Multiplicity of Positive Solutions for a Singular Tempered Fractional Initial-Boundary Value Problem with Changing-Sign Perturbation Term

Abstract

In this paper, we focus on the multiplicity of positive solutions for a singular tempered fractional initial-boundary value problem with a p-Laplacian operator and a changing-sign perturbation term. By introducing a truncation function and combing with the properties of the solution of isomorphic linear equations, we transform the changing-sign tempered fractional initial-boundary value problem into a positive problem, and then the existence results of multiple positive solutions are established by the fixed point theorem in a cone. It is worth noting that the changing-sign perturbation term only satisfies the weaker Carathèodory conditions, which implies that the perturbation term ℏ can be allowed to have an infinite number of singular points; moreover, the value of the changing-sign perturbation term can tend to negative infinity in some singular points.

1. Introduction

In this paper, we focus on the multiplicity of positive solutions for the following singular tempered fractional initial-boundary value problem with the changing-sign perturbation term

$$\left\{\begin{array}{c}{}_0{}^R{D}_t^{a,\eta }\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }u\left(t\right)-\hslash (t,u\left(t\right)\right)\right]=f(t,u\left(t\right)),t\in (0,1),\\ u\left(0\right)=u^{\prime }\left(0\right)={}_0{}^R{D}_t^{b,\eta }u\left(0\right)=0,u\left(1\right)=0,\end{array}\right.$$

(1)

where the nonlinear function $f:[0,1]\times R\to R_{+}$ is continuous, $\hslash :[0,1]\times R\to R$ is a singular changing-sign perturbation term which only satisfies the Carathèodory conditions and can tend to negative infinity in some singular points. $0<a\le 1,2<b\le 3$, and $\eta$ are positive constants, ${\phi }_p$ is a p-Laplacian operator with conjugate relationship ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,p>1,q>1$ and ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s,{\phi }_p^{-1}\left(s\right)={\left|s\right|}^{q-2}s$. ${}_0{}^R{D}_t^{a,\eta },{}_0{}^R{D}_t^{b,\eta }$ are denoted as tempered fractional derivatives with a mathematical relation with the Riemann–Liouville fractional derivative

$${}_0{}^R{D}_t^{a,\eta }x\left(t\right)=e^{-\eta t}{}_0{}^R{\mathbb{D}}_t^a\left(e^{\eta t}x\left(t\right)\right),{}_0{}^R{D}_t^{b,\eta }x\left(t\right)=e^{-\eta t}{}_0{}^R{\mathbb{D}}_t^b\left(e^{\eta t}x\left(t\right)\right),$$

(2)

where the Riemann–Liouville fractional derivative and integral with order $a>0$ are defined by

$$I^{a}x\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(a\right)}}{\int }_0^t{(t-s)}^{a-1}x\left(s\right)ds,$$

(3)

and

$${\mathbb{D}}^{a}x\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-a)}}{\left({\displaystyle \frac{d}{dt}}\right)}^n{\int }_0^t{(t-s)}^{n-a-1}x\left(s\right)ds,$$

(4)

where $n=\left[a\right]+1,$ where $\left[a\right]$ is the greatest integer less than or equal to a. For more details on the definition of Riemann–Liouville fractional derivative and integral and applications in mathematical biochemistry, engineering, economics automatic control, and complex systems, we refer the readers to [1,2,3,4]. In addition, for further consideration, we also give the definition of the tempered Caputo fractional derivative for a function $f\in A{C}^n[0,1]$

$${}_0{}^{C}D_t^{a}f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-a)}}{\int }_0^t{\displaystyle \frac{f^{\left(n\right)}\left(s\right)e^{-\lambda (t-s)}ds}{{(t-s)}^{n-a-1}}},$$

where $n=\left[a\right]+1,$ where $\left[a\right]$ is the greatest integer less than or equal to a.

Equation (1) arises from modeling the random walk of Brownian motion in anomalous diffusion transition. In diffusion, given a random walk of mean zero particle jumps $T\left(n\right)=Y_1+Y_2+\cdots +Y_n$, using the Central Limit Theorem, in distribution sense to obtain $n^{-{\displaystyle \frac{1}{2}}}T\left(nt\right)\Rightarrow B\left(t\right)$, which implies that the Brownian motion limit $B\left(t\right)$ of the probability of the particle jump density function $p(z,t)$ solves the diffusion equation

$${\partial }_{t}p(z,t)=R{\partial }_z^{2}p(z,t).$$

(5)

Equation (5) actually establishes an important connection among Brownian motion, random walks and the diffusion equation with finite variance particle jumps. However, when a cloud of particles spreads in a different manner rather than traditional diffusion, the power law jumps with density possess a finite mean but an infinite variance; we subtract the mean and employ the extended Central Limit Theorem (Theorem 3.37 in [4]) to obtain $n^{-{\displaystyle \frac{1}{\beta }}}T\left(nt\right)\Rightarrow E\left(t\right),1<\beta <2$. Thus, the probability density $p(z,t)$ of the $\beta$-stable Lévy motion $E\left(t\right)$ solves the fractional diffusion equation

$${\partial }_{t}p(z,t)=R{\partial }_z^{\beta }p(z,t).$$

(6)

In particular, in anomalous diffusion transition, the probability density $p(z,t)$ of the $\beta$-stable Lévy motion ultimately relaxes into a traditional diffusion profile with semi-heavy tails at the last time, named transient anomalous diffusion [5]. This heavy-tails phenomenon also occur in finance, insurance, physics, geophysics, cell biology, ecology, medicine, and computer engineering. For example, in finance, the price fluctuations resemble a pure power law at moderate time but converge to a Gaussian at long time [6], i.e., the Brownian motion of the price fluctuations follows the correlations that look like a power law at moderate time but then eventually become short-range dependent at long time. We call this phenomenon semi-heavy tails. In order to temper the heavy-tails phenomenon, it is necessary to multiply the fractional derivative and integral by an exponential factor, which leads to the tempered fractional diffusion equation whose Lévy motion limit $E\left(t\right)$ is determined by a random walk model with an exponentially adjusted power law jump distribution [7]. In addition, Equation (1) is also closely related to the beam vibration equation

$${\displaystyle \frac{{\partial }^a}{\partial x^a}}\left({\displaystyle \frac{EI{\partial }^{a}u}{\partial x^a}}\right)=-R{\displaystyle \frac{{\partial }^{a}u}{\partial t^a}}+q\left(u\right),$$

(7)

where x is the position, t is the time, u is the displacement of the beam in the z direction, E is the elastic modulus, R is the mass per unit length, q is the load on the beam acting in the z-direction, and I is the second moment of area of the beam’s cross-section. In [8], Dias, Jorge, and Prata considered the time-dependent Euler–Bernoulli beam equation with discontinuous and singular coefficients. By using an extension of the Hörmander product of distributions with non-intersecting singular supports, the authors determined the general structure of its separable solutions and proved the existence, uniqueness, and regularity results under quite general conditions. For other related work, we refer the readers to [9,10,11].

Recently, Zhang et al. [12] considered a singular tempered fractional equation with the p-Laplacian operator

$$\left\{\begin{array}{c}{}_0{}^R{\mathbb{D}}_t^{\alpha ,\eta }\left({\phi }_p\left({}_0{}^R{\mathbb{D}}_t^{\beta ,\eta }x\left(t\right)\right)\right)=f(t,x\left(t\right)),\\ x\left(0\right)=0,{}_0{}^R{\mathbb{D}}_t^{\beta ,\eta }x\left(0\right)=0,x\left(1\right)={\int }_0^1{e}^{-\eta (1-t)}x\left(t\right)dt,\end{array}\right.$$

(8)

where $\alpha \in (0,1],\beta \in (1,2]$, $\eta$ is a positive constant, $f\in C\left(\right(0,1)\times (0,+\infty ),[0,+\infty \left)\right)$ is decreasing with respect to the second variable satisfying

$$0<{\int }_0^1{e}^{{\displaystyle \frac{\eta }{\sigma }}t}{f}^{{\displaystyle \frac{1}{\sigma }}}(t,\rho e^{-\eta t}{t}^{\beta -1})dt<+\infty ,$$

for some positive constant $\sigma \in (0,\alpha )$. By applying the upper and lower solutions method, the existence and asymptotic behavior of the positive solution for the singular tempered fractional Equation (8) are established. By the iterative technique and introducing a growth function $\phi :[0,1]\to [0,+\infty )$ and a constant $\theta \in (0,1)$ with $\phi \left(\gamma \right)>{\gamma }^{\theta }$ satisfying for any $\gamma \in (0,1),$

$$f(t,\gamma x,\gamma y)\ge \phi \left(\gamma \right)f(t,x,y),(t,x,y)\in (0,1)\times [0,+\infty )\times [0,+\infty ).$$

In recent years, because of the wide applications of the fractional equation in finance, insurance, physics, geophysics, cell biology, ecology, medicine, and computer engineering, many theoretical and computational tools, including spaces and smooth theories [13,14,15,16,17], operator theories [18,19,20,21], monotone iterative techniques [22,23,24], spectral analysis [25,26], the differential inequality approach [27], complex structure-preserving method [28], Hermitian Clifford analysis [29], different methods [30], and critical point theories [31,32,33] were developed to study various types of fractional equations such as the fractional differential system from eco-economical processes [34], fractional BMO space, Morrey space and Lorentz space problems [35,36,37], fractional nonlinear equation regularity and solvability [38,39], the fractional p-Laplace and the fractional p-convexity [40,41], and nonlocal fractional Schrödinger equations [42,43].

However, little work has been performed for the singular changing-sign differential equation, which originated in the study of chemical combustion reactions [44] and then was applied for the design of long-span cable-stayed bridge construction [45]. In a recent work [46], a third-order changing sign equation on time scales was studied by using some fixed point theorems; under the case of the nonlinearity f being a changing sign, the existence of positive solutions for third-order changing-sign equation on time scales was established. But if the changing-sign perturbation term involves a nonlinear operator such as a p-Laplacian operator, due to the nonlinear impact, it becomes quite difficult for the treatment of the changing-sign factor. Moreover, some nonlinear analysis theories about positivity such as cone theory and the method of upper and lower solutions [12,47,48] cannot be directly used for solving this kind of problem. Thus, to overcome these difficulties, in this paper, we firstly introduce a truncation function, and then establish an a priori estimate of the solution of isomorphic linear equations. By employing these properties, we transform the changing-sign tempered fractional initial-boundary value problem into a positive problem, and the existence results of multiple positive solutions are established by fixed point theorem in a cone.

This paper is structured as follows. In Section 2, we collect some lemmas and preliminaries, and then a series of transformations is made so that the original problem is transformed into a positive problem by the moving plane technique. In Section 3, the existence results of a single positive solution are established and the multiplicity results of a positive solution are stated in Section 4. Some examples are given in Section 5.

2. Preliminaries and Lemmas

In this section, we first introduce an important definition.

Definition 1. 

Suppose that there exists a map $(t,v)\mapsto \hslash (t,v)$ in $[0,1]\times R$ that satisfies the following conditions:

$\left(a\right)$$t\mapsto \hslash (t,v)$ is Lebesgue measurable for all $v\in R$;

$\left(b\right)$$v\mapsto \hslash (t,v)$ is continuous for any $t\in [0,1]$;

$\left(c\right)$ For any $t\in [0,1],v\in R$, there exists a function $\kappa \in L^1[0,1]$ such that $|\hslash (t,v\left)\right|<\kappa \left(t\right)$.

We say the map ℏ satisfies the Carathèodory condition.

Next, we list some basic conditions for the subsequent research.

$\left(P_1\right)$$f\in C\left(\right[0,1]\times (-\infty ,+\infty ),[0,+\infty \left)\right)$.

$\left(P_2\right)$$\hslash :[0,1]\times R\to R$ satisfies the Carathèodory condition, and $\hslash \ge -\kappa \left(t\right)$, with $\kappa \left(0\right)=0$.

Lemma 1 

(see [1]). Suppose $x\left(t\right)\in C[0,1]\cap L^1[0,1]$ and $a>b>0$, and take $n=\left[a\right]+1$, then one has

$$I^a{}_0{}^R{\mathbb{D}}_t^{a}x\left(t\right)=x\left(t\right)+{\gamma }_1{t}^{a-1}+{\gamma }_2{t}^{a-2}+···+{\gamma }_n{t}^{a-n},$$

(9)

where ${\gamma }_i\in R,$ and $i=1,2,3,...,n.$

Meanwhile, we also have

$$I^a{I}^{b}x\left(t\right)=I^{a+b}x\left(t\right),{}_0{}^R{\mathbb{D}}_t^b{I}^{a}x\left(t\right)=I^{a-b}x\left(t\right),{}_0{}^R{\mathbb{D}}_t^b{I}^{b}x\left(t\right)=x\left(t\right).$$

Lemma 2. 

Supposing that $2<b\le 3$ and $\kappa \in L^1[0,1]$, the following linear tempered fractional equation

$$\left\{\begin{array}{c}-{}_0{}^R{D}_t^{b,\eta }u\left(t\right)=\kappa \left(t\right),t\in [0,1],\\ u\left(0\right)=u^{\prime }\left(0\right)=0,u\left(1\right)=0,\end{array}\right.$$

(10)

has a unique solution

$$x\left(t\right)={\int }_0^{1}G(t,s)\kappa \left(s\right)ds,$$

(11)

where

$$G(t,s)=\left\{\begin{array}{c}{\displaystyle \frac{{(1-s)}^{b-1}{t}^{b-1}-{(t-s)}^{b-1}}{\mathrm{\Gamma }\left(b\right)}}{e}^{\eta s}{e}^{-\eta t},0\le s\le t\le 1,\\ {\displaystyle \frac{{(1-s)}^{b-1}{t}^{b-1}}{\mathrm{\Gamma }\left(b\right)}}{e}^{-\eta t}{e}^{\eta s},0\le t\le s\le 1,\end{array}\right.$$

(12)

is the Green function of Equation (10).

Proof. 

It follows from Lemma 2.1 that (10) can be transformed to the following form:

$$e^{\eta t}u\left(t\right)-c_1{t}^{b-1}-c_2{t}^{b-2}-c_3{t}^{b-3}=-{\int }_0^t{\displaystyle \frac{{(t-s)}^{b-1}}{\mathrm{\Gamma }\left(b\right)}}{e}^{\eta s}\kappa \left(s\right)ds,$$

(13)

since $b\in (2,3]$ and $u\left(0\right)=0$, we have $c_3=0$ and then

$$e^{\eta t}u\left(t\right)-c_1{t}^{b-1}-c_2{t}^{b-2}=-{\int }_0^t{\displaystyle \frac{{(t-s)}^{b-1}}{\mathrm{\Gamma }\left(b\right)}}{e}^{\eta s}\kappa \left(s\right)ds.$$

(14)

Integrating (14) with respect to t, we have

$$\eta e^{\eta t}u\left(t\right)+e^{\eta t}{u}^{\prime }\left(t\right)-c_1(b-1)t^{b-2}-c_2(b-2)t^{b-3}=-(b-1){\int }_0^t{\displaystyle \frac{{(t-s)}^{b-2}}{\mathrm{\Gamma }\left(b\right)}}{e}^{\eta s}\kappa \left(s\right)ds,$$

(15)

and it follows from boundary conditions $u\left(0\right)=u^{\prime }\left(0\right)=0$ and the above Equation (15) that $c_2=0$. Consequently,

$$e^{\eta t}u\left(t\right)-c_1{t}^{b-1}=-{\int }_0^t{\displaystyle \frac{{(t-s)}^{b-1}}{\mathrm{\Gamma }\left(b\right)}}{e}^{\eta s}\kappa \left(s\right)ds,$$

(16)

and taking $t=1$, one has

$$c_1={\int }_0^1{\displaystyle \frac{{(1-s)}^{b-1}}{\mathrm{\Gamma }\left(b\right)}}{e}^{\eta s}\kappa \left(s\right)ds,$$

(17)

then by substituting (17) into (16), we obtain the Green function of (10)

$$G(t,s)=\left\{\begin{array}{c}{\displaystyle \frac{{(1-s)}^{b-1}{t}^{b-1}-{(t-s)}^{b-1}}{\mathrm{\Gamma }\left(b\right)}}{e}^{\eta s}{e}^{-\eta t},0\le s\le t\le 1,\\ {\displaystyle \frac{{(1-s)}^{b-1}{t}^{b-1}}{\mathrm{\Gamma }\left(b\right)}}{e}^{-\eta t}{e}^{\eta s},0\le t\le s\le 1,\end{array}\right.$$

and therefore, the proof is completed. □

It follows from Lemma 2.3 of [3] that the following properties of the Green function $G(t,s)$ hold.

Lemma 3. 

(i) $G(t,s)$ is a non-negative continuous function for all $(t,s)\in [0,1]\times [0,1]$.

(ii) For any $t,s\in [0,1]$, and $G(t,s)$ satisfies the following inequalities:

$${\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}{e}^{-\eta t}{t}^{b-1}(1-t)}{\mathrm{\Gamma }\left(b\right)}}\le G(t,s)\le {\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}\mathit{or}{\displaystyle \frac{e^{-\eta t}{t}^{b-1}(1-t)}{\mathrm{\Gamma }\left(b\right)}}.$$

Lemma 4. 

The unique solution of the linear tempered fractional Equation (10) possesses the following estimate:

$$0\le x\left(t\right)\le L{e}^{-\eta t}{t}^{b-1}(1-t),$$

where

$$L={\displaystyle \frac{{\int }_0^1\kappa \left(s\right)ds}{\mathrm{\Gamma }\left(b\right)}}.$$

Proof. 

It follows from Lemma 3 that

$$\begin{array}{c}x\left(t\right)={\int }_0^{1}G(t,s)\kappa \left(s\right)ds\\ \le {\int }_0^1{\displaystyle \frac{e^{-\eta t}{t}^{b-1}(1-t)}{\mathrm{\Gamma }\left(b\right)}}\kappa \left(s\right)ds\\ =L{e}^{-\eta t}{t}^{b-1}(1-t).\end{array}$$

(18)

On the other hand, noticing that $G(t,s)\ge 0,[t,s]\in [0,1\times [0,1]$ and $\kappa \left(t\right)\ge 0,t\in [0,1]$, one has

$$x\left(t\right)={\int }_0^{1}G(t,s)\kappa \left(s\right)ds\ge 0.$$

□

Now, define a positive piecewise function of $𝘍\in C[0,1]$ such that

$${\left[𝘍\left(t\right)\right]}^{*}=\left\{\begin{array}{c}𝘍\left(t\right),𝘍\left(t\right)\ge 0,\\ 0,𝘍\left(t\right)<0.\end{array}\right.$$

(19)

Lemma 5. 

Let $a\in (0,1],b\in (2,3]$, suppose that $\left(P_1\right)$ and $\left(P_2\right)$ hold. Then, Equation (1) has a positive solution equivalent to the following modified equation:

$$\left\{\begin{array}{c}{}_0{}^R{D}_t^{a,\eta }\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }v\left(t\right)-\hslash (t,{[v\left(t\right)-x\left(t\right)]}^{*})-\kappa \left(t\right)\right)\right]=f(t,{[v\left(t\right)-x\left(t\right)]}^{*}),t\in (0,1),\\ v\left(0\right)=v^{\prime }\left(0\right)=0,{}_0{}^R{D}_t^{b,\eta }v\left(0\right)=0,v\left(1\right)=0,\end{array}\right.$$

(20)

has a solution $v\left(t\right)$ satisfying $v\left(t\right)\ge x\left(t\right),t\in [0,1]$.

Moreover, the solution of the modified Equation (20) can be expressed by the following integral form:

$$v\left(t\right)=$$

$${\int }_0^{1}G(t,s)\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds.$$

Proof. 

Suppose that v is a solution of the modified Equation (20) with $v\left(t\right)\ge x\left(t\right)$. Then, by the definitions of the piecewise function, one has

$$\left\{\begin{array}{c}{}_0{}^R{D}_t^{a,\eta }\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }v\left(t\right)-\hslash (t,[v\left(t\right)-x\left(t\right)])-\kappa \left(t\right)\right)\right]=f(t,[v\left(t\right)-x\left(t\right)]),t\in (0,1),\\ v\left(0\right)=v^{\prime }\left(0\right)=0,{}_0{}^R{D}_t^{b,\eta }v\left(0\right)=0,v\left(1\right)=0.\end{array}\right.$$

(21)

Let $u\left(t\right)=v\left(t\right)-x\left(t\right)$, then $u\left(t\right)\ge 0,$ and $u^{\prime }\left(t\right)=v^{\prime }\left(t\right)-x^{\prime }\left(t\right)$. Noting that

$$v\left(0\right)=v^{\prime }\left(0\right)=0,x\left(0\right)=x^{\prime }\left(0\right)=0,v\left(1\right)=0,x\left(1\right)=0,{}_0{}^R{D}_t^{b,\eta }v\left(0\right)=0,{}_0{}^R{D}_t^{b,\eta }x\left(0\right)=0,$$

so, we have

$$u\left(0\right)=u\left(1\right)=0,u^{\prime }\left(0\right)=0,{}_0{}^R{D}_t^{b,\eta }u\left(0\right)=0.$$

Thus, it follows from $-{}_0{}^R{D}_t^{b,\eta }x\left(t\right)=\kappa \left(t\right)$ and $v\left(t\right)=u\left(t\right)+x\left(t\right)$ that

$$\begin{array}{c}{}_0{}^R{D}_t^{a,\eta }\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }v\left(t\right)-\hslash (t,[v\left(t\right)-x\left(t\right)])-\kappa \left(t\right)\right)\right]\\ ={}_0{}^R{D}_t^{a,\eta }\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }u\left(t\right)+\kappa \left(t\right)-\hslash (t,u\left(t\right))-\kappa \left(t\right)\right)\right]\\ ={}_0{}^R{D}_t^{a,\eta }\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }u\left(t\right)-\hslash (t,u\left(t\right))\right)\right]=f(t,[v\left(t\right)-x\left(t\right)])=f(t,u\left(t\right)),\end{array}$$

which implies that the (20) can be converted into (1), and $u\left(t\right)=v\left(t\right)-x\left(t\right)$ is a positive solution of Equation (1).

Conversely, if u is a positive solution of Equation (1), then we have

$$\left\{\begin{array}{c}{}_0{}^R{D}_t^{a,\eta }\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }u\left(t\right)-\hslash (t,u\left(t\right)\right)\right]=f(t,u\left(t\right)),t\in (0,1),\\ u\left(0\right)=u\left(1\right)=0,{}_0{}^R{D}_t^{b,\eta }u\left(0\right)=0,u^{\prime }\left(0\right)=0.\end{array}\right.$$

Let $u\left(t\right)=v\left(t\right)-x\left(t\right)$, then $v\left(t\right)\ge x\left(t\right)$, then

$$\begin{array}{c}{}_0{}^R{D}_t^{a,\eta }\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }u\left(t\right)-\hslash (t,u\left(t\right)\right)\right]\\ ={}_0{}^R{D}_t^{a,\eta }\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }v\left(t\right)-\hslash (t,[v\left(t\right)-x\left(t\right)])-\kappa \left(t\right)\right)\right]\\ ={}_0{}^R{D}_t^{a,\eta }\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }v\left(t\right)-\hslash (t,{[v\left(t\right)-x\left(t\right)]}^{*})-\kappa \left(t\right)\right)\right]\\ =f(t,u\left(t\right))=f(t,[v\left(t\right)-x\left(t\right)])=f(t,{[v\left(t\right)-x\left(t\right)]}^{*}).\end{array}$$

On the other hand, the boundary conditions clearly hold. Thus, Equation (1) is equivalent to the modified problem (20).

Moreover, it follows from (2) and (20) that

$$e^{-\eta t}{}_0{}^R{\mathbb{D}}_t^a\left\{e^{\eta t}\left[{\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }v\left(t\right)-\kappa \left(t\right)-\hslash (t,{[v\left(t\right)-x\left(t\right)]}^{*})\right)\right]\right\}=f(t,{[v\left(t\right)-x\left(t\right)]}^{*}).$$

Notice $a\in (0,1]$, and by Lemma 1, we have

$${\phi }_p\left(-{}_0{}^R{D}_t^{b,\eta }v\left(t\right)-\kappa \left(t\right)-\hslash (t,{[v\left(t\right)-x\left(t\right)]}^{*})\right)=e^{-\eta t}{I}^a\left(e^{\eta t}f(t,{[v\left(t\right)-x\left(t\right)]}^{*})\right),$$

(22)

then

$$-{}_0{}^R{D}_t^{b,\eta }v\left(t\right)-\kappa \left(t\right)-\hslash (t,{[v\left(t\right)-x\left(t\right)]}^{*})={\left({\int }_0^t{\displaystyle \frac{{(t-s)}^{a-1}{e}^{-\eta t}}{\mathrm{\Gamma }\left(a\right)}}{e}^{\eta s}f(s,{[v\left(s\right)-x\left(s\right)]}^{*})ds\right)}^{q-1},$$

that is

$$-{}_0{}^R{D}_t^{b,\eta }v\left(t\right)={\left({\int }_0^t{\displaystyle \frac{{(t-s)}^{a-1}{e}^{-\eta t}}{\mathrm{\Gamma }\left(a\right)}}{e}^{\eta s}f(s,{[v\left(s\right)-x\left(s\right)]}^{*})ds\right)}^{q-1}+\kappa \left(t\right)+\hslash (t,{[v\left(t\right)-x\left(t\right)]}^{*}).$$

(23)

It follows from Lemma 2 that

$$\begin{array}{c}v\left(t\right)=\hfill \\ {\int }_0^{1}G(t,s)\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}{e}^{\eta \tau }f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})d\tau \right)}^{q-1}+\kappa \left(s\right)+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})\right]ds.\hfill \end{array}$$

(24)

□

Remark 1. 

As $\hslash \in R$ satisfies the Carathèodory condition with $\hslash \ge -\kappa \left(t\right),t\in [0,1]$, and the nonlinearity function $f\in R_{+}$, so the sum of the right-hand side functions in (23) is greater than 0, which implies that, by the truncation technique, the changing-sign problem (1) is transformed to a positive problem (20).

Now, define our work space $E=C[0,1]$, which is a Banach space equipped with norm $\left|\right|v\left|\right|={max}_{t\in [0,1]}\left|v\left(t\right)\right|$. And define a cone P and an operator T as follows:

$$P=\left\{v\in E:v\left(t\right)\ge t^{b-1}{e}^{-\eta t}(1-t)\left|\right|v\left|\right|\right\},$$

and

$$\begin{array}{c}Tv\left(t\right)\\ ={\int }_0^{1}G(t,s)\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds.\end{array}$$

(25)

Thus, to seek the solution of the problem (20) is equivalent to finding the fixed point of the following problem:

$$Tv=v.$$

Lemma 6. 

Suppose that $\left(P_1\right)$ and $\left(P_2\right)$ hold, then $T:P\to P$ is a completely continuous operator.

Proof. 

Firstly, for any $v\in P$, there exists a constant $M>0$ and $\left|\right|v\left|\right|\le M$, and thus, one has

$$0\le {[v-x]}^{*}\left(t\right)\le v\left(t\right)\le \left|\right|v\left|\right|\le M.$$

Denote $\mathrm{\Lambda }={max}_{(t,v)\in [0,1]\times [0,M]}f(t,v)$, then $\mathrm{\Lambda }\ge 0$. It follows from Lemma 3, $\left(P_1\right)$ and $\left(P_2\right)$ that

$$\begin{array}{c}\left|\right|Tv\left|\right|\\ =\underset{t\in [0,1]}{max}{\int }_0^{1}G(t,s)\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\right.\\ \left.+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds\\ \le {\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\right.\\ \left.+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds\\ \le {\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}+2\kappa \left(s\right)\right]ds\\ ={\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\\ +2{\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right))}}\kappa \left(s\right)ds\\ \le {\mathrm{\Lambda }}^{q-1}{\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}{e}^{\eta \tau }d\tau \right)}^{q-1}ds+2L{e}^{\eta }\\ \le {\displaystyle \frac{{\mathrm{\Lambda }}^{q-1}{e}^{\eta }}{\mathrm{\Gamma }\left(b\right)}}{\int }_0^1{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}{e}^{\eta \tau }d\tau \right)}^{q-1}ds+2L{e}^{\eta }\\ ={\displaystyle \frac{{\mathrm{\Lambda }}^{q-1}{e}^{\eta }}{{\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}\mathrm{\Gamma }\left(b\right)}}{\int }_0^1{s}^{a(q-1)}ds+2L{e}^{\eta }\\ ={\displaystyle \frac{{\mathrm{\Lambda }}^{q-1}{e}^{\eta }}{{\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}\mathrm{\Gamma }\left(b\right)[a(q-1)+1]}}+2L{e}^{\eta }<+\infty .\end{array}$$

(26)

So, T is well defined and uniformly bounded.

Next, by Lemma 3 and (26), one has

$$\begin{array}{c}Tv\left(t\right)={\int }_0^{1}G(t,s)\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\right.\\ \left.+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds\\ \ge {\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}{e}^{-\eta t}{t}^{b-1}(1-t)}{\mathrm{\Gamma }\left(b\right)}}\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\right.\\ \left.+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds\\ =e^{-\eta t}{t}^{b-1}(1-t){\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\right.\\ \left.+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds\\ \ge e^{-\eta t}{t}^{b-1}(1-t)\left|\right|Tv\left|\right|,\end{array}$$

(27)

which implies that $T\left(P\right)\subset P$.

Obviously, T is equicontinuous in E because of the uniform continuity of G on $[0,1]\times [0,1]$, thus by the Arzela–Ascoli theorem, $T:P\to P$ is completely continuous. □

Lemma 7 

([49]). Supposing that P is a cone of real Banach space E, the bounded open subsets ${\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2$ of E satisfy $\theta \in {\mathrm{\Omega }}_1,{\overline{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2$. Let $T:P\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1)\to P$ be a completely continuous operator such that either

(1) $\parallel Tv\parallel \le \parallel v\parallel ,v\in P\cap \partial {\mathrm{\Omega }}_1$ and $\parallel Tv\parallel \ge \parallel v\parallel ,v\in P\cap \partial {\mathrm{\Omega }}_2,$

or

(2) $\parallel Tv\parallel \ge \parallel v\parallel ,v\in P\cap \partial {\mathrm{\Omega }}_1$ and $\parallel Tv\parallel \le \parallel v\parallel ,v\in P\cap \partial {\mathrm{\Omega }}_2.$

Then, T has a fixed point in $P\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1).$

3. Existence Results

In this section, we firstly denote

$$\beta =\left({\left({\displaystyle \frac{1}{a\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{1}{\mathrm{\Gamma }\left(b\right)\left[a\right(q-1)+1]}}+L\right)e^{2\eta },$$

and then introduce some conditions to be used in the rest of this paper.

(A) There exists a positive constant $n\ge max\{2L,2\beta \}$ and for any $(t,v)\in [0,1]\times [0,n]$ such that

$$f(t,v)\le {\left({\displaystyle \frac{n-2\beta }{\beta }}\right)}^{{\displaystyle \frac{1}{q-1}}}.$$

(B) There exists an interval $[c,d]\subset (0,1)$ and a positive constant $m>n$ such that for any $(t,v)\in [c,d]\times [{\displaystyle \frac{1}{2}}m{e}^{-dt}(1-d)c^{b-1},m]$,

$$f(t,u)\ge {\left({\displaystyle \frac{m{\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{e}^{\frac{\eta }{2}}{2}^b\mathrm{\Gamma }\left(b\right)}{c^{a(q-1)+1}{(1-d)}^{b-1}{e}^{\eta (2c-(q-1\left)d\right)}(d-c)}}\right)}^{{\displaystyle \frac{1}{q-1}}}.$$

(C) There exists $[\omega ,\varpi ]\subset [0,1]$, for any $(t,v)\in [\omega ,\varpi ]\times [3\beta e^{-\eta \varpi }(1-\varpi ){\omega }^{b-1},4\beta ]$ such that

$$f(t,u)\ge {\left({\displaystyle \frac{4\beta {\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{e}^{\frac{\eta }{2}}{2}^b\mathrm{\Gamma }\left(b\right)}{{\omega }^{a(q-1)+1}{(1-\varpi )}^{b-1}{e}^{\eta (\omega -(q-1\left)\varpi \right)}(\varpi -\omega )}}\right)}^{{\displaystyle \frac{1}{q-1}}}.$$

(D) There exists a positive constant $k>4\beta$, and for any $(t,v)\in [0,1]\times [0,k]$ such that

$$f(t,u)\le {\left({\displaystyle \frac{k}{\beta }}\right)}^{{\displaystyle \frac{1}{q-1}}}.$$

Theorem 1. 

Assuming that $\left(P_1\right)$, $\left(P_2\right)$, (A), and (B) hold, Equation (1) with a changing-sign perturbation term has at least one positive solution $u_1\left(t\right)$, which satisfies the following asymptotic properties:

$$L{e}^{-\eta t}(1-t)t^{b-1}\le u_1\left(t\right)\le m,t\in [0,1].$$

Proof. 

Firstly, let

$${\mathrm{\Omega }}_1=\{v\in E:|\left|v\right||<n\},\partial {\mathrm{\Omega }}_1=\{v\in E:|\left|v\right||=n\}.$$

For any $v\in P\cap \partial {\mathrm{\Omega }}_1$, by the definition of piecewise function ${\left[𝘍\left(t\right)\right]}^{*}$, we have

$${[v\left(t\right)-x\left(t\right)]}^{*}\le v\left(t\right)-x\left(t\right)\le \left|\right|v\left|\right|=n.$$

By (A) and (26), we obtain

$$\begin{array}{c}\left|\right|Tv\left|\right|=\underset{t\in [0,1]}{max}{\int }_0^{1}G(t,s)\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\right.\\ \left.+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds\\ \le {\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}+2\kappa \left(s\right)\right]ds\\ ={\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\\ +2{\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}\kappa \left(s\right)ds\\ \le {\left({\displaystyle \frac{1}{\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{e^{\eta }}{\mathrm{\Gamma }\left(b\right)}}{\int }_0^1{\left({\int }_0^s{(s-\tau )}^{a-1}{e}^{-\eta s}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds+2L{e}^{\eta }\\ \le {\left({\displaystyle \frac{1}{\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{(n-2\beta )e^{\eta }}{\beta \mathrm{\Gamma }\left(b\right)}}{\int }_0^1{\left({\int }_0^s{(s-\tau )}^{a-1}{e}^{-\eta s}{e}^{\eta \tau }d\tau \right)}^{q-1}ds+2L{e}^{\eta }\\ \le {\left({\displaystyle \frac{1}{\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{(n-2\beta )e^{2\eta }}{\beta \mathrm{\Gamma }\left(b\right)}}{\int }_0^1{\left({\int }_0^s{(s-\tau )}^{a-1}d\tau \right)}^{q-1}ds+2L{e}^{\eta }\\ ={\left({\displaystyle \frac{1}{a\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{(n-2\beta )e^{2\eta }}{\beta \mathrm{\Gamma }\left(b\right)}}{\int }_0^1{s}^{a(q-1)}ds+2L{e}^{\eta }\\ \le {\left({\displaystyle \frac{1}{a\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{n{e}^{2\eta }}{\beta \mathrm{\Gamma }\left(b\right)\left[a\right(q-1)+1]}}+2L{e}^{2\eta }\\ \le {\left({\displaystyle \frac{1}{a\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{n{e}^{2\eta }}{\beta \mathrm{\Gamma }\left(b\right)\left[a\right(q-1)+1]}}+{\displaystyle \frac{n}{\beta }}L{e}^{2\eta }\\ \le {\displaystyle \frac{n{e}^{2\eta }}{\beta }}\left({\left({\displaystyle \frac{1}{a\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{1}{\mathrm{\Gamma }\left(b\right)\left[a\right(q-1)+1]}}+L\right)\\ =n=\left|\right|v\left|\right|,\end{array}$$

(28)

so, for any $v\in P\cap \partial {\mathrm{\Omega }}_1$, we have $\left|\right|Tv\left|\right|\le \left|\right|v\left|\right|$.

Next, let

$${\mathrm{\Omega }}_2=\{v\in E:|\left|v\right||<m\},\partial {\mathrm{\Omega }}_2=\{v\in E:|\left|v\right||=m\}.$$

For any $v\in P\cap \partial {\mathrm{\Omega }}_2,t\in [0,1]$, we have

$$\begin{array}{c}0\le {\displaystyle \frac{1}{2}}m{e}^{-\eta t}{t}^{b-1}(1-t)={\displaystyle \frac{1}{2}}\left|\right|v\left|\right|e^{-\eta t}{t}^{b-1}(1-t)\le {\displaystyle \frac{1}{2}}v\left(t\right)\le v\left(t\right)-{\displaystyle \frac{Lv\left(t\right)}{m}}\\ \le v\left(t\right)-L{e}^{-\eta t}{t}^{b-1}(1-t)\le v\left(t\right)-x\left(t\right)\le v\left(t\right)\le \left|\right|v\left|\right|=m,\end{array}$$

thus, for $t\in [c,d]$, one has

$${\displaystyle \frac{1}{2}}m{e}^{-\eta d}(1-d)c^{b-1}\le v\left(t\right)-x\left(t\right)\le m.$$

(29)

So for any $v\in P\cap \partial {\mathrm{\Omega }}_2$, it follows from (B) and (32) that

$$\begin{array}{c}\left|\right|Tv\left|\right|=\underset{t\in [0,1]}{max}\left|Tv\left(t\right)\right|\\ \ge {\int }_0^{1}G\left({\displaystyle \frac{1}{2}},s\right)\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\right.\\ \left.+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds\\ \ge {\int }_0^{1}G\left({\displaystyle \frac{1}{2}},s\right){\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\int }_c^{d}G\left({\displaystyle \frac{1}{2}},s\right){\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\int }_c^d{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}{e}^{-\frac{\eta }{2}}{\left(\frac{1}{2}\right)}^b}{\mathrm{\Gamma }\left(b\right)}}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ ={\displaystyle \frac{1}{e^{\frac{\eta }{2}}{2}^b\mathrm{\Gamma }\left(b\right)}}{\int }_c^{d}s{(1-s)}^{b-1}{e}^{\eta s}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\displaystyle \frac{c{(1-d)}^{b-1}{e}^{\eta c}}{e^{\frac{\eta }{2}}{2}^b\mathrm{\Gamma }\left(b\right)}}\left({\displaystyle \frac{m{\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{e}^{\frac{\eta }{2}}{2}^b\mathrm{\Gamma }\left(b\right)}{c^{a(q-1)+1}{(1-d)}^{b-1}{e}^{\eta (c-(q-1\left)d\right)}(d-c)}}\right)\times \\ {\int }_c^d{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\displaystyle \frac{m{\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}}{c^{a(q-1)}{e}^{\eta c-\eta d(q-1)}(d-c)}}{\int }_c^d{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}{e}^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\displaystyle \frac{m{a}^{q-1}}{c^{a(q-1)}{e}^{-\eta d(q-1)}(d-c)}}{\int }_c^d{\left({\int }_0^s{(s-\tau )}^{a-1}{e}^{-\eta s}d\tau \right)}^{q-1}ds\\ ={\displaystyle \frac{m{a}^{q-1}}{c^{a(q-1)}{e}^{-\eta d(q-1)}(d-c)}}{\int }_c^d{e}^{-\eta (q-1)s}{\left({\int }_0^s{(s-\tau )}^{a-1}d\tau \right)}^{q-1}ds\\ ={\displaystyle \frac{m}{c^{a(q-1)}{e}^{-\eta d(q-1)}(d-c)}}{\int }_c^d{e}^{-\eta (q-1)s}{s}^{a(q-1)}ds\\ \ge m=\left|\right|v\left|\right|,\end{array}$$

(30)

so, for any $v\in P\cap \partial {\mathrm{\Omega }}_2$, we have $\left|\right|Tv\left|\right|\ge \left|\right|v\left|\right|$.

Therefore, by Lemma 7, T has a fixed point $v_1\in P\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1)$, which satisfies

$$2L\le n\le \left|\right|v_1\left|\right|\le m.$$

In the following, we prove that $v_1\left(t\right)\ge x\left(t\right),t\in [0,1]$. In fact, by Lemma 5, one has

$$\begin{array}{c}{v}_1\left(t\right)-x\left(t\right)\ge v_1\left(t\right)-L{e}^{-\eta t}(1-t)t^{b-1}\ge v_1\left(t\right)-{\displaystyle \frac{L{v}_1\left(t\right)}{\left|\right|v_1\left|\right|}}\\ =\left(1-{\displaystyle \frac{L}{n}}\right)v_1\left(t\right)\ge {\displaystyle \frac{1}{2}}{v}_1\left(t\right)\ge L{e}^{-\eta t}(1-t)t^{b-1}\ge 0,t\in [0,1].\end{array}$$

(31)

Thus, we have $v_1\left(t\right)\ge x\left(t\right),t\in [0,1]$. By Lemma 5, $u_1\left(t\right)=v_1\left(t\right)-x\left(t\right)$ is a positive solution of the problem (1), which possesses the following asymptotic property:

$$L{e}^{-\eta t}(1-t)t^{b-1}\le u_1\left(t\right)\le m.$$

The proof is completed. □

Theorem 2. 

Assuming that $\left(P_1\right)$, $\left(P_2\right)$, (C), and (D) hold, Equation (1) with a changing-sign perturbation term has at least one positive solution $u_3\left(t\right)$, which satisfies the following asymptotic properties:

$$3L{t}^{b-1}{e}^{-\eta t}(1-t)\le u_3\left(t\right)\le k,t\in [0,1].$$

Proof. 

Firstly, let

$${\mathrm{\Omega }}_4=\{v\in E:|\left|v\right||<4\beta \},\partial {\mathrm{\Omega }}_4=\{v\in E:|\left|v\right||=4\beta \}.$$

Since $\beta >L,$ for any $v\in P\cap \partial {\mathrm{\Omega }}_4,t\in [0,1]$, we have

$$\begin{array}{c}0<3\beta e^{-\eta t}{t}^{b-1}(1-t)={\displaystyle \frac{3}{4}}\left|\right|v\left|\right|e^{-\eta t}{t}^{b-1}(1-t)\le {\displaystyle \frac{3}{4}}v\left(t\right)\le v\left(t\right)-{\displaystyle \frac{Lv\left(t\right)}{4\beta }}\\ \le v\left(t\right)-L{e}^{-\eta t}{t}^{b-1}(1-t)\le v\left(t\right)-x\left(t\right)\le v\left(t\right)\le \left|\right|v\left|\right|=4\beta ,\end{array}$$

(32)

then by (32) and $t\in [\omega ,\varpi ]$, we have

$$3\beta e^{-\eta \varpi }{\omega }^{b-1}(1-\varpi )\le {[v\left(t\right)-x\left(t\right)]}^{*}=v\left(t\right)-x\left(t\right)\le 4\beta .$$

(33)

Thus, for any $v\in P\cap \partial {\mathrm{\Omega }}_4$, by (C) and (33), one has

$$\begin{array}{c}\left|\right|Tv\left|\right|=\underset{t\in [0,1]}{max}\left|Tv\left(t\right)\right|\\ \ge {\int }_0^{1}G\left({\displaystyle \frac{1}{2}},s\right)\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\right.\\ \left.+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds\\ \ge {\int }_0^{1}G\left({\displaystyle \frac{1}{2}},s\right){\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\int }_{\omega }^{\varpi }G\left({\displaystyle \frac{1}{2}},s\right){\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\int }_{\omega }^{\varpi }{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}{e}^{-\frac{\eta }{2}}{\left(\frac{1}{2}\right)}^b}{\mathrm{\Gamma }\left(b\right)}}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ ={\displaystyle \frac{1}{e^{\frac{\eta }{2}}{2}^b\mathrm{\Gamma }\left(b\right)}}{\int }_{\omega }^{\varpi }s{(1-s)}^{b-1}{e}^{\eta s}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\displaystyle \frac{\omega {(1-\varpi )}^{b-1}{e}^{\eta \omega }}{e^{\frac{\eta }{2}}{2}^b\mathrm{\Gamma }\left(b\right)}}{\int }_{\omega }^{\varpi }{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\displaystyle \frac{4\beta {\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}}{{\omega }^{a(q-1)}{e}^{\eta \omega -\eta \varpi (q-1)}(\varpi -\omega )}}{\int }_{\omega }^{\varpi }{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}{e}^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\displaystyle \frac{4\beta a^{q-1}}{{\omega }^{a(q-1)}{e}^{-\eta \varpi (q-1)}(\varpi -\omega )}}{\int }_{\omega }^{\varpi }{\left({\int }_0^s{(s-\tau )}^{a-1}{e}^{-\eta s}d\tau \right)}^{q-1}ds\\ ={\displaystyle \frac{4\beta a^{q-1}}{{\omega }^{a(q-1)}{e}^{-\eta \varpi (q-1)}(\varpi -\omega )}}{\int }_{\omega }^{\varpi }{e}^{-\eta (q-1)s}{\left({\int }_0^s{(s-\tau )}^{a-1}d\tau \right)}^{q-1}ds\\ ={\displaystyle \frac{4\beta }{{\omega }^{a(q-1)}{e}^{-\eta \varpi (q-1)}(\varpi -\omega )}}{\int }_{\omega }^{\varpi }{e}^{-\eta (q-1)s}{s}^{a(q-1)}ds\\ \ge 4\beta =\left|\right|v\left|\right|,\end{array}$$

(34)

thus, for any $v\in P\cap \partial {\mathrm{\Omega }}_4$, we have $\left|\right|Tv\left|\right|\ge \left|\right|v\left|\right|$.

Next, let

$${\mathrm{\Omega }}_5=\{v\in E:|\left|v\right||<k\},\partial {\mathrm{\Omega }}_5=\{v\in E:|\left|v\right||=k\}.$$

For any $v\in P\cap \partial {\mathrm{\Omega }}_5$, by the definition of piecewise function ${\left[𝘍\left(t\right)\right]}^{*}$, we have

$${[v\left(t\right)-x\left(t\right)]}^{*}\le v\left(t\right)-x\left(t\right)\le \left|\right|v\left|\right|=k.$$

Then, for any $v\in P\cap \partial {\mathrm{\Omega }}_5$, one has

$$\begin{array}{c}\left|\right|Tv\left|\right|=\underset{t\in [0,1]}{max}{\int }_0^{1}G(t,s)\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\right.\\ \left.+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds\le \\ {\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}+2\kappa \left(s\right)\right]ds\\ ={\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\\ +2{\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}\kappa \left(s\right)ds\\ \le {\left({\displaystyle \frac{1}{\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{e^{\eta }}{\mathrm{\Gamma }\left(b\right)}}{\int }_0^1{\left({\int }_0^s{(s-\tau )}^{a-1}{e}^{-\eta s}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds+2L{e}^{\eta }\\ \le {\left({\displaystyle \frac{1}{\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{k{e}^{\eta }}{\beta \mathrm{\Gamma }\left(b\right)}}{\int }_0^1{\left({\int }_0^s{(s-\tau )}^{a-1}{e}^{-\eta s}{e}^{\eta \tau }d\tau \right)}^{q-1}ds+2L{e}^{\eta }\\ \le {\left({\displaystyle \frac{1}{\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{k{e}^{2\eta }}{\beta \mathrm{\Gamma }\left(b\right)}}{\int }_0^1{\left({\int }_0^s{(s-\tau )}^{a-1}d\tau \right)}^{q-1}ds+2L{e}^{\eta }\\ ={\left({\displaystyle \frac{1}{a\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{k{e}^{2\eta }}{\beta \mathrm{\Gamma }\left(b\right)}}{\int }_0^1{s}^{a(q-1)}ds+2L{e}^{\eta }\\ \le {\left({\displaystyle \frac{1}{a\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{k{e}^{2\eta }}{\beta \mathrm{\Gamma }\left(b\right)\left[a\right(q-1)+1]}}+2L{e}^{2\eta }\\ \le {\left({\displaystyle \frac{1}{a\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{k{e}^{2\eta }}{\beta \mathrm{\Gamma }\left(b\right)\left[a\right(q-1)+1]}}+{\displaystyle \frac{k}{\beta }}L{e}^{2\eta }\\ \le {\displaystyle \frac{k{e}^{2\eta }}{\beta }}\left({\left({\displaystyle \frac{1}{a\mathrm{\Gamma }\left(a\right)}}\right)}^{q-1}\times {\displaystyle \frac{1}{\mathrm{\Gamma }\left(b\right)\left[a\right(q-1)+1]}}+L\right)\\ =k=\left|\right|v\left|\right|.\end{array}$$

(35)

So, for any $v\in P\cap \partial {\mathrm{\Omega }}_5$, we have $\left|\right|Tv\left|\right|\le \left|\right|v\left|\right|$. Therefore, by Lemma 7, T has a fixed point $v_3\left(t\right)$ satisfying

$$4\beta \le \left|\right|v_3\left|\right|\le k.$$

Letting $u_3\left(t\right)=v_3\left(t\right)-x\left(t\right)$, we have

$$\begin{array}{cc}{u}_3\left(t\right)& =v_3\left(t\right)-x\left(t\right)\ge v_3\left(t\right)-L{e}^{-\eta t}{t}^{b-1}(1-t)\ge v_3\left(t\right)-{\displaystyle \frac{Lv\left(t\right)}{4\beta }}\\ & \ge {\displaystyle \frac{3}{4}}{v}_3\left(t\right)\ge 3k{e}^{-\eta t}{t}^{b-1}(1-t)\ge 3L{e}^{-\eta t}{t}^{b-1}(1-t)\ge 0.\end{array}$$

By Lemma 5, $u_3\left(t\right)=v_3\left(t\right)-x\left(t\right)$ is another positive solution of the problem (1), which possesses the following asymptotic property

$$3L{e}^{-\eta t}(1-t)t^{b-1}\le u_3\left(t\right)\le k.$$

The proof is completed. □

4. Multiplicity Results

In this section, we focus on the multiplicity results of the problem (1). Firstly, we list some assumptions and conditions required for the proof part of this section.

(E)

$$\underset{\left|v\right|\to \infty }{lim}\underset{t\in [0,1]}{max}{\displaystyle \frac{f(t,v)}{{\left|v\right|}^{\frac{1}{q-1}}}}=0.$$

(F) There exists a subinterval $[\mathit{ȷ},\mathit{\ell }]\subset (0,1)$ such that

$$\underset{v\to +\infty }{lim}\underset{t\in [\mathit{ȷ},\mathit{\ell }]}{min}{\displaystyle \frac{f(t,v)}{v^{\frac{1}{q-1}}}}=+\infty .$$

Theorem 3. 

Supposing that $\left(P_1\right)$, $\left(P_2\right)$, (A), (B) and (E) hold, the problem (1) has at least two positive solutions $u_1,u_2$. Moreover, there exists a constant $\tilde{R}>0$ such that the following asymptotic property holds:

$$L{t}^{b-1}{e}^{-\eta t}(1-t)\le u_1\left(t\right)\le m,$$

$${\displaystyle \frac{1}{2}}m{t}^{b-1}{e}^{-\eta t}(1-t)\le u_2\left(t\right)\le \tilde{R}.$$

Proof. 

Firstly, it follows from Theorem 1 that T has one fixed point $v_1$ which satisfies

$$2L\le n\le \left|\right|v_1\left|\right|\le m.$$

In what follows, we seek another fixed point $v_2$. To do this, choose a sufficiently small constant ${ϵ}_2>0$ such that

$${\displaystyle \frac{e^{\eta q}{\left(2{ϵ}_2\right)}^{q-1}}{\mathrm{\Gamma }\left(b\right){\mathrm{\Gamma }}^{q-1}\left(a\right)}}<1.$$

For the above positive constant ${ϵ}_2$, by (E), there exists a positive constant $L_1>m$ such that for any $\left|v\right|>L_1$,

$$f(t,v)\le {ϵ}_2{\left|v\right|}^{{\displaystyle \frac{1}{q-1}}},t\in [0,1].$$

(36)

Take

$$\tilde{R}={\displaystyle \frac{2L{e}^{\eta }}{1-\frac{e^{\eta q}{\left(2{ϵ}_2\right)}^{q-1}}{\mathrm{\Gamma }\left(b\right){\mathrm{\Gamma }}^{q-1}\left(a\right)}}}+L_1+{\left({\displaystyle \frac{{max}_{[0,1]\times [0,L_1]}f(t,u)}{{ϵ}_2}}\right)}^{q-1},$$

(37)

Then, $\tilde{R}>L_1$. Let

$${\mathrm{\Omega }}_3=\{v\in E:|\left|v\right||<\tilde{R}\},\partial {\mathrm{\Omega }}_3=\{v\in E:|\left|v\right||=\tilde{R}\}.$$

For any $v\in P\cap \partial {\mathrm{\Omega }}_3$, by (32), one has

$$\begin{array}{c}\left|\right|Tv\left|\right|=\underset{t\in [0,1]}{max}\left|Tv\left(t\right)\right|\\ \le {\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}+2\kappa \left(s\right)\right]ds\\ \le {\int }_0^1{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds+2L{e}^{\eta }\\ \le {\displaystyle \frac{e^{\eta }}{\mathrm{\Gamma }\left(b\right)}}{\left({\int }_0^1{\displaystyle \frac{{(1-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,v\left(\tau \right)-x\left(\tau \right))e^{\eta \tau }d\tau \right)}^{q-1}+2L{e}^{\eta }\\ \le {\displaystyle \frac{e^{\eta }}{\mathrm{\Gamma }\left(b\right)}}\left({\int }_{0\le v\left(\tau \right)-x\left(\tau \right)\le L_1}{\displaystyle \frac{{(1-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,v\left(\tau \right)-x\left(\tau \right))e^{\eta \tau }d\tau \right.\\ {\left.+{\int }_{L_1\le v\left(\tau \right)-x\left(\tau \right)\le \tilde{R}}{\displaystyle \frac{{(1-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,v\left(\tau \right)-x\left(\tau \right))e^{\eta \tau }d\tau \right)}^{q-1}+2L{e}^{\eta }\\ \le {\displaystyle \frac{e^{\eta }}{\mathrm{\Gamma }\left(b\right)}}{\left(\underset{[0,1]\times [0,L_1]}{max}f(t,u){\int }_0^1{\displaystyle \frac{{(1-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}{e}^{\eta \tau }d\tau +{\int }_0^1{\displaystyle \frac{{(1-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}{ϵ}_2{\tilde{R}}^{{\displaystyle \frac{1}{q-1}}}{e}^{\eta \tau }d\tau \right)}^{q-1}+2L{e}^{\eta }\\ \le {\displaystyle \frac{e^{\eta }}{\mathrm{\Gamma }\left(b\right)}}{\left({ϵ}_2{\tilde{R}}^{{\displaystyle \frac{1}{q-1}}}{\int }_0^1{\displaystyle \frac{{(1-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}{e}^{\eta \tau }d\tau +{\int }_0^1{\displaystyle \frac{{(1-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}{ϵ}_2{\tilde{R}}^{{\displaystyle \frac{1}{q-1}}}{e}^{\eta \tau }d\tau \right)}^{q-1}+2L{e}^{\eta }\\ \le {\displaystyle \frac{e^{\eta q}{\left(2{ϵ}_2\right)}^{q-1}\tilde{R}}{\mathrm{\Gamma }\left(b\right){\mathrm{\Gamma }}^{q-1}\left(a\right)}}+2L{e}^{\eta }\le \tilde{R}.\end{array}$$

(38)

Thus, for any $v\in P\cap \partial {\mathrm{\Omega }}_3$, we have $\left|\right|Tv\left|\right|\le \left|\right|v\left|\right|.$ Therefore, by Lemma 7, T has another fixed point $v_2\left(t\right)$ satisfying

$$2L\le m\le \left|\right|v_2\left|\right|\le \tilde{R}.$$

Letting $u_2\left(t\right)=v_2\left(t\right)-x\left(t\right)$, we have

$$\begin{array}{cc}{u}_2\left(t\right)& =v_2\left(t\right)-x\left(t\right)\ge v_2\left(t\right)-L{e}^{-\eta t}{t}^{b-1}(1-t)\ge v_2\left(t\right)-{\displaystyle \frac{Lv\left(t\right)}{m}}\\ & \ge {\displaystyle \frac{1}{2}}{v}_2\left(t\right)\ge {\displaystyle \frac{1}{2}}m{e}^{-\eta t}{t}^{b-1}(1-t)\ge 0.\end{array}$$

By Lemma 5, $u_2\left(t\right)=v_2\left(t\right)-x\left(t\right)$ is another positive solution of the problem (1), which possesses the following asymptotic property:

$${\displaystyle \frac{1}{2}}m{e}^{-\eta t}(1-t)t^{b-1}\le u_2\left(t\right)\le \tilde{R}.$$

The proof is completed. □

Theorem 4. 

Supposing that $\left(P_1\right)$, $\left(P_2\right)$, (C), (D) and (F) hold, the problem (1) has at least two positive solutions $u_3,u_4$. Moreover, there exists a constant $N>0$ such that the following asymptotic property holds:

$$3L{t}^{b-1}{e}^{-\eta t}(1-t)\le u_3\left(t\right)\le k,$$

$${\displaystyle \frac{1}{2}}k{t}^{b-1}{e}^{-\eta t}(1-t)\le u_4\left(t\right)\le N.$$

Proof. 

By Theorem 2, we know that T has one fixed point $v_3$ which satisfies

$$4\beta \le \left|\right|v_3\left|\right|\le k.$$

In what follows, we seek another fixed point $v_4$. Firstly, it follows from (F) that there exists a constant $L_1>k$ and sufficiently large ${\rm Y}$

$${\rm Y}\ge {\left[{\displaystyle \frac{{(1-\mathit{\ell })}^b{e}^{\eta (\mathit{ȷ}-(q-1)\mathit{\ell })}{\mathit{ȷ}}^{b+a(q-1)}(\mathit{\ell }-\mathit{ȷ})}{{\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{2}^b{e}^{\frac{\eta }{2}}\mathrm{\Gamma }\left(b\right)}}\right]}^{-{\displaystyle \frac{1}{q-1}}},$$

(39)

such that for any $v>L_1$ and $t\in [\mathit{ȷ},\mathit{\ell }]$,

$$f(t,v)\ge {\rm Y}{v}^{{\displaystyle \frac{1}{q-1}}}.$$

(40)

Choosing

$$N>{\displaystyle \frac{2L_1{e}^{\eta \mathit{\ell }}}{{\mathit{ȷ}}^{b-1}(1-\mathit{\ell })}}+k,$$

since $[\mathit{ȷ},\mathit{\ell }]\subset (0,1)$, one has

$$N>k>2L.$$

Let

$${\mathrm{\Omega }}_6=\{v\in E:|\left|v\right||<N\},\partial {\mathrm{\Omega }}_6=\{v\in E:|\left|v\right||=N\}.$$

For any $v\in P\cap \partial {\mathrm{\Omega }}_6,t\in [\mathit{ȷ},\mathit{\ell }]$, by the definitions of the cone P, one has

$$\begin{array}{c}v\left(t\right)-x\left(t\right)\ge v\left(t\right)-L{e}^{-\eta t}(1-t)t^{b-1}\ge v\left(t\right)-{\displaystyle \frac{Lv\left(t\right)}{\left|\right|v\left|\right|}}=\left(1-{\displaystyle \frac{L}{N}}\right)v\left(t\right)\\ \ge {\displaystyle \frac{1}{2}}v\left(t\right)\ge {\displaystyle \frac{1}{2}}N{e}^{-\eta t}(1-t)t^{b-1}\ge {\displaystyle \frac{1}{2}}N{e}^{-\eta \mathit{\ell }}(1-\mathit{\ell }){\mathit{ȷ}}^{b-1}\ge L_1>0.\end{array}$$

(41)

Therefore, for any $v\in P\cap \partial {\mathrm{\Omega }}_6,$ by (39)–(41), we have

$$\begin{array}{c}\left|\right|Tv\left|\right|\ge \left|Tv\left({\displaystyle \frac{1}{2}}\right)\right|\\ =\left|{\int }_0^{1}G\left({\displaystyle \frac{1}{2}},s\right)\left[{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}\right.\right.\\ \left.\left.+\hslash (s,{[v\left(s\right)-x\left(s\right)]}^{*})+\kappa \left(s\right)\right]ds\right|\\ \ge {\int }_0^{1}G\left({\displaystyle \frac{1}{2}},s\right){\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\int }_{\mathit{ȷ}}^{\mathit{\ell }}G\left({\displaystyle \frac{1}{2}},s\right){\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\displaystyle \frac{1}{2^b{e}^{\frac{\eta }{2}}}}{\int }_{\mathit{ȷ}}^{\mathit{\ell }}{\displaystyle \frac{s{(1-s)}^{b-1}{e}^{\eta s}}{\mathrm{\Gamma }\left(b\right)}}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\displaystyle \frac{\mathit{ȷ}{(1-\mathit{\ell })}^{b-1}{e}^{\eta \mathit{ȷ}}}{2^b{e}^{\frac{\eta }{2}}\mathrm{\Gamma }\left(b\right)}}{\int }_{\mathit{ȷ}}^{\mathit{\ell }}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}f(\tau ,{[v\left(\tau \right)-x\left(\tau \right)]}^{*})e^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\displaystyle \frac{{{\rm Y}}^{q-1}\mathit{ȷ}{(1-\mathit{\ell })}^{b-1}{e}^{\eta \mathit{ȷ}}N}{2^b{e}^{\frac{\eta }{2}}\mathrm{\Gamma }\left(b\right)}}{\int }_{\mathit{ȷ}}^{\mathit{\ell }}{\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{a-1}{e}^{-\eta s}}{\mathrm{\Gamma }\left(a\right)}}{e}^{\eta \tau }d\tau \right)}^{q-1}ds\\ \ge {\displaystyle \frac{{{\rm Y}}^{q-1}\mathit{ȷ}{(1-\mathit{\ell })}^{b-1}{e}^{\eta \mathit{ȷ}}N}{{\left(\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{2}^b{e}^{\frac{\eta }{2}}\mathrm{\Gamma }\left(b\right)}}{\int }_{\mathit{ȷ}}^{\mathit{\ell }}{\left({\int }_0^s{(s-\tau )}^{a-1}{e}^{-\eta s}d\tau \right)}^{q-1}ds\\ ={\displaystyle \frac{{{\rm Y}}^{q-1}\mathit{ȷ}{(1-\mathit{\ell })}^{b-1}{e}^{\eta \mathit{ȷ}}N}{{\left(\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{2}^b{e}^{\frac{\eta }{2}}\mathrm{\Gamma }\left(b\right)}}{\int }_{\mathit{ȷ}}^{\mathit{\ell }}{e}^{-\eta (q-1)s}{\left({\int }_0^s{(s-\tau )}^{a-1}d\tau \right)}^{q-1}ds\\ ={\displaystyle \frac{{{\rm Y}}^{q-1}\mathit{ȷ}{(1-\mathit{\ell })}^{b-1}{e}^{\eta \mathit{ȷ}}N}{{\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{2}^b{e}^{\frac{\eta }{2}}\mathrm{\Gamma }\left(b\right)}}{\int }_{\mathit{ȷ}}^{\mathit{\ell }}{e}^{-\eta (q-1)s}{s}^{a(q-1)}ds\\ \ge {\displaystyle \frac{{{\rm Y}}^{q-1}{(1-\mathit{\ell })}^{b}N{e}^{\eta (\mathit{ȷ}-(q-1)\mathit{\ell })}{\mathit{ȷ}}^{b+a(q-1)}(\mathit{\ell }-\mathit{ȷ})}{{\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{2}^b{e}^{\frac{\eta }{2}}\mathrm{\Gamma }\left(b\right)}}\\ \ge N=\left|\right|v\left|\right|,\end{array}$$

(42)

thus, for any $v\in P\cap \partial {\mathrm{\Omega }}_6$, we have $\left|\right|Tv\left|\right|\ge \left|\right|v\left|\right|$. Consequently, by Lemma 7, T has another fixed point $v_4\in P\cap ({\overline{\mathrm{\Omega }}}_6\setminus {\mathrm{\Omega }}_5)$, which satisfies

$$k\le \left|\right|v_4\left|\right|\le N.$$

Letting $u_4\left(t\right)=v_4\left(t\right)-x\left(t\right)$, we have

$$\begin{array}{cc}{u}_4\left(t\right)& =v_4\left(t\right)-x\left(t\right)\ge v_4\left(t\right)-L{e}^{-\eta t}{t}^{b-1}(1-t)\ge v_4\left(t\right)-{\displaystyle \frac{Lv\left(t\right)}{k}}\\ & \ge {\displaystyle \frac{1}{2}}{v}_4\left(t\right)\ge {\displaystyle \frac{1}{2}}k{e}^{-\eta t}{t}^{b-1}(1-t)\ge 0.\end{array}$$

By Lemma 5, $u_4\left(t\right)=v_4\left(t\right)-x\left(t\right)$ is another positive solution of the problem (1), which possesses the following asymptotic property

$${\displaystyle \frac{1}{2}}k{e}^{-\eta t}(1-t)t^{b-1}\le u_4\left(t\right)\le N.$$

The proof is completed. □

Remark 2. 

In particular, if the tempered Riemann–Liouville fractional derivative ${}_0{}^R{D}_t^{b,\eta }u\left(t\right)$ of Equation (1) is replaced by the tempered Caputo fractional derivative ${}_0{}^C{D}_t^{b,\eta }u\left(t\right)$, the results of Theorem 1–4 are still valid. Since the proof is similar, we here omit it.

5. Examples

Example 1. 

Take $a={\displaystyle \frac{2}{3}},b={\displaystyle \frac{8}{3}},\eta =3,q=4$ in (1), we consider the following singular tempered fractional equation with perturbation term

$$\left\{\begin{array}{c}{}_0{}^R{D}_t^{{\displaystyle \frac{2}{3}},3}\left[{\phi }_{{\displaystyle \frac{4}{3}}}\left({}_0{}^R{D}_t^{{\displaystyle \frac{8}{3}},3}u\left(t\right)-{\left({\displaystyle \frac{1}{3}}-t\right)}^{-{\displaystyle \frac{3}{5}}}{(u^2+2)}^{-1}\right)\right]=f(t,u\left(t\right)),\\ u\left(0\right)=u\left(1\right)=0,{}_0{}^R{D}_t^{{\displaystyle \frac{8}{3}},3}u\left(0\right)=0,u^{\prime }\left(0\right)=0,\end{array}\right.$$

(43)

where

$$\hslash (t,u\left(t\right))=\left\{\begin{array}{c}{\displaystyle \frac{t}{100}}{\left({\displaystyle \frac{1}{3}}-t\right)}^{-{\displaystyle \frac{3}{5}}}{(u^2+2)}^{-1},t\in \left[0,{\displaystyle \frac{1}{3}}\right],\\ -{\displaystyle \frac{t}{100}}{\left(t-{\displaystyle \frac{1}{3}}\right)}^{-{\displaystyle \frac{3}{5}}}{(u^2+2)}^{-1},t\in \left({\displaystyle \frac{1}{3}},1\right],\end{array}\right.$$

$$f(t,u\left(t\right))=\left\{\begin{array}{c}{\displaystyle \frac{1}{2^{16}}}\times {10}^{-48}{(1-t)}^2{u}^{16},u\in [0,{2000}^2],\\ 2^{48}\times {10}^{144}{(1-t)}^2{u}^{-16},u>{2000}^2,\end{array}\right.$$

$$\kappa \left(t\right)={\displaystyle \frac{t}{10}}\left|{\left({\displaystyle \frac{1}{3}}-t\right)}^{-{\displaystyle \frac{3}{5}}}\right|.$$

Conclusion. Equation (43) has at least two distinct positive solutions $u_1\left(t\right),u_2\left(t\right)$, which satisfies the following asymptotic properties

$$u_1\left(t\right)\ge 0.02549t^{{\displaystyle \frac{2}{3}}}{e}^{-3t}(1-t),u_2\left(t\right)\ge {10}^6{t}^{{\displaystyle \frac{2}{3}}}{e}^{-3t}(1-t).$$

Proof. 

Obviously, the function $f:[0,1]\times R\to R_{+}$ and $\hslash :[0,1]\times R\to R$ satisfies the Crathèodory condition, and

$$\underset{\left|v\right|\to +\infty }{lim}\underset{t\in [0,1]}{max}{\displaystyle \frac{f(t,v(t\left)\right)}{v^{\frac{1}{q-1}}}}=0.$$

Since

$$0<L={\displaystyle \frac{1}{10\mathrm{\Gamma }\left(b\right)}}{\int }_0^1\kappa \left(s\right)ds=0.0665{\int }_0^{1}s\left|{\left({\displaystyle \frac{1}{3}}-s\right)}^{-{\displaystyle \frac{3}{5}}}\right|ds=0.02549,$$

we have

$$0<\beta =\left({\displaystyle \frac{9}{8\mathrm{\Gamma }\left(\frac{8}{3}\right){\left(\mathrm{\Gamma }\left(\frac{2}{3}\right)\right)}^3}}+0.02549\right)e^6=131.77.$$

Take $n=2000>max\{2L,2\beta \}=263.54$, and for any $f:[0,1]\times [0,2000]$, one has

$$f(t,u\left(t\right))\le {\displaystyle \frac{1}{2^{16}}}\times {10}^{-48}\times {2000}^{16}=1<{\left({\displaystyle \frac{n}{\beta }}-2\right)}^{{\displaystyle \frac{1}{3}}}=2.362,$$

and therefore, $\left(P_1\right),\left(P_2\right),\left(A\right)$ and $\left(E\right)$ are satisfied.

Next, we verify the condition $\left(B\right)$. Take $m=2\times {10}^6>n=2000$ and a subinterval $[{\displaystyle \frac{1}{3}},{\displaystyle \frac{2}{3}}]\subset [0,1]$, then for $(t,v)\in [{\displaystyle \frac{1}{3}},{\displaystyle \frac{2}{3}}]\times [{\displaystyle \frac{1}{2}}\times 2\times {10}^8{e}^{-2}{\left({\displaystyle \frac{1}{3}}\right)}^{{\displaystyle \frac{8}{3}}},2\times {10}^6]$, we have

$$f(t,v)\ge {\displaystyle \frac{1}{2^{16}}}\times {10}^{-48}{\left({\displaystyle \frac{1}{3}}\right)}^2{\left({\displaystyle \frac{1}{2}}\times 2\times {10}^6{e}^{-2}{\left({\displaystyle \frac{1}{3}}\right)}^{{\displaystyle \frac{8}{3}}}\right)}^{16}=56329.3,$$

and

$${\left({\displaystyle \frac{m{\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{e}^{\frac{\eta }{2}}{2}^b\mathrm{\Gamma }\left(b\right)}{c^{a(q-1)+1}{(1-d)}^{b-1}{e}^{\eta (2c-(q-1\left)d\right)}(d-c)}}\right)}^{{\displaystyle \frac{1}{q-1}}}={\left({\displaystyle \frac{{2000}^2{\left(\frac{2}{3}\mathrm{\Gamma }\left(\frac{2}{3}\right)\right)}^3{e}^{\frac{3}{2}}{2}^{\frac{8}{3}}\mathrm{\Gamma }\left(\frac{8}{3}\right)}{{\left(\frac{1}{3}\right)}^3{\left(\frac{1}{3}\right)}^{\frac{5}{3}}{e}^{-4)}\left(\frac{1}{3}\right)}}\right)}^{{\displaystyle \frac{1}{3}}}=15921.71,$$

and thus, if $(t,v)\in [{\displaystyle \frac{1}{3}},{\displaystyle \frac{2}{3}}]\times [{\displaystyle \frac{1}{2}}\times 2\times {10}^8{e}^{-2}{\left({\displaystyle \frac{1}{3}}\right)}^{{\displaystyle \frac{8}{3}}},2\times {10}^6]$, we have

$$f(t,v)\ge {\left({\displaystyle \frac{m{\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{e}^{\frac{\eta }{2}}{2}^b\mathrm{\Gamma }\left(b\right)}{c^{a(q-1)+1}{(1-d)}^{b-1}{e}^{\eta (2c-(q-1\left)d\right)}(d-c)}}\right)}^{{\displaystyle \frac{1}{q-1}}},$$

which implies that the condition (B) holds.

According to Theorems 1 and 3, the singular tempered fractional equation with perturbation term (43) has at least two distinct positive solutions $u_1\left(t\right),u_2\left(t\right)$, which satisfies the following asymptotic properties

$$u_1\left(t\right)\ge 0.02549t^{{\displaystyle \frac{2}{3}}}{e}^{-3t}(1-t),u_2\left(t\right)\ge {10}^6{t}^{{\displaystyle \frac{2}{3}}}{e}^{-3t}(1-t).$$

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Example 2. 

Still taking $a={\displaystyle \frac{2}{3}},b={\displaystyle \frac{8}{3}},\eta =3,q=4$ in (1), we consider the following singular tempered fractional equation with a perturbation term

$$\left\{\begin{array}{c}{}_0{}^R{D}_t^{{\displaystyle \frac{2}{3}},3}\left[{\phi }_{{\displaystyle \frac{4}{3}}}\left({}_0{}^R{D}_t^{{\displaystyle \frac{8}{3}},3}u\left(t\right)-\hslash (t,u\left(t\right))\right)\right]=f(t,u\left(t\right)),\\ u\left(0\right)=u\left(1\right)=0,{}_0{}^R{D}_t^{{\displaystyle \frac{8}{3}},3}u\left(0\right)=0,u^{\prime }\left(0\right)=0,\end{array}\right.$$

(44)

where

$$f(t,u)=10108t^{{\displaystyle \frac{1}{3}}}{(1-t)}^{{\displaystyle \frac{1}{3}}}{u}^{{\displaystyle \frac{2}{3}}},\hslash (t,u)=t^2{({\displaystyle \frac{1}{5}}-2t)}^{-{\displaystyle \frac{1}{3}}}.$$

Conclusion. The singular tempered fractional equation with perturbation term (44) has at least two positive solutions $u_3\left(t\right),u_4\left(t\right)$, which satisfies the following asymptotic properties:

$$u_3\left(t\right)\ge 0.6318\times t^{{\displaystyle \frac{2}{3}}}{e}^{-3t}(1-t),$$

$$u_4\left(t\right)\ge 8\times {10}^{29}\times t^{{\displaystyle \frac{2}{3}}}{e}^{-3t}(1-t).$$

Remark 3. 

Clearly

$$\underset{t\to {{\displaystyle \frac{1}{10}}}^{-}}{lim}\hslash (t,u)=-\infty ,$$

which indicates that the effect of the changing-sign perturbation term on the system at some singular points is enormous. So the appearance of the change-sign perturbation term increases the difficulty of treating the equation.

Proof. 

Obviously, $f:[0,1]\times R\to R_{+}$ and $\hslash :[0,1]\times R\to R$ satisfies the Carathèodory condition with $\kappa \left(t\right)=t^2|{({\displaystyle \frac{1}{5}}-2t)}^{-{\displaystyle \frac{1}{3}}}|$, and thus the conditions (P₁), (P₂) and (F) hold. By simple calculations, we have

$$\begin{array}{c}0<L={\displaystyle \frac{1}{\mathrm{\Gamma }\left(b\right)}}{\int }_0^1\kappa \left(s\right)ds=0.6646{\int }_0^1{s}^2\left|{\left({\displaystyle \frac{1}{5}}-2s\right)}^{-{\displaystyle \frac{1}{3}}}\right|ds\\ =0.6646\left[{\int }_0^{{\displaystyle \frac{1}{10}}}{s}^2{\left({\displaystyle \frac{1}{5}}-2t\right)}^{-{\displaystyle \frac{1}{3}}}ds+{\int }_{{\displaystyle \frac{1}{10}}}^1{s}^2{\left(2t-{\displaystyle \frac{1}{5}}\right)}^{-{\displaystyle \frac{1}{3}}}ds\right]=0.2106,\end{array}$$

and

$$0<\beta =\left({\displaystyle \frac{9}{8\mathrm{\Gamma }\left(\frac{8}{3}\right){\left(\mathrm{\Gamma }\left(\frac{2}{3}\right)\right)}^3}}+0.2106\right)e^6=206.45.$$

In the following, we verify (C) and (D). Taking a subset $[{\displaystyle \frac{1}{4}},{\displaystyle \frac{3}{4}}]\subset [0,1]$, for any

$$(t,u)\in \left[{\displaystyle \frac{1}{4}},{\displaystyle \frac{3}{4}}\right]\times \left[3\times 206.45\times e^{-{\displaystyle \frac{9}{4}}}{\left({\displaystyle \frac{1}{4}}\right)}^{{\displaystyle \frac{8}{3}}},4\times 206.45\right],$$

we have

$$\begin{array}{cc}f(t,u)& \ge 10108{\left({\displaystyle \frac{1}{4}}\right)}^{{\displaystyle \frac{2}{3}}}{\left(3\times 206.45\times e^{-{\displaystyle \frac{9}{4}}}{\left({\displaystyle \frac{1}{4}}\right)}^{{\displaystyle \frac{8}{3}}}\right)}^{{\displaystyle \frac{2}{3}}}=5531.1071\\ & \ge 5173.96={\left({\displaystyle \frac{4\beta {\left(a\mathrm{\Gamma }\left(a\right)\right)}^{q-1}{e}^{\frac{\eta }{2}}{2}^b\mathrm{\Gamma }\left(b\right)}{{\omega }^{a(q-1)+1}{(1-\varpi )}^{b-1}{e}^{\eta (\omega -(q-1\left)\varpi \right)}(\varpi -\omega )}}\right)}^{{\displaystyle \frac{1}{q-1}}},\end{array}$$

so the condition (C) holds.

Next, choose a constant $k=16\times {10}^{29}>4\beta =825.8,$ then we have

$$f(t,u)\le 10108{\left(4\beta \right)}^{{\displaystyle \frac{2}{3}}}=889710.03\le {\left({\displaystyle \frac{k}{\beta }}\right)}^{{\displaystyle \frac{1}{3}}}=1978950958.37,$$

which implies that the condition (D) holds.

Consequently, the singular tempered fractional equation with perturbation term (44) has at least two positive solutions $u_3\left(t\right),u_4\left(t\right)$, which satisfies the following asymptotic properties

$$u_3\left(t\right)\ge 0.6318t^{{\displaystyle \frac{2}{3}}}{e}^{-3t}(1-t),u_4\left(t\right)\ge 8\times {10}^{29}{t}^{{\displaystyle \frac{2}{3}}}{e}^{-3t}(1-t).$$

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6. Conclusions

This paper focuses on the multiplicity of positive solutions for a singular tempered fractional initial-boundary value problem with changing-sign perturbation term. The effect of the changing-sign perturbation term on the system at some singular points is enormous, which leads to great difficulties and challenges for treating the equation. To overcome these difficulties, in Section 2, we firstly introduce a truncation function and then study the properties of the solution of isomorphic linear equations, and transform the changing-sign tempered fractional initial-boundary value problem into a positive problem. In Section 3, we introduce some suitable growth conditions to overcome the singularity of perturbations and then use the fixed point theorem in a cone. The existence and asymptotic properties of the single positive solution of the singular tempered fractional initial-boundary value problem with changing-sign perturbation term are established. In Section 4, the multiplicity of the positive solution is derived. This paper has some new features and contributions.

(i)

The equation is an initial-boundary value problem containing lower order initial value and higher-order boundary value conditions.

(ii)

The changing-sign perturbation term involves a nonlinear p-Laplacian operator.

(ii)

The changing-sign perturbation term only satisfies the weaker Carathèodory conditions.

(iv)

The perturbation term ℏ can have infinite singular points. Moreover, it can tend to negative infinity in some singular points.

(v)

The multiplicity of positive solutions for the target equation is established.

In future work, we can draw on the current research to develop efficient numerical algorithms for this type of problem, such as improved versions of the finite element method, finite difference method, spectral method, or new hybrid algorithms. Consider how to better handle the singular terms and sign-changing perturbation terms to improve the accuracy and stability of the algorithms. For example, design an adaptive finite element algorithm that can automatically adjust the mesh density according to the local characteristics of the solution to more accurately capture the effects of singular points and perturbation terms. On the other hand, conduct an in-depth study of the regularity properties of the solutions, including local and global regularity. Analyze the impact of the sign-changing characteristics of the perturbation terms on the regularity of the solutions, and determine under what conditions the solutions have higher smoothness or differentiability. For instance, explore the variation laws of the solution’s regularity near singular points, and study how to ensure a certain degree of solution regularity by imposing restrictions on the perturbation terms.