# Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems

- Chengbo Zhai
^{1}Email author and - Mengru Hao
^{1}

**2013**:85

**DOI: **10.1186/1687-2770-2013-85

© Zhai and Hao; licensee Springer. 2013

**Received: **22 November 2012

**Accepted: **18 March 2013

**Published: **10 April 2013

## Abstract

This work is concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problem:

where ${D}_{{0}^{+}}^{\nu}$ is the standard Riemann-Liouville fractional derivative of order *ν*, and $n\in N$, $n>3$. Our analysis relies on two new fixed point theorems for mixed monotone operators with perturbation. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate the main result.

**MSC:**26A33, 34B18, 34B27.

### Keywords

Riemann-Liouville fractional derivative fractional differential equation positive solution existence and uniqueness fixed point theorem for mixed monotone operator## 1 Introduction

where ${D}_{{0}^{+}}^{\nu}$ is the standard Riemann-Liouville fractional derivative of order *ν*, and $n\in N$, $n>3$.

Fractional differential equations arise in many fields such as physics, mechanics, chemistry, economics, engineering and biological sciences, *etc.*; see [1–6] for example. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Miller and Ross [3], Podlubny [5], Kilbas *et al.* [6], and the papers [7–16] and the references therein. In these papers, many authors have investigated the existence of positive solutions for nonlinear fractional differential equation boundary value problems. On the other hand, the uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems has been studied by some authors; see [10, 14, 17] for example.

under the assumptions that ${a}_{1}$, ${a}_{2}$, *f*, *g* are nonnegative and continuous. But the uniqueness of positive solutions is not treated in these papers.

Different from the works mentioned above, motivated by the work [20], we will use two fixed point theorems for mixed monotone operators with perturbation to show the existence and uniqueness of positive solutions for FBVP (1.1). To our knowledge, there are still very few to utilize the fixed point results on mixed monotone operators with perturbation to study the existence and uniqueness of a positive solution for nonlinear fractional differential equation boundary value problems. So, it is worthwhile to investigate FBVP (1.1) by using our new fixed point theorems in [20]. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it.

With this context in mind, the outline of this paper is as follows. In Section 2 we recall certain results from the theory of fractional calculus and some definitions, notations and results of mixed monotone operators. In Section 3 we provide some conditions, under which the problem FBVP (1.1) has a unique positive solution. Finally, in Section 4, we provide an example, which explicates the applicability of our result.

## 2 Preliminaries

For the convenience of the reader, we present here some definitions, lemmas and basic results that will be used in the proofs of our theorems.

**Definition 2.1** (See [18])

*ν*th Riemann-Liouville fractional integral is defined to be

*ν*th Riemann-Liouville fractional derivative to be

where $n\in N$ is the unique positive integer satisfying $n-1\le \nu <n$ and $t>a$.

**Lemma 2.2** (See [19])

*Let*$g\in C[0,1]$

*be given*.

*Then the unique solution to problem*$-{D}_{{0}^{+}}^{\nu}y(t)=g(t)$

*together with the boundary conditions*${y}^{(i)}(0)=0={[{D}_{{0}^{+}}^{\alpha}y(t)]}_{t=1}$,

*where*$1\le \alpha \le n-2$

*and*$0\le i\le n-2$,

*is*

*where*

*is the Green function for this problem*.

**Lemma 2.3** (See [19])

*Let*$G(t,s)$

*be as given in the statement of Lemma*2.2.

*Then we have*

- (i)
$G(t,s)$

*is a continuous function on the unit square*$[0,1]\times [0,1]$; - (ii)
$G(t,s)\ge 0$

*for each*$(t,s)\in [0,1]\times [0,1]$.

**Lemma 2.4**

*The function*$G(t,s)$

*defined by*(2.2)

*satisfies the following conditions*:

*Proof*Evidently, the right inequality holds. So, we only need to prove the left inequality. If $0\le s\le t\le 1$, then we have $0\le t-s\le t-ts=(1-s)t$, and thus

So, the proof is complete. □

In the sequel, we present some basic concepts in ordered Banach spaces for completeness and two fixed point theorems which we will be used later. For convenience of readers, we suggest that one refers to [20–22] for details.

Suppose that $(E,\parallel \cdot \parallel )$ is a real Banach space which is partially ordered by a cone $P\subset E$, *i.e.*, $x\le y$ if and only if $y-x\in P$. 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 non-empty 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 $.

*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 $[{x}_{1},{x}_{2}]=\{x\in E\mid {x}_{1}\le x\le {x}_{2}\}$ 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 ${P}_{h}=\{x\in E\mid x\sim h\}$. It is easy to see that ${P}_{h}\subset P$.

$A:P\times P\to P$ 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 $A({u}_{1},{v}_{1})\le A({u}_{2},{v}_{2})$. Element $x\in P$ is called a fixed point of *A* if $A(x,x)=x$.

**Definition 2.6**An operator $A:P\to P$ is said to be sub-homogeneous if it is satisfies

**Definition 2.7**Let $D=P$ and

*β*be a real number with $0\le \beta <1$. An operator $A:D\to D$ is said to be

*β*-concave if it satisfies

**Lemma 2.8** (See Theorem 2.1 in [20])

*Let*$h>\theta $

*and*$\beta \in (0,1)$. $A:P\times P\to P$

*is a mixed monotone operator and satisfies*

*is an increasing sub*-

*homogeneous 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)
$A:{P}_{h}\times {P}_{h}\to {P}_{h}$

*and*$B:{P}_{h}\to {P}_{h}$; - (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*$A(x,x)+Bx=x$*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*${x}_{n}=A({x}_{n-1},{y}_{n-1})+B{x}_{n-1},\phantom{\rule{2em}{0ex}}{y}_{n}=A({y}_{n-1},{x}_{n-1})+B{y}_{n-1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,$

*we have* ${x}_{n}\to {x}^{\ast}$ *and* ${y}_{n}\to {x}^{\ast}$ *as* $n\to \mathrm{\infty}$.

**Lemma 2.9** (See Theorem 2.4 in [20])

*Let*$h>\theta $

*and*$\beta \in (0,1)$. $A:P\times P\to P$

*is a mixed monotone operator and satisfies*

*is an increasing*

*β*-

*concave 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)
$A:{P}_{h}\times {P}_{h}\to {P}_{h}$

*and*$B:{P}_{h}\to {P}_{h}$; - (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*$A(x,x)+Bx=x$*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*${x}_{n}=A({x}_{n-1},{y}_{n-1})+B{x}_{n-1},\phantom{\rule{2em}{0ex}}{y}_{n}=A({y}_{n-1},{x}_{n-1})+B{y}_{n-1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,$

*we have* ${x}_{n}\to {x}^{\ast}$ *and* ${y}_{n}\to {x}^{\ast}$ *as* $n\to \mathrm{\infty}$.

**Remark 2.10** (i) If we take $B=\theta $ in Lemma 2.8, then the corresponding conclusion is still true (see Corollary 2.2 in [20]); (ii) if we take $A=\theta $ in Lemma 2.9, then the conclusion obtained is also true (see Theorem 2.7 in [23]).

## 3 Main results

In this section, we apply Lemma 2.8 and Lemma 2.9 to study FBVP (1.1), and we obtain some new results on the existence and uniqueness of positive solutions. The method used here is relatively new to the literature and so are the existence and uniqueness results to the fractional differential equations.

Set $P=\{x\in C[0,1]\mid x(t)\ge 0,t\in [0,1]\}$, the standard cone. It is clear that *P* is a normal cone in $C[0,1]$ and the normality constant is 1.

**Theorem 3.1**
*Assume that*

(H_{1}) $f:[0,1]\times [0,+\mathrm{\infty})\times [0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ *is continuous and* $g:[0,1]\times [0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ *is continuous*;

(H_{2}) $f(t,u,v)$ *is increasing in* $u\in [0,+\mathrm{\infty})$ *for fixed* $t\in [0,1]$ *and* $v\in [0,+\mathrm{\infty})$, *decreasing in* $v\in [0,+\mathrm{\infty})$ *for fixed* $t\in [0,1]$ *and* $u\in [0,+\mathrm{\infty})$, *and* $g(t,u)$ *is increasing in* $u\in [0,+\mathrm{\infty})$ *for fixed* $t\in [0,1]$;

(H_{3}) $g(t,0)\not\equiv 0$ *and* $g(t,\lambda u)\ge \lambda g(t,u)$ *for* $\lambda \in (0,1)$, $t\in [0,1]$, $u\in [0,+\mathrm{\infty})$, *and there exists a constant* $\beta \in (0,1)$ *such that* $f(t,\lambda u,{\lambda}^{-1}v)\ge {\lambda}^{\beta}f(t,u,v)$, $\mathrm{\forall}t\in [0,1]$, $\lambda \in (0,1)$, $u,v\in [0,+\mathrm{\infty})$;

(H_{4}) *there exists a constant* ${\delta}_{0}>0$ *such that* $f(t,u,v)\ge {\delta}_{0}g(t,u)$, $t\in [0,1]$, $u,v\ge 0$.

*Then*:

- (1)

*where*$h(t)={t}^{\nu -1}$, $t\in [0,1]$

*and*$G(t,s)$

*is given as in*(2.2);

- (2)
*FBVP*(1.1)*has a unique positive solution*${u}^{\ast}$*in*${P}_{h}$; - (3)

*we have* $\parallel {x}_{n}-{u}^{\ast}\parallel \to 0$ *and* $\parallel {y}_{n}-{u}^{\ast}\parallel \to 0$ *as* $n\to \mathrm{\infty}$.

*Proof*To begin with, from Lemma 2.2, FBVP (1.1) has an integral formulation given by

where $G(t,s)$ is given as in (2.2).

It is easy to prove that *u* is the solution of FBVP (1.1) if and only if $u=A(u,u)+Bu$. From (H_{1}), we know that $A:P\times P\to P$ and $B:P\to P$. In the sequel, we check that *A*, *B* satisfy all the assumptions of Lemma 2.8.

*A*is a mixed monotone operator. In fact, for ${u}_{i},{v}_{i}\in P$, $i=1,2$ with ${u}_{1}\ge {u}_{2}$, ${v}_{1}\le {v}_{2}$, we know that ${u}_{1}(t)\ge {u}_{2}(t)$, ${v}_{1}(t)\le {v}_{2}(t)$, $t\in [0,1]$, and by (H

_{2}) and Lemma 2.3,

That is, $A({u}_{1},{v}_{1})\ge A({u}_{2},{v}_{2})$.

_{2}) and Lemma 2.3 that

*B*is increasing. Next we show that

*A*satisfies the condition (2.5). For any $\lambda \in (0,1)$ and $u,v\in P$, by (H

_{3}) we have

*A*satisfies (2.5). Also, for any $\lambda \in (0,1)$, $u\in P$, from (H

_{3}) we know that

*B*is sub-homogeneous. Now we show that $A(h,h)\in {P}_{h}$ and $Bh\in {P}_{h}$. On the one hand, from (H

_{1}), (H

_{2}) and Lemma 2.4, for any $t\in [0,1]$, we have

_{1}), (H

_{2}) and Lemma 2.4, for any $t\in [0,1]$, we obtain

_{2}), (H

_{4}), we have

from $g(t,0)\not\equiv 0$, we easily prove $Bh\in {P}_{h}$. Hence the condition (i) of Lemma 2.8 is satisfied.

_{4}),

satisfy $\parallel {x}_{n}-{u}^{\ast}\parallel \to 0$ and $\parallel {y}_{n}-{u}^{\ast}\parallel \to 0$ as $n\to \mathrm{\infty}$. □

**Theorem 3.2** *Assume* (H_{1}), (H_{2}) *and*

(H_{5}) *there exists a constant* $\beta \in (0,1)$ *such that* $g(t,\lambda u)\ge {\lambda}^{\beta}g(t,u)$, $\mathrm{\forall}t\in [0,1]$, $\lambda \in (0,1)$, $u\in [0,+\mathrm{\infty})$, *and* $f(t,\lambda u,{\lambda}^{-1}v)\ge \lambda f(t,u,v)$ *for* $\lambda \in (0,1)$, $t\in [0,1]$, $u,v\in [0,+\mathrm{\infty})$;

(H_{6}) $f(t,0,1)\not\equiv 0$ *for* $t\in [0,1]$ *and there exists a constant* ${\delta}_{0}>0$ *such that* $f(t,u,v)\le {\delta}_{0}g(t,u)$, $t\in [0,1]$, $u,v\ge 0$.

*Then*:

- (1)

*where*$h(t)={t}^{\nu -1}$, $t\in [0,1]$

*and*$G(t,s)$

*is given as in*(2.2);

- (2)
*FBVP*(1.1)*has a unique positive solution*${u}^{\ast}$*in*${P}_{h}$; - (3)

*we have* $\parallel {x}_{n}-{u}^{\ast}\parallel \to 0$ *and* $\parallel {y}_{n}-{u}^{\ast}\parallel \to 0$ *as* $n\to \mathrm{\infty}$.

*Sketch of the proof*Consider two operators

*A*,

*B*defined in the proof of Theorem 3.1. Similarly, from (H

_{1}), (H

_{2}), we obtain that $A:P\times P\to P$ is a mixed monotone operator and $B:P\to P$ is increasing. From (H

_{5}), we have

_{2}), (H

_{6}), we have

_{6}),

satisfy $\parallel {x}_{n}-{u}^{\ast}\parallel \to 0$ and $\parallel {y}_{n}-{u}^{\ast}\parallel \to 0$ as $n\to \mathrm{\infty}$. □

From Remark 2.10 and similar to the proofs of Theorems 3.1-3.2, we can prove the following conclusions.

**Corollary 3.3**

*Let*$g\equiv 0$.

*Assume that*

*f*

*satisfies the conditions of Theorem*3.1

*and*$f(t,0,1)\not\equiv 0$.

*Then*: (i)

*there exist*${u}_{0},{v}_{0}\in {P}_{h}$

*and*$r\in (0,1)$

*such that*$r{v}_{0}\le {u}_{0}<{v}_{0}$

*and*

*where*$h(t)={t}^{\nu -1}$, $t\in [0,1]$

*and*$G(t,s)$

*is given as in*(2.2); (ii)

*the FBVP*

*has a unique positive solution*${u}^{\ast}$

*in*${P}_{h}$; (iii)

*for any*${x}_{0},{y}_{0}\in {P}_{h}$,

*constructing successively the sequences*

*we have* $\parallel {x}_{n}-{u}^{\ast}\parallel \to 0$ *and* $\parallel {y}_{n}-{u}^{\ast}\parallel \to 0$ *as* $n\to \mathrm{\infty}$.

**Corollary 3.4**

*Let*$f\equiv 0$.

*Assume that*

*g*

*satisfies the conditions of Theorem*3.2

*and*$g(t,0)\not\equiv 0$

*for*$t\in [0,1]$.

*Then*: (i)

*there exist*${u}_{0},{v}_{0}\in {P}_{h}$

*and*$r\in (0,1)$

*such that*$r{v}_{0}\le {u}_{0}<{v}_{0}$

*and*

*where*$h(t)={t}^{\nu -1}$, $t\in [0,1]$

*and*$G(t,s)$

*is given as in*(2.2); (ii)

*the FBVP*

*has a unique positive solution*${u}^{\ast}$

*in*${P}_{h}$; (iii)

*for any*${x}_{0},{y}_{0}\in {P}_{h}$,

*constructing successively the sequences*

*we have* $\parallel {x}_{n}-{u}^{\ast}\parallel \to 0$ *and* $\parallel {y}_{n}-{u}^{\ast}\parallel \to 0$ *as* $n\to \mathrm{\infty}$.

## 4 An example

We now present one example to illustrate Theorem 3.1.

**Example 4.1**

where $c>0$ is a constant, $a,b:[0,1]\to [0,\mathrm{\infty})$ are continuous with $a\not\equiv 0$.

Hence all the conditions of Theorem 3.1 are satisfied. An application of Theorem 3.1 implies that problem (4.1) has a unique positive solution in ${P}_{h}$, where $h(t)={t}^{\nu -1}={t}^{5.3}$, $t\in [0,1]$.

## Declarations

### Acknowledgements

The authors are grateful to the anonymous referee for his/her valuable suggestions. The first author was supported financially by the Youth Science Foundations of China (11201272) and Shanxi Province (2010021002-1).

## Authors’ Affiliations

## References

- Oldham KB, Spanier J:
*The Fractional Calculus*. Academic Press, New York; 1974.Google Scholar - Gaul L, Klein P, Kempffe S: Damping description involving fractional operators.
*Mech. Syst. Signal Process.*1991, 5: 81-88. 10.1016/0888-3270(91)90016-XView ArticleGoogle Scholar - Miller KS, Ross B:
*An Introduction to the Fractional Calculus and Fractional Differential Equations*. Wiley, New York; 1993.Google Scholar - Glockle WG, Nonnenmacher TF: A fractional calculus approach of self-similar protein dynamics.
*Biophys. J.*1995, 68: 46-53. 10.1016/S0006-3495(95)80157-8View ArticleGoogle Scholar - Podlubny I Mathematics in Science and Engineering. In
*Fractional Differential Equations*. Academic Press, New York; 1999.Google Scholar - Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In
*Theory and Applications of Fractional Differential Equations*. Elsevier, Amsterdam; 2006.Google Scholar - Zhang SQ: Existence of positive solution for some class of nonlinear fractional differential equations.
*J. Math. Anal. Appl.*2003, 278: 136-148. 10.1016/S0022-247X(02)00583-8MathSciNetView ArticleGoogle Scholar - Bai ZB, Lü HS: Positive solutions for boundary value problem of nonlinear fractional differential equation.
*J. Math. Anal. Appl.*2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleGoogle Scholar - Lakshmikantham V: Theory of fractional functional differential equations.
*Nonlinear Anal.*2008, 69: 3337-3343. 10.1016/j.na.2007.09.025MathSciNetView ArticleGoogle Scholar - Zhou Y: Existence and uniqueness of fractional functional differential equations with unbounded delay.
*Int. J. Dyn. Syst. Differ. Equ.*2008, 1(4):239-244.MathSciNetGoogle Scholar - Kaufmann ER, Mboumi E: Positive solutions of a boundary value problem for a nonlinear fractional differential equation.
*Electron. J. Qual. Theory Differ. Equ.*2008., 2008: Article ID 3Google Scholar - Kosmatov N: A singular boundary value problem for nonlinear differential equations of fractional order.
*J. Appl. Math. Comput.*2009, 29: 125-135. 10.1007/s12190-008-0104-xMathSciNetView ArticleGoogle Scholar - Xu X, Jiang D, Yuan C: Multiple positive solutions for boundary value problem of nonlinear fractional differential equation.
*Nonlinear Anal.*2009, 71: 4676-4688. 10.1016/j.na.2009.03.030MathSciNetView ArticleGoogle Scholar - Yang L, Chen H: Unique positive solutions for fractional differential equation boundary value problems.
*Appl. Math. Lett.*2010, 23: 1095-1098. 10.1016/j.aml.2010.04.042MathSciNetView ArticleGoogle Scholar - Wang YQ, Liu LS, Wu YH: Positive solutions for a nonlocal fractional differential equation.
*Nonlinear Anal.*2011, 74: 3599-3605. 10.1016/j.na.2011.02.043MathSciNetView ArticleGoogle Scholar - Lizama C: An operator theoretical approach to a class of fractional order differential equations.
*Appl. Math. Lett.*2011, 24: 184-190. 10.1016/j.aml.2010.08.042MathSciNetView ArticleGoogle Scholar - Yang C, Zhai CB: Uniqueness of positive solutions for a fractional differential equation via a fixed point theorem of a sum operator.
*Electron. J. Differ. Equ.*2012., 2012: Article ID 70Google Scholar - Goodrich CS: Existence of a positive solution to a class of fractional differential equations.
*Appl. Math. Lett.*2010, 23: 1050-1055. 10.1016/j.aml.2010.04.035MathSciNetView ArticleGoogle Scholar - Goodrich CS: Existence of a positive solution to systems of differential equations of fractional order.
*Comput. Math. Appl.*2011, 62: 1251-1268. 10.1016/j.camwa.2011.02.039MathSciNetView ArticleGoogle Scholar - Zhai CB, Hao MR: Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems.
*Nonlinear Anal.*2012, 75: 2542-2551. 10.1016/j.na.2011.10.048MathSciNetView ArticleGoogle Scholar - Guo D, Lakshmikantham V:
*Nonlinear Problems in Abstract Cones*. Academic Press, Boston; 1988.Google Scholar - Guo D, Lakskmikantham V: Coupled fixed points of nonlinear operators with applications.
*Nonlinear Anal.*1987, 11(5):623-632. 10.1016/0362-546X(87)90077-0MathSciNetView ArticleGoogle Scholar - Zhai CB, Yang C, Zhang XQ: Positive solutions for nonlinear operator equations and several classes of applications.
*Math. Z.*2010, 266: 43-63. 10.1007/s00209-009-0553-4MathSciNetView ArticleGoogle Scholar

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