R E S E A R C H
Open Access
Existence and uniqueness of fixed point for
mixed monotone ternary operators with
application
Changchang Bu
1,2, Yuqiang Feng
1,2*and Hui Li
1,2*Correspondence:
1School of Science, Wuhan
University of Science and Technology, Wuhan, 430065, P.R. China
2Hubei Province Key Laboratory of
Systems Science in Metallurgical Process, Wuhan, 430065, P.R. China
Abstract
In this paper, partial order theory is used to study the fixed point of a mixed monotone ternary operatorA:P×P×P→P. The existence and uniqueness of a fixed point are obtained without assuming the operators to be compact or continuous. In the end, the application to an integral equation is presented. Our results unify, generalize, and complement various known comparable results from the current literature.
MSC: 47H05; 47H10
Keywords: fixed point; mixed monotone ternary operator; normal cone
1 Introduction
Fixed point theory has fascinated hundreds of researchers since with the celebrated Banach fixed point theorem. It is well known that mixed monotone operators were in-troduced by Guo and Lakshmikantham [] in . Later, Bhaskar and Lakshmikantham [] introduced the notion of a coupled fixed point and proved some coupled fixed point results under certain conditions, in a complete metric space endowed with a partial or-der. Their study has not only important theoretical meaning but also wide applications in engineering, nuclear physics, biological chemistry technology,etc.(see [–] and the references therein).
Very recently, Harjaniet al.[] have established the existence results of coupled fixed point for mixed monotone operators, and further obtained their applications to integral equations. Berinde and Borcut [] have introduced the concept of a triple fixed point and proved some related theorems for contractive type operators in partially ordered metric spaces. Zhai [] has considered mixed monotone operators with convexity and get the existence and uniqueness of a fixed point (A(u,u) =utype) without assuming the operator to be compact or continuous.
Motivated by the work reported in [–], the aim of this paper is to discuss the existence and uniqueness of a fixed point (A(u,u,u) =utype) for mixed monotone ternary operators in the context of ordered metric spaces. Our results unify, generalize, and complement various known comparable results from the current literature.
The rest of the paper is organized as follows. In Section , we recall some basic defi-nitions and notations which will be used in the sequel. The existence and uniqueness of
a fixed point for mixed monotone ternary operators (without assuming the operators to be compact or continuous) are obtained in Section . We also present an application in Section to an integral equation to illustrate our results.
2 Preliminaries
In this section, we recall some standard definitions and notations needed in the following section. For the convenience of the reader, we suggest that one refers to [, , –] for details.
Throughout this paper, unless otherwise specified, suppose that (E,·) is a real Banach space which is partially ordered by a coneP⊂E,i.e.,x≤yif and only ify–x∈P. Ifx≤y
andx=y, then we denotex<yory>x. Byθ we denote the zero element ofE. Recall that a non-empty closed convex setP⊂E is a cone if it satisfies (i)x∈P,λ≥⇒λx∈P; (ii)x∈P, –x∈P⇒x=θ.
Further,Pis called normal if there exists a constantN> such that, for allx,y∈E,θ≤
x≤yimpliesx ≤Ny; in this caseNis called the normality constant ofP. Ifx,x∈E, the set [x,x] ={x∈E|x≤x≤x}is called the order interval betweenxandx.
Definition .(see []) A:P×P→Pis said to be a mixed monotone operator ifA(x,y) is monotone non-decreasing inxand monotone non-increasing iny, that is, for anyx,y∈P,
x,x∈P, x≤x ⇒ A(x,y)≤A(x,y),
y,y∈P, y≤y ⇒ A(x,y)≤A(x,y).
(.)
Definition .(see []) An elementx∈Pis called a fixed point ofA:P×P→Pif
A(x,x) =x.
Definition .(see []) A:P×P×P→Pis said to be a mixed monotone operator if
A(x,y,z) is monotone non-decreasing inx,zand monotone non-increasing iny, that is, for anyx,y,z∈P
x,x∈P, x≤x ⇒ A(x,y,z)≤A(x,y,z),
y,y∈P, y≤y ⇒ A(x,y,z)≥A(x,y,z), (.)
z,z∈P, z≤z ⇒ A(x,y,z)≤A(x,y,z).
Definition . An elementx∈Pis called a fixed point ofA:P×P×P→Pif
A(x,x,x) =x.
3 Main results
In this section we consider the existence and uniqueness of a fixed point for mixed mono-tone ternary operators in ordered Banach spaces. Our first main result is the following.
(H) fort∈(, ),x,y∈P,there existsα(t,x,y)∈(, +∞),such that
A(tx,y,tx)≤tα(t,x,y)A(x,y,x); (.)
(H) there existu,v,m∈P,r∈(, ),such that
u≤rv, m≤rv,
A(u,v,m)≥u, A(v,u,v)≤v, A(m,v,u)≥m, (.)
A(u,v,u)≥u, A(m,v,m)≥m.
Then A has a unique fixed point u∗in[u,rv]∩[m,rv].Moreover,constructing
succes-sively the sequences
xn=A(xn–,yn–,zn–), yn=A(yn–,xn–,yn–),
zn=A(zn–,yn–,xn–), n= , , . . . ,
for any initial values x,y,z∈[u,rv]∩[m,rv],we have
xn–u∗→, yn–u∗→, zn–u∗→
as n→ ∞.
Proof Letw=rv,ε=rα(r,v,u)–. Thenw≥u,ε∈(, ), and
A(w,u,w) =A(rv,u,rv)≤rα(r,v,u)A(v,u,v)
≤rα(r,v,u)v
=rα(r,v,u)–·rv=εw≤w, (.)
A(u,w,m) =A(u,rv,m)≥A(u,v,m)≥u, (.)
A(m,w,u) =A(m,rv,u)≥A(m,v,u)≥m. (.)
Construct successively the sequences
un=A(un–,wn–,mn–), wn=A(wn–,un–,wn–), mn=A(mn–,wn–,un–),
wn= εA
wn–,un–,wn–
, w=w, n= , , . . . .
From (.)-(.) and the mixed monotonicity ofA, we have
u≤u≤u≤ · · · ≤un≤ · · · ≤wn≤ · · · ≤w≤w, (.)
m≤m≤m≤ · · · ≤mn≤ · · · ≤wn≤ · · · ≤w≤w. (.)
Next we prove that
From (.) and (.),
w=
εA(w,u,w)≤
ε·εw=w,
w=
εA(w,u,w)≥
εA(u,v,u)≥
εu≥u,
w= εA
w,u,w
≤
εA(w,u,w)≤
ε·εw=w,
w= εA
w,u,w
≥
εA(u,v,u)≥
εu≥u.
Suppose that whenn=k, we have
u≤wk≤w,
then whenn=k+ , note thatuk≤w=rv≤v, we obtain
wk+= εA
wk,uk,wk≤
εA(w,u,w)≤
ε·εw=w,
wk+= εA
wk,uk,wk
≥
εA(u,v,u)≥
εu≥u.
By mathematical induction, we know that (.) holds. The same procedure may easily be adapted to obtain
m≤wn≤w. (.)
On the other hand, from (.),
w=A(w,u,w) =ε
εA(w,u,w) =εw
,
w=A(w,u,w) =A
εw,u,εw
≤εα(ε,w,u)Aw
,u,w
=εα(ε,w,u)+·
εA
w,u,w
≤εw.
Suppose that whenn=k, we havewk≤εkwk. Then whenn=k+ , in view of (.), we obtain
wk+ =A(wk,uk,wk)≤A
εkwk,uk,εkwk
≤εkα(ε
k,w
k,uk)
Awk,uk,wk
=εkα(εk,wk,uk)+·
εA
wk,uk,wk
≤εk+wk+.
By mathematical induction, we have
By (.)-(.) we get
θ≤wn–un≤εnwn–un≤εnwn–εnun=εn
wn–un
≤εn(w–u),
θ≤un+p–un≤wn–un, θ≤wn–wn+p≤wn–un;
θ≤wn–mn≤εnwn–mn≤εnwn–εnmn=εn
wn–mn
≤εn(w–m),
θ≤mn+p–mn≤wn–mn.
Noting thatPis normal andε∈(, ), we have
wn–un ≤Nεnw–u → (asn→ ∞),
wn–mn ≤Nεnw–m → (asn→ ∞).
Further,
un+p–un ≤Nwn–un → (asn→ ∞),
wn–wn+p ≤Nwn–un → (asn→ ∞),
mn+p–mn ≤Nwn–mn → (asn→ ∞).
HereNis the normality constant.
So, we can claim that{un},{wn}, and{mn}are Cauchy sequences. SinceEis complete, there existu∗,w∗,m∗∈Psuch that
un→u∗, wn→w∗, mn→m∗ (asn→ ∞).
By (.), (.), respectively, we know that
u≤un≤u∗≤w∗≤wn≤w,
m≤mn≤m∗≤w∗≤wn≤w,
and then
θ≤w∗–u∗≤wn–un≤εn(w–u),
θ≤w∗–m∗≤wn–mn≤εn(w–m).
Further,w∗–u∗ ≤Nεnw
–u → (asn→ ∞), and thusw∗=u∗. Similarly, we get
w∗–m∗ ≤Nεnw
–m → (asn→ ∞), and thusw∗=m∗. Consequently,w∗=u∗=
m∗. Then we obtain
un+=A(un,wn,mn)≤A
u∗,u∗,u∗≤A(wn,un,wn) =wn+.
Lettingn→ ∞, then we get
Au∗,u∗,u∗=u∗.
In the following, we prove thatu∗is the unique fixed point ofAin [u,rv]∩[m,rv]. Suppose that there existsx∗∈[u,rv]∩[m,rv] such thatA(x∗,x∗,x∗) =x∗. Thenu≤
x∗≤wandm≤x∗≤w. By mathematical induction and the mixed monotonicity ofA, we have
un+=A(un,wn,mn)≤x∗=A
x∗,x∗,x∗≤A(wn,un,wn) =wn+.
Then from the normality ofP, we havex∗=u∗. Moreover, constructing successively the sequences
xn=A(xn–,yn–,zn–), yn=A(yn–,xn–,yn–),
zn=A(zn–,yn–,xn–), n= , , . . . ,
for any initial valuesx,y,z∈[u,rv]∩[m,rv], we haveun≤xn,wn≥yn,mn≤zn,
n= , , . . . . Lettingn→ ∞yieldsxn→u∗,yn→u∗,zn→u∗asn→ ∞.
Remark . It is evident from (.) that fort∈(, ),x,y∈P, there existsα(t,tx,y)∈ (, +∞), such that
A
tx,y,
tx
≥
tα(t,tx,y)A(x,y,x). (.)
Remark . Letα(t,x,y) be a constantα∈(, +∞), then Theorem . also holds.
Corollary . Let E be a real Banach space and let P be a normal cone in E.A:P×P×P→ P is a mixed monotone ternary operator which satisfies(H)and,for t∈(, ),x,y∈P,there
existsα∈(, +∞),such that A(tx,y,tx)≤tαA(x,y,x).Then A has a unique fixed point u∗ in[u,rv]∩[m,rv].Moreover,constructing successively the sequences
xn=A(xn–,yn–,zn–), yn=A(yn–,xn–,yn–),
zn=A(zn–,yn–,xn–), n= , , . . . ,
for any initial values x,y,z∈[u,rv]∩[m,rv],we have un≤xn,wn≥yn,mn≤zn,
n= , , . . . .Letting n→ ∞yields xn→u∗,yn→u∗,zn→u∗as n→ ∞.
Following the lines of the proof of Theorem ., we obtain an immediate consequence.
Corollary .(see []) Let E be a real Banach space and let P be a normal cone in E.
A:P×P→P is a mixed monotone operator which satisfies the following:
(H) fort∈(, ),x,y∈P,there existsα(t,x,y)∈(, +∞),such that
A(tx,y)≤tα(t,x,y)A(x,y);
(H) there existu,v∈P,r∈(, ),such that
Then A has a unique fixed point u∗ in[u,rv].Moreover, constructing successively the
sequences
xn=A(xn–,yn–), yn=A(yn–,xn–), n= , , . . . ,
for any initial values x,y∈[u,rv],we have
xn–u∗→, yn–u∗→
as n→ ∞.
Theorem . Let E be a real Banach space and let P be a normal cone in E.A:P×P×P→ P is a mixed monotone ternary operator which satisfies(.)and
(H) forR∈(, +∞),x,y,z∈Pthere existα(R,Rx,y,z),α(R,x,y,Rz)∈(, +∞)such that
A(Rx,y,z)≥Rα(R,Rx,y,z)A(x,y,z), (.)
A(x,y,Rz)≥Rα(R,x,y,Rz)A(x,y,z); (.)
(H) there existu,v,m∈P,R∈(, +∞),such that
v≥Ru, v≥Rm,
A(u,v,m)≥u, A(v,u,v)≤v, A(m,v,u)≥m, (.)
A(u,v,u)≥u, A(m,v,m)≥m.
Then the operator equation A(w,w,w) =bw has a unique solution w∗ in [Ru,v]∩ [Rm,v],where b=min{Rα(
R,Ru,v,m)–,Rα(R,m,v,Ru)–}.Moreover,constructing succes-sively the sequences
xn=b–A(xn–,yn–,zn–), yn=b–A(yn–,xn–,yn–),
zn=b–A(zn–,yn–,xn–), n= , , . . . ,
for any initial values x,y,z∈[Ru,v]∩[Rm,v],we have
xn–w∗→, yn–w∗→, zn–w∗→
as n→ ∞.
Remark . Two comments with respect to conditions (.) and (.) are in order:
(a) A sufficient condition onAfor(.)to be satisfied is that fort∈(, ),x,y,z∈P, there existsα(t,x,y,z)∈(, +∞), such that
A(tx,y,z)≤tα(t,x,y,z)A(x,y,z).
(b) A sufficient condition onAfor(.)to be satisfied is that fort∈(, ),x,y,z∈P, there existsα(t,x,y,z)∈(, +∞), such that
Proof of Theorem. Letw=Ru. Thenv≥w. Note thatb> , from (.)-(.),
A(w,v,m) =A(Ru,v,m)≥Rα(
R,Ru,v,m)A(u
,v,m)
=Rα(R,Ru,v,m)–RA(u
,v,m)≥bRu=bw≥w, (.)
A(v,w,v) =A(v,Ru,v)≤A(v,u,v)≤v, (.)
A(m,v,w) =A(m,v,Ru)≥Rα(
R,m,v,Ru)A(m
,v,u)
=Rα(R,m,v,Ru)–RA(m
,v,u)≥bA(m,v,u)≥bm≥m. (.)
SetB(x,y,z) =b–A(x,y,z),x,y,z∈P. Then from the above inequalities, we have
B(w,v,m) =b–A(w,v,m)≥b–bw=w,
B(v,w,v) =b–A(v,w,v)≤b–v≤v, (.)
B(m,v,w) =b–A(m,v,w)≥b–bm=m.
Also, construct successively the sequences
wn=B(wn–,vn–,mn–), vn=B(vn–,wn–,vn–), mn=B(mn–,vn–,wn–),
vn=bBvn–,wn–,vn–
, v=v, n= , , . . . .
From (.) and the mixed monotonicity ofA, we have
w≤w≤w≤ · · · ≤wn≤ · · · ≤vn≤ · · · ≤v≤v, (.)
m≤m≤m≤ · · · ≤mn≤ · · · ≤vn≤ · · · ≤v≤v. (.)
Next we prove that
w≤vn≤v, n= , , . . . . (.)
By (.) and (.), we have
A(w,v,w) =A(Ru,v,Ru)≥Rα(
R,Ru,v)A(u
,v,u)
≥RA(u,v,u)
≥Ru
=w. (.)
From (.)-(.) and (.),
v=bBv,w,v
=bB(v,w,v) =A(v,w,v)≤v,
v=bBv,w,v
=bB(v,w,v)≥bB(w,v,m) =A(w,v,m)≥w,
v=bBv,w,v
≤bB(v,w,v) =A(v,w,v)≤v,
v=bBv,w,v
Suppose that whenn=k, we have
w≤vk≤v,
then whenn=k+ , recalling (.) and (.), we obtain
vk+=bBvk,wk,vk≤bB(v,w,v) =A(v,w,v)≤v,
vk+=bBvk,wk,vk≥bB(w,v,w) =A(w,v,w) =A(Ru,v,Ru)≥w.
By mathematical induction, we know that (.) holds. The same procedure may easily be adapted to obtain
m≤vn≤v, n= , , . . . . (.)
On the other hand, from (.),
v=B(v,w,v) =
bbB(v,w,v) =
bbB
v,w,v
=
bv
,
v=B(v,w,v) =B
bv
,w, bv ≤ b
α(b,v,w)
Bv,w,v
=
b
α(b,v,w)+
bBv,w,v
≤ b
v.
Suppose that whenn=k, we havevk≤(b)kvk. Then whenn=k+ , in view of (.), we obtain
vk+ =B(vk,wk,vk)≤B
b
k
vk,wk,
b
k
vk
≤
b
kα((b)k,v k,wk)
Bvk,wk,vk
≤
b
kα((b)k,vk,wk)+
bBvk,wk,vk
≤
b
k+
vk+.
By mathematical induction, we have
vn≤
b
n
vn, n= , , . . . . (.)
By (.)-(.) we get
θ≤vn–wn≤
b
n
vn–wn≤
b
n
vn–
b n wn = b n
vn–wn
≤ b n
θ≤wn+p–wn≤vn–wn, θ≤vn–vn+p≤vn–wn;
θ≤vn–mn≤
b
n
vn–mn≤
b
n
vn–
b
n
mn
=
b
n
vn–mn
≤
b
n
(v–m),
θ≤mn+p–mn≤vn–mn, θ≤vn–vn+p≤vn–mn.
Note thatPis normal andb> , we have
vn–wn ≤N
b
n
v–w → (asn→ ∞),
vn–mn ≤N
b
n
v–m → (asn→ ∞).
Further,
wn+p–wn ≤Nvn–wn → (asn→ ∞),
vn–vn+p ≤Nvn–wn → (asn→ ∞),
mn+p–mn ≤Nvn–mn → (asn→ ∞).
HereNis the normality constant.
So, we can claim that{wn},{vn}, and{mn}are Cauchy sequences. SinceEis complete, there existw∗,v∗,m∗∈Psuch that
wn→w∗, vn→v∗, mn→m∗ (asn→ ∞).
By (.), (.), respectively, we know that
w≤wn≤w∗≤v∗≤vn≤v,
m≤mn≤m∗≤v∗≤vn≤v,
and then
θ≤v∗–w∗≤vn–wn≤
b
n
(v–w),
θ≤v∗–m∗≤vn–mn≤
b
n
(v–m).
Further,v∗–w∗ ≤N(b)nv
–w → (asn→ ∞), and thusv∗=w∗. Similarly, we get
v∗–m∗ ≤N( b)
nv
–m → (asn→ ∞), and thusv∗=m∗. Consequently,w∗=v∗=
m∗. Then we obtain
wn+=B(wn,vn,mn)≤B
w∗,w∗,w∗≤B(vn,wn,vn) =vn+.
Lettingn→ ∞, we get
That is, the operator equation A(w,w,w) =bw has a unique solution w∗ in [Ru,v]∩ [Rm,v].
In the following, we prove thatw∗is the unique solution ofA(w,w,w) =bwin [Ru,v]∩ [Rm,v]. Suppose that there existsx∗∈[Ru,v]∩[Rm,v] such thatA(x∗,x∗,x∗) =bx∗. Thenw≤x∗≤vandm≤x∗≤v. By mathematical induction and the mixed mono-tonicity ofA, we have
wn+=B(wn,vn,mn)≤x∗=B
x∗,x∗,x∗≤B(vn,wn,vn) =vn+.
Then from the normality ofP, we havex∗=w∗. Moreover, constructing successively the sequences
xn=b–A(xn–,yn–,zn–), yn=b–A(yn–,xn–,yn–),
zn=b–A(zn–,yn–,xn–), n= , , . . . ,
for any initial valuesx,y,z∈[Ru,v]∩[Rm,v], we havewn≤xn,vn≥yn,mn≤zn,
n= , , . . . . Lettingn→ ∞yieldsxn→w∗,yn→w∗,zn→w∗asn→ ∞.
From the proof of Theorem ., we can easily obtain the following conclusion.
Corollary .(see []) Let E be a real Banach space and let P be a normal cone in E.
A:P×P→P is a mixed monotone operator which satisfies(H)and
(H) there existu,v∈P,R∈(, +∞)such that
v≥Ru, A(u,v)≥u, A(v,u)≤v.
Then the operator equation A(w,w) =bw has a unique solution w∗in[Ru,v],where b=
Rα(R,Ru,v)–.Moreover,constructing successively the sequences
xn=b–A(xn–,yn–), yn=b–A(yn–,xn–), n= , , . . . ,
for any initial values x,y∈[Ru,v],we have
xn–w∗→, yn–w∗→
as n→ ∞.
4 Application
As application of our results, we investigate the solvability of the following integral equa-tion:
x(τ) =
k(τ,s)
xα(s)
+xα(s)+x
α(s) ds (.)
withα,α> ,α> .
is continuous and <k(τ,s) ds≤ . In the following, we prove that (.) has a unique solution.
Consider the integral operatorA:P×P×P→Pdefined by
A(u,v,m) =
k(τ,s)
uα(s)
+vα(s)+m α(s) ds
withα,α> ,α> .
It is clear that A(u,v,m) is a mixed monotone ternary operator. We shall show that
A(u,v,m) satisfies (H) and for t∈ (, ),x,y∈P, there exists α ∈(, +∞), such that
A(tx,y,tx)≤tαA(x,y,x).
In fact, letu(τ)≡,m(τ)≡,v(τ)≡, then
A(u,v,u) = ≥u,
A(v,u,v) =
k(τ,s) ds≤.
On the other hand, noting that for anyt∈(, ), letting
α=min{α,α}> ,
we obtain
A(tx,y,tx) =
k(τ,s)
tαxα(s)
+yα(s)+t
αxα(s) ds
≤
k(τ,s)
tαxα(s)
+yα(s)+t α
xα(s) ds
=tα
k(τ,s)
xα(s)
+yα(s)+x α(s) ds
=tαA(x,y,x).
Hence, all the hypotheses of Corollary . are satisfied. The operator
A(u,v,m) =
k(τ,s)
uα(s)
+vα(s)+m α(s) ds
has a unique fixed point in [u,v],i.e., the integral equation (.) has a unique solution in [u,v].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
This research is supported by the Doctoral Fund of Education Ministry of China (20134219120003), the Natural Science Foundation of Hubei Province (2013CFA131), the National Natural Science Foundation of China (71231007) and Hubei Province Key Laboratory of Systems Science in Metallurgical Process (z201302).
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