University of Mazandaran, Iran
http://cjms.journals.umz.ac.ir
ISSN: 1735-0611 CJMS.8 (2)(2019), 137-144
Endpoints of generalized ϕ-contractive multivalued mappings of integral type
Babak Mohammadi1, Esmaeil Alizadeh2
1,2 Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran
Abstract. Recently, some researchers have established some re-sults on existence of endpoints for multivalued mappings. In partic-ular, Mohammadi and Rezapour’s [Endpoints of Suzuki type quasi-contractive multifunctions, U.P.B. Sci. Bull., Series A, 2015] used the technique of α-ψ-contractive mappings, due to Samet et al. (2012), to give some results about endpoints of Suzuki type quasi-contractive multifunctions satisfing property (BS). In this paper, we prove existence and uniqueness of endpoint for multivalued map-pings satisfing the weaker conditions generalizedϕ-contractivity of integral type and property (HS). This result generalize and improve Mohammadi and Rezapour’s result. Also, we give an example to illustrate the usability of the result.
Keywords: endpoint,ϕ-contractions, multivalued mappings, prop-erty (HS), integral type.
2010 Mathematics subject classification: Primery 46T30; Sec-ondary 46C60.
1. Introduction and preliminaries
Let (X, d) be a metric space, 2X the set of all nonempty subsets of X and CB(X) the set of all nonempty closed bounded subsets of X.
1Corresponding author: [email protected] Received: 09 January 2016
Revised: 05 March 2016 Accepted: 29 July 2018
Assume thatH be the Hausdorff metric on CB(X) defined by
H(A, B) =max{sup
x∈A
d(x, B),sup
y∈B
d(y, A)},
for allA, B∈CB(X) whered(x, B) = infy∈Bd(x, y). An elementx∈X
is said to be an endpoint ofT wheneverT x={x}. It is said thatT has the approximate endpoint property whenever infx∈Xsupy∈T xd(x, y) = 0
(see [1]). In 2010, Amini-Harandi [1] proved that some multifunctions have unique endpoint if and only if have approximate endpoint prop-erty . Then, Moradi and Khojasteh [7] generalized Amini-Harandi’s result for generalized contractive multivalued mappings. Denote by Ψ the family of nondecreasing functions ψ : [0,∞) → [0,∞) such that ∑∞n=1ψn(t) < ∞ for all t > 0 ([9]). The technique of α-ψ
-contractive mappings introduced by Samet, Vetro and Vetro in 2012 (see [9]). Then, some authors generalized it in different subjects (see, for example, [3, 5, 8]). It is said that a multifunctionT :X →CB(X) has the property (BS) whenever for eachx∈Xthere existsy∈T xsuch thatH(T x, T y) = supb∈T yd(y, b). Recently, Mohammadi and Rezapour [6] used this technique to give some results about endpoints of Suzuki type quasi-contractive multifunctions satisfing property (BS) without using the approximate endpoint property.
Theorem 1.1. ([6]) Let (X, d) be a complete metric space, ψ ∈ Ψ,
α :X×X → [0,∞) a mapping and T :X → CB(X) an α-admissible such that T has the property (BS) and α(x, y)H(T x, T y) ≤ψ(M(x, y))
for allx, y∈X, where
M(x, y) =max{d(x, y), d(x, T x), d(y, T y),d(x, T y) +d(y, T x)
2 }.
Suppose that there exist x0 ∈ X and x1 ∈T x0 such that α(x0, x1) ≥1.
Let either T is continuous or ψ is right upper semi-continuous and for each sequence {xn} in X with α(xn, xn+1) ≥ 1 for all n and xn → x,
there exists a subsequence {xnk} of {xn} such that α(xnk, x)≥1 for all
k. Then T has an endpoint.
On the other hand, in 2001, Branciari ’s [2] generalized the Banach contraction principle to integral type contractive self-mappings by us-ing an Lebesgue integrable mappus-ing φ : [0,+∞) → [0,+∞), which is summable on each compact subset of [0,+∞), such that ∫0ϵφ(t)dt >0 for anyϵ >0. Througout this paper, we denote by Φ1 the family of all this functions which is bounded on [0,+∞). Denote by Φ2 the collec-tion of all continuous nondecreasing funccollec-tions ϕ : [0,+∞) → [0,+∞) such that ϕ(t) < t for all t > 0. It is well known that ϕ(0) = 0 and limn→∞ϕn(t) = 0 for all t > 0. Also it is known that for any ϕ ∈ Ψ
mapping T : X → CB(X) has the property (HS) whenever for each
x∈X there existsy∈T xsuch that H(T x, T y)≥supb∈T yd(y, b). Note that the property (HS) is weaker than the property (BS). Also, we say that T is a generalized ϕ-contractive multivalued mapping of integral type whenever there existφ∈Φ1 and ϕ∈Φ2 such that
∫ H(T x,T y) 0
φ(t)dt≤ϕ(
∫ M(x,y) 0
φ(t)dt),
for all x, y∈X, where
M(x, y) = max{d(x, y), d(x, T x), d(y, T y),1
2[d(x, T y) +d(y, T x)]}. In this paper, we prove existence and uniqueness of endpoint for such mappings having the conditionsϕ-contractivity and property (HS) which are weaker conditions with respect to Mohammadi and Rezapour’s [6].
2. Main results
The following theorem is the main result of this study.
Theorem 2.1. Let (X, d) be a complete metric space and T : X → CB(X) be a generalized ϕ-contractive multivalued mapping of integral type which has the property (HS). Then T has a unique endpoint.
Proof. Choose a fixe element x0 ∈ X. Since T has the property (HS), there existsx1 ∈T x0 such that H(T x0, T x1)≥supb∈T x1d(x1, b). Con-tinuing this process, we obtain a sequence {xn} such thatxn+1 ∈T xn
andH(T xn, T xn+1)≥supb∈T xn+1d(xn+1, b) for alln≥0. Then we have ∫d(xn,xn+1)
0 φ(t)dt ≤
∫supb∈T xnd(xn,b)
0 φ(t)dt ≤ ∫H(T xn−1,T xn)
0 φ(t)dt ≤ ϕ(∫M(xn−1,xn)
0 φ(t)dt),
(2.1)
for all n≥1. Note that
M(xn−1, xn) = max{d(xn−1, xn), d(xn−1, T xn−1)
, d(xn, T xn),12[d(xn−1, T xn) +d(xn, T xn−1)]} ≤ max{d(xn−1, xn)), d(xn, xn+1),12[d(xn−1, xn+1)]}) ≤ max{d(xn−1, xn)), d(xn, xn+1)},
(2.2) for all n≥ 1. If d(xn, xn+1) > d(xn−1, xn), then, from (2.1) and (2.2),
we have
∫ d(xn,xn+1)
0
φ(t)dt≤ϕ(
∫ d(xn,xn+1)
0
which is a contradiction. Hence,d(xn, xn+1)≤d(xn−1, xn) and so from (2.1) and (2.2),
∫ d(xn,xn+1)
0
φ(t)dt≤ϕ(
∫ d(xn−1,xn)
0
φ(t)dt), (2.3)
for alln≥1. From (2.3), we get∫d(xn,xn+1)
0 φ(t)dt≤ϕ
n(∫d(x0,x1)
0 φ(t)dt)→ 0 as n→ ∞. From this, we conclude that d(xn, xn+1) →0 as n→ ∞. We claim that {xn} is a Cauchy sequence. Suppose to the contrary,
{xn} is not a Cauchy sequence. Then, there exists ε > 0 such that for any i ∈ N there exist natural numbers mi, ni with mi > ni and d(xni, xmi)≥ε. Put
ki= min{mi|mi > ni, d(xni, xmi)≥ε}.
Hence, for any i ∈ N, we have d(xni, xki) ≥ ε and d(xni, xki−1) < ε.
Now
ε ≤ d(xni, xki)
≤ d(xni, T xni) +H(T xni,{xki})
≤ d(xni, T xni) +H(T xni, T xki−1) +H(T xki−1,{xki−1}) +d(xki−1, xki).
(2.4) But we haved(xni, T xni)≤d(xni, xni+1)→0 and
∫H(T xki−1,{xki−1})
0 φ(t)dt =
∫supb∈T xk
i−1d(xki−1,b) 0 φ(t)dt ≤ ∫H(T xki−2,T xki−1)
0 φ(t)dt ≤ ϕ(∫0M(xki−2,xki−1)φ(t)dt.
(2.5)
From inequality (2.2), we conclude thatM(xki−2, xki−1)→0 asi→ ∞.
Hence, from (2.5),H(T xki−1,{xki−1})→0.Now from (2.4), we have ∫ε
0 φ(t)dt ≤
∫d(xni,T xni)+H(T xni,T xki−1)+H(T xki−1,{xki−1})+d(xki−1,xki)
0 φ(t)dt
= ∫0H(T xni,T xki−1)φ(t)dt
+ ∫Hd((xT xni,T xni)+H(T xni,T xki−1)+H(T xki−1,{xki−1})+d(xki−1,xki)
ni,T xki−1) φ(t)dt
≤ ϕ(∫0M(xni,xki−1)φ(t)dt)
+ M[d(xni, T xni) +H(T xki−1,{xki−1}) +d(xki−1, xki)]
(2.6) whereM is a positive number such that φ(t)≤M for allt≥0. Taking lim supi→∞ we obtain
∫ ε
0
φ(t)dt≤lim sup
i→∞ ϕ(
∫ M(xni,xki−1) 0
where
M(xni, xki−1) = max{d(xni, xki−1), d(xni, T xni)
, d(xki−1, T xki−1),
1
2[d(xni, T xki−1) +d(xki−1, T xni)]}
≤ max{d(xni, xki−1), d(xni, xni+1), d(xki−1, xki)
,12[d(xni, xki−1) +d(xki−1, xki) +d(xki−1, xni) +d(xni, xni+1)]}.
(2.8)
Taking lim supi→∞in (2.8), we get lim supi→∞M(xni, xki−1)≤max{ε,0,0, ε}=
ε. Therefore, from (2.7), we obtain ∫0εφ(t)dt ≤ϕ(∫0εφ(t)dt), which is a contradiction. Hence {xn} is a Cauchy sequence. SinceX is complete, there existsx∈X such that limn→∞d(xn, x) = 0. Next, we shall show
thatx is an endpoint ofT. To see this, we have ∫H({x},T x)
0 φ(t)dt ≤
∫d(x,xn)+H({xn},T xn)+H(T xn,T x)
0 φ(t)dt
= ∫H(T xn,T x)
0 φ(t)dt+
∫d(x,xn)+H({xn},T xn)+H(T xn,T x)
H(T xn,T x) φ(t)dt
≤ ϕ(∫M(xn,x)
0 φ(t)dt) +M(d(x, xn) +H({xn}, T xn)). (2.9) Similar to way in (2.5), it is easy to check thatH({xn}, T xn)→0 as
n→ ∞. Taking lim supi→∞ in (2.9), we have ∫ H({x},T x)
0
φ(t)dt≤lim sup
n→∞ ϕ(
∫ M(xn,x)
0
φ(t)dt), (2.10)
where
M(xn, x) = max{d(xn, x), d(xn, T xn), d(x, T x),
1
2[d(xn, T x)+d(x, T xn)]}
≤max{d(xn, x), d(xn, xn+1), d(x, T x), 1
2[d(xn, x)+d(x, T x)+d(x, xn+1)]}. Tendingnto∞, we obtain limn→∞M(xn, x) =d(x, T x). Consequently,
we obtain from (2.10), ∫ H({x},T x)
0
φ(t)dt≤ϕ(
∫ d(x,T x) 0
φ(t)dt)≤ϕ(
∫ H({x},T x) 0
φ(t)dt).
From the above inequality, we conclude that H({x}, T x) = 0 which implies T x = {x}. To show the uniqueness of endpoint of T assume thatx, y are two endpoints ofT such thatx̸=y. Then
∫ d(x,y) 0
φ(t)dt=
∫ H(T x,T y) 0
φ(t)dt≤ϕ(
∫ M(x,y) 0
φ(t)dt) =ϕ( ∫ d(x,y)
0
φ(t)dt),
which is a contradiction.
Example 2.1. LetX ={n1|n∈N}∪{0}with the usual metricd(x, y) = |x−y|. Obviously (X, d) is complete. Define φ: [0,+∞)→[0,+∞) by
φ(t) =t1t[1−lnt
we have ∫0τφ(t)dt=ττ1 (because if we define Φ(t) = {
t1t 0< t < e
0 t= 0 ,
it is easy to check that Φ′(t) = φ(t) for all 0 ≤ t < e). Also, define
T :X →CB(X) by
T x= {
{0} x= 0,
[0,n+11 ]∩X x= 1n, n∈N.
If x= 0 and y= n1 , then ∫H(T x,T y)
0 φ(t)dt = ∫ 1
n+1 0 φ(t)dt = n+11 n+1 = n+11 (n+11 )n ≤ 1
2( 1
n) n
= ϕ(∫0d(x,y)φ(t)dt) ≤ ϕ(∫0M(x,y)φ(t)dt)
where ϕ(t) = 12t.
Ifx, y∈ {n1|n∈N}, then assume thatx= n1 andy= m1. In this case, we have
∫H(T x,T y)
0 φ(t)dt = ∫| 1
n+1− 1
m+1| 0 φ(t)dt = ((n+1)(|m−mn|+1))
(n+1)(m+1)
|m−n|
= ((n+1)(|m−mn|+1))
n+m+1
|m−n| (|m−n|
nm )
nm
|m−n|( mn
(n+1)(m+1))
nm |m−n|
≤ 1 2(1)(
|m−n| nm )
nm |m−n|
= ϕ(∫0d(x,y)φ(t)dt) ≤ ϕ(∫0M(x,y)φ(t)dt).
We see that ∫ H(T x,T y)
0
φ(t)dt≤ϕ(
∫ M(x,y) 0
φ(t)dt) for all x, y∈X.
It is easy to see that T has the property (HS). Therefore, the mapping
T defined in this example satisfies conditions of Theorem 2.1, and so by this theorem, T has a unique endpoint inX. Here, T0 ={0}.
Also, by using [10], we can give some equivalent results for Theorem 2.1. Here, we give only two results in this way as follows.
andη: [0,∞)→[0,∞)satisfyingη−1{0}={0}andlim inft→∞η(t)>0
such that
ψ(
∫ H(T x,T y) 0
φ(t)dt)≤ψ(
∫ M(x,y) 0
φ(t)dt)−η(
∫ M(x,y) 0
φ(t)dt),
for allx, y∈X. Then T has a unique endpoint.
Proposition 2.3. Let (X, d) be a complete metric space and T :X → CB(X) be a multivalued mapping which has the property (HS). Suppose there exist φ ∈ Φ1, ψ : [0,∞) → [0,∞) a nondecreasing function such that Σ∞n=1ψn(t) < ∞ for all t > 0 and ϕ : [0,∞) → [0,∞) a right continuous and nondecreasing function satisfying ϕ(t) < t for all t >0
such that
ψ(
∫ H(T x,T y) 0
φ(t)dt)≤ϕ(ψ(
∫ M(x,y) 0
φ(t)dt)),
for allx, y∈X. Then T has a unique endpoint.
In 2007, Zhang [11] defined a new generalized contractive type con-dition for a pair of mappings in metric spaces. Let A ∈ (0,+∞] and
R+
A= [0, A). Denote by ℑ[0, A) the collection of all functionsF :R
+
A→
Rsatisfing the following conditions:
(i) F(0) = 0 andF(t)>0 for eacht∈(0, A), (ii) F is nondecreasing onR+A,
(iii) F is continuous.
From [11], we know that for any F ∈ ℑ[0, A), limn→+∞F(εn) = 0
(εn∈R+A) implies limn→+∞εn= 0. Denote by Ψ[0, A) the family of all functions ψ : R+A → R+ which is nondecreasing and right upper semi-continuous such that limn→+∞ψn(t) = 0 for eacht∈ (0, A). It is easy to see that for anyψ ∈Ψ[0, A) we haveψ(0) = 0 and ψ(t)< t for each
t∈(0, A).
Regarding the above notations Zhang [11] proved the following theo-rem about existing of common fixed point for a pair of mappings.
Theorem 2.4. Let (X, d) be a complete metric space and let D = sup{d(x, y)|x, y ∈ X}. Set A > D if D < ∞ and A = D if D = ∞. Suppose that T, S : X → X are two mappings, F ∈ ℑ[0, A) and
ψ∈Ψ[0, F(A−))satisfing
F(d(T x, Sy))≤ψ(F(M(x, y))) for each x, y∈X,
where
M(x, y) = max{d(x, y), d(x, T x), d(y, Sy),1
Then T and S have a unique common fixed point in X. Moreover, for each x0 ∈ X, the iterated sequence {xn} with x2n+1 = T x2n and x2n+2 =Sx2n+1 converges to the common fixed point of T and S.
Similar to the proof of Theorem 2.1, one can easily prove the follow-ing endpoint result for multivalued mappfollow-ings, which is a multivalued endpoint version of Theorem 2.4.
Theorem 2.5. Let (X, d) be a complete metric space and T : X → CB(X) be a multivalued mapping satisfing property (HS), F ∈ ℑ[0, A)
and ψ∈Ψ[0, F(A−))such that
F(H(T x, T y))≤ψ(F(M(x, y))) for each x, y∈X.
ThenT has a unique endpoint.
Acknowledgement
This study was supported by Marand Branch, Islamic Azad Univer-sity, Marand, Iran.
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