This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces Fixed Point Theory and Applications 2012, 2012:17 doi:10.1186/1687-1812-2012-17 Chi MING Chen (ming@mail.nhcue.edu.tw) ISSN 1687-1812 Article type Research Submission date 14 November 2011 Acceptance date 16 February 2012 Publication date 16 February 2012 Article URL http://www.fixedpointtheoryandapplications.com/content/2012/1/17 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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Fixed point theory for the cyclic weaker Meir–Keeler function in complete metric spaces Chi-Ming Chen Department of Applied Mathematics, National Hsinchu University of Education, No. 521, Nanda Rd., Hsinchu City 300, Taiwan Email address: ming@mail.nhcue.edu.tw Abstract In this article, we introduce the notions of cyclic weaker φ◦ϕ-contractions and cyclic weaker (φ, ϕ)-contractions in complete metric spaces and prove two theorems which assure the existence and uniqueness of a 1 fixed point for these two types of contractions. Our results generalize or improve many recent fixed point theorems in the literature. MSC: 47H10; 54C60; 54H25; 55M20. Keywords: fixed point theory; weaker Meir–Keeler function; cyclic weaker φ ◦ ϕ-contraction; cyclic weaker (φ, ϕ)-contraction. 1 Introduction and preliminaries Throughout this article, by R + , R we denote the sets of all nonnegative real numb ers and all real numbers, respectively, while N is the set of all natural numb ers. Let (X, d) be a metric space, D be a subset of X and f : D → X be a map. We say f is contractive if there exists α ∈ [0, 1) such that for all x, y ∈ D, d(fx,fy) ≤ α · d(x, y). 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. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. In 1969, Boyd and Wong [2] introduced the notion of Φ-contraction. A mapping f : X → X on a metric space is called Φ-contraction if there exists an upper semi-continuous 2 function Φ : [0, ∞) → [0, ∞) such that d(fx,fy) ≤ Φ(d(x, y)) for all x, y ∈ X. Generalization of the above Banach contraction principle has been a heavily investigated branch research. (see, e.g., [3,4]). In 2003, Kirk et al. [5] introduced the following notion of cyclic representation. Definition 1 [5] Let X be a nonempty set, m ∈ N and f : X → X an operator. Then X = ∪ m i=1 A i is called a cyclic representation of X with respect to f if (1) A i , i =1, 2, ,m are nonempty subsets of X; (2) f (A 1 ) ⊂ A 2 , f(A 2 ) ⊂ A 3 , ,f(A m−1 ) ⊂ A m , f (A m ) ⊂ A 1 . Kirk et al. [5] also proved the below theorem. Theorem 1 [5] Let (X, d) be a complete metric space, m ∈ N, A 1 ,A 2 , ,A m , closed nonempty subsets of X and X = ∪ m i=1 A i . Suppose that f satisfies the following condition. d(fx,fy) ≤ ψ(d(x, y)), for all x ∈ A i ,y∈ A i+1 ,i∈{1, 2, ,m}, where ψ :[0, ∞) → [0, ∞) is upper semi-continuous from the right and 0 ≤ ψ(t) <tfor t>0. Then, f has a fixed point z ∈∩ n i=1 A i . 3 Recently, the fixed theorems for an operator f : X → X that defined on a metric space X with a cyclic representation of X with respect to f had appeared in the literature. (see, e.g., [6–10]). In 2010, Pˇacurar and Rus [7] introduced the following notion of cyclic weaker ϕ-contraction. Definition 2 [7] Let (X, d) be a metric space, m ∈ N, A 1 ,A 2 , ,A m closed nonempty subsets of X and X = ∪ m i=1 A i . An operator f : X → X is called a cyclic weaker ϕ-contraction if (1) X = ∪ m i=1 A i is a cyclic representation of X with respect to f; (2) there exists a continuous, non-decreasing function ϕ :[0, ∞) → [0, ∞) with ϕ(t) > 0 for t ∈ (0, ∞) and ϕ(0) = 0 such that d(fx,fy) ≤ d(x, y) − ϕ(d(x, y)), for any x ∈ A i , y ∈ A i+1 , i =1, 2, ,m where A m+1 = A 1 . And, P ˇacurar and Rus [7] proved the below theorem. Theorem 2 [7] Let (X, d) be a complete metric space, m ∈ N, A 1 ,A 2 , ,A m closed nonempty subsets of X and X = ∪ m i=1 A i . Suppose that f is a cyclic weaker ϕ-contraction. Then, f has a fixed point z ∈∩ n i=1 A i . 4 In this article, we also recall the notion of Meir–Keeler function (see [11]). A function φ :[0, ∞) → [0, ∞) is said to be a Meir–Keeler function if for each η>0, there exists δ>0 such that for t ∈ [0, ∞) with η ≤ t<η+ δ,we have φ(t) <η. We now introduce the notion of weaker Meir–Keeler function φ :[0, ∞) → [0, ∞), as follows: Definition 3 We call φ :[0, ∞) → [0, ∞) a weaker Meir–Keeler function if for each η>0, there exists δ>0 such that for t ∈ [0, ∞) with η ≤ t<η+ δ, there exists n 0 ∈ N such that φ n 0 (t) <η. 2 Fixed point theory for the cyclic weaker φ ◦ ϕ-contractions The main purpose of this section is to present a generalization of Theorem 1. In the section, we let φ :[0, ∞) → [0, ∞) be a weaker Meir–Keeler function satisfying the following conditions: (φ 1 ) φ(t) > 0 for t>0 and φ(0) = 0; (φ 2 ) for all t ∈ (0, ∞), {φ n (t)} n∈N is decreasing; (φ 3 ) for t n ∈ [0, ∞), we have that 5 (a) if lim n→∞ t n = γ>0, then lim n→∞ φ(t n ) <γ, and (b) if lim n→∞ t n = 0, then lim n→∞ φ(t n )=0. And, let ϕ :[0, ∞) → [0, ∞) be a non-decreasing and continuous function satisfying (ϕ 1 ) ϕ(t) > 0 for t>0 and ϕ(0) = 0; (ϕ 2 ) ϕ is subadditive, that is, for every µ 1 ,µ 2 ∈ [0, ∞), ϕ(µ 1 + µ 2 ) ≤ ϕ(µ 1 )+ϕ(µ 2 ); (ϕ 3 ) for all t ∈ (0, ∞), lim n→∞ t n = 0 if and only if lim n→∞ ϕ(t n )=0. We state the notion of cyclic weaker φ ◦ ϕ-contraction, as follows: Definition 4 Let (X, d) be a metric space, m ∈ N, A 1 ,A 2 , ,A m nonempty subsets of X and X = ∪ m i=1 A i . An operator f : X → X is called a cyclic weaker φ ◦ ϕ-contraction if (i) X = ∪ m i=1 A i is a cyclic representation of X with respect to f; (ii) for any x ∈ A i , y ∈ A i+1 , i =1, 2, ,m, ϕ(d(fx,fy)) ≤ φ(ϕ(d(x, y))), where A m+1 = A 1 . 6 Theorem 3 Let (X, d) be a complete metric space, m ∈ N, A 1 ,A 2 , ,A m nonempty subsets of X and X = ∪ m i=1 A i .Letf : X → X be a cyclic weaker φ ◦ ϕ-contraction. Then, f has a unique fixed point z ∈∩ m i=1 A i . Proof. Given x 0 and let x n+1 = fx n = f n+1 x 0 , for n ∈ N∪{0}. If there exists n 0 ∈ N ∪{0} such that x n 0 +1 = x n 0 , then we finished the proof. Suppose that x n+1 = x n for any n ∈ N ∪{0}. Notice that, for any n>0, there exists i n ∈{1, 2, ,m} such that x n−1 ∈ A i n and x n ∈ A i n +1 . Since f : X → X is a cyclic weaker φ ◦ ϕ-contraction, we have that for all n ∈ N ϕ(d(x n ,x n+1 )) = ϕ(d(fx n−1 ,fx n )) ≤ φ(ϕ(d(x n−1 ,x n ))), and so ϕ(d(x n ,x n+1 )) ≤ φ(ϕ(d(x n−1 ,x n ))) ≤ φ(φ(ϕ(d(x n−2 ,x n−1 ))) = φ 2 (ϕ((d(x n−2 ,x n−1 ))) ≤······ ≤ φ n (ϕ(d(x 0 ,x 1 ))). Since {φ n (ϕ(d(x 0 ,x 1 )))} n∈N is decreasing, it must converge to some η ≥ 0. We claim that η = 0. On the contrary, assume that η>0. Then by the 7 definition of weaker Meir–Keeler function φ, there exists δ>0 such that for x 0 ,x 1 ∈ X with η ≤ ϕ(d(x 0 ,x 1 )) <δ+ η, there exists n 0 ∈ N such that φ n 0 (ϕ(d(x 0 ,x 1 ))) <η. Since lim n→∞ φ n (ϕ(d(x 0 ,x 1 ))) = η, there exists p 0 ∈ N such that η ≤ φ p (ϕ(d(x 0 ,x 1 )) <δ+ η, for all p ≥ p 0 . Thus, we conclude that φ p 0 +n 0 (ϕ(d(x 0 ,x 1 ))) <η. So we get a contradiction. Therefore lim n→∞ φ n (ϕ(d(x 0 ,x 1 ))) = 0, that is, lim n→∞ ϕ(d(x n ,x n+1 )) = 0. Next, we claim that {x n } is a Cauchy sequence. We claim that the following result holds: Claim: for each ε>0, there is n 0 (ε) ∈ N such that for all p, q ≥ n 0 (ε), ϕ(d(x p ,x q )) <ε, (*) We shall prove (∗) by contradiction. Suppose that (∗) is false. Then there exists some ε>0 such that for all n ∈ N, there are p n ,q n ∈ N with p n > q n ≥ n satisfying: (i) ϕ( d(x p n ,x q n )) ≥ ε, and (ii) p n is the smallest number greater than q n such that the condition (i) holds. 8 Since ε ≤ ϕ(d(x p n ,x q n )) ≤ ϕ(d(x p n ,x p n −1 )+d(x p n −1 ,x q n )) ≤ ϕ(d(x p n ,x p n −1 )) + ϕ(d(x p n −1 ,x q n )) ≤ ϕ(d(x p n ,x p n −1 )) + ε, hence we conclude lim p→∞ ϕ(d(x p n ,x q n )) = ε. Since ϕ is subadditive and nondecreasing, we conclude ϕ(d(x p n ,x q n )) ≤ ϕ(d(x p n ,x q n +1 )+d(x p n +1 ,x q n )) ≤ ϕ(d(x p n ,x q n +1 )) + ϕ(d(x p n +1 ,x q n )), and so ϕ(d(x p n ,x q n )) − ϕ(d(x p n ,x p n +1 )) ≤ ϕ(d(x p n +1 ,x q n )) ≤ ϕ(d(x p n ,x p n +1 )+d(x p n ,x q n )) ≤ ϕ(d(x p n ,x p n +1 )) + ϕ(d(x p n ,x q n )). Letting n →∞, we also have lim n→∞ ϕ(d(x p n +1 ,x q n )) = ε. 9 [...]... conditions Fixed Point Theory 4(1), 79– 89 (2003) [6] Rus, IA: Cyclic representations and fixed points Ann T Popoviciu, Seminar Funct Eq Approx Convexity 3, 171–178 (2005) [7] Pˇcurar, M, Rus, IA: Fixed point theory for cyclic ϕ-contractions Nona linear Anal 72(3–4), 2683–2693 (2010) [8] Karapinar, E: Fixed point theory for cyclic weaker φ-contraction Appl Math Lett 24(6), 822–825 (2011) [9] Karapinar, E, Sadarangani,... 1 0, x, 0 ; for all x ∈ R; 4 f ((0, y, 0)) = 1 0, 0, y ; for all y ∈ R; 4 f ((0, 0, z)) = 1 z, 0, 0 ; for all z ∈ R 4 11 We define ϕ : [0, ∞) → [0, ∞) by 1 φ(t) = t for t ∈ [0, ∞), 3 and ϕ : [0, ∞) → [0, ∞) by 1 ϕ(t) = t for t ∈ [0, ∞) 2 Then f is a cyclic weaker φ ◦ ϕ-contraction and (0, 0, 0) is the unique fixed point 3 Fixed point theory for the cyclic weaker (φ, ϕ)-contractions The main purpose of... Karapinar, E, Sadarangani, K: Corrigendum to “Fixed point theory for cyclic weaker φ-contraction” [Appl Math Lett Vol.24(6), 822-825.] In Press (2011) [10] Karapinar, E, Sadarangani, K: Fixed point theory for cyclic (φ − ψ)-contractions Fixed Point Theory Appl 2011, 69 (2011) doi:10.1186/1687-1812-2011-69 [11] Meir, A, Keeler, E: A theorem on contraction mappings J Math Anal Appl 28, 326–329 (1969) 22 ... generalization of Theorem 2 In the section, we let φ : [0, ∞) → [0, ∞) be a weaker Meir–Keeler function satisfying the following conditions: (φ1 ) φ(t) > 0 for t > 0 and φ(0) = 0; (φ2 ) for all t ∈ (0, ∞), {φn (t)}n∈N is decreasing; (φ3 ) for tn ∈ [0, ∞), if limn→∞ tn = γ, then limn→∞ φ(tn ) ≤ γ 12 And, let ϕ : [0, ∞) → [0, ∞) be a non-decreasing and continuous function satisfying ϕ(t) > 0 for t > 0 and... W: Coupled fixed point results for (ψ − ϕ)-weakly contractive condition in ordered partial metric spaces Comput Math Appl 62(12), 4449–4460 (2011) [4] Karapinar, E: Weak ϕ-contraction on partial metric spaces and existence of fixed points in partially ordered sets Mathematica Aeterna 1(4), 237– 244 (2011) 21 [5] Kirk, WA, Srinivasan, PS, Veeramani, P: Fixed points for mappings satisfying cyclical contractive... ; for all x ∈ [0, 1]; 8 f ((0, y, 0)) = 1 0, 0, y 2 ; for all y ∈ [0, 1]; 8 f ((0, 0, z)) = 1 2 z , 0, 0 ; for all z ∈ [0, 1] 8 We define ϕ : [0, ∞) → [0, ∞) by φ(t) = t2 for t ∈ [0, ∞), t+1 and ϕ : [0, ∞) → [0, ∞) by ϕ(t) = t2 for t ∈ [0, ∞) t+2 Then f is a cyclic weaker (φ, ϕ)-contraction and (0, 0, 0) is the unique fixed point Competing interests The authors declare that they have no competing interests... + d(xxpn+1 ,pn ) Letting n → ∞, we obtain that lim d(xqn+1 , xpn+1 ) = n→∞ Since xqn and xpn lie in different adjacently labeled sets Ai and Ai+1 for certain 1 ≤ i ≤ m, by using the fact that f is a cyclic weaker (φ, ϕ)contraction, we have d(xqn+1 , xpn+1 ) = d(f xqn , fxpn ) ≤ φ(d(xqn , xpn )) − ϕ(d(xqn , xpn )) Letting n → ∞, by using the condition φ3 of the function φ, we obtain that ≤ − ϕ( ), and... also all converge to ν Since Ai is clsoed for all i = 1, 2, , m, we conclude ν ∈ ∪m Ai , and also we conclude that ∩m Ai = φ Since i=1 i=1 ϕ(d(ν, f ν)) = lim ϕ(d(fxmn , fν)) n→∞ ≤ lim φ(ϕ(d(fxmn−1 , ν))) = 0, n→∞ 10 hence ϕ(d(ν, f ν)) = 0, that is, d(ν, f ν) = 0, ν is a fixed point of f Finally, to prove the uniqueness of the fixed point, let µ be another fixed point of f By the cyclic character of f... f , the sequence {xn } has in nite terms in each Ai for i ∈ {1, 2, , m} Now for all i = 1, 2, , m, we may take a subsequence 17 {xnk } of {xn } with xnk ∈ Ai−1 and also all converge to ν Since d(xnk+1 , fν) = d(fxnk , fν) ≤ φ(d(xnk , ν)) − ϕ(d(xnk , ν)) ≤ φ(d(xnk , ν)) Letting k → ∞, we have d(ν, f ν) ≤ 0, and so ν = fν Finally, to prove the uniqueness of the fixed point, let µ be the another... where Am+1 = A1 Theorem 4 Let (X, d) be a complete metric space, m ∈ N, A1 , A2, , Am nonempty subsets of X and X = ∪m Ai Let f : X → X be a cyclic weaker i=1 (φ, ϕ)-contraction Then f has a unique fixed point z ∈ ∩m Ai i=1 Proof Given x0 and let xn+1 = fxn = f n+1 x0, for n ∈ N∪{0} If there exists n ∈ N ∪ {0} such that xn0 +1 = xn0 , then we finished the proof Suppose that xn+1 = xn for any n ∈ N . below). For information about publishing your research in Fixed Point Theory and Applications go to http://www.fixedpointtheoryandapplications.com/authors/instructions/ For information about other. reproduction in any medium, provided the original work is properly cited. Fixed point theory for the cyclic weaker Meir–Keeler function in complete metric spaces Chi-Ming Chen Department of Applied Mathematics, National. by ϕ(t)= 1 2 t for t ∈ [0, ∞). Then f is a cyclic weaker φ ◦ ϕ-contraction and (0, 0, 0) is the unique fixed point. 3 Fixed point theory for the cyclic weaker (φ, ϕ)-contractions The main purpose