Using viscosity approximation method, we study
strong convergence to a common element of the set of solutions of
an equilibrium problem and the set of common fixed points of a finite
family of multivalued mappings satisfying the condition (E) in the
setting of Hilbert space. Our results improve and extend some recent
results in the literature.
1. Introduction
Let H be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let C be a nonempty closed convex subset H. A subset C⊂H is called proximal if, for each x∈H, there exists an element y∈C such that
(1)∥x-y∥=dist(x,C)=inf{∥x-z∥:z∈C}.
A single-valued mapping T:C→C is said to be nonexpansive, if
(2)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.
Let PC be a nearest point projection of H into C; that is, for x∈H, PCx is a unique nearest point in C with the property
(3)∥x-PCx∥∶=inf{∥x-y∥:y∈C}.
We denote by CB(C), K(C), and P(C) the collection of all nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of C respectively. The Hausdorff metric H on CB(H) is defined by
(4)H(A,B)∶=max{supx∈Adist(x,B),supy∈Bdist(y,A)},
for all A,B∈CB(H).
Let T:H→2H be a multivalued mapping. An element x∈H is said to be a fixed point of T, if x∈Tx and the set of fixed points of T is denoted by F(T).
A multivalued mapping T:H→CB(H) is called
nonexpansive if
(5)H(Tx,Ty)≤∥x-y∥,x,y∈H;
quasi-nonexpansive if F(T)≠∅ and H(Tx,Tp)≤∥x-p∥ for all x∈H and all p∈F(T).
Recently, García-Falset et al. [1] introduced a new condition on single-valued mappings, called condition (E), which is weaker than nonexpansiveness.
Definition 1.
A mapping T:H→H is said to satisfy condition (Eμ) provided that
(6)∥x-Ty∥≤μ∥x-Tx∥+∥x-y∥,x,y∈H.
We say that T satisfies condition (E) whenever T satisfies (Eμ) for some μ≥1.
Recently, Abkar and Eslamian [2, 3] generalized this condition for multivalued mappings as follows.
Definition 2.
A multivalued mapping T:H→CB(H) is said to satisfy condition (E) provided that
(7)H(Tx,Ty)≤μdist(x,Tx)+∥x-y∥,x,y∈H,
for some μ≥1.
It is obvious that every nonexpansive multivalued mapping T:H→CB(H) satisfies the condition (E), and every mapping T:H→CB(H) which satisfies the condition (E) with nonempty fixed point set F(T) is quasi-nonexpansive.
Example 3.
Let us define a mapping T on [0,3] by
(8)T(x)={[0,x3],x≠3[1,2]x=3.
It is easy to see that T satisfies the condition (E) but is not nonexpansive. Indeed, for x,y∈[0,3), H(Tx,Ty)=|(x-y)/3|≤|x-y|. Let x=0 and y=3. Then H(Tx,Ty)=2≤3=|x-y|. If x∈(0,3) and y=3, then, we have dist(x,Tx)=2x/3 and dist(y,Ty)=1; hence
(9)H(Tx,Ty)=2-x3≤3-x+4x3=|x-y|+2dist(x,Tx).
Thus, T satisfies the condition (E). However, T is not nonexpansive; indeed for x=3 and y=7/3, H(Tx,Ty)=11/9>2/3=|x-y|.
Let Ψ:C×C→ℝ be a bifunction. The equilibrium problem associated with the bifunction Ψ and the set C is:
(10)findx∈CsuchthatΨ(x,y)≥0,∀y∈C.
Such a point x∈C is called the solution of the equilibrium problem. The set of solutions is denoted by EP(Ψ).
A broad class of problems in optimization theory, such as variational inequality, convex minimization, and fixed point problems, can be formulated as an equilibrium problem; see [4, 5]. In the literature, many techniques and algorithms have been proposed to analyze the existence and approximation of a solution to equilibrium problem; see [6]. Many researchers have studied various iteration processes for finding a common element of the set of solutions of the equilibrium problems and the set of fixed points of a class of nonlinear mappings. For example, see [7–22].
Fixed points and fixed point iteration process for nonexpansive mappings have been studied extensively by many authors to solve nonlinear operator equations, as well as variational inequalities; see, for example, [23–28]. In the recent years, fixed point theory for multivalued mappings has been studied by many authors; see [29–40] and the references therein.
In this paper, using viscosity approximation method, we study strong convergence to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of multivalued mappings satisfying the condition (E) in the setting of Hilbert space. Our results improve and extend some recent results in the literature.
2. Preliminaries
For solving the equilibrium problem, we assume that the bifunction Ψ satisfies the following conditions:
Ψ(x,x)=0 for any x∈C;
Ψ is monotone; that is, Ψ(x,y)+Ψ(y,x)≤0 for any x,y∈C;
Ψ is upper-hemicontinuous; that is, for each x,y,z∈C,
(11)limsupt→0+Ψ(tz+(1-t)x,y)≤Ψ(x,y);
Ψ(x,.) is convex and lower semicontinuous for each x∈C.
Lemma 4 (see [4]).
Let C be a nonempty closed convex subset of H and let Ψ be a bifunction of C×C into ℝ satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that
(12)Ψ(z,y)+1r〈y-z,z-x〉≥0∀y∈C.
Lemma 5 (see [6]).
Assume that Ψ:C×C→ℝ satisfies (A1)–(A4). For r>0 and x∈H, define a mapping Sr:H→C as follows:(13)Srx={z∈C:Ψ(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}.
Then, the following hold:
Sr is single valued;
Sr is firmly nonexpansive; that is, for any x,y∈H,
(14)∥Srx-Sry∥2≤〈Srx-Sry,x-y〉;
F(Sr)=EP(Ψ);
EP(Ψ) is closed and convex.
Lemma 6 (see [41]).
Let H be a real Hilbert space. Then, for all x,y,z∈H and α,β,γ∈[0,1] with α+β+γ=1 one has
(15)∥αx+βy+γz∥2=α∥x∥2+β∥y∥2+γ∥z∥2-αβ∥x-y∥2-αγ∥x-z∥2-βγ∥z-y∥2.
Lemma 7.
For every x and y in a Hilbert space H, the following inequality holds:
(16)∥x+y∥2≤∥x∥2+2〈y,x+y〉.
Lemma 8 (see [42]).
Let {an} be a sequence of nonnegative real numbers, {αn} a sequence in (0,1) with ∑n=1∞αn=∞, {γn} a sequence of nonnegative real numbers with ∑n=1∞γn<∞, and {βn} a sequence of real numbers with limsupn→∞βn≤0. Suppose that the following inequality holds:
(17)an+1≤(1-αn)an+αnβn+γn,n≥0.
Then, limn→∞an=0.
Lemma 9 (see [43]).
Let {un} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {uni} of {un} such that uni<uni+1 for all i≥0. For every sufficiently large number n≥n0, define an integer sequence {τ(n)} as
(18)τ(n)=max{k≤n:uk<uk+1}.
Then, τ(n)→∞ as n→∞ and for all n≥n0,
(19)max{uτ(n),un}≤uτ(n)+1.
Lemma 10 (see [20]).
Let C be a closed convex subset of a real Hilbert space H. Let T:C→CB(C) be a quasi-nonexpansive multivalued mapping. If F(T)≠∅ and T(p)={p} for all p∈F(T). Then F(T) is closed and convex.
Lemma 11 (see [20]).
Let C be a closed convex subset of a real Hilbert space H. Let T:C→P(C) be a multivalued mapping such that PT is quasi-nonexpansive and F(T)≠∅, where PT(x)={y∈Tx:∥x-y∥=dist(x,Tx)}. Then, F(T) is closed and convex.
Lemma 12 (see [16, 20]).
Let C be a nonempty closed convex subset of a real Hilbert space H. Let T:C→K(C) be a multivalued mapping satisfying the condition (E). If xn converges weakly to v and limn→∞dist(xn,Txn)=0, then v∈Tv.
3. A Strong Convergence TheoremTheorem 13.
Let C be a nonempty closed convex subset of a real Hilbert space H and Ψ a bifunction of C×C into ℝ satisfying (A1)–(A4). Let Ti:C→CB(C) (i=1,2,…,m) be a finite family of multivalued mappings, each satisfying condition (E). Assume further that ℱ=⋂i=1mF(Ti)⋂EP(Ψ)≠∅ and Ti(p)={p},(i=1,2,…,m) for each p∈ℱ. Let f be a k-contraction of C into itself. Let {xn} and {un} be sequences generated the following algorithm:
(20)x0∈C,un∈CsuchthatΨ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈Cyn,1=an,1un+bn,1xn+cn,1zn,1,yn,2=an,2un+bn,2zn,1+cn,2zn,2,yn,3=an,3un+bn,3zn,2+cn,3zn,3⋮yn,m=an,mun+bn,mzn,m-1+cn,mzn,m,xn+1=ϑnf(xn)+(1-ϑn)yn,m,∀n≥0,
where zn,1∈T1(un), zn,k∈Tk(yn,k-1) for k=2,…,m, and {an,i}, {bn,i}, {cn,i}, {ϑn}, and {rn} satisfy the following conditions:
Then, the sequences {xn} and {un} converge strongly to q∈ℱ, where q=Pℱf(q).
Proof.
Let Q=Pℱ. It is easy to see that Qf is a contraction. By Banach contraction principle, there exists a q∈ℱ such that q=Pℱf(q). From Lemma 5 for all n≥0, we have (21)∥un-q∥=∥Srnxn-Srnq∥≤∥xn-q∥.
We show that {xn} is bounded. Since, for each i=1,2,…,m, Ti satisfies the condition (E) and we have
(22)∥yn,1-q∥=∥an,1un+bn,1xn+cn,1zn,1-q∥≤an,1∥un-q∥+bn,1∥xn-q∥+cn,1∥zn,1-q∥=an,1∥un-q∥+bn,1∥xn-q∥+cn,1dist(zn,1,T1q)≤an,1∥un-q∥+bn,1∥xn-q∥+cn,1H(T1un,T1q)≤an,1∥un-q∥+bn,1∥xn-q∥+cn,1∥un-q∥≤∥xn-q∥,(23)∥yn,2-q∥=∥an,2un+bn,2zn,1+cn,2zn,2-q∥≤an,2∥un-q∥+bn,2∥zn,1-q∥+cn,2∥zn,2-q∥=an,2∥un-q∥+bn,2dist(zn,1,T1q)+cn,2dist(zn,2,T2q)≤an,2∥un-q∥+bn,2H(T1un,T1q)+cn,2H(T2yn,1,T2q)≤an,2∥un-q∥+bn,2∥un-q∥+cn,2∥yn,1-q∥≤∥xn-q∥.
By continuing this process, we obtain
(24)∥yn,m-q∥≤∥xn-q∥.
This implies that
(25)∥xn+1-q∥=∥ϑnfxn+(1-ϑn)yn-q∥≤ϑn∥fxn-q∥+(1-ϑn)∥yn-q∥≤ϑn(∥fxn-fq∥+∥fq-q∥)+(1-ϑn)∥xn-q∥≤ϑnk∥xn-q∥+ϑn∥fq-q∥+(1-ϑn)∥xn-q∥=(1-ϑn(1-k))∥xn-q∥+ϑn∥fq-q∥≤max{∥xn-q∥,∥fq-q∥1-k}.
By induction, we get
(26)∥xn-q∥≤max{∥x0-q∥,∥fq-q∥1-k},
for all n∈ℕ. This implies that {xn} is bounded and we also obtain that {un},{yn},{fxn}, and {zn,i} are bounded. Next, we show that limn→∞dist(un,Tiun)=0 for each i∈ℕ. By Lemma 6, we have
(27)∥yn,1-q∥2=∥an,1un+bn,1xn+cn,1zn,1-q∥2≤an,1∥un-q∥2+bn,1∥xn-q∥2+cn,1∥zn,1-q∥2-an,1bn,1∥xn-un∥2-an,1cn,1∥un-zn,1∥2=an,1∥un-q∥2+bn,1∥xn-q∥2+cn,1dist(zn,1,T1q)2-an,1bn,1∥xn-un∥2-an,1cn,1∥un-zn,1∥2≤an,1∥un-q∥2+bn,1∥xn-q∥2+cn,1H(T1un,T1q)2-an,1bn,1∥xn-un∥2-an,1cn,1∥un-zn,1∥2≤an,1∥un-q∥2+bn,1∥xn-q∥2+cn,1∥un-q∥2-an,1bn,1∥xn-un∥2-an,1cn,1∥un-zn,1∥2≤∥xn-q∥2-an,1bn,1∥xn-un∥2-an,1cn,1∥un-zn,1∥2.
Applying Lemma 6 once more, we have
(28)∥yn,2-q∥2=∥an,2un+bn,2zn,1+cn,2zn,2-q∥2≤an,2∥un-q∥2+bn,2∥zn,1-q∥2+cn,2∥zn,2-q∥2-an,2cn,2∥un-zn,2∥2=an,2∥un-q∥2+bn,2dist(zn,1,T1q)2+cn,2dist(zn,2,T2q)2-an,2cn,2∥un-zn,2∥2≤an,2∥un-q∥2+bn,2H(T1un,T1q)2+cn,2H(T1yn,1,T2q)2-an,2cn,2∥un-zn,2∥2≤an,2∥un-q∥2+bn,2∥un-q∥2+cn,2∥yn,1-q∥2-an,2cn,2∥un-zn,2∥2≤∥xn-q∥2-an,2cn,2∥un-zn,2∥2-an,1cn,1cn,2∥un-zn,1∥2-an,1bn,1cn,2∥xn-un∥2.
By continuing this process we have
(29)∥yn,m-q∥2=∥an,mun+bn,mzn,m-1+cn,mzn,m-q∥2≤an,m∥un-q∥2+bn,m∥zn,m-1-q∥2+cn,m∥zn,m-q∥2-an,mcn,m∥un-zn,m∥2=an,m∥un-q∥2+bn,mdist(zn,m-1,Tm-1q)2+cn,mdist(zn,m,Tmq)2-an,mcn,m∥un-zn,m∥2≤an,m∥un-q∥2+bn,mH(Tm-1yn,m-2,Tm-1q)2+cn,mH(Tmyn,m-1,Tmq)2-an,mcn,m∥un-zn,m∥2≤an,m∥un-q∥2+bn,m∥yn,m-2-q∥2+cn,m∥yn,m-1-q∥2-an,mcn,m∥un-zn,m∥2≤∥un-q∥2-an,mcn,m∥un-zn,m∥2-an,m-1cn,m-1cn,m∥un-zn,m-1∥2-⋯-an,1cn,1cn,2…cn,m∥un-zn,1∥2-an,1bn,1cn,2…cn,m∥un-xn∥2,
which implies that
(30)∥xn+1-q∥2=∥ϑnfxn+(1-ϑn)yn,m-q∥2≤ϑn∥fxn-q∥2+(1-ϑn)∥yn,m-q∥2≤ϑn∥fxn-q∥2+(1-ϑn)∥un-q∥2-(1-ϑn)an,mcn,m∥un-zn,m∥2-(1-ϑn)an,m-1cn,m-1cn,m∥un-zn,m-1∥2-⋯-(1-ϑn)an,1cn,1cn,2…cn,m∥un-zn,1∥2-(1-ϑn)an,1bn,1cn,2…cn,m∥un-xn∥2.
Therefore, we have that
(31)(1-ϑn)an,1bn,1cn,2…cn,m∥un-xn∥2≤∥xn-q∥2-∥xn+1-q∥2+ϑn∥γfxn-q∥.
In order to prove that xn→q as n→∞, we consider the following two cases.
Case 1. Suppose that there exists n0 such that {∥xn-q∥} is nonincreasing, for all n≥n0. Boundedness of {∥xn-q∥} implies that ∥xn-q∥ is convergent. From (31) and conditions (i), (ii) we have that
(32)limn→∞∥un-xn∥=0.
By a similar argument, for k=1,2,…,m, we obtain that
(33)limn→∞∥un-zn,k∥=0.
Hence,
(34)limn→∞dist(un,T1un)≤limn→∞∥un-zn,1∥=0,limn→∞dist(un,Tkyn,k-1)≤limn→∞∥un-zn,k∥=0,(k=2,…,m).
Therefore, we have
(35)limn→∞∥un-yn,1∥≤limn→∞bn,1∥un-xn∥+limn→∞cn,1∥un-zn,1∥=0.
For k=2,…,m, we have
(36)limn→∞∥un-yn,k∥≤limn→∞bn,k∥un-zn,k-1∥+limn→∞cn,k∥un-zn,k∥=0.
Using the previous inequality for k=2,…,m, we have
(37)dist(un,Tkun)≤dist(un,Tkyn,k-1)+H(Tkyn,k-1,Tkun)≤dist(un,Tkyn,k-1)+μdist(yn,k-1,Tkyn,k-1)+∥yn,k-1-un∥≤(μ+1)dist(un,Tkyn,k-1)+(μ+1)∥yn,k-1-un∥≤(μ+1)∥un-zn,k∥+(μ+1)∥yn,k-1-un∥⟶0,n⟶∞.
Next, we show that
(38)limsupn→∞〈q-fq,q-xn〉≤0,
where q=Pℱf(q). To show this inequality, we choose a subsequence {xni} of {xn} such that
(39)limi→∞〈q-fq,q-xni〉=limsupn→∞〈q-fq,q-xn〉.
Since {xni} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly to v. Without loss of generality, we can assume that xni converges weakly to v. Since limn→∞∥xn-un∥=0, we have uni converges weakly to v. We show that v∈ℱ. Let us show v∈EP(Ψ). Since un=Srnxn, we have
(40)Ψ(un,y)+1rn〈y-un,un-xn〉≥0∀y∈C.
From (A2), we have
(41)1rn〈y-un,un-xn〉≥Ψ(y,un).
Replacing n with ni, we have
(42)〈y-uni,uni-xnirni〉≥Ψ(y,uni).
From (A4), we have
(43)0≥Ψ(y,v),∀y∈C.
For t∈(0,1] and y∈C, let yt=ty+(1-t)v. Since y,v∈C, and C is convex, we have yt∈C and hence Ψ(yt,v)≤0. So, from (A1) and (A4) we have
(44)0=Ψ(yt,yt)≤tΨ(yt,y)+(1-t)Ψ(yt,v)≤tΨ(yt,y),
which gives 0≤Ψ(yt,y). Letting t→0, we have, for each y∈C, 0≤Ψ(v,y) Also, since uni⇀v and limn→∞dist(un,Tiun)=0, by Lemma 12 we have v∈⋂i=1mF(Ti). Hence, v∈ℱ. Since q=Pℱf(q) and v∈ℱ, it follows that
(45)limsupn→∞〈q-fq,q-xn〉=limi→∞〈q-fq,q-xni〉=〈q-fq,q-v〉≤0.
By using Lemma 7 and inequality (31) we have
(46)∥xn+1-q∥2≤∥(1-ϑn)(yn,m-q)∥2+2ϑn〈fxn-q,xn+1-q〉≤(1-ϑn)2∥yn,m-q∥2+2ϑn〈fxn-fq,xn+1-q〉+2ϑn〈fq-q,xn+1-q〉≤(1-ϑn)2∥xn-q∥2+2ϑnk∥xn-q∥∥xn+1-q∥+2ϑn〈fq-q,xn+1-q〉≤(1-ϑn)2∥xn-q∥2+ϑnk(∥xn-q∥2+∥xn+1-q∥2)+2ϑn〈fq-q,xn+1-q〉≤((1-ϑn)2+ϑnk)∥xn-q∥2+ϑnk∥xn+1-q∥2+2ϑn〈fq-q,xn+1-q〉.
This implies that
(47)∥xn+1-q∥2≤(1-2(1-k)ϑn1-ϑnk)∥xn-q∥2+ϑn21-ϑnk∥xn-q∥2+2ϑn1-ϑnk〈fq-q,xn+1-q〉.
From Lemma 8, we conclude that the sequence {xn} converges strongly to q.
Case 2. Assume that there exists a subsequence {xnj} of {xn} such that
(48)∥xnj-q∥<∥xnj+1-q∥,
for all j∈ℕ. In this case, from Lemma 9, there exists a nondecreasing sequence {τ(n)} of ℕ for all n≥n0 (for some n0 large enough) such that τ(n)→∞ as n→∞ and the following inequalities hold for all n≥n0:(49)∥xτ(n)-q∥≤∥xτ(n)+1-q∥,∥xn-q∥≤∥xτ(n)+1-q∥.
From (31) we obtain limn→∞∥uτ(n)-Tiuτ(n)∥=0, and limn→∞∥uτ(n)-xτ(n)∥=0. Following an argument similar to that in Case 1, we have
(50)limn→∞∥xτ(n)-q∥=0,limn→∞∥xτ(n)+1-q∥=0.
Thus, by Lemma 9 we have
(51)0≤∥xn-q∥≤max{∥xτ(n)-q∥,∥xn-q∥}≤∥xτ(n)+1-q∥.
Therefore, {xn} converges strongly to q=Pℱf(q). This completes the proof.
Now, we remove the condition that T(p)={p} for all p∈ℱ, and state the following theorem.
Theorem 14.
Let C be a nonempty closed convex subset of a real Hilbert space H and Ψ a bifunction of C×C into ℝ satisfying (A1)–(A4). Let, for each 1≤i≤m, Ti:C→P(C) be multivalued mappings such that PTi satisfies the condition (E). Assume that ℱ=⋂i=1mF(Ti)⋂EP(Ψ)≠∅. Let f be a k-contraction of C into itself. Let {xn} and {un} be sequences generated the following algorithm:
(52)x0∈C,un∈CsuchthatΨ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈Cyn,1=an,1un+bn,1xn+cn,1zn,1,yn,2=an,2un+bn,2zn,1+cn,2zn,2,yn,3=an,3un+bn,3zn,2+cn,3zn,3⋮yn,m=an,mun+bn,mzn,m-1+cn,mzn,m,xn+1=ϑnfxn+(1-ϑn)yn,m,∀n≥0,
where zn,1∈PT1(un), zn,k∈PTk(yn,k-1) for k=2,…,m, and {an,i}, {bn,i}, {cn,i}, {ϑn} and, {rn} satisfy the following conditions:
Then, the sequences {xn} and {un} converge strongly to q∈ℱ, where q=Pℱf(q).
Proof.
Let p∈ℱ; then PTi(p)={p},(i=1,2,…,m). Now by substituting PTi instead of Ti, and using a similar argument as in the proof of Theorem 13, the desired result follows.
As a corollary for single-valued mappings, we obtain the following result.
Corollary 15.
Let C be a nonempty closed convex subset of a real Hilbert space H and Ψ a bifunction of C×C into ℝ satisfying (A1)–(A4). Let, for each 1≤i≤m, Ti:C→C be a finite family of mappings satisfying condition (E). Assume that ℱ=⋂i=1mF(Ti)⋂EP(Ψ)≠∅. Let f be a k-contraction of C into itself. Let {xn} and {un} be sequences generated the following algorithm:
(53)x0∈C,un∈CsuchthatΨ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈Cyn,1=an,1un+bn,1xn+cn,1T1un,yn,2=an,2un+bn,2T1un+cn,2T2yn,1⋮yn,m=an,mun+bn,mTm-1yn,m-2+Tmyn,m-1,xn+1=ϑnfxn+(1-ϑn)yn,m,∀n≥0,
where {an,i}, {bn,i},{cn,i}, {ϑn}, and {rn} satisfy the following conditions:
Then, the sequences {xn} and {un} converge strongly to q∈ℱ, where q=Pℱf(q).
Remark 16.
Our results generalize the corresponding results of S. Takahashi and W. Takahashi [9] from a single valued nonexpansive mapping to a finite family of multivalued mappings satisfying the condition (E). Our results also improve the recent results of Eslamian [16].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors therefore, acknowledge with thanks DSR for technical and financial support. The authors are also thankful to the referees for their valuable suggestion/comments.
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