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Uniqueness of positive solutions for several classes of sum operator equations and applications
Journal of Inequalities and Applications volume 2014, Article number: 58 (2014)
Abstract
In this article we study several classes of sum operator equations on ordered Banach spaces and present some new existence and uniqueness results of positive solutions, which extend the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for several classes of sum operator equations. As applications, we utilize the main results obtained in this paper to study two classes nonlinear problems; one is the integral equation u(t)=\lambda {\int}_{a}^{b}G(t,s)f(s,u(s))\phantom{\rule{0.2em}{0ex}}ds, where f and G are both nonnegative, \lambda >0 is a parameter; the other is the elliptic boundary value problem for the LaneEmdenFowler equation \mathrm{\Delta}u=\lambda f(x,u), u(x)>0 in Ω, u(x)=0 on ∂ Ω, where Ω is a bounded domain with smooth boundary in {\mathbf{R}}^{N} (N\ge 1), \lambda >0 and f(x,u) is allowed to be singular on ∂ Ω. The new results on the existence and uniqueness of positive solutions for these problems are given, which complement the existing results of positive solutions for these problems in the literature.
MSC:47H10, 47H07, 45G15, 35J60, 35J65.
1 Introduction and preliminaries
With the development of nonlinear sciences, nonlinear functional analysis has been an active area of research over the past several decades. As an important branch of nonlinear functional analysis, nonlinear operator theory has attracted much attention and has been widely studied, especially nonlinear operators which arise in the connection with nonlinear differential and integral equations have been extensively studied (see for instance [1–12]). It is well known that the existence and uniqueness of positive solutions to nonlinear operator equations is very important in theory and applications. Many authors have studied this problem; for a small sample of such work, we refer the reader to [7, 10, 13–23]. The operator equation considered in this papers is always of the following form:
In [23], Zhao considered the existence of solutions for the sum operator equation
where A is increasing econcave, B is increasing econvex and A+B is a strict set contraction. Motivated by the works [22, 23], Sang et al. considered the operator equation (1.1), where A is {\phi}_{1}concave, B is {\phi}_{2}convex and A+B is also a strict set contraction. However, we can see that the conditions of the main results in [23, 24] are strong and of utmost convenience.
Recently, we considered successively the operator equation (1.1) and the following operator equation:
the operators A, B in (1.1) are increasing, αconcave and subhomogeneous, respectively; the operators A, B in (1.2) are mixed monotone and increasing αconcave (or subhomogeneous), respectively. In [7], by using the properties of cones and a fixed point theorem for increasing general αconcave operators, we established the existence and uniqueness of positive solutions for the operator equation (1.1), and we utilized the main results to present the existence and uniqueness of positive solutions for the following two problems; one is a fourthorder twopoint boundary value problem for elastic beam equations,
where f\in C([0,1]\times \mathbf{R}) and g\in C(\mathbf{R}) are real functions; and the other is an elliptic value problem for LaneEmdenFowler equations
where f(x,u), g(x,u) are allowed to be singular on ∂ Ω. In [21], by using the properties of cones and a fixed point theorem for mixed monotone operators, we established the existence and uniqueness of positive solutions for the operator equation (1.2), and we utilized the results obtained to study the existence and uniqueness of positive solutions for a nonlinear fractional differential equation boundary value problem,
where {D}_{0+}^{\alpha} is the RiemannLiouville fractional derivative of order \alpha >0. These results are useful and interesting. For completeness, in this paper we will further consider the following several classes of sum operators:

(i)
the sum of increasing operators and decreasing operators;

(ii)
the sum of increasing operators and mixed monotone operators;

(iii)
the sum of decreasing operators and mixed monotone operators;

(iv)
the sum of increasing operators, decreasing operators and mixed monotone operators.
Motivated by our works [7, 10, 21], we will study the above cases (i)(iv). So this article is a continuation of our papers [7, 10, 21], and we will present some interesting results on the existence and uniqueness of positive solutions for the above several classes of sum operator equations. To demonstrate the applicability of our abstract results, we give, in the last section of the paper, some applications to nonlinear integral equations and elliptic boundary value problems for the LaneEmdenFowler equations.
In the following two subsections, we state some definitions, notations, and known results. For convenience of the readers, we refer to [7–13, 20–22, 25–27] for details.
1.1 Some basic definitions and notations
Suppose that E is a real Banach space which is partially ordered by a cone P\subset E, i.e., x\le y if and only if yx\in P. If x\le y and x\ne y, then we denote x<y or y>x. By θ we denote the zero element of E. Recall that a nonempty closed convex set P\subset E is a cone if it satisfies (i) x\in P,\lambda \ge 0\Rightarrow \lambda x\in P; (ii) x\in P,x\in P\Rightarrow x=\theta.
Putting \stackrel{\u02da}{P}=\{x\in Px\text{is an interior point of}P\}, a cone P is said to be solid if \stackrel{\u02da}{P} is nonempty. Moreover, P is called normal if there exists a constant N>0 such that, for all x,y\in E, \theta \le x\le y implies \parallel x\parallel \le N\parallel y\parallel; in this case N is called the normality constant of P. If {x}_{1},{x}_{2}\in E, the set [{x}_{1},{x}_{2}]=\{x\in E{x}_{1}\le x\le {x}_{2}\} is called the order interval between {x}_{1} and {x}_{2}. We say that an operator A:E\to E is increasing (decreasing) if x\le y implies Ax\le Ay (Ax\ge Ay).
For all x,y\in E, the notation x\sim y means that there exist \lambda >0 and \mu >0 such that \lambda x\le y\le \mu x. Clearly, ∼ is an equivalence relation. Given h>\theta (i.e., h\ge \theta and h\ne \theta), we denote by {P}_{h} the set {P}_{h}=\{x\in Ex\sim h\}. It is easy to see that {P}_{h}\subset P.
Definition 1.1 Let D=P or D=\stackrel{\u02da}{P} and α be a real number with 0\le \alpha <1. An operator A:P\to P is said to be αconcave if it satisfies
Notice that the definition of an αconcave operator mentioned above is different from that in [26], because we need not require the cone to be solid in general.
Definition 1.2 An operator A:P\to P is said to be subhomogeneous if it satisfies
Definition 1.3 (See [10, 21, 27])
A:P\times P\to P is said to be a mixed monotone operator if A(x,y) is increasing in x and decreasing in y, i.e., {u}_{i},{v}_{i}\phantom{\rule{0.25em}{0ex}}(i=1,2)\in P, {u}_{1}\le {u}_{2}, {v}_{1}\ge {v}_{2} imply A({u}_{1},{v}_{1})\le A({u}_{2},{v}_{2}). An element x\in P is called a fixed point of A if A(x,x)=x.
1.2 Some fixed point theorems and properties
In this subsection, we assume that E is a real Banach space with a partial order introduced by a cone P of E. Take h\in E, h>\theta, {P}_{h} is given as in Section 1.1.
In the paper [7], we considered the existence and uniqueness of positive solutions to the operator equation (1.1) on ordered Banach spaces and established the following conclusion.
Theorem 1.1 (See Theorem 2.2 in [7])
Let P be a normal cone in E, A:P\to P be an increasing αconcave operator and B:P\to P be an increasing subhomogeneous operator. Assume that

(i)
there is h>\theta such that Ah\in {P}_{h} and Bh\in {P}_{h};

(ii)
there exists a constant {\delta}_{0}>0 such that Ax\ge {\delta}_{0}Bx, \mathrm{\forall}x\in P.
Then the operator equation (1.1) has a unique solution {x}^{\ast} in {P}_{h}. Moreover, constructing successively the sequence {y}_{n}=A{y}_{n1}+B{y}_{n1}, n=1,2,\dots for any initial value {y}_{0}\in {P}_{h}, we have {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
In the paper [10], we present the following fixed point theorem for a class of general mixed monotone operators and established some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators.
Theorem 1.2 (See Lemma 2.1 and Theorem 2.1 in [10])
Let P be a normal cone in E. Assume that A:P\times P\to P is a mixed monotone operator and satisfies:

(i)
there exists h\in P with h\ne \theta such that A(h,h)\in {P}_{h};

(ii)
for any u,v\in P and t\in (0,1), there exists \phi (t)\in (t,1] such that A(tu,{t}^{1}v)\ge \phi (t)A(u,v).
Then:

(1)
T:{P}_{h}\times {P}_{h}\to {P}_{h};

(2)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that r{v}_{0}\le {u}_{0}<{v}_{0}, {u}_{0}\le A({u}_{0},{v}_{0})\le A({v}_{0},{u}_{0})\le {v}_{0};

(3)
the operator equation (1.2) has a unique solution {x}^{\ast} in {P}_{h};

(4)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A({x}_{n1},{y}_{n1}),\phantom{\rule{2em}{0ex}}{y}_{n}=A({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast} and {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Theorem 1.3 (See Theorem 2.3 in [10])
Assume that the operator A satisfies the conditions of Theorem 1.2. Let {x}_{\lambda} (\lambda >0) denote the unique solution of nonlinear eigenvalue equation A(x,x)=\lambda x in {P}_{h}. Then we have the following conclusions:
(R_{1}) If \phi (t)>{t}^{\frac{1}{2}} for t\in (0,1), then {x}_{\lambda} is strictly decreasing in λ, that is, 0<{\lambda}_{1}<{\lambda}_{2} implies {x}_{{\lambda}_{1}}>{x}_{{\lambda}_{2}};
(R_{2}) If there exists \beta \in (0,1) such that \phi (t)\ge {t}^{\beta} for t\in (0,1), then {x}_{\lambda} is continuous in λ, that is, \lambda \to {\lambda}_{0} ({\lambda}_{0}>0) implies \parallel {x}_{\lambda}{x}_{{\lambda}_{0}}\parallel \to 0;
(R_{3}) If there exists \beta \in (0,\frac{1}{2}) such that \phi (t)\ge {t}^{\beta} for t\in (0,1), then {lim}_{\lambda \to \mathrm{\infty}}\parallel {x}_{\lambda}\parallel =0, {lim}_{\lambda \to {0}^{+}}\parallel {x}_{\lambda}\parallel =\mathrm{\infty}.
Based on Theorem 1.2, in [21] we considered the operator equation (1.2) and established the following conclusions.
Theorem 1.4 (See Theorem 2.1 in [21])
Let P be a normal cone in E, \alpha \in (0,1). A:P\times P\to P is a mixed monotone operator and satisfies
B:P\to P is an increasing subhomogeneous operator. Assume that

(i)
there is {h}_{0}\in {P}_{h} such that A({h}_{0},{h}_{0})\in {P}_{h} and B{h}_{0}\in {P}_{h};

(ii)
there exists a constant {\delta}_{0}>0 such that A(x,y)\ge {\delta}_{0}Bx, \mathrm{\forall}x,y\in P.
Then:

(1)
A:{P}_{h}\times {P}_{h}\to {P}_{h}, B:{P}_{h}\to {P}_{h};

(2)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A({u}_{0},{v}_{0})+B{u}_{0}\le A({v}_{0},{u}_{0})+B{v}_{0}\le {v}_{0}; 
(3)
the operator equation (1.2) has a unique solution {x}^{\ast} in {P}_{h};

(4)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A({x}_{n1},{y}_{n1})+B{x}_{n1},\phantom{\rule{2em}{0ex}}{y}_{n}=A({y}_{n1},{x}_{n1})+B{y}_{n1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast} and {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Theorem 1.5 (See Theorem 2.4 in [21])
Let P be a normal cone in E, \alpha \in (0,1). A:P\times P\to P is a mixed monotone operator and satisfies
B:P\to P is an increasing αconcave operator. Assume that

(i)
there is {h}_{0}\in {P}_{h} such that A({h}_{0},{h}_{0})\in {P}_{h} and B{h}_{0}\in {P}_{h};

(ii)
there exists a constant {\delta}_{0}>0 such that A(x,y)\le {\delta}_{0}Bx, \mathrm{\forall}x,y\in P.
Then:

(1)
A:{P}_{h}\times {P}_{h}\to {P}_{h}, B:{P}_{h}\to {P}_{h};

(2)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A({u}_{0},{v}_{0})+B{u}_{0}\le A({v}_{0},{u}_{0})+B{v}_{0}\le {v}_{0}; 
(3)
the operator equation (1.2) has a unique solution {x}^{\ast} in {P}_{h};

(4)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A({x}_{n1},{y}_{n1})+B{x}_{n1},\phantom{\rule{2em}{0ex}}{y}_{n}=A({y}_{n1},{x}_{n1})+B{y}_{n1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast} and {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
2 Main results
In this section we consider the existence and uniqueness of positive solutions for several classes of sum operator equations. We always assume that E is a real Banach space with a partial order induced by a cone P of E. Take h\in E, h>\theta and {P}_{h} as given in the Introduction.
2.1 The sum of increasing operators and decreasing operators
Now we first consider the following sum operator equations:
Theorem 2.1 Let P be a normal cone, A:P\to P be an increasing operator and B:P\to P be a decreasing operator. Assume that:
(H_{11}) for any x\in P and t\in (0,1), there exist {\phi}_{i}(t)\in (t,1) (i=1,2) such that
(H_{12}) there exists {h}_{0}\in {P}_{h} such that A{h}_{0}+B{h}_{0}\in {P}_{h}.
Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A{u}_{0}+B{v}_{0}\le A{v}_{0}+B{u}_{0}\le {v}_{0}; 
(ii)
the operator equation (2.1) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A{x}_{n1}+B{y}_{n1},\phantom{\rule{2em}{0ex}}{y}_{n}=A{y}_{n1}+B{x}_{n1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Proof Firstly, from (2.3), we have
Since A{h}_{0}+B{h}_{0}\in {P}_{h}, there exist constants {\lambda}_{1},{\lambda}_{2}>0 such that
Also from {h}_{0}\in {P}_{h}, there exists a constant {t}_{0}\in (0,1) such that
Let \phi (t)=min\{{\phi}_{1}(t),{\phi}_{2}(t)\}, t\in (0,1). Then \phi (t)\in (t,1). From (2.3) and (2.4), we obtain
Note that {\lambda}_{1}\phi ({t}_{0}),\frac{{\lambda}_{2}}{\phi ({t}_{0})}>0, we can get Ah+Bh\in {P}_{h}.
Next we define an operator T=A+B by T(x,y)=Ax+By. Then T:P\times P\to P is a mixed monotone operator and T(h,h)=Ah+Bh\in {P}_{h}. Moreover, for any x,y\in P and t\in (0,1), we have
Hence, the operator T satisfies the condition (ii) in Theorem 1.2. An application of Theorem 1.2 implies that: there are {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that r{v}_{0}\le {u}_{0}<{v}_{0}, {u}_{0}\le T({u}_{0},{v}_{0})\le T({v}_{0},{u}_{0})\le {v}_{0}; operator equation T(x)=x has a unique positive {x}^{\ast}\in {P}_{h}; for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
we have \parallel {x}_{n}{x}^{\ast}\parallel \to 0 and \parallel {y}_{n}{x}^{\ast}\parallel \to 0 as n\to \mathrm{\infty}. That is:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A{u}_{0}+B{v}_{0}\le A{v}_{0}+B{u}_{0}\le {v}_{0}; 
(ii)
the operator equation (2.1) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A{x}_{n1}+B{y}_{n1},\phantom{\rule{2em}{0ex}}{y}_{n}=A{y}_{n1}+B{x}_{n1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
□
Note that h\in {P}_{h}, and we can easily obtain the following conclusions.
Corollary 2.2 Let P be a normal cone, A:{P}_{h}\to {P}_{h} be an increasing operator and B:{P}_{h}\to {P}_{h} be a decreasing operator. Assume that:
(H_{13}) for any x\in {P}_{h} and t\in (0,1), there exist {\phi}_{i}(t)\in (t,1) (i=1,2) such that
Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A{u}_{0}+B{v}_{0}\le A{v}_{0}+B{u}_{0}\le {v}_{0}; 
(ii)
the operator equation (2.1) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A{x}_{n1}+B{y}_{n1},\phantom{\rule{2em}{0ex}}{y}_{n}=A{y}_{n1}+B{x}_{n1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Corollary 2.3 Let {\alpha}_{1},{\alpha}_{2}\in (0,1). Let P be a normal cone, A:P\to P be an increasing {\alpha}_{1}concave operator and B:P\to P be a decreasing {\alpha}_{2}convex operator. Assume that (H_{12}) holds. Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A{u}_{0}+B{v}_{0}\le A{v}_{0}+B{u}_{0}\le {v}_{0}; 
(ii)
the operator equation (2.1) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A{x}_{n1}+B{y}_{n1},\phantom{\rule{2em}{0ex}}{y}_{n}=A{y}_{n1}+B{x}_{n1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Proof Let {\phi}_{1}(t)={t}^{{\alpha}_{1}}, {\phi}_{2}(t)={t}^{{\alpha}_{2}}, t\in (0,1). Then {\phi}_{1}(t),{\phi}_{2}(t)\in (t,1) for t\in (0,1) and
Hence, the conclusions follow from Theorem 2.1. □
Corollary 2.4 Let \alpha \in (0,1) and P be a normal cone. Let {A}_{1}:P\to P be an increasing αconcave operator and {A}_{2}:P\to P be an increasing subhomogeneous operator, B:P\to P be a decreasing operator which satisfies (2.3). Assume that:
(H_{14}) there exists {h}_{0}\in {P}_{h} such that {A}_{1}{h}_{0}+{A}_{2}{h}_{0}\in {P}_{h};
(H_{15}) there exists \delta >0 such that {A}_{1}x\ge \delta {A}_{2}x, x\in P;
(H_{16}) there exists {h}_{1}\in {P}_{h} such that B{h}_{1}\in {P}_{h}.
Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le {A}_{1}{u}_{0}+{A}_{2}{u}_{0}+B{v}_{0}\le {A}_{1}{v}_{0}+{A}_{2}{v}_{0}+B{u}_{0}\le {v}_{0}; 
(ii)
the following operator equation:
{A}_{1}x+{A}_{2}x+Bx=x,(2.5)
has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
\begin{array}{c}{x}_{n}={A}_{1}{x}_{n1}+{A}_{2}{x}_{n1}+B{y}_{n1},\hfill \\ {y}_{n}={A}_{1}{y}_{n1}+{A}_{2}{y}_{n1}+B{x}_{n1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,\hfill \end{array}
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Proof Define an operator A={A}_{1}+{A}_{2} by Ax={A}_{1}x+{A}_{2}x. Then A:P\to P is an increasing operator and A{h}_{0}\in {P}_{h}. Since {h}_{0},{h}_{1}\in {P}_{h}, there exist {t}_{0},{t}_{1}\in (0,1) such that
Then {h}_{0}\le \frac{1}{{t}_{0}}h\le \frac{1}{{t}_{0}{t}_{1}}{h}_{1}, {h}_{0}\ge {t}_{0}h\ge {t}_{0}{t}_{1}{h}_{1}, and thus
Note that {\phi}_{2}({t}_{0}{t}_{1}),\frac{1}{{\phi}_{2}({t}_{0}{t}_{1})}>0 and B{h}_{1}\in {P}_{h}, we can get B{h}_{0}\in {P}_{h}. Hence, A{h}_{0}+B{h}_{0}\in {P}_{h}.
From the proof of Theorem 1.1, there exists {\beta}_{0}(t)\in (\alpha ,1) with respect to t, such that
Let {\phi}_{1}(t)={t}^{{\beta}_{0}(t)}, t\in (0,1). Then {\phi}_{1}(t)\in (t,1) and A(tx)\ge {\phi}_{1}(t)Ax, x\in P.
Therefore, operators A, B satisfy all the conditions of Theorem 2.1. So we easily obtain the following conclusions:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h}, and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le {A}_{1}{u}_{0}+{A}_{2}{u}_{0}+B{v}_{0}\le {A}_{1}{v}_{0}+{A}_{2}{v}_{0}+B{u}_{0}\le {v}_{0}; 
(ii)
the operator equation (2.5) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
\begin{array}{c}{x}_{n}={A}_{1}{x}_{n1}+{A}_{2}{x}_{n1}+B{y}_{n1},\hfill \\ {y}_{n}={A}_{1}{y}_{n1}+{A}_{2}{y}_{n1}+B{x}_{n1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,\hfill \end{array}
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
□
Corollary 2.5 Assume that all the conditions of Theorem 2.1 hold. Let {x}_{\lambda} (\lambda >0) denote the unique solution of operator equation (2.2). Then we have the following conclusions:

(i)
if {\phi}_{i}(t)>{t}^{\frac{1}{2}} (i=1,2) for t\in (0,1), then {x}_{\lambda} is strictly decreasing in λ, that is, 0<{\lambda}_{1}<{\lambda}_{2} implies {x}_{{\lambda}_{1}}>{x}_{{\lambda}_{2}};

(ii)
if there exists \beta \in (0,1) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {x}_{\lambda} is continuous in λ, that is, \lambda \to {\lambda}_{0} ({\lambda}_{0}>0) implies \parallel {x}_{\lambda}{x}_{{\lambda}_{0}}\parallel \to 0;

(iii)
if there exists \beta \in (0,\frac{1}{2}) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {lim}_{\lambda \to \mathrm{\infty}}\parallel {x}_{\lambda}\parallel =0, {lim}_{\lambda \to {0}^{+}}\parallel {x}_{\lambda}\parallel =\mathrm{\infty}.
Proof Define an operator T=A+B by T(x,y)=Ax+By. Then T:P\times P\to P is a mixed monotone operator. From the proof of Theorem 2.1, we have T(h,h)\in {P}_{h}, and
where \phi (t)=min\{{\phi}_{1}(t),{\phi}_{2}(t)\}. Evidently, \phi (t)\in (t,1) for t\in (0,1). Hence, the conclusions follow from Theorem 1.3. □
Similarly, we can easily obtain the following result.
Corollary 2.6 Assume that all the conditions of Corollary 2.3 hold. Let {x}_{\lambda} (\lambda >0) denote the unique solution of operator equation (2.2). Then we have the following conclusions:

(i)
if {\alpha}_{1},{\alpha}_{2}\in (0,\frac{1}{2}), then {x}_{\lambda} is strictly decreasing in λ, that is, 0<{\lambda}_{1}<{\lambda}_{2} implies {x}_{{\lambda}_{1}}>{x}_{{\lambda}_{2}};

(ii)
{x}_{\lambda} is continuous in λ, that is, \lambda \to {\lambda}_{0} ({\lambda}_{0}>0) implies \parallel {x}_{\lambda}{x}_{{\lambda}_{0}}\parallel \to 0;

(iii)
if {\alpha}_{1},{\alpha}_{2}\in (0,\frac{1}{2}), then {lim}_{\lambda \to \mathrm{\infty}}\parallel {x}_{\lambda}\parallel =0, {lim}_{\lambda \to {0}^{+}}\parallel {x}_{\lambda}\parallel =\mathrm{\infty}.
2.2 The sum of increasing operators and mixed monotone operators
Next, we consider the following sum operator equations:
Theorem 2.7 Let P be a normal cone, A:P\to P be an increasing operator and B:P\times P\to P be a mixed monotone operator. Assume that:
(H_{21}) for any x\in P, t\in (0,1), there exists {\phi}_{1}(t)\in (t,1) such that
(H_{22}) for any x,y\in P, t\in (0,1), there exists {\phi}_{2}(t)\in (t,1) such that
(H_{23}) there exists {h}_{0}\in {P}_{h} such that A{h}_{0}+B({h}_{0},{h}_{0})\in {P}_{h}.
Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A{u}_{0}+B({u}_{0},{v}_{0})\le A{v}_{0}+B({v}_{0},{u}_{0})\le {v}_{0}; 
(ii)
the operator equation (2.6) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A{x}_{n1}+B({x}_{n1},{y}_{n1}),\phantom{\rule{2em}{0ex}}{y}_{n}=A{y}_{n1}+B({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Proof From (2.9), we obtain
Since A{h}_{0}+B({h}_{0},{h}_{0})\in {P}_{h}, there exist constants {\lambda}_{1},{\lambda}_{2}>0 such that
Also, from {h}_{0}\in {P}_{h}, there exists a small constant {t}_{0}\in (0,1) such that
Let \phi (t)=min\{{\phi}_{1}(t),{\phi}_{2}(t)\}. Then \phi (t)\in (t,1) for t\in (0,1). From (2.8)(2.10),
Note that {\lambda}_{1}\phi ({t}_{0}),\frac{{\lambda}_{2}}{\phi ({t}_{0})}>0, we can get Ah+B(h,h)\in {P}_{h}.
Next, we define an operator T=A+B by T(x,y)=Ax+B(x,y). Then T:P\times P\to P is a mixed monotone operator and T(h,h)=Ah+B(h,h)\in {P}_{h}. Moreover, for any x,y\in P and t\in (0,1), we have
Hence, all the conditions of Theorem 1.2 are satisfied. An application of Theorem 1.2 implies that: there are {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that r{v}_{0}\le {u}_{0}<{v}_{0}, {u}_{0}\le T({u}_{0},{v}_{0})\le T({v}_{0},{u}_{0})\le {v}_{0}; operator equation T(x,x)=x has a unique solution {x}^{\ast}\in {P}_{h}; for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
we have \parallel {x}_{n}{x}^{\ast}\parallel \to 0 and \parallel {y}_{n}{x}^{\ast}\parallel \to 0 as n\to \mathrm{\infty}. That is:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A{u}_{0}+B({u}_{0},{v}_{0})\le A{v}_{0}+B({v}_{0},{u}_{0})\le {v}_{0}; 
(ii)
the operator equation (2.6) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A{x}_{n1}+B({x}_{n1},{y}_{n1}),\phantom{\rule{2em}{0ex}}{y}_{n}=A{y}_{n1}+B({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
□
Corollary 2.8 Let P be a normal cone, A:{P}_{h}\to {P}_{h} be an increasing operator and B:{P}_{h}\times {P}_{h}\to {P}_{h} be a mixed monotone operator. Assume that:
(H_{24}) for any x\in {P}_{h}, t\in (0,1), there exists {\phi}_{1}(t)\in (t,1) such that
(H_{25}) for any x,y\in {P}_{h}, t\in (0,1), there exists {\phi}_{2}(t)\in (t,1) such that
Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A{u}_{0}+B({u}_{0},{v}_{0})\le A{v}_{0}+B({v}_{0},{u}_{0})\le {v}_{0}; 
(ii)
the operator equation (2.6) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A{x}_{n1}+B({x}_{n1},{y}_{n1}),\phantom{\rule{2em}{0ex}}{y}_{n}=A{y}_{n1}+B({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Corollary 2.9 Let \alpha \in (0,1) and P be a normal cone. Let A:P\to P be an increasing operator which satisfies (H_{21}), {B}_{1}:P\to P be an increasing subhomogeneous operator and {B}_{2}:P\times P\to P be a mixed monotone operator which satisfies
Assume that:
(H_{26}) there exist {h}_{0},{h}_{1}\in {P}_{h} such that
(H_{27}) there exists a constant {\delta}_{0}>0 such that
Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
\begin{array}{c}r{v}_{0}\le {u}_{0}<{v}_{0},\hfill \\ {u}_{0}\le A{u}_{0}+{B}_{1}{u}_{0}+{B}_{2}({u}_{0},{v}_{0})\le A{v}_{0}+{B}_{1}{v}_{0}+{B}_{2}({v}_{0},{u}_{0})\le {v}_{0};\hfill \end{array} 
(ii)
the following operator equation:
Ax+{B}_{1}x+{B}_{2}(x,x)=x(2.12)
has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
\begin{array}{c}{x}_{n}=A{x}_{n1}+{B}_{1}{x}_{n1}+{B}_{2}({x}_{n1},{y}_{n1}),\hfill \\ {y}_{n}=A{y}_{n1}+{B}_{1}{y}_{n1}+{B}_{2}({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,\hfill \end{array}
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Proof Define an operator B={B}_{1}+{B}_{2} by B(x,y)={B}_{1}x+{B}_{2}(x,y). Then B:P\times P\to P is a mixed monotone operator and B({h}_{1},{h}_{1})={B}_{1}{h}_{1}+{B}_{2}({h}_{1},{h}_{1})\in {P}_{h}. Since {h}_{0},{h}_{1}\in {P}_{h}, there exist {t}_{0},{t}_{1}\in (0,1) such that
Then {h}_{1}\le \frac{1}{{t}_{1}}h\le \frac{1}{{t}_{0}{t}_{1}}{h}_{0}, {h}_{1}\ge {t}_{1}h\ge {t}_{0}{t}_{1}{h}_{0}, and thus
Note that {\phi}_{1}({t}_{0}{t}_{1}),\frac{1}{{\phi}_{1}({t}_{0}{t}_{1})}>0 and A{h}_{0}\in {P}_{h}, we can get A{h}_{1}\in {P}_{h}. Hence, A{h}_{1}+B({h}_{1},{h}_{1})\in {P}_{h}.
Note that (H_{27}) and from the proof of Theorem 1.4, there exists {\beta}_{0}(t)\in (\alpha ,1) with respect to t such that
Let {\phi}_{2}(t)={t}^{{\beta}_{0}(t)}, t\in (0,1). Then {\phi}_{2}(t)\in (t,1) and
Therefore, the operators A, B satisfy all the conditions of Theorem 2.7. So we easily obtain the following conclusions:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
\begin{array}{c}r{v}_{0}\le {u}_{0}<{v}_{0},\hfill \\ {u}_{0}\le A{u}_{0}+{B}_{1}{u}_{0}+{B}_{2}({u}_{0},{v}_{0})\le A{v}_{0}+{B}_{1}{v}_{0}+{B}_{2}({v}_{0},{u}_{0})\le {v}_{0};\hfill \end{array} 
(ii)
the operator equation (2.12) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
\begin{array}{c}{x}_{n}=A{x}_{n1}+{B}_{1}{x}_{n1}+{B}_{2}({x}_{n1},{y}_{n1}),\hfill \\ {y}_{n}=A{y}_{n1}+{B}_{1}{y}_{n1}+{B}_{2}({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,\hfill \end{array}
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
□
Corollary 2.10 Let \alpha \in (0,1) and P be a normal cone. Let A:P\to P be an increasing operator which satisfies (H_{21}), {B}_{1}:P\to P be an increasing αconcave operator and {B}_{2}:P\times P\to P be a mixed monotone operator which satisfies
Assume that (H_{26}) holds and
(H_{28}) there exists a constant {\delta}_{0}>0 such that {B}_{2}(x,y)\le {\delta}_{0}{B}_{1}x, \mathrm{\forall}x,y\in P.
Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{1em}{0ex}}{u}_{0}\le A{u}_{0}+{B}_{1}{u}_{0}+{B}_{2}({u}_{0},{v}_{0})\le A{v}_{0}+{B}_{1}{v}_{0}+{B}_{2}({v}_{0},{u}_{0})\le {v}_{0}; 
(ii)
the operator equation (2.12) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
\begin{array}{c}{x}_{n}=A{x}_{n1}+{B}_{1}{x}_{n1}+{B}_{2}({x}_{n1},{y}_{n1}),\hfill \\ {y}_{n}=A{y}_{n1}+{B}_{1}{y}_{n1}+{B}_{2}({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,\hfill \end{array}
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Proof Consider the same operator B defined by the proof of Corollary 2.9, we have B:P\times P\to P is a mixed monotone operator and B({h}_{1},{h}_{1})\in {P}_{h}. From Definition 1.1, we have A(tx)\le \frac{1}{{t}^{\alpha}}Ax, t\in (0,1), x\in P. Since {h}_{0},{h}_{1}\in {P}_{h}, there exist {t}_{0},{t}_{1}\in (0,1) such that
Then {t}_{0}{t}_{1}{h}_{0}\le {h}_{1}\le \frac{1}{{t}_{0}{t}_{1}}{h}_{0}, and thus
Note that {({t}_{0}{t}_{1})}^{\alpha},\frac{1}{{({t}_{0}{t}_{1})}^{\alpha}}>0 and A{h}_{0}\in {P}_{h}, we can get A{h}_{1}\in {P}_{h}. Hence, A{h}_{1}+B({h}_{1},{h}_{1})\in {P}_{h}.
Note that (H_{28}) and from the proof of Theorem 1.5, we know that there exists {\beta}_{0}(t)\in (\alpha ,1) with respect to t such that
Let {\phi}_{2}(t)={t}^{{\beta}_{0}(t)}, t\in (0,1). Then {\phi}_{2}(t)\in (t,1) and B(tx,{t}^{1}y)\ge {\phi}_{2}(t)B(x,y), x,y\in P.
Therefore, the operators A, B satisfy all the conditions of Theorem 2.7. So we easily obtain the following conclusions:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
\begin{array}{c}r{v}_{0}\le {u}_{0}<{v}_{0},\hfill \\ {u}_{0}\le A{u}_{0}+{B}_{1}{u}_{0}+{B}_{2}({u}_{0},{v}_{0})\le A{v}_{0}+{B}_{1}{v}_{0}+{B}_{2}({v}_{0},{u}_{0})\le {v}_{0};\hfill \end{array} 
(ii)
the operator equation (2.12) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
\begin{array}{c}{x}_{n}=A{x}_{n1}+{B}_{1}{x}_{n1}+{B}_{2}({x}_{n1},{y}_{n1}),\hfill \\ {y}_{n}=A{y}_{n1}+{B}_{1}{y}_{n1}+{B}_{2}({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,\hfill \end{array}
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
□
Similar to Corollary 2.5, we have the following result.
Corollary 2.11 Assume that all the conditions of Theorem 2.7 hold. Let {x}_{\lambda} (\lambda >0) denote the unique solution of operator equation (2.7). Then we have the following conclusions:

(i)
if {\phi}_{i}(t)>{t}^{\frac{1}{2}} (i=1,2) for t\in (0,1), then {x}_{\lambda} is strictly decreasing in λ, that is, 0<{\lambda}_{1}<{\lambda}_{2} implies {x}_{{\lambda}_{1}}>{x}_{{\lambda}_{2}};

(ii)
if there exists \beta \in (0,1) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {x}_{\lambda} is continuous in λ, that is, \lambda \to {\lambda}_{0} ({\lambda}_{0}>0) implies \parallel {x}_{\lambda}{x}_{{\lambda}_{0}}\parallel \to 0;

(iii)
if there exists \beta \in (0,\frac{1}{2}) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {lim}_{\lambda \to \mathrm{\infty}}\parallel {x}_{\lambda}\parallel =0, {lim}_{\lambda \to {0}^{+}}\parallel {x}_{\lambda}\parallel =\mathrm{\infty}.
2.3 The sum of decreasing operators and mixed monotone operators
In the following we also consider the operator equations (2.6) and (2.7).
Theorem 2.12 Let P be a normal cone, A:P\to P be a decreasing operator and B:P\times P\to P be a mixed monotone operator. Assume that (H_{22}) and (H_{23}) hold and
(H_{31}) for any x\in P and t\in (0,1), there exists {\phi}_{1}(t)\in (t,1) such that
Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{1em}{0ex}}{u}_{0}\le A{v}_{0}+B({u}_{0},{v}_{0})\le A{u}_{0}+B({v}_{0},{u}_{0})\le {v}_{0}; 
(ii)
the operator equation (2.6) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A{y}_{n1}+B({x}_{n1},{y}_{n1}),\phantom{\rule{2em}{0ex}}{y}_{n}=A{x}_{n1}+B({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Proof From (2.14), we have
Since A{h}_{0}+B({h}_{0},{h}_{0})\in {P}_{h}, there exist constants {\lambda}_{1},{\lambda}_{2}>0 such that
Also from {h}_{0}\in {P}_{h}, there exists a small constant {t}_{0}\in (0,1) such that
Let \phi (t)=min\{{\phi}_{1}(t),{\phi}_{2}(t)\}. Then \phi (t)\in (t,1) for t\in (0,1). From (H_{22}) and (2.14), (2.15),
Note that {\lambda}_{1}\phi ({t}_{0}),\frac{{\lambda}_{2}}{\phi ({t}_{0})}>0, we can get Ah+B(h,h)\in {P}_{h}.
Next, we define an operator T=A+B by T(x,y)=Ay+B(x,y). Then T:P\times P\to P is a mixed monotone operator and T(h,h)=Ah+B(h,h)\in {P}_{h}.
Moreover, for any x,y\in P and t\in (0,1), we have
Hence, all the conditions of Theorem 1.2 are satisfied. Application of Theorem 1.2 implies that: there are {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that r{v}_{0}\le {u}_{0}<{v}_{0}, {u}_{0}\le T({u}_{0},{v}_{0})\le T({v}_{0},{u}_{0})\le {v}_{0}; operator equation T(x,x)=x has a unique solution {x}^{\ast}\in {P}_{h}; for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
we have \parallel {x}_{n}{x}^{\ast}\parallel \to 0 and \parallel {y}_{n}{x}^{\ast}\parallel \to 0 as n\to \mathrm{\infty}. That is,

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A{v}_{0}+B({u}_{0},{v}_{0})\le A{u}_{0}+B({v}_{0},{u}_{0})\le {v}_{0}; 
(ii)
the operator equation (2.6) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A{y}_{n1}+B({x}_{n1},{y}_{n1}),\phantom{\rule{2em}{0ex}}{y}_{n}=A{x}_{n1}+B({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
□
Corollary 2.13 Let P be a normal cone, A:{P}_{h}\to {P}_{h} be a decreasing operator and B:{P}_{h}\times {P}_{h}\to {P}_{h} be a mixed monotone operator. Assume that:
(H_{32}) for any x\in {P}_{h} and t\in (0,1), there exists {\phi}_{1}(t)\in (t,1) such that
(H_{33}) for any x,y\in {P}_{h}, t\in (0,1), there exists {\phi}_{2}(t)\in (t,1) such that
Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
r{v}_{0}\le {u}_{0}<{v}_{0},\phantom{\rule{2em}{0ex}}{u}_{0}\le A{v}_{0}+B({u}_{0},{v}_{0})\le A{u}_{0}+B({v}_{0},{u}_{0})\le {v}_{0}; 
(ii)
the operator equation (2.6) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
{x}_{n}=A{y}_{n1}+B({x}_{n1},{y}_{n1}),\phantom{\rule{2em}{0ex}}{y}_{n}=A{x}_{n1}+B({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Corollary 2.14 Assume that all the conditions of Theorem 2.12 hold. Let {x}_{\lambda} (\lambda >0) denote the unique solution of operator equation (2.7). Then we have the following conclusions:

(i)
if {\phi}_{i}(t)>{t}^{\frac{1}{2}} (i=1,2) for t\in (0,1), then {x}_{\lambda} is strictly decreasing in λ, that is, 0<{\lambda}_{1}<{\lambda}_{2} implies {x}_{{\lambda}_{1}}>{x}_{{\lambda}_{2}};

(ii)
if there exists \beta \in (0,1) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {x}_{\lambda} is continuous in λ, that is, \lambda \to {\lambda}_{0} ({\lambda}_{0}>0) implies \parallel {x}_{\lambda}{x}_{{\lambda}_{0}}\parallel \to 0;

(iii)
if there exists \beta \in (0,\frac{1}{2}) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {lim}_{\lambda \to \mathrm{\infty}}\parallel {x}_{\lambda}\parallel =0, {lim}_{\lambda \to {0}^{+}}\parallel {x}_{\lambda}\parallel =\mathrm{\infty}.
2.4 The sum of increasing operators, decreasing operators, and mixed monotone operators
From the above results, we can easily obtain the following results on operator equations:
By Theorem 2.12 and Corollary 2.9, Corollary 2.10, we have the following conclusions.
Theorem 2.15 Let \alpha \in (0,1) and P be a normal cone. Let A:P\to P be a decreasing operator which satisfies (H_{31}), operators {B}_{1}, {B}_{2} be the same as for Corollary 2.9. Assume that (H_{26}), (H_{27}) hold. Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
\begin{array}{c}r{v}_{0}\le {u}_{0}<{v}_{0},\hfill \\ {u}_{0}\le A{v}_{0}+{B}_{1}{u}_{0}+{B}_{2}({u}_{0},{v}_{0})\le A{u}_{0}+{B}_{1}{v}_{0}+{B}_{2}({v}_{0},{u}_{0})\le {v}_{0};\hfill \end{array} 
(ii)
the operator equation (2.16) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
\begin{array}{c}{x}_{n}=A{y}_{n1}+{B}_{1}{x}_{n1}+{B}_{2}({x}_{n1},{y}_{n1}),\hfill \\ {y}_{n}=A{x}_{n1}+{B}_{1}{y}_{n1}+{B}_{2}({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,\hfill \end{array}
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Theorem 2.16 Let \alpha \in (0,1) and P be a normal cone. Let A:P\to P be a decreasing operator which satisfies (H_{31}), operators {B}_{1}, {B}_{2} be the same as for Corollary 2.10. Assume that (H_{26}), (H_{28}) hold. Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
\begin{array}{c}r{v}_{0}\le {u}_{0}<{v}_{0},\hfill \\ {u}_{0}\le A{v}_{0}+{B}_{1}{u}_{0}+{B}_{2}({u}_{0},{v}_{0})\le A{u}_{0}+{B}_{1}{v}_{0}+{B}_{2}({v}_{0},{u}_{0})\le {v}_{0};\hfill \end{array} 
(ii)
the operator equation (2.16) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
\begin{array}{c}{x}_{n}=A{y}_{n1}+{B}_{1}{x}_{n1}+{B}_{2}({x}_{n1},{y}_{n1}),\hfill \\ {y}_{n}=A{x}_{n1}+{B}_{1}{y}_{n1}+{B}_{2}({y}_{n1},{x}_{n1}),\phantom{\rule{1em}{0ex}}n=1,2,\dots ,\hfill \end{array}
we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
From Corollary 2.9, Corollary 2.10, and Corollary 2.4, we can easily obtain the following results.
Theorem 2.17 Let {\alpha}_{1},{\alpha}_{2}\in (0,1) and P be a normal cone, operators {A}_{1}, {A}_{2} satisfy the conditions of Corollary 2.4, where {A}_{1} is {\alpha}_{1}concave, operators {B}_{1}, {B}_{2} satisfy the conditions of Corollary 2.9, where {B}_{2} satisfies (2.11) with α replaced by {\alpha}_{2}. Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
\begin{array}{c}r{v}_{0}\le {u}_{0}<{v}_{0},\hfill \\ {u}_{0}\le {A}_{1}{u}_{0}+{A}_{2}{u}_{0}+{B}_{1}{u}_{0}+{B}_{2}({u}_{0},{v}_{0})\le {A}_{1}{v}_{0}+{A}_{2}{v}_{0}+{B}_{1}{v}_{0}+{B}_{2}({v}_{0},{u}_{0})\le {v}_{0};\hfill \end{array} 
(ii)
the operator equation (2.17) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
\begin{array}{c}{x}_{n}={A}_{1}{x}_{n1}+{A}_{2}{x}_{n1}+{B}_{1}{x}_{n1}+{B}_{2}({x}_{n1},{y}_{n1}),\hfill \\ {y}_{n}={A}_{1}{y}_{n1}+{A}_{2}{y}_{n1}+{B}_{1}{y}_{n1}+{B}_{2}({y}_{n1},{x}_{n1}),\hfill \end{array}
where n=1,2,\dots , we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
Theorem 2.18 Let {\alpha}_{1},{\alpha}_{2}\in (0,1) and P be a normal cone, operator A:P\to P is {\alpha}_{1}concave, operators {B}_{1}, {B}_{2} satisfy the conditions of Corollary 2.10, where {B}_{1} is {\alpha}_{2}concave. Then:

(i)
there exist {u}_{0},{v}_{0}\in {P}_{h} and r\in (0,1) such that
\begin{array}{c}r{v}_{0}\le {u}_{0}<{v}_{0},\hfill \\ {u}_{0}\le {A}_{1}{u}_{0}+{A}_{2}{u}_{0}+{B}_{1}{u}_{0}+{B}_{2}({u}_{0},{v}_{0})\le {A}_{1}{v}_{0}+{A}_{2}{v}_{0}+{B}_{1}{v}_{0}+{B}_{2}({v}_{0},{u}_{0})\le {v}_{0};\hfill \end{array} 
(ii)
the operator equation (2.17) has a unique solution {x}^{\ast} in {P}_{h};

(iii)
for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
\begin{array}{c}{x}_{n}={A}_{1}{x}_{n1}+{A}_{2}{x}_{n1}+{B}_{1}{x}_{n1}+{B}_{2}({x}_{n1},{y}_{n1}),\hfill \\ {y}_{n}={A}_{1}{y}_{n1}+{A}_{2}{y}_{n1}+{B}_{1}{y}_{n1}+{B}_{2}({y}_{n1},{x}_{n1}),\hfill \end{array}
where n=1,2,\dots , we have {x}_{n}\to {x}^{\ast}, {y}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
3 Some applications
In this section, we will apply the main results to study nonlinear problems which include nonlinear integral equations and nonlinear elliptic boundary value problems for the LaneEmdenFowler equations. And then we will obtain new results on the existence and uniqueness of positive solutions for these problems, which are not consequences of the corresponding fixed point theorems in the literature.
3.1 Applications to nonlinear integral equations
A standard approach, in studying the existence of positive solutions of boundary value problems (BVPs for short) for ordinary differential equations, is to rewrite the problem as an equivalent positivesolution problem for a Hammerstein integral equation of the form
in the space E=C[a,b], where the nonlinearity f and the kernel G (the Green function of the problem) are both nonnegative, \lambda >0 is a parameter. One seeks fixed points of a Hammerstein integral operator in a suitable cone of positive functions.
Set P=\{u\in C[a,b]u(t)\ge 0,t\in [a,b]\}, the standard cone. It is easy to see that P is a normal cone of which the normality constant is 1. Then {P}_{h}=\{x\in P\text{there are}{\lambda}_{2}(x)\ge {\lambda}_{1}(x)0\text{such that}{\lambda}_{1}(x)h(t)\le x(t)\le {\lambda}_{2}(x)h(t),t\in [a,b]\}. Assume that G(t,s):[a,b]\times [a,b]\to [0,+\mathrm{\infty}) is continuous with G(t,s)\not\equiv 0 and there exist h,m,n\in C([a,b],[0,+\mathrm{\infty})) with h(t),m(t),n(t)\not\equiv 0, such that
Theorem 3.1 Assume that f(t,x)={f}_{1}(t,x)+{f}_{2}(t,x)\not\equiv 0 and
(H_{31}) {f}_{i}:[a,b]\times [0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) is continuous (i=1,2), {f}_{1}(t,x) is increasing in x\in [0,+\mathrm{\infty}) for fixed t\in [a,b] and {f}_{2}(t,x) is decreasing in x\in [0,+\mathrm{\infty}) for fixed t\in [a,b];
(H_{32}) for \eta \in (0,1), there exist {\phi}_{i}(\eta )\in (\eta ,1) (i=1,2) such that
Then, for any given \lambda >0, the integral equation (3.1) has a unique positive solution {u}_{\lambda}^{\ast} in {P}_{h}. Moreover, for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences:
we have {x}_{n}\to {u}_{\lambda}^{\ast}, {y}_{n}\to {u}_{\lambda}^{\ast} as n\to +\mathrm{\infty}. Further, (i) if {\phi}_{i}(t)>{t}^{\frac{1}{2}} (i=1,2) for t\in (0,1), then {u}_{\lambda}^{\ast} is strictly increasing in λ, that is, 0<{\lambda}_{1}<{\lambda}_{2} implies {u}_{{\lambda}_{1}}^{\ast}<{u}_{{\lambda}_{2}}^{\ast}; (ii) if there exists \beta \in (0,1) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {u}_{\lambda}^{\ast} is continuous in λ, that is, \lambda \to {\lambda}_{0} ({\lambda}_{0}>0) implies \parallel {u}_{\lambda}^{\ast}{u}_{{\lambda}_{0}}^{\ast}\parallel \to 0; (iii) if there exists \beta \in (0,\frac{1}{2}) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {lim}_{\lambda \to {0}^{+}}\parallel {u}_{\lambda}^{\ast}\parallel =0, {lim}_{\lambda \to +\mathrm{\infty}}\parallel {u}_{\lambda}^{\ast}\parallel =+\mathrm{\infty}.
Proof Define two operators A:P\to E and B:P\to E by
It is easy to see that u is the solution of (3.1) if and only if u=\lambda (Au+Bu). From (H_{31}), we know that A:P\to P is increasing and B:P\to P is decreasing. Further, from (H_{32}), we can prove that A, B satisfy (H_{11}). Next we prove that Ah+Bh\in {P}_{h}. Set {h}_{\mathrm{max}}={max}_{t\in [a,b]}h(t), {h}_{\mathrm{min}}={min}_{t\in [a,b]}h(t). Then {h}_{\mathrm{max}}\ge {h}_{\mathrm{min}}>0.
For any t\in [a,b], from (H_{31}) and (3.2), we have
Let {r}_{1}={\int}_{a}^{b}m(s)[{f}_{1}(s,{h}_{\mathrm{min}})+{f}_{2}(s,{h}_{\mathrm{max}})]\phantom{\rule{0.2em}{0ex}}ds, {r}_{2}={\int}_{a}^{b}n(s)[{f}_{1}(s,{h}_{\mathrm{max}})+{f}_{2}(s,{h}_{\mathrm{min}})]\phantom{\rule{0.2em}{0ex}}ds. Note that f={f}_{1}+{f}_{2}\ge 0 is continuous with f\not\equiv 0 and from (3.2), we get 0<{r}_{1}\le {r}_{2} and in consequence, {r}_{1}h\le Ah+Bh\le {r}_{2}h. That is, Ah+Bh\in {P}_{h}. Hence, all the conditions of Theorem 2.1 are satisfied. It follows from Theorem 2.1 and Corollary 2.5 that the operator equation Au+Bu=\frac{1}{\lambda}u has a unique solution {u}_{\lambda}^{\ast} in {P}_{h}, that is, \lambda (A{u}_{\lambda}^{\ast}+B{u}_{\lambda}^{\ast})={u}_{\lambda}^{\ast}. So {u}_{\lambda}^{\ast} is a unique positive solution of the integral equation (3.1) in {P}_{h} for given \lambda >0. From Corollary 2.5, we have (i) if {\phi}_{i}(t)>{t}^{\frac{1}{2}} (i=1,2) for t\in (0,1), then {u}_{\lambda}^{\ast} is strictly increasing in λ, that is, 0<{\lambda}_{1}<{\lambda}_{2} implies {u}_{{\lambda}_{1}}^{\ast}<{u}_{{\lambda}_{2}}^{\ast}; (ii) if there exists \beta \in (0,1) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {u}_{\lambda}^{\ast} is continuous in λ, that is, \lambda \to {\lambda}_{0} ({\lambda}_{0}>0) implies \parallel {u}_{\lambda}^{\ast}{u}_{{\lambda}_{0}}^{\ast}\parallel \to 0; (iii) if there exists \beta \in (0,\frac{1}{2}) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {lim}_{\lambda \to {0}^{+}}\parallel {u}_{\lambda}^{\ast}\parallel =0, {lim}_{\lambda \to +\mathrm{\infty}}\parallel {u}_{\lambda}^{\ast}\parallel =+\mathrm{\infty}.
Let {A}_{\lambda}=\lambda A, {B}_{\lambda}=\lambda B. Then {A}_{\lambda}, {B}_{\lambda} also satisfy the conditions of Theorem 2.1. By Theorem 2.1, for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences
we have {x}_{n}\to {u}_{\lambda}^{\ast}, {y}_{n}\to {u}_{\lambda}^{\ast} as n\to +\mathrm{\infty}. That is,
as n\to +\mathrm{\infty}. □
Theorem 3.2 Assume that f(t,x)={f}_{1}(t,x)+{f}_{2}(t,x,x)\not\equiv 0 with {f}_{1}(t,x) satisfies (H_{31}) and
(H_{33}) {f}_{2}(t,x,y):[a,b]\times [0,+\mathrm{\infty})\times [0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) is continuous, increasing in x\in [0,+\mathrm{\infty}) for fixed t\in [a,b], y\in [0,+\mathrm{\infty}), decreasing in y\in [0,+\mathrm{\infty}) for fixed t\in [a,b], x\in [0,+\mathrm{\infty});
(H_{34}) for \eta \in (0,1), there exist {\phi}_{i}(\eta )\in (\eta ,1) (i=1,2) such that
Then, for any given \lambda >0, the integral equation (3.1) has a unique positive solution {u}_{\lambda}^{\ast} in {P}_{h}. Moreover, for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences:
we have {x}_{n}\to {u}_{\lambda}^{\ast}, {y}_{n}\to {u}_{\lambda}^{\ast} as n\to +\mathrm{\infty}. Further, the conclusions (i), (ii), and (iii) in Theorem 3.1 also hold.
Proof Define two operators A:P\to E and B:P\times P\to E by
It is easy to see that u is the solution of (3.1) if and only if u=\lambda [Au+B(u,u)]. From (H_{31}) and (H_{33}), we know that A:P\to P is increasing and B:P\times P\to P is mixed monotone. Further, from (H_{34}), we can prove that A, B satisfy (H_{21}) and (H_{22}). Next we prove that Ah+B(h,h)\in {P}_{h}.
For any t\in [a,b], from (H_{31}), (H_{33}), and (3.2), we have
Let {r}_{1}={\int}_{a}^{b}m(s)[{f}_{1}(s,{h}_{\mathrm{min}})+{f}_{2}(s,{h}_{\mathrm{min}},{h}_{\mathrm{max}})]\phantom{\rule{0.2em}{0ex}}ds, {r}_{2}={\int}_{a}^{b}n(s)[{f}_{1}(s,{h}_{\mathrm{max}})+{f}_{2}(s,{h}_{\mathrm{max}},{h}_{\mathrm{min}})]\phantom{\rule{0.2em}{0ex}}ds. Note that f={f}_{1}+{f}_{2} is nonnegative and continuous with f\not\equiv 0 and from (3.2), we get 0<{r}_{1}\le {r}_{2} and in consequence, {r}_{1}h\le Ah+B(h,h)\le {r}_{2}h. That is, Ah+B(h,h)\in {P}_{h}. Hence, all the conditions of Theorem 2.7 are satisfied. It follows from Theorem 2.7 and Corollary 2.11 that the operator equation Au+B(u,u)=\frac{1}{\lambda}u has a unique solution {u}_{\lambda}^{\ast} in {P}_{h}, that is, \lambda [A{u}_{\lambda}^{\ast}+B({u}_{\lambda}^{\ast},{u}_{\lambda}^{\ast})]={u}_{\lambda}^{\ast}. So {u}_{\lambda}^{\ast} is a unique positive solution of the integral equation (3.1) in {P}_{h} for given \lambda >0. From Corollary 2.11, we have (i) if {\phi}_{i}(t)>{t}^{\frac{1}{2}} (i=1,2) for t\in (0,1), then {u}_{\lambda}^{\ast} is strictly increasing in λ, that is, 0<{\lambda}_{1}<{\lambda}_{2} implies {u}_{{\lambda}_{1}}^{\ast}<{u}_{{\lambda}_{2}}^{\ast}; (ii) if there exists \beta \in (0,1) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {u}_{\lambda}^{\ast} is continuous in λ, that is, \lambda \to {\lambda}_{0} ({\lambda}_{0}>0) implies \parallel {u}_{\lambda}^{\ast}{u}_{{\lambda}_{0}}^{\ast}\parallel \to 0; (iii) if there exists \beta \in (0,\frac{1}{2}) such that {\phi}_{i}(t)\ge {t}^{\beta} (i=1,2) for t\in (0,1), then {lim}_{\lambda \to {0}^{+}}\parallel {u}_{\lambda}^{\ast}\parallel =0, {lim}_{\lambda \to +\mathrm{\infty}}\parallel {u}_{\lambda}^{\ast}\parallel =+\mathrm{\infty}.
Let {A}_{\lambda}=\lambda A, {B}_{\lambda}=\lambda B. Then {A}_{\lambda}, {B}_{\lambda} also satisfy the conditions of Theorem 2.7. By Theorem 2.7, for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences {x}_{n}={A}_{\lambda}{x}_{n1}+{B}_{\lambda}({x}_{n1},{y}_{n1}), {y}_{n}={A}_{\lambda}{y}_{n1}+{B}_{\lambda}({y}_{n1},{x}_{n1}), n=1,2,\dots , we have {x}_{n}\to {u}_{\lambda}^{\ast}, {y}_{n}\to {u}_{\lambda}^{\ast} as n\to +\mathrm{\infty}. That is,
as n\to +\mathrm{\infty}. □
Theorem 3.3 Assume that f(t,x)={f}_{1}(t,x)+{f}_{2}(t,x,x)\not\equiv 0 with {f}_{1}(t,x) satisfies all the conditions of {f}_{2}(t,x) in Theorem 3.1 and {f}_{2}(t,x,y) satisfies (H_{33}) and (H_{34}). Then, for any given \lambda >0, the integral equation (3.1) has a unique positive solution {u}_{\lambda}^{\ast} in {P}_{h}. Moreover, for any initial values {x}_{0},{y}_{0}\in {P}_{h}, constructing successively the sequences:
we have {x}_{n}\to {u}_{\lambda}^{\ast}, {y}_{n}\to {u}_{\lambda}^{\ast} as n\to +\mathrm{\infty}. Further, the conclusions (i), (ii), and (iii) in Theorem 3.1 also hold.
Proof Similar to the proofs of Theorem 3.1 and Theorem 3.2, the conclusions follow from Theorem 2.12 and Corollary 2.14. □
3.2 Applications to nonlinear elliptic BVPs for the LaneEmdenFowler equations
Let Ω be a bounded domain with smooth boundary in {\mathbf{R}}^{N} (N\ge 1). Consider the following singular Dirichlet problem for the LaneEmdenFowler equation:
where \lambda >0 and the nonlinear term f(x,u) is allowed to be singular on ∂ Ω.
The problem (3.3) arises in the study of nonNewtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrically materials (see [28–32]). The theory of singular elliptic boundary value problems for partial differential equations has become an important area of investigation in the past three decades, see [28–