Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems

Abstract

The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.

Introduction

Mixed monotone operators were introduced by Guo and Lakshmikantham [1]. Thereafter, many authors have investigated these kinds of operators in Banach spaces and obtained a lot of interesting and important results (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] and the references therein). Their study has not only important theoretical meaning but also wide applications in engineering, nuclear physics, biology, chemistry, technology, etc. That is, they are used extensively in nonlinear differential and integral equations. For a small sample of such work, we refer the reader to works [13], [15], [16], [17], [18], [19], [20].

In [2], Guo proved the following result.

Theorem A See [2]

Let $P$ be a normal, solid cone of Banach space $E,\mathring{P}$ be the interior point set of $P$, and let $A:\mathring{P}\times \mathring{P}\to \mathring{P}$ be a mixed monotone operator; suppose that there exists $\alpha \in (0,1)$ such that

$$A(tu,t^{-1}v)\ge t^{\alpha }A(u,v)\text{,}\forall u,v\in \mathring{P},t\in (0,1)\text{.}$$

Then operator $A$ has a unique fixed point $x^{\ast }$ in $\mathring{P}$ . Moreover, for any initial $x_0,y_0\in \mathring{P}$, constructing successively the sequences

$$x_n=A(x_{n-1},y_{n-1})\text{,}{y}_n=A(y_{n-1},x_{n-1})\text{,}n=1,2,\dots \text{,}$$

we have $\Vert x_n-x^{\ast }\Vert \to 0$ and $\Vert y_n-x^{\ast }\Vert \to 0$ as $n\to \infty$.

This theorem has plenty of extensions and generalizations. We refer to Chen [3], Liang et al. [7], Lian and Li [10], Wu [11], Wang et al. [12], Zhao [15], Zhai and Zhang [17] and references therein. In this paper, we study the existence and uniqueness of positive fixed points for mixed monotone operators with perturbation. In other words, we consider the existence and uniqueness of positive solutions to the following operator equation on ordered Banach spaces

$$A(x,x)+Bx=x\text{,}$$

where $A$ is a mixed monotone operator, $B$ is an increasing sub-homogeneous operator or $\alpha$-concave operator. To our knowledge, the fixed point results on mixed monotone operators with perturbation are still very few. So it is worthwhile to investigate the operator Eq. (1.1). In this paper, using the properties of cones and a fixed point theorem for mixed monotone operators, we obtain some existence and uniqueness results of positive solutions for the operator Eq. (1.1) without assuming operators to be continuous or compact, which extend the corresponding results in [2], [3], [7], [10], [12], [15]. Even in the special case of $B$ being a null operator, the conditions of our result(Corollary 2.2) are also weak. That is, our result extends Theorem A. To demonstrate the applicability of our abstract results, we give, in the last section of the paper, some simple applications to nonlinear fractional differential equation boundary value problems.

For the discussion of the following sections, we state here some definitions, notations and known results. For convenience of readers, we suggest that one refer to [1], [2], [17], [21], [22] for details.

Suppose that $(E,\Vert \cdot \Vert )$ 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 $\theta$ 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) $x\in P,-x\in P\Rightarrow x=\theta$.

Putting $\mathring{P}=\{x\in P|x\text{ is an interior point of }P\}$, a cone $P$ is said to be solid if its interior $\mathring{P}$ is non-empty. 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 $\Vert x\Vert \le N\Vert y\Vert$; 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, $\sim$ is an equivalence relation. Given $h>\theta$ (i.e., $h\ge \theta$ and $h\ne \theta$), we denote by $P_h$ the set $P_h=\{x\in E|x\sim h\}$. It is easy to see that $P_h\subset P$. Take $h\in \mathring{P}$, then $P_h=\mathring{P}$.

Definition 1.1 See [1], [2]

$A:P\times P\to P$ is said to be a mixed monotone operator if $A(x,y)$ is increasing in $x$ and decreasing in $y$, i.e., $u_i,v_i(i=1,2)\in P,u_1\le u_2,v_1\ge v_2$ 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 1.2

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

$$A\left(tx\right)\ge tAx\text{,}\forall t\in (0,1),x\in P\text{.}$$

Definition 1.3

Let $D=P$ or $D=\mathring{P}$ and $\alpha$ be a real number with $0\le \alpha <1$. An operator $A:D\to D$ is said to be $\alpha$-concave if it satisfies

$$A\left(tx\right)\ge t^{\alpha }Ax\text{,}\forall t\in (0,1),x\in D\text{.}$$

Notice that the definition of an $\alpha$-concave operator mentioned above is different from that in [22], because we need not require the cone to be solid in general.

Lemma 1.4 See Lemma 2.1 and Theorem 2.1 in [17]

Let $P$ be a normal cone in $E$ . Assume that $T:P\times P\to P$ is a mixed monotone operator and satisfies:

• there exists $h\in P$ with $h\ne \theta$ such that $T(h,h)\in P_h$ ;

• for any $u,v\in P$ and $t\in (0,1)$, there exists $\phi \left(t\right)\in (t,1]$ such that $T(tu,t^{-1}v)\ge \phi \left(t\right)T(u,v)$.

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 $r{v}_0\le u_0<v_0,u_0\le T(u_0,v_0)\le T(v_0,u_0)\le v_0$ ;

(3) $T$ has a unique fixed point $x^{\ast }$ in $P_h$ ;

(4) for any initial values $x_0,y_0\in P_h$, constructing successively the sequences

$$x_n=T(x_{n-1},y_{n-1})\text{,}{y}_n=T(y_{n-1},x_{n-1})\text{,}n=1,2,\dots \text{,}$$

we have $x_n\to x^{\ast }$ and $y_n\to x^{\ast }$ as $n\to \infty$.

Remark 1.5

Under the conditions $\left(A_1\right),\left(A_2\right)$, this lemma cannot only guarantee the existence of upper–lower solutions for the operator $T$ and the existence of a unique fixed point, but also construct successively some sequences for approximating the fixed point.