Theory and Modern Applications

# Existence of positive solutions for a discrete fractional boundary value problem

## Abstract

This paper is concerned with the existence of positive solutions to a discrete fractional boundary value problem. By using the Krasnosel’skii and Schaefer fixed point theorems, the existence results are established. Additionally, examples are provided to illustrate the effectiveness of the main results.

MSC:26A33, 39A10, 47H07.

## 1 Introduction

Fractional differential equations have received increasing attention within the last ten years or so. The theory of fractional differential equations has been a new important mathematical branch due to its wide applications in different research areas and engineering, such as physics, chemistry, economics, control of dynamical etc. For more details, see  and the references therein. On the other hand, accompanied with the development of the theory for fractional calculus, fractional difference equations have attracted increasing attention slowly but steadily in the past three years or so. Some research papers have appeared, see . For example, Atici and Eloe  analyzed the conjugate discrete fractional boundary value problem (FBVP) with delta derivative:

$\left\{\begin{array}{l}-{\mathrm{\Delta }}^{v}y\left(t\right)=f\left(t+v-1,y\left(t+v-1\right)\right),\phantom{\rule{1em}{0ex}}t\in {\left[0,b\right]}_{{\mathbf{N}}_{0}},\\ y\left(v-2\right)=y\left(v+b+1\right)=0,\phantom{\rule{1em}{0ex}}1

Goodrich  studied the discrete fractional boundary value problems:

$\left\{\begin{array}{l}{\mathrm{\Delta }}^{v}y\left(t\right)=\lambda f\left(t+v-1,y\left(t+v-1\right)\right),\phantom{\rule{1em}{0ex}}t\in {\left[0,T\right]}_{Z},\\ y\left(v-1\right)=y\left(v+T\right)+{\sum }_{i=1}^{N}F\left({t}_{i},y\left({t}_{i}\right)\right),\phantom{\rule{1em}{0ex}}0

In , Lv discussed the existence of solutions for discrete fractional boundary value problems with a p-Laplacian operator:

$\left\{\begin{array}{l}{\mathrm{\Delta }}_{c}^{\beta }\left[{\varphi }_{p}\left({\mathrm{\Delta }}_{c}^{\alpha }u\right)\right]\left(t\right)=f\left(t+\alpha +\beta -1,u\left(t+\alpha +\beta -1\right)\right),\phantom{\rule{1em}{0ex}}t\in {\left[0,b\right]}_{{\mathbf{N}}_{0}},\\ {\mathrm{\Delta }}_{c}^{\alpha }u\left(t\right){|}_{t=\beta -1}+{\mathrm{\Delta }}_{c}^{\alpha }u\left(t\right){|}_{t=\beta +b}=0,\\ u\left(\alpha +\beta -2\right)+u\left(\alpha +\beta +b\right)=0,\phantom{\rule{1em}{0ex}}0<\alpha ,\beta \le 1,1<\alpha +\beta \le 2.\end{array}$

They obtained a series of excellent results of discrete fractional boundary value problems. Motivated by the aforementioned works, in this paper we consider a discrete fractional boundary value problem (FBVP):

$\left\{\begin{array}{l}{\mathrm{\Delta }}^{v}y\left(t\right)=f\left(t+v-1,y\left(t+v-1\right)\right),\phantom{\rule{1em}{0ex}}t\in {\left[0,b\right]}_{{\mathbf{N}}_{0}},\\ y\left(v-2\right)=0,\phantom{\rule{2em}{0ex}}\mathrm{\Delta }y\left(v-2\right)=\mathrm{\Delta }y\left(v+b-1\right),\end{array}$
(1.1)

where $1, ${\mathrm{\Delta }}^{v}$ denotes the Riemann-Liouville fractional difference operator, ${\mathbf{N}}_{a}=\left\{a,a+1,a+2,\dots \right\}$ and ${I}_{{\mathbf{N}}_{a}}=I\cap {\mathbf{N}}_{a}$ for any number $a\in \mathbf{R}$ and each interval I of R, $b\in {\mathbf{N}}_{1}$. We appeal to the convention that ${\sum }_{s=k}^{k-1}y\left(s\right)=0$ for any $k\in {\mathbf{N}}_{a}$, where y is a function defined on ${\mathbf{N}}_{a}$. By using the Krasnosel’skii and Schaefer fixed point theorems, the existence results are established and two examples are also provided to illustrate the effectiveness of the main results.

The rest of the paper is organized as follows. In Section 2, we introduce some lemmas and definitions which will be used later. In Section 3, the existence of positive solutions for the boundary value problem (1.1) is investigated. In Section 4, two examples are provided to illustrate the effectiveness of the main results.

## 2 Basic definitions and preliminaries

Firstly we present here some necessary definitions and lemmas which are used throughout this paper.

Definition 2.1 [13, 14]

Define ${t}^{\underline{v}}:=\frac{\mathrm{\Gamma }\left(t+1\right)}{\mathrm{\Gamma }\left(t+1-v\right)}$ for any t and v for which the right-hand side is defined. If $t+1-v$ is a pole of the gamma function and $t+1$ is not a pole, then ${t}^{\underline{v}}=0$.

Definition 2.2 

The v th fractional sum of a function f, for $v>0$, is defined to be

${\mathrm{\Delta }}^{-v}f\left(t\right)={\mathrm{\Delta }}^{-v}f\left(t;a\right):=\frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=a}^{t-v}{\left(t-s-1\right)}^{\underline{v-1}}f\left(s\right)$
(2.1)

for $t\in \left\{a+v,a+v+1,\dots \right\}:={\mathbf{N}}_{a+v}$. Define the v th fractional difference for $v>0$ by ${\mathrm{\Delta }}^{v}f\left(t\right):={\mathrm{\Delta }}^{N}{\mathrm{\Delta }}^{v-N}f\left(t\right)$, $t\in {\mathbf{N}}_{a+v}$ and $N\in \mathbf{N}$ satisfies $0\le N-1.

Lemma 2.3 

Let t and v be any numbers for which ${t}^{\underline{v}}$ and ${t}^{\underline{v-1}}$ are defined. Then $\mathrm{\Delta }{t}^{\underline{v}}=v{t}^{\underline{v-1}}$.

Lemma 2.4 

Assume that $0\le N-1. Then

${\mathrm{\Delta }}^{-v}{\mathrm{\Delta }}^{v}y\left(t\right)=y\left(t\right)+{C}_{1}{t}^{\underline{v-1}}+{C}_{2}{t}^{\underline{v-2}}+\cdots +{C}_{N}{t}^{\underline{v-N}}$
(2.2)

for some ${C}_{i}\in \mathbf{R}$, with $1\le i\le N$.

Lemma 2.5 (The nonlinear alternative of Leray and Schauder )

Let E be a Banach space with $C\subseteq \mathbf{E}$ closed and convex. Let U be a relatively open subset of C with $0\in U$ and $T:\overline{U}\to C$ be a continuous and compact mapping. Then either

1. (a)

the mapping T has a fixed point in $\overline{U}$; or

2. (b)

there exist $u\in \partial U$ and $\lambda \in \left(0,1\right)$ with $u=\lambda Tu$.

Lemma 2.6 

Let B be a Banach space and let $\mathbf{K}\subseteq \mathbf{B}$ be a cone. Assume that ${\mathrm{\Omega }}_{1}$ and ${\mathrm{\Omega }}_{2}$ are bounded open sets contained in B such that $0\in {\mathrm{\Omega }}_{1}$ and ${\overline{\mathrm{\Omega }}}_{1}\subseteq {\mathrm{\Omega }}_{2}$. Assume further that $T:\mathbf{K}\cap \left({\overline{\mathrm{\Omega }}}_{2}\setminus {\mathrm{\Omega }}_{1}\right)\to \mathbf{K}$ is a completely continuous operator. If either

1. (i)

$\parallel Ty\parallel \le \parallel y\parallel$ for $y\in \mathbf{K}\cap \partial {\mathrm{\Omega }}_{1}$ and $\parallel Ty\parallel \ge \parallel y\parallel$ for $y\in \mathbf{K}\cap \partial {\mathrm{\Omega }}_{2}$; or

2. (ii)

$\parallel Ty\parallel \ge \parallel y\parallel$ for $y\in \mathbf{K}\cap \partial {\mathrm{\Omega }}_{1}$ and $\parallel Ty\parallel \le \parallel y\parallel$ for $y\in \mathbf{K}\cap \partial {\mathrm{\Omega }}_{2}$;

then T has at least one fixed point in $\mathbf{K}\cap \left({\overline{\mathrm{\Omega }}}_{2}\setminus {\mathrm{\Omega }}_{1}\right)$.

We state next the structural assumptions that we impose on (1.1).

(H1) Assume that the nonlinearity function $f:{\left[v-1,v+b-1\right]}_{{\mathbf{N}}_{v-1}}×\mathbf{R}\to \left[0,+\mathrm{\infty }\right)$ is continuous.

(H2) Assume that there exist nonnegative continuous functions ${a}_{1}\left(t\right)$, ${a}_{2}\left(t\right)$, $t\in {\left[v-1,v+b-1\right]}_{{\mathbf{N}}_{v-1}}$ such that $|f\left(t,y\right)|\le {a}_{1}\left(t\right)+{a}_{2}\left(t\right)|y|$, $\mathrm{\forall }t\in {\left[v-1,v+b-1\right]}_{{\mathbf{N}}_{v-1}}$, $y\in \mathbf{R}$.

(H3) Assume that ${lim}_{y\to {0}^{+}}\frac{f\left(t,y\right)}{y}=0$ uniformly for $t\in {\left[v-1,v+b-1\right]}_{{\mathbf{N}}_{v-1}}$.

(H4) Assume that ${lim}_{y\to +\mathrm{\infty }}\frac{f\left(t,y\right)}{y}=+\mathrm{\infty }$ uniformly for $t\in {\left[v-1,v+b-1\right]}_{{\mathbf{N}}_{v-1}}$.

## 3 Existence results

In this section, we will establish the existence of at least one positive solution for problem (1.1). At first, we state and prove some preliminary lemmas.

Lemma 3.1 Let $h:{\left[v-1,v+b-1\right]}_{{\mathbf{N}}_{v-1}}\to \mathbf{R}$ be given. Then the unique solution of the discrete fractional boundary value problem

$\left\{\begin{array}{l}{\mathrm{\Delta }}^{v}y\left(t\right)=h\left(t+v-1\right),\phantom{\rule{1em}{0ex}}t\in {\left[0,b\right]}_{{\mathbf{N}}_{0}},\\ y\left(v-2\right)=0,\phantom{\rule{2em}{0ex}}\mathrm{\Delta }y\left(v-2\right)=\mathrm{\Delta }y\left(v+b-1\right),\end{array}$
(3.1)

is

$y\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}G\left(t,s\right)h\left(s+v-1\right).$
(3.2)

Here, for $\left(t,s\right)\in {\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}×{\left[0,b\right]}_{{\mathbf{N}}_{0}}$, $G\left(t,s\right)$ is defined by

$G\left(t,s\right)=\left\{\begin{array}{ll}\frac{{\left(b+v-s-2\right)}^{\underline{v-2}}{t}^{\underline{v-1}}}{\mathrm{\Gamma }\left(v-1\right)-{\left(b+v-1\right)}^{\underline{v-2}}}+{\left(t-s-1\right)}^{\underline{v-1}},& 0\le s\le t-v\le b,\\ \frac{{\left(b+v-s-2\right)}^{\underline{v-2}}{t}^{\underline{v-1}}}{\mathrm{\Gamma }\left(v-1\right)-{\left(b+v-1\right)}^{\underline{v-2}}},& 0\le t-v
(3.3)

Proof Suppose that $y\left(t\right)$ defined on ${\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}$ is a solution of (3.1). Using Lemma 2.4, for some constants ${C}_{1},{C}_{2}\in \mathbf{R}$, we have

$y\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{t-v}{\left(t-s-1\right)}^{\underline{v-1}}h\left(s+v-1\right)+{C}_{1}{t}^{\underline{v-1}}+{C}_{2}{t}^{\underline{v-2}},\phantom{\rule{1em}{0ex}}t\in {\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}.$
(3.4)

By $y\left(v-2\right)=0$ and Definition 2.1, we obtain ${C}_{2}=0$.

Then, for all $t\in {\left[v-2,v+b-1\right]}_{{\mathbf{N}}_{v-2}}$, we obtain 

$\mathrm{\Delta }y\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(v-1\right)}\sum _{s=0}^{t-\left(v-1\right)}{\left(t-s-1\right)}^{\underline{v-2}}h\left(s+v-1\right)+{C}_{1}\left(v-1\right){t}^{\underline{v-2}}.$
(3.5)

In view of $\mathrm{\Delta }y\left(v-2\right)=\mathrm{\Delta }y\left(v+b-1\right)$, we have

${C}_{1}=\frac{1}{\mathrm{\Gamma }\left(v-1\right)\left[\mathrm{\Gamma }\left(v\right)-\left(v-1\right){\left(b+v-1\right)}^{\underline{v-2}}\right]}\sum _{s=0}^{b}{\left(b+v-s-2\right)}^{\underline{v-2}}h\left(s+v-1\right).$

Substituting the values of ${C}_{1}$ and ${C}_{2}$ in (3.4), we have

$\begin{array}{rcl}y\left(t\right)& =& \frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{t-v}{\left(t-s-1\right)}^{\underline{v-1}}h\left(s+v-1\right)\\ +\sum _{s=0}^{b}\frac{{\left(b+v-s-2\right)}^{\underline{v-2}}{t}^{\underline{v-1}}}{\mathrm{\Gamma }\left(v-1\right)\left[\mathrm{\Gamma }\left(v\right)-\left(v-1\right){\left(b+v-1\right)}^{\underline{v-2}}\right]}h\left(s+v-1\right)\\ =& \frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}G\left(t,s\right)h\left(s+v-1\right),\phantom{\rule{1em}{0ex}}t\in {\left[v-2,v+b-1\right]}_{{\mathbf{N}}_{v-2}}.\end{array}$

□

Lemma 3.2 The function $G\left(t,s\right)$ given in (3.3) satisfies the following:

1. (1)

$0\le G\left(t,s\right)\le \frac{D{\left(v+b\right)}^{\underline{v-1}}}{{\left(s+v-1\right)}^{\underline{v-1}}}G\left(s+v-1,s\right)$, $\left(t,s\right)\in {\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}×{\left[0,b\right]}_{{\mathbf{N}}_{0}}$;

2. (2)

${min}_{t\in {\left[v-1,v+b\right]}_{{\mathbf{N}}_{v-1}}}G\left(t,s\right)\ge \frac{\mathrm{\Gamma }\left(v\right)}{{\left(s+v-1\right)}^{\underline{v-1}}}G\left(s+v-1,s\right)>0$.

Here,

$\begin{array}{rcl}D& =& \underset{s\in \left[0,b\right]}{max}\left\{1+\frac{\mathrm{\Gamma }\left(v-1\right)-{\left(v+b-1\right)}^{\underline{v-2}}}{{\left(v+b-s-2\right)}^{\underline{v-2}}}\right\}\\ =& 1+\frac{\mathrm{\Gamma }\left(v-1\right)-{\left(v+b-1\right)}^{\underline{v-2}}}{{\left(v+b-2\right)}^{\underline{v-2}}}.\end{array}$
(3.6)

Proof First of all, (3.3) implies that $G\left(s+v-1,s\right)=\frac{{\left(v+b-s-2\right)}^{\underline{v-2}}{\left(s+v-1\right)}^{\underline{v-1}}}{\mathrm{\Gamma }\left(v-1\right)-{\left(v+b-1\right)}^{\underline{v-2}}}$. Note that $\mathrm{\Gamma }\left(v-1\right)-{\left(v+b-1\right)}^{\underline{v-2}}>0$, we know $G\left(s+v-1,s\right)>0$.

Second of all, by (3.3) and the definition of D in (3.6), we obtain

$\begin{array}{rcl}0& \le & G\left(t,s\right)\le \frac{{\left(b+v-s-2\right)}^{\underline{v-2}}{\left(v+b\right)}^{\underline{v-1}}}{\mathrm{\Gamma }\left(v-1\right)-{\left(b+v-1\right)}^{\underline{v-2}}}+{\left(v+b\right)}^{\underline{v-1}}\\ \le & \frac{D{\left(b+v\right)}^{\underline{v-1}}}{{\left(s+v-1\right)}^{\underline{v-1}}}G\left(s+v-1,s\right).\end{array}$
(3.7)

On the other hand,

$\begin{array}{rcl}\underset{t\in {\left[v-1,v+b\right]}_{{\mathbf{N}}_{v-1}}}{min}G\left(t,s\right)& =& \underset{t\in {\left[v-1,v+b\right]}_{{\mathbf{N}}_{v-1}}}{min}\frac{{\left(b+v-s-2\right)}^{\underline{v-2}}{t}^{\underline{v-1}}}{\mathrm{\Gamma }\left(v-1\right)-{\left(b+v-1\right)}^{\underline{v-2}}}\\ \ge & \frac{{\left(v-1\right)}^{\underline{v-1}}}{{\left(s+v-1\right)}^{\underline{v-1}}}\cdot \frac{{\left(b+v-s-2\right)}^{\underline{v-2}}{\left(s+v-1\right)}^{\underline{v-1}}}{\mathrm{\Gamma }\left(v-1\right)-{\left(v+b-1\right)}^{\underline{v-2}}}\\ =& \frac{\mathrm{\Gamma }\left(v\right)}{{\left(s+v-1\right)}^{\underline{v-1}}}G\left(s+v-1,s\right)>0.\end{array}$
(3.8)

The proof of Lemma 3.2 is completed. □

Let B be the collection of all functions $y:{\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}\to \mathbf{R}$ with the norm $\parallel y\parallel =max\left\{|y\left(t\right)|:t\in {\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}\right\}$.

Define the operator $T:\mathbf{B}\to \mathbf{B}$ by

$Ty\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}G\left(t,s\right)f\left(s+v-1,y\left(s+v-1\right)\right).$
(3.9)

In view of the continuity of f, it is easy to know that T is continuous. Furthermore, it is not difficult to verify that T maps bounded sets into bounded sets and equi-continuous sets. Therefore, in the light of the well-known Arzelá-Ascoli theorem, we know that T is a compact operator (see [11, 12]).

Let $\mathbf{E}=\left\{y\in \mathbf{B}|y\left(t\right)\ge 0,t\in {\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}\right\}$ and set

$\begin{array}{c}A=\sum _{s=0}^{b}\frac{D{\left(v+b\right)}^{\underline{v-1}}G\left(s+v-1,s\right)}{\mathrm{\Gamma }\left(v\right){\left(s+v-1\right)}^{\underline{v-1}}}\parallel {a}_{2}\parallel ,\hfill \\ B=\sum _{s=0}^{b}\frac{D{\left(v+b\right)}^{\underline{v-1}}G\left(s+v-1,s\right)}{\mathrm{\Gamma }\left(v\right){\left(s+v-1\right)}^{\underline{v-1}}}\parallel {a}_{1}\parallel .\hfill \end{array}$

We have the following theorem.

Theorem 3.3 Assume that (H1) and (H2) hold. Then system (1.1) has at least one positive solution provided that

$\sum _{s=0}^{b}\frac{D{\left(v+b\right)}^{\underline{v-1}}G\left(s+v-1,s\right)}{\mathrm{\Gamma }\left(v\right){\left(s+v-1\right)}^{\underline{v-1}}}\parallel {a}_{2}\parallel <1.$
(3.10)

Proof Let $\mathrm{\Omega }=\left\{y\in \mathbf{E}|\parallel y\parallel with $r=\frac{B}{1-A}>0$. If $y\in \overline{\mathrm{\Omega }}$, that is, $\parallel y\parallel \le r$. From (H1), (H2) and (3.9), we have

$\begin{array}{rcl}\parallel Ty\left(t\right)\parallel & =& \underset{t\in {\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}}{max}|\frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}G\left(t,s\right)f\left(s+v-1,y\left(s+v-1\right)\right)|\\ \le & \frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}\frac{D{\left(v+b\right)}^{\underline{v-1}}}{{\left(s+v-1\right)}^{\underline{v-1}}}G\left(s+v-1,s\right)\left(|{a}_{1}\left(t\right)|+|{a}_{2}\left(t\right)||y\left(t\right)|\right)\\ \le & \sum _{s=0}^{b}\frac{{\left(v+b\right)}^{\underline{v-1}}}{\mathrm{\Gamma }\left(v\right)}\frac{DG\left(s+v-1,s\right)}{{\left(s+v-1\right)}^{\underline{v-1}}}\parallel {a}_{1}\parallel \\ +\sum _{s=0}^{b}\frac{{\left(v+b\right)}^{\underline{v-1}}}{\mathrm{\Gamma }\left(v\right)}\frac{DG\left(s+v-1,s\right)}{{\left(s+v-1\right)}^{\underline{v-1}}}\parallel {a}_{2}\parallel \parallel y\parallel \\ =& B+A\parallel y\parallel \le r,\end{array}$

which shows that $Ty\in \overline{\mathrm{\Omega }}$.

Consider the eigenvalue problem

$y=\lambda Ty,\phantom{\rule{1em}{0ex}}\lambda \in \left(0,1\right).$
(3.11)

Assume that y is a solution of (3.11), we obtain

$\parallel y\parallel =\parallel \lambda Ty\parallel <\parallel Ty\parallel \le r.$
(3.12)

It shows that $y\notin \partial \mathrm{\Omega }$. By Lemma 2.5, T has a fixed point in $\overline{\mathrm{\Omega }}$. The proof is completed. □

We define the cone $\mathbf{K}\subseteq \mathbf{B}$ by

$\mathbf{K}=\left\{y\in \mathbf{E}:\underset{t\in {\left[v-1,v+b\right]}_{{\mathbf{N}}_{v-1}}}{min}y\left(t\right)\ge \frac{\mathrm{\Gamma }\left(v\right)}{D{\left(v+b\right)}^{\underline{v-1}}}\parallel y\parallel \right\}.$
(3.13)

Lemma 3.4 Let T be the operator defined in (3.9) and K be the cone defined in (3.13). Then $T:\mathbf{K}\to \mathbf{K}$.

Proof Note that for each $t\in {\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}$, we have

$\begin{array}{rcl}\parallel Ty\left(t\right)\parallel & =& \underset{t\in {\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}}{max}|\frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}G\left(t,s\right)f\left(s+v-1,y\left(s+v-1\right)\right)|\\ \le & \frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}\frac{D{\left(v+b\right)}^{\underline{v-1}}}{{\left(s+v-1\right)}^{\underline{v-1}}}G\left(s+v-1,s\right)|f\left(s+v-1,y\left(s+v-1\right)\right)|\\ =& \sum _{s=0}^{b}\frac{D{\left(v+b\right)}^{\underline{v-1}}G\left(s+v-1,s\right)}{\mathrm{\Gamma }\left(v\right){\left(s+v-1\right)}^{\underline{v-1}}}|f\left(s+v-1,y\left(s+v-1\right)\right)|.\end{array}$

Therefore, it holds that

$\begin{array}{r}\underset{t\in {\left[v-1,v+b\right]}_{{\mathbf{N}}_{v-1}}}{min}\left(Ty\right)\left(t\right)\\ \phantom{\rule{1em}{0ex}}=\underset{t\in {\left[v-1,v+b\right]}_{{\mathbf{N}}_{v-1}}}{min}\frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}G\left(t,s\right)f\left(s+v-1,y\left(s+v-1\right)\right)\\ \phantom{\rule{1em}{0ex}}\ge \frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}\frac{\mathrm{\Gamma }\left(v\right)}{{\left(s+v-1\right)}^{\underline{v-1}}}G\left(s+v-1,s\right)f\left(s+v-1,y\left(s+v-1\right)\right)\\ \phantom{\rule{1em}{0ex}}=\frac{\mathrm{\Gamma }\left(v\right)}{D{\left(v+b\right)}^{\underline{v-1}}}\sum _{s=0}^{b}\frac{D{\left(v+b\right)}^{\underline{v-1}}G\left(s+v-1,s\right)}{\mathrm{\Gamma }\left(v\right){\left(s+v-1\right)}^{\underline{v-1}}}f\left(s+v-1,y\left(s+v-1\right)\right)\\ \phantom{\rule{1em}{0ex}}\ge \frac{\mathrm{\Gamma }\left(v\right)}{D{\left(v+b\right)}^{\underline{v-1}}}\parallel Ty\parallel .\end{array}$

The conclusion of Lemma 3.4 holds. □

Theorem 3.5 Suppose that conditions (H1), (H3) and (H4) hold. Then problem (1.1) has at least one positive solution.

Proof We have already shown $T\left(\mathbf{K}\right)\subseteq \mathbf{K}$ in Lemma 3.4. By condition (H3), we can select ${\eta }_{1}>0$ sufficiently small so that both $|f\left(t,y\right)|\le {\eta }_{1}\parallel y\parallel$ and ${\eta }_{1}{\sum }_{s=0}^{b}\frac{{\left(v+b\right)}^{\underline{v-1}}}{\mathrm{\Gamma }\left(v\right)}\frac{DG\left(s+v-1,s\right)}{{\left(s+v-1\right)}^{\underline{v-1}}}<1$ hold for all $t\in {\left[v-1,v+b-1\right]}_{{\mathbf{N}}_{v-1}}$ and $0, where ${r}_{1}:={r}_{1}\left({\eta }_{1}\right)$.

Let ${\mathrm{\Omega }}_{1}=\left\{y\in \mathbf{B}:\parallel y\parallel <{r}_{1}\right\}$. Then, for $y\in \partial {\mathrm{\Omega }}_{1}\cap \mathbf{K}$, we have

$\begin{array}{rl}\parallel Ty\left(t\right)\parallel & =\underset{t\in {\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}}{max}|\frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}G\left(t,s\right)f\left(s+v-1,y\left(s+v-1\right)\right)|\\ \le \frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}\frac{D{\left(v+b\right)}^{\underline{v-1}}}{{\left(s+v-1\right)}^{\underline{v-1}}}G\left(s+v-1,s\right)|f\left(s+v-1,y\left(s+v-1\right)\right)|\\ \le \sum _{s=0}^{b}\frac{D{\left(v+b\right)}^{\underline{v-1}}G\left(s+v-1,s\right)}{\mathrm{\Gamma }\left(v\right){\left(s+v-1\right)}^{\underline{v-1}}}{\eta }_{1}\parallel y\parallel <\parallel y\parallel .\end{array}$

It implies that T is a cone contraction on $y\in \partial {\mathrm{\Omega }}_{1}\cap \mathbf{K}$.

On the other hand, from condition (H4), we may select a number ${\eta }_{2}>0$ such that both $|f\left(t,y\right)|>{\eta }_{2}\parallel y\parallel$ and ${\eta }_{2}{\sum }_{s=0}^{b}\frac{G\left(s+v-1,s\right)}{{\left(s+v-1\right)}^{\underline{v-1}}}>1$ hold for all $t\in {\left[v-1,v+b-1\right]}_{{\mathbf{N}}_{v-1}}$ and $0, where ${r}_{2}:={r}_{2}\left({\eta }_{2}\right)$ and ${r}_{2}>{r}_{1}>0$. Define ${\mathrm{\Omega }}_{2}=\left\{y\in \mathbf{B}:\parallel y\parallel <{r}_{2}\right\}$, we obtain

$\begin{array}{rcl}\parallel Ty\left(t\right)\parallel & =& \underset{t\in {\left[v-2,v+b\right]}_{{\mathbf{N}}_{v-2}}}{max}\frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}G\left(t,s\right)|f\left(s+v-1,y\left(s+v-1\right)\right)|\\ \ge & \frac{1}{\mathrm{\Gamma }\left(v\right)}\sum _{s=0}^{b}\underset{t\in {\left[v-1,v+b\right]}_{{\mathbf{N}}_{v-1}}}{min}G\left(t,s\right)|f\left(s+v-1,y\left(s+v-1\right)\right)|\\ >& \sum _{s=0}^{b}\frac{G\left(s+v-1,s\right)}{{\left(s+v-1\right)}^{\underline{v-1}}}{\eta }_{2}\parallel y\parallel >\parallel y\parallel ,\end{array}$

whenever $y\in \partial {\mathrm{\Omega }}_{2}\cap \mathbf{K}$, so that T is a cone expansion on $\partial {\mathrm{\Omega }}_{2}\cap \mathbf{K}$.

In summary, we may invoke Lemma 2.6 to deduce the existence of a function ${y}_{0}\in \mathbf{K}\cap \left({\overline{\mathrm{\Omega }}}_{2}\setminus {\mathrm{\Omega }}_{1}\right)$ such that $T{y}_{0}={y}_{0}$, where ${y}_{0}$ is a positive solution to problem (1.1). The proof is completed. □

## 4 Example

Example 4.1 Consider the fractional difference boundary value problem

$\left\{\begin{array}{l}{\mathrm{\Delta }}^{\frac{3}{2}}y\left(t\right)=f\left(t+\frac{1}{2},y\left(t+\frac{1}{2}\right)\right),\phantom{\rule{1em}{0ex}}t\in {\left[0,3\right]}_{{\mathbf{N}}_{0}},\\ y\left(-\frac{1}{2}\right)=0,\phantom{\rule{2em}{0ex}}\mathrm{\Delta }y\left(-\frac{1}{2}\right)=\mathrm{\Delta }y\left(\frac{7}{2}\right).\end{array}$
(4.1)

Set ${a}_{1}\left(t\right)=1$, ${a}_{2}\left(t\right)=\frac{t}{30}$, $f\left(t,y\right)=\frac{t|y|}{30}+|sint|$, $t\in {\left[\frac{1}{2},\frac{7}{2}\right]}_{{\mathbf{N}}_{\frac{1}{2}}}$. We have

$|f\left(t,y\right)|\le 1+\frac{t}{30}|y|.$

By a simple computation, we can obtain $D\approx 1.7750$, ${\sum }_{s=0}^{3}\frac{{\left(\frac{9}{2}\right)}^{\underline{\frac{1}{2}}}G\left(s+\frac{1}{2},s\right)}{\mathrm{\Gamma }\left(\frac{3}{2}\right){\left(s+\frac{1}{2}\right)}^{\underline{\frac{1}{2}}}}\approx 4.7688$, $\parallel {a}_{2}\parallel =\frac{7}{60}$. Therefore, $A\approx 0.9875<1$. The conditions of Theorem 3.3 hold, the boundary value problem (4.1) has at least one positive solution.

Example 4.2 Consider the fractional difference boundary value problem

$\left\{\begin{array}{l}{\mathrm{\Delta }}^{\frac{3}{2}}y\left(t\right)=f\left(t+\frac{1}{2},y\left(t+\frac{1}{2}\right)\right),\phantom{\rule{1em}{0ex}}t\in {\left[0,11\right]}_{{\mathbf{N}}_{0}},\\ y\left(-\frac{1}{2}\right)=0,\phantom{\rule{2em}{0ex}}\mathrm{\Delta }y\left(-\frac{1}{2}\right)=\mathrm{\Delta }y\left(\frac{23}{2}\right).\end{array}$
(4.2)

Set $f\left(t,y\right)=t{y}^{2}$, $t\in {\left[\frac{1}{2},\frac{23}{2}\right]}_{{\mathbf{N}}_{\frac{1}{2}}}$. We have

$\left(1\right)\phantom{\rule{1em}{0ex}}\underset{y\to {0}^{+}}{lim}\frac{f\left(t,y\right)}{y}=\underset{y\to {0}^{+}}{lim}\frac{t{y}^{2}}{y}=0,\phantom{\rule{2em}{0ex}}\left(2\right)\phantom{\rule{1em}{0ex}}\underset{y\to +\mathrm{\infty }}{lim}\frac{f\left(t,y\right)}{y}=\underset{y\to +\mathrm{\infty }}{lim}\frac{t{y}^{2}}{y}=+\mathrm{\infty }.$

The conditions of Theorem 3.5 hold, the boundary value problem (4.2) has at least one positive solution.

## Misc

Equal contributors

## References

1. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, Yverdon; 1993.

2. Podlubny I Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, New York; 1999.

3. 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.2005.02.052

4. Zhang SQ: Positive solutions for boundary value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006., 2006: Article ID 36

5. Jafari H, Gejji VD: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 2006, 180: 700-706. 10.1016/j.amc.2006.01.007

6. Dehghani R, Ghanbari K: Triple positive solutions for boundary value problem of a nonlinear fractional differential equation. Bull. Iran. Math. Soc. 2007, 33(2):1-14.

7. Wang JH, Xiang HJ, Liu ZG: Positive solutions for three-point boundary value problems of nonlinear fractional differential equations with P -Laplacian. Far East J. Appl. Math. 2009, 37(1):33-47.

8. Wang JH, Xiang HJ, Liu ZG: Upper and lower solutions method for a class of singular fractional boundary value problems with P -Laplacian. Abstr. Appl. Anal. 2010., 2010: Article ID 971824

9. Cai G: Positive solutions for boundary value problems of fractional differential equations with P -Laplacian operator. Bound. Value Probl. 2012., 2012: Article ID 18

10. Atici FM, Eloe PW: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 2011, 17: 445-456. 10.1080/10236190903029241

11. Goodrich CS: On a first-order semipositone discrete fractional boundary value problem. Arch. Math. 2012, 99: 509-518. 10.1007/s00013-012-0463-2

12. Lv WD: Existence of solutions for discrete fractional boundary value problems with a P -Laplacian operator. Adv. Differ. Equ. 2012., 2012: Article ID 163

13. Goodrich CS: Existence of a positive solution to a system of discrete fractional boundary value problems. Appl. Math. Comput. 2011, 217: 4740-4753. 10.1016/j.amc.2010.11.029

14. Atici FM, Eloe PW: A transform method in discrete fractional calculus. Int. J. Differ. Equ. 2007, 2(2):165-176.

15. Goodrich CS: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 2012, 385: 111-124. 10.1016/j.jmaa.2011.06.022

16. Goodrich CS: Positive solutions to boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 2012, 75: 417-432. 10.1016/j.na.2011.08.044

17. Goodrich CS: On discrete fractional three-point boundary value problems. J. Differ. Equ. Appl. 2012, 18: 397-415. 10.1080/10236198.2010.503240

18. Goodrich CS: Nonlocal systems of BVPs with asymptotically superlinear boundary conditions. Comment. Math. Univ. Carol. 2012, 53: 79-97.

19. Atici FM, Sengul S: Modeling with fractional difference equations. J. Math. Anal. Appl. 2010, 369: 1-9. 10.1016/j.jmaa.2010.02.009

20. Zeidler E: Nonlinear Functional Analysis and Applications, I: Fixed Point Theorems. Springer, New York; 1986.

21. Atici FM, Eloe PW: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 2009, 137: 981-989.

## Acknowledgements

This work was jointly supported by the Natural Science Foundation of China under Grants 11471278, the Natural Science Foundation of Hunan Province under Grants 13JJ3120 and 14JJ2133, and the Construct Program of the Key Discipline in Hunan Province. We would like to show our great thanks to the anonymous referee for his/her valuable suggestions and comments, which improve the former version of this paper and make us rewrite the paper in a more clear way.

## Author information

Authors

### Corresponding author

Correspondence to Hongjun Xiang.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

## Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions 