𝑛+1,π‘₯𝑛𝑑π‘₯β‰€πœ“π‘›,π‘₯𝑛+1ξ€·π‘₯ξ€Έξ€Έ<𝑑𝑛,π‘₯𝑛+1ξ€Έ,(2.7) a contradiction. Thus, for all 𝑛β‰₯1, we have 𝑑π‘₯max𝑛,π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝑑𝑛,π‘₯𝑛+1ξ€·π‘₯ξ€Έξ€Ύ=𝑑𝑛,π‘₯π‘›βˆ’1ξ€Έ.(2.8) Using (2.6) and (2.8), we get that 𝑑π‘₯𝑛+1,π‘₯𝑛𝑑π‘₯β‰€πœ“π‘›,π‘₯π‘›βˆ’1,ξ€Έξ€Έ(2.9) for all 𝑛β‰₯1. By induction, we get 𝑑π‘₯𝑛+1,π‘₯π‘›ξ€Έβ‰€πœ“π‘›ξ€·π‘‘ξ€·π‘₯1,π‘₯0ξ€Έξ€Έ,βˆ€π‘›β‰₯1.(2.10) From (2.10) and using the triangular inequality, for all π‘˜β‰₯1, we have 𝑑π‘₯𝑛,π‘₯𝑛+π‘˜ξ€Έξ€·π‘₯≀𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯+β‹―+𝑑𝑛+π‘˜βˆ’1,π‘₯𝑛+π‘˜ξ€Έβ‰€π‘›+π‘˜βˆ’1𝑝=π‘›πœ“π‘›ξ€·π‘‘ξ€·π‘₯1,π‘₯0≀+βˆžξ“π‘=π‘›πœ“π‘›ξ€·π‘‘ξ€·π‘₯1,π‘₯0ξ€Έξ€ΈβŸΆ0asπ‘›β†’βˆž.(2.11) This implies that {π‘₯𝑛} is a Cauchy sequence in (𝑋,𝑑). Since (𝑋,𝑑) is complete, there exists π‘’βˆˆπ‘‹ such that limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑛,𝑒=0.(2.12) Since 𝑇 is continuous, we obtain from (2.12) that limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑛+1ξ€Έ,𝑇𝑒=limπ‘›β†’βˆžπ‘‘ξ€·π‘‡π‘₯𝑛,𝑇𝑒=0.(2.13) From (2.12), (2.13) and the uniqueness of the limit, we get immediately that 𝑒 is a fixed point of 𝑇, that is, 𝑇𝑒=𝑒.

The next theorem does not require the continuity of 𝑇.

Theorem 2.4. Let (𝑋,𝑑) be a complete metric space. Suppose that π‘‡βˆΆπ‘‹β†’π‘‹ is a generalized 𝛼-πœ“ contractive mapping and the following conditions hold: (i)𝑇 is 𝛼 admissible; (ii)there exists π‘₯0βˆˆπ‘‹ such that 𝛼(π‘₯0,𝑇π‘₯0)β‰₯1; (iii)if {π‘₯𝑛} is a sequence in 𝑋 such that 𝛼(π‘₯𝑛,π‘₯𝑛+1)β‰₯1 for all 𝑛 and π‘₯𝑛→π‘₯βˆˆπ‘‹ as π‘›β†’βˆž, then there exists a subsequence {π‘₯𝑛(π‘˜)} of {π‘₯𝑛} such that 𝛼(π‘₯𝑛(π‘˜),π‘₯)β‰₯1 for all π‘˜. Then there exists π‘’βˆˆπ‘‹ such that 𝑇𝑒=𝑒.

Proof. Following the proof of Theorem 2.3, we know that the sequence {π‘₯𝑛} defined by π‘₯𝑛+1=𝑇π‘₯𝑛 for all 𝑛β‰₯0, converges for some π‘’βˆˆπ‘‹. From (2.3) and condition (iii), there exists a subsequence {π‘₯𝑛(π‘˜)} of {π‘₯𝑛} such that 𝛼(π‘₯𝑛(π‘˜),𝑒)β‰₯1 for all π‘˜. Applying (2.1), for all π‘˜, we get that 𝑑π‘₯𝑛(π‘˜)+1ξ€Έξ€·,𝑇𝑒=𝑑𝑇π‘₯𝑛(π‘˜)ξ€Έξ€·π‘₯,𝑇𝑒≀𝛼𝑛(π‘˜)𝑑,𝑒𝑇π‘₯𝑛(π‘˜)𝑀π‘₯,π‘‡π‘’β‰€πœ“π‘›(π‘˜).,𝑒(2.14) On the other hand, we have 𝑀π‘₯𝑛(π‘˜)𝑑π‘₯,𝑒=max𝑛(π‘˜)ξ€Έ,𝑑π‘₯,𝑒𝑛(π‘˜),π‘₯𝑛(π‘˜)+1ξ€Έ+𝑑(𝑒,𝑇𝑒)2,𝑑π‘₯𝑛(π‘˜)ξ€Έξ€·,𝑇𝑒+𝑑𝑒,π‘₯𝑛(π‘˜)+1ξ€Έ2ξƒ°.(2.15) Letting π‘˜β†’βˆž in the above equality, we get that limπ‘˜β†’βˆžπ‘€ξ€·π‘₯𝑛(π‘˜)ξ€Έ=,𝑒𝑑(𝑒,𝑇𝑒)2.(2.16) Suppose that 𝑑(𝑒,𝑇𝑒)>0. From (2.16), for π‘˜ large enough, we have 𝑀(π‘₯𝑛(π‘˜),𝑒)>0, which implies that πœ“(𝑀(π‘₯𝑛(π‘˜),𝑒))<𝑀(π‘₯𝑛(π‘˜),𝑒). Thus, from (2.14), we have 𝑑π‘₯𝑛(π‘˜)+1ξ€Έξ€·π‘₯,𝑇𝑒<𝑀𝑛(π‘˜)ξ€Έ.,𝑒(2.17) Letting π‘˜β†’βˆž in the above inequality, using (2.16), we obtain that 𝑑(𝑒,𝑇𝑒)≀𝑑(𝑒,𝑇𝑒)2,(2.18) which is a contradiction. Thus we have 𝑑(𝑒,𝑇𝑒)=0, that is, 𝑒=𝑇𝑒.

With the following example, we will show that hypotheses in Theorems 2.3 and 2.4 do not guarantee uniqueness of the fixed point.

Example 2.5. Let 𝑋={(1,0),(0,1)}βŠ‚β„2 be endowed with the Euclidean distance 𝑑((π‘₯,𝑦),(𝑒,𝑣))=|π‘₯βˆ’π‘’|+|π‘¦βˆ’π‘£| for all (π‘₯,𝑦),(𝑒,𝑣)βˆˆπ‘‹. Obviously, (𝑋,𝑑) is a complete metric space. The mapping 𝑇(π‘₯,𝑦)=(π‘₯,𝑦) is trivially continuous and satisfies for any πœ“βˆˆΞ¨π›Ό((π‘₯,𝑦),(𝑒,𝑣))𝑑(𝑇(π‘₯,𝑦),𝑇(𝑒,𝑣))β‰€πœ“(𝑀((π‘₯,𝑦),(𝑒,𝑣))),(2.19) for all (π‘₯,𝑦),(𝑒,𝑣)βˆˆπ‘‹, where ξ‚»1𝛼((π‘₯,𝑦),(𝑒,𝑣))=if0(π‘₯,𝑦)=(𝑒,𝑣),if(π‘₯,𝑦)β‰ (𝑒,𝑣).(2.20) Thus 𝑇 is a generalized 𝛼-πœ“ contractive mapping. On the other hand, for all (π‘₯,𝑦),(𝑒,𝑣)βˆˆπ‘‹, we have 𝛼((π‘₯,𝑦),(𝑒,𝑣))β‰₯1⟢(π‘₯,𝑦)=(𝑒,𝑣)βŸΆπ‘‡(π‘₯,𝑦)=𝑇(𝑒,𝑣)βŸΆπ›Ό(𝑇(π‘₯,𝑦),𝑇(𝑒,𝑣))β‰₯1.(2.21) Thus 𝑇 is 𝛼 admissible. Moreover, for all (π‘₯,𝑦)βˆˆπ‘‹, we have 𝛼((π‘₯,𝑦),𝑇(π‘₯,𝑦))β‰₯1. Then the assumptions of Theorem 2.3 are satisfied. Note that the assumptions of Theorem 2.4 are also satisfied; indeed if {(π‘₯𝑛,𝑦𝑛)} is a sequence in 𝑋 that converges to some point (π‘₯,𝑦)βˆˆπ‘‹β€‰β€‰with 𝛼((π‘₯𝑛,𝑦𝑛),(π‘₯𝑛+1,𝑦𝑛+1))β‰₯1 for all 𝑛, then, from the definition of 𝛼, we have (π‘₯𝑛,𝑦𝑛)=(π‘₯,𝑦) for all 𝑛, which implies that 𝛼((π‘₯𝑛,𝑦𝑛),(π‘₯,𝑦))=1 for all 𝑛. However, in this case, 𝑇 has two fixed points in 𝑋.

For the uniqueness of a fixed point of a generalized 𝛼-πœ“ contractive mapping, we will consider the following hypothesis. (H) For all π‘₯,π‘¦βˆˆFix(𝑇), there exists π‘§βˆˆπ‘‹ such that 𝛼(π‘₯,𝑧)β‰₯1 and 𝛼(𝑦,𝑧)β‰₯1.

Here, Fix(𝑇) denotes the set of fixed points of 𝑇.

Theorem 2.6. Adding condition (H) to the hypotheses of Theorem 2.3 (resp., Theorem 2.4), one has obtains that 𝑒 is the unique fixed point of 𝑇.

Proof. Suppose that 𝑣 is another fixed point of 𝑇. From (H), there exists π‘§βˆˆπ‘‹ such that 𝛼(𝑒,𝑧)β‰₯1,𝛼(𝑣,𝑧)β‰₯1.(2.22) Since 𝑇 is 𝛼 admissible, from (2.22), we have 𝛼(𝑒,𝑇𝑛𝑧)β‰₯1,𝛼(𝑣,𝑇𝑛𝑧)β‰₯1,βˆ€π‘›.(2.23) Define the sequence {𝑧𝑛} in 𝑋 by 𝑧𝑛+1=𝑇𝑧𝑛 for all 𝑛β‰₯0 and 𝑧0=𝑧. From (2.23), for all 𝑛, we have 𝑑𝑒,𝑧𝑛+1ξ€Έξ€·=𝑑𝑇𝑒,𝑇𝑧𝑛≀𝛼𝑒,𝑧𝑛𝑑𝑇𝑒,π‘‡π‘§π‘›ξ€Έξ€·π‘€ξ€·β‰€πœ“π‘’,𝑧𝑛.ξ€Έξ€Έ(2.24) On the other hand, we have 𝑀𝑒,𝑧𝑛𝑑=max𝑒,𝑧𝑛,𝑑𝑧𝑛,𝑧𝑛+1ξ€Έ2,𝑑𝑒,𝑧𝑛+1𝑧+𝑑𝑛,𝑒2𝑑≀max𝑒,𝑧𝑛,𝑑𝑧𝑛,𝑒+𝑑𝑒,𝑧𝑛+1ξ€Έ2𝑑≀max𝑒,𝑧𝑛,𝑑𝑒,𝑧𝑛+1.ξ€Έξ€Ύ(2.25) Using the above inequality, (2.24) and the monotone property of πœ“, we get that 𝑑𝑒,𝑧𝑛+1ξ€Έξ€·ξ€½π‘‘ξ€·β‰€πœ“max𝑒,𝑧𝑛,𝑑𝑒,𝑧𝑛+1,ξ€Έξ€Ύξ€Έ(2.26) for all 𝑛. Without restriction to the generality, we can suppose that 𝑑(𝑒,𝑧𝑛)>0 for all 𝑛. If max{𝑑(𝑒,𝑧𝑛),𝑑(𝑒,𝑧𝑛+1)}=𝑑(𝑒,𝑧𝑛+1), we get from (2.26) that 𝑑𝑒,𝑧𝑛+1ξ€Έξ€·π‘‘ξ€·β‰€πœ“π‘’,𝑧𝑛+1ξ€·ξ€Έξ€Έ<𝑑𝑒,𝑧𝑛+1ξ€Έ,(2.27) which is a contradiction. Thus we have max{𝑑(𝑒,𝑧𝑛),𝑑(𝑒,𝑧𝑛+1)}=𝑑(𝑒,𝑧𝑛), and 𝑑𝑒,𝑧𝑛+1ξ€Έξ€·π‘‘ξ€·β‰€πœ“π‘’,𝑧𝑛,ξ€Έξ€Έ(2.28) for all 𝑛. This implies that 𝑑𝑒,π‘§π‘›ξ€Έβ‰€πœ“π‘›ξ€·π‘‘ξ€·π‘’,𝑧0ξ€Έξ€Έ,βˆ€π‘›β‰₯1.(2.29) Letting π‘›β†’βˆž in the above inequality, we obtain that limπ‘›β†’βˆžπ‘‘ξ€·π‘§π‘›ξ€Έ,𝑒=0.(2.30) Similarly, one can show that limπ‘›β†’βˆžπ‘‘ξ€·π‘§π‘›ξ€Έ,𝑣=0.(2.31) From (2.30) and (2.31), it follows that 𝑒=𝑣. Thus we proved that 𝑒 is the unique fixed point of 𝑇.

Example 2.7. Let 𝑋=[0,1] be endowed with the standard metric 𝑑(π‘₯,𝑦)=|π‘₯βˆ’π‘¦| for all π‘₯,π‘¦βˆˆπ‘‹. Obviously, (𝑋,𝑑) is a complete metric space. Define the mapping π‘‡βˆΆπ‘‹β†’π‘‹ by ξƒ―1𝑇π‘₯=4if[0π‘₯∈0,1),ifπ‘₯=1.(2.32) In this case, 𝑇 is not continuous. Define the mapping π›ΌβˆΆπ‘‹Γ—π‘‹β†’[0,∞) by 𝛼1(π‘₯,𝑦)=if1(π‘₯,𝑦)βˆˆξ‚€ξ‚ƒ0,4×14βˆͺ1,14×1,10,4,0otherwise.(2.33) We will prove that (A)π‘‡βˆΆπ‘‹β†’π‘‹ is a generalized 𝛼-πœ“ contractive mapping, where πœ“(𝑑)=𝑑/2 for all 𝑑β‰₯0; (B)𝑇 is 𝛼-admissible; (C)there exists π‘₯0βˆˆπ‘‹ such that 𝛼(π‘₯0,𝑇π‘₯0)β‰₯1; (D)if {π‘₯𝑛} is a sequence in 𝑋 such that 𝛼(π‘₯𝑛,π‘₯𝑛+1)β‰₯1 for all 𝑛 and π‘₯𝑛→π‘₯βˆˆπ‘‹ as 𝑛→+∞, then there exists a subsequence {π‘₯𝑛(π‘˜)} of {π‘₯𝑛} such that 𝛼(π‘₯𝑛(π‘˜),π‘₯)β‰₯1 for all π‘˜; (E)condition (H) is satisfied.

Proof of (A). To show (A), we have to prove that (2.1) is satisfied for every π‘₯,π‘¦βˆˆπ‘‹. If π‘₯∈[0,1/4] and 𝑦=1, we have |||1𝛼(π‘₯,𝑦)𝑑(𝑇π‘₯,𝑇𝑦)=𝑑(𝑇π‘₯,𝑇𝑦)=4|||=1βˆ’04𝑑(𝑦,𝑇𝑦)β‰€πœ“(𝑀(π‘₯,𝑦)).(2.34) Then (2.1) holds. If π‘₯=1 and π‘¦βˆˆ[0,1/4], we have |||1𝛼(π‘₯,𝑦)𝑑(𝑇π‘₯,𝑇𝑦)=𝑑(𝑇π‘₯,𝑇𝑦)=0βˆ’4|||=14𝑑(π‘₯,𝑇π‘₯)β‰€πœ“(𝑀(π‘₯,𝑦)).(2.35) Then (2.1) holds also in this case. The other cases are trivial. Thus (2.1) is satisfied for every π‘₯,π‘¦βˆˆπ‘‹.

Proof of (B). Let (π‘₯,𝑦)βˆˆπ‘‹Γ—π‘‹ such that 𝛼(π‘₯,𝑦)β‰₯1. From the definition of 𝛼, we have two cases.
Case  1 (if (π‘₯,𝑦)∈[0,1/4]Γ—[1/4,1]). In this case, we have (𝑇π‘₯,𝑇𝑦)∈[1/4,1]Γ—[0,1/4], which implies that 𝛼(𝑇π‘₯,𝑇𝑦)=1.
Case  2 (if (π‘₯,𝑦)∈[1/4,1]Γ—[0,1/4]). In this case, we have (𝑇π‘₯,𝑇𝑦)∈[0,1/4]Γ—[1/4,1], which implies that 𝛼(𝑇π‘₯,𝑇𝑦)=1.
So, in all cases, we have 𝛼(𝑇π‘₯,𝑇𝑦)β‰₯1. Thus 𝑇 is 𝛼 admissible.

Proof of (C). Taking π‘₯0=0, we have 𝛼(π‘₯0,𝑇π‘₯0)=𝛼(0,1/4)=1.

Proof of (D). Let {π‘₯𝑛} be a sequence in 𝑋 such that 𝛼(π‘₯𝑛,π‘₯𝑛+1)β‰₯1 for all 𝑛 and π‘₯𝑛→π‘₯ as 𝑛→+∞ for some π‘₯βˆˆπ‘‹. From the definition of 𝛼, for all 𝑛, we have ξ€·π‘₯𝑛,π‘₯𝑛+1ξ€Έβˆˆ10,4×14βˆͺ1,14×1,10,4.(2.36) Since ([0,1/4]Γ—[1/4,1])βˆͺ([1/4,1]Γ—[0,1/4]) is a closed set with respect to the Euclidean metric, we get that 1(π‘₯,π‘₯)βˆˆξ‚€ξ‚ƒ0,4×14βˆͺ1,14×1,10,4,(2.37) which implies that π‘₯=1/4. Thus we have 𝛼(π‘₯𝑛,π‘₯)β‰₯1 for all 𝑛.

Proof of (E). Let (π‘₯,𝑦)βˆˆπ‘‹Γ—π‘‹. It is easy to show that, for 𝑧=1/4, we have 𝛼(π‘₯,𝑧)=𝛼(𝑦,𝑧)=1. So, condition (H) is satisfied.

Conclusion. Now, all the hypotheses of Theorem 2.6 are satisfied; thus 𝑇 has a unique fixed point π‘’βˆˆπ‘‹. In this case, we have 𝑒=1/4.

3. Consequences

Now, we will show that many existing results in the literature can be deduced easily from our Theorem 2.6.

3.1. Standard Fixed Point Theorems

Taking in Theorem 2.6, 𝛼(π‘₯,𝑦)=1 for all π‘₯,π‘¦βˆˆπ‘‹, we obtain immediately the following fixed point theorem.

Corollary 3.1. Let (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a given mapping. Suppose that there exists a function πœ“βˆˆΞ¨ such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ“(𝑀(π‘₯,𝑦)),(3.1) for all π‘₯,π‘¦βˆˆπ‘‹. Then 𝑇 has a unique fixed point.

The following fixed point theorems follow immediately from Corollary 3.1.

Corollary 3.2 (see Berinde [2]). Let (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a given mapping. Suppose that there exists a function πœ“βˆˆΞ¨ such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ“(𝑑(π‘₯,𝑦)),(3.2) for all π‘₯,π‘¦βˆˆπ‘‹. Then 𝑇 has a unique fixed point.

Corollary 3.3 (see Δ†iriΔ‡ [3]). Let (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a given mapping. Suppose that there exists a constant πœ†βˆˆ(0,1) such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†max𝑑(π‘₯,𝑦),𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦)2,𝑑(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯)2ξ‚Ό,(3.3) for all π‘₯,π‘¦βˆˆπ‘‹. Then 𝑇 has a unique fixed point.

Corollary 3.4 (see Hardy and Rogers [4]). Let (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a given mapping. Suppose that there exist constants 𝐴,𝐡,𝐢β‰₯0 with (𝐴+2𝐡+2𝐢)∈(0,1) such that [][],𝑑(𝑇π‘₯,𝑇𝑦)≀𝐴𝑑(π‘₯,𝑦)+𝐡𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦)+𝐢𝑑(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯)(3.4) for all π‘₯,π‘¦βˆˆπ‘‹. Then 𝑇 has a unique fixed point.

Corollary 3.5 (see Banach Contraction Principle [5]). Let (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a given mapping. Suppose that there exists a constant πœ†βˆˆ(0,1) such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†π‘‘(π‘₯,𝑦),(3.5) for all π‘₯,π‘¦βˆˆπ‘‹. Then 𝑇 has a unique fixed point.

Corollary 3.6 (see Kannan [6]). Let (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a given mapping. Suppose that there exists a constant πœ†βˆˆ(0,1/2) such that [],𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†π‘‘(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦)(3.6) for all π‘₯,π‘¦βˆˆπ‘‹. Then 𝑇 has a unique fixed point.

Corollary 3.7 (see Chatterjea [7]). Let (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a given mapping. Suppose that there exists a constant πœ†βˆˆ(0,1/2) such that [],𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†π‘‘(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯)(3.7) for all π‘₯,π‘¦βˆˆπ‘‹. Then 𝑇 has a unique fixed point.

3.2. Fixed Point Theorems on Metric Spaces Endowed with a Partial Order

Recently there have been so many exciting developments in the field of existence of fixed point on metric spaces endowed with partial orders. This trend was started by Turinici [8] in 1986. Ran and Reurings in [9] extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. The obtained result in [9] was further extended and refined by many authors (see, e.g., [10–15] and the references cited therein). In this section, from our Theorem 2.6, we will deduce very easily various fixed point results on a metric space endowed with a partial order. At first, we need to recall some concepts.

Definition 3.8. Let (𝑋,βͺ―) be a partially ordered set and π‘‡βˆΆπ‘‹β†’π‘‹ be a given mapping. We say that 𝑇 is nondecreasing with respect to βͺ― if π‘₯,π‘¦βˆˆπ‘‹,π‘₯βͺ―π‘¦βŸΉπ‘‡π‘₯βͺ―𝑇𝑦.(3.8)

Definition 3.9. Let (𝑋,βͺ―) be a partially ordered set. A sequence {π‘₯𝑛}βŠ‚π‘‹ is said to be nondecreasing with respect to βͺ― if π‘₯𝑛βͺ―π‘₯𝑛+1 for all 𝑛.

Definition 3.10. Let (𝑋,βͺ―) be a partially ordered set and 𝑑 be a metric on 𝑋. We say that (𝑋,βͺ―,𝑑) is regular if for every nondecreasing sequence {π‘₯𝑛}βŠ‚π‘‹ such that π‘₯𝑛→π‘₯βˆˆπ‘‹ as π‘›β†’βˆž, there exists a subsequence {π‘₯𝑛(π‘˜)} of {π‘₯𝑛} such that π‘₯𝑛(π‘˜)βͺ―π‘₯ for all π‘˜.

We have the following result.

Corollary 3.11. Let (𝑋,βͺ―) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let π‘‡βˆΆπ‘‹β†’π‘‹ be a nondecreasing mapping with respect to βͺ―. Suppose that there exists a function πœ“βˆˆΞ¨ such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ“(𝑀(π‘₯,𝑦)),(3.9) for all π‘₯,π‘¦βˆˆπ‘‹ with π‘₯≽𝑦. Suppose also that the following conditions hold: (i)there exists π‘₯0βˆˆπ‘‹ such that π‘₯0βͺ―𝑇π‘₯0; (ii)𝑇 is continuous or (𝑋,βͺ―,𝑑) is regular. Then 𝑇 has a fixed point. Moreover, if for all π‘₯,π‘¦βˆˆπ‘‹ there exists π‘§βˆˆπ‘‹ such that π‘₯βͺ―𝑧 and 𝑦βͺ―𝑧, one has uniqueness of the fixed point.

Proof. Define the mapping π›ΌβˆΆπ‘‹Γ—π‘‹β†’[0,∞) by ξ‚»1𝛼(π‘₯,𝑦)=ifπ‘₯βͺ―𝑦or0π‘₯βͺ°π‘¦,otherwise.(3.10) Clearly, 𝑇 is a generalized 𝛼-πœ“ contractive mapping, that is, 𝛼(π‘₯,𝑦)𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ“(𝑀(π‘₯,𝑦)),(3.11) for all π‘₯,π‘¦βˆˆπ‘‹. From condition (i), we have 𝛼(π‘₯0,𝑇π‘₯0)β‰₯1. Moreover, for all π‘₯,π‘¦βˆˆπ‘‹, from the monotone property of 𝑇, we have 𝛼(π‘₯,𝑦)β‰₯1⟹π‘₯βͺ°π‘¦orπ‘₯βͺ―π‘¦βŸΉπ‘‡π‘₯βͺ°π‘‡π‘¦or𝑇π‘₯βͺ―π‘‡π‘¦βŸΉπ›Ό(𝑇π‘₯,𝑇𝑦)β‰₯1.(3.12) Thus 𝑇 is 𝛼 admissible. Now, if 𝑇 is continuous, the existence of a fixed point follows from Theorem 2.3. Suppose now that (𝑋,βͺ―,𝑑) is regular. Let {π‘₯𝑛} be a sequence in 𝑋 such that 𝛼(π‘₯𝑛,π‘₯𝑛+1)β‰₯1 for all 𝑛 and π‘₯𝑛→π‘₯βˆˆπ‘‹ as π‘›β†’βˆž. From the regularity hypothesis, there exists a subsequence {π‘₯𝑛(π‘˜)} of {π‘₯𝑛} such that π‘₯𝑛(π‘˜)βͺ―π‘₯ for all π‘˜. This implies from the definition of 𝛼 that 𝛼(π‘₯𝑛(π‘˜),π‘₯)β‰₯1 for all π‘˜. In this case, the existence of a fixed point follows from Theorem 2.4. To show the uniqueness, and let π‘₯,π‘¦βˆˆπ‘‹. By hypothesis, there exists π‘§βˆˆπ‘‹ such that π‘₯βͺ―𝑧 and 𝑦βͺ―𝑧, which implies from the definition of 𝛼 that 𝛼(π‘₯,𝑧)β‰₯1 and 𝛼(𝑦,𝑧)β‰₯1. Thus we deduce the uniqueness of the fixed point by Theorem 2.6.

The following results are immediate consequences of Corollary 3.11.

Corollary 3.12. Let (𝑋,βͺ―) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let π‘‡βˆΆπ‘‹β†’π‘‹ be a nondecreasing mapping with respect to βͺ―. Suppose that there exists a function πœ“βˆˆΞ¨ such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ“(𝑑(π‘₯,𝑦)),(3.13) for all π‘₯,π‘¦βˆˆπ‘‹β€‰β€‰with π‘₯≽𝑦. Suppose also that the following conditions hold: (i)there exists π‘₯0βˆˆπ‘‹ such that π‘₯0βͺ―𝑇π‘₯0; (ii)𝑇 is continuous or (𝑋,βͺ―,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all π‘₯,π‘¦βˆˆπ‘‹ there exists π‘§βˆˆπ‘‹ such that π‘₯βͺ―𝑧 and 𝑦βͺ―𝑧, one has uniqueness of the fixed point.

Corollary 3.13. Let (𝑋,βͺ―) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let π‘‡βˆΆπ‘‹β†’π‘‹ be a nondecreasing mapping with respect to βͺ―. Suppose that there exists a constant πœ†βˆˆ(0,1) such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†max𝑑(π‘₯,𝑦),𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦)2,𝑑(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯)2ξ‚Ό,(3.14) for all π‘₯,π‘¦βˆˆπ‘‹β€‰β€‰with π‘₯≽𝑦. Suppose also that the following conditions hold: (i)there exists π‘₯0βˆˆπ‘‹ such that π‘₯0βͺ―𝑇π‘₯0; (ii)𝑇 is continuous or (𝑋,βͺ―,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all π‘₯,π‘¦βˆˆπ‘‹ there exists π‘§βˆˆπ‘‹ such that π‘₯βͺ―𝑧 and 𝑦βͺ―𝑧, one has uniqueness of the fixed point.

Corollary 3.14. Let (𝑋,βͺ―) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let π‘‡βˆΆπ‘‹β†’π‘‹ be a nondecreasing mapping with respect to βͺ―. Suppose that there exist constants 𝐴,𝐡,𝐢β‰₯0 with (𝐴+2𝐡+2𝐢)∈(0,1) such that [][],𝑑(𝑇π‘₯,𝑇𝑦)≀𝐴𝑑(π‘₯,𝑦)+𝐡𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦)+𝐢𝑑(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯)(3.15) for all π‘₯,π‘¦βˆˆπ‘‹ with π‘₯≽𝑦. Suppose also that the following conditions hold: (i)there exists π‘₯0βˆˆπ‘‹ such that π‘₯0βͺ―𝑇π‘₯0; (ii)𝑇 is continuous or (𝑋,βͺ―,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all π‘₯,π‘¦βˆˆπ‘‹ there exists π‘§βˆˆπ‘‹ such that π‘₯βͺ―𝑧 and 𝑦βͺ―𝑧, one has uniqueness of the fixed point.

Corollary 3.15 (see Ran and Reurings [9], Nieto and LΓ³pez [16]). Let (𝑋,βͺ―) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let π‘‡βˆΆπ‘‹β†’π‘‹ be a nondecreasing mapping with respect to βͺ―. Suppose that there exists a constant πœ†βˆˆ(0,1) such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†π‘‘(π‘₯,𝑦),(3.16) for all π‘₯,π‘¦βˆˆπ‘‹ with π‘₯≽𝑦. Suppose also that the following conditions hold: (i)there exists π‘₯0βˆˆπ‘‹ such that π‘₯0βͺ―𝑇π‘₯0; (ii)𝑇 is continuous or (𝑋,βͺ―,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all π‘₯,π‘¦βˆˆπ‘‹ there exists π‘§βˆˆπ‘‹ such that π‘₯βͺ―𝑧 and 𝑦βͺ―𝑧, one has uniqueness of the fixed point.

Corollary 3.16. Let (𝑋,βͺ―) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let π‘‡βˆΆπ‘‹β†’π‘‹ be a nondecreasing mapping with respect to βͺ―. Suppose that there exists a constant πœ†βˆˆ(0,1/2) such that [],𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†π‘‘(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦)(3.17) for all π‘₯,π‘¦βˆˆπ‘‹ with π‘₯≽𝑦. Suppose also that the following conditions hold: (i)there exists π‘₯0βˆˆπ‘‹ such that π‘₯0βͺ―𝑇π‘₯0; (ii)𝑇 is continuous or (𝑋,βͺ―,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all π‘₯,π‘¦βˆˆπ‘‹ there exists π‘§βˆˆπ‘‹ such that π‘₯βͺ―𝑧 and 𝑦βͺ―𝑧, one has uniqueness of the fixed point.

Corollary 3.17. Let (𝑋,βͺ―) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let π‘‡βˆΆπ‘‹β†’π‘‹ be a nondecreasing mapping with respect to βͺ―. Suppose that there exists a constant πœ†βˆˆ(0,1/2) such that [],𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†π‘‘(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯)(3.18) for all π‘₯,π‘¦βˆˆπ‘‹ with π‘₯≽𝑦. Suppose also that the following conditions hold: (i)there exists π‘₯0βˆˆπ‘‹ such that π‘₯0βͺ―𝑇π‘₯0; (ii)𝑇 is continuous or (𝑋,βͺ―,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all  π‘₯,π‘¦βˆˆπ‘‹ there exists π‘§βˆˆπ‘‹ such that π‘₯βͺ―𝑧 and 𝑦βͺ―𝑧, one has uniqueness of the fixed point.

3.3. Fixed Point Theorems for Cyclic Contractive Mappings

One of the remarkable generalizations of the Banach Contraction Mapping Principle was reported by Kirk et al. [17] via cyclic contraction. Following the paper [17], many fixed point theorems for cyclic contractive mappings have appeared (see, e.g., [18–23]). In this section, we will show that, from our Theorem 2.6, we can deduce some fixed point theorems for cyclic contractive mappings.

We have the following result.

Corollary 3.18. Let {𝐴𝑖}2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and π‘‡βˆΆπ‘Œβ†’π‘Œ be a given mapping, where π‘Œ=𝐴1βˆͺ𝐴2. Suppose that the following conditions hold: (I)𝑇(𝐴1)βŠ†π΄2 and 𝑇(𝐴2)βŠ†π΄1; (II)there exists a function πœ“βˆˆΞ¨ such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ“(𝑀(π‘₯,𝑦)),βˆ€(π‘₯,𝑦)∈𝐴1×𝐴2.(3.19)Then 𝑇 has a unique fixed point that belongs to 𝐴1∩𝐴2.

Proof. Since 𝐴1 and 𝐴2 are closed subsets of the complete metric space (𝑋,𝑑), then (π‘Œ,𝑑) is complete. Define the mapping π›ΌβˆΆπ‘ŒΓ—π‘Œβ†’[0,∞) by ξ‚»1𝛼(π‘₯,𝑦)=if𝐴(π‘₯,𝑦)∈1×𝐴2ξ€Έβˆͺ𝐴2×𝐴1ξ€Έ,0otherwise.(3.20) From (II) and the definition of 𝛼, we can write 𝛼(π‘₯,𝑦)𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ“(𝑀(π‘₯,𝑦)),(3.21) for all π‘₯,π‘¦βˆˆπ‘Œ. Thus 𝑇 is a generalized 𝛼-πœ“ contractive mapping.
Let (π‘₯,𝑦)βˆˆπ‘ŒΓ—π‘Œ such that 𝛼(π‘₯,𝑦)β‰₯1. If (π‘₯,𝑦)∈𝐴1×𝐴2, from (I), (𝑇π‘₯,𝑇𝑦)∈𝐴2×𝐴1, which implies that 𝛼(𝑇π‘₯,𝑇𝑦)β‰₯1. If (π‘₯,𝑦)∈𝐴2×𝐴1, from (I), (𝑇π‘₯,𝑇𝑦)∈𝐴1×𝐴2, which implies that 𝛼(𝑇π‘₯,𝑇𝑦)β‰₯1. Thus in all cases, we have 𝛼(𝑇π‘₯,𝑇𝑦)β‰₯1. This implies that 𝑇 is 𝛼-admissible.
Also, from (I), for any π‘Žβˆˆπ΄1, we have (π‘Ž,π‘‡π‘Ž)∈𝐴1×𝐴2, which implies that 𝛼(π‘Ž,π‘‡π‘Ž)β‰₯1.
Now, let {π‘₯𝑛} be a sequence in 𝑋 such that 𝛼(π‘₯𝑛,π‘₯𝑛+1)β‰₯1 for all 𝑛 and π‘₯𝑛→π‘₯βˆˆπ‘‹ as π‘›β†’βˆž. This implies from the definition of 𝛼 that ξ€·π‘₯𝑛,π‘₯𝑛+1ξ€Έβˆˆξ€·π΄1×𝐴2ξ€Έβˆͺ𝐴2×𝐴1ξ€Έ,βˆ€π‘›.(3.22) Since (𝐴1×𝐴2)βˆͺ(𝐴2×𝐴1) is a closed set with respect to the Euclidean metric, we get that 𝐴(π‘₯,π‘₯)∈1×𝐴2ξ€Έβˆͺ𝐴2×𝐴1ξ€Έ,(3.23) which implies that π‘₯∈𝐴1∩𝐴2. Thus we get immediately from the definition of 𝛼 that 𝛼(π‘₯𝑛,π‘₯)β‰₯1 for all 𝑛.
Finally, let π‘₯,π‘¦βˆˆFix(𝑇). From (I), this implies that π‘₯,π‘¦βˆˆπ΄1∩𝐴2. So, for any π‘§βˆˆπ‘Œ, we have 𝛼(π‘₯,𝑧)β‰₯1 and 𝛼(𝑦,𝑧)β‰₯1. Thus condition (H) is satisfied.
Now, all the hypotheses of Theorem 2.6 are satisfied, and we deduce that 𝑇 has a unique fixed point that belongs to 𝐴1∩𝐴2 (from (I)).

The following results are immediate consequences of Corollary 3.18.

Corollary 3.19 (see Pacurar and Rus [21]). Let {𝐴𝑖}2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and π‘‡βˆΆπ‘Œβ†’π‘Œ be a given mapping, where π‘Œ=𝐴1βˆͺ𝐴2. Suppose that the following conditions hold: (I)𝑇(𝐴1)βŠ†π΄2 and 𝑇(𝐴2)βŠ†π΄1; (II)there exists a function πœ“βˆˆΞ¨ such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ“(𝑑(π‘₯,𝑦)),βˆ€(π‘₯,𝑦)∈𝐴1×𝐴2.(3.24)Then 𝑇 has a unique fixed point that belongs to 𝐴1∩𝐴2.

Corollary 3.20. Let {𝐴𝑖}2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and π‘‡βˆΆπ‘Œβ†’π‘Œ be a given mapping, where π‘Œ=𝐴1βˆͺ𝐴2. Suppose that the following conditions hold: (I)𝑇(𝐴1)βŠ†π΄2 and 𝑇(𝐴2)βŠ†π΄1; (II)there exists a constant πœ†βˆˆ(0,1) such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†max𝑑(π‘₯,𝑦),𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦)2,𝑑(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯)2ξ‚Ό,βˆ€(π‘₯,𝑦)∈𝐴1×𝐴2.(3.25)Then 𝑇 has a unique fixed point that belongs to 𝐴1∩𝐴2.

Corollary 3.21. Let {𝐴𝑖}2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and π‘‡βˆΆπ‘Œβ†’π‘Œ be a given mapping, where π‘Œ=𝐴1βˆͺ𝐴2. Suppose that the following conditions hold: (I)𝑇(𝐴1)βŠ†π΄2 and 𝑇(𝐴2)βŠ†π΄1; (II)there exist constants 𝐴,𝐡,𝐢β‰₯0 with (𝐴+2𝐡+2𝐢)∈(0,1) such that [][],βˆ€π‘‘(𝑇π‘₯,𝑇𝑦)≀𝐴𝑑(π‘₯,𝑦)+𝐡𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦)+𝐢𝑑(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯)(π‘₯,𝑦)∈𝐴1×𝐴2.(3.26)Then 𝑇 has a unique fixed point that belongs to 𝐴1∩𝐴2.

Corollary 3.22 (see Kirk et al. [17]). Let {𝐴𝑖}2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and π‘‡βˆΆπ‘Œβ†’π‘Œ be a given mapping, where π‘Œ=𝐴1βˆͺ𝐴2. Suppose that the following conditions hold: (I)𝑇(𝐴1)βŠ†π΄2 and 𝑇(𝐴2)βŠ†π΄1; (II)there exists a constant πœ†βˆˆ(0,1) such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†π‘‘(π‘₯,𝑦),βˆ€(π‘₯,𝑦)∈𝐴1×𝐴2.(3.27)Then 𝑇 has a unique fixed point that belongs to 𝐴1∩𝐴2.

Corollary 3.23. Let {𝐴𝑖}2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and π‘‡βˆΆπ‘Œβ†’π‘Œ be a given mapping, where π‘Œ=𝐴1βˆͺ𝐴2. Suppose that the following conditions hold: (I)𝑇(𝐴1)βŠ†π΄2 and 𝑇(𝐴2)βŠ†π΄1; (II)there exists a constant πœ†βˆˆ(0,1/2) such that []𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†π‘‘(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦),βˆ€(π‘₯,𝑦)∈𝐴1×𝐴2.(3.28)Then 𝑇 has a unique fixed point that belongs to 𝐴1∩𝐴2.

Corollary 3.24. Let {𝐴𝑖}2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and π‘‡βˆΆπ‘Œβ†’π‘Œ be a given mapping, where π‘Œ=𝐴1βˆͺ𝐴2. Suppose that the following conditions hold: (I)𝑇(𝐴1)βŠ†π΄2 and 𝑇(𝐴2)βŠ†π΄1; (II)there exists a constant πœ†βˆˆ(0,1/2) such that []𝑑(𝑇π‘₯,𝑇𝑦)β‰€πœ†π‘‘(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯),βˆ€(π‘₯,𝑦)∈𝐴1×𝐴2.(3.29)Then 𝑇 has a unique fixed point that belongs to 𝐴1∩𝐴2.

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