Some results on fixed points of multifunctions on abstract metric spaces

Mathematical and Computer Modelling 53 (2011) 746–754

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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm

Some results on fixed points of multifunctions on abstract metric spaces Stojan Radenović a , Zoran Kadelburg b,∗ a

University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Beograd, Serbia

b

University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Beograd, Serbia

article

info

Article history: Received 12 June 2010 Accepted 15 October 2010 Keywords: Cone metric space Normal and nonnormal cone Multifunctions Symmetric space

abstract Recently, Kunze et al. [H.E. Kunze, D. La Torre, E.R. Vrscay, Contraction multifunctions, fixed point inclusions and iterated multifunction system, J. Math. Anal. Appl. 330 (2007) 159–173] proved some fixed point results for multifunctions in metric spaces. Rezapour and Haghi [Sh. Rezapour, R.H. Haghi, Fixed point of multifunctions on cone metric spaces, Numer. Funct. Anal. Optim. 30 (2009) 825–832] adapted these results to the case of abstract (cone) metric spaces when the underlying cone is normal with normal constant M = 1. The aim of this paper is to show that these results remain valid in the case when M > 1. Introducing new contraction conditions our results generalize fixed point theorems of Covitz and Nadler, Kunze et al. and Rezapour and Haghi. An example is given to distinguish our results from the known ones. In addition, the case when two mappings are considered is treated. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton’s approximation method [1–4] and in optimization theory [5]. K -metric and K -normed spaces were introduced in the mid20th century by using an ordered Banach space instead of the set of real numbers, as the codomain for a metric ([2]; see also [3,4]). Huang and Zhang [6] re-introduced such spaces under the name of cone metric spaces, but went further, defining convergent and Cauchy sequences in terms of interior points of the underlying cone. These and other authors proved some fixed point and common fixed point theorems for contractive-type mappings in cone metric spaces. Fixed point problems for multifunctions (set-valued functions) in cone metric spaces were treated, e.g., in [7–12]. In particular, in [10], the authors proved two fixed point results for multifunctions that are cone metric versions of results from [13] for metric spaces. These results were obtained under the assumption that the underlying cone is normal with the normal constant M = 1. We shall prove here more general results. The generalization goes in three directions. Firstly, we shall prove that results of [10] remain valid for all M ≥ 1. Secondly, our contraction conditions will be weaker than those in [10]. Finally, we shall treat the case when two mappings are concerned. We shall use our results on cone symmetric spaces from [14]. An example is given to distinguish our results from the known ones. Note that it was shown recently in [15–17] that some of the fixed point results in cone metric spaces can be directly reduced to the respective metric results. However, the results of the present paper do not fall into this category, since they are new even in the context of metric spaces.

∗

Corresponding author. Tel.: +381 11 3234967; fax: +381 11 3036819. E-mail addresses: [email protected] (S. Radenović), [email protected] (Z. Kadelburg).

0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.10.012

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2. Preliminaries 2.1. Normal cones and cone metric spaces Consistent with [6], the following definitions and results will be needed in what follows. Let E be a real Banach space. A subset P of E is called a cone if: (a) P is closed, nonempty and P ̸= {0}; (b) a, b ∈ R, a, b ≥ 0, and x, y ∈ P imply ax + by ∈ P; (c) P ∩ (−P ) = {0}. Given a cone P ⊂ E, a partial ordering ≼ with respect to P is defined by x ≼ y if and only if y − x ∈ P. We shall write x ≺ y to indicate that x ≼ y but x ̸= y, while x ≪ y will stand for y − x ∈ int P (interior of P). Definition 2.1. A cone P in a Banach space E is called: 1◦ normal if inf{‖x + y‖ : x, y ∈ P , ‖x‖ = ‖y‖ = 1} > 0; 2 semi-monotone if there exists M > 0 such that for all x, y ∈ E, ◦

0 ≼ x ≼ y implies ‖x‖ ≤ M ‖y‖;

(2.1)

3◦ monotone if for all x, y ∈ E, 0 ≼ x ≼ y implies ‖x‖ ≤ ‖y‖, i.e. it is semi-monotone with M = 1. The following lemma contains results about cones in ordered Banach spaces that are rather old (dating 1940s). It is interesting that most of the authors that worked with cone spaces after 2007 do not mention this result, with the help of which most of the results in spaces with normal cones can be deduced from the known ones. Lemma 2.2. The following conditions are equivalent for a cone P in a Banach space E: 1◦ P is normal; 2◦ For arbitrary sequences {xn }, {yn }, {zn } in E,

(∀n)xn ≼ yn ≼ zn and

lim xn = lim zn = x imply lim yn = x;

n→∞

n→∞

n→∞

◦

3 P is semi-monotone; 4◦ There exists a norm ‖ · ‖1 on E, equivalent with the given norm ‖ · ‖, such that the cone P is monotone w.r.t. ‖ · ‖1 . Proof of this important assertion can be found in [18–20]. The least positive number M satisfying the inequality in (2.1) is called the normal constant of P. It is clear that M ≥ 1. From [5] we know that there exists a cone P which is not normal but satisfies int P ̸= ∅ (cones satisfying the last condition are called solid). Remark 2.3. Recall that equivalent norms generate the same convergent and Cauchy sequences, and also the same collections of bounded, resp. compact subsets. Also, the interior of the cone is the same in both equivalent norms. Hence, taking the previous lemma into account, we can always assume that the given normal cone has the normal constant M = 1 (otherwise we can renorm the space E). See also Remarks 2.6 and 2.9 further on. Definition 2.4 ([6]). Let X be a nonempty set. Suppose that the mapping d : X × X → E satisfies: (d1) 0 ≼ d(x, y) for all x, y ∈ X and d(x, y) = 0 if and only if x = y; (d2) d(x, y) = d(y, x) for all x, y ∈ X ; (d3) d(x, y) ≼ d(x, z ) + d(z , y) for all x, y, z ∈ X . Then d is called a cone metric on X and (X , d) is called a cone metric space. The concept of a cone metric space is more general than that of a metric space. Definition 2.5 ([6]). Let (X , d) be a cone metric space and {xn } a sequence in X . We say that {xn } is: (e) a Cauchy sequence if for every c in E with 0 ≪ c, there is an integer N such that for all n, m > N, d(xn , xm ) ≪ c; (f) a convergent sequence if for every c in E with 0 ≪ c, there is an N such that for all n > N , d(xn , x) ≪ c for some fixed x in X . A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X . It is known that in the case of a normal cone, {xn } converges to x ∈ X if and only if ‖d(xn , x)‖ → 0 as n → ∞.

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Remark 2.6. Taking into account Lemma 2.2, it follows that results for normal cone metric spaces can be derived from results for metric spaces. Namely, if d is a cone metric on X and the norm ‖ · ‖ is monotone on the cone, then the composition ‖ · ‖ ◦ d is a usual metric on X . Clearly, nonexpansive and contractive (with respect to d) self-mappings of X are nonexpansive and contractive (respectively) with respect to the metric ‖ · ‖ ◦ d. This also shows the importance of the notion of a normal cone, used firstly by Kre˘ın (see, e.g. [21]) in the theory of positive operators on ordered Banach spaces. A lot of fixed point results in the frame of normal cone metric spaces (or K -metric spaces as they were called earlier) can be derived in an easier manner—in the same way as in the frame of metric spaces. In the next subsection we shall show how it can be done effectively. 2.2. Symmetric and cone symmetric spaces We shall recall in this subsection some results about cone symmetric spaces from [14] which will be used in what follows. Definition 2.7 ([14]). Let X be a nonempty set. Suppose that a mapping d : X × X → E satisfies: (s1) 0 ≼ d(x, y), for all x, y ∈ X and d(x, y) = 0 if and only if x = y; (s2) d(x, y) = d(y, x), for all x, y ∈ X . Then d is called a cone symmetric on X , and (X , d) is called a cone symmetric space. Obviously, each cone metric space is a cone symmetric space, but the converse does not hold. Definitions of convergent and Cauchy sequences are the same as for cone metric spaces. Let d be a cone symmetric on a nonempty set X . For x ∈ X and c ∈ P , 0 ≪ c, let Bc (x) = {y ∈ X : d(x, y) ≪ c }. The topology td on X is defined as follows: U ∈ td if and only if for each x ∈ U, there exists c ∈ P , 0 ≪ c, such that Bc (x) ⊂ U. A subset S of X is a neighborhood of x ∈ X if there exists U ∈ td such that x ∈ U ⊂ S. For the given cone symmetric space (X , d) one can construct a symmetric space (X , D) where ‘‘symmetric’’ (in the sense of [22]) D : X × X → R is given by D(x, y) = ‖d(x, y)‖. Definition 2.8 ([14]). The space (X , D) is called the symmetric space associated with the cone symmetric space (X , d). Remark 2.9. In the case when d is a cone metric and the underlying cone is normal, the triangle inequality d(x, y) ≼ d(x, z ) + d(z , y) for each x, y, z ∈ X , implies that the symmetric D = ‖d‖ satisfies the condition D(x, y) = ‖d(x, y)‖ ≤ M ‖d(x, z ) + d(z , y)‖ ≤ M (D(x, z ) + D(z , y)) where M ≥ 1 is the normal constant of P. So, the symmetric D satisfies (s3): for each x, y, z ∈ X D(x, y) ≤ M (D(x, z ) + D(z , y)). Hence, in this case the symmetric space (X , D) is ‘‘almost’’ a metric space. It is a metric space if the normal constant M = 1 (recall that Lemma 2.2 implies that this can always be obtained by renorming the space E). Now, for x ∈ X and ε > 0 let Bε (x) = {y ∈ X : D(y, x) < ε}. Let tD be the topology on X generated by the balls of the form Bε (x), x ∈ X , ε > 0. Theorem 2.10 ([14]). Let (X , d) be a cone metric space with a solid normal cone P and let D be the associated symmetric. Then td = tD . In other words, the spaces (X , d) and (X , D) have the same collections of open, closed and compact sets, and also the same convergent and Cauchy sequences and the same continuous functions. 2.3. Some known results Let us first recall the following important result of Covitz and Nadler about multi-valued contractions. Theorem 2.11 ([23]). Let (X , d) be a complete metric space and suppose that T : X → K (X ) (the collection of nonempty compact subsets of X ) is such that dh (Tx, Ty) ≤ cd(x, y) for some c ∈ (0, 1) and all x, y ∈ X . Then there exists x¯ ∈ X such that x¯ ∈ T x¯ .

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Here, dh (A, B) is the Hausdorff distance between sets A, B ∈ K (X ) defined by dh (A, B) = max{max d′ (x, B), max d′ (y, A)}, x∈A

y∈B

where d′ (t , C ) = min d(t , x).

(2.2)

x∈C

In what follows we will denote h(A, B) = maxx∈A d′ (x, B) so that dh (A, B) = max{h(A, B), h(B, A)}. In [13] the authors proved (among others) the following results. Lemma 2.12 ([13]). Let (X , d) be a metric space. 1◦ For x, y ∈ X and C ⊂ X , we have d′ (x, C ) ≤ d(x, y) + d′ (y, C ). 2◦ For all x, y ∈ X and A, B ⊂ X we have d′ (x, A) ≤ d(x, y) + d′ (y, B) + h(B, A). 3◦ For all x ∈ X and A, B ⊂ X we have d′ (x, A) ≤ d′ (x, B) + h(B, A). Given a point x ∈ X and a compact set A ⊂ X , it is known that the function A ∋ a → d(x, a) has at least one minimum point a¯ ∈ A, i.e. d(x, a¯ ) ≤ d(x, a) for all a ∈ A (of course such a¯ is not necessarily unique). Following [13], we will call (any such) point a¯ a projection of the point x on the set A and denote it as a¯ = πx A. Define the following projection function associated with the multifunction T : P (x) = πx (Tx). Theorem 2.13 ([13]). Let (X , d) be a complete metric space and suppose that T : X → K (X ) is such that dh (Tx, Ty) ≤ cd(x, y) for some c ∈ (0, 1) and all x, y ∈ X . Then, 1◦ for all x0 ∈ X there exists a point x¯ ∈ X such that xn+1 = P (xn ) → x¯ when n → ∞; 2◦ x¯ is a fixed point of T , i.e. x¯ ∈ T x¯ . Adapting these results to the setting of cone metric spaces, Rezapour and Haghi proved in [10] the following results (td and D have the meaning from Section 2.2). Lemma 2.14 ([10]). Let (X , d) be a cone metric space with the underlying cone P having the normal constant M = 1, and let A, B be td -compact sets. Then, 1◦ for every x ∈ X there exists a0 ∈ A such that D(x, a0 ) = infa∈A D(x, a) (this number shall be denoted as d′ (x, A)); 2◦ supx∈B d′ (x, A) < ∞. As a consequence, the Hausdorff distance dh (A, B) can be defined in a similar way as in the setting of metric spaces, only d′ (t , C ) = min ‖d(t , x)‖ = min D(t , x) x∈C

x∈C

should stay instead of (2.2). Theorem 2.15 ([10]). Let (X , d) be a cone metric space with the underlying cone P having the normal constant M = 1, and let a multifunction T : X → Kd (X ) satisfy 1◦ there exists c ∈ (0, 1/2) such that for all x, y ∈ X , dh (Tx, Ty) ≤ c (d′ (Tx, x) + d′ (Ty, y)) or 2◦ there exists c ∈ (0, 1/2) such that for all x, y ∈ X , dh (Tx, Ty) ≤ c (d′ (Tx, y) + d′ (Ty, x)). Then T has a fixed point. 3. Results In what follows we always assume that E is a Banach space, P is a normal cone in E with normal constant M ≥ 1 and with int P ̸= ∅. ≼ will be the partial ordering induced by P.

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First we note that the proof of Lemma 2.14 does not depend on the assumption M = 1, taking into account (Theorem 2.10) that the spaces (X , d) and (X , D) (D is the associated symmetric; see Definition 2.8) have the same collections of compact sets. Hence, it remains valid in the case M ≥ 1 and Hausdorff distance can be defined in the same way. Also note that, according to Lemma 2.2 and Remark 2.9, we can always renorm the space E with an equivalent norm such that the normal constant of P becomes M = 1. Hence, we shall always assume that this has been done. Also note that when u, v ∈ X , we shall put d′ (u, v) := d′ (u, {v}) = D(u, v). Theorem 3.1. Let (X , d) be a complete normal cone metric space and let T : X → Kd (X ) be a mapping satisfying the condition dh (Tx, Ty) ≤ pd′ (x, y) + qd′ (x, Tx) + rd′ (y, Ty) + sd′ (x, Ty) + td′ (y, Tx)

(3.1)

for some p, q, r , s, t ≥ 0 such that p + q + r + s + t < 1 and for all x, y ∈ X . Then T has a fixed point, i.e. there exists x¯ ∈ X such that x¯ ∈ T x¯ . Proof. Take an arbitrary x0 ∈ X and choose one of the closest points x1 ∈ Tx0 , i.e. a point satisfying D(x0 , x1 ) = d′ (x0 , Tx0 ). Inductively, for the chosen xn choose xn+1 ∈ Txn such that D(xn , xn+1 ) = d′ (xn , Txn ). We shall prove that {xn } is a Cauchy sequence in the space (X , D). For each n ∈ N we have that D(xn , xn+1 ) = d′ (xn , Txn ) ≤ h(Txn−1 , Txn ) ≤ dh (Txn−1 , Txn )

≤ pd′ (xn−1 , xn ) + qd′ (xn−1 , Txn−1 ) + rd′ (xn , Txn ) + sd′ (xn−1 , Txn ) + td′ (xn , Txn−1 ) ≤ pD(xn−1 , xn ) + qD(xn−1 , xn ) + rD(xn , xn+1 ) + sD(xn−1 , xn+1 ) ≤ pD(xn−1 , xn ) + qD(xn−1 , xn ) + rD(xn , xn+1 ) + sD(xn−1 , xn ) + sD(xn , xn+1 ), and so

(1 − r − s)D(xn xn+1 ) ≤ (p + q + s)D(xn−1 , xn ). Changing places for xn and xn+1 and using the symmetry of D, we obtain that

(1 − q − t )D(xn+1 , xn ) ≤ (p + r + t )D(xn , xn−1 ). Hence, D(xn , xn+1 ) ≤ α D(xn−1 , xn ) for n ∈ N, where

α = min



p+q+s p+r +t

,



1−r −s 1−q−t

< 1.

It is now easy to prove in the standard way that D(xn , xn+1 ) ≤ α n D(x0 , x1 ) and that {xn } is a Cauchy sequence in (X , D). Since (X , D) (together with (X , d)) is complete, there exists x¯ ∈ X such that xn → x¯ (in topology tD ) when n → ∞. We have to prove that x¯ ∈ T x¯ . We have (using 3◦ of Lemma 2.12 and (3.1)) that d′ (¯x, T x¯ ) ≤ d′ (¯x, Txn ) + h(Txn , T x¯ ) ≤ d′ (¯x, Txn ) + dh (Txn , T x¯ )

≤ d′ (¯x, Txn ) + pd′ (xn , x¯ ) + qd′ (xn , Txn ) + rd′ (¯x, T x¯ ) + sd′ (xn , T x¯ ) + td′ (¯x, Txn ) = (1 + t )d′ (¯x, Txn ) + pd′ (xn , x¯ ) + qd′ (xn , xn+1 ) + rd′ (¯x, T x¯ ) + sd′ (xn , T x¯ ). Since (see 1◦ of Lemma 2.12) d′ (¯x, Txn ) ≤ d′ (¯x, xn+1 ) + d′ (xn+1 , Txn ) = D(¯x, xn+1 ), d′ (xn , T x¯ ) ≤ d′ (xn , x¯ ) + d′ (¯x, T x¯ ), we obtain that d′ (¯x, T x¯ ) ≤ (1 + t )D(¯x, xn+1 ) + pD(xn , x¯ ) + qD(xn , xn+1 ) + rd′ (¯x, T x¯ ) + sD(xn , x¯ ) + sd′ (¯x, T x¯ ), i.e.

(1 − r − s)d′ (¯x, T x¯ ) ≤ (1 + t )D(¯x, xn+1 ) + (p + s)D(xn , x¯ ) + qD(xn , xn+1 ). Passing to the limit when n → ∞ and taking into account that 0 < 1 − r − s < 1, we obtain that d′ (¯x, T x¯ ) = 0, i.e. x¯ ∈ T x¯ . Thus, x¯ is a fixed point of mapping T .  As corollaries we obtain (a) Theorems 2.11 and 2.13 (taking q = r = s = t = 0); (b) Theorem 2.15, 1◦ and 2◦ (taking q = r ̸= 0, s = t = p = 0, resp. s = t ̸= 0, p = q = r = 0). In the next example (which is obtained by adapting an example from [24]) we show that Theorem 3.1 is a proper generalization of these results.

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Example 3.2. Let X = {1, 2, 3} and d1 : X × X → R be defined by

d1 (x, y) = d1 (y, x) =

0,   2   ,   7 1

,    2    1 , 4

x = y, x, y ∈ X \ {2}, x, y ∈ X \ {3}, x, y ∈ X \ {1}.

Further, let P = {(x, y) : x, y ≥ 0} ⊂ R2 and define a cone metric d : X × X → R2 by d(x, y) = (d1 (x, y), 0). Then, with the earlier notation, D(x, y) = ‖d(x, y)‖ = d1 (x, y). Finally, let T : X → 2X be defined by T 2 = {1} and Tx = {3} for x ̸= 2. Obviously, 3 ∈ T 3 = {3}, so 3 is a fixed point of T . But one cannot conclude about its existence using Theorem 2.11 or Theorem 2.15. Indeed, for x = 3, y = 2 we have 2 dh (T 3, T 2) = dh ({3}, {1}) = d′ (3, 1) = d1 (3, 1) = ; 7 on the other hand, for p, q, s ≥ 0, p + 2q + 2s < 1 we have pd′ (3, 2) + q[d′ (3, T 3) + d′ (2, T 2)] + s[d′ (3, T 2) + d′ (2, T 3)]

= pd1 (3, 2) + q[d1 (3, 3) + d1 (2, 1)] + s[d1 (3, 1) + d1 (2, 3)] =

p 4

+

q 2

+

15 28

s<

2 7

⇔

7 8

p+

7 4

q+

15 8

s < 1.

The last inequality holds true since 87 p + 47 q + 15 s < p + 2q + 2s < 1. 8 On the other hand, taking in relation (3.1) p = q = s = t = 0, r = 47 , x = 3, y = 2, it reduces to 2 7

≤

4 7

·

1 2

=

2 7

which obviously holds true; the same is true for other values of x and y. Hence, the existence of a fixed point follows from Theorem 3.1. Under the conditions of Theorem 3.1 the set XT of fixed points of the mapping T is nonempty. Additional information is given by Proposition 3.3. Under the conditions of Theorem 3.1 the set XT is closed. Proof. Let xn ∈ XT and xn → x¯ , n → ∞ in topology tD (i.e. td ). Then, using 2◦ of Lemma 2.12 and d′ (xn , Txn ) = 0 we obtain that d′ (¯x, T x¯ ) ≤ d′ (¯x, xn ) + d′ (xn , Txn ) + h(Txn , T x¯ )

≤ d′ (¯x, xn ) + dh (Txn , T x¯ ). Now the contractive condition (3.1) implies that d′ (¯x, T x¯ ) ≤ d′ (¯x, xn ) + pd′ (xn , x¯ ) + qd′ (xn , Txn ) + rd′ (¯x, T x¯ ) + sd′ (xn , T x¯ ) + td′ (¯x, Txn ) Property 1◦ of Lemma 2.12 implies that d′ (xn , T x¯ ) ≤ d′ (xn , x¯ ) + d′ (¯x, T x¯ ), and we obtain that d′ (¯x, T x¯ ) ≤ d′ (¯x, xn ) + pd′ (xn , x¯ ) + rd′ (¯x, T x¯ ) + sd′ (xn , x¯ ) + sd′ (¯x, T x¯ ) + td′ (¯x, xn ) and finally

(1 − r − s)d′ (¯x, T x¯ ) ≤ (1 + p + s + t )d′ (xn , x¯ ). Passing to the limit when n → ∞, we get (1 − r − s)d′ (¯x, T x¯ ) ≤ 0 and (since 0 < 1 − r − s < 1), x¯ ∈ T x¯ . Hence the set XT is closed.  In our next result we consider two mappings S and T and give sufficient conditions for the existence of their common fixed point. Theorem 3.4. Let (X , d) be a normal complete cone metric space and let S , T : X → K (X ) be two mappings satisfying the condition dh (Sx, Ty) ≤ pd′ (x, y) + qd′ (x, Sx) + rd′ (y, Ty) + sd′ (x, Ty) + td′ (y, Sx),

(3.2)

for some real numbers p, q, r , s, t ≥ 0 such that p + q + r + s + t < 1 and q = r or s = t, and for all x, y ∈ X . Then S and T have a common fixed point in X .

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Proof. Let x0 be an arbitrary point in X . Choose x1 ∈ Sx0 such that D(x0 , x1 ) = d′ (x0 , Sx0 ); then choose x2 ∈ Tx1 such that D(x1 , x2 ) = d′ (x1 , Tx1 ). By induction, after choosing x1 , x2 , . . . , x2n , choose x2n+1 ∈ Sx2n such that D(x2n , x2n+1 ) = d′ (x2n , Sx2n ) and x2n+2 ∈ Tx2n+1 such that D(x2n+1 , x2n+2 ) = d′ (x2n+1 , Tx2n+1 ). Using condition (3.2) we obtain D(x2n+1 , x2n+2 ) = d′ (x2n+1 , Tx2n+1 ) ≤ h(Sx2n , Tx2n+1 ) ≤ dh (Sx2n , Tx2n+1 )

≤ pd′ (x2n , x2n+1 ) + qd′ (x2n , Sx2n ) + rd′ (x2n+1 , Tx2n+1 ) + sd′ (x2n , Tx2n+1 ) + td′ (x2n+1 , Sx2n ) ≤ pD(x2n , x2n+1 ) + qD(x2n , x2n+1 ) + rD(x2n+1 , x2n+2 ) + sD(x2n , x2n+2 ) + t · 0, where from (1 − r − s)D(x2n+1 , x2n+2 ) ≤ (p + q + s)D(x2n , x2n+1 ) and p+q+s

D(x2n+1 , x2n+2 ) ≤

1−r −s

· D(x2n , x2n+1 ) = A · D(x2n , x2n+1 ).

(3.3)

· D(x2n+1 , x2n+2 ) = B · D(x2n+1 , x2n+2 ).

(3.4)

In a similar way one obtains p+r +t

D(x2n+2 , x2n+3 ) ≤

1−q−t

Then we obtain by induction from (3.3) and (3.4) that D(x2n+1 , x2n+2 ) ≤ A(AB)n D(x0 , x1 ) and

D(x2n+2 , x2n+3 ) ≤ (AB)n+1 D(x0 , x1 ).

In the case q = r, AB =

p+q+s 1−q−s

·

p+r +t 1−q−t

p+q+s

=

1−q−t

·

p+r +t 1−r −s

< 1 · 1 = 1,

and if s = t, AB =

p+q+s 1−r −s

·

p+r +s 1−q−t

< 1 · 1 = 1.

Now, for n < m we have D(x2n+1 , x2m+1 ) ≤ D(x2n+1 , x2n+2 ) + · · · + D(x2n , x2m+1 )

≤

 m −1 − A

(AB) + i

i=n

m −



(AB) D(x0 , x1 ) i

i=n+1

 (AB)n+1 D(x0 , x1 ) 1 − AB 1 − AB A(AB)n D(x0 , x1 ). = (1 + B) 1 − AB 

≤

A(AB)n

+

Similarly, we obtain

(AB)n D(x0 , x1 ), 1 − AB (AB)n D(x2n , x2m ) ≤ (1 + A) D(x0 , x1 ) 1 − AB D(x2n , x2m+1 ) ≤ (1 + A)

and D(x2n+1 , x2m ) ≤ (1 + B)

A(AB)n 1 − AB

D(x0 , x1 ).

Hence, for n < m



D(xn , xm ) ≤ max (1 + B)

A(AB)n 1 − AB

, (1 + A)

 (AB)n D(x0 , x1 ) = λn D(x0 , x1 ), 1 − AB

where λn → 0, as n → ∞. It follows that {xn } is a Cauchy sequence (in tD and td ) and so there exists u ∈ X such that xn → u when n → ∞. We shall prove that u ∈ Su ∩ Tu. Estimating d′ (u, Su), using Lemma 2.12, we get d′ (u, Su) ≤ d′ (u, Tx2n+1 ) + h(Tx2n+1 , Su) ≤ d′ (u, Tx2n+1 ) + dh (Tx2n+1 , Su)

≤ d′ (u, x2n+1 ) + d′ (x2n+1 , Tx2n+1 ) + dh (Su, Tx2n+1 ) = d′ (u, x2n+1 ) + d′ (x2n+1 , x2n+2 ) + dh (Su, Tx2n+1 ) ≤ d′ (u, x2n+1 ) + d′ (x2n+1 , x2n+2 ) + pd′ (u, x2n+1 ) + qd′ (u, Su) + rd′ (x2n+1 , Tx2n+1 ) + sd′ (u, Tx2n+1 ) + td′ (x2n+1 , Su).

S. Radenović, Z. Kadelburg / Mathematical and Computer Modelling 53 (2011) 746–754

753

Since d′ (u, Tx2n+1 ) ≤ d′ (u, x2n+1 ) + d′ (x2n+1 , Tx2n+1 ), d′ (x2n+1 , Su) ≤ d′ (x2n+1 , u) + d′ (u, Su), finally we obtain d′ (u, Su) ≤ d′ (u, x2n+1 ) + pd′ (u, x2n+1 ) + qd′ (u, Su) + rd′ (x2n+1 , x2n+2 )

+ sd′ (u, x2n+1 ) + sd′ (x2n+1 , x2n+2 ) + td′ (x2n+1 , u) + td′ (u, Su), i.e.

(1 − q − t )d′ (u, Su) ≤ (1 + p + s + t )D(u, x2n+1 ) + (1 + r + s)D(x2n+1 , x2n+2 ). Passing to the limit when n → ∞ it follows that d′ (u, Su) = 0 and, since Su is a compact set, u ∈ Su. One can prove in a similar way that u ∈ Tu. Hence, u ∈ Su ∩ Tu, i.e. u is a common fixed point for multifunctions S and T.  If, in addition to data of Example 3.2, we define Sx = {3} for each x ∈ X , we obtain two multifunctions S and T on X satisfying conditions of the previous theorem for p = q = s = t = 0, r = 47 . Hence, there exists a common fixed point of S and T (obviously, 3 is such a point). Corollary 3.5. Let (X , d) be a normal complete cone metric space and let S , T : X → K (X ) be two mappings satisfying the condition dh (Sx, Ty) ≤ pD(x, y) + q[d′ (x, Sx) + d′ (y, Ty)] + s[d′ (x, Ty) + d′ (y, Sx)] for some p, q, s ≥ 0 such that p + 2q + 2s < 1 and for all x, y ∈ X . Then S and T have a common fixed point in X . This is a set-valued variant of a result of Abbas and Rhoades from [25]. Corollary 3.6. Let (X , d) be a normal complete cone metric space and let S , T : X → K (X ) be two mappings satisfying the condition dh (Sx, Ty) ≤ pD(x, y) = p‖d(x, y)‖, for some p ∈ (0, 1) and for all x, y ∈ X . Then S and T have a common fixed point in X . In particular, for S = T we get a cone metric version of Theorem 2.11 of Covitz and Nadler. On the other hand, putting S = T and 1◦ p = s = t = 0 and q = r, resp. 2◦ p = q = r = 0 and s = t, one obtains Theorem 2.15 of Rezapour and Haghi. Yet another possibility is to consider conditions like those in Ćirić’s quasicontractions [26]. Theorem 3.7. Let (X , d) be a complete normal cone metric space and let T : X → Kd (X ) be a mapping satisfying the condition dh (Tx, Ty) ≤ λ max{d′ (x, y), d′ (x, Tx), d′ (y, Ty), d′ (x, Ty), d′ (y, Tx)}

(3.5)

for some λ ∈ [0, 1/2) and for all x, y ∈ X . Then T has a fixed point, i.e. there exists x¯ ∈ X such that x¯ ∈ T x¯ . Proof. Take an arbitrary x0 ∈ X and choose one of the closest points x1 ∈ Tx0 , i.e. a point satisfying D(x0 , x1 ) = d′ (x0 , Tx0 ). Inductively, for the chosen xn choose xn+1 ∈ Txn such that D(xn , xn+1 ) = d′ (xn , Txn ). In order prove that {xn } is a Cauchy sequence in the space (X , D), apply condition (3.5) to obtain D(xn , xn+1 ) = d′ (xn , Txn ) ≤ h(Txn−1 , Txn ) ≤ dh (Txn−1 , Txn )

  ≤ λ max d′ (xn−1 , xn ), d′ (xn−1 , Txn−1 ), d′ (xn .Txn ), d′ (xn−1 , Txn ), d′ (xn , Txn−1 ) . Consider the following five possible cases. λ 1◦ D(xn , xn+1 ) ≤ λD(xn−1 , xn ) ≤ 1−λ D(xn−1 , xn ).

λ 2◦ D(xn , xn+1 ) ≤ λd′ (xn−1 , Txn−1 ) = λD(xn−1 , xn ) ≤ 1−λ D(xn−1 , xn ). ◦ ′ 3 D(xn , xn+1 ) ≤ λd (xn , Txn ) = λD(xn , xn+1 ); it follows that D(xn , xn+1 ) = 0. 4◦ D(xn , xn+1 ) ≤ λd′ (xn−1 , Txn ) ≤ λD(xn−1 , xn ) + λd′ (xn , Txn ) = λD(xn−1 , xn ) + λD(xn , xn+1 ); it follows that D(xn , xn+1 ) ≤ λ D(xn−1 , xn ). 1−λ 5◦ D(xn , xn+1 ) ≤ λd′ (xn , Txn−1 ) = 0.

Hence, in all possible cases we obtain that D(xn , xn+1 ) ≤ hD(xn−1 , xn ),

h=

λ 1−λ

∈ [0, 1),

because λ ∈ [0, 1/2). It follows that D(xn , xn+1 ) ≤ hn D(x0 , x1 ) for each n ∈ N and the conclusion that {xn } is a Cauchy sequence follows in the usual way.

754

S. Radenović, Z. Kadelburg / Mathematical and Computer Modelling 53 (2011) 746–754

Thus, there exists u ∈ X such that xn → u, n → ∞ in tD = td . We have to prove that u ∈ Tu, i.e. (since Tu is compact) that d′ (u, Tu) = 0. We have (Lemma 2.12) d′ (u, Tu) ≤ d′ (u, xn ) + d′ (xn , Tu) ≤ d′ (u, xn ) + d′ (xn , Txn ) + h(Txn , Tu)

≤ d′ (u, xn ) + d′ (xn , xn+1 ) + dh (Txn , Tu). For the last summand dh (Txn , Tu), by (3.5), we obtain that dh (Txn , Tu) ≤ λ max{d′ (xn , u), d′ (xn , Txn ), d′ (u, Tu), d′ (xn , Tu), d′ (u, Txn )}

≤ λ max{d′ (xn , u), d′ (xn , xn+1 ), d′ (u, Tu), d′ (xn , u) + d′ (u, Tu), d′ (u, xn ) + d′ (xn , xn+1 )}. Consider the following possible cases: 1◦ d′ (u, Tu) ≤ D(u, xn ) + D(xn , xn+1 ) + λD(u, xn ). 2◦ d′ (u, Tu) ≤ D(u, xn ) + D(xn , xn+1 ) + λD(xn , xn+1 ) = D(u, xn ) + (1 + λ)D(xn , xn+1 ). 1 3◦ d′ (u, Tu) ≤ D(u, xn ) + D(xn , xn+1 ) + λd′ (u, Tu); it follows that d′ (u, Tu) ≤ 1−λ (D(u, xn ) + D(xn , xn+1 )).

+λ 1 4◦ d′ (u, Tu) ≤ D(u, xn ) + D(xn , xn+1 ) + λD(xn , u) + λd′ (u, Tu); it follows that d′ (u, Tu) ≤ 11−λ D(xn , u) + 1−λ D(xn , xn+1 ). ◦ ′ 5 d (u, Tu) ≤ D(u, xn ) + D(xn , xn+1 ) + λD(u, xn ) + λD(xn , xn+1 ) = (1 + λ)(D(u, xn ) + D(xn , xn+1 )).

In all these cases, passing to the limit when n → ∞, we get that d′ (u, Tu) = 0, i.e. u ∈ Tu. The proof is complete.



Similarly as Proposition 3.3 one can prove Proposition 3.8. Under the conditions of Theorem 3.7 the set XT is closed. Acknowledgements The authors are thankful to the Ministry of Science and Technology of Serbia. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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