math.math-ap math.math-ca
J Vanterler 
  
Da C Sousa 
  
Rubens F Camargo 
  
E Capelas De Oliveira 
  
Gast Ão 
  
S F Frederico 
email: gastao.frederico@ua.pt
  
  
PSEUDO-FRACTIONAL DIFFERENTIAL EQUATIONS AND GENERALIZED g-LAPLACE TRANSFORM

Keywords: 2020 Mathematics Subject Classification. 34A08, 34A12, 47G30, 44A10 Pseudo-fractional operator, existence and uniqueness, g-Laplace transform

In this article, we introduce a generalized g-Laplace transform and discuss some essential results of integral transform theory, in particular, involving a ψ-Hilfer pseudo-fractional derivative and function convolution. In this sense, we investigated the existence and uniqueness of known solutions for a pseudo-fractional differential equation.

Introduction and motivation

Over the years, one of the issues discussed in the scientific community, is what is fractional calculus? What is its importance to the academic community? When the first ideas about fractional calculus were discussed, there was no dimension of the importance and relevance that their theory and applications in physics, chemistry, biology, engineering, medicine, among other areas of knowledge, [START_REF] Agahi | Stochastic g-fractional Integrals and their Bounds for Convex Stochastic Processes[END_REF][START_REF] Aghayan | Stability analysis of fractional order neutral-type systems considering time varying delays, nonlinear perturbations, and input saturation[END_REF][START_REF] Almeida | Modeling some real phenomena by fractional differential equations[END_REF][START_REF] Almeida | Leitmann's direct method for fractional optimization problems[END_REF][START_REF] Lazo | Noether-type theorem for fractional variational problems depending on fractional derivatives of functions[END_REF][START_REF] Sayevand | A survey on fractional asymptotic expansion method: A forgotten theory[END_REF][START_REF] Sousa | Fractional calculus and the ESR test[END_REF][START_REF] Sousa | Validation of a fractional model for erythrocyte sedimentation rate[END_REF], and recently, fractional derivatives have been used to generalize and refine models that describe the epidemic disease of coronavirus [START_REF] Atangana | Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination?[END_REF][START_REF] Khan | Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative[END_REF]. Nowadays we have countless definitions of derivatives and fractional integrals [START_REF] Almeida | A Caputo fractional derivative of a function with respect to another function[END_REF][START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF][START_REF] Sousa | ψ-Hilfer pseudo-fractional operator: new results about fractional calculus[END_REF][START_REF] Sousa | On the ψ-fractional integral and applications[END_REF][START_REF] Teodoro | A review of definitions of fractional derivatives and other operators[END_REF].

In 2018 when Sousa and Oliveira [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF], introduced the fractional derivative ψ-Hilfer motivated by solving the vast number of fractional derivatives in a single operator, it was already known, which would be a possible version for the Laplace transform with respect to another function, that is, ψ. Thus, in 2019 Jarad and Abdeljawad [START_REF] Jarad | Generalized fractional derivatives and Laplace transform[END_REF], introduced a version for the Laplace transform with ψ. However, the reverse version was still missing. In 2020 Fahad et al. [START_REF] Fahad | On Laplace transforms with respect to functions and their applications to fractional differential equations[END_REF], introduced the inverse of the Laplace transform with ψ. In this sense, from the Laplace transform with ψ, it became possible to discuss properties such as existence, uniqueness, stability, controllability of mild solutions of fractional differential equations, involving the fractional derivative ψ-Hilfer. Some works involving the fractional derivative ψ-Hilfer, can be consulted in [START_REF] Sousa | Attractivity for differential equations of fractional order and ψ-Hilfer type[END_REF][START_REF] Sousa | Ulam-Hyers stabilities of fractional functional differential equations[END_REF][START_REF] Sousa | On the fractional functional differential equation with abstract Volterra operator[END_REF][START_REF] Sousa | On the Ulam-Hyers stabilities of the solutions of ψ-Hilfer fractional differential equation with abstract Volterra operator[END_REF][START_REF] Sousa | Stability of ψ-Hilfer impulsive fractional differential equations[END_REF].

On the other hand, Endre Pap [START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] being one of the main researchers in the area of g-calculus, with numerous cutting-edge works and applications, started to discuss problems involving partial differential equations via pseudo-analysis. Many works in the area come from Pap works. Recently, some researchers began to discuss more closely, the idea of unifying the g-calculus theory, pseudo-analysis with fractional calculation, since it is still a new field and because there are open questions [START_REF] Agahi | Generalized expectation with general kernels on g-semirings and its applications[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Ralević | The pseudo-linear superposition principle for nonlinear partial differential equations and representation of their solution by the pseudo-integral[END_REF][START_REF] Sousa | ψ-Hilfer pseudo-fractional operator: new results about fractional calculus[END_REF][START_REF] Sousa | Reachability of fractional dynamical systems using ψ-Hilfer pseudo-fractional derivative[END_REF]. There are some works on the g-calculus theory with fractional calculus, in particular, involving inequalities, but it still needs more research and future contributions to the development and growth of the area [START_REF] Agahi | Pseudo-fractional integral inequality of Chebyshev type[END_REF][START_REF] Agahi | On pseudo-Mittag-Leffler functions and applications[END_REF][START_REF] Agahi | Stochastic g-fractional Integrals and their Bounds for Convex Stochastic Processes[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Yadollahzadeh | Hermite-Hadamard's inequality for pseudo-fractional integral operators[END_REF]. Aiming to unify these two areas more, in 2020 Sousa et al. [START_REF] Sousa | ψ-Hilfer pseudo-fractional operator: new results about fractional calculus[END_REF], extended the fractional derivative ψ-Hilfer to pseudo-operators and discussed some basic properties.

In 2005, Pap [START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] discussed a theory of generalized functions in analogy to Mikusinski's operators, which allows the construction of a generalized solution of the Burgers equation. Considering the extensions of operations ⊕ and for non-commutative and non-associative cases, some non-linear partial differential equations were addressed using the pseudo-linear superposition principle. Other works in the same top, involving different equations via pseudo-analysis, can be obtained in the following works [START_REF] Pap | Pseudo-analysis approach to nonlinear partial differential equations[END_REF][START_REF] Pap | Applications of pseudo-analysis on models with nonlinear partial differential equations[END_REF][START_REF] Pap | Noncommutative and nonassociative pseudo-analysis and its applications on nonlinear partial differential equations[END_REF][START_REF] Ralević | The pseudo-linear superposition principle for nonlinear partial differential equations and representation of their solution by the pseudo-integral[END_REF][START_REF] Sousa | Reachability of fractional dynamical systems using ψ-Hilfer pseudo-fractional derivative[END_REF][START_REF] Sousa | Existence and uniqueness of global solution in g-variational calculus[END_REF].

In 2020 Sousa et al. [START_REF] Sousa | Existence and uniqueness of global solution in g-variational calculus[END_REF], investigated the existence and uniqueness of global solutions of the Cauchy problem associated with data (t 0 , x 0 ) any solution (I := [a, b], x) is given by

   d ⊕ dt x(t) = F(t, x)
x(t 0 ) = x 0 with t 0 ∈ I.

On the other hand, also in 2020 Sousa et al. [START_REF] Sousa | Reachability of fractional dynamical systems using ψ-Hilfer pseudo-fractional derivative[END_REF], discussed the reachability of linear and non-linear systems in the sense of the ψ-Hilfer pseudo-fractional derivative in g-calculus by means of the Mittag-Leffler functions (one and two parameters) with the form

H α,β ;ψ ⊕, ,t 0 + x(t) = Ax(t) ⊕ Bu(t), t ∈ J := [t 0 ,t 1 ] (1) 
I 1-γ;ψ ⊕, ,t 0 + x(t 0 ) = 0 and H α,β ;ψ ⊕, ,t 0 + x(t) = Ax(t) ⊕ Bu(t) ⊕ f (t, x(t), u(t)), t ∈ J (2) I 1-γ;ψ ⊕, ,t 0 + x(t 0 ) = 0 where H α,β ;ψ ⊕, ,t 0 + (•) is the ψ-Hilfer pseudo-fractional derivative of order 0 < α < 1 and type 0 ≤ β ≤ 1, γ = α + β (1 -α), I 1-γ;ψ ⊕, ,t 0 + (•)
is the Riemann-Liouville pseudo-fractional integral with respect to another function 1γ, the state vector x ∈ R n , the control vector u ∈ R m and A and B are the constant matrices of dimension n × n and n × m respectively and the nonlinear function f :

J × R n × R m → R n is continuous, respectively.
To date, the studies Sousa et al. [START_REF] Sousa | Existence and uniqueness of global solution in g-variational calculus[END_REF][START_REF] Sousa | Reachability of fractional dynamical systems using ψ-Hilfer pseudo-fractional derivative[END_REF], are the first results in this area involving the reachability approach of pseudo-differential equations involving ψ-Hilfer pseudo-fractional derivative. Motivated by the above works and by issues that are still open in the theory of seven-sector and quasi-sector operators, we will briefly describe the main contributions of this article, in order to make it clear throughout the article. as a result, we have:

(1) The generalized g-Laplace transform, is given by

(3) L ⊕ ψ ( f (t)) = g -1 L ψ (g( f (t))) . (2)
We discuss some properties of Eq.( 3), specially related to the ψ-convolution product. In addition, we discussed the calculation of the pseudo-fractional derivative ψ-Hilfer and the pseudo-fractional derivative ψ-Riemann-Liouville.

Finally, we consider the following ψ-Hilfer pseudo-fractional differential equation given by ( 4)

H α,β ;ψ ⊕, ,t 0 + x (t) = Ax (t) ⊕ f (t, x(t)) , t ∈ J I 1-γ;ψ ⊕, ,t 0 + x (t 0 ) = x 0 in which H α,β ;ψ ⊕, ,t 0 + (•) is the pseudo-fractional ψ-Hilfer derivative with order 0 < α ≤ 1 and type 0 ≤ β ≤ 1, I 1-γ;ψ ⊕, ,t 0 + (•) is the pseudo-fractional integral with order 1 -γ (γ = α -β (1 -α)), n × n matrix A and f : [t 0 , ∞) × R n → R n be a continuous function.
The second main objective of this article is to investigate the existence and uniqueness of the solution of Eq.( 4), given by the following Theorems 1 and 2:

Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION Theorem 1. Let f : [t 0 , ∞) × R n → R n be continuous function satisfying the Lipschitz condition || f (t, x 1 (t)) f (t, x 2 (t))|| g ≤ g L ||x 1 (t) x 2 (t)|| g ,t ∈ J := [t 0 ,t 1 ], L > 0.
Then, the initial value Eq.( 4) has a unique solution whenever

M L g -1 (ψ(t) -ψ(t 0 )) α α < 1 Theorem 2. Let f : [t 0 , ∞) × R n → R n be a continuous function such that there exist a positive constant L, such that || f (t, x(t))|| g ≤ g L ||x(t)|| g , t ∈ J.
Then, the initial value Eq.( 4) has a solution on J.

The paper is organized as follows: In section 2, we present some definitions and essential results for the good development of the article. In section 3, we discuss the first main result of the article, that is, we introduce a new extension for the g-Laplace transform with respect to the ψ function. From this result, some results were discussed, in particular, involving the pseudo-fractional derivative ψ-Hilfer. In section 4, we discussed the convolution of functions. Finally, the second main result of this article is discussed in section 5, that is, we investigate the existence and uniqueness of mild solutions for Eq.( 4) via Theorem 1 and Theorem 2. We conclude the article with open questions and problems.

Preliminaries

In this section, we will discuss some concepts and results, essential for the development of this paper.

The Mittag-Leffler function E α,β (z) is a complex function which depends on two complex parameters, and it is defined by [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF] (5)

E α,β (z) = ∞ ∑ k=0 z k Γ (αk + β ) , α, β > 0 .
The function E α,β (•) converges for all values of the argument z. For a n × n matrix A, the matrix extension of the above Mittag-Leffler function is

(6) E α,β (A) = ∞ ∑ k=0 A k Γ (αk + β ) .
Definition 3. [START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] A binary operator ⊕ on J is pseudo-addition if it is commutative, nondecreasing, with respect to , continuous; associative, and with a zero (neutral) element denoted by 0. Let

J + = {x|, x ∈ [a, b], 0 ≤ x} for a, b ∈ R + .
Definition 4. [START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] A binary operation on J is pseudo-multiplication if it is commutative, positively non-decreasing, i.e., x ≤ y implies x z ≤ y z for all z ∈ [a, b] + , associative and with a unit element 1 ∈ [a, b], i.e., for each x ∈ [a, b], 1 x = x. Also, 0 x = 0 and that is distributive over ⊕, i.e., x (y ⊕ z) = (x y) ⊕ (x ) z.

The structure (J, ⊕, ) is a semiring [START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF].

Definition 5. [START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] An important class of pseudo-operations ⊕ and is when these are defined by a monotone and continuous function g : J -→ [0, ∞], i.e., pseudo-operations ⊕ and are given by [START_REF] Almeida | Modeling some real phenomena by fractional differential equations[END_REF] x ⊕ y = g -1 (g(x) + g(y)) and x y = g -1 (g(x)g(y)).

Definition 6. [START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] Let X be a non-empty set and A be a σ -algebra of subsets of a set X. A set function µ : A -→ [a, b] is called a σ -⊕-measure if the following conditions are satisfy:

Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION

(1) µ ( / 0) = 0; (2) µ ( ∞ i=1 A i ) = ∞ i=1 µ(A i )
for any sequence {A } i∈N of pairwise disjoint sets from A . Definition 7. [START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] Let pseudo-operations ⊕ and be defined a monotone and continuous function g : J -→ [0, ∞].

(1) The g-integral for a measurable function f :

[c, d] → J is given by ⊕ [c,d] f dx = g -1 d c g( f (x)) dx .
(2) The g-Laplace of a function f is defined by [START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] Let g be the additive generator of the strict-pseudo-addition ⊕ on J such that g is continuous differentiable on (a, b). The corresponding pseudo-multiplication will always be defined as u v = g -1 (g(u) • g(v)). If the function f is differentiable on (c, d) and has the same monotonicity as the function g, then the g-derivative of f at the point x ∈ (c, d) is defined by

L ⊕ [ f (x)] = g -1 (L [g( f (x))]) • Definition 8.
d ⊕ dx f (x) = g -1 d dx g( f (x)) •
Also, if there exists the n-g-derivative of f , then

d (n)⊕ dx f (x) = g -1 d n dx n g( f (x)) • Definition 9
. [START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] Let g be a generator of a pseudo-addition ⊕ on interval [-∞, +∞]. Binary operations and on [-∞, +∞] are defined by the expressions x y = g -1 (g(x)g(y)) and x y = g -1 g(x) g(y) .

If the expressions g(x)g(y) and g(x) g(y) have sense are said to be the pseudo-subtraction and pseudodivision consistent with the pseudo-addition ⊕.

Definition 10. [START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] Let g : [-∞, +∞] -→ [-∞, +∞] be a continuous, strictly increasing and odd function such that g(0) = 0, g(1) = 1 and g(+∞) = +∞. The system of pseudo-arithmetical operations {⊕, , , } generated by these functions is said to be the consistent system.

Definition 11. [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF][START_REF] Sousa | Leibniz type rule: ψ-Hilfer fractional operator[END_REF] 

Let J := [a, b] (-∞ ≤ a < b ≤ ∞)
be a finite or infinite interval of the real line R and α > 0. Also let ψ(x) be an increasing and positive monotone function on (a, b], having a continuous derivative ψ (x) on J. The left-sided and right-sided fractional integrals of a function f with respect to another function ψ, are defined by

(8) I α;ψ a+ f (x) = 1 Γ (α) x a ψ (t) (ψ (x) -ψ (t)) α-1 f (t)dt. and (9) 
I α;ψ b-f (x) = 1 Γ (α) b x ψ (t) (ψ (t) -ψ (x)) α-1 f (t)dt.
Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION Definition 12. [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF][START_REF] Sousa | Leibniz type rule: ψ-Hilfer fractional operator[END_REF] Let n -1 < α < n, with n ∈ N, J is an interval such that -∞ ≤ a < b ≤ +∞ and f , ψ ∈ C n (J, R) are two functions such that ψ is increasing and ψ(x) = 0, for all x ∈ J. The ψ-Hilfer fractional derivative left-sided and right-sided, denoted by H D α,β ;ψ a + (•) of a function f of order α and type 0 ≤ β ≤ 1, is defined by

(10) H D α,β ;ψ a + f (x) = I β (n-α);ψ a + 1 ψ (x) d dx n I (1-β )(n-α);ψ a + f (x)
and

(11) H D α,β ;ψ b - f (x) = I β (n-α);ψ b - - 1 ψ (x) d dx n I (1-β )(n-α);ψ b - f (x)
where

I α;ψ a + (•) and I α;ψ b -(•)
by defined in Eq.( 8) and Eq.( 9), respectively. Definition 13. [START_REF] Sousa | ψ-Hilfer pseudo-fractional operator: new results about fractional calculus[END_REF] Let a generator g : J → [0, ∞] of the pseudo-addition ⊕ and the pseudo-multiplication be an increasing function. Also let ψ de an increasing and positive function on (a, b], having a continuous derivative ψ (x). The left-sided and the right-sided ψ-Riemann-Liouville pseudo-fractional integrals of order α > 0 of a measurable function f : J → J with respect to function ψ on J are defined by:

I α;ψ ⊕, ,a+ f (x) = g -1 I α;ψ a+ g ( f (x)) = ⊕ [a,x] g -1 ψ (t) (ψ (x) -ψ (t)) α-1 Γ (α) f (t) dt ( 12 
)
and 8) and Eq.( 9), respectively. Definition 14. [START_REF] Sousa | ψ-Hilfer pseudo-fractional operator: new results about fractional calculus[END_REF] Let a generator g : J → [0, ∞] of the pseudo-addition ⊕ and the pseudo-multiplication be an increasing function. Also let ψ ∈ C n (J, R), a function such that ψ be an increasing and positive function on (a, b] having a continuous derivative ψ and ψ (x) = 0 for all x ∈ J. The left-sided and rightsided ψ-Hilfer pseudo-fractional derivative of order n -1 < α < n and type 0 ≤ β ≤ 1, of a measurable function f : J → J is defined by 10) and Eq. [START_REF] Bede | The theory of pseudo-linear operators[END_REF].

I α;ψ ⊕, ,b-f (x) = g -1 I α;ψ b-g ( f (x)) = ⊕ [x,b] g -1 ψ (t) (ψ (t) -ψ (x)) α-1 Γ (α) f (t) dt ( 
H α,β ;ψ ⊕, ,a+ f (x) = g -1 H D α,β ;ψ a+ g ( f (x)) = I β (n-α);ψ ⊕, ,a+ g -1 D ψ (x) n I 1-γ;ψ ⊕, ,a+ f (x) (14) and H α,β ;ψ ⊕, ,b-f (x) = g -1 H D α,β ;ψ b-g ( f (x)) = I β (n-α);ψ ⊕, ,b-g -1 - D ψ (x) n I 1-γ;ψ ⊕, ,b-f (x) (15)

Note that

H α,β ;ψ ⊕, ,a+ f (x) = g -1 I γ-α;ψ a+ RL D γ;ψ a+ g ( f (x)) = I γ-α;ψ ⊕, ,a+ RL D γ;ψ ⊕, ,a+ f (x) (16) 
Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION and

H α,β ;ψ ⊕, ,b-f (x) = g -1 I γ-α;ψ b- RL D γ;ψ b-g ( f (x)) = I γ-α;ψ ⊕, ,b- RL D γ;ψ ⊕, ,b-f (x) (17) where γ = α + β (n -α). If α > 0 and A is n × n matrix, then (18) 
H α,β ;ψ ⊕, ,a+ E α,β A (ψ (x) -ψ (a)) α = AE α,β A (ψ (x) -ψ (a)) α .
Theorem 15.

[27] Let f : J → J be a measurable functions. If n ∈ N, then we have

(1) H 0,β ;ψ ⊕, ,a+ f (x) = f (x); (2) H 1,β ;ψ ⊕, ,a+ f (x) = g -1 D ψ (x) g ( f (x)) ; (3) H n,β ;ψ ⊕, ,a+ f (x) = D ψ (x) (n)⊕ f (x); (4) H α,β ;ψ ⊕, ,a+ ( f 1 (x) ⊕ f 2 (x)) = H α,β ;ψ ⊕, ,a+ f 1 (x) ⊕ H α,β ;ψ ⊕, ,a+ f 2 (x); (5) H α,β ;ψ ⊕, ,a+ (λ f (x)) = λ H α,β ;ψ ⊕, ,a+ f (x); (6) H α,β ;ψ ⊕, ,a+ I α;ψ ⊕, ,a+ f (x) = f (x); (7) I α;ψ ⊕, ,a+ H α,β ;ψ ⊕, ,a+ f (x) = f (x) n k=1 g -1 (C k ) g -1 (ψ (x) -ψ (a)) γ-k with γ = α + β (n -α) and C k = (g • f ) [n-k] I n-γ;ψ ⊕, ,a+ g ( f (a)) Γ (γ -k + 1)
.

Definition 16.

[15] Let f , ψ : [0, ∞) -→ R be real valued functions such that ψ be a non negative increasing function with ψ(0) = 0. Then, the Laplace transform of f with respect to ψ is defined by

L ψ ( f (t)) = F(s) = ∞ 0 e -sψ(t) ψ (t) f (t) dt
for all s ∈ C such that this integral converges. Here, L ψ (•) denotes the Laplace transform with respect to ψ, which we call a generalized Laplace transform.

Definition 17. [START_REF] Jarad | Generalized fractional derivatives and Laplace transform[END_REF] Assume that the function f is defined for f ≥ 0. Then, the Laplace transform of f , denoted by L ( f ), is defined by the improper integral

(19) L ( f (t)) ≡ F(s) = ∞ 0 e -st f (t) dt
provided that the integral in Eq.( 19) exists, i.e., that the integral is convergent. The corresponding inverse Laplace transform is given by

(20) L -1 ( f (t)) = 1 2πi c+i∞ c-i∞ e st F(s) ds with s ∈ C such that Re(s) = c. Let m -1 < α < m, m ∈ N, -∞ ≤ a < b ≤ ∞, 0 ≤ β ≤ 1, and f , ψ ∈ C m (J, R
) be functions such that ψ is increasing and ψ (x) = 0 for all x ∈ J. Then, the ψ-Hilfer fractional derivative of order α and type β is given by [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF][START_REF] Sousa | Leibniz type rule: ψ-Hilfer fractional operator[END_REF] (21)

H D α,β ,ψ a + f (x) = I β (m-α),ψ a + 1 ψ (x) d dx m I (1-β )(m-α),ψ a + f (x).
Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION Taking β → 1 in Eq.( 21), we obtain the Caputo fractional derivative given by [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF] (22

) C D α,ψ a + f (x) = I m-α,ψ a + 1 ψ (x) d dx m f (x).
Also, taking β → 0 in Eq.( 21), we get the Riemann-Liouville fractional derivative given by ( 23)

RL D α,ψ a + f (x) = 1 ψ (x) d dx m I m-α,ψ a + f (x).
These generalized fractional operators can be written as the conjugation of the standard fractional operators with the operation of composition with ψ or ψ -1 [12]

I α,ψ a + = Q ψ • I α,ψ(a) t + • Q ψ -1 , RL D α,ψ a + = Q ψ • RL D α,ψ(a) t + • Q ψ -1 , ( 24 
) C D α,ψ a + = Q ψ • C D α,ψ(a) t + • Q ψ -1 , and (25) 
H D α,β ,ψ a + = Q ψ • H D α,β ,ψ(a) t + • Q ψ -1 ,
where the functional operator Q ψ is defined by

(26) Q ψ f (x) = f (ψ(x)).
Theorem 18. [START_REF] Fahad | On Laplace transforms with respect to functions and their applications to fractional differential equations[END_REF] The generalized Laplace transform may be written as a combination of the classical Laplace transform with the operation of composition with ψ or ψ -1 , as follows

(27) L ψ = L • Q -1 ψ
where the fundamental operator Q ψ is defined in Eq.( 26).

Corollary 19. [15]

The inverse generalized Laplace transform may be written as a combination of the inverse classical Laplace transform with the operation of composition with ψ or ψ -1 , as follows

(28) L -1 ψ = Q ψ • L -1 or, in other words (29) L -1 ψ (F(s)) = 1 2πi c+i∞ c-i∞ e sψ(t) F(s) ds. Corollary 20. [15] If f (t) is a function whose classical Laplace transform is F(s), the generalized Laplace transform of the function f • ψ = f (ψ(t)) is also F(s), L ( f (t)) = F(s) =⇒ L ψ ( f (ψ(t))) = F(s).
Below some particular cases, presented as examples [START_REF] Fahad | On Laplace transforms with respect to functions and their applications to fractional differential equations[END_REF][START_REF] Jarad | Generalized fractional derivatives and Laplace transform[END_REF]. 

Example 21. Let µ ∈ C be such that Re(µ) > -1, then L ψ ((ψ(t)) µ ) = Γ(µ + 1) s µ+1 for Re(s) > 0. Example 22. Let λ ∈ R, then L ψ (e λ ψ(t) ) = 1 s -λ for Re(s) > λ .
L ψ (E µ (λ ψ(t)) µ ) = s µ-1 s µ -λ for |λ /s µ | < 1.
Example 24. Let µ ∈ C be such that Re(µ) > 0 and E µ,µ (•) be the two-parameter Mittag-Leffler function, then 

L ψ (E µ,µ (λ ψ(t)) µ ) = 1 s µ -λ for |λ /s µ | < 1.
L ψ ((ψ(t)) ν-1 E γ µ,ν (λ ψ(t)) µ ) = s µγ-ν (s µ -λ ) γ .
Theorem 26. [START_REF] Fahad | On Laplace transforms with respect to functions and their applications to fractional differential equations[END_REF][START_REF] Jarad | Generalized fractional derivatives and Laplace transform[END_REF] Let µ > 0 and let f be a function of ψ-exponential order, piecewise continuous over each finite interval [0, T ]. Then,

L ψ I µ,ψ a + f (t) = s -µ L ψ ( f (t)). Theorem 27. [12, 15] Assume m-1 < µ < 1, 0 ≤ ν ≤ 1 and f a function such that f (t), D j,ψ I (1-ν)(m-µ),ψ 0 + f (t) ∈ C[0, ∞) are of ψ-exponential order for j = 0, 1, 2, . . . , m -1, while H D µ,ν,ψ 0 + f (t) is piecewise continuous on [0, ∞). Then, ( 30 
) L ψ H D µ,ν,ψ 0 + f (t) = s µ L ψ ( f (t)) - m-1 ∑ i=0 s m(1-ν)+µν-i-1 I (1-ν)(m-µ)-ν 0 + f (0) .
Definition 28. [START_REF] Jarad | Generalized fractional derivatives and Laplace transform[END_REF] Let f and h be of ψ-exponential order, piecewise continuous functions over each finite interval [0, T ]. Then, the ψ-convolution of f and h, denoted by f * ψ h is given by

(31) f * ψ h = t 0 f ψ -1 (ψ(t) -ψ(τ)) ψ (τ)h(τ) dτ.
Theorem 29.

[12] Let f , p, h be of ψ-exponential order, piecewise continuous functions over each finite interval [0, T ], and let a and b be constants. Then,

* ψ p = p * ψ f . ( (1) f 
) ( f * ψ p) * ψ h = f * ψ (p * ψ h). ( 2 
) f * ψ (ap + bh) = a f * ψ p + b f * ψ h. 3 
Theorem 30. [START_REF] Fahad | On Laplace transforms with respect to functions and their applications to fractional differential equations[END_REF] Assume that f and p are piecewise continuous functions on [0, T ] and of ψ-exponential order c > 0. Then,

(32) L ψ ( f * ψ p) = L ψ ( f )L ψ (p)
Definition 31. [START_REF] Babakhani | Some properties of pseudo-fractional operators[END_REF][START_REF] Hosseini | On pseudo-fractional integral inequalities related to Hermite-Hadamard type[END_REF][START_REF] Pap | Pseudo-additive measures and their applications[END_REF][START_REF] Pap | Applications of the generated pseudo-analysis to nonlinear partial differential equations[END_REF] Let the pseudo-operations ⊕ and be defined through a monotone and continuous function g : J → [0, ∞]:

(1) The g-integral of a measurable function f :

[c, d] → [a, b) is given by ⊕ [c,d] f dt = g -1 d c g( f (t))dt , (2) 
The g-Laplace transform of a function f is defined by

L ⊕ ( f (t)) = g -1 (L (g( f (t))) .
Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION

Generalized g-Laplace transform

In this section, we will present a generalization for the g-Laplace transform (g-calculus) with respect to another function ψ and its respective inverse. In this perspective we discuss some properties, in particular, we obtain the g-Laplace transform of the ψ-Hilfer fractional derivative and the ψ-Riemann-Liouville fractional integral.

Definition 32. Let a generator g : J → [0, ∞] of the pseudo-addition ⊕ and the pseudo-multiplication be an increasing function. Let f , ψ : [0, ∞) → R be real valued functions such that ψ be a non-negative increasing function such that ψ(0) = 0. Then, the g-Laplace transform of f with respect to another function ψ is defined by

L ⊕ ψ ( f (t)) = g -1 L ψ (g( f (t))) where L ψ (•) is defined in Definition 16.
Theorem 33. Let a generator g : J → [0, ∞] of the pseudo-addition ⊕ and the pseudo-multiplication be an increasing function. Let f , ψ : [0, ∞) → R be real valued functions such that ψ be increasing function. The inverse generalized Laplace transform may be written as a combination of the inverse classical Laplace transform with the operation of composition with ψ and ψ -1 as follows

L ⊕ ψ = L ⊕ • Q -1 ψ where the functional operator Q -1 ψ f (t) = f (ψ -1 (t)). Proof. Let f : t → f (t). On the other hand, Q -1 ψ f (t) : t → Q -1 ψ f (t) = f (ψ -1 (t)
). So, using the definition of the generalized g-Laplace transform, we have

L ⊕ • Q -1 ψ ( f ) : t → L ⊕ ψ • Q -1 ψ f (t) = L ⊕ Q -1 ψ f (t) = g -1 L g Q -1 ψ f (t) (t) = g -1 ∞ 0 e -st g Q -1 ψ f (t) dt = g -1 ∞ 0 e -st g f (ψ (t)) (t) dt = g -1 ∞ 0 e -sψ(u) g( f (u))ψ (u) du
thus, the result follows.

At this point, we are motivated to present the inverse generalized g-Laplace transform, based on the following corollary.

Corollary 34. The inverse generalized g-Laplace transform with respect to another function, ψ, may be written as a combination of the inverse classical g-Laplace transform with the operation of composition with ψ or ψ -1 as follows

L ⊕ -1 ψ = Q ψ • L ⊕ -1
or, in other words

L ⊕ -1 ψ {F(s)} = g -1 L -1 ψ g (F(s)) = g -1 1 2πi c+i∞ c-i∞
e sψ(t) g(F(s)) ds

Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION

As a result we can write

L ⊕ ψ {λ f (t) ⊕ h(t)} = λ L ⊕ ψ { f (t)} ⊕ L ⊕ ψ {g(t)
} whose proof follows from the definition. Definition 35. [START_REF] Bede | The theory of pseudo-linear operators[END_REF] Given a pseudo-linear space V over (J, ⊕, ), a generalized norm (g-norm) is a mapping || • || g : V → J + such that (1) ||v|| g ≥ g 0, ∀v ∈ V and ||v|| g = 0 if and only

if v = 0 (2) ||c v|| g = |c| g ||v|| g , ∀c ∈ [0, b] and v ∈ V (3) ||v ⊕ w|| g ≤ g ||v|| g ⊕ ||w|| g , ∀v, w ∈ V . A function f : [0, ∞) → R n is said to be of ψ-exponential order c > 0, if there exist positive constants M, c, T , such that || f || g ≤ M e cψ(t) , for t ≥ T . Theorem 36. If f : [0, ∞) → R is a piecewise continuous function and is of ψ-exponential order c > 0,
where ψ is a non-negative increasing function with ψ(0) = 0, then the generalized g-Laplace transform of f exists for s > c.

Proof. For the ψ-exponentially bounded function f , we have

L ⊕ ψ { f (t)} = g -1 L ψ g( f (t)) ≤ g -1 M s -c where L ψ { f (t)} ≤ M s -c
and in the last step we used the assumption that s > c. Therefore, the integral is convergent.

Theorem 37. Let α > 0 and f be a piecewise continuous function on each interval [0,t] and of ψexponential order. Then

L ⊕ ψ I α,ψ ⊕, ,0 + f (t) = g -1 (s -α ) L ⊕ ψ f (t)
Proof. In fact, from Definition 32 and remembering that

L ψ I α,ψ 0 + f (t) = s -α L ψ ( f (t)) we have L ⊕ ψ I α,ψ ⊕, ,0 + f (t) = g -1 L ψ g I α,ψ ⊕, ,0 + f (t) = g -1 L ψ g g -1 I α,ψ 0 + g( f (t)) = g -1 L ψ I α,ψ 0 + g( f (t)) = g -1 s -α L ψ g( f (t)) = g -1 (s -α ) g -1 L ψ g( f (t)) = g -1 (s -α ) L ⊕ ψ f (t) which conclude the proof.
Theorem 38. Let g be the same as in Definition 32, m -1 < α < m, 0 ≤ β ≤ 1, and s ∈ R. Let f be a function such that f (t), D j,ψ

0 + I (1-β )(n-α),ψ 0 + f (t) ∈ C[0, ∞) are of ψ-exponential order for j = 0, 1, 2, . . . , m -1, while H D α,β ,ψ ⊕, ,0 + f (t) is piecewise continuous on [0, ∞).
Then, the generalized g-Laplace Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION transform of the ψ-Hilfer fractional derivative of order α and type β , with 0 ≤ β ≤ 1 is given by

L ⊕ ψ H D α,β ,ψ ⊕, ,0 + f (t) = g -1 (s α ) L ⊕ ψ f (t) m-1 i=1 g -1 s m(1-β )+αβ -i-1 I (1-β )(m-α)-i,ψ ⊕, ,0 + f (0) .
Proof. In fact, from Definition 32 and remembering that

L ψ H D α,β ,ψ 0 + f (t) = s α L ψ [ f (t)] - m-1 ∑ i=0 s m(1-β )+αβ -i-1 I (1-β )(m-α)-i,ψ 0 + f (0)
we have

L ⊕ ψ H D α,β ,ψ ⊕, ,0 + f (t) = g -1 L ψ g H D α,β ,ψ ⊕, ,0 + f (t) = g -1 L ψ H D α,β ,ψ ⊕, ,0 + g( f (t)) = g -1 s α L ψ [g( f (t))] - m-1 ∑ i=0 s m(1-β )+αβ -i-1 I (1-β )(m-α)-i,ψ 0 + f (0) = g -1 g g -1 s α L ψ (g( f (t))) -g g -1 s m(1-β )+αβ -i-1 C 0 ⊕ • • • ⊕ g -1 s m(1-β )+αβ -m C m-1 = g -1 (s α ) L ⊕ ψ f (t) m-1 i=0 g -1 s m(11-β )+αβ -i-1 I (1-β )(m-α)-i,ψ ⊕, ,0 + f (0)
which complete the proof.

ψ-convolution

Definition 39. Let a generator g : J → [0, ∞] of the pseudo-addition, ⊕, and the pseudo-multiplication, , be an increasing function. Let f and h be of ψ-exponential order, piecewise continuous functions over each finite interval [0, T ]. Then, the ψ-convolution of f and h is a function, f ⊗ ψ h, defined by

(33) ( f ⊗ ψ h)(t) = ⊕ [0,t] f ψ -1 (ψ(t) -ψ(x)) g -1 (ψ (x)) h(x) dx = g -1 t 0 g f ψ -1 (ψ(t) -ψ(x)) ψ (x)g(h(x)) dx .
Remark 40. Some particular cases of the ψ-convolution (Definition 39), given by from choosing of g(•) and ψ(•):

(1) Note that, taking g(x) = x in Definition 39, we have the classical convolution with respect to another function, (ψ), given by the following relation

( f * ψ h)(t) = t 0 f ψ -1 (ψ(t) -ψ(x)) ψ (x)h(x) dx.
(2) (2) Taking ψ(x) = x we get the definition of convolution in the sense of g-calculus, given by

( f ⊗ h)(t) = ⊕ [0,t] f (t -x) h(x) dx.
(3) Finally, taking g(x) = x = ψ(x) Definition 39, we obtain the classical convolution given by

( f * h)(t) = t 0 f (t -x)h(x) dx.
Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION (4) Making the same choice for Theorem 41 and Theorem 42, we obtain their respective particular cases.

Theorem 41. Let a generator g : J → [0, ∞] of the pseudo-addition, ⊕, and the pseudo-multiplication, be an increasing function. Let f and h be ψ-exponential order, piecewise continuous functions over each finite interval [0, T ]. Then, the following relation

f ⊗ ψ h = Q ψ Q -1 ψ f ⊗ Q -1 ψ f or in other words f ⊗ ψ h = ( f • ψ -1 ) ⊗ (g • ψ -1 ) • ψ
Proof. Using the classical g-condition and substituting x = ψ(u) we get

f ⊗ ψ h(t) = ⊕ [0,t] f (t -x) h(x) dx = g -1 t 0 g( f (t -x))g(h(x)) dx = g -1 ψ -1 (t) 0 g( f (t -ψ(u)))g(h(ψ(u)))ψ (u) du
Applying Q π we have

( f ⊗ ψ h) • ψ(t) = t 0 g ( f (ψ(t) -ψ(u))) g(h(ψ(u)))ψ (u) du = g -1 t 0 g ( f • ψ)(ψ -1 (ψ(t) -ψ(u))) g[(h • ψ)(u)]ψ (u) du = g(•ψ) ⊗ ψ (g • ψ)
which conclude the proof.

If we interpret convolution as a binary operation acting on two functions, and use alternative notation ⊗( f , g) and ⊗ ψ ( f , g) instead of f ⊗ g and f ⊗ ψ g, then the results of Theorem 41 can be written, like Eq.( 2) for ψ-fractional derivatives and integrals, as a conjugation of operators

⊗ ψ = Q ψ • ⊗ • Q -1 ψ , Q -1 ψ namely ⊗ ψ ( f , g) = Q ψ ⊗ Q -1 ψ f , Q -1 ψ g . Theorem 42
. Let a generator g : J → [0, ∞] of the pseudo-addition ⊕ and the pseudo-multiplication be an increasing function. Let f and h be piecewise continuous functions on [0, T ] and of ψ-exponential order c > 0. Then,

L ⊕ ψ f ⊗ ψ h = L ⊕ ψ { f }L ⊕ ψ {h}. Proof.
In fact, note that by Theorem 41 and Theorem 33, we have

(L ⊕ ψ ) • (⊗ ψ ) = L ⊕ • Q -1 ψ • Q ψ • ⊗ • Q -1 ψ , Q -1 ψ = (L ⊕ • ⊗) • Q -1 ψ , Q -1 ψ therefore (L ⊕ ψ )( f , h) = (L ⊕ • ⊗) • (Q -1 ψ f , Q -1 ψ h) = L ⊕ (Q -1 ψ f )L ⊕ (Q -1 ψ h) = L ⊕ ψ { f }L ⊕ ψ {h} which conclude the proof.
The proof of Theorem 43 follows directly from Theorem 41.

Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION Theorem 43. Let a generator g : J → [0, ∞] of the pseudo-addition ⊕ and the pseudo multiplication be an increasing function. Let ψ, h and p be of ψ-exponential order, piecewise continuous function over each finite interval [0, T ], and let a and b constants. Then,

⊗ ψ h = h ⊗ ψ f . (2) ( f ⊗ ψ h) ⊗ ψ p = f ⊗ ψ h ⊗ ψ p . ( (1) f 
) f ⊗ ψ (ah ⊕ bp) = a f ⊗ ψ h ⊕ b f ⊗ ψ p. 3 

Existence and uniqueness

In this section, we will discuss the existence and uniqueness of the solution of the nonlinear pseudofractional differential equation in terms of the Mittag-Leffler function of one and two parameters.

Lemma 44. For 0 < α ≤ 1, the solution of Eq.( 4) is

x (t) = x 0 g -1 (ψ (t) -ψ (t 0 )) γ-1 ⊕ [t 0 ,t] g -1 ψ (s) E α,γ g (A) (ψ (t) -ψ (s)) α ds ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 E α,α g (A) (ψ (t) -ψ (s)) α f (s, x(s)) ds.
Proof. Applying the operator I 1-γ;ψ ⊕, ,t 0 + (•) on both sides of the Eq.( 4) and using the Theorem 15, one has

x (t) g -1 (C 1 ) g -1 (ψ (t) -ψ (a)) γ-k = I 1-γ;ψ ⊕, ,t 0 + (Ax (t) ⊕ f (t, x(t))
). This implies (34)

x (t) = g -1 (C 1 ) g -1 (ψ (t) -ψ (t 0 )) γ-k ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) (Ax (s) ⊕ f (s, x(s))) ds
where

C 1 = I 1-γ;ψ t 0 + g (x (t)) Γ (γ) .
Note that, the Eq.( 34) can be rewritten in the form (35)

x (t) = I 1-γ;ψ ⊕, ,t 0 + x (t 0 ) Γ (γ) g -1 (ψ (t) -ψ (t 0 )) γ-k ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) (Ax (s) ⊕ f (s, x(s))) ds.
Now, through successive approximations, let's get an expression for Eq.( 35). Let for this set [START_REF] Sousa | Existence and uniqueness of global solution in g-variational calculus[END_REF] x

0 (t) = x 0 Γ (γ) g -1 (ψ (t) -ψ (t 0 )) γ-1
and

x m (t) = x 0 Γ (γ) g -1 (ψ (t) -ψ (t 0 )) γ-1 (37) ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) (Ax m-1 (s) ⊕ f (s, x(s))) ds.
Using the Eq.( 36) and Eq.(37), we find

x 1 (t) = x 0 (t) ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) (Ax 0 (s) ⊕ f (s, x(s))) ds = x 0 Γ (γ) g -1 (ψ (t) -ψ (t 0 )) γ-1 ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α)
Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION

A x 0 Γ (γ) g -1 (ψ (s) -ψ (t 0 )) γ-1 ⊕ f (s, x(s)) ds = x 0 Γ (γ) g -1 (ψ (t) -ψ (t 0 )) γ-1 ⊕ A x 0 Γ (γ) ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) g -1 (ψ (s) -ψ (t 0 )) γ-1 ds ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) f (s, x(s)) ds = x 0 Γ (γ) g -1 (ψ (t) -ψ (t 0 )) γ-1 ⊕ A x 0 Γ (γ) I α;ψ ⊕, ,t 0 + g -1 (ψ (t) -ψ (t 0 )) γ-1 ⊕I α;ψ ⊕, ,t 0 + f (t, x(t)).
On the other hand, we have

x 2 (t) = x 0 Γ (γ) g -1 (ψ (t) -ψ (t 0 )) γ-1 ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) (Ax 1 (s) ⊕ f (s, x(s))) ds = x 0 Γ (γ) g -1 (ψ (t) -ψ (t 0 )) γ-1 ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) A x 0 Γ (γ) g -1 (ψ (s) -ψ (t 0 )) γ-1 ds ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) A 2 x 0 Γ (γ) I α;ψ ⊕, ,t 0+ g -1 (ψ (t) -ψ (s)) γ-1 ds ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) A I α;ψ ⊕, ,t 0+ f (s, x(s)) ds ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 Γ (α) f (s) ds = x 0 ⊕ [t 0 ,t] 2 ∑ k=0 A k g -1 ψ (s) (ψ (t) -ψ (s)) αk-1 Γ (αk) g -1 (ψ (s) -ψ (t 0 )) γ-1 Γ (γ) ds ⊕ ⊕ [t 0 ,t] 2 ∑ k=1 A k g -1 ψ (s) (ψ (t) -ψ (s)) αk-1 Γ (αk) f (s, x(s)) ds. ( 38 
)
Continuing this process, we derive the following relation

x m (t) = x 0 ⊕ [t 0 ,t] m ∑ k=0 A k g -1 ψ (s) (ψ (t) -ψ (s)) αk-1 Γ (αk) g -1 (ψ (s) -ψ (t 0 )) γ-1 Γ (γ) ds ⊕ ⊕ [t 0 ,t] m ∑ k=0 A k g -1 ψ (s) (ψ (t) -ψ (s)) αk+α-1 Γ (αk + α) f (s, x(s)) ds. ( 39 
)
Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION Taking limit as m → ∞ on both sides of the Eq.( 39), we have

lim m→∞ x m (t) = x 0 ⊕ [t 0 ,t] ∞ ∑ k=0 A k g -1 ψ (s) (ψ (t) -ψ (s)) αk-1 Γ (αk + γ) g -1 (ψ (s) -ψ (t 0 )) γ-1 ds ⊕ ⊕ [t 0 ,t] ∞ ∑ k=0 A k g -1 ψ (s) (ψ (t) -ψ (s)) αk+α-1 Γ (αk + α) f (s, x(s)) ds = x 0 g -1 (ψ (t) -ψ (t 0 )) γ-1 ⊕ [t 0 ,t] g -1 ψ (s) E α,γ g (A) (ψ (t) -ψ (s)) α ds ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 E α,α g (A) (ψ (t) -ψ (s)) α f (s, x(s)) ds.
Therefore, we conclude that

x (t) = x 0 g -1 (ψ (t) -ψ (t 0 )) γ-1 ⊕ [t 0 ,t] g -1 ψ (s) E α,γ g (A) (ψ (t) -ψ (s)) α ds ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ (t) -ψ (s)) α-1 E α,α g (A) (ψ (t) -ψ (s)) α f (s, x(s)) ds. (40) 
The norm an n-vector function x(t) = (x 1 (t), x 2 (t), . . . , x n (t)) in R n is defined by

||x(t)|| = n ∑ k=1 |x k (t)| 2 1 2
.

On the other hand, we have ||x(t)|| g = g -1 ||g(x(t))||. Let

X = x, x ∈ C, H α,β ,ψ ⊕, ,t + 0 x ∈ C .
Then X is a Banach space endowed with the norm ||x|| X,g = ||x|| g .

In view of Lemma 44, we convert the initial value problem Eq.( 4)

Fx = x, x ∈ C, H α,β ,ψ ⊕, ,t + 0 x ∈ C
where F : X → X is defined by ( 41)

Fx(t) = x 0 g -1 (ψ(t) -ψ(t 0 )) γ-1 ⊕ [t 0 ,t] g -1 (ψ (s)) E α,γ (g(A)(ψ(t) -ψ(s)) α ) ds⊕ ⊕ [t 0 ,t] g -1 ψ (s)(ψ(t) -ψ(s)) α-1 E α,α (g(A)(ψ(t) -ψ(s)) α ) f (s, x(s)) ds Theorem 45. Let f : [t 0 , ∞) × R n → R n be a continuous function satisfying the Lipschitz condition || f (t, x 1 (t)) f (t, x 2 (t))|| g ≤ g L ||x 1 (t) x 2 (t)|| g ,t ∈ J = [t 0 ,t 1 ], L > 0.
Then, the initial value problem Eq.( 4) has a unique solution whenever

M L g -1 (ψ(t) -ψ(t 0 )) α α < 1 
.

Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION

Proof. The proof consists of two steps.

Step 1. We show that F(B r ) ⊂ B r , i.e., we show that F(x) ∈ B r , where B r is a closed ball of radius r > 0 in X, that is, || f (t, 0)|| g and

E α,i g (A) (ψ (t) -ψ (t 0 )) α g ≤ g M, i = α, γ and r > max 2 ||x 0 || g g -1 ((ψ(T ) -ψ(t 0 )) γ-1 ) M ⊕ M L g -1 (ψ(t) -ψ(t 0 )) α α .
For x ∈ B r , we have

Fx (t) ≤ x 0 g -1 (ψ(t) -ψ(t 0 )) γ-1 ⊕ [t 0 ,t] g -1 (ψ (s)) E α,γ (g(A)(ψ(t) -ψ(s)) α ) ds ⊕ ⊕ [t 0 ,t] g -1 ψ (s) (ψ(t) -ψ(s)) α-1 E α,α (g(A)(ψ(t) -ψ(s)) α ) f (s, x(s)) ds ≤ x 0 g -1 (ψ(T ) -ψ(t 0 )) γ-1 M ⊕ M L g -1 (ψ(t) -ψ(t 0 )) α α (42)
Thus, Fx g ≤ r, then Fx ∈ B r , whenever x ∈ B r .

Step 2. We next show that F is contraction mapping on X. For x, y ∈ X and for each t ∈ J we obtain

(Fx) (t) (Fy) (t) g = ⊕ [t 0 ,t] g -1 ψ (s) (ψ(t) -ψ(s)) α-1 E α,α (g(A)(ψ(t) -ψ(s)) α ) f (s, x(s)) ds ⊕ [t 0 ,t] g -1 ψ (s) (ψ(t) -ψ(s)) α-1 E α,α (g(A)(ψ(t) -ψ(s)) α ) f (s, y(s)) ds g ≤ g ⊕ [t 0 ,t] g -1 ψ (s) (ψ(t) -ψ(s)) α-1 g E α,α (g(A)(ψ(t) -ψ(s)) α ) g f (s, x (s)) f (s, y(s)) g ds ≤ g M ⊕ [t 0 ,t] g -1 ψ (s) (ψ(t) -ψ(s)) α-1 L x(s) y(s) g ds ≤ g M L x y g ⊕ [t 0 ,t] g -1 ψ (s) (ψ(t) -ψ(s)) α-1 ds = M L ||x y|| g g -1 t [t 0 ,t] ψ (s) (ψ(t) -ψ(s)) α-1 ds = M L x y g g -1 (ψ(t) -ψ(t 0 )) α α = M L g -1 (ψ(t) -ψ(t 0 )) α α x y g . (43) We conclude that F is a contraction since (44) M L g -1 (ψ(t) -ψ(t 0 )) α α < 1 
and the statement of the theorem follows by the classical Banach fixed point theorem.

Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION Theorem 46. Let f : [t 0 , ∞) × R n → R n be a continuous function such that there exist a positive constant L, such that

(45) f (t, x(t)) g ≤ g L x(t) g , t ∈ J = [t 0 ,t 1 ].
Then, the initial value Eq.( 4) has a solution on J.

Proof. We shall use the Schauder fixed point theorem to prove that F defined by Eq.( 41) has a fixed point. The proof will be given in three steps.

Step 1. F is continuous.

Let n n be a sequence in R n such that lim n→∞ x n = x in X. Keeping in mind the previous bounds of Mittag-Leffler functions, we have

(Fx n )(t) (Fx)(t) g ≤ g ⊕ [t 0 ,t] g -1 ψ (s) (ψ(t) -ψ(s)) α-1 E α,γ (g(A)(ψ(t) -ψ(s)) α ) g f (s, x n (s)) f (s, x(s)) g ds ≤ g M ⊕ [t 0 ,t] g -1 ψ (s) (ψ(t) -ψ(s)) α-1 f (s, x n (s)) f (s, x(s)) g ds.
Since f is continuous and bounded, g is continuous then by dominated convergent theorem, we deduce that the integrate tends to zero as n approaches to infinity. Thus,

(46) lim n→∞ (Fx n )(t) (Fx)(t) h = 0
which implies that F is continuous.

Step 2. F maps bounded set. We show that F maps bounded set. Indeed, it is enough to show that for any r > 0, there exists a positive constant L > 0, such that for each

x ∈ B r = x ∈ X : x g ≤ r we have Fx(t) g ≤ g x 0 g g -1 ((ψ(t) -ψ(t 0 )) γ-1 ) ⊕ [t 0 ,t] g -1 (ψ (s)) E α,γ (g(A)(ψ(t) -ψ(s)) α ) g ds ⊕ ⊕ [t 0 ,t] g -1 (ψ (s) (ψ(t) -ψ(s)) α-1 ) E α,α (g(A)(ψ(t) -ψ(s)) α ) g f (s, x(s)) g ds ≤ g x 0 g g -1 ((ψ(t) -ψ(t 0 )) γ-1 ) M g -1 t t 0 ψ (s)ds ⊕M L ⊕ [t 0 ,t] g -1 (ψ (s) (ψ(t) -ψ(s)) α-1 ) ||x(s)|| g ds ≤ g x 0 g g -1 ((ψ(t) -ψ(t 0 )) γ-1 ) M g -1 (ψ(t) -ψ(t 0 ) ⊕M L ||x|| g g -1 (ψ(t) -ψ(t 0 ) α α = L (47)
Therefore, Fx g ≤ g L.

Step 3. F maps bounded set into equicontinuous set. We show that F maps bounded set into equicontinuous set of X. Let t 1 ,t 2 ∈ J with t 1 < t 2 and B r be a bounded set of X as in Step 2. Let x ∈ B r , then

(Fx)(t 2 ) (Fx)(t 1 ) g = x 0 g -1 (ψ(t 2 ) -ψ(t 0 )) γ-1 ⊕ [t 0 ,t 2 ] g -1 (ψ (s)) E α,γ (g(A)(ψ(t 2 ) -ψ(s)) α ) ds ⊕ ⊕ [t 0 ,t 2 ] g -1 (ψ (s)(ψ(t 2 ) -ψ(s)) α-1 ) E α,α (g(A)(ψ(t 2 ) -ψ(s)) α ) F(s, x(s)) ds x 0 g -1 (ψ(t 1 ) -ψ(t 0 )) γ-1 ⊕ [t 0 ,t 1 ] g -1 (ψ (s)) E α,γ (g(A)(ψ(t 1 ) -ψ(s)) α ) ds ⊕ [t 0 ,t 1 ] g -1 ψ (s)(ψ(t 1 ) -ψ(ts)) α-1 E α,α (g(A)(ψ(t 1 ) -ψ(s)) α ) f (s, x(s)) ds g = x 0 g -1 (ψ(t 2 ) -ψ(t 0 )) γ-1 ⊕ [t 0 ,t 1 ] g -1 (ψ (s)) E α,γ (g(A)(ψ(t 2 ) -ψ(s)) α ) ds ⊕ ⊕ [t 1 ,t 2 ] g -1 (ψ (s)) E α,γ (g(A)(ψ(t 2 ) -ψ(s)) α ) ds ⊕ ⊕ [t 0 ,t 1 ] g -1 (ψ (s)(ψ(t 2 ) -ψ(s)) α-1 ) E α,α (g(A)(ψ(t 2 ) -ψ(s)) α ) f (s, x(s)) ds ⊕ ⊕ [t 1 ,t 2 ] g -1 (ψ (s)(ψ(t 2 ) -ψ(s)) α-1 ) E α,α (g(A)(ψ(t 2 ) -ψ(s)) α ) f (s, x(s)) ds x 0 g -1 (ψ(t 1 ) -ψ(t 0 )) γ-1 ⊕ [t 0 ,t 1 ] g -1 (ψ (s)) E α,γ (g(A)(ψ(t 1 ) -ψ(s)) α ) ds ⊕ [t 0 ,t 1 ]
g -1 (ψ (s)(ψ(t 1 )ψ(s)) α-1 ) E α,α (g(A)(ψ(t 1 )ψ(s)) α ) f (s, x(s)) ds g ≤ g ||x 0 || g g -1 (ψ(t 2 )ψ(t 0 )) γ-1 (ψ(t 1 )ψ(t 0 )) γ-1

⊕ [t 0 ,t 1 ]
g -1 (ψ (s)) E α,γ (g(A)(ψ(t 2 )ψ(s)) α ) E α,γ (g(A)(ψ(t 1 )ψ(s)) α ) g ds

⊕ ⊕ [t 1 ,t 2 ]
g -1 (ψ (s)) E α,γ (g(A)(ψ(t 2 )ψ(s)) α ) g ds

⊕ ⊕ [t 0 ,t 1 ]
g -1 (ψ (s)(ψ(t 2 )ψ(s)) α-1 ) E α,α (g(A)(ψ(t 2 )ψ(s)) α ) g -1 (ψ (s)(ψ(t 1 )ψ(s)) α-1 ) E α,α (g(A)(ψ(t 1 )ψ(s)) α ) g f (s, x(s)) g ds

⊕ ⊕ [t 1 ,t 2 ]
g -1 (ψ (s)(ψ(t 2 )ψ(s)) α-1 ) ||E α,α (g(A)(ψ(t 2 )ψ(s)) α )|| g || f (s, x(s))|| g ds ≤ g ||x 0 || g g -1 (ψ(t 2 )ψ(t 0 )) γ-1 ) g -1 ((ψ(t 1 )ψ(t 0 )) g -1 ψ (s)(ψ(t 2 )ψ(s)) α-1 L 1 x(s) g ds ≤ g

x 0 g g -1 (ψ(t 2 )ψ(t 0 )) γ-1 g -1 (ψ(t 1 )ψ(t 0 )) γ-1

M ⊕ [t 0 ,t 1 ]
g -1 (ψ -1 (ψ(t 1 )ψ(t 2 )) ⊕ L 1 x g

⊕ [t 0 ,t 1 ]
g -1 (ψ (s)(ψ(t 2 )ψ(s)) α-1 ) g ∞ ∑ k=0

(ψ(t 2 )ψ(s)) αk g Γ(αk + α)

g -1 (ψ (s)(ψ(t 1 )ψ(s)) α-1 ) g ∞ ∑ k=0

(ψ(t 1 )ψ(s)) αk g Γ(αk + α) ds ⊕M L 1 x g g -1 (ψ(t 2 )ψ(t 1 )) α α ≤ g x 0 g g -1 (ψ(t 2 )ψ(t 0 )) γ-1 g -1 (ψ(t 1 )ψ(t 0 )) γ-1 M g -1 (ψ(t 1 )ψ(t 0 )) ⊕ M g -1 (ψ(t)ψ(t 0 ))

⊕L 1 x g ⊕ [t 0 ,t 1 ]
g -1 (ψ (s)(ψ(t 2 )ψ(s)) α-1 ) g ∞ ∑ k=0

(ψ(t 2 )ψ(s)) αk g Γ(αk + α)

g -1 (ψ (s)(ψ(t 1 )ψ(s)) α-1 ) g ∞ ∑ k=0

(ψ(t 1 )ψ(s)) αk g Γ(αk + α) ds ⊕M L 1 x g g -1 (ψ(t)ψ(t 0 )) α α ≤ g x 0 g g -1 (ψ(t)ψ(t 0 )) γ-1 M ⊕ M g -1 (ψ(t)ψ(t 0 ))

⊕M L 1 x g g -1 (ψ(t)ψ(t 0 )) α α .

Since the Mittag-Leffler function is uniformly bounded, hence we interchange the summation and integration. As t 1 → t 2 , we deduce Fx(t 2 ) -Fx(t 1 ) g → 0.

As a consequence of steps 1 to 3 together with the Arzelà-Ascoli theorem, we can conclude that F : X → X is continuous and completely continuous. Therefore F is completely continuous. By Schauder fixed point theorem the IPVs, Eq.( 4) has a solution.

Concluding remark and open problem

From the generalized g-Laplace will be possible to investigate some open questions in the theory of pseudo-fractional differential equations, in particular, the discussion of mild solutions involving sectoral and quasi-sectoral operators in Banach spaces and their consequences. Besides, traditional problems, such as obtaining analytical solution of linear pseudo-differential equations that model natural phenomena, will be allowed possible discussions via generalized g-Laplace transform. On the other hand, the discussion of the existence and uniqueness of the mild solutions of the pseudo-fractional differential equation in the sense of the pseudo-fractional derivative ψ-Hilfer, On the other hand, the discussion of the existence Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION and uniqueness of the mild solution of the pseudo-fractional differential equation in the sense of the pseudo-fractional derivative ψ-Hilfer, will allow a new way to discuss new and important results.

  •) are given by Eq.(

  are ψ-Hilfer fractional derivative are given by Eq.(

  Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION Example 23. Let µ ∈ C be such that Re(µ) > 0 and E µ (•) be the one-parameter Mittag-Leffler function, then

Example 25 .

 25 Assume that µ ∈ C be such that Re(µ) > 0 and |λ /s µ | < 1. If E γ µ,ν (•) denotes the threeparameter Mittag-Leffler function, we evaluate the Laplace transform of the so-called Prabhakar function, then we have

  Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION

  γ-1 )Submitted to Numerical Methods for Partial Differential Equations -NOT THE PUBLISHED VERSION g -1 ψ (s)(ψ(t 1 )ψ(s)) α-1

			g	∞ ∑ k=0	g(A) k	g	||(ψ(t 1 ) -ψ(s)) αk || g Γ(αk + α)	L 1 x(s) g ds
		⊕				
	⊕M					
		[t 1 ,t 2 ]				
							⊕	g -1 (ψ (s))
							[t 0 ,t 1 ]
	∞ ∑ k=0	g(A) k	g ||(ψ(t g	∞ ∑	g(A) k
						k=0

2 )ψ(s)) αk (ψ(t 1 )ψ(s)) αk || g Γ(αk + γ) ds ⊕ M ⊕ [t 1 ,t 2 ] g -1 (ψ (s)) ds ⊕ ⊕ [t 0 ,t 1 ] g -1 ψ (s)(ψ(t 2 )ψ(s)) α-1 g (ψ(t 2 )ψ(s)) αk g Γ(αk + α)