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.
Let ψ:[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 Theorem 1 by providing the following result [3].
Theorem 2.
Let ψ:[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 introduce β-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 if T 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 a β-generalized weak contractive multifunction is not necessarily a generalized weak contractive multifunction.
Example 4.
Let X=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 Theorem 3, while they are not generalized weak contractive multifunctions.
Example 5.
Let X=[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. Let X 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 Theorem 3, 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 Example 5, 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 define ψ(t)=kt for all t≥0. Then, Theorem 3 and Corollary 6 guarantee the result.
It has been proved that lower semicontinuity of the multifunction T 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 put ψ(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 define β: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 define β: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.
Define β(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.
Define β(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 Theorem 15.
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 Theorem 16.
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.