Fixed points of Edelstein-type multivalued maps ...

Fixed points of Edelstein-type multivalued maps

Nayyar Mehmood, Akbar Azam & Ismat Beg

Rendiconti del Circolo Matematico di Palermo (1952 -) ISSN 0009-725X Volume 63 Number 3 Rend. Circ. Mat. Palermo (2014) 63:399-407 DOI 10.1007/s12215-014-0166-6

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Author's personal copy Rend. Circ. Mat. Palermo (2014) 63:399–407 DOI 10.1007/s12215-014-0166-6

Fixed points of Edelstein-type multivalued maps Nayyar Mehmood · Akbar Azam · Ismat Beg

Received: 5 June 2014 / Accepted: 27 August 2014 / Published online: 9 September 2014 © Springer-Verlag Italia 2014

Abstract In this article we prove the existence of common fixed points for Edelstein type locally contractive multivalued mappings in complete cone metric spaces, without assumption of normality of cone. We generalize/extend several remarkable and useful results in the existing literature. Keywords

Fixed point · Multivalued maps · Locally contractive maps · Cone metric space

Mathematics Subject Classification

46S40 · 47H10 · 54H25

1 Introduction Edelstein [9] proved the following result for locally contractive mapping: Theorem 1 Let X be a complete ε-chainable metric space and f : X → X is an (ε, λ)uniformly locally contractive mapping, then there exists x ∈ X such that x = f (x) . Nadler [20, Theorem 6] generalized this result for multivalued mappings. Theorem 2 Let X be a complete ε-chainable metric space and F : X → C B (X ) is an (ε, λ)-uniformly locally contractive mapping, then F has a fixed point.

N. Mehmood · A. Azam Department of Mathematics, COMSATS Institute of InformationTechnology, Islamabad 44000, Pakistan e-mail: [email protected] A. Azam e-mail: [email protected] I. Beg (B) Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan e-mail: [email protected]

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One of generalizations of Nadler’s theorem [20, Theorem 5] for globally contractive mappings was given by Mizoguchi and Takahashi [18, Theorem 5] as: Theorem 3 Let (X, d) be a complete metric space and let T : X → 2 X be multivalued map such that T x is closed bounded subset of X , for all x ∈ X . If there exists a function ϕ : (0, ∞) → [0, 1) such that lim sup ϕ(r ) < 1 for all t ∈ [0, ∞) and if r →t +

H (T x, T y) ≤ ϕ(d(x, y))(d(x, y)) f or all x, y(x  = y) ∈ X Then T has a fixed point in X. Suzuki [27], proved that Mizoguchi and Takahashi theorem is a real generalization of Nadler’s theorem. In [9] the author remarked that globally contractive mapping can be regarded as a (∞, λ)-uniformly contractive mapping, which depicts the novelty of the result. After the remarkable result of Nadler, many authors generalized/extended the above mentioned results in various directions (see [1,5,7,8,10,12,13,16,19,21,22,24,25,28,29]). One of the real generalization of the above result of Nadler was presented by Hu in [11]. Azam and Arshad [2] generalized the result of [11] for a sequence of multivalued locally contractive mappings and generalized several results. Huang and Zhang [14] introduced cone metric space with normal cone, as a generalization of metric space. Cho and Bae [6] generalized the result of [18] for multivalued mappings in cone metric spaces with normal cone. Rezapour and Hamlbarani [23] extended the results of [14] for the case of cone metric space without normality of cone. It is worthy to mention that the work in cone metric spaces with non-normal cones is significant (see [15]). Subsequently some authors generalized many results in this setting without assumption of normality of cone (see [3,4,15,17,26]). Aim of this paper is to prove existence of common fixed point for a sequence of locally contractive multivalued mappings in the setting of cone metric spaces without the assumption of normality of cone. Our results generalize and extend several useful results present in the existing literature.

2 Preliminaries Let E be a real Banach space and K be a subset of E. By θ we denote the zero element of E and by int K the interior of K . The subset K is called a cone if and only if: (i) K is closed, nonempty, and K  = {θ }; (ii) a, b ∈ R, a, b ≥ 0, x, y ∈ K implies ax + by ∈ K ; (iii) K ∩ (−K ) = {θ }. For a given cone K ⊆ E,we define a partial ordering  with respect to K by x  y if and only if y − x ∈ K ; x ≺ y will stand for x  y and x  = y,while x  y stands for y − x ∈ int K . The cone K is said to be solid if it has a nonempty interior. Definition 4 [14] Let X be a nonempty set and E be a real Banach space with a cone K . A function ρ : X × X → E is said to be a cone metric, if the following conditions hold: (C1) θ  ρ(x, y) for all x, y ∈ X and ρ(x, y) = θ if and only if x = y; (C2) ρ(x, y) = ρ(y, x) for all x, y ∈ X ; (C3) ρ(x, z)  ρ(x, y) + ρ(y, z) for all x, y, z ∈ X. The pair (X, ρ) is then called a cone metric space.

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Definition 5 [14] Let(X, ρ) be a cone metric space, x ∈ X,let {xn } be a sequence in X. Then (i) {xn } converges to x whenever for every c ∈ E with θ  c there is a natural number n 0 such that ρ(xn , x)  c, for all n ≥ n 0 . We denote this by limn→∞ xn = x; (ii) {xn } is a Cauchy sequence whenever for every c ∈ E with θ  c there is a natural number n 0 such that ρ(xn , xm )  c, for all n, m ≥ n 0 ; (iii) (X, ρ) is complete cone metric if every Cauchy sequence in X is convergent. In the sequel K denotes a cone, C(X ), C B(X ) denote closed, closed and bounded subsets of X respectively, and ρ will be a cone metric on X. Definition 6 A cone metric space (X, ρ) is said to be a ε-chainable cone metric space if for every x, y ∈ X, there exists an ε-chain for some θ  ε, that is a finite set of points x = x0 , x1 , x2 , . . . , xn = y, (n may depend on both x and y) such that ρ(xi−1 , xi ) ≺ ε, for i = 1, 2, 3, . . . , n. Remark 7 [15] The results concerning fixed points and other results, in case of cone spaces with non-normal solid cones, cannot be provided by reducing to metric spaces, because in this case neither of the conditions of the [14, lemma 1–4] hold. Further, the vector cone metric is not continuous in the general case, i.e., from xn → x, yn → y it need not follow that ρ(xn , yn ) → ρ(x, y). Let E be an ordered Banach space with a positive cone K . Then the following properties hold [15]: (P T 1) If u  v and v  w, then u  w. (P T 2) If θ  u  c for each c ∈ int K , then u = θ. (P T 3) Let {an } be a sequence in E. If c ∈ int K and an → θ (as n → ∞), then there exists n 0 ∈ N such that for all n ≥ n 0 , we have an  c.

3 Main result According to [6] we denote for p ∈ E, s ( p) = {q ∈ E : p  q} for q ∈ E and s (a, B) = ∪ s (d (a, b)) = ∪ {x ∈ E : d (a, b)  x} for a ∈ X and B ∈ C(X ). b∈B

b∈B

For A, B ∈ C(X ), we denote s (A, B) =



   ∩ s (a, B) ∩ ∩ s (b, A) .

a∈A

b∈B

Define for some θ  ε, the set K ∗ = {ω ∈ E : θ  u  ε} . Definition 8 Let (X, ρ) be a cone metric space. A mapping F : X → C B(X ) is said to be an (ε, λ)-uniformly locally contractive multivalued mapping provided that, if x, y ∈ X and θ ≺ ρ(x, y) ≺ ε implies λρ(x, y) ∈ s(F x, F y), where θ  ε and 0 ≤ λ < 1.

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The following lemma from [6] will be often used. Lemma 9 Let (X, d) be a cone metric space with a cone K . Then we have: (i) (ii) (iii) (iv)

Let p, q ∈ E. If p  q, then s(q) ⊂ s( p). Let x ∈ X and A ∈ C(X ). If θ ∈ s (x, A) , then x ∈ A. Let q ∈ K and let A, B ∈ C(X ) and a ∈ A. If q ∈ s (A, B) , then q ∈ s (a, B) . For all q ∈ K and A, B ∈ C(X ). Then q ∈ s(A, B) if and only if there exist a ∈ A and b ∈ B such that d(a, b)  q.

Remark 10 Let (X, ρ) be a cone metric space. If E = R and K = [0, +∞), then (X, ρ) is a metric space. Moreover, for A, B ∈ C B(X ), H (A, B) = in f s(A, B) is the Hausdorff distance induced by ρ. Now, we present our main result. Theorem 11 Let (X, ρ) be an ε-chainable complete cone metric space for some θ  ε, and ∞ be a sequence of multivalued mappings from X into C(X ). If {Fi }i=1 u, v ∈ X, θ ≺ ρ(u, v) ≺ ε implies (ρ(u, v))ρ(u, v) ∈ s(Fi (u), F j (v)) for all i, j = 1, 2, . . . ,where  :

K∗

(1)

→ [0, 1) is a function such that

lim sup (rn ) < 1, n→∞

for any decreasing sequence {rn } in K ∗ . Then there exists a point v ∗ ∈ X such that {v ∗ } ⊂ ∞ F (v ∗ ). ∩i=1 i Proof Let v0 be an arbitrary, but fixed element of X, find v1 ∈ X such that v1 ∈ F1 (v0 ). Let v0 = u (1,0) , u (1,1) , u (1,2) , . . . , u (1,m) = v1 ∈ F1 (v0 ), be an ε-chain from v0 to v1 . Rename v1 as u (2,0) . Since u (1,0) and u (1,1) are consecutive elements of an ε-chain, so consider (ρ(u (1,0) , u (1,1) ))ρ(u (1,0) , u (1,1) ) ∈ s(F1 (u (1,0) ), F2 (u (1,1) )). Since u (2,0) ∈ F1 (u (1,0) ), using Lemma 9 we obtain u (2,1) ∈ F2 (u (1,1) ) such that (ρ(u (1,0) , u (1,1) ))ρ(u (1,0) , u (1,1) ) ∈ s(ρ(u (2,0) , u (2,1) )). It further implies ρ(u (2,0) , u (2,1) )  (ρ(u (1,0) , u (1,1) ))ρ(u (1,0) , u (1,1) )  ≺ (ρ(u (1,0) , u (1,1) ))ρ(u (1,0) , u (1,1) ) ≺ ρ(u (1,0) , u (1,1) ) ≺ ε. Since ρ(u (1,1) , u (1,2) )  ε, so we have (ρ(u (1,1) , u (1,2) ))ρ(u (1,1) , u (1,2) ) ∈ s(F1 (u (1,1) ), F2 (u (1,2) )). Since u (2,1) ∈ F2 (u (1,1) ), using Lemma 9 we obtain u (2,2) ∈ F2 (u (1,2) ) such that (ρ(u (1,1) , u (1,2) ))ρ(u (1,1) , u (1,2) ) ∈ s(ρ(u (2,1) , u (2,2) )).

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Therefore ρ(u (2,1) , u (2,2) )  (ρ(u (1,1) , u (1,2) ))ρ(u (1,1) , u (1,2) )  ≺ (ρ(u (1,1) , u (1,2) ))ρ(u (1,1) , u (1,2) ) ≺ ρ(u (1,1) , u (1,2) ) ≺ ε. Thus we obtain a finite set of points u (2,0) , u (2,1) , u (2,2) , . . . , u (2,m) such that u (2,0) ∈ F1 (u (1,0) ) and u (2, j) ∈ F2 (u (1, j) ) for j = 1, 2, . . . , m, with ρ(u (2, j) , u (2, j+1) )  (ρ(u (1, j) , u (1, j+1) ))ρ(u (1, j) , u (1, j+1) ) ≺ ρ(u (1, j) , u (1, j+1) ) ≺ ε, for j = 1, 2, . . . , m − 1. Let u (2,m) = v2 then the set of points v1 = u (2,0) , u (2,1) , u (2,2) , . . . , u (2,m) = v2 ∈ F2 (v1 ) is an ε-chain from v0 to v1 . Rename v2 as u (3,0) , then by the same procedure we obtain an ε-chain v2 = u (3,0) , u (3,1) , u (3,2) , . . . , u (3,m) = v3 ∈ F3 (v2 ) from v2 to v3 . Inductively, we obtain vn = u (n+1,0) , u (n+1,1) , u (n+1,2) , . . . , u (n+1,m) = vn+1 ∈ Fn+1 (vn ) with ρ(u (n+1, j) , u (n+1, j+1) )  (ρ(u (n, j) , u (n, j+1) ))ρ(u (n, j) , u (n, j+1) ) ≺ ρ(u (n, j) , u (n, j+1) )

(3)

 ε, for j = 1, 2, . . . , m − 1. Consequently, we obtain a sequence {vn } of points of X with v1 = u (1,m) = u (2,0) ∈ F1 (v0 ), v2 = u (2,m) = u (3,0) ∈ F2 (v1 ), v3 = u (3,m) = u (4,0) ∈ F3 (v2 ), .. . vn+1 = u (n+1,m) = u (n+2,0) ∈ Fn+1 (vn ), for n = 0, 1, 2, . . . . Then from (3) we see that {ρ(u (n, j) , u (n, j+1) ) : n ≥ 0} is decreasing sequence in K ∗ , therefore by definition of , lim sup (ρ(u (n, j) , u (n, j+1) )) = l < 1. n→∞

So there exists some n 0 ∈ N such that for all n ≥ n 0 , (ρ(u (n, j) , u (n, j+1) )) < l whenever n ≥ n 0 . Then ρ(u (n, j) , u (n, j+1) )  (ρ(u (n−1, j) , u (n−1, j+1) ))ρ(u (n−1, j) , u (n−1, j+1) )   n0  max max (ρ(u (i, j) , u (i, j+1) )), l ρ(u (n−1, j) , u (n−1, j+1) ) i=1   n n0  max max (ρ(u (i, j) , u (i, j+1) )), l ρ(u (0, j) , u (0, j+1) ). i=1

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  N (ρ(u Put maxi=1 (i, j) , u (i, j+1) )), l = λ j , then λ j < 1, for all j and for all n ≥ n 0 , we have ρ(vn−1 , vn ) = ρ(u (n,0) , u (n,m) ) 

m−1

ρ(u (n, j) , u (n, j+1) )

j=0



m−1



λj

n

ρ(u (0, j) , u (0,

j+1) ).

j=0

Thus ρ(vn , v p )  ρ(vn , vn+1 ) + ρ(vn+1 , vn+2 ) + · · · + ρ(v p−1 , v p ) m−1





λj

n+1

ρ(u (0, j) , u (0,

j+1) ) + · · · +

m−1

j=0



λj

p

ρ(u (0, j) , u (0,

j+1) ).

j=0

as n, p → ∞, using the (P T 3), one can find some m 0 such that m−1



λj

n+1

ρ(u (0, j) , u (0,

j+1) ) + · · · +

j=0

m−1



λj

p

ρ(u (0, j) , u (0,

j+1) )

j=0

 c, f or all n, p > m 0 . So we have ρ(u m , u n ) 

m−1



λj

n+1

ρ(u (0, j) , u (0,

j+1) ) + · · · +

j=0

m−1



λj

p

ρ(u (0, j) , u (0,

j+1) )

j=0

 c, f or all n, p > m 0 . It implies that {vn } is a Cauchy sequence. Since X is complete, therefore vn → v ∗ ∈ X . Therefore there exists an integer k > 0 such that n ≥ k implies ρ(vn , v ∗ ) < 2c . From (1) we have (ρ(vn , v ∗ ))ρ(vn , v ∗ ) ∈ s(Fn+1 (vn ), F j (v ∗ )). Using Lemma 9 we obtain vk ∈ F j (v ∗ ) such that (ρ(vn , v ∗ ))ρ(vn , v ∗ ) ∈ s(ρ(vn+1 , vk )), it implies that ρ(vn+1 , vk )  (ρ(vn , v ∗ ))ρ(vn , v ∗ )  ≺ (ρ(vn , v ∗ ))ρ(vn , v ∗ ) ≺ ρ(vn , v ∗ ) ≺ ε. Now consider ρ(vk , v ∗ )  ρ(vk , vn+1 ) + ρ(vn+1 , v ∗ ) ≺ ρ(vn , v ∗ ) + ρ(vn+1 , v ∗ ) c

for all n ≥ k.

∞ F (v ∗ ).This completes Thus vk → v ∗ ∈ F j (v ∗ ), since F j (v ∗ ) is closed. Hence {v ∗ } ⊂ ∩i=1 j the proof.

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If we take (t) = λ, where λ ∈ [0, 1) is a constant, we obtain the following result. Theorem 12 Let (X, ρ) be a ε-chainable complete cone metric space for some θ  ε, and ∞ be a sequence of multivalued mappings from X into C(X ). If there exist μ ∈ [0, 1) {Fi }i=1 such that u, v ∈ X, θ ≺ ρ(u, v) ≺ ε ⇒ μρ(u, v) ∈ s(Fi (u), F j (v)), f or all i, j = 1, 2, . . . , ∞ F (v ∗ ). Then there exists a point v ∗ ∈ X such that {v ∗ } ⊂ ∩i=1 i

Remark 13 A globally contractive mapping is an (ε, λ) uniformly locally contractive mapping for all ε ∈ int K . Whereas converse is not true in general. This can be seen in Example 17 (see also [9]). In the view of above remark, the following theorem extended the main result of [14]. Theorem 14 Let (X, ρ) be an ε-chainable complete cone metric space and let T : X −→ C B (X ) be a multivalued mapping. If there exists a function ϕ : K → [0, 1) such that lim sup ϕ(rn ) < 1 n→∞

for any decreasing sequence {rn } in K , and if x, y ∈ X, θ ≺ ρ(x, y) ≺ ε implies ϕ(ρ (x, y))ρ (x, y) ∈ s (T x, T y) for all x, y ∈ X , then T has a fixed point in X. Corollary 15 [18] Let (X, d) be a complete metric space and let T : X → 2 X be a multivalued map such that T x is closed bounded subset of X , for all x ∈ X . If there exists a function ϕ : (0, ∞) → [0, 1) such that lim sup ϕ(r ) < 1 for all t ∈ [0, ∞) and if r →t +

H (T x, T y) ≤ ϕ(d(x, y))(d(x, y))

f or all x, y(x  = y) ∈ X.

Then T has a fixed point in X. The following corollary is about the Nadler’s result for multivalued (ε, λ) uniformly locally contractive mappings in complete metric spaces. Corollary 16 [20] Let X be a complete ε-chainable metric space and T be a multivalued mapping from X into C B(X ) satisfying the following condition: u, v ∈ X, 0 < d(u, v) < ε, implies H (T (u), T (v)) ≤ λd(u, v), where λ ∈ (0, 1).Then there exists a point v ∗ ∈ X such that v ∗ ∈ T (v ∗ ). In the next example the multivalued mapping is locally contractive but not globally contractive. Example 17 Let X = {eiθ : 0 ≤ θ ≤ 2π} , E = C 1R [0, 2π] with norm x = x∞ +  x and ∞ K = {x ∈ E : x(s) ≥ 0, f or s ∈ [0, 2π]}. Then, K is a non-normal solid cone. Define d : X × X → E by  (d(eiθ1 , eiθ2 ))(s) = (cos(θ1 ) − (cos(θ2 ))2 + (sin(θ1 ) − sin(θ2 ))2 es

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where 0 ≤ s ≤ 2π. Then d is a complete cone metric on X. Consider a mapping T : X → C(X ) defined by θ

¯ i 2 , 1) (T ) (x) = B(e which satisfies all the conditions of Theorem 14 to obtain the fixed point. In the following we furnish a simple example to support our main theorem. Example 18 Let X = [0, 1] and E be the set of all real valued functions on X which also have continuous derivatives on X. Then E is a vector space over R under the following operations: (x + y) (t) = x (t) + y (t), (αx) (t) = αx (t),

for all x, y ∈ E, α ∈ R. That is E = C 1R [0, 1] with norm y = y∞ + y  ∞ and K = {x ∈ E : θ  x},

where θ (t) = 0 for all t ∈ X,

then K is a non-normal cone. Define d : X × X → E as follows: (d (x, y)) (t) = |x − y| et . Define Fk : X → C B(X ) by

x  k = 1, 2, 3, . . . . Fk (x) = 0, πk

As

    x y  t  s(Fk (x), F j (y)) = s  e , for all k, j = 1, 2, 3, . . . . − πk πj 

Since

   x y  t 1 t   πk − π j  e  π |x − y| e

then for (t) =

1 π

we have

(d(x, y))d(x, y) ∈ s(Fk (x), F j (y)) for k, j = 1, 2, 3, . . . . ∞ F v. Thus there exists v ∈ ∩i=1 i

Acknowledgments The authors sincerely thank the anonymous referees and editor for their constructive comments which contributed to the improvement of the paper.

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