Fixed Point Theory of Weak Contractions in Partially Ordered Metric ...

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Correspondence should be addressed to Chao-Hung Chen; chen@webmail.mlc.edu.tw. Received 12 March 2013; Accepted 20 April 2013. Academic Editor:Β ...
Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 302438, 7 pages http://dx.doi.org/10.1155/2013/302438

Research Article Fixed Point Theory of Weak Contractions in Partially Ordered Metric Spaces Chi-Ming Chen,1 Jin-Chirng Lee,2 and Chao-Hung Chen2 1 2

Department of Applied Mathematics, National Hsinchu University of Education, No. 521, Nandah Road, Hsinchu City, Taiwan Department of Applied Mathematics, Chung Yuan Christian University, Taiwan

Correspondence should be addressed to Chao-Hung Chen; [email protected] Received 12 March 2013; Accepted 20 April 2013 Academic Editor: Erdal KarapΔ±nar Copyright Β© 2013 Chi-Ming Chen 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. We prove two new fixed point theorems in the framework of partially ordered metric spaces. Our results generalize and improve many recent fixed point theorems in the literature.

1. Introduction and Preliminaries Throughout this paper, by R+ , we denote the set of all nonnegative real numbers, while N is the set of all natural numbers. Let (𝑋, 𝑑) be a metric space, 𝐷 a subset of 𝑋, and 𝑓 : 𝐷 β†’ 𝑋 a map. We say 𝑓 is contractive if there exists 𝛼 ∈ [0, 1) such that, for all π‘₯, 𝑦 ∈ 𝐷, 𝑑 (𝑓π‘₯, 𝑓𝑦) ≀ 𝛼 β‹… 𝑑 (π‘₯, 𝑦) .

(1)

The well-known Banach’s 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. In nonlinear analysis, the study of fixed points of given mappings satisfying certain contractive conditions in various abstract spaces has been investigated deeply. The Banach contraction principle [1] is one of the initial and crucial results in this direction. Also, this principle has many generalizations. For instance, Alber and Guerre-Delabriere in [2] suggested a generalization of the Banach contraction mapping principle by introducing the concept of weak contraction in Hilbert spaces. In [2], the authors also proved that the result of Eslamian and Abkar [3] is equivalent to the result of Dutta and Choudhury [4]. Later, weakly contractive mappings and mappings satisfying other weak contractive inequalities have been discussed in several works, some of which are noted in [4–16]. In 2008, Dutta and Choudhury proved the following theorem.

Theorem 1 (see [4]). Let (𝑋, 𝑑) be a complete metric space, and let 𝑓 : 𝑋 β†’ 𝑋 be such that πœ“ (𝑑 (𝑓π‘₯, 𝑓𝑦)) ≀ πœ“ (𝑑 (π‘₯, 𝑦)) βˆ’ πœ™ (𝑑 (π‘₯, 𝑦)) , π‘“π‘œπ‘Ÿ π‘’π‘Žπ‘β„Ž π‘₯, 𝑦 ∈ 𝑋,

(2)

where πœ“, πœ™ : R+ β†’ R+ are continuous and nondecreasing, and πœ“(𝑑) = πœ™(𝑑) = 0 if and only if 𝑑 = 0. Then 𝑓 has a fixed point in 𝑋. Recently, Eslamian and Abkar [3] proved the following theorem. Theorem 2 (see [3]). Let (𝑋, 𝑑) be a complete metric space, and let 𝑓 : 𝑋 β†’ 𝑋 be such that πœ“ (𝑑 (𝑓π‘₯, 𝑓𝑦)) ≀ 𝛼 (𝑑 (π‘₯, 𝑦)) βˆ’ 𝛽 (𝑑 (π‘₯, 𝑦)) , π‘“π‘œπ‘Ÿ π‘’π‘Žπ‘β„Ž π‘₯, 𝑦 ∈ 𝑋,

(3)

where πœ“, 𝛼, 𝛽 : R+ β†’ R+ are such that πœ“ is continuous and nondecreasing, 𝛼 is continuous, 𝛽 is lower semicontinuous, and πœ“ (𝑑) βˆ’ 𝛼 (𝑑) + 𝛽 (𝑑) > 0 βˆ€π‘‘ > 0, πœ“ (𝑑) = 0 𝑖𝑓 π‘Žπ‘›π‘‘ π‘œπ‘›π‘™π‘¦ 𝑖𝑓 𝑑 = 0, Then 𝑓 has a fixed point in 𝑋.

𝛼 (0) = 𝛽 (0) = 0.

(4)

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Journal of Applied Mathematics

In the recent, fixed point theory has developed rapidly in partially ordered metric spaces (e.g., [17–22]). In 2012, Choudhury and Kundu [23] proved the following fixed point theorem as a generalization of Theorem 2. Theorem 3 (see [23]). Let (𝑋, βŠ‘) 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 𝑓 : 𝑋 β†’ 𝑋 be a nondecreasing mapping such that πœ“ (𝑑 (𝑓π‘₯, 𝑓𝑦)) ≀ 𝛼 (𝑑 (π‘₯, 𝑦)) βˆ’ 𝛽 (𝑑 (π‘₯, 𝑦)) , π‘“π‘œπ‘Ÿ π‘’π‘Žπ‘β„Ž π‘₯, 𝑦 ∈ 𝑋 π‘ π‘’π‘β„Ž π‘‘β„Žπ‘Žπ‘‘ π‘₯ βŠ‘ 𝑦,

(5)

where πœ“, 𝛼, 𝛽 : R+ β†’ R+ are such that πœ“ is continuous and nondecreasing, 𝛼 is continuous, 𝛽 is lower semicontinuous, and πœ“ (𝑑) βˆ’ 𝛼 (𝑑) + 𝛽 (𝑑) > 0 βˆ€π‘‘ > 0, πœ“ (𝑑) = 0 𝑖𝑓 π‘Žπ‘›π‘‘ π‘œπ‘›π‘™π‘¦ 𝑖𝑓 𝑑 = 0,

𝛼 (0) = 𝛽 (0) = 0.

(6)

Also, if any nondecreasing sequence {π‘₯𝑛 } in 𝑋 converges to ], then one assumes that π‘₯𝑛 βŠ‘ ]

βˆ€π‘› ∈ N.

(7)

If there exists π‘₯0 ∈ 𝑋 with π‘₯0 βŠ‘ 𝑓π‘₯0 , then 𝑓 and 𝑔 have a coincidence point in 𝑋. In this paper, we prove two new fixed point theorems in the framework of partially ordered metric spaces. Our results generalize and improve many recent fixed point theorems in the literature.

Θ = {πœ‘ : R+ β†’ R+ Ξ = {πœ‰ : R+ β†’ R+

such that πœ‘ is continuous} ,

such that πœ‰ is lower continuous} . (9)

Let 𝑋 be a nonempty set, and let (𝑋, βŠ‘) be a partially ordered set endowed with a metric 𝑑. Then, the triple (𝑋, βŠ‘, 𝑑) is called a partially ordered complete metric space. We now state the main fixed point theorem for (πœ‘, πœ“, πœ™, πœ‰)-contractions in partially ordered metric spaces, as follows. Theorem 5. Let (𝑋, βŠ‘, 𝑑) be a partially ordered complete metric space. Let 𝑓 : 𝑋 β†’ 𝑋 be monotone nondecreasing, and πœ‘ (𝑑 (𝑓π‘₯, 𝑓𝑦)) ≀ πœ“ (πœ™ (𝑑 (π‘₯, 𝑦)) , πœ™ (𝑑 (π‘₯, 𝑓π‘₯)) , πœ™ (𝑑 (𝑦, 𝑓𝑦))) βˆ’ πœ‰ (max {𝑑 (π‘₯, 𝑦) , 𝑑 (π‘₯, 𝑓π‘₯) , 𝑑 (𝑦, 𝑓𝑦)}) , (10) for all comparable π‘₯, 𝑦 ∈ 𝑋, where πœ‘ ∈ Θ, πœ“ ∈ Ξ¨, πœ™ ∈ Ξ¦, and πœ‰ ∈ Ξ, and πœ‘ (𝑑) βˆ’ πœ™ (𝑑) + πœ‰ (𝑑) > 0 βˆ€π‘‘ > 0, πœ‘ (𝑑) = 0 𝑖𝑓 π‘Žπ‘›π‘‘ π‘œπ‘›π‘™π‘¦ 𝑖𝑓 𝑑 = 0,

πœ™ (0) = πœ‰ (0) = 0.

(11)

Suppose that either (a) 𝑓 is continuous or

2. Fixed Point Results (I)

(b) if any nondecreasing sequence {π‘₯𝑛 } in 𝑋 converges to ], then one assumes that

We start with the following definition. Definition 4. Let (𝑋, βŠ‘) be a partially ordered set and 𝑓 : 𝑋 β†’ 𝑋. Then 𝑓 is said to be monotone nondecreasing if, for π‘₯, 𝑦 ∈ 𝑋, π‘₯ βŠ‘ 𝑦 󳨐⇒ 𝑓π‘₯ βŠ‘ 𝑓𝑦.

And we denote the following sets of functions:

(8)

Let (𝑋, βŠ‘) be a partially ordered set. π‘₯, 𝑦 ∈ 𝑋 are said to be comparable if either π‘₯ βŠ‘ 𝑦 or 𝑦 βŠ‘ π‘₯ holds. In the section, we denote by Ξ¨ the class of functions πœ“ : +3 R β†’ R+ satisfying the following conditions: (πœ“1 ) πœ“ is an increasing, continuous function in each coordinate; (πœ“2 ) for all 𝑑 ∈ R+ , πœ“(𝑑, 𝑑, 𝑑) ≀ 𝑑, πœ“(0, 0, 𝑑) ≀ 𝑑 and πœ“(𝑑, 0, 0) ≀ 𝑑. Next, we denote by Ξ¦ the class of functions πœ™ : R+ β†’ R+ satisfying the following conditions: (πœ™1 ) πœ™ is a continuous, nondecreasing function; (πœ™2 ) πœ™(𝑑) > 0 for 𝑑 > 0 and πœ™(0) = 0; (πœ™3 ) πœ™ is subadditive; that is, πœ™(𝑑1 + 𝑑2 ) ≀ πœ™(𝑑1 ) + πœ™(𝑑2 ) for all 𝑑1 , 𝑑2 > 0.

π‘₯𝑛 βŠ‘ ] βˆ€π‘› ∈ N.

(12)

If there exists π‘₯0 ∈ 𝑋 with π‘₯0 βŠ‘ 𝑓π‘₯0 , then 𝑓 has a fixed point in 𝑋. Proof. Since 𝑓 is nondecreasing, by induction, we construct the sequence {π‘₯𝑛 } recursively as π‘₯𝑛 = 𝑓𝑛 π‘₯0 = 𝑓π‘₯π‘›βˆ’1

βˆ€π‘› ∈ N.

(13)

Thus, we also conclude that π‘₯0 ⊏ π‘₯1 = 𝑓π‘₯0 βŠ‘ π‘₯2 = 𝑓π‘₯1 βŠ‘ β‹… β‹… β‹… βŠ‘ π‘₯𝑛 = 𝑓π‘₯π‘›βˆ’1 βŠ‘ β‹… β‹… β‹… . (14) If any two consecutive terms in (14) are equal, then the 𝑓 has a fixed point, and hence the proof is completed. So we may assume that 𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 ) =ΜΈ 0,

βˆ€π‘› ∈ N.

(15)

Now, we claim that 𝑑(π‘₯𝑛 , π‘₯𝑛+1 ) ≀ 𝑑(π‘₯π‘›βˆ’1 , π‘₯𝑛 ) for all 𝑛 ∈ N. If not, we assume that 𝑑(π‘₯π‘›βˆ’1 , π‘₯𝑛 ) < 𝑑(π‘₯𝑛 , π‘₯𝑛+1 ) for some 𝑛 ∈ N;

Journal of Applied Mathematics

3

substituting π‘₯ = π‘₯𝑛 and 𝑦 = π‘₯𝑛+1 in (10) and using the definition of the function πœ“, we have

By letting 𝑛 β†’ ∞. we get that lim 𝑑 (π‘₯𝑝𝑛 , π‘₯π‘žπ‘› ) = πœ–.

πœ“ (πœ™ (𝑑 (π‘₯𝑛 , π‘₯𝑛+1 )) , πœ™ (𝑑 (π‘₯𝑛 , 𝑓π‘₯𝑛 )) , πœ™ (𝑑 (π‘₯𝑛+1 , 𝑓π‘₯𝑛+1 )))

π‘›β†’βˆž

= πœ“ (πœ™ (𝑑 (π‘₯𝑛 , π‘₯𝑛+1 )) , πœ™ (𝑑 (π‘₯𝑛 , π‘₯𝑛+1 )) , πœ™ (𝑑 (π‘₯𝑛+1 , π‘₯𝑛+2 ))) ≀ πœ™ (𝑑 (π‘₯𝑛+1 , π‘₯𝑛+2 )) ,

(24)

On the other hand, we have 𝑑 (π‘₯𝑝𝑛 , π‘₯π‘žπ‘› )

πœ‰ (max {𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ) , 𝑑 (π‘₯𝑛 , 𝑓π‘₯𝑛 ) , 𝑑 (π‘₯𝑛+1 , 𝑓π‘₯𝑛+1 )})

≀ 𝑑 (π‘₯𝑝𝑛 , π‘₯𝑝𝑛 βˆ’1 ) + 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) + 𝑑 (π‘₯π‘žπ‘› βˆ’1 , π‘₯π‘žπ‘› ) ,

= πœ‰ (max {𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ) , 𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ) , 𝑑 (π‘₯𝑛+1 , π‘₯𝑛+2 )})

𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 )

= πœ‰ (𝑑 (π‘₯𝑛+1 , π‘₯𝑛+2 )) , (16)

(25)

≀ 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯𝑝𝑛 ) + 𝑑 (π‘₯𝑝𝑛 , π‘₯π‘žπ‘› ) + 𝑑 (π‘₯π‘žπ‘› , π‘₯π‘žπ‘› βˆ’1 ) . Letting 𝑛 β†’ ∞, then we get

and hence πœ‘ (𝑑 (π‘₯𝑛+1 , π‘₯𝑛+2 ))

lim 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) = πœ–.

= πœ‘ (𝑑 (𝑓π‘₯𝑛 , 𝑓π‘₯𝑛+1 ))

(17)

≀ πœ™ (𝑑 (π‘₯𝑛+1 , π‘₯𝑛+2 )) βˆ’ πœ‰ (𝑑 (π‘₯𝑛+1 , π‘₯𝑛+2 )) . Since πœ‘(𝑑) βˆ’ πœ™(𝑑) + πœ‰(𝑑) > 0 for all 𝑑 > 0, we have that 𝑑(π‘₯𝑛+1 , π‘₯𝑛+2 ) = 0, which contradicts (15). Therefore, we conclude that 𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ) ≀ 𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )

βˆ€π‘› ∈ N.

(18)

From the previous argument, we also have that for each 𝑛 ∈ N πœ‘ (𝑑 (π‘₯𝑛 , π‘₯𝑛+1 )) ≀ πœ™ (𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )) βˆ’ πœ‰ (𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )) . (19) It follows in (18) that the sequence {𝑑(π‘₯𝑛 , π‘₯𝑛+1 )} is monotone decreasing; it must converge to some πœ‚ β‰₯ 0. Taking limit as 𝑛 β†’ ∞ in (19) and using the continuities of πœ‘ and πœ™ and the lower semicontinuous of πœ‰, we get πœ‘ (πœ‚) ≀ πœ™ (πœ‚) βˆ’ πœ‰ (πœ‚) ,

(20)

which implies that πœ‚ = 0. So we conclude that lim 𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ) = 0.

π‘›β†’βˆž

(21)

(22)

Further, corresponding to π‘žπ‘› β‰₯ 𝑛, we can choose 𝑝𝑛 in such a way that it the smallest integer with 𝑝𝑛 > π‘žπ‘› β‰₯ 𝑛 and 𝑑(π‘₯π‘žπ‘› , π‘₯𝑝𝑛 ) β‰₯ πœ–. Therefore 𝑑(π‘₯π‘žπ‘› , π‘₯𝑝𝑛 βˆ’1 ) < πœ–. Now we have that for all 𝑛 ∈ N πœ– ≀ 𝑑 (π‘₯𝑝𝑛 , π‘₯π‘žπ‘› ) ≀ 𝑑 (π‘₯𝑝𝑛 , π‘₯𝑝𝑛 βˆ’1 ) + 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› ) < 𝑑 (π‘₯𝑝𝑛 , π‘₯𝑝𝑛 βˆ’1 ) + πœ–.

(26)

By (14), we have that the elements π‘₯𝑝𝑛 and π‘₯π‘žπ‘› are comparable. Substituting π‘₯ = π‘₯𝑝𝑛 βˆ’1 and 𝑦 = π‘₯π‘žπ‘› βˆ’1 in (10), we have that, for all 𝑛 ∈ N, πœ“ (πœ™ (𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 )) , πœ™ (𝑑 (π‘₯𝑝𝑛 βˆ’1 , 𝑓π‘₯𝑝𝑛 βˆ’1 )) , πœ™ (𝑑 (π‘₯π‘žπ‘› βˆ’1 , 𝑓π‘₯π‘žπ‘› βˆ’1 ))) ≀ πœ“ (πœ™ (𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 )) , πœ™ (𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯𝑝𝑛 )) , πœ™ (𝑑 (π‘₯π‘žπ‘› βˆ’1 , π‘₯π‘žπ‘› ))) , 𝑀 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 )

(27)

= max {𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) , 𝑑 (π‘₯𝑝𝑛 βˆ’1 , 𝑓π‘₯𝑝𝑛 βˆ’1 ) , 𝑑 (π‘₯π‘žπ‘› βˆ’1 , 𝑓π‘₯π‘žπ‘› βˆ’1 )} = max {𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) , 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯𝑝𝑛 ) ,

We next claim that {π‘₯𝑛 } is a Cauchy sequence; that is, for every πœ€ > 0, there exists 𝑛 ∈ N such that if 𝑝, π‘ž β‰₯ 𝑛, then 𝑑(π‘₯𝑝 , π‘₯π‘ž ) < πœ€. Suppose, on the contrary, that there exists πœ– > 0 such that, for any 𝑛 ∈ N, there are 𝑝𝑛 , π‘žπ‘› ∈ N with 𝑝𝑛 > π‘žπ‘› β‰₯ 𝑛 satisfying 𝑑 (π‘₯π‘žπ‘› , π‘₯𝑝𝑛 ) β‰₯ πœ–.

π‘›β†’βˆž

𝑑 (π‘₯π‘žπ‘› βˆ’1 , π‘₯π‘žπ‘› )} . By the previous argument and using inequality (10), we can conclude that πœ‘ (πœ–) ≀ πœ“ (πœ™ (πœ–) , 0, 0) βˆ’ πœ‰ (πœ–) ≀ πœ™ (πœ–) βˆ’ πœ‰ (πœ–) ,

(28)

which implies that πœ– = 0, a contradiction. Therefore, the sequence {π‘₯𝑛 } is a Cauchy sequence. Since 𝑋 is complete, there exists ] ∈ 𝑋 such that lim π‘₯ π‘›β†’βˆž 𝑛

= ].

(29)

Suppose that (a) holds. Then (23)

] = lim π‘₯𝑛+1 = lim 𝑓π‘₯𝑛 = 𝑓]. π‘›β†’βˆž

π‘›β†’βˆž

Thus, ] is a fixed point in 𝑋.

(30)

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Journal of Applied Mathematics

Suppose that (b) holds; that is, π‘₯𝑛 βŠ‘ ] for all 𝑛 ∈ N. Substituting π‘₯ = π‘₯𝑛 and 𝑦 = ] in (10), we have that

Next, we denote by Θ the class of functions πœ‘ : R+ β†’ R+ satisfying the following conditions:

πœ‘ (𝑑 (π‘₯𝑛+1 , 𝑓])) = πœ‘ (𝑑 (𝑓π‘₯𝑛 , 𝑓])) ≀ πœ“ (πœ™ (𝑑 (π‘₯𝑛 , ])) , πœ™ (𝑑 (π‘₯𝑛 , 𝑓π‘₯𝑛 )) , πœ™ (𝑑 (], 𝑓])))

(31)

Taking limit as 𝑛 β†’ ∞ in equality (31), we have πœ‘ (𝑑 (], 𝑓])) ≀ πœ“ (πœ™ (0) , πœ™ (π‘œ) , πœ™ (𝑑 (], 𝑓]))) βˆ’ πœ‰ (𝑑 (], 𝑓])) ≀ πœ™ (𝑑 (], 𝑓])) βˆ’ πœ‰ (𝑑 (], 𝑓])) , (32) which implies that 𝑑(], 𝑓]) = 0; that is ] = 𝑓]. So we complete the proof. If we let πœ“ (πœ™ (𝑑 (π‘₯, 𝑦)) , πœ™ (𝑑 (π‘₯, 𝑓π‘₯)) , πœ™ (𝑑 (𝑦, 𝑓𝑦))) = max {πœ™ (𝑑 (π‘₯, 𝑦)) , πœ™ (𝑑 (π‘₯, 𝑓π‘₯)) , πœ™ (𝑑 (𝑦, 𝑓𝑦))} ,

(πœ‘1 ) πœ‘ is continuous and nondecreasing; (πœ‘2 ) for 𝑑 > 0, πœ‘(𝑑) > 0 and πœ‘(0) = 0. And we denote by Ξ¦ the class of functions πœ™ : R+ β†’ R+ satisfying the following conditions.

βˆ’ πœ‰ (max {𝑑 (π‘₯𝑛 , ]) , 𝑑 (π‘₯𝑛 , 𝑓π‘₯𝑛 ) , 𝑑 (], 𝑓])}) .

(33)

(πœ™1 ) πœ™ is continuous; (πœ™2 ) for 𝑑 > 0, πœ™(𝑑) > 0 and πœ™(0) = 0. We now state the main fixed point theorem for the (πœ“, πœ‘, πœ™)-contractions in partially ordered metric spaces, as follows. Theorem 7. Let (𝑋, βŠ‘, 𝑑) be a partially ordered complete metric space, and let 𝑓 : 𝑋 β†’ 𝑋 be monotone nondecreasing, and πœ‘ (𝑑 (𝑓π‘₯, 𝑓𝑦)) ≀ πœ“ (πœ‘ (𝑑 (π‘₯, 𝑦)) , πœ‘ (𝑑 (π‘₯, 𝑓π‘₯)) , πœ‘ (𝑑 (𝑦, 𝑓𝑦)))

it is easy to get the following theorem. Theorem 6. Let (𝑋, βŠ‘, 𝑑) be a partially ordered complete metric space. Let 𝑓 : 𝑋 β†’ 𝑋 be monotone nondecreasing, and

(37)

βˆ’ πœ™ (𝑀 (π‘₯, 𝑦)) + 𝐿 β‹… π‘š (π‘₯, 𝑦) , for all comparable π‘₯, 𝑦 ∈ 𝑋 and πœ“ ∈ Ξ¨, πœ‘ ∈ Θ, πœ™ ∈ Ξ¦, where 𝐿 > 0 and 𝑀 (π‘₯, 𝑦) = max {𝑑 (π‘₯, 𝑦) , 𝑑 (π‘₯, 𝑓π‘₯) , 𝑑 (𝑦, 𝑓𝑦)} ,

πœ‘ (𝑑 (𝑓π‘₯, 𝑓𝑦)) ≀ max {πœ™ (𝑑 (π‘₯, 𝑦)) , πœ™ (𝑑 (π‘₯, 𝑓π‘₯)) , πœ™ (𝑑 (𝑦, 𝑓𝑦))} (34)

for all comparable π‘₯, 𝑦 ∈ 𝑋, where πœ‘ ∈ Θ, πœ™ ∈ Ξ¦, and πœ‰ ∈ Ξ, and πœ‘ (𝑑) βˆ’ πœ™ (𝑑) + πœ‰ (𝑑) > 0 βˆ€π‘‘ > 0, 𝑖𝑓 π‘Žπ‘›π‘‘ π‘œπ‘›π‘™π‘¦ 𝑖𝑓 𝑑 = 0,

πœ™ (0) = πœ‰ (0) = 0.

π‘š (π‘₯, 𝑦) = min {𝑑 (π‘₯, 𝑦) , 𝑑 (π‘₯, 𝑓π‘₯) , 𝑑 (𝑦, 𝑓𝑦) , 𝑑 (π‘₯, 𝑓𝑦) ,

βˆ’ πœ‰ (max {𝑑 (π‘₯, 𝑦) , 𝑑 (π‘₯, 𝑓π‘₯) , 𝑑 (𝑦, 𝑓𝑦)}) ,

πœ‘ (𝑑) = 0

(πœ“2 ) for 𝑑 ∈ R+ , πœ™(𝑑, 𝑑, 𝑑) ≀ 𝑑, πœ™(𝑑, 0, 0) ≀ 𝑑, and πœ™(0, 0, 𝑑) ≀ 𝑑.

(35)

Suppose that either

𝑑 (𝑦, 𝑓π‘₯)} . Suppose that either (a) 𝑓 is continuous or (b) if any nondecreasing sequence {π‘₯𝑛 } in 𝑋 converges to ], then one assumes that π‘₯𝑛 βŠ‘ ] βˆ€π‘› ∈ N.

(a) 𝑓 is continuous or (b) if any nondecreasing sequence {π‘₯𝑛 } in 𝑋 converges to ], then one assumes that π‘₯𝑛 βŠ‘ ]

βˆ€π‘› ∈ N.

(36)

If there exists π‘₯0 ∈ 𝑋 with π‘₯0 βŠ‘ 𝑓π‘₯0 , then 𝑓 has a fixed point in 𝑋.

3. Fixed Point Results (II) In the section, we denote by Ξ¨ the class of functions πœ“ : 3 R+ β†’ R+ satisfying the following conditions: (πœ“1 ) πœ“ is an increasing and continuous function in each coordinate;

(38)

(39)

If there exists π‘₯0 ∈ 𝑋 with π‘₯0 βŠ‘ 𝑓π‘₯0 , then 𝑓 has a fixed point in 𝑋. Proof. If 𝑓π‘₯0 = π‘₯0 , then the proof is finished. Suppose that π‘₯0 ⊏ 𝑓π‘₯0 . Since 𝑓 is nondecreasing, by induction, we construct the sequence {π‘₯𝑛 } recursively as π‘₯𝑛 = 𝑓𝑛 π‘₯0 = 𝑓π‘₯π‘›βˆ’1

βˆ€π‘› ∈ N.

(40)

Thus, we also conclude that π‘₯0 ⊏ π‘₯1 = 𝑓π‘₯0 βŠ‘ π‘₯2 = 𝑓π‘₯1 βŠ‘ β‹… β‹… β‹… βŠ‘ π‘₯𝑛 = 𝑓π‘₯π‘›βˆ’1 βŠ‘ β‹… β‹… β‹… . (41) We now claim that lim 𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ) = 0.

π‘›β†’βˆž

(42)

Journal of Applied Mathematics

5

Put π‘₯ = π‘₯π‘›βˆ’1 and 𝑦 = π‘₯𝑛 in (37). Note that

Suppose, on the contrary, that there exists πœ– > 0 such that, for any 𝑛 ∈ N, there are 𝑝𝑛 , π‘žπ‘› ∈ N with 𝑝𝑛 > π‘žπ‘› β‰₯ 𝑛 satisfying

π‘š (π‘₯π‘›βˆ’1 , π‘₯𝑛 )

𝑑 (π‘₯π‘žπ‘› , π‘₯𝑝𝑛 ) β‰₯ πœ–.

= min {𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 ) , 𝑑 (π‘₯π‘›βˆ’1 , 𝑓π‘₯π‘›βˆ’1 ) , 𝑑 (π‘₯𝑛 , 𝑓π‘₯𝑛 ) , 𝑑 (π‘₯π‘›βˆ’1 , 𝑓π‘₯𝑛 ) , 𝑑 (π‘₯𝑛 , 𝑓π‘₯π‘›βˆ’1 )} = min {𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 ) , 𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 ) , 𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ) ,

(43)

𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛+1 ) , 𝑑 (π‘₯𝑛 , π‘₯𝑛 )}

(50)

Further, corresponding to π‘žπ‘› β‰₯ 𝑛, we can choose 𝑝𝑛 in such a way that it the smallest integer with 𝑝𝑛 > π‘žπ‘› β‰₯ 𝑛 and 𝑑(π‘₯π‘žπ‘› , π‘₯𝑝𝑛 ) β‰₯ πœ–. Therefore 𝑑(π‘₯π‘žπ‘› , π‘₯𝑝𝑛 βˆ’1 ) < πœ–. By the rectangular inequality, we have

= 0.

πœ– ≀ 𝑑 (π‘₯𝑝𝑛 , π‘₯π‘žπ‘› )

So, we obtain that

≀ 𝑑 (π‘₯𝑝𝑛 , π‘₯𝑝𝑛 βˆ’1 ) + 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› )

πœ‘ (𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ))

(51)

< 𝑑 (π‘₯𝑝𝑛 , π‘₯𝑝𝑛 βˆ’1 ) + πœ–.

= πœ‘ (𝑑 (𝑓π‘₯π‘›βˆ’1 , 𝑓π‘₯𝑛 )) ≀ πœ“ (πœ‘ (𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )) , πœ‘ (𝑑 (π‘₯π‘›βˆ’1 , 𝑓π‘₯π‘›βˆ’1 )) ,

Letting 𝑛 β†’ ∞, then we get

πœ‘ (𝑑 (π‘₯𝑛 , 𝑓π‘₯𝑛 ))) βˆ’ πœ™ (𝑀 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )) lim 𝑑 (π‘₯𝑝𝑛 , π‘₯π‘žπ‘› ) = πœ–.

≀ πœ“ (πœ‘ (𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )) , πœ‘ (𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )) , πœ‘ (𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ))) βˆ’ πœ™ (𝑀 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )) , (44)

π‘›β†’βˆž

(52)

On the other hand, we have 𝑑 (π‘₯𝑝𝑛 , π‘₯π‘žπ‘› )

where 𝑀 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )

≀ 𝑑 (π‘₯𝑝𝑛 , π‘₯𝑝𝑛 βˆ’1 ) + 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) + 𝑑 (π‘₯π‘žπ‘› βˆ’1 , π‘₯π‘žπ‘› ) ,

= max {𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 ) , 𝑑 (π‘₯π‘›βˆ’1 , 𝑓π‘₯π‘›βˆ’1 ) , 𝑑 (π‘₯𝑛 , 𝑓π‘₯𝑛 )} (45) = max {𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 ) , 𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 ) , 𝑑 (π‘₯𝑛 , π‘₯𝑛+1 )} .

βˆ€π‘› ∈ N.

(46)

If not, we assume that 𝑑(π‘₯π‘›βˆ’1 , π‘₯𝑛 ) ≀ 𝑑(π‘₯𝑛 , π‘₯𝑛+1 ); then πœ‘(𝑑(π‘₯π‘›βˆ’1 , π‘₯𝑛 )) ≀ πœ‘(𝑑(π‘₯𝑛 , π‘₯𝑛+1 )), since πœ‘ is non-decreasing. Using inequality (44) and the conditions of the function πœ“, we have that, for each 𝑛 ∈ N, πœ‘ (𝑑 (π‘₯𝑛 , π‘₯𝑛+1 )) ≀ πœ‘ (𝑑 (π‘₯𝑛 , π‘₯𝑛+1 )) βˆ’ πœ™ (𝑑 (π‘₯𝑛 , π‘₯𝑛+1 )) , (47) which implies that πœ™(𝑑(π‘₯𝑛 , π‘₯𝑛+1 )) = 0, and hence 𝑑(π‘₯𝑛 , π‘₯𝑛+1 ) = 0. This contradicts our initial assumption. From the previous argument, we have that, for each 𝑛 ∈ N, πœ‘ (𝑑 (π‘₯𝑛 , π‘₯𝑛+1 )) ≀ πœ‘ (𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )) βˆ’ πœ™ (𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 )) , 𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ) < 𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 ) .

(48)

And since the sequence {𝑑(π‘₯𝑛 , π‘₯𝑛+1 )} is decreasing, it must converge to some πœ‚ β‰₯ 0. Taking limit as 𝑛 β†’ ∞ in (48) and by the continuity of πœ‘ and πœ™, we get πœ‘ (πœ‚) ≀ πœ‘ (πœ‚) βˆ’ πœ™ (πœ‚) ,

(53)

≀ 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯𝑝𝑛 ) + 𝑑 (π‘₯𝑝𝑛 , π‘₯π‘žπ‘› ) + 𝑑 (π‘₯π‘žπ‘› , π‘₯π‘žπ‘› βˆ’1 ) .

We now claim that 𝑑 (π‘₯𝑛 , π‘₯𝑛+1 ) < 𝑑 (π‘₯π‘›βˆ’1 , π‘₯𝑛 ) ,

𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 )

(49)

and so we conclude that πœ™(πœ‚) = 0 and πœ‚ = 0. We next claim that {π‘₯𝑛 } is Cauchy; that is, for every πœ€ > 0, there exists 𝑛 ∈ N such that if 𝑝, π‘ž β‰₯ 𝑛, then 𝑑(π‘₯𝑝 , π‘₯π‘ž ) < πœ€.

By letting 𝑛 β†’ ∞, we get that lim 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) = πœ–.

π‘›β†’βˆž

(54)

Using inequalities (37), (52), and (54) and putting π‘₯ = π‘₯𝑝𝑛 βˆ’1 and 𝑦 = π‘₯π‘žπ‘› βˆ’1 , we have that πœ‘ (𝑑 (π‘₯𝑝𝑛 , π‘₯π‘žπ‘› )) = πœ‘ (𝑑 (𝑓π‘₯𝑝𝑛 βˆ’1 , 𝑓π‘₯π‘žπ‘› βˆ’1 )) ≀ πœ“ (πœ‘ (𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 )) , πœ‘ (𝑑 (π‘₯𝑝𝑛 βˆ’1 , 𝑓π‘₯𝑝𝑛 βˆ’1 )) , πœ‘ (𝑑 (π‘₯π‘žπ‘› βˆ’1 , 𝑓π‘₯π‘žπ‘› βˆ’1 ))) βˆ’ πœ™ (𝑀 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 )) + 𝐿 β‹… π‘š (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) = πœ“ (πœ‘ (𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 )) , πœ‘ (𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯𝑝𝑛 )) , πœ‘ (𝑑 (π‘₯π‘žπ‘› βˆ’1 , π‘₯π‘žπ‘› ))) βˆ’ πœ™ (𝑀 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 )) + 𝐿 β‹… π‘š (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) ,

(55)

6

Journal of Applied Mathematics

where

If we let πœ“ (πœ‘ (𝑑 (π‘₯, 𝑦)) , πœ‘ (𝑑 (π‘₯, 𝑓π‘₯)) , πœ‘ (𝑑 (𝑦, 𝑓𝑦)))

𝑀 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) = max {𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) , 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯𝑝𝑛 ) , 𝑑 (π‘₯π‘žπ‘› βˆ’1 , π‘₯π‘žπ‘› )} ,

= max {πœ‘ (𝑑 (π‘₯, 𝑦)) , πœ‘ (𝑑 (π‘₯, 𝑓π‘₯)) , πœ‘ (𝑑 (𝑦, 𝑓𝑦))} ,

(63)

it is easy to get the following theorem.

π‘š (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) = min {𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) , 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯𝑝𝑛 ) , 𝑑 (π‘₯π‘žπ‘› βˆ’1 , π‘₯π‘žπ‘› ) , 𝑑 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› ) , 𝑑 (π‘₯π‘žπ‘› βˆ’1 , π‘₯𝑝𝑛 )} . (56)

βˆ’ πœ™ (𝑀 (π‘₯, 𝑦)) + 𝐿 β‹… π‘š (π‘₯, 𝑦) ,

lim 𝑀 (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) = πœ–,

π‘›β†’βˆž

lim π‘š (π‘₯𝑝𝑛 βˆ’1 , π‘₯π‘žπ‘› βˆ’1 ) = 0,

π‘›β†’βˆž

(57)

This implies that πœ™(πœ–) = 0, and hence πœ– = 0. So we get a contraction. Therefore {π‘₯𝑛 } is a Cauchy sequence. Since 𝑋 is complete, there exists ] ∈ 𝑋 such that lim π‘₯𝑛 = ].

(58)

π‘›β†’βˆž

for all comparable π‘₯, 𝑦 ∈ 𝑋 and πœ‘ ∈ Θ, πœ™ ∈ Ξ¦, where 𝐿 > 0 and 𝑀 (π‘₯, 𝑦) = max {𝑑 (π‘₯, 𝑦) , 𝑑 (π‘₯, 𝑓π‘₯) , 𝑑 (𝑦, 𝑓𝑦)} ,

πœ‘ (πœ–) ≀ πœ“ (πœ‘ (πœ–) , 0, 0) βˆ’ πœ™ (πœ–) ≀ πœ‘ (πœ–) βˆ’ πœ™ (πœ–) .

Suppose that (a) holds. Then

π‘š (π‘₯, 𝑦) = min {𝑑 (π‘₯, 𝑦) , 𝑑 (π‘₯, 𝑓π‘₯) , 𝑑 (𝑦, 𝑓𝑦) , 𝑑 (π‘₯, 𝑓𝑦) , 𝑑 (𝑦, 𝑓π‘₯)} . (65) Suppose that either (a) 𝑓 is continuous or

] = lim π‘₯𝑛+1 = lim 𝑓π‘₯𝑛 = 𝑓]. π‘›β†’βˆž

(59)

Thus, ] is a fixed point in 𝑋 Suppose that (b) holds; that is, π‘₯𝑛 βŠ‘ ] for all 𝑛 ∈ N. Substituting π‘₯ = π‘₯𝑛 and 𝑦 = ] in (37), we have that πœ‘ (𝑑 (π‘₯𝑛+1 , 𝑓])) = πœ‘ (𝑑 (𝑓π‘₯𝑛 , 𝑓])) ≀ πœ“ (πœ‘ (𝑑 (π‘₯𝑛 , ])) , πœ‘ (𝑑 (π‘₯𝑛 , 𝑓π‘₯𝑛 )) , πœ‘ (𝑑 (], 𝑓])))

(60)

where 𝑀 (π‘₯𝑛 , ]) = max {𝑑 (π‘₯𝑛 , ]) , 𝑑 (π‘₯𝑛 , 𝑓π‘₯𝑛 ) , 𝑑 (], 𝑓])} , π‘š (π‘₯𝑛 , ]) = min {𝑑 (π‘₯𝑛 , ]) , 𝑑 (π‘₯𝑛 , 𝑓π‘₯𝑛 ) , 𝑑 (], 𝑓]) , 𝑑 (π‘₯𝑛 , 𝑓])

(b) if any nondecreasing sequence {π‘₯𝑛 } in 𝑋 converges to ], then one assumes that π‘₯𝑛 βŠ‘ ] βˆ€π‘› ∈ N.

(66)

If there exists π‘₯0 ∈ 𝑋 with π‘₯0 βŠ‘ 𝑓π‘₯0 , then 𝑓 has a fixed point in 𝑋.

Acknowledgment The authors would like to thank the referee(s) for the many useful comments and suggestions for the improvement of the paper.

βˆ’ πœ™ (𝑀 (π‘₯𝑛 , ])) + 𝐿 β‹… π‘š (π‘₯𝑛 , ]) ,

References (61)

𝑑 (], 𝑓π‘₯𝑛 )} . Letting 𝑛 β†’ ∞, then we obtain that 𝑀 (π‘₯𝑛 , ]) 󳨀→ 𝑑 (], 𝑓]) ,

πœ‘ (𝑑 (𝑓π‘₯, 𝑓𝑦)) ≀ max {πœ‘ (𝑑 (π‘₯, 𝑦)) , πœ‘ (𝑑 (π‘₯, 𝑓π‘₯)) , πœ‘ (𝑑 (𝑦, 𝑓𝑦))} (64)

Letting 𝑛 β†’ ∞, then we obtain that

π‘›β†’βˆž

Theorem 8. Let (𝑋, βŠ‘, 𝑑) be a partially ordered complete metric space, and let 𝑓 : 𝑋 β†’ 𝑋 be monotone nondecreasing, and

π‘š (π‘₯𝑛 , ]) 󳨀→ 0,

πœ‘ (𝑑 (], 𝑓])) ≀ πœ“ (πœ‘ (0) , πœ‘ (0) , πœ‘ (𝑑 (], 𝑓]))) βˆ’ πœ™ (𝑑 (], 𝑓])) ≀ πœ‘ (𝑑 (], 𝑓])) βˆ’ πœ™ (𝑑 (], 𝑓])) , (62) which implies that 𝑑(], 𝑓]) = 0; that is, ] = 𝑓]. So we complete the proof.

[1] S. Banach, β€œSur les operations dans les ensembles abstraits et leur application aux equations integerales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922. [2] I. Ya. Alber and S. Guerre-Delabriere, β€œPrinciples of weakly contractive maps in Hilbert spaces,” in New Results in Operator Theory, I. Gohberg and Yu. Lyubich, Eds., vol. 98 of Advances and Applications, pp. 7–22, Birkhauser, Basel, Switzerland, 1997. [3] M. Eslamian and A. Abkar, β€œA fixed point theorem for generalized weakly contractive mappings in complete metric space ,” Italian Journal of Pure and Applied Mathematics. In press. [4] P. N. Dutta and B. S. Choudhury, β€œA generalisation of contraction principle in metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 406368, 8 pages, 2008. [5] H. Aydi, E. KarapΔ±nar, and B. Samet, β€œRemarks on some recent fixed point theorems,” Fixed Point Theory and Applications, vol. 2012, article 76, 2012.

Journal of Applied Mathematics [6] H. Aydi, E. KarapΔ±nar, and W. Shatanawi, β€œCoupled fixed point results for (πœ‘, πœ“)-weakly contractive condition in ordered partial metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 12, pp. 4449–4460, 2011. [7] W.-S. Du, β€œOn coincidence point and fixed point theorems for nonlinear multivalued maps,” Topology and its Applications, vol. 159, no. 1, pp. 49–56, 2012. [8] W.-S. Du, β€œOn generalized weakly directional contractions and approximate fixed point property with applications,” Fixed Point Theory and Applications, vol. 2012, article 6, 2012. [9] W.-S. Du and S.-X. Zheng, β€œNonlinear conditions for coincidence point and fixed point theorems,” Taiwanese Journal of Mathematics, vol. 16, no. 3, pp. 857–868, 2012. [10] W.-S. Du and S.-X. Zheng, β€œNew nonlinear conditions and inequalities for the existence of coincidence points and fixed points,” Journal of Applied Mathematics, vol. 2012, Article ID 196759, 12 pages, 2012. [11] E. Karapinar and K. Sadarangani, β€œFixed point theory for cyclic (πœ‘, πœ“)-contractions,” Fixed Point Theory and Applications, vol. 2011, article 69, 2011. [12] E. KarapΔ±nar, β€œFixed point theory for cyclic weak πœ™contraction,” Applied Mathematics Letters, vol. 24, no. 6, pp. 822–825, 2011. [13] E. KarapΔ±nar and I. S. Yuce, β€œFixed point theory for cyclic generalized weak πœ™-contraction on partial metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 491542, 12 pages, 2012. [14] B. E. Rhoades, β€œSome theorems on weakly contractive maps,” Nonlinear Analysis, vol. 47, no. 4, pp. 2683–2693, 2001. [15] B. Djafari Rouhani and S. Moradi, β€œCommon fixed point of multivalued generalized πœ‘-weak contractive mappings,” Fixed Point Theory and Applications, Article ID 708984, 201013 pages, 2010. [16] Q. Zhang and Y. Song, β€œFixed point theory for generalized πœ™weak contractions,” Applied Mathematics Letters, vol. 22, no. 1, pp. 75–78, 2009. Β΄ c, N. CakiΒ΄c, M. RajoviΒ΄c, and J. S. Ume, β€œMonotone [17] L. B. CiriΒ΄ generalized nonlinear contractions in partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 131294, 11 pages, 2008. [18] T. G. Bhaskar and V. Lakshmikantham, β€œFixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 65, no. 7, pp. 1379–1393, 2006. [19] J. Harjani, B. LΒ΄opez, and K. Sadarangani, β€œFixed point theorems for weakly C-contractive mappings in ordered metric spaces,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 790–796, 2011. [20] J. Harjani, B. LΒ΄opez, and K. Sadarangani, β€œA fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space,” Abstract and Applied Analysis, vol. 2010, Article ID 190701, 8 pages, 2010. [21] J. J. Nieto and R. RodrΒ΄Δ±guez-LΒ΄opez, β€œContractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005. [22] A. C. M. Ran and M. C. B. Reurings, β€œA fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004. [23] B. S. Choudhury and A. Kundu, β€œ(πœ“, 𝛼, 𝛽)-weak contractions in partially ordered metric spaces,” Applied Mathematics Letters, vol. 25, no. 1, pp. 6–10, 2012.

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