R E S E A R C H
Open Access
A class of
ϕ
-concave operators and
applications
Yanbin Sang
**Correspondence:
syb6662004@163.com
Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, P.R. China
Abstract
In this paper, by means of the concept of
ϕ
-concave operators, which was introduced by Li and Liang (J. Systems Sci. Math. Sci. 14(4):355-360, 1994 (in Chinese)), we obtain some new existence and uniqueness theorems of a fixed point of mixed monotone operators with such concavity. Moreover, we apply the main theorem to a class of Hammerstein integral equations.Keywords: cone and partial order;
ϕ
-concave operator; mixed monotone operator; fixed point1 Introduction
It is well known that concave operators are a class of important operators that are exten-sively used in nonlinear differential and integral equations (see [–]). In [], Krasnosel-skii introduced in detail many important ideas and results about concave operators. A problem is that there are various concepts of concave operators, such asu-concave oper-ators [], ordered concave operoper-ators [] andα-concave operators [], which is somewhat confusing. We should mention that Lianget al.[] proved that both ordered concave operators andα-concave operators areu-concave operators and gave necessary and suf-ficient conditions on whichu-concave operators have a unique fixed point. Furthermore, Zhaiet al.[] considered an operator equation with generalα-concave or homogeneous operator, their results improved previous results (see Corollary . in []).
We note that Li and Liang [] introduced the definition ofϕ-concave operator, which provided a general method to copy with such a class of operators together. Furthermore, [, ] extended the concept ofϕ-concave operator toφ-concave(–ψ) convex operators. In [], Zhao weakened some conditions and strengthened the conclusions. Mixed mono-tone operators were introduced by Guo and Lakshmikantham [] in . Thereafter, many authors have focused on various existence (and uniqueness) theorems of fixed points for mixed monotone operators; for details, see [, , , –] and references therein. In [], Bhaskar and Lakshmikantham established some coupled fixed point theorems for mixed monotone operators in partially ordered metric spaces and discussed the existence and uniqueness of a solution for a periodic boundary value problem. Instead of using a direct proof as in [], Driciet al.[] employed the notion of reflection operator and in-vestigated fixed point theorems for mixed monotone operators by weakening the require-ments in the contractive assumption and strengthening the metric space utilized with a partial order. Sintunavarat and Kumam [] extended classical coupled fixed point theo-rems of Bhaskar and Lakshmikantham [] to the coupled common fixed point theotheo-rems
for mappings satisfying a new non-commuting condition. These theorems are generaliza-tions of the results of []. Furthermore, Harjaniet al.[] improved the main results of [] using the altering distance functions. Sintunavaratet al.[] established some cou-pled fixed point theorems for a contraction mapping induced by the cone ball metric in partially ordered spaces and also discussed the condition claim of the uniqueness of a coupled fixed point. Very recently, Chandoket al.[] were concerned with some coupled coincidence point theorems for a pair of mappings having a mixedg-monotone property in partially orderedG-metric spaces. Also, they presented a result on the existence and uniqueness of coupled common fixed points. In [], the authors introduced the concept ofW-compatible mappings. Based on this notion, a tripled coincidence point and a com-mon tripled fixed point for mappingsF:X×X×X→Xandg:X→Xwere obtained, where (X,d) is a cone metric space. We should point out that their results do not rely on the assumption of normality condition of the cone.
On the other hand, there is much attention paid to mixed monotone operators with cer-tain concavity and convexity; for example, see [, , , , , , –]. In [], Zhang and Wang modified the methods in [, ] to obtain some new existence and uniqueness results of a positive fixed point of mixed monotone operators. Recently, in [], Zhai and Zhang presented a new fixed point theorem for a class of general mixed monotone opera-tors, which extends the existing corresponding results. Moreover, they investigated some pleasant results of nonlinear eigenvalue problems with mixed monotone properties. Based on them, the local existence-uniqueness of positive solutions for nonlinear boundary value problems which include Neumann boundary value problems, three-point boundary value problems and elliptic boundary value problems for Lane-Emden-Fowler equations was proved.
In this paper, by means of the concept ofϕ-concave operator, some new theorems of fixed points of mixed monotone operators with such concavity are obtained. Our theo-rems unify and extend some previous results, then we apply these results in discussing a class of Hammerstein integral equations and give the new condition for determining the unique solution to this equation.
2 Preliminaries
Some definitions, notations and known results are from Refs. [, , ]. LetE be a real Banach space. A nonempty convex closed setPis called a cone if it satisfies the following conditions:
(i) x∈P,λ≥impliesλx∈P;
(ii) x∈Pand–x∈Pimplyx=θ, whereθdenotes the zero element ofE.
LetEbe partially ordered by a coneP ofE,i.e.,x≤y, if and only ify–x∈Pfor any
x,y∈E. Recall that the conePis said to be solid if the interiorP◦is nonempty, andPis said to be normal if there exists a positive constantMsuch thatθ≤x≤y(x,y∈E) implies
x ≤My, whereMis the normal constant ofP. LetD⊂E. An operatorA:D×D→E
is said to be mixed monotone ifA(x,y) is nondecreasing inxand nonincreasing iny, that is, for anyx,y∈D,
and
y,y∈D, y≤y ⇒ A(x,y)≥A(x,y).
An elementx∗∈Dis called a fixed point ofAif it satisfiesA(x∗,x∗) =x∗.A:D⊂E→Eis said to be convex if forx,y∈Dwithx≤yand eacht∈[, ], we have
Atx+ ( –t)y≤tAx+ ( –t)Ay;
Ais said to be concave if –Ais convex.
Leth>θ, writePh={x∈E| ∃λ,μ> , such thatλh≤x≤μh}.
Lete>θ. An operatorA:P→Pis said to bee-concave if it satisfies the following two conditions:
(i) Aise-positive,i.e.,A(P–{θ})⊂Pe; (ii) ∀x∈Pe,∀ <t< ,∃η=η(t,x) > such that
A(tx)≥( +η)tAx,
whereη=η(t,x)is called the characteristic function ofA.
The operatorT:P◦→P◦is said to beα-concave(–α-convex) (≤α< ) if
T(tx)≥tαT(x) T(tx)≤t–αT(x), ∀x∈P◦,∀t∈(, ].
Letu,v∈Ewithu≤v. Write [u,v] ={x∈E|u≤x≤v},
where [u,v] is called an ordering interval.
Definition .(see []) LetA:Ph→Phbe aϕ-concave operator if there exists a function ϕ: (, ]×Ph→(, ] such thatt∈(, ) impliest<ϕ(t,x), andAsatisfies the following
condition:
A(tx)≥ϕ(t,x)Ax, <t≤,x∈Ph.
Definition .(see []) Let S be a bounded set of a real Banach spaceE. Let
α(S) =inf
δ> S=
m
i=
Siwith diam(Si)≤δ,i= , , . . . ,m
.
Clearly, ≤α(S) <∞.α(S) is called the Kuratowski measure of noncompactness. Definition .(see [, ]) LetD⊂E,A:D→Ebe an operator. ThenSis said to be a generalized condensing operator if for anyS⊂D,α(S)= implies α(A(S)) <α(S). Lemma .(see [, ]) Let S,T be bounded subsets of E.Then
(iii) α(S) =α(S);
(iv) α(S∪T) =max{α(S),α(T)};
(v) α(coS) =α(S),wherecoSdenotes the convex hull ofS.
3 Main results
Theorem . Let E be a real Banach space and P be a normal cone of E.Let u,v∈E
with u≤v and A: [u,v]×P→P be a mixed monotone operator.For fixed v∈P,
A(·,v) : [u,v]→P is aϕ-concave operator.Suppose that
(i) there exists a real positive numberrsuch thatu≥rv;
(ii) uandvare such that
u≤A(u,v), A(v,u)≤v;
(iii) there exists an elementw∈[u,v]such that
ϕ(t,x)≥ϕ(t,w), ∀(t,x)∈(, )×[u,v],
and
lim
s→t–ϕ(s,w) >t, ∀t∈(, ); (iv) for fixedu∈[u,v],∃N> such that
A(u,·) :P→P, A(u,v) –A(u,v)≥–N(v–v), ∀v≥v,v,v∈P.
Then A has exactly one fixed point x∗in[u,v].
Proof We divide the proof into three steps.
Step . We prove that for any fixedu∈[u,v],A(u,·) has exactly one fixed pointT(u)∈ [A(u,v),A(v,u)] such thatA(u,T(u)) =T(u). Our proof is the same as Theorem . in []. For completeness, we list it as follows.
For fixedu∈[u,v], there existsN> such thatA(u,v) +Nvis increasing inv. Let
B(u,v) =A(u,v) +Nv
N+ ,
thenB(u,v) =vis equivalent toA(u,v) =v.
Letx=A(u,v),y=A(v,u) andxn+=B(u,xn),yn+=B(u,yn). Byu≤u≤v,x=
A(u,v)≤v,u≤vandy=A(v,u)≥u, we have
A(u,x)≥A(u,v) =x,
A(u,y)≤A(v,u) =y, thus
x=B(u,x) =
A(u,x) +Nx
N+ ≥
x+Nx
N+ =x,
y=B(u,y) =
A(u,y) +Ny
N+ ≤
y+Ny
that is,x≤x≤y≤y. By induction, we havexn≤xn+≤yn+≤yn. Thus we get that
θ≤yn–xn=
A(u,yn–) +Nyn––A(u,xn–) –Nxn–
N+ ≤
N
N+ (yn––xn–).
SincePis a normal cone and by induction, we get that
yn–xn ≤C
N N+
n
y–x, (.)
whereCis the normal constant ofP. Moreover (for any natural numberp),
xn+p–xn≤yn+p–xn≤yn–xn, (.)
xn+p–xn ≤Cyn–xn ≤C
N N+
n
y–x, (.)
which implies thatxnis a Cauchy sequence. NoticingEis complete,xnconverges to some
element, we denote it byT(u). Equation (.) implies thatynalso converges toT(u). By
xn≤T(u)≤yn, (.)
xn+=B(u,xn)≤B
u,T(u)≤B(u,yn)≤yn+, (.) using the normality of the coneP, we can conclude thatxn,ynalso converge toB(u,T(u)).
ThusB(u,T(u)) =T(u).
IfB(u,x) =x, then we have for any integern,xn≤x≤yn, which implies by taking the
limit thatx=T(u),i.e.,T(u) is the unique fixed point ofA(u,·) in [A(u,v),A(v,u)]. Step . We prove thatT(u) is increasing inu.
Ifu,u∈[u,v],u≤u, then we letx=x=A(u,v) in Step . SinceB(u,v) is increas-ing in both variables, we have
x=B(u,x)≤B
u,x=x.
By induction we knowxn≤xn. Taking the limit, we haveT(u)≤T(u).
Step . We prove thatT(·) has a unique fixed point in [u,v].
Letun+=T(un),vn+=T(vn), we haveu≤u,v≤v. Thus, by conclusion of Step , we getun≤un+,vn+≤vn, andun≤vn. By condition (i), we have
u =A
u,T(u)
≥Arv,T(u)
≥ϕ(r,v)A
v,T(u)
≥ϕ(r,v)A
v,T(v)
=ϕ(r,v)v,
that is
Letrn=ϕ(rn–,vn–),n= , , . . . . By induction, we know
un≥rnvn, n= , , . . . .
Obviously, the sequence{rn} is increasing with{rn} ⊂(, ]. Supposern→r(n→ ∞),
thenr= . Otherwise, we have <r< . Thus, by condition (iii), we have
rn=ϕ(rn–,vn–)≥ϕ(rn–,w). (.)
Letn→ ∞in (.), we have
r≥ lim
s→r–ϕ(s,w) >r,
which is a contradiction. Hence, we haver= . Therefore,∀n,p≥, we get
θ≤vn–un≤vn–rnvn= ( –rn)vn≤( –rn)v,
and
θ≤un+p–un≤vn+p–un≤vn–un, θ≤vn–vn+p≤vn–un+p≤vn–un.
Thus, by the normality ofP, it is easy to see thatvn–un→ (n→ ∞), and hence{un},{vn}
are Cauchy sequences. Therefore, there existu∗,v∗such thatun→u∗,vn→v∗(n→ ∞)
andu∗=v∗. Writex∗=u∗=v∗.
Now we show thatT(x∗) =x∗. It is easy to see that
Tx∗≥T(un) =un+→x∗,
and soT(x∗)≥x∗. On the other hand, we have
Tx∗≤T(vn) =vn+→x∗,
so we obtainT(x∗)≤x∗. HenceT(x∗) =x∗.
The proof of the uniqueness is the same as above in Step . Finally, from Step we get thatx∗is the fixed point ofAin [u,v] such thatT(x∗) =x∗andA(x∗,x∗) =x∗.
By the construction we can see that it is unique. IfxsatisfiesA(x,x) =x. By Step ,T(x) is the unique fixed point ofA(x,·), thusT(x) =x. By the uniqueness of the fixed point of
T(·), we getx=x∗. This ends the proof of Theorem .. Remark . In fact, by Theorem . in [] and Theorem . in [], we can know that conditionlims→t–ϕ(s,w) >tin Theorem . can be deleted, while the conclusions remain
Theorem . Let E be a real Banach space and P be a cone of E.Let u,v∈E with u≤v
and A: [u,v]×P→P be a generalized condensing and mixed monotone operator.For
fixed v∈P,A(·,v) : [u,v]→P is aϕ-concave operator.Suppose that conditions(i)-(iv)of
Theorem.hold.Then A has exactly one fixed point x∗in[u,v].
Proof For fixedu∈[u,v], letS={xn|n= , , , . . .}. We show thatS⊂co{x,A(S)}. It is easy to see thatx∈co{x,A(S)}. Ifxk∈co{x,A(S)}(k≥), then for fixedu∈[u,v],
xk+=
A(u,xk) +Nxk
N+ =
A(u,xk) N+ +
–
N+ xk.
Noticing thatA(u,xk),xk∈co{x,A(S)}, we have thatxk+∈co{x,A(S)}. By induction, we have thatS⊂co{x,A(S)}. It follows from Lemma . that
α(S)≤αcox,A(S)
=αA(S)<α(S).
This is a contradiction. Consequently, we find thatα(S) = . In view of Lemma ., we have thatSis a compact set inE. This implies that there exists a subsequence{xni}of{xn},
which converges tox∗inE.
Now, we prove that{xn}itself also converges tox∗. If not, there exists another
subse-quence{xnj}of{xn}, which converges to another pointx∗=x∗(x∗∈E). Thus, for any fixed
{xni}, and ifnjis large enough, thenxni≤xnj. Letnj→ ∞, we get thatxni ≤x∗. Since xni is any fixed element of{xni}, then for any elementxniof{xni}, we have thatxni≤x∗.
Letni→ ∞, we get thatx∗≤x∗. Similarly, we can prove thatx∗≤x∗. Thus,x∗=x∗. This
is a contradiction. Therefore,xn→x∗. In the same method, we can know that there exists y∗∈Esuch thatyn→y∗.
Sinceθ≤yn–xn≤(NN+)n(y–x),n= , , , . . . . Letn→ ∞, we have thaty∗=x∗. In the following, we show thatx∗is the fixed point ofA. In fact, ifm≥, for any fixed integern,xn≤xn+m≤yn. Letm→ ∞, we have thatxn≤x∗≤yn. Thus,
Au,x∗=Au,x∗–A(u,xn–) +A(u,xn–)
≥–Nx∗–xn–
+A(u,xn–)
= –Nx∗+Nxn–+A(u,xn–)
= –Nx∗+ (N+ )xn→x∗ (n→ ∞).
On the other hand,
Au,x∗=Au,x∗–A(u,yn–) +A(u,yn–)
≤–Nx∗–yn–
+A(u,yn–)
= –Nx∗+Nyn–+A(u,yn–)
= –Nx∗+ (N+ )yn→x∗ (n→ ∞).
Therefore, for fixedu∈[u,v],A(u,x∗) =x∗. The rest of the proof is the same as that of
Corollary .(see Theorem . in []) Let P be a normal cone of E,and let A:P×P→P be a mixed monotone operator.Suppose that
. For fixedv∈P,A(·,v) :P→Pis concave;for fixedu∈P,∃N> such that A(u,·) :P→P,A(u,v) –A(u,v)≥–N(v–v),∀v≥v,v,v∈P.
. ∃¯v>θ, <c≤such thatθ<A(v¯,θ)≤ ¯vand A(θ,v¯)≥cA(v¯,θ).
Then A has exactly one fixed point u∗in[θ,¯v]and A(θ,v¯)≤u∗≤A(v¯,θ).
Proof Set
un=A(un–,vn–), vn=A(vn–,un–), n= , , . . . ,
u=θ,v=¯v, it is easy to show that
u<A(v,u)≤v, A(u,v)≥cA(v,u),
u≥cv, u=A(u,v)≤A(u,v),
A(v,u)≤v=A(v,u).
Now let us prove thatA(·,v) : [u,v]→Pis aϕ-concave operator for fixedv∈[u,v]. It suffices to showA(·,v) : [u,v]→Pis aϕ-concave operator for fixedv∈[u,v].
For eachu∈[u,v],t∈(, ), we see
A(tu,v) =Atu+ ( –t)θ,v
≥tA(u,v) + ( –t)A(θ,v)
≥tA(u,v) + ( –t)A(θ,v)
≥tA(u,v) +c( –t)A(v,u)
≥tA(u,v) +c( –t)A(u,v).
Setϕ(t,x) =t+c( –t), thenϕ: (, ]×[u,v]→(, ],ϕ(t,x) >t,∀t∈(, ),lims→t–ϕ(s,
w) =t+c( –t) >t.
So, by Theorem ., we see thatAhas exactly one fixed pointx∗in [u,v]. Similarly, ifPis a solid cone, we have the following corollary.
Corollary .(see Theorem . in []) Let P be a normal solid cone of E,and let A:
P◦×P◦→P◦be a mixed monotone operator.Suppose that
. For fixedv∈P◦,A(·,v) :P◦→P◦is concave;for fixedu∈P◦,∃N> such that A(u,·) :P◦→P◦,A(u,v) –A(u,v)≥–N(v–v),∀v≥v,v,v∈P◦.
. ∃u∈P◦,v∈P◦such thatu≤v,uA(u,v),A(v,u)≤v.
Then A has exactly one fixed point in[u,v].
Proof It is easy to show that there exists a real numberr> such thatu≥rvsince
Corollary .(see Remark . in []) Let P be a normal solid cone of E,and let A:P◦× P→P◦be a mixed monotone operator.Suppose that
. For fixedv∈P,A(·,v) :P◦→P◦isα-concave;for fixedu∈P,∃N> such that A(u,·) :P→P,A(u,v) –A(u,v)≥–N(v–v),∀v≥v,v,v∈P.
. ∃u∈P◦,v∈P◦such thatu≤v,uA(u,v),A(v,u)≤v.
Then A has exactly one fixed point in[u,v].
Proof Takeϕ(t,x) =tα,∀t∈(, ). Thus, by Corollary ., we easily see that the
conclu-sions of Corollary . hold.
Corollary .(see Theorem . in [] or Theorem . in []) Assume that the operator A satisfies the following conditions:
(H) A:Ph→Phis increasing inPh;
(H) for∀x∈Phandt∈(, ),there existsα(t)∈(, )such that A(tx)≥tα(t)Ax.
Then A has a unique solution in Ph.
Proof IfA(u,v) is independent ofv, then takeϕ(t,x) =tα(t),∀t∈(, ). It is easy to check that conditions (i), (iii) in Theorem . hold. By Lemma . in [], assume that the operator
Asatisfies conditions (H) and (H), we can know that there areu,v∈Phsuch thatu<
v,u≤Au≤Av≤v. Thus, condition (ii) in Theorem . holds. Therefore, it follows from Theorem . that the conclusions of Corollary . hold.
Corollary .(see Theorem . in []) Assume that the operator A satisfies the following conditions:
(H) A:Ph→Phis increasing inPh; (H)
A(tx)≥tα(t,x)Ax, ∀x∈P
h,t∈(, ),
whereα: (, )×Ph→(, )is increasing inxfor fixedt∈(, ); (H) there existst∈(, )such that
th≤Ah≤
t–α(t, th)
h.
ThenAhas a unique solution inPh.
Proof By Corollary ., we only need to check condition (ii) in Theorem .. By the proof of Theorem . in [], choosek∈Rsuch thatk>–α(t
). Putu=t k
h,v=
tk h, set
un=Aun–, vn=Avn–, n= , , . . . ,
Theorem . Let P be a normal cone of the real Banach space E,e>θand u,v∈P with
u≤v,and let A:P×P→P be a mixed monotone operator.Suppose that
(i) there exists a real positive numberrsuch thatu≥rv;
(ii) u≤A(u,v),A(v,u)≤v;
(iii) for fixedv,A(·,v) :P→Pise-concave with its characteristic function,η(t,x)is monotone inxand continuous intfrom left;
(iv) for fixedu∈P,∃N> such that
A(u,·) :P→P, A(u,v) –A(u,v)≥–N(v–v), ∀v≥v,v,v∈P.
Then A has exactly one fixed point x∗in[u,v].
Corollary .(see Theorem . in []) Assume that the operator A satisfies the following conditions:
(H) A:Ph→Phis increasing inPh;
(H) for eachx∈Phandt∈(, ),there existsη(t)∈(, )such that A(tx)≥t +η(t)Ax.
Then A has a unique solution in Ph.
Proof Setη(t,x) =η(t), by Corollary . and Theorem ., we can know that the
conclu-sions of Corollary . hold.
4 Application
In this section, we present an example to explain our results.
Example . Consider the following nonlinear integral equation:
x(t) = (Ax)(t) =
RNK(t,s)
x(s) +x(s) +x–(s)ds. (.)
Conclusion . Suppose thatK:RN×RN →Ris nonnegative and continuous with
≤
RN
K(t,s)ds≤
. (.)
Then Eq. (.) has a unique positive solutionx∗(t) satisfying –≤x∗(t)≤.
Proof We use Theorem . to prove Conclusion .. LetE=CB(RN) denote the set of all
bounded continuous functions onRN; we definex=sup
t∈RN|x(t)|, and thenEis a real
Banach space. LetP=C+
B(RN) denote the set of all nonnegative functions ofCB(RN). Then Pis a normal cone ofE. Obviously, Eq. (.) can be written in the formx=A(x,x), where
A(x,y) =A(x) +A(y),
A(x) =
RN
K(t,s)x(s) +x(s)ds,
A(y) =
RNK(t,s)y
Now, let us show that the operatorAsatisfies all the conditions in Theorem .. In fact, setu= –,v= . It is easy to see thatA:P×P→Pis a mixed monotone operator. It is obvious thatu,v∈P,u<vand there exists a real number> such thatu≥v. By (.) we can easily get
A(u,v) =
RNK(t,s)
–+ –+ ds≥–=u,
and
A(v,u) =
RNK(t,s)
+ + ds≤ =v.
For fixedy,∀t∈(, ),∃η=η(t,x) = (
√
t–t)√x
t(√x+x) > such that
A(tx,y)≥( +η)tA(x,y),
whereη(t,x) is decreasing inxand continuous intfrom left. For fixedx,∃N= such that
A(x,v) –A(x,v)≥–
(v–v), ∀v≥v,v,v∈
–, .
Therefore, we see Conclusion . holds by means of Theorem ..
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author was supported financially by the National Natural Science Foundation of China, Tianyuan Foundation (11226119), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi, the Youth Science Foundation of Shanxi Province (2013021002-1), and Shandong Provincial Natural Science Foundation, China (ZR2012AQ024).
Received: 4 August 2013 Accepted: 9 October 2013 Published:08 Nov 2013 References
1. Amann, H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev.18, 620-709 (1976)
2. Guo, DJ: Nonlinear Functional Analysis. Shandong Science and Technology Press, Jinan (2001) (in Chinese) 3. Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988) 4. Krasnoselskii, MA: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)
5. Krasnoselskii, MA, Zabreiko, PP: Geometrical Methods of Nonlinear Analysis. Springer, Berlin (1984)
6. Liang, Z, Lian, X, Zhang, M: A class of concave operators with applications. Nonlinear Anal.68, 2507-2515 (2008) 7. Potter, AJB: Applications of Hilbert projective metric to certain class of nonhomogeneous operators. Q. J. Math.28,
93-99 (1977)
8. Xu, SY, Jia, BG: Fixed-point theorems ofφ-concave(–ψ)convex mixed monotone operators and applications. J. Math. Anal. Appl.295, 645-657 (2004)
9. Xu, SY, Zeng, CY, Zhu, CX: Existence and uniqueness for the fixed points ofφ-concave(–ψ)convex mixed monotone operators and applications. Acta Math. Sin.48, 1055-1064 (2005) (in Chinese)
10. Zhai, CB, Wang, WX, Zhang, LL: Generalizations for a class of concave and convex operators. Acta Math. Sin.51, 529-540 (2008) (in Chinese)
11. Zhai, CB, Yang, C, Guo, CM: Positive solutions of operator equations on ordered Banach spaces and applications. Comput. Math. Appl.56, 3150-3156 (2008)
12. Zhao, ZQ, Du, X: Fixed points of generalizede-concave (generalizede-convex) operators and their applications. J. Math. Anal. Appl.334, 1426-1438 (2007)
13. Karapinar, E, Luong, NV, Thuan, NX: Coupled coincidence points for mixed monotone operators in partially ordered metric spaces. Arab. J. Math.1, 329-339 (2012)
15. Li, FY, Liang, ZD: Fixed point ofϕ-concave(–ϕ-convex) operator and application. J. Syst. Sci. Math. Sci.14(4), 355-360 (1994)
16. Guo, DJ, Lakshmikantham, V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. TMA
11(5), 623-632 (1987)
17. Abbas, M, Ali, B, Sintunavarat, W, Kumam, P: Tripled fixed point and tripled coincidence point theorems in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl.2012, 187 (2012)
18. Abbas, M, Sintunavarat, W, Kumam, P: Coupled fixed point of generalized contractive mappings on partially ordered
G-metric spaces. Fixed Point Theory Appl.2012, 31 (2012)
19. Aydi, H, Abbas, M, Sintunavarat, W, Kumam, P: Tripled fixed point ofW-compatible mappings in abstract metric spaces. Fixed Point Theory Appl.2012, 134 (2012)
20. Gnana Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. TMA65(7), 1379-1393 (2006)
21. Chandok, S, Sintunavarat, W, Kumam, P: Some coupled common fixed points for a pair of mappings in partially orderedG-metric spaces. Math. Sci.7, 24 (2013)
22. Drici, Z, McRae, FA, Vasundhara Devi, J: Fixed point theorems for mixed monotone operators with PPF dependence. Nonlinear Anal. TMA69, 632-636 (2008)
23. Guo, DJ: Existence and uniqueness of positive fixed point for mixed monotone operators and applications. Appl. Anal.46, 91-100 (1992)
24. Guo, DJ: Fixed points for mixed monotone operators with application. Appl. Anal.34, 215-224 (1988)
25. Harjani, J, López, B, Sadarangani, K: Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Anal. TMA74, 1749-1760 (2011)
26. Karapinar, E, Kumam, P, Sintunavarat, W: Coupled fixed point theorems in cone metric spaces with ac-distance and applications. Fixed Point Theory Appl.2012, 194 (2012)
27. Wu, YS, Li, GZ: Existence and uniqueness theorems of fixed points for mixed monotone operators and their applications. Acta Math. Sin. (Chin. Ser.)46(1), 161-166 (2003) (in Chinese)
28. Zhai, CB: Fixed point theorems for a class of mixed monotone operators with convexity. Fixed Point Theory Appl.
2013, 119 (2013)
29. Zhai, CB, Zhang, LL: New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems. J. Math. Anal. Appl.382, 594-614 (2011)
30. Zhang, ZT: Fixed point theorems of mixed monotone operators and its applications. Acta Math. Sin.41(6), 1121-1126 (1998) (in Chinese)
31. Zhang, ZT: New fixed point theorems of mixed monotone operators and applications. J. Math. Anal. Appl.204(1), 307-319 (1996)
32. Zhang, ZT, Wang, K: On fixed point theorems of mixed monotone operators and applications. Nonlinear Anal. TMA
70, 3279-3284 (2009)
33. Zhao, ZQ: Uniqueness and existence of fixed points on some mixed monotone mappings in ordered linear spaces. J. Syst. Sci. Math. Sci.19(2), 217-224 (1999)
34. Sintunavarat, W, Kumam, P: Coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces. Thai J. Math.10(3), 551-563 (2012)
35. Sintunavarat, W, Cho, YJ, Kumam, P: Coupled fixed-point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces. Fixed Point Theory Appl.2012, 128 (2012)
36. Guo, YX: Iterative solution on a class of nonlinear operator equation. Acta Anal. Funct. Appl.1, 45-50 (1999) 37. Sun, JX: Iterative solution of nonlinear operator equation (II). J. Shandong Univ.3, 281-288 (1992) (in Chinese)
10.1186/1687-1812-2013-274
Cite this article as:Sang:A class ofϕ-concave operators and applications.Fixed Point Theory and Applications
