Fractional p-Laplacian Equations with Sandwich Pairs

Abstract

The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational methods in $\psi$-fractional space ${\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)$. The results obtained in this paper are the first to make use of the theory of $\psi$-Hilfer fractional operators with p-Laplacian.

1. Introduction and Motivation

In this paper, we consider a new class of fractional differential equations with p-Laplacian given by

$${}_C^{\mathbf{H}}{\mathfrak{D}}_T^{\alpha ,\beta ;\psi }\left({\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|}^{p-2}{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right)=f(\xi ,\varphi ),\mathrm{in}\mathsf{\Omega }$$

(1)

where $\mathsf{\Omega }=[0,T]$ is a bounded domain in $\mathbb{R}$ (where $\mathbb{R}$ is the real line), ${}_C^{\mathbf{H}}{\mathfrak{D}}_T^{\alpha ,\beta ;\psi }(·)$ and ${}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }(·)$ are the Hilfer–Caputo and $\psi$-Hilfer fractional derivative of order $\frac{1}{p}<\alpha <1$ and type $0\le \beta \le 1$, $1<p<\infty$ and f is a Caratheodory function on $[0,T]\times \mathbb{R}$ with subcritical growth. Note that, for $p>2$ the (1) degenerates and $1<p<2$ is singular, at points where $\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|=0$. For $p=2$, we only obtain the usual, that is, ${}_C^{\mathbf{H}}{\mathfrak{D}}_T^{\alpha ,\beta ;\psi }\left({}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right)$. In this sense, we include the condition $\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|\ne 0$.

Over the last few years, the theory of fractional differential equations has attracted much attention; from problems involving a theoretical approach, to problems involving applications [1,2]. We highlight problems in the controllability theory of differential equation solutions [3,4,5,6,7], which are relevant problems that contribute significantly to this area.

First, before commenting on sandwich pairs, it is worth mentioning the importance and relevance of the p-Laplacian equations and their contribution. Over the years, the impact and importance of p-Laplacian equations to the theory of differential equations, especially elliptical ones, has been remarkable and undeniable. Problems of the existence and multiplicity are indeed interesting and have drawn much attention over these decades; especially in this last decade, with double phase problems [8,9,10,11]. We can highlight some applications: in the mechanics of nanostructures, fluids, diffusion processes, and asymptotic dynamics [12,13,14,15,16,17,18] and the references therein. On the other hand, it is worth highlighting the problems of differential equations with p-Laplacian via variational methods and $\psi$-Hilfer fractional operators, which began in mid 2021 and has been gaining ground in this area [19,20,21,22,23,24,25,26,27]. The theory is still new and under construction. In this sense, the results in this area are still quite restricted. Consequently, there are two aspects: the first is that there are few results to use and; in the vast majority of cases, it is necessary to build them. On the other hand, this allows a range of options to work with and numerous open problems arise as the theory grows.

The first sandwich pairs were built using eigenspaces and were used to find critical points of a functional. This approach was taken by Schechter [28]. We can also highlight the works on solved quasilinear problems using cones as sandwich pairs [29,30]. See the works in [31,32,33].

Many problems arising in science and engineering call for the solving of Euler equations of functionals, i.e., equations of the form $G^{\prime }\left(u\right)=0$, where $G\left(u\right)$ is a $C^1$ functional arising from the given data. Since the development of the calculus of variations, there has been interest in finding the critical points of functionals. This was intensified by the fact that for many equations arising in practice, the solutions are critical points. See some interesting papers on sandwich pairs and applications for more details [34,35,36,37].

Perera and Scheter [29] discussed the boundary value problem for the p-Laplacian using the notion of sandwich pairs; that is, they addressed the following problem:

$$\left\{\begin{array}{ccc}-{\Delta }_{p}u& =& f(x,u),\mathrm{in}\mathsf{\Omega }\\ u& =& 0,on\partial \mathsf{\Omega }.\end{array}\right.$$

(2)

For more details about this problem (2), see [29].

The sandwich pairs used until 2007 were introduced using the eigenspaces of a semilinear operator and are therefore unsuitable for dealing with quasilinear problems where there are no eigenspaces. In this sense, in 2008, Perera and Scheter [30], showed that the method could be modified to be applied in the problem (2).

Perera and Schechter [38] discussed the solution of problems of the following type:

$$\left\{\begin{array}{ccc}-{\Delta }_{p}u& =& {\nabla }_{u}F(x,u),\mathrm{in}\mathsf{\Omega }\\ u& =& 0,on\partial \mathsf{\Omega }.\end{array}\right.$$

(3)

For more details about this problem (3), see [38].

We say shall that a pair of subsets $\mathcal{A},\mathcal{B}$ of a Banach space W form a sandwich pair, if for any ${\mathcal{E}}^{\alpha ,\beta ;\psi }(·)\in C^1(W,\mathbb{R})$ the inequality [37]

$$-\infty <b:=\underset{\mathcal{B}}{inf}{\mathcal{E}}^{\alpha ,\beta ;\psi }(·)\le a:=\underset{\mathcal{A}}{sup}{\mathcal{E}}^{\alpha ,\beta ;\psi }(·)<\infty$$

(4)

implies that there is a sequence $\left({\varphi }_j\right)\subset W$, such that

$${\mathcal{E}}^{\alpha ,\beta ;\psi }\left({\varphi }_j\right)\to c,{\left({\mathcal{E}}^{\alpha ,\beta ;\psi }\right)}^{\prime }\left({\varphi }_j\right)\to 0$$

(5)

for some $c\in [b,a]$.

Note that the sequence satisfying (4) is called a Palais–Smale sequence at the level c and ${\mathcal{E}}^{\alpha ,\beta ;\psi }$ satisfies the compactness condition ${\left(PS\right)}_c$ if every such sequence has a convergent subsequence.

Motivated by these works [29,30,38], in this paper we tried to find a solution to the problem (1) via variational methods and sandwich pairs. We will discuss the existence of solutions through two theorems, one with the lower limit condition, and the other with the upper limit condition and from the function

$$\mathsf{\Theta }(\xi ,t)=\mathfrak{F}(\xi ,t)-tf(\xi ,t),$$

where ${\displaystyle \mathfrak{F}(\xi ,t)={\int }_0^{t}f(x,s)ds}$; in other words, we are interested in discussing the following results:

Theorem 1.

If

$$({\lambda }_l+\epsilon ){\left|t\right|}^p-\mathcal{V}\left(\xi \right)\le \mathfrak{F}(\xi ,t)\le {\lambda }_{l+1}{\left|t\right|}^p+\mathcal{V}\left(\xi \right)$$

(6)

for some $l ,\epsilon >0$ and $\mathcal{V}\in L_{\psi }^1\left(\mathsf{\Omega }\right)$, and

$$\mathsf{\Theta }(\xi ,t)\le C\left(\right|t|+1),\overline{\mathsf{\Theta }}\left(\xi \right):=\underset{\left|t\right|\to \infty }{\overline{lim}}\frac{\mathsf{\Theta }(\xi ,t)}{\left|t\right|}<0,a.e$$

(7)

then the problem(1)has a solution.

Theorem 2.

If

$${\lambda }_l{\left|t\right|}^p-\mathcal{V}\left(\xi \right)\le \mathfrak{F}(\xi ,t)\le ({\lambda }_{l+1}-\epsilon ){\left|t\right|}^p+\mathcal{V}\left(\xi \right)$$

(8)

for some $l,\epsilon >0$ and $\mathcal{V}\in L_{\psi }^1\left(\mathsf{\Omega }\right)$, and

$$\mathsf{\Theta }(\xi ,t)\ge -C\left(\right|t|+1),\underline{\mathsf{\Theta }}\left(\xi \right):=\underset{\left|t\right|\to \infty }{\underline{lim}}\frac{\mathsf{\Theta }(\xi ,t)}{\left|t\right|}>0,a.e$$

(9)

then the problem(1)has a solution.

A natural consequence when working with fractional operators is to obtain the classic case, as a particular case, this is of paramount importance and relevance for the investigated results. Here, in this work, it is possible to obtain such a property, in addition to obtaining other possible particular cases from the choice of the parameters $\beta \to 1$ or $\beta \to 0$ and from the function $\psi (·)$. Some cases will be discussed at the end of the paper, as “special cases”. However, one of the limitations of this work is that it cannot choose the function $\psi \left(\xi \right)=ln\xi$ as a particular case, since the problem (1) is covered in the space $\psi$-fractional ${\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)$. However, it is possible to discuss this case, but it is necessary to work with the weight space of ${\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)$. Furthermore, we can rule out that the results obtained here are the first in the area of fractional differential equations with p-Laplacian and $\psi$-Hilfer fractional operators. Certainly, the results presented in this work will draw attention to future work; in particular, a natural continuation of this work, as highlighted at the end of the paper.

In Section 2, we present definitions and results about the theory of fractional operators and sandwich pairs. Finally, in Section 3, we investigate the main results of the article, i.e., the proof of Theorems 1 and 2. In this sense, we present some special cases.

2. Mathematical Background: Preliminaries

In this section, we present some definitions of p-integrable spaces, $\psi$-fractional space, and results of the $\psi$-Hilfer fractional derivative. In addition, we end the section with the sandwich pair results.

The space of the p-integrable function with respect to a function $\psi$ is defined as

$$\begin{array}{c}\hfill L_{\psi }^p([a,b],\mathbb{R})=\left\{\varphi :[a,b]\to \mathbb{R}:{\int }_a^b{\left|\varphi \left(x\right)\right|}^p{\psi }^{\prime }\left(x\right)dx<\infty \right\}\end{array}$$

(10)

with norm

$${\left|\right|\varphi \left|\right|}_{L_{\psi }^p([a,b],\mathbb{R})}={\left({\int }_a^b{\psi }^{\prime }\left(x\right){\left|\varphi \left(x\right)\right|}^{p}dx\right)}^{1/p}.$$

Choosing $p=1$ in Equation (10), we have the integrable space $L_{\psi }^1([a,b],\mathbb{R})$ with its respective norm

$${\left|\right|\varphi \left|\right|}_{L_{\psi }^1([a,b],\mathbb{R})}={\int }_a^b{\psi }^{\prime }\left(x\right)\left|\varphi \left(x\right)\right|dx.$$

Let $n-1<\alpha <n$ with $n\in \mathbb{N}$, $I=[a,b]$ is the interval such that $-\infty \le a<b\le \infty$ and there exist two functions $f,\psi \in C^n([a,b],\mathbb{R})$, such that $\psi$ increasing and ${\psi }^{\prime }\left(\xi \right)\ne 0$, for all $\xi \in I$. The $\psi$-Hilfer fractional derivatives left-sided and right-sided ${{}^{\mathbf{H}}\mathfrak{D}}_a^{\alpha ,\beta ;\psi }(·)$$\left({{}^{\mathbf{H}}\mathfrak{D}}_b^{\alpha ,\beta ;\psi }(·)\right)$ of order $\alpha$ and type $0\le \beta \le 1$ are defined by [39]:

$${{}^{\mathbf{H}}\mathfrak{D}}_a^{\alpha ,\beta ;\psi }\varphi \left(\xi \right)={\mathbf{I}}_a^{\beta (n-\alpha ),\psi }\left(\frac{1}{{\psi }^{\prime }\left(\xi \right)}\frac{d}{d\xi }\right){\mathbf{I}}_a^{(1-\beta )(n-\alpha ),\psi }\varphi \left(\xi \right)$$

(11)

and

$${{}^{\mathbf{H}}\mathfrak{D}}_b^{\alpha ,\beta ;\psi }\varphi \left(\xi \right)={\mathbf{I}}_b^{\beta (n-\alpha ),\psi }\left(-\frac{1}{{\psi }^{\prime }\left(\xi \right)}\frac{d}{d\xi }\right){\mathbf{I}}_b^{(1-\beta )(n-\alpha ),\psi }\varphi \left(\xi \right)$$

(12)

where

$${\mathbf{I}}_a^{\alpha ,\psi }\varphi \left(\xi \right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_a^{\xi }{\psi }^{\prime }\left(s\right){(\psi \left(\xi \right)-\psi \left(s\right))}^{\alpha -1}\varphi \left(s\right)ds,toa<s<\xi$$

and

$${\mathbf{I}}_b^{\alpha ,\psi }\varphi \left(\xi \right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_{\xi }^b{\psi }^{\prime }\left(s\right){(\psi \left(s\right)-\psi \left(\xi \right))}^{\alpha -1}\varphi \left(s\right)ds,to\xi <s<b$$

are the fractional integrals of $\varphi$, with respect to the function $\psi$. The definition of ${}_C^{\mathbf{H}}{\mathfrak{D}}_T^{\alpha ,\beta ;\psi }(·)$ is the commutation between the integral operators ${\mathbf{I}}_a^{\beta (n-\alpha ),\psi }(·)$ and ${\mathbf{I}}_a^{(1-\beta )(n-\alpha ),\psi }(·)$ of Equation (12). Furthermore, $\left({\displaystyle \frac{d}{d\xi }}\right)$ and ${\psi }^{\prime }\left(\xi \right)$ are classical derivatives.

Note that the function $\psi (·)$ is part of the kernel of the $\psi$-Riemann–Liouville fractional integral and, consequently, of the $\psi$-Hilfer fractional derivative. The motivation for introducing the $\psi$-Hilfer fractional derivative comes from the fractional derivatives: Caputo, Riemann–Liouville, and Hilfer, to unify a wide class of fractional operators in a unique operator. The restriction on the function $\psi \left(\xi \right)$ is that $\psi \left(\xi \right)\ne 0$ for all $\xi \in I$, since ${\displaystyle \frac{1}{{\psi }^{\prime }\left(\xi \right)}}$, as stated in the definition itself. A priori, the physical meaning of $\psi (·)$ is still unknown. However, what can be discussed are their respective particular cases, based on the choice of the function $\psi (·)$ and the limits of $\beta \to 1$ or $\to 0$. For example, in the particular choice of $\psi \left(\xi \right)=\xi$, $\psi \left(\xi \right)={\xi }^{\rho }$ with $\rho >0$ and $\psi \left(\xi \right)=ln\left(\xi \right)$, classic fractional derivatives are obtained; for example Caputo, Riemann–Liouville, Hadamard, Caputo–Hadamard, Hilfer, among others.

Let $\alpha \in (0,1)$, $p\in (1,\infty )$ and $\frac{1}{p}+\frac{1}{q}\le 1+\alpha$. If $\varphi \in L_{\psi }^q\left([a,b]\right)$ and $\phi \in L_{\psi }^p\left([a,b]\right)$, then the following integration by parts [19]:

$${\int }_a^b\left({\mathbf{I}}_a^{\alpha ;\psi }\phi \left(\xi \right)\right)\varphi \left(\xi \right){\psi }^{\prime }\left(\xi \right)d\xi ={\int }_a^b\phi \left(\xi \right){\mathbf{I}}_b^{\alpha ;\psi }\varphi \left(\xi \right){\psi }^{\prime }\left(\xi \right)d\xi .$$

(13)

On the other hand, let $\alpha \in (0,1)$, $\beta \in [0,1]$, $\varphi ,\phi \in AC[a,b]$ and $-{\displaystyle \frac{1}{{\psi }^{\prime }(·)}}{\mathbf{I}}_b^{\beta (1-\alpha );\psi }\varphi \in L_{\psi }^2[a,b]$, then [21]

$$\begin{array}{ccc}\hfill & & {\int }_a^b{}_c^{\mathbf{H}}{\mathfrak{D}}_b^{\alpha ,\beta ;\psi }\varphi \left(\xi \right)\phi \left(\xi \right){\psi }^{\prime }\left(\xi \right)d\xi \hfill \\ & & =-\underset{x\to b}{lim}{\mathbf{I}}_{0+}^{(1-\alpha )(1-\beta );\psi }\phi \left(\xi \right){\mathbf{I}}_b^{\beta (1-\alpha );\psi }\varphi \left(\xi \right)\hfill \\ & & +\underset{x\to 0+}{lim}{\mathbf{I}}_{0+}^{(1-\alpha )(1-\beta );\psi }\phi \left(\xi \right){\mathbf{I}}_b^{\beta (1-\alpha );\psi }\varphi \left(\xi \right)+{\int }_0^b\varphi {\left(\xi \right)}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\phi \left(\xi \right){\psi }^{\prime }\left(\xi \right)d\xi .\hfill \end{array}$$

(14)

The $\psi$-fractional space is given by [19]

$${\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)=\left\{\varphi \in L_{\psi }^p\left(\mathsf{\Omega }\right):\left|{}^{\mathrm{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|\in L_{\psi }^p\left(\mathsf{\Omega }\right),\varphi =0on\partial \mathsf{\Omega }\right\}$$

with the norm

$$\left|\right|\varphi \left|\right|={\left|\right|\varphi \left|\right|}_{{\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)}={\left|\right|\varphi \left|\right|}_{L_{\psi }^p\left(\mathsf{\Omega }\right)}+{\left|\right|}^{\mathrm{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi {\left|\right|}_{L_{\psi }^p\left(\mathsf{\Omega }\right)}.$$

The space $C_0^{\infty }\left(\mathsf{\Omega }\right)$ is dense in ${\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)$. The space ${\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)$ is a separable and reflexive Banach space [19].

Indeed, for $\phi \in C_0^{\infty }\left([0,T]\right)$ and taking the integral in both sides of Equation (1) yields

$${\int }_0^T{}_C^{\mathbf{H}}{\mathfrak{D}}_T^{\alpha ,\beta ;\psi }\left({\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|}^{p-2}{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right)\phi \left(\xi \right){\psi }^{\prime }\left(\xi \right)d\xi ={\int }_0^{T}f(\xi ,\varphi )\phi \left(\xi \right){\psi }^{\prime }\left(\xi \right)d\xi .$$

(15)

Using the relation Equation (14) and taking

$$\begin{array}{c}\hfill \underset{x\to 0+}{lim}{\mathbf{I}}^{(1-\alpha )(1-\beta );\psi }\phi \left(\xi \right)=0=\underset{x\to T}{lim}{\mathbf{I}}^{(1-\alpha )(1-\beta );\psi }\phi \left(\xi \right),\end{array}$$

we have

$$\begin{array}{ccc}\hfill & & {\int }_0^T{}_C^{\mathbf{H}}{\mathfrak{D}}_T^{\alpha ,\beta ;\psi }\left({\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|}^{p-2}{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right)\phi \left(\xi \right){\psi }^{\prime }\left(\xi \right)d\xi \hfill \\ & & ={\int }_0^T{\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|}^{p-2}{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi {}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\phi \left(\xi \right){\psi }^{\prime }\left(\xi \right)d\xi .\hfill \end{array}$$

(16)

Therefore, from Equations (15) and (16) yield

$${\int }_0^T{\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|}^{p-2}{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi {}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\phi \left(\xi \right){\psi }^{\prime }\left(\xi \right)d\xi ={\int }_0^{T}f(\xi ,\varphi )\phi \left(\xi \right){\psi }^{\prime }\left(\xi \right)d\xi .$$

(17)

Consider $\phi =\varphi$, we have

$${\int }_0^T{\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|}^p{\psi }^{\prime }\left(\xi \right)d\xi ={\int }_0^{T}f(\xi ,\varphi )\varphi {\psi }^{\prime }\left(\xi \right)d\xi .$$

(18)

Now, we define the Euler functional ${\mathcal{E}}^{\alpha ,\beta ;\psi }:{\mathbb{H}}_p^{\alpha ,\beta ;\psi }([0,T],\mathbb{R})\to \mathbb{R}$ on ${\mathbb{H}}_p^{\alpha ,\beta ;\psi }([0,T],\mathbb{R})$, given by

$${\mathcal{E}}^{\alpha ,\beta ;\psi }\left(\varphi \right):=\frac{1}{p}{\int }_0^T{\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|}^p{\psi }^{\prime }\left(\xi \right)d\xi -{\int }_0^T\mathfrak{F}(\xi ,\varphi \left(\xi \right)){\psi }^{\prime }\left(\xi \right)d\xi$$

(19)

where ${\displaystyle \mathfrak{F}(\xi ,t)={\int }_0^{t}f(x,s)ds}$.

The solution of (1) coincides with the critical points of the $C^1$ functional Equation (19).

Consider the result of the sandwich pairs:

Proposition 1

([29,38]). Γ be the class of maps $\gamma \in C\left(W\times [0,1],W\right)$ such that:

$\left(a\right){\gamma }_0=id$;

$\left(b\right)\underset{(\varphi ,t)\in W\times [0,1]}{sup}∥{\gamma }_t\left(\varphi \right)-\varphi ∥<\infty$ where ${\gamma }_t=\gamma (·,t)$. Assume that for any $\gamma \in \mathsf{\Gamma }$,

$$\gamma \left(\mathcal{A}\right)\cap \mathcal{B}\ne \varnothing .$$

(20)

Then $\mathcal{A},\mathcal{B}$ forms a sandwich pair.

Let ${\mathcal{A}}_1\ne \varnothing$ and ${\mathcal{B}}_1\ne \varnothing$ be subsets of the interval J in a Banach space W, such that $dist({\mathcal{A}}_1,{\mathcal{B}}_1)>0$. We say that ${\mathcal{A}}_1$ links ${\mathcal{B}}_1$ if for any ${\mathcal{E}}^{\alpha ,\beta ;\psi }(·)\in C^1(J,\mathbb{R})$

$$-\infty <\underset{{\mathcal{A}}_1}{sup}{\mathcal{E}}^{\alpha ,\beta ;\psi }(·)=:a_0<b_0:=\underset{{\mathcal{B}}_1}{inf}{\mathcal{E}}^{\alpha ,\beta ;\psi }(·)<\infty$$

(21)

implies that there exists a sequence $\left({\varphi }_j\right)\subset J$, such that Equation (21) holds for some $c\ge b_0$.

Proposition 2

([40]). ${\mathcal{A}}_1$ links ${\mathcal{B}}_1$ in J if for any $\phi \in C(C{\mathcal{A}}_1,J)$ such that $\phi (·,0)=i{d}_{{\mathcal{A}}_1}$

$$\phi \left(C{\mathcal{A}}_1\right)\cap {\mathcal{B}}_1\ne \varnothing$$

(22)

where $C{\mathcal{A}}_1=({\mathcal{A}}_1\times [0,1])/({\mathcal{A}}_1\times \left\{1\right\})$ is a subset on ${\mathcal{A}}_1$.

Proposition 3

([29,38]). If ${\mathcal{A}}_1$ and ${\mathcal{B}}_1$ satisfy the hypotheses of Proposition 2 in J, then

$$\mathcal{A}={\pi }^{-1}\left({\mathcal{A}}_1\right)\cup \left\{0\right\},\mathcal{B}={\pi }^{-1}\left({\mathcal{B}}_1\right)\cup \left\{0\right\}$$

(23)

forms a sandwich pair, where $\pi :W\setminus \left\{0\right\}\to J$.

Consider the nonlinear eigenvalue fractional problem

$$\left\{\begin{array}{ccc}{}_C^{\mathbf{H}}{\mathfrak{D}}_T^{\alpha ,\beta ;\psi }\left({\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|}^{p-2}{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right)& =& {\lambda \left|\varphi \right|}^{p-2},\mathrm{in}\mathsf{\Omega }\\ \varphi & =& 0,on\partial \mathsf{\Omega }.\end{array}\right.$$

(24)

Its eigenvalues are similar to the critical values of the $C^1$ functional

$${\mathcal{I}}_{\psi }\left(\varphi \right)=\frac{1}{{\displaystyle {\int }_0^T{\psi }^{\prime }\left(\xi \right){\left|\varphi \right|}^{p}d\xi }}$$

(25)

on the interval J in ${\mathbb{H}}_p^{\alpha ,\beta ;\psi }([0,T],\mathbb{R})$. Let ${\mathsf{\Gamma }}^l$ be the class of odd continuous maps $\gamma$ from the interval $J_{l-1}$ in $\mathbb{R}$ to J and set

$${\lambda }_l=\underset{\gamma \in {\mathsf{\Gamma }}_l}{inf}\underset{\varphi \in \gamma \left(J_{l-1}\right)}{max}{\mathcal{I}}_{\psi }\left(\varphi \right).$$

(26)

Then $0<{\lambda }_1<{\lambda }_2\le ···\to \infty$ are eigenvalues of the problem (24). Consider

$$\mathsf{\Theta }(\xi ,t)=\mathfrak{F}(\xi ,t)-tf(\xi ,t).$$

We emphasize that the resonance is considered only concerning the specific variational eigenvalues given by (26) and not with respect to other possible nonvariational eigenvalues or variational eigenvalues, which are given by different methods.

3. Main Results

In this section, we investigate the main results of the paper, i.e., the proof of Theorems 1 and 2 through the results presented in Section 2.

Thus, we start with the proof of the first result according to the theorem below:

Proof of Theorem 1.

Using Equation (26), there is a $\gamma \in {\mathsf{\Gamma }}_l$ such that ${\mathcal{I}}_{\psi }\le {\lambda }_l+{\displaystyle \frac{\epsilon }{2}}$ on ${\mathcal{A}}_1=\gamma \left(J_{l-1}\right)$. Let ${\mathcal{B}}_1=\left\{\varphi \in J:{\mathcal{I}}_{\psi }\left(\varphi \right)\right\}\ge {\lambda }_{l+1}$. Since ${\lambda }_l+{\displaystyle \frac{\epsilon }{2}}<{\lambda }_{l+1}$ by inequality (6), ${\mathcal{A}}_1$ and ${\mathcal{B}}_1$ are disjoint. Since ${\mathcal{A}}_1$ is compact and ${\mathcal{B}}_1$ is closed, it follows that $dist({\mathcal{A}}_1,{\mathcal{B}}_1)>0$. We claim that ${\mathcal{A}}_1$ links ${\mathcal{B}}_1$ in J. Given $\phi \in C(C{\mathcal{A}}_1,J)$ such that $\phi (·,0)=i{d}_{{\mathcal{A}}_1}$, writing $\xi \in J_l$ as $({\xi }^{\prime },{\xi }_{l+1})\in \mathbb{R}\oplus \mathbb{R}$, define $\overline{\gamma }\in {\mathsf{\Gamma }}_{l+1}$ by

$$\overline{\gamma }\left(\xi \right)=\left\{\begin{array}{c}\phi \left(\gamma \right({\xi }^{\prime }/|x^{\prime }\left|\right),{\xi }_{l+1})for0\le {\xi }_{l+1}<1\\ \phi ({\mathcal{A}}_1\times \left\{1\right\})for{\xi }_{l+1}=1\\ \overline{\gamma }({\xi }^{\prime },-{\xi }_{l+1})for{\xi }_{l+1}<0\end{array}.\right.$$

Then, $\overline{\gamma }\left(J_l\right)\cap {\mathcal{B}}_1\ne \varnothing$ by definition of ${\lambda }_{l+1}$ so Equation (22) holds, as ${\mathcal{B}}_1$ is symmetric. Hence $\mathcal{A},\mathcal{B}$ given by (23) forms a sandwich pair through Proposition 3. Let ${\mathcal{E}}^{\alpha ,\beta ;\psi }$ given by Equation (19). Since

$${\int }_0^T{\psi }^{\prime }\left(\xi \right){\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|}^{p}d\xi \ge {\lambda }_{l+1}{\int }_0^T{\psi }^{\prime }\left(\xi \right){\left|\varphi \right|}^{p}d\xi ,\varphi \in \mathcal{B}$$

and

$${\int }_0^T{\psi }^{\prime }\left(\xi \right){\left|{}^{\mathbf{H}}{\mathfrak{D}}_{0+}^{\alpha ,\beta ;\psi }\varphi \right|}^{p}d\xi \le ({\lambda }_{l+1}+\epsilon ){\int }_0^T{\psi }^{\prime }\left(\xi \right){\left|\varphi \right|}^{p}d\xi ,\varphi \in \mathcal{A},$$

(6) implies

$$-{\int }_0^T{\psi }^{\prime }\left(0\right)\mathcal{V}\left(\xi \right)d\xi \le \underset{\mathcal{B}}{inf}{\mathcal{E}}^{\alpha ,\beta ;\psi }(·)\le \underset{\mathcal{A}}{sup}{\mathcal{E}}^{\alpha ,\beta ;\psi }\le {\int }_0^T{\psi }^{\prime }\left(\xi \right)\mathcal{V}\left(\xi \right)d\xi .$$

Hence, there exists a sequence $\left({\varphi }_j\right)\subset {\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)$ satisfying Equation (5).

Since $\left({\varphi }_j\right)$ is bounded, there exists a convergent subsequence ${\rho }_j=\left|\right|{\varphi }_j\left|\right|\to \infty$, a subsequence of ${\overline{\varphi }}_j={\displaystyle \frac{{\varphi }_j}{{\rho }_j}}\rightharpoonup \overline{\varphi }$ in ${\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)$, strongly in $L_{\psi }^p\left(\mathsf{\Omega }\right)$ and almost everywhere in $\mathsf{\Omega }$. Then, using Equation (5) yields

$$\begin{array}{ccc}\hfill {\int }_0^T{\psi }^{\prime }\left(\xi \right)\frac{\mathsf{\Theta }(\xi ,{\varphi }_j)}{{\rho }_j}d\xi & & ={\int }_0^T{\psi }^{\prime }\left(\xi \right)\left(\frac{\mathfrak{F}(\xi ,{\varphi }_j)-tf(\xi ,{\varphi }_j)}{{\rho }_j}\right)d\xi \hfill \\ & & ={\int }_0^T{\psi }^{\prime }\left(\xi \right)\frac{\mathfrak{F}(\xi ,{\varphi }_j)}{{\rho }_j}d\xi {\int }_0^T{\psi }^{\prime }\left(\xi \right)\frac{tf(\xi ,{\varphi }_j)}{{\rho }_j}d\xi \hfill \\ & & ={\displaystyle \frac{{\displaystyle \frac{{\left({\mathcal{E}}^{\alpha ,\beta ;\psi }\left({\varphi }_j\right)\right)}^{\prime }{\varphi }_j}{p^2}}-{\mathcal{E}}^{\alpha ,\beta ;\psi }\left({\varphi }_j\right)}{{\rho }_j}}\to 0.\hfill \end{array}$$

Using ${\overline{\varphi }}_j={\displaystyle \frac{{\varphi }_j}{{\rho }_j}}$ and inequality (7), it follows that

$$\overline{lim}{\int }_0^T{\psi }^{\prime }\left(\xi \right)\frac{\mathsf{\Theta }(\xi ,{\varphi }_j)}{{\rho }_j}d\xi \le {\int }_0^T{\psi }^{\prime }\left(\xi \right)\overline{lim}\frac{\mathsf{\Theta }(\xi ,{\varphi }_j)}{|{\varphi }_j|}|{\overline{\varphi }}_j|d\xi ={\int }_0^T{\psi }^{\prime }\left(\xi \right)\overline{\mathsf{\Theta }}\left(\xi \right)\left|\overline{\varphi }\right|d\xi \le 0.$$

Since $\overline{\mathsf{\Theta }}<0$ almost everywhere, we have that $\overline{\varphi }=0$ almost everywhere. In this sense, passing to the limit, yields

$$1-\frac{{\mathcal{E}}^{\alpha ,\beta ;\psi }\left({\varphi }_j\right)}{{\rho }_j^p}={\int }_0^T{\psi }^{\prime }\left(\xi \right)\frac{\mathfrak{F}(\xi ,{\varphi }_j)}{{\rho }_j^p}d\xi \le {\int }_0^T{\psi }^{\prime }\left(\xi \right)\left({\lambda }_{l+1}{\left|\overline{{\varphi }_j}\right|}^p+\frac{\mathcal{V}}{{\rho }_j^p}\right)d\xi$$

and gives a contraction. □

Now, we will prove the second main result of this paper.

Proof of Theorem 2.

Let a sequence $\left({\epsilon }_j\right)\subset (0,\epsilon ]$ decreasing to 0 and

$${\mathcal{E}}_j^{\alpha ,\beta ;\psi }\left(\varphi \right)={\mathcal{E}}^{\alpha ,\beta ;\psi }\left(\varphi \right)-{\epsilon }_j{\int }_0^T{\psi }^{\prime }\left(\xi \right){\left|\varphi \right|}^{p}d\xi .$$

Then, using Equation (8), we obtain

$$\left(({\lambda }_l+{\epsilon }_j){\left|t\right|}^p-\mathcal{V}\left(\xi \right)\right)\le \left(\mathfrak{F}(\xi ,t)+{\epsilon }_j{\left|t\right|}^p\right)\le \left({\lambda }_{l+1}{\left|t\right|}^p+\mathcal{V}\left(\xi \right)\right).$$

(27)

In this sense, there exists a sequence $\left({\varphi }_j\right)\subset {\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)$ such that ${\mathcal{E}}_j^{\alpha ,\beta ;\psi }\left({\varphi }_j\right)$ is bounded ${\left({\mathcal{E}}_j^{\alpha ,\beta ;\psi }\right)}^{\prime }\left({\varphi }_j\right)\to 0$ (see proof of Theorem 1). Since $\left({\varphi }_j\right)$ is bounded, there exists a subsequence that converges to a critical point of ${\mathcal{E}}_j^{\alpha ,\beta ;\psi }(·)$. If ${\rho }_j=\left|\right|{\varphi }_j{\left|\right|}_{}\psi \to \infty$, a subsequence of ${\overline{\varphi }}_j={\displaystyle \frac{{\varphi }_j}{{\rho }_j}}\rightharpoonup \overline{\varphi }$ in ${\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)$, strongly in $L_{\psi }^p\left(\mathsf{\Omega }\right)$, and almost everywhere $\mathsf{\Omega }=[0,T]$. Then using Equation (8) yields

$$\begin{array}{ccc}\hfill {\int }_0^T{\psi }^{\prime }\left(\xi \right)\frac{\mathsf{\Theta }(\xi ,{\varphi }_j)}{{\rho }_j}d\xi & & ={\int }_0^T\left(\frac{\mathfrak{F}(\xi ,{\varphi }_j)-tf(\xi ,{\varphi }_j)}{{\rho }_j}\right)d\xi \hfill \\ & & ={\int }_0^T{\psi }^{\prime }\left(\xi \right)\frac{\mathfrak{F}(\xi ,{\varphi }_j)}{{\rho }_j}d\xi -{\int }_0^T{\psi }^{\prime }\left(\xi \right)\frac{tf(\xi ,{\varphi }_j)}{{\rho }_j}d\xi \hfill \\ & & ={\displaystyle \frac{{\displaystyle \frac{{\left({\mathcal{E}}^{\alpha ,\beta ;\psi }\left({\varphi }_j\right)\right)}^{\prime }{\varphi }_j}{p^2}}-{\mathcal{E}}^{\alpha ,\beta ;\psi }\left({\varphi }_j\right)}{{\rho }_j}}\to 0.\hfill \end{array}$$

On the other hand, using Equation (9) we obtain

$$\begin{array}{ccc}& & \underline{lim}{\int }_0^T{\psi }^{\prime }\left(\xi \right)\frac{\mathsf{\Theta }(\xi ,{\varphi }_j,{\varphi }_j)}{{\rho }_j}d\xi \hfill \\ & & \ge {\int }_0^T{\psi }^{\prime }\left(\xi \right)\underline{lim}\frac{\mathsf{\Theta }(\xi ,{\varphi }_j)}{|{\varphi }_j|}\left|{\overline{\varphi }}_j\right|d\xi \hfill \\ & & \ge {\int }_0^T{\psi }^{\prime }\left(\xi \right)\underline{\mathsf{\Theta }}\left(\xi \right)\left|\overline{\varphi }\right|\ge 0.\hfill \end{array}$$

Since $\underline{\mathsf{\Theta }}>0$ almost everywhere, then $\overline{\varphi }=0$ almost everywhere. In this sense, passing to the limit in

$$1-\frac{{\mathcal{E}}^{\alpha ,\beta ;\psi }\left({\varphi }_j\right)}{{\rho }_j^p}={\int }_0^T{\psi }^{\prime }\left(\xi \right)\left(\frac{\mathfrak{F}(\xi ,{\varphi }_j)}{{\rho }_j^p}+{\epsilon }_j{\left|{\varphi }_j\right|}^p\right)d\xi \le {\int }_0^T{\psi }^{\prime }\left(\xi \right)\left({\lambda }_{l+1}{\left|\overline{{\varphi }_j}\right|}^p+\frac{\mathcal{V}}{{\rho }_j^p}\right)d\xi$$

gives a contradiction. Therefore, we have concluded the proof. □

Special Cases

A natural consequence of the results obtained above is the freedom to present a wide class of possible particular cases, especially the integer case. In this sense, we present three cases below:

(Special case.) Taking the limit $\alpha \to 1$ and $\psi \left(\xi \right)=\xi$, we obtain the problem in its classic version, given by

$${\left({\left|{\varphi }^{\prime }\right|}^{p-2}{\varphi }^{\prime }\right)}^{\prime }=f(\xi ,\varphi ),\mathrm{in}\mathsf{\Omega }.$$

(28)

(Caputo fractional operator case.) Taking $\beta =1$ and $\psi \left(\xi \right)=\xi$, we have

$${}_C{\mathfrak{D}}_T^{\alpha }\left({\left|{}^{\mathbf{C}}{\mathfrak{D}}_{0+}^{\alpha }\varphi \right|}^{p-2}{}^{\mathbf{C}}{\mathfrak{D}}_{0+}^{\alpha }\varphi \right)=f(\xi ,\varphi ),\mathrm{in}\mathsf{\Omega }.$$

(29)

(Riemann–Liouville fractional operator case.) Taking $\beta =0$ and $\psi \left(\xi \right)=\xi$, we obtain

$${}_C^{\mathbf{RL}}{\mathfrak{D}}_T^{\alpha }\left({\left|{}^{\mathbf{RL}}{\mathfrak{D}}_{0+}^{\alpha }\varphi \right|}^{p-2}{}^{\mathbf{RL}}{\mathfrak{D}}_{0+}^{\alpha }\varphi \right)=f(\xi ,\varphi ),\mathrm{in}\mathsf{\Omega }.$$

(30)

Once we obtain the particular cases, i.e., the problems (28)–(30), the results investigated here, Theorems 1 and 2, are valid for such a case. It is possible to notice that the freedom of choice of the function $\psi (·)$ allows obtaining other examples involving fractional operators; however, we restrict ourselves to the ones previously discussed. See other formulations of fractional operators that can be obtained by choosing the function $\psi (·)$ in [39].

4. Conclusions and Future Work

At the end of this work, we were able to show the existence of a solution for a new class of fractional differential equations with p-Laplacian and sandwich pairs via variational methods. Although the results investigated here are new, there is still a much to be investigated, which shows that the theory is still under construction and, consequently, there are some open problems a priori; we can highlight these as follows:

(1)

Taking the problem (1), we can work with the $p\left(x\right)$-Laplacian that has a more complex nonlinearity, which raises some essential difficulties; for example, it is inhomogeneous.

(2)

An interesting issue that can also be worked on is to discuss the problem (1) with a double phase.

(3)

Finally, a possible discussion of the problem (1) with the Kirchhoff problem would be interesting.

In this sense, we conclude this work in the certainty that it will pave the way for new results.