A Direct Proof of Caristi s Fixed Point Theorem

Transcription

1 Applied Mathematical Sciences, Vol. 10, 2016, no. 46, HIKARI Ltd, A Direct Proof of Caristi s Fixed Point Theorem Wei-Shih Du Department of Mathematics National Kaohsiung Normal University Kaohsiung 82444, Taiwan Copyright c 2016 Wei-Shih Du. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The main aim of this paper is to give a new and simple proof of Caristi s fixed point theorem which is different from the known proofs in the literature. Mathematics Subject Classification: 47H09, 47H10 Keywords: Caristi s fixed point theorem, transfinite induction, Brézis- Browder order principle, Zorn s lemma 1. Introduction and preliminaries It is well known that Banach contraction principle plays an important role in various fields of nonlinear functional analysis and applied mathematical analysis. Among these generalizations of the Banach contraction principle, Caristi s fixed point theorem [4] is undoubtedly one of the most valuable one in nonlinear analysis. In fact, Caristi [4] proved his famous fixed point theorem by using transfinite induction. Several elegant proofs of original Caristi s fixed point theorem were given; see, for example, [1-3, 5-20] and references therein. In this paper, we give a new, simple and direct proof of Caristi s fixed point theorem which is different from the known proofs in the literature.

2 2290 Wei-Shih Du 2. A new and direct proof of Caristi s fixed point theorem In this section, we present a new, simple and direct proof of Caristi s fixed point theorem as follows. Theorem 2.1 (Caristi [4]). Let (X, d) be a complete metric space and f : X R be a lower semicontinuous and bounded below function. Suppose that T is a Caristi type mapping on X dominated by f; that is, T satisfies Then T has a fixed point in X. d(x, T x) f(x) f(t x) for each x X. (2.1) Proof. For any x X, define a set-valued mapping S : X 2 X (the power set of X) by S(x) = {y X : d(x, y) f(x) f(y)}. Clearly, x S(x) and hence S(x) for all x X. We claim that for each y S(x), we have f(y) f(x) and S(y) S(x). Let y S(x) be given. Then d(x, y) f(x) f(y). So we have f(y) f(x). Since S(y), let z S(y). Thus d(y, z) f(y) f(z). It follows that and hence f(z) f(y) f(x) d(x, z) d(x, y) + d(y, z) f(x) f(z). So z Γ(x). Therefore we prove S(y) S(x). We shall construct a sequence {x n } in X by induction, starting with any point x 1 X. Suppose that x n X is known. Then choose x n+1 S(x n ) such that f(x n+1 ) For any n N, since x n+1 S(x n ), we have inf f(z) + 1, n N. (2.2) z S(x n) n d(x n, x n+1 ) f(x n ) f(x n+1 ). (2.3) So f(x n+1 ) f(x n ) for each n N. Since f is bounded below, γ := lim f(x n ) = inf n N f(x n) exists. (2.4) For m > n with m, n N, by (2.3) and (2.4), we obtain d(x n, x m ) m 1 j=n d(x j, x j+1 ) f(x n ) γ.

3 A direct proof of Caristi s fixed point theorem 2291 Since lim f(x n ) = γ, we obtain lim sup{d(x n, x m ) : m > n} = 0. Hence {x n } is a Cauchy sequence in X. By the completeness of X, there exists v X such that x n v as n. Since f is lower semicontinuous, by (2.4), we get f(v) lim inf f(x n) = inf n N f(x n) f(x j ) for all j N. (2.5) Next, we prove that n=1 S(x n) = {v}. For m > n with m, n N, by (2.3) and (2.5), we obtain d(x n, x m ) m 1 j=n Since x m v as m, the inequality (2.6) implies d(x j, x j+1 ) f(x n ) f(v). (2.6) d(x n, v) f(x n ) f(v) for all n N. (2.7) By (2.7), we know v n=1 S(x n). Hence n=1 S(x n) and S(v) S(x n ). n=1 For any w n=1 S(x n), by (2.2), we have d(x n, w) f(x n ) f(w) f(x n ) inf f(z) z S(x n) f(x n ) f(x n+1 ) + 1 n for all n N. Hence lim d(x n, w) = 0 or, equivalently, x n w as n. By the uniqueness of limit of a sequence, we have w = v. So we show n=1 S(x n) = {v}. Since S(v) and S(v) S(x n ) = {v} n=1 we get S(v) = {v}. On the other hand, by (2.1), we know T v S(v). Hence it must be T v = v. Therefore T has a fixed point v in X. The proof is completed.

4 2292 Wei-Shih Du Remark 2.2. (a) Classic proofs of Caristi s theorem involve assigning a partial order on X by setting x y d(x, y) f(x) f(y), and then either using Zorn s lemma or the Brézis-Browder order principle with the set {y X : x y} = {y X : d(x, y) f(x) f(y)} for x X. However, in our proof, we do not define any partial order on X, and then Zorn s lemma or the Brézis-Browder order principle are not applicable here. (b) In [7], we proved the Caristi s fixed point theorem with a proof by contradiction by using the set Γ(x) = {y X : y x, d(x, y) f(x) f(y)} for x X. So our proof in this paper is different from [7]. Acknowledgements. This research was supported by grant no. MOST M of the Ministry of Science and Technology of the Republic of China. References [1] H. Brézis, F.E. Browder, A general principle on ordered sets in nonlinear functional analysis, Adv. Math., 21 (1976), no. 3, [2] A. Brøndsted, Fixed points and partial orders, Proc. Am. Math. Soc., 60 (1976), [3] F. E. Browder, On a theorem of Caristi and Kirk, in Proceedings of the Seminar on Fixed Point Theory and Its Applications, June 1975, 23-27, Academic Press, New York, [4] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc., 215 (1976),

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6 2294 Wei-Shih Du [16] L.-J. Lin, W.-S. Du, On maximal element theorems, variants of Ekeland s variational principle and their applications, Nonlinear Anal., 68 (2008), [17] L. Pasicki, A short proof of the Caristi theorem, Ann. Soc. Math. Polon. Series I: Comm. Math., 22 (1978), [18] T. Suzuki, Generalized Caristi s fixed point theorems by Bae and others, J. Math. Anal. Appl., 302 (2005), [19] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, [20] C.S. Wong, On a fixed point theorem of contractive type, Proc. Amer. Math. Soc., 57 (1976), no. 2, Received: June 29, 2016; Published: July 11, 2016