Báo cáo sinh học: "Maximal and minimal point theorems and Caristi's fixed point theorem"

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Fixed Point Theory and Applications 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 © 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, T x)) ≤ 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 ϕ(t) + ε} for some ε > 0. Assume t∈X 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: (H1 ) 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; (H2 ) for any increasing sequence {xn }n≥1 ⊂ X , if there exists some x ∈ X such that xn → x as n → ∞, then xn x for each n ≥ 1; (H3 ) for each x ∈ X , the set {y ∈ X : x y } is closed; (H4 ) η is nondecreasing; (H5 ) η is continuous and lim inf η (t) > 0; t→+∞ (H6 ) 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; (H7 ) for any decreasing sequence {xn }n≥1 ⊂ X , if there exists some x ∈ X such that xn → x as n → ∞, then x xn for each n ≥ 1; (H8 ) 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 (H1 ) and (H2 ) hold, and (H4 ) or (H5 ) is satisﬁed, then (X, ) has a maximal point. Proof. Case 1. (H4 ) is satisﬁed. Let {xα }α∈Γ ⊂ F be an increasing chain 5
with respect to the partial order . From (2) we ﬁnd that {ϕ(xα )}α∈Γ is a decreasing net of reals, where Γ is a directed set. Since ϕ is bounded below, then inf ϕ(xα ) is meaningful. Let {αn } be an increasing sequence α∈Γ of elements from Γ such that (3) lim ϕ(xαn ) = inf ϕ(xα ). n→∞ α∈ Γ We claim that {xαn }n≥1 is a Cauchy sequence. Otherwise, there exists a subsequence {xαni }i≥1 ⊂ {xαn }n≥1 and δ > 0 such that xαni xαni+1 for each i ≥ 1 and (4) d(xαni , xαni+1 ) ≥ δ, ∀ i ≥ 1. By (4) and (H4 ), we have (5) η (d(xαni , xαni+1 )) ≥ η (δ ), ∀ i ≥ 1. Therefore from (2) and (5) we have ϕ(xαni ) − ϕ(xαni+1 ) ≥ η (δ ), ∀ i ≥ 1, which indicates that (6) ϕ(xαni+1 ) ≤ ϕ(xαn1 ) − iη (δ ), ∀ i ≥ 1. Let i → ∞ in (6), by (3) and η −1 ({0}) = {0} we have inf ϕ(xα ) = lim ϕ(xαni ) ≤ −∞. i→∞ α∈Γ 6
This is a contradiction, and consequently, {xαn }n≥1 is a Cauchy sequence. Therefore by the completeness of X , there exists x ∈ X such that xαn → x as n → ∞. Moreover, (H2 ) forces that (7) xαn x, ∀ n ≥ 1. In the following, we show that {xα }α∈Γ has an upper bound. In fact, for each α ∈ Γ, if there exists some n ≥ 1 such that xα xαn , by (7) we get xα xαn x, i.e., x is an upper bound of {xα }α∈Γ . Otherwise, there exists some β ∈ Γ such that xαn xβ for each n ≥ 1. From (2) we ﬁnd that ϕ(xβ ) ≤ ϕ(xαn ) for each n ≥ 1. This together with (3) implies that ϕ(xβ ) = inf ϕ(xα ) and hence ϕ(xβ ) ≤ ϕ(xα ) for each α ∈ Γ. Note that α∈Γ {ϕ(xα )}α∈Γ is a decreasing chain, then we have β ≥ α for each α ∈ Γ. Since {xα }α∈Γ is an increasing chain, then xα xβ for each α ∈ Γ. This shows that xβ is an upper bound of {xα }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. (H5 ) is satisﬁed. By lim inf η (t) > 0, there exists l > δ and c1 > 0 t→+∞ such that η (t) ≥ c1 , ∀ t ≥ l. Since η is continuous and η −1 ({0}) = {0}, then c2 = min η (t) > 0. Let t∈[δ,l] 7
c = min{c1 , c2 }, then by (4) we have η (d(xαni , xαni+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 (H6 ) and (H7 ) hold, and (H4 ) or (H5 ) is satisﬁed, then (X, ) has a minimal point. Proof. Let be an inverse partial order of , i.e., x y⇔y x 1 1 for each x, y ∈ X . Let φ(x) = −ϕ(x). Then, φ is bounded below since ϕ is bounded above, and hence from (H6 ) and (H7 ) we ﬁnd that both (H1 ) and (H2 ) 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 (H1 ) and (H3 ) hold, and (H4 ) or (H5 ) 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 n0 ≥ 1, {y ∈ X : xαn0 y} is closed by (H3 ). From (2) we know that xαn0 xαn as n ≥ n0 and 8
hence xαn ∈ {y ∈ X : xαn0 y } for all n ≥ n0 . Therefore, we have x ∈ {y ∈ X : xαn0 y }, i.e., xαn0 x. Finally, the arbitrary property of n0 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 (H6 ) and (H8 ) hold, and (H4 ) or (H5 ) 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 (H1 ) holds, and (H2 ) or (H3 ) is satisﬁed. If (H4 ) or (H5 ) 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 (H6 ) holds, and (H7 ) or (H8 ) is satisﬁed. 9
If (H4 ) or (H5 ) is satisﬁed, then 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 fol- lows we shall show that Theorem 5 implies Caristi’s ﬁxed point theorem. The following lemma shows that there does exist some partial order on X such that (H3 ) is satisﬁed. Lemma 1. Let (X, d) be a metric space and the relationship deﬁned by (1) be a partial order on X . If η : [0, +∞) → [0, +∞) is continuous and ϕ : X → (−∞, +∞) is lower semicontinuous, then (H3 ) holds. Proof. For arbitrary x ∈ X , let {xn }n≥1 ⊂ {y ∈ X : x y } be a sequence such that xn → x∗ as n → ∞ for some x∗ ∈ X . From (1) we have (8) η (d(x, xn )) ≤ ϕ(x) − ϕ(xn ). Let n → ∞ in (8), then lim sup η (d(x, xn )) ≤ lim sup(ϕ(x) − ϕ(xn )) ≤ ϕ(x) − lim inf ϕ(xn ). n→∞ n→∞ n→∞ 10
Moreover, by the continuity of η and the lower semicontinuity of ϕ we get η (d(x, x∗ )) ≤ ϕ(x) − ϕ(x∗ ), x∗ , i.e., x∗ ∈ {y ∈ X : x which implies that x y }. Therefore, {y ∈ X : x y } is closed for each x ∈ X . The proof is complete. By Theorem 5 and Lemma 1 we have the following result. Corollary 1. Let (X, d) be a complete metric space and the relation- ship deﬁned by (1) be a partial order on X . Let T : X → X be a Caristi-type mapping and ϕ be a lower semicontinuous and bounded be- low 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 com- plete metric space and T : X → X be a 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, 12–14] could be obtained by Theorem 5. Contrarily, Theorem 5 could not be derived from Caristi’s ﬁxed point theorem. Hence, Theorem 5 indeed improve Caristi’s ﬁxed point theorem. 1 Example 1. Let X = {0} ∪ { n : n = 2, 3, . . .} with the usual metric d(x, y ) = |x − y | and the partial order as follows x y ⇐⇒ y ≤ x. Let ϕ(x) = x2 and    0,  x = 0, Tx =  1 1   , x= , n = 2, 3, . . . . n+1 n Clearly, (X, d) is a complete metric space, (H2 ) is satisﬁed, and ϕ 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 y and x = y , we have two possible cases. 1 Case 1. When x = n , n ≥ 2 and y = 0, we have 1 η (d(x, y )) = = ϕ(x) − ϕ(y ). n2 12
1 1 Case 2. When x = n , n ≥ 2 and y = ,m > n, we have m (m − n)2 m2 − n2 η (d(x, y )) = < = ϕ(x) − ϕ(y ). m2 n2 m2 n2 Therefore, (2) holds for each x, y ∈ X with x y and hence (H1 ) is satisﬁed. Finally, the existence of ﬁxed point follows from Theorem 5. 1 While for each x = n , n ≥ 2, we have 2n + 1 1 ϕ(x) − ϕ(T x) = < = d(x, T x), n2 (n 2 + 1) n(n + 1) which implies that corresponding to the function ϕ(x) = x2 , T is not a Caristi-type mapping. Therefore, we can conclude that for some given function ϕ and some given mapping T , there may exist some function η such that all the conditions of Theorem 5 are satisﬁed even though T may not be a Caristi-type mapping corresponding to the function ϕ. 4 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 The authors declare that they have no competing interests. Authors’ contributions ZL carried out the main part of this article. All authors read and ap- proved 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 Foun- dation of Jiangxi Educational Department (GJJ11420). References 1. Kirk WA, Caristi J: Mapping theorems in metric and Banach spaces. Bull. Acad. Polon. Sci. 23, 891–894 (1975) 2. Kirk 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: Lak- shmikantham, V (ed.) Applied Nonlinear Analysis, pp. 479–483. Academic Press, New York (1979) 5. Brondsted A: Fixed point and partial orders. Proc. Am. Math. Soc. 60, 365–368 (1976) 6. Downing D, Kirk WA: A generalization of Caristi’s theorem with applications to nonlinear mapping theory. Paciﬁc J. Math. 69, 339– 345 (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 multivalued maps. J. Math. Anal. Appl. 284, 690–697 (2003) 11. Suzuki T: 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. Nonlinear Anal. 73, 3751–3755 (2010) 15. Agarwal RP, Khamsi MA: Extension of Caristi’s ﬁxed point theorem to vector valued metric space. Nonlinear Anal. 74, 141–145 (2011) 16