R E S E A R C H
Open Access
A class of nonlinear mixed ordered inclusion
problems for ordered
(α
A
,
λ)
-ANODM
set-valued mappings with strong comparison
mapping
A
Hong Gang Li
1*, Li Pei Li
2and Mao Ming Jin
2*Correspondence: [email protected] 1Faculty of Mathematics and
Physics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China Full list of author information is available at the end of the article
Abstract
The purpose of this paper is to introduce and study a new class of nonlinear mixed ordered inclusion problems in ordered Banach spaces and to obtain an existence theorem and a comparability theorem of the resolvent operator. Further, by using fixed point theory and the resolvent operator, the authors constructed and studied an approximation algorithm for this kind of problems, and they show the relation between the first valuedx0and the solution of the problems. The results obtained seem to be general in nature.
MSC: 49J40; 47H06
Keywords: a new class of nonlinear mixed ordered inclusion problems; ordered (
α
A,ωλ)-ANODMset-valued mapping; strong comparison mapping; ordered Banach spaces; convergence1 Introduction
Well-known, generalized nonlinear inclusion (variational inequality and equation) have wide applications in many fields, including, for example, mathematics, physics, optimiza-tion and control, nonlinear programming, economic, and engineering sciences [–]. In , the number of solutions of nonlinear equations had been introduced and studied by Amann [], and in recent years, nonlinear mapping fixed point theory and applications have been intensively studied in ordered Banach space [–]. Therefore, it is very im-portant and natural that generalized nonlinear ordered variational inequalities (ordered equations) are studied and discussed.
From , the authors introduced and studied the approximation algorithm and the approximation solution theory for the generalized nonlinear ordered variational inclusion problems (inequalities, systems, and equations) in ordered Banach spaces; for example, in , Li has introduced and studied the approximation algorithm and the approxima-tion soluapproxima-tion for a class of generalized nonlinear ordered variaapproxima-tional inequality and or-dered equation in oror-dered Banach spaces []. In , by using theB-restricted-accretive method of mappingAwith constantsα,α, Li has introduced and studied an existence
theorem and an approximation algorithm of solutions for a new class of general nonlin-ear ordered variational inequalities and equations in ordered Banach spaces []. In ,
Li has introduced and studied a class of nonlinear inclusion problems for orderedRME set-valued mappings in order Hilbert spaces []; in , Li has introduced and studied a class of nonlinear inclusion problems for ordered (α,λ)-NODMset-valued mappings, and then, applying the resolvent operator associated with the set-valued mappings, es-tablished an existence theorem on the solvability and a general algorithm applied to the approximation solvability of the nonlinear inclusion problem of this class of nonlinear in-clusion problems in ordered Hilbert space [], and have proved a sensitivity analysis of the solution for a new class of general nonlinear ordered parametric variational inequalities in []. Recently, Liet al.have studied the characterizations of ordered (αA,λ
)-weak-ANODDset-valued mappings, which was applied to solving approximate solution for a new class of general nonlinear mixed order quasi-variational inclusions involving the⊕ operator, and a new class of generalized nonlinear mixed order variational inequalities systems with order Lipschitz continuous mappings in ordered Banach spaces [, ].
In this field, the obtained results seem to be general in nature. As regards new devel-opments, it is exceedingly of interest to study the problems: for w∈Xandω> , find x∈Xsuch thatw∈f(x) +ωM(x). A new class of nonlinear mixed ordered inclusion prob-lems for ordered (αA,λ)-ANODMset-valued mappings with strong comparison mapping
Aand characterizations of ordered (αA,ωλ)-ANODMset-valued mappings are introduced
in ordered Banach spaces. An existence theorem and a comparability theorem of the re-solvent operator associated to a (αA,λω)-ANODMset-valued mapping are established. By
using fixed point theory and the resolvent operator associated for the (αA,ωλ)-ANODM
set-valued mapping, an existence theorem of solutions and an approximation algorithm for this kind of problems are studied, and the relation of between the first valuedxand the
solution of the problems is discussed. The results obtained seem to be general in nature. For details, we refer the reader to [–] and the references therein.
LetXbe a real ordered Banach space with norm·, a zeroθ, a normal coneP, a normal constantNofPand a partial ordered relation≤defined by the coneP[, ]. Letf :X→ Xbe a single-valued ordered compression mapping, andM:X→Xand
f(x) +M(x) =y|y=f(x) +u,∀x∈X,u∈M(x):X→X
be two set-valued mappings. We consider the following problem. Forw∈X, and anyω> , findx∈Xsuch that
w∈f(x) +ωM(x). (.)
The problem (.) is called a nonlinear mixed ordered inclusion problems for the ordered ANODMset-valued mappingMin an ordered Banach space.
Remark . We have the following special cases of the problem (.):
(i) IfM(x) =F(g(x))be a single-valued mapping,ω= ,f = andw=θ, then the problem (.) in [] can be obtained by the problem (.).
(ii) Ifω= ,f = andw=θ, then the problem (.) changes to the problem (.) in [] and [].
2 Preliminaries
LetXbe a real ordered Banach space with norm · , a zeroθ, a normal coneP, normal constantNand a partial ordered relation≤defined by the coneP. For arbitraryx,y∈X,
lub{x,y}andglb{x,y}express the least upper bound of the set{x,y}and the greatest lower bound of the set{x,y}on the partial ordered relation≤, respectively. Supposelub{x,y}and
glb{x,y}exist. Let us recall some concepts and results.
Definition .[, ] LetXbe a real Banach space with norm · ,θ be a zero element in theX.
(i) A nonempty closed convex subsetsPofXis said to be a cone, if () for anyx∈Pand anyλ> , we haveλx∈P,
() x∈Pand–x∈P, thenx=θ;
(ii) Pis said to be a normal cone if and only if there exists a constantN> , and a normal constant ofPsuch that forθ≤x≤y, we havex ≤Ny; (iii) for arbitraryx,y∈X,x≤yif and only ifx–y∈P;
(iv) forx,y∈X,xandyare said to be a comparison between each other, if and only if we havex≤y(ory≤x) (denoted byx∝yforx≤yandy≤x).
Lemma .[] If x∝y,thenlub{x,y},andglb{x,y}exist,x–y∝y–x,andθ≤(x–y)∨ (y–x).
Lemma . If for any natural number n,x∝yn,and yn→y∗(n→ ∞),then x∝y∗.
Proof If for any natural numbern,x∝ynandyn→y∗(n→ ∞), thenx–yn∈Poryn–x∈
P for any natural numbern. SincePis a nonempty closed convex subsets ofXso that x–y∗=limn→∞(x–yn)∈Pory∗–x=limn→∞(yn–x)∈P. Therefore,x∝y∗.
Lemma .[, , , ] Let X be an ordered Banach space,Pbe a cone of X,and≤ be a relation defined by the conePin Definition.(iii).For x,y,v,u∈X,then we have the following relations:
() the relation≤inXis a partial ordered relation inX; () x⊕y=y⊕x;
() x⊕x=θ; () θ≤x⊕θ;
() letλbe a real,then(λx)⊕(λy) =|λ|(x⊕y);
() ifx,y,andwcan be comparative each other,then(x⊕y)≤x⊕w+w⊕y; () let(x+y)∨(u+v)exist,and ifx∝u,vandy∝u,v,then
(x+y)⊕(u+v)≤(x⊕u+y⊕v)∧(x⊕v+y⊕u); () ifx,y,z,wcan be compared with each other,then
(x∧y)⊕(z∧w)≤((x⊕z)∨(y⊕w))∧((x⊕w)∨(y⊕z)); () ifx≤yandu≤v,thenx+u≤y+v;
() ifx∝θ,then–x⊕θ≤x≤x⊕θ;
() ifx∝y,then(x⊕θ)⊕(y⊕θ)≤(x⊕y)⊕θ=x⊕y; () (x⊕θ) – (y⊕θ)≤(x–y)⊕θ;
() ifθ≤xandx=θ,andα> ,thenθ≤αxandαx=θ.
Proof ()-() come from Lemma . in [] and Lemma . in [], and ()-() directly
Definition . LetXbe a real ordered Banach space,A:X→Xbe a single-valued map-ping, andM:X→Xbe a set-valued mapping. Then:
() a single-valued mappingAis said to be aγ-ordered non-extended mapping, if there exists a constantγ> such that
γ(x⊕y)≤A(x)⊕A(y), ∀x,y∈X;
() a single-valued mappingAis said to be a strong comparison mapping, ifAis a comparison mapping, andA(x)∝A(y), thenx∝yfor anyx,y∈X;
() a comparison mappingMis said to be anαA-non-ordinary difference mapping with
respect toA, if there exists a constantαA> such that for eachx,y∈X,vx∈M(x),
andvy∈M(y),
(vx⊕vy)⊕αA
A(x)⊕A(y)=θ;
() a comparison mappingMis said to be aλ-ordered monotone mapping with respect toB, if there exists a constantλ> such that
λ(vx–vy)≥x–y, ∀x,y∈X,vx∈M
B(x),vy∈M
B(y);
() a comparison mappingMis said to be a(αA,λ)-ANODMmapping, ifMis
aαA-non-ordinary difference mapping with respect toAand aλ-ordered monotone
mapping with respect toB, and(A+λM)(X) =XforαA,λ> .
Lemma . Let X be a real ordered Banach space.If A is aγ-ordered non-extended map-ping,and M is aλ-ordered monotone mapping and anαA-non-ordinary difference mapping
with respect to A,then for anyω> ,ωM is aωλ-ordered monotone and anαA-non-ordinary
difference mapping with respect to A.
Proof Let a comparison mappingMbe aλ-ordered monotone mapping with respect to A, then it is obvious thatωMis a λ
ω-ordered monotone mapping with respect toA. IfM
is anαA-non-ordinary difference mapping with respect toA, then there exists a constant
αA> such that for eachx,y∈X, andvx∈ωM(x) andvy∈ωM(y) (vx=ωux,vy=ωuy,
ux∈M(x),uy∈M(y)) we have
(ux⊕uy)⊕αA
A(x)⊕A(y)=θ
and
ω(ux⊕uy)⊕αA
A(x)⊕A(y)=ωθ=θ.
By Lemma . andω> , we have
(ωux⊕ωuy)⊕αA
A(x)⊕A(y)=θ.
Therefore,
(vx⊕vy)⊕αA
It follows that ωMis aαA-non-ordinary difference mapping with respect toAfor any
ω> .
Lemma . Let X be a real ordered Banach space.If A is aγ-ordered non-extended map-ping and a comparison mapmap-ping M is a(αA,λ)-ANODM mapping,thenωM is a(αA,ωλ)
-ANODM mapping.
Proof LetXbe a real ordered Banach space, letAbe aγ-ordered non-extended mapping and a comparison mappingMbe a (αA,λ)-ANODMmapping, then (A+λM)(X) =Xfor
αA,λ> . It is follows that (A+ωλ(ωM))(X) =XforαA,ωλ> . Therefore,ωMis a (αA,λω
)-ANODMmapping by Lemma ..
Lemma .[] Let X be a real ordered Banach space.If A is aγ-ordered non-extended mapping and M is aαA-non-ordinary difference mapping with respect to A,then an inverse
mapping JMA,λ= (A+λM)–:X→Xof(A+λM)is a single-valued mapping(αA,λ> ).
Lemma . Let X be a real ordered Banach space.If A is aγ-ordered non-extended map-ping and M is aαA-non-ordinary difference mapping with respect to A,then an inverse
mapping JA
ωM,λω = (A+ λ ω(ωM))
–:X→X of(A+ λ
ω(ωM))is a single-valued mapping
(αA,λ> ).
Proof This directly follows from Lemma ., Lemma ., and Lemma ..
Lemma .[] Let X be a real ordered Banach space with norm · ,a zeroθ,a normal coneP,a normal constant N ofP and a partial ordered relation≤defined by the cone P,and the operator⊕be a XOR operator.If A is a strong comparison mapping,and M: X→Xis aλ-ordered monotone mapping with respect to JA
M,λ,then the resolvent operator
JA
M,λ:X→X is a comparison mapping.
Lemma .[] Let X be a real ordered Banach space with norm · ,a zeroθ,a normal coneP,a normal constant N ofP and a partial ordered relation≤defined by the cone P,and the operator⊕be a XOR operator.If A is a strong comparison mapping,and M: X→Xis aλ-ordered monotone mapping with respect to JA
M,λ,then the resolvent operator
JA
ωM,λ ω
:X→X is a comparison mapping.
Proof This directly follows from Lemma ., Lemma ., and Lemma ..
Lemma .[] Let X be a real ordered Banach space with norm · ,a zeroθ,a normal coneP,a normal constant N ofPand a partial ordered relation≤defined by the coneP.If A is aγ-ordered non-extended mapping,and M:X→Xis a(α
A,λ)-ANODM mapping,
which is aαA-non-ordinary difference mapping with respect to A andλ-ordered monotone
mapping with respect to JMA,λ,then for the resolvent operator JMA,λ:X→X,the following
relation holds:
JMA,λ(x)⊕JMA,λ(y)≤ γ(αAλ– )
(x⊕y), (.)
Lemma . Let X be a real ordered Banach space with norm · ,a zeroθ,a normal coneP,a normal constant N ofPand a partial ordered relation≤defined by the coneP.If A is aγ-ordered non-extended mapping and M:X→Xis a(αA,λ)-ANODM mapping,
which is aαA-non-ordinary difference mapping with respect to A andλ-ordered monotone
mapping with respect to JMA,λ,then for the resolvent operator JωAM,λ ω
:X→X,the following relation holds:
JωA
M,λω(x)⊕J
A
ωM,ωλ(y)≤
ω γ(αAλ–ω)
(x⊕y), (.)
whereαA>ωλ> .
Proof LetXbe a real ordered Banach space,Pbe a normal cone with the normal constant Nin theX,≤be a ordered relation defined by the coneP. Forx,y∈X, letux=JωAM,λ
ω
(x)∝
uy=JωAM,λ ω
(y) andvx=ωλ(x–A(ux))∈ωM(ux),vy= ω
λ(y–A(uy))∈ωM(uy). SinceωM:X→
Xis a (αA,λω)-ANODMmapping with respect toAso that the following relations hold by
() in Lemma . and the condition (vx⊕vy)⊕αA(A(ux)⊕A(uy)) =θ:
ω λ
(x⊕y) +A(ux)⊕A(uy)
≥ωvx⊕ωvy
=αA
A(ux)⊕A(uy)
.
It follows that (ωλαA– )(A(ux)⊕A(uy))≤(x⊕y) from the conditionsαA> ωλ > and
A(ux)⊕A(uy)≥γ(ux⊕uy), andAis aγ-ordered non-extended mapping. The proof is
completed.
Remark . It is clear that Lemma ., Theorem ., and Theorem . in [] are special cases of Lemma ., Lemma ., and Lemma ., respectively, whenA=I, the identity mapping inX.
3 Main results
In this section, we will show the algorithm of the approximation sequences for finding a solution of the problem (.), and we discuss the convergence and the relation between the first valuedxand the solution of the problem (.) inX, a real Banach space.
Theorem . Let X be a real ordered Banach space with norm · ,a zeroθ,a normal coneP,a normal constant N ofPand a partial ordered relation≤defined by the coneP, and the operator⊕be a XOR operator.Let A,f :X→X be two single-valued ordered com-pression mappings and A∝f,f ∝θ.If A is aγ-ordered non-extended strong comparison mapping and M:X→Xis aα
A-non-ordinary difference mapping with respect to A,then
the inclusion problem(.)has a solution x∗if and only if x∗=JA
ωM,ωλ
(A+λ
ω(w–f))(x
∗)in X.
Proof This directly follows from the definition of the resolvent operatorJA
ωM,λ ω
ofωM(x).
A,f:X→X be two single-valuedβ,ξordered compression mappings,respectively,A be a γ non-extended and strong compression mapping,and M:X→Xbe a(αA,λ)-ANODM
mapping,which is aαA-non-ordinary difference mapping with respect to A andλ-ordered
monotone mapping with respect to JA
M,λ.If A∝f,w∝A,f,M,αA>ωλ> ,and
βω+γ ω+λξ<γ λαA (.)
(whereβ,ξ> ),then the sequence{xn}converges strongly to x∗,the solution of the problem
(.),which is generated by following algorithm. For any given x∈X,let x=JωAM,λ
ω
(A+λω(w–f))(x),and for n> and <ϕ< ,set
xn+= ( –ϕ)xn+ϕJωAM,λ ω
A+λ
ω(w–f)
(xn).
For any x∈X,we have
x∗–x≤
–N
– γ(αAλ–ω) ϕ(αAγ λ– (βω+γ ω+λξ))
×JωA
M,λω
A+λ
ω(w–f)
(x) –x
. (.)
Proof LetX be a real ordered Banach space, let Pbe a normal cone with the normal constantNin theX, let≤be a partial ordered relation defined by the coneP. For any x∈X, we setx= ( –ϕ)x+ϕJωAM,λ
ω
(A+λω(w–f))(x). By using Lemma ., Lemma .,
Lemma .,ωλ-monotonicity ofωM, (A+ωλωM)(X) =X, and the comparability ofJA
ωM,λω,
we know that x ∝x. Further, we can obtain a sequence {xn}, and xn+ ∝xn (where
n= , , , . . .). Using Lemma ., Lemma ., Lemma ., and Lemma ., we have
θ ≤xn+⊕xn
≤
( –ϕ)xn+ϕJωAM,λ ω
A+λ
ω(w–f)
(xn)
⊕
( –ϕ)xn–+ϕJωAM,λ ω
A+λ
ω(w–f)
(xn–)
≤ϕ ω
γ(αAλ–ω)
A+λ
ω(w–f)
(xn)⊕
A+λ
ω(w–f)
(xn–)
+ ( –ϕ)(xn–⊕xn)
≤ϕ ω
γ(αAλ–ω)
A(xn)⊕A(xn–) +
λ
ω(w–f)(xn)⊕ λ
ω(w–f)(xn–)
+ ( –ϕ)(xn–⊕xn)
≤ϕ
γ(αAλ– )
A(xn)⊕A(xn–) +λf(xn)⊕f(xn–) +λ(w⊕w)
+ ( –ϕ)(xn–⊕xn)
≤ϕ ω
γ(αAλ–ω)
β(xn⊕xn–) +
λ
ωξ(xn⊕xn–)
+ ( –ϕ)(xn–⊕xn)
≤
–ϕ+ϕ βω+λξ γ(αAλ–ω)
(xn⊕xn–)
≤
–ϕ+ϕ βω+λξ γ(αAλ–ω)
n
by Lemma . and Definition .(ii), we obtain
xn+–xn ≤( –ϕ+ϕδ)nNx–x, (.)
whereδ=γ(βωα+λξ
Aλ–ω). Hence, for anym>n> , we have
xm–xn ≤ m–
i=n
xi+–xi ≤Nx–x
m–
i=n
( –ϕ+ϕδ)i.
It follows from the condition (.) that <δ< , andxm–xn →, asn→ ∞, and so
{xn}is a Cauchy sequence in the complete spaceX. Letxn→x∗asn→ ∞(x∗∈X). By the
conditions, we have
x∗= lim
n→∞xn+
= lim
n→∞J A
ωM,λω
A+λ
ω(w–f)(xn)
=JA
ωM,λω
A+λ
ω(w–f)
x∗.
We know thatx∗ is a solution of the inclusion problem (.). It follows that (JA
ϕM,λ ω
(A+
λ
ω(w–f))(xn))∝x
∗(n= , , , . . .) from Lemma . and (.). Also we have
x∗–x= lim
n→∞xn–x
≤ lim
n→∞ n
i=
xi+–xi ≤ lim n→∞N
n
i=
( –ϕ+ϕδ)n–x–x+x–x
≤
+ (N– )( –ϕ+ϕδ) – ( –ϕ+ϕδ)
JωA
M,ωλ
A+λ
ω(w–f)
(x) –x
≤
–N
– γ(αAλ–ω) ϕ(αAγ λ– (βω+γ ω+λξ))
×JωA
M,λω
A+λ
ω(w–f)
(x) –x
.
This completes the proof.
Remark . Though the method of solving problem by the resolvent operator is the same as in [, , ], and [] for a nonlinear inclusion problem, the character of the ordered (αA,λ)-ANODMset-valued mapping is different from the one of the (A,η)-accretive
map-ping [], (H,η)-monotone mapping [], (G,η)-monotone mapping [], and monotone mapping [].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Author details
1Faculty of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China. 2Changjiang Normal University, Fuling, Chongqing, 400803, China.
Received: 7 November 2013 Accepted: 5 March 2014 Published:26 Mar 2014
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Cite this article as:Li et al.:A class of nonlinear mixed ordered inclusion problems for ordered(αA,λ)-ANODM
