# Endpoints of Multivalued Contraction Operators

Authors: Bhagwati Prasad; Ritu Sahni
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

The existence of the endpoints and approximate endpoints are studied in a general setting for the operators satisfying various contractive conditions. Some recent results are also derived as special cases.


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## Body

## 1. Introduction

Among several generalizations of celebrated Banach fixed point theorem, one interesting extension is Nadler’s [1] fixed point theorem for multivalued contraction. He exactly proved that a multivalued contraction has a fixed point in a complete metric space. Subsequently, it received great attention in applicable mathematics and was extended and generalized on various settings. Indeed, these extensions and generalizations have been influenced by the applications of the multivalued fixed point theory in mathematical economics, game theory, differential inclusions, interval arithmetic, Hammerstein equations, convex optimization, duality theory in optimization, variational inequalities and control theory, nonlinear evolution equations and nonlinear semigroups, quasivariational inequalities, and elasticity and plasticity theory (see, for instance, [2–8]  and several references thereof). The results related to existence of endpoints or strict fixed-points were first given by Rus [9] in 2003. Thereafter, a number of authors established interesting results concerning existence and uniqueness of endpoints for multivalued contractions in different settings; see,  for example, [10–15]. The main purpose of this paper is to establish some existence and uniqueness results for endpoints using different multivalued contractions. Our results include some recent results.


## 2. Preliminaries

LetT:X→2X be a multivalued mapping. An element x∈X is said to be an endpoint of T if Tx={x}. We say that a multivalued mapping T:X→2X has the approximate endpoint property (AEPP)  if  infx∈Xsupy∈Txd(x,y)=0 (also see [3, 10]). Throughout the paper, let (X,d) be a b-metric space, and let P(X) denote the family of all nonempty subsets of X and Cl(X) the family of all nonempty closed subsets of X. For any A,B∈P(X), the Hausdorff metric is defined as
(1)H(A,B)=max{supa∈Ad(a,B),supb∈Bd(b,A)},
where d(a,B)=inf{d(a,b):b∈B} is the distance from the point a to the set B.LetI:X→X be a single-valued mapping and T:X→Cl(X) a multivalued contraction. We say that the mappings I and T have an AEPP provided infx∈Xsupy∈Txd(Ix,y)=0.A pointx∈X is called an endpoint of I and T if Tx={Ix}. For each ε>0, let Eε(I,T)={x∈X:supy∈Txd(Ix,y)≤ε} be the set of all approximate endpoints of the mappings I and T.Example 1.
LetX=(-∞,∞) with Euclidean norm. Assume that K=[0,c] and T:K→Cl(K) defined by
(2)Tx={[x,c-x],x∈[0,c2)[c-x,x],x∈[c2,c],
where c is a positive constant. Clearly, Fix(T)=[0,c] and End(T)={c/2}.Definition 2 (see [16]).
LetX be a nonempty set and b≥1 a given real number. A function  d:X×X→ℜ+ (set of nonnegative real numbers) is said to be a b-metric iff for all x,y,z∈X the following conditions are satisfied:(i)
d(x,y)=0iffx=y,(ii)
d(x,y)=d(y,x),(iii)
d(x,z)≤b[d(x,y)+d(y,z)].A pair (X,d) is called a b-metric space.The class ofb-metric spaces is effectively larger than that of metric spaces, since a b-metric space is a metric space when b=1 in the above condition (iii). The following example shows that a b-metric on X need not be a metric on X (see also [16, page 264]).Example 3 (see [17]).
LetX={x1,x2,x3,x4} and d(x1,x2)=k≥2,d(x1,x3)=d(x1,x4)
=
d
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1, d(xi,xj)=d(xj,xi) for all i,j=1,2,3,4 and d(xi,xi)=0, i=1,2,3,4. Then,
(3)d(xi,xj)≤k2[d(xi,xn)+d(xn,xj)]hhhhforn,i,j=1,2,3,4,
and if k>2, the ordinary triangle inequality does not hold.Definition 4 (see [16]).
Let(X,d) be a b-metric space. Then, a sequence {xn}n∈N in X is called(a)
convergent if and only if there exists x∈X such that d(xn,X)→0 as n→∞. In this case, one writes limn→∞xn=x,(b)
Cauchy if and only if d(xn,xm)→0 as m,n→∞.Remark 5 (see [16]).
In ab-metric space (X,d), the following assertions hold.(i)
A convergent sequence has a unique limit.(ii)
Each convergent sequence is Cauchy.(iii)
In general, ab-metric is not continuous.Definition 6 (see [16]).
Let(X,d) be a b-metric space. If Y is a nonempty subset of X, then the closure Y- of Y is the set of limits of all convergent sequences of points in Y, i.e.,
(4)Y-={limn⟶∞x∈X:there exists a sequence{xn}n∈Nhsuch thatlimn→∞xn=x}.Definition 7 (see [16]).
Let(X,d) be a b-metric space. Then, a subset Y⊂X is called(a)
closed if and only if for each sequence {xn}n∈N in Y  which converges to an element x, one has x∈Y,(b)
compact if and only if for every sequence of elements of Y, there exists a subsequence that converges to an element of Y,(c)
bounded if and only if δ(Y)=sup{d(a,b):a,b∈Y}<∞.Definition 8 (see [16]).
Theb-metric space (X,d) is complete iff every Cauchy sequence in X converges.Definition 9.
LetX and Y be two Hausdorff topological spaces and T:X→P(Y), a multivalued mapping with nonempty values. Then, T is said to be(i)
upper semicontinuous (u.s.c.) if, for each closed setB⊂Y, T-1(B)={x∈X:T(x)∩B≠ϕ} is closed in X;(ii)
lower semicontinuous (l.s.c.) if, for each open setB⊂Y,  T-1(B)={x∈X:T(x)∩B≠ϕ} is open in X;(iii)
continuous if it is both u.s.c. and l.s.c.;(iv)
closed if its graphGr(T)={(x,y)∈X×Y:y∈T(x)} is closed;(v)
compact if  closure ofT(X) is a compact subset of Y.Definition 10 (see [5]).
LetI:X→X be a single-valued mapping and T:X→Cl(X) a multivalued mapping. Then, T is called(i)
a multivaluedI-contraction  if there exists a number α∈(0,1):(I-mc)H(Tx,Ty)≤αd(Ix,Iy),∀x,y∈X,(ii)
a multivaluedI-Kannan contraction if there exists a number β∈(0,1/2):(I-mkc)H(Tx,Ty)≤β[d(Ix,Tx)+d(Iy,Ty)],hhhhhhhhhhhhhh∀x,y∈X,(iii)
a multivaluedI-Chatterjea contraction if there exists a number γ∈(0,1/2):(I-mcc)H(Tx,Ty)≤γ[d(Ix,Ty)+d(Iy,Tx)],hhhhhhhhhhhh∀x,y∈X,(iv)
a multivaluedI-quasi-contraction  if
(I-mqc)H(Tx,Ty)≤k·max{d(Ix,Iy),d(Ix,Tx),d(Iy,Ty),hhhhhhhhd(Ix,Ty),d(Iy,Tx)}for some0≤k<1 and all x,y in X,(v)
a multivaluedI-weak or almost contraction  if there exist α∈(0,1) and L≥0:(I-mac)H(Tx,Ty)≤αd(Ix,Iy)+Ld(Iy,Tx),∀x,y∈X,(vi)
a multivalued  generalizedI-almost contraction if there exists a function α:[0,∞)→[0,1) satisfying limsupr→t+α(r)<1 for every t∈[0,∞) such that
(I-gmac)H(Tx,Ty)≤α(d(Ix,Iy))d(Ix,Iy)+Ld(Iy,Tx),hhhhhhhhhhhhhhhhhhhhhhhhhhhh∀x,y∈X.Remark 11.
A multivalued mappingT:X→Cl(X) is called a multivalued I-Zamfirescu (I-mzc) operator if it satisfies at least one of the conditions (i), (ii), and (iii).We have used following Cantor’s intersection theorem in our results.Theorem 12.
LetX be a compact space, and let C1⊃C2⊃C3⋯ be a nested chain of nonempty closed subsets of X. Then, ∩Cn≠ϕ.


## 3. Main Results

Lemma 13.
Let(X,d) be a b-metric space with the b-metric as a continuous functional. Let I:X→X be a single-valued mapping such that rd(x,y)≤d(Ix,Iy) for all x,y∈X, where r>0 is a constant. If T:X→Cl(X) satisfies (I-mc) with rαb2<1, then
(5)δ(Eε(I,T))≤bε(1+b)r(1-αb2),∀ε>0.Proof.
For anyx,y∈Eε(I,T), we have
(6)d(Ix,Iy)=H({Ix},{Iy})≤b[H({Ix},Tx)+H(Tx,{Iy})]≤bε+b2[H(Tx,Ty)+H(Ty,{Iy})]≤bε+b2H(Tx,Ty)+b2ε≤bε+b2ε+αb2d(Ix,Iy).
So,
(7)d(Ix,Iy)≤bε(1+b)1-αb2.
Since rd(x,y)≤d(Ix,Iy), we have
(8)δ(Eε(I,T))≤bε(1+b)r(1-αb2),∀ε>0.Lemma 14.
Let(X,d) be a b-metric space with the b-metric as a continuous functional. Let I:X→X be a continuous single-valued mapping. If T:X→Cl(X) is a lower semicontinuous multivalued mapping. Then, for each ε>0,  Eε(I,T) is closed.Proof.
The proof follows from Lemma16 of Hussain et al. [11].Theorem 15.
Let(X,d) be a complete b-metric space with the b-metric as a continuous functional. Let I:X→X be a continuous single-valued mapping such that rd(x,y)≤d(Ix,Iy), where r>0 is a constant. Let T:X→Cl(X) be a lower semicontinuous map satisfying (I-mc). Then, I and T have a unique endpoint if and only if I and T have the AEPP.Proof.
It is clear that ifI and T have an endpoint, then I and T have the AEPP.
Then,(9)Cn={x∈X:supy∈Txd(Ix,y)≤1n}≠ϕ,∀n∈N.
Also, we have for each n∈N, Cn⊇Cn+1. By Lemma 14, Cn is closed for each  n∈N. Since I and T satisfy AEPP, then Cn≠ϕ for each n∈N. Now, we show that limn→∞δ(Cn)=0. To show this, let x,y∈Cn. Then, from Lemma 13,
(10)δ(Cn)=δ(E1/n(I,T))≤b(1+b)(1/n)r(1-αb2)
and so limn→∞δ(Cn)=0. It follows from the Cantor intersection theorem that
(11)⋂n∈NCn={x0}.
Thus, x0 is the unique endpoint of I and T.Lemma 16.
Let(X,d) be a b-metric space with the b-metric as a continuous functional. Let I:X→X be a single-valued mapping such that rd(x,y)≤d(Ix,Iy) for all x,y∈X, where r>0 is a constant. If T:X→Cl(X) satisfies (I-mkc), then
(12)δ(Eε(I,T))≤bεr(1+b+2βb),∀ε>0.Proof.
For anyx,y∈Eε(I,T), we have
(13)d(Ix,Iy)=H({Ix},{Iy})≤b[H({Ix},Tx)+H(Tx,{Iy})]≤bε+b2[H(Tx,Ty)+H(Ty,{Iy})]≤bε+b2H(Tx,Ty)+b2ε≤bε+b2ε+b2β[d(Ix,Tx)+d(Iy,Ty)]≤bε+b2ε+b22βε≤bε(1+b+2βb).
So,
(14)d(Ix,Iy)≤bε(1+b+2βb).
Since rd(x,y)≤d(Ix,Iy), we have
(15)δ(Eε(I,T))≤bεr(1+b+2βb),∀ε>0.Theorem 17.
Let(X,d) be a complete b-metric space with the b-metric as a continuous functional. Let I:X→X be a continuous single-valued mapping such that rd(x,y)≤d(Ix,Iy), where r>0 is a constant. Let T:X→Cl(X) be a lower semicontinuous map satisfying (I-mkc). Then, I and T have a unique endpoint if and only if I and T have the AEPP.Proof.
It is clear that ifI and T have an endpoint, then I and T have AEPP. Then,
(16)Cn={x∈X:supy∈Txd(Ix,y)≤1n}≠ϕ,∀n∈N.
Also, we have for each n∈N, Cn⊇Cn+1. By Lemma 14, Cn is closed for each  n∈N. Since I and T satisfy AEPP, then Cn≠ϕ for each n∈N. Now, we show that limn→∞δ(Cn)=0. To show this, let x,y∈Cn. Then, from Lemma 16,
(17)δ(Cn)=δ(E1/n(I,T))≤b(1/n)r(1+b+2βb)
and so limn→∞δ(Cn)=0. It follows from the Cantor intersection theorem that
(18)⋂n∈NCn={x0}.
Thus, x0 is the unique endpoint of I and T.Lemma 18.
Let(X,d) be a b-metric space with the b-metric as a continuous functional. Let I:X→X be a single-valued mapping such that rd(x,y)≤d(Ix,Iy) for all x,y∈X, where r>0 is a constant. If T:X→Cl(X) satisfies (I-mcc) with 2b2rγ<1, then
(19)δ(Eε(I,T))≤bε(1+b+2γb)r(1-2b2γ),∀ε>0.Proof.
For anyx,y∈Eε(I,T), we have
(20)d(Ix,Iy)=H({Ix},{Iy})≤b[H({Ix},Tx)+H(Tx,{Iy})]≤bε+b2[H(Tx,Ty)+H(Ty,{Iy})]≤bε+b2H(Tx,Ty)+b2ε≤bε+b2ε+b2γ[d(Ix,Ty)+d(Iy,Tx)]≤bε+b2ε+b2γ[d(Ix,Iy)+d(Iy,Ty)]+b2γ[d(Iy,Ix)+d(Ix,Tx)]=bε+b2ε+b2γ[H({Ix},{Iy})+H({Iy},Ty)]+b2γ[H({Iy},{Ix})+H({Ix},Tx)]≤bε+b2ε+2b2γH({Ix},{Iy})+2b2γε=bε+b2ε+2b2γd(Ix,Iy)+2b2γε≤bε(1+b+2γb)1-2b2γ.
So,
(21)d(Ix,Iy)≤bε(1+b+2γb)1-2b2γ.
Since rd(x,y)≤d(Ix,Iy), we have
(22)δ(Eε(I,T))≤bε(1+b+2γb)r(1-2b2γ),∀ε>0.Theorem 19.
Let(X,d) be a complete b-metric space with the b-metric as a continuous functional. Let I:X→X be a continuous single-valued mapping such that rd(x,y)≤d(Ix,Iy), where r>0 is a constant. Let T:X→Cl(X) be a lower semicontinuous map satisfying (I-mcc). Then, I and T have a unique endpoint if and only if I and T have the AEPP.Proof.
It is clear that ifI and T have an endpoint, then I and T have the AEPP.
Then,(23)Cn={x∈X:supy∈Txd(Ix,y)≤1n}≠ϕ,∀n∈N.
Also, we have for each n∈N, Cn⊇Cn+1. By Lemma 14, Cn is closed for each n∈N. Since I and T satisfy AEPP, then Cn≠ϕ for each n∈N. Now, we show that limn→∞δ(Cn)=0. To show this, let x,y∈Cn. Then, from Lemma 18,
(24)δ(Cn)=δ(E1/n(I,T))≤b(1/n)(1+b+2γb)r(1-2b2γ)
and so limn→∞δ(Cn)=0. It follows from the Cantor intersection theorem that
(25)⋂n∈NCn={x0}.
Thus, x0 is the unique endpoint of I and T.Lemma 20.
Let(X,d) be a  b-metric space with the b-metric as a continuous functional. Let I:X→X be a single-valued mapping such that rd(x,y)≤d(Ix,Iy)  for all x,y∈X, where r>0 is a constant. If  T:X→Cl(X) satisfies (I-mzc) with δ=max{α,βb/(1-βb2),β(1+b2)/2(1-βb2),γb/(1-γb)}, rδ<1, then
(26)δ(Eε(I,T))≤bε(1+b+2δb)r(1-δ),∀ε>0.Proof.
Suppose thatT satisfies (I-mkc), then we have
(27)H(Tx,Ty)≤β[d(Ix,Tx)+d(Iy,Ty)]≤β[H(Ix,Tx)+H(Iy,Ty)]≤βH(Ix,Tx)+βbH(Iy,Ix)+βbH(Ix,Ty)≤βH(Ix,Tx)+βbH(Iy,Ix)+βb2H(Ix,Tx)+βb2H(Tx,Ty)≤(β+βb2)H(Ix,Tx)+βbd(Iy,Ix)+βb2H(Tx,Ty).
So, we have
(28)H(Tx,Ty)≤(β+βb2)1-βb2H(Ix,Tx)+βb1-βb2d(Ix,Iy).
If T satisfies (I-mcc), then we have
(29)H(Tx,Ty)≤γ[d(Ix,Ty)+d(Iy,Tx)]≤γ[H(Ix,Ty)+H(Iy,Tx)]≤γb[H(Ix,Tx)+H(Tx,Ty)]+γb[H(Iy,Ix)+H(Ix,Tx)]≤2γbH(Ix,Tx)+γbd(Ix,Iy)+γbH(Tx,Ty)H(Tx,Ty)≤2γb1-γbH(Ix,Tx)+γb1-γbd(Ix,Iy).
Let δ=max{α,(βb)/(1-βb^2),(β(1+b^2))/(2(1-βb^2)),(γb)/(1-γb)}. Then,
(30)H(Tx,Ty)≤2δH(Ix,Tx)+δd(Ix,Iy).
Thus, for any x,y∈Eε(I,T), we have
(31)d(Ix,Iy)=H({Ix},{Iy})≤b[H({Ix},Tx)+H(Tx,{Iy})]≤bε+b2[H(Tx,Ty)+H(Ty,{Iy})]≤bε+b2H(Tx,Ty)+b2ε≤bε+b2ε+b22δd(Ix,Tx)+δd(Ix,Iy)≤bε+b2ε+b22δε+δd(Ix,Iy).
So,
(32)d(Ix,Iy)≤bε(1+b+2δb)1-δ.
Since rd(x,y)≤d(Ix,Iy), we have
(33)δ(Eε(I,T))≤bε(1+b+2δb)r(1-δ),∀ε>0.Theorem 21.
Let(X,d) be a complete b-metric space with the b-metric as a continuous functional. Let I:X→X be a continuous single-valued mapping such that rd(x,y)≤d(Ix,Iy), where r>0 is a constant. Let T:X→Cl(X) be a lower semicontinuous map  satisfying (I-mzc). Then, I and T have a unique endpoint if and only if I and T have the AEPP.Proof.
It is clear that ifI and T have an endpoint, then I and T have the AEPP. Then,
(34)Cn={x∈X:supy∈Txd(Ix,y)≤1n}≠ϕ,∀n∈N.
Further, we have for each n∈N, Cn⊇Cn+1. By Lemma 14, Cn is closed for each n∈N. Since I and T satisfy AEPP, then Cn≠ϕ for each n∈N. Now, we show that limn→∞δ(Cn)=0. To show this, let x,y∈Cn. Then, from Lemma 20,
(35)δ(Cn)=δ(E1/n(I,T))≤b(1/n)(1+b+2δb)r(1-δ).
and so limn→∞δ(Cn)=0. It follows from the Cantor intersection theorem that
(36)⋂n∈NCn={x0}.
Thus, x0 is the unique endpoint ofI and T.Lemma 22.
Let(X,d) be a b-metric space with the b-metric as a continuous functional. Let I:X→X be a single-valued mapping such that rd(x,y)≤d(Ix,Iy) for all x,y∈X, where r>0 is a constant. If T:X→Cl(X) satisfies (I-mqc) with k<1/(r(b^2+b)), then
(37)δ(Eε(I,T))≤bε(1+b+kb)r(1-kb2),∀ε>0.Proof.
For anyx,y∈Eε(I,T), we have
(38)d(Ix,Iy)=H({Ix},{Iy})≤b[H({Ix},Tx)+d(Tx,{Iy})]≤bε+b2[H(Tx,Ty)+H(Ty,{Iy})]≤bε+b2H(Tx,Ty)+b2ε≤bε+b2ε+kb2·max{d(Ix,Iy),d(Ix,Tx),d(Iy,Ty),d(Ix,Ty),d(Iy,Tx)}≤bε+b2ε+kb2{d(Ix,Iy)+ε}.
So,
(39)d(Ix,Iy)≤bε(1+b+kb)1-kb2.
Since rd(x,y)≤d(Ix,Iy), we have
(40)δ(Eε(I,T))≤bε(1+b+kb)r(1-kb2),∀ε>0.Theorem 23.
Let(X,d) be a complete b-metric space with the b-metric as a continuous functional. Let I:X→X be a continuous single-valued mapping such that rd(x,y)≤d(Ix,Iy), where r>0 is a constant. Let T:X→Cl(X) be a lower semicontinuous map satisfying (I-mqc). Then, I and T have a unique endpoint if and only if I and T have the AEPP.Proof.
It is clear that ifI and T have an endpoint, then I and T have the AEPP.
Then,(41)Cn={x∈X:supy∈Txd(Ix,y)≤1n}≠ϕ,∀n∈N.
Also, it is clear that, for each n∈N, Cn⊇Cn+1. By the above Lemma 14, Cn is closed for each n∈N. Since I and T satisfy AEPP, then Cn≠ϕ for each n∈N. Now, we show that limn→∞δ(Cn)=0. To show this, let x,y∈Cn. Then, from Lemma 22,
(42)δ(Cn)=δ(E1/n(I,T))≤b(1/n)(1+b+kb)r(1-kb2)
and so limn→∞δ(Cn)=0. It follows from the Cantor intersection theorem that
(43)⋂n∈NCn={x0}.
Thus, x0 is the unique endpoint of I and T.Lemma 24.
Let(X,d) be a b-metric space and I:X→X a single-valued mapping such that rd(x,y)≤d(Ix,Iy) for all x,y∈X, where r>0 is a constant. If   T:X→Cl(X) satisfies (I-mac) with rb2(α+bL)<1, then
(44)δ(Eε(I,T))≤bε(1+b+Lb2)r(1-b2(α+bL)),∀ε>0.Proof.
For anyx,y∈Eε(I,T), we have
(45)d(Ix,Iy)=H({Ix},{Iy})≤b[H({Ix},Tx)+d(Tx,{Iy})]≤bε+b2[H(Tx,Ty)+H(Ty,{Iy})]≤bε+b2H(Tx,Ty)+b2ε≤bε+b2ε+b2[αd(Ix,Iy)+Ld(Iy,Tx)]≤bε+b2ε+b2αd(Ix,Iy)+b3L[d(Ix,Iy)+d(Ix,Tx)]≤bε+b2ε+b2αd(Ix,Iy)+b3Ld(Ix,Iy)+b3Lε≤bε(1+b+Lb2)+b2(α+sL)d(Ix,Iy).
So,
(46)d(Ix,Iy)≤bε(1+b+Lb2)1-b2(α+bL).
Since rd(x,y)≤d(Ix,Iy), we have
(47)δ(Eε(I,T))≤bε(1+b+Lb2)r(1-b2(α+bL)),∀ε>0.Theorem 25.
Let(X,d) be a complete b-metric space with the b-metric as a continuous functional. Let I:X→X be a continuous single-valued mapping such that rd(x,y)≤d(Ix,Iy), where r>0 is a constant. Let T:X→Cl(X) be a lower semicontinuous map satisfying (I-mac). Then, I and T have a unique endpoint if and only if I and T have the AEPP.Proof.
It is clear that ifI and T have an endpoint, then I and T have the AEPP. Then,
(48)Cn={x∈X:supy∈Txd(Ix,y)≤1n}≠ϕ,∀n∈N.
Also, we have for each n∈N, Cn⊇Cn+1. By Lemma 14, Cn is closed for each n∈N. Since I and T satisfy AEPP, then Cn≠ϕ for each n∈N. Now, we show that limn→∞δ(Cn)=0. To show this, let x,y∈Cn. Then, from Lemma 24,
(49)δ(Cn)=δ(E1/n(I,T))≤b(1/n)(1+b+Lb2)r(1-b2(α+bL))
and so limn→∞δ(Cn)=0. It follows from the Cantor intersection theorem that
(50)⋂n∈NCn={x0}.
Thus, x0 is the unique endpoint of I and T.On puttingb=1 in the above Theorem 25, we obtain the following result of [11].Corollary 26 (see [11]).
Let(X,d) be a complete metric space. Let I:X→X be a continuous single-valued mapping such that rd(x,y)≤d(Ix,Iy), where r>0 is a constant. Let T:X→Cl(X) be a lower semicontinuous  map satisfying (I-mac). Then, I and T have a unique endpoint if and only if I and T have the AEPP.IfI is the identity mapping on X and b=1, then the above result reduces to the following results:Corollary 27 (see [11, Corollary 3.5]).
Let(X,d) be a metric space, and let T:X→Cl(X) satisfy (I-mac) with α+L<1. Then, for each ε>0,
(51)δ(Eε(T))≤ε(2+L)1-(α+L),
where Eε(T)={x∈X:supy∈Txd(x,y)≤ε}.Corollary 28 (see [11, Corollary 3.6]).
Let(X,d) be a complete metric space. Let T:X→Cl(X) be a lower semicontinuous map satisfying (I-mac) with α+L<1. Then, T has a unique endpoint if and only if T has the AEPP.IfL=0, in almost contraction, then we have following result in metric space.Corollary 29 ([10, Corollary 2.2]).
Let(X,d) be a complete metric space. Let T:X→Cl(X) satisfy (mc). Then, T has a unique endpoint if and only if T has the AEPP.



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*Source: 542302-2013-11-17.xml*