Skip to main content

Advertisement

We’d like to understand how you use our websites in order to improve them. Register your interest.

Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions

Abstract

This paper is concerned with the following boundary value problem of nonlinear fractional differential equation with integral boundary conditions:

$$ \textstyle\begin{cases} ({}^{C}D_{0+}^{q}u)(t)+f(t,u(t))=0,\quad t\in [0,1], \\ u^{\prime \prime }(0)=0, \\ \alpha u(0)-\beta u^{\prime }(0)=\int _{0}^{1}h_{1}(s)u(s)\,ds, \\ \gamma u(1)+\delta ({}^{C}D_{0+}^{\sigma }u)(1) =\int _{0}^{1}h_{2}(s)u(s)\,ds, \end{cases} $$

where \(2< q\leq 3\), \(0<\sigma \leq 1\), \(\alpha , \gamma , \delta \geq 0\), and \(\beta >0\) satisfying \(0<\rho :=(\alpha +\beta )\gamma + \frac{\alpha \delta }{\varGamma (2-\sigma )}<\beta [\gamma + \frac{\delta \varGamma (q)}{\varGamma (q-\sigma )} ]\). \({}^{C}D_{0+}^{q}\) denotes the standard Caputo fractional derivative. First, Green’s function of the corresponding linear boundary value problem is constructed. Next, some useful properties of the Green’s function are obtained. Finally, existence results of at least one positive solution for the above problem are established by imposing some suitable conditions on f and \(h_{i}\) (\(i=1,2\)). The method employed is Guo–Krasnoselskii’s fixed point theorem. An example is also included to illustrate the main results of this paper.

Introduction

Fractional calculus has widespread applications in many fields of science and engineering, for example, physics, viscoelasticity, continuum mechanics, bioengineering, rheology, electrical networks, control theory of dynamical systems, optics and signal processing, and so on [1, 2].

Since the discussion of many problems can be summed up in the study of boundary value problems (BVPs for short) to nonlinear fractional differential equations, recently, the existence of solutions or positive solutions of BVPs for nonlinear fractional differential equations has received considerable attention from many authors, see [326] and the references therein.

In particular, in 2009, by using nonlinear alternative of Leray–Schauder type and Guo–Krasnoselskii’s fixed point theorem, Bai and Qiu [5] obtained the existence of a positive solution to the singular BVP

$$ \textstyle\begin{cases} ({}^{C}D_{0+}^{q}u)(t)+f(t,u(t))=0, \quad t\in (0,1), \\ u(0)=u^{\prime }(1)=u^{\prime \prime }(0)=0, \end{cases} $$
(1)

where \(2< q\leq 3 \) is a real number, \(f:(0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous, and \(\lim _{t\rightarrow 0^{+}}f(t,\cdot )=+\infty \).

In 2012, Cabada and Wang [7] studied the existence of a positive solution for the BVP with integral boundary conditions

$$ \textstyle\begin{cases} ({}^{C}D_{0+}^{q}u)(t)+f(t,u(t))=0, \quad t\in (0,1), \\ u(0)=u^{\prime \prime }(0)=0,\quad\quad u(1)=\lambda \int ^{1}_{0}u(s)\,ds, \end{cases} $$
(2)

where \(2< q<3\), \(0<\lambda <2\), and \(f:[0,1] \times [0,+\infty )\rightarrow [0,+\infty )\) is continuous. Their analysis relied on Guo–Krasnoselskii’s fixed point theorem.

In 2014, Cabada and Hamdi [25] investigated the BVP with integral boundary conditions

$$ \textstyle\begin{cases} (D_{0+}^{q}u)(t)+f(t,u(t))=0,\quad t\in (0,1), \\ u(0)=u^{\prime }(0)=0,\quad\quad u(1)=\lambda \int _{0}^{1} u(s)\,ds, \end{cases} $$
(3)

where \(2< q\leq 3\), \(D_{0+}^{q}\) denotes the Riemann–Liouville fractional derivative, \(0<\lambda <q\), and \(f:[0,1] \times [0,+\infty )\rightarrow [0,+\infty )\) is a continuous function. The authors proved the existence of a positive solution to BVP (3) by employing Guo–Krasnoselskii’s fixed point theorem.

As it has been stated in [7], BVPs with integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermo-elasticity, underground water flow, population dynamics, and so forth. Motivated by the above-mentioned works, in this paper, we consider the existence of a positive solution for the following BVP of nonlinear fractional differential equation with integral boundary conditions:

$$ \textstyle\begin{cases} ({}^{C}D_{0+}^{q}u)(t)+f(t,u(t))=0, \quad t\in [0,1], \\ u^{\prime \prime }(0)=0, \\ \alpha u(0)-\beta u^{\prime }(0)=\int _{0}^{1}h_{1}(s)u(s)\,ds, \\ \gamma u(1)+\delta ({}^{C}D_{0+}^{\sigma }u)(1) =\int _{0}^{1}h_{2}(s)u(s)\,ds. \end{cases} $$
(4)

Throughout this paper, we always assume that \(2< q\leq 3\), \(0<\sigma \leq 1\), \(\alpha , \gamma , \delta \geq 0\), and \(\beta >0\) satisfying \(0<\rho :=(\alpha +\beta )\gamma + \frac{\alpha \delta }{\varGamma (2-\sigma )}<\beta [\gamma + \frac{\delta \varGamma (q)}{\varGamma (q-\sigma )} ]\), \(f :[0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) and \(h_{i}\) (\(i=1,2\)): \([0,1] \rightarrow [0,+\infty )\) are continuous.

The main tool used is the following well-known Guo–Krasnoselskii’s fixed point theorem [27, 28].

Theorem 1.1

LetEbe a Banach space andKbe a cone in E. Assume that\(\varOmega _{1}\)and\(\varOmega _{2}\)are bounded open subsets ofEsuch that\(0 \in \varOmega _{1}\), \(\overline{\varOmega }_{1}\subset \varOmega _{2}\), and let\(T:K\cap (\overline{\varOmega }_{2}\backslash \varOmega _{1} ) \rightarrow K\)be a completely continuous operator such that either

  1. (1)

    \(\Vert Tu \Vert \leq \Vert u \Vert \)for\(u\in K\cap \partial \varOmega _{1}\)and\(\Vert Tu \Vert \geq \Vert u \Vert \)for\(u\in K\cap \partial \varOmega _{2}\), or

  2. (2)

    \(\Vert Tu \Vert \geq \Vert u \Vert \)for\(u\in K\cap \partial \varOmega _{1}\)and\(\Vert Tu \Vert \leq \Vert u \Vert \)for\(u\in K\cap \partial \varOmega _{2}\).

ThenThas a fixed point in\(K\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1} )\).

Preliminaries

Let \([a,b] \) (\(-\infty < a< b<+\infty \)) be a finite interval on the real axis \(\mathbb{R}\), \(\mathbb{N}=\{1, 2, 3,\ldots\}\), \(\mu >0\) and \([\mu ]\) be the integer part of μ.

First, we present definitions of some spaces.

Let \(\operatorname{AC}[a,b]\) be the space of functions u which are absolutely continuous on \([a,b]\). For \(n\in \mathbb{N}\), we denote by \(\operatorname{AC}^{n}[a,b]\) the space of functions u which have continuous derivatives up to order \(n-1\) on \([a,b]\) such that \(u^{(n-1)}\in \operatorname{AC}[a,b]\). In particular, \(\operatorname{AC}^{1}[a,b]=\operatorname{AC}[a,b]\).

For \(m\in \mathbb{N}_{0}=\{0, 1, 2,\ldots\}\), we denote by \(C^{m}[a,b]\) the space of functions u which are m times continuously differentiable on \([a,b]\). In particular, for \(m=0\), \(C^{0}[a,b]=C[a,b]\) is the space of continuous functions u on \([a,b]\).

Next, we give the definitions of the Riemann–Liouville fractional integrals and fractional derivatives and the Caputo fractional derivatives on \([a,b]\), which may be found in [1].

Definition 2.1

The Riemann–Liouville fractional integrals \(I_{a+}^{\mu }u\) and \(I_{b-}^{\mu }u\) of order μ are defined by

$$ \bigl(I_{a+}^{\mu }u\bigr) (t):=\frac{1}{\varGamma (\mu )} \int _{a}^{t} \frac{u(s)\,ds}{(t-s)^{1-\mu }} \quad (t>a) $$

and

$$ \bigl(I_{b-}^{\mu }u\bigr) (t):=\frac{1}{\varGamma (\mu )} \int _{t}^{b} \frac{u(s)\,ds}{(s-t)^{1-\mu }} \quad (t< b), $$

respectively, where

$$ \varGamma (\mu )= \int _{0}^{+\infty }s^{\mu -1}e^{-s}\,ds. $$

Definition 2.2

The Riemann–Liouville fractional derivatives \(D_{a+}^{\mu }u\) and \(D_{b-}^{\mu }u\) of order μ are defined by

$$ \bigl(D_{a+}^{\mu }u\bigr) (t):= \biggl(\frac{d}{dt} \biggr)^{n}\bigl(I_{a+}^{n-\mu }u\bigr) (t)= \frac{1}{\varGamma (n-\mu )} \biggl(\frac{d}{dt} \biggr)^{n} \int _{a}^{t} \frac{u(s)\,ds}{(t-s)^{\mu -n+1}} \quad (t>a) $$

and

$$ \bigl(D_{b-}^{\mu }u\bigr) (t):= \biggl(-\frac{d}{dt} \biggr)^{n}\bigl(I_{b-}^{n-\mu }u\bigr) (t)= \frac{1}{\varGamma (n-\mu )} \biggl(-\frac{d}{dt} \biggr)^{n} \int _{t}^{b} \frac{u(s)\,ds}{(s-t)^{\mu -n+1}} \quad (t< b), $$

respectively, where \(n=[\mu ]+1\).

Definition 2.3

Let \(D_{a+}^{\mu }[u(s)](t)\equiv (D_{a+}^{\mu }u)(t)\) and \(D_{b-}^{\mu }[u(s)](t)\equiv (D_{b-}^{\mu }u)(t)\) be the Riemann–Liouville fractional derivatives of order μ, respectively. The Caputo fractional derivatives \({}^{C}D_{a+}^{\mu }u\) and \({}^{C}D_{b-}^{\mu }u\) of order μ on \([a,b]\) are defined by

$$ \bigl({}^{C}D_{a+}^{\mu }u\bigr) (t):= \Biggl(D_{a+}^{\mu } \Biggl[u(s)-\sum _{k=0}^{n-1} \frac{u^{(k)}(a)}{k!}(s-a)^{k} \Biggr] \Biggr) (t) $$

and

$$ \bigl({}^{C}D_{b-}^{\mu }u\bigr) (t):= \Biggl(D_{b-}^{\mu } \Biggl[u(s)-\sum _{k=0}^{n-1} \frac{u^{(k)}(b)}{k!}(b-s)^{k} \Biggr] \Biggr) (t), $$

respectively, where

$$ n= \textstyle\begin{cases} [\mu ]+1,&\mu \notin \mathbb{N}, \\ \mu , &\mu \in \mathbb{N}. \end{cases} $$
(5)

Lemma 2.1

(see [2])

Let\(\nu >\mu \). Then the equation\(({}^{C}D_{0+}^{\mu }I_{0+}^{\nu }u)(t)=(I_{0+}^{\nu -\mu }u)(t)\), \(t \in [0,1]\)is satisfied for\(u\in C [0,1 ]\).

Lemma 2.2

(see [1])

Letnbe given by (5). Then the following relations hold:

  1. (1)

    for\(k\in \{0,1,2,\ldots,n-1\}\), \({}^{C}D_{0+}^{\mu }t^{k}=0\);

  2. (2)

    if\(\nu >n\), \({}^{C}D_{0+}^{\mu }t^{\nu -1}= \frac{\varGamma (\nu )}{\varGamma (\nu -\mu )}t^{\nu -\mu -1}\).

Lemma 2.3

(see [1])

Letnbe given by (5). If\(u\in \operatorname{AC}^{n}[0,1]\)or\(u\in C^{n}[0,1]\), then

$$ \bigl(I_{0+}^{\mu }{{}^{C}D_{0+}^{\mu }}u \bigr) (t)=u(t)-\sum_{k=0}^{n-1} \frac{u^{k}(0)}{k!}t^{k}. $$

For convenience, we denote

$$\begin{aligned}& P_{i}=\frac{1}{\rho } \int _{0}^{1}(\alpha s+\beta )h_{i}(s) \,ds \end{aligned}$$

and

$$\begin{aligned}& Q_{i}=\frac{1}{\rho \varGamma (2-\sigma )} \int _{0}^{1}\bigl[\gamma \varGamma (2- \sigma ) (1-s)+\delta \bigr]h_{i}(s)\,ds,\quad i=1,2. \end{aligned}$$

Lemma 2.4

Let\((1-Q_{1})(1-P_{2})\neq P_{1}Q_{2}\). Then, for any\(y\in C [0,1 ]\), the BVP

$$ \textstyle\begin{cases} ({}^{C}D_{0+}^{q}u)(t)+y(t)=0, \quad t\in [0,1], \\ u^{\prime \prime }(0)=0, \\ \alpha u(0)-\beta u^{\prime }(0)=\int _{0}^{1}h_{1}(s)u(s)\,ds, \\ \gamma u(1)+\delta ({}^{C}D_{0+}^{\sigma }u)(1)=\int _{0}^{1}h_{2}(s)u(s)\,ds \end{cases} $$
(6)

has a unique solution

$$ u(t) = \int _{0}^{1} H(t,s) y(s) \,ds,\quad t \in [0,1], $$

here

$$\begin{aligned}& H(t,s)=G(t,s)+\sum_{i=1}^{2}\phi _{i}(t) \int _{0}^{1} G(\tau ,s)h_{i}( \tau )\,d \tau ,\quad (t,s)\in [0,1]\times [0,1], \end{aligned}$$

where

$$\begin{aligned}& G(t,s)=\frac{\alpha t+\beta }{\rho } \biggl[ \frac{\gamma (1-s)^{q-1}}{\varGamma (q)}+ \frac{\delta (1-s)^{q-\sigma -1}}{\varGamma (q-\sigma )} \biggr]- \textstyle\begin{cases} \frac{(t-s)^{q-1}}{\varGamma (q)},& 0\leq s\leq t\leq 1, \\ 0, &0\leq t\leq s\leq 1, \end{cases}\displaystyle \\& \phi _{1}(t)= \frac{\varGamma (2-\sigma )Q_{2}(\alpha t+\beta )+(1-P_{2}) [\gamma \varGamma (2-\sigma )(1-t)+\delta ]}{\rho \varGamma (2-\sigma ) [(1-Q_{1})(1-P_{2})-P_{1}Q_{2} ]}, \quad t\in [0,1], \end{aligned}$$

and

$$\begin{aligned}& \phi _{2}(t)= \frac{\varGamma (2-\sigma )(1-Q_{1})(\alpha t+\beta )+P_{1}[\gamma \varGamma (2-\sigma )(1-t)+\delta ]}{\rho \varGamma (2-\sigma )[(1-Q_{1})(1-P_{2})-P_{1}Q_{2}]}, \quad t\in [0,1]. \end{aligned}$$

Proof

In view of the equation in (6), Lemma 2.3, and \(u^{\prime \prime }(0)=0\), we have

$$ \begin{aligned} u(t)=-\bigl(I^{q}_{0+}y\bigr) (t)+u(0)+u^{\prime }(0)t,\quad t\in [0,1]. \end{aligned} $$
(7)

By (7), Lemma 2.1, and Lemma 2.2, we obtain

$$ \bigl({}^{C}D_{0+}^{\sigma }u\bigr) (t)=- \bigl(I_{0+}^{q-\sigma }y\bigr) (t)+ \frac{u^{\prime }(0)}{\varGamma (2-\sigma )}t^{1-\sigma }, \quad t\in [0,1]. $$
(8)

It follows from (7), (8), and the boundary conditions in (6) that

$$ \begin{aligned} u(0)={}&\frac{1}{\rho } \biggl[\beta \gamma \bigl(I_{0+}^{q}y \bigr) (1)+\beta \delta \bigl(I_{0+}^{q- \sigma }y\bigr) (1)+ \frac{\gamma \varGamma (2-\sigma )+\delta }{\varGamma (2-\sigma )} \int _{0}^{1}h_{1}(s)u(s)\,ds \\ &{}+ \beta \int _{0}^{1}h_{2}(s)u(s)\,ds \biggr] \end{aligned} $$

and

$$ u^{\prime }(0)=\frac{1}{\rho } \biggl[\alpha \gamma \bigl(I_{0+}^{q}y \bigr) (1)+ \alpha \delta \bigl(I_{0+}^{q-\sigma }y\bigr) (1)-\gamma \int _{0}^{1}h_{1}(s)u(s)\,ds+ \alpha \int _{0}^{1}h_{2}(s)u(s)\,ds \biggr], $$

which together with (7) shows that

$$\begin{aligned} u(t)={}& \int _{0}^{t} \biggl\{ -\frac{(t-s)^{q-1}}{\varGamma (q)}+ \frac{\alpha t+\beta }{\rho } \biggl[ \frac{\gamma (1-s)^{q-1}}{\varGamma (q)}+ \frac{\delta (1-s)^{q-\sigma -1}}{\varGamma (q-\sigma )} \biggr] \biggr\} y(s)\,ds \\ & {} + \int _{t}^{1} \biggl\{ \frac{\alpha t+\beta }{\rho } \biggl[ \frac{\gamma (1-s)^{q-1}}{\varGamma (q)}+ \frac{\delta (1-s)^{q-\sigma -1}}{\varGamma (q-\sigma )} \biggr] \biggr\} y(s)\,ds \\ & {} + \frac{\gamma \varGamma (2-\sigma )(1-t)+\delta }{\rho \varGamma (2-\sigma )} \int _{0}^{1}h_{1}(s)u(s)\,ds + \frac{\alpha t+\beta }{\rho } \int _{0}^{1}h_{2}(s)u(s)\,ds \\ ={}& \int _{0}^{1}G(t,s)y(s)\,ds+ \frac{\gamma \varGamma (2-\sigma )(1-t)+\delta }{\rho \varGamma (2-\sigma )} \int _{0}^{1}h_{1}(s)u(s)\,ds \\ & {} +\frac{\alpha t+\beta }{\rho } \int _{0}^{1}h_{2}(s)u(s)\,ds,\quad t\in [0,1]. \end{aligned}$$
(9)

From (9), we get

$$ (1-Q_{1}) \int _{0}^{1}h_{1}(s)u(s) \,ds-P_{1} \int _{0}^{1}h_{2}(s)u(s)\,ds= \int _{0}^{1}h_{1}(s) \int _{0}^{1}G(s,\tau )y(\tau )\,d\tau \,ds $$

and

$$ -Q_{2} \int _{0}^{1}h_{1}(s)u(s) \,ds+(1-P_{2}) \int _{0}^{1}h_{2}(s)u(s)\,ds= \int _{0}^{1}h_{2}(s) \int _{0}^{1}G(s,\tau )y(\tau )\,d\tau \,ds, $$

and so,

$$\begin{aligned}& \int _{0}^{1}h_{1}(s)u(s)\,ds \\& \quad = \frac{(1-P_{2})\int _{0}^{1}h_{1}(s)\int _{0}^{1}G(s,\tau )y(\tau )\,d\tau \,ds+P_{1}\int _{0}^{1}h_{2}(s)\int _{0}^{1}G(s,\tau )y(\tau )\,d\tau \,ds}{(1-Q_{1})(1-P_{2})- P_{1}Q_{2}} \end{aligned}$$

and

$$\begin{aligned}& \int _{0}^{1}h_{2}(s)u(s)\,ds \\& \quad = \frac{Q_{2}\int _{0}^{1}h_{1}(s)\int _{0}^{1}G(s,\tau )y(\tau )\,d\tau \,ds+(1-Q_{1})\int _{0}^{1}h_{2}(s)\int _{0}^{1}G(s,\tau )y(\tau )\,d\tau \,ds}{(1-Q_{1})(1-P_{2})-P_{1}Q_{2}}, \end{aligned}$$

which together with (9) implies that

$$ \begin{aligned} u(t)={}& \int _{0}^{1}G(t,s)y(s)\,ds+\sum _{i=1}^{2}\phi _{i}(t) \int _{0}^{1}h_{i}(s) \int _{0}^{1}G(s,\tau )y(\tau )\,d\tau \,ds \\ ={}& \int _{0}^{1}G(t,s)y(s)\,ds+\sum _{i=1}^{2}\phi _{i}(t) \int _{0}^{1}h_{i}( \tau ) \int _{0}^{1}G(\tau ,s)y(s)\,ds \,d\tau \\ ={}& \int _{0}^{1}G(t,s)y(s)\,ds+\sum _{i=1}^{2}\phi _{i}(t) \int _{0}^{1}y(s) \int _{0}^{1}G(\tau ,s)h_{i}(\tau )\,d \tau \,ds \\ ={}& \int _{0}^{1} \Biggl[G(t,s)+\sum _{i=1}^{2}\phi _{i}(t) \int _{0}^{1} G( \tau ,s)h_{i}(\tau )\,d \tau \Biggr]y(s)\,ds \\ ={}& \int _{0}^{1}H(t,s)y(s)\,ds,\quad t\in [0,1]. \end{aligned} $$

 □

In what follows, we let

$$ \begin{aligned} g(s)=\frac{\alpha +\beta }{\rho } \biggl[ \frac{\gamma (1-s)^{q-1}}{\varGamma (q)}+ \frac{\delta (1-s)^{q-\sigma -1}}{\varGamma (q-\sigma )} \biggr],\quad s\in [0,1] \end{aligned} $$

and

$$ \begin{aligned} \eta (s)= \frac{\beta \delta \varGamma (q)-\varGamma (q-\sigma )(\rho -\beta \gamma )(1-s)^{\sigma }}{(\alpha +\beta ) [\gamma \varGamma (q-\sigma )+\delta \varGamma (q) ]}, \quad s\in [0,1]. \end{aligned} $$

Lemma 2.5

\(G(t, s)\)satisfies the following properties:

  1. (1)

    \(G(t,s)\leq g(s)\), \((t,s)\in [0,1]\times [0,1]\);

  2. (2)

    \(G(t,s)\geq \eta (s)g(s)\), \((t,s)\in [0,1]\times [0,1]\).

Proof

Since (1) is obvious, we only need to prove that (2) holds.

First, it is clear that \(G(t,1)\geq \eta (1)g(1)\) for \(t\in [0,1]\).

Now, we verify that \(G(t,s)\geq \eta (s)g(s)\) for \((t,s)\in [0,1]\times [0,1)\). In fact, if \(s\leq t\), then

$$\begin{aligned}& \begin{aligned} \frac{G(t,s)}{g(s)}&= \frac{(\alpha t+\beta )[\gamma \varGamma (q-\sigma )(1-s)^{q-1}+\delta \varGamma (q)(1-s)^{q-\sigma -1}]-\rho \varGamma (q-\sigma )(t-s)^{q-1}}{(\alpha +\beta ) [\gamma \varGamma (q-\sigma )(1-s)^{q-1}+\delta \varGamma (q)(1-s)^{q-\sigma -1}]} \\ &\geq \frac{\beta \gamma \varGamma (q-\sigma )(1-s)^{\sigma } +\beta \delta \varGamma (q)-\rho \varGamma (q-\sigma )(1-s)^{\sigma }}{(\alpha +\beta )[\gamma \varGamma (q-\sigma )(1-s)^{\sigma }+\delta \varGamma (q)]} \\ &\geq \frac{\beta \delta \varGamma (q)-\varGamma (q-\sigma )(\rho -\beta \gamma )(1-s)^{\sigma }}{(\alpha +\beta ) [\gamma \varGamma (q-\sigma )+\delta \varGamma (q) ]} \\ &=\eta (s) \end{aligned} \end{aligned}$$

and if \(t\leq s\), then

$$\begin{aligned}& \begin{aligned} \frac{G(t,s)}{g(s)}&= \frac{(\alpha t+\beta )[\gamma \varGamma (q-\sigma )(1-s)^{q-1}+\delta \varGamma (q)(1-s)^{q-\sigma -1}]}{(\alpha +\beta ) [\gamma \varGamma (q-\sigma )(1-s)^{q-1}+\delta \varGamma (q)(1-s)^{q-\sigma -1}]} \\ &\geq \frac{\beta \gamma \varGamma (q-\sigma )(1-s)^{\sigma }+\beta \delta \varGamma (q)}{(\alpha +\beta )[\gamma \varGamma (q-\sigma )(1-s)^{\sigma }+\delta \varGamma (q)]} \\ &\geq \frac{\beta \gamma \varGamma (q-\sigma )(1-s)^{\sigma } +\beta \delta \varGamma (q)-\rho \varGamma (q-\sigma )(1-s)^{\sigma }}{(\alpha +\beta )[\gamma \varGamma (q-\sigma )(1-s)^{\sigma }+\delta \varGamma (q)]} \\ &\geq \frac{\beta \delta \varGamma (q)-\varGamma (q-\sigma )(\rho -\beta \gamma )(1-s)^{\sigma }}{(\alpha +\beta ) [\gamma \varGamma (q-\sigma )+\delta \varGamma (q) ]} \\ &=\eta (s). \end{aligned} \end{aligned}$$

 □

By the definition of η and the condition \(0<\rho <\beta [\gamma + \frac{\delta \varGamma (q)}{\varGamma (q-\sigma )} ]\), we may obtain the following remark.

Remark 2.1

η is increasing on \([0,1]\) and \(0<\eta (s)<1\) for \(s\in [0,1]\).

In the remainder of this paper, we always assume that the following conditions are satisfied:

$$ Q_{1}< 1,\qquad P_{2}< 1\quad \text{and}\quad (1-Q_{1}) (1-P_{2})> P_{1}Q_{2}. $$

Lemma 2.6

\(H(t,s)\)has the following property:

$$\begin{aligned}& \begin{aligned} m\eta (s)g(s)\leq H(t,s)\leq M g(s),\quad (t,s)\in [0,1]\times [0,1], \end{aligned} \end{aligned}$$

where

$$ \begin{aligned} m=1+\sum_{i=1}^{2} \min_{t\in [0,1]}\phi _{i}(t) \int _{0}^{1} h_{i}(\tau )\,d\tau \end{aligned} $$

and

$$ \begin{aligned} M=1+\sum_{i=1}^{2} \max_{t\in [0,1]}\phi _{i}(t) \int _{0}^{1} h_{i}(\tau )\,d\tau . \end{aligned} $$

Proof

On the one hand, in view of (1) of Lemma 2.5, we have

$$ \begin{aligned} H(t,s)&=G(t,s)+\sum_{i=1}^{2} \phi _{i}(t) \int _{0}^{1} G(\tau ,s)h_{i}( \tau )\,d \tau \\ &\leq \Biggl(1+\sum_{i=1}^{2}\phi _{i}(t) \int _{0}^{1}h_{i}(\tau )\,d \tau \Biggr)g(s) \\ &\leq Mg(s),\quad (t,s)\in [0,1]\times [0,1]. \end{aligned} $$

On the other hand, by (2) of Lemma 2.5, we get

$$ \begin{aligned} H(t,s)&=G(t,s)+\sum_{i=1}^{2} \phi _{i}(t) \int _{0}^{1} G(\tau ,s)h_{i}( \tau )\,d \tau \\ &\geq \Biggl(1+\sum_{i=1}^{2}\phi _{i}(t) \int _{0}^{1} h_{i}(\tau )\,d \tau \Biggr)\eta (s)g(s) \\ &\geq m\eta (s)g(s),\quad (t,s)\in [0,1]\times [0,1]. \end{aligned} $$

 □

Let \(E=C[0,1]\) be equipped with norm \(\Vert u \Vert =\max_{t\in [0,1]} \vert u(t) \vert \) and

$$ \begin{aligned} K=\bigl\{ u\in E:u(t)\geq \theta \Vert u \Vert , t\in [0,1]\bigr\} , \end{aligned} $$

where \(0<\theta =\frac{m\eta (0)}{M}<1\). Then it is easy to check that E is a Banach space and K is a cone in E.

Now, we define an operator T on K by

$$ (Tu) (t)= \int _{0}^{1}H(t,s)f\bigl(s,u(s)\bigr)\,ds,\quad u \in K, t\in [0,1]. $$

Obviously, if u is a fixed point of T, then u is a nonnegative solution of BVP (4).

Lemma 2.7

\(T:K\rightarrow K\)is completely continuous.

Proof

Let \(u\in K\). Then, in view of Lemma 2.6, we have

$$ \begin{aligned} & \Vert Tu \Vert \leq M \int _{0}^{1}g(s)f\bigl(s,u(s)\bigr)\,ds, \end{aligned} $$

which together with Lemma 2.6 and Remark 2.1 implies that

$$ \begin{aligned} (Tu) (t)&\geq m \int _{0}^{1}\eta (s)g(s)f\bigl(s,u(s)\bigr)\,ds \\ &\geq m\eta (0) \int _{0}^{1}g(s)f\bigl(s,u(s)\bigr)\,ds \\ &\geq \theta \Vert Tu \Vert ,\quad t\in [0,1]. \end{aligned} $$

This indicates that \(Tu\in K\). Furthermore, it is easy to prove that T is completely continuous by an application of Arzela–Ascoli theorem [29]. □

Main results

Define

$$\begin{aligned}& f^{0}=\limsup_{u\rightarrow 0^{+}}\max_{t\in [0,1]} \frac{f(t,u)}{u}, \quad\quad f^{\infty }=\limsup_{u\rightarrow +\infty } \max_{t\in [0,1]} \frac{f(t,u)}{u}, \\& f_{0}=\liminf_{u\rightarrow 0^{+}}\min_{t\in [0,1]} \frac{f(t,u)}{u}, \quad\quad f_{\infty }=\liminf_{u\rightarrow +\infty } \min_{t\in [0,1]} \frac{f(t,u)}{u}. \end{aligned}$$

Theorem 3.1

Suppose that one of the following conditions is satisfied:

  1. (i)

    \(f_{0}=+\infty \)and\(f^{\infty }=0\), or

  2. (ii)

    \(f^{0}=0\)and\(f_{\infty }=+\infty \).

Then BVP (4) has at least one positive solution.

Proof

First, we consider case (i): \(f_{0}=+\infty \) and \(f^{ \infty }=0\).

In view of \(f_{0}=+\infty \), there exists \(r_{1}>0\) such that

$$ \begin{aligned} f(t,u)\geq G_{1} u,\quad (t,u)\in [0,1] \times [0,r_{1}], \end{aligned} $$
(10)

where \(G_{1}\geq \frac{1}{m\theta \int _{0}^{1}\eta (s)g(s)\,ds}\).

Let \(\varOmega _{1}=\{u\in E: \Vert u \Vert < r_{1}\}\). Then, for any \(u\in K\cap \partial \varOmega _{1}\), by Lemma 2.6 and (10), we get

$$ \begin{aligned} (Tu) (t)&\geq m \int _{0}^{1}\eta (s)g(s)f\bigl(s,u(s)\bigr)\,ds \\ &\geq mG_{1} \int _{0}^{1}\eta (s)g(s)u(s)\,ds \\ &\geq mG_{1}\theta \Vert u \Vert \int _{0}^{1}\eta (s)g(s)\,ds \\ &\geq \Vert u \Vert ,\quad t\in [0,1], \end{aligned} $$

which shows that

$$ \Vert Tu \Vert \geq \Vert u \Vert \quad \text{for } u\in K\cap \partial \varOmega _{1}. $$
(11)

On the other hand, since \(f^{\infty }=0\), there exists \(U_{1}>0\) such that

$$ \begin{aligned} f(t,u)\leq \varepsilon _{1} u,\quad (t,u) \in [0,1]\times (U_{1},+\infty ), \end{aligned} $$

where \(\varepsilon _{1}>0\) satisfies \(\varepsilon _{1}\leq \frac{1}{2M\int _{0}^{1}g(s)\,ds}\).

Let \(M^{*}=\max_{(t,u)\in [0,1]\times [0,U_{1}]}f(t,u)\). Then we have

$$ \begin{aligned} f(t,u)\leq M^{*}+\varepsilon _{1}u,\quad (t,u)\in [0,1]\times [0,+\infty ). \end{aligned} $$
(12)

If we choose \(r_{2}=\max {\{2r_{1},2MM^{*}\int _{0}^{1}g(s)\,ds\}}\) and let \(\varOmega _{2}=\{u\in E: \Vert u \Vert < r_{2}\}\), then for any \(u\in K\cap \partial \varOmega _{2}\), from Lemma 2.6 and (12), we obtain

$$ \begin{aligned} (Tu) (t)&\leq M \int _{0}^{1}g(s)f\bigl(s,u(s)\bigr)\,ds \\ &\leq MM^{*} \int _{0}^{1}g(s)\,ds+M\varepsilon _{1} \Vert u \Vert \int _{0}^{1}g(s)\,ds \\ &\leq \frac{ \Vert u \Vert }{2}+\frac{ \Vert u \Vert }{2} \\ &= \Vert u \Vert ,\quad t\in [0,1], \end{aligned} $$

which indicates that

$$ \Vert Tu \Vert \leq \Vert u \Vert \quad \text{for } u\in K\cap \partial \varOmega _{2}. $$
(13)

Therefore, it follows from Theorem 1.1, Lemma 2.7, (11), and (13) that T has a fixed point \(u\in K\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1} )\), which is a desired positive solution of BVP (4).

Next, we consider case (ii): \(f^{0}=0\) and \(f_{ \infty }=+\infty \).

In view of \(f^{0}=0\), there exists \(r_{3}>0\) such that

$$ \begin{aligned} f(t,u)\leq \varepsilon _{2}u,\quad (t,u) \in [0,1]\times [0,r_{3}], \end{aligned} $$
(14)

where \(\varepsilon _{2}>0\) satisfies \(\varepsilon _{2}\leq \frac{1}{M\int _{0}^{1}g(s)\,ds}\).

Let \(\varOmega _{3}=\{u\in E: \Vert u \Vert < r_{3}\}\). Then, for any \(u\in K\cap \partial \varOmega _{3}\), by Lemma 2.6 and (14), we get

$$ \begin{aligned} (Tu) (t)&\leq M \int _{0}^{1}g(s)f\bigl(s,u(s)\bigr)\,ds \\ &\leq M\varepsilon _{2} \int _{0}^{1}g(s)u(s)\,ds \\ &\leq M\varepsilon _{2} \Vert u \Vert \int _{0}^{1}g(s)\,ds \\ &\leq \Vert u \Vert ,\quad t\in [0,1], \end{aligned} $$

which shows that

$$ \begin{aligned} \Vert Tu \Vert \leq \Vert u \Vert \quad \text{for } u\in K\cap \partial \varOmega _{3}. \end{aligned} $$
(15)

On the other hand, since \(f_{\infty }=+\infty \), there exists \(U_{2}>0\) such that

$$ \begin{aligned} f(t,u)\geq G_{2}u,\quad (t,u)\in [0,1] \times [U_{2},+\infty ), \end{aligned} $$
(16)

where \(G_{2}\geq \frac{1}{m\theta \int _{0}^{1}\eta (s)g(s)\,ds}\).

If we choose \(r_{4}=\max \{\frac{U_{2}}{\theta },2r_{3}\}\) and let \(\varOmega _{4}=\{u\in E: \Vert u \Vert < r_{4}\}\), then for any \(u\in K\cap \partial \varOmega _{4}\), we know

$$ \begin{aligned} u(t)\geq \theta \Vert u \Vert =\theta r_{4}\geq U_{2},\quad t\in [0,1], \end{aligned} $$

which together with Lemma 2.6 and (16) implies that

$$ \begin{aligned} (Tu) (t)&\geq m \int _{0}^{1}\eta (s)g(s)f\bigl(s,u(s)\bigr)\,ds \\ &\geq mG_{2} \int _{0}^{1}\eta (s)g(s)u(s)\,ds \\ &\geq mG_{2}\theta \Vert u \Vert \int _{0}^{1}\eta (s)g(s)\,ds \\ &\geq \Vert u \Vert ,\quad t\in [0,1]. \end{aligned} $$

This indicates that

$$ \begin{aligned} \Vert Tu \Vert \geq \Vert u \Vert \quad \text{for } u\in K\cap \partial \varOmega _{4}. \end{aligned} $$
(17)

Therefore, it follows from Theorem 1.1, Lemma 2.7, (15), and (17) that T has a fixed point \(u\in K\cap (\overline{\varOmega }_{4}\setminus \varOmega _{3} )\), which is a desired positive solution of BVP (4). □

Example 3.1

Consider the following BVP:

$$ \textstyle\begin{cases} ({}^{C}D^{\frac{5}{2}}_{0+}u)(t)+ [\sin (\frac{\pi t}{2} )+1 ]u^{2}(t)=0,\quad t\in [0,1], \\ u^{\prime \prime }(0)=0, \\ u(0)-4u^{\prime }(0)=\int _{0}^{1}su(s)\,ds, \\ u(1)+({}^{C}D_{0+}^{\frac{1}{2}}u)(1)=\int _{0}^{1}(1-s)u(s)\,ds. \end{cases} $$
(18)

In view of \(q=\frac{5}{2}\), \(\sigma =\frac{1}{2}\), \(\alpha =\gamma =\delta =1\), \(\beta =4\), \(h_{1}(s)=s\), and \(h_{2}(s)=1-s\), \(s\in [0,1]\), a simple calculation shows that

$$ \begin{aligned} &0< \rho =5+\frac{2}{\pi }\sqrt{\pi }< \beta \biggl[\gamma + \frac{\delta \varGamma (q)}{\varGamma (q-\sigma )} \biggr]=4+3\sqrt{\pi } \end{aligned} $$

and

$$\begin{aligned}& P_{1}=\frac{7\sqrt{\pi }}{3(5\sqrt{\pi }+2)},\qquad P_{2}= \frac{13\sqrt{\pi }}{6(5\sqrt{\pi }+2)}, \\& Q_{1}= \frac{\sqrt{\pi }+6}{6(5\sqrt{\pi }+2)}, \qquad Q_{2}= \frac{\sqrt{\pi }+3}{3(5\sqrt{\pi }+2)}. \end{aligned}$$

Obviously, \(Q_{1}<1\), \(P_{2}<1\) and

$$ \begin{aligned} (1-Q_{1}) (1-P_{2})= \frac{(29\sqrt{\pi }+6)(17\sqrt{\pi }+12)}{36(5\sqrt{\pi }+2)^{2}}>P_{1}Q_{2}= \frac{7\sqrt{\pi }(\sqrt{\pi }+3)}{9(5\sqrt{\pi }+2)^{2}}. \end{aligned} $$

Moreover, since \(f(t,u)= [\sin (\frac{\pi t}{2})+1 ]u^{2}\), \((t,u)\in [0,1] \times [0,+\infty )\), it is easy to know that \(f:[0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous and

$$ \begin{aligned} f^{0}=0,\quad\quad f_{\infty } =+ \infty . \end{aligned} $$

Therefore, it follows from Theorem 3.1 that BVP (18) has at least one positive solution.

Conclusion

In this paper, by applying Guo–Krasnoselskii’s fixed point theorem, we obtain the existence of at least one positive solution for a class of nonlinear boundary value problems involving fractional differential equation and integral boundary conditions. An illustrative example is also given to show the effectiveness of theoretical results.

References

  1. 1.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

  2. 2.

    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)

  3. 3.

    Agarwal, R.P., Ahmad, B., Garout, D., Alsaedi, A.: Existence results for coupled nonlinear fractional differential equations equipped with nonlocal coupled flux and multi-point boundary conditions. Chaos Solitons Fractals 102, 149–162 (2017)

  4. 4.

    Agarwal, R.P., Benchohra, M., Hamani, S.: Boundary value problems for fractional differential equations. Georgian Math. J. 16, 401–411 (2009)

  5. 5.

    Bai, Z., Qiu, T.: Existence of positive solution for singular fractional differential equation. Appl. Math. Comput. 215, 2761–2767 (2009)

  6. 6.

    Cabada, A., Aleksić, S., Tomović, T.V., Dimitrijević, S.: Existence of solutions of nonlinear and non-local fractional boundary value problems. Mediterr. J. Math. 16, Article ID 119 (2019)

  7. 7.

    Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403–411 (2012)

  8. 8.

    Guezane-Lakoud, A., Bensebaa, S.: Solvability of a fractional boundary value problem with fractional derivative condition. Arab. J. Math. 3, 39–48 (2014)

  9. 9.

    Guezane-Lakoud, A., Khaldi, R.: Existence results for a fractional boundary value problem with fractional Lidstone conditions. J. Appl. Math. Comput. 49, 261–268 (2015)

  10. 10.

    Guezane-Lakoud, A., Khaldi, R.: Solvability of a fractional boundary value problem with fractional integral condition. Nonlinear Anal. 75, 2692–2700 (2012)

  11. 11.

    Ji, Y., Guo, Y., Qiu, J., Yang, L.: Existence of positive solutions for a boundary value problem of nonlinear fractional differential equations. Adv. Differ. Equ. 2015, Article ID 13 (2015)

  12. 12.

    Jiang, D., Yuan, C.: The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. 72, 710–719 (2010)

  13. 13.

    Li, C.F., Luo, X.N., Zhou, Y.: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59, 1363–1375 (2010)

  14. 14.

    Ma, W., Cui, Y.: The eigenvalue problem for Caputo type fractional differential equation with Riemann–Stieltjes integral boundary conditions. J. Funct. Spaces 2018, Article ID 2176809 (2018)

  15. 15.

    Su, C.-M., Sun, J.-P., Zhao, Y.-H.: Existence and uniqueness of solutions for BVP of nonlinear fractional differential equation. Int. J. Differ. Equ. 2017, Article ID 4683581 (2017)

  16. 16.

    Xie, W., Xiao, J., Luo, Z.: Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions. Appl. Math. Lett. 41, 46–51 (2015)

  17. 17.

    Xu, J., Wei, Z., Dong, W.: Uniqueness of positive solutions for a class of fractional boundary value problems. Appl. Math. Lett. 25, 590–593 (2012)

  18. 18.

    Yang, W.: Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions. J. Appl. Math. Comput. 44, 39–59 (2014)

  19. 19.

    Yue, J.-R., Sun, J.-P., Zhang, S.: Existence of positive solution for BVP of nonlinear fractional differential equation. Discrete Dyn. Nat. Soc. 2015, Article ID 736108 (2015)

  20. 20.

    Zhang, S.: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006, Article ID 36 (2006)

  21. 21.

    Zhang, X., Wang, L., Sun, Q.: Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter. Appl. Math. Comput. 226, 708–718 (2014)

  22. 22.

    Zhao, K., Gong, P.: Existence of positive solutions for a class of higher-order Caputo fractional differential equation. Qual. Theory Dyn. Syst. 14, 157–171 (2015)

  23. 23.

    Chen, P., Gao, Y.: Positive solutions for a class of nonlinear fractional differential equations with nonlocal boundary value conditions. Positivity 22, 761–772 (2018)

  24. 24.

    He, Y.: Existence and multiplicity of positive solutions for singular fractional differential equations with integral boundary value conditions. Adv. Differ. Equ. 2016, Article ID 31 (2016)

  25. 25.

    Cabada, A., Hamdi, Z.: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251–257 (2014)

  26. 26.

    Wang, W.: Properties of Green’s function and the existence of different types of solutions for nonlinear fractional BVP with a parameter in integral boundary conditions. Bound. Value Probl. 2019, Article ID 76 (2019)

  27. 27.

    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)

  28. 28.

    Krasnoselskii, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)

  29. 29.

    Fréchet, M.: Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo 22, 1–74 (1906)

Download references

Acknowledgements

The authors wish to express their sincere thanks to anonymous referees for their detailed comments and valuable suggestions which have improved the paper greatly.

Availability of data and materials

Not applicable.

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11661049).

Author information

Affiliations

Authors

Contributions

All authors contributed equally to this manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Jian-Ping Sun.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

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 http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, M., Sun, J. & Zhao, Y. Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions. Adv Differ Equ 2020, 177 (2020). https://doi.org/10.1186/s13662-020-02618-9

Download citation

MSC

  • 34A08
  • 34B18

Keywords

  • Fractional differential equation
  • Integral boundary condition
  • Boundary value problem
  • Positive solution
  • Existence