R E S E A R C H
Open Access
Some new results for single-valued and
multi-valued mixed monotone operators of
Rhoades type
Farshid Khojasteh
**Correspondence: [email protected] Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran
Abstract
In (2008), Zhang proved the existence of fixed points of mixed monotone operators along with certain convexity and concavity conditions. In this paper, mixed
monotone single-valued and multi-valued operators of Rhoades type are defined and two fixed point theorems are proved.
MSC: 47H10; 47H07
Keywords: mixed monotone operator; Rhoades type; multi-valued; increasing inward mappings;L-function
1 Introduction and preliminaries
In (), mixed monotone operators were introduced by Guo and Lakshmikantham []. Then many authors studied them in Banach spaces and obtained lots of interesting results (see [, ] and [–]).
On the other hand, in (), Rhoades [] introduced a new fixed point theorem as a generalization of Banach fixed point theorem.
Theorem .(Rhoades []) Let(X,d)be a complete metric space.Suppose that T:X→X is a single-valued mapping that satisfies
d(Tx,Ty)≤d(x,y) –ψd(x,y) ()
for each x,y∈X,whereψ: [, +∞)→[, +∞)is continuous,nondecreasing andψ–() =
{}(i.e.,weakly contractive mappings).Then T has a fixed point.
In this paper, a weak mixed monotone single-valued and multi-valued operator of Rhoades type is defined. Then two fixed point theorems for this kind of operators are proved.
LetEbe a real Banach space. The zero element ofEis denoted byθ. A subsetPofEis called a cone if and only if:
• Pis closed, nonempty andP={θ},
• a,b∈R,a,b≥andx,y∈Pimply thatax+by∈P, • x∈Pand–x∈Pimply thatx=θ.
Given a coneP⊂E, a partial ordering≤with respect toPis defined byx≤yif and only if
y–x∈P. We writex<yto indicate thatx≤ybutx=y, whilexystands fory–x∈intP, whereintPdenotes the interior ofP. The conePis called normal if there exists a number
K> such thatθ≤x≤yimpliesx ≤Kyfor everyx,y∈E. The least positive number satisfying this is called the normal constant ofP.
Assume thatu,v∈Eandu≤v. The set{x∈E:u≤x≤v}is denoted by [u,v].
Now, we recall the following definitions from [, ].
Definition . LetPbe a cone of a real Banach spaceE. Suppose thatD⊂P andα∈
(–∞, +∞). An operatorA:D→Dis said to beα-convex (α-concave) if it satisfiesA(tx)≤
tαAx(A(tx)≥tαAx) for (t,x)∈(, )×D.
Definition . LetEbe an ordered Banach space andD⊂E. An operator is called mixed monotone onD×Dif A:D×D→E andA(x,y)≤A(x,y) for anyx,x,y,y∈D,
wherex≤xandy≥y. Also,x*∈Dis called a fixed point ofAifA(x*,x*) =x*.
LetC(E) be a collection of all closed subsets ofE.
Definition . For two subsetsX,YofE, we write
• XYif for allx∈X, there existsy∈Ysuch thatx≤y, • x≺Xif there existsz∈Xsuch thatxz,
• X≺xif for allz∈X,zx.
Definition . LetDbe a nonempty subset ofE.T:D→C(E) is called increasing (de-creasing) upward ifu,v∈D,u≤vandx∈T(u) imply there existsy∈T(v) such thatx≤y
(x≥y). Similarly,T :D→C(E) is called increasing (decreasing) downward ifu,v∈D,
u≤vandy∈T(v) imply there existsx∈T(u) such thatx≤y(x≥y).Tis called increas-ing (decreasincreas-ing) ifTis an increasing (decreasing) upward and downward.
Definition . LetDbe a nonempty subset ofE. A multi-valued operatorT:D×D→
C(E) is said to be mixed monotone upward ifT(x,y) is increasing upward inxand decreas-ing upward iny,i.e.,
(A) for eachy∈Dand any x,x ∈Dwithx≤x, if u∈T(x,y), then there exists a
u∈T(x,y)such thatu≤u;
(A) for eachx∈Dand any y,y∈Dwithy ≤y, if v∈T(x,y), then there exists a
v∈T(x,y)such thatv≥v.
Definition . x*∈Dis called a fixed point ofTifx*∈T(x*,x*).
Definition . [] A function : [, )×P×P×E→E is called anL-function if
(t,x,y, ) = ,(t,x,y,s) fors, and (t,x,y,z) <zfor all (t,x,y,z)∈[, )×
P×P×E.
Definition .[] Suppose thatPis a cone inE. Leta,b∈Eanda<b. Define
[a,b] :=x∈E:x=tb+ ( –t)afor somet∈[, ] ()
and
[a,b) :=x∈E:x=tb+ ( –t)afor somet∈[, ). ()
Definition .[] The set{a=x,x, . . . ,xn=b}is called a partition for [a,b] if and only
if the intervals{[xi–,xi)}ni=are pairwise disjoint and [a,b] ={
n
i=[xi–,xi)} ∪ {b}. Denote P[a,b] as the collection of all partitions of [a,b].
Definition . [] For each partition Q of [a,b] and each increasing function φ : [a,b]→E, we define cone lower summation and cone upper summation as
LConn (φ,Q) =
n–
i=
φ(xi)xi–xi+ ()
and
UnCon(φ,Q) =
n–
i=
φ(xi+)xi–xi+, ()
respectively. Also, we denote(Q)=sup{xi–xi–,xi∈Q}.
Definition .[] Suppose thatPis a cone inE.φ: [a,b]→Eis called an integrable function on [a,b] with respect to a conePor, to put it simply, a cone integrable function if and only if for all partitionQof [a,b],
lim (Q)→L
Con
n (φ,Q) =SCon=lim
(Q)→U Con
n (φ,Q),
whereSConmust be unique.
We show the common valueSConby
b
a
φ(x)dP(x) or to simplicity
b
a φdp.
We denote the set of all cone integrable functionsφ: [a,b]→EbyL([a,b],E).
Lemma .[] Let M be a subset of P.The following conditions hold:
() If[a,b]⊆[a,c]⊂M,then abf dp≤ acf dpforf ∈L(M,P).
() ab(αf +βg)dp=α abf dp+β abg dpforf,g∈L(M,P)andα,β∈R.
2 Main results
In this section, we introduce some new fixed point theorems in the class of mixed mono-tone operators. Due to this, the following definition is presented.
Definition . A mixed monotone operatorA:D×D→Eis said to be a Weak Mixed Monotone single-valued operator of Rhoades type (WMR property for short) if
A(tx,y)≤A(x,ty) –t,x,y,A(x,ty) ()
for all (x,y)∈D×D, where: [, )×P×P×E→Eis anL-function.
Theorem . Let P be a cone of E,let S be a completely ordered closed subset of E with S=S\{θ} ⊂intP and letλS⊂S for allλ∈[, ].Let u,v∈S,A:P×P→E be a weak
mixed monotone operator of Rhoades type with A(([θ,v]∩S)×([θ,v]∩S))⊂S satisfying
the following conditions:
(I) there existsr> such thatu≥rv,
(II) A(u,v)uvA(v,u),
(III) foru,v∈[u,v]∩SwithA(u,v)uv,there existsu∈Ssuch that
u≤A(u,v)uv;similarly,foru,v∈[u,v]∩SwithuvA(v,u),there
existsv∈Ssuch thatuvA(v,u)≤v.
Then A has at least one fixed point x*∈[u,v]∩S.
Proof By the above condition (III), there existsu∈Ssuch thatu≤A(u,v)u
v. Then there exists v ∈S such thatu v A(v,u)≤v. Likewise, there exists
u∈Ssuch thatu≤A(u,v)uv. Then there existsv∈Ssuch thatuv
A(v,u)≤v. In general, there existsun∈S such that un–≤A(un,vn–)unvn–.
Then there existsvn∈Ssuch thatunvnA(vn,un)≤vn–(n= , , . . .).
Take rn=sup{r∈ (, ) :un≥ rvn}, thus < r <r <· · · <rn< rn+ <· · · < and
limn→∞rn=sup{rn:n= , , , . . .}=r*∈(, ]. Sincern+>rn=sup{r∈(, ) :un≥rvn},
thusun≥rn+vn. In addition,Sis completely ordered andλS⊂Sfor allλ∈[, ], then
un<rn+vn. Now, one can prover*= . Otherwise,r*∈(, ).
Sinceun<rn+vnandrn+<r*, henceun<r*vn, and we have
A(un+,vn+)≤A
r*un+,r *v
n+
≤A(un+,vn+) –
r*,
r*un+,vn+,A(un+,vn+)
<A(un+,vn+), ()
which is a contradiction. Thus,r*= . Let be given. Chooseδ> such that +
Nδ()⊆P, whereNδ() ={y∈E:y<δ}. Sincern→, one can choose a natural number
Nsuch that ( –rn)v∈Nδ() for alln≥N. Therefore ( –rn)v . Also,vn≤vand
<vn–un≤( –rn)vn≤( –rn)v . ()
For alln,p≥, applying the same argument, we have
<vn–vn+p≤vn–un . ()
Also,
<un+p–un≤vn–un . ()
Hence,{un}and{vn}are Cauchy sequences inE, then there existu*,v*∈Esuch thatun→
u*,v
n→v*(n→ ∞) andu*=v*. Writex*=u*=v*.
It is easy to seeu≤un≤u*≤vn≤vfor alln= , , . . . . In addition,Sis closed, then
u*∈[u
n,vn]∩S⊂[u,v]∩S(n= , , , . . .).
Finally, by the mixed monotone property ofA,
un–≤A(un,vn)≤A
x*,x*≤A(un,vn)≤un–. ()
On taking limit on both sides of (), whenn→ ∞, we haveA(x*,x*) =x*. This meansx*
is a fixed point ofAin [u,v]∩S.
Corollary . Let P be a cone of E,let S be a completely ordered closed subset of E with S=S\{θ} ⊂intP and letλS⊂S for allλ∈[, ].Let u,v∈S,A:P×P→E satisfy
tx
y
φdP≤
x
ty
φdP–
t,x,y,
x
ty φdP
()
for all(x,y)∈D×D,where: [, )×P×P×E→E is anL-function,and letφ:P→P be a non-vanishing,cone integrable mapping on each[a,b]⊂P such that for each ,
φdpand the mappingθ(x) =
x
φdPfor(x≥)has a continuous inverse at zero.
Also,A(([θ,v]∩S)×([θ,v]∩S))⊂S satisfies the following conditions:
(I) there existsr> such thatu≥rv,
(II) A(u,v)uvA(v,u),
(III) foru,v∈[u,v]∩SwithA(u,v)uv,there existsu∈Ssuch that
u≤A(u,v)uv;similarly,foru,v∈[u,v]∩SwithuvA(v,u),there
existsv∈Ssuch thatuvA(v,u)≤v.
Then A has at least one fixed point x*∈[u
,v]∩S.
Proof Define
A(x,y) =
x
y φdP.
Ais a mixed monotone operator, and one can easily see that all conditions of Theorem . hold. Thus we obtain the desired result.
3 M3R property
Definition . A mixed monotone operatorT:D×D→C(E) is said to be a Mixed Mono-tone Multi-valued operator of Rhoades type (MR property for short) if
T(tx,y)T(x,ty) –t,x,y,T(tx,y) () for each (x,y)∈D×D, where: [, )×P×P×E→Eis anL-function.
Theorem . Let P be a cone of E,let S be a completely ordered closed subset of E with S=
S\{θ} ⊂intP and letλS⊂S for allλ∈[, ].Let u,v∈S,T:P×P→C(E)be a mixed
monotone multi-valued operator of Rhoades type with T(([θ,v]∩S)×([θ,v]∩S))⊂S
satisfying the following conditions:
(I) there existsr> such thatu≥rv,
(II) T(u,v)≺uv≺T(v,u),
(III) foru,v∈[u,v]∩SwithT(u,v)≺uv,there existsu∈Ssuch that
uT(u,v)≺uv;similarly,foru,v∈[u,v]∩Swithuv≺T(v,u),there
existsv∈Ssuch thatuv≺T(v,u)v.
Then T has at least one fixed point x*∈[u
,v]∩S.
Proof By the above condition (III), there existsu∈S such thatuT(u,v)≺u
v. Then there exists v ∈S such that u v ≺T(v,u)v. Likewise, there exists
u∈Ssuch thatuT(u,v)≺uv. Then there existsv∈Ssuch thatuv≺
T(v,u)≤v. In general, there exists un∈S such thatun–T(un,vn–)≺unvn–.
Then there existsvn∈Ssuch thatunvn≺T(vn,un)vn–(n= , , . . .).
Take rn =sup{r ∈(, ) :un≥ rvn}, thus <r<r <· · · <rn <rn+ < · · ·< , and
limn→∞rn=sup{rn:n= , , , . . .}=r*∈(, ]. Sincern+>rn=sup{r∈(, ) :un≥rvn},
thusun≥rn+vn. In addition,Sis completely ordered andλS⊂Sfor allλ∈[, ], then
un<rn+vn. Now, one can prover*= . Otherwise,r*∈(, ). We claim
T(un+,vn+)T
/r*un+,r*vn+
. ()
Suppose that x∈T(un+,vn+) is arbitrary. We haveun+≤(/r*)un+. If x=un+, x=
(/r*)u
n+ andy=vn+, then by (A) of Definition ., there existsz∈T((/r*)un+,vn+)
such thatx≤z. Thus,T(un+,vn+)T((/r*)un+,vn+).
Also, ify=r*vn+,y=vn+ andx= (/r*)un+, then forw∈T((/r*)un+,r*vn+), there
existsh∈T((/r*)u
n+,vn+) such thatw≥h. It means that
T/r*un+,vn+
T/r*un+,r*vn+
. ()
Thus,
T(un+,vn+)T
/r*un+,r*vn+
T(un+,vn+) –
r*,un+,r *v
n+,T(un+,vn+)
≺T(un+,vn+), ()
and this is a contradiction. Therefore,r*= . Let be given. Chooseδ> such that
Figure 1 A(u0,v0)u0u1 ··· un ··· 1 ··· vn ··· v1v0A(v0,u0).
numberNsuch that ( –rn)v∈Nδ() for alln≥N. Therefore ( –rn)v . Also,vn≤v
and
<vn–un≤( –rn)vn≤( –rn)v . ()
By Remark .,limn→∞un=limn→∞vn.
For alln,p≥, applying the same argument, we have
<vn–vn+p≤vn–un . ()
Also,
<un+p–un≤vn–un . ()
Hence,{un}and{vn}are Cauchy sequences inE, then there existu*,v*∈Esuch thatun→
u*,vn→v*(n→ ∞) andu*=v*. Writex*=u*=v*.
It is easy to see thatunT(un+,vn+)T(x*,x*)T(vn+,un+)vnfor alln= , , . . . .
Thus, there existszn∈T(x*,x*) such thatun≤zn≤vn. By taking limit on both sides of
(),
<zn–un≤( –rn)vn≤( –rn)v . ()
So,zn→x*. SinceT has closed values, thenx*∈T(x*,x*) and
x*∈[un,vn]∩S⊂[u,v]∩S.
Remark . One can see easily that Theorem . should be included as a corollary of Theorem ..
Example . LetE=R,P= [, +∞) andS=P. ThenS=int(P) = (, +∞).
DefineA: [, +∞)×[, +∞)→Ras
A(x,y) =
x
y, (x,y)= (, ),
, (x,y) = (, ).
Ais a mixed monotone operator. Now suppose that : [, )×P×P×E→E is as
(t,x,y,s) = ( –t)s. Thenis anL-function. Moreover, A(tx,y)≤A(x,ty) –t,x,y,A(x,ty)
for eachx,y∈S. Also, by takingu=,v= andr=, we have
(II) A(u,v) =uvA(v,u) = ,
(III) foru,v∈[u,v]∩SwithA(u,v)uv, there existsu∈Ssuch that
u≤A(u,v)uv; similarly, foru,v∈[u,v]∩SwithuvA(v,u), there
existsv∈Ssuch thatuvA(v,u)≤v.
For further explanation on (III), sinceA(u,v) =uv, by (III) there existsu∈S
such that uA(u,v)u v. It means that u
u . Thus u must
be greater than . Therefore we can set u =
+
. Similarly, since
=u v=
A(v,u) =, thus by (III) there existsv∈Ssuch thatuvA(v,u)≤v. It means
thatvmust be less than . We can setv=
+
. By the continuity of such ways, we can
consider the following reflexive sequences:
u=
, v=
, un=
un–vn–+
and vn=
vn–un+
,
which satisfy (I), (II) and (III) (see Figure ). Moreover,un→ andvn→ andA(, ) = . 4 Application
The following result is given by Zhang [] and is obtained by our main result.
Corollary . Let P be a normal cone of E,let S be a completely ordered closed subset of E with S=S\{θ} ⊂intP and letλS⊂S for allλ∈[, ].Let u,v∈S,A:P×P→E be
a mixed monotone operator with A(([θ,v]∩S)×([θ,v]∩S))⊂S and A(u,v)u
vA(v,u).Assume that there exists a functionφ: (, )×([u,v]∩S)×([u,v]∩
S)→(, +∞)such that A(tx,y)≤φ(t,x,y)A(x,ty),where <φ(t,x,x) <t for all(t,x,y)∈ (, )×([u,v]∩S)×([u,v]∩S).Suppose that
(I) foru,v∈[u,v]∩SwithA(u,v)uv,there existsu∈Ssuch that
u≤A(u,v)uv;similarly,foru,v∈[u,v]∩SwithuvA(v,u),there
existsv∈Ssuch thatuvA(v,u)≤v.
(II) there exists an elementw∈[u,v]∩Ssuch thatφ(t,x,x)≤φ(t,w,w)for all
(t,x)∈(, )×([u,v]∩S),andlims→t–φ(s,w,w) <tfor allt∈(, ).
Then A has at least one fixed point x*∈[u,v]∩S.
Proof Set(t,x,y,z) = ( –φ(t,x,y))z. Thenis anL-function, and we have
A(tx,y)≤φ(t,x,y)A(x,ty) =A(x,ty) –t,x,y,A(x,ty).
Thus, by Theorem . the desired result is obtained.
Competing interests
The author declares that they have no competing interests.
Received: 11 May 2012 Accepted: 1 March 2013 Published: 28 March 2013
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