The Eigenvalue Problem of a Singular Tempered Fractional Equation with the Riemann–Stieltjes Integral Boundary Condition

Abstract

In this paper, we investigate the existence of positive solutions of the eigenvalue problem for a singular tempered fractional equation with a Riemann–Stieltjes integral boundary condition and signed measures. By establishing the Green function and its properties, an eigenvalue interval for the existence of positive solutions is outlined based on Schauder’s fixed-point theorem and the upper and lower solutions method. An interesting feature of this paper is that f may be singular in both the time and space variables, and the Riemann–Stieltjes integral may involve signed measures.

1. Introduction

Let $0<\alpha \le 1,1<\beta \le 2$, $\mu$ be a positive constant and $\varkappa$ be a bounded variation function. We consider the existence of positive solutions for the following eigenvalue problem of a singular tempered fractional equation with a Riemann–Stieltjes integral boundary condition and signed measures:

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }z\left(t\right)\right]={\lambda }^{{\displaystyle \frac{1}{q-1}}}g\left(t,z\left(t\right)\right),\hfill \\ z\left(0\right)=0,{\mathbb{D}}_t^{\beta ,\mu }\left(z\left(0\right)\right)=0,z\left(1\right)={\int }_0^{1}z\left(t\right)d\varkappa \left(t\right),\hfill \end{array}\right.$$

(1)

where ${\phi }_p={\left|s\right|}^{p-2}s$ is a p-Laplacian operator with $p,q>1$ and ${}_0{}^R\mathbb{D}_t^{\alpha ,\mu },{}_0{}^R\mathbb{D}_t^{\beta ,\mu }$ are tempered fractional-order derivatives (see Definition 1). The boundary condition ${\int }_0^{1}z\left(t\right)d\varkappa \left(t\right)$ is the Riemann–Stieltjes integral type, which can be signed measures; $g(t,z):(0,1)\times (0,+\infty )\to [0,+\infty )$ is continuous and may be singular at $t=0,1$ and $z=0$.

Tempered fractional equations can replace classical fractional-order equations [1,2,3,4,5,6] in describing semi-heavy-tailed phenomena, such as power-law memory or jump behavior with exponential tempering, where heavy tails are truncated and all moments remain finite. In particular, in anomalous diffusion and porous media, transitions between sub- and super-diffusion lead to diffusion profiles with semi-heavy-tailed waiting times or jump sizes, representing finite-range heterogeneity [7,8]. These long waiting times are exponentially truncated; that is, particles may wait for long periods but not infinitely long, making the model more realistic for physical systems as all processes eventually relax.

Recently, Sabzikar [9] adopted a fractional Fourier transform with an exponential factor to study anomalous diffusion equations with tempered semi-heavy tails. Fractional Fourier transforms have also been widely applied in heat conduction [10], image processing [11,12], and related fields [13,14]. In addition to their applications in anomalous diffusion [15,16,17,18,19,20,21], tempered fractional equations have been used to model tempered Lévy flight diffusion [22], geophysical processes [23], financial models [24], problems in applied mathematics [25,26], and chemical graph theory [27].

It is well known that fractional-order integral and derivative operators have nonlocal characteristics [28,29,30,31,32,33], which provide more accurate tools for describing the viscoelastic and hereditary properties of natural phenomena [34,35,36,37,38,39,40,41,42]. In tempered fractional derivative operator (4), if $\mu \to 0$, it reduces to the standard fractional diffusion equation [43,44,45] (pure power-law memory), and, when $\mu \to 1$, it reduces to the classical diffusion equation [46,47,48,49,50,51,52,53,54,55,56]. For finite $\lambda >0,$ memory decays roughly as $t^{-\alpha }{e}^{-\lambda t}$ [57]. This demonstrates that the tempered fractional equation encompasses both classical convection–diffusion and fractional advection models as special cases. Due to the fact that the tempered fractional equation provides a more general framework, Chen et al. [58] recently established the iterative properties of a unique positive solution for the following coupled tempered fractional system:

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\lambda }\left({\phi }_p\left({}_0{}^R\mathbb{D}_t^{\beta ,\lambda }u\left(t\right)\right)\right)=f_1(t,v\left(t\right)),\hfill \\ {}_0{}^R\mathbb{D}_t^{j,\lambda }\left({\phi }_p\left({}_0{}^R\mathbb{D}_t^{\mathsf{\ell },\lambda }v\left(t\right)\right)\right)=f_2(t,u\left(t\right)),\hfill \\ u\left(0\right)=v\left(0\right)=0,\hfill \\ {}_0{}^R\mathbb{D}_t^{\beta ,\lambda }u\left(0\right)={}_0{}^R\mathbb{D}_t^{\mathsf{\ell },\lambda }v\left(0\right)=0,\hfill \\ u\left(1\right)={\int }_0^1{e}^{-\lambda (1-t)}u\left(t\right)dt,v\left(1\right)={\int }_0^1{e}^{-\lambda (1-t)}v\left(t\right)dt.\hfill \end{array}\right.$$

(2)

based on iterative technique, where constants $\beta ,\mathsf{\ell }\in (1,2)$, $\alpha ,j\in (0,1)$, ${\phi }_p\left(t\right)$ is p-Laplacian operator, $f_1$ is increasing, and $f_2$ is decreasing with respect to the second variable, respectively. By employing the fixed-point theorem, Zhang et al. [59] considered the existence of multiple positive solutions for the following singular perturbation tempered fractional equation:

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

(3)

where $f:[0,1]\times R\to R_{+}$, $\hslash :[0,1]\times R\to R$ is a singular changing sign perturbation term. In fact, in addition to the fixed-point theorem and iterative technique, many other analytical theories and numerical approaches, such as space theory [60,61,62,63,64], regularity theory [65], operator theories [66,67,68], spectral theories [69,70], finite volume method [71,72,73,74,75], variational method [76,77,78], and finite-difference method [79], have also been developed to solve various nonlinear differential equations.

In the existing literature, relatively few studies have addressed tempered fractional diffusion equations with Riemann–Stieltjes integral boundary conditions and nonlinearities exhibiting singular behavior in the spatial variables. Motivated by this gap, and by employing the analytical theory and numerical approaches described above, this paper first establishes the Green function associated with the problem and investigates its fundamental properties. We then study the upper and lower solutions of the corresponding tempered fractional eigenvalue problem. Based on these upper and lower solutions, an amended fractional eigenvalue problem is constructed, and new existence results for positive solutions are obtained via the Schauder fixed-point theorem.

The contributions of this paper are threefold:

(1) Generalization of tempered fractional models

This paper studies tempered fractional diffusion equations with nonlinearities exhibiting strong singular behavior in the spatial variables. By incorporating exponential tempering, the proposed framework extends classical fractional models while preserving nonlocal memory effects and ensuring finite moments, making it more realistic for physical and biological applications.

(2) Riemann–Stieltjes integral boundary conditions with signed measures

Unlike most existing works that impose standard pointwise boundary conditions, this study considers Riemann–Stieltjes integral boundary conditions involving signed measures, as in [80]. This formulation captures distributed and weighted boundary interactions, providing a flexible and physically meaningful modeling framework.

(3) Analytical framework and existence results

The Green function and its fundamental properties are established for the associated tempered fractional eigenvalue problem. Using the method of upper and lower solutions, an amended eigenvalue problem is constructed, and new existence results for positive solutions are obtained via the Schauder fixed-point theorem.

The rest of this paper is structured as follows. In Section 2, we introduce some properties of Riemann–Liouville fractional calculus and provide the definitions of lower and upper solutions for the eigenvalue problem and some preliminaries. The main results and an example are presented in Section 3 and Section 4. Finally, Section 5 concludes the paper with a brief summary and potential directions for future research.

2. Preliminaries and Lemmas

Here, we first introduce the definitions of tempered Riemann–Liouville fractional derivative and Jordan decomposition.

Definition 1

(Tempered Riemann–Liouville fractional derivative). Let $\alpha >0$, $\mu \ge 0$, $n=\lceil \alpha \rceil$, and $z\in C^n\left([0,1]\right)$. The tempered Riemann–Liouville derivative of order α is defined by

$${\mathbb{D}}_t^{\alpha ,\mu }z\left(t\right)=e^{-\mu t}{\mathcal{D}}_t^{\alpha ,\mu }\left(e^{\mu t}z\left(t\right)\right),$$

(4)

where ${\mathcal{D}}_t^{\alpha ,\mu }z\left(t\right)={\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}\left({\mathcal{I}}_t^{n-\alpha }z\left(t\right)\right),$ refers to the Riemann–Liouville fractional derivative and

$${\mathcal{I}}_t^{n-\alpha }z\left(t\right)={\int }_0^t{(t-s)}^{\alpha -1}z\left(s\right)\mathrm{d}s,$$

is the Riemann–Liouville fractional integral operator.

Definition 2

(Jordan decomposition). Let ϰ be a signed measure of bounded variation on a measurable space $(X,\mathcal{F})$. For any measurable set $A\in \mathcal{F}$, define

$${\varkappa }^{+}\left(A\right):=sup\{\varkappa \left(B\right):B\subset A,B\in \mathcal{F}\},$$

$${\varkappa }^{-}\left(A\right):=-inf\{\varkappa \left(B\right):B\subset A,B\in \mathcal{F}\}.$$

Then, ${\varkappa }^{+}$ and ${\varkappa }^{-}$ are positive measures, called the positive variation and negative variation of ϰ, respectively, and satisfy

$$\varkappa ={\varkappa }^{+}-{\varkappa }^{-},\left|\varkappa \right|={\varkappa }^{+}+{\varkappa }^{-},$$

where $\left|\varkappa \right|$ denotes the total variation measure of ϰ. The decomposition $\varkappa ={\varkappa }^{+}-{\varkappa }^{-}$ is called the Jordan decomposition of ϰ and is unique.

It is well known that the Riemann–Liouville fractional calculus has the following properties.

Lemma 1

([81]). Let $\alpha >\beta >0$ and $m=\left[\alpha \right]+1$; if $z\left(t\right)\in C[0,1]\cap L^1[0,1]$, then

(i) 

${\mathcal{I}}_t^{\gamma }{\mathcal{D}}_t^{\gamma }\left(z\left(t\right)\right)=z\left(t\right)+a_1{t}^{\gamma -1}+a_2{t}^{\gamma -2}+\cdots +a_m{t}^{\gamma -m},$ where $a_i\in \mathbb{R},i=1,2,3,\dots ,m.$

(ii) 

${\mathcal{I}}_t^{\alpha }{\mathcal{I}}_t^{\beta }\left(z\left(t\right)\right)={\mathcal{I}}_t^{\alpha +\beta }z\left(t\right),{\mathcal{D}}_t^{\beta }{\mathcal{I}}_t^{\alpha }\left(z\left(t\right)\right)={\mathcal{I}}_t^{\alpha -\beta }z\left(t\right),{\mathcal{D}}_t^{\beta }{\mathcal{I}}_t^{\beta }\left(z\left(t\right)\right)=z\left(t\right).$

Lemma 2

([82]). Suppose that $h\left(t\right)\in C\left(\right[0,1],[0,+\infty \left)\right)$. Then, for any $\lambda >0$, the following eigenvalue problem for a tempered fractional equation with Dirichlet boundary conditions

$$\left\{\begin{array}{c}-{}_0{}^R\mathbb{D}_t^{\beta ,\mu }z\left(t\right)={\lambda }^{{\displaystyle \frac{1}{q-1}}}h\left(t\right),\hfill \\ z\left(0\right)=z\left(1\right)=0,\hfill \end{array}\right.$$

(5)

admits a unique positive solution given by

$$z\left(t\right)={\lambda }^{{\displaystyle \frac{1}{q-1}}}{\int }_0^{1}H(t,s)h\left(s\right)ds,$$

(6)

where

$$H(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{t^{\beta -1}{(1-s)}^{\beta -1}-{(t-s)}^{\beta -1}}{\Gamma \left(\beta \right)}}{e}^{-\mu t}{e}^{\mu s},\hfill & 0\le s\le t\le 1,\hfill \\ {\displaystyle \frac{t^{\beta -1}{(1-s)}^{\beta -1}}{\Gamma \left(\beta \right)}}{e}^{-\mu t}{e}^{\mu s},\hfill & 0\le t\le s\le 1.\hfill \end{array}\right.$$

(7)

The function $H(t,s)$ is the Green function associated with the tempered fractional eigenvalue problem (5).

Moreover, the Green function $H(t,s)$ satisfies the following properties:

(i) 

$H(t,s)$ is non-negative and continuous for all $(t,s)\in [0,1]\times [0,1]$.

(ii) 

For any $t,s\in [0,1]$, $H(t,s)$ satisfies the inequalities

$${\displaystyle \frac{s{(1-s)}^{\beta -1}{e}^{\mu s}{e}^{-\mu t}{t}^{\beta -1}(1-t)}{\Gamma \left(\beta \right)}}\le H(t,s)\le {\displaystyle \frac{s{(1-s)}^{\beta -1}{e}^{\mu s}}{\Gamma \left(\beta \right)}}\mathit{or}{\displaystyle \frac{e^{-\mu t}{t}^{\beta -1}(1-t)}{\Gamma \left(\beta \right)}}.$$

In this paper, we adopt the following basic assumption to guarantee the non-negativity of the Green function.

(G0) 

Let $\varkappa ={\varkappa }^{+}-{\varkappa }^{-}$ be the Jordan decomposition of the bounded variation measure $\varkappa$ satisfying, for every $s\in [0,1]$,

$${\int }_0^{1}H(t,s)d{\varkappa }^{+}\left(t\right)\ge {\int }_0^{1}H(t,s)d{\varkappa }^{-}\left(t\right),$$

and $0<\varpi <1$.

Remark 1.

To guarantee the non-negativity of Green’s function G, we impose the structural condition that ϰ is a non-negative measure (or, alternatively, that ϰ is absolutely continuous with non-negative density) such as  (i)   and  (ii)   and signed measure  (iii)   , etc.:

(i) 

ϰ is a non-negative measure (equivalently nondecreasing);

(ii) 

ϰ is absolutely continuous with non-negative density;

(iii) 

(Jordan-dominance) writing $\varkappa ={\varkappa }^{+}-{\varkappa }^{-}$, we require

$${\int }_0^{1}H(t,s)d{\varkappa }^{+}\left(t\right)\ge {\int }_0^{1}H(t,s)d{\varkappa }^{-}\left(t\right)\mathit{for}\mathit{all}s\in [0,1],$$

which is equivalent to ${\mathcal{H}}_{\varkappa }\left(s\right)\ge 0$.

It follows from Lemma 1 and straightforward computation that the following equation

$$\left\{\begin{array}{c}{-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\left(z\left(t\right)\right)=0,0<t<1,\hfill \\ z\left(0\right)=0,z\left(1\right)=1,\hfill \end{array}\right.$$

has a unique solution $e^{\mu (1-t)}{t}^{\beta -1}$. Let

$$\varpi ={\int }_0^1{e}^{\mu (1-t)}{t}^{\beta -1}d\varkappa \left(t\right),{\mathcal{H}}_{\varkappa }\left(s\right)={\int }_0^{1}H(t,s)d\varkappa \left(t\right).$$

(8)

According to [83], the Green function of the tempered fractional equation with the Riemann–Stieltjes integral boundary condition

$$\left\{\begin{array}{c}{-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\left(z\left(t\right)\right)={\lambda }^{{\displaystyle \frac{1}{q-1}}}h\left(t\right),\hfill \\ z\left(0\right)=0,z\left(1\right)={\int }_0^{1}z\left(t\right)d\varkappa \left(t\right),\hfill \end{array}\right.$$

(9)

is

$$G(t,s)={\displaystyle \frac{e^{\mu (1-t)}{t}^{\beta -1}}{1-\varpi }}{\mathcal{H}}_{\varkappa }\left(s\right)+H(t,s).$$

(10)

Consequently, Equation (9) can be expressed as an equivalent integral form

$$z\left(t\right)={\lambda }^{{\displaystyle \frac{1}{q-1}}}{\int }_0^{1}G(t,s)h\left(s\right)ds.$$

(11)

Lemma 3.

Suppose $\left(\mathbf{G}\mathbf{0}\right)$ holds; then, Green’s function defined in (10) possesses the following properties:

(1) 

$G(t,s)\ge 0$, for all $t,s\in [0,1].$

(2) 

$e^{-\mu t}{t}^{\beta -1}(1-t)\vartheta \left(s\right)\le G(t,s)\le \vartheta \left(s\right),t,s\in [0,1],$

where

$$\vartheta \left(s\right)={\displaystyle \frac{e^{\mu }}{1-\varpi }}{\mathcal{H}}_{\varkappa }\left(s\right)+{\displaystyle \frac{{(1-s)}^{\beta -1}s{e}^{\mu s}}{\Gamma \left(\beta \right)}}.$$

Proof. 

(1) Firstly, it follows from (7) that $H(t,s)\ge 0,t,s\in [0,1]$; thus, the condition $\left(\mathbf{G}\mathbf{0}\right)$ guarantees the non-negativity of Green’s function G.

In what follows, we estimate Green’s function G from (7) and (10). In fact, for any $t,s\in [0,1]$, it follows from (7), (10), and Lemma 2 that

$$\begin{array}{cc}G(t,s)\hfill & ={\displaystyle \frac{e^{\mu (1-t)}{t}^{\beta -1}}{1-\varpi }}{\mathcal{H}}_{\varkappa }\left(s\right)+H(t,s)\ge {\displaystyle \frac{e^{\mu (1-t)}{t}^{\beta -1}(1-t)}{1-\varpi }}{\mathcal{H}}_{\varkappa }\left(s\right)+{\displaystyle \frac{t^{\beta -1}(1-t)e^{-\mu t}{(1-s)}^{\beta -1}s{e}^{\mu s}}{\Gamma \left(\beta \right)}}\hfill \\ & =\left({\displaystyle \frac{e^{\mu }}{1-\varpi }}{\mathcal{H}}_{\varkappa }\left(s\right)+{\displaystyle \frac{{(1-s)}^{\beta -1}s{e}^{\mu s}}{\Gamma \left(\beta \right)}}\right)e^{-\mu t}{t}^{\beta -1}(1-t)=e^{-\mu t}{t}^{\beta -1}(1-t)\vartheta \left(s\right).\hfill \end{array}$$

At the same time, because $1<\beta \le 2$, one has

$$G(t,s)={\displaystyle \frac{e^{\mu (1-t)}{t}^{\beta -1}}{1-\varpi }}{\mathcal{H}}_{\varkappa }\left(s\right)+H(t,s)\le {\displaystyle \frac{e^{\mu }}{1-\varpi }}{\mathcal{H}}_{\varkappa }\left(s\right)+{\displaystyle \frac{(\beta -1){(1-s)}^{\beta -1}s{e}^{\mu s}}{\Gamma \left(\beta \right)}}=\vartheta \left(s\right).$$

□

Lemma 4.

Let $h\left(t\right)\in L^1(0,1)$, and then the linear eigenvalue problem of the tempered fractional equation with the Riemann–Stieltjes integral boundary condition

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }z\left(t\right)\right]={\lambda }^{{\displaystyle \frac{1}{q-1}}}h\left(t\right),\hfill \\ z\left(0\right)=0,{\mathbb{D}}_t^{\beta ,\mu }\left(z\left(0\right)\right)=0,z\left(1\right)={\int }_0^{1}z\left(t\right)d\varkappa \left(t\right),\hfill \end{array}\right.$$

(12)

has one unique solution

$$z\left(t\right)=\lambda {\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}h\left(\tau \right)e^{\mu \tau }d\tau \right)}^{q-1}ds.$$

(13)

Proof. 

Let

$$\Lambda \left(t\right)={-}_0^R{\mathbb{D}}_t^{\beta ,\mu }z\left(t\right),y\left(t\right)={\phi }_p(\Lambda \left(t\right)).$$

Consider the following linear eigenvalue problem of the tempered fractional equation with initial value condition

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }y\left(t\right)={\lambda }^{{\displaystyle \frac{1}{q-1}}}h\left(t\right),t\in [0,1],\hfill \\ y\left(0\right)=0.\hfill \end{array}\right.$$

(14)

Integrate both sides of (14) to obtain

$$e^{\mu t}y\left(t\right)={\lambda }^{{\displaystyle \frac{1}{q-1}}}{}_0\mathcal{I}_t^{\alpha }{e}^{\mu t}h\left(t\right)-b_1{t}^{\alpha -1}={\lambda }^{{\displaystyle \frac{1}{q-1}}}{\int }_0^t{\displaystyle \frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}h\left(s\right)e^{\mu s}ds-b_1{t}^{\alpha -1}.$$

It follows from $y\left(0\right)=0,\alpha \in (0,1]$ that $b_1=0$, and thus we have

$$y\left(t\right)={\lambda }^{{\displaystyle \frac{1}{q-1}}}{\int }_0^t{\displaystyle \frac{{(t-s)}^{\alpha -1}{e}^{-\mu t}}{\Gamma \left(\alpha \right)}}h\left(s\right)e^{\mu s}ds.$$

Consequently,

$$y\left(t\right)={\phi }_p\left({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }z\left(t\right)\right)={\lambda }^{{\displaystyle \frac{1}{q-1}}}{\int }_0^t{\displaystyle \frac{{(t-s)}^{\alpha -1}{e}^{-\mu t}}{\Gamma \left(\alpha \right)}}h\left(s\right)e^{\mu s}ds,$$

which implies that the linear eigenvalue problem (12) can be transformed to the following boundary value problem:

$$\left\{\begin{array}{c}-{}_0{}^R\mathbb{D}_t^{\beta ,\mu }z\left(t\right)={\phi }_p^{-1}\left({\lambda }^{{\displaystyle \frac{1}{q-1}}}{\int }_0^t{\displaystyle \frac{{(t-s)}^{\alpha -1}{e}^{\mu t}}{\Gamma \left(\alpha \right)}}h\left(s\right)e^{\mu s}ds\right),\hfill \\ z\left(0\right)=0,z\left(1\right)={\int }_0^{1}z\left(t\right)d\varkappa \left(t\right).\hfill \end{array}\right.$$

Therefore, it follows from (11) that the linear eigenvalue problem (12) is equivalent to the following integral form:

$$\begin{array}{cc}\hfill z\left(t\right)& ={\int }_0^{1}G(t,s){\phi }_p^{-1}\left({\lambda }^{{\displaystyle \frac{1}{q-1}}}{\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}h\left(\tau \right)e^{\mu \tau }d\tau \right)ds\hfill \\ & =\lambda {\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}h\left(\tau \right)e^{\mu \tau }d\tau \right)}^{q-1}ds.\hfill \end{array}$$

□

Now we introduce the definitions of lower and upper solution for the eigenvalue problem of the tempered fractional equation with the Riemann–Stieltjes integral boundary condition (1).

Definition 3.

If continuous function $\Psi \left(t\right)$ satisfies

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Psi \left(t\right)\right]\le {\lambda }^{{\displaystyle \frac{1}{q-1}}}g\left(t,z\left(t\right)\right),\hfill \\ \Psi \left(0\right)\ge 0,{\mathbb{D}}_t^{\beta ,\mu }(\Psi \left(0\right))\ge 0,z\left(1\right)\ge {\int }_0^1\Psi \left(t\right)d\varkappa \left(t\right),\hfill \end{array}\right.$$

then $\Psi \left(t\right)$ is called a lower solution of the eigenvalue problem (1).

Definition 4.

If continuous function $\Phi \left(t\right)$ satisfies

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Phi \left(t\right)\right]\ge {\lambda }^{{\displaystyle \frac{1}{q-1}}}g\left(t,z\left(t\right)\right),\hfill \\ \Phi \left(0\right)\le 0,{\mathbb{D}}_t^{\beta ,\mu }(\Phi \left(0\right))\le 0,z\left(1\right)\le {\int }_0^1\Phi \left(t\right)d\varkappa \left(t\right),\hfill \end{array}\right.$$

then $\Phi \left(t\right)$ is called an upper solution of the eigenvalue problem (1).

Lemma 5.

Suppose $\left(\mathbf{G}\mathbf{0}\right)$ holds if $z\in C\left(\right[0,1],R)$ satisfies ${\mathcal{D}}_t^{\alpha ,\mu }z\left(t\right)\ge 0,t\in [0,1]$ and

$$z\left(0\right)=0,z\left(1\right)={\int }_0^{1}z\left(s\right)d\varkappa \left(s\right).$$

Then, $z\left(t\right)\ge 0,t\in [0,1].$

Proof. 

Let $h\left(t\right)\in (L^1(0,1),[0,+\infty )),$ and

$$-{\mathcal{D}}_t^{\alpha ,\mu }z\left(t\right)=h\left(t\right),$$

subject to the Riemann–Stieltjes integral boundary condition

$$z\left(0\right)=0,z\left(1\right)={\int }_0^{1}z\left(s\right)d\varkappa \left(s\right).$$

It follows from (11) that

$$z\left(t\right)={\int }_0^{1}H(t,s)h\left(s\right)ds\ge 0,t\in [0,1].$$

□

Lemma 6

(Schauder fixed-point theorem). Assume T is a continuous and compact mapping of Banach space E into itself if

$$\{z\in E:z=\sigma Tz,\mathit{for}\mathit{some}0\le \sigma \le 1\},$$

(15)

is bounded. Then, T has a fixed point in E.

3. Main Results

In this paper, we choose $E=C[0,1]$ with the norm $\left|\right|z\left|\right|={max}_{0\le t\le 1}\left|z\left(t\right)\right|$ as our working space, which is a Banach space with a partial-order relation

$$z\le y\iff z\left(t\right)\le y\left(t\right),z,y\in C[0,1].$$

We now list an assumption that will be used throughout the rest of the paper.

• $\left(\mathbf{G}\mathbf{1}\right)$  $g\in C\left(\right(0,1)\times (0,+\infty ),[0,+\infty \left)\right)$, and g is non-increasing in space variable; moreover, for any constant $c\ge 1$ and for any $(t,z)\in (0,1]\times (0,+\infty )$, there exists a constant $\kappa >0$ such that

$$g(t,cz)\ge c^{-\kappa }g(t,z).$$

(16)

Remark 2.

From (16), if $0<c<1$, then the following inequality holds:

$$g(t,cz)\le c^{-\kappa }g(t,z).$$

(17)

In fact, if $0<c<1$, then ${\displaystyle \frac{1}{c}}>1$. It follows from (16) that

$$g(t,z)=g\left(t,{\displaystyle \frac{1}{c}}\left(cy\right)\right)\ge {\left({\displaystyle \frac{1}{c}}\right)}^{-\kappa }g(t,cz),$$

which implies that (17) holds.

Remark 3.

The condition $\left(\mathbf{G}\mathbf{1}\right)$ not only includes a large number of nonlinear functions such as

$$\begin{array}{c}\left(1\right)g(t,z)=a\left(t\right)z^r,-\kappa \le r<0,a\left(t\right)\ge 0,\hfill \\ \left(2\right)g(t,z)=a\left(t\right)z^{r}log{(1+z)}^l,-\kappa \le r<0,l\ge 0,\hfill \\ \left(3\right)g(t,z)=a\left(t\right){\displaystyle \sum _{i=1}^n}\left(z^{r_i}+z^{l_i}\right),-\kappa \le min\{r_i,l_i\}<0,i=1,2,...n,\hfill \end{array}$$

but also the condition $\left(\mathbf{G}\mathbf{1}\right)$ permits g to be singular at $t=0,1$ and $z=0$; for example, $g(t,z)={\displaystyle \frac{z^{-\frac{1}{3}}}{t^{\frac{1}{5}}{(1-t)}^{\frac{1}{10}}}}.$

Define the function

$$\varrho \left(s\right)=g\left(s,e^{-\mu s}{s}^{\beta -1}(1-s)\right),s\in (0,1),$$

and a Lebesgue space $L^{{\displaystyle \frac{1}{\rho }}}(0,1)(\rho >0)$ equipped with the norm

$${\left|\right|\varrho \left|\right|}_{L^{{\displaystyle \frac{1}{\rho }}}}={\left({\int }_0^1{\varrho }^{{\displaystyle \frac{1}{\rho }}}\left(s\right)ds\right)}^{\rho }.$$

Theorem 1.

Assume that hypotheses  (G0)   and  (G1)   hold. Moreover, suppose that there exists a constant $\rho \in (0,\alpha )$ such that

$$\varrho \in L^{{\displaystyle \frac{1}{\rho }}}(0,1).$$

(18)

Then, there exists a positive constant ${\lambda }^{*}$ such that, for any $\lambda \in [{\lambda }^{*},+\infty )$, the eigenvalue problem for the tempered fractional equation with the Riemann–Stieltjes integral boundary condition (1) admits at least one positive solution $z^{*}\left(t\right)$ satisfying

$$l{e}^{-\mu t}{t}^{\beta -1}(1-t)\le z^{*}\left(t\right)\le l^{-1},t\in [0,1],$$

where l is a constant with $0<l<1$.

Proof. 

First, define the cone P in E by

$$P=\{z\in E:z(t)\ge 0,t\in [0,1\left]\right\}.$$

Next, choose a subset $K\subset P$ defined by

$$K=\left\{z\in E:\mathrm{there}\mathrm{exists}\mathrm{a}\mathrm{constant}0<l<1\mathrm{such}\mathrm{that}l{e}^{-\mu t}{t}^{\beta -1}(1-t)\le z\left(t\right)\le l^{-1},t\in [0,1]\right\}.$$

For any $t\in [0,1]$, we have $e^{-\mu t}{t}^{\beta -1}(1-t)\le 1$, which implies $e^{-\mu t}{t}^{\beta -1}(1-t)\in K$. Hence, K is nonempty.

Now, define the operator $T_{\lambda }:E\to E$ by

$$\left(T_{\lambda }z\right)\left(t\right)=\lambda {\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}g(\tau ,z\left(\tau \right))e^{\mu \tau }d\tau \right)}^{q-1}ds,t\in [0,1].$$

(19)

We next show that $T_{\lambda }$ is well defined and satisfies $T_{\lambda }\left(K\right)\subseteq K$.

Indeed, for any $z\in K$, by the definition of K, there exists a constant $0<l_z<1$ such that

$$l_z{e}^{-\mu t}{t}^{\beta -1}(1-t)\le z\left(t\right)\le l_z^{-1},t\in [0,1].$$

(20)

Since $0<\rho <\alpha$, by Lemma 3, the Hölder inequality, assumption (G1), and conditions (17) and (18), we obtain

$$\begin{array}{cc}\hfill \left(T_{\lambda }z\right)\left(t\right)& \le \lambda {\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}g\left(\tau ,l_z{e}^{-\mu \tau }{\tau }^{\beta -1}(1-\tau )\right)e^{\mu \tau }d\tau \right)}^{q-1}ds\hfill \\ \hfill & \le {\displaystyle \frac{\lambda l_z^{-\kappa (q-1)}}{\Gamma \left(\alpha \right)}}{\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,e^{-\mu \tau }{\tau }^{\beta -1}(1-\tau )\right)e^{\mu \tau }d\tau \right)}^{q-1}ds\hfill \\ \hfill & \le {\displaystyle \frac{\lambda l_z^{-\kappa (q-1)}\parallel \vartheta \parallel e^{\mu (q-1)}}{\Gamma \left(\alpha \right)}}{\int }_0^1{\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,e^{-\mu \tau }{\tau }^{\beta -1}(1-\tau )\right)d\tau \right)}^{q-1}ds\hfill \\ & \le {\displaystyle \frac{\lambda l_z^{-\kappa (q-1)}\parallel \vartheta \parallel e^{\mu (q-1)}}{\Gamma \left(\alpha \right)}}{\int }_0^1{\left[{\left({\int }_0^s{(s-\tau )}^{{\displaystyle \frac{\alpha -1}{1-\rho }}}d\tau \right)}^{1-\rho }{\left({\int }_0^s{\varrho }^{{\displaystyle \frac{1}{\rho }}}\left(\tau \right)d\tau \right)}^{\rho }\right]}^{q-1}ds\hfill \\ & \le {\displaystyle \frac{\lambda l_z^{-\kappa (q-1)}\parallel \vartheta \parallel e^{\mu (q-1)}}{\Gamma \left(\alpha \right)}}{\left|\right|\varrho \left|\right|}_{L^{{\displaystyle \frac{1}{\rho }}}}^{q-1}{\int }_0^1{\left({\int }_0^s{(s-\tau )}^{{\displaystyle \frac{\alpha -1}{1-\rho }}}d\tau \right)}^{(1-\rho )(q-1)}ds\hfill \\ \hfill & \le {\displaystyle \frac{\lambda l_z^{-\kappa (q-1)}\parallel \vartheta \parallel e^{\mu (q-1)}}{\Gamma \left(\alpha \right)}}{\parallel \varrho \parallel }_{L^{{\displaystyle \frac{1}{\rho }}}}^{q-1}{\left({\displaystyle \frac{1-\rho }{\alpha -\rho }}\right)}^{(q-1)(1-\rho )}<+\infty .\hfill \end{array}$$

(21)

Hence, $T_{\lambda }z$ is well defined for all $z\in K$.

On the other hand, for any $s\in (0,1)$, we have $0<e^{-\mu s}{s}^{\beta -1}(1-s)<1$. By assumption (G1) and $\varrho \in L^{{\displaystyle \frac{1}{\rho }}}(0,1)$, it follows that

$$0<{\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g(\tau ,1)d\tau \right)}^{q-1}ds<+\infty .$$

Therefore, by (16) and Lemma 3, we obtain

$$\begin{array}{cc}\hfill \left(T_{\lambda }z\right)\left(t\right)& \ge {\displaystyle \frac{\lambda e^{-\mu t}{t}^{\beta -1}(1-t)}{{\Gamma }^{q-1}\left(\alpha \right)e^{\mu (q-1)}}}{\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,l_z^{-1}\right)d\tau \right)}^{q-1}ds\hfill \\ \hfill & \ge {\displaystyle \frac{\lambda l_z^{\kappa (q-1)}{e}^{-\mu t}{t}^{\beta -1}(1-t)}{{\Gamma }^{q-1}\left(\alpha \right)e^{\mu (q-1)}}}{\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g(\tau ,1)d\tau \right)}^{q-1}ds.\hfill \end{array}$$

(22)

Define

$$\begin{array}{cc}\hfill {\tilde{l}}_z=min\{& {\displaystyle \frac{1}{2}},{\left[{\displaystyle \frac{\lambda l_z^{-\kappa (q-1)}\parallel \vartheta \parallel e^{\mu (q-1)}}{\Gamma \left(\alpha \right)}}{\parallel \varrho \parallel }_{L^{{\displaystyle \frac{1}{\rho }}}}^{q-1}{\left({\displaystyle \frac{1-\rho }{\alpha -\rho }}\right)}^{(q-1)(1-\rho )}\right]}^{-1},\hfill \\ \hfill & {\displaystyle \frac{\lambda l_z^{\kappa (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)e^{\mu (q-1)}}}{\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g(\tau ,1)d\tau \right)}^{q-1}ds\}.\hfill \end{array}$$

(23)

Then, from (21)–(23), we conclude that

$${\tilde{l}}_z{e}^{-\mu t}{t}^{\beta -1}(1-t)\le \left(T_{\lambda }z\right)\left(t\right)\le {\tilde{l}}_z^{-1},t\in [0,1].$$

Hence, $T_{\lambda }\left(K\right)\subseteq K$,

Thus, from (12) and (13) in Lemma 4, it is clear that the following equation is valid:

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\left(T_{\lambda }z\right)\left(t\right)\right]={\lambda }^{{\displaystyle \frac{1}{q-1}}}g\left(t,z\left(t\right)\right),\hfill \\ \left(T_{\lambda }z\right)\left(0\right)=0,{\mathbb{D}}_t^{\beta ,\mu }\left(\left(T_{\lambda }z\right)\left(0\right)\right)=0,\left(T_{\lambda }z\right)\left(1\right)={\int }_0^1\left(T_{\lambda }z\right)\left(t\right)d\varkappa \left(t\right).\hfill \end{array}\right.$$

(24)

In what follows, we try to construct the upper and lower solutions of the eigenvalue problem of the tempered fractional equation with the Riemann–Stieltjes integral boundary condition (1). To achieve this, let

$$\chi \left(t\right)={\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{e}^{\mu \tau }g\left(\tau ,e^{-\mu \tau }{\tau }^{\beta -1}(1-\tau )\right)d\tau \right)}^{q-1}ds.$$

It follows from Lemma 3 that

$$\begin{array}{cc}\hfill \chi \left(t\right)& \ge {\displaystyle \frac{e^{-\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}{e}^{-\mu t}{t}^{\beta -1}(1-t){\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,e^{-\mu \tau }{\tau }^{\beta -1}(1-\tau )\right)d\tau \right)}^{q-1}ds\hfill \\ & \ge {\displaystyle \frac{e^{-\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}{e}^{-\mu t}{t}^{\beta -1}(1-t){\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,1\right)d\tau \right)}^{q-1}ds,\forall t\in [0,1].\hfill \end{array}$$

(25)

Now, take a fixed constant

$${\lambda }_1=max\left\{2,{\left({\displaystyle \frac{e^{-\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}{\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,1\right)d\tau \right)}^{q-1}ds\right)}^{-1}\right\}>1,$$

and then, by (25), we have

$${\lambda }_1\chi \left(t\right)\ge e^{-\mu t}{t}^{\beta -1}(1-t),\forall t\in [0,1].$$

(26)

On the other hand, for any $\lambda >{\lambda }_1$, it follows from the fact that the operator $T_{\lambda }$ is decreasing and (26) that

$$\begin{array}{c}{\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{e}^{\mu \tau }g\left(\tau ,\lambda \chi \left(\tau \right)\right)d\tau \right)}^{q-1}ds\hfill \\ \le {\displaystyle \frac{e^{\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}{\int }_0^{1}G(t,s){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,{\lambda }_1\chi \left(\tau \right)\right)d\tau \right)}^{q-1}ds\hfill \\ \le {\displaystyle \frac{e^{\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}{\int }_0^{1}G(t,s){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,e^{-\mu \tau }{\tau }^{\beta -1}(1-\tau )\right)d\tau \right)}^{q-1}ds\hfill \\ <+\infty ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \chi \left(t\right)& \le {\displaystyle \frac{e^{\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}\left|\right|\vartheta \left|\right|{\int }_0^1{\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,e^{-\mu \tau }{\tau }^{\beta -1}(1-\tau )\right)d\tau \right)}^{q-1}ds\hfill \\ & \le {\displaystyle \frac{e^{\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}\left|\right|\vartheta \left|\right|{\left({\displaystyle \frac{1-\rho }{\alpha -\rho }}\right)}^{(q-1)(1-\rho )}{\left|\right|\varrho \left|\right|}_{L^{{\displaystyle \frac{1}{\rho }}}}^{q-1}<+\infty .\hfill \end{array}$$

Let

$$\varsigma ={\displaystyle \frac{e^{\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}\left|\right|\vartheta \left|\right|{\left({\displaystyle \frac{1-\rho }{\alpha -\rho }}\right)}^{(q-1)(1-\rho )}{\left|\right|\varrho \left|\right|}_{L^{{\displaystyle \frac{1}{\rho }}}}^{q-1}+1,$$

and choose a constant

$${\lambda }^{*}=max\left\{{\lambda }_1,{\left[{\displaystyle \frac{e^{-\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}{\varsigma }^{-\kappa (q-1)}{\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,1\right)d\tau \right)}^{q-1}ds\right]}^{{\displaystyle \frac{1}{\kappa (q-1)-1}}}\right\}.$$

Thus, by employing Lemma 3 and (16), one has

$$\begin{array}{c}+\infty >{\lambda }^{*}{\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{e}^{\mu \tau }g\left(\tau ,{\lambda }^{*}\chi \left(\tau \right)\right)d\tau \right)}^{q-1}ds\hfill \\ \ge {\displaystyle \frac{e^{-\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}{\left({\lambda }^{*}\right)}^{1-\kappa (q-1)}{e}^{-\mu t}{t}^{\beta -1}(1-t){\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,\chi \left(\tau \right)\right)d\tau \right)}^{q-1}ds\hfill \\ \ge {\displaystyle \frac{e^{-\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}{\left({\lambda }^{*}\right)}^{1-\kappa (q-1)}{e}^{-\mu t}{t}^{\beta -1}(1-t){\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,\varsigma \right)d\tau \right)}^{q-1}ds\hfill \\ \ge {\displaystyle \frac{e^{-\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}{\left({\lambda }^{*}\right)}^{1-\kappa (q-1)}{\varsigma }^{-\kappa (q-1)}{e}^{-\mu t}{t}^{\beta -1}(1-t){\int }_0^1\vartheta \left(s\right){\left({\int }_0^s{(s-\tau )}^{\alpha -1}g\left(\tau ,1\right)d\tau \right)}^{q-1}ds\hfill \\ \ge e^{-\mu t}{t}^{\beta -1}(1-t),\forall t\in [0,1].\hfill \end{array}$$

(27)

Let

$$\Phi \left(t\right)={\lambda }^{*}\chi \left(t\right),\Psi \left(t\right)={\lambda }^{*}{\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{e}^{\mu \tau }{(s-\tau )}^{\alpha -1}g\left(\tau ,{\lambda }^{*}\chi \left(\tau \right)\right)d\tau \right)}^{q-1}ds,$$

and then

$$\Phi \left(t\right)=T_{{\lambda }^{*}}\left(e^{-\mu t}{t}^{\beta -1}(1-t)\right),\Psi \left(t\right)=T_{{\lambda }^{*}}(\Phi \left(t\right)).$$

(28)

Thus, for any $t\in [0,1]$, (26) and (27) imply that

$$\begin{array}{c}\Phi \left(t\right)={\lambda }^{*}{\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{e}^{\mu \tau }g\left(\tau ,e^{-\mu \tau }{\tau }^{\beta -1}(1-\tau )\right)d\tau \right)}^{q-1}ds\hfill \\ ={\lambda }^{*}\chi \left(t\right)\ge {\lambda }_1\chi \left(t\right)\ge e^{-\mu t}{t}^{\beta -1}(1-t),\hfill \\ \Psi \left(t\right)={\lambda }^{*}{\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{e}^{\mu \tau }g\left(\tau ,{\lambda }^{*}\chi \left(\tau \right)\right)d\tau \right)}^{q-1}ds\hfill \\ \ge e^{-\mu t}{t}^{\beta -1}(1-t).\hfill \end{array}$$

(29)

Moreover, by using (24), (28), and (21)–(23), one has

$$\begin{array}{c}\Phi \left(0\right)=0,{\mathbb{D}}_t^{\beta ,\mu }(\Phi \left(0\right))=0,\Phi \left(1\right)={\int }_0^1\Phi \left(t\right)d\varkappa \left(t\right),\hfill \\ \Psi \left(0\right)=0,{\mathbb{D}}_t^{\beta ,\mu }(\Psi \left(0\right))=0,\Psi \left(1\right)={\int }_0^1\Psi \left(t\right)d\varkappa \left(t\right),\hfill \end{array}$$

(30)

and $\Phi \left(t\right),\Psi \left(t\right)\in K$. Thus, it follows from (29) that

$$\Psi \left(t\right)=(T_{{\lambda }^{*}}\Phi )\left(t\right)\ge e^{-\mu t}{t}^{\beta -1}(1-t),\Phi \left(t\right)\ge e^{-\mu t}{t}^{\beta -1}(1-t),\forall t\in [0,1],$$

which implies that, for any $t\in [0,1]$, we have

$$\begin{gathered} \Psi \left(t\right)=(T_{{\lambda }^{*}}\Phi )\left(t\right)={\lambda }^{*}{\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{e}^{\mu \tau }g\left(\tau ,{\lambda }^{*}\chi \left(\tau \right)\right)d\tau \right)}^{q-1}ds \\ \le {\lambda }^{*}{\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{e}^{\mu \tau }g\left(\tau ,e^{-\mu \tau }{\tau }^{\beta -1}(1-\tau )\right)d\tau \right)}^{q-1}ds=\Phi \left(t\right). \end{gathered}$$

(31)

On the other hand, by using the monotonicity of g in $z>0$ and (24), (28), (29), and (31), one gets

$$\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Psi \left(t\right))\right]-{{\lambda }^{*}}^{\frac{1}{q-1}}g(t,\Psi \left(t\right))\hfill \\ {=}_0^R{\mathbb{D}}_t^{\alpha ,\mu }\left[{\phi }_p\left({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }(T_{{\lambda }^{*}}\Phi )\left(t\right)\right)\right]-{{\lambda }^{*}}^{\frac{1}{q-1}}g(t,\Psi \left(t\right))\hfill \\ {\le }_0^R{\mathbb{D}}_t^{\alpha ,\mu }\left[{\phi }_p\left({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }(T_{{\lambda }^{*}}\Phi )\left(t\right)\right)\right]-{{\lambda }^{*}}^{\frac{1}{q-1}}g(t,\Phi \left(t\right))\hfill \\ ={{\lambda }^{*}}^{\frac{1}{q-1}}g(t,\Phi \left(t\right))-{{\lambda }^{*}}^{\frac{1}{q-1}}g(t,\Phi \left(t\right))=0,\hfill \end{array}$$

(32)

$$\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Phi \left(t\right))\right]-{{\lambda }^{*}}^{\frac{1}{q-1}}g(t,\Phi \left(t\right))\hfill \\ {=}_0^R{\mathbb{D}}_t^{\alpha ,\mu }\left[{\phi }_p\left({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }{T}_{{\lambda }^{*}}\left(e^{-\mu t}{t}^{\beta -1}(1-t)\right)\right)\right]-{{\lambda }^{*}}^{\frac{1}{q-1}}g(t,\Phi \left(t\right))\hfill \\ ={{\lambda }^{*}}^{\frac{1}{q-1}}g(t,e^{-\mu t}{t}^{\beta -1}(1-t))-{{\lambda }^{*}}^{\frac{1}{q-1}}g(t,\Phi \left(t\right))\hfill \\ \ge {{\lambda }^{*}}^{\frac{1}{q-1}}g(t,e^{-\mu t}{t}^{\beta -1}(1-t))-{{\lambda }^{*}}^{\frac{1}{q-1}}g(t,e^{-\mu t}{t}^{\beta -1}(1-t))=0.\hfill \end{array}$$

(33)

Thus, (31)–(33) assure that $\Phi \left(t\right),\Psi \left(t\right)$ are a couple of upper and lower solutions of the eigenvalue problem of the tempered fractional equation with the Riemann–Stieltjes integral boundary condition (1) and $\Psi \left(t\right),\Phi \left(t\right)\in K$.

In order to establish the existence of the positive solution for the eigenvalue problem of the tempered fractional equation with the Riemann–Stieltjes integral boundary condition (1), we construct a function $g^{*}$

$$g^{*}(t,z)=\left\{\begin{array}{c}g(t,\Psi (t\left)\right),z<\Psi \left(t\right),\hfill \\ g(t,z(t\left)\right),\Psi \left(t\right)\le z\le \Phi \left(t\right),\hfill \\ g(t,\Phi (t\left)\right),z>\Phi \left(t\right),\hfill \end{array}\right.$$

(34)

and obviously $g^{*}:(0,1)\times [0,+\infty )\to [0,+\infty )$ is continuous. In this case, we consider the following amended tempered fractional equation with the Riemann–Stieltjes integral boundary condition

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }z\left(t\right)\right]={{\lambda }^{*}}^{\frac{1}{q-1}}{g}^{*}\left(t,z\left(t\right)\right),\hfill \\ z\left(0\right)=0,{\mathbb{D}}_t^{\beta ,\mu }\left(z\left(0\right)\right)=0,z\left(1\right)={\int }_0^{1}z\left(t\right)d\varkappa \left(t\right),\hfill \end{array}\right.$$

(35)

and define an amended operator ${\mathcal{D}}_{{\lambda }^{*}}$ by

$$\left({\mathcal{D}}_{{\lambda }^{*}}z\right)\left(t\right)={\lambda }^{*}{\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{g}^{*}(\tau ,z\left(\tau \right))e^{\mu \tau }d\tau \right)}^{q-1}ds,t\in [0,1].$$

(36)

Clearly, the amended operator ${\mathcal{D}}_{{\lambda }^{*}}:C[0,1]\to C[0,1]$, and a fixed point of the operator ${\mathcal{D}}_{{\lambda }^{*}}$ is a solution of the amended tempered fractional equation with the Riemann–Stieltjes integral boundary condition (35).

For all $z\in E$, it follows from the monotonicity of g in z and (34) that

$$\begin{array}{c}g(t,\Phi \left(t\right))\le g^{*}(t,z\left(t\right))\le g(t,\Psi \left(t\right)),\mathrm{if}\Psi \left(t\right)\le z\le \Phi \left(t\right),\hfill \\ g^{*}(t,z\left(t\right))=g(t,\Psi \left(t\right)),\mathrm{if}z<\Psi \left(t\right),\hfill \\ g^{*}(t,z\left(t\right))=g(t,\Phi \left(t\right)),\mathrm{if}z>\Phi \left(t\right),\hfill \end{array}$$

which implies that

$$g(t,\Phi \left(t\right))\le g^{*}(t,z\left(t\right))\le g(t,\Psi \left(t\right)),\forall z\in E.$$

(37)

Thus, by using (29) and (37), the following inequality holds:

$$g(t,\Phi \left(t\right))\le g^{*}(t,z\left(t\right))\le g(t,\Psi \left(t\right))\le g(t,e^{-\mu t}{t}^{\beta -1}(1-t))=\varrho \left(t\right),\forall z\in E.$$

(38)

Consequently, for any $z\in E$, Lemma 3 and (38) guarantee that

$$\begin{array}{cc}\hfill {\mathcal{D}}_{{\lambda }^{*}}z\left(t\right)& ={\lambda }^{*}{\int }_0^{1}G(t,s){\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{g}^{*}(\tau ,z\left(\tau \right))e^{\mu \tau }d\tau \right)}^{q-1}ds\hfill \\ & \le {\lambda }^{*}\left|\right|\vartheta \left|\right|{\displaystyle \frac{e^{\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}{\int }_0^1{\left({\int }_0^s{(s-\tau )}^{\alpha -1}g(\tau ,e^{-\mu \tau }{\tau }^{\beta -1}(1-\tau ))d\tau \right)}^{q-1}ds\hfill \\ & \le {\lambda }^{*}{\displaystyle \frac{e^{\mu (q-1)}}{{\Gamma }^{q-1}\left(\alpha \right)}}\left|\right|\vartheta \left|\right|{\left({\displaystyle \frac{1-\rho }{\alpha -\rho }}\right)}^{(q-1)(1-\rho )}{\left|\right|\varrho \left|\right|}_{L^{\frac{1}{\rho }}}^{q-1}<+\infty .\hfill \end{array}$$

(39)

Consequently, for the fixed ${\lambda }^{*}$, the amended operator ${\mathcal{D}}_{{\lambda }^{*}}$ is uniformly bounded. In addition, according to the continuity of $g^{*}$ and Green function G, we get that ${\mathcal{D}}_{{\lambda }^{*}}:E\to E$ is continuous.

Let $\Omega$ be a bounded subset of E, and then we have $\left|\right|z\left|\right|\le N$ for some positive constant $N>0$ and all $z\in \Omega$. Let $L={max}_{0\le t\le 1,0\le z\le N}\left|g^{*}(t,z)\right|+1.$ It follows from the uniform continuity of $G(t,s)$ that, for any $ϵ>0$ and $s\in [0,1]$, there exists $\sigma >0$ such that

$$\mid G(t_1,s)-G(t_2,s)\mid <{\displaystyle \frac{ϵ}{{\left(\frac{L{e}^{\mu }}{\alpha \Gamma \left(\alpha \right)}\right)}^{q-1}}},$$

for $\mid t_1-t_2\mid <\sigma$. Then,

$$\begin{array}{cc}\hfill \mid {\mathcal{D}}_{{\lambda }^{*}}z\left(t_1\right)-{\mathcal{D}}_{{\lambda }^{*}}z\left(t_2\right)\mid & \le {\int }_0^1\mid G(t_1,s)-G(t_2,s)\mid \left|{\left({\int }_0^s{\displaystyle \frac{e^{-\mu s}}{\Gamma \left(\alpha \right)}}{(s-\tau )}^{\alpha -1}{g}^{*}(\tau ,z\left(\tau \right))e^{\mu \tau }d\tau \right)}^{q-1}\right|ds\hfill \\ & \le {\int }_0^1\mid G(t_1,s)-G(t_2,s)\mid {\left({\displaystyle \frac{L{e}^{\mu }}{\alpha \Gamma \left(\alpha \right)}}\right)}^{q-1}ds<ϵ,\hfill \end{array}$$

which implies that ${\mathcal{D}}_{{\lambda }^{*}}(\Omega )$ is equicontinuous. Thus, according to the Arzela–Ascoli theorem, ${\mathcal{D}}_{{\lambda }^{*}}:E\to E$ is a completely continuous operator.

Noticing that (39) indicates that

$$\{z\in E:z=\sigma Tz,\mathrm{for}\mathrm{some}0\le \sigma \le 1\},$$

is bounded. Consequently, from the Schauder fixed-point theorem, we know that ${\mathcal{D}}_{{\lambda }^{*}}$ has at least one fixed point $z^{*}$ satisfying $z^{*}={\mathcal{D}}_{{\lambda }^{*}}{z}^{*}$.

To prove the fixed point $z^{*}$ of ${\mathcal{D}}_{{\lambda }^{*}}$ is also the fixed point of $T_{{\lambda }^{*}}$, by (34), we only need to show

$$\Psi \left(t\right)\le z^{*}\left(t\right)\le \Phi \left(t\right),t\in [0,1].$$

(40)

In fact, first, it follows from the fact that $z^{*}$ is a fixed point of ${\mathcal{D}}_{{\lambda }^{*}}$ and (30) that

$$\begin{array}{c}{z}^{*}\left(0\right)=0,{\mathbb{D}}_t^{\beta ,\mu }\left(z^{*}\left(0\right)\right)=0,z^{*}\left(1\right)={\int }_0^1{z}^{*}\left(t\right)d\varkappa \left(t\right),\hfill \\ \Phi \left(0\right)=0,{\mathbb{D}}_t^{\beta ,\mu }(\Phi \left(0\right))=0,\Phi \left(1\right)={\int }_0^1\Phi \left(t\right)d\varkappa \left(t\right).\hfill \end{array}$$

(41)

On the other hand, since $z^{*}$ is a fixed point of ${\mathcal{D}}_{{\lambda }^{*}}$, by using (24) and (38), one gets

$$\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Phi \left(t\right))-{\phi }_p\left({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }{z}^{*}\left(t\right)\right)\right]\hfill \\ {=}_0^R{\mathbb{D}}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Phi \left(t\right))\right]{-}_0^R{\mathbb{D}}_t^{\alpha ,\mu }\left[{\phi }_p\left({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }{z}^{*}\left(t\right)\right)\right]\hfill \\ ={{\lambda }^{*}}^{\frac{1}{q-1}}g(t,e^{-\mu t}{t}^{\beta -1}(1-t)-{{\lambda }^{*}}^{\frac{1}{q-1}}{g}^{*}(t,z^{*}\left(t\right))\ge 0,\forall t\in [0,1].\hfill \end{array}$$

(42)

Let $\zeta \left(t\right)={\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Phi \left(t\right))-{\phi }_p\left({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }{z}^{*}\left(t\right)\right),$ and it follows from (41) and (42) that

$$\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\alpha ,\mu }\zeta \left(t\right){=}_0^R{\mathbb{D}}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Phi \left(t\right))\right]{-}_0^R{\mathbb{D}}_t^{\alpha ,\mu }\left[{\phi }_p\left({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }{z}^{*}\left(t\right)\right)\right]\ge 0,t\in [0,1]\hfill \\ {,}_0^R{\mathbb{D}}_t^{\alpha ,\mu }\zeta \left(0\right){=}_0^R{\mathbb{D}}_t^{\alpha ,\mu }\left[{\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Phi \left(0\right))\right]{-}_0^R{\mathbb{D}}_t^{\alpha ,\mu }\left[{\phi }_p\left({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }{z}^{*}\left(0\right)\right)\right]=0.\hfill \end{array}$$

By Lemma 4, we have

$$\zeta \left(t\right)\ge 0,$$

that is

$${\phi }_p({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Phi \left(t\right))-{\phi }_p\left({-}_0^R{\mathbb{D}}_t^{\beta ,\mu }{z}^{*}\left(t\right)\right)\ge 0,$$

which implies that

$${-}_0^R{\mathbb{D}}_t^{\beta ,\mu }\Phi \left(t\right)\ge {-}_0^R{\mathbb{D}}_t^{\beta ,\mu }{z}^{*}\left(t\right),\mathrm{i}.\mathrm{e}.,{}_0{}^R\mathbb{D}_t^{\beta ,\mu }(\Phi -z^{*})\left(t\right)\le 0.$$

Thus, (41) and Lemma 4 yield that

$$\Phi \left(t\right)-z^{*}\left(t\right)\ge 0.$$

So, $z^{*}\left(t\right)\le \Phi \left(t\right)$ on $[0,1].$ Similarly, one also easily gets $z^{*}\left(t\right)\ge \Psi \left(t\right)$ on $[0,1].$ Thus, the inequality (40) holds, which ensures

$$g^{*}(t,z^{*}\left(t\right))=g(t,z^{*}\left(t\right)),t\in [0,1].$$

Consequently, $z^{*}\left(t\right)$ is a positive solution of the eigenvalue problem of the tempered fractional equation with the Riemann–Stieltjes integral boundary condition (1).

In the end, it follows from (21)–(22) and $\Phi ,\Psi \in K$ that

$$l{e}^{-\mu t}{t}^{\beta -1}(1-t)\le \Psi \left(t\right)\le z^{*}\left(t\right)\le \Phi \left(t\right)\le l^{-1}.$$

□

4. Example

Tempered fractional equations arise naturally in physical and engineering models of anomalous transport where memory effects persist over finite ranges. Such phenomena occur, for instance, in turbulent diffusion, transport in porous or heterogeneous media, and subdiffusive processes with environmental damping. Compared with classical fractional models, exponential tempering suppresses unrealistically long jumps and heavy-tailed waiting times while preserving essential nonlocal behavior.

The general model (1) can therefore be interpreted as describing a transition from subdiffusive to effectively normal diffusion under truncation or damping effects. The presence of Riemann–Stieltjes integral boundary conditions allows the boundary response to depend on cumulative or weighted interactions, which arise in systems with distributed feedback or boundary control. To illustrate the applicability of our existence results, we present a concrete example involving a singular nonlinearity and a signed boundary measure, together with numerical evidence supporting the theoretical conclusions.

Example 1.

Consider a nonlocal tempered fractional turbulent-flow equation

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\frac{1}{2},\frac{1}{3}}\left[{\phi }_{\frac{5}{2}}({-}_0^R{\mathbb{D}}_t^{\frac{3}{2},\frac{1}{3}}z\left(t\right)\right]={\displaystyle \frac{{\lambda }^{\frac{3}{2}}{z}^{-\frac{1}{3}}}{t^{\frac{1}{5}}{(1-t)}^{\frac{1}{10}}}},\hfill \\ z\left(0\right)=0,{\mathbb{D}}_t^{\frac{1}{2},\frac{1}{3}}\left(z\left(0\right)\right)=0,z\left(1\right)={\int }_0^{1}z\left(t\right)d\varkappa \left(t\right),\hfill \end{array}\right.$$

(43)

where

$$\varkappa \left(t\right)=\left\{\begin{array}{c}0,t\in \left[0,{\displaystyle \frac{1}{2}}\right),\hfill \\ 1,t\in \left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}}\right),\hfill \\ -{\displaystyle \frac{1}{4}},t\in \left[{\displaystyle \frac{3}{4}},1\right].\hfill \end{array}\right.$$

Then, there exists a positive constant ${\lambda }^{*}$ such that, for any $\lambda \in [{\lambda }^{*},+\infty )$, the eigenvalue problem of the tempered fractional equation with the Riemann–Stieltjes integral boundary condition (43) has at least one positive solution $z^{*}\left(t\right)$ satisfying

$$l{e}^{-\frac{t}{3}}{t}^{\frac{1}{2}}(1-t)\le z^{*}\left(t\right)\le l^{-1},$$

where $0<l<1$ is a constant.

Remark 4.

The fractional order $\alpha =\frac{1}{2}$ corresponds to subdiffusive transport, while the tempering parameter $\mu =\frac{1}{3}$ introduces exponential damping that limits long-range memory effects. The exponent $p=\frac{5}{2}$ characterizes the intensity of the nonlinear diffusion, and the singular term $z^{-1/3}$ represents localized source or sink effects commonly encountered in heterogeneous media. These parameter values are chosen to reflect physically meaningful transport regimes while ensuring that the assumptions of Theorem 1 are satisfied.

Proof. 

First, by straightforward calculation, the tempered fractional Equation (43) reduces to the following form:

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{\frac{1}{2},\frac{1}{3}}\left[{\phi }_{\frac{5}{2}}({-}_0^R{\mathbb{D}}_t^{\frac{3}{2},\frac{1}{3}}z\left(t\right)\right]={\displaystyle \frac{{\lambda }^{\frac{3}{2}}{z}^{-\frac{1}{3}}}{t^{\frac{1}{5}}{(1-t)}^{\frac{1}{10}}}},\hfill \\ z\left(0\right)=0,{\mathbb{D}}_t^{\beta ,\frac{1}{3}}\left(z\left(0\right)\right)=0,z\left(1\right)=z\left(\frac{1}{2}\right)-{\displaystyle \frac{5}{4}}z\left(\frac{3}{4}\right).\hfill \end{array}\right.$$

(44)

Let

$$\alpha ={\displaystyle \frac{1}{2}},\beta ={\displaystyle \frac{3}{2}},\mu ={\displaystyle \frac{1}{3}},p={\displaystyle \frac{5}{2}}.$$

For the signed measure $\varkappa$, the Green kernel associated with (43) takes the form

$$\begin{array}{cc}\hfill {\mathcal{H}}_{\varkappa }\left(s\right)& =H(\frac{1}{2},s)-{\displaystyle \frac{1}{4}}H(\frac{3}{4},s){\mathcal{H}}_{\varkappa }\left(s\right)\hfill \\ & \approx \left\{\begin{array}{cc}{\displaystyle \left[0.956\left({(1-s)}^{\frac{1}{2}}-0.7071{(\frac{1}{2}-s)}^{\frac{1}{2}}\right)-0.245\left({(1-s)}^{\frac{1}{2}}-0.8660{(\frac{3}{4}-s)}^{\frac{1}{2}}\right)\right]e^{\frac{s}{3}},}\hfill & 0\le s\le \frac{1}{2},\hfill \\ {\displaystyle \left[0.956{(1-s)}^{\frac{1}{2}}-0.245\left({(1-s)}^{\frac{1}{2}}-0.8660{(\frac{3}{4}-s)}^{\frac{1}{2}}\right)\right]e^{\frac{s}{3}},}\hfill & \frac{1}{2}\le s\le \frac{3}{4},\hfill \\ {\displaystyle 0.711{(1-s)}^{\frac{1}{2}}{e}^{\frac{s}{3}},}\hfill & \frac{3}{4}\le s\le 1.\hfill \end{array}\right.\hfill \end{array}$$

(45)

Hence, ${\mathcal{H}}_{\varkappa }\left(s\right)\ge 0$.

To examine the validity of condition (G0) and the applicability of Theorem 1, we compute the associated Green kernel ${\mathcal{H}}_{\varkappa }\left(s\right)$ numerically. The values of

Table 1. Numerical values of ${\mathcal{H}}_{\varkappa }\left(s\right)$ on $[0,1]$.

s	0.00	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.75	0.85	0.95	1.00
${\mathcal{H}}_{\varkappa }\left(s\right)$	0.0000	0.2748	0.2817	0.2778	0.2678	0.2590	0.2254	0.1814	0.1369	0.0861	0.0335	0.0000

reported below are obtained by direct evaluation of the explicit Green function expression using standard floating-point arithmetic. The discretization of the interval $[0,1]$ is sufficiently fine to ensure numerical stability and accuracy.

On the other hand, we have

$$\begin{array}{cc}\hfill 0<\varpi & ={\int }_0^1{e}^{\frac{1}{3}(1-t)}{t}^{\frac{1}{2}}d\varkappa \left(t\right)=e^{\frac{1}{3}(1-\frac{1}{2})}{\left(\frac{1}{2}\right)}^{\frac{1}{2}}-\frac{1}{4}{e}^{\frac{1}{3}(1-\frac{3}{4})}{\left(\frac{3}{4}\right)}^{\frac{1}{2}}\hfill \\ & =e^{\frac{1}{6}}·\frac{1}{\sqrt{2}}-\frac{1}{4}{e}^{\frac{1}{12}}·\frac{\sqrt{3}}{2}\approx 0.8353-0.2352\approx 0.6001<1.\hfill \end{array}$$

(46)

Thus, condition (G0) holds.

Take $\kappa =4$ since

$$g(t,z)={\displaystyle \frac{z^{-\frac{1}{3}}}{t^{\frac{1}{5}}{(1-t)}^{\frac{1}{10}}}},$$

and clearly $g\in C\left(\right(0,1)\times (0,+\infty ),[0,+\infty \left)\right)$, and g is non-increasing in $z>0$; moreover, for any constant $c\ge 1$ and for any $(t,z)\in (0,1]\times (0,+\infty )$,

$$g(t,cz)={\displaystyle \frac{c^{-\frac{1}{3}}{z}^{-\frac{1}{3}}}{t^{\frac{1}{5}}{(1-t)}^{\frac{1}{10}}}}\ge {\displaystyle \frac{c^{-4}{z}^{-\frac{1}{3}}}{t^{\frac{1}{5}}{(1-t)}^{\frac{1}{10}}}}=c^{-\kappa }g(t,z).$$

which implies that $\left(\mathbf{G}\mathbf{1}\right)$ holds.

In the end, let

$$\varrho \left(s\right)=g\left(s,e^{-\frac{s}{3}}{s}^{\frac{1}{2}}(1-s)\right)=e^{\frac{s}{9}}{s}^{-\frac{1}{10}}{(1-s)}^{-\frac{3}{10}},s\in (0,1),$$

and choose $\rho =\frac{7}{5}$; we verify

$$\varrho \in L^{\frac{5}{7}}(0,1).$$

In fact,

$${\left|\right|\varrho \left|\right|}_{L^{\frac{5}{7}}}={\left({\int }_0^1{\varrho }^{\frac{5}{7}}\left(s\right)ds\right)}^{\frac{7}{5}}={\left({\int }_0^1{e}^{\frac{5}{63}s}{s}^{-\frac{5}{70}}{(1-s)}^{-\frac{15}{70}}ds\right)}^{\frac{7}{5}}={\left(1.4161497611\right)}^{\frac{7}{5}}\approx 1.655.$$

Thus, $\varrho \in L^{\frac{5}{7}}(0,1).$

According to Theorem 1, there exists a positive constant ${\lambda }^{*}$ such that, for any $\lambda \in [{\lambda }^{*},+\infty )$, the eigenvalue problem of the tempered fractional equation with the Riemann–Stieltjes integral boundary condition (43) has at least one positive solution $z^{*}\left(t\right)$ satisfying

$$l{e}^{-\frac{t}{3}}{t}^{\frac{1}{2}}(1-t)\le z^{*}\left(t\right)\le l^{-1},$$

where $0<l<1$ is a constant. □

The numerical approximation of the positive solution $z^{*}\left(t\right)$ is obtained using a standard tempered fractional finite-difference scheme combined with the upper–lower solution iteration. The step size is chosen to balance computational efficiency and accuracy, and the parameters are selected to satisfy the assumptions of Theorem 1. Numerical experiments indicate that the solution remains stable under small variations regarding the discretization parameters.

Figure 1 shows the numerical plot of ${\mathcal{H}}_{\varkappa }\left(s\right)$, demonstrating positivity despite $\varkappa$ including the signed measure.

Using a standard tempered fractional difference scheme with upper–lower solution iteration, a positive solution $z^{*}\left(t\right)$ of (43) can be approximated. Figure 2 shows an illustrative numerical solution, satisfying

$$0.1e^{-t/3}{t}^{1/2}(1-t)\lesssim z^{*}\left(t\right)\lesssim 10.$$

Remark 5.

We provide a brief physical interpretation for the example (43):

(i) 

The fractional order $\alpha =\frac{1}{2}$ and tempering parameter $\mu =\frac{1}{3}$ model tempered anomalous diffusion, where long-range memory effects persist but extreme transport events are suppressed.

(ii) 

The singular nonlinearity $z^{-1/3}$ reflects localized sources or sinks, which are commonly encountered in turbulent or heterogeneous transport processes.

(iii) 

The Riemann–Stieltjes boundary condition with a signed measure represents nonlocal boundary feedback, allowing both reinforcing and dissipative effects at the boundary.

(iv) 

The numerical solution $z^{*}\left(t\right)$ exhibits smooth growth and decay, illustrating how the tempered operator moderates long-time behavior.

Remark 6.

Despite the signed measure, the positivity of ${\mathcal{H}}_{\varkappa }\left(s\right)$ ensures that condition  (G0)  is satisfied, validating the existence results and providing a clear physical picture.

5. Conclusions

Tempered fractional equations generalize both ordinary diffusion and pure fractional diffusion, providing flexible models for semi-heavy-tailed phenomena that arise in anomalous diffusion processes. In this paper, we investigated the existence of positive solutions to the eigenvalue problem for a singular tempered fractional equation with Riemann–Stieltjes integral boundary conditions involving signed measures. By constructing appropriate upper and lower solutions, we established an eigenvalue interval guaranteeing the existence of positive solutions and obtained estimates on their bounds. A notable feature of our results is that the nonlinearity f is allowed to be singular in both the temporal and spatial variables, and the boundary condition incorporates a Riemann–Stieltjes integral with signed measures.

Potential extensions of this work include developing numerical schemes to approximate the positive solutions, exploring higher-order or multi-term tempered fractional operators, and studying analogous eigenvalue problems under different types of integral or nonlocal boundary conditions. Such directions may further deepen the understanding of tempered fractional models and broaden their applicability in complex diffusion phenomena.