Fractional differential equations arise in many engineering and scientific disciplines. Recently, more and more scholars are interested in fractional differential equations, see [1-7]. For example, Arafa, Rida and Khalil [6] used the following fractional order model to describe the efficacy of anti-viral drugs in the treatment of human immunodeficiency virus type 1 (HIV-1):

$ \begin{array}{l} \left\{ \begin{array}{l} {D^{{\alpha _1}}}(x) = s - \mu x - \beta xz, \hfill \\ {D^{{\alpha _2}}}(y) = \beta xz - \varepsilon y, \hfill \\ {D^{{\alpha _3}}}(z) = cy - \gamma z, \hfill \\ \end{array} \right. \end{array} $

where $ {D^{{\alpha _1}}}, {D^{{\alpha _2}}}, {D^{{\alpha _3}}} $ are Caputo fractional derivatives with $ 0<{\alpha _1}, {\alpha _2}, {\alpha _3}\leq 1 $, all parameters and variables are non-negative, $ x $, $ y $ is the number of uninfected and infected CD4+ T-cells, respectively, $ z $ is the number of virions in plasma, $ s $ is the assumed constant rate of production of CD4+ T-cells, $ l $ is their per capita death rate, $ b $ is the rate of infection of CD4+ T-cells by virus, $ e $ is the per capita rate of disappearance of infected cells, and $ c $ is the death rate of virus particles. Ates and Zegeling [7] investigated the fractional-order advection-diffusion reaction boundary value problems:

$ \begin{array}{l} \left\{ \begin{array}{l} \varepsilon {}^C{D^\alpha }u + \gamma u' + f\left( u \right) = S\left( x \right), \quad x \in \left[ {0, 1} \right], \hfill \\ u\left( 0 \right) = {u_L}, u\left( 1 \right) = {u_R}, \hfill \\ \end{array} \right. \end{array} $

where $ 1 < \alpha \le 2, \, 0 < \varepsilon \le 1, \, \gamma \in \mathbb{R}, \, {}^C{D^\alpha } $ is the Caputo fractional derivative.

In recent years, more and more attention are being paid to the existence of solutions for fractional $ p $-Laplacian problems. And many important results have been achieved in this regard, see [8-14]. The $ p $-Laplacian equation was derived from the following nonlinear diffusion equation proposed by Leibenson [15] in 1983, when studying the one-dimensional variable turbulent flow of gases through porous media

$ u_t=\frac{\partial}{\partial x} \left(\frac{\partial u^m}{\partial x}\left|\frac{\partial u^m}{\partial x}\right|^{\mu-1}\right), \ \ m=n+1. $

When $ m>1 $, the above equation is the porous media equation, when $ 0<m<1 $, the above equation is the fast diffusion equation, and when $ m=1 $, the above equation is the heat equation, while when $ m=1, \mu\neq1 $, such equation often appears in the study of non-Newton fluids. Given the importance of such equations, the above equation is abstracted into the following $ p $-Laplacian equation

$ (\phi _p (u'))' = f(t, u), $

where $ \phi_p(s)=|s|^{p-2}s\ (s\neq0), \phi_p(0)=0, p>1. $ Note that when $ p=2 $, $ p $-Laplacian equation degenerates into a classical second-order differential equation. Naturally, in view of its significance in theory and practice, more and more people are concerned about the existence of solutions for fractional $ p $-Laplacian problems. For example, Wang [16] studied $ p $-Laplacian problems:

$ \begin{array}{l} \left\{ \begin{array}{l} D_{0 + }^\gamma {\varphi _p}\left( {D_{0 + }^\alpha x\left( t \right)} \right) = f\left( {t, x\left( t \right)} \right), 0 < t < 1, \hfill \\ x\left( 0 \right) = 0, x\left( 1 \right) = ax\left( \xi \right), D_{0 + }^\alpha x\left( 0 \right) = 0, D_{0 + }^\alpha x\left( 1 \right) = bD_{0 + }^\alpha x\left( \eta \right), \hfill \\ \end{array} \right. \end{array} $

where $ 1<\alpha, \gamma \leq2 $, $ 0 \le a, b \le 1 $, $ 0<\xi, \eta<1, $ $ D_{0 + }^\alpha $ is Riemann-Liouville fractional derivative. The existence results of positive solution for the problem were obtained by lower and upper solutions method. Tian [17] considered the following $ p $-Laplacian problems:

$ \begin{array}{l} \left\{ \begin{array}{l} D_{0 + }^\alpha {\varphi _p}\left( {D_{0 + }^\beta x\left( t \right)} \right) + f\left( {t, x\left( t \right)} \right) = 0, 0 < t < 1, \hfill \\ x\left( 0 \right) = 0, D_{0 + }^\gamma x\left( 1 \right) = \lambda D_{0 + }^\gamma x\left( \xi \right), D_{0 + }^\beta x\left( 0 \right) = 0, \hfill \\ \end{array} \right. \end{array} $

where $ 0<\alpha<1 $, $ 1<\beta\leq2 $, $ 0< \gamma \le 1 $, $ 0<\xi<1, $ $ 1+\gamma \le \beta $, $ \lambda \in[0, \infty) $, $ D_{0 + }^\alpha $ is Riemann-Liouville fractional derivative. By using the fixed point theorem on the cone, the existence results of positive solution for this problem were obtained. Chen and Liu [18] discussed the following problems:

$ \begin{array}{l} \left\{ \begin{array}{l} {}^CD_{0 + }^\beta {\phi _p}\left( {{}^CD_{0 + }^\alpha x} \right) = f\left( {t, x} \right), t \in [0, 1], \hfill \\ x\left( 0 \right) = - x\left( 1 \right), {}^CD_{0 +}^\alpha x\left( 0 \right) = - {}^CD_{0 + }^\alpha x\left( 1 \right), \hfill \\ \end{array} \right. \end{array} $

where $ 0 < \alpha $, $ \beta \le 1 $, $ 1<\alpha, \beta\leq2 $, $ ^CD_{0 + }^\alpha $ is Caputo fractional derivative, $ f:\left[ {0, 1} \right] \times {\mathbb{R}} \to {\mathbb{R}} $ is continuous, $ {\phi _p}\left( \cdot \right) $ is $ p $-Laplacian operator defined by $ {\phi _p}\left( s \right) = {\left| s \right|^{p\!-\! 2}}s (s \ne 0, p > 1), {\phi _p}\left( 0 \right) = 0 $. Note that, when $ p=2 $, the nonlinear operator $ {}^CD_{0 + }^\beta {\phi _p}\left( {{}^CD_{0 + }^\alpha } \right) $ reduces to the linear operator. By Schaefer's fixed point theorem, the existence results of solutions for the problem were obtained.

An interesting and effective method used to prove the existence of positive solutions for fractional differential problems at resonance is Leggett-Williams norm-type theorem. Many existence results of positive solution for fractional boundary value problems at resonance with the linear derivative operator have been obtained, see literature [19-28]. However, as far as we know, only Jiang [29] studied the existence of positive solutions for the following fractional problems with $ p $-Laplacian operator at resonance:

$ \begin{array}{l} \left\{ \begin{array}{l} {}^CD_{0 + }^\beta {\varphi _p}\left( {{}^CD_{0 + }^\alpha x\left( t \right)} \right) = f\left( {t, {}^CD_{0 + }^\alpha x\left( t \right)} \right), t \in \left( {0, 1} \right), \hfill \\ {}^CD_{0 + }^\alpha x\left( 0 \right) ={}^C D_{0 + }^\alpha x\left( 1 \right), {x^{\left( i \right)}}{\rm{(0) = 0, }} i = 0, 1, 2, \cdots , n - 1, \hfill \\ \end{array} \right. \end{array} $

where $ 0<\beta <1 $, $ n-1<\alpha \leq n $, $ {}^C D_{0 + }^\alpha $ is Caputo fractional derivative, $ {\varphi _{p}}(s) = |s{|^{p - 2}} s, $ $ p>1 $. By using Leggett-Williams norm-type theorem, the existence results of positive solutions for the problem with a nonlinear derivative operator at resonance were obtained.

Inspired by the above excellent results, first, this paper will study the existence of positive solutions for the following $ p $-Laplacian boundary value problem

$ \begin{equation} {}^CD_{0 + }^\beta {\varphi _{p}}({}^CD_{0 + }^\alpha x(t)) = f(t, {}^CD_{0 + }^\alpha x(t)), t \in (0, 1), \end{equation} $

(1.1)

$ \begin{equation} {}^CD_{0 + }^\alpha x(1) = {}^CD_{0 + }^\alpha x(\delta ), {x^{(i)}}(0) = 0, i = 0, 1, 2, \cdots , n - 1, \end{equation} $

(1.2)

where $ n - 1 < \alpha \le n $, $ 0 < \beta < 1 $, $ {}^CD_{0 + }^\alpha $, $ {}^CD_{0 + }^\beta $ are Caputo fractional derivatives, $ 0 < \delta < 1, $ $ {\varphi _{p}}(s) = |s{|^{p - 2}} s, $ $ p>1, $ $ f:\left[ {0, 1} \right] \times \mathbb{R} \to \mathbb{R} $ is continuous.

On the other hand, we discuss problem (1.1) with the following integral boundary conditions:

$ \begin{equation} x(0) = 0, \quad {\varphi _p}{(^C}D_{0 + }^\alpha x(1)) = \int_0^1 {h(t){\varphi _p}{(^C}D_{0 + }^\alpha x(t))} dt, \end{equation} $

(1.3)

where $ {}^CD_{0 + }^\alpha $, $ {}^CD_{0 + }^\beta $ are Caputo fractional derivatives, $ 0 < \alpha, $ $ \beta<1, $ $ h(t) \ge 0, \int_0^1 {h(t)dt}=1, $ $ {\varphi _{p}}(s) = s|s{|^{p - 2}}, $ $ p>1, $ $ f:\left[ {0, 1} \right] \times \mathbb{R} \to \mathbb{R} $ is continuous.

Let us emphasize the contribution of our article: firstly, as far as we know, there is no paper on the existence of positive solutions for fractional $ p $-Laplacian boundary value problems (1.1)(1.2) and (1.1)(1.3) at resonance, so our article enriches some existing articles. Secondly, our article serves as a further development for the result of [29]. When $ \delta = 0 $, the results of [29] will be a special case of our result.

To facilitate understanding, this section introduces some concepts and lemmas related to this article. For more details, please refer to the references hereunder (see [20, 30, 31]).

Definition 2.1 ([30]) Let $ X $, $ Y $ be real Banach spaces, and $ L:\mbox{dom}L\subset X\rightarrow Y $ be a linear map. If $ \dim {\rm{Ker}}L = {\rm{codim}} {\mathop{\rm Im}\nolimits} L < + \infty $ and $ {\mathop{\rm Im}\nolimits} L $ is a closed subset in $ Y $, then the map $ L $ is a Fredholm operator with index zero. If there exists the continuous projections $ P:X\rightarrow X $ and $ Q:Y\rightarrow Y $ satisfying $ {\mathop{\rm \mbox{Im}}\nolimits}P =\mbox{Ker}L $ and $ \mbox{Ker}Q={\mathop{\rm \mbox{Im}}\nolimits}L $, then $ L \mid_{\mbox{dom}L\cap \mbox{Ker}P}:\mbox{dom}L\cap \mbox{Ker}P\rightarrow \mbox{Im}L $ is reversible. We denote the inverse of this map by $ K_P $, i.e. $ {K_P} = L_P^{ - 1} $ and $ {K_{P, Q}} = {K_P}\left( {I - Q} \right) $. Moreover, since $ \dim {\rm{Im}} Q = {\rm{codimIm}}L $, there exists an isomorphism $ J:{\rm{Im}} Q \rightarrow {\rm{Ker}} L $. It is known that the operator equation $ Lx=Nx $ is equivalent to

$ x=(P+JQN)x+K_P(I-Q)Nx, $

where $ N:X \rightarrow Y $ be a nonlinear operator. If $ \Omega $ is an open bounded subset of $ X $ and $ {\rm{dom}}L \cap \Omega \ne \varnothing $, then the map $ N $ is $ L $-compact on $ \overline \Omega $ when $ QN:\overline \Omega \to Y $ is bounded and $ {K_P}\left( {I - Q} \right)N:\overline \Omega \to X $ is compact.

Let $ C $ be a cone in $ X $. Then $ C $ induces a partial order in $ X $ by $ x \le y\quad {\rm{iff}} \quad y - x \in C. $

Lemma 2.1 ([20]) Let $ C $ be a cone in $ X $. Then for every $ u \in C\backslash \{ 0\} $ there exists a positive number $ \sigma (u) $ such that $ \left\| {x + u} \right\| \ge \sigma (u)\left\| x \right\| $ for all $ x \in C. $ Let $ \gamma :X \to C $ be a retraction, that is, a continuous mapping such that $ \gamma (x)=x $ for all $ x \in C $. Set

$ \Psi :=P+JQN+K_P(I-Q)N \quad {\rm{and}} \quad \Psi_\gamma :=\Psi \circ \gamma. $

Lemma 2.2 ([20]) Let $ C $ be a cone in $ X $ and $ \Omega_1, \Omega_2 $ be open bounded subsets of $ X $ with $ \overline {{\Omega _1}} \subset \Omega_2 $ and $ C \cap (\overline {{\Omega _2}} \backslash {\Omega _1}) \ne \emptyset $. Assume that the following conditions are satisfied:

(1) $ L:\mbox{dom}L\subset X\rightarrow Y $ be a Fredholm operator of index zero and $ N:X \rightarrow Y $ be $ L $-compact on every bounded subset of $ X $.

(2) $ Lx\neq \lambda Nx $ for every $ (x, \lambda ) \in [C \cap \partial {\Omega _2} \cap {\rm{dom}}L] \times (0, 1). $

(3) $ \gamma $ maps subsets of $ \overline {{\Omega _2}} $ into bounded subsets of $ C $.

(4) $ {\rm{deg}}([I - (P + JQN)\gamma ]{|_{{\rm{Ker}}L}}, {\rm{Ker}}L \cap {\Omega _2}, 0) \ne 0. $

(5) there exists $ u_0 \in C\backslash \{ 0\} $ such that $ \left\| x \right\| \le \sigma ({u_0})\left\| \Psi x \right\| $ for $ x \in C(u_0) \cap \partial {\Omega _1} $, where $ C(u_0)=\{ x \in C: \mu u_0 \leq x\} $ for some $ \mu >0 $ and $ \sigma (u_0) $ are such that $ \left\| {x + u_0} \right\| \ge \sigma (u_0)\left\| x \right\| $ for every $ x \in C $.

(6) $ (P+JQN)\gamma (\partial {\Omega _2})\subset C. $

(7) $ \Psi_\gamma (\overline {{\Omega _2}} \backslash {\Omega _1})\subset C. $

Then the equation $ Lx=Nx $ has at least one solution in $ C\cap (\overline {{\Omega _2}} \backslash {\Omega _1}) $.

Definition 2.2 ([31]) The Riemann-Liouville fractional integral of order $ \alpha (\alpha > 0) $ for the function $ x:(0, + \infty ) \to \mathbb{R} $ is defined as

$ I_{0 + }^\alpha x(t) = \frac{1}{{\Gamma (\alpha )}}\int_0^t {{{(t - s)}^{\alpha - 1}}x(s)\mbox{d}s, } $

provided that the right-hand side integral is defined on $ (0, + \infty) $.

Definition 2.3 ([31]) The Captuo fractional derivative of order $ \alpha (\alpha > 0) $ for the function $ x:(0, + \infty ) \to \mathbb{R}: $ is defined as

$ ^{C}D_{0 + }^\alpha x(t) =I_{0 + }^{n - \alpha } \frac{{{d^n}x(t)}}{{d{t^n}}} = \frac{1}{{\Gamma (n - \alpha )}}\int_0^t {{{(t - s)}^{n - \alpha - 1}}x^{(n)}(s)\mbox{d}s, } $

where $ n = [ \alpha ]+1 $, provided that the right-hand side integral is defined on $ (0, + \infty) $.

Lemma 2.3 ([31]) Assume $ x\in L[0, 1], \alpha > \beta \ge 0, \alpha > 1 $, then $ {}^CD_{0 + }^\beta I_{0 + }^\alpha x(t) = I_{0 + }^{\alpha - \beta }x(t), {}^CD_{0 + }^\beta I_{0 + }^\beta x(t) = x(t). $

Lemma 2.4 ([31]) Let $ n-1<\alpha \leq n $, if $ ^{C}D_{0+}^{\alpha }x(t)\in C[0, 1], $ then $ I_{0+}^{\alpha }\, ^{C}D_{0+}^{\alpha }x(t)=x(t)+c_{0}+c_{1}t+c_{2}t^{2}+\cdots +c_{n-1}t^{n-1}, $ where $ c_i \in \mathbb{R}, i = 0, 1, \cdots , n-1, n = [ \alpha ]+1. $

Since $ {}^CD_{0 + }^\beta [{\varphi _{p}}({}^CD_{0 + }^\alpha (\cdot) )] $ is a nonlinear operator, so we can't solve problem (1.1)(1.2) by Lemma 2.2. Hence, we provide the following lemma.

Lemma 3.1 $ u(t) $ is a solution of the following problem:

$ \begin{equation} \left\{ \begin{array}{l} {}^CD_{0 + }^\beta u(t) = f(t, \varphi _{q}(u(t))), t \in (0, 1), \hfill \\ u(1) = u(\delta ), \hfill \\ \end{array} \right. \end{equation} $

(3.1)

if and only if $ x(t) $ is a solution of problem (1.1)(1.2), where $ x(t) = I_{0 + }^\alpha \varphi _{q}(u(t)) $, $ \frac{1}{p} + \frac{1}{q} = 1 $.

Proof If $ u(t) $ is a solution of problem (3.1) and $ x(t) = I_{0 + }^\alpha \varphi _{q}(u(t)) $, then $ u(t) = {\varphi _{p}}({}^CD_{0 + }^\alpha x(t)) $ and $ {x^{(i)}}(0) = 0, i = \overline {0, n - 1} $. Replacing $ u(t) $ with $ {\varphi _{p}}({}^CD_{0 + }^\alpha x(t)) $ in problem (3.1), we can find that $ x(t) $ is a solution of problem (1.1)(1.2).

On the other hand, if $ x(t) $ is a solution of problem (1.1)(1.2) and $ u(t) = {\varphi _{p}}({}^CD_{0 + }^\alpha x(t)) $, substituting $ u(t) $ for $ {\varphi _{p}}({}^CD_{0 + }^\alpha x(t)) $ in problem (1.1)(1.2), we can find that $ u(t) $ satisfies problem (3.1).

Let $ X = Y = C\left[ {0, 1} \right] $ with the norm $ {\left\| u \right\| } = \mathop {\max }\limits_{t \in \left[ {0, 1} \right]} \left| {u\left( t \right)} \right| $. Set a cone $ C = \{ u(t) \in X|u(t) \ge 0, t \in [0, 1]\} . $ Define operators $ L:{\rm{dom}}L \subset X \to Y $ and $ N:X \to Y $ as follows:

$ \begin{equation} Lu(t) = {}^CD_{0 + }^\beta u(t), Nu(t) = f(t, \varphi _{q}(u(t))), \end{equation} $

(3.2)

where $ {\rm{dom}}L = \{ u(t)|u(t), {}^CD_{0 + }^\beta u(t) \in X, u(1) = u(\delta )\} . $ So problem (3.1) can be written by $ Lu = Nu, u \in {\rm{dom}}L. $

For simplicity of notation, we set

$ \begin{array}{l} l(s)=\left\{ \begin{array}{l} {{{(1 - s)}^{\beta - 1}} - {{(\delta - s)}^{\beta - 1}}, 0 \le s \le \delta < 1}, \\ {{{(1 - s)}^{\beta - 1}}, \qquad \qquad \qquad 0 \le \delta \le s < 1}, \end{array} \right. \end{array} $

and

$ {G_1}(t, s) = \left\{ \begin{array}{l} \frac{{{{(t - s)}^{\beta - 1}}}}{{\Gamma (\beta )}} - \frac{{{{(1 - s)}^\beta }}}{{\Gamma (\beta + 1)}} + \frac{{\beta (\Gamma (\beta + 2) + 1 - (\beta + 1){t^\beta })}}{{(1 - {\delta ^\beta })\Gamma (\beta + 2)}}l(s), 0 \le s < t \le 1\\ - \frac{{{{(1 - s)}^\beta }}}{{\Gamma (\beta + 1)}} + \frac{{\beta (\Gamma (\beta + 2) + 1 - (\beta + 1){t^\beta })}}{{(1 - {\delta ^\beta })\Gamma (\beta + 2)}}l(s), 0 \le t \le s \le 1. \end{array} \right. $

We denote

$ {K_1} = \min \{ 1, \frac{{1 - {\delta ^\beta }}}{{\beta \mathop {\max }\limits_{s \in [0, 1]} l(s)}}, \frac{1}{{\mathop {\max }\limits_{t, s \in [0, 1]} {G_1}(t, s)}}\}. $

Thus, one has

$ \begin{equation} 1 - \frac{{{K_1}\beta l(s)}}{{1 - {\delta ^\beta }}} \ge 0, 1 - {K_1}{G_1}(t, s) \ge 0. \end{equation} $

(3.3)

First, we give the main results of existence of positive solution for problem (1.1)(1.2).

Theorem 3.1 Suppose the following conditions hold.

$ \rm {(H_1)} $ There exists a constant $ R_0>0 $ such that $ f(t, u)<0, t\in [0, 1], u>R_0 $.

$ \rm {(H_2)} $ There exist nonnegative functions $ a(t), b(t)\in C[0, 1] $ with

$ \mathop {\max }\limits_{t \in [0, 1]} \int_0^t {{{(t - s)}^{\beta - 1}}a(s)ds: = A < + \infty }, \mathop {\max }\limits_{t \in [0, 1]} \int_0^t {{{(t - s)}^{\beta - 1}}b(s)ds: = B < \frac{{\Gamma (\beta )}}{2}}, $

such that

$ |f(t, u)| \le a(t) + b(t)\varphi _{p}(|u|), \forall t \in [0, 1]. $

$ \rm {(H_3)} $ $ f(t, u)\ge - {K_1}{\varphi _{p}}(u), t \in [0, 1], u > 0. $

$ \rm {(H_4)} $ There exist $ r>0, t_0 \in [0, 1] $ and $ M_0 \in (0, 1) $ such that

$ {G_1}({t_0}, s)f(s, u) \ge \frac{{1 - {M_0}}}{{{M_0}}}{\varphi _{p}}(u), s \in [0, 1), {M_0}r \le u \le r. $

Then problem (1.1)(1.2) has at least one positive solution.

Next, we give some important lemmas related to Theorem 3.1.

Lemma 3.2 Let $ L $ be defined by (3.2), then

$ \begin{equation} {\rm{Ker}}L = \{ u \in X|u(t) = c, c\in\mathbb{R}, \forall t\in[0, 1]\} , \end{equation} $

(3.4)

$ \begin{equation} {\mathop{\rm Im}\nolimits} L = \{ y \in Y|\int_0^1 {l(s)y(s)ds = 0} \}. \end{equation} $

(3.5)

Proof By Lemma 2.4, we can obtain (3.4). If $ y \in {\mathop{\rm Im}\nolimits} L $, there exists $ u \in {\rm{dom}}L $ such that $ y=Lu\in Y $. From Lemma 2.4, we have

$ u(t) = \frac{1}{{\Gamma (\beta )}}\int_0^t {{{(t - s)}^{\beta - 1}}y(s)ds} + c, c \in \mathbb{R}. $

Combined with boundary conditions of problem (3.1), we get

$ \int_0^1 {{{(1 - s)}^{\beta - 1}}y(s)ds} = \int_0^\delta {{{(\delta - s)}^{\beta - 1}}y(s)ds}, $

that is, $ \int_0^1 l(s)y(s)ds = 0 . $

On the other hand, if $ \int_0^1 l(s)y(s)ds = 0 $ for $ y\in Y $, let $ u(t) = I_{0 + }^\beta y(t) $, then $ u \in {\rm{dom}}L $ and $ {}^CD_{0 + }^\beta u(t) = y(t) $. Hence, $ y \in {\mathop{\rm Im}\nolimits} L $.

Lemma 3.3 Let $ L $ be defined by (3.2), then $ L $ is a Fredholm operator of index zero. The linear projection operators $ P:X \to Y $ and $ Q:Y \to Y $ can be defined as follows:

$ Pu(t) = \int_0^1 {u(t)dt} , Qy(t) = \frac{\beta }{{1 - {\delta ^\beta }}}\int_0^1 l (s)y(s)ds, \forall t \in [0, 1], $

and $ {K_P}:{\mathop{\rm Im}\nolimits} L \to {\rm{dom}}L \cap {\rm{Ker}} P $ is defined as

$ {K_P}y(t) = \int_0^1 k (t, s)y(s)ds, \forall t \in [0, 1], $

where

$ \begin{array}{l} k(t, s)=\left\{ \begin{aligned} {\frac{1}{{\Gamma (\beta )}}{{(t - s)}^{\beta - 1}} - \frac{1}{{\Gamma (\beta + 1)}}{{(1 - s)}^\beta }, 0 \le s \le t \le 1}, \\ { - \frac{1}{{\Gamma (\beta + 1)}}{{(1 - s)}^\beta }, \quad\qquad \qquad \qquad 0 \le t \le s \le 1}. \end{aligned} \right. \end{array} $

Proof Clearly, $ {\mathop{\rm Im}\nolimits} P = {\rm{Ker}} L $ and $ Pu^2=Pu $. By $ u=(u-Pu)+Pu $, we have $ X = {\rm{Ker}} P + {\rm{Ker}} L $. By a simple calculation, we obtain $ {\rm{Ker}} L \cap {\rm{Ker}} P = \{ 0\} $. Hence, $ X = {\rm{Ker}} L \oplus {\rm{Ker}} P. $ It is clear that $ {\mathop{\rm Im}\nolimits} L \subset {\rm{Ker}} Q $. On the other hand, if $ y(t) \in {\rm{Ker}} Q \subset Y $, then

$ {Q^2}y = Q(Qy) = Qy \cdot \frac{\beta }{{1 - {\delta ^\beta }}}\int_0^1 l (s)ds = Qy. $

If $ y\in Y $, let $ y=(y-Qy)+Qy $, where $ y-Qy\in {\rm{Ker}} Q, Qy\in {\mathop{\rm Im}\nolimits} Q $. It follows from $ {\rm{Ker}} Q = {\mathop{\rm Im}\nolimits} L $ and $ Q^2y=Qy $ that $ {\mathop{\rm Im}\nolimits} Q \cap {\mathop{\rm Im}\nolimits} L = \{ 0\} $. Then, we obtain $ Y = {\mathop{\rm Im}\nolimits} L \oplus {\mathop{\rm Im}\nolimits} Q $. Thus, $ \dim {\rm{Ker}} L = \dim {\mathop{\rm Im}\nolimits} Q = {\rm{codim}}{\mathop{\rm Im}\nolimits} L = 1 < \infty. $ It implies that $ L $ is a Fredholm operator of index zero.

For $ y\in {\mathop{\rm Im}\nolimits} L $, we have $ {K_P}y \in {\rm{dom}}L \cap {\rm{Ker}} P $ and $ LK_Py=y $. On the other hand, if $ u\in {\rm{dom}}L \cap {\rm{Ker}} P $, by Lemma 2.4, one has

$ \begin{array}{l} \begin{aligned} & {K_P}Lu(t) = \frac{1}{{\Gamma (\beta )}}[\int_0^t {{{(t - s)}^{\beta - 1}}Lu(s)ds} - \frac{1}{\beta }\int_0^1 {{{(1 - s)}^\beta }Lu(s)ds} ]\hfill \\ &= I_{0 + }^\beta {}^CD_{0 + }^\beta u(t) - I_{0 + }^{\beta + 1}{}^CD_{0 + }^\beta u(t){|_{t = 1}} = u(t) + c - I_{0 + }^{\beta + 1}{}^CD_{0 + }^\beta u(1).\hfill \\ \end{aligned} \end{array} $

So, $ \int_0^1 {{K_P}Lu(t)dt = } \int_0^1 {u(t)dt + c - } I_{0 + }^{\beta + 1}{}^CD_{0 + }^\beta u(1) $. It follows from $ u\in {\rm{Ker}} P $ and $ K_PLu\in {\rm{Ker}} P $ that $ c=I_{0 + }^{\beta +1}{}^CD_{0 + }^\beta u(1) $. Hence, we have $ K_PLu=u, u \in {\rm{dom}}L \cap {\rm{Ker}} P $.

Lemma 3.4 $ QN:X\to Y $ is continuous and bounded and $ {K_P}(I - Q)N:\overline \Omega \to X $ is compact, where $ \Omega \subset X $ is bounded.

Proof By the continuity of $ f $, we see that $ QN(\overline \Omega ) $ and $ {K_P}(I - Q)N(\overline \Omega ) $ are bounded. That is, there exist constants $ {M_1}, {M_2} > 0 $ such that $ |(I - Q)Nu)| \le {M_1} $ and $ |{K_P}(I - Q)Nu)| \le {M_2}, \forall u \in \overline \Omega , t \in [0, 1] $. Thus, one need only prove that $ {K_P}(I - Q)N(\overline \Omega ) \subset X $ is equicontinuous. Let $ K_{P, Q}=K_P(I-Q)N $, for $ 0\le t_1<t_2\le 1, u\in \overline \Omega $, we get

$ \begin{array}{l} \begin{aligned} &\quad |{K_{P, Q}}u({t_2}) - {K_{P, Q}}u({t_1})|\hfill \\ & = \frac{1}{{\Gamma (\beta )}}|\int_0^{{t_2}} {{{({t_2} - s)}^{\beta - 1}}(I - Q)Nu(s)ds} - \int_0^{{t_1}} {{{({t_1} - s)}^{\beta - 1}}(I - Q)Nu(s)ds} |\hfill \\ & = \frac{1}{{\Gamma (\beta )}}|\int_0^{{t_1}} {[{{({t_2} - s)}^{\beta - 1}} - {{({t_1} - s)}^{\beta - 1}}](I - Q)Nu(s)ds} + \int_{{t_1}}^{{t_2}} {{{({t_2} - s)}^{\beta - 1}}(I - Q)Nu(s)ds} |\hfill \\ & \le \frac{{{M_1}}}{{\Gamma (\beta )}}[\int_0^{{t_1}} {[{{({t_1} - s)}^{\beta - 1}} - {{({t_2} - s)}^{\beta - 1}}]ds} + \int_{{t_1}}^{{t_2}} {{{({t_2} - s)}^{\beta - 1}}ds} \hfill \\ & = \frac{{{M_1}}}{{\Gamma (\beta +1)}}[t_1^\beta - t_2^\beta + 2{({t_2} - {t_1})^\beta }].\hfill \\ \end{aligned} \end{array} $

It follows from the uniform continuity of $ t^\beta $ and $ t $ on $ [0, 1] $ that $ {K_P}(I - Q)N(\overline \Omega) $ are equicontinuous on $ [0, 1] $. By Arzela-Ascoli theorem, we show that $ {K_P}(I - Q)N(\overline \Omega ) $ is compact.

Lemma 3.5 If the condition $ \rm {(H_1)} $ and $ \rm {(H_2)} $ hold, the set

$ {\Omega _0} = \{ u(t)|Lu(t) = \lambda Nu(t), u(t) \in C \cap {\rm{dom}}L, \lambda \in (0, 1)\} $

is bounded.

Proof For $ u(t) \in {\Omega _0} $, we have $ QNu(t)=0 $. By $ \rm {(H_1)} $ and $ QNu(t)=0 $, there exists $ t_0\in [0, 1] $ such that $ \varphi _{q}(u({t_0})) \le {R_0} $, i.e. $ u({t_0}) \le {\varphi _{p}}({R_0}) $. By $ u(t) = I{_{0 + }^\beta }{}^C D_{0 + }^\beta u(t) + c $, one has

$ \begin{array}{l} |c| \le |u(t)| + |I{_{0 + }^\beta }{}^C D_{0 + }^\beta u(t)| \le |u(t_0)| + |I{_{0 + }^\beta }{}^C D_{0 + }^\beta u(t_0)| \le {\varphi _{p}}({R_0}) + |I{_{0 + }^\beta }{}^C D_{0 + }^\beta u(t_0)| , \end{array} $

and

$ \begin{equation} \left\| u \right\| \le \varphi _p (R_0)+ |I{_{0 + }^\beta }{}^C D_{0 + }^\beta u(t_0)|+|I{_{0 + }^\beta }{}^C D_{0 + }^\beta u(t)|. \end{equation} $

(3.6)

From $ Lu=\lambda Nu $, we get $ ^CD_{0 + }^\beta u(t) = \lambda f(t, \varphi _{q}(u(t))) $. By $ \rm {(H_2)} $ and $ \lambda \in (0, 1) $, one has

$ \begin{array}{l} \begin{aligned} & \left\| u \right\| \le {\varphi _p}({R_0}) + |I_{0 + }^\beta f(t, {\varphi _q}(u(t))){|_{{t_0}}}| + |I_{0 + }^\beta f(t, {\varphi _q}(u(t)))|\hfill \\ & \le {\varphi _p}({R_0})\! +\! \frac{1}{{\Gamma (\beta )}}\int_0^{{t_0}} {{{({t_0} \!-\! s)}^{\beta \! -\! 1}}|f(s, {\varphi _q}(u(t)))|ds} \!+\! \frac{1}{{\Gamma (\beta )}}\int_0^t {{{(t \!-\! s)}^{\beta \!-\! 1}}|f(s, {\varphi _q}(u(s)))|ds} \hfill \\ &\le {\varphi _p}({R_0}) \!+\! \frac{1}{{\Gamma (\beta )}}\int_0^{{t_0}} {{{({t_0} - s)}^{\beta - 1}}[a(s) \!+\! b(s)u(s)]ds} \!+\! \frac{1}{{\Gamma (\beta )}}\int_0^t {{{(t\! -\! s)}^{\beta \!-\! 1}}[a(s) \!+\! b(s)u(s)]ds}\hfill \\ &\le {\varphi _p}({R_0}) + \frac{2}{{\Gamma (\beta )}}(A + B\left\| u \right\|).\hfill \\ \end{aligned} \end{array} $

Thus,

$ \left\| u \right\| \le \frac{{{\varphi _p}({R_0}) + \frac{{2A}}{{\Gamma (\beta )}}}}{{1 - \frac{{2B}}{{\Gamma (\beta )}}}}:=N < + \infty . $

Hence, $ {\Omega _0} $ is bounded.

Proof of Theorem 3.1 Set

$ {\Omega _1} = \{ u \in X|{M_0}\left\| u \right\| < \left| {u(t)} \right| < r < R, t \in [0, 1]\} , {\Omega _2} = \{ u \in X|\left\| u \right\| < R\} , $

where $ R = \max \{ \varphi_p(R_0), {N}\} + 1 $. It is clear that $ {\Omega _1} $ and $ {\Omega _2} $ are open bounded sets of $ X, \overline {{\Omega}}_1 \subset {\Omega _2} $ and $ C \cap (\overline {{\Omega}}_2 \backslash {\Omega _1}) \ne \phi $. By Lemma 3.2, 3.3, 3.4 and 3.5, we know that $ L $ is a Fredholm operator of index zero and the conditions (1), (2) of Lemma 2.2 are fulfilled.

Define $ \gamma:X\to C $ as $ \gamma u(t)=|u(t)|, u(t)\in X $ and $ J:{\mathop{\rm Im}\nolimits} Q \to {\rm{Ker}} L $ as $ J(c)=c, c\in \mathbb{R} $. Then $ \gamma :X\to C $ is a retraction and (3) of Lemma 2.2 holds. For $ u(t) \in {\rm{Ker}} L \cap \partial {\Omega _2} $, then $ u(t) \equiv c. $ Let

$ H(c, \lambda ) = c - \lambda |c| - \frac{{\lambda \beta }}{{1 - {\delta ^\beta }}}\int_0^1 {l(s)f(s, \varphi _{q}(|c|))ds} , $

where $ \lambda \in [0, 1] $. Suppose $ H(c, \lambda)=0 $, by $ \rm {(H_3)} $, we have

$ \begin{array}{l} \begin{aligned} &c = \lambda |c| + \frac{{\lambda \beta }}{{1 - {\delta ^\beta }}}\int_0^1 {l(s)f(s, \varphi _{q}(|c|))ds} \ge \lambda |c| - \frac{{\lambda \beta }}{{1 - {\delta ^\beta }}}\int_0^1 {l(s){K_1}|c|ds} \hfill \\ & = \lambda |c|(1 - {K_1}) \ge 0. \hfill \\ \end{aligned} \end{array} $

Thus $ H(c, \lambda)=0 $ implies $ c \ge 0. $ Clearly, $ H(R, 0)\neq 0 $. Moreover, if $ H(R, \lambda)=0, \lambda \in (0, 1] $, we get

$ 0 \le R(1 - \lambda ) = \frac{{\lambda \beta }}{{1 - {\delta ^\beta }}}\int_0^1 {l(s)f(s, \varphi _{q}(R))ds} , $

which contradicts to condition $ \rm {(H_1)} $. Hence $ H(u, \lambda)\neq 0 $ for $ u \in {\rm{Ker}} L \cap \partial {\Omega _2}, \lambda \in [0, 1] $. Therefore,

$ \begin{array}{l} \begin{aligned} &\quad \deg ([I - (P + JQN)\gamma]{|_{{\rm{Ker}} L}}, {\rm{Ker}} L \cap {\Omega _2}, 0) \hfill \\ & = \deg (H(x, 1), {\rm{Ker}} L \cap {\Omega _2}, 0) = \deg (H(x, 0), {\rm{Ker}} L \cap {\Omega _2}, 0) \hfill \\ & = \deg (I, {\rm{Ker}} L \cap {\Omega _2}, 0) = 1 \ne 0. \hfill \\ \end{aligned} \end{array} $

Then, (4) of Lemma 2.2 holds.

Set $ {u_0}(t) = 1, t \in [0, 1] $, then $ {u_0} \in C\backslash \{ 0\} , C({u_0}) = \{ u \in C|u(t) > 0, t \in [0, 1]\} $. Take $ \sigma (u_0)=1 $ and $ u \in C({u_0}) \cap \partial {\Omega _1} $, then $ {M_0}r \le u(t) \le r, t \in [0, 1] $. By $ \rm {(H_4)} $, we have

$ \begin{array}{l} \begin{aligned} \Psi u({t_0}) &= \int_0^1 {u(s)ds} + \frac{\beta }{{1 - {\delta ^\beta }}}\int_0^1 {l(s)f(s, \varphi _{q}(u(s)))ds} \hfill \\ & \quad + \int_0^1 {k({t_0}, s)[f(s, \varphi _{q}(u(s))) - \frac{\beta }{{1 - {\delta ^\beta }}}\int_0^1 {l(\tau )f(\tau , \varphi _{q}(u(\tau )))d\tau } ]ds} \hfill \\ & = \int_0^1 {u(s)ds} + \int_0^1 {{G_1}({t_0}, s)f(s, \varphi _{q}(u(s)))ds} \hfill \\ & \ge \int_0^1 {u(s)ds} + \int_0^1 {\frac{{1 - {M_0}}}{{{M_0}}}u(s)ds} \hfill \\ & \ge {M_0}r + (1 - {M_0})r = r = \left\| u \right\|. \hfill \\ \end{aligned} \end{array} $

So, $ \left\| u \right\| \le \sigma ({u_0})\left\| {\Psi u} \right\| $, for $ u \in C({u_0}) \cap \partial {\Omega _1} $. Hence, (5) of Lemma 2.2 holds.

For $ u(t) \in \partial {\Omega _2}, t \in [0, 1] $, by $ \rm {(H_3)} $ and (3.3), we get

$ \begin{array}{l} \begin{aligned} (P + JQN)\gamma(u) &= \int_0^1 {|u(s)|ds} + \frac{\beta }{{1 - {\delta ^\beta }}}\int_0^1 {l(s)f(s, \varphi _{q}(|u(s)|))ds} \hfill \\ & \ge \int_0^1 {(1 - \frac{{\beta {K_1}}}{{1 - {\delta ^\beta }}}l(s))|u(s)|ds} \ge 0. \hfill \\ \end{aligned} \end{array} $

Thus, $ (P + JQN)\gamma(\partial {\Omega _2}) \subset C $. So (6) of Lemma 2.2 holds.

For $ u(t) \in {\overline \Omega _2}\backslash {\Omega _1}, t\in [0, 1] $, by $ \rm {(H_3)} $ and (3.3), we have

$ \begin{array}{l} \begin{aligned} {\Psi _\gamma}(u(t)) &= \int_0^1 {|u(s)|ds} + \int_0^1 {{G_1}(t, s)f(s, \varphi _{q}(|u(s)|))ds} \hfill \\ & \ge \int_0^1 {|u(s)|ds} - {K_1}\int_0^1 {{G_1}(t, s)|u(s)|ds} \hfill \\ & = \int_0^1 {(1 - {K_1}{G_1}(t, s))|u(s)|ds} \ge 0. \hfill \\ \end{aligned} \end{array} $

Hence, (7) of Lemma 2.2 holds.

By Lemma 2.2, we see that the equation $ Lu=Nu $ has a positive solution $ u $. By Lemma 3.1, problem (1.1)(1.2) has at least one positive solution.

Since $ {}^CD_{0 + }^\beta [{\varphi _{p}}({}^CD_{0 + }^\alpha (\cdot) )] $ is a nonlinear operator, so we can't solve problem (1.1)(1.3) by Lemma 2.2. Hence, we provide the following lemma.

Lemma 4.1 $ u(t) $ is a solution of the following problem:

$ \begin{equation} \left\{ \begin{array}{l} {}^CD_{0 + }^\beta u(t) = f(t, \varphi _{q}(u(t))), t \in (0, 1), \hfill \\ u(1) = \int_0^1 {h(t)u(t)dt}, \hfill \\ \end{array} \right. \end{equation} $

(4.1)

if and only if $ x(t) $ is a solution of problem (1.1)(1.3), where $ x(t) = I_{0 + }^\alpha \varphi _{q}(u(t)) $, $ \frac{1}{p} + \frac{1}{q} = 1 $.

Proof The proof process is similar to Lemma 3.1, which is omitted here.

Let $ X = Y = C\left[ {0, 1} \right] $ with the norm $ {\left\| u \right\| } = \mathop {\max }\limits_{t \in \left[ {0, 1} \right]} \left| {u\left( t \right)} \right| $. Take a cone $ C = \{ u(t) \in X|u(t) \ge 0, t \in [0, 1]\} . $ Define operators $ L:{\rm{dom}}L \subset X \to Y $ and $ N:X \to Y $ as follows:

$ \begin{equation} Lu(t) = {}^CD_{0 + }^\beta u(t), Nu(t) = f(t, \varphi _{q}(u(t))), \end{equation} $

(4.2)

where $ {\rm{dom}}L = \{ u(t)|u(t), {}^CD_{0 + }^\beta u(t) \in X, u(1)=\int_0^1 {h(t)u(t)dt}\} . $ Then problem (4.1) can be written by $ Lu = Nu, u \in {\rm{dom}}L. $ For the simplicity of notation, let

$ \begin{array}{l} {G_2}(t, s)=\left\{ \begin{aligned} & \frac{{{{(t - s)}^{\beta - 1}}}}{{\Gamma (\beta )}} - \frac{{{{(1 - s)}^\beta }}}{{\Gamma (\beta + 1)}} + \frac{{\beta (1 - \frac{{{t^\beta }}}{{\Gamma (\beta + 1)}} + \frac{1}{{\Gamma (\beta + 2)}})}}{{1 - \int_0^1 {h(t){t^\beta }dt} }}\\ & \times [{(1 - s)^{\beta - 1}} - \int_s^1 h(t){(t - s)^{\beta - 1}}dt], 0 \le s < t < 1, \\ & - \! \frac{{{{(1 \! - \! s)}^\beta }}}{{\Gamma (\beta \! + \! 1)}} \!+ \! \frac{\beta (1 \! - \!\frac{{{t^\beta }}}{{\Gamma (\beta \! + \! 1)}} \! + \!\frac{1}{{\Gamma (\beta + 2)}}) }{{1 \!- \! \int_0^1 {h(t){t^\beta }dt} }}[{(1 \! - \! s)^{\beta \! - \!1}} \! - \! \int_s^1 \!{h(t){{(t \!- \!s)}^{\beta \! - \! 1}}dt} ], 0 \le t \le s < 1, \end{aligned} \right. \end{array} $

$ {K_2} = \min \{ 1, \frac{{1 - \int_0^1 {h(t){t^\beta }dt} }}{{\beta \mathop {\max }\limits_{t, s \in [0, 1]} [{{(1 - s)}^{\beta - 1}} - \int_s^1 {h(t){{(t - s)}^{\beta - 1}}dt} ]}}, \frac{1}{{\mathop {\max }\limits_{t, s \in [0, 1]} {G_2}(t, s)}}\} . $

Thus, one has

$ \begin{equation} 1 - \frac{{{K_2}\beta }}{{1 - \int_0^1 {h(t){t^\beta }dt} }}[{(1 - s)^{\beta - 1}} - \int_s^1 {h(t){{(t - s)}^{\beta - 1}}dt} ] \ge 0, 1 - {K_2}{G_2}(t, s) \ge 0. \end{equation} $

(4.3)

Below, we first give the main results of existence of positive solution for problem (1.1)(1.3).

Theorem 4.1 Assume that the conditions $ {(H_1)} $-$ {(H_2)} $ hold. And the following conditions are satisfied.

$ \rm {(H_5)} $ $ f(t, u)\ge - {K_2}{\varphi _{p}}(u), t \in [0, 1], u > 0. $

$ \rm {(H_6)} $ There exist $ r>0, t_0 \in [0, 1] $ and $ M_0 \in (0, 1) $ such that

$ {G_2}({t_0}, s)f(s, u) \ge \frac{{1 - {M_0}}}{{{M_0}}}{\varphi _{p}}(u), s \in [0, 1), {M_0}r \le u \le r. $

Then problem (1.1)(1.3) has at least one positive solution.

Next, we give some important lemmas related to Theorem 4.1.

Lemma 4.2. Let $ L $ be defined by (4.2), then

$ \begin{array}{l} {\rm{Ker}}L = \{ u \in X|u(t) = c, c\in\mathbb{R}, \forall t\in[0, 1]\} , \end{array} $

$ \begin{array}{l} {\mathop{\rm Im}\nolimits} L = \{ y \in Y|\int_0^1 {[{{(1 - s)}^{\beta - 1}} - \int_s^1 {h(t){{(t - s)}^{\beta - 1}}dt} ]y(s)ds} = 0 \}. \end{array} $

The linear projection operators $ P:X \to Y $ and $ Q:Y \to Y $ can be defined as follows:

$ Pu(t) = \int_0^1 {u(t)dt} , $

$ Qy(t) = \frac{\beta }{{1 - \int_0^1 {h(t){t^\beta }dt} }}\int_0^1 {[{{(1 - s)}^{\beta - 1}} - \int_s^1 {h(t){{(t - s)}^{\beta - 1}}dt} ]} y(s)ds, \forall t \in [0, 1], $

and $ {K_P}:{\mathop{\rm Im}\nolimits} L \to {\rm{dom}}L \cap {\rm{Ker}} P $ is defined as

$ {K_P}y(t) = \int_0^1 k (t, s)y(s)ds, \forall t \in [0, 1], $

where

$ \begin{array}{l} k(t, s)=\left\{ \begin{aligned} {\frac{1}{{\Gamma (\beta )}}{{(t - s)}^{\beta - 1}} - \frac{1}{{\Gamma (\beta + 1)}}{{(1 - s)}^\beta }, 0 \le s \le t \le 1}, \\ { - \frac{1}{{\Gamma (\beta + 1)}}{{(1 - s)}^\beta }, \quad\qquad \qquad \qquad 0 \le t \le s \le 1}. \end{aligned} \right. \end{array} $

Proof The proof is similar to that of Lemma 3.2, 3.3 and is omitted.

Lemma 4.3 $ QN:X\to Y $ is continuous and bounded and $ {K_P}(I - Q)N: $$ \overline \Omega $$ \to $$ X $ is compact, where $ \Omega \subset X $ is bounded.

Proof The proof is similar to that of Lemma 3.4 and is omitted.

Lemma 4.4 If the condition $ \rm {(H_1)} $ and $ \rm {(H_2)} $ hold, the set

$ {\Omega _0} = \{ u(t)|Lu(t) = \lambda Nu(t), u(t) \in C \cap {\rm{dom}}L, \lambda \in (0, 1)\} $

is bounded.

Proof The proof is similar to that of Lemma 3.5 and is omitted.

Proof of Theorem 4.1 Set

$ {\Omega _1} = \{ u \in X|{M_0}\left\| u \right\| < \left| {u(t)} \right| < r < R, t \in [0, 1]\} , {\Omega _2} = \{ u \in X|\left\| u \right\| < R\} , $

where $ R = \max \{ \varphi_p(R_0), {N}\} + 1 $. It is clear that $ {\Omega _1} $ and $ {\Omega _2} $ are open bounded sets of $ X, \overline {{\Omega}}_1 \subset {\Omega _2} $ and $ C \cap (\overline {{\Omega}}_2 \backslash {\Omega _1}) \ne \phi $. By Lemma 4.2, 4.3 and 4.4, we know that $ L $ is a Fredholm operator of index zero and the conditions (1), (2) of Lemma 2.2 are fulfilled.

Define $ \gamma:X\to C $ as $ \gamma u(t)=|u(t)|, u(t)\in X $ and $ J:{\mathop{\rm Im}\nolimits} Q \to {\rm{Ker}} L $ as $ J(c)=c, c\in \mathbb{R} $. Then $ \gamma :X\to C $ is a retraction and (3) of Lemma 2.2 holds.

For $ u(t) \in {\rm{Ker}} L \cap \partial {\Omega _2} $, then $ u(t) \equiv c. $ Let

$ H(c, \lambda ) = c - \lambda |c| - \frac{{\lambda \beta }}{{1 - \int_0^1 {h(t){t^\beta }dt} }}\int_0^1 {[{{(1 - s)}^{\beta - 1}} - \int_s^1 {h(t){{(t - s)}^{\beta - 1}}dt} ]} f(s, {\varphi _q}(|c|))ds, $

where $ \lambda \in [0, 1] $. Suppose $ H(c, \lambda)=0 $, by $ \rm {(H_5)} $, we have

$ \begin{array}{l} \begin{aligned} c &= \lambda |c| +\frac{{\lambda \beta }}{{1 - \int_0^1 {h(t){t^\beta }dt} }}\int_0^1 {[{{(1 - s)}^{\beta - 1}} - \int_s^1 {h(t){{(t - s)}^{\beta - 1}}dt} ]} f(s, {\varphi _q}(|c|))ds \hfill \\ & \ge \lambda |c| - \frac{{\lambda \beta }}{{1 - \int_0^1 {h(t){t^\beta }dt} }}\int_0^1 {[{{(1 - s)}^{\beta - 1}} - \int_s^1 {h(t){{(t - s)}^{\beta - 1}}dt} ]} {K_2}|c|ds \hfill \\ & = \lambda |c|(1 - {K_2}) \ge 0. \hfill \\ \end{aligned} \end{array} $

Thus $ H(c, \lambda)=0 $ implies $ c \ge 0. $ Clearly, $ H(R, 0)\neq 0 $. Moreover, if $ H(R, \lambda)=0, \lambda \in (0, 1] $, we get

$ 0 \le R(1 - \lambda ) = \frac{{\lambda \beta }}{{1 - \int_0^1 {h(t){t^\beta }dt} }}\int_0^1 {[{{(1 - s)}^{\beta - 1}} - \int_s^1 {h(t){{(t - s)}^{\beta - 1}}dt} ]} f(s, {\varphi _q}(R))ds , $

which contradicts to condition $ \rm {(H_1)} $. Hence $ H(u, \lambda)\neq 0 $ for $ u \in {\rm{Ker}} L \cap \partial {\Omega _2}, \lambda \in [0, 1] $. Therefore,

$ \begin{array}{l} \begin{aligned} &\quad \deg ([I - (P + JQN)\gamma]{|_{{\rm{Ker}} L}}, {\rm{Ker}} L \cap {\Omega _2}, 0) \hfill \\ & = \deg (H(x, 1), {\rm{Ker}} L \cap {\Omega _2}, 0) = \deg (H(x, 0), {\rm{Ker}} L \cap {\Omega _2}, 0) \hfill \\ & = \deg (I, {\rm{Ker}} L \cap {\Omega _2}, 0) = 1 \ne 0. \hfill \\ \end{aligned} \end{array} $

Then, (4) of Lemma 2.2 holds.

Set $ {u_0}(t) = 1, t \in [0, 1] $, then $ {u_0} \in C\backslash \{ 0\} , C({u_0}) = \{ u \in C|u(t) > 0, t \in [0, 1]\} $. Take $ \sigma (u_0)=1 $ and $ u \in C({u_0}) \cap \partial {\Omega _1} $, then $ {M_0}r \le u(t) \le r, t \in [0, 1] $. By $ \rm {(H_6)} $, we have

$ \begin{array}{l} \begin{aligned} \Psi u({t_0}) &= \int_0^1 {u(s)ds} + \frac{\beta }{{1 - \int_0^1 {h(t){t^\beta }dt} }}\int_0^1 {[{{(1 - s)}^{\beta - 1}} - \int_s^1 {h(t){{(t - s)}^{\beta - 1}}dt} ]} f(s, {\varphi _q}(u(s)))ds \hfill \\ & \quad \!+\! \int_{\rm{0}}^{\rm{1}} {k({t_0}, s)} \{ f(s, {\varphi _q}(u(s))) - \frac{\beta }{{1 - \int_0^1 {h(t){t^\beta }dt} }} \times \hfill \\ & \quad \int_0^1 {[{{(1 - \tau )}^{\beta - 1}} - \int_\tau ^1 {h(t){{(t - \tau )}^{\beta - 1}}dt} ]} f(\tau , {\varphi _q}(u(\tau )))d\tau \} ds \hfill \\ & = \int_0^1 {u(s)ds} + \int_0^1 {{G_2}({t_0}, s)f(s, \varphi _{q}(u(s)))ds} \ge \int_0^1 {u(s)ds} + \int_0^1 {\frac{{1 - {M_0}}}{{{M_0}}}u(s)ds} \hfill \\ & \ge {M_0}r + (1 - {M_0})r = r = \left\| u \right\|. \hfill \\ \end{aligned} \end{array} $

So, $ \left\| u \right\| \le \sigma ({u_0})\left\| {\Psi u} \right\| $, for $ u \in C({u_0}) \cap \partial {\Omega _1} $. Hence, (5) of Lemma 2.2 holds.

For $ u(t) \in \partial {\Omega _2}, t \in [0, 1] $, by $ \rm {(H_5)} $ and (4.3), we get

$ \begin{array}{l} \begin{aligned} &(P + JQN)\gamma(u) \hfill \\ &= \int_0^1 {|u(s)|ds} \! +\! \frac{{ \beta }}{{1\! -\! \int_0^1 {h(t){t^\beta }dt} }}\int_0^1 {[{{(1 \!- \! s)}^{\beta \! -\! 1}}\! - \!\int_s^1 {h(t){{(t \!-\! s)}^{\beta \! -\! 1}}dt} ]} f(s, {\varphi _q}(|u(s)|))ds \hfill \\ & \ge \int_0^1 \{ 1 - \frac{{\beta {K_2}}}{{1 - \int_0^1 {h(t){t^\beta }dt} }}[{(1 - s)^{\beta - 1}} - \int_s^1 {h(t){{(t - s)}^{\beta - 1}}dt} ]\}|u(s)|ds \ge 0. \hfill \\ \end{aligned} \end{array} $

Thus, $ (P + JQN)\gamma(\partial {\Omega _2}) \subset C $. So (6) of Lemma 2.2 holds.

For $ u(t) \in {\overline \Omega _2}\backslash {\Omega _1}, t\in [0, 1] $, by $ \rm {(H_5)} $ and (4.3), we have

$ \begin{array}{l} \begin{aligned} & {\Psi _\gamma}(u(t)) = \int_0^1 {|u(s)|ds} + \int_0^1 {{G_2}(t, s)f(s, \varphi _{q}(|u(s)|))ds} \hfill \\ & \ge \int_0^1 {|u(s)|ds} - {K_2}\int_0^1 {{G_2}(t, s)|u(s)|ds} = \int_0^1 {(1 - {K_2}{G_2}(t, s))|u(s)|ds} \ge 0. \hfill \\ \end{aligned} \end{array} $

Hence, (7) of Lemma 2.2 holds. By Lemma 2.2, we see that the equation $ Lu=Nu $ has a positive solution $ u $. By Lemma 4.1, problem (1.1)(1.3) have at least one positive solution.

Example 5.1 Consider the following problem

$ \begin{equation} \left\{ \begin{array}{l} ^CD_{0 + }^{\frac{1}{2}}\varphi_2{(^C}D_{0 + }^{\frac{1}{2}}x(t)) = \frac{1}{4} - \frac{1}{{20}}{|^C}D_{0 + }^{\frac{1}{2}}x(t)|^{\frac{1}{2}}, \quad t \in (0, 1), \hfill \\ x(0) = 0, {\quad ^C}D_{0 + }^{\frac{1}{2}}x(1){ = ^C}D_{0 + }^{\frac{1}{2}}x(\frac{1}{2}), \hfill \\ \end{array} \right. \end{equation} $

(5.1)

where $ \alpha =\beta= \delta= \frac{1}{2}, p=2, q=2, f(t, { ^C}D_{0 + }^{\frac{1}{2}}x(t)) = \frac{1}{4} - \frac{1}{{20}}{|^C}D_{0 + }^{\frac{1}{2}}x(t){|^{\frac{1}{2}}}. $

By Lemma 3.1, we have

$ \begin{equation} \left\{ \begin{array}{l} ^CD_{0 + }^{\frac{1}{2}}u(t) = \frac{1}{4}- \frac{1}{{20}}|u(t)|^{\frac{1}{2}}, \hfill \\ u(1) = u(\frac{1}{2}).\hfill \\ \end{array} \right. \end{equation} $

(5.2)

So, we get

$ \begin{array}{l} l(s)=\left\{ \begin{aligned} \frac{1}{{\sqrt {1 - s} }} - \frac{1}{{\sqrt {\frac{1}{2} - s} }}, 0 \le s \le \frac{1}{2}, \hfill\\ \frac{1}{{\sqrt {1 - s} }}, \quad \quad\quad\quad\quad \frac{1}{2} \le s < 1.\hfill \\ \end{aligned} \right. \end{array} $

$ \begin{array}{l} {G_1}(t, s)=\left\{ \begin{aligned} \frac{1}{{\Gamma (\frac{1}{2})}}{(t - s)^{ - \frac{1}{2}}} - \frac{1}{{\Gamma (\frac{3}{2})}}{(1 - s)^{\frac{1}{2}}} + \frac{{\frac{1}{2}(\Gamma (\frac{5}{2}) + 1 - \frac{3}{2}{t^{\frac{1}{2}}})}}{{(1 - {{\frac{1}{2}}^{\frac{1}{2}}})\Gamma (\frac{5}{2})}}l(s), 0 \le s < t \le 1, \hfill \\ - \frac{1}{{\Gamma (\frac{3}{2})}}{(1 - s)^{\frac{1}{2}}} + \frac{{\frac{1}{2}(\Gamma (\frac{5}{2}) + 1 - \frac{3}{2}{t^{\frac{1}{2}}})}}{{(1 - {{\frac{1}{2}}^{\frac{1}{2}}})\Gamma (\frac{5}{2})}}l(s), \quad \qquad \qquad \qquad 0 \le t \le s \le 1.\hfill \\ \end{aligned} \right. \end{array} $

Take $ {R_0} = 25, a(t) = \frac{1}{4}, b(t) = \frac{1}{{20}}, {K_1} \approx 0.23641, {M_0} = 0.7, r = 0.04, {t_0} = 0. $ By simple calculations, we can see that

$ \begin{array}{l} \begin{aligned} &f(t, u) = \frac{1}{4} - \frac{1}{{20}}|u{|^{\frac{1}{2}}} < 0, u > 25, \hfill \\ &|f(t, u)| \le {a(t) + b(t)\varphi_p(|u|)}, \hfill \\ &A = \mathop {\max }\limits_{t \in [0, 1]} \int_0^t {{{(t - s)}^{ - \frac{1}{2}}} \cdot \frac{1}{4}ds = \frac{1}{2} < + \infty , } \hfill \\ &B = \mathop {\max }\limits_{t \in [0, 1]} \int_0^t {{{(t - s)}^{ - \frac{1}{2}}} \cdot \frac{1}{20}ds = \frac{1}{{10}} < \frac{{\Gamma (\frac{1}{2})}}{2}, } \hfill \\ &f(t, u) \ge - 0.23641{u}, u > 0, \hfill \\ &{G_1}({t_0}, s)f(s, u) \ge 0.375 - 0.4583{u} > \frac{{0.3}}{{0.7}}{u}, 0.028 \le u \le 0.04, s \in [0, 1).\hfill \\ \end{aligned} \end{array} $

So the conditions $ \rm {(H_1)} $-$ \rm {(H_4)} $ of Theorem 3.1 hold. By Theorem 3.1, we can conclude that problem (5.1) has at least one positive solution.

Example 5.2 Consider the following problem

$ \begin{equation} \left\{ \begin{array}{l} ^CD_{0 + }^{\frac{1}{2}}\varphi_2{(^C}D_{0 + }^{\frac{1}{2}}x(t)) = \frac{1}{4} - \frac{1}{{20}}{|^C}D_{0 + }^{\frac{1}{2}}x(t)|^{\frac{1}{2}}, \quad t \in (0, 1), \hfill \\ x(0) = 0, \quad {\varphi _2}{(^C}D_{0 + }^{\frac{1}{2}}x(1)) = \int_0^1 {{\varphi _2}{(^C}D_{0 + }^{\frac{1}{2}}x(t))dt}, \hfill \\ \end{array} \right. \end{equation} $

(5.3)

where $ \alpha =\beta= \frac{1}{2}, p=2, q=2, f(t, { ^C}D_{0 + }^{\frac{1}{2}}x(t)) = \frac{1}{4} - \frac{1}{{20}}{|^C}D_{0 + }^{\frac{1}{2}}x(t){|^{\frac{1}{2}}}, h(t)=1>0, \int_0^1 {h(t)dt}=1. $

By Lemma 4.1, we have

$ \begin{equation} \left\{ \begin{array}{l} ^CD_{0 + }^{\frac{1}{2}}u(t) = \frac{1}{4}- \frac{1}{{20}}|u(t)|^{\frac{1}{2}}, \hfill \\ u(1) =\int_0^1 {u(t)dt}.\hfill \\ \end{array} \right. \end{equation} $

(5.4)

So, we get

$ \begin{array}{l} {G_2}(t, s)=\left\{ \begin{aligned} &- \frac{2}{{\sqrt \pi }}{(1 - s)^{\frac{1}{2}}} + \frac{3}{2}[{(1 - s)^{ - \frac{1}{2}}} - 2{(1 - s)^{\frac{1}{2}}}](1 - \frac{{2{t^{\frac{1}{2}}}}}{{\sqrt \pi }} + \frac{4}{{3\sqrt \pi }}), 0 \le t \le s < 1, \hfill \\ &\frac{{{{(t - s)}^{ - \frac{1}{2}}}}}{{\sqrt \pi }} \!- \! \frac{{2{{(1 - s)}^{\frac{1}{2}}}}}{{\sqrt \pi }} \!+ \! \frac{3}{2}[{(1 - s)^{ \!-\! \frac{1}{2}}} \!- \!2{(1 \!-\! s)^{\frac{1}{2}}}](1 \!- \!\frac{{2{t^{\frac{1}{2}}}}}{{\sqrt \pi }} \!+\! \frac{4}{{3\sqrt \pi }}), 0 \le s < t < 1.\hfill \\ \end{aligned} \right. \end{array} $

Take $ {R_0} = 25, a(t) = \frac{1}{4}, b(t) = \frac{1}{{20}}, {K_2} \approx 0.14356, {M_0} = 0.6, r = 0.02, {t_0} = 0. $ By simple calculations, we can see that all conditions of Theorem 4.1 are satisfied. Hence, by Theorem 4.1, we can conclude that problem (5.3) has at least one positive solution.