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Li and Jiang Fixed Point Theory and Applications (2016) 2016:40 DOI 10.1186/s13663-016-0526-3

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Fixed point theorems of JS-quasi-contractions Zhilong Li1,2 and Shujun Jiang3* *

Correspondence: [email protected] 3 Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, 330013, China Full list of author information is available at the end of the article

Abstract In this paper, we introduce the concept of JS-quasi-contraction and prove some fixed point results for JS-quasi-contractions in complete metric spaces under the assumption that the involving function is nondecreasing and continuous. These fixed point results extend and improve many existing results since some assumptions made there are removed or weakened. In addition, we present some examples showing the usability of our results. MSC: 47H10; 54H25 Keywords: fixed point theorem; JS-quasi-contraction

1 Introductions Recall the Banach contraction principle [], which states that each Banach contraction T : X → X (i.e., there exists k ∈ [, ) such that d(Tx, Ty) ≤ kd(x, y) for all x, y ∈ X) has a unique fixed point, provided that (X, d) is a complete metric space. According to its importance and simplicity, this principle have been extended and generalized in various directions (see [–]). For example, the concepts of Ćirić contraction [], quasi-contraction [], JScontraction [], and JS-Ćirić contraction [] have been introduced, and many interesting generalizations of the Banach contraction principle are obtained. Following Hussain et al. [], we denote by  the set of all nondecreasing functions ψ : [, +∞) → [, +∞) satisfying the following conditions: () () () ()

ψ(t) =  if and only if t = ; for each sequence {tn } ⊂ (, +∞), limn→∞ ψ(tn ) =  if and only if limn→∞ tn = ; there exist r ∈ (, ) and l ∈ (, +∞] such that limt→+ ψ(t)– = l; tr ψ(t + s) ≤ ψ(t)ψ(s) for all t, s > .

For convenience, we set:   = ψ : (, +∞) → (, +∞) : ψ is a nondecreasing function satisfying  () and () ,    = ψ : (, +∞) → (, +∞) : ψ is a nondecreasing continuous function ,   = ψ : [, +∞) → [, +∞) : ψ is a nondecreasing continuous function satisfying  () , © 2016 Li and Jiang. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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  = ψ : [, +∞) → [, +∞) : ψ is a nondecreasing continuous function satisfying  () and () . t

Example  Let f (t) = ete for t ≥ . Then f ∈  ∩  , but f ∈/  ∪  ∪  since t ete – t→+ t r

lim

s+t

=  for each r ∈ (, ) and e(s+t)e

s

t

> ese ete for all s, t > .

a

Example  Let g(t) = et for t ≥ , where a > . When a ∈ (, ), g ∈  ∩ ∩ ∩ ∩ . t When a = , g ∈  ∩  ∩  , but g ∈/  ∪  since limt→+ e t– r =  for each r ∈ (, ). When a > , g ∈  ∩  , but g ∈/  ∪  ∪  since limt→+ a a a and e(t+s) > et es for all s, t > .

a

et – tr

=  for each r ∈ (, )

Example  Let h(t) =  for t ∈ [, a] and h(t) = et–a for t > a, where a > . Then h ∈  , but h ∈/  ∪  ∪  ∪  since neither () nor () is satisfied. √

t

. p ∈  ∩  ∩  , but p ∈/  ∪  since Example  Let p(t) = e te for √ √t ≥ √ Then √ √ s t (t +s )e(t +s ) = e e > e e = e t e  e s e  whenever t = s = . e Remark  (i) Clearly,  ⊆  and  ⊆  ⊆  . Moreover, from Examples - it follows that  ⊂  and  ⊂  ⊂  . (ii) From Examples - we can conclude that  ⊂  ,  ⊂ ,  ∩  = ∅, and  ∩  = ∅. Definition  Let (X, d) be a metric space. A mapping T : X → X is said to be: (i) a Ćirić contraction [] if there exist nonnegative numbers q, r, s, t with q + r + s + t <  such that   d(Tx, Ty) ≤ qd(x, y) + rd(x, Tx) + sd(y, Ty) + t d(x, Ty) + d(y, Tx) ,

∀x, y ∈ X;

(ii) a quasi-contraction [] if there exists λ ∈ [, ) such that d(Tx, Ty) ≤ λMd (x, y),

∀x, y ∈ X,

where Md (x, y) = max{d(x, y), d(x, Tx), d(y, Ty), d(x,Ty)+d(y,Tx) };  (iii) a JS-contraction [] if there exist ψ ∈  and λ ∈ [, ) such that    λ ψ d(Tx, Ty) ≤ ψ d(x, y) ,

∀x, y ∈ X with Tx = Ty;

()

(iv) a JS-Ćirić contraction [] if there exist ψ ∈  and nonnegative numbers q, r, s, t with q + r + s + t <  such that r  s  t    q  ψ d(Tx, Ty) ≤ ψ d(x, y) ψ d(x, Tx) ψ d(y, Ty) ψ d(x, Ty) + d(y, Tx) , ∀x, y ∈ X.

()

In the s, Ćirić [, ] established the following two well-known generalizations of the Banach contraction principle.

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Theorem  (see []) Let (X, d) be a complete metric space, and T : X → X a Ćirić contraction. Then T has a unique fixed point in X. Theorem  (see []) Let (X, d) be a complete metric space, and T : X → X a quasicontraction. Then T has a unique fixed point in X. Recently, Jleli and Samet [] proved the following fixed point result of JS-contractions, which is a real generalization of the Banach contraction principle. Theorem  (see [], Corollary .) Let (X, d) be a complete metric space, and T : X → X a JS-contraction with ψ ∈  . Then T has a unique fixed point in X. Remark  The Banach contraction principe follows immediately from Theorem . In√ √ deed, let T : X → X be a JS-contraction. Then, if we choose ψ(t) = e t ∈  and λ = k in  √  (), then we get d(Tx, Ty) ≤ k d(x, y), that is, d(Tx, Ty) ≤ kd(x, y),

∀x, y ∈ X,

which means that T is a Banach contraction. Remark that Theorem  is a real generalization of the Banach contraction principle (see Example in []), but the Banach contraction principle is not a particular case of Theorem  with ψ(t) = et since et ∈/ . Recently, Hussain et al. [] presented the following extension of Theorem  and Theorem . Theorem  (see [], Theorem .) Let (X, d) be a complete metric space, and T : X → X a continuous JS-Ćirić contraction with ψ ∈ . Then T has a unique fixed point in X. Remark  It is clear that Theorem  is not a particular case of Theorem  since, in Theo√ rem , a mapping T does not have to be continuous. In addition, letting ψ(t) = e t in (), we can only obtain 

    d(Tx, Ty) ≤ q d(x, y) + r d(x, Tx) + s d(y, Ty) + t d(x, Ty) + d(y, Tx),

∀x, y ∈ X,

which does not imply that T is a Ćirić contraction whenever qr +rs+st = , and hence Theorem  cannot be derived from Theorem  by using the same method as in []. Therefore, Theorem  may not be a real generalization of Theorem . In this paper, we generalize and improve Theorems - and remove or weaken the assumptions made on ψ appearing in [, ].

2 Main results Definition  Let (X, d) be a metric space. A mapping T : X → X is said to be a JS-quasicontraction if there exist a function ψ : (, +∞) → (, +∞) and λ ∈ (, ) such that λ    ψ d(Tx, Ty) ≤ ψ Md (x, y) ,

∀x, y ∈ X with Tx = Ty.

()

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Remark  (i) Each quasi-contraction is a JS-quasi-contraction with ψ(t) = et . (ii) Each JS-contraction is a JS-quasi-contraction whenever ψ is nondecreasing. (iii) Let T : X → X and ψ : [, +∞) → [, +∞) be such that

   q  r  s d(x, Ty) + d(y, Tx) t ψ d(Tx, Ty) ≤ ψ d(x, y) ψ d(x, Tx) ψ d(y, Ty) ψ ,  ∀x, y ∈ X,

()

where q, r, s, t are nonnegative numbers with q + r + s + t < . Then T is a JS-quasicontraction with λ = p + r + s + t, provided that () is satisfied. (iv) Let T : X → X and ψ : [, +∞) → [, +∞) be such that () is satisfied. Suppose that ψ is a nondecreasing function such that () is satisfied. Then, ψ(d(x, Ty) + d(y, Tx))t ≤ ψ( d(x,Ty)+d(y,Tx) )t for all x, y ∈ X, and so () holds. Moreover, if () is satisfied, then it  follows from (iii) that T is a JS-quasi-contraction with λ = p + r + s + t. Therefore, a JSĆirić contraction with ψ ∈  or ψ ∈  is certainly a JS-quasi-contraction. Theorem  Let (X, d) be a complete metric space, and T : X → X a JS-quasi-contraction with ψ ∈  . Then T has a unique fixed point in X. Proof Fix x ∈ X and let xn = T n x for each n. We first show that T has a fixed point. We may assume that d(xn , xn+ ) > ,

∀n.

()

Otherwise, there exists some positive integer p such that xp = xp+ and so xp is a fixed point of T, and the proof is complete. Note that   max d(xn– , xn ), d(xn , xn+ )  d(xn– , xn+ ) + d(xn , xn ) ≤ max d(xn– , xn ), d(xn , xn+ ),   d(xn– , xn ) + d(xn , xn+ ) = Md (xn– , xn ) ≤ max d(xn– , xn ), d(xn , xn+ ),    = max d(xn– , xn ), d(xn , xn+ ) , ∀n. Then by () and () we get      λ  ψ d(xn , xn+ ) ≤ ψ Md (xn– , xn ) = ψ max d(xn– , xn ), d(xn , xn+ ) ,

∀n.

If there exists some m such that d(xm , xm + ) > d(xm – , xm ), then by () we get    λ   ψ d(xm , xm + ) ≤ ψ d(xm , xm + ) < ψ d(xm , xm + ) , a contradiction. Consequently, we obtain d(xn , xn+ ) ≤ d(xn– , xn ),

∀n,

()

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which implies that {d(xn , xn+ )} is a decreasing sequence of nonnegative reals, and so there exists α ≥  such that limn→∞ d(xn , xn+ ) = α and d(xn , xn+ ) ≥ α.

()

Suppose that α > . By () and (), since ψ is nondecreasing, we get    λ   λn  < ψ(α) ≤ ψ d(xn , xn+ ) ≤ ψ d(xn– , xn ) ≤ · · · ≤ ψ d(x , x ) ,

∀n.

()

Letting n → ∞ in this inequality, we get ψ(α) = , which contradicts the assumption that ψ(t) >  for each t > . Consequently, we have α = , that is, lim d(xn , xn+ ) = .

()

n→∞

We claim that lim d(xn , xm ) = .

()

m,n→∞

Otherwise, there exist β >  and two subsequences {xmk } and {xnk } of {xn } such that mk is the smallest index with mk > nk > k for which d(xnk , xmk ) ≥ β,

()

which indicates that d(xnk , xmk – ) < β.

()

By (), (), and the triangle inequality we get β ≤ d(xnk , xmk ) ≤ d(xnk , xmk – ) + d(xmk – , xmk ) < β + d(xmk – , xmk ),

∀mk > nk > k.

Letting k → ∞ in this inequality, by () we obtain lim d(xnk , xmk ) = β.

()

k→∞

Also by the triangle inequality we get d(xnk , xmk ) – d(xnk + , xnk ) – d(xmk , xmk + ) ≤ d(xnk + , xmk + ) ≤ d(xnk + , xnk ) + d(xnk , xmk ) + d(xmk , xmk + ),

∀mk > nk > k.

Letting k → ∞ in this inequality, by () and () we obtain lim d(xnk + , xmk + ) = β.

k→∞

()

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In analogy to (), by () and () we can prove that lim d(xnk , xmk + ) = lim d(xnk + , xmk ) = β.

k→∞

()

k→∞

It follows (), (), and () that lim Md (xnk , xmk ) = β,

()

k→∞

where β ≤ d(xnk , xmk ) ≤ Md (xnk , xmk )  d(xnk , xmk + ) + d(xnk + , xmk ) . = max d(xnk , xmk )d(xnk , xnk + ), d(xmk , xmk + ),  Note that () and () implies that there exists a positive integer k such that d(xnk + , xmk + ) >  and

Md (xnk , xmk ) > ,

∀k ≥ k .

Thus, by () we get     λ  ψ d(xnk + , xmk + ) = ψ d(Txnk , Txmk ) ≤ ψ Md (xnk , xmk ) ,

∀mk > nk > k ≥ k .

Letting k → ∞ in this inequality, by (), (), and the continuity of ψ we obtain    λ ψ(β) = lim ψ d(xnk + , xmk + ) ≤ lim ψ Md (xnk , xmk ) = ψ(β)λ < ψ(β), k→∞

k→∞

a contradiction. Consequently, () holds, that is, {xn } is a Cauchy sequence in X. Moreover, by the completeness of (X, d) there exists x∗ ∈ X such that   lim d xn , x∗ = .

()

n→∞

Suppose that d(x∗ , Tx∗ ) > . It follows from () and () that there exists a positive integer n such that     d xn , x∗ ≤ d x∗ , Tx∗

  and d(xn , xn+ ) ≤ d x∗ , Tx∗ ,

∀n ≥ n .

()

Denoting       d(xn , Tx∗ ) + d(x∗ , xn+ )  Md xn , x∗ = max d xn , x∗ , d(xn , xn+ ), d x∗ , Tx∗ ,  for each n, by () we get     Md xn , x∗ = d x∗ , Tx∗ ,

∀n ≥ n .

()

From the continuity of d it follows that     lim d xn+ , Tx∗ = d x∗ , Tx∗ ,

n→∞

()

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which implies that there exists a positive integer n such that   d xn+ , Tx∗ > ,

∀n ≥ n .

()

Thus, by () we get         λ ψ d xn+ , Tx∗ = ψ d Txn , Tx∗ ≤ ψ Md xn , x∗ ,

∀n ≥ n ,

and so, by (),      λ ψ d xn+ , Tx∗ ≤ ψ d x∗ , Tx∗ ,

∀n ≥ max{n , n }.

()

Letting n → ∞ in this inequality, by () and the continuity of ψ we obtain         λ    ψ d x∗ , Tx∗ = ψ d xn+ , Tx∗ ≤ ψ d x∗ , Tx∗ < ψ d x∗ , Tx∗ , a contradiction. Consequently, d(x∗ , Tx∗ ) = , that is, x∗ = Tx∗ . Let x be another fixed point of T. Suppose that d(x, x∗ ) > . Then by () we get         λ ψ d x, x∗ = ψ d Tx, Tx∗ ≤ ψ Md x, x∗ , where Md (x, x∗ ) = max{d(x, x∗ ), d(x,x

∗ )+d(x∗ ,x)



} = d(x, x∗ ). Thus, we obtain

     λ    ψ d x, x∗ ≤ ψ d x, x∗ < ψ d x, x∗ , a contradiction. Consequently, we have x = x∗ . This shows that x∗ is the unique fixed point of T. The proof is completed.  Remark  In view of Example  and (i) of Remark , Theorem  is a particular case of Theorem  with ψ(t) = et ∈  . The following example shows that Theorem  is a real generalization of Theorem . Example  Let X = {τn } and d(x, y) = |x – y|, where τn = n(n+)(n+) for each n. Clearly, (X, d)  is a complete metric space. Define the mapping T : X → X by Tτ = τ and Tτn = τn– for each n ≥ . t

We show that T is a JS-quasi-contraction with ψ(t) = ete . In fact, it suffices to show that there exists λ ∈ (, ) such that, for all x, y ∈ X with Tx = Ty, d(Tx, Ty)ed(Tx,Ty)–Md (x,y) ≤ λ. Md (x, y) In the case of m >  and n = , we have d(Tτ , Tτm ) = d(τ , τm– ) =

(m–)m(m+)– 

 d(τ , τm– ) + d(τm , τ ) Md (τ , τm ) = max d(τ , τm ), d(τ , τ ), d(τm , τm– ),  = d(τ , τm ) =

m(m + )(m + ) –  , 

and

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and so d(Tτ , Tτm )ed(Tτ ,Tτm )–Md (τ ,τm ) (m – )m(m + ) –  –m(m+) – = e n > , we have d(Tτn , Tτm ) = d(τn– , τm– ) =

(m – n)(m + n + mn – ) 

and  d(τn , τm– ) + d(τm , τn– ) Md (τn , τm ) = max d(τn , τm ), d(τn , τn– ), d(τm , τm– ),   τm + τm– – τn – τn– = τm – τn = max τm – τn ,  =

(m – n)(m + n + mn + (m + n) + ) , 

and so d(Tτn , Tτm )ed(Tτn ,Tτm )–Md (τn ,τm ) m + n + mn –  =  e(n–m)(m+n+) Md (τn , τm ) m + n + mn + (m + n) +  ≤ e(n–m) ≤ e– . t

This shows that T is a JS-quasi-contraction with ψ(t) = ete and λ ∈ [e– , ). Note that t ete ∈  by Example . Then from Theorem  we know that T has a unique fixed point τ . For each m > , we have lim

m→∞

d(Tτ , Tτm ) (m – )m(m + ) –  = lim = , Md (τ , τm ) m→∞ m(m + )(m + ) – 

which implies that T is not a quasi-contraction. Hence, Theorem  is not applicable here. On the other hand, it is not hard to check that there exists λ ∈ (, ) (resp. nonnegative t numbers q, r, s, t with q + r + s + t < ) such that () (resp. ()) is satisfied with ψ(t) = ete . t But neither Theorem  nor Theorem  is applicable in this situation since ete ∈/  ∪  by Example . Example  Let X = {, , } and d(x, y) = |x – y|. Clearly, (X, d) is a complete metric space. Define the mapping T : X → X by T = T =  and T = . √

t

We show that T is a JS-quasi-contraction with ψ(t) = e te . In fact, it suffices to show that there exists λ ∈ (, ) such that, for all x, y ∈ X with Tx = Ty, d(Tx, Ty)ed(Tx,Ty)–Md (x,y) ≤ λ . Md (x, y) Then, we only need to consider the cases x = , y =  and x = , y = . For both cases, we have d(T, T) = d(T, T) =  and Md (, ) = Md (, ) = , and so

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d(T, T)ed(T,T)–Md (,) d(T, T)ed(T,T)–Md (,) e– = = , Md (, ) Md (, )  √

which implies that T is a JS-quasi-contraction with ψ(t) = e tet

that e

tet

and λ ∈ [

e– , ). 

Note

∈  by Example . Then from Theorem  we know T has a unique fixed point

x = . When x =  and y = , we have d(T, T) = d(, ) =  and hence √

which implies that T is not a JS-contraction with ψ(t) = e

tet

d(T,T)ed(T,T)–d(,) d(,)

= ,

. Therefore, Theorem  is not

applicable here. In addition, it is not hard to check that there exist nonnegative numbers q, r, s, t with √ tet

q + r + s + t <  such that () is satisfied with ψ(t) = e √

applicable here since e

tet

. However, Theorem  is not

∈/  by Example .

Theorem  Let (X, d) be a complete metric space, and T : X → X. Assume that there exist ψ ∈  and nonnegative numbers q, r, s, t with q + r + s + t <  such that () is satisfied. Then T has a unique fixed point in X. Proof In view of (iii) of Remark , T is a JS-quasi-contraction with λ = q + r + s + t. In the case where q + r + s + t = , by () we have ψ(d(Tx, Ty)) =  for all x, y ∈ X. Moreover, by () we get d(Tx, Ty) =  for all x, y ∈ X. This shows that y = Tx is a fixed point of T. Let z be another fixed point of T. Then d(y, z) = d(Ty, Tz) = , and hence y = z, that is, T has a unique fixed point. In the case where  < q + r + s + t < , the conclusion immediately follows from Theorem . The proof is completed.



Remark  Theorem  and Theorem  are respectively particular cases of Theorem  and Theorem  with ψ(t) = et , whereas they are not particular cases of Theorem  and Theorem  with ψ(t) = et since et ∈  ∩  but et ∈/  ∪  . Hence, Theorem  and Theorem  are new generalizations of Theorem  and Theorem . In view of (ii) and (iv) of Remark , we have the following two corollaries of Theorem  and Theorem . Corollary  Let (X, d) be a complete metric space, and T : X → X a JS-contraction with ψ ∈  . Then T has a unique fixed point in X. Corollary  Let (X, d) be a complete metric space, and T : X → X a JS-Ćirić contraction with ψ ∈  . Then T has a unique fixed point in X. Remark  Conditions () and () assumed in Theorems  and  are removed from Corollaries  and  at the expense that ψ is continuous. Thus, Corollaries  and  partially improve Theorems  and . a

Taking ψ(t) = et (a > ) in Theorem , we have the following new generalization of Theorem .

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Corollary  Let (X, d) be a complete metric space, and T : X → X. Assume that there exist a >  and nonnegative numbers q, r, s, t with q + r + s + t <  such that d(Tx, Ty)a ≤ qd(x, y)a + rd(x, Tx)a + sd(y, Ty)a + t

d(x, Ty) + d(y, Tx) 

a ,

∀x, y ∈ X.

()

Then T has a unique fixed point in X. Remark  Theorem  is a particular case of Corollary  with a = . Corollary  (see [], Theorem . and Corollary .) Let (X, d) be a complete metric space, and T : X → X. Assume that there exist nonnegative numbers q, r, s, t with q + r + s + t <  such that  a d(Tx, Ty)a ≤ qd(x, y)a +rd(x, Tx)a +sd(y, Ty)a +t d(x, Ty)+d(y, Tx) , where a =

 

∀x, y ∈ X, ()

or a = n . Then T has a unique fixed point in X.

Proof For each a ∈ (, ], we have (d(x, Ty) + d(y, Tx))a ≤ ( d(x,Ty)+d(y,Tx) )a , and so () im mediately follows from (). Thus, by Corollary , T has a unique fixed point. The proof is completed.  Remark  Theorem . and Corollary . of [] are consequences of Theorem . In fact, let a ∈ (, ] and D(x, y) = d(x, y)a for all x, y ∈ X. Then (X, D) is a complete metric space by the completeness of (X, d). Note that (d(x, Ty) + d(y, Tx))a ≤ d(x, Ty)a + d(y, Tx)a for all x, y ∈ X. Then () implies   D(Tx, Ty) ≤ qD(x, y) + rD(x, Tx) + sD(y, Ty) + t D(x, Ty) + D(y, Tx) ,

∀x, y ∈ X,

that is, T is a Ćirić contraction in (X, D). Therefore, Theorem . and Corollary . of [] immediately follow from Theorem . However, Corollary  cannot be derived from Theorem  by the previous method since the pair (X, D) is not a metric space whenever a > .

Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors have contributed in this work on an equal basis. All authors read and approved the final manuscript. Author details 1 School of Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, China. 2 Research Center of Applied Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, China. 3 Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, 330013, China. Acknowledgements The work was supported by the Natural Science Foundation of China (11161022, 11561026, 71462015), the Natural Science Foundation of Jiangxi Province (20142BCB23013, 20143ACB21012, 20151BAB201003, 20151BAB201023), the Natural Science Foundation of Jiangxi Provincial Education Department (KJLD14034, GJJ150479). Received: 25 November 2015 Accepted: 9 March 2016

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