Existence of Positive Solutions for a Class of Nabla Fractional Boundary Value Problems

Abstract

In this manuscript, we study a class of equations with two different Riemann–Liouville-type orders of nabla difference operators. We show some new and fundamental properties of the related Green’s function. Depending on the values of the orders of the operators, we split our research into two main cases, and for each one of them, we obtain suitable conditions under which we prove that the considered problem possesses a positive solution. We consider the latter to be the main novelty in this work. Our main tool in both cases of our study is Guo–Krasnoselskii’s fixed point theorem. In the end, we give particular examples in order to offer a concrete demonstration of our new theoretical findings, as well as some possible future work in this direction.

1. Introduction

Lately, fractional differential and difference equations and their applications have generated much attention [1,2,3,4]. One of the main reasons for this is the fast development of the theories of fractional and discrete fractional calculus, since they are widely used in biology, chemistry, mechanics and medicine. More precisely, some examples of models in environmental science, uncertainty, approximation or control theory, stability analysis, quantum physics or astrophysics, signal and image processing and many others can be found in [5,6,7,8]. Moreover, various real-life models can be modeled by both fractional operators of the Riemann–Liouville type or the Liouville–Caputo type [9,10,11,12,13]. We also refer the reader to some important and some very recent results about both fractional delta difference operators [14,15,16,17] and fractional nabla difference equations [18,19,20,21].

In particular, nabla problems provide a powerful tool for describing the nonlocal memory of different viscoelastic materials. The clear physical background of these studies opens up a new directions of scientific research, including both theoretical analysis and numerical methods. And while in the last few years different solvers for systems of fractional boundary value problems with boundary or initial conditions have been developed, the analysis of the existence and uniqueness of solutions to the fractional difference equations concerning boundary value problems is still paramount in comprehending discrete fractional calculus. Thus, all being said, finding new positive solutions to fractional problems is an undoubtedly important task.

Recently, in [22], the author studied the two-point nabla fractional problem

$$\begin{array}{c}\left({\nabla }_{d_1}^{\alpha }u\right)\left(\varrho \right)+h\left(\varrho \right)=0,1<\alpha <2,\varrho \in {\mathbb{N}}_{d_1+2}^{d_2},\\ u\left(d_1\right)=\left({\nabla }_{d_1}^{\beta }u\right)\left(d_2\right)=0,0\le \beta \le 1\end{array}$$

and managed to obtain a Lyapunov-type inequality for it.

Inspired by this work, we continue this research as we consider

$$\begin{array}{c}-\left({\nabla }_{\rho \left(d_1\right)}^{\nu }u\right)\left(\varrho \right)=\lambda f\left(\varrho \right)g\left(u\left(\varrho \right)\right),\varrho \in {\mathbb{N}}_{d_1+2}^{d_2},\end{array}$$

(1)

$$\begin{array}{c}u\left(d_1\right)=\left({\nabla }_{\rho \left(d_1\right)}^{\mu }u\right)\left(d_2\right)=0,\end{array}$$

(2)

where $f:{\mathbb{N}}_{d_1+2}^{d_2}\to (0,\infty )$ and $g:[0,\infty )\to [0,\infty )$ are continuous functions, while $0\le \mu \le 1<\nu <2$, ${\nabla }_{\rho \left(d_1\right)}^{\nu }$ and ${\nabla }_{\rho \left(d_1\right)}^{\mu }$ are $\nu$th- and $\mu$th-order nabla difference operators of the Riemann–Liouville type, respectively. Here, we use the standard notation ${\mathbb{N}}_e^f=\{e,e+1,e+2,\dots ,f\}$ for any real numbers e and f such that $f-e\in \mathbb{N}$.

Our aim is to extend the findings given in [22]. We shall use some of the results from there and we shall deduce new fundamental properties of the related Green function. Then, using our new findings, we are going to obtain existence and nonexistence results for the considered problem (1) and (). To the best of our knowledge, this has never been carried out so far in the existing literature, which we consider to be the main novelty of this manuscript.

The rest of this paper is structured as follows: In Section 2, we deduce the exact expression and we recall some of the properties of the Green function that is related to the linear problem. Then, we split our study into two main cases depending on the values of $\nu$ and $\mu$. In each one of them, we impose some suitable conditions on the right-hand side of our problem, in order to obtain existence results, presented in Section 3 and Section 4, respectively. To validate our theoretical results, some numerical examples are given in Section 5. Finally, in Section 6, we finish with a summary of our results and some possible future directions for expanding this research.

2. Preliminaries

In this section, we recall some previous results that we will extend in the next sections and a classical theorem, which will be our main tool for establishing our new results. Let us consider the following linear problem:

$$\begin{array}{c}-\left({\nabla }_{\rho \left(d_1\right)}^{\nu }u\right)\left(\varrho \right)=h\left(\varrho \right),\varrho \in {\mathbb{N}}_{d_1+2}^{d_2},\end{array}$$

(3)

$$\begin{array}{c}u\left(d_1\right)=\left({\nabla }_{\rho \left(d_1\right)}^{\mu }u\right)\left(d_2\right)=0.\end{array}$$

(4)

Here, $0\le \mu \le 1<\nu <2$ and $h:{\mathbb{N}}_{d_1+2}^{d_2}\to \mathbb{R}$ are continuous functions.

For any $\alpha$, $\beta \in \mathbb{R}$, we denote the generalized rising function as

$${\alpha }^{\overline{\beta }}={\displaystyle \frac{\mathsf{\Gamma }(\alpha +\beta )}{\mathsf{\Gamma }\left(\alpha \right)}}$$

and for $\nu \in \mathbb{R}\setminus \{\dots ,-2,-1\}$, the $\nu$th-order nabla fractional Taylor monomial is denoted as

$$H_{\nu }(\alpha ,\beta )={\displaystyle \frac{{(\alpha -\beta )}^{\overline{\nu }}}{\mathsf{\Gamma }(\nu +1)}}.$$

Recall the following theorem:

 Theorem 1 

(Theorem 3.9 in [22]). The linear boundary value problem (3) and (4) has a unique solution given in the form

$$u\left(\varrho \right)=\sum _{s=d_1+2}^{d_2}\mathfrak{W}(\varrho ,s)h\left(s\right),\varrho \in {\mathbb{N}}_{d_1}^{d_2},$$

where

$$\mathfrak{W}(\varrho ,s)=\left\{\begin{array}{c}{\mathfrak{W}}_1(\varrho ,s),\varrho \in {\mathbb{N}}_{d_1}^{\rho \left(s\right)},\hfill \\ {\mathfrak{W}}_2(\varrho ,s),\varrho \in {\mathbb{N}}_s^{d_2},\hfill \end{array}\right.$$

(5)

with

$${\mathfrak{W}}_1(\varrho ,s)={\displaystyle \frac{H_{\nu -1}(\varrho ,d_1)H_{\nu -\mu -1}(d_2,\rho \left(s\right))}{H_{\nu -\mu -1}(d_2,d_1)}}$$

and

$${\mathfrak{W}}_2(\varrho ,s)=G_1(\varrho ,s)-H_{\nu -1}(\varrho ,\rho \left(s\right)).$$

Moreover, the maximum of the nonnegative Green function $\mathfrak{W}\left(\varrho ,s\right)$ defined in (5) is given by

$$\underset{\left(\varrho ,s\right)\in {\mathbb{N}}_{d_1}^{d_2}\times {\mathbb{N}}_{d_1+2}^{d_2}}{max}\mathfrak{W}\left(\varrho ,s\right)=\left\{\begin{array}{cc}\underset{s\in {\mathbb{N}}_{d_1+2}^{d_2}}{max}\mathfrak{W}\left(s-1,s\right),& 0\le \mu \le \nu -1,\\ \underset{s\in {\mathbb{N}}_{d_1+2}^{d_2}}{max}\mathfrak{W}\left(s,s\right),& \nu -1<\mu <1.\end{array}\right.$$

Based on the findings in the above theorem, we will split our study of the existence of positive solutions of (1) and (2) into two main cases. Our main tool for both will be the classical Guo–Krasnoselskii fixed point theorem in cones [23].

 Theorem 2. 

Let $B=\left(B,\parallel ·\parallel \right)$ be a Banach space, and let $K\subseteq B$ be a cone. Assume that ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_2$ are bounded open subsets contained in B such that $0 \in$${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_1\subseteq$${\mathsf{\Omega }}_2$. Assume further that $T:K\cap \left(\overline{{\mathsf{\Omega }}_2}\setminus {\mathsf{\Omega }}_1\right)\to K$ is a completely continuous operator. Here, either

(1) $∥Ty∥\le ∥y∥$ for $y\in K\cap \partial {\mathsf{\Omega }}_1$, and $∥Ty∥\ge ∥y∥$ for $y\in K\cap \partial {\mathsf{\Omega }}_2$; or

(2) $∥Ty∥\ge ∥y∥$ for $y\in K\cap \partial {\mathsf{\Omega }}_1$, and $∥Ty∥\le ∥y∥$ for $y\in K\cap \partial {\mathsf{\Omega }}_2$.

Then, T has at least one fixed point in $K\cap \left(\overline{{\mathsf{\Omega }}_2}\setminus {\mathsf{\Omega }}_1\right)$.

3. Case I: $\mathbf{0}\le \mathit{\mu }<\mathit{\nu }-\mathbf{1}$

First, let us study the case $0\le \mu \le \nu -1$. Following the idea given in [14], we have the next result.

 Lemma 1. 

There is ${\gamma }_1\in (0,1)$ such that $\mathfrak{W}(\varrho ,s)$ given in (5) has the following property:

$$\underset{d_1+2\le \varrho \le d_2}{min}\mathfrak{W}\left(\varrho ,s\right)\ge {\gamma }_1\mathfrak{W}\left(s-1,s\right)\mathrm{for}\mathrm{all}s\in {\mathbb{N}}_{d_1+2}^{d_2}.$$

 Proof. 

In this case,

$${\displaystyle \frac{\mathfrak{W}\left(\varrho ,s\right)}{\mathfrak{W}\left(s-1,s\right)}}=\left\{\begin{array}{cc}{\displaystyle \frac{{\left(\varrho -d_1\right)}^{\overline{\nu -1}}}{{\left(s-d_1-1\right)}^{\overline{\nu -1}}},}& \varrho \in {\mathbb{N}}_{d_1}^{\rho \left(s\right)},\\ {\displaystyle \frac{{\left(d_2-s+1\right)}^{\overline{\nu -\mu -1}}{\left(\varrho -d_1\right)}^{\overline{\nu -1}}-{\left(d_2-d_1\right)}^{\overline{\nu -\mu -1}}{\left(t-s+1\right)}^{\overline{\nu -1}}}{{\left(d_2-s+1\right)}^{\overline{\nu -\mu -1}}{\left(s-d_1-1\right)}^{\overline{\nu -1}}},}& \varrho \in {\mathbb{N}}_s^{d_2}.\end{array}\right.$$

For $\varrho \in {\mathbb{N}}_{d_1+1}^{\rho \left(s\right)},$ we find that

$${\displaystyle \frac{\mathfrak{W}\left(\varrho ,s\right)}{\mathfrak{W}\left(s-1,s\right)}}={\displaystyle \frac{{\left(\varrho -d_1\right)}^{\overline{\nu -1}}}{{\left(s-d_1-1\right)}^{\overline{\nu -1}}}}\ge {\displaystyle \frac{2^{\overline{\nu -1}}}{{\left(d_2-d_1-1\right)}^{\overline{\nu -1}}}}.$$

(6)

For $\varrho \in {\mathbb{N}}_s^{d_2},$ we know that $\mathfrak{W}\left(\varrho ,s\right)$ is decreasing with respect to $\varrho$, which implies that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathfrak{W}\left(\varrho ,s\right)}{\mathfrak{W}\left(s-1,s\right)}}& =& {\displaystyle \frac{{\left(d_2-s+1\right)}^{\overline{\nu -\mu -1}}{\left(\varrho -d_1\right)}^{\overline{\nu -1}}-{\left(d_2-d_1\right)}^{\overline{\nu -\mu -1}}{\left(\varrho -s+1\right)}^{\overline{\nu -1}}}{{\left(d_2-s+1\right)}^{\overline{\nu -\mu -1}}{\left(s-d_1-1\right)}^{\overline{\nu -1}}}}\hfill \\ & \ge & {\displaystyle \frac{{\left(d_2-s+1\right)}^{\overline{\nu -\mu -1}}{\left(d_2-d_1\right)}^{\overline{\nu -1}}-{\left(d_2-d_1\right)}^{\overline{\nu -\mu -1}}{\left(d_2-s+1\right)}^{\overline{\nu -1}}}{{\left(d_2-s+1\right)}^{\overline{\nu -\mu -1}}{\left(s-d_1-1\right)}^{\overline{\nu -1}}}}.\hfill \end{array}$$

Denote

$$\phi \left(s\right)={\displaystyle \frac{1}{{\left(s-d_1-1\right)}^{\overline{\nu -1}}}}\left[{\left(d_2-d_1\right)}^{\overline{\nu -1}}-{\displaystyle \frac{{\left(d_2-s+1\right)}^{\overline{\nu -1}}{\left(d_2-d_1\right)}^{\overline{\nu -\mu -1}}}{{\left(d_2-s+1\right)}^{\overline{\nu -\mu -1}}}}\right].$$

One can check that

$${\displaystyle \frac{{\left(d_2-s+1\right)}^{\overline{\nu -1}}{\left(d_2-d_1\right)}^{\overline{\nu -\mu -1}}}{{\left(d_2-s+1\right)}^{\overline{\nu -\mu -1}}}}$$

is decreasing for $d_1+2\le s\le d_2+1.$ Then,

$$\begin{array}{ccc}\hfill \phi \left(s\right)& \ge & {\displaystyle \frac{1}{{\left(s-d_1-1\right)}^{\overline{\nu -1}}}}\left[{\left(d_2-d_1\right)}^{\overline{\nu -1}}-{\displaystyle \frac{{\left(d_2-d_1-1\right)}^{\overline{\nu -1}}{\left(d_2-d_1\right)}^{\overline{\nu -\mu -1}}}{{\left(d_2-d_1-1\right)}^{\overline{\nu -\mu -1}}}}\right]\hfill \\ & \ge & {\displaystyle \frac{1}{{\left(d_2-d_1\right)}^{\overline{\nu -1}}}}\left[{\left(d_2-d_1\right)}^{\overline{\nu -1}}-{\displaystyle \frac{{\left(d_2-d_1-1\right)}^{\overline{\nu -1}}{\left(d_2-d_1\right)}^{\overline{\nu -\mu -1}}}{{\left(d_2-d_1-1\right)}^{\overline{\nu -\mu -1}}}}\right].\hfill \end{array}$$

The last inequality, combined with (6), shows us that for each $s\in {\mathbb{N}}_{d_1+2}^{d_2},$

$$\underset{d_1+2\le \varrho \le d_2}{min}\mathfrak{W}\left(\varrho ,s\right)\ge {\gamma }_1\mathfrak{W}\left(s-1,s\right),$$

where

$${\gamma }_1=min\left\{{\displaystyle \frac{1}{{\left(d_2-d_1\right)}^{\overline{\nu -1}}}}\left[{\left(d_2-d_1\right)}^{\overline{\nu -1}}-{\displaystyle \frac{{\left(d_2-d_1-1\right)}^{\overline{\nu -1}}{\left(d_2-d_1\right)}^{\overline{\nu -\mu -1}}}{{\left(d_2-d_1-1\right)}^{\overline{\nu -\mu -1}}}}\right],{\displaystyle \frac{2^{\overline{\nu -1}}}{{\left(d_2-d_1-1\right)}^{\overline{\nu -1}}}}\right\}.$$

Moreover, it is clear from the above expression that ${\gamma }_1<1$. □

Define the Banach space B by

$$B=\left\{y:{\mathbb{N}}_{d_1+2}^{d_2}\to \mathbb{R}:y\left(d_1\right)=\left({\nabla }_{\rho \left(d_1\right)}^{\mu }y\right)\left(d_2\right)=0\right\},$$

coupled with

$$∥y∥=\underset{\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}}{max}\left|y\left(\varrho \right)\right|.$$

Set the cone

$$K_1=\left\{y\in B,y\left(\varrho \right)\ge 0,\underset{d_1+2\le \varrho \le d_2}{min}y\left(\varrho \right)\ge {\gamma }_1∥y∥,\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}\right\}$$

and the operator $T_{\lambda }:K_1\to B$ by

$$\left(T_{\lambda }y\right)\left(\varrho \right)=\lambda \sum _{s=d_1+2}^{d_2}\mathfrak{W}\left(\varrho ,s\right)f\left(s\right)g\left(y\left(s\right)\right).$$

By using Lemma 1, we have

$$\begin{array}{ccc}\hfill \underset{d_1+2\le \varrho \le d_2}{min}\left(T_{\lambda }y\right)\left(\varrho \right)& \ge & \lambda \sum _{s=d_1+2}^{d_2}\underset{d_1+2\le \varrho \le d_2}{min}\mathfrak{W}\left(\varrho ,s\right)f\left(s\right)g\left(y\left(s\right)\right)\hfill \\ & \ge & \lambda {\gamma }_1\sum _{s=d_1+2}^{d_2}\underset{d_1+2\le \varrho \le d_2}{max}\mathfrak{W}\left(\varrho ,s\right)f\left(s\right)g\left(y\left(s\right)\right)\hfill \\ & \ge & {\gamma }_1∥T_{\lambda }y∥,\hfill \end{array}$$

which gives us $T_{\lambda }:K_1\to K_1$.

3.1. Positive Solutions

Here, we will establish some suitable conditions that will allow us to confirm that (1) and (2) has a positive solution.

Let us assume the following conditions about function g:

(G1) $\underset{y\to 0^{+}}{lim}{\displaystyle \frac{g\left(y\right)}{y}}=0$ and $\underset{y\to +\infty }{lim}{\displaystyle \frac{g\left(y\right)}{y}}=+\infty ;$

(G2) $\underset{y\to 0^{+}}{lim}{\displaystyle \frac{g\left(y\right)}{y}}=+\infty$ and $\underset{y\to +\infty }{lim}{\displaystyle \frac{g\left(y\right)}{y}}=0.$

Define

$${\mathfrak{W}}^{*}=\underset{\left(\varrho ,s\right)\in {\mathbb{N}}_{d_1}^{d_2}\times {\mathbb{N}}_{d_1+2}^{d_2}}{max}\mathfrak{W}\left(\varrho ,s\right)=\underset{s\in {\mathbb{N}}_{d_1+2}^{d_2}}{max}\mathfrak{W}\left(s-1,s\right).$$

Then, for $0\le \mu <\nu -1$,

$${\mathfrak{W}}^{*}=\mathfrak{W}\left(s^{*}-1,s^{*}\right),$$

(7)

where

$$s^{*}=⌊{\displaystyle \frac{(d_2+1)(\nu -1)+(d_1+2)(\nu -\mu -1)}{2(\nu -1)-\mu }}⌋.$$

For $\mu =\nu -1$,

$${\mathfrak{W}}^{*}=\mathfrak{W}\left(d_2-1,d_2\right).$$

(8)

In particular,

$$\mathfrak{W}\left(s-1,s\right)={\displaystyle \frac{H_{\nu -1}(s-1,d_1)H_{\nu -\mu -1}(d_2,\rho \left(s\right))}{H_{\nu -\mu -1}(d_2,d_1)}}={\displaystyle \frac{\mathsf{\Psi }\left(s\right)\mathsf{\Gamma }(d_2-d_1)}{\mathsf{\Gamma }\left(\nu \right)\mathsf{\Gamma }(d_2-d_1+\nu -\mu -1)}},$$

with

$$\mathsf{\Psi }\left(s\right)={\displaystyle \frac{\mathsf{\Gamma }(s-d_1+\nu -2)\mathsf{\Gamma }(d_2-s+\nu -\mu )}{\mathsf{\Gamma }(d_2-s+1)\mathsf{\Gamma }(s-d_1-1)}}.$$

Consider, for $s\in {\mathbb{N}}_{d_1+3}^{d_2}$,

$$\left(\nabla \mathsf{\Psi }\right)\left(s\right)={\displaystyle \frac{\left((1-\nu +\mu )(s-d_1-2)+(\nu -1)(d_2-s+1)\right)\mathsf{\Gamma }(s-d_1+\nu -3)\mathsf{\Gamma }(d_2-s+\nu -\mu )}{\mathsf{\Gamma }(d_2-s+2)\mathsf{\Gamma }(s-d_1-1)}}.$$

Then, the equation $(\nabla \mathsf{\Psi })\left(s\right)=0$ has a unique solution, and so we set $s^{*}$ as the critical point of $\mathsf{\Psi }$. If $s\le s^{*}$, the term

$${\displaystyle \frac{(d_2+1)(\nu -1)+(d_1+2)(\nu -\mu -1)}{2(\nu -1)-\mu }}$$

is positive, and thus $\mathsf{\Psi }$ is increasing. On the other hand, if $s\ge s^{*}$, the same term is negative, and thus $\mathsf{\Psi }$ is decreasing. As a result, we obtain (7).

If $\mu =\nu -1$, $\mathsf{\Gamma }(s-d_1+\nu -3)>0$, since $(1-\nu +\mu )(s-d_1-2)+(\nu -1)(d_2-s+1)>0$, then $(\nabla \mathsf{\Psi })\left(s\right)>0$ implies that $\mathsf{\Psi }$ is increasing. Thus, we obtain (8).

Also, take

$${\mathcal{F}}_{*}=\underset{\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}}{min}f\left(\varrho \right)\mathrm{and}{\mathcal{F}}^{*}=\underset{\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}}{max}f\left(\varrho \right).$$

Our first main result in this section is as follows:

 Theorem 3. 

Let condition (G1) hold. Moreover, if there is a sufficiently small positive constant ϵ and a sufficiently large constant $C_1$, ${\mathcal{F}}^{*}ϵ<C_1{\mathcal{F}}_{*}$ holds. Then, for each

$$\lambda \in \left({\left(C_1(d_2-d_1-1){\mathcal{F}}_{*}{\mathfrak{W}}^{*}\right)}^{-1},{\left((d_2-d_1-1){\mathcal{F}}^{*}{\mathfrak{W}}^{*}ϵ\right)}^{-1}\right),$$

the boundary value problem (1) and (2) has at least one positive solution.

 Proof. 

From the first limit in (G1), there is $r_1>0$ and a sufficiently small constant $ϵ>0$ satisfying $g\left(y\right)\le ϵr_1$ for all $y\in \left(0,r_1\right]$. Thus, for any $y\in K_1$ with $∥y∥=r_1$,

$$\left(T_{\lambda }y\right)\left(\varrho \right)\le \lambda {\mathfrak{W}}^{*}\sum _{s=d_1+2}^{d_2}f\left(s\right)g\left(y\left(s\right)\right)\le \lambda \left(d_2-d_1-1\right){\mathfrak{W}}^{*}{\mathcal{F}}^{*}ϵr_1\le r_1=∥y∥.$$

Therefore, if we set ${\mathsf{\Omega }}_1=\left\{y\in B:∥y∥<r_1\right\}$, the above inequality implies

$$∥T_{\lambda }y∥\le ∥y∥\mathrm{for}y\in K_1\cap \partial {\mathsf{\Omega }}_1.$$

Moreover, from the second limit in condition (G1), one can show that there is $r_2>r_1>0$ and a sufficiently large constant $C_1$ satisfying $g\left(y\right)\ge {\displaystyle \frac{C_1{r}_2}{{\gamma }_1^2}}$ for every $y\ge r_2$. Set $r_2^{*}={\displaystyle \frac{r_2}{{\gamma }_1}}>r_2$ and ${\mathsf{\Omega }}_2=\left\{y\in B:∥y∥<r_2^{*}\right\}$. Hence, for every $y\in K_1$ with $∥y∥=r_2^{*}$,

$$\underset{d_1+2\le \varrho \le d_2}{min}y\left(\varrho \right)\ge {\gamma }_1∥y∥={\gamma }_1{r}_2^{*}=r_2.$$

As a result, one can verify

$$\left(T_{\lambda }y\right)\left(\varrho \right)=\lambda \sum _{s=d_1+2}^{d_2}\mathfrak{W}\left(\varrho ,s\right)f\left(s\right)g\left(y\left(s\right)\right)\ge \lambda (d_2-d_1-1){\gamma }_1{\mathfrak{W}}^{*}{\mathcal{F}}_{*}{\displaystyle \frac{C_1{r}_2}{{\gamma }_1^2}}\ge r_2^{*}=∥y∥.$$

This gives us

$$∥T_{\lambda }y∥\ge ∥y∥\mathrm{for}y\in K_1\cap \partial {\mathsf{\Omega }}_2.$$

From Theorem 2, we find that the operator $T_{\lambda }$ possesses a fixed point $y\in K_1\cap \left(\overline{{\mathsf{\Omega }}_2}\setminus {\mathsf{\Omega }}_1\right)$ with $r_1\le ∥y∥\le$$r_2^{*}.$ □

Our second main existence result states the following:

 Theorem 4. 

Let (G2) hold. Furthermore, if there is a sufficiently large constant $C_2$ such that ${\mathcal{F}}^{*}<C_2{\mathcal{F}}_{*}$ holds, yhen, for each

$$\lambda \in \left({\left(C_2(d_2-d_1-1){\mathcal{F}}_{*}{\mathfrak{W}}^{*}\right)}^{-1},{\left((d_2-d_1-1){\mathcal{F}}^{*}{\mathfrak{W}}^{*}\right)}^{-1}\right),$$

problems (1) and (2) possess at least one positive solution.

 Proof. 

From the first limit in (G2), there is $r_3>0$ and a sufficiently large constant $C_2>0$ satisfying $g\left(y\right)>{\displaystyle \frac{C_2{r}_3}{{\gamma }_1}}$ for $y\in \left(0,r_3\right)$. If we choose ${\mathsf{\Omega }}_1=\left\{y\in B:∥y∥<r_3\right\}$, for each $y\in {\mathsf{\Omega }}_1$,

$$\left(T_{\lambda }y\right)\left(\varrho \right)=\lambda \sum _{s=d_1+2}^{d_2}\mathfrak{W}\left(\varrho ,s\right)f\left(s\right)g\left(y\left(s\right)\right)\ge \lambda (d_2-d_1-1){\gamma }_1{\mathfrak{W}}^{*}{\mathcal{F}}_{*}{\displaystyle \frac{C_2{r}_3}{{\gamma }_1}}\ge r_3=∥y∥,$$

which gives us

$$∥T_{\lambda }y∥\ge ∥y∥\mathrm{for}y\in K_1\cap \partial {\mathsf{\Omega }}_1.$$

Next, we consider two cases in order to set ${\mathsf{\Omega }}_2$.

Case 1. Let g be bounded. In other words, there is some $R_1>r_3$ such that $g\left(y\right)\le R_1$ for $y\in K_1$. Hence, for $y\in K_1$ with $∥y∥=R_1$,

$$\left(T_{\lambda }y\right)\left(\varrho \right)\le \lambda {\mathfrak{W}}^{*}\sum _{s=d_1+2}^{d_2}f\left(s\right)g\left(y\left(s\right)\right)\le \lambda {\mathfrak{W}}^{*}(d_2-d_1-1){\mathcal{F}}^{*}{R}_1\le R_1=∥y∥.$$

Case 2. On the other hand, if g is unbounded, there is some $R_2$ and a sufficiently small ${ϵ}_2$ with $g\left(y\right)\le {ϵ}_{2}y$for $y\ge R_2$. Set $R=max\{R_1,R_2\}$ and ${\mathsf{\Omega }}_2=\left\{y\in B:∥y∥<R\right\}$. Hence, $g\left(R\right)\le {ϵ}_{2}R$ and $\lambda <{\displaystyle \frac{1}{{ϵ}_2(d_2-d_1-1){\mathcal{F}}^{*}{\mathfrak{W}}^{*}}}$. As a consequence,

$$\left(T_{\lambda }y\right)\left(\varrho \right)\le \lambda {\mathfrak{W}}^{*}\sum _{s=d_1+2}^{d_2}f\left(s\right)g\left(y\left(s\right)\right)\le \lambda {\mathfrak{W}}^{*}(d_2-d_1-1){\mathcal{F}}^{*}{ϵ}_{2}R\le R=∥y∥,$$

which means that in both cases, we have

$$∥T_{\lambda }y∥\le ∥y∥\mathrm{for}y\in K_1\cap \partial {\mathsf{\Omega }}_2.$$

Thus, Theorem 2 ensures us that the operator $T_{\lambda }$ has a fixed point $y\in K_1\cap \left(\overline{{\mathsf{\Omega }}_2}\setminus {\mathsf{\Omega }}_1\right)$ with $r_3\le ∥y∥\le$R. □

3.2. Nonexistence

Now, we will establish some sufficient conditions that will allow us to show when problems (1) and (2) do not possess any positive solutions.

Suppose that the following conditions are satisfied:

(G3) $\underset{y\to 0^{+}}{lim}sup{\displaystyle \frac{g\left(y\right)}{y}}=g_0,\underset{y\to +\infty }{lim}sup{\displaystyle \frac{g\left(y\right)}{y}}=g_{\infty },$

(G4) $\underset{y\to 0^{+}}{lim}inf{\displaystyle \frac{g\left(y\right)}{y}}=g_0^{*},\underset{y\to +\infty }{lim}inf{\displaystyle \frac{g\left(y\right)}{y}}=g_{\infty }^{*}.$

 Theorem 5. 

Suppose that (G3) holds. Moreover, assuming that both $g_0<+\infty$ and $g_{\infty }<+\infty$, then there is a ${\lambda }_1$ such that for each $\lambda \in (0,{\lambda }_1)$, problems (1) and (2) do not possess any positive solutions.

 Proof. 

From $g_0<+\infty$ and $g_{\infty }<+\infty$, it follows that there are some positive $m_1,m_2,r_4$ and $r_5$ such that $r_4<r_5$, $g\left(y\right)\le m_{1}y$ for $y\in [0,r_4]$ and $g\left(y\right)\le m_{2}y$ for $y\in [r_5,+\infty )$. Choose

$$m=max\left\{m_1,m_2,\underset{r_4\le y\le r_5}{max}{\displaystyle \frac{g\left(y\right)}{y}}\right\}.$$

Hence, $g\left(y\right)\le my$. Now, let $y_1$ be a positive solution of (1) and (2). In other words, $T{y}_1\left(\varrho \right)=y_1\left(\varrho \right)$ for $\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$ and

$$∥y_1∥=∥T{y}_1∥\le \lambda {\mathfrak{W}}^{*}{\mathcal{F}}^{*}\sum _{s=d_1+2}^{d_2}g\left(y_1\left(s\right)\right)\le \lambda {\mathfrak{W}}^{*}(d_2-d_1-1){\mathcal{F}}^{*}m{y}_1<∥y_1∥,$$

which is a contradiction if we choose ${\lambda }_1={\displaystyle \frac{1}{{\mathfrak{W}}^{*}(d_2-d_1-1){\mathcal{F}}^{*}m}}$. Therefore, (1) and (2) have no positive solutions for every $\lambda \in (0,{\lambda }_1)$. □

 Theorem 6. 

Let (G4) hold. Furthermore, if $g_0^{*}>0$ and $g_{\infty }^{*}>0$, then there is a ${\lambda }_2$ such that for all $\lambda >{\lambda }_2$, the boundary value problem (1) and (2) has no positive solution.

 Proof. 

From $g_0^{*}>0$ and $g_{\infty }^{*}>0$, it follows that there are positive numbers $m_3,m_4,r_6,r_7$ such that $r_6<r_7$, $g\left(y\right)\ge m_{3}y$, for $y\in [0,r_6]$, and $g\left(y\right)\ge m_{4}y$, for $y\in [r_7,+\infty )$. This time, choose

$$m=min\left\{m_3,m_4,\underset{r_6\le y\le r_7}{max}{\displaystyle \frac{g\left(y\right)}{y}}\right\}.$$

We have $g\left(y\right)\ge my$ for all $y>0$. Suppose $y_2$ is a positive solution of (1) and (2). Then, $T{y}_2\left(\varrho \right)=y_2\left(\varrho \right)$ for $\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$ and

$$∥y_2∥=∥T{y}_2∥\ge \lambda \sum _{s=d_1+2}^{d_2}\mathfrak{W}\left(\varrho ,s\right)f\left(s\right)g\left(y_2\left(s\right)\right)\ge \lambda (d_2-d_1-1){\gamma }_1{\mathfrak{W}}^{*}{\mathcal{F}}_{*}m{y}_2>∥y_2∥,$$

which is a contradiction if we set ${\lambda }_2={\displaystyle \frac{1}{{\mathfrak{W}}^{*}(d_2-d_1-1){\mathcal{F}}_{*}m}}$. Therefore, (1) and (2) have no positive solutions for every $\lambda >{\lambda }_2$. □

4. Case II: $\mathit{\nu }-\mathbf{1}<\mathit{\mu }\le \mathbf{1}$

In the considered case, we will prove the existence of positive solutions of the following more general equation:

$$\begin{array}{c}-\left({\nabla }_{\rho \left(d_1\right)}^{\nu }u\right)\left(\varrho \right)=\lambda f(\varrho ,y\left(\varrho \right)),\varrho \in {\mathbb{N}}_{d_1+2}^{d_2},\end{array}$$

(9)

coupled with boundary conditions (2), where $f:{\mathbb{N}}_{d_1+2}^{d_2}\times \mathbb{R}\to \mathbb{R}$ is continuous and $f(\varrho ,y)\ge 0$ for all $\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$ and $y\ge 0$. We know that in this case

$$M=\underset{\left(\varrho ,s\right)\in {\mathbb{N}}_{d_1}^{d_2}\times {\mathbb{N}}_{d_1+2}^{d_2}}{max}\mathfrak{W}\left(\varrho ,s\right)=\underset{s\in {\mathbb{N}}_{d_1+2}^{d_2}}{max}\mathfrak{W}\left(s,s\right).$$

In particular,

$$\mathfrak{W}\left(s,s\right)={\displaystyle \frac{H_{\nu -1}(s,d_1)H_{\nu -\mu -1}(d_2,\rho \left(s\right))}{H_{\nu -\mu -1}(d_2,d_1)}}-1={\displaystyle \frac{\psi \left(s\right)\mathsf{\Gamma }(d_2-d_1)}{\mathsf{\Gamma }\left(\nu \right)\mathsf{\Gamma }(d_2-d_1+\nu -\mu -1)}}-1,$$

with

$$\psi \left(s\right)={\displaystyle \frac{\mathsf{\Gamma }(s-d_1+\nu -1)\mathsf{\Gamma }(d_2-s+\nu -\mu )}{\mathsf{\Gamma }(d_2-s+1)\mathsf{\Gamma }(s-d_1)}}.$$

Consider, for $s\in {\mathbb{N}}_{d_1+3}^{d_2}$,

$$\left(\nabla \psi \right)\left(s\right)={\displaystyle \frac{\left((1-\nu +\mu )(s-d_1-1)+(\nu -1)(d_2-s+1)\right)\mathsf{\Gamma }(s-d_1+\nu -2)\mathsf{\Gamma }(d_2-s+\nu -\mu )}{\mathsf{\Gamma }(d_2-s+2)\mathsf{\Gamma }(s-d_1)}}.$$

Thus, $\left(\nabla \psi \right)\left(s\right)>0,$ implying that

$$\underset{s\in {\mathbb{N}}_{d_1+2}^{d_2}}{max}\psi \left(s\right)=\psi \left(d_2\right).$$

Consequently,

$$M=\underset{s\in {\mathbb{N}}_{d_1+2}^{d_2}}{max}\mathfrak{W}(s,s)=\mathfrak{W}(d_2,d_2)={\displaystyle \frac{\mathsf{\Gamma }(d_2-d_1+\nu -1)\mathsf{\Gamma }(\nu -\mu )}{\mathsf{\Gamma }\left(\nu \right)\mathsf{\Gamma }(d_2-d_1+\nu -\mu -1)}}-1.$$

Moreover, since $\mathfrak{W}\left(\varrho ,s\right)$ is increasing for $\varrho \in \left[d_1+1,s-1\right]$ and decreasing for $\varrho \in \left[s+1,d_2\right]$, one proves that

$$m=\underset{\left(\varrho ,s\right)\in {\mathbb{N}}_{d_1+1}^{d_2}\times {\mathbb{N}}_{d_1+2}^{d_2}}{min}\mathfrak{W}\left(\varrho ,s\right)=\underset{s\in {\mathbb{N}}_{d_1+2}^{d_2}}{min}\left\{\mathfrak{W}\left(d_1+1,s\right),\mathfrak{W}\left(d_2,s\right)\right\}.$$

In particular,

$$\underset{s\in {\mathbb{N}}_{d_1+2}^{d_2}}{min}\mathfrak{W}\left(d_1+1,s\right)={\displaystyle \frac{H_{\nu -\mu -1}(d_2,a+1)}{H_{\nu -\mu -1}(d_2,d_1)}}={\displaystyle \frac{d_2-d_1-1}{d_2-d_1+\nu -\mu -1}}$$

and

$$\underset{s\in {\mathbb{N}}_{d_1+2}^{d_2}}{min}\mathfrak{W}\left(d_2,s\right)={\displaystyle \frac{\mathsf{\Gamma }(d_2-d_1+\nu -1)\mathsf{\Gamma }(\nu -\mu )}{\mathsf{\Gamma }(d_2-d_1+\nu -\mu -1)\mathsf{\Gamma }\left(\nu \right)}}-{\displaystyle \frac{\mathsf{\Gamma }(d_2-d_1+\nu -2)}{\mathsf{\Gamma }(d_2-d_1-1)\mathsf{\Gamma }\left(\nu \right)}}.$$

Therefore,

$$m=min\left\{{\displaystyle \frac{d_2-d_1-1}{d_2-d_1+\nu -\mu -1}},{\displaystyle \frac{\mathsf{\Gamma }(d_2-d_1+\nu -1)\mathsf{\Gamma }(\nu -\mu )}{\mathsf{\Gamma }(d_2-d_1+\nu -\mu -1)\mathsf{\Gamma }\left(\nu \right)}}-{\displaystyle \frac{\mathsf{\Gamma }(d_2-d_1+\nu -2)}{\mathsf{\Gamma }(d_2-d_1-1)\mathsf{\Gamma }\left(\nu \right)}}\right\}.$$

Define a different cone than before, namely

$$K_2=\left\{y\in B,y\left(\varrho \right)\ge 0,miny\left(\varrho \right)\ge {\displaystyle \frac{m}{M}}∥y∥,\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}\right\},$$

and an operator $A_{\lambda }:K_2\to B,$

$$\left(A_{\lambda }y\right)\left(\varrho \right)=\lambda \sum _{s=d_1+2}^{d_2}\mathfrak{W}\left(\varrho ,s\right)f(s,y\left(s\right)).$$

 Lemma 2. 

If $y\in K_2$, then $A_{\lambda }y\in K_2.$

 Proof. 

Indeed, let $y\in K_2$. From the definition of the operator and from (5), we deduce that $A_{\lambda }y\ge 0$ for all $\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$. Moreover,

$$\underset{\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}}{min}{A}_{\lambda }y\left(\varrho \right)\ge \lambda m\sum _{s=d_1+2}^{d_2}f\left(s,y\left(s\right)\right)\ge \lambda {\displaystyle \frac{m}{M}}\sum _{s=d_1+2}^{d_2}\underset{\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}}{max}\mathfrak{W}\left(\varrho ,s\right)f\left(s,y\left(s\right)\right)={\displaystyle \frac{m}{M}}∥A_{\lambda }y∥.$$

□

Now, our first existence result for this case is

 Theorem 7. 

Let $f_0\left(\varrho \right)$ and $f_{\infty }\left(\varrho \right)$ be nonnegative functions for all $\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$, and there is $0<r<R$ such that for $\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$,

$$f\left(\varrho ,s\right)\le s{f}_0\left(\varrho \right)\mathit{for}s\in \left[0,r\right]\mathit{and}f\left(\varrho ,s\right)\ge s{f}_{\infty }\left(\varrho \right)\mathit{for}s\ge R$$

and

$$M^2\sum _{s=d_1+2}^{d_2}{f}_0\left(s\right)\le m^2\sum _{s=d_1+2}^{d_2}{f}_{\infty }\left(s\right).$$

Then, for each

$${\displaystyle \frac{M}{m^2{\sum }_{s=d_1+2}^{d_2}{f}_{\infty }\left(s\right)}}\le \lambda \le {\displaystyle \frac{1}{M{\sum }_{s=d_1+2}^{d_2}{f}_0\left(s\right)}},$$

(10)

problem (9) and (10) possess a positive solution y. Moreover, for all $\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$,

$${\displaystyle \frac{mr}{M}}\le y\left(\varrho \right)\le {\displaystyle \frac{MR}{m}}.$$

(11)

 Proof. 

Let $\lambda$ be such that (10) holds, and let $y\in K_2$ with $∥y∥=r$. For $s\in \left[0,r\right],$ one can deduce that

$$A_{\lambda }y\left(\varrho \right)\le \lambda M\sum _{s=d_1+2}^{d_2}f\left(s,y\left(s\right)\right)\le \lambda M\sum _{s=d_1+2}^{d_2}y\left(s\right)f_0\left(s\right)\le \lambda M∥y∥\sum _{s=d_1+2}^{d_2}{f}_0\left(s\right)\le ∥y∥.$$

As a result, we prove that $∥A_{\lambda }y∥\le ∥y∥$ for $y\in K_2\cap \partial {\mathsf{\Omega }}_1,$ with ${\mathsf{\Omega }}_1=\left\{y\in B,∥y∥<r\right\}.$

Next, set $R_1={\displaystyle \frac{MR}{m}}$ and ${\mathsf{\Omega }}_2=\left\{y\in B,∥y∥<R_1\right\}.$ It is easy to verify that for $y\in K_2\cap \partial {\mathsf{\Omega }}_2,$

$$\underset{\varrho \in {\mathbb{N}}_{d_1+1}^{d_2}}{min}y\left(\varrho \right)\ge {\displaystyle \frac{m}{M}}∥y∥={\displaystyle \frac{m}{M}}{R}_1=R.$$

Hence,

$$A_{\lambda }y\left(\varrho \right)\ge \lambda m\sum _{s=d_1+2}^{d_2}f\left(s,y\left(s\right)\right)\ge \lambda m\sum _{s=d_1+2}^{d_2}y\left(s\right)f_{\infty }\left(s\right)\ge \lambda {\displaystyle \frac{m^2}{M}}∥y∥\sum _{s=d_1+2}^{d_2}{f}_{\infty }\left(s\right)\ge ∥y∥.$$

In other words, $∥A_{\lambda }y∥\ge ∥y∥$ for $y\in K_2\cap \partial {\mathsf{\Omega }}_2.$ Using Theorem 2, it follows that A has a fixed point in $y\in K_2\cap \left({\overline{\mathsf{\Omega }}}_2\setminus {\mathsf{\Omega }}_1\right)$, which is a solution of (9) and (2) satisfying (11). □

 Corollary 1. 

Suppose that

$$\underset{s\to 0^{+}}{lim}{\displaystyle \frac{f\left(\varrho ,s\right)}{s}}=f_0\left(\varrho \right)\mathrm{and}\underset{s\to +\infty }{lim}{\displaystyle \frac{f\left(\varrho ,s\right)}{s}}=f_{\infty }\left(\varrho \right)\mathrm{for}\varrho \in {\mathbb{N}}_{d_1+2}^{d_2},$$

and

$$M^2\sum _{s=d_1+2}^{d_2}{f}_0\left(s\right)<m^2\sum _{s=d_1+2}^{d_2}{f}_{\infty }\left(s\right).$$

Hence, for each

$${\displaystyle \frac{M}{m^2{\sum }_{s=d_1+2}^{d_2}{f}_{\infty }\left(s\right)}}<\lambda <{\displaystyle \frac{1}{M{\sum }_{s=d_1+2}^{d_2}{f}_0\left(s\right)}},$$

problem (9) and (2) has at least one positive solution y.

 Proof. 

Suppose that $\lambda$ is the interval stated above and set as $ϵ>0$ such that

$${\displaystyle \frac{M}{m^2{\sum }_{s=d_1+2}^{d_2}\left(f_{\infty }\left(s\right)-\delta \left(s\right)\right)}}\le \lambda \le {\displaystyle \frac{1}{M{\sum }_{s=d_1+2}^{d_2}\left(f_0\left(s\right)+ϵ\right)}},$$

with $\delta \left(s\right)=min\left\{ϵ,f_{\infty }\left(s\right)\right\}.$ Hence, for this choice of $\delta ,$ there is $0<r<R$ such that

$$f\left(\varrho ,s\right)\le s\left(f_0\left(\varrho \right)+ϵ\right)\mathrm{for}s\in \left[0,r\right],\mathrm{and}f\left(\varrho ,s\right)\ge s\left(f_{\infty }\left(\varrho \right)-\delta \left(\varrho \right)\right)\mathrm{for}s\ge R.$$

Having these conditions, one can use Theorem 7 in order to verify that there is a positive solution y for (9) and (2). □

Our second main existence result states the following:

 Theorem 8. 

If $f_0\left(\varrho \right)$ and $f_{\infty }\left(\varrho \right)$ are nonnegative functions for all $\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$ and there is $0<r<R$ such that for $\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$,

$$f\left(\varrho ,s\right)\ge s{f}_0\left(\varrho \right)\mathit{for}s\in \left[0,r\right],\mathit{and}f\left(\varrho ,s\right)\le s{f}_{\infty }\left(\varrho \right)\mathit{for}s\ge R$$

and

$$M^2\sum _{s=d_1+2}^{d_2}{f}_{\infty }\left(s\right)\le m^2\sum _{s=d_1+2}^{d_2}{f}_0\left(s\right).$$

Then, for each

$${\displaystyle \frac{M}{m^2{\sum }_{s=d_1+2}^{d_2}{f}_0\left(s\right)}}\le \lambda \le {\displaystyle \frac{1}{M{\sum }_{s=d_1+2}^{d_2}{f}_{\infty }\left(s\right)}},$$

problem (9) and (2) has at least one positive solution y such that for all $\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$, we have ${\displaystyle \frac{mr}{M}}\le y\left(\varrho \right)\le {\displaystyle \frac{MR}{m}}.$

 Proof. 

One can easily verify this result using similar arguments as the ones given for the proof of Theorem 7. □

 Corollary 2. 

Suppose that

$$\underset{s\to 0^{+}}{lim}{\displaystyle \frac{f\left(\varrho ,s\right)}{s}}=f_0\left(\varrho \right)\mathit{and}\underset{s\to +\infty }{lim}{\displaystyle \frac{f\left(\varrho ,s\right)}{s}}=f_{\infty }\left(\varrho \right)\mathit{for}\varrho \in {\mathbb{N}}_{d_1+2}^{d_2}$$

and

$$M^2\sum _{s=d_1+2}^{d_2}{f}_{\infty }\left(s\right)\le m^2\sum _{s=d_1+2}^{d_2}{f}_0\left(s\right).$$

Then, for each

$${\displaystyle \frac{M}{m^2{\sum }_{s=d_1+2}^{d_2}{f}_0\left(s\right)}}<\lambda <{\displaystyle \frac{1}{M{\sum }_{s=d_1+2}^{d_2}{f}_{\infty }\left(s\right)}},$$

we find that (9) and (2) possesses a positive solution y.

 Proof. 

We omit it, as it follows from Theorem 8. □

5. Examples

Now, we are going to establish three numerical examples to validate our theoretical findings.

 Example 1. 

Let us study (1) and (2) with $d_1=0$, $d_2=5$, $\nu =1.5$, $\mu =0.5$, $g\left(u\right)=u^2$ and $f\left(\varrho \right)=\varrho$. Clearly, g satisfies condition (G1). Then, ${\mathcal{F}}_{*}=2$ and ${\mathcal{F}}^{*}=5$, so

$${\mathfrak{W}}^{*}={\displaystyle \frac{(d_2-d_1-1)\mathsf{\Gamma }(d_2-d_1+\nu -2)\mathsf{\Gamma }(\nu -\mu )}{\mathsf{\Gamma }\left(\nu \right)\mathsf{\Gamma }(d_2-d_1+\nu -\mu -1)}}={\displaystyle \frac{2\mathsf{\Gamma }\left(4.5\right)}{\mathsf{\Gamma }\left(1.5\right)}}=26.25.$$

If we choose $ϵ={\displaystyle \frac{1}{2}}$ and $C_1=3$, then ${\mathcal{F}}^{*}ϵ<C_1{\mathcal{F}}_{*}$ is true. Thus, by Theorem 3, for each

$$\lambda \in \left({\displaystyle \frac{1}{630}},{\displaystyle \frac{1}{262.5}}\right),$$

we deduce that (1) and (2) possess a positive solution.

 Example 2. 

Consider (1) and (2) with $d_1=0$, $d_2=5$, $\nu =1.5$, $\mu =0.5$, $g\left(u\right)=u{e}^{-u}$ and $f\left(\varrho \right)=\varrho$. Clearly, g satisfies condition (G3) with $g_0=1<+\infty$ and $g_{\infty }=0<+\infty$. Then, $m=1$, ${\mathcal{F}}_{*}=2$, ${\mathcal{F}}^{*}=5$, and ${\mathfrak{W}}^{*}=26.25$, so

$${\lambda }_1={\displaystyle \frac{1}{525}}.$$

Therefore, by Theorem 5, for all $\lambda \in (0,{\lambda }_1)$, problem (1) and (2) have no positive solutions.

 Example 3. 

Consider (9) and (2) with $d_1=0$, $d_2=5$, $\nu =1.5$, $\mu =0.75$, and $f=u\left(e^{-\varrho }+e^{-u}\right)$. Hence,

$$f_0=e^{-\varrho }+1\mathrm{and}{f}_{\infty }=e^{-\varrho }.$$

Also,

$$M={\displaystyle \frac{\mathsf{\Gamma }\left(5.5\right)\mathsf{\Gamma }\left(0.75\right)}{\mathsf{\Gamma }\left(1.5\right)\mathsf{\Gamma }\left(4.75\right)}}-1=3.3636$$

and

$$m=min\left\{{\displaystyle \frac{\left(4\right)}{\left(4.75\right)}},{\displaystyle \frac{\mathsf{\Gamma }\left(5.5\right)\mathsf{\Gamma }\left(0.75\right)}{\mathsf{\Gamma }\left(1.5\right)\mathsf{\Gamma }\left(4.75\right)}}-{\displaystyle \frac{\mathsf{\Gamma }\left(4.5\right)}{\mathsf{\Gamma }\left(4\right)\mathsf{\Gamma }\left(1.5\right)}}\right\}=0.8421.$$

Furthermore,

$${\left(3.3636\right)}^2\sum _{s=2}^5{e}^{-s}\le {\left(0.8421\right)}^2\sum _{s=2}^5(e^{-s}+1).$$

Thus, by Corollary 2, for each

$${\displaystyle \frac{3.3636}{{\left(0.8421\right)}^2{\sum }_{s=2}^5(e^{-s}+1)}<\lambda <\frac{1}{\left(3.3636\right){\sum }_{s=2}^5{e}^{-s}},}$$

that is,

$$\lambda \in \left(1.1266,1.4144\right),$$

problem (9) and (2) has at least one positive solution.

6. Conclusions

In this work we were able to deduce new important properties of the Green’s function related to the considered problem (1) and (2). Depending on the values of $\nu$ and $\mu$, we studied two cases, and for each one of them, we obtained suitable conditions, under which we have shown some existence results. In the end, we were able to show the applicability of these theoretical findings with some particular examples. As far as we know, this is the first research study where such results are established for this problem.

According to us, the above-mentioned results can be extended in some future works, where the authors may study both cases, and using different methods, they may obtain different existence results or multiplicity.