Solvability of Generalized Hilfer Fractional p-Laplacian Differential Problems in Orlicz Spaces

Abstract

This paper investigates non-fractional operators, a type of nonlocal operator, within the framework of Orlicz spaces. Using inclusions between certain function spaces, we prove the continuity and/or compactness of generalized operators in Orlicz spaces and show that solutions exist for integral equations of fractional order. We also introduce a generalized Hilfer-type derivative and examine the equivalence of differential and integral problems. Finally, we relate these results to the study of compositional p-Laplacian fractional problems involving generalized Hilfer fractional derivatives. Among other things, we prove the existence of solutions to such problems in Orlicz and Orlicz–Sobolev spaces.

1. Introduction and Preliminaries

This paper addresses issues related to generalized Hilfer fractional operators, bringing together several strands of current research. It examines the properties of these operators in Orlicz spaces. More specifically, it investigates fractional p-Laplacian differential problems in these spaces. Building upon research into classical p-Laplacian problems, for which solutions are found in Orlicz–Sobolev spaces [1], this study considers the one-dimensional case.

However, to address the shortcomings of the classical theory of fractional-order equations, we consider the one-dimensional case here. These equations also fall within the field of research on nonlocal equations, for which fractional-order operators are a well-known example. By considering an operator whose special cases arise in previous problems, we extend the scope of existing research in this area, beginning naturally with Hilfer operators. We define a combination of fractional and non-fractional integral operators. A key focus of the paper is to obtain forms equivalent to integral and differential operators in Orlicz spaces and to apply these to some generalized compositional p-Laplacian problems.

Having examined the properties of operators acting on Orlicz spaces (initiated in [2] for the Riemann–Liouville integral operator; see also [3,4]), we investigate p-Laplacian fractional problems involving generalized Hilfer fractional derivatives. As well as solving problems in Orlicz–Sobolev spaces (see, for example, [5]), we analyze existing cases and address gaps in the study of the parameters of these types of equations.

The operator L is considered nonlocal if its value at point x always depends on the value of the function u outside a neighborhood of t. Usually, it is represented as a singular integral operator:

$$\left(Lu\right)\left(t\right)={\mathit{\int }}_{{\mathbb{R}}^n}\left(u\left(t\right)-u\left(s\right)\right)K(t,s)ds.$$

The most well-known example of a nonlocal operator is the fractional Laplacian, denoted by ${(-\Delta )}^s$, where the value at a given point is the average of the jumps, weighted according to their distance. This is our starting point for research connecting this topic with classical operators that appear in fractional calculus. This paper does not cover research conducted using Fourier analysis; however, we recommend ref. [6] for those interested in the relationships between fractional and pseudo-differential operators.

A significant part of our results relates to selecting the appropriate function spaces for our research. Our aim is to maximize the regularity of the solutions to the fractional-order problems under investigation (with the formulation in terms of the considered class of operators), thereby surpassing the scope of conventional studies in function spaces with weighted norms. Taking inspiration from solutions in Orlicz–Sobolev spaces for p-Laplacian, we also examine generalized fractional operators in Orlicz spaces. Current research on problems in generalized Orlicz–Sobolev spaces involves both problems with appropriate fractional-order operators [7], including the study of weak solutions to a Kirchhoff-type fractional problem [8], and extends to interesting practical applications that lead to both generalized fractional-order operators and Orlicz spaces [9].

Studying integral operators in this class of spaces enables us to extend previous findings while preserving the regularity of solutions (in terms of both the domain and the range of the operators under consideration). In ref. [10], we investigated the basic properties of operators that are combinations of classical fractional-order operators and the identity operator acting on Hölder spaces. In particular, we examined certain improvement properties of integral operators and studied the problem of the mutual invertibility of differential and integral operators, which forms the basis for applying such operators to differential problems. Continuing these studies, we prove additional properties of the operators currently under investigation in this paper and discuss their applications in certain generalizations of equations involving the fractional p-Laplacian. As noted above, these operators should be studied within a framework of spaces of discontinuous functions, so we continue this study when the operators act on Orlicz spaces.

The paper is structured as follows: Section 1 contains all the auxiliary results, particularly those relating to Orlicz spaces.

In Section 2, we examine some recently introduced generalized integral nonlocal operators, which combine fractional and non-fractional operators. Given their applications, our focus is on properties related to the domain and range of these operators.

Section 3 is devoted to extending the properties of the considered operators within the context of Orlicz spaces. This crucial aspect enables us to examine the regularity of solutions to problems defined in terms of the studied operators, which are extensions of the specific cases examined previously. These results are also used to demonstrate the existence of solutions to certain fractional-order generalized integral equations in Orlicz spaces.

Section 4 is devoted to studying the inverse Hilfer-type differential operators of the integral operators examined in the preceding sections. We provide a brief discussion of the introduced notion in relation to other fractional derivatives and examine p-fractional problems involving generalized Hilfer fractional derivatives in order to demonstrate the importance of the notions and results introduced. We prove that the solutions lie in certain Orlicz spaces and, for that matter, in certain Orlicz–Sobolev spaces. This also establishes a link between studying problems governed by p-Laplacian operators and studying fractional problems involving nonlocal derivatives.

For the paper to be self-contained, we must first recall and summarize the key facts about this class of spaces.

In this section, we also need to summarize the relevant definitions and results concerning important function spaces, specifically Orlicz classes. A function $\psi :[0,\infty )\to [0,\infty )$ is a Young function if it is continuous, even, convex, and non-decreasing, as well as if $\psi \left(0\right)=0$. If the function satisfies some additional conditions (see [11,12]), it is called an N-function. If $\psi ,\tilde{\psi }$ are two mutually complementary N-functions (see [11,12]), then the Young inequality holds. If $\psi$ and $\tilde{\psi }$ are two mutually complementary N-functions, then

$$uv\le \psi \left(u\right)+\tilde{\psi }\left(v\right),u,v\in \mathbb{R}(\mathrm{Young} \mathrm{inequality}).$$

Define

1.

${\mathfrak{L}}_{\psi }[a,b]:$ The class of measurable real-valued functions $f:[a,b]\to \mathbb{R}$ for which

$$\langle \psi ,f\rangle :={\mathit{\int }}_{[a,b]}\psi \left(f\left(t\right)\right)dt<\infty .$$

2.

$L_{\psi }[a,b]:$ The class of measurable real-valued functions $f:[a,b]\to \mathbb{R}$ for which

$${\mathit{\int }}_{[a,b]}f\left(t\right)g\left(t\right)dt<\infty ,\mathrm{for} \mathrm{all} g\in {\mathfrak{L}}_{\psi }[a,b].$$

3.

$E_{\psi }[a,b]:$ The closure of $L_{\infty }[a,b]$ in $L_{\psi }[a,b]$.

The pair $\left[L_{\psi }[a,b],{∥·∥}_{\psi }\right]$, where

$${∥f∥}_{\psi }:=\underset{\langle \tilde{\psi },g\rangle \le 1,g\in {\mathfrak{L}}_{\tilde{\psi }}}{sup}\left|{\mathit{\int }}_{[a,b]}f\left(s\right)g\left(s\right)ds\right|<\infty ,$$

forms a Banach space called an Orlicz space. Note that every integrable function on $[a,b]$ belongs to an Orlicz class (see, e.g., Lemma 9.2 in [11]), and that

$$〈\psi ,{\displaystyle \frac{f}{{∥f∥}_{\psi }}}〉\le 1.$$

(1)

Also recall (see Theorem 8.2 [11]) that if $\psi$ satisfies the ${\Delta }_2$-condition, i.e., if there exists a constant $K>0$ and $x_0\ge 0$ such that $\psi \left(2x\right)\le K\psi \left(x\right)$ for all $x\ge x_0$, then for any $uu,v\in L_{\psi }[a,b]$, we have

$$〈\psi ,\alpha u+\beta v〉\le {\displaystyle \frac{1}{2}}〈\psi ,2\alpha u〉+{\displaystyle \frac{1}{2}}〈\psi ,2\beta v〉,\alpha ,\beta \in \mathbb{R}.$$

(2)

In addition, Hölder’s inequality (see page 74 in [11]) states that

$$\left|{\mathit{\int }}_{[a,b]}f\left(s\right)g\left(s\right)ds\right|\le {∥f∥}_{\psi }{∥g∥}_{\tilde{\psi }},f\in L_{\psi }[a,b],g\in L_{\tilde{\psi }}.$$

Further, in accordance with Young’s inequality, it is clear that $E_{\psi }\subseteq L_{\psi }[a,b]\subseteq {\mathfrak{L}}_{\psi }[a,b]$ and that

$${∥f∥}_{\psi }\le \underset{\langle \tilde{\psi },g\rangle \le 1,g\in {\mathfrak{L}}_{\tilde{\psi }}}{sup}\left|{\mathit{\int }}_{[a,b]}\left[\psi \left(f\left(s\right)\right)+\tilde{\psi }\left(g\left(s\right)\right)\right]ds\right|\le \langle \psi ,f\rangle +1.$$

(3)

However, due to Theorem 10.5 in [11], we have

$${∥f∥}_{\psi }=\underset{k>0}{inf}\left\{{\displaystyle \frac{1}{k}}\left[1+\langle \psi ,kf\rangle \right]\right\},\mathrm{for} \mathrm{any} f\in L_{\psi }[a,b].$$

(4)

Let us recall that a function $f\in L_{\psi }[a,b]$ is said to have an absolutely continuous norm if, for every $\epsilon >0$, there exists $\delta >0$ such that

$${∥f{\mathrm{\Lambda }}_{\mathfrak{P}}∥}_{\psi }=\underset{<\tilde{\psi },g>\le 1}{sup}\left|{\mathit{\int }}_{\mathfrak{P}}f\left(t\right)g\left(t\right)dt\right|<\epsilon ,$$

whenever $\mathrm{meas}\left(\mathfrak{P}\right)<\delta$, $g\in \mathfrak{L}\tilde{\psi }[a,b]$, where $\mathrm{\Lambda }\mathfrak{P}$ denotes the characteristic function of a measurable subset $\mathfrak{P}\subseteq [a,b]$.

A family of functions $\mathrm{\Omega }\subset L_{\psi }[a,b]$ is said to have equi-absolutely continuous norms if, for every $\epsilon >0$, there exists $\delta >0$ such that for all $f\in \mathrm{\Omega }$, we have

$${∥f{\mathrm{\Lambda }}_{\mathfrak{P}}∥}_{\psi }<\epsilon ,\mathrm{provided}\mathrm{meas}\left(\mathfrak{P}\right)<\delta .$$

Recall that in refs. [11,12], it is proven that a necessary and sufficient condition for a function $f\in L_{\psi }[a,b]$ to belong to $E_{\psi }[a,b]$ is for its norm to be absolutely continuous. However, $E_{\psi }[a,b]\equiv L_{\psi }[a,b]$ if $\psi$ satisfies the ${\Delta }_2$-condition (see Chapter II §10 in [11]).

To obtain the best possible assumptions, we consider several types of convergence in Orlicz spaces. However, this requires indicating the relationships among them.

The following lemmas provide the necessary and sufficient conditions under which a sequence of functions converges in norm to a function in $E_{\psi }[a,b]$. To this end, we recall certain types of convergence of sequences of measurable functions. Denote by $S=S\left(\right[a,b\left]\right)$ the set of measurable functions on $[a,b]$. Identifying functions that are equal almost everywhere, the set S is endowed with the metric $d(x,y)={inf}_{k>0}[a+meas\{s:|x\left(s\right)-y\left(s\right)|\ge k\left\}\right]$, and we obtain a complete metric space. Moreover, convergence in measure on $[a,b]$ is equivalent to convergence with respect to metric d. Compactness in such spaces is called “compactness in measure” (cf. [13]).

The modular is defined by ${\rho }_{\psi }\left(f\right)={\mathit{\int }}_{[a,b]}\psi \left(\right|f\left|\right)\left(s\right)ds$. A sequence $\left\{f_n\right\}\subset L^{\psi }$ is said to converge in modular (or to be modular convergent) to $f\in L_{\psi }[a,b]$ if there exists $\lambda >0$ such that

$${\mathit{\int }}_{[a,b]}\psi \left({\displaystyle \frac{|f_n-f|}{\lambda }}\right)\left(s\right)ds\to 0\mathrm{as}n\to \infty .$$

Equivalently, ${\rho }_{\psi }\left({\displaystyle \frac{f_n-f}{\lambda }}\right)\to 0$ as $n\to \infty$ for some instances where $\lambda >0$. As shown in Proposition 5.1 [13], under the assumption that $\psi$ satisfies the ${\Delta }_2$-condition, bounded sets that are additionally compact in measure are compact in $L_{\psi }[a,b]$ if and only if they have equi-absolutely continuous norms. In regular ideal spaces ([13]), such as $L_{\psi }[a,b]$, compactness in measure is equivalent to weak compactness. Therefore, by considering $E_{\psi }[a,b]$, we obtain the following useful lemma:

Lemma 1 

(Lemma 11.2 [11]). A sequence of functions $f_n\in E_{\psi }[a,b]$$(n=1,2,3,\dots )$, which converges in measure, converges in norm if and only if it has equi-absolutely continuous norms.

Since, from each sequence of the family $\mathrm{\Omega }\subset E_{\psi }$, we can select a subsequence that converges in measure, by virtue of Lemma 1, we obtain the following compactness criterion:

Lemma 2 

(Theorem 11.3 [11]). The family $\mathrm{\Omega }\subset E_{\psi }$ is compact in $L_{\psi }[a,b]$ if it has equi-absolutely continuous norms and is compact in measure.

In addition to norm convergence, we also require a characterization of weak convergence within this class of spaces.

Definition 1. 

A sequence of functions $\left\{f_n\right\}\subset L_{\psi }[a,b]$ is said to be $E_{\tilde{\psi }}$-weakly convergent if

$$\left\{\phi \left(f_n\right):={\mathit{\int }}_{[a,b]}{f}_n\left(t\right)\phi \left(t\right)dt\right\}convergesforevery\phi \in E_{\tilde{\psi }}[a,b].$$

It should be noted that Definition 1 coincides with the usual definition of weak convergence if both $\psi$ and $\tilde{\psi }$ satisfy the ${\Delta }_2$-condition (see page 130 in [11]). Recall that (see page 226 in [11]; also see [14]) the $E_{\tilde{\psi }}$-weak convergence of a sequence of functions from $L_{\psi }[a,b]$ implies the boundedness of the norms of the elements of the sequence.

Besides the norm $∥·∥\psi$ on $L_{\psi }[a,b]$, we also have an equivalent norm, known as the Luxemburg norm:

$${∥f∥}_{L_{\psi }}:=inf\left\{k>0:{\mathit{\int }}_a^b\psi \left({\displaystyle \frac{\left|f\right(t\left)\right|}{k}}\right)dt\le 1\right\},{∥f∥}_{L_{\psi }}\le {\rho }_{\psi }\left(f\right)\le 2{∥f∥}_{L_{\psi }},$$

(5)

for any $f\in L_{\psi }[a,b]$.

The following stronger versions of the Hölder inequality can be obtained:

$$\left|{\mathit{\int }}_a^{b}f\left(s\right)g\left(s\right)ds\right|\le min\left\{{\rho }_{\psi }\left(f\right){∥g∥}_{L_{\tilde{\psi }}},{\rho }_{\tilde{\psi }}\left(g\right){∥f∥}_{L_{\psi }}\right\}\le 2{∥f∥}_{L_{\psi }}{∥g∥}_{L_{\tilde{\psi }}},$$

for $f\in L_{\psi }[a,b]$, $g\in L_{\tilde{\psi }}[a,b]$.

Let $r,r^{*},B>0$ and assume that $\psi ,{\psi }^{*}$ are two Young functions. Recall that refs. [11,15] state that if a Carathéodory function $f(·,·):[a,b]\times \mathbb{R}\to \mathbb{R}$ satisfies

$$\psi \left({\displaystyle \frac{\left|f\right(t,x\left)\right|}{r}}\right)\le c\left(t\right)+B{\psi }^{*}\left({\displaystyle \frac{\left|x\right|}{r^{*}}}\right),$$

(6)

for $x\in \mathbb{R}$, $t\in [a,b]$, $c\in L^1[a,b]$, then the superposition operator $F\left(x\right)\left(t\right):=f(t,x(t\left)\right)$, $t\in [a,b]$ is bounded from $U_{{\psi }^{*}}:=\left\{x\in L_{{\psi }^{*}}:∥x∥{\psi }^{*}<r^{*}\right\}$ into $L_{\psi }[a,b]$. It is also continuous if $\psi$ satisfies the ${\Delta }_2$-condition.

Remark 1. 

We should note that, according to Formula (1.20) in [11], there exists $a_r\ge 0$ such that the growth condition (6) can be written as follows:

$$\left|f(t,x)\right|\le c_r\left(t\right)+a_r{\psi }^{-1}\left({\psi }^{*}\left({\displaystyle \frac{\left|x\right|}{r^{*}}}\right)\right),c_r(·)=r{\psi }^{-1}\left(c(·)\right)\in L_{\psi }[a,b].$$

(7)

Indeed, since $c\in L^1[a,b]$, we deduce that

$$〈\psi ,{\psi }^{-1}\left(c\right)〉={\mathit{\int }}_a^b\psi \left({\psi }^{-1}\left(c\left(t\right)\right)\right)dt\le {\mathit{\int }}_a^{b}c\left(t\right))dt<\infty .$$

Hence, in view of (3), we have that ${∥{\psi }^{-1}\left(c\right)∥}_{\psi }<\infty$ and so ${\psi }^{-1}\left(c\right)\in L_{\psi }[a,b]$. It follows that $c_r(·)\in L_{\psi }[a,b]$. It is also quite obvious that (7) implies (6) when ψ satisfies the ${\Delta }_2$-condition.

Now, we need to describe the properties of operators acting on different function spaces. First, we need an acting condition.

Lemma 3 

(Theorem 17.5 [11]). Assume that $f(·,·):[a,b]\times \mathbb{R}\to \mathbb{R}$ be Carathéodory function. Then,

$${\psi }_2\left(f(s,x)\right)\le k\left(s\right)+l·{\psi }_1\left(x\right),$$

where $l\ge 0$ and $k\in L^1[a,b]$, if and only if the superposition operator F acts from $L_{{\psi }_1}[a,b]$ to $L_{{\psi }_2}[a,b]$.

Unfortunately, in Orlicz spaces, there is no automatic continuity of superposition operators, as in $L^p$ spaces, so the ${\Delta }_2$-condition is useful (though not necessary, as claimed in ref. [13]). From Lemma 3.5 in [13], we can obtain the continuity of F when $F:L_{{\psi }_1}[a,b]\to E_{{\psi }_2}[a,b]$. The next theorem provides, in general, sufficient conditions that ensure the continuity and boundedness of the superposition operator $F\left(x\right)\left(t\right):=f(t,x(t\left)\right)$, $t\in [a,b]$ in Orlicz spaces.

Theorem 1 

(Theorem 17.6 [11] (see also [15])). Assume that $f(·,·):[a,b]\times \mathbb{R}\to \mathbb{R}$ be Carathéodory function and $r^{*}>0$. Assume that $\psi ,{\psi }^{*}$ be two Young functions, where ψ satisfies the ${\Delta }_2$-condition, and f satisfies the growth condition

$$\left|f(t,x)\right|\le c\left(t\right)+B{\psi }^{-1}\left({\psi }^{*}\left({\displaystyle \frac{\left|x\right|}{r^{*}}}\right)\right),x\in \mathbb{R},t\in [a,b],c\in L_{\psi }[a,b],B>0.$$

(8)

Then, F maps $\{x\in L_{{\psi }^{*}}:{∥x∥}_{{\psi }^{*}}\le r^{*}\}$ to $L_{\psi }[a,b]$, and it is always continuous and bounded.

Remark 2. 

Orlicz spaces provide greater flexibility, a more systematic treatment of singularities, and sharper operator estimates, all within a unified functional-analytic framework—something that classical $L^p$ or $C[a,b]$ approaches cannot easily achieve. In particular, as claimed in ref. [2], they are useful for analyzing integral operators with weakly singular or nonlinear kernels, allowing one to go beyond the limitations of classical $L^p$ or $C[a,b]$ spaces.

Orlicz spaces differ from $L^p$ spaces. In $L^p$ spaces, a singular kernel may require restrictive conditions on p or α. Orlicz spaces can include functions that grow more quickly than polynomials, such as exponential or logarithmic functions, which allows a wider range of singularities and weight functions. The modular structure, together with the properties of N-functions, allows one to determine norms and verify the continuity of operators more easily. This is important for proving that an operator is compact. Many classical results in Lebesgue or continuous function spaces thus appear as special cases.

Moreover, modular convergence and Hölder’s inequality in Orlicz spaces provide a natural way to establish equicontinuity of operator images. Kernels with weak singularities, such as ${(t-s)}^{\alpha -1}$, are easier to handle because the complementary function $\tilde{\psi }$ can ‘absorb’ the singularity integrably. Therefore, the Orlicz framework allows for a more systematic and efficient treatment of fractional, weighted, and nonlinear integral operators.

The following theorem forms an important tool in our existence results.

Theorem 2 

([16]). (Rothe fixed-point theorem) Let U be an open and bounded subset of a Banach space E; let $T:\overline{U}\to E$ be completely continuous operator. Then, T has a fixed point if the following condition holds:

$$T(\partial U)\subseteq \overline{U}where\overline{U}\equiv the closure ofUonE,\partial U\equiv the boundary ofU.$$

(9)

2. Generalized Fractional Integral and Differential Operators

Throughout this paper, we assume that the functions ${\mathit{\ell }}_1,{\mathit{\ell }}_2:[0,1]\times [a,b]\to [0,\infty )$ are continuous, and that the following conditions hold: ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L^1[a,b]$ for all $\rho \in (0,1]$, and

$$\left\{\begin{array}{c}{\displaystyle \underset{\rho \to 0}{lim}{\mathit{\ell }}_2(\rho ,t)=1,\underset{\rho \to 0}{lim}{\mathit{\ell }}_1(\rho ,t)=0\mathrm{for} \mathrm{all}t\in [a,b],}\hfill \\ {\displaystyle \underset{\rho \to 1}{lim}{\mathit{\ell }}_2(\rho ,t)=0,\underset{\rho \to 1}{lim}{\mathit{\ell }}_1(\rho ,t)=1\mathrm{for} \mathrm{all}t\in [a,b],}\hfill \\ {\displaystyle {\mathit{\ell }}_2(\rho ,t)\ne 0,\rho \in [0,1),{\mathit{\ell }}_1(\rho ,t)\ne 0,\rho \in (0,1]\mathrm{for} \mathrm{all}t\in [a,b].}\hfill \end{array}\right.$$

(10)

A simple example of such a system is ${\mathit{\ell }}_1(\rho ,t)=\rho t^{1-\rho }$, ${\mathit{\ell }}_2(\rho ,t)=(1-\rho )t^{\rho }$, $\rho \in [0,1]$ and $t\in [0,1]$. Define

$$\mathit{\ell }(\rho ,t):={\mathit{\ell }}_2(\rho ,t)/{\mathit{\ell }}_1(\rho ,t)={\displaystyle \frac{1-\rho }{\rho }}{t}^{2\rho -1},\rho \in (0,1].$$

Obviously, we have ${\mathit{\ell }}_2(\rho ,·)\in C[0,1]$ and ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L_{{\psi }_p}[0,1]$, $\rho \in (0,1]$, where ${\psi }_p(·)={\displaystyle \frac{{|·|}^p}{p}},p\in (1,{(1-\rho )}^{-1})$. The assumption $p>1$ is the standard requirement for Orlicz spaces, as ${\psi }_p(·)$ should be the N-function. Since

$$〈{\psi }_p,{\displaystyle \frac{1}{{\mathit{\ell }}_1(\rho ,·)}}〉={\displaystyle \frac{1}{p\rho }}{\mathit{\int }}_0^1{t}^{p(\rho -1)}dt<\infty ,\mathrm{for} \mathrm{any}p\in \left(1,{\displaystyle \frac{1}{1-\rho }}\right),$$

according to (3), for a given $\rho \in (0,1]$, it follows that ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L_{{\psi }_p}[0,1],p\in (1,{(1-\rho )}^{-1})$.

Let $\varphi \in C^1\left[[a,b],\mathbb{R}\right]$ be a positive increasing function such that ${\varphi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$, with $\varphi \left(a\right)=0$. Define the non-fractional differentiation operators

$${\Delta }_{\varphi }^{\rho ,\mathit{\ell }}f:=\left({\mathit{\ell }}_2(\rho ,t)+{\mathit{\ell }}_1(\rho ,t)\left[\mu +{\displaystyle \frac{1}{{\varphi }^{\prime }}}\mathfrak{D}\right]\right)f,\mu \ge 0,\rho \in [0,1],$$

where f is a differentiable function on $[a,b]$. Although this is not a fractional derivative, it can correspond to the limits or generators of integer-order fractional operators. It can also be used to define new fractional-type operators and corresponds to the integer-order limit of fractional derivatives, such as the Hilfer derivative (Section 4).

• Limit $\rho \to 0$: ${\Delta }_{\varphi }^{\rho ,\mathit{\ell }}f\to f$.
• Limit $\rho \to 1$: ${\Delta }_{\varphi }^{\rho ,\mathit{\ell }}f\to \mu f+{\displaystyle \frac{1}{{\varphi }^{\prime }\left(t\right)}}{\displaystyle \frac{df}{dt}}$.
• Case $\mu =0$: ${\Delta }_{\varphi }^{\rho ,\mathit{\ell }}f={\mathit{\ell }}_2(\rho ,t)f+{\mathit{\ell }}_1(\rho ,t){\displaystyle \frac{1}{{\varphi }^{\prime }\left(t\right)}}{\displaystyle \frac{df}{dt}}$.
• Classical derivative ($\varphi \left(t\right)=t$): ${\Delta }_{\varphi }^{\rho ,\mathit{\ell }}f=\Delta f=({\mathit{\ell }}_2+\mu {\mathit{\ell }}_1)f+{\mathit{\ell }}_1{f}^{\prime }$.
• Pure multiplication (formally ${\mathit{\ell }}_1=0$): ${\Delta }_{\varphi }^{\rho ,\mathit{\ell }}={\mathit{\ell }}_2(\rho ,t)f$.
• Stieltjes $\varphi$-derivative (formally ${\mathit{\ell }}_2=0$): ${\Delta }_{\varphi }^{\rho ,\mathit{\ell }}=\mu f+{\displaystyle \frac{1}{{\varphi }^{\prime }\left(t\right)}}{\displaystyle \frac{df}{dt}}$.

Example:

$$\mathit{\ell }(\rho ,t)={\displaystyle \frac{{\mathit{\ell }}_2}{{\mathit{\ell }}_1}},\Delta f={\mathit{\ell }}_1(\rho ,t)\left({\displaystyle \frac{1}{{\varphi }^{\prime }\left(t\right)}}{\displaystyle \frac{df}{dt}}(\mu +\mathit{\ell }(\rho ,t))f\right).$$

We do not consider a fractional order here, since this operator replaces the standard differentiation operator in the definition of the Hilfer type (see Section 4). The corresponding fractional-order integral operator, necessary for defining the generalized Hilfer derivative, requires a separate definition. This operator will be constructed to allow the generalization of classical problems, including those involving the fractional p-Laplacian.

The following general definition of an integral operator was first established by the authors in [10].

Definition 2 

([10]). Let $\varphi \in C^1\left[[a,b],\mathbb{R}\right]$ be a positive, increasing function such that ${\varphi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\varphi \left(a\right)=0$. Let $\alpha >0,\mu \ge 0,\rho \in (0,1]$ and assume that Assumption (10) is satisfied. We define the combination of the fractional and non-fractional integral operators acting on $f\in L^1[a,b]$ by

$$K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\left(t\right):={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\mathit{\int }}_a^t{e}^{-\left(\chi \right(t)-\chi (s\left)\right)}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)f\left(s\right)ds,$$

where

$$\omega (·):={\mathit{\int }}_a^{(·)}{\displaystyle \frac{{\varphi }^{\prime }\left(\theta \right)d\theta }{{\mathit{\ell }}_1(\rho ,\theta )}},\chi (·):={\mathit{\int }}_0^{\varphi (·)}\left(\mathit{\ell }\left(\rho ,{\varphi }^{-1}\left(u\right)\right)+\mu \right)du.$$

For the sake of completeness, we define $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\left(a\right):=0$ and $K_{\rho ,\varphi }^{0,\mathit{\ell },\mu }f:=f$.

Our considerations in this paper do not concern the codification through axioms of the properties of operators called fractional (cf. [17,18,19]). Some properties of classical operators are generally accepted, and operators that do not have the full set of classical operator properties have been called non-fractional for some time. We maintain this terminology; however, we note that this results in the study of the properties of the operators considered here without reference to classical results.

• Riemann–Liouville fractional integral: if $\varphi \left(t\right)=t,{\mathit{\ell }}_1=1,{\mathit{\ell }}_2=0,\mu =0$, then $\omega \left(t\right)=t,\chi \left(t\right)=0$, $Kf\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\mathit{\int }}_a^t{(t-s)}^{\alpha -1}f\left(s\right)ds$.
• Exponentially tempered fractional integral: if $phi\left(t\right)=t,{\mathit{\ell }}_1=1,{\mathit{\ell }}_2=0,\mu >0$, then $\omega \left(t\right)=t,\chi \left(t\right)=\mu (t-a)$, $Kf\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\mathit{\int }}_a^t{e}^{-\mu (t-s)}{(t-s)}^{\alpha -1}f\left(s\right)ds$.
• $\varphi$-Riemann–Liouville fractional integral: if ${\mathit{\ell }}_1=1,{\mathit{\ell }}_2=0,\mu =0$, then $\omega \left(t\right)=\varphi \left(t\right),\chi \left(t\right)=0$, $Kf\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\mathit{\int }}_a^t{(\varphi \left(t\right)-\varphi \left(s\right))}^{\alpha -1}{\varphi }^{\prime }\left(s\right)f\left(s\right)ds$.
• Tempered $\varphi$-fractional integral: if ${\mathit{\ell }}_1=1,{\mathit{\ell }}_2=0,\mu >0$, then $\omega \left(t\right)=\varphi \left(t\right),\chi \left(t\right)=\mu \varphi \left(t\right)$, $Kf\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\mathit{\int }}_a^t{e}^{-\mu \left(\varphi \right(t)-\varphi (s\left)\right)}{(\varphi \left(t\right)-\varphi \left(s\right))}^{\alpha -1}{\varphi }^{\prime }\left(s\right)f\left(s\right)ds$.
• Katugampola-type fractional integral: if $\varphi \left(t\right)=t,{\mathit{\ell }}_1(\rho ,t)=t^{1-\rho },\mu =0$, then $\omega \left(t\right)={\mathit{\int }}_a^t{\theta }^{\rho -1}d\theta \sim t^{\rho },\chi \left(t\right)=0$, $Kf\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\mathit{\int }}_a^t{(t^{\rho }-s^{\rho })}^{\alpha -1}{s}^{\rho -1}f\left(s\right)ds$.
• Tempered Katugampola-type integral: if $\varphi \left(t\right)=t,{\mathit{\ell }}_1(\rho ,t)=t^{1-\rho },\mu >0$, then $\omega \left(t\right)\sim t^{\rho },\chi \left(t\right)=\mu t$, $Kf\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\mathit{\int }}_a^t{e}^{-\mu (t-s)}{(t^{\rho }-s^{\rho })}^{\alpha -1}{s}^{\rho -1}f\left(s\right)ds$.
• Hadamard-type fractional integral (logarithmic case): if $\varphi \left(t\right)=lnt,{\mathit{\ell }}_1=1,\mu =0$, then $\omega \left(t\right)=lnt,\chi \left(t\right)=0$, $Kf\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\mathit{\int }}_a^t{\left(ln{\displaystyle \frac{t}{s}}\right)}^{\alpha -1}{\displaystyle \frac{f\left(s\right)}{s}}ds$.

In recent years, there has been growing interest in nonlocal operators and problems based on such operators, including fractional Laplacian operators. This has led us to study the entire class of these operators (see [20,21]). It is important to introduce a combination operator for local and nonlocal p-Laplacian operators (see, for example, ref. [22]). However, to date, research has focused on specific fractional-order Laplace operators and fractional Sobolev spaces. In this paper, we propose a more general form of these operators and examine their properties in Orlicz spaces (see also [23]). Due to the broad scope of this topic, we only partially examine the action of such operators in Orlicz–Sobolev spaces.

It is a general nonlocal operator. The basic properties of this operator are discussed in detail in refs. [10,24], with a focus on its role as a generalization of classical operators (including fractional-order operators) and its ability to handle differential and integral problems under weaker assumptions.

Because the value at t is calculated by summing (integrating) the values of $f(·)$ from the starting point a to t, the operator has memory. It is nonlocal because it considers the entire history of f on the interval $[a,t]$, not just the value of f at t. This is an excellent tool for modeling anomalous diffusion where the medium is not uniform (variable ${\mathit{\ell }}_1$) and where there is a physical constraint on how far a “jump” can influence the future (the tempering $\chi$); see refs. [25,26] or [27] for special forms of operators considered in Orlicz spaces.

The importance of such operators can be seen in studies of the Langevin ([28]) or Bagley-Torvik problems ([24]). Even in the case of vector-valued functions, these operators are natural, and the properties of the solutions are determined by the parameters of the operator. In this paper, we focus on mathematical aspects, so for examples of the application of such operators, we refer the reader to Section 4 in [24] for the Bagley–Torvik equation.

The following remark summarizes what is currently known about $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }$. We generalize or strengthen these properties for operators acting on Orlicz spaces.

Remark 3. 

Obviously,

1. 

Since ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L^1[a,b],\rho \in (0,1]$, then $\omega \in C[a,b]$ and ${\omega }^{\prime }\in L^1[a,b].$

2. 

$K_{\rho ,\psi }^{\alpha ,\mathit{\ell },\mu }f,\alpha \in (0,1),$ is satisfied in any point $t\in [a,b]$ for every $f/{\mathit{\ell }}_1(\rho ,·)\in C[a,b]$ and in almost every point for $f\in L_{1,\mathit{\ell }}[a,b]:=\left\{f\in L^1[a,b]:{\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}f(·)\in L^1[a,b]\right\}$.

3. 

$K_{\rho ,\psi }^{\alpha ,\mathit{\ell },\mu }$ maps $L_{1,\mathit{\ell }}[a,b]$ into $L_{1,\mathit{\ell }}[a,b]$: Let $f\in L_{1,\mathit{\ell }}[a,b]$ and note that (after changing the order of integration and using the substitution $u=\omega \left(t\right)-\omega \left(s\right)$)

$$\begin{array}{ccc}\hfill {\mathit{\int }}_a^b\left|{\displaystyle \frac{K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\left(t\right)}{{\mathit{\ell }}_1(\rho ,t)}}\right|dt& \le & {\mathit{\int }}_a^b{\displaystyle \frac{{∥{\varphi }^{\prime }∥}_{\infty }}{\mathrm{\Gamma }\left(\alpha \right)}}{\displaystyle \frac{\left|f\right(s\left)\right|}{{\mathit{\ell }}_1(\rho ,s)}}{\mathit{\int }}_0^{\omega \left(b\right)-\omega \left(s\right)}{u}^{\alpha -1}{\displaystyle \frac{du}{{\omega }^{\prime }\left(t\right)}}ds\hfill \\ & \le & {\mathit{\int }}_a^b{\displaystyle \frac{{∥{\varphi }^{\prime }∥}_{\infty }}{\mathrm{\Gamma }\left(\alpha \right)}}{\displaystyle \frac{\left|f\right(s\left)\right|}{{\mathit{\ell }}_1(\rho ,s)}}{\mathit{\int }}_0^{\omega \left(b\right)-\omega \left(s\right)}{u}^{\alpha -1}{\displaystyle \frac{du}{{\varphi }^{\prime }\left(t\right)}}ds.\hfill \end{array}$$

Hence,

$${∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f∥}_{L_{1,\mathit{\ell }}}={\mathit{\int }}_a^b\left|{\displaystyle \frac{K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\left(t\right)}{{\mathit{\ell }}_1(\rho ,t)}}\right|dt\le {\displaystyle \frac{{∥\omega ∥}_{\infty }^{\alpha }{∥{\varphi }^{\prime }∥}_{\infty }}{{min}_{t\in [a,b]}\left|{\varphi }^{\prime }\left(t\right)\right|\mathrm{\Gamma }(1+\alpha )}}{∥{\displaystyle \frac{f(·)}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{L_1},$$

(11)

for $f\in L_{1,\mathit{\ell }}[a,b]$. Also (see Lemma 4 in [10]), for any ${\alpha }_1,{\alpha }_2>0$, we have

$$K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }{K}_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }f=K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }{K}_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }f=K_{\rho ,\varphi }^{{\alpha }_1+{\alpha }_2,\mathit{\ell },\mu }f,$$

where $f\in L_{1,\mathit{\ell }}[a,b]$ holds true almost everywhere on $[a,b]$.

4. 

For any $\alpha >0$, we have

$$\left|K_{\rho ,\psi }^{\alpha ,\mathit{\ell },\mu }f\left(t\right)\right|\le {\displaystyle \frac{{∥f∥}_{\infty }}{\mathrm{\Gamma }\left(\alpha \right)}}{\mathit{\int }}_a^t{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)ds={\displaystyle \frac{{∥f∥}_{\infty }}{\mathrm{\Gamma }(1+\alpha )}}{\left(\omega \left(t\right)\right)}^{\alpha },$$

(12)

where $f\in C[a,b]$. Therefore, $K_{\rho ,\psi }^{\alpha ,\mathit{\ell },\mu }f\left(a\right)=0$. Therefore, our definition that $K_{\rho ,\psi }^{\alpha ,\mathit{\ell },\mu }f\left(a\right)=0$, for any f in $L^1[a,b]$ is meaningful.

Calculating $K_{\rho ,\psi }^{\alpha ,\mathit{\ell },\mu }\left[e^{-\chi \left(t\right)}{\left(\omega \left(t\right)\right)}^{\gamma }\right]$, after setting $u={\displaystyle \frac{\omega \left(s\right)}{\omega \left(t\right)}}$, we immediately obtain a natural generalization of the classical fractional integral operators:

Lemma 4. 

If $\alpha >0,\gamma >-1$, then

$$K_{\rho ,\psi }^{\alpha ,\mathit{\ell },\mu }\left[e^{-\chi \left(t\right)}{\left(\omega \left(t\right)\right)}^{\gamma }\right]={\displaystyle \frac{\mathrm{\Gamma }(1+\gamma )}{\mathrm{\Gamma }(1+\gamma +\alpha )}}{e}^{-\chi \left(t\right)}{\left(\omega \left(t\right)\right)}^{\alpha +\gamma }.$$

In addition to the combination of fractional and non-fractional integral operators, we also define several special operators.

Definition 3. 

1.

The Riemann–Liouville-type combination of the fractional and non-fractional derivatives of order $n+\alpha$, with parameters $\rho \in (0,1]$ and $\mu \ge 0$, is defined as follows:

$${\mathfrak{D}}_{\varphi ,\mu }^{n+\alpha ,\rho ,\mathit{\ell }}f:={\left({\Delta }_{\varphi }^{\rho ,\mathit{\ell }}\right)}^n{\mathfrak{D}}_{\varphi ,\mu }^{\alpha ,\rho ,\mathit{\ell }}f,where{\mathfrak{D}}_{\varphi ,\mu }^{\alpha ,\rho ,\mathit{\ell }}:={\Delta }_{\psi }^{\rho ,\mathit{\ell }}{K}_{\rho ,\varphi }^{1-\alpha ,\mathit{\ell },\mu }.$$

2.

The Caputo-type combination of the fractional and non-fractional derivatives of order $n+\alpha$, with parameters $\rho \in (0,1$] and $\mu \ge 0$, is defined as follows:

$${}^C\mathfrak{D}_{\varphi ,\mu }^{n+\alpha ,\rho ,\mathit{\ell }}:={\left({\Delta }_{\varphi }^{\rho ,\mathit{\ell }}\right)}^n{}^C\mathfrak{D}_{\varphi ,\mu }^{\alpha ,\rho ,\mathit{\ell }}f,where{}^C\mathfrak{D}_{\varphi ,\mu }^{\alpha ,\rho ,\mathit{\ell }}:=K_{\rho ,\varphi }^{1-\alpha ,\mathit{\ell },\mu }{\Delta }_{\varphi }^{\rho ,\mathit{\ell }}.$$

3. Generalized Fractional Integrals in Orlicz Spaces

An important goal of this work is to maintain the best possible regularity of the operators (see ref. [29] for an interesting survey on the mapping properties of fractional operators). Rather than considering Lebesgue spaces, we focus on smaller function spaces, namely Orlicz spaces, where we can expect additional regularity properties of the operator values.

First, however, we need to define the Orlicz-type spaces associated with the operators under consideration.

Let $\psi$ be an Orlicz function and define a new space:

$$L_{\psi ,\mathit{\ell }}[a,b]:=\left\{f\in L_1[a,b]:f(·)/{\mathit{\ell }}_1(\rho ,·)\in L_{\psi }[a,b]\right\}.$$

Therefore,

$${∥f∥}_{\psi ,\mathit{\ell }}:={∥{\displaystyle \frac{f(·)}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{\psi }=\underset{<\tilde{\psi },g>\le 1,g\in {\mathfrak{L}}_{\tilde{\psi }}}{sup}\left|{\mathit{\int }}_{[a,b]}{\displaystyle \frac{f\left(s\right)}{{\mathit{\ell }}_1(\rho ,s)}}g\left(s\right)ds\right|,$$

for $\rho \in (0,1]$. As in the classical case, we define a norm on this space:

$${∥f∥}_{L_{\psi },\mathit{\ell }}:={∥{\displaystyle \frac{f(·)}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{L_{\psi }}=inf\left\{k>0:〈\psi ,{\displaystyle \frac{f}{k{\mathit{\ell }}_1(\rho ,·)}}〉\le 1\right\},$$

for $\rho \in (0,1]$. We note that, for any $f\in C[a,b],$ we have (in view of the property of $\psi$: $\psi \left(\lambda u\right)\le \lambda \psi \left(u\right)$, $\lambda \in (0,1],u\in \mathbb{R}$)

$$〈\psi ,k{\displaystyle \frac{f(·)}{{\mathit{\ell }}_1(\rho ,·)}}〉\le {\mathit{\int }}_a^b{\displaystyle \frac{\left|f\right(t\left)\right|}{{∥f∥}_{\infty }}}\psi \left(k{\displaystyle \frac{{∥f∥}_{\infty }}{{\mathit{\ell }}_1(\rho ,t)}}\right)dt\le {\mathit{\int }}_a^b\psi \left({\displaystyle \frac{k{∥f∥}_{\infty }}{{\mathit{\ell }}_1(\rho ,t)}}\right)dt,k>0.$$

Following (4), for any $f\in C[a,b]$, ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L_{\psi }[a,b]$, we obtain that

$${∥f∥}_{\psi ,\mathit{\ell }}=\underset{k>0}{inf}\left\{{\displaystyle \frac{1}{k}}\left[1+〈\psi ,{\displaystyle \frac{kf(·)}{{\mathit{\ell }}_1(\rho ,·)}}〉\right]\right\}\le \underset{k>0}{inf}\left\{{\displaystyle \frac{{∥f∥}_{\infty }}{{∥f∥}_{\infty }k}}\left[1+〈\psi ,{\displaystyle \frac{k{∥f∥}_{\infty }}{{\mathit{\ell }}_1(\rho ,·)}}〉\right]\right\}={∥{\displaystyle \frac{{∥f∥}_{\infty }}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{\psi }.$$

Consequently, $f\in L_{\psi ,\mathit{\ell }}[a,b]$ and

$${∥f∥}_{\psi ,\mathit{\ell }}\le {∥{\displaystyle \frac{{∥f∥}_{\infty }}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{\psi }\le {∥{\displaystyle \frac{1}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{\psi }{∥f∥}_{\infty },$$

(13)

where $f\in C[a,b]$, ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L_{\psi }[a,b]$. Also, in view of (5), we deduce that

$${∥f∥}_{L_{\psi },\mathit{\ell }}\le 2{∥{\displaystyle \frac{1}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{\psi }{∥f∥}_{\infty }\le 4{∥{\displaystyle \frac{1}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{L_{\psi }}{∥f∥}_{\infty },$$

(14)

where $f\in C[a,b]$, ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L_{\psi }[a,b]$. Moreover, since we have $C[a,b]\subset L_{\psi ,\mathit{\ell }}[a,b]$ for any Orlicz function $\psi$, then for any $f\in C[a,b]$ and $k>0,$ we can estimate

$$〈\psi ,{\displaystyle \frac{f}{k}}〉={\mathit{\int }}_a^b\psi \left({\displaystyle \frac{\left|f\right(t\left)\right|}{k}}\right)dt\le {\mathit{\int }}_a^b\psi \left({\displaystyle \frac{{∥f∥}_{\infty }}{k}}\right)dt=\psi \left({\displaystyle \frac{{∥f∥}_{\infty }}{k}}\right)(b-a).$$

Therefore, $〈\psi ,{\displaystyle \frac{f}{k}}〉\le 1$ whenever k satisfies

$$\psi \left({\displaystyle \frac{{∥f∥}_{\infty }}{k}}\right)\le {\displaystyle \frac{1}{(b-a)}}⟹k\ge {\displaystyle \frac{{∥f∥}_{\infty }}{{\psi }^{-1}\left(\frac{1}{(b-a)}\right)}}.$$

Finally,

$${∥f∥}_{L_{\psi }}=inf\left\{k>0:〈\psi ,{\displaystyle \frac{f}{k}}〉\le 1\right\}={\displaystyle \frac{{∥f∥}_{\infty }}{{\psi }^{-1}\left(\frac{1}{(b-a)}\right)}},{∥f∥}_{\psi }\le {\displaystyle \frac{2{∥f∥}_{\infty }}{{\psi }^{-1}\left(\frac{1}{(b-a)}\right)}}.$$

(15)

We shall now search for conditions ensuring the continuity of the operator

$$K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi ,\mathit{\ell }}[a,b]\to L_{{\psi }^{*},\mathit{\ell }}[a,b],$$

where $\psi$ and ${\psi }^{*}$ are appropriate Orlicz functions. Specifically, $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }$ satisfies the condition

$$\parallel K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f\parallel }_{{\psi }^{*},\mathit{\ell }}\le \parallel K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{\parallel \parallel f\parallel }_{\psi ,\mathit{\ell }},\parallel K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }\parallel \ge 0.$$

We begin with an acting condition on the considered Orlicz-type spaces. It is important to choose the Orlicz function $\psi$ so that the operator is not only well-defined but also compact. For any fixed $\alpha \in (0,1)$, this condition can be formulated in terms of the growth of the conjugate function $\tilde{\psi }$. A sufficient condition is obtained by assuming the convergence of certain integrals, i.e., the modular integrability of the kernel of this operator. We follow the approach in the proof of Theorem 1 in [10].

Lemma 5. 

Let $\varphi \in C^1([a,b],\mathbb{R})$ be a positive increasing function with ${\varphi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ and $\varphi \left(a\right)=0$. Let $\alpha \in (0,1]$, $\rho \in (0,1]$, $\mu \ge 0$, and $\tilde{\psi }$ be the complementary N-function of a given Orlicz function ψ. Assume that this function $\tilde{\psi }$ satisfies

$${\mathit{\int }}_0^t\tilde{\psi }\left(s^{\alpha -1}\right)ds<\infty ,t>0.$$

(16)

Then, the operator

$$K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi ,\mathit{\ell }}[a,b]\to C[a,b]$$

is completely continuous and satisfies

$$\parallel K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f\parallel }_{\infty }\le 2\mathbf{K}{\tilde{\mathrm{\Psi }}}_{\alpha }{(\parallel \omega \parallel }_{\infty }{)\parallel f\parallel }_{L_{\psi ,\mathit{\ell }}},$$

where

$$\mathbf{K}:={\displaystyle \frac{4e^{\chi \left(b\right)}{\parallel {\varphi }^{\prime }\parallel }_{\infty }}{\mathrm{\Gamma }\left(\alpha \right)}},$$

and ${\tilde{\mathrm{\Psi }}}_{\alpha }$ is the continuous increasing function with ${\tilde{\mathrm{\Psi }}}_{\alpha }\left(0\right)=0$ defined by

$${\tilde{\mathrm{\Psi }}}_{\alpha }\left(y\right):=inf\left\{k>0:{\displaystyle \frac{1}{\parallel {\omega }^{\prime }{\parallel }_1}}{\mathit{\int }}_0^{k^{{\displaystyle \frac{1}{1-\alpha }}}y}\tilde{\psi }\left(s^{\alpha -1}\right)ds\le k^{{\displaystyle \frac{1}{1-\alpha }}}\right\}.$$

Proof. 

Let $f\in L_{\psi ,\mathit{\ell }}[a,b]$. The operator has the Volterra-type representation

$$\left(K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\right)\left(t\right)={\mathit{\int }}_a^{t}k(t,s)f\left(s\right){\varphi }^{\prime }\left(s\right)ds+\mathcal{P}\left(t\right),$$

where $k(t,s)={\displaystyle \frac{e^{-\mu \varphi \left(t\right)}}{\mathrm{\Gamma }\left(\alpha \right)}}{(\varphi \left(t\right)-\varphi \left(s\right))}^{\alpha -1}{e}^{\mu \varphi \left(s\right)}$ and $\mathcal{P}\left(t\right)$ accounts for initial conditions.

Step 1: Uniform boundedness of the operator. Applying Hölder’s inequality in Orlicz spaces,

$$\left|\left(Kf\right)\left(t\right)\right|\le {\parallel f\parallel }_{L_{\psi ,\mathit{\ell }}}{\parallel k(t,·)\parallel }_{\tilde{\psi }}.$$

From Assumption (16), for all $t\in [a,b]$:

$${\parallel k(t,·)\parallel }_{\tilde{\psi }}\le \mathbf{K}{\tilde{\mathrm{\Psi }}}_{\alpha }{(\parallel \omega \parallel }_{\infty })<\infty ,$$

so

$${\parallel Kf\parallel }_{\infty }\le 2\mathbf{K}{\tilde{\mathrm{\Psi }}}_{\alpha }{(\parallel \omega \parallel }_{\infty }{)\parallel f\parallel }_{L_{\psi ,\mathit{\ell }}}.$$

(17)

Hence, the image of the unit ball is uniformly bounded in $C[a,b]$.

Step 2: Equicontinuity via modular convergence. For $t_1,t_2\in [a,b]$:

$$|Kf\left(t_2\right)-Kf\left(t_1\right)|\le {\mathit{\int }}_a^{t_2}|k(t_2,s)-k(t_1,s)\left|\right|f\left(s\right)|{\varphi }^{\prime }\left(s\right)ds.$$

Applying Hölder’s inequality, we obtain:

$$|Kf\left(t_2\right)-Kf\left(t_1\right){|\le 2\parallel f\parallel }_{L_{\psi ,\mathit{\ell }}}{\parallel k(t_2,·)-k(t_1,·)\parallel }_{\tilde{\psi }}.$$

(18)

Now, consider the modular of the kernel difference:

$${\rho }_{\tilde{\psi }}(k(t_2,·)-k(t_1,·))={\mathit{\int }}_a^{t_2}\tilde{\psi }\left(\right|k(t_2,s)-k(t_1,s)\left|\right)ds.$$

Split the integral into two parts ${\mathit{\int }}_a^{t_2}={\mathit{\int }}_a^{t_1}+{\mathit{\int }}_{t_1}^{t_2}=:I_1+I_2$.

Small interval $I_2$. For $s\in [t_1,t_2]$:

$$|k(t_2,s)|\le {\displaystyle \frac{e^{-\mu \varphi \left(t_2\right)}}{\mathrm{\Gamma }\left(\alpha \right)}}{(\varphi \left(t_2\right)-\varphi \left(s\right))}^{\alpha -1}{e}^{\mu \varphi \left(s\right)}\le C{(t_2-t_1)}^{\alpha -1}.$$

Hence, by monotonicity of $\tilde{\psi }$,

$$I_2\le {\mathit{\int }}_0^{t_2-t_1}\tilde{\psi }\left(C{u}^{\alpha -1}\right)du\to 0\mathrm{as}{t}_2\to t_1.$$

Main interval $I_1$. For $s\in [a,t_1]$, we have pointwise convergence $k(t_2,s)\to k(t_1,s)$. Moreover, from Assumption (16), the family $|k(t_2,s)-k(t_1,s)|$ is dominated in the modular sense by an integrable function.

By modular convergence in Orlicz spaces (see Chapter 2 in [12]):

$$I_1={\mathit{\int }}_a^{t_1}\tilde{\psi }\left(\right|k(t_2,s)-k(t_1,s)\left|\right)ds\to 0\mathrm{as}{t}_2\to t_1.$$

Equicontinuity: Combining $I_1$ and $I_2$, we obtain

$$\parallel k(t_2,·)-k(t_1,·){\parallel }_{\tilde{\psi }}\to 0\mathrm{as}{t}_2\to t_1.$$

Thus, according to (18), the family $\mathcal{P}:=\{Kf:\parallel f{\parallel }_{L_{\psi ,\mathit{\ell }}}\le 1\}$ is equicontinuous in $C[a,b]$.

From Step 1 (uniform boundedness) and Step 2 (equicontinuity), the Arzelá–Ascoli theorem implies that any sequence in $\mathcal{P}$ has a uniformly convergent subsequence. Hence, $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }$ is compact.

Step 4: Continuity. Continuity follows immediately from (18) and the continuity of $t\mapsto k(t,·)$ in $L_{\tilde{\psi }}$-norm. Therefore, the operator $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }$ is completely continuous, and the norm estimate (17) holds:

$$\parallel K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f\parallel }_{\infty }\le 2\mathbf{K}{\tilde{\mathrm{\Psi }}}_{\alpha }{(\parallel \omega \parallel }_{\infty }{)\parallel f\parallel }_{L_{\psi ,\mathit{\ell }}}.$$

(19)

□

Corollary 1. 

1.

Lemma 5 is still valid in the particular case when ψ denotes N-function. Also, in view of (5) and (19)), we obtain that

$${∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f∥}_{\infty }\le 4\mathbf{K}{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right){∥f∥}_{\psi ,\mathit{\ell }}.$$

2.

Since, for any $\alpha \ge 1$, we have

$${\mathit{\int }}_0^t\tilde{\psi }\left(s^{\alpha -1}\right)ds\le t\tilde{\psi }\left(t^{\alpha -1}\right)<\infty ,0<t<\infty ,$$

then condition (16) is automatically satisfied. Therefore, Lemma 5 is still valid for any Orlicz (or N-) function ψ, whenever $\alpha \ge 1$.

3.

If ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L_{\psi }[a,b]$, for some Orlicz function ψ, whose complementary function satisfies (16), then $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:C[a,b]\to C[a,b]$ is completely continuous and, in view of (14),

$${∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f∥}_{\infty }\le 4\mathbf{K}{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right)·{∥{\displaystyle \frac{1}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{L_{\psi }}·{∥f∥}_{\infty }.$$

In this connection, since $C[a,b]$ is a proper subset of any Orlicz space, we can now prove the continuity and compactness of $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }$ as a map from $L_{\psi ,\mathit{\ell }}[a,b]$ to $L_{{\psi }^{*}}[a,b]$ for any Orlicz function ${\psi }^{*}$. Indeed, we have the following:

Corollary 2. 

Let $\varphi \in C^1[[a,b],\mathbb{R}]$ be a positive increasing function such that ${\varphi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ and $\varphi \left(a\right)=0$. Assume that $\alpha \in (0,1]$, $\rho \in (0,1]$, and $\mu \ge 0$, and ψ is an Orlicz function whose complementary function $\tilde{\psi }$ satisfies (16).

Then, the operator $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }$ continuously maps $L_{\psi ,\mathit{\ell }}[a,b]$ into an arbitrary Orlicz space $L_{{\psi }^{*}}[a,b]$, and

$${∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f∥}_{{\psi }^{*}}\le {\displaystyle \frac{4\mathbf{K}{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right)}{{\left({\psi }^{*}\right)}^{-1}\left(\frac{1}{b-a}\right)}}{∥f∥}_{\psi ,\mathit{\ell }},$$

(20)

where ${\tilde{\mathrm{\Psi }}}_{\alpha }$ is a continuous increasing function with ${\tilde{\mathrm{\Psi }}}_{\alpha }\left(0\right)=0$ defined by

$${\tilde{\mathrm{\Psi }}}_{\alpha }\left(y\right):=inf\left\{k>0:{\displaystyle \frac{1}{\parallel {\omega }^{\prime }{\parallel }_1}}{\mathit{\int }}_0^{k^{{\displaystyle \frac{1}{1-\alpha }}}y}\tilde{\psi }\left(s^{\alpha -1}\right)ds\le k^{{\displaystyle \frac{1}{1-\alpha }}}\right\}.$$

This operator is also compact if ${\psi }^{*}$ satisfies the ${\Delta }_2$-condition.

Proof. 

We proceed with the proof in a manner similar to that in proof of Theorem 16.3 [11]. The continuity (and hence the boundedness) of $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi ,\mathit{\ell }}[a,b]\to L_{{\psi }^{*}}[a,b]$ follows directly from Lemma 5, since, in view of (15),

$${∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f∥}_{{\psi }^{*}}\le 2{∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f∥}_{L_{{\psi }^{*}}}\le {\displaystyle \frac{2{∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f∥}_{\infty }}{{\left({\psi }^{*}\right)}^{-1}\left(\frac{1}{b-a}\right)}}\le {\displaystyle \frac{4\mathbf{K}{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right)}{{\left({\psi }^{*}\right)}^{-1}\left(\frac{1}{b-a}\right)}}{∥f∥}_{\psi ,\mathit{\ell }}.$$

Thus, it is clear that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi ,\mathit{\ell }}[a,b]\to L_{{\psi }^{*}}[a,b]$ is continuous. Now assume that the function ${\psi }^{*}$ satisfies the ${\Delta }_2$-condition (hence $L_{{\psi }^{*}}[a,b]\equiv E_{{\psi }^{*}}[a,b]$). Define

$$U_{\{\psi ,\mathit{\ell }\}}:=\left\{f\in L_{\psi ,\mathit{\ell }}:{∥f∥}_{\psi ,\mathit{\ell }}<r\right\},$$

for some $r>0$, and assume that $\left\{f_n\right\}\subseteq U_{\{\psi ,\mathit{\ell }\}}$ is $E_{\tilde{\psi }}$-weakly convergent to $f_0$. From the continuity of $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi ,\mathit{\ell }}[a,b]\to L_{{\psi }^{*}}[a,b]$, it follows that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_n$ converges almost everywhere (hence in measure) to $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_0$.

Since, in view of (19),

$$\left|K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_n\left(t\right)\right|\le 2\mathbf{K}{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right){∥f_n∥}_{\psi ,\mathit{\ell }},\mathbf{K}:={\displaystyle \frac{4e^{\chi \left(b\right)}{∥{\varphi }^{\prime }∥}_{\infty }}{\mathrm{\Gamma }\left(\alpha \right)}},$$

we deduce that, for the characteristic function ${\mathrm{\Lambda }}_{\mathfrak{P}}$ where $\mathfrak{P}\subseteq [a,b]$ is a measurable subset, we have

$$\begin{array}{ccc}\hfill {∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_n{\mathrm{\Lambda }}_{\mathfrak{P}}∥}_{{\psi }^{*}}& =& \underset{<\widetilde{{\psi }^{*}},g>\le 1,g\in {\mathfrak{L}}_{\widetilde{{\psi }^{*}}}}{sup}\left|{\mathit{\int }}_{\mathfrak{P}}{K}_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_n\left(s\right)g\left(s\right)ds\right|\hfill \\ & \le & 2{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right)\mathbf{K}{∥f_n∥}_{\psi ,\mathit{\ell }}\underset{{∥g∥}_{\widetilde{{\psi }^{*}}}\le 2,g\in {\mathfrak{L}}_{\widetilde{{\psi }^{*}}}}{sup}{\mathit{\int }}_{\mathfrak{P}}\left|g\left(s\right)\right|ds\hfill \\ & \le & 2{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right)\mathbf{K}r\underset{{∥g∥}_{\widetilde{{\psi }^{*}}}\le 2,g\in {\mathfrak{L}}_{\widetilde{{\psi }^{*}}}}{sup}{\mathit{\int }}_{\mathfrak{P}}\left|g\left(s\right)\right|ds.\hfill \end{array}$$

Consequently, for all $f_n\in U_{\{\psi ,\mathit{\ell }\}}$,

$${∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_n{\mathrm{\Lambda }}_{\mathfrak{P}}∥}_{{\psi }^{*}}<\epsilon ,\epsilon \mathrm{independent}\mathrm{of}n,\mathrm{whenever}\mathrm{meas}\left(\mathfrak{P}\right)<\delta .$$

It follows that the functions $\left\{K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_n\right\}$ have equi-absolutely continuous norms. Hence, in virtue of Lemma 1, the sequence $\left\{K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_n\right\}\subset L_{{\psi }^{*}}[a,b]$ converges in norm. According to Lemma 2, the operator is also compact. □

We can now also prove the continuity and compactness of $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }$ as a map from $L_{\psi ,\mathit{\ell }}[a,b]$ into $L_{{\psi }^{*},\mathit{\ell }}[a,b]$, where the given N-function ${\psi }^{*}$ satisfies the ${\Delta }_2$-condition:

Corollary 3. 

Let $\varphi \in C^1[[a,b],\mathbb{R}]$ be a positive increasing function such that ${\varphi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ and $\varphi \left(a\right)=0$. Assume that $\alpha \in (0,1]$, $\rho \in (0,1]$, and $\mu \ge 0$, and that the Orlicz function ψ is such that its complementary function $\tilde{\psi }$ satisfies (16). Then, the operator $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }$ continuously maps $L_{\psi ,\mathit{\ell }}[a,b]$ into $L_{{\psi }^{*},\mathit{\ell }}[a,b]$ for any ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L_{{\psi }^{*}}[a,b]$,

$${∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f∥}_{{\psi }^{*},\mathit{\ell }}\le {∥{\displaystyle \frac{1}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{{\psi }^{*}}4\mathit{K}{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right){∥f∥}_{\psi ,\mathit{\ell }}.$$

This operator is also compact if the function ${\psi }^{*}$ satisfies the ${\Delta }_2$-condition.

Proof. 

Since ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L_{{\psi }^{*}}[a,b]$, $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\in C[a,b]$ for $f\in L_{\psi ,\mathit{\ell }}[a,b]$, we deduce, in view of (14) and Lemma 5, that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\in L_{{\psi }^{*},\mathit{\ell }}[a,b]$ and

$${∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f∥}_{{\psi }^{*},\mathit{\ell }}\le {∥{\displaystyle \frac{{∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f∥}_{\infty }}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{{\psi }^{*}}\le {∥{\displaystyle \frac{1}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{{\psi }^{*}}4\mathbf{K}{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right){∥f∥}_{\psi ,\mathit{\ell }}.$$

Thus, it is clear that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi ,\mathit{\ell }}[a,b]\to L_{{\psi }^{*},\mathit{\ell }}[a,b]$ is continuous. Now, in view of the Hölder inequality and (19), we have for all $f_n\in U_{\{\psi ,\mathit{\ell }\}}$,

$$\begin{array}{ccc}\hfill {∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_n{\mathrm{\Lambda }}_{\mathfrak{P}}∥}_{{\psi }^{*},\mathit{\ell }}& =& {∥{\displaystyle \frac{K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_n{\mathrm{\Lambda }}_{\mathfrak{P}}}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{{\psi }^{*}}=\underset{<\widetilde{{\psi }^{*}},g>\le 1,g\in {\mathfrak{L}}_{\widetilde{{\psi }^{*}}}}{sup}\left|{\mathit{\int }}_{\mathfrak{P}}{\displaystyle \frac{K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{f}_n\left(s\right)g\left(s\right)}{{\mathit{\ell }}_1(\rho ,s)}}ds\right|\hfill \\ & \le & 2{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right)\mathbf{K}{∥f_n∥}_{\psi ,\mathit{\ell }}\underset{{∥g∥}_{\widetilde{{\psi }^{*}}}\le 2}{sup}{\mathit{\int }}_{\mathfrak{P}}{\displaystyle \frac{\left|g\right(s\left)\right|}{|{\mathit{\ell }}_1(\rho ,s)|}}ds\hfill \\ & \le & 4{\tilde{\mathrm{\Psi }}}_{\alpha }\left({∥\omega ∥}_{\infty }\right)\mathbf{K}{∥f_n∥}_{\psi ,\mathit{\ell }}\underset{{∥g∥}_{\widetilde{{\psi }^{*}}}\le 2}{sup}\left\{{∥{\displaystyle \frac{1}{{\mathit{\ell }}_1(\rho ,·)}}∥}_{L_{{\psi }^{*}}[a,b]}{∥g∥}_{L_{\widetilde{{\psi }^{*}}}\left[\mathfrak{P}\right]}\right\}\hfill \\ & <& \epsilon ,\epsilon independentonn,whenevermeas\left(\mathfrak{P}\right)<\delta .\hfill \end{array}$$

We can now continue the proof in the same way as in the proof of Corollary 2. Consequently, $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi }[a,b]\to L_{{\psi }^{*},\mathit{\ell }}[a,b]$ is compact, as required. □

Example 1. 

Let ${\psi }_p(·)={|·|}^p/p$, $p>1$, and ${\tilde{\psi }}_p(·)={\psi }_q(·)={|·|}^q/q$, with $1/p+1/q=1$. It can be easily seen that Assumption (16) of Lemma 5 holds for any $p>max\{1,{\alpha }^{-1}\}$. For such p, we have $q(\alpha -1)+1>0$, and so

$${\mathit{\int }}_0^t{\tilde{\psi }}_p\left(s^{\alpha -1}\right)ds={\displaystyle \frac{t^{q(\alpha -1)+1}}{q\left(q(\alpha -1)+1\right)}}<\infty ,\mathit{for}\mathit{any}t>0.$$

In this case, we obtain

$$\begin{array}{ccc}\hfill {\tilde{\mathrm{\Psi }}}_{\alpha }\left(t\right)& =& inf\left\{k>0:{\displaystyle \frac{1}{q{∥{\omega }^{\prime }∥}_1}}{\mathit{\int }}_0^{k^{{\displaystyle \frac{1}{1-\alpha }}}t}{s}^{q(\alpha -1)}ds\le k^{{\displaystyle \frac{1}{1-\alpha }}}\right\}\hfill \\ & =& inf\left\{k>0:{\displaystyle \frac{1}{q{∥{\omega }^{\prime }∥}_1}}{\displaystyle \frac{{\left(k^{\frac{1}{1-\alpha }}t\right)}^{q(\alpha -1)+1}}{q(\alpha -1)+1}}\le k^{{\displaystyle \frac{1}{1-\alpha }}}\right\}\hfill \\ & =& {\left[{\displaystyle \frac{1}{q(\alpha -1+1){∥{\omega }^{\prime }∥}_1}}\right]}^{{\displaystyle \frac{1}{q}}}{t}^{\alpha -{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

Since ${\psi }_p(·)={|·|}^p/p$, $p>max\{1,{\alpha }^{-1}\}$ satisfies the ${\Delta }_2$-condition, we can conclude (in view of Lemma 5 and Corollary 2) that

$$K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{{\psi }_p,\mathit{\ell }}[a,b]\to L_{{\psi }^{*}}[a,b]$$

is completely continuous for any given N-function ${\psi }^{*}$.

In this connection, we are able to prove an important result.

Proposition 1. 

Define ${\psi }_p(·)={|·|}^p/p$, $p>1$. Then,

$${\left|K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\left(t\right)\right|}^p\le {\displaystyle \frac{{∥\omega ∥}^{\alpha (p-1)}}{{\left(\mathrm{\Gamma }(1+\alpha )\right)}^{p-1}}}{K}_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{\left|f\left(t\right)\right|}^p,f\in L_{{\psi }_p,\mathit{\ell }}[a,b],p>max\left\{1,{\alpha }^{-1}\right\}.$$

(21)

$$K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{\left|f\left(t\right)\right|}^p\le {\displaystyle \frac{{∥\omega ∥}^{\alpha (1-p)}}{{\left(\mathrm{\Gamma }(1+\alpha )\right)}^{1-p}}}{\left|K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\left(t\right)\right|}^p,f\in L_{{\psi }_{{\displaystyle \frac{1}{p}},\mathit{\ell }}}[a,b],p<min\left\{1,\alpha \right\}.$$

(22)

Proof. 

Firstly, by Lemma 5, we note that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f$ exists and is continuous on $[a,b]$. Also, given the continuity of ${\mathit{\ell }}_1(\rho ,·),t\in [a,b]$, it follows that

$${\displaystyle \frac{{\left|f(·)\right|}^p}{|{\mathit{\ell }}_1(\rho ,·)|}}={\left|{\displaystyle \frac{f(·)}{{\mathit{\ell }}_1(\rho ,·)}}\right|}^p{\left|{\mathit{\ell }}_1(\rho ,·)\right|}^{p-1}\in L_1[a,b],p>1.$$

Hence, we see that ${\psi }_p\left(f(·)\right)\in L_{1,\mathit{\ell }}[a,b]$. According to Remark 3, we deduce that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{\psi }_p\left(f(·)\right)$ exists. Thus, in view of Jensen-type inequality for convex functions (see, e.g., ref. [30]), we have

$${\psi }_p\left({\displaystyle \frac{K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\left(t\right)}{\mathrm{W}}}\right)={\psi }_p\left({\displaystyle \frac{{\mathit{\int }}_a^{b}h\left(s\right)f\left(s\right)ds}{\mathrm{W}}}\right)\le {\displaystyle \frac{{\mathit{\int }}_a^{b}h\left(s\right){\psi }_p\left(f\left(s\right)\right)ds}{\mathrm{W}}}={\displaystyle \frac{K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{\psi }_p\left(f\left(t\right)\right)}{\mathrm{W}}},$$

where

$$\mathrm{W}:={\mathit{\int }}_a^{b}h\left(s\right)ds\le {\displaystyle \frac{{∥\omega ∥}^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}},$$

$$h\left(s\right):={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\left\{\begin{array}{cc}{e}^{-\left(\chi \right(t)-\chi (s\left)\right)}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)\hfill & s<t,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Consequently,

$${\left|K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\left(t\right)\right|}^p\le {\mathrm{W}}^{p-1}{K}_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{\left|f\left(t\right)\right|}^p={\displaystyle \frac{{∥\omega ∥}^{\alpha (p-1)}}{{\left(\mathrm{\Gamma }(1+\alpha )\right)}^{p-1}}}{K}_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{\left|f\left(t\right)\right|}^p,$$

for $f\in L_{{\psi }_p,\mathit{\ell }}[a,b]$ and $p>max\left\{1,{\displaystyle \frac{1}{\alpha }}\right\}$. Also, since the Jensen-type inequality changes direction for the concave function ${\psi }_p^{-1}$, $p\in (0,1)$, we deduce that

$$K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{\left|f\left(t\right)\right|}^p\le {\displaystyle \frac{{∥\omega ∥}^{\alpha (1-p)}}{{\left(\mathrm{\Gamma }(1+\alpha )\right)}^{1-p}}}{\left|K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f\left(t\right)\right|}^p,$$

for $f\in L_{{\psi }_{{\displaystyle \frac{1}{p}},\mathit{\ell }}}[a,b]$, $p<min\left\{1,\alpha \right\}$. □

Example 2. 

There is an Orlicz (or Young) function, particularly N-function, such that Assumption (16) of Lemma 5 is satisfied for all $\alpha >0$. Indeed, we consider the Orlicz (or Young) function

$$\psi \left(t\right):=e^{\left|t\right|}-\left|t\right|-1,(havingacomplementaryfunction\tilde{\psi }\left(t\right)=(1+|t\left|\right)ln\left(\right|t|+1)-\left|t\right|).$$

When $\alpha \ge 1$, we obtain

$${\mathit{\int }}_0^t\tilde{\psi }\left(s^{\alpha -1}\right)ds\le t\tilde{\psi }\left(t^{\alpha -1}\right)<\infty ,forallt\in (0,\infty ).$$

When $\alpha \in (0,1)$, we define

$$\begin{array}{ccc}\hfill I_{ϵ}& :=& {\mathit{\int }}_{ϵ}^t\tilde{\psi }\left(s^{\alpha -1}\right)ds={\mathit{\int }}_{ϵ}^t\left[\left(1+s^{\alpha -1}\right)ln\left(1+s^{\alpha -1}\right)-s^{\alpha -1}\right]ds\hfill \\ & \le & {\left[\left(s+{\displaystyle \frac{s^{\alpha }}{\alpha }}\right)ln\left(1+s^{\alpha -1}\right)\right]}_{ϵ}^t+(1-\alpha ){\mathit{\int }}_{ϵ}^t\left(1+{\displaystyle \frac{s^{\alpha -1}}{\alpha }}\right)ds-{\displaystyle \frac{t^{\alpha }-{ϵ}^{\alpha }}{\alpha }}.\hfill \end{array}$$

Since

$$\underset{ϵ\to 0}{lim}\left(ϵ+{\displaystyle \frac{{ϵ}^{\alpha }}{\alpha }}\right)ln\left(1+{ϵ}^{\alpha -1}\right)=0,$$

it follows that

$${\mathit{\int }}_0^t\tilde{\psi }\left(s^{\alpha -1}\right)ds=\underset{ϵ\to 0}{lim}{\mathit{\int }}_{ϵ}^t\tilde{\psi }\left(s^{\alpha -1}\right)ds<\infty ,forallt>0asclamed.$$

Hence (in view of Lemma 5), we deduce that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi ,\mathit{\ell }}[a,b]\to L_{{\psi }^{*}}[a,b]$ (for any given N-function ${\psi }^{*}$) is continuous (but not necessary completely continuous because ψ does not satisfy the ${\Delta }_2$-condition).

The following result follows from Theorem 1:

Proposition 2. 

Assume that $f(·,·):[a,b]\times \mathbb{R}\to \mathbb{R}$ be Carathéodory type function and $r^{*}>0$. Assume that $\psi ,{\psi }^{*}$ be two Young functions such that ψ satisfies the ${\Delta }_2$-condition and, for some $\alpha >0$, $\tilde{\psi }$ satisfies (16). If f satisfies the growth condition

$${\displaystyle \frac{\left|f\right(t,x\left)\right|}{|\mathit{\ell }(\rho ,t\left)\right|}}\le c\left(t\right)+B{\psi }^{-1}\left({\psi }^{*}\left({\displaystyle \frac{\left|K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }x\right|}{r^{*}}}\right)\right),$$

(23)

for $x\in \mathbb{R}$, $t\in [a,b]$, $c\in L_{\psi }[a,b]$, $B>0$, then F maps $\left\{x\in L_{\psi ,\mathit{\ell }}[a,b]:{∥x∥}_{\psi ,\mathit{\ell }}\le {\displaystyle \frac{r^{*}}{∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥}}\right\}$ to $L_{\psi ,\mathit{\ell }}[a,b]$, and it is continuous and bounded.

Proof. 

Define $U_{\{r,\psi ,\mathit{\ell }\}}:=\left\{x\in L_{\psi ,\mathit{\ell }}:{∥x∥}_{\psi ,\mathit{\ell }}<r\right\}$ and define $F_{\mathit{\ell }}(·):={\displaystyle \frac{f(t,·)}{\mathit{\ell }(\rho ,t)}},t\in [a,b]$. From Corollary 2, we know that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi ,\mathit{\ell }}\to L_{{\psi }^{*}}$ is continuous and bounded since

$${∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }x∥}_{{\psi }^{*}}\le ∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥{∥x∥}_{\psi ,\mathit{\ell }},$$

it follows ${∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }x∥}_{{\psi }^{*}}\le r^{*}$ for any $x\in U_{\left\{r^{*},\psi ,\mathit{\ell }\right\}}$. Hence, from Theorem 1, we conclude $F_{\mathit{\ell }}$, acting from $U_{\left\{{\displaystyle \frac{r^{*}}{∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥}},\psi ,\mathit{\ell }\right\}}$ to $L_{\psi }[a,b]$, is continuous and bounded. Hence, F maps $\left\{x\in L_{\psi ,\mathit{\ell }}[a,b]:{∥x∥}_{\psi ,\mathit{\ell }}\le {\displaystyle \frac{r^{*}}{∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥}}\right\}$ into $L_{\psi ,\mathit{\ell }}[a,b]$, and it is therefore continuous and bounded. □

Theorem 1 may be combined with Corollary 1 in order to assure the complete continuity of the mapping $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }F$, and to investigate the existence of Orlicz solutions to the following fractional order integral equation (and the corresponding fractional order initial or boundary problem):

$$x=h+K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }F.$$

As a summary of the research carried out so far, we can now prove the existence of solutions to the investigated integral equation in Orlicz spaces.

Theorem 3. 

Let $\alpha >0$, $\rho \in (0,1]$, $\mu \ge 0$ and $\varphi \in C^1\left[[a,b],\mathbb{R}\right]$ be a positive, increasing function such that ${\varphi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\varphi \left(a\right)=0$. Assume that the two N-functions $\psi ,{\psi }^{*}$ satisfy the ${\Delta }_2$-condition such that the complementary function $\tilde{\psi }$ to ψ satisfies (16). Assume that $f(·,·):[a,b]\times \mathbb{R}$ is a Carathéodory function such that, for each $r>0$, $c_r\in L_{\psi }[a,b]$ and that $a_r\ge 0$ such that

$${\displaystyle \frac{\left|f\right(t,x\left)\right|}{|\mathit{\ell }(\rho ,t\left)\right|}}\le c_r\left(t\right)+a_r{\psi }^{-1}\left({\psi }^{*}\left({\displaystyle \frac{\left|x\right|}{r}}\right)\right),$$

for $x\in \mathbb{R}$, $t\in [a,b]$. If $h\in L_{{\psi }^{*}}[a,b]$ and

$$\underset{r\in (0,\infty )}{sup}{\displaystyle \frac{r}{{∥h(·)∥}_{{\psi }^{*}}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left({∥c_r∥}_{\psi }+2a_r\right)}}\ge 1,$$

(24)

then the integral equation

$$x\left(t\right)=h\left(t\right)+K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f(t,x\left(t\right)),$$

(25)

for $t\in [a,b]$, admits a solution $x\in L_{{\psi }^{*}}[a,b]$.

Proof. 

We verify the hypotheses of Rothe’s fixed-point theorem. At the beginning, it should be noted that, according to (24), there exists $r_0>0$ such that

$${∥h(·)∥}_{{\psi }^{*}}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left({∥c_{r_0}∥}_{\psi }+2a_{r_0}\right)\le r_0,$$

(26)

Accordingly, define $F_{\mathit{\ell }}(·):={\displaystyle \frac{f(t,·)}{\mathit{\ell }(\rho ,t)}},t\in [a,b]$ and the operator $\mathbf{T}:\overline{U}\to L_{{\psi }^{*}}[a,b]$ by

$$\mathbf{T}x\left(t\right):=h\left(t\right)+K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }Fx\left(t\right),F(·)=f(t,·),$$

$t\in [a,b]$, where

$$U:=U_{\{r_0,{\psi }^{*}\}}\equiv \left\{x\in L_{{\psi }^{*}}:{∥x∥}_{{\psi }^{*}}<r_0\right\}.$$

(27)

From Corollary 2, we know that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi ,\mathit{\ell }}[a,b]\to L_{{\psi }^{*}}[a,b]$ is completely continuous. Also, from Theorem 1, we have $F_{\mathit{\ell }}:\overline{U}\to L_{\psi }[a,b]$ (hence, $F:\overline{U}\to L_{\psi .\mathit{\ell }}[a,b]$), which is continuous and maps bounded sets into bounded sets. Thus, $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }F:\overline{U}\to L_{{\psi }^{*}}[a,b]$ is completely continuous. Now, for any $x\in \overline{U}$, we have (in view of (20))

$$\begin{array}{ccc}\hfill {∥\mathbf{T}x(·)∥}_{{\psi }^{*}}& \le & {∥h(·)∥}_{{\psi }^{*}}+{∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f(·,x(·))∥}_{{\psi }^{*}}\le {∥h(·)∥}_{{\psi }^{*}}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥{∥f(·,x(·\left)\right)∥}_{\psi ,\mathit{\ell }}\hfill \\ & \le & {∥h(·)∥}_{{\psi }^{*}}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left({∥c_{r_0}∥}_{\psi }+a_{r_0}{∥{\psi }^{-1}\left({\psi }^{*}\left({\displaystyle \frac{\left|x\right(·\left)\right|}{r_0}}\right)\right)∥}_{\psi }\right).\hfill \end{array}$$

Thus, by virtue of (4) and (1), we obtain

$$\begin{array}{ccc}\hfill {∥\mathbf{T}x(·)∥}_{{\psi }^{*}}& \le & {∥h(·)∥}_{{\psi }^{*}}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left[{∥c_{r_0}∥}_{\psi }+a_{r_0}\left(1+〈\psi ,{\psi }^{-1}\left({\psi }^{*}\left({\displaystyle \frac{\left|x\right(·\left)\right|}{r_0}}\right)\right)〉\right)\right]\hfill \\ & \le & {∥h(·)∥}_{{\psi }^{*}}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left({∥c_{r_0}∥}_{\psi }+a_{r_0}\left[1+〈{\psi }^{*},{\displaystyle \frac{\left|x\right(·\left)\right|}{r_0}}〉\right]\right)\hfill \\ & \le & {∥h(·)∥}_{{\psi }^{*}}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left({∥c_{r_0}∥}_{\psi }+2a_{r_0}\right).\hfill \end{array}$$

From this, it follows (in view of (26)) that for any $x\in \partial U$, we have ${\parallel \mathbf{T}x(·)\parallel }_{{\psi }^{*}}\le r_0$. Let us summarize: we have the Banach space $L_{{\psi }^{*}}[a,b]$ and its open and bounded subset U (see (27)) with $0\in U$ (by (24)). We proved from Corollary 2 that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }:L_{\psi ,\mathit{\ell }}[a,b]\to L_{{\psi }^{*}}[a,b]$ is completely continuous. Moreover, $F:\overline{U}\to L_{\psi ,\mathit{\ell }}[a,b]$ is continuous and bounded. Therefore, $K\circ F:\overline{U}\to L_{{\psi }^{*}}[a,b]$ is completely continuous. Adding the constant function h preserves complete continuity, so $\mathbf{T}$ is completely continuous on $\overline{U}$. Take $x\in \partial U$, i.e., ${\parallel x\parallel }_{{\psi }^{*}}=r_0$. Using the growth condition on f and the properties of complementary N-functions, we obtain ${\parallel \mathbf{T}x(·)\parallel }_{{\psi }^{*}}\le r_0$. All hypotheses of Rothe’s fixed-point theorem (Theorem 2) are satisfied. Therefore, $\mathbf{T}$ has a fixed point in $L_{{\psi }^{*}}[a,b]$, which solves (25). □

Recall that Proposition 1 [31], if $\alpha ,{\alpha }^{*}\in (0,1)$ and the complementary function $\tilde{\psi }$ to the given Orlicz function $\psi$ satisfies

$${\mathit{\int }}_0^t\tilde{\psi }\left(s^{min\{\alpha ,{\alpha }^{*}\}-1}\right)ds<\infty ,t>0,$$

then

$${\mathit{\int }}_0^t\tilde{\psi }\left(s^{\alpha -1}\right)ds<\infty \mathrm{and}{\mathit{\int }}_0^t\tilde{\psi }\left(s^{{\alpha }^{*}-1}\right)ds<\infty ,\mathrm{for}\mathrm{any}t>0.$$

The above assertion, together with Proposition 2, yields the following result in $L_{\psi ,\mathit{\ell }}[a,b]$, in view of Theorem 3:

Theorem 4. 

Let $\alpha ,{\alpha }^{*}>0$, $\rho \in (0,1]$, $\mu \ge 0$ and $\varphi \in C^1\left[[a,b],\mathbb{R}\right]$ be a positive, increasing function such that ${\varphi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\varphi \left(a\right)=0$. Assume that $\psi ,{\psi }^{*}$ are two N-functions satisfying the ${\Delta }_2$-condition, such that the complementary function $\tilde{\psi }$ to ψ satisfies

$${\mathit{\int }}_0^t\tilde{\psi }\left(s^{min\{\alpha ,{\alpha }^{*}\}-1}\right)ds<\infty ,t>0.$$

(28)

Assume that $f(·,·):[a,b]\times \mathbb{R}$, is a Carathéodory function such that, for each $r>0$, there is $C_r\in L_{\psi }[a,b]$ and that $A_r\ge 0$ such that

$${\displaystyle \frac{\left|f\right(t,x\left)\right|}{|\mathit{\ell }(\rho ,t\left)\right|}}\le C_r\left(t\right)+A_r{\psi }^{-1}\left({\psi }^{*}\left({\displaystyle \frac{K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }\left|x\right|}{r}}\right)\right),x\in \mathbb{R},t\in [a,b].$$

(29)

If ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L_{\psi }[a,b],h\in L_{\psi ,\mathit{\ell }}$ and

$$\underset{r\in (0,\infty )}{sup}{\displaystyle \frac{r}{{∥h(·)∥}_{\psi ,\mathit{\ell }}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left({∥C_r∥}_{\psi }+2A_r\right)}}\ge ∥K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }∥,$$

(30)

then the integral equation

$$x\left(t\right)=h\left(t\right)+K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f(t,x\left(t\right)),t\in [a,b],$$

(31)

admits a solution $x\in L_{\psi ,\mathit{\ell }}[a,b]$.

Proof. 

Define the operator $\mathbf{T}:\overline{U}\to L_{\psi ,\mathit{\ell }}[a,b]$ by $\mathbf{T}x\left(t\right):=h\left(t\right)+K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f(t,x\left(t\right)),t\in [a,b]$, where

$$U:=U_{\left\{{\displaystyle \frac{r_0}{∥K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }∥}},\psi ,\mathit{\ell }\right\}}\equiv \left\{x\in L_{\psi ,\mathit{\ell }}:{∥x∥}_{\psi ,\mathit{\ell }}<{\displaystyle \frac{r_0}{∥K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }∥}}\right\}.$$

To show that $0\in U$, we need the following property ${\displaystyle \frac{r_0}{\parallel K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }\parallel }}>0$, i.e., $r_0>0$ and $\parallel K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }\parallel <\infty$. But from (30), there exists $r_0>0$ such that

$${\parallel h\parallel }_{\psi ,\mathit{\ell }}+\parallel K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }\parallel (\parallel C_{r_0}{\parallel }_{\psi }+2A_{r_0})\le {\displaystyle \frac{r_0}{\parallel K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }\parallel }}.$$

In view of Corollary 2, we know that $K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }:\overline{U}\to L_{{\psi }^{*}}[a,b]$ is continuous (hence bounded) and that ${∥K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }x∥}_{{\psi }^{*}}\le r_0$ for any $x\in \overline{U}$. For $x\in \overline{U}$, the growth condition (29) together with $\parallel K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }{x\parallel }_{{\psi }^{*}}\le r_0$ implies $F\left(x\right)\in L_{\psi ,\mathit{\ell }}$.

From Proposition 2, we also know that F maps continuously $\overline{U}$ to $L_{\psi ,\mathit{\ell }}[a,b]$. Since ${\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in L_{\psi }[a,b]$, it follows from Corollary 3 that $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }$ maps continuously $L_{\psi ,\mathit{\ell }}[a,b]$ into $L_{\psi ,\mathit{\ell }}[a,b]$ and that it is compact (since $\psi$ satisfies the ${\Delta }_2$-condition). Therefore, $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }F:\overline{U}\to L_{\psi }[a,b]$ is completely continuous. The proof follows a similar approach to that used in the proof of Theorem 3. For any $x\in \overline{U}$, we have (in view of (20))

$$\begin{array}{ccc}\hfill {∥\mathbf{T}x(·)∥}_{\psi ,\mathit{\ell }}& \le & {∥h(·)∥}_{\psi ,\mathit{\ell }}+{∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f(·,x(·))∥}_{\psi ,\mathit{\ell }}\le {∥h(·)∥}_{\psi ,\mathit{\ell }}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥{∥f(·,x(·\left)\right)∥}_{\psi ,\mathit{\ell }}\hfill \\ & \le & {∥h(·)∥}_{\psi ,\mathit{\ell }}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left({∥C_{r_0}∥}_{\psi }+A_{r_0}{∥{\psi }^{-1}\left({\psi }^{*}\left({\displaystyle \frac{|K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }x(·)|}{r_0}}\right)\right)∥}_{\psi }\right).\hfill \end{array}$$

Consequently, in view of (4) and (1), we obtain (in view ${∥K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }x(·)|∥}_{\psi ,\mathit{\ell }}\le ∥K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }∥{∥x∥}_{\psi ,\mathit{\ell }}\le r_0$, for any $x\in \overline{U}$):

$$\begin{array}{ccc}\hfill {∥\mathbf{T}x(·)∥}_{\psi ,\mathit{\ell }}& \le & {∥h(·)∥}_{\psi ,\mathit{\ell }}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left[{∥C_{r_0}∥}_{\psi }\right.\hfill \\ & & \left.+A_{r_0}\left(1+〈\psi ,{\psi }^{-1}\left({\psi }^{*}\left({\displaystyle \frac{|K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }x(·)|}{r_0}}\right)\right)〉\right)\right]\hfill \\ & \le & {∥h(·)∥}_{\psi ,\mathit{\ell }}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left({∥C_{r_0}∥}_{\psi }+A_{r_0}\left[1+〈{\psi }^{*},{\displaystyle \frac{|K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }x(·)|}{r_0}}〉\right]\right)\hfill \\ & \le & {∥h(·)∥}_{\psi ,\mathit{\ell }}+∥K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }∥\left({∥C_{r_0}∥}_{\psi }+2A_{r_0}\right).\hfill \end{array}$$

Making a similar argument as in the previous theorem, we can summarize that all the assumptions of Theorem 2 are satisfied. In particular, in view of (30), it follows that for any $x\in \partial U$, we have

$${∥\mathbf{T}x(·)∥}_{\psi ,\mathit{\ell }}\le {\displaystyle \frac{r_0}{∥K_{\rho ,\varphi }^{{\alpha }^{*},\mathit{\ell },\mu }∥}}.$$

Therefore, $\mathbf{T}$ has a fixed point in $L_{\psi ,\mathit{\ell }}[a,b]$, which solves (31). □

4. $\mathit{p}$-Laplacian Fractional Problems with Generalized Hilfer Fractional Derivatives

Having recalled the definition of the combination of fractional and non-fractional integral operators, we now extend the notation to generalized Hilfer fractional derivatives of arbitrary orders. Following the approach in Section 3, we define these derivatives to include integer-order limits, variable kernels, and tempering parameters, allowing us to handle a broader class of p-Laplacian fractional problems in Orlicz spaces.

This approach generalizes classical Hilfer derivatives and enables the study of nonlinear problems with variable growth conditions, memory effects, and tempered singularities, all within the Orlicz space framework.

Definition 4. 

Let $\psi \in C^1[a,b]$ be a positive increasing function such that ${\varphi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ and $f\in L^1[a,b]$. For $\alpha \in (0,1)$ and $n\in \mathbb{N},$ we define the Hilfer-type combination of the fractional and non-fractional derivatives of order $n+\alpha$, with parameters $\rho \in (0,1]$, $\mu \ge 0$ and type $\beta \in [0,1]$, by

$${}_p{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{n+\alpha ,\beta ,\mathit{\ell }}f:={\left({\Delta }_{\varphi }^{\rho ,\mathit{\ell }}\right)}^n{}_p{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}f,$$

(32)

where

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}:=K_{\rho ,\varphi }^{\beta (1-\alpha ),\mathit{\ell },\mu }{\Delta }_{\psi }^{\rho ,\mathit{\ell }}{K}_{\rho ,\varphi }^{(1-\beta )(1-\alpha ),\mathit{\ell },\mu }.$$

Obviously,

$${\mathfrak{D}}_{\varphi ,\mu }^{n+\alpha ,\rho ,\mathit{\ell }}f={}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{n+\alpha ,0,\mathit{\ell }}f,{}^C\mathfrak{D}_{\varphi ,\mu }^{n+\alpha ,\rho ,\mathit{\ell }}f={}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{n+\alpha ,1,\mathit{\ell }}f.$$

(33)

Some special cases are as follows:

• Riemann–Liouville-type derivative: if $\beta =0$, then ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{n+\alpha ,0,\mathit{\ell }}={\mathfrak{D}}_{\varphi ,\mu }^{n+\alpha ,\rho ,\mathit{\ell }}$.
• Caputo-type derivative: if $\beta =1$, then ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{n+\alpha ,1,\mathit{\ell }}={}^C\mathfrak{D}_{\varphi ,\mu }^{n+\alpha ,\rho ,\mathit{\ell }}$.
• Classical Hilfer operator: if $\varphi \left(t\right)=t,{\mathit{\ell }}_1=1,{\mathit{\ell }}_2=0,\mu =0$, then ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}$ is a Hilfer derivative.
• Classical Riemann–Liouville operators: if $phi\left(t\right)=t,{\mathit{\ell }}_1=1,{\mathit{\ell }}_2=0,\mu =0,\beta =0$, then ${}^H\mathrm{D}^{\alpha ,0}$ is a Riemann–Liouville fractional derivative.
• Classical Caputo derivative: if $\varphi \left(t\right)=t,{\mathit{\ell }}_1=1,{\mathit{\ell }}_2=0,\mu =0,\beta =1$, then ${}^H\mathrm{D}^{\alpha ,1}$ is a Caputo fractional derivative,
• $\varphi$-Hilfer derivative: if ${\mathit{\ell }}_1=1,{\mathit{\ell }}_2=0,\mu =0$, then ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}$ is a $\varphi$-Hilfer derivative.
• Tempered Hilfer-type derivative: if ${\mathit{\ell }}_1=1,{\mathit{\ell }}_2=0,\mu >0$, then ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}$ is a tempered Hilfer derivative.
• Katugampola–Hilfer-type derivative: if $\varphi \left(t\right)=t,{\mathit{\ell }}_1(\rho ,t)=t^{1-\rho },\mu =0$, then ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}$ is a Katugampola-type Hilfer derivative.
• Hadamard–Hilfer-type derivative: if $\varphi \left(t\right)=lnt,{\mathit{\ell }}_1=1,\mu =0$, then ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}$ is a Hadamard-type Hilfer derivative.

The following inverse problem is investigated in relation to our differential one:

Proposition 3 

(Lemma 11 [10]). For any $f\in AC[a,b]$ and $\alpha >0$, we have

$$K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}f\left(t\right)=\left\{\begin{array}{c}{\displaystyle f\left(t\right),\beta \in [0,1)}\hfill \\ {\displaystyle f\left(t\right)-f\left(a\right)e^{-\chi \left(t\right)},\beta =1.}\hfill \end{array}\right.,$$

(34)

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}{K}_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }f=f,forall\beta \in [0,1].$$

In the study of nonlinear fractional differential equations involving p-Laplacian operators, the classical Orlicz spaces $L_{\psi }$ are often insufficient to capture the full regularity of the solution. Since the operator involves a composition of the form ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}{\mathrm{\Phi }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\right)$, the solution must possess a fractional derivative that is itself integrable.

Definition 5. 

Let ψ be an N-function satisfying the ${\Delta }_2$-condition. The weighted fractional Orlicz–Sobolev space $W_{\varphi ,\rho ,\mathit{\ell }}^{\alpha ,\psi }[a,b]$ is defined as the set of functions:

$$W_{\varphi ,\rho ,\mathit{\ell }}^{\alpha ,\psi }[a,b]:=\left\{x\in L_{\psi }[a,b]:{}^{H}D_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}x\in L_{\psi }[a,b]\right\}.$$

(35)

This space becomes a Banach space when it is equipped with the following Luxemburg-type norm (cf. [11,12,32]):

$${\parallel x\parallel }_{W_{\varphi ,\rho ,\mathit{\ell }}^{\alpha ,\psi }}:={\parallel x\parallel }_{\psi }+{∥{}^{H}D_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}x∥}_{\psi },$$

(36)

where ${\parallel ·\parallel }_{\psi }$ is the Luxemburg norm.

It is a deeply nonlocal operator, and its relationship with classical operators is entirely determined by the choice of the parameters $\alpha$ and $\beta$. This represents a system where the memory of the process is filtered twice: once before the rate of change is measured, and once after.

The study of non-symmetric fractional p-Laplacians, often referred to as ’one-sided’ or ’directional’ operators, is a significant and active area of research within fractional calculus. While the standard fractional p-Laplacian defined on ${\mathbb{R}}^n$ is symmetric and involves a singular integral over the entire space, the generalized Hilfer derivative naturally falls within the non-symmetric framework. In the one-dimensional case on a bounded interval $[a,b]$, the operator is defined using left-sided or right-sided integrals, representing systems with causality or ’memory’ that only looks backward in time. The mathematical behavior of one-sided operators differs significantly from that of the symmetric case. In particular, the variational method is often replaced by a fixed-point approach, due to the lack of symmetry and the fact that the corresponding Sobolev spaces need not be reflexive. Since left-sided integrals are Volterra operators, the problem can be transformed into a Volterra integral equation, which is typically easier to solve than the Fredholm equations arising from symmetric operators.

What are the physical motivations for studying this non-symmetric case? Two simple arguments are causality and anisotropy. In viscoelasticity and signal processing, the future cannot affect the past (causality). In fluid flow through porous media, the pressure gradient may depend solely on flow from a particular direction (upstream), representing anisotropy.

Remark 4. 

The differential operator ${\mathrm{D}}_{\varphi ,\mu }^{n+\alpha ,\rho ,\mathit{\ell }}$ introduced in (32) is motivated by constructions from fractional calculus. To establish an analog with nonlocal operators such as the fractional Laplacian on bounded domains, it is natural to consider a symmetrized version of this operator.

Fractional operators are typically defined in a one-sided (directional) manner. The left-sided operator is associated with causal dynamics, as it depends on the past history over the interval $a<s<t$, whereas the corresponding right-sided operator incorporates values over $t<s<b$ and may therefore be interpreted as anti-causal.

In the present setting, this directionality is encoded through kernels of the form

$$K(t,s)\sim {(g\left(t\right)-g\left(s\right))}^{\alpha -1}{e}^{-\mu \left(g\right(t)-g(s\left)\right)}{g}^{\prime }\left(s\right),$$

which depend explicitly on the oriented difference $g\left(t\right)-g\left(s\right)$ for some function g. While this structure is appropriate for evolution-type problems, it introduces an intrinsic asymmetry.

To remove this directional bias and obtain an operator with properties analogous to the symmetric fractional Laplacian, one introduces the right-sided counterpart and combines it with the left-sided operator. An appropriate symmetrization—commonly given by a weighted average—leads to a Riesz-type operator, whose kernel depends on $\left|g\right(t)-g(s\left)\right|$, thereby restoring symmetry (cf. [33]).

From our definition, the right-handed generalized derivative (cf. (32) for the left-handed derivative) is:

$${}_{b^{-}}{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}:=K_{\rho ,\varphi ,b^{-}}^{\beta (1-\alpha ),\mathit{\ell },\mu }{\Delta }_{\psi }^{\rho ,\mathit{\ell }}{K}_{\rho ,\varphi ,b^{-}}^{(1-\beta )(1-\alpha ),\mathit{\ell },\mu },$$

$$K_{\rho ,\varphi ,b^{-}}^{\alpha ,\mathit{\ell },\mu }f\left(t\right):={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\mathit{\int }}_t^b{e}^{-\left(\chi \right(t)-\chi (s\left)\right)}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)f\left(s\right)ds.$$

The symmetric version of the Riesz–Hilfer operator is the following:

$${\mathbb{D}}_{\mathrm{Riesz}}^{\alpha ,\beta }f\left(t\right):={\displaystyle \frac{1}{2}}\left({}_{a^{+}}{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}f\left(t\right)+{}_{b^{-}}{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}f\left(t\right)\right).$$

Why is this referred to as the ’fractional Laplacian’ ([34])? Its relationship with the classical fractional Laplacian, defined as ${(-\Delta )}^{\alpha }$, on a bounded interval can be understood through its spectral and symmetry properties. The fractional Laplacian is also symmetric (self-adjoint).

When the parameters are set to the simplest case ($\varphi \left(t\right)=t$, $\mu =0$, ${\mathit{\ell }}_1=1$), the Riesz–Hilfer operator reduces to the Riesz derivative. On the whole real line, the Riesz derivative of order $2\alpha$ coincides precisely with the classical fractional Laplacian.

The Riesz–Hilfer formulation on a bounded interval accounts for the fact that the ’jumps’ of the process cannot extend beyond $[a,b]$. This introduces a natural limitation in establishing a direct equivalence with the classical fractional Laplacian. Nevertheless, this formulation provides a flexible framework that allows for a significant expansion of research based on fractional calculus problems.

Remark 5. 

The operator analyzed in this paper is of the following compositional form:

$$\mathcal{L}\left(x\right):={}^{H}D_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}{\mathrm{\Theta }}_p\left({}^{H}D_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right),{\mathrm{\Theta }}_p\left(y\right)={\left|y\right|}^{p-2}y,$$

where $p>1$, ${\alpha }_1,{\alpha }_2>0$, $\rho \in (0,1]$, and $\beta \in [0,1]$.

This operator defines a generalized fractional nonlinear p-Laplacian, providing a unified framework that bridges classical local differential equations with nonlocal models.

By selecting the appropriate limits for the parameters ${\alpha }_1,{\alpha }_2,\rho ,$ and $\varphi$, we can derive several well-known operators from the literature.

• Local p-Laplacian: For ${\alpha }_1={\alpha }_2=1,\rho =1,\mu =0$, and $\varphi \left(t\right)=t$, the operator reduces to the classical 1D p-Laplacian: ${\displaystyle \frac{d}{dt}}{\mathrm{\Theta }}_p\left({\displaystyle \frac{d}{dt}}\right)={\displaystyle \frac{d}{dt}}\left(|x^{\prime }{|}^{p-2}{x}^{\prime }\right)$ ([35]).
• Standard fractional p-Laplacian type: To obtain the standard p-Laplacian of Riemann–Liouville ($\beta =0$) or Caputo ($\beta =1$) types, set ${\alpha }_1={\alpha }_2=\alpha \in (0,1)$, $\varphi \left(t\right)=t,\rho =1,\mu =0,\mathit{\ell }=1$, with the operator representing a nested fractional p-Laplacian structure $\mathcal{L}\left(x\right)\to {\mathrm{D}}^{\alpha }{\mathrm{\Theta }}_p\left({\mathrm{D}}^{\alpha }x\left(t\right)\right)$ ([20,21,36,37]).
• Mixed compositional type: For ${\alpha }_1=1$ (local divergence) and ${\alpha }_2=\alpha \in (0,1)$ (fractional gradient), the operator models a nonlocal flux within a local conservation law: ${\displaystyle \frac{d}{dt}}{\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}x\left(t\right)\right)$ ([38,39]).

The operator under investigation provides a unified link between local diffusion and nonlocal fractional dynamics. To understand its role, we compare it to standard p-Laplacian forms.

Note that the compositional fractional p-Laplacian (of differential type, see

Table 1. Comparison of p-Laplacian operators.

Operator Type	Structure	Nature
Classical p-Laplacian	${\Delta }_{p}u={\mathrm{div}\left(\right|\nabla u|}^{p-2}\nabla u)$	Local (Pointwise)
Fractional p-Laplacian	${(-\Delta )}_p^{s}u\left(x\right)=C\mathit{\int }{\displaystyle \frac{{\mathrm{\Theta }}_p(u\left(x\right)-u\left(y\right))}{{|x-y|}^{n+sp}}}dy$	Nonlocal (Integral)
Compositional p-Fractional	${}^{H}D^{\alpha }{\mathrm{\Theta }}_p({}^{H}D^{\alpha }x\left(t\right))$	Nonlocal (Differential)

) is distinct from the integral fractional p-Laplacian ${(-\Delta )}_p^s$, which is defined via a singular integral over the whole space rather than an iteration of derivatives. It is essential to distinguish this compositional model

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}\left[{\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)\right]=f\left(t\right)$$

from the widely studied mixed local–nonlocal p-Laplacian of the form $-{\Delta }_{p}u+\mu {(-{\Delta }_p)}^{\alpha }u=f$. While the latter represents the additive interaction of two distinct diffusion processes (e.g., Brownian motion and Lévy flights), the current operator suggests a fractionalization of the flux itself. In classical mechanics, the flux $\mathbf{q}$ is proportional to the gradient $\mathbf{q}={\mathrm{\Phi }}_p(\nabla u)$. Our model replaces this with

$${\mathbf{q}}_{\mathrm{nonloc}.}={\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{\alpha ,\beta ,\mathit{\ell }}u\left(t\right)\right).$$

The nonlocal structure of the generalized fractional p-Laplacian operator implies that the ’flow’ at point t is not just determined by the slope at t but by the medium’s accumulated state as well. This occurs in non-homogeneous media or viscoelastic fluids, where the material remembers its previous deformations. In spatial domains, this corresponds to the local divergence of a nonlocal p-nonlinear flux, a structure that is particularly relevant in non-homogeneous media or censored Lévy processes, where the constitutive relation between gradient and flow is history-dependent (i.e., jumps outside the domain are restricted or ignored).

The presence of the parameters $\varphi ,\rho ,\mathit{\ell }$, and $\mu$ allows us to include various weight functions and generalized kernels, accommodating different fractional derivatives such as Hilfer, Prabhakar, and others. Consequently, this operator belongs to a broad class of fractional Sobolev settings with variable order. The specific choice of parameters determines whether the system exhibits purely local, purely nonlocal, or hybrid multi-scale characteristics. Further investigation is required for cases not mentioned above.

The generalized compositional form allows for a “memory-sensitive” flux. By using the Hilfer parameter $\beta \in [0,1]$, the operator interpolates between Riemann–Liouville and Caputo behaviors, providing a flexible framework for modeling non-Newtonian anomalous diffusion.

4.1. The Generalized Differential Problem

The exact differential problem considered in this framework is given by:

$$\left\{\begin{array}{cc}\mathcal{L}\left(x\right(t\left)\right)=g(t,x(t\left)\right),\hfill & t\in [a,b]\hfill \\ {}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(a\right)=0\hfill & \\ \mathcal{P}\left(x\right)=0\hfill & (\mathrm{Boundary}\mathrm{Conditions})\hfill \end{array}\right.$$

(37)

where the operator $\mathcal{L}$ is defined as:

$$\mathcal{L}\left(x\right):={}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}{\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)$$

with ${\mathrm{\Theta }}_p\left(s\right):={\left|s\right|}^{p-2}s$ for $p>1$, and generalized orders ${\alpha }_1,{\alpha }_2>0$.

To rigorously analyze the existence of solutions, we define the generalized fractional Orlicz–Sobolev space $\mathbb{W}:={\mathbb{W}}_{\varphi ,\rho ,\mu ,\mathit{\ell }}^{{\alpha }_2,\beta ,\psi }[a,b]$ (cf. (35)). This space consists of functions x such that both the function and its (generalized) derivative are in $L_{\psi }[a,b]$:

$$\mathbb{W}=\left\{x\in L_{\psi }[a,b]:{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\in L_{\psi }[a,b]\right\}.$$

The space is equipped with the norm introduced earlier in (36).

We can consider two different notions of solutions: Weak solution is a function $x\in \mathbb{W}$ such that for all test functions $v\in {\mathbb{W}}_0$:

$${\mathit{\int }}_a^b{\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)·\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}v\left(t\right)\right)dt={\mathit{\int }}_a^{b}g(t,x\left(t\right))v\left(t\right)dt.$$

Strong solution is a function $x\left(t\right)$ that satisfies our equation almost everywhere in $[a,b]$, implying that the composition of the nonlinear term ${\mathrm{\Theta }}_p$ and the derivative lies within the domain of the outer operator.

In the context of solutions to integral equations in Orlicz spaces, and given the Carathéodory conditions imposed on the right-hand side, we can expect to find weak solutions to differential forms of the problem.

Recall that a function ${\mathrm{\Theta }}_p$, ($p>1$) is called p-Laplace operator [40,41] if ${\mathrm{\Theta }}_p$ satisfies the following relationships:

• If $p\in (1,2],u,v\ge n>0$, we have

  $$\left|{\mathrm{\Theta }}_p\left(u\right)-{\mathrm{\Theta }}_p\left(v\right)\right|\le (p-1)n^{p-2}|u-v|.$$
• If $p\in (2,\infty ),u,v>0,\left|u\right|,\left|v\right|\le m$, we have

  $$\left|{\mathrm{\Theta }}_p\left(u\right)-{\mathrm{\Theta }}_p\left(v\right)\right|\le (p-1)m^{p-2}|u-v|.$$

Lemma 6. 

Let $y\in AC[a,b]$, then for the Laplacian operator ${\mathrm{\Theta }}_p,p\ge 2,$ we have ${\mathrm{\Theta }}_p\left(y\right)\in AC[a,b].$

Proof. 

Let $y\in AC[a,b],ϵ>0$ and assume that $\left\{(a_i,b_i)\right\}$ be a collection of n-disjoint subintervals of $[a,b]$ such that ${\sum }_{k=1}^n|b_k-a_k|<\delta ,\delta >0$. Since

$$\sum _{k=1}^n\left|{\mathrm{\Theta }}_p\left(y\left(b_k\right)\right)-{\mathrm{\Theta }}_p\left(y\left(a_k\right)\right)\right|\le (p-1){\left({∥y∥}_{\infty }\right)}^{p-2}\sum _{k=1}^n|y\left(b_k\right)-y\left(a_k\right)|\le (p-1){\left({∥y∥}_{\infty }\right)}^{p-2}ϵ,$$

it follows ${\mathrm{\Theta }}_p\left(y\right)\in AC[a,b]$ as required. □

We now analyze the following fractional differential equations with a generalized compositional p-Laplacian operator:

$$\left\{\begin{array}{c}{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}{\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)=g\left(t,x\left(t\right)\right),{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(a\right)=0,t\in [a,b],\hfill \\ {\displaystyle {\mathrm{\Theta }}_p(·):=(·){|·|}^{p-2},p>1,{\alpha }_1,{\alpha }_2>0,\beta \in [0,1],}\hfill \end{array}\right.$$

(38)

combined with additional initial, terminal, nonlocal, or boundary conditions.

We have already discussed this problem. Now, let us take a look at its structure. The outer layer (${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}$) is the generalized Hilfer derivative. It acts as the final differential filter. Because $\beta \in [0,1]$, this part determines whether the system behaves more like a Riemann–Liouville process (better for initial power-law growth) or a Caputo process (better for constant initial conditions). The nonlinear core (${\mathrm{\Theta }}_p(·)$) is the standard 1D p-Laplacian. It introduces non-Newtonian physics. When $p>2$, the diffusion is “slow”; when $1<p<2$, it is “fast”. The inner layer (${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)$) is the fractional gradient of the state $x\left(t\right)$. It captures the history of the variable x before it ever hits the nonlinear ${\mathrm{\Theta }}_p$ filter.

The presence of the parameters $\varphi ,\rho ,\mu ,\mathit{\ell }$ makes it a generalized equation. By adjusting these parameters, we can alter the geometry of the time/space being modeled. The parameters $\varphi$ and $\rho$ scale the kernel, enabling the recovery of Hadamard or Katugampola derivatives, for instance. The parameter $\mu$ is a weight used to model systems with exponential decay or growth in their memory. Finally, parameter ℓ can be used to adjust the scale of the domain or the density of the medium.

The Orlicz–Sobolev space is critical for this specific equation because the ${\mathrm{\Theta }}_p$ term grows like ${|·|}^{p-1}$. For a solution to exist in the sense of the abstract, the integral of the state x and its fractional derivative must converge when raised to the power of p. Some specific boundary conditions are adjusted in accordance with the real-world problem and depend on the order of the equation. In these types of problems, the solution is usually a fixed point in an Orlicz–Sobolev space.

Example 3. 

Antiperiodic ψ-Hilfer p-Laplacian BVP (cf. [28] for the p-Laplacian Langevin problems with a ψ-Hilfer fractional derivative with $l_2=0,l_1=1$)). The problem is typically stated as in our problem:

$$\begin{array}{ccc}\hfill {}^H\mathrm{D}_{\psi ,1,0}^{{\alpha }_1,\beta ,\mathit{\ell }}{\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\psi ,1,0}^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)+\lambda x\left(t\right)& =& f(t,x(t\left)\right),t\in [a,b]\hfill \\ \hfill x\left(a\right)& =& -x\left(b\right),\hfill \\ \hfill {}^H\mathrm{D}_{\psi ,1,0}^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(a\right)& =& -{}^H\mathrm{D}_{\psi ,1,0}^{{\alpha }_2,\beta ,\psi }x\left(b\right).\hfill \end{array}$$

Remark 6. 

We remark that

1.

Since ${\mathrm{\Theta }}_p\left(x\right)={\left|x\right|}^{p-1}sign\left(x\right)$, $p>1$, it follows ${\mathrm{\Theta }}_p\left(0\right)=0,$ for all $p>1$.

2.

Since for any $p,q\in (1,\infty )$, where we have

$${\mathrm{\Theta }}_q\left({\mathrm{\Theta }}_p\left(x\right)\right)={\left|{\left|x\right|}^{p-2}x\right|}^{q-2}{\left|x\right|}^{p-2}x={\left|x\right|}^{(q-2)(p-1)+p-2}x={\left|\right|}^{pq-(p+q)}x,$$

it can be easily seen that

$${\mathrm{\Theta }}_q={\left({\mathrm{\Theta }}_p\right)}^{-1},1/p+1/q=1,p,q\in (1,\infty ).$$

3.

Since, for any $a_i\in \mathbb{R},(i=1,2,\dots ,n)$, we have

$$\left|\sum _{i=1}^n{a}_i\right|\le \sum _{i=1}^n|a_i|\le n\underset{i}{max}\left\{\right|a_i\left|\right\}.$$

Thus, for any $\eta \ge 0$, we have

$${\left|\sum _{i=1}^n{a}_i\right|}^{\eta }\le n^{\eta }\underset{i}{max}\left\{\right|a_i{|}^{\eta }\}\le n^{\eta }\left[\sum _{i=1}^n{\left|a_i\right|}^{\eta }\right],\eta \ge 0.$$

In our investigation, we consider the following different cases:

• Case $\left(I\right)$: When ${\alpha }_1\in (1,2],{\alpha }_2\in (0,1].$
• Case $\left(II\right)$: When ${\alpha }_1\in (0,1],{\alpha }_2\in (1,2].$

The problem (38) and some of its special cases have attracted the attention of many authors (see, for example, refs. [40,42,43,44,45] or [46,47,48], as well as the references therein for the background on these topics). Compared to the results on within these references, our assumptions are more natural. Unlike these references, we consider a solution in Orlicz spaces under the assumption that a Carathéodory function g has a special growth condition.

Let us formally convert the problem (38) into its corresponding integral form. Let $\beta \in [0,1]$, ${\alpha }_1\in (1,2]$ and ${\alpha }_2\in (0,1]$ (namely, Case $\left(I\right)$), and assume that g is sufficiently smooth. Clearly, in view of the definition, with ${\alpha }_1\in (1,2]$ in the Hilfer fractional derivative, the problem (38) can be written as follows:

$$\left({\Delta }_{\varphi }^{\rho ,\mathit{\ell }}\right){}_p{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1-1,\beta ,\mathit{\ell }}{\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)=g\left(t,x\left(t\right)\right)$$

for $t\in [a,b],p\in (1,\infty )$. Since from Proposition 1 [10], where $K_{\rho ,\varphi }^{1,\mathit{\ell },\mu }{\Delta }_{\varphi }^{\rho ,\mathit{\ell }}y\left(t\right)=y\left(t\right)-e^{-\chi \left(t\right)}y\left(a\right)y\in AC[a,b]$, it follows that

$${}_p{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1-1,\beta ,\mathit{\ell }}{\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)=C_I{e}^{-\chi \left(t\right)}+K_{\rho ,\varphi }^{1,\mathit{\ell },\mu }g\left(t,x\left(t\right)\right),$$

for $p\in (1,\infty )$, $t\in [a,b]$. Applying the operator $K_{\rho ,\varphi }^{{\alpha }_1-1,\mathit{\ell },\mu }$ to both sides of the final equation, we obtain

$${}_p{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1-1,\beta ,\mathit{\ell }}\left({\displaystyle \frac{e^{-\chi \left(t\right)}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(2-{\alpha }_1)}}}\right)=0,$$

so

$${\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)={\displaystyle \frac{e^{-\chi \left(t\right)}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(2-{\alpha }_1)}}}{C}_I^{\prime }+C_I{e}^{-\chi \left(t\right)}{\left(\omega \left(t\right)\right)}^{{\alpha }_1-1}+K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g\left(t,x\left(t\right)\right),$$

for $p\in (1,\infty )$, $t\in [a,b]$, where $C_I,C_I^{\prime }$ are constants that depend only on the initial, terminal, or boundary conditions. From the continuity of ${\mathrm{\Theta }}_p,p>1$ (cf. Remark 6), it follows that (in view of the initial condition ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(a\right)=0$) $C_I^{\prime }=0$. Hence, we obtain

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)={\mathrm{\Theta }}_q\left(C_I{e}^{-\chi \left(t\right)}{\left(\omega \left(t\right)\right)}^{{\alpha }_1-1}+K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g\left(t,x\left(t\right)\right)\right),$$

for $q\in (1,\infty )$, $t\in [a,b]$. Similarly, by applying the operator $K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }$ to both sides of the last equations leads, in view of

$${}_p{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}\left({\displaystyle \frac{e^{-\chi \left(t\right)}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_2)}}}\right)=0,$$

it leads us to the following integral form, which corresponding to the problem (38)

$$x\left(t\right)={\displaystyle \frac{e^{-\chi \left(t\right)}{C}_I^{\prime \prime }}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_2)}}}+K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }{f}_I(t,x\left(t\right)),$$

(39)

where $t\in [a,b]$, $\beta \in [0,1]$, ${\alpha }_1\in (1,2]$, ${\alpha }_2\in (0,1]$. Here, $C_I^{″}$ is a constant that depends only on the initial, terminal, or boundary conditions, and

$$f_I(t,x):={\mathrm{\Theta }}_q\left(C_I{e}^{-\chi \left(t\right)}{\left(\omega \left(t\right)\right)}^{{\alpha }_1-1}+K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g\left(t,x\right)\right),$$

(40)

for $t\in [a,b],q\in (1,\infty )$.

Next, we consider the case when $\beta \in [0,1]$, ${\alpha }_1\in (0,1],{\alpha }_2\in (1,2]$. This is Case $\left(II\right)$). Assume that g is sufficiently smooth. Operating by $K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }$ on both sides of (38), in view of

$${}_p{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}\left({\displaystyle \frac{e^{-\chi \left(t\right)}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_1)}}}\right)=0,$$

leads to

$${\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)={\displaystyle \frac{e^{-\chi \left(t\right)}{C}_{II}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_1)}}}+K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g\left(t,x\left(t\right)\right),$$

$p\in (1,\infty )$, $t\in [a,b]$, where $C_{II}$ is constant depending only on initial, terminal, or boundary conditions. From the continuity of ${\mathrm{\Theta }}_p$, $p>1$ (cf. Remark 6), it follows that $C_{II}=0$ in view of the initial condition ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(a\right)=0$. Thus,

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)={\mathrm{\Theta }}_q\left(K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g\left(t,x\left(t\right)\right)\right),$$

where $q\in (1,\infty )$, $t\in [a,b]$. This reads as:

$$\left({\Delta }_{\varphi }^{\rho ,\mathit{\ell }}\right){}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2-1,\beta ,\mathit{\ell }}x\left(t\right)={\mathrm{\Theta }}_q\left(K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g\left(t,x\left(t\right)\right)\right),$$

where $q\in (1,\infty )$. From Proposition 1 [10], $K_{\rho ,\varphi }^{1,\mathit{\ell },\mu }{\Delta }_{\varphi }^{\rho ,\mathit{\ell }}y\left(t\right)=y\left(t\right)-e^{-\chi \left(t\right)}y\left(a\right)$, where $y\in AC[a,b]$, it follows that

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2-1,\beta ,\mathit{\ell }}x\left(t\right)=e^{-\chi \left(t\right)}{C}_{II}^{\prime }+K_{\rho ,\varphi }^{1,\mathit{\ell },\mu }{\mathrm{\Theta }}_q\left(K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g\left(t,x\left(t\right)\right)\right),$$

where $q\in (1,\infty )$, where $C_{II}^{\prime }$ is a constant depending only on the initial, terminal, or boundary conditions. Consequently, the following integral form that corresponds to (38) when $\beta \in [0,1]$, ${\alpha }_2\in (1,2]$ and ${\alpha }_1\in (0,1]$ is of the form

$$x\left(t\right)={\displaystyle \frac{C_{II}^{\prime \prime }{e}^{-\chi \left(t\right)}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(2-{\alpha }_2)}}}+e^{-\chi \left(t\right)}{C}_{II}^{\prime }{\left(\omega \left(t\right)\right)}^{{\alpha }_2-1}+K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }{f}_{II}(t,x\left(t\right)),$$

(41)

for $t\in [a,b]$, $\beta \in [0,1]$, ${\alpha }_2\in (1,2]$, ${\alpha }_1\in (0,1],$ where

$$f_{II}(t,x):={\mathrm{\Theta }}_q\left(K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g\left(t,x\right)\right),$$

for $t\in [a,b]$ and $q\in (1,\infty )$.

Now, we are in a position to state and prove the following main result. Motivated by fractional calculus, we focus on the problem involving the integral operator. According to our results, we demonstrate that this problem has a solution in the relevant Orlicz space.

Theorem 5. 

Let ${\alpha }_1\in (1,2]$, ${\alpha }_2\in (0,1]$, $\beta \in [0,1]$, $\zeta \in (0,1)$, $\rho \in (0,1]$, $\mu \ge 0$ be such that $(1-\beta )(1-{\alpha }_2)<\zeta <1$. Let $\varphi \in C^1([a,b],\mathbb{R})$ be a positive increasing function such that ${\varphi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ and $\varphi \left(a\right)=0$.

Assume that ${\mathit{\ell }}_1(\rho ,·),{\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in C[a,b]$, and let $g:[a,b]\times \mathbb{R}\to \mathbb{R}$ be a Carathéodory function such that for each $r>0$ there exist $c_r\in L_{1,\mathit{\ell }}[a,b]$ and $a_r\ge 0$, satisfying the following:

1.

If $p\in (1,2]$ (so $q\in [2,\infty )$), then for all $x\in \mathbb{R}$,

$$\left|g(t,x)\right|\le c_r\left(t\right)+a_r{\left({\displaystyle \frac{\left|x\right|}{r}}\right)}^{\zeta },t\in [a,b].$$

2.

If $p\in (2,\infty )$, then for all $x\in \mathbb{R}$ and $\gamma \in ({\alpha }_1-1,{\alpha }_1)$,

$$\left|g(t,x)\right|\le c_r\left(t\right)+a_r{\left({\displaystyle \frac{\left|x\right|}{r}}\right)}^{\zeta },t\in [a,b],$$

and

$${\displaystyle \frac{\mathrm{\Gamma }(1+\gamma )}{\mathrm{\Gamma }(1+\gamma -{\alpha }_1)}}{\left(\omega \left(t\right)\right)}^{\gamma -{\alpha }_1}\le e^{\chi \left(t\right)}g(t,x),t\in (a,b].$$

Define

$$\psi =\left\{\begin{array}{cc}{\psi }_{{\displaystyle \frac{1}{\zeta }}},\hfill & p\in (1,2),\hfill \\ {\psi }_{{\displaystyle \frac{p}{\zeta }}},\hfill & p\in [2,\infty ),\hfill \end{array}\right.h\left(t\right):={\displaystyle \frac{e^{-\chi \left(t\right)}{C}_I^{″}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_2)}}}.$$

Assume that there exists $r>0$ such that

$${\parallel h\parallel }_{\psi ,\mathit{\ell }}+\parallel K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }\parallel \left(\parallel C_r{\parallel }_{\psi }+2A_r\right)\le r,$$

(42)

where $C_r(·)$ and $A_r$ are defined as above.

Then, the problem (38) admits a solution $x\in L_{{\displaystyle \frac{1}{\zeta }},\mathit{\ell }}[a,b]$.

Proof. 

Define the operator T by

$$\left(Tx\right)\left(t\right):=h\left(t\right)+\left(K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }\left(g(·,x)\right)\right)\left(t\right).$$

Step 1: Compactness and continuity. From Lemma 5, the operator $K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }$ is completely continuous. Since g is a Carathéodory function, the superposition operator $x\mapsto g(·,x(·\left)\right)$ is continuous from $L_{\psi ,\mathit{\ell }}[a,b]$ into itself. Hence, T is continuous and compact.

Step 2: Invariant set. Let

$$U_r:=\{x\in L_{\psi ,\mathit{\ell }}{[a,b]:\parallel x\parallel }_{\psi ,\mathit{\ell }}\le r\}.$$

Then, $U_r$ is closed, bounded, and convex. For $x\in U_r$, using the assumptions on g together with the estimates leading to the definitions of $C_r$ and $A_r$, we obtain

$${\parallel g(·,x)\parallel }_{\psi }\le {\parallel C_r\parallel }_{\psi }+2A_r.$$

Therefore,

$${\parallel Tx\parallel }_{\psi ,\mathit{\ell }}\le {\parallel h\parallel }_{\psi ,\mathit{\ell }}+\parallel K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }\parallel \left(\parallel C_r{\parallel }_{\psi }+2A_r\right).$$

From (42), it follows that

$${\parallel Tx\parallel }_{\psi ,\mathit{\ell }}\le r,$$

hence $T\left(U_r\right)\subset U_r$.

Step 3: Rothe boundary condition. Let $x\in \partial U_r$, i.e., ${\parallel x\parallel }_{\psi ,\mathit{\ell }}=r$, and let $\lambda \in (0,1)$. Suppose that

$$x=\lambda Tx.$$

Then,

$${\parallel x\parallel }_{\psi ,\mathit{\ell }}=\lambda {\parallel Tx\parallel }_{\psi ,\mathit{\ell }}\le \lambda r<r,$$

which contradicts ${\parallel x\parallel }_{\psi ,\mathit{\ell }}=r$.

Thus,

$$x\ne \lambda Tx\mathrm{for}\mathrm{all}x\in \partial U_r,\lambda \in (0,1).$$

Step 4: Conclusion. From the Rothe fixed-point theorem, the operator T has a fixed point $x\in U_r$, which is a solution of problem (38). □

Example 4. 

Let $g(t,x)={\left({\left(\omega \left(t\right)\right)}^{-\lambda }ln\left(\nu +\omega \left(t\right)\right)+{\left[t^{\lambda }x\right]}^2\right)}^{{\displaystyle \frac{2}{5}}}$, where $t\in [0,1]$, $x\in \mathbb{R}$, $\lambda \in (0,1)$, $\nu >1$. Clearly, g satisfies the assumptions of Theorem 5 with $\zeta ={\displaystyle \frac{4}{5}}$. Indeed, it is clear that

1.

$g(t,·),t\in [0,1]$ is continuous on $\mathbb{R}$.

2.

$g(·,x),x\in \mathbb{R}$ is measurable on $[0,1]$.

Also, for any $t\in [0,1]$, $x\in \mathbb{R}$, we have

$$\begin{array}{ccc}\hfill \left|g\right(t,x\left)\right|& \le & {\left(|{\left(\omega \left(t\right)\right)}^{-\lambda }ln\left(\nu +\omega \left(t\right)\right)|\right)}^{{\displaystyle \frac{2}{5}}}+{\left(t^{\lambda }\left|x\right|\right)}^{{\displaystyle \frac{4}{5}}}\hfill \\ & \le & {\left(|{\left(\omega \left(t\right)\right)}^{-\lambda }ln\left(\nu +\omega \left(t\right)\right)|\right)}^{{\displaystyle \frac{2}{5}}}+r^{{\displaystyle \frac{4}{5}}}{\left({\displaystyle \frac{\left|x\right|}{r}}\right)}^{{\displaystyle \frac{4}{5}}},\hfill \end{array}$$

$r>0$. Moreover, for any $t\in (0,1],x\in \mathbb{R}$, we have

$$g(t,x)\ge {\left(\omega \left(t\right)\right)}^{-{\displaystyle \frac{2\lambda }{5}}}{\left(ln\left(\nu +\omega \left(t\right)\right)\right)}^{{\displaystyle \frac{2}{5}}}⟹{\left(\omega \left(t\right)\right)}^{{\displaystyle \frac{2\lambda }{5}}}g(t,x)\ge {\left(ln\left(\nu +\omega \left(t\right)\right)\right)}^{{\displaystyle \frac{2}{5}}}\ge {\left(ln\left(\nu \right)\right)}^{{\displaystyle \frac{2}{5}}}.$$

Consequently,

$$e^{\chi \left(t\right)}g(t,x)\ge g(t,x)\ge {\left(\omega \left(t\right)\right)}^{-{\displaystyle \frac{2\lambda }{5}}}{\left(ln\left(\nu \right)\right)}^{{\displaystyle \frac{2}{5}}}.$$

So, we can find a value $\gamma \in \left({\alpha }_1-1,{\alpha }_1)\right)$ such that

$${\displaystyle \frac{2\lambda }{5}}={\alpha }_1-\gamma \in (0,1),e^{{\left({\displaystyle \frac{\mathrm{\Gamma }(1+\gamma )}{\mathrm{\Gamma }(1+\gamma -{\alpha }_1)}}\right)}^{{\displaystyle \frac{5}{2}}}}\le \nu .$$

For example, if $\lambda =1.5$, we can choose $\gamma =0.75$. In this case, we obtain ${\displaystyle \frac{2\lambda }{5}}=0.3\in (0,1)$, and

$$\nu \ge e^{{\left({\displaystyle \frac{\mathrm{\Gamma }(1+\gamma )}{\mathrm{\Gamma }(1+\gamma -{\alpha }_1)}}\right)}^{{\displaystyle \frac{5}{2}}}}=e^{{\left({\displaystyle \frac{\mathrm{\Gamma }\left(1.75\right)}{\mathrm{\Gamma }\left(0.25\right)}}\right)}^{{\displaystyle \frac{5}{2}}}}=e^{0.{278}^{{\displaystyle \frac{5}{2}}}}=1.0414.$$

Now, we can prove the second of our main results regarding solutions to the problem under study in Orlicz spaces. We consider another set of assumptions, namely the ranges of ${\alpha }_1$ and ${\alpha }_2$, the condition involving $\zeta$, and the integrability exponent required for the function $c_r$. The key difference is that, instead of the previously considered case ${\alpha }_1\in (1,2],{\alpha }_2\in (0,1]$, we will assume that ${\alpha }_1\in (0,1],{\alpha }_2\in (1,2]$.

Theorem 6. 

Let ${\alpha }_1\in (0,1],{\alpha }_2\in (1,2],\beta \in [0,1],\zeta \in (0,1),\rho \in (0,1],\mu \ge 0$ such that $(1-\beta )(2-{\alpha }_2)<\zeta <{\alpha }_1$. Let $\varphi \in C^1\left[[a,b],\mathbb{R}\right]$ be a positive, increasing function such that ${\varphi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\varphi \left(a\right)=0$. Let $p,q>1$ such that $1/p+1/q=1$ and assume that ${\mathit{\ell }}_1(\rho ,·),{\left({\mathit{\ell }}_1(\rho ,·)\right)}^{-1}\in C[a,b]$, and $g(·,·):[a,b]\times \mathbb{R}\to \mathbb{R}$ be a Carathéodory function such that, for each $r>0$, there is $c_r\in L_{q,\mathit{\ell }}[a,b]$ and $a_r\ge 0$ such that

1.

If $p\in (1,2]$$(q\in [2,\infty \left)\right)$, then for any $x\in \mathbb{R}$, we have

$$\left|g(t,x)\right|\le c_r\left(t\right)+a_r{\left({\displaystyle \frac{\left|x\right|}{r}}\right)}^{\zeta },t\in [a,b].$$

2.

If $p\in (2,\infty )$, then for any $x\in \mathbb{R}$, and $\gamma \in ({\alpha }_1-1,{\alpha }_1)$, we have

$$\left|g(t,x)\right|\le c_r\left(t\right)+a_r{\left({\displaystyle \frac{\left|x\right|}{r}}\right)}^{\zeta },t\in [a,b],\zeta \in (0,1),$$

$${\displaystyle \frac{\mathrm{\Gamma }(1+\gamma )}{\mathrm{\Gamma }(1+\gamma -{\alpha }_1)}}{\left(\omega \left(t\right)\right)}^{\gamma -{\alpha }_1}\le e^{\chi \left(t\right)}g(t,x),t\in (a,b].$$

If

$$\underset{r\in (0,\infty )}{sup}{\displaystyle \frac{r}{{∥h(·)∥}_{\psi ,\mathit{\ell }}+∥K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }∥\left({∥C_r∥}_{\psi }+2A_r\right)}}\ge ∥K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }∥,$$

$$\psi =\left\{\begin{array}{c}{\psi }_{{\displaystyle \frac{1}{\zeta }}},p\in \left(1,2\right),\hfill \\ {\psi }_{{\displaystyle \frac{p}{\zeta }}},p\in [2,\infty ),\hfill \end{array}\right.,$$

where

$$h(·):={\displaystyle \frac{C_{II}^{\prime \prime }{e}^{-\chi (·)}}{{\left(\omega (·)\right)}^{(1-\beta )(2-{\alpha }_2)}}}+e^{-\chi (·)}{C}_{II}^{\prime }{\left(\omega (·)\right)}^{{\alpha }_2-1},$$

$$C_r(·):={\displaystyle \frac{2^{\frac{q}{p}}}{|{\mathit{\ell }}_1(\rho ,t)|}}{\left|K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }{c}_r(·)\right|}^{{\displaystyle \frac{q}{p}}},$$

$$A_r:={\displaystyle \frac{2^{\frac{q}{p}}{\left|a_r\right|}^{\frac{q}{p}}{∥\omega ∥}^{{\alpha }_1(1-\zeta )\frac{q}{p}}}{min|{\mathit{\ell }}_1(\rho ,·)|{\left(\mathrm{\Gamma }(1+{\alpha }_1)\right)}^{1-\zeta }}}\times \left\{\begin{array}{c}{\left({\displaystyle \frac{q}{p}}\right)}^{\zeta },p\in (1,2),\hfill \\ {\left(q\right)}^{{\displaystyle \frac{\zeta }{p}}},p\in [2,\infty ),\hfill \end{array}\right.$$

Then, for all $p>1,$ the problem (38) admits a solution $x\in L_{{\displaystyle \frac{1}{\zeta }},\mathit{\ell }}[a,b]$.

Proof. 

The proof follows the same structure as in Theorem 5.

Step 1. $f_{II}$ is a Carathéodory function that satisfies the assumptions of Theorem 4. For each $x\in \mathbb{R}$, the function $g(·,x)$ is measurable on $[a,b]$. Since the function ${\mathrm{\Theta }}_q$ is continuous and $K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }$ preserves measurability, it follows that the function $f_{II}(·,x)$ is measurable. Fix $t\in [a,b]$. Since $g(t,·)$ is continuous and satisfies the growth condition, it is locally bounded. Therefore, $K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g(t,·)$ is continuous.

If $q\ge 2$, then the function $f_{II}(t,x)$ is continuous because the function $f_q(t,x)$ is locally Lipschitz on bounded sets. If $q\in (1,2)$, then, by assumption, the argument of the function is bounded away from zero; therefore, the function is locally Lipschitz on the open interval $(0,\infty )$. Therefore, $f_{II}$ is a Carathéodory function.

Using $p(q-1)=q$, we obtain

$$|f_{II}(t,x)|={\left|K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g(t,x)\right|}^{{\displaystyle \frac{q}{p}}}.$$

By the growth condition on g and inequality (22),

$$|f_{II}(t,x)|\le C_r\left(t\right)+A_r{\left({\displaystyle \frac{K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }\left|x\right|}{r}}\right)}^{{\displaystyle \frac{q\zeta }{p}}}.$$

Hence,

$${\displaystyle \frac{|f_{II}(t,x)|}{|{\mathit{\ell }}_1(\rho ,t)|}}\le C_r\left(t\right)+A_r{\psi }^{-1}\left({\psi }^{*}\left({\displaystyle \frac{K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }\left|x\right|}{r}}\right)\right).$$

Since $c_r\in L_{q,\mathit{\ell }}[a,b]$ and $\zeta <{\alpha }_1$, standard estimates for fractional integrals yield

$$C_r\in L_{\psi }[a,b].$$

Thus, Assumption (29) holds.

Step 2. Existence of an Orlicz solution of (41). Define

$$h\left(t\right)={\displaystyle \frac{C_{II}^{″}{e}^{-\chi \left(t\right)}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(2-{\alpha }_2)}}}+C_{II}^{\prime }{e}^{-\chi \left(t\right)}{\left(\omega \left(t\right)\right)}^{{\alpha }_2-1}.$$

Since $(1-\beta )(2-{\alpha }_2)<\zeta$ and $\omega$ is increasing, we have

$${\left(\omega (·)\right)}^{-(1-\beta )(2-{\alpha }_2)}\in L^{{\displaystyle \frac{1}{\zeta }}}[a,b],$$

and clearly ${\left(\omega (·)\right)}^{{\alpha }_2-1}\in C[a,b]$. Hence, $h\in L_{{\displaystyle \frac{1}{\zeta }},\mathit{\ell }}[a,b]$. From the hypothesis,

$$\underset{r>0}{sup}{\displaystyle \frac{r}{{\parallel h\parallel }_{\psi ,\mathit{\ell }}+\parallel K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }\parallel \left(\parallel C_r{\parallel }_{\psi }+2A_r\right)}}\ge \parallel K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }\parallel ,$$

there exists $r>0$ such that Assumption (30) holds. Therefore, from Theorem 4, Equation (41) admits a solution

$$x\in L_{{\displaystyle \frac{1}{\zeta }},\mathit{\ell }}[a,b].$$

Step 3. Equivalence with problem (38). Let $x\in L_{{\displaystyle \frac{1}{\zeta }},\mathit{\ell }}[a,b]$ solve (41). Then,

$$x=K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }{f}_{II}.$$

Applying ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}$ and using Proposition 3, we obtain

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x=f_{II}.$$

Hence,

$${\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\right)={\mathrm{\Theta }}_q\left(K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g\right)=K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g.$$

Applying ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}$, we obtain

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}{\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\right)=g(t,x\left(t\right)).$$

The boundary condition

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(a\right)=0$$

follows directly from (41). Thus, x solves (38). □

4.2. Embedding Theorem for Orlicz Spaces

To demonstrate that the solutions exhibit higher regularity and belong to specific Orlicz–Sobolev spaces (see, e.g., [5,49,50,51]), it is first necessary to first establish certain properties concerning the inclusion of these spaces (see [52]).

The following property of the kernel of an integral operator is to be considered:

Lemma 7. 

Let $\alpha \in (0,1]$ and define

$$k(t,s)=e^{-\left(\chi \right(t)-\chi (s\left)\right)}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right){\chi }_{[a,t]}\left(s\right),$$

where $\omega \in C^1\left([a,b]\right)$ is strictly increasing with ${\omega }^{\prime }\left(t\right)>0$ on $[a,b]$, and $\chi \in C\left(\right[a,b\left]\right)$. Assume that

$${\mathit{\int }}_0^{\delta }\tilde{\psi }\left(u^{\alpha -1}\right)du<\infty \mathit{for}\mathit{some}\delta >0.$$

Then, the mapping

$$t\mapsto k(t,·)$$

is continuous from $[a,b]$ into $L_{\tilde{\psi }}\left([a,b]\right)$, i.e.,

$$\parallel k(t_2,·)-k(t_1,·){\parallel }_{L_{\tilde{\psi }}}\to 0\mathrm{as}{t}_2\to t_1.$$

Proof. 

Let $t_2>t_1$. We decompose

$$k(t_2,s)-k(t_1,s)=\left\{\begin{array}{cc}A(t_2,t_1,s),\hfill & s\in [a,t_1],\hfill \\ B(t_2,s),\hfill & s\in (t_1,t_2],\hfill \\ 0,\hfill & s>t_2,\hfill \end{array}\right.$$

where

$$A(t_2,t_1,s)=e^{-(\chi \left(t_2\right)-\chi \left(s\right))}{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)-e^{-(\chi \left(t_1\right)-\chi \left(s\right))}{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right),$$

and

$$B(t_2,s)=e^{-(\chi \left(t_2\right)-\chi \left(s\right))}{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right).$$

Step 1: Estimate on $(t_1,t_2]$. Since $\chi$ is continuous on $[a,b]$, the exponential factor is bounded. Moreover, since ${\omega }^{\prime }\in C[a,b]$ and ${\omega }^{\prime }\left(s\right)>0$, it is bounded. Hence, there exists $C>0$ such that

$$|B(t_2,s)|\le C{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1},s\in (t_1,t_2].$$

Using the monotonicity of $\omega$, there exists $c>0$ such that

$$\omega \left(t_2\right)-\omega \left(s\right)\ge c(t_2-s),$$

and therefore,

$$|B(t_2,s)|\le C{(t_2-s)}^{\alpha -1}.$$

Thus,

$${\mathit{\int }}_{t_1}^{t_2}\tilde{\psi }\left(\right|B(t_2,s)\left|\right)ds\le {\mathit{\int }}_{t_1}^{t_2}\tilde{\psi }\left(C{(t_2-s)}^{\alpha -1}\right)ds.$$

Following the change in variables $u=t_2-s$, this becomes

$${\mathit{\int }}_0^{t_2-t_1}\tilde{\psi }\left(C{u}^{\alpha -1}\right)du,$$

which tends to 0 as $t_2\to t_1$ by the assumption.

• Step 2: Estimate on $[a,t_1]$. For $s\in [a,t_1]$, we have pointwise convergence

  $$k(t_2,s)\to k(t_1,s)\mathrm{as}{t}_2\to t_1.$$

We estimate the difference $A(t_2,t_1,s)$. Since $\chi$ is continuous and ${\omega }^{\prime }$ is bounded, we may write

$$|A(t_2,t_1,s)|\le C\left|{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}-{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}\right|.$$

Using the mean value theorem for the function $x\mapsto x^{\alpha -1}$, we obtain

$$\left|{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}-{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}\right|\le C|\omega \left(t_2\right)-\omega \left(t_1\right)|{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -2}.$$

Since $\omega \in C^1$ and ${\omega }^{\prime }>0$, we have

$$|\omega \left(t_2\right)-\omega \left(t_1\right)|\le \parallel {\omega }^{\prime }{\parallel }_{\infty }|t_2-t_1|,$$

and

$$\omega \left(t_1\right)-\omega \left(s\right)\ge c(t_1-s),$$

for some $c>0$. Hence,

$$|A(t_2,t_1,s)|\le C|t_2-t_1|{(t_1-s)}^{\alpha -2}.$$

Thus, for $t_2$ sufficiently close to $t_1$, we obtain a dominating function

$$g\left(s\right)=C{(t_1-s)}^{\alpha -2}.$$

From the assumption on $\tilde{\psi }$, it follows that

$${\mathit{\int }}_a^{t_1}\tilde{\psi }\left(g\left(s\right)\right)ds<\infty .$$

Therefore, by the standard modular convergence argument in Orlicz spaces (see Theorem 2.1 in [12]),

$${\mathit{\int }}_a^{t_1}\tilde{\psi }\left(\right|A(t_2,t_1,s)\left|\right)ds\to 0.$$

Combining the estimates on $[a,t_1]$ and $(t_1,t_2]$, we conclude that

$${\mathit{\int }}_a^b\tilde{\psi }\left(\right|k(t_2,s)-k(t_1,s)\left|\right)ds\to 0,$$

which implies

$$\parallel k(t_2,·)-k(t_1,·){\parallel }_{L_{\tilde{\psi }}}\to 0.$$

□

Proposition 4. 

Let ψ be an N-function satisfying the ${\Delta }_2$-condition. The fractional Orlicz–Sobolev space $W_{\varphi ,\rho ,\mathit{\ell }}^{{\alpha }_2,\psi }[a,b]$ is continuously embedded in $L_{\psi ,\mathit{\ell }}[a,b]$ and compactly embedded in $L_{1/\zeta ,\mathit{\ell }}[a,b]$ for any $\zeta \in (0,1)$ satisfying $(1-\beta )(1-{\alpha }_2)<\zeta$.

Proof. 

Let $\mathcal{P}=\{x\in W_{\varphi ,\rho ,\mathit{\ell }}^{{\alpha }_2,\psi }[a,b]:\parallel x{\parallel }_W\le 1\}$ be the unit ball in our space. By definition, for any $x\in \mathcal{P}$, both x and $v={}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x$ are bounded in $L_{\psi ,\mathit{\ell }}$.

1. Continuous embedding ($W_{\varphi ,\rho ,\mathit{\ell }}^{{\alpha }_2,\psi }[a,b]\hookrightarrow L_{\psi ,\mathit{\ell }}$). From the fundamental property of the generalized Hilfer derivative, x can be reconstructed via the fractional integral operator $K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }$:

$$x\left(t\right)={\displaystyle \frac{e^{-\left(\chi \right(t\left)\right)}}{\mathrm{\Gamma }\left({\alpha }_2\right)}}{\mathit{\int }}_a^t{e}^{\chi \left(s\right))}{(\omega \left(t\right)-\omega \left(s\right))}^{{\alpha }_2-1}{\omega }^{\prime }\left(s\right)f\left(s\right)ds+\mathcal{P}\left(t\right),$$

where $\mathcal{P}\left(t\right)$ represents the initial conditions at $t=a$. The operator $K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }$ is a Volterra integral operator with a weakly singular kernel $k(t,s)\approx {(\omega \left(t\right)-\omega \left(s\right))}^{{\alpha }_2-1}$. Since $\varphi$ is strictly increasing and $C^1$, the kernel is integrable. In Orlicz spaces, such operators are bounded linear mappings from $L_{\psi ,\mathit{\ell }}$ to $L_{\psi ,\mathit{\ell }}$. Thus,

$${∥x∥}_{L_{\psi }}\le {∥K_{\rho ,\varphi }^{v,\mathit{\ell },\mu }∥}_{L_{\psi }}+{∥\mathcal{P}∥}_{L_{\psi }}\le M{∥v∥}_{L_{\psi }}+{∥\mathcal{P}∥}_{L_{\psi }}\le C{\parallel x\parallel }_W.$$

This proves the continuity of the embedding.

2. Compact Embedding ($W_{\varphi ,\rho ,\mathit{\ell }}^{{\alpha }_2,\psi }[a,b]\hookrightarrow L_{1/\zeta ,\mathit{\ell }}$). To prove compactness, we must show that $\mathcal{P}$ is relatively compact in $L_{1/\zeta ,\mathit{\ell }}$. We use the Arzelà–Ascoli Theorem to show it is relatively compact in $C[a,b]$.

From Part 1, ${∥x∥}_{L_{\psi }}\le C$. Since $[a,b]$ is bounded and $\varphi$ is continuous, $x\left(t\right)$ is pointwise bounded for all $t\in [a,b]$, so we have uniform boundedness.

By Lemma 7, the mapping

$$t\mapsto k(t,·)\in L_{\tilde{\psi }}\left([a,b]\right)$$

is continuous. That is,

$$\parallel k(t_2,·)-k(t_1,·){\parallel }_{\tilde{\psi }}\to 0\mathrm{as}{t}_2\to t_1.$$

Thus, by Hölder’s inequality in Orlicz spaces,

$$|x\left(t_2\right)-x\left(t_1\right){|\le 2\parallel v\parallel }_{\psi }{\parallel k(t_2,·)-k(t_1,·)\parallel }_{\tilde{\psi }}.$$

Since ${\parallel v\parallel }_{\psi }\le C$, it follows that

$$\underset{x\in \mathcal{P}}{sup}|x\left(t_2\right)-x\left(t_1\right)|\to 0\mathrm{as}{t}_2\to t_1,$$

i.e., $\mathcal{P}$ is equicontinuous.

By Arzelà–Ascoli, any sequence in $\mathcal{P}$ has a subsequence converging uniformly in $C[a,b]$. Since $C[a,b]$ is continuously embedded in $L_{1/\zeta ,\mathit{\ell }}[a,b]$ for any $\zeta >0$ on a bounded interval, the subsequence also converges in $L_{1/\zeta ,\mathit{\ell }}$. This completes the proof of compactness. □

We should recall that we announced that the solutions of the differential problems under study are, in fact, in some Orlicz–Sobolev spaces. In this equation,

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}{\mathrm{\Phi }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\right)=g(t,x)$$

the embedding is what allows us to handle the nonlinear term $g(t,x)$.

To ensure the operator T maps the fractional Orlicz–Sobolev space into itself and is compact, we assume the following:

(H1)

The function $g:[a,b]\times \mathbb{R}\to \mathbb{R}$ satisfies the Carathéodory conditions.

(H2)

There exists a Sobolev conjugate N-function ${\psi }_{{\alpha }_2}^{*}$, i.e., $W^{{\alpha }_2,\psi }\hookrightarrow L_{{\psi }_{{\alpha }_2}^{*}}$, such that g satisfies the growth condition

$$\left|g(t,x)\right|\le a\left(t\right)+b\psi \left(\mathcal{K}\left|x\right|\right),\mathrm{for}\mathrm{a}.\mathrm{e}.t\in [a,b],$$

where $a\in L_{\psi }[a,b]$ and $\mathcal{K}$ is the constant from the fractional embedding $W^{{\alpha }_2,\psi }\hookrightarrow L_{{\psi }_{{\alpha }_2}^{*}}$.

(H3)

To ensure the compactness of the operator T, the following limit holds:

$$\underset{\left|x\right|\to \infty }{lim}{\displaystyle \frac{g(t,x)}{{\mathrm{\Phi }}_p\left(x\right)}}=0,$$

uniformly for $t\in [a,b]$. This condition implies that the nonlinearity is sub-p-harmonic, allowing for the application of the Schauder fixed-point theorem in $W_{\varphi ,\rho ,\mathit{\ell }}^{{\alpha }_2,\psi }$.

Finally, we present a result that demonstrate some similarities between the previous theorems in the differential form and multidimensional studies of problems with Orlicz–Sobolev solutions. Below is a regularity result for solutions to the problem under study.

Proposition 5. 

Let the parameters ${\alpha }_1,{\alpha }_2,\beta ,\rho ,\mu ,\zeta$ and the N-function ψ be as defined in Theorem 5. Suppose $x\in L_{1/\zeta ,\mathit{\ell }}[a,b]$ is a solution to the generalized fractional p-Laplacian boundary value problem:

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathit{\ell }}{\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)=g(t,x\left(t\right)).$$

where g is a Carathéodory function that satisfies the growth condition $\left|g(t,x)\right|\le c_r\left(t\right)+a_r{\left({\displaystyle \frac{\left|x\right|}{r}}\right)}^{\zeta }$. Then, the solution x is a weak solution and possesses the higher regularity, i.e., belongs to Orlicz–Sobolev space $W_{\varphi ,\rho ,\mathit{\ell }}^{{\alpha }_2,\psi }[a,b]$.

Proof. 

The proof relies on the sequential inversion of the operators using the unified integral family $K_{\rho ,\varphi }^{\alpha ,\mathit{\ell },\mu }$. When we apply the integral operator of order ${\alpha }_1$, denoted as $K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }$, to the governing equation, we can isolate the flux term:

$${\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right)=K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g(t,x\left(t\right))+{\mathcal{P}}_1\left(t\right).$$

Following the growth conditions in Theorem 5, $g(·,x)\in L_{\psi ,\mathit{\ell }}$. Since $K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }$ is bounded on Orlicz spaces, the term ${\mathrm{\Theta }}_p\left({}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\right)$ is in $L_{\psi ,\mathit{\ell }}$. Applying now the inverse p-Laplacian ${\mathrm{\Theta }}_q\left(·\right)$, we obtain

$${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)={\mathrm{\Theta }}_q\left(K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }g(t,x\left(t\right))+{\mathcal{P}}_1\left(t\right)\right).$$

The properties of the N-function $\psi$ (specifically the ${\Delta }_2$-condition) ensure that the q-power mapping ${\mathrm{\Theta }}_q\left(·\right)$ preserves the required integrability. Thus, ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\in L_{\psi ,\mathit{\ell }}$. Finally, we apply the integral operator of order ${\alpha }_2$, denoted as $K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }$, to reconstruct the solution:

$$x\left(t\right)=K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }\left[{}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\left(t\right)\right]+{\mathcal{P}}_2\left(t\right).$$

Since ${}^H\mathrm{D}_{\varphi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathit{\ell }}x\in L_{\psi ,\mathit{\ell }}$ and $K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }$ are bounded, it follows that $x\in L_{\psi ,\mathit{\ell }}$. From the definition of the Orlicz–Sobolev space as the set of functions whose fractional derivatives remain in the base Orlicz space, we conclude $x\in W_{\varphi ,\rho ,\mathit{\ell }}^{{\alpha }_2,\psi }[a,b]$. □

The fractional integral operator $K_{\rho ,\varphi }^{\alpha }$ is inherently a regularity-improving mapping. While our prior existence results conclude that $x\in L_{{\psi }^{*}}[a,b]$, yielding the boundedness of the solution, we must also ensure that the structure of the function is compatible with the original fractional differential equation.

Remark 7. 

While the fixed-point argument ensures that $x\in L_{{\psi }^{*}}[a,b]$, the operator structure of the boundary value problem implies higher regularity. Specifically, the solution $x\left(t\right)$ admits the integral representation $x=K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }\left[{\mathrm{\Theta }}_q\left(v\right)\right]$, where $v\left(t\right)$ is the fractional integral of the nonlinear term g. From the fundamental mapping properties of the fractional integral operator, if $g(t,x)\in L_{\psi }$, then the solution x automatically gains generalized fractional derivatives of order ${\alpha }_2$. It follows that the generalized Hilfer derivative ${}^{H}D^{{\alpha }_2}x$ exists and belongs to $L_{\psi }[a,b]$. Consequently, the solution $x\left(t\right)$ strictly resides in the fractional Orlicz–Sobolev space $W_{\varphi ,\rho ,\mathit{\ell }}^{{\alpha }_2,\psi }[a,b]$, ensuring that the p-Laplacian term ${\mathrm{\Theta }}_p\left({}^{H}D^{{\alpha }_2}x\right)$ is mathematically well-defined.

Corollary 4. 

Let ${\alpha }_1\in (1,2]$, ${\alpha }_2\in (0,1]$, $\beta \in [0,1]$, $\rho \in (0,1]$, $\mu \ge 0$, and let $\varphi \in C^1([a,b],\mathbb{R})$ be a positive increasing function. Assume ψ is an N-function satisfying the ${\Delta }_2$-condition such that the weakly singular kernel is integrable:

$${\mathit{\int }}_0^t\tilde{\psi }\left(s^{min\{{\alpha }_1,{\alpha }_2\}-1}\right)ds<\infty ,\mathrm{for}t>0.$$

Suppose the nonlinear term $g:[a,b]\times \mathbb{R}\to \mathbb{R}$ satisfies the Carathéodory conditions and the sub-linear growth estimates defined previously. If the following radius condition holds:

$$\underset{r\in (0,\infty )}{sup}{\displaystyle \frac{r}{{\parallel h\parallel }_{\psi ,\mathit{\ell }}+\parallel K_{\rho ,\varphi }^{{\alpha }_2,\mathit{\ell },\mu }\parallel \left(\parallel c_r{\parallel }_{\psi ,\mathit{\ell }}+2A_r\right)}}\ge \parallel K_{\rho ,\varphi }^{{\alpha }_1,\mathit{\ell },\mu }\parallel ,$$

then the boundary value problem admits at least one solution $x\in W_{\varphi ,\rho ,\mathit{\ell }}^{{\alpha }_2,\psi }[a,b]$.

We can also prove the regularity of the solutions to Theorem 6 by the same method, so we omit the details here.

5. Conclusions

In this paper, we propose a rigorous framework for proving the existence and regularity of solutions to a generalized compositional p-Laplacian problem.

To obtain these results, we examine the images of Orlicz spaces via certain integral operators, which combine fractional and non-fractional operators—that is to say, generalizations of many classical operators. We then investigated the invertibility of these operators on the corresponding Orlicz spaces.

We study the solvability of certain differential equations that have been reformulated in integral form. These equations are, in fact, generalizations of one-dimensional equations defined for p-Laplacian operators. By combining Hilfer-type fractional derivatives with the flexibility of Orlicz–Sobolev theory, we showed that the nested operator structure, representing a nonlocal flux within a broader conservation law, remains mathematically well-posed under sub-critical growth conditions. The transition from a ’weak’ existence result in the weighted Orlicz space to a ’strong’ regularity result in the fractional Orlicz–Sobolev space $W_{\varphi ,\rho ,\mathit{\ell }}^{{\alpha }_2,\psi }[a,b]$ is an important observation as it confirms that the topology of the operator is perfectly compatible with its differential components.

This research builds on classical fractional-order equations and p-Laplacian equations in one dimension, as well as more general problems involving nonlocal operators. This is an interesting area for further research, as it unifies many previous studies. What currently limits the use of this type of operator in practical issues and is an open problem? Our construction starts with one-dimensional fractional-order operators. A wider application for partial differential equations requires examining the appropriate definition in symmetric form and fractional order derivatives for distributions.