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. Maximal and minimal point theorems and Caristi's fixed point theorem Fixed Point Theory and Applications 2011, 2011:103 doi:10.1186/1687-1812-2011-103 Zhilong Li (lzl771218@sina.com) Shujun Jiang (jiangshujun_s@yahoo.com.cn) ISSN 1687-1812 Article type Research Submission date 8 August 2011 Acceptance date 21 December 2011 Publication date 21 December 2011 Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/103 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice 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 SpringerOpen publications go to http://www.springeropen.com Fixed Point Theory and Applications © 2011 Li and Jiang ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Maximal and minimal point theorems and Caristi’s fixed point theorem Zhilong Li ∗ and Shujun Jiang Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China ∗ Corresponding author: lzl771218@sina.com E-mail address: SJ: jiangshujun s@yahoo.com.cn Abstract This study is concerned with the existence of fixed points of Caristi-type mappings motivated by a problem stated by Kirk. First, several existence theorems of maximal and minimal points are established. By using them, some generalized Caristi’s fixed 1 point theorems are proved, which improve Caristi’s fixed point theorem and the results in the studies of Jachymski, Feng and Liu, Khamsi, and Li. MSC 2010: 06A06; 47H10. Keywords: maximal and minimal point; Caristi’s fixed point theorem; Caristi-type mapping; partial order. 1 Introduction In the past decades, Caristi’s fixed point theorem has been generalized and extended in several directions, and the proofs given for Caristi’s result varied and used different techniques, we refer the readers to [1–15]. Recall that T : X → X is said to be a Caristi-type mapping [14] pro- vided that there exists a function η : [0, +∞) → [0, +∞) and a function ϕ : X → (−∞, +∞) such that η(d(x, Tx)) ≤ ϕ(x) − ϕ(T x), ∀ x ∈ X, where (X, d) is a complete metric space. Let  be a relationship defined on X as follows (1) x  y ⇐⇒ η(d(x, y)) ≤ ϕ(x) − ϕ(y), ∀ x, y ∈ X. Clearly, x  T x for each x ∈ X provided that T is a Caristi-type mapping. Therefore, the existence of fixed points of Caristi-type map- 2 pings is equivalent to the existence of maximal point of (X, ). Assume that η is a continuous, nondecreasing, and subadditive function with η −1 ({0}) = {0}, then the relationship defined by (1) is a partial order on X. Feng and Liu [12] proved each Caristi-type mapping has a fixed point by investigating the existence of maximal point of (X, ) provided that ϕ is lower semicontinuous and bounded below. The additivity of η appearing in [12] guarantees that the relationship  defined by (1) is a partial order on X. However, if η is not subadditive, then the relation- ship  defined by (1) may not be a partial order on X, and consequently the method used there becomes invalid. Recently, Khamsi [13] removed the additivity of η by introducing a partial order on Q as follows x  ∗ y ⇐⇒ cd(x, y) ≤ ϕ(x) − ϕ(y), ∀ x, y ∈ Q, where Q = {x ∈ X : ϕ(x) ≤ inf t∈X ϕ(t) + ε} for some ε > 0. Assume that ϕ is lower semicontinuous and bounded below, η is continuous and nondecreasing, and there exists δ > 0 and c > 0 such that η(t) ≥ ct for each t ∈ [0, δ]. He showed that (Q,  ∗ ) has a maximal point which is exactly the maximal point of (X, ) and hence each Caristi-type mapping has a fixed point. Very recently, the results of [9, 12, 13] were improved by Li [14] in which the continuity, subadditivity and nondecreasing property of η are removed at the expense that 3 (H) there exists c > 0 and ε > 0 such that η(t) ≥ ct for each t ∈ {t ≥ 0 : η(t) ≤ ε}. From [14, Theorem 2 and Remark 2] we know that the assumptions made on η in [12, 13] force that (H) is satisfied. In other words, (H) is necessarily assumed in [12–14]. Meanwhile, ϕ is always assumed to be lower semicontinuous there. In this study, we shall show how the condition (H) and the lower semicontinuity of ϕ could be removed. We first proved several existence theorems of maximal and minimal points. By using them, we obtained some fixed point theorems of Caristi-type mappings in a partially ordered complete metric space without the lower semicontinuity of ϕ and the condition (H). 2 Maximal and minimal point theorems For the sake of convenience, we in this section make the following assumptions: (H 1 ) there exists a bounded below function ϕ : X → (−∞, +∞) and a function η : [0, +∞) → [0, +∞) with η −1 ({0}) = {0} such that (2) η(d(x, y)) ≤ ϕ(x) − ϕ(y), 4 for each x, y ∈ X with x  y; (H 2 ) for any increasing sequence {x n } n≥1 ⊂ X, if there exists some x ∈ X such that x n → x as n → ∞, then x n  x for each n ≥ 1; (H 3 ) for each x ∈ X, the set {y ∈ X : x  y} is closed; (H 4 ) η is nondecreasing; (H 5 ) η is continuous and lim inf t→+∞ η(t) > 0; (H 6 ) there exists a bounded above function ϕ : X → (−∞, +∞) and a function η : [0, +∞) → [0, +∞) with η −1 ({0}) = {0} such that (2) holds for each x, y ∈ X with x  y; (H 7 ) for any decreasing sequence {x n } n≥1 ⊂ X, if there exists some x ∈ X such that x n → x as n → ∞, then x  x n for each n ≥ 1; (H 8 ) for each x ∈ X, the set {y ∈ X : y  x} is closed. Recall that a point x ∗ ∈ X is said to be a maximal (resp. minimal) point of (X, ) provided that x = x ∗ for each x ∈ X with x ∗  x (resp. x  x ∗ ). Theorem 1. Let (X, d, ) be a partially ordered complete metric space. If (H 1 ) and (H 2 ) hold, and (H 4 ) or (H 5 ) is satisfied, then (X, ) has a maximal point. Proof. Case 1. (H 4 ) is satisfied. Let {x α } α∈Γ ⊂ F be an increasing chain 5 with respect to the partial order . From (2) we find that {ϕ(x α )} α ∈ Γ is a decreasing net of reals, where Γ is a directed set. Since ϕ is bounded below, then inf α∈Γ ϕ(x α ) is meaningful. Let {α n } be an increasing sequence of elements from Γ such that (3) lim n→∞ ϕ(x α n ) = inf α∈Γ ϕ(x α ). We claim that {x α n } n≥1 is a Cauchy sequence. Otherwise, there exists a subsequence {x α n i } i≥1 ⊂ {x α n } n≥1 and δ > 0 such that x α n i  x α n i+1 for each i ≥ 1 and (4) d(x α n i , x α n i+1 ) ≥ δ, ∀ i ≥ 1. By (4) and (H 4 ), we have (5) η(d(x α n i , x α n i+1 )) ≥ η(δ), ∀ i ≥ 1. Therefore from (2) and (5) we have ϕ(x α n i ) − ϕ(x α n i+1 ) ≥ η(δ), ∀ i ≥ 1, which indicates that (6) ϕ(x α n i+1 ) ≤ ϕ(x α n 1 ) − iη(δ), ∀ i ≥ 1. Let i → ∞ in (6), by (3) and η −1 ({0}) = {0} we have inf α∈Γ ϕ(x α ) = lim i→∞ ϕ(x α n i ) ≤ −∞. 6 This is a contradiction, and consequently, {x α n } n ≥ 1 is a Cauchy sequence. Therefore by the completeness of X, there exists x ∈ X such that x α n → x as n → ∞. Moreover, (H 2 ) forces that (7) x α n  x, ∀ n ≥ 1. In the following, we show that {x α } α∈Γ has an upper bound. In fact, for each α ∈ Γ, if there exists some n ≥ 1 such that x α  x α n , by (7) we get x α  x α n  x, i.e., x is an upper bound of {x α } α∈Γ . Otherwise, there exists some β ∈ Γ such that x α n  x β for each n ≥ 1. From (2) we find that ϕ(x β ) ≤ ϕ(x α n ) for each n ≥ 1. This together with (3) implies that ϕ(x β ) = inf α∈Γ ϕ(x α ) and hence ϕ(x β ) ≤ ϕ(x α ) for each α ∈ Γ. Note that {ϕ(x α )} α∈Γ is a decreasing chain, then we have β ≥ α for each α ∈ Γ. Since {x α } α∈Γ is an increasing chain, then x α  x β for each α ∈ Γ. This shows that x β is an upper bound of {x α } α∈Γ . By Zorn’s lemma we know that (X, ) has a maximal point x ∗ , i.e., if there exists x ∈ X such that x ∗  x, we must have x = x ∗ . Case 2. (H 5 ) is satisfied. By lim inf t→+∞ η(t) > 0, there exists l > δ and c 1 > 0 such that η(t) ≥ c 1 , ∀ t ≥ l. Since η is continuous and η −1 ({0}) = {0}, then c 2 = min t∈[δ,l] η(t) > 0. Let 7 c = min{c 1 , c 2 }, then by (4) we have η(d(x α n i , x α n i+1 )) ≥ c, ∀ i ≥ 1. In analogy to Case 1, we know that (X, ) has a maximal point. The proof is complete. Theorem 2. Let (X, d, ) be a partially ordered complete metric space. If (H 6 ) and (H 7 ) hold, and (H 4 ) or (H 5 ) is satisfied, then (X, ) has a minimal point. Proof. Let  1 be an inverse partial order of , i.e., x  y ⇔ y  1 x for each x, y ∈ X. Let φ(x) = −ϕ(x). Then, φ is bounded below since ϕ is bounded above, and hence from (H 6 ) and (H 7 ) we find that both (H 1 ) and (H 2 ) hold for (X, d,  1 ) and φ. Finally, Theorem 2 forces that (X,  1 ) has a maximal point which is also the minimal point of (X, ). The proof is complete. Theorem 3. Let (X, d, ) be a partially ordered complete metric space. If (H 1 ) and (H 3 ) hold, and (H 4 ) or (H 5 ) is satisfied, then (X, ) has a maximal point. Proof. Following the proof of Theorem 1, we only need to show that (7) holds. In fact, for arbitrarily given n 0 ≥ 1, {y ∈ X : x α n 0  y} is closed by (H 3 ). From (2) we know that x α n 0  x α n as n ≥ n 0 and 8 hence x α n ∈ {y ∈ X : x α n 0  y} for all n ≥ n 0 . Therefore, we have x ∈ {y ∈ X : x α n 0  y}, i.e., x α n 0  x. Finally, the arbitrary property of n 0 implies that (7) holds. The proof is complete. Similarly, we have the following result. Theorem 4. Let (X, d, ) be a partially ordered complete metric space. If (H 6 ) and (H 8 ) hold, and (H 4 ) or (H 5 ) is satisfied, then (X, ) has a minimal point. 3 Caristi’s fixed point theorem Theorem 5. Let (X, d, ) be a partially ordered complete metric space and T : X → X. Suppose that (H 1 ) holds, and (H 2 ) or (H 3 ) is satisfied. If (H 4 ) or (H 5 ) is satisfied, then T has a fixed point provided that x  Tx for each x ∈ X. Proof. From Theorems 1 and 3, we know that (X, ) has a maximal point. Let x ∗ be a maximal point of (X, ), then x∗  T x ∗ . The maximality of x ∗ forces x ∗ = T x ∗ , i.e., x ∗ is a fixed point of T . The proof is complete. Theorem 6. Let (X, d, ) be a partially ordered complete metric space and T : X → X. Suppose that (H 6 ) holds, and (H 7 ) or (H 8 ) is satisfied. 9 [...]... T has a fixed point provided that T x x for each x ∈ X Proof From Theorems 2 and 4, we know that (X, ) has a minimal point Let x∗ be a minimal point of (X, ), then T x∗ x∗ The mini- mality of x∗ forces x∗ = T x∗ , i.e., x∗ is a fixed point of T The proof is complete Remark 1 The lower semicontinuity of ϕ and (H) necessarily assumed in [9, 12–14] are no longer necessary for Theorems 5 and 6 In what... Conclusions In this article, some new fixed point theorems of Caristi-type mappings have been proved by establishing several maximal and minimal point theorems As one can see through Remark 2, many recent results could be obtained by Theorem 5, but Theorem 5 could not be derived from Caristi’s fixed point theorem Therefore, the fixed point theorems indeed improve Caristi’s fixed point theorem 13 Competing interests... WA: Caristi’s fixed -point theorem and metric convexity Colloq Math 36, 81–86 (1976) 3 Caristi J: Fixed point theorems for mappings satisfying inwardness 14 conditions Trans Am Math Soc 215, 241–251 (1976) 4 Caristi J: Fixed point theory and inwardness conditions In: Lakshmikantham, V (ed.) Applied Nonlinear Analysis, pp 479–483 Academic Press, New York (1979) 5 Brondsted A: Fixed point and partial orders... Generalized Caristi’s fixed point theorems by Bae and others J Math Anal Appl 302, 502–508 (2005) 12 Feng YQ, Liu SY: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi-type mappings J Math Anal 15 Appl 317, 103–112 (2006) 13 Khamsi MA: Remarks on Caristi’s fixed point theorem Nonlinear Anal 71, 227–231 (2009) 14 Li Z: Remarks on Caristi’s fixed point theorem and Kirk’s problem... (1977) 7 Downing D, Kirk WA: Fixed point theorems for set-valued mappings in metric and Banach spaces Math Japon 22, 99–112 (1977) 8 Khamsi MA, Misane D: Compactness of convexity structures in metrics paces Math Japon 41, 321–326 (1995) 9 Jachymski J: Caristi’s fixed point theorem and selection of set-valued contractions J Math Anal Appl 227, 55–67 (1998) 10 Bae JS: Fixed point theorems for weakly contractive... semicontinuous and bounded below function If η is a continuous function with η −1 ({0}) = {0}, and (H4 ) or lim inf η(t) > 0 is satisfied, then T has a fixed point t→+∞ It is clear that the relationship defined by (1) is a partial order on X for when η(t) = t Then, we obtain the famous Caristi’s fixed point theorem by Corollary 1 Corollary 2 (Caristi’s fixed point theorem) Let (X, d) be a complete metric space and. .. All authors read and approved the final manuscript Acknowledgments This study was supported by the National Natural Science Foundation of China (10701040,11161022,60964005), the Natural Science Foundation of Jiangxi Province (2009GQS0007), and the Science and Technology Foundation of Jiangxi Educational Department (GJJ11420) References 1 Kirk WA, Caristi J: Mapping theorems in metric and Banach spaces... derived from Caristi’s fixed point theorem Hence, Theorem 5 indeed improve Caristi’s fixed point theorem Example 1 1 Let X = {0} ∪ { n : n = 2, 3, } with the usual metric d(x, y) = |x − y| and the partial order x as follows y ⇐⇒ y ≤ x Let ϕ(x) = x2 and    0,  Tx = x = 0,    x= 1 , n+1 1 , n = 2, 3, n Clearly, (X, d) is a complete metric space, (H2 ) is satisfied, and ϕ is bounded below For... is bounded below For each x ∈ X, we have x ≥ T x and hence x T x Let η(t) = t2 Then η −1 ({0}) = {0}, (H4 ) and (H5 ) are satisfied Clearly, (2) holds for each x, y ∈ X with x = y For each x, y ∈ X with x and x = y, we have two possible cases 1 Case 1 When x = n , n ≥ 2 and y = 0, we have η(d(x, y)) = 1 = ϕ(x) − ϕ(y) n2 12 y 1 Case 2 When x = n , n ≥ 2 and y = η(d(x, y)) = 1 ,m m > n, we have (m − n)2... Caristi-type mapping with η(t) = t If ϕ is lower semicontinuous and bounded below, then T has a fixed point Remark 2 From [14, Remarks 1 and 2] we find that [14, Theorem 1] 11 includes the results appearing in [3, 4, 9, 12, 13] Note that [14, Theorem 1] is proved by Caristi’s fixed point theorem, then the results of [9, 12–14] are equivalent to Caristi’s fixed point theorem Therefore, all the results of [3, 4, 9, . Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Maximal and minimal point theorems and Caristi's. use, distribution, and reproduction in any medium, provided the original work is properly cited. Maximal and minimal point theorems and Caristi’s fixed point theorem Zhilong Li ∗ and Shujun Jiang Department. fixed 1 point theorems are proved, which improve Caristi’s fixed point theorem and the results in the studies of Jachymski, Feng and Liu, Khamsi, and Li. MSC 2010: 06A06; 47H10. Keywords: maximal and