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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 756453, 10 pages
http://dx.doi.org/10.1155/2012/756453
Research Article

Fixed Point Theorems for πœ“-Contractive Mappings in Ordered Metric Spaces

Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu 30014, Taiwan

Received 19 September 2011; Accepted 25 November 2011

Academic Editor: YanshengΒ Liu

Copyright Β© 2012 Chi-Ming Chen. 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 obtain some new fixed point theorems for πœ“-contractive mappings in ordered metric spaces. Our results generalize or improve many recent fixed point theorems in the literature (e.g., Harjani et al., 2011 and 2010).

1. Introduction and Preliminaries

Throughout this paper, by ℝ+, we denote the set of all real nonnegative numbers, while β„• is the set of all natural numbers. Let (𝑋,𝑑) be a metric space, 𝐷 a subset of 𝑋; and π‘“βˆΆπ·β†’π‘‹ a map. We say 𝑓 is contractive if there exists π›Όβˆˆ[0,1) such that for all π‘₯,π‘¦βˆˆπ·,𝑑(𝑓π‘₯,𝑓𝑦)≀𝛼⋅𝑑(π‘₯,𝑦).(1.1)

The well-known Banach fixed point theorem asserts that if 𝐷=𝑋, 𝑓 is contractive and (𝑋,𝑑) is complete, then 𝑓 has a unique fixed point in 𝑋. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, a mapping π‘“βˆΆπ‘‹β†’π‘‹ is called a quasicontraction if there exists π‘˜<1 such that𝑑(𝑓π‘₯,𝑓𝑦)β‰€π‘˜β‹…max{𝑑(π‘₯,𝑦),𝑑(π‘₯,𝑓π‘₯),𝑑(𝑦,𝑓𝑦),𝑑(π‘₯,𝑓𝑦),𝑑(𝑦,𝑓π‘₯)},(1.2)

for any π‘₯,π‘¦βˆˆπ‘‹. In 1974, Δ†iriΔ‡ [2] introduced these maps and proved an existence and uniqueness fixed point theorem.

In 1972, Chatterjea [3] introduced the following definition.

Definition 1.1. Let (𝑋,𝑑) be a metric space. A mapping π‘“βˆΆπ‘‹β†’π‘‹ is said to be a π’ž-contraction if there exists π›Όβˆˆ(0,1/2) such that for all π‘₯,π‘¦βˆˆπ‘‹, the following inequality holds: 𝑑(𝑓π‘₯,𝑓𝑦)≀𝛼⋅(𝑑(π‘₯,𝑓𝑦)+𝑑(𝑦,𝑓π‘₯)).(1.3)

Choudhury [4] introduced a generalization of π’ž-contraction as follows.

Definition 1.2. Let (𝑋,𝑑) be a metric space. A mapping π‘“βˆΆπ‘‹β†’π‘‹ is said to be a weakly π’ž-contraction if for all π‘₯,π‘¦βˆˆπ‘‹, 1𝑑(𝑓π‘₯,𝑓𝑦)≀2(𝑑(π‘₯,𝑓𝑦)+𝑑(𝑦,𝑓π‘₯)βˆ’πœ™(𝑑(π‘₯,𝑓𝑦),𝑑(𝑦,𝑓π‘₯))),(1.4) where πœ™βˆΆβ„+2→ℝ+ is a continuous function such that πœ™(π‘₯,𝑦)=0 if and only if π‘₯=𝑦=0.

In [3, 4], the authors proved some fixed point results for the π’ž-contractions. In [5], Harjani et al. proved some fixed point results for weakly π’ž-contractive mappings in a complete metric space endowed with a partial order.

In the following, we assume that the function πœ“βˆΆβ„+5→ℝ+ satisfies the following conditions:(C1)πœ“ is a strictly increasing and continuous function in each coordinate, and(C2)for all π‘‘βˆˆβ„+⧡{0}, πœ“(𝑑,𝑑,𝑑,0,2𝑑)<𝑑, πœ“(𝑑,𝑑,𝑑,2𝑑,0)<𝑑, πœ“(0,0,𝑑,𝑑,0)<𝑑, and πœ“(𝑑,0,0,𝑑,𝑑)<𝑑.

Example 1.3. Let πœ“βˆΆβ„+5→ℝ+ denote πœ“ξ€·π‘‘1,𝑑2,𝑑3,𝑑4,𝑑5𝑑=π‘˜β‹…max1,𝑑2,𝑑3,𝑑42,𝑑52ξ‚Ό,forπ‘˜βˆˆ(0,1).(1.5) Then, πœ“ satisfies the above conditions (C1) and (C2).

Now, we define the following notion of a πœ“-contractive mapping in metric spaces.

Definition 1.4. Let (𝑋,≀) be a partially ordered set and suppose that there exists a metric 𝑑 in 𝑋 such that (𝑋,𝑑) is a metric space. The mapping π‘“βˆΆπ‘‹β†’π‘‹ is said to be a πœ“-contractive mapping, if 𝑑(𝑓π‘₯,𝑓𝑦)β‰€πœ“(𝑑(π‘₯,𝑦),𝑑(π‘₯,𝑓π‘₯),𝑑(𝑦,𝑓𝑦),𝑑(π‘₯,𝑓𝑦),𝑑(𝑦,𝑓π‘₯)),(βˆ—) for π‘₯β‰₯𝑦.

Using Example 1.3, it is easy to get the following examples of πœ“-contractive mappings.

Example 1.5. Let 𝑋=ℝ+ endowed with usual ordering and with the metric π‘‘βˆΆπ‘‹Γ—π‘‹β†’β„+ given by ||||,𝑑(π‘₯,𝑦)=π‘₯βˆ’π‘¦forπ‘₯,π‘¦βˆˆπ‘‹.(1.6) Let πœ“βˆΆβ„+5→ℝ+ denote πœ“ξ€·π‘‘1,𝑑2,𝑑3,𝑑4,𝑑5ξ€Έ=34𝑑⋅max1,𝑑2,𝑑3,𝑑42,𝑑52ξ‚Ό,(1.7) where 𝑑1=𝑑(π‘₯,𝑦), 𝑑2=𝑑(π‘₯,𝑓π‘₯), 𝑑3=𝑑(𝑦,𝑓𝑦), 𝑑4=𝑑(π‘₯,𝑓𝑦),and 𝑑5=𝑑(𝑦,𝑓π‘₯), for all π‘₯,π‘¦βˆˆπ‘‹. Let π‘“βˆΆπ‘‹β†’π‘‹ denote 1𝑓(π‘₯)=3π‘₯.(1.8) Then, 𝑓 is a πœ“-contractive mapping.

Example 1.6. Let 𝑋=ℝ+×ℝ+ endowed with the coordinate ordering (i.e., (π‘₯,𝑦)≀(𝑧,𝑀)⇔π‘₯≀𝑧 and 𝑦≀𝑀) and with the metric π‘‘βˆΆπ‘‹Γ—π‘‹β†’β„+ given by 𝑑||π‘₯(π‘₯,𝑦)=1βˆ’π‘¦1||+||π‘₯2βˆ’π‘¦2||,forξ€·π‘₯π‘₯=1,π‘₯2𝑦,𝑦=1,𝑦2ξ€Έβˆˆπ‘‹.(1.9) Let πœ“βˆΆβ„+5→ℝ+ denote πœ“ξ€·π‘‘1,𝑑2,𝑑3,𝑑4,𝑑5ξ€Έ=34𝑑⋅max1,𝑑2,𝑑3,𝑑42,𝑑52ξ‚Ό,(1.10) where 𝑑1=𝑑(π‘₯,𝑦), 𝑑2=𝑑(π‘₯,𝑓π‘₯), 𝑑3=𝑑(𝑦,𝑓𝑦), 𝑑4=𝑑(π‘₯,𝑓𝑦), and 𝑑5=𝑑(𝑦,𝑓π‘₯), for all π‘₯,π‘¦βˆˆπ‘‹. Let π‘“βˆΆπ‘‹β†’π‘‹ denote 1𝑓(π‘₯)=3π‘₯.(1.11) Then, 𝑓 is a πœ“-contractive mapping.

In this paper, we obtain some new fixed point theorems for πœ“-contractive mappings in ordered metric spaces. Our results generalize or improve many recent fixed point theorems in the literature (e.g., [5, 6]).

2. Main Results

We start with the following definition.

Definition 2.1. Let (𝑋,≀) be a partially ordered set and π‘“βˆΆπ‘‹β†’π‘‹. Then one says that 𝑓 is monotone nondecreasing if, for π‘₯,π‘¦βˆˆπ‘‹, π‘₯β‰€π‘¦βŸΉπ‘“π‘₯≀𝑓𝑦.(2.1)

We now state the main fixed point theorem for πœ“-contractive mappings in ordered metric spaces when the operator is nondecreasing, as follows.

Theorem 2.2. Let (𝑋,≀) be a partially ordered set and suppose that there exists a metric 𝑑 in 𝑋 such that (𝑋,𝑑) is a complete metric space, and let π‘“βˆΆπ‘‹β†’π‘‹ be a continuous and nondecreasing πœ“-contractive mapping. If there exists π‘₯0βˆˆπ‘‹ with π‘₯0≀𝑓π‘₯0, then 𝑓 has a fixed point in 𝑋.

Proof. If 𝑓(π‘₯0)=π‘₯0, then the proof is finished. Suppose that π‘₯0<𝑓(π‘₯0). Since 𝑓 is nondecreasing mapping, by induction, we obtain that π‘₯0<𝑓π‘₯0≀𝑓2π‘₯0≀𝑓3π‘₯0≀⋯≀𝑓𝑛π‘₯0≀𝑓𝑛+1π‘₯0≀⋯.(2.2) Put π‘₯𝑛+1=𝑓π‘₯𝑛=𝑓𝑛+1π‘₯0 for π‘›βˆˆβ„•βˆͺ{0}. Then, for each π‘›βˆˆβ„•, from (βˆ—), and, as the elements π‘₯𝑛 and π‘₯π‘›βˆ’1 are comparable, we get 𝑑π‘₯𝑛+1,π‘₯𝑛𝑑π‘₯β‰€πœ“π‘›,π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝑑𝑛,𝑓π‘₯𝑛π‘₯,π‘‘π‘›βˆ’1,𝑓π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝑑𝑛,𝑓π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,π‘‘π‘›βˆ’1,𝑓π‘₯𝑛𝑑π‘₯ξ€Έξ€Έβ‰€πœ“π‘›,π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛π‘₯,𝑑𝑛,π‘₯𝑛π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛+1𝑑π‘₯ξ€Έξ€Έβ‰€πœ“π‘›,π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛π‘₯,0,π‘‘π‘›βˆ’1,π‘₯𝑛π‘₯+𝑑𝑛,π‘₯𝑛+1,ξ€Έξ€Έ(2.3) and so we can deduce that, for each π‘›βˆˆβ„•, 𝑑π‘₯𝑛+1,π‘₯𝑛π‘₯≀𝑑𝑛,π‘₯π‘›βˆ’1ξ€Έ.(2.4) Let we denote π‘π‘š=𝑑(π‘₯π‘š+1,π‘₯π‘š). Then, π‘π‘š is a nonincreasing sequence and bounded below. Thus, it must converge to some 𝑐β‰₯0. If 𝑐>0, then by the above inequalities, we have 𝑐≀𝑐𝑛+1ξ€·π‘β‰€πœ“π‘›,𝑐𝑛,𝑐𝑛,0,2𝑐𝑛.(2.5) Passing to the limit, as π‘›β†’βˆž, we have π‘β‰€π‘β‰€πœ“(𝑐,𝑐,𝑐,0,2𝑐)<𝑐,(2.6) which is a contradiction. So 𝑐=0.
We next claim that that the following result holds.
For each 𝛾>0, there is 𝑛0(𝛾)βˆˆβ„• such that for all π‘š>𝑛>𝑛0(𝛾),𝑑π‘₯π‘š,π‘₯𝑛<𝛾.(βˆ—) We will prove (βˆ—) by contradiction. Suppose that (βˆ—) is false. Then, there exists some 𝛾>0 such that for all π‘˜βˆˆβ„•, there exist π‘šπ‘˜ and π‘›π‘˜ with π‘šπ‘˜>π‘›π‘˜>π‘˜ such that 𝑑π‘₯π‘šπ‘˜,π‘₯π‘›π‘˜ξ€Έξ€·π‘₯β‰₯𝛾,π‘‘π‘šπ‘˜βˆ’1,π‘₯π‘›π‘˜ξ€Έ<𝛾.(2.7) Using the triangular inequality: ξ€·π‘₯π›Ύβ‰€π‘‘π‘šπ‘˜,π‘₯π‘›π‘˜ξ€Έξ€·π‘₯β‰€π‘‘π‘šπ‘˜,π‘₯π‘šπ‘˜βˆ’1ξ€Έξ€·π‘₯+π‘‘π‘šπ‘˜βˆ’1,π‘₯π‘›π‘˜ξ€Έξ€·π‘₯<𝛾+π‘‘π‘šπ‘˜,π‘₯π‘šπ‘˜βˆ’1ξ€Έ,(2.8) and letting π‘˜β†’βˆž, we have limπ‘˜β†’βˆžπ‘‘ξ€·π‘₯π‘šπ‘˜,π‘₯π‘›π‘˜ξ€Έ=𝛾.(2.9)
Since 𝑓 is a πœ“-contractive mapping, we also haveξ€·π‘₯π›Ύβ‰€π‘‘π‘šπ‘˜,π‘₯π‘›π‘˜ξ€Έξ€·=𝑑𝑓π‘₯π‘šπ‘˜βˆ’1,𝑓π‘₯π‘›π‘˜βˆ’1𝑑π‘₯β‰€πœ“π‘šπ‘˜βˆ’1,π‘₯π‘›π‘˜βˆ’1ξ€Έξ€·π‘₯,π‘‘π‘šπ‘˜βˆ’1,π‘₯π‘šπ‘˜ξ€Έξ€·π‘₯,π‘‘π‘›π‘˜βˆ’1,π‘₯π‘›π‘˜ξ€Έξ€·π‘₯,π‘‘π‘šπ‘˜βˆ’1,π‘₯π‘›π‘˜ξ€Έξ€·π‘₯,π‘‘π‘›π‘˜βˆ’1,π‘₯π‘šπ‘˜ξ€·π‘ξ€Έξ€Έβ‰€πœ“π‘šπ‘˜βˆ’1ξ€·π‘₯+π‘‘π‘šπ‘˜,π‘₯π‘›π‘˜ξ€Έ+π‘π‘›π‘˜βˆ’1,π‘π‘šπ‘˜βˆ’1,π‘π‘›π‘˜βˆ’1,π‘π‘šπ‘˜βˆ’1ξ€·π‘₯+π‘‘π‘šπ‘˜,π‘₯π‘›π‘˜ξ€Έξ€·π‘₯,π‘‘π‘šπ‘˜,π‘₯π‘›π‘˜ξ€Έ+π‘π‘›π‘˜βˆ’1ξ€Έ.(2.10) Letting π‘˜β†’βˆž. Then, we get π›Ύβ‰€πœ“(𝛾,0,0,𝛾,𝛾)<𝛾,(2.11) a contradiction. It follows from (βˆ—) that the sequence {π‘₯𝑛} must be a Cauchy sequence.
Similary, we also conclude that for each π‘›βˆˆβ„•, 𝑑π‘₯𝑛,π‘₯𝑛+1𝑑π‘₯β‰€πœ“π‘›βˆ’1,π‘₯𝑛π‘₯,π‘‘π‘›βˆ’1,𝑓π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝑑𝑛,𝑓π‘₯𝑛π‘₯,π‘‘π‘›βˆ’1,𝑓π‘₯𝑛π‘₯,𝑑𝑛,𝑓π‘₯π‘›βˆ’1𝑑π‘₯ξ€Έξ€Έβ‰€πœ“π‘›βˆ’1,π‘₯𝑛π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛π‘₯,𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛+1ξ€Έξ€·π‘₯,𝑑𝑛,π‘₯𝑛𝑑π‘₯ξ€Έξ€Έβ‰€πœ“π‘›βˆ’1,π‘₯𝑛π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛π‘₯,𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛π‘₯+𝑑𝑛,π‘₯𝑛+1ξ€Έξ€Έ,,0(2.12) and so we have that for each π‘›βˆˆβ„•, 𝑑π‘₯𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯β‰€π‘‘π‘›βˆ’1,π‘₯𝑛.(2.13)
Let us denote π‘π‘š=𝑑(π‘₯π‘š,π‘₯π‘š+1). Then, π‘π‘š is a nonincreasing sequence and bounded below. Thus, it must converge to some 𝑏β‰₯0. If 𝑏>0, then by the above inequalities, we have 𝑏≀𝑏𝑛+1ξ€·π‘β‰€πœ“π‘›,𝑏𝑛,𝑏𝑛,2𝑏𝑛.,0(2.14)Passing to the limit, as π‘›β†’βˆž, we have π‘β‰€π‘β‰€πœ“(𝑏,𝑏,𝑏,2𝑏,0)<𝑏,(2.15)which is a contradiction. So 𝑏=0. By the above argument, we also conclude that {π‘₯𝑛} is a Cauchy sequence.
Since 𝑋 is complete, there exists πœ‡βˆˆπ‘‹ such that limπ‘›β†’βˆžπ‘₯𝑛=πœ‡. Moreover, the continuity of 𝑓 implies that πœ‡=limπ‘›β†’βˆžπ‘₯𝑛+1=limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛=𝑓(πœ‡).(2.16)
So we complete the proof.

In what follows, we prove that Theorem 2.2 is still valid for 𝑓 not necessarily continuous, assuming the following hypothesis in 𝑋(which appears in Theorem 1 of [7]).If {π‘₯𝑛} is a nondecreasing sequence in 𝑋, such that π‘₯π‘›βŸΆπ‘₯,thenπ‘₯𝑛≀π‘₯βˆ€π‘›βˆˆβ„•.(βˆ—βˆ—)

Theorem 2.3. Let (𝑋,≀) be a partially ordered set and suppose that there exists a metric 𝑑 in 𝑋 such that (𝑋,𝑑) is a complete metric space. Assume that 𝑋 satisfies (βˆ—βˆ—), and let π‘“βˆΆπ‘‹β†’π‘‹ be a nondecreasing πœ“-contractive mapping. If there exists π‘₯0βˆˆπ‘‹ with π‘₯0≀𝑓(π‘₯0), then 𝑓 has a fixed point in 𝑋.

Proof. Following the proof of Theorem 2.2, we only have to check that 𝑓(πœ‡)=πœ‡. As {π‘₯𝑛} is a nondecreasing sequence in 𝑋 and π‘₯π‘›β†’πœ‡, then the condition (βˆ—βˆ—) gives us that π‘₯π‘›β‰€πœ‡ for every π‘›βˆˆβ„•. Since π‘“βˆΆπ‘‹β†’π‘‹ is a nondecreasing πœ“-contractive mapping, we have 𝑑π‘₯𝑛+1ξ€Έξ€·,π‘“πœ‡=𝑑𝑓π‘₯𝑛𝑑π‘₯,π‘“πœ‡β‰€πœ“π‘›ξ€Έξ€·π‘₯,πœ‡,𝑑𝑛,𝑓π‘₯𝑛π‘₯,𝑑(πœ‡,π‘“πœ‡),𝑑𝑛,π‘“πœ‡,π‘‘πœ‡,𝑓π‘₯𝑛𝑑π‘₯ξ€Έξ€Έβ‰€πœ“π‘›ξ€Έξ€·π‘₯,πœ‡,𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯,𝑑(πœ‡,π‘“πœ‡),𝑑𝑛,π‘“πœ‡,π‘‘πœ‡,π‘₯𝑛+1.ξ€Έξ€Έ(2.17) Letting π‘›β†’βˆž and using the continuity of πœ“, we have 𝑑(πœ‡,π‘“πœ‡)β‰€πœ“(0,0,𝑑(πœ‡,π‘“πœ‡),𝑑(πœ‡,π‘“πœ‡),0)<𝑑(πœ‡,π‘“πœ‡),(2.18) and this is a contraction unless 𝑑(πœ‡,π‘“πœ‡)=0, or equivalently, πœ‡=π‘“πœ‡.

In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorems 2.2 and 2.3. This condition is the following and it appears in [8]:forπ‘₯,π‘¦βˆˆπ‘‹,thereexistsalowerboundoranupperbound.(2.19) In [7], it is proved that the above-mentioned condition is equivalent to the following: forπ‘₯,π‘¦βˆˆπ‘‹,thereexistsπ‘§βˆˆπ‘‹whichiscomparabletoπ‘₯and𝑦.(βˆ—βˆ—βˆ—)

Theorem 2.4. Adding condition (βˆ—βˆ—βˆ—) to the hypothesis of Theorem 2.2 (or Theorem 2.3) and the condition for all π‘‘βˆˆβ„+, πœ“(𝑑,0,2𝑑,𝑑,𝑑)<𝑑(or, πœ“(𝑑,2𝑑,0,0,𝑑)<𝑑) to the function πœ“, one obtains the uniqueness of the fixed point of 𝑓.

Proof. Suppose that there exist πœ‡,πœˆβˆˆπ‘‹ which are fixed points of 𝑓. We distinguish two cases.
Case 1. If πœ‡ and 𝜈 are comparable and πœ‡β‰ πœˆ, then π‘“π‘›πœ‡=πœ‡ is comparable to π‘“π‘›πœˆ=𝜈 for all π‘›βˆˆβ„•, and 𝑑(πœ‡,𝜈)=𝑑(π‘“π‘›πœ‡,π‘“π‘›ξ€·π‘‘ξ€·π‘“πœˆ)β‰€πœ“π‘›βˆ’1πœ‡,π‘“π‘›βˆ’1πœˆξ€Έξ€·π‘“,π‘‘π‘›βˆ’1πœ‡,π‘“π‘›πœ‡ξ€Έξ€·π‘“,π‘‘π‘›βˆ’1𝜈,π‘“π‘›πœˆξ€Έξ€·π‘“,π‘‘π‘›βˆ’1πœ‡,π‘“π‘›πœˆξ€Έξ€·π‘“,π‘‘π‘›βˆ’1𝜈,π‘“π‘›πœ‡ξ€Έξ€Έβ‰€πœ“(𝑑(πœ‡,𝜈),𝑑(πœ‡,πœ‡),𝑑(𝜈,𝜈),𝑑(πœ‡,𝜈),𝑑(𝜈,πœ‡))=πœ“(𝑑(πœ‡,𝜈),0,0,𝑑(πœ‡,𝜈),𝑑(𝜈,πœ‡))<𝑑(πœ‡,𝜈),(2.20) and this is a contradiction unless 𝑑(πœ‡,𝜈)=0, that is, πœ‡=𝜈.Case 2. If πœ‡ and 𝜈 are not comparable, then there exists π‘₯βˆˆπ‘‹ comparable to πœ‡ and 𝜈. Monotonicity of 𝑓 implies that 𝑓𝑛π‘₯ is comparable to π‘“π‘›πœ‡ and π‘“π‘›πœˆ for all π‘›βˆˆβ„•. We also distinguish two cases.Subcase 2.1. If there exists 𝑛0βˆˆβ„• with 𝑓𝑛0π‘₯=πœ‡, then we have 𝑓𝑑(πœ‡,𝜈)=𝑑(π‘“πœ‡,π‘“πœˆ)=𝑑𝑛0+1π‘₯,𝑓𝑛0+1πœˆξ€Έξ€·π‘‘β‰€πœ“(𝑓𝑛0π‘₯,𝑓𝑛0πœˆξ€·π‘“),𝑑𝑛0π‘₯,𝑓𝑛0+1π‘₯𝑓,𝑑𝑛0𝜈,𝑓𝑛0+1πœˆξ€Έξ€·π‘“,𝑑𝑛0π‘₯,𝑓𝑛0+1πœˆξ€Έξ€·π‘“,𝑑𝑛0𝜈,𝑓𝑛0+1π‘₯ξ€Έξ€Έ=πœ“(𝑑(πœ‡,𝜈),𝑑(πœ‡,π‘“πœ‡),𝑑(𝜈,𝜈),𝑑(πœ‡,𝜈),𝑑(𝜈,π‘“πœ‡))=πœ“(𝑑(πœ‡,𝜈),0,0,𝑑(πœ‡,𝜈),𝑑(𝜈,πœ‡))<𝑑(πœ‡,𝜈),(2.21) and this is a contradiction unless 𝑑(πœ‡,𝜈)=0, that is, πœ‡=𝜈.Subcase 2.2. For all π‘›βˆˆβ„• with 𝑓𝑛π‘₯β‰ πœ‡, since 𝑓 is a nondecreasing πœ“-contractive mapping, we have 𝑑(πœ‡,𝑓𝑛π‘₯)=𝑑(π‘“π‘›πœ‡,𝑓𝑛𝑑𝑓π‘₯)β‰€πœ“π‘›βˆ’1πœ‡,π‘“π‘›βˆ’1π‘₯𝑓,π‘‘π‘›βˆ’1πœ‡,π‘“π‘›πœ‡ξ€Έξ€·π‘“,π‘‘π‘›βˆ’1π‘₯,𝑓𝑛π‘₯𝑓,π‘‘π‘›βˆ’1πœ‡,𝑓𝑛π‘₯𝑓,π‘‘π‘›βˆ’1π‘₯,π‘“π‘›πœ‡ξ€·π‘‘ξ€·ξ€Έξ€Έβ‰€πœ“πœ‡,π‘“π‘›βˆ’1π‘₯𝑓,𝑑(πœ‡,πœ‡),π‘‘π‘›βˆ’1π‘₯,𝑓𝑛π‘₯ξ€Έ,𝑑(πœ‡,𝑓𝑛𝑓π‘₯),π‘‘π‘›βˆ’1𝑑π‘₯,πœ‡ξ€Έξ€Έβ‰€πœ“πœ‡,π‘“π‘›βˆ’1π‘₯𝑓,0,π‘‘π‘›βˆ’1ξ€Έπ‘₯,πœ‡+𝑑(πœ‡,𝑓𝑛π‘₯),𝑑(πœ‡,𝑓𝑛𝑓π‘₯),π‘‘π‘›βˆ’1.π‘₯,πœ‡ξ€Έξ€Έ(2.22) Using the above inequality, we claim that for each π‘›βˆˆβ„•, 𝑑(πœ‡,𝑓𝑛π‘₯ξ€·)<π‘‘πœ‡,π‘“π‘›βˆ’1π‘₯ξ€Έ.(2.23) If not, we assume that 𝑑(πœ‡,π‘“π‘›βˆ’1π‘₯)≀𝑑(πœ‡,𝑓𝑛π‘₯), then by the definition of πœ“ and πœ“(𝑑,0,2𝑑,𝑑,𝑑)<𝑑, we have 𝑑(πœ‡,𝑓𝑛π‘₯𝑑)β‰€πœ“πœ‡,π‘“π‘›βˆ’1π‘₯𝑓,0,π‘‘π‘›βˆ’1ξ€Έπ‘₯,πœ‡+𝑑(πœ‡,𝑓𝑛π‘₯),𝑑(πœ‡,𝑓𝑛π‘₯𝑓),π‘‘π‘›βˆ’1π‘₯,πœ‡ξ€Έξ€Έβ‰€πœ“(𝑑(πœ‡,𝑓𝑛π‘₯),0,2𝑑(𝑓𝑛π‘₯,πœ‡),𝑑(πœ‡,𝑓𝑛π‘₯),𝑑(𝑓𝑛π‘₯,πœ‡))<𝑑(πœ‡,𝑓𝑛π‘₯),(2.24) which implies a contradiction. Therefore, our claim is proved.
This proves that the nonnegative decreasing sequence {𝑑(πœ‡,𝑓𝑛π‘₯)} is convergent. Put limπ‘›β†’βˆžπ‘‘(πœ‡,𝑓𝑛π‘₯)=πœ‚, πœ‚β‰₯0. We now claim that πœ‚=0. If πœ‚>0, then making π‘›β†’βˆž, we getπœ‚=limπ‘›β†’βˆžπ‘‘(πœ‡,𝑓𝑛π‘₯)β‰€πœ“(πœ‚,0,2πœ‚,πœ‚,πœ‚)<πœ‚,(2.25)this is a contradiction. So πœ‚=0, that is, limπ‘›β†’βˆžπ‘‘(πœ‡,𝑓𝑛π‘₯)=0.
Analogously, it can be proved that limπ‘›β†’βˆžπ‘‘(𝜈,𝑓𝑛π‘₯)=0.
Finally, the uniqueness of the limit gives us πœ‡=𝜈.
This finishes the proof.

In the following, we present a fixed point theorem for a πœ“-contractive mapping when the operator 𝑓 is nonincreasing. We start with the following definition.

Definition 2.5. Let (𝑋,≀) be a partially ordered set and π‘“βˆΆπ‘‹β†’π‘‹. Then one says that 𝑓 is monotone nonincreasing if, for π‘₯,π‘¦βˆˆπ‘‹, π‘₯β‰€π‘¦βŸΉπ‘“π‘₯β‰₯𝑓𝑦.(2.26)
Using a similar argument to that in the proof of Theorem 3.1 of [5], we get the following point results.

Theorem 2.6. Let (𝑋,≀) be a partially ordered set satisfying condition (βˆ—βˆ—βˆ—) and suppose that there exists a metric 𝑑 in 𝑋 such that (𝑋,𝑑) is a complete metric space, and let 𝑓 be a nonincreasing πœ“-contractive mapping. If there exists π‘₯0βˆˆπ‘‹ with π‘₯0≀𝑓π‘₯0 or π‘₯0β‰₯𝑓π‘₯0, then inf{𝑑(π‘₯,𝑓π‘₯)∢π‘₯βˆˆπ‘‹}=0. Moreover, if in addition, 𝑋 is compact and 𝑓 is continuous, then 𝑓 has a unique fixed point in 𝑋.

Proof. If 𝑓π‘₯0=π‘₯0, then it is obvious that inf{𝑑(π‘₯,𝑓π‘₯)∢π‘₯βˆˆπ‘‹}=0. Suppose that π‘₯0<𝑓π‘₯0 (the same argument serves for π‘₯0<𝑓π‘₯0). Since 𝑓 is nonincreasing the consecutive terms of the sequence {𝑓𝑛π‘₯0} are comparable, we have 𝑑𝑓𝑛+1π‘₯0,𝑓𝑛π‘₯0ξ€Έξ€·π‘‘ξ€·π‘“β‰€πœ“π‘›π‘₯0,π‘“π‘›βˆ’1π‘₯0𝑓,𝑑𝑛π‘₯0,𝑓𝑛+1π‘₯0𝑓,π‘‘π‘›βˆ’1π‘₯0,𝑓𝑛π‘₯0𝑓,π‘‘π‘›βˆ’1π‘₯0,𝑓𝑛+1π‘₯0𝑓,𝑑𝑛π‘₯0,𝑓𝑛π‘₯0ξ€·π‘‘ξ€·π‘“ξ€Έξ€Έβ‰€πœ“π‘›π‘₯0,π‘“π‘›βˆ’1π‘₯0𝑓,𝑑𝑛π‘₯0,𝑓𝑛+1π‘₯0𝑓,π‘‘π‘›βˆ’1π‘₯0,𝑓𝑛π‘₯0𝑓,π‘‘π‘›βˆ’1π‘₯0,𝑓𝑛+1π‘₯0𝑑𝑓,0β‰€πœ“π‘›π‘₯0,π‘“π‘›βˆ’1π‘₯0𝑓,𝑑𝑛π‘₯0,𝑓𝑛+1π‘₯0𝑓,π‘‘π‘›βˆ’1π‘₯0,𝑓𝑛π‘₯0𝑓,π‘‘π‘›βˆ’1π‘₯0,𝑓𝑛π‘₯0𝑓+𝑑𝑛π‘₯0,𝑓𝑛+1π‘₯0,ξ€Έξ€Έ(2.27) and so we conclude that for each π‘›βˆˆβ„•, 𝑑𝑓𝑛+1π‘₯0,𝑓𝑛π‘₯0𝑓<𝑑𝑛π‘₯0,π‘“π‘›βˆ’1π‘₯0ξ€Έ.(2.28)
Thus, {𝑑(𝑓𝑛+1π‘₯0,𝑓𝑛π‘₯0)} is a decreasing sequence and bounded below, and it must converge to πœ‚β‰₯0. We claim that πœ‚=0. If πœ‚>0, then by the above inequalities and the continuity of πœ“, letting π‘›β†’βˆž, we have πœ‚=limπ‘›β†’βˆžπ‘‘ξ€·π‘“π‘›+1π‘₯0,𝑓𝑛π‘₯0ξ€Έβ‰€πœ“(πœ‚,πœ‚,πœ‚,2πœ‚,0)<πœ‚,(2.29) which is a contradiction. So πœ‚=0, that is, limπ‘›β†’βˆžπ‘‘(𝑓𝑛+1π‘₯0,𝑓𝑛π‘₯0)=0. Consequently, inf{𝑑(π‘₯,𝑓π‘₯)∢π‘₯βˆˆπ‘‹}=0.
Further, since 𝑓 is continuous and 𝑋 is compact, we can find πœ‡βˆˆπ‘‹ such that 𝑑(πœ‡,π‘“πœ‡)=inf{𝑑(π‘₯,𝑓π‘₯)∢π‘₯βˆˆπ‘‹}=0,(2.30)
and, therefore, πœ‡ is a fixed point of 𝑓.
The uniqueness of the fixed point is proved as in Theorem 2.4.

Acknowledgment

The authors would like to thank the referees whose helpful comments and suggestions led to much improvement in this paper.

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