Cantor s intersection theorem for K-metric spaces with a solid cone and a contraction principle

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1 J. Fixe Point Theory Appl. 18 (2016) DOI /s Publishe online August 20, 2016 Journal of Fixe Point Theory 2016 The Author(s) an Applications This article is publishe with open access at Springerlink.com Cantor s intersection theorem for -metric spaces with a soli cone an a contraction principle Jacek Jachymski an Jakub lima With great respect an amiration for Professor Anrzej Granas Abstract. We establish an extension of Cantor s intersection theorem for a -metric space (X, ), where is a generalize metric taking values in a soli cone in a Banach space E. This generalizes a recent result of Alnafei, Raenović an Shahza (2011) obtaine for a -metric space over a soli strongly miniheral cone. Next we show that our Cantor s theorem yiels a special case of a generalization of Banach s contraction principle given very recently by Cvetković an Rakočević (2014): we assume that a mapping T satisfies the conition (T x, T y) Λ((x, y)) for x, y X, where is a partial orer inuce by, an Λ: E E is a linear positive operator with the spectral raius less than one. We also obtain new characterizations of convergence in the sense of Huang an Zhang in a -metric space. Mathematics Subject Classification. Primary 47H09, 47H10, 54H25; Seconary 46B40, 54E35. eywors. -metric space, cone metric space, soli cone, Cantor s intersection theorem, fixe point, spectral raius, contraction principle. 1. Introuction It is well known that the classical Banach fixe point principle can be erive from Cantor s intersection theorem. This observation is ue to Boy an Wong [3] an their proof can also be foun in [10, p. 8] or [11, p. 2]. Actually, Cantor s theorem has a number of applications in fixe point theory; see, e.g., the papers of Dugunji [8] on positive efinite functions, Goebel [9] on nonexpansive mappings, irk [22] on asymptotic contractions, Jachymski [16] on generalize contractions, or the monograph of Granas an Dugunji [12, pp. 17 an 27], where, in particular, an elegant proof of the Bishop Phelps theorem is given.

2 446 J. Jachymski an J. lima JFPTA On the other han, Granas an Horvath [13] (see also [12, pp ]) establishe the so-calle orer-theoretic Cantor theorem, from which they erive a lot of results relate to the Bishop Phelps theorem. Recently, another extension of Cantor s theorem was given by Alnafei, Raenović an Shahza [2], who use it to obtain a fixe point theorem for mappings on -metric spaces. Recall that a -metric space is a pair (X, ), where X is a nonempty set an is a vector-value function from X X to a close cone in a Banach space E, satisfying three well-known axioms of a metric with respect to the following partial orer in E: for a, b E, a b if an only if b a. (1.1) This notion was first introuce in 1934 by urepa [24], who use the term pseuoistance for. Following Zabreĭko we, however, use the term metric instea of pseuoistance since now the latter term has a ifferent meaning. -metric spaces were reiscovere in 2007 by Huang an Zhang [15] uner the name cone metric spaces. They also establishe an extension of Banach s contraction principle for mappings T satisfying the following conition: (Tx,Ty) λ(x, y) for x, y X, (1.2) where λ [0, 1). Subsequently, their result was generalize by a number of authors; see, e.g., the survey paper [19] an 100 references therein. Our purpose here is to establish an extension of Cantor s theorem for -metric spaces. Our result is more general than that given in [2] since we omit the extra assumption use in [2] that a cone is strongly miniheral, i.e., every subset of E which is boune from above with respect to the partial orer efine by (1.1) has a supremum; cf., e.g., [5, p. 219]. Moreover, our proof is completely ifferent: instea of aapting an argument from the classical proof as one in [2], we use a remetrization technique. This is precee by results on a characterization of some types of convergence in a Banach space E, inuce by a soli cone, i.e., the cone with a nonempty interior. Hence, as an immeiate consequence, we also get a characterization of convergence in the sense of Huang an Zhang [15] in a -metric space. At last, with the help of our Cantor s theorem, we prove an extension of Banach s contraction principle for mappings T satisfying the conition (Tx,Ty) Λ ( (x, y) ) for x, y X, (1.3) where Λ: E E is a linear boune operator, which is positive, i.e., Λ(), an r(λ), the spectral raius of Λ, is less than one. Clearly, (1.2) is a particular case of (1.3) with Λ := λi, where I is the ientity mapping on E. Contractive conitions of type (1.3) were stuie by many mathematicians, mainly from the former Soviet Union, starting from the paper of Perov [25], in which the case when E = R m was consiere. A more general fixe point theorem for such mappings may be foun in the monograph of rasnosel skiĭ et al. [23, pp ]. Here a cone is assume to be normal, i.e., inf { x + y : x, y, x = y =1 } > 0.

3 Vol. 18 (2016) Cantor s intersection theorem 447 In our theorem we allow to be nonnormal, however, is assume to be soli. In fact, as pointe out to us by the referee, the theorem is a special case of the result obtaine very recently by Cvetković an Rakočević [4], who consiere a more general contractive conition than (1.3). However, our argument use in the proof is completely ifferent from that in [4]. Extensions of Banach s contraction principle using conitions of type (1.3) have many interesting applications to functional or ifferential equations. In particular, in our opinion, the most elegant proof of the classical Picar Linelöf theorem is that one which uses Perov s fixe point theorem. The etails of this proof can be foun in our recent paper [18]; we were inspire by the paper [6]. In fact, the original proof is probably given in Perov s paper [25], however, it seems that this article is now unavailable. 2. On c-uniform convergence in E an the Huang Zhang convergence in X Throughout this section we assume that E is a Banach space an is a soli cone in E, i.e., int. Then the orere Banach space E is calle a rein space (see, e.g, [1, Definition 2.62]). Let (X, ) bea-metric space, i.e., : X X satisfies the three axioms of a metric with respect to the partial orer efine by (1.1). For a, b E, a b stans for b a int, an a b means that a b an a b. Huang an Zhang [15] introuce the following efinition of convergence in (X, ): a sequence (x n ) is -convergent to x X, if for any c 0, there is k N such that (x n,x) c for n k. We enote it by x n x. In a similar way, as in a metric setting, they efine Cauchy s sequences an then the completeness of (X, ). Recently, Du [7] an aelburg, Raenović an Rakočević [20] have prove (in ifferent ways) that there exists a realvalue metric ρ on X such that for any sequence (x n ), (x n ) is -convergent if an only if it is ρ-convergent, an (x n ) is a -Cauchy sequence if an only if it is a ρ-cauchy sequence. In particular, it means that the topology inuce by on X via the limit operator is metrizable. In this section we propose yet another approach to the problem of metrizability of (X, ). Namely, first we give a characterization of some type of convergence in a Banach space E. Then, as an immeiate consequence, we get the above metrizability result as well as other equivalent conitions for the convergence in the sense of Huang an Zhang in (X, ). Given a, b E with a b, we enote by [a, b] the orer interval, i.e., [a, b] := {c E : a c b}.

4 448 J. Jachymski an J. lima JFPTA For c, the ieal generate by c (see [1, Definition 2.53]) is the linear subspace E c := n N[ nc, nc]. We can efine the Minkowski functional on E c by setting a c := inf { λ>0:a [ λc, λc] } for a E c. (2.1) For the following result, see, e.g., [1, Theorem 2.55] an take into account that E is Archimeean (see [1, Definition 1.10]). Theorem 2.1. For any c, we have the following. (1) The Minkowski functional c is a monotone norm on E c. (2) The close unit ball in (E c, c ) coincies with the orer interval [ c, c]. (3) The cone E c is c -close. Now if c int, then by [1, Lemma 2.5], 0 int[ c, c], so [ c, c] is absorbing, an hence E c = E. Thus Theorem 2.1 an [27, Theorem 1.36] yiel the following corollary. Corollary 2.2. Let be a soli cone in a Banach space (E, ) an let c int. Then the Minkowski functional c is a monotone norm on E an is continuous with respect to the norm. Moreover, is c -close an [ c, c] ={a E : a c 1}. Remark 2.3. Let us observe that the Minkowski functional c is a norm on E only if c int. Inee, if it is a norm on E, then E c = E, so by [1, Lemma 2.54 (b)] c is an orer unit of E (cf. the efinition in [1, p. 5]) an by [1, Theorem 2.8], c is an interior point of. Now let c 0 an (a n ) a sequence in E. Following [1, Definition 2.56] c we say that (a n ) is c-uniformly convergent to a E (in symbols, a n a), if for any ε>0, there is k N such that εc a n a εc for n k. c Lemma 2.4. a n a if an only if a n a E c for sufficiently large n an a n a c 0. Proof. Observe that the conition εc a n a εc is equivalent to (1/ε)(a n a) [ c, c], which in turn means in view of Theorem 2.1(2) that 1 ε (a n a) 1, c i.e., a n a c ε. This yiels the result. Inspire by the Huang Zhang efinition, we say that a sequence (a n ) in E with a soli cone is -convergent to a E (in symbols, a n a), if for any c 0, there is k N such that c a n a c for n k.

5 Vol. 18 (2016) Cantor s intersection theorem 449 Below we give a list of equivalent conitions for this convergence. The last of them is relate to the notion of a funamental sequence in the antorovich sense (see [28, p. 830]). Theorem 2.5. Let be a soli cone in a Banach space E, (a n ) a sequence in E an a E. The following statements are equivalent: (i) a n a; (ii) for any c 0, there is k N such that c a n a c for n k; (iii) for any c 0, there is k N such that a n a c 1 for n k; c (iv) for any c 0, a n a; (v) for any c 0, a n a c 0; c (vi) there exists c 0 such that a n a; (vii) there exists c 0 such that a n a c 0; (viii) there exists a sequence (c n ) in such that c n 0 an c n a n a c n for any n N. Proof. (i) (ii) is obvious since a b implies a b for a, b E. (ii) (iii) follows immeiately from Corollary 2.2, by which the conition c a n a c is equivalent to a n a c 1. (ii) (iv): Let c 0 an ε>0. Then εc 0, so by (ii), there is k N such that εc a n a εc for n k. That means (iv) hols. (iv) (v) an (vi) (vii) follow from Lemma 2.4. (iv) (vi) is obvious. c (vi) (viii): By (vi), there is c 0 such that a n a. Hence there exists an increasing sequence (k n ) of positive integers such that 1 n c a j a 1 n c for j>k n. We have that for j =1,...,k 1, a j a E = n N [ nc, nc], so there is p N such that pc a j a pc for j =1,...,k 1. Set c n := pc for n =1,...,k 1, an c n := (1/m)c for m N an n N with k m <n k m+1. It is easily seen that (c n ) has the properties as in (viii). (viii) (i): Let (c n ) be as in (viii). Fix c 0. Then (1/2)c 0, so 0 int[( 1/2)c, (1/2)c]. Hence, since c n 0, there is k N such that

6 450 J. Jachymski an J. lima JFPTA ( 1/2)c c n (1/2)c for n k, so by (viii), we obtain that Thus (i) hols. c 1 2 c a n a 1 c c for n k. 2 Now let (X, ) be a -metric space with a soli cone. For any c 0, set ρ c (x, y) := (x, y) c for x, y X. (2.2) It follows from Corollary 2.2 that ρ c is a (real-value) metric on X. Since the Huang Zhang convergence of a sequence (x n ) to x X means that (x n,x) 0, Theorem 2.5 yiels the following corollary. Corollary 2.6. Let (X, ) be a -metric space over a soli cone, (x n ) a sequence in X an x X. The following statements are equivalent: (i) x n x (the Huang Zhang convergence); (ii) for any c 0, there is k N such that (x n,x) c for n k; (iii) for any c 0, there is k N such that ρ c (x n,x) 1 for n k; (iv) for any c 0, (x n,x) c 0; (v) for any c 0, ρ c (x n,x) 0; (vi) there exists c 0 such that (x n,x) c 0; (vii) there exists c 0 such that ρ c (x n,x) 0; (viii) there exists a sequence (c n ) in such that c n 0 an (x n,x) c n for any n N. Corollary 2.6 shows that a -metric is equivalent to the metric ρ c for any c 0. It turns out that an ρ c are also Cauchy equivalent (i.e., every -Cauchy sequence is a ρ c -Cauchy sequence an vice versa) as state in the following corollary. Corollary 2.7. Let (X, ) be a -metric space over a soli cone in E an (x n ) a sequence in X. The following statements are equivalent: (i) (x n ) is a -Cauchy sequence; (ii) for any c 0, there is k N such that (x n,x m ) c for n, m k; (iii) for any subsequence (x kn ) of (x n ), (x n,x kn ) 0; (iv) for any c 0, (x n ) is a ρ c -Cauchy sequence; (v) there exists c 0 such that (x n ) is a ρ c -Cauchy sequence; (vi) there exists a sequence (c n ) in E such that c n 0 an (x i,x j ) c n for i, j n.

7 Vol. 18 (2016) Cantor s intersection theorem 451 Proof. (i) (ii) an (ii) (iii) are obvious. (iii) (iv) an (iv) (v) follow from Theorem 2.5, if we invoke the fact that (x n ) is a ρ c -Cauchy sequence if an only if ρ c (x n,x kn ) 0 for any subsequence (x kn )of(x n ). To show (iii) (i) suppose, on the contrary, that there exist c 0 an increasing sequences (k n ) an (m n ) of positive integers such that the conition (x kn,x mn ) c oes not hol for any n N. By (iii), we have that (x kn,x mn ) (x kn,x n )+(x n,x mn ) c for sufficiently large n, which yiels a contraiction. Finally, (i) (vi) can be prove by a similar argument as in the proof of (vi) (viii) an (viii) (i) of Theorem 2.5. Let us notice that Corollary 2.7 also implies that a -metric an a metric ρ c are equivalent. This is a consequence of the following result, a particular case of which was prove in our paper [18]. Proposition 2.8. Let an ρ be -metrics or real-value metrics on X. If an ρ are Cauchy equivalent, then we have the following: (1) an ρ are equivalent; (2) (X, ) is complete if an only if (X, ρ) is complete. Proof. For example, assume that is a -metric an ρ is a real-value metric. Let (x n ) be -convergent to x X. For n N, set y 2n 1 := x n an y 2n := x. Then y n x, so by [15, Lemma 3], (yn ) is a -Cauchy sequence. By hypothesis, (y n ) is a ρ-cauchy sequence, so in particular, ρ(y 2n 1,y 2n ) 0, which means that (x n ) is ρ-convergent to x. The same argument shows that ρ-convergence implies -convergence. Now statement (2) is obvious. 3. Cantor s intersection theorem for -metric spaces In a -metric space (X, ) it is natural to consier the following two notions of bouneness of a set. We say that A X is orer-boune if there exists c such that (x, y) c for x, y A. A is calle norm-boune if its norm-iameter iam A := sup (x, y) x,y A is finite. In general, the two notions o not coincie. However, some connections between them can be establishe uner aitional assumptions on a cone as will be shown in what follows. Let us notice that if A is orer-boune an a cone is strongly miniheral, then the orer-iameter of A can be efine as iam A := sup{(x, y) :x, y A}.

8 452 J. Jachymski an J. lima JFPTA In this case, Cantor s intersection theorem for -metric spaces can be formulate as its classical version. Recently, this has been one by Alnafei, Raenović an Shahza [2] in the following form. Theorem 3.1. Let (X, ) be a -metric space over a soli an strongly miniheral cone. Then (X, ) is complete if an only if any ecreasing sequence (A n ) of nonempty close an orer-boune subsets of X, with iam A n 0, has a nonempty intersection. However, in general, it is not possible to attribute the orer-iameter to any orer-boune set as shown in the following example. Example 3.2. Let E := C([0, 2]) be enowe with the max-norm an let be the positive cone in E, i.e., := {f E : f 0}. It is known that is not strongly miniheral (cf. [14, Example 1.3.1, p. 12]). Set X := E an for f,g X, (f,g) := f g. It is easily seen that is a -metric. Set A := { f C([0, 2]) : 0 f(t) 1 for t [0, 1] an 0 f(t) 2 for t (1, 2] }. Then A X an (f,g) h 0 for f,g A, where h 0 (t) := 2 for t [0, 2], so A is orer-boune. It is easily seen that {(f,g) :f,g A} = A. An elementary argument shows that sup A oes not exist. Thus the set {(f,g) :f,g A} has no supremum. Fortunately, Cantor s theorem can be formulate without referring to iameters of sets. We omit an obvious proof of the following proposition. Proposition 3.3. Let (X, ρ) be a metric space an let (A n ) be a sequence of boune subsets of X. Then iam A n 0 if an only if there exists a sequence (α n ) of reals such that α n 0 an ρ(x, y) α n for x, y A n an n N. In a -metric setting we may consier the following two versions of the latter conition of Proposition 3.3, which, however, turn out to be equivalent. Proposition 3.4. Let (X, ) be a -metric space over a soli cone in a Banach space E, an let (A n ) be a sequence of subsets of X. The following statements are equivalent: (i) there exists a sequence (c n ) in E such that c n 0 an (x, y) cn for x, y A n an n N; (ii) there exists a sequence (c n ) in E such that c n 0 an (x, y) c n for x, y A n an n N. Proof. (i) (ii): Assume that (c n ) is as in (i). By Theorem 2.5 ((i) (viii)), there exists a sequence ( n ) such that n 0 an c n n. Obviously, (x, y) n for x, y A n an n N, so (ii) hols.

9 Vol. 18 (2016) Cantor s intersection theorem 453 (ii) (i): Now let (c n ) be as in (ii). By Theorem 2.5 ((viii) (i)), with a n := c n an a := 0, we get that so (i) hols. c n 0, It turns out that the convergence of norm-iameters of sets to 0 is a stronger assumption than (i) of Proposition 3.4. Proposition 3.5. Let (X, ) be a -metric space over a soli cone in X. Then we have the following. (1) If a set A X is norm-boune, then it is orer-boune. (2) If (A n ) is a sequence of subsets of X an iam A n = sup (x, y) 0, x,y A n then there exists a sequence (c n ) in E such that c n 0 an (x, y) c n for x, y A n an n N. Proof. Let A X an iam A<. Fix c 0. Then for some α>0, the close ball B(c, α) is containe in. Hence for any λ>0, λb(c, α) =B(λc, λα). In particular, for λ := iam A/α we get that ( ) iam A B c, iam A. α Hence, if a E an a iam A, then iam A c a, α i.e., a (iam A/α)c. This yiels (x, y) iam A c for x, y A. α Now assume that A n X for n N an iam A n 0. If c 0 an α is as above, then by the proof of point (1), we have that (x, y) c n for x, y A n an n N, where c n := (iam A n /α)c. Clearly, c n 0 which completes the proof. The following example shows that in general the converse of Proposition 3.5(2) is false. Example 3.6. Let E := C 1 ([0, 1]) be enowe with the norm f := max f(t) + max f (t) for f E. t [0,1] t [0,1] Let be the positive cone in E, X := an for f,g X, { f + g if f =g, (f,g) := 0 if f = g.

10 454 J. Jachymski an J. lima JFPTA For n N, set A n := {f X : (f,0) c n }, where c n (t) := t n /n for t [0, 1]. Then 0 A n an c n A n, so iam A n (0,c n ) = c n = 1 n +1, an hence iam A n 0. On the other han, for f,g A n, (f,g) 2c n n, where n (t) := 2/n for t [0, 1]. Obviously, n 0. Observe that the cone in Example 3.6 is not normal. In fact, it turns out that for any nonnormal cone, there exist a -metric space an a sequence of its subsets with the properties as in Example 3.6. Namely, we have the following characterization of normal cones. Proposition 3.7. Let be an arbitrary cone in a Banach space E. The following statements are equivalent: (i) is normal; (ii) for any -metric space (X, ) an any A X, if A is orer-boune, then it is norm-boune; (iii) for any -metric space (X, ) an any ecreasing sequence (A n ) of subsets of X, if there exists a sequence (c n ) in E such that c n 0 an (x, y) c n for x, y A n an n N, then iam A n 0. Proof. (i) (ii) an (i) (iii) follow from the fact that is normal if an only if the norm on E is semimonotone (cf. [1, Theorem 2.38]), i.e., there is γ>0 such that for a, b E, 0 a b implies a γ b. Hence, if (A n ) is as in (iii), then (x, y) γ c n for x, y A n an n N, so iam A n 0. Now we show simultaneously implications (ii) (i) an (iii) (i). Our proof is partially inspire by the proof of [1, Theorem 2.40]. Suppose, on the contrary, that is not normal. Then there exist sequences (a n ) an (b n ) in E such that for n N, 0 a n b n an a n >n 3 b n. Hence a n > 0, so a n 0, which yiels b n 0. Thus we may set for n N, a n := 1 n 2 b n a n an b 1 n := n 2 b n b n. Then 0 a n b n. The series n=1 b n is absolutely convergent in a Banach space E, so we may efine b n := b i for n N. i=n Then b n 0an b n since is close. Now set X := an for x, y X, { x + y if x =y, (x, y) := 0 if x = y.

11 Vol. 18 (2016) Cantor s intersection theorem 455 Then is a -metric on X. For n N, efine A n := {a k : k n} {0}. Clearly, (A n ) is ecreasing an for any k n, iam A n (a k, 0) = a k = a k k 2 b k > k, so iam A n =. On the other han, if x, y A n, then either (x, y) = 0, (x, y) =a k, or (x, y) =a k + a m for some k, m n with k m. In each case, (x, y) b k + b m b n, which contraicts (iii) since b n 0 an iam A n 0. Moreover, each A n is orer-boune an is not norm-boune, which in turn contraicts (ii). Given x X an a, we efine the close ball B(x, a) := {y X : (x, y) a}. The following two results will be use in the proof of Cantor s theorem. Proposition 3.8. Let (X, ) be a -metric space over a soli cone. Then every ball B(x, a) is -close. Proof. Let y n B(x, a) for n N an y n y. Then (x, y) (x, y n )+(y n,y) a + (y n,y). (3.1) Fix c 0. By Theorem 2.5 ((i) (v)), (y n,y) 0 implies that (y n,y) c 0. Since by Corollary 2.2, is c -close, (3.1) yiels that (x, y) a, i.e., y B(x, a). Proposition 3.9. Let (X, ) be a -metric space over a soli cone an let (x n ) be a -Cauchy sequence in X. Then (x n ) is -convergent if an only if it contains a -convergent subsequence. Proof. The part only if is trivial. So assume that x kn x. Then 0 (x n,x) ( ) ( x n,x kn + xkn,x ). (3.2) Fix c 0. By Theorem 2.5 ((i) (v)), imply that (x n,x kn ) 0 an (x kn,x) 0 ( x n,x kn ) + ( xkn,x ) c 0. By Corollary 2.2, is c -normal, so (3.2) implies that (x n,x) c 0. Now Corollary 2.6 ((vii) (i)) yiels that x n x. The following is Cantor s intersection theorem for -metric spaces.

12 456 J. Jachymski an J. lima JFPTA Theorem Let (X, ) be a -metric space over a soli cone in a Banach space E. The following statements are equivalent: (i) (X, ) is complete; (ii) every ecreasing sequence (A n ) of nonempty close subsets of X such that there exists a sequence (c n ) in E with c n 0 an (x, y) cn for x, y A n an n N, has a nonempty intersection an n N A n is a singleton; (iii) every ecreasing sequence (A n ) of nonempty close subsets of X such that there exists a sequence (c n ) in E with c n 0 an (x, y) c n for x, y A n an n N, has a nonempty intersection an n N A n is a singleton; (iv) every ecreasing sequence (B(x n,a n )) of close balls in X with a n 0 an a n 0 has a nonempty intersection. Proof. (i) (ii): Let a sequence (A n ) be as in (ii). By Theorem 2.5 ((i) (viii)), there exists a sequence ( n ) in E such that n 0 an c n n. Then we have (x, y) n for x, y A n. (3.3) Fix c 0. By Corollary 2.2, c is monotone an -continuous, so (3.3) yiels that (x, y) c n c for x, y A n, which implies that the iameters of sets A n with respect to the metric ρ c efine by (2.2) converge to 0. Since by Corollary 2.7, an ρ c are Cauchy equivalent, we may conclue using Proposition 2.8 that each A n is ρ c -close an (X, ρ c ) is complete. Thus, by the classical Cantor s intersection theorem, n N A n is a singleton. The equivalence (ii) (iii) follows from Proposition 3.4. (iii) (iv): Let a sequence (B(x n,a n )) be as in (iv). By Proposition 3.8, each B(x n,a n ) is -close. Moreover, for x, y B(x n,a n ), (x, y) (x, x n )+(x n,y) 2a n. Since a n 0, (iii) yiels that n N B(x n,a n ) is nonempty. (iv) (i): Here we use a similar argument as in the classical proof of Cantor s theorem. Let (x n ) be a Cauchy sequence in (X, ). Fix c 0. Then there exists a subsequence (x kn ) such that ( ) c x kn,x kn+1 2 n+1. Set B n := B(x kn, c/2 n ) for n N. If x B n+1, then ( ) ( ) ( ) c x, x kn x, xkn+1 + xkn+1,x kn 2 n+1 + c 2 n+1 = c 2 n, so x B n, an hence B n+1 B n. Thus (iv) yiels the existence of x 0 in n N B n. Then (x 0,x kn ) c 2 n,

13 Vol. 18 (2016) Cantor s intersection theorem 457 so by Corollary 2.6, x kn x 0. Now Proposition 3.9 ensures that also (x n ) converges to x 0. Consequently, (X, ) is complete. 4. A contraction principle for -metric spaces via Cantor s theorem Janković, aelburg an Raenović [19] have observe that a number of fixe point theorems for mappings on -metric spaces can be erive from the corresponing results in metric spaces with the help of the following rein s characterization of normal cones (cf. [1, Theorem 2.38]). Theorem 4.1 (rein). A cone in a Banach space E is normal if an only if E amits an equivalent monotone norm. Following their approach, we present here a proof of the Huang Zhang theorem [15] via the classical contraction principle. A similar argument can be use in proofs of many other fixe point results mentione in [19]. Theorem 4.2 (Huang Zhang). Let (X, ) be a complete -metric space over a soli normal cone in a Banach space E. Let a mapping T : X X be such that for some λ [0, 1), (T x, T y) λ(x, y) for x, y X. Then T has a unique fixe point x an for any x 0 X, T n x 0 x. Proof. Let be an equivalent monotone norm on E. Set ρ(x, y) := (x, y) for x, y X. Then ρ is a metric on X an it can easily be shown, using the monotonicity of the norm, that for any (x n ) in X, (x n ) is a -Cauchy sequence if an only if (x n,x m ) 0 as n, m, i.e., an ρ are Cauchy equivalent. Hence, by Proposition 2.8, an ρ are equivalent, an (X, ρ) is complete. Moreover, the monotonicity of implies that for x, y X, (Tx,Ty) λ(x, y) = λ (x, y), i.e., T is a ρ-contraction. Now it suffices to apply the classical contraction principle. Rezapour an Hamlbarani [26] have obtaine an extension of Theorem 4.2 by omitting the assumption of normality of. Then, however, as observe by aelburg, Raenović an Rakočević [20], it is possible to repeat the above proof only replacing the norm by the Minkowski functional c efine by (2.1), where c 0. So also this result is subsume by the classical Banach s fixe point theorem. (Let us also notice that interesting applications of the Minkowski functional technique to Caristi-type fixe point results were foun recently by hamsi an Wojciechowski [21].)

14 458 J. Jachymski an J. lima JFPTA However, the problem is more elicate if we consier a mapping T satisfying conition (1.3), i.e., (Tx,Ty) Λ ( (x, y) ) for x, y X, where Λ is a linear continuous positive operator on E with the spectral raius less than 1. If is normal an ρ is a metric from the proof of Theorem 4.2, then we obtain only that ρ(t x, T y) Λ ρ(x, y). In general, Λ 1, but since r(λ)<1, there exists p N such that Λ p <1. Then, by monotonicity of Λ, we may infer that T p is a ρ-contraction. Since T is continuous, [17, Theorem 2.1] implies that there exists a complete metric ρ equivalent to ρ such that T is a ρ -contraction, so the classical contraction principle is applicable. In fact, this argument can be extene to mappings satisfying (1.3) with a nonlinear operator Λ (see [18, Theorem 6]). Another argument (in the linear case) is given in [6]. It seems that for nonnormal cones the problem is yet more ifficult. In this case, using a metric ρ inuce on X by the norm c, we get from (1.3) by monotonicity of c that for any n N, ρ ( T n x, T n y ) Λ n ((x, y)) c Λ n c ρ(x, y) (it can be shown that Λ is c -continuous), but now it is not clear if the spectral raius of Λ with respect to c is less than 1 since by [1, Theorem 2.63], the norms c an are not equivalent an, moreover, (E, c ) is not a Banach space. Here we show that a fixe point theorem for mappings on a -metric space over a soli (not necessarily normal) cone can be erive from our Cantor s intersection theorem. The result is a special case of the Cvetković Rakočević theorem [4], in which a mapping T is such that for any x, y X, there exists u {(x, y),(x, T x),(y, Ty),(x, T y),(y, Tx)} with (T x, T y) Λ(u). Theorem 4.3. Let (X, ) be a complete -metric space over a soli cone in a Banach space E. Let T : X X be a mapping such that (Tx,Ty) Λ ( (x, y) ) for x, y X, where Λ: E E is a linear positive operator with the spectral raius r(λ) less than 1. Then T has a unique fixe point x an for any x 0 X, T n x 0 x. The proof of Theorem 4.3 will be precee by some auxiliary results. Also, let us notice that an operator Λ as in Theorem 4.3 is automatically continuous, so r(λ) is well efine. Inee, by [5, Proposition 19.1], every soli cone is generating, i.e., = E, so by [1, Theorem 2.32], every linear positive operator from E to E is then continuous. Janković, aelburg an Raenović [19, Example 6.1] have presente that a -metric nee not be continuous (even with respect to one variable):

15 Vol. 18 (2016) Cantor s intersection theorem 459 it may happen that x n x, but (x n,x) 0. However, is continuous in other sense accoring to the following proposition. Proposition 4.4. Let (X, ) be a -metric space over a soli cone, x, y X an let (x n ), (y n ) be sequences in X. If then x n x an y n y, (x n,y n ) (x, y). Proof. By Corollary 2.6 ((i) (viii)), there exist sequences (c n )an( n ) in E such that c n 0, n 0, (x n,x) c n, (y n,y) n. Hence, by the triangle inequality, we get (x n,y n ) (x, y) (x n,x)+(y n,y) c n + n. Interchanging x n with x an y n with y, an multiplying by 1 yiels c n n (x n,x) (y n,y) (x n,y n ) (x, y). Thus we obtain that (c n + n ) (x n,y n ) (x, y) c n + n. Since c n + n 0, Theorem 2.5 ((viii) (i)) implies that (x n,y n ) (x, y). Lemma 4.5. Let T be as in Theorem 4.3. Then T is -continuous. Proof. Let x n x. By Corollary 2.6 ((i) (viii)), there exists a sequence (c n ) such that c n 0 an (x n,x) c n. By hypothesis, Λ is positive, so is monotone. Hence we get (Tx n,tx) Λ ( (x n,x) ) Λ(c n ). Since Λ is -continuous, we have that Λ(c n ) 0, so by Corollary 2.6 ((viii) (i)), Tx n T x. Lemma 4.6. Let T be as in Theorem 4.3 an c 0. Set A := {x X : (x, T x) c}. Then A is nonempty an close, an for any x, y A, (x, y) 2(I Λ) 1 (c). (4.1)

16 460 J. Jachymski an J. lima JFPTA Proof. We show that A is nonempty. Fix x 0 X. By monotonicity of Λ, ( T n x 0,T n+1 x 0 ) Λ n ( (x 0,Tx 0 ) ) for n N. Since r(λ) < 1, we have that Λ n ( (x 0,Tx 0 ) ) 0. Thus by Theorem 2.5 ((viii) (i)), we get that ( T n x 0,T n+1 ) x 0 0. Hence there is p N such that T p x 0 A. We show that A is close. Set φ(x) := (x, T x) for x X. By Proposition 4.4 an Lemma 4.5, φ is continuous in the sense that if then x n φ(x n ) x, φ(x). By Theorem 2.5, the latter convergence is equivalent to the convergence φ(x n ) φ(x) c 0. The continuity of φ implies the closeness of A since A = φ 1 ([ c, c]) an by Corollary 2.2, [ c, c] is c -close. Finally, fix x, y A. Then (x, y) (x, T x)+(tx,ty)+(y, Ty) 2c +Λ ( (x, y) ) an hence (I Λ) ( (x, y) ) 2c. (4.2) Since r(λ) < 1, the operator I Λ is invertible an (I Λ) 1 can be expresse by the Neumann series: (I Λ) 1 = Λ n. Since Λ is increasing, so is each iterate Λ n. Hence, given a, b E with a b, we may infer that n n Λ k a Λ k b. k=0 n=0 k=0 By closeness of, letting n ten to, we get Λ k a Λ k b, k=0 i.e., (I Λ) 1 a (I Λ) 1 b, which means that (I Λ) 1 is increasing. Hence, by (4.2), we obtain that (4.1) hols. k=0

17 Vol. 18 (2016) Cantor s intersection theorem 461 Proof of Theorem 4.3. Fix c 0 an for n N, set A n := {x X : (x, T x) 1n } c. Clearly, (A n ) is ecreasing. For any n N, (1/n)c 0, so by Lemma 4.6, each A n is nonempty an close, an (x, y) 2 n (I Λ) 1 (c) for x, y A n. Set c n := (2/n)(I Λ) 1 (c). Clearly, c n 0, so Theorem 3.10 implies that A n = {x } for some x X. n N By closeness of, we easily get that Fix T = n N A n, so x is the unique fixe point of T. Now fix x 0 X. For any n N, we have ( T n x 0,x ) = ( T n x 0,T n x ) Λ n ( (x 0,x ) ). Since Λ n ((x 0,x )) 0, Corollary 2.6 yiels that T n x 0 x. Finally, let us notice that it is also possible to give another proof of Theorem 4.3 by moifying the classical proof of the contraction principle. Acknowlegment We are grateful to the referee for calling our attention to the papers [4, 21]. References [1] C. D. Aliprantis an R. Tourky, Cones an Duality. Gra. Stu. Math. 84, Amer. Math. Soc., Provience, RI, [2] S. H. Alnafei, S. Raenović an N. Shahza, Fixe point theorems for mappings with convex iminishing iameters on cone metric spaces. Appl. Math. Lett. 24 (2011), [3] D. W. Boy an J. S. W. Wong, On nonlinear contractions. Proc. Amer. Math. Soc. 20 (1969), [4] M. Cvetković an Rakočević, Quasi-contraction of Perov type. Appl. Math. Comput. 237 (2014), [5]. Deimling, Nonlinear Functional Analysis. Springer-Verlag, Berlin, [6] E. De Pascale an L. De Pascale, Fixe points for some non-obviously contractive operators. Proc. Amer. Math. Soc. 130 (2002), [7] W.-S. Du, A note on cone metric fixe point theory an its equivalence. Nonlinear Anal. 72 (2010), [8] J. Dugunji, Positive efinite functions an coinciences. Fun. Math. 90 (1975/76),

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19 Vol. 18 (2016) Cantor s intersection theorem 463 [28] P. P. Zabreĭko, -metric an -norme linear spaces: Survey. Collect. Math. 48 (1997), Jacek Jachymski Institute of Mathematics Lóź University of Technology Wólczańska Lóź Polan jacek.jachymski@p.loz.pl Jakub lima Institute of Mathematics Lóź University of Technology Wólczańska Lóź Polan jakub.klima.87@gmail.com Open Access This article is istribute uner the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricte use, istribution, an reprouction in any meium, provie you give appropriate creit to the original author(s) an the source, provie a link to the Creative Commons license, an inicate if changes were mae.