Positive Solutions for a Semipositone Singular ψ–Riemann–Liouville Fractional Boundary Value Problem

Abstract

We explore the existence of positive solutions to a $\psi$–Riemann–Liouville fractional differential equation with a parameter and a sign-changing singular nonlinearity, supplemented with nonlocal boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. To establish our main results, we use the Guo–Krasnosel’skii fixed-point theorem.

1. Introduction

We consider the fractional differential equation

$$D_{a+}^{\alpha ,\psi }\mathfrak{u}\left(\mathfrak{t}\right)+\lambda \mathfrak{f}(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (a,b),$$

(1)

subject to the nonlocal boundary conditions

$${\displaystyle {\mathfrak{u}}^{\left(i\right)}\left(a\right)=0,i=0,1,\dots ,n-2;D_{a+}^{\varsigma ,\psi }\mathfrak{u}\left(b\right)=\sum _{i=1}^m{\int }_a^b{D}_{a+}^{{\varrho }_i,\psi }\mathfrak{u}\left(\mathfrak{s}\right)d{\mathcal{H}}_i\left(\mathfrak{s}\right),}$$

(2)

where $0\le a<b$, $\alpha >0$, $\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n\ge 3$, $\psi \in C^n[a,b]$ with ${\psi }^{\prime }\left(\mathfrak{t}\right)>0$ for all $\mathfrak{t}\in [a,b]$, $D_{a+}^{k,\psi }u$ denotes the $\psi$–Riemann–Liouville fractional derivative of function $\mathfrak{u}$ of order k, for $k\in \{\alpha ,\varsigma ,{\varrho }_i,i=1,\dots ,m\}$, $m\in \mathbb{N}$, $1\le \varsigma <\alpha -1$, $0\le {\varrho }_i\le \varsigma$, $i=1,\dots ,m$, $\lambda$ is a positive parameter, the function $\mathfrak{f}$ may change sign and may be singular at the points $\mathfrak{t}=a$ and/or $\mathfrak{t}=b$, and ${\mathcal{H}}_i:[a,b]\to \mathbb{R}$, $i=1,\dots ,m$ are bounded variation functions.

In this paper we give intervals for the parameter $\lambda$ such that the problem (1)–(2) has at least one positive solution. Since $\mathfrak{f}$ can have negative values, our problem is called a semipositone fractional boundary value problem. In the proofs of the main results we apply the Guo–Krasnosel’skii fixed-point theorem. The $\psi$–Riemann–Liouville fractional derivative generalizes the Riemann–Liouville derivative (for $\psi \left(\mathfrak{t}\right)=\mathfrak{t})$, and the Hadamard derivative (for $\psi \left(\mathfrak{t}\right)=ln\mathfrak{t})$ (see the definitions in Section 2). In the following, we present some papers that investigate various fractional differential equations and systems which are connected to our problem. In [1], the authors studied the nonlinear fractional differential equation

$$D_{0+}^{\alpha }\mathfrak{u}\left(\mathfrak{t}\right)+\lambda \mathfrak{f}(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (0,1),$$

(3)

with the multi-point boundary conditions

$${\displaystyle \mathfrak{u}\left(0\right)={\mathfrak{u}}^{\prime }\left(0\right)=\cdots ={\mathfrak{u}}^{(n-2)}\left(0\right)=0,D_{0+}^p\mathfrak{u}\left(1\right)=\sum _{i=1}^m{a}_i{D}_{0+}^q\mathfrak{u}\left({\xi }_i\right),}$$

(4)

where $\alpha \in \mathbb{R}$, $\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n\ge 3$, $m\in \mathbb{N}$, $0<{\xi }_1<\cdots <{\xi }_m<1$, $p,q\in \mathbb{R}$, $1\le p\le n-2$, $q\in [0,p]$, $\lambda$ is a positive parameter, the function $\mathfrak{f}$ may change sign and may be singular at $\mathfrak{t}=0$ and/or $\mathfrak{t}=1$, and $D_{0+}^{\kappa }$ denotes the Riemann–Liouville derivative of order $\kappa$, for $\kappa \in \{\alpha ,p,q\}$. With the aid of the Guo–Krasnosel’skii fixed-point theorem, they proved the existence of positive solutions for problem (3)–(4). In [2], the author investigated the fractional differential equation

$$D_{0+}^{\alpha }\mathfrak{u}\left(\mathfrak{t}\right)+\mathfrak{f}(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (0,1),$$

(5)

supplemented with the integral boundary conditions

$${\displaystyle \mathfrak{u}\left(0\right)={\mathfrak{u}}^{\prime }\left(0\right)=\cdots ={\mathfrak{u}}^{(n-2)}\left(0\right)=0,D_{0+}^p\mathfrak{u}\left(1\right)={\int }_0^1{D}_{0+}^q\mathfrak{u}\left(t\right)d\mathcal{H}\left(t\right),}$$

(6)

where $n\in \mathbb{N}$, $n\ge 3$, $\alpha \in \mathbb{R}$, $\alpha \in (n-1,n]$, $p,q\in \mathbb{R}$, $1\le p\le n-2$, $0\le q\le p$, the function $\mathfrak{f}$ may change sign and may be singular at the points $\mathfrak{t}=0$, $\mathfrak{t}=1$, and/or $\mathfrak{u}=0$, and $\mathcal{H}$ is a bounded variation function. By way of various height functions of the nonlinearity of Equation (5) defined on special bounded sets, and the Guo–Krasnosel’skii fixed-point theorem, they proved the existence and multiplicity of positive solutions of problem (5)–(6). In [3], the authors examined the existence of multiple positive solutions for the fractional differential Equation (5) subject to the boundary conditions

$${\displaystyle \mathfrak{u}\left(0\right)={\mathfrak{u}}^{\prime }\left(0\right)=\cdots ={\mathfrak{u}}^{(n-2)}\left(0\right)=0,D_{0+}^{{\beta }_0}\mathfrak{u}\left(1\right)=\sum _{i=1}^m{\int }_0^1{D}_{0+}^{{\beta }_i}\mathfrak{u}\left(t\right)d{\mathcal{H}}_i\left(t\right),}$$

where $n,m\in \mathbb{N}$, $n\ge 3$, $\alpha \in \mathbb{R}$, $\alpha \in (n-1,n]$, ${\beta }_i\in \mathbb{R}$ for all $i=0,\dots ,m$, $1\le {\beta }_0<\alpha -1$, $0\le {\beta }_1<{\beta }_2<\cdots <{\beta }_m\le {\beta }_0$, the function $\mathfrak{f}$ may change sign and may be singular at the points $\mathfrak{t}=0$, $\mathfrak{t}=1$, and/or $\mathfrak{u}=0$, and ${\mathcal{H}}_i,i=1\dots ,m$ are bounded variation functions. For the proof of the main results, they used varied height functions of $\mathfrak{f}$ defined on special bounded sets, and two theorems from the fixed-point index theory. We also mention the papers [4,5,6,7,8,9,10], where the authors explored the existence of positive solutions for various semipositone or singular Riemann–Liouville fractional differential equations with different boundary conditions. In [11], the authors studied the existence of at least three positive solutions for a class of Riemann–Liouville fractional differential equations with p-Laplacian operator and singular sign-changing nonlinearities, supplemented with integral boundary conditions containing a positive parameter. In [12], by using the Guo–Krasnosel’skii and Leggett–Williams fixed-point index theorems, the authors analyzed the existence of positive solutions for a singular Riemann–Liouville fractional differential equation with both generalized Laplacian and positive parameter subject to nonlocal boundary conditions. In [13], the authors examined the system of nonlinear fractional differential equations

$$\left\{\begin{array}{c}{D}_{0+}^{\alpha }\mathfrak{u}\left(t\right)+\lambda \mathfrak{f}(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{v}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (0,1),\hfill \\ D_{0+}^{\beta }\mathfrak{v}\left(\mathfrak{t}\right)+\mu \mathfrak{g}(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{v}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (0,1),\hfill \end{array}\right.$$

(7)

with the coupled integral boundary conditions

$$\left\{\begin{array}{c}\mathfrak{u}\left(0\right)={\mathfrak{u}}^{\prime }\left(0\right)=\cdots ={\mathfrak{u}}^{(n-2)}\left(0\right)=0,D_{0+}^p\mathfrak{u}\left(1\right)={\int }_0^1\mathfrak{v}\left(s\right)d\mathcal{H}\left(s\right),\hfill \\ \mathfrak{v}\left(0\right)={\mathfrak{v}}^{\prime }\left(0\right)=\cdots ={\mathfrak{v}}^{(m-2)}\left(0\right)=0,D_{0+}^q\mathfrak{v}\left(1\right)={\int }_0^1\mathfrak{u}\left(s\right)d\mathcal{K}\left(s\right),\hfill \end{array}\right.$$

(8)

where $n,m\in \mathbb{N}$, $n,m\ge 3$, $\alpha ,\beta \in \mathbb{R}$, $\alpha \in (n-1,n]$, $\beta \in (m-1,m]$, $p,q\in \mathbb{N}$, $1\le p\le n-2$, $1\le q\le m-2$, $\lambda$ and $\mu$ are positive parameters, the functions $\mathfrak{f}$ and $\mathfrak{g}$ are sign-changing continuous functions which may be singular at $\mathfrak{t}=0$ and/or $\mathfrak{t}=1$, and $\mathcal{H}$ and $\mathcal{K}$ are bounded variation functions. They gave intervals for the parameters $\lambda$ and $\mu$ such that the problem (7)–(8) has at least one positive solution. Other systems of Riemann–Liouville fractional differential equations supplemented with varied nonlocal boundary conditions were explored in the papers [14,15,16,17,18]. For detailed studies of various Riemann–Liouville, Caputo and Hadamard fractional differential equations and systems with various applications, we refer the reader to the monographs [19,20,21,22]. Applications of fractional calculus in physics and engineering can be found in the papers [23,24,25,26]. $\psi$–Caputo fractional differential equations or inclusions with initial or boundary conditions were studied in the papers [27,28,29,30,31]. The generalizations of $\psi$–Riemann–Liouville and $\psi$–Caputo fractional derivatives, namely $\psi$–Hilfer fractional derivatives with their properties were introduced in [32]. Another generalization of the $\psi$–Hilfer fractional derivative, that is, the $(k,\psi )$–Hilfer fractional derivative, was defined in [33]. Some fractional boundary value problems containing $(k,\psi )$–Hilfer fractional derivatives were investigated in [34,35]. Discrete fractional boundary value problems were studied in the papers [36,37,38,39,40] (with nabla discrete fractional differences), [41,42,43,44] (with delta discrete fractional differences), and [45,46] (with delta-nabla discrete fractional differences). The novelty of our paper consists in the presence of the $\psi$–Riemann–Liouville fractional derivatives in Equation (1) and the boundary conditions (2). In addition, the nonlinearity $\mathfrak{f}$ from (1) is not a non-negative function, but it may change sign and may be singular at some points. Lastly, the condition from (2) with Riemann–Stieltjes integrals is a general condition containing multi-point boundary conditions (when ${\mathcal{H}}_i,i=1,\dots ,m$ are step functions), classical integral conditions (when ${\mathcal{H}}_i,i=1,\dots ,m$ are differentiable functions), and combinations of them.

The paper is organized as follows. Section 2 introduces the definitions and key properties of $\psi$–Riemann–Liouville fractional integrals and derivatives. We also examine the linear counterpart of problem (1)–(2), together with the associated Green function and its properties. Section 3 is devoted to the main existence theorems for positive solutions of problem (1)–(2). In Section 4, we provide two illustrative examples that demonstrate the applicability of our results, while Section 5 concludes this paper.

2. Auxiliary Results

In this section we will firstly present the definitions for the $\psi$–Riemann–Liouville fractional integral and derivative and some of their properties.

Let $0\le a<b<\infty$, $\alpha >0$, $\alpha \in (n-1,n]$, and $\psi \in C^n[a,b]$ with ${\psi }^{\prime }\left(\mathfrak{t}\right)>0$ for all $\mathfrak{t}\in [a,b]$.

Definition 1

([47]-pag.99–100; [48]-pag.325). The (left-sided) Riemann–Liouville fractional integral of function $\mathfrak{u}:[a,b]\to \mathbb{R}$ with respect to function ψ on $[a,b]$ of order $\alpha >0$, or the (left-sided) ψ–Riemann–Liouville fractional integral of function $\mathfrak{u}:[a,b]\to \mathbb{R}$ of order $\alpha >0$, is defined by

$${\displaystyle I_{a+}^{\alpha ,\psi }\mathfrak{u}\left(\mathfrak{t}\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_a^{\mathfrak{t}}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -1}\mathfrak{u}\left(\mathfrak{s}\right)d\mathfrak{s},\mathfrak{t}>a;I_{a+}^{\alpha ,\psi }\mathfrak{u}\left(a\right)=0.}$$

Definition 2

([47]-pag.101–102, [48]-pag.326). The (left-sided) Riemann–Liouville fractional derivative of function $\mathfrak{u}:[a,b]\to \mathbb{R}$ with respect to function ψ on $[a,b]$ of order $\alpha >0$, or the (left-sided) ψ–Riemann–Liouville fractional derivative of function $\mathfrak{u}:[a,b]\to \mathbb{R}$ of order $\alpha >0$, is defined by

$$\begin{array}{c}{D}_{a+}^{\alpha ,\psi }\mathfrak{u}\left(\mathfrak{t}\right)={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(\mathfrak{t}\right)}\frac{d}{d\mathfrak{t}}}\right)}^n{I}_{a+}^{n-\alpha ,\psi }\mathfrak{u}\left(\mathfrak{t}\right)\hfill \\ {\displaystyle ={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(\mathfrak{t}\right)}\frac{d}{d\mathfrak{t}}}\right)}^n\frac{1}{\Gamma (n-\alpha )}{\int }_a^{\mathfrak{t}}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{n-\alpha -1}\mathfrak{u}\left(\mathfrak{s}\right)d\mathfrak{s},\mathfrak{t}>a,}\hfill \end{array}$$

where $n=\lfloor \alpha \rfloor +1$.

The notation $\lfloor \alpha \rfloor$ stands for the largest integer not greater than $\alpha$. For $\mathfrak{u}\in C^m[a,b]$, if $\alpha =m\in \mathbb{N}$ then $D_{a+}^{m,\psi }\mathfrak{u}\left(\mathfrak{t}\right)={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(\mathfrak{t}\right)}}{\displaystyle \frac{d}{d\mathfrak{t}}}\right)}^m\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{t}>a,$ and if $\alpha =0$ then $D_{a+}^{0,\psi }\mathfrak{u}\left(\mathfrak{t}\right)=\mathfrak{u}\left(\mathfrak{t}\right)$ for $\mathfrak{t}>0$.

We observe that the $\psi$–Riemann–Liouville integral and derivative are generalizations of various fractional integrals and derivatives; indeed:

-

If $\psi \left(\mathfrak{t}\right)=\mathfrak{t}$, $\mathfrak{t}\in [a,b]$, $a\ge 0$, then we obtain the Riemann–Liouville fractional integral/derivative of order $\alpha$;

-

If $\psi \left(\mathfrak{t}\right)=ln\mathfrak{t}$, $\mathfrak{t}\in [a,b]$, $a>0$, then we get the Hadamard fractional integral/derivative of order $\alpha$;

-

If $\psi \left(\mathfrak{t}\right)={\displaystyle \frac{{\mathfrak{t}}^{\varrho }}{\varrho }}$, $\mathfrak{t}\in [a,b]$, $a>0$, with $\varrho >0$, then we find the Katugampola fractional integral/derivative of order $\alpha$; and so on.

Lemma 1

([47]-Lemma 2.26; [48]-pag.326). Let $\alpha ,\beta >0$. Then for a function $u:[a,b]\to \mathbb{R}$, $u\in L^p(a,b)$, $1\le p\le \infty$, we have

$$(I_{a+}^{\alpha ,\psi }{I}_{a+}^{\beta ,\psi }u)\left(t\right)=(I_{a+}^{\alpha +\beta ,\psi }u)\left(t\right),fora.e.t\in (a,b).$$

Lemma 2.

Let $\alpha >\beta >0$. Then for $u\in L^p(a,b)$, $1\le p\le \infty$, we have

$$(D_{a+}^{\beta ,\psi }{I}_{a+}^{\alpha ,\psi }u)\left(t\right)=(I_{a+}^{\alpha -\beta ,\psi }u)\left(t\right),fora.e.t\in (a,b).$$

The proof of Lemma 2 is similar to the proof of Property 2.2 from [47].

Lemma 3

([47]-pag.100, 103; [48]-pag.326). Let $\alpha ,\beta >0$. Then, we have

(a) ${\displaystyle (I_{a+}^{\alpha ,\psi }({[\psi (x)-\psi (a)]}^{\beta -1}))(t)=\frac{\Gamma (\beta )}{\Gamma (\alpha +\beta )}{[\psi (t)-\psi (a)]}^{\alpha +\beta -1},t>a}$;

(b) ${\displaystyle D_{a+}^{\alpha ,\psi }({[\psi (x)-\psi (a)]}^{\beta -1})(t)=\frac{\Gamma (\beta )}{\Gamma (\beta -\alpha )}{[\psi (t)-\psi (a)]}^{\beta -\alpha -1},t>a}$;

(c) $D_{a+}^{\alpha ,\psi }({[\psi (x)-\psi (a)]}^{\alpha -j})(t)=0,$ for $j=1,2,\dots ,p$, where $p=n$ if $\alpha \notin \mathbb{N}$ with $\alpha \in (n-1,n)$, and $p=n$ if $\alpha =n\in \mathbb{N}$.

Lemma 4.

Let $\alpha >0$. Then for $f\in L^p(a,b)$, $1\le p\le \infty$; we have

$$D_{a+}^{\alpha ,\psi }{I}_{a+}^{\alpha ,\psi }f\left(t\right)=f\left(t\right),fora.e.t\in (a,b).$$

Proof. 

We obtain

$$\begin{array}{c}{D}_{a+}^{\alpha ,\psi }{I}_{a+}^{\alpha ,\psi }f\left(t\right)={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{I}_{a+}^{n-\alpha ,\psi }{I}_{a+}^{\alpha ,\psi }f\left(t\right)={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{I}_{a+}^{n,\psi }f\left(t\right)\hfill \\ ={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n\left({\displaystyle \frac{1}{\Gamma \left(n\right)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{n-1}f\left(s\right)ds\right)\hfill \\ ={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n-1}\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{\Gamma \left(n\right)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{n-1}f\left(s\right)ds\right)\right)\hfill \\ ={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n-1}{\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{1}{\Gamma \left(n\right)}}\left({\int }_a^t{\psi }^{\prime }\left(s\right)(n-1){(\psi \left(t\right)-\psi \left(s\right))}^{n-2}{\psi }^{\prime }\left(t\right)f\left(s\right)ds\right.\hfill \\ \left.+{\psi }^{\prime }\left(t\right){(\psi \left(t\right)-\psi \left(t\right))}^{n-1}f\left(t\right)\right)\hfill \\ ={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n-1}\left({\displaystyle \frac{1}{(n-2)!}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{n-2}f\left(s\right)ds\right)\hfill \\ ={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n-2}\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{(n-2)!}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{n-2}f\left(s\right)ds\right)\right)\hfill \\ ={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n-2}\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{1}{(n-2)!}}\left({\int }_a^t{\psi }^{\prime }\left(s\right)(n-2){(\psi \left(t\right)-\psi \left(s\right))}^{n-3}{\psi }^{\prime }\left(t\right)f\left(s\right)ds\right.\right.\hfill \\ \left.\left.+{\psi }^{\prime }\left(t\right){(\psi \left(t\right)-\psi \left(t\right))}^{n-2}f\left(t\right)\right)\right)\hfill \\ ={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n-2}\left({\displaystyle \frac{1}{(n-3)!}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{n-3}f\left(s\right)ds\right)\hfill \\ =\dots ={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^2\left({\displaystyle \frac{1}{1!}}{\int }_a^t{\psi }^{\prime }\left(s\right)(\psi \left(t\right)-\psi \left(s\right))f\left(s\right)ds\right)\hfill \\ =\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\left({\int }_a^t{\psi }^{\prime }\left(s\right)(\psi \left(t\right)-\psi \left(s\right))f\left(s\right)ds\right)\right)\hfill \\ ={\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}\left({\int }_a^t{\psi }^{\prime }\left(s\right){\psi }^{\prime }\left(t\right)f\left(s\right)ds+{\psi }^{\prime }\left(t\right)(\psi \left(t\right)-\psi \left(t\right))f\left(t\right)\right)\right)\hfill \\ =\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)\left({\int }_a^t{\psi }^{\prime }\left(s\right)f\left(s\right)ds\right)={\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\psi }^{\prime }\left(t\right)f\left(t\right)=f\left(t\right),fora.e.t\in (a,b).\hfill \end{array}$$

□

Lemma 5.

Let $\alpha >0$, $n=\lfloor \alpha \rfloor +1$ if $\alpha \notin \mathbb{N}$, and $n=\alpha$ if $\alpha \in \mathbb{N}$. If $f\in L^1(a,b)$ and $I_{a+}^{n-\alpha ,\psi }f\in A{C}^n[a,b]$, then

$${\displaystyle I_{a+}^{\alpha ,\psi }{D}_{a+}^{\alpha ,\psi }f\left(t\right)=f\left(t\right)-\sum _{j=1}^n\frac{{\mathcal{D}}^{n-j}\left(I_{a+}^{n-\alpha ,\psi }f\right)\left(a\right)}{\Gamma (\alpha -j+1)}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -j},fora.e.t\in (a,b),}$$

where $\mathcal{D}={\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}$.

Proof. 

We find

$$I_{a+}^{\alpha ,\psi }{D}_{a+}^{\alpha ,\psi }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{D}_{a+}^{\alpha ,\psi }f\left(s\right)ds={\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{dA}{dt}}\left(t\right),$$

where

$$A\left(t\right)={\displaystyle \frac{1}{\Gamma (\alpha +1)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }{D}_{a+}^{\alpha ,\psi }f\left(s\right)ds.$$

For $A\left(t\right)$ we obtain

$$\begin{array}{c}A\left(t\right)={\displaystyle \frac{1}{\Gamma (\alpha +1)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^n(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\hfill \\ ={\displaystyle \frac{1}{\Gamma (\alpha +1)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }{\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\hfill \\ ={\displaystyle \frac{1}{\Gamma (\alpha +1)}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }{\displaystyle \frac{d}{ds}}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\hfill \\ ={\displaystyle \frac{1}{\Gamma (\alpha +1)}}\left[{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}^{s=t}\right.\hfill \\ -\left.{\int }_a^t\alpha {(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}(-{\psi }^{\prime }\left(s\right)){\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\right]\hfill \\ ={\displaystyle \frac{1}{\Gamma (\alpha +1)}}\left[0-{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\right.\hfill \\ +\left.\alpha {\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{\psi }^{\prime }\left(s\right){\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\right]\hfill \\ =-{\displaystyle \frac{1}{\Gamma (\alpha +1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\hfill \\ =-{\displaystyle \frac{1}{\Gamma (\alpha +1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\left[{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right))\right]ds\hfill \\ =-{\displaystyle \frac{1}{\Gamma (\alpha +1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left[{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}^{s=t}\right.\hfill \\ -\left.{\int }_a^t(\alpha -1){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -2}(-{\psi }^{\prime }\left(s\right)){\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\right]\hfill \\ =-{\displaystyle \frac{1}{\Gamma (\alpha +1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left[0-{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\right.\hfill \\ +\left.(\alpha -1){\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -2}{\psi }^{\prime }\left(s\right){\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\right]\hfill \\ =-{\displaystyle \frac{1}{\Gamma (\alpha +1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ +{\displaystyle \frac{1}{\Gamma (\alpha -1)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -2}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds.\hfill \end{array}$$

We continue with these computations, and we find

$$A\left(t\right)=B\left(t\right)+{\displaystyle \frac{1}{\Gamma (\alpha -n+2)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -n+1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^1(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds,$$

where

$$\begin{array}{c}B\left(t\right)=-{\displaystyle \frac{1}{\Gamma (\alpha +1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -\dots -{\displaystyle \frac{1}{\Gamma (\alpha -n+3)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n+2}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^1(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}.\hfill \end{array}$$

So we deduce

$$\begin{array}{c}A\left(t\right)=B\left(t\right)+{\displaystyle \frac{1}{\Gamma (\alpha -n+2)}}\left[{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -n+1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}^{s=t}\right.\hfill \\ \left.-{\int }_a^t(\alpha -n+1){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -n}(-{\psi }^{\prime }\left(s\right))(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\right]\hfill \\ =B\left(t\right)+{\displaystyle \frac{1}{\Gamma (\alpha -n+2)}}\left[0-{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n+1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\right.\hfill \\ +\left.{\int }_a^t(\alpha -n+1){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -n}{\psi }^{\prime }\left(s\right)(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\right]\hfill \\ =B\left(t\right)-{\displaystyle \frac{1}{\Gamma (\alpha -n+2)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n+1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ +{\displaystyle \frac{1}{\Gamma (\alpha -n+1)}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -n}{\psi }^{\prime }\left(s\right)(I_{a+}^{n-\alpha ,\psi }f\left(s\right))ds\hfill \\ =B\left(t\right)-{\displaystyle \frac{1}{\Gamma (\alpha -n+2)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n+1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}+I_{a+}^{\alpha -n+1,\psi }(I_{a+}^{n-\alpha ,\psi }f\left(t\right))\hfill \\ =B\left(t\right)-{\displaystyle \frac{1}{\Gamma (\alpha -n+2)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n+1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}+I_{a+}^{1,\psi }f\left(t\right).\hfill \end{array}$$

Therefore, we conclude

$$\begin{array}{c}{I}_{a+}^{\alpha ,\psi }{D}_{a+}^{\alpha ,\psi }f\left(t\right)={\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{dA}{dt}}\left(t\right)\hfill \\ ={\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\left[-{\displaystyle \frac{1}{\Gamma (\alpha +1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\right.\hfill \\ -{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -\dots -{\displaystyle \frac{1}{\Gamma (\alpha -n+3)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n+2}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^1(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -\left.{\displaystyle \frac{1}{\Gamma (\alpha -n+2)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n+1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}+I_{a+}^{1,\psi }f\left(t\right)\right]\hfill \\ =-{\displaystyle \frac{\alpha }{\Gamma (\alpha +1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -{\displaystyle \frac{\alpha -1}{\Gamma \left(\alpha \right)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -2}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -\dots -{\displaystyle \frac{\alpha -n+2}{\Gamma (\alpha -n+3)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n+1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^1(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -{\displaystyle \frac{\alpha -n+1}{\Gamma (\alpha -n+2)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^0(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ +\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)(I_{a+}^{1,\psi }f\left(t\right))\hfill \\ =-{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-1}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -{\displaystyle \frac{1}{\Gamma (\alpha -1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -2}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n-2}(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -\dots -{\displaystyle \frac{1}{\Gamma (\alpha -n+2)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n+1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^1(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ -{\displaystyle \frac{1}{\Gamma (\alpha -n+1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -n}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^0(I_{a+}^{n-\alpha ,\psi }f\left(s\right)){|}_{s=a}\hfill \\ +\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)\left({\displaystyle \frac{1}{\Gamma \left(1\right)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{0}f\left(s\right)ds\right)\hfill \\ =-\sum _{j=1}^n{\displaystyle \frac{{\mathcal{D}}^{n-j}(I_{a+}^{n-\alpha ,\psi }f)\left(a\right)}{\Gamma (\alpha -j+1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -j}+{\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\left({\int }_a^t{\psi }^{\prime }\left(s\right)f\left(s\right)ds\right)\hfill \\ =-\sum _{j=1}^n{\displaystyle \frac{{\mathcal{D}}^{n-j}(I_{a+}^{n-\alpha ,\psi }f)\left(a\right)}{\Gamma (\alpha -j+1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha -j}+f\left(t\right).\hfill \end{array}$$

□

In what follows we will present the linear problem associated with our problem (1)–(2), its solution, and the corresponding Green function with its properties. We consider the fractional differential equation

$$D_{a+}^{\alpha ,\psi }\mathfrak{u}\left(\mathfrak{t}\right)+\mathfrak{K}\left(\mathfrak{t}\right)=0,\mathfrak{t}\in (a,b),$$

(9)

with the boundary conditions (2), where $\mathfrak{K}\in C(a,b)\cap L^1(a,b)$. We denote by

$${\displaystyle \Delta =\frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha -\varsigma )}{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}-\sum _{i=1}^m\frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha -{\varrho }_i)}{\int }_a^b{(\psi \left(s\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(s\right).}$$

(10)

Lemma 6.

If $\Delta \ne 0$, then the solution of problem (9)–(2) is

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)=-\frac{1}{\Gamma \left(\alpha \right)}{\int }_a^{\mathfrak{t}}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle +\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\left[{\displaystyle \frac{1}{\Gamma (\alpha -\varsigma )}{\int }_a^b{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right.}\hfill \\ -\left.{\displaystyle \sum _{i=1}^m\frac{1}{\Gamma (\alpha -{\varrho }_i)}{\int }_a^b\left({\displaystyle {\int }_a^{\mathfrak{s}}{\psi }^{\prime }\left(\tau \right){(\psi \left(\mathfrak{s}\right)-\psi \left(\tau \right))}^{\alpha -{\varrho }_i-1}\mathfrak{K}\left(\tau \right)d\tau }\right)d{\mathcal{H}}_i\left(\mathfrak{s}\right)}\right],\mathfrak{t}\in [a,b].\hfill \end{array}$$

(11)

Proof. 

By Lemma 5, the solutions of Equation (9) are

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)=-I_{a+}^{\alpha ,\psi }\mathfrak{K}\left(\mathfrak{t}\right)+\sum _{j=1}^n{c}_j{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -j}}\hfill \\ {\displaystyle =-\frac{1}{\Gamma \left(\alpha \right)}{\int }_a^{\mathfrak{t}}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}+\sum _{j=1}^n{c}_j{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -j},\mathfrak{t}\in [a,b],}\hfill \end{array}$$

(12)

with $c_i\in \mathbb{R},i=1,\dots ,n$.

Because $\mathfrak{u}\left(a\right)={\mathfrak{u}}^{\prime }\left(a\right)=\cdots ={\mathfrak{u}}^{(n-2)}\left(a\right)=0$, then we find $c_n=c_{n-1}=\cdots =c_2=0$. So Equation (12) gives us

$$\begin{array}{c}\mathfrak{u}\left(\mathfrak{t}\right)=-I_{a+}^{\alpha ,\psi }\mathfrak{K}\left(\mathfrak{t}\right)+c_1{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}\hfill \\ {\displaystyle =-\frac{1}{\Gamma \left(\alpha \right)}{\int }_a^{\mathfrak{t}}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}+c_1{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1},\mathfrak{t}\in [a,b].}\hfill \end{array}$$

(13)

We compute now the $\psi$-fractional derivatives of function $\mathfrak{u}$ given by (13), and we obtain

$$\begin{array}{c}{D}_{a+}^{k,\psi }\mathfrak{u}\left(\mathfrak{t}\right)=-D_{a+}^{k,\psi }{I}_{a+}^{\alpha ,\psi }\mathfrak{K}\left(\mathfrak{t}\right)+c_1{D}_{a+}^{k,\psi }{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}\hfill \\ {\displaystyle =-I_{a+}^{\alpha -k,\psi }\mathfrak{K}\left(\mathfrak{t}\right)+c_1\frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha -k)}{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -k-1},\mathfrak{t}\in (a,b),}\hfill \end{array}$$

for $k=\varsigma ,{\varrho }_i,i=1,\dots ,m$.

Therefore, the condition $D_{a+}^{\varsigma ,\psi }\mathfrak{u}\left(b\right)={\sum }_{i=1}^m{\int }_a^b{D}_{a+}^{{\varrho }_i,\psi }\mathfrak{u}\left(\mathfrak{s}\right)d{\mathcal{H}}_i\left(\mathfrak{s}\right)$ gives us

$$\begin{array}{c}{\displaystyle -I_{a+}^{\alpha -\varsigma ,\psi }\mathfrak{K}\left(b\right)+c_1\frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha -\varsigma )}{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}}\hfill \\ {\displaystyle =\sum _{i=1}^m{\int }_a^b\left[{\displaystyle -I_{a+}^{\alpha -{\varrho }_i,\psi }\mathfrak{K}\left(\mathfrak{s}\right)+c_1\frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha -{\varrho }_i)}{(\psi \left(\mathfrak{s}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}}\right]d{\mathcal{H}}_i\left(\mathfrak{s}\right).}\hfill \end{array}$$

Then we deduce

$$\begin{array}{c}{c}_1\left[{\displaystyle \frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha -\varsigma )}{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}-\sum _{i=1}^m\frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha -{\varrho }_i)}{\int }_a^b{(\psi \left(\mathfrak{s}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(\mathfrak{s}\right)}\right]\hfill \\ {\displaystyle =I_{a+}^{\alpha -\varsigma ,\psi }\mathfrak{K}\left(b\right)-\sum _{i=1}^m{\int }_a^b{I}_{a+}^{\alpha -{\varrho }_i,\psi }\mathfrak{K}\left(\mathfrak{s}\right)d{\mathcal{H}}_i\left(\mathfrak{s}\right),}\hfill \end{array}$$

and so

$$\begin{array}{c}{\displaystyle c_1=\frac{1}{\Delta \Gamma (\alpha -\varsigma )}{\int }_a^b{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle -\frac{1}{\Delta }\sum _{i=1}^m{\int }_a^b\frac{1}{\Gamma (\alpha -{\varrho }_i)}\left({\displaystyle {\int }_a^{\mathfrak{s}}{\psi }^{\prime }\left(\tau \right){(\psi \left(\mathfrak{s}\right)-\psi \left(\tau \right))}^{\alpha -{\varrho }_i-1}\mathfrak{K}\left(\tau \right)d\tau }\right)d{\mathcal{H}}_i\left(s\right),}\hfill \end{array}$$

where $\Delta$ is given by (10). By replacing $c_1$ above in relation (13), we obtain the solution $\mathfrak{u}$ of problem (9)–(2) given by (11). □

Lemma 7.

If $\Delta \ne 0$, then the solution of problem (9)–(2) given by (11) can be written as

$${\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)={\int }_a^b\mathfrak{G}(\mathfrak{t},\mathfrak{s})\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s},\mathfrak{t}\in [a,b],}$$

(14)

where

$${\displaystyle \mathfrak{G}(\mathfrak{t},\mathfrak{s})={\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})+\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\sum _{i=1}^m\left({\displaystyle {\int }_a^b{\mathfrak{g}}_{2i}(\tau ,\mathfrak{s})d{\mathcal{H}}_i\left(\tau \right)}\right),\mathfrak{t},\mathfrak{s}\in [a,b],}$$

(15)

and

$$\begin{array}{c}{\displaystyle {\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})=\frac{1}{\Gamma \left(\alpha \right)}\left\{\begin{array}{c}{\psi }^{\prime }\left(\mathfrak{s}\right)\left[{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}-{(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -1}\right],\hfill \\ a\le \mathfrak{s}\le \mathfrak{t}\le b,\hfill \\ {\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1},a\le \mathfrak{t}\le \mathfrak{s}\le b,\hfill \end{array}\right.}\hfill \\ {\displaystyle {\mathfrak{g}}_{2i}(\tau ,\mathfrak{s})=\frac{1}{\Gamma (\alpha -{\varrho }_i)}\left\{\begin{array}{c}{\psi }^{\prime }\left(\mathfrak{s}\right)\left[{(\psi \left(\tau \right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}\right.\hfill \\ -{(\psi \left(\tau \right)-\psi \left(\mathfrak{s}\right))}^{\alpha -{\varrho }_i-1}],a\le \mathfrak{s}\le \tau \le b,\hfill \\ {\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\tau \right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1},\hfill \\ a\le \tau \le \mathfrak{s}\le b,\hfill \end{array}\right.}\hfill \end{array}$$

(16)

for $i=1,\dots ,m$.

Proof. 

By Lemma 6 and relation (11), we deduce

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)=\frac{1}{\Gamma \left(\alpha \right){(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}}\left\{{\displaystyle {\int }_a^{\mathfrak{t}}\left[{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\right.}\right.}\hfill \\ \left.-{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}{(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -1}{\psi }^{\prime }\left(\mathfrak{s}\right)\right]\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}\hfill \\ +\left.{\displaystyle {\int }_{\mathfrak{t}}^b{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right\}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\alpha \right){(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}}{\int }_a^b{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle +\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\left[{\displaystyle \frac{1}{\Gamma (\alpha -\varsigma )}{\int }_a^b{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right.}\hfill \\ \left.{\displaystyle -\sum _{i=1}^m\frac{1}{\Gamma (\alpha -{\varrho }_i)}{\int }_a^b\left({\displaystyle {\int }_a^{\mathfrak{s}}{\psi }^{\prime }\left(\tau \right){(\psi \left(\mathfrak{s}\right)-\psi \left(\tau \right))}^{\alpha -{\varrho }_i-1}\mathfrak{K}\left(\tau \right)d\tau }\right)d{\mathcal{H}}_i\left(\mathfrak{s}\right)}\right]\hfill \\ {\displaystyle ={\int }_a^b{\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}\times \frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha -\varsigma )\Delta }\left({\displaystyle {\int }_a^b{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)}\hfill \\ {\displaystyle +\frac{1}{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}}\times \frac{1}{\Gamma \left(\alpha \right)\Delta }\left({\displaystyle \sum _{i=1}^m\frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha -{\varrho }_i)}{\int }_a^b{(\psi \left(\mathfrak{s}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(\mathfrak{s}\right)}\right)}\hfill \\ \times \left({\displaystyle {\int }_a^b{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\hfill \\ {\displaystyle +\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\left[{\displaystyle \frac{1}{\Gamma (\alpha -\varsigma )}{\int }_a^b{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right.}\hfill \\ -\left.{\displaystyle \sum _{i=1}^m\frac{1}{\Gamma (\alpha -{\varrho }_i)}{\int }_a^b\left({\displaystyle {\int }_a^{\mathfrak{s}}{\psi }^{\prime }\left(\tau \right){(\psi \left(\mathfrak{s}\right)-\psi \left(\tau \right))}^{\alpha -{\varrho }_i-1}\mathfrak{K}\left(\tau \right)d\tau }\right)d{\mathcal{H}}_i\left(\mathfrak{s}\right)}\right]\hfill \\ {\displaystyle ={\int }_a^b{\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}+\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\left[{\displaystyle \frac{1}{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}}}\right.}\hfill \\ \times \left({\displaystyle \sum _{i=1}^m\frac{1}{\Gamma (\alpha -{\varrho }_i)}{\int }_a^b{(\psi \left(\mathfrak{s}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(\mathfrak{s}\right)}\right)\hfill \\ \times \left({\displaystyle {\int }_a^b{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\hfill \\ -\left.{\displaystyle \sum _{i=1}^m\frac{1}{\Gamma (\alpha -{\varrho }_i)}{\int }_a^b\left({\displaystyle {\int }_a^{\mathfrak{s}}{\psi }^{\prime }\left(\tau \right){(\psi \left(\mathfrak{s}\right)-\psi \left(\tau \right))}^{\alpha -{\varrho }_i-1}\mathfrak{K}\left(\tau \right)d\tau }\right)d{\mathcal{H}}_i\left(\mathfrak{s}\right)}\right].\hfill \end{array}$$

Then we obtain

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)={\int }_a^b{\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}+\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }}\hfill \\ {\displaystyle \times \sum _{i=1}^m\frac{1}{\Gamma (\alpha -{\varrho }_i)}\left[\left({\displaystyle {\int }_a^b\frac{1}{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}}{(\psi \left(\tau \right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(\tau \right)}\right)\right.}\hfill \\ \times \left({\displaystyle {\int }_a^b{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\hfill \\ -\left.{\displaystyle {\int }_a^b\left({\displaystyle {\int }_{\tau }^b{(\psi \left(\mathfrak{s}\right)-\psi \left(\tau \right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(\mathfrak{s}\right)}\right){\psi }^{\prime }\left(\tau \right)\mathfrak{K}\left(\tau \right)d\tau }\right]\hfill \\ {\displaystyle ={\int }_a^b{\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}+\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }}\hfill \\ {\displaystyle \times \sum _{i=1}^m\frac{1}{\Gamma (\alpha -{\varrho }_i)}\left[\left({\displaystyle {\int }_a^b\frac{1}{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}}{(\psi \left(\tau \right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(\tau \right)}\right)\right.}\hfill \\ \times \left({\displaystyle {\int }_a^b{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\hfill \\ \left.{\displaystyle -{\int }_a^b\left({\displaystyle {\int }_{\mathfrak{s}}^b{(\psi \left(\tau \right)-\psi \left(\mathfrak{s}\right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(\tau \right)}\right){\psi }^{\prime }\left(\mathfrak{s}\right)\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right]\hfill \\ {\displaystyle ={\int }_a^b{\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}+\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }}\hfill \\ {\displaystyle \times \sum _{i=1}^m\frac{1}{\Gamma (\alpha -{\varrho }_i)}\left[{\displaystyle {\int }_a^b\left({\displaystyle {\int }_a^b\frac{1}{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -\varsigma -1}}{(\psi \left(\tau \right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}{\psi }^{\prime }\left(\mathfrak{s}\right)}\right.}\right.}\hfill \\ \times {(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}d{\mathcal{H}}_i\left(\tau \right))\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}\hfill \\ -\left.{\displaystyle {\int }_a^b\left({\displaystyle {\int }_{\mathfrak{s}}^b{(\psi \left(\tau \right)-\psi \left(\mathfrak{s}\right))}^{\alpha -{\varrho }_i-1}{\psi }^{\prime }\left(\mathfrak{s}\right)d{\mathcal{H}}_i\left(\tau \right)}\right)\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\right]\hfill \\ {\displaystyle ={\int }_a^b{\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}+\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\sum _{i=1}^m{\int }_a^b\left({\displaystyle {\int }_a^b{\mathfrak{g}}_{2i}(\tau ,\mathfrak{s})d{\mathcal{H}}_i\left(\tau \right)}\right)\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}.}\hfill \end{array}$$

Therefore, we find

$${\displaystyle u\left(\mathfrak{t}\right)={\int }_a^b\mathfrak{G}(\mathfrak{t},\mathfrak{s})\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s},\mathfrak{t}\in [a,b],}$$

where $\mathfrak{G}$, $\mathfrak{g}$, ${\mathfrak{g}}_{2i},i=1,\dots ,m$ are given by (15) and (16). So we obtain formula (14) for the solution $\mathfrak{u}$ of problem (9)–(2). □

Lemma 8.

The functions ${\mathfrak{g}}_1,{\mathfrak{g}}_{2i},i=1,\dots ,m$ have the following properties for all $\mathfrak{t},\mathfrak{s}\in [a,b]$:

(a) ${\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\le {\mathfrak{h}}_1\left(\mathfrak{s}\right),$ where

$${\mathfrak{h}}_1\left(\mathfrak{s}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}[{(\psi \left(b\right)-\psi \left(a\right))}^{\varsigma }-{(\psi \left(b\right)-\psi \left(s\right))}^{\varsigma }],\mathfrak{s}\in [a,b].$$

(b) ${\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\mathfrak{h}}_1\left(\mathfrak{s}\right),$ where

$$\gamma \left(\mathfrak{t}\right)={\displaystyle \frac{\psi \left(\mathfrak{t}\right)-\psi \left(a\right)}{\psi \left(b\right)-\psi \left(a\right)}},\mathfrak{t}\in [a,b].$$

(c) ${\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\le {\displaystyle \frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma \left(\alpha \right)}}{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}.$

(d) ${\mathfrak{g}}_{2i}(\mathfrak{t},\mathfrak{s})\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -{\varrho }_i-1}{\mathfrak{h}}_{2i}\left(\mathfrak{s}\right),$ where

$${\mathfrak{h}}_{2i}\left(\mathfrak{s}\right)={\displaystyle \frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma (\alpha -{\varrho }_i)}}{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\left[{(\psi \left(b\right)-\psi \left(a\right))}^{\varsigma -{\varrho }_i}-{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\varsigma -{\varrho }_i}\right],\mathfrak{s}\in [a,b],$$

(17)

for $i=1,\dots ,m.$

(e) ${\mathfrak{g}}_{2i}(\mathfrak{t},\mathfrak{s})\le {\displaystyle \frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma (\alpha -{\varrho }_i)}}{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1},i=1,\dots ,m$.

(f) ${\mathfrak{g}}_1,{\mathfrak{g}}_{2i},i=1,\dots ,m$ are continuous on $[a,b]\times [a,b]$; ${\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\ge 0,{\mathfrak{g}}_{2i}(\mathfrak{t},\mathfrak{s})\ge 0$ for all $\mathfrak{t},\mathfrak{s}\in [a,b],$ ${\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})>0,{\mathfrak{g}}_{2i}(\mathfrak{t},\mathfrak{s})>0$, for all $\mathfrak{t},\mathfrak{s}\in (a,b),i=1,\dots ,m$.

Proof. 

(a) The function ${\mathfrak{g}}_1$ is nondecreasing in the first variable. Indeed, for $\mathfrak{s}\le \mathfrak{t}$, we have

$$\begin{array}{c}{\displaystyle \frac{\partial {\mathfrak{g}}_1}{\partial \mathfrak{t}}(\mathfrak{t},\mathfrak{s})=\frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma \left(\alpha \right)}\left[(\alpha -1){\psi }^{\prime }\left(\mathfrak{t}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -2}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}\right.}\hfill \\ -(\alpha -1){\psi }^{\prime }\left(\mathfrak{t}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -2}]\hfill \\ {\displaystyle =\frac{{\psi }^{\prime }\left(\mathfrak{s}\right){\psi }^{\prime }\left(\mathfrak{t}\right)}{\Gamma (\alpha -1)}\left[{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -2}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}-{(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -2}\right]}\hfill \\ {\displaystyle \ge \frac{{\psi }^{\prime }\left(\mathfrak{s}\right){\psi }^{\prime }\left(\mathfrak{t}\right)}{\Gamma (\alpha -1)}\left[{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -2}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -2}-{(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -2}\right]}\hfill \\ {\displaystyle =\frac{{\psi }^{\prime }\left(\mathfrak{s}\right){\psi }^{\prime }\left(\mathfrak{t}\right)}{\Gamma (\alpha -1)}\left[{\left({\displaystyle \frac{\left(\psi \right(\mathfrak{t})-\psi (a\left)\right)\left(\psi \right(b)-\psi (\mathfrak{s}\left)\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -2}-{(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -2}\right]\ge 0.}\hfill \end{array}$$

The last inequality is true because

$$\begin{array}{c}{\displaystyle \frac{\left(\psi \right(\mathfrak{t})-\psi (a\left)\right)\left(\psi \right(b)-\psi (\mathfrak{s}\left)\right)}{\psi \left(b\right)-\psi \left(a\right)}\ge \psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right)}\hfill \\ \iff \left(\psi \right(\mathfrak{t})-\psi (a\left)\right)\left(\psi \right(b)-\psi (\mathfrak{s}\left)\right)\ge \left(\psi \right(b)-\psi (a\left)\right)\left(\psi \right(\mathfrak{t})-\psi (\mathfrak{s}\left)\right)\hfill \\ \iff \psi \left(\mathfrak{t}\right)\psi \left(b\right)-\psi \left(a\right)\psi \left(b\right)-\psi \left(\mathfrak{t}\right)\psi \left(\mathfrak{s}\right)+\psi \left(a\right)\psi \left(\mathfrak{s}\right)\hfill \\ \ge \psi \left(b\right)\psi \left(\mathfrak{t}\right)-\psi \left(a\right)\psi \left(\mathfrak{t}\right)-\psi \left(b\right)\psi \left(\mathfrak{s}\right)+\psi \left(a\right)\psi \left(\mathfrak{s}\right)\hfill \\ \iff -\psi \left(a\right)\left(\psi \right(b)-\psi (\mathfrak{t}\left)\right)+\psi \left(\mathfrak{s}\right)(-\psi (\mathfrak{t})+\psi (b\left)\right)\ge 0\hfill \\ \iff \left(\psi \right(b)-\psi (\mathfrak{t}\left)\right)\left(\psi \right(\mathfrak{s})-\psi (a\left)\right)\ge 0,\mathfrak{t},\mathfrak{s}\in [a,b].\hfill \end{array}$$

(18)

Hence ${\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\le {\mathfrak{g}}_1(b,\mathfrak{s})$ for all $\mathfrak{t},\mathfrak{s}\in [a,b]$ with $\mathfrak{s}\le \mathfrak{t}.$

For $\mathfrak{s}\ge \mathfrak{t}$, we obtain

$${\displaystyle \frac{\partial {\mathfrak{g}}_1}{\partial \mathfrak{t}}(\mathfrak{t},\mathfrak{s})={\psi }^{\prime }\left(\mathfrak{s}\right)(\alpha -1){\psi }^{\prime }\left(\mathfrak{t}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -2}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}\ge 0.}$$

So ${\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\le {\mathfrak{g}}_1(\mathfrak{s},\mathfrak{s})$ for all $\mathfrak{t},\mathfrak{s}\in [a,b]$ with $\mathfrak{s}\ge \mathfrak{t}$.

Therefore, we infer that ${\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\le {\mathfrak{h}}_1\left(\mathfrak{s}\right)$ for all $\mathfrak{t},\mathfrak{s}\in [a,b]$, where

$$\begin{array}{c}{\mathfrak{h}}_1\left(s\right)={\mathfrak{g}}_1(b,\mathfrak{s})\hfill \\ {\displaystyle =\frac{1}{\Gamma \left(\alpha \right)}{\psi }^{\prime }\left(\mathfrak{s}\right)\left[{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}-{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -1}\right]}\hfill \\ {\displaystyle =\frac{1}{\Gamma \left(\alpha \right)}{\psi }^{\prime }\left(\mathfrak{s}\right)\left[{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}{(\psi \left(b\right)-\psi \left(a\right))}^{\varsigma }-{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -1}\right]}\hfill \\ {\displaystyle =\frac{1}{\Gamma \left(\alpha \right)}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\left[{(\psi \left(b\right)-\psi \left(a\right))}^{\varsigma }-{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\varsigma }\right]\ge 0.}\hfill \end{array}$$

(b) For $\mathfrak{s}\le \mathfrak{t}$, by also using (18), we have

$$\begin{array}{c}{\displaystyle {\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})=\frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma \left(\alpha \right)}\left[{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}-{(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -1}\right]}\hfill \\ {\displaystyle \ge \frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma \left(\alpha \right)}\left[{\displaystyle {(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}-\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -1}}{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -1}}}\right]}\hfill \\ {\displaystyle =\frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma \left(\alpha \right)}{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}\left[{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}-{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -1}\right]}\hfill \\ {\displaystyle =\frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma \left(\alpha \right)}{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}\left(1-{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\varsigma }\right)}\hfill \\ {\displaystyle =\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\psi }^{\prime }\left(\mathfrak{s}\right)}{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -1}\Gamma \left(\alpha \right)}{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}({(\psi \left(b\right)-\psi \left(a\right))}^{\varsigma }-{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\varsigma })}\hfill \\ ={\left({\displaystyle \frac{\psi \left(\mathfrak{t}\right)-\psi \left(a\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -1}{\mathfrak{h}}_1\left(\mathfrak{s}\right)={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\mathfrak{h}}_1\left(\mathfrak{s}\right).\hfill \end{array}$$

For $\mathfrak{s}\ge \mathfrak{t}$, we find

$$\begin{array}{c}{\displaystyle {\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})=\frac{1}{\Gamma \left(\alpha \right)}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}}\hfill \\ {\displaystyle \ge \frac{1}{\Gamma \left(\alpha \right)}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}\left(1-{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\varsigma }\right)}\hfill \\ {\displaystyle ={\left({\displaystyle \frac{\psi \left(\mathfrak{t}\right)-\psi \left(a\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -1}\frac{1}{\Gamma \left(\alpha \right)}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}\left[{(\psi \left(b\right)-\psi \left(a\right))}^{\varsigma }-{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\varsigma }\right]}\hfill \\ ={\left({\displaystyle \frac{\psi \left(\mathfrak{t}\right)-\psi \left(a\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -1}{\mathfrak{h}}_1\left(\mathfrak{s}\right)={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\mathfrak{h}}_1\left(\mathfrak{s}\right).\hfill \end{array}$$

Therefore, we deduce that ${\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\mathfrak{h}}_1\left(\mathfrak{s}\right)$, for all $\mathfrak{t},\mathfrak{s}\in [a,b]$.

(c) For all $\mathfrak{t},\mathfrak{s}\in [a,b]$, we have

$$\begin{array}{c}{\displaystyle {\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})\le \frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma \left(\alpha \right)}{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}\le \frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma \left(\alpha \right)}{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}.}\hfill \end{array}$$

(d) For $\mathfrak{s}\le \mathfrak{t}$, we obtain

$$\begin{array}{c}{\displaystyle {\mathfrak{g}}_{2i}(\mathfrak{t},\mathfrak{s})=\frac{1}{\Gamma (\alpha -{\varrho }_i)}{\psi }^{\prime }\left(\mathfrak{s}\right)\left[{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}\right.}\hfill \\ -{(\psi \left(\mathfrak{t}\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -{\varrho }_i-1}]\hfill \\ {\displaystyle \ge \frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma (\alpha -{\varrho }_i)}\left[{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}\right.}\hfill \\ -\left.{\displaystyle \frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -{\varrho }_i-1}}{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}}}\right]\hfill \\ {\displaystyle =\frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma (\alpha -{\varrho }_i)}{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}\left[{\displaystyle {\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(s\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}-\frac{{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -{\varrho }_i-1}}{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}}}\right]}\hfill \\ {\displaystyle =\frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma (\alpha -{\varrho }_i)}{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}\left(1-{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\varsigma -{\varrho }_i}\right)}\hfill \\ {\displaystyle =\frac{{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}}{\Gamma (\alpha -{\varrho }_i){(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}}{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}}\hfill \\ \times \left({(\psi \left(b\right)-\psi \left(a\right))}^{\varsigma -{\varrho }_i}-{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\varsigma -{\varrho }_i}\right)\hfill \\ ={\left({\displaystyle \frac{\psi \left(\mathfrak{t}\right)-\psi \left(a\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -{\varrho }_i-1}{\mathfrak{h}}_{2i}\left(\mathfrak{s}\right)={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -{\varrho }_i-1}{\mathfrak{h}}_{2i}\left(\mathfrak{s}\right),\hfill \end{array}$$

where ${\mathfrak{h}}_{2i}\left(\mathfrak{s}\right)$ is given by (17).

If $\mathfrak{t}\le \mathfrak{s}$, we infer

$$\begin{array}{c}{\displaystyle {\mathfrak{g}}_{2i}(\mathfrak{t},\mathfrak{s})=\frac{1}{\Gamma (\alpha -{\varrho }_i)}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}}\hfill \\ {\displaystyle \ge \frac{1}{\Gamma (\alpha -{\varrho }_i)}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}\left(1-{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\varsigma -{\varrho }_i}\right)}\hfill \end{array}$$

$$\begin{array}{c}{\displaystyle ={\left({\displaystyle \frac{\psi \left(\mathfrak{t}\right)-\psi \left(a\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -{\varrho }_i-1}\frac{1}{\Gamma (\alpha -{\varrho }_i)}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\alpha -\varsigma -1}}\hfill \\ \times \left[{(\psi \left(b\right)-\psi \left(a\right))}^{\varsigma -{\varrho }_i}-{(\psi \left(b\right)-\psi \left(\mathfrak{s}\right))}^{\varsigma -{\varrho }_i}\right]\hfill \\ ={\left({\displaystyle \frac{\psi \left(\mathfrak{t}\right)-\psi \left(a\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -{\varrho }_i-1}{\mathfrak{h}}_{2i}\left(\mathfrak{s}\right)={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -{\varrho }_i-1}{\mathfrak{h}}_{2i}\left(\mathfrak{s}\right).\hfill \end{array}$$

Hence, we deduce $g_{2i}(\mathfrak{t},\mathfrak{s})\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -{\varrho }_i-1}{\mathfrak{h}}_{2i}\left(\mathfrak{s}\right)$, for all $\mathfrak{t},\mathfrak{s}\in [a,b]$.

(e) We have

$$\begin{array}{c}{\displaystyle {\mathfrak{g}}_{2i}(\mathfrak{t},\mathfrak{s})\le \frac{1}{\Gamma (\alpha -{\varrho }_i)}{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \frac{\psi \left(b\right)-\psi \left(\mathfrak{s}\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -\varsigma -1}}\hfill \\ {\displaystyle \le \frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma (\alpha -{\varrho }_i)}{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1},\forall \mathfrak{t},\mathfrak{s}\in [a,b].}\hfill \end{array}$$

(f) This property follows from the definition of functions ${\mathfrak{g}}_1$ and ${\mathfrak{g}}_{2i},i=1,\dots ,m$, and from the properties (b) and (d) above. □

Lemma 9.

Assume that $\Delta >0$ and ${\mathcal{H}}_i:[a,b]\to \mathbb{R}$, $i=1,\dots ,m$ are nondecreasing functions. Then the function $\mathfrak{G}$ given by (15) is a continuous function on $[a,b]\times [a,b]$ and satisfies the following conditions.

(a) $\mathfrak{G}(\mathfrak{t},\mathfrak{s})\le \mathfrak{J}\left(\mathfrak{s}\right),$ for all $\mathfrak{t},\mathfrak{s}\in [a,b]$, where

$${\displaystyle \mathfrak{J}\left(\mathfrak{s}\right)={\mathfrak{h}}_1\left(\mathfrak{s}\right)+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\sum _{i=1}^m\left({\displaystyle {\int }_a^b{\mathfrak{g}}_{2i}(\tau ,\mathfrak{s})d{\mathcal{H}}_i\left(\tau \right)}\right),\mathfrak{s}\in [a,b].}$$

(b) $\mathfrak{G}(\mathfrak{t},\mathfrak{s})\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\mathfrak{J}\left(\mathfrak{s}\right),$ for all $\mathfrak{t},\mathfrak{s}\in [a,b]$.

(c) $\mathfrak{G}(\mathfrak{t},\mathfrak{s})\le \theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1},$ for all $\mathfrak{t},\mathfrak{s}\in [a,b]$, where

$$\begin{array}{c}\theta =C_0{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -1}\hfill \\ \times \left[{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}+\frac{1}{\Delta }\sum _{i=1}^m\left({\displaystyle {\int }_a^b\frac{1}{\Gamma (\alpha -{\varrho }_i)}{(\psi \left(\tau \right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(\tau \right)}\right)}\right]>0,\hfill \end{array}$$

(19)

with $C_0={sup}_{\mathfrak{s}\in [a,b]}{\psi }^{\prime }\left(\mathfrak{s}\right)$.

Proof. 

By the definition of function $\mathfrak{G}$, we deduce that $\mathfrak{G}$ is a continuous function. In addition, by using Lemma 8, we obtain the following for all $\mathfrak{t},\mathfrak{s}\in [a,b]$:

(a)

$$\begin{array}{c}{\displaystyle \mathfrak{G}(\mathfrak{t},\mathfrak{s})={\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})+\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\sum _{i=1}^m\left({\displaystyle {\int }_a^b{\mathfrak{g}}_{2i}(\tau ,\mathfrak{s})d{\mathcal{H}}_i\left(\tau \right)}\right)}\hfill \\ {\displaystyle \le {\mathfrak{h}}_1\left(\mathfrak{s}\right)+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\sum _{i=1}^m\left({\displaystyle {\int }_a^b{\mathfrak{g}}_{2i}(\tau ,\mathfrak{s})d{\mathcal{H}}_i\left(\tau \right)}\right)=\mathfrak{J}\left(\mathfrak{s}\right).}\hfill \end{array}$$

(b)

$$\begin{array}{c}{\displaystyle \mathfrak{G}(\mathfrak{t},\mathfrak{s})\ge {\left({\displaystyle \frac{\psi \left(\mathfrak{t}\right)-\psi \left(a\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -1}{\mathfrak{h}}_1\left(\mathfrak{s}\right)+\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\sum _{i=1}^m\left({\displaystyle {\int }_a^b{\mathfrak{g}}_{2i}(\tau ,\mathfrak{s})d{\mathcal{H}}_i\left(\tau \right)}\right)}\hfill \\ ={\left({\displaystyle \frac{\psi \left(\mathfrak{t}\right)-\psi \left(a\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -1}\left[{\displaystyle {\mathfrak{h}}_1\left(\mathfrak{s}\right)+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\sum _{i=1}^m\left({\displaystyle {\int }_a^b{\mathfrak{g}}_{2i}(\tau ,\mathfrak{s})d{\mathcal{H}}_i\left(\tau \right)}\right)}\right]\hfill \\ ={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\mathfrak{J}\left(\mathfrak{s}\right).\hfill \end{array}$$

(c)

$$\begin{array}{c}{\displaystyle \mathfrak{G}(\mathfrak{t},\mathfrak{s})\le \frac{{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}+\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }\sum _{i=1}^m\left({\displaystyle {\int }_a^b{\mathfrak{g}}_{2i}(\tau ,\mathfrak{s})d{\mathcal{H}}_i\left(\tau \right)}\right)}\hfill \\ {\displaystyle \le \frac{{\psi }^{\prime }\left(\mathfrak{s}\right){(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}+\frac{{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}}{\Delta }}\hfill \\ {\displaystyle \times \sum _{i=1}^m\left({\displaystyle {\int }_a^b\frac{{\psi }^{\prime }\left(\mathfrak{s}\right)}{\Gamma (\alpha -{\varrho }_i)}{(\psi \left(\tau \right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(\tau \right)}\right)}\hfill \\ \le C_0{(\psi \left(\mathfrak{t}\right)-\psi \left(a\right))}^{\alpha -1}\left[{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}+\frac{1}{\Delta }\sum _{i=1}^m\left({\displaystyle {\int }_a^b\frac{1}{\Gamma (\alpha -{\varrho }_i)}{(\psi \left(\tau \right)-\psi \left(a\right))}^{\alpha -{\varrho }_i-1}d{\mathcal{H}}_i\left(\tau \right)}\right)}\right]\hfill \\ =\theta {\left({\displaystyle \frac{\psi \left(\mathfrak{t}\right)-\psi \left(a\right)}{\psi \left(b\right)-\psi \left(a\right)}}\right)}^{\alpha -1}=\theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1},\hfill \end{array}$$

where $C_0={sup}_{\mathfrak{s}\in [a,b]}{\psi }^{\prime }\left(\mathfrak{s}\right)$ and $\theta$ is given by (19). □

Lemma 10.

Assume that $\Delta >0$, ${\mathcal{H}}_i:[a,b]\to \mathbb{R}$, $i=1,\dots ,m$ are nondecreasing functions, and $\mathfrak{K}\left(\mathfrak{t}\right)\ge 0$ for all $\mathfrak{t}\in (a,b)$. Then the solution $\mathfrak{u}$ of problem (9)–(2) given by (14) satisfies the inequality $\mathfrak{u}\left(\mathfrak{t}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\mathfrak{u}\left(\zeta \right)$ for all $\mathfrak{t},\zeta \in [a,b]$.

Proof. 

By Lemma 9, we obtain the following for all $\mathfrak{t},\zeta \in [a,b]$:

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)={\int }_a^b\mathfrak{G}(\mathfrak{t},\mathfrak{s})\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}\ge {\int }_a^b{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\mathfrak{J}\left(\mathfrak{s}\right)\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle ={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{G}(\zeta ,\mathfrak{s})\mathfrak{K}\left(\mathfrak{s}\right)d\mathfrak{s}={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\mathfrak{u}\left(\zeta \right).}\hfill \end{array}$$

□

In Section 3 we will use the Guo–Krasnosel’skii fixed-point theorem presented below (see [49]).

Theorem 1.

Let $\mathfrak{X}$ be a Banach space and let $\mathfrak{P}\subset \mathfrak{X}$ be a cone in $\mathfrak{X}$. Assume ${\mathrm{Y}}_1$ and ${\mathrm{Y}}_2$ are bounded open subsets of $\mathfrak{X}$ with $0\in {\mathrm{Y}}_1\subset \overline{{\mathrm{Y}}_1}\subset {\mathrm{Y}}_2$ and let $\mathfrak{L}:\mathfrak{P}\cap (\overline{{\mathrm{Y}}_2}\setminus {\mathrm{Y}}_1)\to \mathfrak{P}$ be a completely continuous operator such that either

(i) $\parallel \mathfrak{L}\mathfrak{u}\parallel \le \parallel \mathfrak{u}\parallel ,\mathfrak{u}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_1$, and $\parallel \mathfrak{L}\mathfrak{u}\parallel \ge \parallel \mathfrak{u}\parallel ,\mathfrak{u}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_2$;

(ii) or $\parallel \mathfrak{L}\mathfrak{u}\parallel \ge \parallel \mathfrak{u}\parallel ,\mathfrak{u}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_1,$ and $\parallel \mathfrak{L}\mathfrak{u}\parallel \le \parallel \mathfrak{u}\parallel ,\mathfrak{u}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_2$.

Then $\mathfrak{L}$ has a fixed point in $\mathfrak{P}\cap (\overline{{\mathrm{Y}}_2}\setminus {\mathrm{Y}}_1)$.

3. Existence of Positive Solutions

In this section we examine the existence of positive solutions to our problem (1)–(2). Below we present the assumptions that we will use in the sequel.

(A1)

$\alpha >0$, $\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n\ge 3$, $\psi \in C^n\left([a,b]\right)$ with ${\psi }^{\prime }\left(\mathfrak{t}\right)>0$ for all $\mathfrak{t}\in [a,b]$, $1\le \varsigma <\alpha -1$, $0\le {\varrho }_i\le \varsigma$ for all $i=1,\dots ,m$, $\lambda >0$, ${\mathcal{H}}_i:[a,b]\to \mathbb{R}$, $i=1,\dots ,m$ are nondecreasing functions, and $\Delta >0$ (given by (10)).

(A2)

The function $\mathfrak{f}\in C((a,b)\times {\mathbb{R}}_{+},\mathbb{R})$ may be singular at $\mathfrak{t}=a$ and/or $\mathfrak{t}=b$, and there exist the functions $\mathfrak{p},\mathfrak{q}\in C((a,b),{\mathbb{R}}_{+})$, $\mathfrak{w}\in C([a,b]\times {\mathbb{R}}_{+},{\mathbb{R}}_{+})$ such that

$$-\mathfrak{q}\left(\mathfrak{t}\right)\le \mathfrak{f}(\mathfrak{t},\mathfrak{y})\le \mathfrak{p}\left(\mathfrak{t}\right)\mathfrak{w}(\mathfrak{t},\mathfrak{y}),\forall \mathfrak{t}\in (a,b),\mathfrak{y}\in {\mathbb{R}}_{+},$$

with $0<{\int }_a^b\mathfrak{q}\left(\mathfrak{t}\right)d\mathfrak{t}<\infty$ and $0<{\int }_a^b\mathfrak{p}\left(\mathfrak{t}\right)d\mathfrak{t}<\infty$, (${\mathbb{R}}_{+}=[0,\infty )$).

(A3)

There exist $c,d\in (a,b)$, $c<d$ such that ${\displaystyle {\mathfrak{f}}_{\infty }=\underset{\mathfrak{y}\to \infty ,\mathfrak{t}\in [c,d]}{lim}\frac{\mathfrak{f}(\mathfrak{t},\mathfrak{y})}{\mathfrak{y}}=\infty .}$

(A4)

There exist $\tilde{c},\tilde{d}\in (a,b)$, $\tilde{c}<\tilde{d}$ such that ${\displaystyle {lim\; inf}_{\mathfrak{y}\to \infty }\underset{\mathfrak{t}\in [\tilde{c},\tilde{d}]}{min}\mathfrak{f}(\mathfrak{t},\mathfrak{y})>{\Lambda }_0,}$ with

$${\Lambda }_0=\left({\displaystyle 2\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right){\left({\displaystyle {\left(\gamma \left(\tilde{c}\right)\right)}^{\alpha -1}{\int }_{\tilde{c}}^{\tilde{d}}\mathfrak{J}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)}^{-1},$$

and ${\displaystyle {\mathfrak{w}}_{\infty }=\underset{\mathfrak{y}\to \infty }{lim}\underset{\mathfrak{t}\in [a,b]}{max}\frac{\mathfrak{w}(\mathfrak{t},\mathfrak{y})}{\mathfrak{y}}=0}$, where $\mathfrak{J}$ and $\theta$ are given in Lemma 9.

We consider the fractional differential equation

$$D_{a+}^{\alpha ,\psi }\mathfrak{v}\left(\mathfrak{t}\right)+\lambda (\mathfrak{f}(\mathfrak{t},{[\mathfrak{v}\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)]}^{*})+\mathfrak{q}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (a,b),$$

(20)

with the boundary conditions

$${\displaystyle {\mathfrak{v}}^{\left(i\right)}\left(a\right)=0,i=0,1,\dots ,n-2;D_{a+}^{\varsigma ,\psi }\mathfrak{v}\left(b\right)=\sum _{i=1}^m{\int }_a^b{D}_{a+}^{{\varrho }_i,\psi }\mathfrak{v}\left(\mathfrak{s}\right)d{\mathcal{H}}_i\left(\mathfrak{s}\right),}$$

(21)

where $\vartheta {\left(\mathfrak{t}\right)}^{*}=\vartheta \left(\mathfrak{t}\right)$ if $\vartheta \left(\mathfrak{t}\right)\ge 0$, and $\vartheta {\left(\mathfrak{t}\right)}^{*}=0$ if $\vartheta \left(\mathfrak{t}\right)<0$.

Here, $\mathfrak{z}\left(\mathfrak{t}\right)={\int }_a^b\mathfrak{G}(\mathfrak{t},\mathfrak{s})\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s},\mathfrak{t}\in [a,b]$ is the solution of the problem

$$\left\{\begin{array}{c}{D}_{a+}^{\alpha ,\psi }\mathfrak{z}\left(\mathfrak{t}\right)+\mathfrak{q}\left(\mathfrak{t}\right)=0,\mathfrak{t}\in (a,b),\hfill \\ {\displaystyle {\mathfrak{z}}^{\left(i\right)}\left(a\right)=0,i=0,1,\dots ,n-2;D_{a+}^{\varsigma ,\psi }\mathfrak{z}\left(b\right)=\sum _{i=1}^m{\int }_a^b{D}_{a+}^{{\varrho }_i,\psi }\mathfrak{z}\left(\mathfrak{s}\right)d{\mathcal{H}}_i\left(\mathfrak{s}\right).}\hfill \end{array}\right.$$

Under assumptions $\left(A1\right)$ and $\left(A2\right)$, we have $\mathfrak{z}\left(\mathfrak{t}\right)\ge 0$ for all $\mathfrak{t}\in [a,b]$. We will show that there exists a solution $\mathfrak{v}$ of problem (20)–(21), with $\mathfrak{v}\left(\mathfrak{t}\right)\ge \lambda \mathfrak{z}\left(\mathfrak{t}\right)$ on $[a,b]$, and $\mathfrak{v}\left(\mathfrak{t}\right)>\lambda \mathfrak{z}\left(\mathfrak{t}\right)$ on $(a,b)$. In this case, $\mathfrak{u}=\mathfrak{v}-\lambda \mathfrak{z}$ represents a positive solution of problem (1)–(2). Hence, in what follows, we will study the problem (20)–(21).

By using Lemma 2, $\mathfrak{v}$ is a solution of problem (20)–(21) if and only if $\mathfrak{v}$ is a solution of equation

$${\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)=\lambda {\int }_a^b\mathfrak{G}(\mathfrak{t},\mathfrak{s})(\mathfrak{f}(\mathfrak{s},{[\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right)]}^{*})+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s},\mathfrak{t}\in [a,b].}$$

(22)

We consider the Banach space $\mathfrak{X}=C[a,b]$ with the supremum norm $\parallel \mathfrak{v}\parallel ={sup}_{\mathfrak{t}\in [a,b]}|\mathfrak{v}\left(\mathfrak{t}\right)|$, and we define the cone

$$\mathfrak{P}=\left\{\mathfrak{v}\in \mathfrak{X},\mathfrak{v}\left(\mathfrak{t}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\parallel \mathfrak{v}\parallel ,\forall \mathfrak{t}\in [a,b]\right\}.$$

For $\lambda >0$, we introduce the operator

$${\displaystyle \mathfrak{L}\mathfrak{v}\left(\mathfrak{t}\right)=\lambda {\int }_a^b\mathfrak{G}(\mathfrak{t},\mathfrak{s})(\mathfrak{f}(\mathfrak{s},{[\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right)]}^{*})+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s},}$$

for $\mathfrak{t}\in [a,b],$ and $\mathfrak{v}\in \mathfrak{X}$.

It is clear that $\mathfrak{v}$ is a solution of Equation (22) (or equivalently of problem (20)–(21)) if and only if $\mathfrak{v}$ is a fixed point of operator $\mathfrak{L}$.

Lemma 11.

If $\left(A1\right)$ and $\left(A2\right)$ hold, then the operator $\mathfrak{L}:\mathfrak{P}\to \mathfrak{P}$ is a completely continuous operator.

Proof. 

Let $\mathfrak{v}\in \mathfrak{P}$ be fixed. By using $\left(A1\right)$ and $\left(A2\right)$, we infer that $\mathfrak{L}\mathfrak{v}\left(\mathfrak{t}\right)<\infty$. In addition, by Lemma 9, we deduce for all $\mathfrak{t},\zeta \in [a,b]$ that

$${\displaystyle \mathfrak{L}\mathfrak{v}\left(\mathfrak{t}\right)\le \lambda {\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)(\mathfrak{f}(\mathfrak{s},{[\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right)]}^{*})+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s},}$$

and

$$\begin{array}{c}{\displaystyle \mathfrak{L}\mathfrak{v}\left(\mathfrak{t}\right)\ge \lambda {\int }_a^b{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\mathfrak{J}\left(\mathfrak{s}\right)(\mathfrak{f}(\mathfrak{s},{[\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right)]}^{*})+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s}\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\mathfrak{L}\mathfrak{v}\left(\zeta \right).}\hfill \end{array}$$

So $\mathfrak{L}\mathfrak{v}\left(\mathfrak{t}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\parallel \mathfrak{L}\mathfrak{v}\parallel$ for all $\mathfrak{t}\in [a,b]$. We conclude that $\mathfrak{L}\mathfrak{v}\in \mathfrak{P}$, and then $\mathfrak{L}\left(\mathfrak{P}\right)\subset \mathfrak{P}$. By using standard arguments, we deduce that the operator $\mathfrak{L}$ is completely continuous (that is, continuous, and maps bounded sets into relatively compact sets). □

Theorem 2.

Assume that $\left(A1\right)$, $\left(A2\right)$ and $\left(A3\right)$ hold. Then there exits ${\lambda }_1>0$ such that for all $\lambda \in (0,{\lambda }_1]$, the boundary value problem (1)–(2) has at least one positive solution $\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{t}\in [a,b]$.

Proof. 

We choose a positive number $r_1>\theta {\int }_a^{b}q\left(\mathfrak{s}\right)d\mathfrak{s}$, and we define the set ${\mathrm{Y}}_1=\{\mathfrak{v}\in \mathfrak{X},\parallel \mathfrak{v}\parallel <r_1\}$. We consider

$${\lambda }_1=min\left\{1,r_1{\left({\displaystyle {\Xi }_1{\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)(\mathfrak{p}\left(\mathfrak{s}\right)+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s}}\right)}^{-1}\right\},$$

with ${\Xi }_1=max\{{max}_{\mathfrak{t}\in [a,b],\mathfrak{y}\in [0,r_1]}\mathfrak{w}(\mathfrak{t},\mathfrak{y}),1\}$.

Let $\lambda \in (0,{\lambda }_1]$. We have

$${\displaystyle \mathfrak{z}\left(\mathfrak{t}\right)={\int }_a^b\mathfrak{G}(\mathfrak{t},\mathfrak{s})\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}\le \theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s},\forall \mathfrak{t}\in [a,b].}$$

Let $\mathfrak{v}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_1$. Then for any $\mathfrak{t}\in [a,b]$ we infer

$${[\mathfrak{v}\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)]}^{*}\le \mathfrak{v}\left(\mathfrak{t}\right)\le \parallel \mathfrak{v}\parallel =r_1,$$

and

$$\begin{array}{c}{\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\parallel \mathfrak{v}\parallel -\lambda \theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ ={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle \parallel \mathfrak{v}\parallel -\lambda \theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle r_1-{\lambda }_1\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\hfill \\ \ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle r_1-\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\ge 0.\hfill \end{array}$$

Therefore we find

$$\begin{array}{c}{\displaystyle \mathfrak{L}\mathfrak{v}\left(\mathfrak{t}\right)\le \lambda {\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)(\mathfrak{p}\left(\mathfrak{s}\right)\mathfrak{w}(\mathfrak{s},{[\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right)]}^{*})+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s}}\hfill \\ {\displaystyle \le {\lambda }_1{\Xi }_1{\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)(\mathfrak{p}\left(\mathfrak{s}\right)+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s}\le r_1=\parallel \mathfrak{v}\parallel ,\forall \mathfrak{t}\in [a,b].}\hfill \end{array}$$

So we deduce

$$\parallel \mathfrak{L}\mathfrak{v}\parallel \le \parallel \mathfrak{v}\parallel ,\forall \mathfrak{v}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_1.$$

(23)

Next, for c and d given by assumption $\left(A3\right)$, we choose a constant $M_0>0$ such that

$$M_0\ge 2{\left({\displaystyle \lambda {\left(\gamma \left(c\right)\right)}^{2(\alpha -1)}{\int }_c^d\mathfrak{J}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)}^{-1}.$$

In addition, by $\left(A3\right)$, we deduce that there exists a constant $M_1>0$ such that

$$\mathfrak{f}(\mathfrak{t},\mathfrak{y})\ge M_0\mathfrak{y},\forall \mathfrak{t}\in [c,d],\mathfrak{y}\ge M_1.$$

(24)

Now we define $r_2=max\{2r_1,2M_1{\left(\gamma \left(c\right)\right)}^{1-\alpha }\}$, and let ${\mathrm{Y}}_2=\{\mathfrak{v}\in \mathfrak{X},\parallel \mathfrak{v}\parallel <r_2\}$.

Let $\mathfrak{v}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_2$. Then we obtain

$$\begin{array}{c}{\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\parallel \mathfrak{v}\parallel -\lambda \theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ \ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle r_2-\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle r_1-\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\ge 0,\forall \mathfrak{t}\in [a,b].\hfill \end{array}$$

Therefore we find

$$\begin{array}{c}{[\mathfrak{v}\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)]}^{*}=\mathfrak{v}\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle r_2-\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\hfill \\ {\displaystyle \ge {\left(\gamma \left(c\right)\right)}^{\alpha -1}\frac{r_2}{2}\ge M_1,\forall \mathfrak{t}\in [c,d].}\hfill \end{array}$$

(25)

Then by (24) and (25), we deduce

$$\begin{array}{c}{\displaystyle \mathfrak{L}\mathfrak{v}\left(\mathfrak{t}\right)\ge \lambda {\int }_c^d\mathfrak{G}(\mathfrak{t},\mathfrak{s})(\mathfrak{f}(\mathfrak{s},{[\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right)]}^{*})+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s}}\hfill \\ {\displaystyle \ge \lambda {\int }_c^d\mathfrak{G}(\mathfrak{t},\mathfrak{s})M_0{[\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right)]}^{*}d\mathfrak{s}=\lambda M_0{\int }_c^d\mathfrak{G}(\mathfrak{t},\mathfrak{s})(\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right))d\mathfrak{s}}\hfill \\ {\displaystyle \ge \lambda M_0{\int }_c^d{\left(\gamma \left(\mathfrak{s}\right)\right)}^{\alpha -1}\mathfrak{J}\left(\mathfrak{s}\right){\left(\gamma \left(c\right)\right)}^{\alpha -1}\frac{r_2}{2}d\mathfrak{s}}\hfill \\ {\displaystyle \ge \lambda M_0\frac{r_2}{2}{\left(\gamma \left(c\right)\right)}^{2(\alpha -1)}{\int }_c^d\mathfrak{J}\left(\mathfrak{s}\right)d\mathfrak{s}\ge r_2=\parallel \mathfrak{v}\parallel ,\forall \mathfrak{t}\in [c,d].}\hfill \end{array}$$

So we conclude

$$\parallel \mathfrak{L}\mathfrak{v}\parallel \ge \parallel \mathfrak{v}\parallel ,\forall \mathfrak{v}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_2.$$

(26)

Then, by (23), (26) and Theorem 1 (i), we deduce that operator $\mathfrak{L}$ has a fixed point ${\mathfrak{v}}_1\in \mathfrak{P}\cap (\overline{{\mathrm{Y}}_2}\setminus {\mathrm{Y}}_1)$, that is, $r_1\le \parallel {\mathfrak{v}}_1\parallel \le r_2$.

Because $\parallel {\mathfrak{v}}_1\parallel \ge r_1$, we find

$$\begin{array}{c}{\displaystyle {\mathfrak{v}}_1\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\parallel {\mathfrak{v}}_1\parallel -\lambda \theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ ={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle \parallel {\mathfrak{v}}_1\parallel -\lambda \theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\hfill \\ \ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle r_1-\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)={\Theta }_1{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1},\forall \mathfrak{t}\in [a,b],\hfill \end{array}$$

and so ${\mathfrak{v}}_1\left(\mathfrak{t}\right)\ge \lambda \mathfrak{z}\left(\mathfrak{t}\right)+{\Theta }_1{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}$ for all $\mathfrak{t}\in [a,b]$, where ${\Theta }_1=r_1-\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}>0$. Let ${\mathfrak{u}}_1\left(\mathfrak{t}\right)={\mathfrak{v}}_1\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)$ for all $\mathfrak{t}\in [a,b]$. Then, ${\mathfrak{u}}_1$ is a positive solution of problem (1)–(2) with ${\mathfrak{u}}_1\left(\mathfrak{t}\right)\ge {\Theta }_1{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}$ for all $\mathfrak{t}\in [a,b]$. □

Theorem 3.

Suppose that $\left(A1\right)$, $\left(A2\right)$, and $\left(A4\right)$ hold. Then there exists ${\lambda }_2>0$ such that for any $\lambda \ge {\lambda }_2$, the boundary value problem (1)–(2) has at least one positive solution $\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{t}\in [a,b]$.

Proof. 

By $\left(A4\right)$, there exists $M_2>0$ such that

$$\mathfrak{f}(\mathfrak{t},\mathfrak{y})\ge {\Lambda }_0,\forall \mathfrak{t}\in [\tilde{c},\tilde{d}],\mathfrak{y}\ge M_2.$$

We define

$${\lambda }_2=M_2{\left({\displaystyle {\left(\gamma \left(\tilde{c}\right)\right)}^{\alpha -1}\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)}^{-1}.$$

Let $\lambda \ge {\lambda }_2$. We consider $r_3=2\lambda \theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}$, and ${\mathrm{Y}}_3=\{\mathfrak{v}\in \mathfrak{X},\parallel \mathfrak{v}\parallel \le r_3\}$. Let $\mathfrak{v}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_3$. Then we deduce

$$\begin{array}{c}{\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\parallel \mathfrak{v}\parallel -\lambda \theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle ={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle r_3-\lambda \theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\lambda \theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle \ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\lambda }_2\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}=\frac{{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{M}_2}{{\left(\gamma \left(\tilde{c}\right)\right)}^{\alpha -1}}\ge 0,\forall \mathfrak{t}\in [a,b].}\hfill \end{array}$$

Therefore, we find

$${\displaystyle {[\mathfrak{v}\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)]}^{*}=\mathfrak{v}\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)\ge \frac{{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{M}_2}{{\left(\gamma \left(\tilde{c}\right)\right)}^{\alpha -1}}\ge M_2,\forall \mathfrak{t}\in [\tilde{c},\tilde{d}].}$$

So we conclude

$$\begin{array}{c}{\displaystyle \mathfrak{L}\mathfrak{v}\left(\mathfrak{t}\right)\ge \lambda {\int }_{\tilde{c}}^{\tilde{d}}\mathfrak{G}(\mathfrak{t},\mathfrak{s})\mathfrak{f}(\mathfrak{s},{[\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right)]}^{*})d\mathfrak{s}}\hfill \\ {\displaystyle \ge \lambda {\int }_{\tilde{c}}^{\tilde{d}}{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\mathfrak{J}\left(\mathfrak{s}\right)\mathfrak{f}(\mathfrak{s},\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right))d\mathfrak{s}\ge \lambda {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_{\tilde{c}}^{\tilde{d}}\mathfrak{J}\left(\mathfrak{s}\right){\Lambda }_{0}d\mathfrak{s}}\hfill \\ {\displaystyle =\lambda {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_{\tilde{c}}^{\tilde{d}}\mathfrak{J}\left(\mathfrak{s}\right)d\mathfrak{s}\left({\displaystyle 2\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right){\left({\displaystyle {\left(\gamma \left(\tilde{c}\right)\right)}^{\alpha -1}{\int }_{\tilde{c}}^{\tilde{d}}\mathfrak{J}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)}^{-1}}\hfill \\ {\displaystyle =\frac{{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}}{{\left(\gamma \left(\tilde{c}\right)\right)}^{\alpha -1}}{r}_3\ge r_3,\forall \mathfrak{t}\in [\tilde{c},\tilde{d}].}\hfill \end{array}$$

Hence we obtain

$$\parallel \mathfrak{L}\mathfrak{v}\parallel \ge \parallel \mathfrak{v}\parallel ,\forall \mathfrak{v}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_3.$$

(27)

On the other hand, we consider the positive number $ϵ={\left(2\lambda {\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)\mathfrak{p}\left(\mathfrak{s}\right)d\mathfrak{s}\right)}^{-1}$. Then by $\left(A4\right)$, we deduce that there exists $M_3>0$ such that

$$\mathfrak{w}(\mathfrak{t},\mathfrak{y})\le ϵ\mathfrak{y},\forall \mathfrak{t}\in [a,b],\mathfrak{y}\ge M_3.$$

Then we find $\mathfrak{w}(\mathfrak{t},\mathfrak{y})\le M_4+ϵ\mathfrak{y}$, for all $\mathfrak{t}\in [a,b]$ and $\mathfrak{y}\ge 0$, where $M_4={max}_{\mathfrak{t}\in [a,b],\mathfrak{y}\in [0,M_3]}\mathfrak{w}(\mathfrak{t},\mathfrak{y})$.

We now define $r_4>max\{r_3,2\lambda max\{M_4,1\}{\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)(\mathfrak{p}\left(\mathfrak{s}\right)+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s}\}$, and ${\mathrm{Y}}_4=\{\mathfrak{v}\in \mathfrak{X},\parallel \mathfrak{v}\parallel <r_4\}$.

Let $\mathfrak{v}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_4$. Therefore, we obtain

$$\begin{array}{c}{\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\parallel \mathfrak{v}\parallel -\lambda \theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ ={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle r_4-\lambda \theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle r_3-\lambda \theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)\hfill \\ {\displaystyle ={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\lambda \theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}\ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\lambda }_2\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}=\frac{M_2{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}}{{\left(\gamma \left(\tilde{c}\right)\right)}^{\alpha -1}}\ge 0,\forall \mathfrak{t}\in [a,b].}\hfill \end{array}$$

Then we find

$$\begin{array}{c}{\displaystyle \mathfrak{L}\mathfrak{v}\left(\mathfrak{t}\right)\le \lambda {\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)[\mathfrak{p}\left(\mathfrak{s}\right)\mathfrak{w}(\mathfrak{s},{[\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right)]}^{*})+\mathfrak{q}\left(\mathfrak{s}\right)]d\mathfrak{s}}\hfill \\ {\displaystyle \le \lambda {\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)[\mathfrak{p}\left(\mathfrak{s}\right)(M_4+ϵ(\mathfrak{v}\left(\mathfrak{s}\right)-\lambda \mathfrak{z}\left(\mathfrak{s}\right)))+\mathfrak{q}\left(\mathfrak{s}\right)]d\mathfrak{s}}\hfill \\ {\displaystyle \le \lambda max\{M_4,1\}{\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)(\mathfrak{p}\left(\mathfrak{s}\right)+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s}+\lambda ϵr_4{\int }_a^b\mathfrak{J}\left(\mathfrak{s}\right)\mathfrak{p}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle \le \frac{r_4}{2}+\frac{r_4}{2}=r_4=\parallel \mathfrak{v}\parallel ,\forall \mathfrak{t}\in [a,b].}\hfill \end{array}$$

Hence we obtain

$$\parallel \mathfrak{L}\mathfrak{v}\parallel \le \parallel \mathfrak{v}\parallel ,\forall \mathfrak{v}\in \mathfrak{P}\cap \partial {\mathrm{Y}}_4.$$

(28)

By (27), (28) and Theorem 1 (ii), we deduce that $\mathfrak{L}$ has a fixed point ${\mathfrak{v}}_2\in \mathfrak{P}\cap (\overline{{\mathrm{Y}}_4}\setminus {\mathrm{Y}}_3)$, so $r_3\le \parallel {\mathfrak{v}}_2\parallel \le r_4.$

In addition, we find

$$\begin{array}{c}{\displaystyle {\mathfrak{v}}_2\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)\ge {\mathfrak{v}}_2\left(\mathfrak{t}\right)-\lambda \theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle \ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\parallel {\mathfrak{v}}_2\parallel -\lambda \theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle \ge {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{r}_3-\lambda \theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle ={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\left({\displaystyle r_3-\lambda \theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\right)={\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}\lambda \theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle \ge {\lambda }_2\theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}=M_2\frac{1}{{\left(\gamma \left(\tilde{c}\right)\right)}^{\alpha -1}\theta {\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\theta {\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}{\int }_a^b\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}}\hfill \\ {\displaystyle =M_2\frac{{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}}{{\left(\gamma \left(\tilde{c}\right)\right)}^{\alpha -1}},\forall \mathfrak{t}\in [a,b].}\hfill \end{array}$$

We take ${\mathfrak{u}}_2\left(\mathfrak{t}\right)={\mathfrak{v}}_2\left(\mathfrak{t}\right)-\lambda \mathfrak{z}\left(\mathfrak{t}\right)$ for all $\mathfrak{t}\in [a,b]$. Then ${\mathfrak{u}}_2\left(\mathfrak{t}\right)\ge {\Theta }_2{\left(\gamma \left(\mathfrak{t}\right)\right)}^{\alpha -1}$ for all $\mathfrak{t}\in [a,b]$, where ${\Theta }_2=M_2{\left(\gamma \left(\tilde{c}\right)\right)}^{1-\alpha }$. Therefore, we conclude that ${\mathfrak{u}}_2$ is a positive solution of problem (1)–(2). □

By using similar arguments as those from the proof of Theorem 3, we obtain the following theorem.

Theorem 4.

Assume that $\left(A1\right)$ and $\left(A2\right)$ hold, and

• $(\widetilde{A4})$ There exist $\tilde{c},\tilde{d}\in (a,b)$, $\tilde{c}<\tilde{d}$ such that

  $${\displaystyle {\tilde{\mathfrak{f}}}_{\infty }=\underset{\mathfrak{y}\to \infty }{lim}\underset{\mathfrak{t}\in [\tilde{c},\tilde{d}]}{min}\mathfrak{f}(\mathfrak{t},\mathfrak{y})=\infty \mathit{and}{\mathfrak{w}}_{\infty }=\underset{\mathfrak{y}\to \infty }{lim}\underset{\mathfrak{t}\in [a,b]}{max}\frac{\mathfrak{w}(\mathfrak{t},\mathfrak{y})}{\mathfrak{y}}=0.}$$

Then, there exists ${\tilde{\lambda }}_2>0$ such that for any $\lambda \ge {\tilde{\lambda }}_2$, the boundary value problem (1)–(2) has at least one positive solution $\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{t}\in [a,b]$.

4. Examples

Example 1.

Let $\psi \left(\mathfrak{t}\right)=ln\mathfrak{t}$, $\mathfrak{t}\in [a,b]=[1,e]$, $\alpha ={\displaystyle \frac{7}{2}}$, $n=4$, $m=2$, $\varsigma ={\displaystyle \frac{9}{4}}$, ${\varrho }_1={\displaystyle \frac{1}{6}}$, ${\varrho }_2={\displaystyle \frac{8}{5}}$, and

$${\mathcal{H}}_1\left(\mathfrak{s}\right)=\left\{\begin{array}{c}{\left(\mathfrak{s}-{\displaystyle \frac{4}{9}}\right)}^{5/3},\mathfrak{s}\in \left[1,{\displaystyle \frac{19}{12}}\right),\hfill \\ {\left({\displaystyle \frac{41}{36}}\right)}^{5/3},\mathfrak{s}\in \left[{\displaystyle \frac{19}{12}},e\right],\hfill \end{array}\right.{\mathcal{H}}_2\left(\mathfrak{s}\right)=\left\{\begin{array}{c}1,\mathfrak{s}\in [1,2),\hfill \\ {\displaystyle \frac{15}{11}},\mathfrak{s}\in [2,e].\hfill \end{array}\right.$$

We consider the Hadamard fractional differential equation

$$D_{1+}^{7/2,\psi }\mathfrak{u}\left(\mathfrak{t}\right)+\lambda \mathfrak{f}(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (1,e),$$

(29)

with the boundary conditions

$$\left\{\begin{array}{c}\mathfrak{u}\left(1\right)={\mathfrak{u}}^{\prime }\left(1\right)={\mathfrak{u}}^{″}\left(1\right)=0,\hfill \\ {\displaystyle D_{1+}^{9/4,\psi }\mathfrak{u}\left(e\right)=\frac{5}{3}{\int }_1^{19/12}{\left(\mathfrak{s}-\frac{4}{9}\right)}^{2/3}{D}_{1+}^{1/6,\psi }\mathfrak{u}\left(\mathfrak{s}\right)d\mathfrak{s}+\frac{4}{11}{D}_{1+}^{8/5,\psi }\mathfrak{u}\left(2\right).}\hfill \end{array}\right.$$

(30)

We have $\gamma \left(\mathfrak{t}\right)=ln\mathfrak{t}$, $\mathfrak{t}\in [1,e]$, and $\Delta \approx 2.69941187>0$. So assumption $\left(A1\right)$ is satisfied. In addition we find

$$\begin{array}{c}{\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})={\displaystyle \frac{1}{\Gamma (7/2)}}\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathfrak{s}}}\left[{(ln\mathfrak{t})}^{5/2}{(1-ln\mathfrak{s})}^{1/4}-{(ln\mathfrak{t}-ln\mathfrak{s})}^{5/2}\right],1\le \mathfrak{s}\le \mathfrak{t}\le e,\hfill \\ {\displaystyle \frac{1}{\mathfrak{s}}}{(ln\mathfrak{t})}^{5/2}{(1-ln\mathfrak{s})}^{1/4},1\le \mathfrak{t}\le \mathfrak{s}\le e,\hfill \end{array}\right.\hfill \\ {\mathfrak{g}}_{21}(\tau ,\mathfrak{s})={\displaystyle \frac{1}{\Gamma (10/3)}}\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathfrak{s}}}\left[{(ln\tau )}^{7/3}{(1-ln\mathfrak{s})}^{1/4}-{(ln\tau -ln\mathfrak{s})}^{7/3}\right],1\le \mathfrak{s}\le \tau \le e,\hfill \\ {\displaystyle \frac{1}{\mathfrak{s}}}{(ln\tau )}^{7/3}{(1-ln\mathfrak{s})}^{1/4},1\le \tau \le \mathfrak{s}\le e,\hfill \end{array}\right.\hfill \\ {\mathfrak{g}}_{22}(\tau ,\mathfrak{s})={\displaystyle \frac{1}{\Gamma (19/10)}}\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathfrak{s}}}\left[{(ln\tau )}^{9/10}{(1-ln\mathfrak{s})}^{1/4}-{(ln\tau -ln\mathfrak{s})}^{9/10}\right],1\le \mathfrak{s}\le \tau \le e,\hfill \\ {\displaystyle \frac{1}{\mathfrak{s}}}{(ln\tau )}^{9/10}{(1-ln\mathfrak{s})}^{1/4},1\le \tau \le \mathfrak{s}\le e,\hfill \end{array}\right.\hfill \end{array}$$

and

$$\mathfrak{G}(\mathfrak{t},\mathfrak{s})={\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})+{\displaystyle \frac{{(ln\mathfrak{t})}^{5/2}}{\Delta }}\left[{\displaystyle \frac{5}{3}}{\int }_1^{19/12}{\left(\tau -{\displaystyle \frac{4}{9}}\right)}^{2/3}{\mathfrak{g}}_{21}(\tau ,\mathfrak{s})d\mathfrak{s}+{\displaystyle \frac{4}{11}}{\mathfrak{g}}_{22}(2,\mathfrak{s})\right],$$

for all $\mathfrak{t},\mathfrak{s}\in [1,e]$. We also obtain

$$\begin{array}{c}{\mathfrak{h}}_1\left(\mathfrak{s}\right)={\displaystyle \frac{1}{\mathfrak{s}\Gamma (7/2)}}\left[{(1-ln\mathfrak{s})}^{1/4}-{(1-ln\mathfrak{s})}^{5/2}\right],\mathfrak{s}\in [1,e],\hfill \\ {\mathfrak{h}}_{21}\left(\mathfrak{s}\right)={\displaystyle \frac{1}{\mathfrak{s}\Gamma (10/3)}}\left[{(1-ln\mathfrak{s})}^{1/4}-{(1-ln\mathfrak{s})}^{7/3}\right],\mathfrak{s}\in [1,e],\hfill \\ {\mathfrak{h}}_{22}\left(\mathfrak{s}\right)={\displaystyle \frac{1}{\mathfrak{s}\Gamma (19/10)}}\left[{(1-ln\mathfrak{s})}^{1/4}-{(1-ln\mathfrak{s})}^{9/10}\right],\mathfrak{s}\in [1,e].\hfill \end{array}$$

In addition, we find $\theta \approx 0.40870482$, and

$$\begin{array}{c}\mathfrak{J}\left(\mathfrak{s}\right)={\mathfrak{h}}_1\left(\mathfrak{s}\right)+{\displaystyle \frac{1}{\Delta }}\left[{\int }_1^e{\mathfrak{g}}_{21}(\tau ,\mathfrak{s})d{\mathcal{H}}_1\left(\tau \right)+{\int }_1^e{\mathfrak{g}}_{22}(\tau ,\mathfrak{s})d{\mathcal{H}}_2\left(\mathfrak{s}\right)\right]\hfill \\ ={\mathfrak{h}}_1\left(\mathfrak{s}\right)+{\displaystyle \frac{1}{\Delta }}\left[{\displaystyle \frac{5}{3}}{\int }_1^{19/12}{\left(\tau -{\displaystyle \frac{4}{9}}\right)}^{2/3}{\mathfrak{g}}_{21}(\tau ,\mathfrak{s})d\tau +{\displaystyle \frac{4}{11}}{\mathfrak{g}}_{22}(2,\mathfrak{s})\right]\hfill \\ =\left\{\begin{array}{c}{\mathfrak{h}}_1\left(\mathfrak{s}\right)+{\displaystyle \frac{1}{\Delta }}\left\{{\displaystyle \frac{5}{3}}\left[{\int }_1^{\mathfrak{s}}{\left(\tau -{\displaystyle \frac{4}{9}}\right)}^{2/3}{\displaystyle \frac{1}{\mathfrak{s}\Gamma (10/3)}}{(ln\tau )}^{7/3}{(1-ln\mathfrak{s})}^{1/4}d\tau \right.\right.\hfill \\ +\left.{\int }_{\mathfrak{s}}^{19/12}{\left(\tau -{\displaystyle \frac{4}{9}}\right)}^{2/3}{\displaystyle \frac{1}{\mathfrak{s}\Gamma (10/3)}}\left({(ln\tau )}^{7/3}{(1-ln\mathfrak{s})}^{1/4}-{(ln\tau -ln\mathfrak{s})}^{7/3}\right)d\tau \right]\hfill \\ \left.+{\displaystyle \frac{4}{11\mathfrak{s}\Gamma (19/10)}}\left[{(ln2)}^{9/10}{(1-ln\mathfrak{s})}^{1/4}-{(ln2-ln\mathfrak{s})}^{9/10}\right]\right\},1\le \mathfrak{s}<{\displaystyle \frac{19}{12}},\hfill \\ {\mathfrak{h}}_1\left(\mathfrak{s}\right)+{\displaystyle \frac{1}{\Delta }}\left\{{\displaystyle \frac{5}{3}}\left[{\int }_1^{19/12}{\left(\tau -{\displaystyle \frac{4}{9}}\right)}^{2/3}{\displaystyle \frac{1}{\mathfrak{s}\Gamma (10/3)}}{(ln\tau )}^{7/3}{(1-ln\mathfrak{s})}^{1/4}d\tau \right]\right.\hfill \\ \left.+{\displaystyle \frac{4}{11\mathfrak{s}\Gamma (19/10)}}\left[{(ln2)}^{9/10}{(1-ln\mathfrak{s})}^{1/4}-{(ln2-ln\mathfrak{s})}^{9/10}\right]\right\},{\displaystyle \frac{19}{12}}\le \mathfrak{s}<2,\hfill \\ {\mathfrak{h}}_1\left(\mathfrak{s}\right)+{\displaystyle \frac{1}{\Delta }}\left\{{\displaystyle \frac{5}{3}}\left[{\int }_1^{19/12}{\left(\tau -{\displaystyle \frac{4}{9}}\right)}^{2/3}{\displaystyle \frac{1}{\mathfrak{s}\Gamma (10/3)}}{(ln\tau )}^{7/3}{(1-ln\mathfrak{s})}^{1/4}d\tau \right]\right.\hfill \\ \left.+{\displaystyle \frac{4}{11\mathfrak{s}\Gamma (19/10)}}{(ln2)}^{9/10}{(1-ln\mathfrak{s})}^{1/4}\right\},2\le \mathfrak{s}\le e.\hfill \end{array}\right.\hfill \end{array}$$

We consider the function

$${\displaystyle \mathfrak{f}(\mathfrak{t},\mathfrak{u})=\frac{3{\mathfrak{u}}^2+\mathfrak{u}+1}{\sqrt[4]{{(\mathfrak{t}-1)}^3(e-\mathfrak{t})}}-\frac{2}{\sqrt[5]{\mathfrak{t}-1}},\mathfrak{t}\in (1,e),\mathfrak{u}\ge 0.}$$

(31)

We have $\mathfrak{q}\left(\mathfrak{t}\right)={\displaystyle \frac{2}{\sqrt[5]{\mathfrak{t}-1}}}$, $\mathfrak{p}\left(\mathfrak{t}\right)={\displaystyle \frac{1}{\sqrt[4]{{(\mathfrak{t}-1)}^3(e-\mathfrak{t})}}}$ for all $\mathfrak{t}\in (1,e)$, $\mathfrak{w}(\mathfrak{t},\mathfrak{u})=3{\mathfrak{u}}^2+\mathfrak{u}+1$ for all $\mathfrak{t}\in [1,e]$ and $\mathfrak{u}\ge 0$. We obtain ${\int }_1^e\mathfrak{q}\left(\mathfrak{t}\right)d\mathfrak{t}\approx 3.85492139$, and ${\int }_1^e\mathfrak{p}\left(\mathfrak{t}\right)d\mathfrak{t}\approx 4.44288294$. Therefore, assumption $\left(A2\right)$ is satisfied. In addition, for $c=3/2$ and $d=2$, the assumption $\left(A3\right)$ is also satisfied, because ${\mathfrak{f}}_{\infty }=\infty$.

After some computations using the Mathematica program, we deduce that ${\int }_1^e\mathfrak{J}\left(\mathfrak{s}\right)(\mathfrak{p}\left(\mathfrak{s}\right)+\mathfrak{q}\left(\mathfrak{s}\right))d\mathfrak{s}\approx 0.76857453$. We choose $r_1=2$ ($>\theta {\int }_1^e\mathfrak{q}\left(\mathfrak{s}\right)d\mathfrak{s}\approx 1.57552$), and we obtain ${\Xi }_1=15$ and ${\lambda }_1\approx 0.1735$. Then, by Theorem 2 we conclude that for any $\lambda \in (0,{\lambda }_1]$, the problem (29)–(30) with the nonlinearity (31) has at least one positive solution $\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{t}\in [1,e]$.

Example 2.

Let $\psi \left(\mathfrak{t}\right)={\mathfrak{t}}^3,\mathfrak{t}\in [a,b]=[1,5]$, $\alpha ={\displaystyle \frac{8}{3}}$, $n=3$, $m=2$, ${\varrho }_1={\displaystyle \frac{1}{7}}$, ${\varrho }_2={\displaystyle \frac{6}{11}}$, and

$${\mathcal{H}}_1\left(\mathfrak{s}\right)=\left\{\begin{array}{c}2,\mathfrak{s}\in [1,3),\hfill \\ {\displaystyle \frac{51}{25}},\mathfrak{s}\in [3,5],\hfill \end{array}\right.{\mathcal{H}}_2\left(\mathfrak{s}\right)={\left(\mathfrak{s}-{\displaystyle \frac{1}{2}}\right)}^{1/41},\mathfrak{s}\in [1,5].$$

We consider the ψ-fractional differential equation

$$D_{1+}^{8/3,\psi }\mathfrak{u}\left(\mathfrak{t}\right)+\lambda \mathfrak{f}(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (1,5)$$

(32)

with the boundary conditions

$$\left\{\begin{array}{c}\mathfrak{u}\left(1\right)={\mathfrak{u}}^{\prime }\left(1\right)=0,\hfill \\ {\displaystyle D_{1+}^{5/4,\psi }\mathfrak{u}\left(5\right)=\frac{1}{25}{D}_{1+}^{1/7,\psi }\mathfrak{u}\left(3\right)+\frac{1}{41}{\int }_1^5{\left({\displaystyle \mathfrak{s}-\frac{1}{2}}\right)}^{-40/41}{D}_{1+}^{6/11,\psi }\mathfrak{u}\left(\mathfrak{s}\right)d\mathfrak{s}.}\hfill \end{array}\right.$$

(33)

We have $\gamma \left(\mathfrak{t}\right)={\displaystyle \frac{{\mathfrak{t}}^3-1}{124}},\mathfrak{t}\in [1,5]$, and $\Delta \approx 3.61577002>0$. So assumption $\left(A1\right)$ is satisfied. Besides, we find

$$\begin{array}{c}{\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})={\displaystyle \frac{1}{\Gamma (8/3)}}\left\{\begin{array}{c}3{\mathfrak{s}}^2\left[{({\mathfrak{t}}^3-1)}^{5/3}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12}-{({\mathfrak{t}}^3-{\mathfrak{s}}^3)}^{5/3}\right],1\le \mathfrak{s}\le \mathfrak{t}\le 5,\hfill \\ 3{\mathfrak{s}}^2{({\mathfrak{t}}^3-1)}^{5/3}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12},1\le \mathfrak{t}\le \mathfrak{s}\le 5,\hfill \end{array}\right.\hfill \\ {\mathfrak{g}}_{21}(\tau ,\mathfrak{s})={\displaystyle \frac{1}{\Gamma (53/21)}}\left\{\begin{array}{c}3{\mathfrak{s}}^2\left[{({\tau }^3-1)}^{32/21}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12}-{({\tau }^3-{\mathfrak{s}}^3)}^{32/21}\right],1\le \mathfrak{s}\le \tau \le 5,\hfill \\ 3{\mathfrak{s}}^2{({\tau }^3-1)}^{32/21}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12},1\le \tau \le \mathfrak{s}\le 5,\hfill \end{array}\right.\hfill \\ {\mathfrak{g}}_{22}(\tau ,\mathfrak{s})={\displaystyle \frac{1}{\Gamma (70/33)}}\left\{\begin{array}{c}3{\mathfrak{s}}^2\left[{({\tau }^3-1)}^{37/33}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12}-{({\tau }^3-{\mathfrak{s}}^3)}^{37/33}\right],1\le \mathfrak{s}\le \tau \le 5,\hfill \\ 3{\mathfrak{s}}^2{({\tau }^3-1)}^{37/33}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12},1\le \tau \le \mathfrak{s}\le 5,\hfill \end{array}\right.\hfill \end{array}$$

and

$$\mathfrak{G}(\mathfrak{t},\mathfrak{s})={\mathfrak{g}}_1(\mathfrak{t},\mathfrak{s})+{\displaystyle \frac{{({\mathfrak{t}}^3-1)}^{5/3}}{\Delta }}\left[{\displaystyle \frac{1}{25}}{\mathfrak{g}}_{21}(3,\mathfrak{s})+{\displaystyle \frac{1}{41}}{\int }_1^5{\left(\tau -{\displaystyle \frac{1}{2}}\right)}^{-40/41}{\mathfrak{g}}_{22}(\tau ,\mathfrak{s})d\mathfrak{s}\right],$$

for all $\mathfrak{t},\mathfrak{s}\in [1,5].$

We also obtain

$$\begin{array}{c}{\mathfrak{h}}_1\left(\mathfrak{s}\right)={\displaystyle \frac{1}{\Gamma (8/3)}}3{\mathfrak{s}}^2\left[{\left(124\right)}^{5/4}{(125-{\mathfrak{s}}^3)}^{5/12}-{(125-{\mathfrak{s}}^3)}^{5/3}\right],\mathfrak{s}\in [1,5],\hfill \\ {\mathfrak{h}}_{21}\left(\mathfrak{s}\right)={\displaystyle \frac{3{\mathfrak{s}}^2}{\Gamma (53/21)}}\left[{\left(124\right)}^{31/28}{(125-{\mathfrak{s}}^3)}^{5/12}-{(125-{\mathfrak{s}}^3)}^{32/21}\right],\mathfrak{s}\in [1,5],\hfill \\ {\mathfrak{h}}_{22}\left(\mathfrak{s}\right)={\displaystyle \frac{3{\mathfrak{s}}^2}{\Gamma (70/33)}}\left[{\left(124\right)}^{31/44}{(125-{\mathfrak{s}}^3)}^{5/12}-{(125-{\mathfrak{s}}^3)}^{37/33}\right],\mathfrak{s}\in [1,5].\hfill \end{array}$$

In addition, we find $\theta \approx 537632.68981746$, and

$$\begin{array}{c}\mathfrak{J}\left(\mathfrak{s}\right)={\mathfrak{h}}_1\left(\mathfrak{s}\right)+{\displaystyle \frac{{\left(124\right)}^{5/3}}{\Delta }}\left({\int }_1^5{\mathfrak{g}}_{21}(\tau ,\mathfrak{s})d{\mathcal{H}}_1\left(\tau \right)+{\int }_1^5{\mathfrak{g}}_{22}(\tau ,\mathfrak{s})d{\mathcal{H}}_2\left(\tau \right)\right)\hfill \\ ={\mathfrak{h}}_1\left(\mathfrak{s}\right)+{\displaystyle \frac{{\left(124\right)}^{5/3}}{\Delta }}\left({\displaystyle \frac{1}{25}}{\mathfrak{g}}_{21}(3,\mathfrak{s})+{\displaystyle \frac{1}{41}}{\int }_1^5{\left(\tau -{\displaystyle \frac{1}{2}}\right)}^{-40/41}{\mathfrak{g}}_{22}(\tau ,\mathfrak{s})d\tau \right)\hfill \end{array}$$

$$\begin{array}{c}=\left\{\begin{array}{c}{\mathfrak{h}}_1\left(\mathfrak{s}\right)+{\displaystyle \frac{{\left(124\right)}^{5/3}}{\Delta }}\left\{{\displaystyle \frac{3{\mathfrak{s}}^2}{25\Gamma (53/21)}}\left[{\left(26\right)}^{32/21}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12}-{(27-{\mathfrak{s}}^3)}^{32/21}\right]\right.\hfill \\ +{\displaystyle \frac{1}{41\Gamma (70/33)}}{\int }_1^{\mathfrak{s}}3{\mathfrak{s}}^2{({\tau }^3-1)}^{37/33}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12}{\left(\tau -{\displaystyle \frac{1}{2}}\right)}^{-40/41}d\tau \hfill \\ +\left.{\displaystyle \frac{1}{41\Gamma (70/33)}}{\int }_{\mathfrak{s}}^5{\left(\tau -{\displaystyle \frac{1}{2}}\right)}^{-40/41}3{\mathfrak{s}}^2\left[{({\tau }^3-1)}^{37/33}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12}-{({\tau }^3-{\mathfrak{s}}^3)}^{37/33}\right]d\tau \right\},\hfill \\ 1\le \mathfrak{s}<3,\hfill \\ {\mathfrak{h}}_1\left(\mathfrak{s}\right)+{\displaystyle \frac{{\left(124\right)}^{5/3}}{\Delta }}\left\{{\displaystyle \frac{3{\mathfrak{s}}^2}{25\Gamma (53/21)}}{\left(26\right)}^{32/21}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12}\right.\hfill \\ +{\displaystyle \frac{1}{41\Gamma (70/33)}}{\int }_1^{\mathfrak{s}}3{\mathfrak{s}}^2{({\tau }^3-1)}^{37/33}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12}{\left(\tau -{\displaystyle \frac{1}{2}}\right)}^{-40/41}d\tau \hfill \\ +\left.{\displaystyle \frac{1}{41\Gamma (70/33)}}{\int }_{\mathfrak{s}}^5{\left(\tau -{\displaystyle \frac{1}{2}}\right)}^{-40/41}3{\mathfrak{s}}^2\left[{({\tau }^3-1)}^{37/33}{\left({\displaystyle \frac{125-{\mathfrak{s}}^3}{124}}\right)}^{5/12}-{({\tau }^3-{\mathfrak{s}}^3)}^{37/33}\right]d\tau \right\},\hfill \\ 3\le \mathfrak{s}\le 5.\hfill \end{array}\right.\hfill \end{array}$$

We consider the function

$${\displaystyle \mathfrak{f}(\mathfrak{t},\mathfrak{u})=\frac{\sqrt{\mathfrak{u}+1}}{\sqrt[3]{(\mathfrak{t}-1){(5-\mathfrak{t})}^2}}-\frac{1}{\sqrt[4]{{(5-\mathfrak{t})}^3}},\mathfrak{t}\in (1,5),\mathfrak{u}\ge 0.}$$

(34)

Here we have $\mathfrak{q}\left(\mathfrak{t}\right)={\displaystyle \frac{1}{\sqrt[4]{{(5-\mathfrak{t})}^3}}}$, $\mathfrak{p}\left(\mathfrak{t}\right)={\displaystyle \frac{1}{\sqrt[3]{(\mathfrak{t}-1){(5-\mathfrak{t})}^2}}}$ for all $\mathfrak{t}\in (1,5)$, $\mathfrak{w}(\mathfrak{t},\mathfrak{u})=\sqrt{\mathfrak{u}+1}$ for all $\mathfrak{t}\in [1,5]$ and $\mathfrak{u}\ge 0$. We find ${\int }_1^5\mathfrak{q}\left(\mathfrak{t}\right)d\mathfrak{t}\approx 5.65685425$ and ${\int }_1^5\mathfrak{p}\left(\mathfrak{t}\right)d\mathfrak{t}\approx 3.62759873$. So assumption $\left(A2\right)$ is satisfied.

For $\tilde{c}=2$ and $\tilde{d}=4$, the assumption $\left(A4\right)$ is also satisfied, because ${lim\; inf}_{\mathfrak{y}\to \infty }{min}_{\mathfrak{t}\in [2,4]}\mathfrak{f}(\mathfrak{t},\mathfrak{y})=\infty$ and ${\mathfrak{w}}_{\infty }=0$. After some computations by using the Mathematica program, we obtain ${\int }_2^4\mathfrak{J}\left(\mathfrak{s}\right)ds\approx 230960$, and ${\Lambda }_0\approx 3170.25.$ For the above ${\Lambda }_0$, we find that $\mathfrak{f}(\mathfrak{t},\mathfrak{y})\ge {\Lambda }_0$ for all $\mathfrak{t}\in [2,4]$ and $\mathfrak{y}\ge M_2$, with $M_2\approx 4.5037\times {10}^7$. In addition, we deduce ${\lambda }_2\approx 1783$. Therefore, by Theorem 3, we conclude that for any $\lambda \ge {\lambda }_2$, the problem (32)–(33) with the nonlinearity (34) has at least one positive solution $\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{t}\in [1,5]$.

5. Conclusions

In this paper, we investigate the existence of positive solutions to the $\psi$–Riemann–Liouville fractional differential Equation (1), involving a positive parameter and a singular, sign-changing nonlinearity. The equation is complemented by nonlocal boundary conditions (2), which incorporate Riemann–Stieltjes integrals together with $\psi$–Riemann–Liouville fractional derivatives of different orders. We first construct the Green function corresponding to problem (1)–(2) and analyze its fundamental properties. Building on this, we establish our main existence results by applying the Guo–Krasnosel’skii fixed-point theorem. In the future we intend to extend the results obtained in this work to systems of $\psi$-fractional differential equations subject to various coupled nonlocal boundary conditions.