Monotone iterative method for a p-Laplacian boundary value problem with fractional conformable derivatives

Abstract

By using monotone iterative method, the extremal solutions and the unique solution are obtained for a nonlinear fractional p-Laplacian boundary value problem involving fractional conformable derivatives and nonlocal integral boundary conditions. Comparison theorems related to the proposed study are also proved. The paper concludes with an illustrative example for the main result.

1 Introduction

Fractional calculus provides powerful tools to deal with complex phenomena occurring in various areas of applied and technical sciences such as control theory, optical and thermal systems, rheology, materials and mechanical systems, robotics, etc. Numerous researchers have investigated different aspects (existence, uniqueness, stability, etc.) of fractional differential equations involving Caputo, Riemann–Liouville, Hadamard type derivatives, for instance, see [1,2,3,4,5,6,7,8,9,10]. For some recent results on Riemann–Liouville fractional differential equations, we refer the reader to the articles [11,12,13,14,15] and the references cited therein. Fractional p-Laplacian boundary value problems also received considerable attention, for example, see [16,17,18,19,20,21,22,23,24,25,26]. The literature on fractional differential equations equipped with integral boundary conditions also contains a variety of interesting results [27,28,29,30,31,32].

Monotone iterative method is found to be an important and efficient method to obtain sequences of monotone solutions for initial and boundary value problems. For some applications of this technique to nonlinear fractional differential equations, see [15, 33,34,35,36,37,38,39,40,41,42,43]. In 2017, Jarad et. al. [44] proposed a new fractional derivative, which is known as fractional conformable derivative (see definition (2.4)). To the best of the authors’ knowledge, the fractional p-Laplacian problem involving fractional conformable derivatives is yet to be investigated. In this paper, we apply monotone iterative method to prove the existence of extremal and uniqueness of solutions for the following nonlinear fractional p-Laplacian problem involving fractional conformable derivatives and nonlocal integral boundary condition:

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }(\phi _{p}({}^{\gamma }_{0}D^{\alpha }h(t)))=f(t, h(t), {}^{\gamma }_{0}D^{\alpha }h(t)), \quad t\in (0,d], d>0, \\ t^{\frac{\alpha (1-\beta )}{p-1}} {{}^{\gamma }_{0}D^{\alpha }h(t)}|_{t=0}= \int _{0}^{\tau }a(s)h(s)\,ds, \quad g(\tilde{h}(0),\tilde{h}(d))=0, \tau \in (0,d), \end{cases} $$

(1.1)

where $0<\alpha , \gamma , \beta \leq 1$, $\phi _{p}(t)=|t|^{p-2}t$, $a\in C([0,d],[0,\infty ))$, $f\in C([0,d]\times \mathbb{R}^{2}, \mathbb{R})$, $g\in C(\times \mathbb{R}^{2},\mathbb{R})$, $\phi _{p}$, $p>1$, denotes the p-Laplacian operator and $\phi _{p}^{-1}=\phi _{q}$, $\frac{1}{p}+\frac{1}{q}=1$, $\tilde{h}(0)=t^{\alpha (1-\gamma )}h(t)|_{t=0}$, $\tilde{h}(d)=t^{\alpha (1-\gamma )}h(t)|_{t=d}$, and ${}^{\gamma }_{0}D^{\alpha }$ is the fractional conformable derivative of order γ.

We emphasize that the results obtained for problem (1.1) are new and significantly contribute to the existing literature on p-Laplacian problems with fractional conformable derivatives. In order to establish the desired results, we prove two comparison theorems related to the problem at hand, which are presented in Sect. 2. The main results are presented in Sect. 3.

2 Preliminaries and lemmas

For $\alpha , \gamma \in (0,1)$, we denote by $C_{\alpha (1- \gamma )}([0,d],\mathbb{R})$ a Banach space

$$ \bigl\{ h\in C((0,d],\mathbb{R}):t^{\alpha (1-\gamma )}h\in C\bigl([0,d], \mathbb{R} \bigr)\bigr\} , $$

(2.1)

endowed with the norm $\|h\|_{C_{\alpha (1-\gamma )}}=\sup_{t\in [0,d]}t^{\alpha (1-\gamma )}|h(t)|$.

Let

$$ Y=\bigl\{ h(t)\in C_{\alpha (1-\gamma )}\bigl([0,d],\mathbb{R}\bigr): {}^{\gamma }_{0}D ^{\alpha }h(t)\in C_{k} \bigl([0,d],\mathbb{R}\bigr) \text{ and } t^{k}{{}^{\gamma }_{0}D^{\alpha }h(t)|_{t=0}}= \varepsilon \bigr\} , $$

(2.2)

where $0<\alpha , \gamma <1$, $k=\frac{\alpha (1-\beta )}{p-1}$, $\varepsilon =\int _{0}^{\tau }a(s)h(s)\,ds$, be a Banach space equipped with the norm $\|h\|_{Y}=\max \{\sup_{t\in [0,d]}t^{ \alpha (1-\gamma )}|h(t)|, \sup_{t\in [0,d]}|{}^{\gamma }_{0}D ^{\alpha }h(t)| \}$.

Definition 2.1

([44])

The Riemann–Liouville type fractional conformable integral of order $\gamma \in \mathbb{C}$, $\operatorname{Re}(\gamma )\geq 0$ is defined by

$$ {}_{a}^{\gamma }I^{\alpha }h(t)=\frac{1}{\varGamma (\gamma )} \int _{a}^{t} \biggl(\frac{(t-a)^{\alpha }-(s-a)^{\alpha }}{\alpha } \biggr)^{\gamma -1}h(s)\frac{ds}{(s-a)^{1- \alpha }}. $$

(2.3)

Definition 2.2

([44])

The fractional conformable derivative of Riemann–Liouville type of order $\gamma \in \mathbb{C}$, $\operatorname{Re}(\gamma ) \geq 0$ is defined by

$$ \begin{aligned} {}_{a}^{\gamma }D^{\alpha }h(t)&={}_{a}^{n} \mathcal{T}^{\alpha }\bigl({}^{n- \gamma }_{a}I^{\alpha } \bigr)h(t) \\ &=\frac{{}_{a}^{n}\mathcal{T}^{\alpha }}{\varGamma (n-\gamma )} \int _{a} ^{t} \biggl(\frac{(t-a)^{\alpha }-(s-a)^{\alpha }}{\alpha } \biggr)^{n- \gamma -1}h(s)\frac{ds}{(s-a)^{1-\alpha }}, \end{aligned} $$

(2.4)

where

$$ n=\bigl[\operatorname{Re}(\gamma )\bigr]+1, \qquad {}_{a}^{n} \mathcal{T}^{\alpha }=\underbrace{{}_{a}\mathcal{T}^{\alpha } {}_{a}\mathcal{T}^{\alpha }\cdots {}_{a} \mathcal{T}^{\alpha }} _{n \text{ times}}, $$

(2.5)

and ${}_{a}\mathcal{T}^{\alpha }$ is the conformable differential operator [45]

$$ {}_{a}\mathcal{T}^{\alpha }h(t)=(t-a)^{1-\alpha }h^{\prime }(t). $$

(2.6)

Lemma 2.1

([44])

Let $0<\operatorname{Re}(\gamma )<1$, $n=-[- \operatorname{Re}(\gamma )]$, $f\in L((0,d),\mathbb{R})$. Then

$$ {}^{\gamma }_{a}I^{\alpha }\bigl({}^{\gamma }_{a}D^{\alpha }h(t) \bigr)=h(t)-\frac{ {}_{a}^{\gamma -1}D^{\alpha }h(a)}{\alpha ^{\gamma -1}\varGamma (\gamma )}(t-a)^{ \alpha \gamma -\alpha }. $$

Let us first consider the problem

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }(\phi _{p}({}^{\gamma }_{0}D^{\alpha }h(t)))=f(t, h(t), {}^{\gamma }_{0}D^{\alpha }h(t)), \quad t\in (0,d], \\ t^{\frac{\alpha (1-\beta )}{p-1}} {{}^{\gamma }_{0}D^{\alpha }h(t)}|_{t=0}= \int _{0}^{\tau }a(s)h(s)\,ds, \quad \tilde{h}(0)=r, \tau \in (0,d). \end{cases} $$

(2.7)

Applying Lemma 2.1 to problem (2.7) with $l(t)=\phi _{p}({}^{\gamma }_{0}D^{\alpha }h(t))$ and $\tilde{h}(0)=r$, we obtain

$$ h(t)=rt^{\alpha (\gamma -1)}+\frac{1}{\varGamma (\gamma )} \int _{0}^{t} \biggl(\frac{t^{\alpha }-s^{\alpha }}{\alpha } \biggr)^{\gamma -1}\phi _{q}\bigl(l(s)\bigr) \frac{ds}{s ^{1-\alpha }}=:Bl(t) $$

(2.8)

and

$$ \phi _{p}\bigl(t^{\frac{\alpha (1-\beta )}{p-1}} {{}^{\gamma }_{0}D^{\alpha }h(t)}\bigr)=t ^{\alpha (1-\beta )}\phi _{p}\bigl({}^{\gamma }_{0}D^{\alpha }h(t) \bigr)=t^{\alpha (1-\beta )}l(t). $$

(2.9)

Thus problem (2.7) takes the form

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }l(t)=f(t, Bl(t), \phi _{q}(l(t))), \quad t\in (0,d], \\ t^{\alpha (1-\beta )}l(t)|_{t=0}=\phi _{p}[\int _{0}^{\tau }a(s)h(s)\,ds], \quad \tau \in (0,d). \end{cases} $$

(2.10)

If (2.10) has a solution $l(t)$, then we get a solution $h(t)$ of Eq. (2.7) after inserting $l(t)$ in Eq.(2.8). This shows the existence of a solution for problem (2.10).

In the following lemma, we use $\|h\|_{*}=\sup_{t\in [0,d]}|h(t)|$.

Lemma 2.2

Suppose that $f\in C([0,d]\times \mathbb{R}^{2},\mathbb{R})$, $0<\alpha , \beta <1$, and there exists a nonnegative bounded integrable function M on $[0,d]$ such that

$$ \bigl\vert f(t,h_{1},h_{2})-f(t,l_{1},l_{2}) \bigr\vert \leq M(t) \bigl\vert \phi _{p}(l_{2})- \phi _{p}(h _{2}) \bigr\vert , \quad t\in (0,d]. $$

Then problem (2.10) has a unique solution $l(t)\in C _{\alpha (1-\beta )}([0,d],\mathbb{R})$, if

$$ \frac{d^{\alpha (\beta -1)}\xi ^{p-2}\eta ^{q-2}}{\varGamma (\gamma +1)} \int _{0}^{\tau }a(s) \biggl(\frac{s^{\alpha }}{\alpha } \biggr)^{\gamma -1}\,ds+\frac{Md ^{\alpha \beta }}{\varGamma (\beta +1)\alpha ^{\beta }}< 1, $$

(2.11)

where ξ takes the values between $\int _{0}^{\tau }a(s){{}_{0}^{ \gamma }I^{\alpha }\phi _{q}(h(s))}\,ds$ and $\int _{0}^{\tau }a(s){{}_{0} ^{\gamma }I^{\alpha }\phi _{q}(l(s))}\,ds$, the values of η remain between $h(u)$ and $l(u)$, and $M=\sup_{t\in [0,d]}|M(t)|$.

Proof

According to Lemma 2.1 and $t^{\alpha (1-\beta )}l(t)|_{t=0}= \phi _{p}(\int _{0}^{\tau }a(s)h(s)\,ds)$, problem (2.10) is equivalent to the following integral equation:

$$ \begin{aligned}[b] l(t)&=\phi _{p} \biggl[ \int _{0}^{\tau }a(s)Bl(s)\,ds \biggr]t^{\alpha (\beta -1)}\\ &\quad {}+ \frac{1}{ \varGamma (\beta )} \int _{0}^{t} \biggl(\frac{t^{\alpha }-s^{\alpha }}{ \alpha } \biggr)^{\beta -1}f\bigl(s, Bl(s), \phi _{q}\bigl(l(s)\bigr) \bigr)\frac{ds}{s^{1- \alpha }}\\ &:=Al(t). \end{aligned} $$

(2.12)

For any $h, l\in C_{\alpha (1-\beta )}([0,d],\mathbb{R})$, we have

$$\begin{aligned} & \Vert Ah-Al \Vert _{*} \\ &\quad\leq \sup_{t\in [0,d]}t^{\alpha (\beta -1)} \biggl\vert \phi _{p}\biggl[ \int _{0}^{\tau }a(s)Bh(s)\,ds\biggr]-\phi _{p}\biggl[ \int _{0}^{\tau }a(s)Bl(s)\,ds\biggr] \biggr\vert \\ &\qquad {}+\sup_{t\in [0,d]}\frac{1}{\varGamma (\beta )} \int _{0}^{t} \biggl(\frac{t^{\alpha }-s^{\alpha }}{\alpha } \biggr)^{\beta -1} \bigl\vert f\bigl(s, Bh(s), \phi _{q} \bigl(h(s)\bigr)\bigr)-f\bigl(s, Bl(s), \phi _{q}\bigl(l(s)\bigr) \bigr) \bigr\vert \frac{ds}{s^{1-\alpha }} \\ &\quad\leq \sup_{t\in [0,d]}t^{\alpha (\beta -1)} \biggl\vert \phi _{p} \biggl[ \int _{0}^{\tau }a(s) \biggl(rs^{\alpha (\gamma -1)}+ \frac{1}{ \varGamma (\gamma )} \int _{0}^{s} \biggl(\frac{s^{\alpha }-u^{\alpha }}{ \alpha } \biggr)^{\gamma -1}\phi _{q}\bigl(h(u)\bigr) \frac{du}{u^{1-\alpha }} \biggr) \,ds \biggr] \\ &\qquad {}-\phi _{p} \biggl[ \int _{0}^{\tau }a(s) \biggl(rs^{\alpha (\gamma -1)}+ \frac{1}{ \varGamma (\gamma )} \int _{0}^{s} \biggl(\frac{s^{\alpha }-u^{\alpha }}{ \alpha } \biggr)^{\gamma -1}\phi _{q}\bigl(l(u)\bigr) \frac{du}{u^{1-\alpha }} \biggr)\,ds \biggr] \biggr\vert \\ &\qquad {}+\sup_{t\in [0,d]}\frac{1}{\varGamma (\beta )} \int _{0}^{t} \biggl(\frac{t^{\alpha } -s^{\alpha }}{\alpha } \biggr)^{\beta -1}M(s) \bigl\vert h(s)-l(s) \bigr\vert \frac{ds}{s ^{1-\alpha }} \\ &\quad\leq \sup_{t\in [0,d]}t^{\alpha (\beta -1)} \biggl\vert \biggl[ \int _{0}^{\tau }a(s) \bigl(rs^{\alpha (\gamma -1)} \bigr)\,ds\\ &\qquad {}+ \int _{0}^{\tau }a(s)\frac{1}{ \varGamma (\gamma )} \int _{0}^{s} \biggl(\frac{s^{\alpha }-u^{\alpha }}{ \alpha } \biggr)^{\gamma -1} \bigl\vert \phi _{q}\bigl(h(u)\bigr) \bigr\vert \frac{du}{u^{1-\alpha }})\,ds \biggr]^{p-1} \\ &\qquad {}- \biggl[ \int _{0}^{\tau }a(s) \bigl(rs^{\alpha (\gamma -1)} \bigr)\,ds+ \int _{0}^{ \tau }a(s)\frac{1}{\varGamma (\gamma )} \int _{0}^{s} \biggl(\frac{s^{\alpha }-u^{\alpha }}{\alpha } \biggr)^{\gamma -1} \bigl\vert \phi _{q}\bigl(l(u)\bigr) \bigr\vert \frac{du}{u ^{1-\alpha }})\,ds \biggr]^{p-1} \biggr\vert \\ &\qquad{}+\sup_{t\in [0,d]}\frac{1}{\varGamma (\beta )} \int _{0}^{t} \biggl(\frac{t^{\alpha }-s^{\alpha }}{\alpha } \biggr)^{\beta -1}M(s)\frac{ds}{s ^{1-\alpha }} \Vert h-l \Vert _{*} \\ &\quad\leq \sup_{t\in [0,d]}t^{\alpha (\beta -1)} \biggl\vert (p-1)\xi ^{p-2} \biggl[ \int _{0}^{\tau }a(s) \bigl(rs^{\alpha (\gamma -1)} \bigr)\,ds+ \int _{0}^{ \tau }a(s)\frac{1}{\varGamma (\gamma )} \int _{0}^{s} \biggl(\frac{s^{\alpha }-u^{\alpha }}{\alpha } \biggr)^{\gamma -1} \\ &\qquad{}\times\phi _{q}\bigl(h(u)\bigr) \frac{du}{u^{1-\alpha }}\,ds - \int _{0}^{\tau }a(s) \bigl(rs ^{\alpha (\gamma -1)} \bigr)\,ds\\ &\qquad {} - \int _{0}^{\tau }a(s)\frac{1}{\varGamma (\gamma )} \int _{0}^{s} \biggl(\frac{s^{\alpha }-u^{\alpha }}{\alpha } \biggr)^{ \gamma -1}\phi _{q}\bigl(l(u)\bigr) \frac{du}{u^{1-\alpha }}\,ds \biggr] \biggr\vert \\ &\qquad{}+\sup_{t\in [0,d]}\frac{M}{\varGamma (\beta +1)} \biggl( \frac{t ^{\alpha }}{\alpha } \biggr)^{\beta } \Vert h-l \Vert _{*} \\ &\quad\leq d^{\alpha (\beta -1)}(p-1)\xi ^{p-2} \biggl\vert \int _{0}^{\tau }a(s)\frac{1}{ \varGamma (\gamma )} \int _{0}^{s} \biggl(\frac{s^{\alpha }-u^{\alpha }}{ \alpha } \biggr)^{\gamma -1} \bigl(\phi _{q}\bigl(h(u)\bigr)-\phi _{q}\bigl(l(u)\bigr)\bigr) \frac{du}{u ^{1-\alpha }}\,ds \biggr\vert \\ &\qquad{}+\frac{Md^{\alpha \beta }}{\varGamma (\beta +1)\alpha ^{\beta }} \Vert h-l \Vert _{*} \\ &\quad\leq d^{\alpha (\beta -1)}(p-1)\xi ^{p-2}(q-1)\eta ^{q-2} \int _{0} ^{\tau }a(s)\frac{1}{\varGamma (\gamma )} \int _{0}^{s} \biggl(\frac{s^{ \alpha }-u^{\alpha }}{\alpha } \biggr)^{\gamma -1}\frac{du}{u^{1-\alpha }}\,ds \Vert h-l \Vert _{*} \\ &\qquad{}+\frac{Md^{\alpha \beta }}{\varGamma (\beta +1)\alpha ^{\beta }} \Vert h-l \Vert _{*} \\ &\quad= \biggl(\frac{d^{\alpha (\beta -1)}\xi ^{p-2}\eta ^{q-2}}{\varGamma ( \gamma +1)} \int _{0}^{\tau }a(s) \biggl(\frac{s^{\alpha }}{\alpha } \biggr)^{\gamma -1}\,ds+\frac{Md^{\alpha \beta }}{\varGamma (\beta +1) \alpha ^{\beta }} \biggr) \Vert h-l \Vert _{*}, \end{aligned}$$

which, in view of (2.11), implies that the operator A has a unique fixed point by the Banach fixed point theorem. In consequence, problem (2.10) has a unique solution. □

Lemma 2.3

If $0<\alpha , \gamma , \beta <1$, $\psi \in C_{\alpha (1-\beta )}([0,d], \mathbb{R})$, and M is a nonnegative bounded integrable function on $[0,d]$, then the following problem

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }(\phi _{p}({}^{\gamma }_{0}D^{\alpha }h(t)))+M(t) \phi _{p}({}^{\gamma }_{0}D^{\alpha }h(t))=\psi (t), \quad t\in (0,d], \\ t^{\frac{\alpha (1-\beta )}{p-1}} {{}^{\gamma }_{0}D^{\alpha }h(t)}|_{t=0}=c, \quad \tilde{h}(0)=r, \end{cases} $$

(2.13)

has a unique solution $h\in Y$, provided that $Md^{\alpha \beta }< \varGamma (\beta +1)\alpha ^{\beta }$.

Proof

Letting $l(t)=\phi _{p}({}^{\gamma }_{0}D^{\alpha }h(t))$, we have

$$ \textstyle\begin{cases} {}^{\gamma }_{0}D^{\alpha }h(t)=\phi _{q}(l(t)), \quad t\in (0,d], \\ \tilde{h}(0)=r, \end{cases} $$

(2.14)

and

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }l(t)+M(t)l(t)=\psi (t), \quad t\in (0,d], \\ t^{\alpha (1-\beta )}l(t)|_{t=0}=\phi _{p}(c). \end{cases} $$

(2.15)

Let $f(t, Bl(t), \phi _{q}(l(t)))=\psi (t)-M(t)l(t)$. For $l_{1}, l _{2}\in C_{\alpha (1-\beta )}([0,d],\mathbb{R})$, we have

$$ \bigl\vert f\bigl(t, Bl_{1}, \phi _{q}(l_{1}) \bigr)-f\bigl(t, Bl_{2}, \phi _{q}(l_{2}) \bigr) \bigr\vert = \bigl\vert M(t) \bigr\vert \vert l _{2}-l_{1} \vert \leq M \vert l_{2}-l_{1} \vert . $$

Thus, problem (2.15) has a unique solution $l\in C_{ \alpha (1-\beta )}([0,d],\mathbb{R})$ by Lemma 2.2, and ${}^{\gamma }_{0}D^{\alpha }h\in C_{\frac{\alpha (1-\beta )}{p-1}}([0,d], \mathbb{R})$. Moreover, problem (2.14) has a solution $h\in C_{\alpha (1-\gamma )}([0,d],\mathbb{R})$ by Lemma 2.1. By inserting l in h, we get a unique solution $h\in Y$ of problem (2.13). □

Definition 2.3

If $h\in Y$ is a lower solution of (1.1), then

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }(\phi _{p}({}^{\gamma }_{0}D^{\alpha }h(t))) \leq f(t, h(t), {}^{\gamma }_{0}D^{\alpha }h(t)), \quad t\in (0,d], d>0, \\ t^{\frac{\alpha (1-\beta )}{p-1}} {{}^{\gamma }_{0}D^{\alpha }h(t)}|_{t=0} \leq \int _{0}^{\tau }a(s)h(s)\,ds, \quad g(\tilde{h}(0),\tilde{h}(d))\leq 0, \tau \in (0,d). \end{cases} $$

(2.16)

If $l\in Y$ is an upper solution of (1.1), then

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }(\phi _{p}({}^{\gamma }_{0}D^{\alpha }l(t))) \geq f(t, l(t), {}^{\gamma }_{0}D^{\alpha }l(t)), \quad t\in (0,d], d>0, \\ t^{\frac{\alpha (1-\beta )}{p-1}} {{}^{\gamma }_{0}D^{\alpha }l(t)}|_{t=0} \geq \int _{0}^{\tau }a(s)h(s)\,ds, \quad g(\tilde{l}(0),\tilde{l}(d))\geq 0, \tau \in (0,d). \end{cases} $$

(2.17)

Lemma 2.4

(Comparison theorem)

$(C_{1})$ :

Let M be a nonnegative bounded integrable function on $[0,d]$. If $m\in C_{\alpha (1-\beta )}([0,d],\mathbb{R})$ satisfies

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }m(t)+M(t)m(t)\geq 0, \quad t\in (0,d], \\ t^{\alpha (1-\beta )}m(t)|_{t=0}\geq 0, \end{cases} $$

then $m(t)\geq 0$, $t\in (0,d]$.

$(C_{2})$ :

Assume that $n\in C_{\alpha (1-\gamma )}([0,d],\mathbb{R})$ satisfies

$$ \textstyle\begin{cases} {}^{\gamma }_{0}D^{\alpha }n(t)\geq 0, \quad t\in (0,d], \\ t^{\alpha (1-\gamma )}n(t)|_{t=0}\geq 0. \end{cases} $$

Then $n(t)\geq 0$, $t\in (0,d]$.

Proof

Assume that $m(t)\geq 0$ is not true. Then there exist $t_{1}, t_{2} \in (0,d]$ such that $m(t_{2})<0$, $m(t_{1})=0$ and $m(t)\geq 0$ for $t\in (0,t_{1})$ and $m(t)<0$ for $t\in (t_{1},t_{2})$. Since $M(t)\geq 0$, $\forall t\in [0,d]$, we have ${}^{\beta }_{0}D^{\alpha }m(t) \geq 0$, $\forall t\in (t_{1},t_{2})$.

According to

$$ {}^{\beta }_{0}D^{\alpha }m(t)=t^{1-\alpha } \frac{d}{dt}{{}_{0}^{1- \beta }I^{\alpha }m(t)}, $$

we obtain that ${}_{0}^{1-\beta }I^{\alpha }m(t)$ is nondecreasing on $(t_{1},t_{2})$. Hence ${}_{0}^{1-\beta }I^{\alpha }m(t)-{{}_{0}^{1- \beta }I^{\alpha }m(t_{1})}\geq 0$, $t\in (t_{1},t_{2})$. On the other hand, we have

$$ \begin{aligned} &{}_{0}^{1-\beta }I^{\alpha }m(t)-{{}_{0}^{1-\beta }I^{\alpha }m(t _{1})} \\ &\quad =\frac{1}{\varGamma (1-\beta )} \int _{0}^{t} \biggl(\frac{t^{\alpha }-s ^{\alpha }}{\alpha } \biggr)^{-\beta }m(s)\frac{ds}{s^{1-\alpha }}-\frac{1}{ \varGamma (1-\beta )} \int _{0}^{t_{1}} \biggl(\frac{{t_{1}}^{\alpha }-s^{ \alpha }}{\alpha } \biggr)^{-\beta }m(s)\frac{ds}{s^{1-\alpha }} \\ &\quad =\frac{1}{\varGamma (1-\beta )} \int _{0}^{t_{1}} \biggl[ \biggl( \frac{t^{ \alpha }-s^{\alpha }}{\alpha } \biggr)^{-\beta }- \biggl(\frac{{t_{1}}^{ \alpha }-s^{\alpha }}{\alpha } \biggr)^{-\beta } \biggr]m(s) \frac{ds}{s ^{1-\alpha }} \\ &\qquad{}+\frac{1}{\varGamma (1-\beta )} \int _{t_{1}}^{t} \biggl(\frac{{t}^{\alpha }-s^{\alpha }}{\alpha } \biggr)^{-\beta }m(s)\frac{ds}{s^{1-\alpha }} \\ &\quad < 0, \quad \forall t\in (t_{1},t_{2}), \end{aligned} $$

which is a contradiction. Therefore, $m(t)\geq 0$, $\forall t\in (0,d]$.

Obviously, the conclusion of $(C_{2})$ holds. It follows from (2.8) that $n(t)\geq 0$, $\forall t\in (0,d]$. □

3 Main results

Theorem 3.1

Assume that

$(L_{1})$ :

$h_{0}, l_{0}\in Y$ are lower and upper solutions of (1.1), respectively with $h_{0}(t)\leq l_{0}(t)$, $t\in (0,d]$;

$(L_{2})$ :

there exists a function $M\in C([0,d],\mathbb{R})$, $t\in [0,d]$ such that

$$ f\bigl(t, l(t), {}^{\gamma }_{0}D^{\alpha }l(t)\bigr)-f \bigl(t, h(t), {}^{\gamma }_{0}D ^{\alpha }h(t)\bigr)\geq -M(t)\bigl[\phi _{p}\bigl({}^{\gamma }_{0}D^{\alpha }l(t) \bigr)-\phi _{p}\bigl({}^{\gamma }_{0}D^{\alpha }h(t) \bigr)\bigr] $$

for $h_{0}(t)\leq h(t)\leq l(t)\leq l_{0}(t)$, $t\in (0,d]$;

$(L_{3})$ :

the function g satisfies

$$ g(m_{2},n_{2})-g(m_{1},n_{1}) \geq m_{2}-m_{1} $$

for $\tilde{h}_{0}(0)\leq m_{2}\leq m_{1}\leq \tilde{l}_{0}(0)$, $\tilde{h}_{0}(d)\leq n_{2}\leq n_{1}\leq \tilde{l}_{0}(d)$, if $M(t)d^{\alpha \beta }<\varGamma (\beta +1)\alpha ^{\beta }$.

Then there exist sequences $\{h_{n}\}, \{l_{n}\}\in Y$ such that (1.1) has extremal solutions $m(t)$, $n(t)$ in $[h_{0},l_{0}]=\{h\in Y:h_{0}(t)\leq h(t)\leq l_{0}(t), t\in (0,d]\}$ satisfying

$$ \textstyle\begin{cases} h_{0}(t)\leq h_{1}(t)\leq \cdots \leq h_{n}(t)\leq \cdots \leq m(t)\leq n(t)\leq \cdots \leq l_{n}(t)\leq \cdots \leq l_{1}(t)\leq l_{0}(t),\\ {}^{\gamma }_{0}D^{\alpha }h_{0}\leq {{}^{\gamma }_{0}D^{\alpha }h_{1}} \leq \cdots \leq {{}^{\gamma }_{0}D^{\alpha }h_{n}}\leq \cdots \leq {{}^{\gamma }_{0}D^{\alpha }m}\leq {{}^{\gamma } _{0}D^{\alpha }n}\leq \cdots \\ \hphantom{{}^{\gamma }_{0}D^{\alpha }h_{0}} \leq {{}^{\gamma }_{0}D^{\alpha }l_{n}}\leq \cdots \leq {{}^{\gamma }_{0}D^{\alpha }l_{1}} \leq {{}^{\gamma }_{0}D^{\alpha }l_{0}}, \\ \phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{0})\leq {\phi _{p}({}^{\gamma } _{0}D^{\alpha }h_{1})}\leq \cdots \leq {\phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{n})}\leq \cdots\leq {\phi _{p}({}^{\gamma }_{0}D^{\alpha }m)}\leq {\phi _{p}({}^{\gamma }_{0}D^{\alpha }n)}\leq \cdots \\ \hphantom{\phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{0})} \leq {\phi _{p}({}^{\gamma }_{0}D^{\alpha }l_{n})} \leq \cdots \leq {\phi _{p}({}^{\gamma }_{0}D^{\alpha }l _{1})}\leq {\phi _{p}({}^{\gamma }_{0}D^{\alpha }l_{0})}, \end{cases} $$

for $t\in (0,d]$, $n=1, 2, 3, \dots$.

Proof

Let $F(h(t))=f(t, h(t), {}^{\gamma }_{0}D^{\alpha }h(t))$. For $n=1,2, \dots$, we define

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }(\phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{n}(t)))+M(t) \phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{n}(t)) \\ \quad =F(h_{n-1}(t))+M(t)\phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{n-1}(t)),\quad t\in (0,d], \\ t^{\frac{\alpha (1-\beta )}{p-1}} {{}^{\gamma }_{0}D^{\alpha }h_{n}(t)}|_{t=0}= \int _{0}^{\tau }a(s)h_{n-1}(s)\,ds, \\ \tilde{h}_{n}(0)=\tilde{h}_{n-1}(0)-g(\tilde{h}_{n-1}(0), \tilde{h} _{n-1}(d)),\quad \tau \in (0,d), \end{cases} $$

(3.1)

and

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }(\phi _{p}({}^{\gamma }_{0}D^{\alpha }l_{n}(t)))+M(t) \phi _{p}({}^{\gamma }_{0}D^{\alpha }l_{n}(t)) \\ \quad =F(l_{n-1}(t))+M(t)\phi _{p}({}^{\gamma }_{0}D^{\alpha }l_{n-1}(t)),\quad t\in (0,d], \\ t^{\frac{\alpha (1-\beta )}{p-1}} {{}^{\gamma }_{0}D^{\alpha }l_{n}(t)}|_{t=0}= \int _{0}^{\tau }a(s)l_{n-1}(s)\,ds, \\ \tilde{l}_{n}(0)=\tilde{l}_{n-1}(0)-g(\tilde{l}_{n-1}(0), \tilde{l} _{n-1}(d)),\quad \tau \in (0,d). \end{cases} $$

(3.2)

Notice that the functions $h_{1}$, $l_{1}$ are well defined in Y by Lemma 2.3.

Now, we prove that $h_{0}(t)\leq h_{1}(t)\leq l_{1}(t)\leq l_{0}(t)$, ${}^{\gamma }_{0}D^{\alpha }h_{0}(t)\leq {{}^{\gamma }_{0}D^{\alpha }h _{1}(t)}\leq {{}^{\gamma }_{0}D^{\alpha }l_{1}(t)}\leq {{}^{\gamma }_{0}D ^{\alpha }l_{0}(t)}$, $t\in (0,d]$, and $\tilde{h}_{0}(0)\leq \tilde{h}_{1}(0)\leq \tilde{l}_{1}(0)\leq \tilde{l}_{0}(0)$. Let $\lambda (t)=\phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{1}(t))-\phi _{p}({}^{ \gamma }_{0}D^{\alpha }h_{0}(t))$. From (2.9), (3.1), and $(L_{1})$, we have

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }\lambda (t)+M(t)\lambda (t)=F(h_{0}(t))-{{}^{ \beta }_{0}D^{\alpha }(\phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{0}(t)))} \geq 0, \\ t^{\alpha (1-\beta )}\lambda (t)|_{t=0}=\phi _{p}(t^{\frac{\alpha (1- \beta )}{p-1}}{{}^{\gamma }_{0}D^{\alpha }h_{1}(t))}|_{t=0} -\phi _{p}(t ^{\frac{\alpha (1-\beta )}{p-1}}{{}^{\gamma }_{0}D^{\alpha }h_{0}(t))}|_{t=0} \\ \hphantom{t^{\alpha (1-\beta )}\lambda (t)|_{t=0}}\geq \int _{0}^{\tau }a(s)h_{0}(s)\,ds-\int _{0}^{\tau }a(s)h_{0}(s)\,ds=0. \end{cases} $$

By $(C_{1})$ of Lemma 2.4, we obtain $\lambda (t)\geq 0$, $t\in (0,d]$, which means $\phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{1}(t)) \geq \phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{0}(t))$. The monotone increasing property of $\phi _{p}(t)$ ensures that ${}^{\gamma }_{0}D ^{\alpha }h_{1}(t)\geq {{}^{\gamma }_{0}D^{\alpha }h_{0}(t)}$. Thus, ${}^{\gamma }_{0}D^{\alpha }(h_{1}(t)-h_{0}(t))\geq 0$. According to $\tilde{h}_{1}(0)-\tilde{h}_{0}(0)=-g(\tilde{h}_{0}(0), \tilde{h}_{0}(d)) \geq 0$, we have $h_{1}(t)\geq h_{0}(t)$, $t\in (0,d]$ by $(C_{2})$ of Lemma 2.4. In a similar manner, we can obtain that $l_{1}(t)\leq l_{0}(t)$, ${}^{\gamma }_{0}D^{\alpha }h_{1}(t)\leq {{}^{ \gamma }_{0}D^{\alpha }h_{0}(t)}$, $t\in (0,d]$, and $\tilde{l}_{1}(0) \leq \tilde{l}_{0}(0)$.

Setting $\eta (t)=\phi _{p}({}^{\gamma }_{0}D^{\alpha }l_{1}(t))-\phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{1}(t))$ and using $(L_{2})$, we have

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }\eta (t)+M(t)\eta (t)=F(l_{0}(t))-F(h_{0}(t))+M(t)[ \phi _{p}({}^{\gamma }_{0}D^{\alpha }l_{0}(t)) -\phi _{p}({}^{\gamma }_{0}D ^{\alpha }h_{0}(t))]\geq 0, \\ t^{\alpha (1-\beta )}\eta (t)|_{t=0}=\phi _{p}(t^{\frac{\alpha (1- \beta )}{p-1}}{{}^{\gamma }_{0}D^{\alpha }l_{1}(t))}|_{t=0} -\phi _{p}(t ^{\frac{\alpha (1-\beta )}{p-1}}{{}^{\gamma }_{0}D^{\alpha }h_{1}(t))}|_{t=0} \geq 0. \end{cases} $$

By $(C_{1})$ of Lemma 2.4, we obtain $\eta (t)\geq 0$, $t\in (0,d]$. Then $\phi _{p}({}^{\gamma }_{0}D^{\alpha }l_{1}(t))\geq \phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{1}(t))$, and ${}^{\gamma }_{0}D ^{\alpha }l_{1}(t)\geq {{}^{\gamma }_{0}D^{\alpha }h_{1}(t)}$. By $(L_{3})$, we have

$$ \begin{aligned} \tilde{l}_{1}(0)-\tilde{h}_{1}(0)&= \tilde{l}_{0}(0)-g\bigl(\tilde{l}_{0}(0), \tilde{l}_{0}(d)\bigr)-\tilde{h}_{0}(0)+g\bigl( \tilde{h}_{0}(0), \tilde{h}_{0}(d)\bigr) \\ &=\tilde{l}_{0}(0)-\tilde{h}_{0}(0)+g\bigl( \tilde{h}_{0}(0), \tilde{h} _{0}(d)\bigr)-g\bigl( \tilde{l}_{0}(0), \tilde{l}_{0}(d)\bigr) \\ &\geq \tilde{l}_{0}(0)-\tilde{h}_{0}(0)+ \tilde{h}_{0}(0)-\tilde{l} _{0}(0)=0. \end{aligned} $$

Thus, $l_{1}(t)\geq h_{1}(t)$, $t\in (0,d]$ by $(C_{2})$ of Lemma 2.4.

Next, we show that $h_{1}$, $l_{1}$ are lower and upper solutions of (1.1), respectively. By (3.1) and $(L_{2})$, we obtain

$$ \begin{aligned} &{}^{\beta }_{0}D^{\alpha } \bigl(\phi _{p}\bigl({}^{\gamma }_{0}D^{\alpha }h_{1}(t) \bigr)\bigr)\\ &\quad =F\bigl(h _{0}(t)\bigr)-M(t)\bigl[\phi _{p} \bigl({}^{\gamma }_{0}D^{\alpha }h_{1}(t) \bigr) -\phi _{p}\bigl({}^{ \gamma }_{0}D^{\alpha }h_{0}(t) \bigr)\bigr]-F\bigl(h_{1}(t)\bigr)+F\bigl(h_{1}(t)\bigr) \\ &\quad \leq M(t)\bigl[\phi _{p}\bigl({}^{\gamma }_{0}D^{\alpha }h_{1}(t) \bigr)-\phi _{p}\bigl({}^{ \gamma }_{0}D^{\alpha }h_{0}(t) \bigr)\bigr] -M(t)\bigl[\phi _{p}\bigl({}^{\gamma }_{0}D^{ \alpha }h_{1}(t) \bigr) \\ &\qquad{}-\phi _{p}\bigl({}^{\gamma }_{0}D^{\alpha }h_{0}(t) \bigr)\bigr]+F\bigl(h_{1}(t)\bigr) \\ &\quad =F\bigl(h_{1}(t)\bigr). \end{aligned} $$

By $(L_{3})$, we have

$$ \begin{aligned} 0&=g\bigl(\tilde{h}_{0}(0), \tilde{h}_{0}(d)\bigr)-g\bigl(\tilde{h}_{1}(0), \tilde{h}_{1}(d)\bigr)+g\bigl(\tilde{h}_{1}(0), \tilde{h}_{1}(d)\bigr)+\tilde{h}_{1}(0)- \tilde{h}_{0}(0) \\ &\geq \tilde{h}_{0}(0)-\tilde{h}_{1}(0)+g\bigl( \tilde{h}_{1}(0), \tilde{h}_{1}(d)\bigr)+ \tilde{h}_{1}(0)-\tilde{h}_{0}(0) \\ &=g\bigl(\tilde{h}_{1}(0), \tilde{h}_{1}(d)\bigr), \end{aligned} $$

and

$$ t^{\frac{\alpha (1-\beta )}{p-1}} {{}^{\gamma }_{0}D^{\alpha }h_{1}(t)}|_{t=0}= \int _{0}^{\tau }a(s)h_{0}(s)\,ds\leq \int _{0}^{\tau }a(s)h_{1}(s)\,ds, $$

(3.3)

which imply that $h_{1}$ is a lower solution of (1.1). Analogously, we can verify that $l_{1}$ is an upper solution of (1.1).

Using the mathematical induction, we have

$$\begin{aligned}& \begin{aligned} &h_{0}(t)\leq h_{1}(t)\leq \cdots \leq h_{n}(t)\leq h_{n+1}(t) \leq l_{n+1}(t)\leq l_{n}(t)\leq \cdots \leq l_{1}(t)\leq l_{0}(t), \\ &{}^{\gamma }_{0}D^{\alpha }h_{0}\leq {{}^{\gamma }_{0}D^{\alpha }h_{1}} \leq \cdots \leq {{}^{\gamma }_{0}D^{\alpha }h_{n}} \leq {{}^{\gamma }_{0}D^{\alpha }h_{n+1}}\leq {{}^{\gamma }_{0}D^{\alpha }l _{n+1}}\leq {{}^{\gamma }_{0}D^{\alpha }l_{n}}\leq \cdots \\ &\hphantom{{}^{\gamma }_{0}D^{\alpha }h_{0}}\leq {{}^{\gamma }_{0}D^{\alpha }l_{1}} \leq {{}^{\gamma }_{0}D^{\alpha }l _{0}}, \\ &\tilde{h}_{0}(0)\leq \tilde{h}_{1}(0)\leq \cdots \leq \tilde{h}_{n}(0)\leq \tilde{h}_{n+1}(0)\leq \tilde{l}_{n+1}(0)\leq \tilde{l}_{n}(0) \leq \cdots \leq \tilde{l}_{1}(0)\leq \tilde{l}_{0}(0) \end{aligned} \end{aligned}$$

(3.4)

for $t\in (0,d]$, $n=1, 2, 3, \dots$.

By the standard analysis, we can get that the sequences $\{t^{\alpha (1-\gamma )}h_{n}\}$ and $\{t^{\alpha (1-\gamma )}l_{n}\}$ are uniformly bounded and equicontinuous. Thus, in view of Arzela–Ascoli theorem, we obtain

$$ \begin{aligned} &\lim_{n\rightarrow \infty }h_{n}(t)=m(t), \qquad \lim_{n\rightarrow \infty }l_{n}(t)=n(t), \quad t\in (0,d], \\ &\lim_{n\rightarrow \infty }{{}^{\gamma }_{0}D^{\alpha }h_{n}(t)}= {{}^{\gamma }_{0}D^{\alpha }m(t)}, \qquad \lim _{n\rightarrow \infty }{{}^{\gamma }_{0}D^{\alpha }l_{n}(t)} ={{}^{ \gamma }_{0}D^{\alpha }n(t)}, \quad t\in (0,d]. \end{aligned} $$

Hence, $h_{0}(t)\leq m(t)\leq n(t)\leq l_{0}(t)$ on $(0,d]$ and $m(t)$, $n(t)$ are solutions of (1.1).

Moreover, we show that $m(t)$, $n(t)$ are extremal solutions of (1.1). Let $h\in [h_{0},l_{0}]$ be any solution of (1.1). Let $h_{n}(t)\leq h(t)\leq l_{n}(t)$, $t\in (0,d]$ and that

$$ j(t)=\phi _{p}\bigl({}^{\gamma }_{0}D^{\alpha }h(t) \bigr)-\phi _{p}\bigl({}^{\gamma }_{0}D ^{\alpha }h_{n+1}(t)\bigr), \qquad k(t)=\phi _{p} \bigl({}^{\gamma }_{0}D^{\alpha }l_{n+1}(t) \bigr)-\phi _{p}\bigl({}^{\gamma }_{0}D^{\alpha }h(t) \bigr). $$

By $(L_{2})$, we obtain

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }j(t)+M(t)j(t)=F(h(t))-F(h_{n}(t))+M(t)[\phi _{p}({}^{\gamma }_{0}D^{\alpha }h(t)) -\phi _{p}({}^{\gamma }_{0}D^{\alpha }h_{n}(t))]\geq 0, \\ t^{\alpha (1-\beta )}j(t)|_{t=0}=\phi _{p}(t^{ \frac{\alpha (1-\beta )}{p-1}}{{}^{\gamma }_{0}D^{\alpha }h(t))}|_{t=0} -\phi _{p}(t^{\frac{\alpha (1-\beta )}{p-1}}{{}^{\gamma }_{0}D^{\alpha }h _{n+1}(t))}|_{t=0}\geq 0, \end{cases} $$

and

$$ \textstyle\begin{cases} {}^{\beta }_{0}D^{\alpha }k(t)+M(t)k(t)=F(l_{n}(t))-F(h(t))+M(t)[\phi _{p}({}^{\gamma }_{0}D^{\alpha }l_{n}(t)) -\phi _{p}({}^{\gamma }_{0}D^{ \alpha }h(t))]\geq 0, \\ t^{\alpha (1-\beta )}k(t)|_{t=0}=\phi _{p}(t^{ \frac{\alpha (1-\beta )}{p-1}}{{}^{\gamma }_{0}D^{\alpha }l_{n+1}(t))}|_{t=0} -\phi _{p}(t^{\frac{\alpha (1-\beta )}{p-1}}{{}^{\gamma }_{0}D^{\alpha }h(t))}|_{t=0} \geq 0. \end{cases} $$

Thus, by $(C_{1})$ of Lemma 2.4, we have $j(t)\geq 0$, $k(t) \geq 0$. Then $\phi _{p}({}^{\gamma }_{0}D^{\alpha }h(t))\geq \phi _{p}({}^{ \gamma }_{0}D^{\alpha }h_{n+1}(t))$, $\phi _{p}({}^{\gamma }_{0}D^{ \alpha }l_{n+1}(t))\geq \phi _{p}({}^{\gamma }_{0}D^{\alpha }h(t))$. Hence, ${}^{\gamma }_{0}D^{\alpha }(h(t)-h_{n+1}(t))\geq 0$, ${}^{\gamma }_{0}D ^{\alpha }(l_{n+1}(t)-h(t))\geq 0$.

By $(L_{3})$, we have

$$ \begin{aligned} \tilde{h}(0)-\tilde{h}_{n+1}(0)&= \tilde{h}(0)-\tilde{h}_{n}(0)+g\bigl( \tilde{h}_{n}(0), \tilde{h}_{n}(d)\bigr)-g\bigl(\tilde{h}(0), \tilde{h}(d)\bigr) \\ &\geq \tilde{h}(0)-\tilde{h}_{n}(0)+\tilde{h}_{n}(0)- \tilde{h}(0) \\ &=0 \end{aligned} $$

and

$$ \begin{aligned} \tilde{l}_{n+1}(0)-\tilde{h}(0)&= \tilde{l}_{n}(0)-\tilde{h}(0)-g\bigl( \tilde{l}_{n}(0), \tilde{l}_{n}(d)\bigr)+g\bigl(\tilde{h}(0), \tilde{h}(d)\bigr) \\ &\geq \tilde{l}_{n}(0)-\tilde{h}(0)+\tilde{h}(0)- \tilde{l}_{n}(0) \\ &=0. \end{aligned} $$

Hence, $h_{n+1}(t)\leq h(t)\leq l_{n+1}(t)$, $t\in (0,d]$ by $(C_{2})$ of Lemma 2.4, which, on taking the limit $n\rightarrow \infty$, yields $m(t)\leq h(t)\leq n(t)$. Therefore, $m(t)$, $n(t)$ are extremal solutions of (1.1). □

Theorem 3.2

If the hypotheses of Theorem 3.1 hold, $a(t)=0$, and there exists a function $L(t)\geq 0$ such that

$$ L(t)\bigl[\phi _{p}\bigl({}_{0}^{\gamma }D^{\alpha }l(t) \bigr)-\phi _{p}\bigl({}_{0}^{\gamma }D ^{\alpha }h(t)\bigr)\bigr]\leq f\bigl(t,h(t),{{}_{0}^{\gamma }D^{\alpha }h(t)} \bigr)-f\bigl(t,l(t), {{}_{0}^{\gamma }D^{\alpha }l(t)} \bigr) $$

(3.5)

for $h_{0}(t)\leq h(t)\leq l(t)\leq l_{0}(t)$, $t\in (0,d]$ and $\tilde{h}_{0}(0)=\tilde{l}_{0}(0)$, then (1.1) has a unique solution in $[h_{0},l_{0}]$.

Proof

It follows by Theorem 3.1 that $m(t)$ and $n(t)$ are extremal solutions such that $m(t)\leq n(t)$, $t\in (0,d]$. Then we just need to prove $m(t)\geq n(t)$, $t\in (0,d]$. Letting $\lambda (t)=\phi _{p}({}_{0} ^{\gamma }D^{\alpha }m(t))-\phi _{p}({}_{0}^{\gamma }D^{\alpha }n(t))$, $t \in (0,d]$ and using (3.5), we obtain

$$ \textstyle\begin{cases} _{0}^{\beta }D^{\alpha }\lambda (t)=F(m(t))-F(n(t))\geq L(t)[\phi _{p}({}_{0} ^{\gamma }D^{\alpha }n(t))-\phi _{p}({}_{0}^{\gamma }D^{\alpha }m(t))]=-L(t) \lambda (t), \\ t^{\alpha (1-\beta )}\lambda (t)|_{t=0}=\phi _{p}(t^{\frac{\alpha (1- \beta )}{p-1}}{{}^{\gamma }_{0}D^{\alpha }m(t))}|_{t=0} -\phi _{p}(t^{\frac{ \alpha (1-\beta )}{p-1}}{{}^{\gamma }_{0}D^{\alpha }n(t))}|_{t=0}=0. \end{cases} $$

(3.6)

Then, by $(C_{1})$ of Lemma 2.4, we have $\lambda (t)\geq 0$. Thus, $\phi _{p}({}_{0}^{\gamma }D^{\alpha }m(t))\geq \phi _{p}({}_{0}^{ \gamma }D^{\alpha }n(t))$. Since $\phi _{p}(t)$ is nondecreasing, we have ${}^{\gamma }_{0}D^{\alpha }m(t)\geq {{}^{\gamma }_{0}D^{\alpha }n(t)}$, $t\in (0,d]$. Then, by $(C_{2})$ of Lemma 2.4, we obtain $m(t)\geq n(t)$. Furthermore, we have $\tilde{m}(0)=\tilde{n}(0)$ by $\tilde{h}_{0}(0)=\tilde{l}_{0}(0)$ and (3.4). Therefore, we have $m=n$. The proof is completed. □

4 Example

Consider the following problem:

$$ \textstyle\begin{cases} {}^{\frac{2}{3}}_{0}D^{{\frac{1}{2}}}(\phi _{3}({}^{\frac{1}{2}}_{0}D^{ \frac{1}{2}}h(t)))=f(t, h(t),{}^{\frac{1}{2}}_{0}D^{\frac{1}{2}}h(t)), \quad t\in (0,1], \\ t^{\frac{1}{12}} {{}^{\frac{1}{2}}_{0}D^{\frac{1}{2}}h(t)}|_{t=0}=\int _{0}^{\tau }a(s)h(s)\,ds, \qquad \frac{1}{2}\tilde{h}(0)-3\tilde{h}(0)\tilde{h}(1)=0, \end{cases} $$

(4.1)

where $\alpha =\frac{1}{2}$, $\gamma =\frac{1}{2}$, $\beta = \frac{2}{3}$, $d=1$, $p=3$, $a(t)=0$, $\tau =1$, and $f(t, h(t),{}^{ \frac{1}{2}}_{0}D^{\frac{1}{2}}h(t))=\frac{1}{2}t+h(t)-2{}^{ \frac{1}{2}}_{0}D^{\frac{1}{2}}h(t)$, $g(m,n)=\frac{1}{2}m-3mn$. Let $h_{0}(t)=0$, $l_{0}(t)=\varGamma (\frac{1}{2})t^{\frac{1}{2}}$. Then we have ${}^{\frac{1}{2}}_{0}D^{\frac{1}{2}}h_{0}(t)=0$, ${}^{\frac{1}{2}} _{0}D^{\frac{1}{2}}l_{0}(t)=2^{\frac{1}{2}}t^{\frac{1}{4}}$, and

$$\begin{aligned}& {}^{\frac{2}{3}}_{0}D^{{\frac{1}{2}}}\bigl(\phi _{3} \bigl({}^{\frac{1}{2}}_{0}D^{ \frac{1}{2}}h_{0}(t) \bigr)\bigr)=0\leq \frac{1}{2}t=f\bigl(t, h_{0}(t),{}^{ \frac{1}{2}}_{0}D^{\frac{1}{2}}h_{0}(t) \bigr), \quad t\in (0,1], \\& t^{\frac{1}{12}} {{}^{\frac{1}{2}}_{0}D^{\frac{1}{2}}h_{0}(t)}|_{t=0}=0, \qquad g\bigl(\tilde{h}_{0}(0),\tilde{h}_{0}(1) \bigr)=0, \\& \begin{aligned} {}^{\frac{2}{3}}_{0}D^{{\frac{1}{2}}}\bigl(\phi _{3} \bigl({}^{\frac{1}{2}}_{0}D^{ \frac{1}{2}}l_{0}(t) \bigr)\bigr)&={{}^{\frac{2}{3}}_{0}D^{{\frac{1}{2}}} \bigl(2t^{ \frac{1}{2}}\bigr)} =\frac{3\cdot 2^{\frac{1}{3}}}{\varGamma (\frac{1}{3})}t ^{\frac{1}{6}} \geq \frac{1}{2}t+\varGamma \biggl(\frac{1}{2}\biggr)t^{\frac{1}{2}}-2^{ \frac{3}{2}}t^{\frac{1}{4}}\\ &=f \bigl(t, l_{0}(t),{}^{\frac{1}{2}}_{0}D^{ \frac{1}{2}}l_{0}(t) \bigr), \end{aligned} \\& t^{\frac{1}{12}} {{}^{\frac{1}{2}}_{0}D^{\frac{1}{2}}l_{0}(t)}|_{t=0}=0, \qquad g\bigl(\tilde{l}_{0}(0),\tilde{l}_{0}(1) \bigr)=0. \end{aligned}$$

Thus, $h_{0}$ and $l_{0}$ are lower and upper solutions of (4.1), respectively, and $h_{0}\leq l_{0}$ on $[0,1]$.

In addition, for $h_{0}\leq h\leq l\leq l_{0}$, we have

$$ \begin{aligned} &f\bigl(t,h(t),{}^{\frac{1}{2}}_{0}D^{\frac{1}{2}}h(t) \bigr)-f\bigl(t,l(t),{}^{ \frac{1}{2}}_{0}D^{\frac{1}{2}}l(t) \bigr) \\ &\quad =h(t)-l(t)-2{{}^{\frac{1}{2}}_{0}D^{\frac{1}{2}}h(t)}+2{{}^{\frac{1}{2}} _{0}D^{\frac{1}{2}}l(t)} \\ &\quad \leq 2\bigl[^{\frac{1}{2}}_{0}D^{\frac{1}{2}}l(t)-{{}^{\frac{1}{2}}_{0}D ^{\frac{1}{2}}h(t)\bigr]} \\ &\quad \leq M(t)\bigl[\phi _{3}\bigl({}^{\frac{1}{2}}_{0}D^{\frac{1}{2}}l(t) \bigr)-\phi _{3}\bigl({}^{ \frac{1}{2}}_{0}D^{\frac{1}{2}}h(t) \bigr)\bigr], \end{aligned} $$

where $M(t)=2$.

For $\tilde{h}_{0}(0)\leq m_{2}\leq m_{1}\leq \tilde{l}_{0}(0)$, $\tilde{h}_{0}(1)\leq n_{2}\leq n_{1}\leq \tilde{l}_{0}(1)$, we have

$$ \begin{aligned} g(m_{1},n_{1})-g(m_{2},n_{2})&= \frac{1}{2}m_{1}-3m_{1}n_{1}- \frac{1}{2}m_{2}+3m_{2}n_{2} \\ &\leq \frac{1}{2}(m_{1}-m_{2}) \leq m_{1}-m_{2}. \end{aligned} $$

Hence, assumptions $(L_{1})$, $(L_{2})$, and $(L_{3})$ hold. According to Theorem 3.1, there exist monotone iterative sequences $\{h_{n}\}$, $\{l_{n}\}$ such that $\lim_{n\rightarrow \infty }h_{n}=m$, $\lim_{n\rightarrow \infty }l_{n}=n$ on $(0,1]$ and m, n are the extremal solutions on $[h_{0},l_{0}]$ of (4.1).