Existence Results for Some p-Laplacian Langevin Problems with a ψ-Hilfer Fractional Derivative with Antiperiodic Boundary Conditions

Abstract

In this work, we establish the existence of at least one solution for a p-Laplacian Langevin differential equation involving the $\psi$-Hilfer fractional derivative with antiperiodic boundary conditions. More precisely, we transform the studied problem into a Hammerstein integral equation, and after that, we use the Schafer fixed point theorem to prove the existence of at least one solution. Two examples are provided to validate the main result.

1. Introduction and Motivation

Fractional calculus is a branch of mathematical analysis that extends the concept of classical derivatives and integrals to non-integer-order ones. It is an interesting area of research that has several applications across various fields, such as Physics (Diffusion processes and viscoelastic materials; see [1,2]), engineering (control systems, where fractional-order controllers can enhance performance; see [3]), and finance (modelling complex systems and processes that exhibit anomalous diffusion, see [4,5]). Due to its importance and widespread applications in many fields, several researchers have concentrated on the development of many fractional derivatives and solved fractional problems involving these derivatives. We cite, for example, the papers [6,7] for the Riemann–Liouville derivative, the papers [8,9] for the fractional Caputo derivative, and the papers [10] for problems involving the Hilfer derivative.

Recently, several authors have developed a new fractional derivative called the $\psi$-Hilfer fractional derivative, which is a generalization of the previous fractional derivatives. It is defined using a function $\psi$ and a fractional order $\alpha$. There are several works involving the fractional $\psi$-Hilfer derivative; for the interested readers, we cite, for example, the works [11,12,13] and the references therein.

In this work, we continue to investigate a problem involving this new derivative. Precisely, we consider a Langevin problem involving a $\psi$-Hilfer derivative and the p-Laplacian operator. It is noted that, in the last few years, great attention has been paid to the study of Langevin equations with different fractional derivatives and using different methods; for example, Ahmad et al. [14,15] used the Krasnoselskii fixed-point theorem to prove the existence of solutions for some Langevin equations with Caputo fractional derivatives, while Almaghamsi et al. [16] proved the existence of weak solutions for fractional Langevin inclusions involving the Katugampola–Caputo derivative; they used the Mönch fixed-point theorem combined with a weak noncompactness. Lim et al. [17] considered a Langevin problem involving the weyl and the Riemann–Liouville fractional derivatives, by studying Gaussian processes, and their relation with fractional Brownian motion provided proof for several existing results. Very recently, Almaghamsi et al. [18] combined a variational method with the iterative method to prove the existence of a solution for a Langevin equation involving the $\psi$-Hilfer derivative. We refer to the papers [19,20,21,22] for other interesting works.

Through this paper, we denote by ${\Phi }_p$ the $p$-Laplacian operator which is given by ${\Phi }_p\left(s\right)={\left|s\right|}^{p-2}s$, $p>1$ and ${\Phi }_p^{-1}={\Phi }_q$ where ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$.

In [23], using the Schauder fixed-point theorem and other mathematical analysis techniques, the authors establish the existence result of the nonlinear fractional differential equations with a p-Laplacian operator and integral boundary conditions

$$\left\{\begin{array}{cc}{}^{C}D_{0^{+}}^{\beta }\left({\Phi }_p{(}^C{D}_{0^{+}}^{\alpha }y\left(t\right))\right)+f(t,y\left(t\right))=0,0\le t\in [0,1]\hfill & \\ y\left(1\right)=\lambda {\int }_0^{1}y\left(s\right)ds,y^{\prime }\left(1\right)=0,{}^{C}D_{0^{+}}^{\alpha }y\left(1\right)=b{}^{C}D_{0^{+}}^{\alpha }y\left(\xi \right)\hfill \end{array}\right.$$

where $1<\alpha \le 2$, $0<\beta \le 1$, $0<\xi ,b,\lambda <1$, $f(t,u):[0,1]\times [0,\infty )\to [0,\infty )$ is a given continuous function, ${}^{C}D_{0^{+}}^{\alpha }$ and ${}^{C}D_{0^{+}}^{\beta }$ are the Caputo fractional derivative.

With the help of Leray-Schafer’s fixed point theorem, the authors of [24] provide existence criteria for the solutions of p-Laplacian fractional Langevin differential equations with antiperiodic boundary conditions:

$$\left\{\begin{array}{cc}{}^{C}D_{0^{+}}^{\beta }{\Phi }_p\left({}^{C}D_{0^{+}}^{\alpha }\left(x\right)\left(t\right)+\lambda x\left(t\right)\right)=f(t,x\left(t\right),{}^{C}D_{0^{+}}^{\alpha }\left(x\right)\left(t\right)),\hfill & t\in [0,1],\hfill \\ x\left(0\right)=-x\left(1\right),{}^{C}D_{0^{+}}^{\alpha }x\left(0\right)=-{}^{C}D_{0^{+}}^{\alpha }x\left(1\right),\hfill \end{array}\right.$$

where $0<\alpha ,\beta \le 1$, $\lambda \ge 0$, $f:[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ is a given continuous function and ${}^{C}D_{0^{+}}^{\alpha }$ and ${}^{C}D_{0^{+}}^{\beta }$ are the Caputo fractional derivatives.

In this work, we are motivated by the above papers to prove the existence of at least one solution to the following problem:

$$\left(\mathcal{P}\right)\left\{\begin{array}{cc}{}^{\psi }D_a^{{\nu }_2,{\mu }_2}{\Phi }_p\left(r\left(t\right){}^{\psi }D_a^{{\nu }_1,{\mu }_1}\left(x\right)\left(t\right)+s\left(t\right)x\left(t\right)\right)=f(x,t),\hfill & t\in [a,b],\hfill \\ x\left(\eta \right)=-x\left(b\right)=A,\hfill & \eta \in (a,b),\hfill \\ {}^{\psi }D_a^{{\nu }_1,{\mu }_1}x\left(\eta \right)=-{}^{\psi }D_a^{{\nu }_1,{\mu }_1}x\left(b\right)=B,\hfill & \end{array}\right.$$

where ${}^{\psi }D_a^{{\nu }_i,{\mu }_i},i=1,2$ is the $\psi$-Hilfer fractional derivative with order $0<{\nu }_i<1$ and type $0<{\mu }_i<1$. The functions $s,r$ are nontrivial in $C\left(\right[a,b],\mathbb{R})$, and satisfy

$$\begin{array}{ccc}\hfill \left(r\right(\eta ),s(\eta \left)\right)& =& (-r(b),-s(b\left)\right),\hfill \end{array}$$

(1)

and ${\Phi }_p$ is the $p$-Laplacian operator which is defined by ${\Phi }_p\left(t\right)={\left|t\right|}^{p-2}t$ which also satisfies ${\Phi }_p^{-1}={\Phi }_{{\displaystyle \frac{p}{p-1}}}.$

Hereafter, we assume that f satisfies the following hypotheses:

• $\left(H_1\right)$ There exist two positive constants, $f_0$ and $f_1$, such that

  $$\left|f(t,x)\right|\le f_0+f_1\left|x\right|, \mathrm{for} \mathrm{each} (t,x)\in [a,b]\times \mathbb{R},$$
• $\left(H_2\right)$ There exists a positive constant L, such that

  $$|f(t,x)-f(t,y)|\le L|x-y|, \mathrm{for} \mathrm{each} t\in [a,b], \mathrm{an} \mathrm{each}(x,y)\in {\mathbb{R}}^2.$$

Throughout this paper, we consider $J=[a,b]$,

$${\displaystyle m=\underset{t\in J}{inf}\left|r\left(t\right)\right|,\mathrm{and}M=\underset{t\in J}{sup}\left|s\left(t\right)\right|.}$$

For $i=1,2$, $0<{\nu }_i,{\mu }_i<1$, we wrote

$${\rho }_i={\mu }_i(1-{\nu }_i)+{\nu }_i.$$

$\psi$ denotes a positive continuous function on J, which has a decreasing continuous derivative ${\psi }^{\prime }$ on J. Finally, we define the function ${\psi }_a$ for $t\in J$, by

$${\psi }_a\left(t\right)=\psi \left(t\right)-\psi \left(a\right),$$

and we denote by ${}_{\psi }\partial _x={\displaystyle \frac{1}{{\psi }^{\prime }\left(x\right)}}{\displaystyle \frac{d}{dx}}$.

In the next section, we present some results related to the $\psi$-Hilfer derivative. In Section 3, we present and prove the main results of this paper, and we finish this work by presenting an illustrative example.

2. Preliminary

In this section, we present a few well-known concepts of $\psi$-fractional calculus. For this purpose, let $\nu >0$, $0<\mu <1$, $n=\left[\nu \right]+1$ and $\rho =\mu (1-\nu )+\nu$.

Definition 1.

We recall from [13] the following useful definitions.

• The integral of a function u with respect to a function ψ is defined by

  $$\begin{array}{cc}\hfill I_a^{\nu ;\psi }u\left(x\right)& ={\displaystyle \frac{1}{\Gamma \left(\nu \right)}}{\int }_a^x{\psi }^{\prime }\left(t\right){\psi }_x^{\nu -1}\left(t\right)u\left(t\right)dt,\hfill \end{array}$$

  (2)

  provided that the integral exists.
• The ψ–Riemann–Liouville fractional derivative ${}^{RL}D_a^{\nu ,\psi }(.)$ of order ν is given by

  $$\begin{array}{cc}\hfill {}^{RL}D_a^{\nu ,\psi }\left(u\right)\left(x\right)& :={}^{\psi }\partial _x^n{I}_a^{n-\nu ,\psi }f\left(u\right).\hfill \end{array}$$

  (3)
• The ψ-Hilfer fractional derivative ${}^{H}D_a^{\nu ,\mu ,\psi }(.)$ of order ν and type $0\le \mu \le 1$, is defined by

  $$\begin{array}{ccc}\hfill {}^{H}D_a^{\nu ,\mu ,\psi }u\left(x\right)& =& I_a^{\mu (n-\nu ),\psi }{}^{\psi }\partial _x^n{I}_a^{(1-\mu )(n-\nu ),\psi }u\left(x\right)\hfill \end{array}$$

  (4)

  $$\begin{array}{ccc}& =& I_a^{\rho -\nu ,\psi }{}^{RL}D_a^{\rho ,\psi }\left(u\right)\left(x\right).\hfill \end{array}$$

  (5)

We note that the Riemann–Liouville and the Caputo fractional derivative can be deduced from the ψ-Hilfer fractional derivative when $\mu \to 0$, resp $\mu =1$ and $\psi \left(x\right)=x$.

Lemma 1

([12,25]). Let $\upsilon >0$, while the ψ–Riemann–Liouville fractional integral and derivative of a power function are given by

• $I_a^{\nu ,\psi }\left({\psi }_a^{\upsilon -1}\right)={\displaystyle \frac{\Gamma \left(\upsilon \right)}{\Gamma (\upsilon +\nu )}}{\psi }_a^{\upsilon +\nu -1}$.
• ${}^{RL}D_a^{\nu ,\psi }\left({\psi }_a^{\upsilon -1}\right)={\displaystyle \frac{\Gamma \left(\upsilon \right)}{\Gamma (\upsilon -\nu )}}{\psi }_a^{\upsilon -\nu -1}$.

Next, we recall from [13] the following definition.

Definition 2.

We denote by $C[J,\mathbb{R}]$ the well-known Banach space of all continuous functions from an interval J into $\mathbb{R}$ with the uniform norm and for an integer n and a function $u\in C[J,\mathbb{R}]$, $u^{[n-1]}$ is defined by

$$u^{[n-1]}={\left({}^{\psi }\partial _x\right)}^{n-1}u.$$

We define the weighted space $C_{\delta ,\psi }(J,\mathbb{R})$, for $\delta >0$, by

$$C_{\delta ,\psi }(J,\mathbb{R})=\left\{u:J\to \mathbb{R}suchthat{\left({\psi }_a\left(t\right)\right)}^{\delta }u\left(t\right)\in C(J,\mathbb{R})\right\},$$

which is equipped with the norm

$${\displaystyle {\parallel u\parallel }_{C_{\delta ,\psi }}=\parallel {\left({\psi }_a\left(t\right)\right)}^{\delta }u\left(t\right)\parallel =\underset{t\in J}{sup}\left\{\right|{\left({\psi }_0\left(t\right)\right)}^{\delta }u\left(t\right)\left|\right\}.}$$

We mention that for $\delta =0,$ we have $C_{0,\psi }(J,\mathbb{R})=C(J,\mathbb{R})$.

For $n\in \mathbb{N}$, we define $C_{\delta ,\psi }^n(J,\mathbb{R})$ by:

$$C_{\delta ,\psi }^n(J,\mathbb{R})=\left\{u:J\to \mathbb{R}suchthat{u}^{\left[n\right]}\in C_{\delta ,\psi }(J,\mathbb{R})\right\},$$

where $C_{0,\psi }^n(J,\mathbb{R})=C^n(J,\mathbb{R})$.

Lemma 2

([26]). Let $0<\rho <\nu$, and $u\in C_{\nu ,\psi }[J,\mathbb{R}]$, then we have

$${}^{RL}D_a^{\rho ,\psi }{I}_a^{\nu ;\psi }u\left(t\right)=I_a^{\nu -\rho ;\psi }u\left(t\right),\forall t\in J.$$

(6)

In particular case, for any positive integer $m\le \left[\nu \right]+1$, we obtain

$$D_a^m{I}_a^{\nu ,\psi }u\left(t\right)=I_a^{\nu -m,\psi }u\left(t\right),\forall t\in J.$$

(7)

Lemma 3

([12]). Let $0<\mu \le 1$, and $u\in C^n[J,\mathbb{R}]$, then we have

$${\displaystyle I_a^{\nu ,\psi }\left({}^{H}D_a^{\nu ,\mu ,\psi }u\right)\left(t\right)=u\left(t\right)-\sum _{k=1}^n\frac{{\psi }_a^{\rho -k}\left(a\right)}{\Gamma (\rho -k+1)}{c}_k,\forall t\in J}$$

(8)

where $c_k,k=1,\dots ,n$ is given by

$$c_k={}_{\psi }\partial _x^k\left(I_a^{(1-\rho )(k-\nu ),\psi }u\right)\left(a\right).$$

In particular, if $0<\nu <1$, then we obtain

$$I_a^{\nu ,\psi }\left({}^{H}D_a^{\nu ,\mu ,\psi }u\left(t\right)\right)=u\left(t\right)-{\displaystyle \frac{{\psi }_a^{\rho -1}\left(t\right)}{\Gamma \left(\rho \right)}}u\left(a\right),\forall t\in J.$$

(9)

Finally, we recall the following elementary inequality for the p-Laplacian operator.

Lemma 4

([22]). If $p>2$, the $p$-Laplacian operator ${\Phi }_p$ is a convex operator. Moreover, if $max\left(\right|x|,|y\left|\right)\le \rho$, then we have

$$|{\Phi }_p\left(x\right)-{\Phi }_p\left(y\right)|\le (p-1){\rho }^{p-2}|x-y|.$$

(10)

3. Main Results and Their Proofs

In this section, we prove the existence and uniqueness of a solution to the problem $\left(\mathcal{P}\right)$. For this purpose, we begin by proving that our problem $\left(\mathcal{P}\right)$ and a Volterra-type integral equation are equivalent. Thus, we obtain the following result.

Theorem 1.

The solution for the problem $\left(\mathcal{P}\right)$ is given by

$$x\left(t\right)=I_a^{{\nu }_1}g(t,x)-{\psi }_a^{{\rho }_1-1}\left(t\right)C_x,$$

(11)

with g is a function defined as follows:

$$g(t,x\left(t\right))={\displaystyle \frac{1}{r\left(t\right)}}\left[{\Phi }_q\left(I_a^{{\nu }_2}f(t,x)-{\psi }_a^{{\rho }_2-1}\left(t\right)C_x^{\prime }\right)-s\left(t\right)x\left(t\right)\right],$$

(12)

where $C_x$ and $C_x^{\prime }$ are given by

$$C_x={\displaystyle \frac{I_a^{{\nu }_1}g(b,x)+I_a^{{\nu }_1}g(\eta ,x)}{{\psi }_a^{{\rho }_1-1}\left(b\right)+{\psi }_a^{{\rho }_1-1}\left(\eta \right)}},$$

(13)

and

$$C_x^{\prime }={\displaystyle \frac{I_a^{{\nu }_2}f(b,x)+I_a^{{\nu }_2}f(\eta ,x)}{{\psi }_a^{{\rho }_2-1}\left(b\right)+{\psi }_a^{{\rho }_2-1}\left(\eta \right)}}.$$

(14)

Proof. 

From Lemma 3, we see that the problem $\left(\mathcal{P}\right)$ is equivalent to the following equation:

$${\Phi }_p\left(r{\left(t\right)}^{\psi }{D}_a^{{\nu }_1,{\mu }_1}\left(x\right)\left(t\right)+s\left(t\right)x\left(t\right)\right)-{\displaystyle \frac{{\psi }_a^{{\rho }_2-1}\left(t\right)}{\Gamma \left({\rho }_2\right)}}{C}_{1,x}^{\prime }{=}^{\psi }{I}_a^{{\nu }_2}f(t,x).$$

(15)

On the other hand, the initial conditions imply that

$$\left\{\begin{array}{cc}{\Phi }_p\left(r\left(\eta \right)B+s\left(\eta \right)A\right)-{\displaystyle \frac{{\psi }_a^{{\rho }_2-1}\left(\eta \right)}{\Gamma \left({\rho }_2\right)}}{C}_{1,x}^{\prime }={}^{\psi }I_a^{{\nu }_2}f(\eta ,x),\hfill & \\ {\Phi }_p\left(-r\left(b\right)B-s\left(b\right)A\right)-{\displaystyle \frac{{\psi }_a^{{\rho }_2-1}\left(b\right)}{\Gamma \left({\rho }_2\right)}}{C}_{1,x}^{\prime }={}^{\psi }I_a^{{\nu }_2}f(b,x).\hfill \end{array}\right.$$

So, based on the fact that ${\Phi }_p(-s)=-{\Phi }_p\left(s\right)$ and conditions (1), we obtain

$$C_{1,x}^{\prime }=-\Gamma \left({\rho }_2\right){\displaystyle \frac{I_a^{{\nu }_2}f(b,x)+I_a^{{\nu }_2}f(\eta ,x)}{{\psi }_a^{{\rho }_2-1}\left(b\right)+{\psi }_a^{{\rho }_2-1}\left(\eta \right)}}.$$

Now, by putting

$$C_x^{\prime }={\displaystyle \frac{I_a^{{\nu }_2}f(b,x)+I_a^{{\nu }_2}f(\eta ,x)}{{\psi }_a^{{\rho }_2-1}\left(b\right)+{\psi }_a^{{\rho }_2-1}\left(\eta \right)}},$$

we can write Equation (15) as follows:

$${}^{\psi }D_a^{{\nu }_1,{\mu }_1}\left(x\right)\left(t\right)={\displaystyle \frac{1}{r\left(t\right)}}\left[{\Phi }_q\left(I_a^{{\nu }_2}f(t,x)-{\psi }_a^{{\rho }_2-1}\left(t\right)C_x^{\prime }\right)-s\left(t\right)x\left(t\right)\right]=:g(t,x).$$

Using Lemma 3, the above equation has a solution given by

$$x\left(t\right)-{\displaystyle \frac{{\psi }_a^{{\rho }_1-1}\left(t\right)}{\Gamma \left({\rho }_1\right)}}{C}_{1,x}=I_a^{{\nu }_1}g(t,x).$$

On the other hand, from the initial conditions and Equation (1), we obtain

$${\displaystyle \frac{{\psi }_a^{{\rho }_1-1}\left(\eta \right)}{\Gamma \left({\rho }_1\right)}}{C}_{1,x}+I_a^{{\nu }_1}g(\eta ,x)=-{\displaystyle \frac{{\psi }_a^{{\rho }_1-1}\left(b\right)}{\Gamma \left({\rho }_1\right)}}{C}_{1,x}-I_a^{{\nu }_1}g(b,x).$$

So, we conclude that

$$C_{1,x}=-\Gamma \left({\rho }_1\right){\displaystyle \frac{I_a^{{\nu }_1}g(b,x)+I_a^{{\nu }_1}g(\eta ,x)}{{\psi }_a^{{\rho }_1-1}\left(b\right)+{\psi }_a^{{\rho }_1-1}\left(\eta \right)}},$$

which completes the proof by taking $C_x={\displaystyle \frac{I_a^{{\nu }_1}g(b,x)+I_a^{{\nu }_1}g(\eta ,x)}{{\psi }_a^{{\rho }_1-1}\left(b\right)+{\psi }_a^{{\rho }_1-1}\left(\eta \right)}}$. □

Before presenting the main result of this paper, we need to prove three technical lemmas. Through this paper, we give the following notation:

$${\psi }_a^{\alpha }(b,\eta )={\psi }_a^{\alpha }\left(b\right)+{\psi }_a^{\alpha }\left(\eta \right),\forall \alpha >0.$$

Lemma 5.

Under hypothesis $\left(H_1\right)$, the function g and the constants $C_x$ and $C_x^{\prime }$ defined, respectively, by Equations (12)–(14) satisfied the following properties:

$$|C_x^{\prime }|\le {\displaystyle \frac{(f_0+f_1{\parallel x\parallel }^{p-1}){\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1){\psi }_a^{{\rho }_2-1}(b,\eta ))}},$$

(16)

$$\begin{array}{cc}\hfill \left|g\right(t,x\left)\right|& \le {\displaystyle \frac{1}{m}}\left[M\parallel x\parallel +{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{2(f_0+f_1{\parallel x\parallel }^{p-1}{)}^{q-1}}{\Gamma ({\nu }_2+1)}}\right)}^{q-1}\left({\psi }_a^{(q-1){\nu }_2}\left(t\right)\right.\right.\hfill \\ & \left.\left.+{\left({\displaystyle \frac{{\psi }_a^{{\nu }_2}(b,\eta )}{{\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)}^{q-1}{\psi }_a^{(q-1)({\rho }_2-1)}\left(t\right)\right)\right],\hfill \end{array}$$

(17)

and

$$\begin{array}{cc}\hfill & |C_x|\le {\displaystyle \frac{{\psi }_a^{{\nu }_1-{\rho }_1+1}(b,\eta )}{m}}\left[{\displaystyle \frac{M}{\Gamma ({\nu }_1+1)}}\parallel x\parallel +{\rm Y}{\displaystyle \frac{(f_0+f_1{\parallel x\parallel }^{p-1}{)}^{q-1}}{2}}{\left({\displaystyle \frac{2{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)}}\right)}^{q-1}\right]\hfill \end{array}$$

(18)

where Υ is a positive constant given by

$${\rm Y}={\displaystyle \frac{\Gamma ((q-1){\nu }_2+1)}{\Gamma ({\nu }_1+(q-1){\nu }_2+1)}}+{\displaystyle \frac{\Gamma ((q-1)({\rho }_2-1)+1)}{\Gamma ({\nu }_1+(q-1)({\rho }_2-1)+1)}}.$$

Proof. 

Using hypothesis $\left(H_1\right)$ and Equations (2) and (14), we obtain

$$\begin{array}{cc}\hfill |C_x^{\prime }|& \le {\displaystyle \frac{|I_a^{{\nu }_2}f(b,x)+I_a^{{\nu }_2}f(\eta ,x)|}{{\psi }_a^{{\rho }_2-1}\left(b\right)+{\psi }_a^{{\rho }_2-1}\left(\eta \right)}}\hfill \\ & \le {\displaystyle \frac{I_a^{{\nu }_2}\left|f(b,x)\right|+I_a^{{\nu }_2}\left|f(\eta ,x)\right|}{{\psi }_a^{{\rho }_2-1}\left(b\right)+{\psi }_a^{{\rho }_2-1}\left(\eta \right)}}\hfill \\ & \le {\displaystyle \frac{(f_0+f_1{\parallel x\parallel }^{p-1})({\psi }_a^{{\nu }_2}\left(b\right)+{\psi }_a^{{\nu }_2}\left(\eta \right))}{\Gamma ({\nu }_2+1){\psi }_a^{{\rho }_2-1}(b,\eta )}}\hfill \\ & \le {\displaystyle \frac{(f_0+f_1{\parallel x\parallel }^{p-1}){\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1){\psi }_a^{{\rho }_2-1}(b,\eta ))}}.\hfill \end{array}$$

So, inequality (16) holds.

Now, from the above estimation, Equation (12) and the convexity property of the p-Laplacian operator, we obtain

$$\begin{array}{cc}\hfill & \left|g(t,x\left(t\right))\right|\le {\displaystyle \frac{1}{m}}\left[2^{q-2}\left({\left(I_a^{{\nu }_2}f(t,x)\right)}^{q-1}+{\psi }_a^{(q-1)({\rho }_2-1)}\left(t\right){\left(C_x^{\prime }\right)}^{q-1}\right)+M\parallel x\parallel \right]\hfill \\ & \le {\displaystyle \frac{1}{m}}\left[{\displaystyle \frac{2^{q-2}(f_0+f_1{\parallel x\parallel }^{p-1}{)}^{q-1}}{{\Gamma }^{q-1}({\nu }_2+1)}}\left({\psi }_a^{(q-1){\nu }_2}\left(t\right)+{\left({\displaystyle \frac{{\psi }_a^{{\rho }_2-1}\left(t\right)({\psi }_a^{{\nu }_2}\left(b\right)+{\psi }_a^{{\nu }_2}\left(\eta \right))}{{\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)}^{q-1}\right)+M\parallel x\parallel \right]\hfill \\ & \le {\displaystyle \frac{M}{m}}\parallel x\parallel +{\displaystyle \frac{(f_0+f_1{\parallel x\parallel }^{p-1}{)}^{q-1}}{2m}}{\left({\displaystyle \frac{2{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)}}\right)}^{q-1}\left({\psi }_a^{(q-1){\nu }_2}\left(t\right)+{\left({\displaystyle \frac{{\psi }_a^{{\nu }_2}(b,\eta )}{{\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)}^{q-1}{\psi }_a^{(q-1)({\rho }_2-1)}\left(t\right)\right),\hfill \end{array}$$

which implies that inequality (17) holds.

Since the proof of (18) is very similar to the last proof, we omit it. □

Lemma 6.

If hypothesis $\left(H_2\right)$ holds, then we have

$$|C_x^{\prime }-C_y^{\prime }|\le {\displaystyle \frac{{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1){\psi }_a^{{\rho }_2-1}(b,\eta )}}L\parallel x-y\parallel ,$$

(19)

$$\begin{array}{ccc}\hfill \left|g\right(t,x)-g(t,y\left)\right|& \le & \left[{\displaystyle \frac{(q-1){\rho }^{q-2}L}{m\Gamma ({\nu }_2+1)}}\left({\psi }_a^{{\nu }_2}\left(t\right)+{\displaystyle \frac{{\psi }_a^{{\rho }_2-1}\left(t\right){\psi }_a^{{\nu }_2}(b,\eta )}{{\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)+{\displaystyle \frac{M}{m}}\right]\parallel x-y\parallel ,\hfill \end{array}$$

(20)

and

$$\begin{array}{ccc}\hfill |C_x-C_y|& \le & \left[{\displaystyle \frac{(q-1){\rho }^{q-2}L}{\Gamma ({\nu }_2+1)}}\left({\displaystyle \frac{\Gamma ({\nu }_2+1)}{\Gamma ({\nu }_1+{\nu }_2+1)}}{\psi }_a^{{\nu }_1+{\nu }_2}(b,\eta )+{\displaystyle \frac{{\psi }_a^{{\rho }_2-1}(b,\eta ){\psi }_a^{{\nu }_1+{\rho }_2-1}(b,\eta )}{\Gamma ({\nu }_1+{\rho }_2){\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)\right.\hfill \\ & +& \left.{\displaystyle \frac{M{\psi }_a^{{\nu }_1}(b,\eta )}{\Gamma ({\nu }_1+1)}}\right]{\displaystyle \frac{\parallel x-y\parallel }{m{\psi }_a^{{\rho }_1-1}(b,\eta )}},\hfill \end{array}$$

(21)

where g, $C_x$ and $C_x^{\prime }$ are given, respectively, by Equations (12)–(14).

Proof. 

Using Equation (14) and properties of the fractional integral, we obtain

$$\begin{array}{cc}\hfill |C_x^{\prime }-C_y^{\prime }|& \le {\displaystyle \frac{I_a^{{\nu }_2}\left|f(b,x)-f(b,y)\right|+I_a^{{\nu }_2}\left|f(\eta ,x)-f(\eta ,y)\right|}{{\psi }_a^{{\rho }_2-1}(b,\eta )}}\hfill \\ \hfill & \le {\displaystyle \frac{{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1){\psi }_a^{{\rho }_2-1}(b,\eta )}}L\parallel x-y\parallel .\hfill \end{array}$$

(22)

By the same argument as in (22) and using Lemma 4, we prove Equation (20).

To complete the proof of this Lemma, we use the same argument as in (22) and using Equation (12). □

Next, we use the Volterra-type integral to prove the existence and the uniqueness of a solution.

Theorem 2.

Assume that the function f satisfies hypotheses $\left(H_1\right)$ and $\left(H_2\right)$, if, in addition, we have

$${\left[m-{\displaystyle \frac{2M{\psi }_a^{{\nu }_1}\left(b\right)}{\Gamma ({\nu }_1+1)}}\right]}^{p-1}\le {\displaystyle \frac{2f_1{{\rm Y}}^{p-1}{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)}}.$$

(23)

Then, the problem $\left(\mathcal{P}\right)$ has at least one solution defined in the interval J.

Proof. 

We begin by defining the following set:

$$U=\left\{x\in C(J,\mathbb{R}) \mathrm{such} \mathrm{that} D_a^{{\nu }_2,{\mu }_2}{D}_a^{{\nu }_1,{\mu }_1}x\in C(J,\mathbb{R})\right\},$$

and we define the map F as follows:

$$\begin{array}{cc}F:U& \to U\\ x& \mapsto & I_a^{{\nu }_1}g(t,x)+{\displaystyle \frac{{\psi }_a^{{\rho }_1-1}\left(t\right)}{\Gamma \left({\rho }_1\right)}}{C}_x.\end{array}$$

(24)

The proof will be divided into four steps.

First step: In this step, we will prove that F is a continuous map. For this purpose, let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be a sequence that converges to x in U. Using hypothesis $H_2$ and Equations (20) and (21), we obtain

$$\begin{array}{cc}\hfill & \left|F\left(x_n\left(t\right)\right)-F\left(x\left(t\right)\right)\right|\le I_a^{{\nu }_1}\left[|g(t,x_n)-g(t,x)|\right]+{\psi }_a^{{\rho }_1-1}\left(t\right)|C_{x_n}-C_x|\hfill \\ & \le I_a^{{\nu }_1}\left[\left[{\displaystyle \frac{(q-1){\rho }^{q-2}L}{m\Gamma ({\nu }_2+1)}}\left({\psi }_a^{{\nu }_2}\left(t\right)+{\displaystyle \frac{{\psi }_a^{{\rho }_2-1}\left(t\right){\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma \left({\rho }_2\right){\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)+{\displaystyle \frac{M}{m}}\right]\parallel x_n-x\parallel \right]\hfill \\ & +{\displaystyle \frac{{\psi }_a^{{\rho }_1-1}\left(t\right)}{\Gamma \left({\rho }_1\right)}}\left[{\displaystyle \frac{(q-1){\rho }^{q-2}L}{\Gamma ({\nu }_2+1)}}\left({\displaystyle \frac{\Gamma ({\nu }_2+1)}{\Gamma ({\nu }_1+{\nu }_2+1)}}{\psi }_a^{{\nu }_1+{\nu }_2}(b,\eta )+{\displaystyle \frac{{\psi }_a^{{\rho }_2-1}(b,\eta ){\psi }_a^{{\nu }_1+{\rho }_2-1}(b,\eta )}{\Gamma ({\nu }_1+{\rho }_2){\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)\right.\hfill \\ & +\left.{\displaystyle \frac{M{\psi }_a^{{\nu }_1}(b,\eta )}{\Gamma ({\nu }_1+1)}}\right]{\displaystyle \frac{\parallel x_n-x\parallel }{m{\psi }_a^{{\rho }_1-1}(b,\eta )}}\hfill \\ & \le \left[{\displaystyle \frac{(q-1){\rho }^{q-2}L}{\Gamma ({\nu }_2+1)}}\left({\displaystyle \frac{\Gamma ({\nu }_2+1)}{\Gamma ({\nu }_1+{\nu }_2+1)}}{\psi }_a^{{\nu }_1+{\nu }_2}\left(t\right)+{\displaystyle \frac{{\psi }_a^{{\rho }_2-1}(b,\eta ){\psi }_a^{{\nu }_1+{\rho }_2-1}(b,\eta )}{\Gamma ({\nu }_1+{\rho }_2){\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)\right.\hfill \\ & +\left.{\displaystyle \frac{M{\psi }_a^{{\nu }_1}\left(t\right)}{\Gamma ({\nu }_1+1)}}\right]{\displaystyle \frac{\parallel x_n-x\parallel }{m}}\hfill \\ & +{\displaystyle \frac{{\psi }_a^{{\rho }_1-1}\left(t\right)}{\Gamma \left({\rho }_1\right)}}\left[{\displaystyle \frac{(q-1){\rho }^{q-2}L}{\Gamma ({\nu }_2+1)}}\left({\displaystyle \frac{\Gamma ({\nu }_2+1)}{\Gamma ({\nu }_1+{\nu }_2+1)}}{\psi }_a^{{\nu }_1+{\nu }_2}(b,\eta )+{\displaystyle \frac{{\psi }_a^{{\rho }_2-1}(b,\eta ){\psi }_a^{{\nu }_1+{\rho }_2-1}(b,\eta )}{\Gamma ({\nu }_1+{\rho }_2){\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)\right.\hfill \\ & +\left.{\displaystyle \frac{M{\psi }_a^{{\nu }_1}(b,\eta )}{\Gamma ({\nu }_1+1)}}\right]{\displaystyle \frac{\parallel x_n-x\parallel }{m{\psi }_a^{{\rho }_1-1}(b,\eta )}},\hfill \end{array}$$

So, it follows that

$$\begin{array}{cc}\hfill & {\displaystyle \parallel F\left(x_n\right)-F\left(x\right)\parallel =\underset{t\in J}{max}\left|F\left(x_n\left(t\right)\right)-F\left(x\left(t\right)\right)\right|}\hfill \\ & \le \left[{\displaystyle \frac{(q-1){\rho }^{q-2}L}{\Gamma ({\nu }_2+1)}}\left({\displaystyle \frac{\Gamma ({\nu }_2+1)}{\Gamma ({\nu }_1+{\nu }_2+1)}}{\psi }_a^{{\nu }_1+{\nu }_2}(b,\eta )+{\displaystyle \frac{{\psi }_a^{{\rho }_2-1}(b,\eta ){\psi }_a^{{\nu }_1+{\rho }_2-1}(b,\eta )}{\Gamma ({\nu }_1+{\rho }_2){\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)\right.\hfill \\ & +\left.{\displaystyle \frac{M{\psi }_a^{{\nu }_1}(b,\eta )}{\Gamma ({\nu }_1+1)}}\right]\left[1+{\displaystyle \frac{{\psi }_a^{{\rho }_1-1}\left(b\right)}{\Gamma \left({\rho }_1\right){\psi }_a^{{\rho }_1-1}(b,\eta )}}\right]{\displaystyle \frac{\parallel x_n-x\parallel }{m}}\underset{n\to \infty }{\to }0.\hfill \end{array}$$

This last piece of information implies that F is continuous.

Second step: In this step, we prove the uniform boundedness of F. Let $r>0,$ and we introduce the closed ball $B_r$, which is defined by

$$B_r=\{x\in U:\parallel x\parallel \le r\}.$$

From hypothesis $\left(H_1\right)$, Lemma 5 and the fact that ${\psi }_a\left(b\right)\le {\psi }_a(b,\eta )$, we obtain

$$\begin{array}{cc}\hfill & \parallel F\left(x\right)\parallel =\underset{t\in J}{max}\left|I_a^{{\nu }_1}g(t,x)-{\psi }_a^{{\rho }_1-1}\left(t\right)C_x\right|\hfill \\ & \le {\displaystyle \frac{1}{m}}\underset{t\in J}{max}\left\{I_a^{{\nu }_1}\left[M\parallel x\parallel +{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{2(f_0+f_1\parallel x{\parallel }^{p-1})}{\Gamma ({\nu }_2+1)}}\right)}^{q-1}\left({\psi }_a^{(q-1){\nu }_2}\left(t\right)\right.\right.\right.\hfill \\ & \left.\left.+{\left({\displaystyle \frac{{\psi }_a^{{\nu }_2}(b,\eta )}{{\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)}^{q-1}{\psi }_a^{(q-1)({\rho }_2-1)}\left(t\right)\right)\right]\hfill \\ & +\left.{\psi }_a^{{\rho }_1-1}\left(t\right)\left[{\psi }_a^{{\nu }_1-{\rho }_1+1}(b,\eta )\left[{\displaystyle \frac{M}{\Gamma ({\nu }_1+1)}}\parallel x\parallel +{\rm Y}{\displaystyle \frac{(f_0+f_1{\parallel x\parallel }^{p-1}{)}^{q-1}}{2}}{\left({\displaystyle \frac{2{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)}}\right)}^{q-1}\right]\right]\right\}\hfill \\ & \le {\displaystyle \frac{{\psi }_a^{{\nu }_1}\left(b\right)}{m}}\left[{\displaystyle \frac{2M}{\Gamma ({\nu }_1+1)}}\parallel x\parallel +{\rm Y}(f_0+f_1{\parallel x\parallel }^{p-1}{)}^{q-1}{\left({\displaystyle \frac{2{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)}}\right)}^{q-1}\right]\hfill \\ & \le {\displaystyle \frac{{\psi }_a^{{\nu }_1}\left(b\right)}{m}}\left[{\displaystyle \frac{2M}{\Gamma ({\nu }_1+1)}}r+{\rm Y}{(f_0+f_{1}r)}^{q-1}{\left({\displaystyle \frac{2{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)}}\right)}^{q-1}\right]:=r^{\prime }.\hfill \end{array}$$

(25)

Since $r^{\prime }$ is independent of x and t, F is uniformly bounded.

Third step: In this step, we checked that F maps the bounded set into an equi-continuous set. For this goal, let $r>0$. Then, for all $x\in B_r$ and $t\in J$, we conclude from Lemma 5 that

$$\begin{array}{cc}\hfill & \left|g\right(t,x\left)\right|\hfill \\ & \le {\displaystyle \frac{1}{m}}\left[M\parallel x\parallel +{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{2(f_0+f_1{\parallel x\parallel }^{p-1}{)}^{q-1}}{\Gamma ({\nu }_2+1)}}\right)}^{q-1}\left({\psi }_a^{(q-1){\nu }_2}\left(t\right)+{\left({\displaystyle \frac{{\psi }_a^{{\nu }_2}(b,\eta )}{{\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)}^{q-1}{\psi }_a^{(q-1)({\rho }_2-1)}\left(t\right)\right)\right]\hfill \\ & \le \underset{:=c_g}{\underbrace{{\displaystyle \frac{1}{m}}\left[Mr+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{2{(f_0+f_1{r}^{p-1})}^{q-1}}{\Gamma ({\nu }_2+1)}}\right)}^{q-1}\left({\psi }_a^{(q-1){\nu }_2}\left(b\right)+{\left({\displaystyle \frac{{\psi }_a^{{\nu }_2}(b,\eta )}{{\psi }_a^{{\rho }_2-1}(b,\eta )}}\right)}^{q-1}{\psi }_a^{(q-1)({\rho }_2-1)}\left(b\right)\right)\right]}},\hfill \end{array}$$

Let $t<t^{\prime }$ be two fixed real numbers in J. Then, for all $x\in B_r$, we have

$$\begin{array}{cc}\hfill & I_a^{{\nu }_1}\left(g(t^{\prime },x)-g(t,x)\right)\hfill \\ & \le {\displaystyle \frac{1}{\Gamma \left({\nu }_1\right)}}\left[{\int }_a^t{\psi }^{\prime }\left(s\right)\left({\psi }_{t^{\prime }}^{{\nu }_1}\left(s\right)-{\psi }_t^{{\nu }_1}\left(s\right)\right)g(s,x)ds+{\int }_t^{t^{\prime }}{\psi }^{\prime }\left(s\right){\psi }_{t^{\prime }}^{{\nu }_1}\left(s\right)g(s,x)ds\right]\hfill \\ & \le {\displaystyle \frac{C_g}{\Gamma \left({\nu }_1\right)}}\left[{\int }_a^t{\psi }^{\prime }\left(s\right)\left({\psi }_{t^{\prime }}^{{\nu }_1}\left(s\right)-{\psi }_t^{{\nu }_1}\left(s\right)\right)ds+{\int }_t^{t^{\prime }}{\psi }^{\prime }\left(s\right){\psi }_{t^{\prime }}^{{\nu }_1}\left(s\right)ds\right]\hfill \\ & \le {\displaystyle \frac{C_g}{\Gamma ({\nu }_1+1)}}\left[{\psi }_{t^{\prime }}^{{\nu }_1}\left(a\right)-{\psi }_t^{{\nu }_1}\left(a\right)\right].\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill & \left|F\left(x\left(t^{\prime }\right)\right)-F\left(x\left(t\right)\right)\right|\le \left|I_a^{{\nu }_1}\left(g(t^{\prime },x)-g(t,x)\right)\right|+{\displaystyle \frac{{\psi }_a^{{\rho }_1-1}\left(t^{\prime }\right)-{\psi }_a^{{\rho }_1-1}\left(t\right)}{\Gamma \left({\rho }_1\right)}}\left|C_x\right|\hfill \\ \hfill & \le {\displaystyle \frac{C_g}{\Gamma ({\nu }_1+1)}}\left({\psi }_{t^{\prime }}^{{\nu }_1}\left(a\right)-{\psi }_t^{{\nu }_1}\left(a\right)\right)+{\displaystyle \frac{{\psi }_a^{{\rho }_1-1}\left(t^{\prime }\right)-{\psi }_a^{{\rho }_1-1}\left(t\right)}{\Gamma \left({\rho }_1\right)}}\left|C_x\right|\underset{t^{\prime }\to t}{\to }0\hfill \end{array}$$

Thus, $F\left(x\left(t^{\prime }\right)\right)$ converge to $F\left(x\right(t\left)\right)$, therefore F maps the bounded set into an equicontinuous set in U.

Fourth step: In this step, we prove that the set T defined by

$$T=\{x\in U,s.t.,x=\mu F(x),\mu \in (0,1\left)\right\},$$

is bounded. For this, let $x\in T$ and $t\in J$, then by using Inequality (25), we have the following:

$$\begin{array}{cc}\hfill \parallel x\parallel & =\mu \parallel F\left(x\right)\parallel <\parallel F\left(x\right)\parallel \hfill \\ & {\displaystyle \le \underset{t\in J}{sup}\left[\left|I_a^{{\nu }_1}\left(g(t,x)\right)\right|+\frac{{\psi }_a^{{\rho }_1-1}\left(t\right)}{\Gamma \left({\rho }_1\right)}\left|C_x\right|\right]}\hfill \\ & \le {\displaystyle \frac{{\psi }_a^{{\nu }_1}\left(b\right)}{m}}\left[{\displaystyle \frac{2M}{\Gamma ({\nu }_1+1)}}\parallel x\parallel +{\rm Y}(f_0+f_1{\parallel x\parallel }^{p-1}{)}^{q-1}{\left({\displaystyle \frac{2{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)}}\right)}^{q-1}\right]\hfill \end{array}$$

It follows that

$$\left({\left[m-{\displaystyle \frac{2M{\psi }_a^{{\nu }_1}\left(b\right)}{\Gamma ({\nu }_1+1)}}\right]}^{p-1}-{\displaystyle \frac{2f_1{{\rm Y}}^{p-1}{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)}}\right){\parallel x\parallel }^{p-1}\le \left({\displaystyle \frac{2f_0{{\rm Y}}^{p-1}{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)}}\right){\parallel x\parallel }^{p-1}.$$

It is not difficult to see that if

$${\left[m-{\displaystyle \frac{2M{\psi }_a^{{\nu }_1}\left(b\right)}{\Gamma ({\nu }_1+1)}}\right]}^{p-1}\le {\displaystyle \frac{2f_1{{\rm Y}}^{p-1}{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)}},$$

that we obtain

$$\parallel x\parallel \le {\left({\displaystyle \frac{2f_0\Gamma ({\nu }_1+1){{\rm Y}}^{p-1}{\psi }_a^{{\nu }_2}(b,\eta )}{\Gamma ({\nu }_2+1)\left[m\Gamma ({\nu }_1+1)-2M{\psi }_a^{{\nu }_1}\left(b\right)\right]}}\right)}^{q-1}$$

Therefore, T is bounded.

Using Steps 1 to 3 and the Ascoli–Arzela Theorem, we deduce that $F\left(B_r\right)$ is included in a compact set and therefore, F is completely continuous. Using the fourth step and the Schaefer fixed point Theorem, we conclude that F has a fixed point. Using Theorem 1 and the definition of the map F given in (24), we deduce that problem $\left(\mathcal{P}\right)$ has at least one solution on J. □

4. Application

To demonstrate the applicability of the proposed results, let us consider the following simple examples.

4.1. Example 1

Considering the following $\psi$–Langevin fractional equations with antiperiodic boundary condition involving p-Laplacian operator:

$$\left(\mathcal{P}\right)\left\{\begin{array}{cc}{}^{\psi }D_0^{{\nu }_2,{\mu }_2}{\Phi }_{5/2}\left((3+cos\left(t\pi \right)){}^{\psi }D_0^{{\nu }_1,{\mu }_1}\left(x\right)\left(t\right)+sin\left(t\pi \right)x\left(t\right)\right)={\displaystyle \frac{e^t}{1+e^t}}(1+{sin}^2\left(x\left(t\right)\right)),\hfill & t\in [0,1],\hfill \\ x\left({\displaystyle \frac{1}{4}}\right)=-x\left(1\right)=A,\hfill & \\ {}^{\psi }D_0^{{\nu }_1,{\mu }_1}x\left({\displaystyle \frac{1}{4}}\right)=-{}^{\psi }D_0^{{\nu }_1,{\mu }_1}x\left(1\right)=B.\hfill \end{array}\right.$$

It is clear that f satisfies $\left(H_1\right)$ and $\left(H_2\right)$ for $L={\displaystyle \frac{2e}{1+e}}$, $f_0=f_1={\displaystyle \frac{e}{1+e}}$, $m=3$ and $M=1$. For $\psi \left(t\right)=t$, if ${\mu }_i\to 0$ or ${\mu }_i=1$ for $i=1,2$, the condition (23) is verified. Then, according to Theorem 2, the Langevin Riemann–Liouville and the Langevin Caputo fractional problem, respectively, has at least one solution on $[0,{\displaystyle \frac{\pi }{2}}]$.

4.2. Example 2

Considering the following Langevin fractional equations with an antiperiodic boundary condition involving p-Laplacian operator:

$$\left(\mathcal{P}\right)\left\{\begin{array}{cc}{}^{\psi }D_0^{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}{\Phi }_{3/2}\left({}^{\psi }D_0^{{\displaystyle \frac{4}{5}},{\displaystyle \frac{1}{5}}}\left(x\right)\left(t\right)+{\displaystyle \frac{1}{40}}x\left(t\right)\right)={\displaystyle \frac{1}{4}}x\left(t\right)+sin\left(t\right),\hfill & t\in [0,1],\hfill \\ x\left(0\right)=-x\left(1\right)=A,\hfill & \\ {}^{\psi }D_0^{{\displaystyle \frac{4}{5}},{\displaystyle \frac{1}{5}}}x\left(0\right)=-{}^{\psi }D_0^{{\displaystyle \frac{4}{5}},{\displaystyle \frac{1}{5}}}x\left(1\right)=B.\hfill \end{array}\right.$$

It is easy to verify that f satisfies $\left(H_1\right)$ and $\left(H_2\right)$ for $L={\displaystyle \frac{1}{10}}$, $f_0=1$, $f_1={\displaystyle \frac{1}{4}}$, $m=1$, $M={\displaystyle \frac{1}{40}}$ and ${\rm Y}=1.0918711879$. For $\psi \left(t\right)=e^t$, the condition (23) is verified. Then, from Theorem 2, the p-Laplacian Langevin with $\psi$–Hilfer fractional derivative and antiperiodic boundary conditions problem has at least one solution on $[0,1]$.

5. Conclusions

In this paper, we demonstrated the existence of at least one solution for a p-Laplacian Langevin differential equation involving the $\psi$-Hilfer fractional derivative with antiperiodic boundary conditions. Specifically, we define an operator F associated with our problem. We then prove that this operator is continuous, uniform, and bounded. Additionally, we verify that F maps bounded sets into equicontinuous sets. This suggests that by applying the Ascoli–Arzelà Theorem, F is completely continuous. Later, we prove that the set $T=\{x\in U,s.t.,x=\mu F(x),\mu \in (0,1\left)\right\}$ is bounded. We conclude by using the Schaefer fixed theorem that the maps F have a fixed point. This proves that our problem has at least one solution. Finally, we give an example to illustrate the applicability of our results. Comparing our work with the paper [27], we consider the $\psi$–Hilfer fractional Langevin equations with $p$-Laplacian operator. We notice that the $\psi$–Hilfer fractional generalizes many fractional derivatives; for example, the Riemann–Liouville derivative and Caputo derivative. So, taking $\psi \left(x\right)=ln\left(x\right)$ and ${\mu }_1,{\mu }_2\to 0$, we obtain the differential equation studied in [27].

In our next work, we will discuss the uniqueness of the solution. Also, we will validate our results by adding numerical examples.