Some Generalizations of Caristi s Fixed Point Theorem with Applications

Transcription

1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 12, Some Generalizations of Caristi s Fixed Point Theorem with Applications Seong-Hoon Cho Department of Mathematics Hanseo University, Seosan Chungnam, , South Korea shcho@hanseo.ac.kr Abstract In this paper, we prove some generalizations of Caristi s fixed point theorem, and then we give applications to Ekeland s variational principle and maximal element theorem. Mathematics Subject Classification: 47H10, 54H25 Keywords: Caristi s fixed point theorem, Maximal element, Lower semicontinuous 1 Introduction and preliminaries Let (X, d) be a complete metric space, and let T : X X be a map which is not assumed to be continuous. Caristi s fixed point theorem [4] states that T has a fixed point provided that there exists a lower semi-continuous function φ : X [0, ) such that d(x, T x) φ(x) φ(tx) for all x X. Because its importance for mathematical theory, above result is generalized by many authors, see [2, 5, 6, 7, 8, 10, 12], etc. Recently, the author [1] obtained a geralization of Caristi s fixed point theorem, and then gave applications to the fixed point theory of weakly contractive set-valued maps and minimization problem. In this paper, we generalize the results in [1]. We give applications to Ekeland s variational principle and maximal element theorem. From now on, let Γ be the class of all nondecreasing functions γ :[0, ) [0, ) such that

2 558 S.H. Cho (1) γ is subadditive; (2) γ is continuous at 0; (3) γ 1 (0) = 0. Remark 1.1. (1) If γ Γ, then γ is continuous on [0, ) (see [8]). (2) If d is a metric on a nonempty set X, then γ d is a metric on X, where γ Γ. Lemma 1.1. Let (X, d) be a metric space, and let {x n } be a sequence in X. Then, {x n } is d-cauchy if and only if it is γ d -Cauchy, where γ Γ. Lemma 1.2. Let (X, ) be a partial ordered set, and let φ : X X R be a function. Suppose that the following are satisfied. (1) every nondecreasing sequence in X is bounded; (2) there exists ˆx X such that φ(ˆx, ) is a nondecreasing function. Then, for each x 0 X, there exists x X with x 0 x such that x x implies φ(ˆx, x) =φ(ˆx, x ). Proof. Let θ(x) =φ(ˆx, x) for all x X, where ˆx X such that φ(ˆx, ) is a nondecreasing function. By Brézis-Browder principle [3], we have desired result. 2 Caristi s fixed point theorem Let (X, d) be a metric space, and let Ψ(X) be the family of all map ψ : X X R satisfying the following conditions: (1) there exists ˆx X such that ψ(ˆx, ) is lower semi-continuous and bounded from below; (2) ψ(x, y)+ψ(y, z) ψ(x, z), for each x, y, z X. We denote Ψ (X) by the family of all map ψ Ψ(X) such that ψ(x, x) =0 for all x Z. In similar way in [8, 9] and in [11], we obtain the following Lemma 2.1. Here, we give detailed proof of Lemma 2.1.

3 Fixed points 559 Lemma 2.1. Let (X, d) be a complete metric space, and let γ Γ and ψ Ψ(X). Define a relation on X by x γ y if and only if γ(d(x, y)) ψ(ˆx, x) ψ(ˆx, y) (2.1) for all x, y X, where ˆx is the point in X such that ψ(ˆx, ) is lower semicontinuous and bounded from below. Then (X, γ ) is a partial odered set which has a maximal element. Proof. It is easy to see that (X, γ ) is a partial odered set. Let φ(ˆx, x) = ψ(ˆx, x) for all x X. Then φ(ˆx, ) is upper semi-continuous and bounded from above. If x γ y, then from (2.1) we have φ(ˆx, x) φ(ˆx, y). Thus, φ(ˆx, ) is nondecreasing on X. Now, let {x n } be a nondecreasing sequence in X. Then {φ(ˆx, x n )} is bounded from above and nondecreasing in R. Thus, {φ(ˆx, x n )} is convergent, and hence it is a Cauchy sequence. For n m, we have x n γ x m. Hence γ(d(x n,x m )) ψ(ˆx, x n ) ψ(ˆx, x m ), and hence γ(d(x n,x m )) φ(ˆx, x m ) φ(ˆx, x n ) (2.2) which implies {x n } is a γ d-cauchy sequence in X. By Lemma 1.1, it is a d-cauchy sequence in X. Since X is complete, there exists x X such that lim n x n = x. Because φ(ˆx, ) is upper semi-continuous, lim m φ(ˆx, x m ) φ(ˆx, x). Letting m in above (2.2), we have γ(d(x n, x)) lim φ(ˆx, x m) φ(ˆx, x n ) m φ(ˆx, x) φ(ˆx, x n ) =ψ(ˆx, x n ) ψ(ˆx, x). Thus, x n γ x for all n N. Hence {x n } is bounded. By Lemma 1.2, for each x 0 X, there exists x X with x 0 γ x such that x γ x implies φ(ˆx, x) =φ(ˆx, x ). (2.3) Since x γ x, from (2.1) we obtain γ(d(x,x)) ψ(ˆx, x ) ψ(ˆx, x) φ(ˆx, x) φ(ˆx, x ). In view of (2.3), it follows γ(d(x,x)) = 0, so we obtain x = x. Thus, x is a maximal element in (X, γ ).

4 560 S.H. Cho Theorem 2.1. Let (X, d) be a complete metric space. Suppose that a mapping T : X X satisfies for all x X, where γ Γ and ψ Ψ(X). Then T has a fixed point in X. γ(d(x, T x)) ψ(tx,x) (2.4) Proof. By Lemma 2.1, the partial odered set (X, γ ) has a maximal element, say x X. From (2.4) we have γ(d(x, T x)) ψ(tx,x) ψ(ˆx, x) ψ(ˆx, T x) for all x X. Hence x γ Tx for all x X, and so x γ Tx. By maximality of x, we have x = Tx. Let Ω be the family of all function η : [0, ) [0, ) satisfying the following conditions: (1) η(0) = 0; (2) there exist γ Γ and ɛ>0 such that γ(t) η(t) for t {t 0:η(t) ɛ}. Theorem 2.2. Let (X, d) be a complete metric space, and let ψ Ψ (X). Suppose that a mapping T : X X satisfies η(d(x, T x)) ψ(tx,x) (2.5) for all x X, where η Ω. Then there exists a nonempty subset Y of X such that T has a fixed point in Y, and hence T has a fixed point in X. Proof. Let ˆx X be the point such that ψ(ˆx, ) is lower semi-continuous and bounded from below on X. Let α = inf{ψ(ˆx, x) :x X}, and let ɛ>0 be such that α + ɛ 0. Let Y = {x X : ψ(ˆx, x) α + ɛ}. Then Y is nonempty closed set, because ψ(ˆx, ) is lower semi-continuous. Obviously, (Y,d) is complete metric subspace of (X, d). Since ψ(ˆx, ˆx) =0 α + ɛ, ˆx Y. ψ(ˆx, ) is lower semi-continuous and bounded from below on Y. Obviously, ψ(x, y)+ψ(y, z) ψ(x, z), for each x, y, z Y. Thus, ψ Ψ(Y ). We show that T (Y ) Y. Let x Y. From (2.5) we obtain η(d(x, T x)) ψ(ˆx, x) ψ(ˆx, T x). (2.6) Thus, we have α ψ(ˆx, T x) ψ(ˆx, x) α + ɛ. (2.7)

5 Fixed points 561 Hence, Tx Y, and so T (Y ) Y. From (2.6) and (2.7) we have 0 η(d(x, T x)) ψ(ˆx, x) ψ(ˆx, T x) ɛ. Since η Ω, there exists γ Γ such that γ(d(x, T x)) η(d(x, T x)) ψ(tx,x) for all x Y. By Theorem 2.1, T has a fixed point in Y. Remark 2.1. (1) condition η(0) = 0 is the essential condition for the existence of a fixed point for map T satisfying (2.5) (see [12]). In fact, if x = Tx and η(0) > 0, then from (2.5) we have 0 <η(d(x, T x)) ψ(tx,x)=ψ(x, x) = 0, which is a contradiction. (2) Theorem 2.2 is a generalization of a result in [1]. The author [1] proved the existence of a fixed point under assumptions of Theorem 3.4, and assumption of the following additional condition: ψ(,x) is upper semi-continuous for each x X. (3) In Theorem 2.2, if we have ψ(x, y) =f(y) f(x), where f : X [0, ) is lower semi-continuous, and γ(t) =ct, where c (0, ), then Theorem 2.3 reduces to a result in [10]. Corollary 2.3. Let (X, d) be a complete metric space, and let η Ω. Assume that a set-valued mapping F : X X with nonempty values satisfies η(d(x, y)) ψ(y, x) (2.8) for all x X, where y Fx and ψ Ψ (X). Then F has a fixed point in X, i.e. there exists x X such that x F x. Proof. For each x X, put y = Tx. Then Tx Fx for each x X. From (2.8) we have η(d(x, T x)) ψ(tx,x). By Theorem 2.2, there exists x X such that x = T x. Hence, x = T x F x. Remark 2.2. Theorem 2.2 and Corollary 2.3 are equivalent. Remark 2.3. In Corollary 2.3, if we have ψ(y, x) =f(x) f(y) for all x, y X, where f : X R is bounded from below and lower semi-continuous, then Corollary 2.3 is a generalization of Theorem 4.2 in [7].

6 562 S.H. Cho 3 Applications As a application of generalized Caristi s fixed point theorem, we derive the following Ekeland s variational principle. Theorem 3.1. (Ekeland-type variational principle) Let (X, d) be a complete metric space, and let η Ω and ψ Ψ (X). Then there exists x X such that for all x X with x x. η(d(x, x)) >ψ(x, x) (3.1) Proof. On the contrary, assume that for each x X, there exists y X with x y such that η(d(x, y)) ψ(y, x). We define a set-valued mapping F : X X by Fx = {y X : y x, η(d(x, y)) ψ(y, x)}. By Corollary 2.3, there exists x X such that x F x. But x F x, which is a contradiction. Remark 3.1. (1) Theorem 3.1 implies Theorem 2.2. In fact, suppose that Theorem 3.1 and assumption of Theorem 2.2 are satisfied. On the contrary, assume that x Tx for all x X. Then x T x. From (3.1) and (2.5) we have η(d(x, T x)) >ψ(tx, x) and η(d(x, T x)) ψ(t x, x) which is a contradiction. (2) Theorem 3.1 and Theorem 2.2 are equivalent. Theorem 3.2. (Maximal element theorem for a set-valued map) Let (X, d) be a complete metric space, and let η Ω and ψ Ψ (X). Suppose that a setvalued mapping F : X X satisfies the following condition: for each x X with Fx, there exists y = y(x) X with y x such that η(d(x, y)) ψ(y, x). Then there exists x X such that F x =. Proof. From Theorem 3.1 there exists x X such that for all x X with x x. We show that F x =. η(d(x, x)) >ψ(x, x)

7 Fixed points 563 On the contrary, assume that F x. By assumption, there exists y = y(x) X with y x such that η(d(x, y)) ψ(y, x). It follows that η(d(x, y)) ψ(y, x) <η(d(x, y)) which is a contradiction. Hence F x =. Remark 3.2. Theorem 3.2 implies Theorem 3.1. In fact, assume that for each x X, there exists y X with y x such that η(d(x, y)) ψ(y, x). We define a set-valued mapping F : X X by Fx = {y X : y x, η(d(x, y)) ψ(y, x)}. Then Fx for all x X. By Theorem 3.2, there exists x X such that F x =, which is a contradiction. Hence Theorem 3.2 implies Theorem 3.1, and hence Theorem 3.2 and Theorem 3.1 are equivalent. References [1] A. Amini-Harandi, Some generalizations of Caristi s fixed point theorem with applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem, Nonlinear Analysis 72(2010), [2] J.S. Bae, E.W. Cho, S.H. Yeom, A generalization of the Caristi-Kirk fixed point theorem and its application to mapping theorems, J. Korean Math. Soc. 31(1994) [3] H. Brézis, F. Browder, A general principle on ordered sets in nonlinear functional analysis, Adv. in Math. 21(1976), [4] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215(1976), [5] Lj. Ćirić, A generalization of Caristi s fixed point theorem, Mathematica Pannonica 3(2) (1992), [6] D. Downing, W.A. Kirk, A generalization of Caristi s theorem with applications to nonlinear mappings theory, Pacific J. Math. 69(1977), [7] Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317(2006), [8] J.R. Jachymski, Caristi fixed point theorem and selections of set-valued contractions, J. Math. Anal. Appl. 227, 55-67(1998).

8 564 S.H. Cho [9] J.R. Jachymski, Converges to fixed point theorem of Zeremlo and Caristi, Nonlinear Anal. 52(2003), [10] M.A. Khamsi, Remarks on Caristi s fixed point theorem, Nonlinear Anal. TMA 70(2009), [11] W.A. Kirk, Caristi s fixed-point theorem and metric convexity, Colloq. Math. 36(1976), [12] L. Zhilong, Remarks on Caristi s fixed point theorem and Kirk s problem, Nonlinear Analysis. 73(2010), Received: October, 2012