# Some Endpoint Results forβ-Generalized Weak Contractive Multifunctions

Authors: H. Alikhani; D. Gopal; M. A. Miandaragh; Sh. Rezapour; N. Shahzad
Journal: The Scientific World Journal
 (2013)

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## Abstract

We introduceβ-generalized weak contractive multifunctions and give some
results about endpoints of the multifunctions. Also, we give some results about role of a
point in the existence of endpoints.


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## Body

## 1. Introduction

Let(X,d) be a metric space, CB(X) the collection of all nonempty bounded and closed subsets of X, and H the Hausdorff metric with respect to d; that is, H(A,B)=max{supx∈Ad(x,B),supy∈Bd(y,A)} for all A,B∈CB(X), where d(x,B)=infy∈Bd(x,y). Let T:X→2X be a multifunction. An element x∈X is said to be a fixed point of T whenever x∈Tx. Also, an element x∈X is said to be an endpoint of T whenever Tx={x} [1]. We say that T has the approximate endpoint property whenever infx∈Xsupy∈Txd(x,y)=0 [1]. Let f:X→X be a mapping. We say that f has the approximate endpoint property whenever infx∈Xd(x,fx)=0 [1]. Also, the function g:ℝ→ℝ is called upper semicontinuous whenever limsupn→∞g(λn)≤g(λ) for all sequences {λn}n≥1 with λn→λ [2]. In 2010, Amini-Harandi defined the concept of approximate endpoint property for multifunctions and proved the following result (see [1]).Theorem 1.
Letψ:[0,∞)→[0,∞) be an upper semicontinuous function such that ψ(t)<t and liminft→∞(t-ψ(t))>0 for all t>0, (X,d) a complete metric space, and T:X→CB(X) a multifunction satisfing H(Tx,Ty)≤ψ(d(x,y)) for all x,y∈X. Then T has a unique endpoint if and only if T has the approximate endpoint property.Then Moradi and Khojasteh introduced the concept of generalized weak contractive multifunctions and improved Theorem1 by providing the following result [3].Theorem 2.
Letψ:[0,∞)→[0,∞) be an upper semicontinuous function such that ψ(t)<t and liminft→∞(t-ψ(t))>0 for all t>0, (X,d) a complete metric space, and T:X→CB(X) a generalized weak contractive multifunction; that is, T satisfies H(Tx,Ty)≤ψ(N(x,y)) for all x,y∈X, where N(x,y)=max{d(x,y),d(x,Tx),d(y,Ty),(d(x,Ty)+d(y,Tx))/2}. Then T has a unique endpoint if and only if T has the approximate endpoint property.In this paper, we introduceβ-generalized weak contractive multifunctions, and by adding some conditions to assumptions of the results, we give some results about endpoints of β-generalized weak contractive multifunctions. In 2012, the technique of α-ψ-contractive mappings was introduced by Samet et al. [4]. Later, some authors used it for some subjects in fixed point theory (see for example [5–8]) or generalized it by using the method of β-ψ-contractive multifunctions (see e.g., [9–12]).Let(X,d) be a metric space and β:2X×2X→[0,∞) a mapping. A multifunction T:X→2X is called β-generalized weak contraction whenever there exists a nondecreasing, upper, semicontinuous function ψ:[0,+∞)→[0,+∞) such that ψ(t)<t for all t>0 and
(1)β(Tx,Ty)H(Tx,Ty)≤ψ(N(x,y))
for all x,y∈X. We say that T is β-admissible whenever β(A,B)≥1 implies that β(Tx,Ty)≥1 for all x∈A, and y∈B, where A and B are subsets of X. We say that T has the property (R) whenever for each convergent sequence {xn} in X with xn→x and β(Txn-1,Txn)≥1 for all n≥1, we have β(Txn,Tx)≥1. One can find idea of the property (R) for mappings in [13]. We say that T has the property (K) whenever for each sequence {xn} in X with β(Txn-1,Txn)≥1 for all n≥1, there exists a natural number k such that β(Txm,Txn)≥1 for all m>n≥k. Finally, we say that T has the property (H) whenever for each ɛ>0, there exists z∈X such that supa∈Tzd(z,a)<ɛ implies that for every x∈X there exists y∈Tx such that H(Tx,Ty)=supb∈Tyd(y,b). A multifunction T:X→2X is called lower semicontinuous at x0∈X whenever for each sequence {xn} in X with xn→x0 and every y∈Tx0, there exists a sequence {yn} in X with yn∈Txn for all n≥1 such that yn→y [14].


## 2. Main Results

Now, we are ready to state and prove our main results.Theorem 3.
Let(X,d) be a complete metric space, β:2X×2X→[0,∞) a mapping, and T:X→CB(X) a β-admissible, β-generalized weak contractive multifunction which has the properties (R), (K), and (H). Suppose that there exist a subset A of X and x0∈A such that β(A,Tx0)≥1. Then T has an endpoint if and only if T has the approximate endpoint property.Proof.
It is clear that ifT has an endpoint, then T has the approximate endpoint property. Conversely, suppose that T has the approximate endpoint property. Choose A⊂X and x0∈A such that β(A,Tx0)≥1. Since T has the approximate endpoint property, for each ɛ>0, there exists z∈X such that supa∈Tzd(z,a)<ɛ. Now by using the condition (H), choose x1∈Tx0 such that H(Tx0,Tx1)=supa∈Tx1d(x1,a). Also, choose x2∈Tx1 such that H(Tx1,Tx2)=supa∈Tx2d(x2,a), and by continuing this process, we find a sequence {xn} in X such that xn∈Txn-1 and
(2)H(Txn-1,Txn)=supa∈Txnd(xn,a),
for all n≥1. Since β(A,Tx0)≥1 and T is β-admissible, β(Tx0,Tx1)≥1. By using induction, it is easy to see that β(Txn-1,Txn)≥1 for all n≥1. Thus, we obtain
(3)d(xn,xn+1)≤supa∈Txnd(xn,a)=H(Txn-1,Txn)≤β(Txn-1,Txn)H(Txn-1,Txn)≤ψ(N(xn-1,xn))
for all n≥1. If N(xn-1,xn)=d(xn-1,xn), then
(4)d(xn,xn+1)≤ψ(d(xn-1,xn)).
If N(xn-1,xn)=d(xn-1,Txn-1), then
(5)d(xn,xn+1)≤ψ(d(xn-1,Txn-1))≤ψ(d(xn-1,xn)).
If N(xn-1,xn)=d(xn,Txn), then
(6)d(xn,xn+1)≤ψ(d(xn,Txn))≤ψ(d(xn,xn+1)),
and so d(xn,xn+1)=0. Thus, d(xn,xn+1)≤ψ(d(xn-1,xn)). If
(7)N(xn-1,xn)=d(xn,Txn-1)+d(xn-1,Txn)2=d(xn-1,Txn)2d(xn-1,Txn)2≤d(xn-1,xn+1)2≤d(xn-1,xn)+d(xn,xn+1)2≤max{d(xn-1,xn),d(xn,xn+1)},
then d(xn,xn+1)≤ψ(d(xn-1,xn)) (other case implies that d(xn,xn+1)=0). Thus,
(8)d(xn,xn+1)≤ψ(d(xn-1,xn))
for all n≥1. We claim that ψ(0)=0. If ψ(0)>0, then ψ2(0)≥ψ(0)>0 because ψ is nondecreasing. On the other hand, since ψ(t)<t for all t>0, we have ψ2(0)<ψ(0) which is a contradiction. Hence, ψ(0)=0. Let dn=d(xn,xn+1) for all n. If there exists a natural number n0 such that dn0=0, then it is easy to see that dn=0 for all n≥n0, and so limn→∞dn=0. Now suppose that dn≠0 for all n. In this case, we have dn≤ψ(dn-1)<dn-1 for all n. Hence, {dn} is a decreasing sequence, and so there exists d≥0 such that limn→∞dn=d. If d>0, then dn>0 for all n, and so dn≤ψ(dn-1)<dn-1 for all n. Since ψ is upper and semicontinuous, we obtain d=limn→∞dn≤limn→∞ψ(dn-1)≤ψ(limn→∞dn-1)=ψ(d)<d which is a contradiction. Thus, limn→∞dn=0. Now, we prove that {xn} is a Cauchy sequence. If {xn} is not a Cauchy sequence, then there exist ɛ>0 and natural numbers mk,nk such that mk>nk≥k and d(xmk,xnk)≥ɛ for all k≥1. Also, we choose mk as small as possible such that
(9)d(xmk-1,xnk)<ɛ.
Thus, ɛ≤d(xmk,xnk)≤d(xmk,xmk-1)+d(xmk-1,xnk)≤dmk-1+ɛ for all k. Hence, limk→∞d(xmk,xnk)=ɛ. Since T has the property (K), we obtain
(*)d(xmk,xnk)≤d(xmk,xmk+1)+d(xmk+1,xnk+1)+d(xnk+1,xnk)≤dmk+H(Txmk,Txnk)+dnk≤dmk+β(Txmk,Txnk)×H(Txmk,Txnk)+dnk≤dmk+ψ(N(xmk,xnk))+dnk
for all k. Since limk→∞d(xmk,xnk)=ɛ, limk→∞N(xmk,xnk)=ɛ. In fact,
(10)d(xmk,xnk)≤N(xmk,xnk)=max{d(xmk,Txnk)+d(xnk,Txmk)2d(xmk,xnk),d(xmk,Txmk),d(xnk,Txnk),d(xmk,Txnk)+d(xnk,Txmk)2}≤max{d(xmk,xnk+1)+d(xnk,xmk+1)2d(xmk,xnk),d(xmk,xmk+1),d(xnk,xnk+1),d(xmk,xnk+1)+d(xnk,xmk+1)2}≤max{d(xmk,xnk),d(xmk,xmk+1),d(xnk,xnk+1),(d(xmk,xnk)+d(xnk,xnk+1)+d(xnk,xmk)+d(xmk,xmk+1))×(2)-1d(xmk,xnk)},
and so ɛ=limk→+∞d(xmk,xnk)≤limk→∞N(xmk,xnk)≤ɛ. Since ψ is upper semicontinuous, by using (*) we obtain
(11)ɛ=limk→∞d(xmk,xnk)≤limk→∞ψ(N(xmk,xnk))≤ψ(ɛ)<ɛ,
which is a contradiction, and so {xn} is a Cauchy sequence. Choose x*∈X such that xn→x*. Now, note that
(12)H({xn},Txn)=max{d(xn,Txn),supy∈Txnd(xn,y)}=H(Txn-1,Txn)
for all n, and so
(13)H({xn},Txn)=H(Txn-1,Txn)≤β(Txn-1,Txn)H(Txn-1,Txn)≤ψ(N(xn-1,xn))≤ψ(d(xn-1,xn))≤d(xn-1,xn)
for all n, and so limn→∞H({xn},Txn)=0. Since T has the property (R), we obtain
(14)H({x*},Tx*)≤d(x*,xn)+H({xn},Txn)+H(Txn,Tx*)≤d(x*,xn)+H({xn},Txn)+β(Txn,Tx*)H(Txn,Tx*)≤d(x*,xn)+H({xn},Txn)+ψ(N(xn,x*))
for all n. If N(xn,x*)=d(x*,Tx*), then we have
(15)H({x*},Tx*)≤d(x*,xn)+H({xn},Txn)+ψ(H({x*},Tx*))
for all n. This implies that H({x*},Tx*)≤ψ(H({x*},Tx*)), and so
(16)H({x*},Tx*)=0.
If N(xn,x*)=d(xn,x*) or N(xn,x*)≤d(xn,xn+1), then it is easy to see that H({x*},Tx*)=0. Thus, x* is an endpoint of T.Next example shows that aβ-generalized weak contractive multifunction is not necessarily a generalized weak contractive multifunction.Example 4.
LetX=ℝ. Define T:X→CB(X) by Tx=[x,x+2] for all x∈X. Suppose that ψ:[0,+∞)→[0,+∞) is an arbitrary upper semicontinuous function such that ψ(t)<t for all t>0. If x=0 and y=2, then H(Tx,Ty)=H([0,2],[2,4])=2 and N(x,y)=2. Hence,
(17)H(Tx,Ty)=2≩ψ(2)=ψ(N(x,y)).
Thus, T is not a generalized weak contractive multifunction. Now, suppose that ψ(t)=t/2 for all t≥0 and define β:2X×2X→[0,∞) by β(A,B)=1/2 for all subsets A and B of X. Then, we have
(18)β(Tx,Ty)H(Tx,Ty)=12d(x,y)=ψ(d(x,y))=ψ(N(x,y))
for all x,y∈ℝ. Thus, T is a β-generalized weak contractive multifunction.Next example shows that there are multifunctions which satisfy the conditions of Theorem3, while they are not generalized weak contractive multifunctions.Example 5.
LetX=[0,9/2] and let d(x,y)=|x-y|. Define T:X→CB(X) by
(19)Tx={{x2}0≤x≤1{4x-32}1<x≤32{0}32<x≤92.
If x=1 and y=3/2, then
(20)H(Tx,Ty)=H({12},{92})=4>3=N(x,y)>ψ(N(x,y)),
where ψ:[0,+∞)→[0,+∞) is an arbitrary upper semicontinuous function such that ψ(t)<t for all t>0. Thus, T is not a generalized weak contractive multifunction. Now, we show that T satisfies all conditions of Theorem 3. For this aim, define ψ(t)=t/2 and β(A,B)=1 whenever A and B are subsets of [0,1] and β(A,B)=0 otherwise.  First suppose that x∉[0,1] or that y∉[0,1]. If x,y∈(3/2,9/2], then Tx,Ty⊂[0,1] and β(Tx,Ty)=1. But, H(Tx,Ty)=0, and so β(Tx,Ty)H(Tx,Ty)≤ψ(N(x,y)). If x∈(1,3/2] or y∈(1,3/2], then Tx⊈[0,1], or Ty⊈[0,1] and so β(Tx,Ty)=0. Hence, β(Tx,Ty)H(Tx,Ty)≤ψ(N(x,y)). Now, suppose that x,y∈[0,1]. In this case, we have β(Tx,Ty)≥1, H(Tx,Ty)=H({x/2},{y/2})=(1/2)d(x,y), and N(x,y)=max{d(x,y),x/2,y/2,(d(x,y/2)+d(y,x/2))/2}. Thus, d(x,y)≤N(x,y), and so
(21)β(Tx,Ty)H(Tx,Ty)=12d(x,y)≤ψ(d(x,y))≤ψ(N(x,y)).
Therefore, T is a β-generalized weak contractive multifunction. Now, we show that T is β-admissible. If β(A,B)≥1, then A,B⊂[0,1], and so Tx={x/2}∈[0,1] and Ty={y/2}∈[0,1] for all x∈A and y∈B. Thus, β(Tx,Ty)≥1 for all x∈A and y∈B. Now, suppose A=[0,1/2] and x0=1/4. Then, Tx0={1/8}∈[0,1] and [0,1/2]⊂[0,1]. Hence, β(A,Tx0)≥1. Now, we show that T satisfies the condition (H). First note that, for each ɛ>0, there exists z∈X such that supa∈Tzd(z,a)<ɛ. Now, we show that for each x∈X there exists y∈Tx such that H(Tx,Ty)=supb∈Tyd(y,b). If 0≤x≤1, then Tx={x/2}, T(x/2)={x/4}, and
(22)H(Tx,T(x2))=H({x2},{x4})=x4=supb∈T(x/2)d(x2,b).
Since for 1<x≤3/2 we have 5/2<4x-(3/2)≤9/2, T(4x-(3/2))={0}. Thus,
(23)H(Tx,T(4x-32))=H({4x-32},{0})=4x-32=supb∈T(4x-3/2)d(4x-32,b).
If 3/2<x≤9/2, then Tx={0} and T(0)={0}. Hence,
(24)H(Tx,T(0))=H({0},{0})=0=supb∈T(0)d(0,b).
It is easy to check that T satisfies the conditions (R) and (K). Note that, 0 is the endpoint of T.Now, we add an assumption to obtain uniqueness of endpoint. In this way, we introduce a new notion. LetX be a set and β:2X×2X→[0,∞) a map. We say that the set X has the property (Gβ) whenever β(A,B)≥1 for all subsets A and B of X with A⊈B or B⊈A.Corollary 6.
Let(X,d) be a complete metric space, β:2X×2X→[0,∞) a mapping, and T:X→CB(X) a β-admissible, β-generalized weak contractive multifunction which has the properties (R), (K), and (H). Suppose that there exist a subset A of X and x0∈A such that β(A,Tx0)≥1. If T has the approximate endpoint property and X has the property (Gβ), then T has a unique endpoint.Proof.
By using Theorem3, T has a endpoint. If T has two distinct endpoints x* and y*, then β(Tx*,Ty*)=β({x*},{y*})≥1 because X has the property (Gβ). Hence,
(25)d(x*,y*)≤H(Tx*,Ty*)≤β(Tx*,Ty*)H(Tx*,Ty*)≤ψ(N(x*,y*))<N(x*,y*)=d(x*,y*),
which is a contradiction. Thus, T has a unique endpoint.In Example5, T has a unique endpoint, while X does has not the property (Gβ). Also, T has the property (R), while T is not lower semicontinuous. To see this, consider the sequence {xn} defined by
(26)xn={1-1nn=2k1+1nn=2k-1
for k≥1 and put y=1/2 and x0=1. Then xn→1 and y∈Tx0={1/2}. Let {yn} be an arbitrary sequence in X such that yn∈Txn for all n≥1. Then, y2k-1∈Tx2k-1 and y2k∈Tx2k for all k. But, y2k-1=4x2k-1-(3/2) for sufficiently large k and y2k=x2k/2 for all k since y2k-1→5/2, yn↛1/2. This implies that T is not lower semicontinuous.Corollary 7.
Let(X,d) be a complete metric space, β:2X×2X→[0,∞) a mapping, and T:X→CB(X) a β-admissible multifunction which has the properties (R), (K), and (H). Suppose that X has the property (Gβ), and there exist a subset A of X, x0∈A and k∈[0,1) such that β(A,Tx0)≥1 and β(Tx,Ty)H(Tx,Ty)≤kN(x,y) for all x,y∈X. Then T has a unique endpoint if and only if T has the approximate endpoint property.Proof.
It is sufficient that we defineψ(t)=kt for all t≥0. Then, Theorem 3 and Corollary 6 guarantee the result.It has been proved that lower semicontinuity of the multifunctionT and the property (R) are independent conditions [9]. We can replace lower semicontinuity of the multifunction instead of the property (R) to obtain the next result. Its proof is similar to the proof of Theorem 3.Theorem 8.
Let(X,d) be a complete metric space, β:2X×2X→[0,∞) a mapping, and T:X→CB(X) a lower semicontinuous, β-admissible, β-generalized weak contractive multifunction which has the properties (K) and (H). Suppose that there exist a subset A of X and x0∈A such that β(A,Tx0)≥1. Then T has the approximate endpoint property if and only if T has an endpoint.Corollary 9.
Let(X,d) be a complete metric space, β:2X×2X→[0,∞) a mapping, and T:X→CB(X) a lower semicontinuous, β-admissible, β-generalized weak contractive multifunction which has the properties (K) and (H). Suppose that there exist a subset A of X and x0∈A such that β(A,Tx0)≥1. If T has the approximate endpoint property and X has the property (Gβ), then T has a unique endpoint.Corollary 10.
Let(X,d) be a complete metric space, β:2X×2X→[0,∞) a mapping, and T:X→CB(X) a β-admissible multifunction which has the properties (R), (K), and (H). Suppose that X has the property (Gβ), and there exist a subset A of X, x0∈A and k∈[0,1) such that β(A,Tx0)≥1 and β(Tx,Ty)H(Tx,Ty)≤kN(x,y) for all x,y∈X. If T has the approximate endpoint property, then Fix(T)=End(T)={x}.Proof.
If we putψ(t)=kt, then, by using Theorem 2.10 in [9], T has a fixed point. Since T has the approximate endpoint property, by using Corollary 7, T has a unique endpoint such x. Let y∈Fix(T). If Tx=Ty, then y=x. If Tx≠Ty, then β(Tx,Ty)≥1 because X has the property (Gβ). Also, we have
(27)d(x,y)≤H({x},Ty)=H(Tx,Ty)≤β(Tx,Ty)H(Tx,Ty)≤kN(x,y).
But, N(x,y)=max{d(x,y),d(x,Tx),d(y,Ty),(d(x,Ty)+d(y,Tx))/2}=d(x,y). Thus, d(x,y)=0, and so Fix(T)=End(T)={x}.Next corollary shows us the role of a point in the existence of endpoints.Corollary 11.
Let(X,d) be a complete metric space, x*∈X a fixed element, and T:X→CB(X) a multifunction such that T has the property (H) and x*∈Tx∩Ty for all subsets A and B of X with x*∈A∩B and all x∈A and y∈B. Assume that H(Tx,Ty)≤ψ(N(x,y)) for all x,y∈X with x*∈Tx∩Ty, where ψ:[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that ψ(t)<t for all t>0. Suppose that there exist a subset A0 of X and x0∈A0 such that x*∈A0∩Tx0. Assume that for each convergent sequence {xn} in X with xn→x and x*∈Txn-1∩Txn, for all n≥1, one has x*∈Txn∩Tx. Also, for each sequence {xn} in X with x*∈Txn-1∩Txn for all n≥1, there exists a natural number k such that x*∈Txm∩Txn for all m>n≥k. Then T has an endpoint if and only if T has the approximate endpoint property.Proof.
It is sufficient we defineβ:2X×2X→[0,∞) by β(A,B)=1 whenever x*∈A∩B and β(A,B)=0 otherwise, and then we use Theorem 3.Corollary 12.
Let(X,d) be a complete metric space, x*∈X a fixed element and T:X→CB(X) a lower semicontinuous multifunction such that T has the property (H) and x*∈Tx∩Ty for all subsets A and B of X with x*∈A∩B and all x∈A and y∈B. Assume that
(28)H(Tx,Ty)≤ψ(N(x,y))
for all x,y∈X with x*∈Tx∩Ty, where ψ:[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that ψ(t)<t for all t>0. Suppose that there exist a subset A0 of X and x0∈A0 such that x*∈A0∩Tx0. Assume that for each convergent sequence {xn} in X with xn→x and x*∈Txn-1∩Txn for all n≥1, we have x*∈Txn∩Tx. Then T has an endpoint if and only if T has the approximate endpoint property.Proof.
It is sufficient to defineβ:2X×2X→[0,∞) by β(A,B)=1 whenever x*∈A∩B and β(A,B)=0 otherwise, and then we use Theorem 8.Let(X,d,≤) be an ordered metric space. Define the order ⪯ on arbitrary subsets A and B of X by A⪯B if and only if for each a∈A there exists b∈B such that a≤b. It is easy to check that (CB(X),⪯) is a partially ordered set.Theorem 13.
Let(X,d,≤) be a complete ordered metric space and T a closed and bounded valued multifunction on X such that T has the property (H) and Tx⪯Ty for all subsets A and B of X with A⪯B and all x∈A and y∈B. Assume that H(Tx,Ty)≤ψ(N(x,y)) for all x,y∈X with Tx⪯Ty, where ψ:[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that ψ(t)<t for all t>0. Suppose that there exist a subset A0 of X and x0∈A0 such that A0⪯Tx0. Assume that for each convergent sequence {xn} in X with xn→x and Txn-1⪯Txn, for all n≥1, one has Txn⪯Tx. Also, for each sequence {xn} in X with Txn-1⪯Txn for all n≥1, there exists a natural number k such that Txm⪯Txn for all m>n≥k. Then T has an endpoint if and only if T has the approximate endpoint property.Proof.
Defineβ(A,B)=1 whenever A⪯B and β(A,B)=0 otherwise, and then we use Theorem 3.Corollary 14.
Let(X,d,≤) be a complete ordered metric space and T a closed and bounded valued multifunction on X such that T has the property (H) and Tx⪯Ty for all subsets A and B of X with A⪯B, all x∈A, and y∈B. Assume that H(Tx,Ty)≤ψ(N(x,y)) for all x,y∈X with Tx⪯Ty, where ψ:[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that ψ(t)<t for all t>0. Suppose that there exist a subset A0 of X and x0∈A0 such that A0⪯Tx0. Assume that for each convergent sequence {xn} in X with xn→x and Txn-1⪯Txn, for all n≥1, one has Txn⪯Tx. Also, for each sequence {xn} in X with Txn-1⪯Txn for all n≥1, there exists a natural number k such that Txm⪯Txn for all m>n≥k. If T has the approximate endpoint property and A⪯B for all subsets A and B of X with A⊈B or B⊈A, then T has a unique endpoint.Proof.
Defineβ(A,B)=1 whenever A⪯B and β(A,B)=0 otherwise, and then we use Corollary 6.Let(X,d) be a metric space and T:X→2X a multifunction. We say that T is an HS-multifunction whenever for each x∈X there exists y∈Tx such that H(Tx,Ty)=supb∈Tyd(y,b). It is obvious that each HS-multifunction is an multifunction which has the property (H). Thus, one can conclude similar results to above ones for HS-multifunctions. Here, we provide some ones. Although by considering HS-multifunction we restrict ourselves, we obtain strange results with respect to above ones. One can prove the following by reading exactly the proofs of similar above results.Theorem 15.
Let(X,d) be a complete metric space, β:2X×2X→[0,∞) a mapping, and T:X→CB(X) a β-admissible, β-generalized weak contractive HS-multifunction which has the properties (R) and (K). Suppose that there exist a subset A of X and x0∈A such that β(A,Tx0)≥1. Then T has an endpoint, and so T has the approximate endpoint property.Theorem 16.
Let(X,d) be a complete metric space, β:2X×2X→[0,∞) a mapping, and T:X→CB(X) a lower semicontinuous, β-admissible, and β-generalized weak contractive HS-multifunction which has the property (K). Suppose that there exist a subset A of X and x0∈A such that β(A,Tx0)≥1. Then T has an endpoint, and so T has the approximate endpoint property.The next result is a consequence of Theorem15.Corollary 17.
Let(X,d) be a complete metric space, x*∈X a fixed element, and T:X→CB(X) an HS-multifunction such that x*∈Tx∩Ty for all subsets A and B of X with x*∈A∩B, all x∈A, and y∈B. Assume that H(Tx,Ty)≤ψ(N(x,y)) for all x,y∈X with x*∈Tx∩Ty, where ψ:[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that ψ(t)<t for all t>0. Suppose that there exist a subset A0 of X and x0∈A0 such that x*∈A0∩Tx0. Assume that for each convergent sequence {xn} in X with xn→x and x*∈Txn-1∩Txn for all n≥1 one has x*∈Txn∩Tx. Also, for each sequence {xn} in X with x*∈Txn-1∩Txn for all n≥1, there exists a natural number k such that x*∈Txm∩Txn for all m>n≥k. Then T has an endpoint, and so T has the approximate endpoint property.The next result is a consequence of Theorem16.Corollary 18.
Let(X,d,≤) be a complete ordered metric space and T a closed and bounded valued lower semicontinuous HS-multifunction on X such that Tx⪯Ty for all subsets A and B of X with A⪯B, all x∈A, and y∈B. Assume that H(Tx,Ty)≤ψ(N(x,y)) for all x,y∈X with Tx⪯Ty, where ψ:[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that ψ(t)<t for all t>0. Suppose that there exist a subset A0 of X and x0∈A0 such that A0⪯Tx0. Assume that for each sequence {xn} in X with Txn-1⪯Txn for all n≥1, there exists a natural number k such that Txm⪯Txn for all m>n≥k. Then T has an endpoint, and so T has the approximate endpoint property.



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*Source: 948472-2013-11-04.xml*