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TRANSFORM
José Vanterler da C. Sousa, Rubens Camargo, E Capelas de Oliveira, Gastão Frederico
To cite this version:
José Vanterler da C. Sousa, Rubens Camargo, E Capelas de Oliveira, Gastão Frederico. PSEUDO- FRACTIONAL DIFFERENTIAL EQUATIONS AND GENERALIZED g-LAPLACE TRANSFORM.
2021. �hal-03189233�
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS Vol. , No. , YEAR
https://doi.org/jie.YEAR..PAGE
PSEUDO-FRACTIONAL DIFFERENTIAL EQUATIONS AND GENERALIZED g-LAPLACE TRANSFORM
J. VANTERLER DA C. SOUSA∗, RUBENS F. CAMARGO1, E. CAPELAS DE OLIVEIRA2, AND GAST ˜AO S. F. FREDERICO3
ABSTRACT. In this article, we introduce a generalizedg-Laplace transform and discuss some essential results of integral transform theory, in particular, involving aψ-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.
1. 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, [3,5,7,8,17,24,33,34], and recently, fractional derivatives have been used to generalize and refine models that describe the epidemic disease of coronavirus [9, 16]. Nowadays we have countless definitions of derivatives and fractional integrals [6,10,25,27,35,38].
In 2018 when Sousa and Oliveira [25], introduced the fractional derivativeψ-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 [15], introduced a version for the Laplace transform withψ. However, the reverse version was still missing. In 2020 Fahad et al. [12], introduced the inverse of the Laplace transform withψ. In this sense, from the Laplace transform withψ, it became possible to discuss properties such as existence, uniqueness, stability, controllability of mild solutions of fractional differential equations, involving the fractional derivativeψ-Hilfer. Some works involving the fractional derivativeψ-Hilfer, can be consulted in [28,29,30,31,32].
On the other hand, Endre Pap [18,19] being one of the main researchers in the area ofg-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 theg-calculus theory, pseudo-analysis with fractional calculation, since it is still a new field and because there are open questions [4,13,23,27,37]. There are some works on theg-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 [1,2,3,14,39].
Aiming to unify these two areas more, in 2020 Sousa et al. [27], extended the fractional derivativeψ-Hilfer to pseudo-operators and discussed some basic properties.
In 2005, Pap [19] 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⊕andfor 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,
2020Mathematics Subject Classification. 34A08,34A12,47G30,44A10.
Key words and phrases. Pseudo-fractional operator, existence and uniqueness,g-Laplace transform.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
1
involving different equations via pseudo-analysis, can be obtained in the following works [20,21,22,23, 37,36].
In 2020 Sousa et al. [36], investigated the existence and uniqueness of global solutions of the Cauchy problem associated with data(t0,x0)any solution(I:= [a,b],x)is given by
d⊕
dtx(t) = F(t,x) x(t0) = x0
witht0∈I.
On the other hand, also in 2020 Sousa et al. [37], discussed the reachability of linear and non-linear systems in the sense of theψ-Hilfer pseudo-fractional derivative ing−calculus by means of the Mittag- Leffler functions (one and two parameters) with the form
Hα,β;ψ⊕,,t0+x(t) = Ax(t)⊕Bu(t), t∈J:= [t0,t1] (1)
I1−γ;ψ⊕,,t0+x(t0) = 0 and
Hα,β⊕,,t;ψ0+x(t) = Ax(t)⊕Bu(t)⊕f(t,x(t),u(t)), t∈J (2)
I1−γ;ψ⊕,,t0+x(t0) = 0
whereHα,β;ψ⊕,,t0+(·)is theψ-Hilfer pseudo-fractional derivative of order 0<α<1 and type 0≤β ≤1,γ= α+β(1−α), I1−γ;ψ⊕,,t0+(·)is the Riemann-Liouville pseudo-fractional integral with respect to another function 1−γ, the state vectorx∈Rn, the control vectoru∈RmandAandBare the constant matrices of dimensionn×nandn×mrespectively and the nonlinear function f :J×Rn×Rm→Rn is continuous, respectively.
To date, the studies Sousa et al. [36,37], are the first results in this area involving the reachability approach of pseudo-differential equations involvingψ-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 generalizedg-Laplace transform, is given by
(3) Lψ⊕(f(t)) =g−1 Lψ(g(f(t))) .
(2) We discuss some properties of Eq.(3), specially related to theψ-convolution product. In addition, we discussed the calculation of the pseudo-fractional derivativeψ-Hilfer and the pseudo-fractional derivativeψ-Riemann-Liouville.
Finally, we consider the followingψ-Hilfer pseudo-fractional differential equation given by (4)
(
Hα,β;ψ⊕,,t0+x(t) = Ax(t)⊕f(t,x(t)), t∈J I1−γ;ψ⊕,,t0+x(t0) = x0
in whichHα,β;ψ⊕,,t0+(·)is the pseudo-fractionalψ-Hilfer derivative with order 0<α≤1 and type 0≤β ≤1, I1−γ;ψ⊕,,t0+(·)is the pseudo-fractional integral with order 1−γ (γ =α−β(1−α)), n×n matrixA and
f :[t0,∞)×Rn→Rnbe 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 Theorems1and2:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Theorem 1. Let f :[t0,∞)×Rn→Rnbe continuous function satisfying the Lipschitz condition
||f(t,x1(t)) f(t,x2(t))||g≤gL ||x1(t) x2(t)||g,t∈J:= [t0,t1],L>0.
Then, the initial valueEq.(4)has a unique solution whenever MLg−1
(ψ(t)−ψ(t0))α α
<1
Theorem 2. Let f :[t0,∞)×Rn→Rn be a continuous function such that there exist a positive constant L, such that
||f(t,x(t))||g≤gL ||x(t)||g, t∈J.
Then, the initial valueEq.(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 theg-Laplace transform with respect to theψfunction. From this result, some results were discussed, in particular, involving the pseudo-fractional derivativeψ-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 Theorem1 and Theorem2. We conclude the article with open questions and problems.
2. Preliminaries
In this section, we will discuss some concepts and results, essential for the development of this paper.
The Mittag-Leffler functionEα,β(z)is a complex function which depends on two complex parameters, and it is defined by [25]
(5) Eα,β(z) =
∞ k=0
∑
zk
Γ(αk+β), α,β >0.
The functionEα,β(·)converges for all values of the argument z. For a n×n matrixA, the matrix extension of the above Mittag-Leffler function is
(6) Eα,β(A) =
∞ k=0
∑
Ak Γ(αk+β).
Definition 3. [10, 13, 18,19]A binary operator ⊕ onJ is pseudo-addition if it is commutative, non- decreasing, with respect to, continuous; associative, and with a zero (neutral) element denoted by0. Let J+={x|,x∈[a,b],0≤x}for a,b∈R+.
Definition 4. [10,13,18,19]A binary operationonJ is pseudo-multiplication if it is commutative, positively non-decreasing, i.e., x≤y implies xz≤yz for all z∈[a,b]+, associative and with a unit element1∈[a,b], i.e., for each x∈[a,b],1x=x. Also,0x=0and thatis distributive over⊕, i.e., x(y⊕z) = (xy)⊕(x)z.
The structure(J,⊕,)is a semiring [10,13,18,19].
Definition 5. [10,13,18,19]An important class of pseudo-operations⊕andis when these are defined by a monotone and continuous function g:J7−→[0,∞], i.e., pseudo-operations⊕andare given by (7) x⊕y=g−1(g(x) +g(y))and xy=g−1(g(x)g(y)).
Definition 6. [10,13,18,19]Let X be a non-empty set andA be aσ-algebra of subsets of a set X . A set functionµ:A 7−→[a,b]is called aσ-⊕-measure if the following conditions are satisfy:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
(1) µ(/0) =0;
(2) µ(S∞i=1Ai) =L∞i=1µ(Ai)
for any sequence{A}i∈Nof pairwise disjoint sets fromA.
Definition 7. [10,13,18, 19]Let pseudo-operations⊕ and be defined a monotone and continuous function g:J7−→[0,∞].
(1) The g-integral for a measurable function f :[c,d]→Jis given by Z ⊕
[c,d]
fdx=g−1 Z d
c
g(f(x))dx
. (2) The g-Laplace of a function f is defined by
L⊕[f(x)] =g−1(L[g(f(x))])·
Definition 8. [10,13,18,19]Let g be the additive generator of the strict-pseudo-addition⊕onJsuch that g is continuous differentiable on(a,b). The corresponding pseudo-multiplicationwill always be defined as uv=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
d⊕
dx f(x) =g−1 d
dxg(f(x))
· Also, if there exists the n-g-derivative of f , then
d(n)⊕
dx f(x) =g−1 dn
dxng(f(x))
·
Definition 9. [10,13,18,19]Let g be a generator of a pseudo-addition⊕on interval[−∞,+∞]. Binary operations andon[−∞,+∞]are defined by the expressions
x y=g−1(g(x)−g(y)) and xy=g−1 g(x)
g(y)
.
If the expressions g(x)−g(y)andg(x)
g(y)have sense are said to be the pseudo-subtraction and pseudo- division consistent with the pseudo-addition⊕.
Definition 10. [10,13,18,19]Let g:[−∞,+∞]7−→[−∞,+∞]be a continuous, strictly increasing and odd function such that g(0) =0, g(1) =1and g(+∞) = +∞. The system of pseudo-arithmetical operations {⊕,,, }generated by these functions is said to be the consistent system.
Definition 11. [25,26]LetJ:= [a,b] (−∞≤a<b≤∞)be a finite or infinite interval of the real lineR andα>0. Also letψ(x)be an increasing and positive monotone function on(a,b], having a continuous derivativeψ0(x)onJ. The left-sided and right-sided fractional integrals of a function f with respect to another functionψ, are defined by
(8) Ia+α;ψf(x) = 1
Γ(α) Z x
a
ψ0(t) (ψ(x)−ψ(t))α−1f(t)dt.
and
(9) Ib−α;ψf(x) = 1
Γ(α) Z b
x ψ0(t) (ψ(t)−ψ(x))α−1 f(t)dt.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Definition 12. [25,26]Let n−1<α<n, with n∈N,Jis an interval such that−∞≤a<b≤+∞and f,ψ∈Cn(J,R)are two functions such thatψ is increasing andψ(x)6=0, for all x∈J. Theψ-Hilfer fractional derivative left-sided and right-sided, denoted byHDα,β;ψa+ (·)of a function f of orderαand type 0≤β ≤1, is defined by
(10) HDα,β;ψa+ f(x) =Iaβ(n−α);ψ+
1 ψ0(x)
d dx
n
I(1−β)(n−α);ψ
a+ f(x)
and
(11) HDα,β;ψb− f(x) =Ibβ−(n−α);ψ
− 1 ψ0(x)
d dx
n
Ib(1−β− )(n−α);ψf(x) whereIaα;ψ+ (·)andIbα;ψ− (·)by defined inEq.(8)andEq.(9), respectively.
Definition 13. [27]Let a generator g:J→[0,∞]of the pseudo-addition⊕and the pseudo-multiplication be an increasing function. Also letψ de an increasing and positive function on (a,b], having a continuous derivativeψ0(x). The left-sided and the right-sidedψ-Riemann-Liouville pseudo-fractional integrals of orderα>0of a measurable function f :J→Jwith respect to functionψonJare defined by:
Iα;ψ⊕,,a+f(x) = g−1 Ia+α;ψg(f(x))
= Z ⊕
[a,x]
"
g−1 ψ0(t) (ψ(x)−ψ(t))α−1 Γ(α)
! f(t)
# dt (12)
and
Iα;ψ⊕,,b−f(x) = g−1 Ib−α;ψg(f(x))
= Z ⊕
[x,b]
"
g−1 ψ0(t) (ψ(t)−ψ(x))α−1 Γ(α)
! f(t)
# dt (13)
whereIa+α;ψ(·)andIb−α;ψ(·)are given by Eq.(8)and Eq.(9), respectively.
Definition 14. [27]Let a generator g:J→[0,∞]of the pseudo-addition⊕and the pseudo-multiplication be an increasing function. Also letψ∈Cn(J,R), a function such thatψbe an increasing and positive function on(a,b]having a continuous derivativeψ0 andψ0(x)6=0for all x∈J.The left-sided and right- sidedψ-Hilfer pseudo-fractional derivative of order n−1<α<n and type0≤β ≤1, of a measurable function f :J→Jis defined by
Hα,β;ψ⊕,,a+f(x) = g−1 H
Dα,β;ψa+ g(f(x))
= Iβ⊕,,a+(n−α);ψg−1
D ψ0(x)
n
I1−γ⊕,,a+;ψ f(x) (14)
and
Hα,β⊕,,b−;ψ f(x) = g−1 H
Dα,βb−;ψg(f(x))
= Iβ(n−α);ψ⊕,,b− g−1
− D ψ0(x)
n
I1−γ;ψ⊕,,b−f(x) (15)
whereHDα,βa+;ψ(·)andHDα,βb−;ψ(·)areψ-Hilfer fractional derivative are given byEq.(10)andEq.(11).
Note that
Hα,β;ψ⊕,,a+f(x) = g−1
Ia+γ−α;ψ RL
Dγ;ψa+g(f(x))
= Iγ−α;ψ⊕,,a+RL
Dγ;ψ⊕,,a+f(x) (16)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
and
Hα,β;ψ⊕,,b−f(x) = g−1
Ib−γ−α;ψ RLDγ;ψb−g(f(x))
= Iγ−α;ψ⊕,,b−RLDγ;ψ⊕,,b−f(x) (17)
whereγ=α+β(n−α).
Ifα>0 andAisn×nmatrix, then
(18) Hα,β;ψ⊕,,a+Eα,β A(ψ(x)−ψ(a))α
=AEα,β A(ψ(x)−ψ(a))α . Theorem 15. [27]Let f :J→Jbe a measurable functions. If n∈N, then we have
(1) H0,β;ψ⊕,,a+f(x) = f(x);
(2) H1,β;ψ⊕,,a+f(x) =g−1
D ψ0(x)
g(f(x))
; (3) Hn,β;ψ⊕,,a+f(x) =
D ψ0(x)
(n)⊕
f(x);
(4) Hα,β⊕,,a+;ψ (f1(x)⊕f2(x)) =Hα,β⊕,,a+;ψ f1(x)⊕Hα,β⊕,,a+;ψ f2(x);
(5) Hα,β⊕,,a+;ψ (λf(x)) =λHα,β;ψ⊕,,a+f(x);
(6) Hα,β⊕,,a+;ψ Iα;ψ⊕,,a+f(x) = f(x);
(7) Iα;ψ⊕,,a+ Hα,β;ψ⊕,,a+f(x) = f(x) n
L
k=1
g−1(Ck)g−1(ψ(x)−ψ(a))γ−k
with γ =α+β(n−α) and Ck=(g◦f)[n−k]In−γ⊕,,a+;ψ 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 withψ(0) =0. Then, the Laplace transform of f with respect toψis defined by
Lψ(f(t)) =F(s) = Z ∞
0
e−sψ(t)ψ0(t)f(t)dt
for all s∈Csuch that this integral converges. Here,Lψ(·)denotes the Laplace transform with respect to ψ, which we call a generalized Laplace transform.
Definition 17. [15]Assume that the function f is defined for f ≥0. Then, the Laplace transform of f , denoted byL(f), is defined by the improper integral
(19) L(f(t))≡F(s) =
Z ∞
0
e−stf(t)dt
provided that the integral inEq.(19)exists, i.e., that the integral is convergent. The corresponding inverse Laplace transform is given by
(20) L−1(f(t)) = 1
2πi Z c+i∞
c−i∞ estF(s)ds with s∈Csuch thatRe(s) =c.
Letm−1<α<m,m∈N,−∞≤a<b≤∞, 0≤β ≤1, and f,ψ∈Cm(J,R)be functions such that ψ is increasing andψ0(x)6=0 for allx∈J. Then, theψ-Hilfer fractional derivative of orderαand typeβ is given by [25,26]
(21) HDα,β,ψa+ f(x) =Iaβ(m−α),ψ+
1 ψ0(x)
d dx
m
I(1−β)(m−α),ψ
a+ f(x).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Takingβ →1 in Eq.(21), we obtain the Caputo fractional derivative given by [25]
(22) CDα,ψa+ f(x) =Iam−α,ψ+ 1
ψ0(x) d dx
m
f(x).
Also, takingβ →0 in Eq.(21), we get the Riemann-Liouville fractional derivative given by
(23) RLDα,ψa+ f(x) =
1 ψ0(x)
d dx
m
Iam−α,ψ+ f(x).
These generalized fractional operators can be written as the conjugation of the standard fractional operators with the operation of composition withψorψ−1[12]
Iaα,ψ+ =Qψ◦Itα,ψ(a)+ ◦ Qψ−1
,
RLDα,ψa+ =Qψ◦RLDα,ψ(a)t+ ◦ Qψ
−1
, (24) CDα,ψa+ =Qψ◦CDα,ψ(a)t+ ◦ Qψ−1
, and
(25) HDα,βa+ ,ψ=Qψ◦HDα,βt+ ,ψ(a)◦ Qψ−1
, where the functional operatorQψ is defined by
(26) Qψf
(x) = f(ψ(x)).
Theorem 18. [12]The generalized Laplace transform may be written as a combination of the classical Laplace transform with the operation of composition withψ orψ−1, as follows
(27) Lψ=L ◦Q−1ψ
where the fundamental operator Qψ is defined inEq.(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 withψorψ−1, as follows
(28) Lψ−1=Qψ◦L−1
or, in other words
(29) Lψ−1(F(s)) = 1
2πi Z c+i∞
c−i∞ esψ(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) =⇒ Lψ(f(ψ(t))) =F(s).
Below some particular cases, presented as examples [12,15].
Example 21. Letµ∈Cbe such thatRe(µ)>−1, then
Lψ((ψ(t))µ) = Γ(µ+1) sµ+1 forRe(s)>0.
Example 22. Letλ ∈R, then
Lψ(eλ ψ(t)) = 1 s−λ forRe(s)>λ.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Example 23. Letµ∈Cbe such thatRe(µ)>0andEµ(·)be the one-parameter Mittag-Leffler function, then
Lψ(Eµ(λ ψ(t))µ) = sµ−1 sµ−λ for|λ/sµ|<1.
Example 24. Letµ∈Cbe such thatRe(µ)>0andEµ,µ(·)be the two-parameter Mittag-Leffler function, then
Lψ(Eµ,µ(λ ψ(t))µ) = 1 sµ−λ for|λ/sµ|<1.
Example 25. Assume thatµ∈Cbe such thatRe(µ)>0and|λ/sµ|<1. IfEγµ,ν(·)denotes the three- parameter Mittag-Leffler function, we evaluate the Laplace transform of the so-called Prabhakar function, then we have
Lψ((ψ(t))ν−1Eγµ,ν(λ ψ(t))µ) = sµ γ−ν (sµ−λ)γ.
Theorem 26. [12,15]Letµ>0and let f be a function ofψ-exponential order, piecewise continuous over each finite interval[0,T]. Then,
Lψ Iaµ,ψ+ f(t)
=s−µLψ(f(t)).
Theorem 27.[12,15]Assume m−1<µ<1,0≤ν≤1and f a function such that f(t), Dj,ψI(1−ν)(m−µ),ψ
0+ f(t)∈
C[0,∞)are ofψ-exponential order for j=0,1,2, . . . ,m−1, whileHD0µ,ν,ψ+ f(t)is piecewise continuous on[0,∞). Then,
(30) Lψ
H
D0µ,ν,ψ+ f(t)
=sµLψ(f(t))−
m−1 i=0
∑
sm(1−ν)+µ ν−i−1
I(1−ν)(m−µ)−ν
0+ f(0)
.
Definition 28. [15]Let f and h be ofψ-exponential order, piecewise continuous functions over each finite interval[0,T]. Then, theψ-convolution of f and h, denoted by f∗ψh is given by
(31) f∗ψh=
Z t 0
f ψ−1(ψ(t)−ψ(τ))
ψ0(τ)h(τ)dτ.
Theorem 29. [12]Let f,p,h be ofψ-exponential order, piecewise continuous functions over each finite interval[0,T], and let a and b be constants. Then,
(1) f∗ψp= p∗ψ f .
(2) (f∗ψp)∗ψh= f∗ψ(p∗ψh).
(3) f∗ψ(ap+bh) =a f∗ψp+b f∗ψh.
Theorem 30. [12]Assume that f and p are piecewise continuous functions on[0,T]and ofψ-exponential order c>0. Then,
(32) Lψ(f∗ψp) =Lψ(f)Lψ(p)
Definition 31. [10,13,18,19]Let the pseudo-operations⊕andbe 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 Z ⊕
[c,d]
fdt=g−1 Z 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))).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
3. Generalizedg-Laplace transform
In this section, we will present a generalization for theg-Laplace transform (g-calculus) with respect to another functionψand its respective inverse. In this perspective we discuss some properties, in particular, we obtain theg-Laplace transform of theψ-Hilfer fractional derivative and theψ-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,∞)→Rbe real valued functions such thatψbe a non-negative increasing function such that ψ(0) =0. Then, the g-Laplace transform of f with respect to another functionψis defined by
Lψ⊕(f(t)) =g−1 Lψ(g(f(t))) whereLψ(·)is defined inDefinition16.
Theorem 33. Let a generator g:J→[0,∞]of the pseudo-addition⊕ and the pseudo-multiplication be an increasing function. Let f,ψ:[0,∞)→Rbe real valued functions such thatψ 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 withψandψ−1as follows
Lψ⊕=L⊕◦Q−1ψ where the functional operator Q−1ψ f
(t) = f(ψ−1(t)).
Proof. Let f :t→ f(t). On the other hand,
Q−1ψ f(t):t→ Q−1ψ f
(t) = f(ψ−1(t)).
So, using the definition of the generalizedg-Laplace transform, we have L⊕◦Q−1ψ (f):t→Lψ⊕◦ Q−1ψ f
(t)
= L⊕ Q−1ψ f
(t) =g−1 Lg Q−1ψ f(t) (t)
= g−1 Z ∞
0
e−stg Q−1ψ f (t)dt
= g−1 Z ∞
0
e−stg
f(ψ0(t)) (t)dt
= g−1 Z ∞
0
e−sψ(u)g(f(u))ψ0(u)du
thus, the result follows.
At this point, we are motivated to present the inverse generalizedg-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 withψorψ−1as follows
Lψ⊕−1 =Qψ◦L⊕−1 or, in other words
Lψ⊕−1{F(s)} = g−1
Lψ−1g(F(s))
= g−1 1
2πi Z c+i∞
c−i∞ esψ(t)g(F(s))ds
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
As a result we can write
Lψ⊕{λf(t)⊕h(t)}=λLψ⊕{f(t)} ⊕Lψ⊕{g(t)}
whose proof follows from the definition.
Definition 35. [11]Given a pseudo-linear space V over (J,⊕,), a generalized norm (g-norm)is a mapping|| · ||g:V →J+such that
(1) ||v||g≥g0,∀v∈V and||v||g=0if and only if v=0 (2) ||cv||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,∞)→Rnis said to be ofψ-exponential orderc>0, if there exist positive constants M,c,T, such that||f||g≤M ecψ(t), fort≥T.
Theorem 36. If f :[0,∞)→Ris a piecewise continuous function and is ofψ-exponential order c>0, whereψis a non-negative increasing function withψ(0) =0, then the generalized g-Laplace transform of
f exists for s>c.
Proof. For theψ-exponentially bounded function f, we have
Lψ⊕{f(t)}
=
g−1 Lψg(f(t))
≤ g−1 M
s−c
where
Lψ{f(t)}
≤ M
s−c and in the last step we used the assumption thats>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 ψ- exponential order. Then
Lψ⊕n
Iα,ψ⊕,,0+f(t) o
= g−1(s−α)Lψ⊕f(t) Proof. In fact, from Definition32and remembering that
Lψ I0α,ψ+ f(t)
=s−αLψ(f(t)) we have
Lψ⊕n
Iα,ψ⊕,,0+f(t)o
= g−1 Lψg
Iα,ψ⊕,,0+f(t)
= g−1
Lψg g−1 I0α,ψ+ g(f(t))
= g−1
Lψ I0α,ψ+ g(f(t))
= g−1
s−αLψg(f(t))
= g−1(s−α)g−1 Lψg(f(t))
= g−1(s−α)Lψ⊕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), D0j,ψ+I(1−β)(n−α),ψ
0+ f(t)∈C[0,∞) are of ψ-exponential order for j= 0,1,2, . . . ,m−1, whileHDα,β,ψ⊕,,0+f(t)is piecewise continuous on[0,∞). Then, the generalized g-Laplace
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
transform of theψ-Hilfer fractional derivative of orderαand typeβ, with0≤β ≤1is given by Lψ⊕h
HDα,β⊕,,0,ψ+f(t)i
= g−1(sα)Lψ⊕f(t) h
Lm−1 i=1 g−1
sm(1−β)+α β−i−1
I(1−β⊕,,0)(m−α)−i,ψ+ f(0) i
. Proof. In fact, from Definition32and remembering that
Lψ
nH
Dα,β,ψ0+ f(t) o
= sαLψ[f(t)]−
m−1 i=0
∑
sm(1−β)+α β−i−1
I0(1−β+ )(m−α)−i,ψf(0)
we have Lψ⊕
hH
Dα,β,ψ⊕,,0+f(t) i
= g−1 hLψg
H
Dα,β,ψ⊕,,0+f(t) i
= g−1 hLψh
HDα,β,ψ⊕,,0+g(f(t))ii
= g−1 (
sαLψ[g(f(t))]−
m−1 i=0
∑
sm(1−β)+α β−i−1
I(1−β)(m−α)−i,ψ
0+ f(0)
)
= g−1
g g−1 sαLψ(g(f(t)))
−g g−1
sm(1−β)+α β−i−1C0
⊕ · · · ⊕g−1
sm(1−β)+α β−mCm−1 i
= g−1(sα)Lψ⊕f(t)
"m−1 M
i=0
g−1
sm(11−β)+α β−i−1
I(1−β⊕,,0)(m−α)−i,ψ+ f(0)
#
which complete the proof.
4. 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 ofψ-exponential order, piecewise continuous functions over each finite interval[0,T]. Then, theψ-convolution of f and h is a function, f⊗ψh, defined by
(33)
(f⊗ψh)(t) = Z ⊕
[0,t]
f ψ−1(ψ(t)−ψ(x))
g−1(ψ0(x))h(x)dx
= g−1 Z t
0
g f ψ−1(ψ(t)−ψ(x))
ψ0(x)g(h(x))dx
.
Remark 40. Some particular cases of theψ-convolution(Definition39), given by from choosing of g(·) andψ(·):
(1) Note that, taking g(x) =x in Definition39, we have the classical convolution with respect to another function,(ψ), given by the following relation
(f∗ψh)(t) = Z t
0
f ψ−1(ψ(t)−ψ(x))
ψ0(x)h(x)dx.
(2) (2)Takingψ(x) =x we get the definition of convolution in the sense of g-calculus, given by (f⊗h)(t) =
Z ⊕ [0,t]
f(t−x)h(x)dx.
(3) Finally, taking g(x) =x=ψ(x)Definition39, we obtain the classical convolution given by (f∗h)(t) =
Z t 0
f(t−x)h(x)dx.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
(4) Making the same choice forTheorem41andTheorem42, 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 beψ-exponential order, piecewise continuous functions over each finite interval[0,T]. Then, the following relation
f⊗ψh=Qψ Q−1ψ f
⊗ Q−1ψ f or in other words
f⊗ψh= (f◦ψ−1)⊗(g◦ψ−1)
◦ψ Proof. Using the classicalg-condition and substitutingx=ψ(u)we get
f⊗ψh(t) = Z ⊕
[0,t]
f(t−x)h(x)dx
= g−1 Z t
0
g(f(t−x))g(h(x))dx
= g−1
Z ψ−1(t) 0
g(f(t−ψ(u)))g(h(ψ(u)))ψ0(u)du
!
ApplyingQπ we have (f⊗ψh)◦ψ(t) =
Z t
0
g(f(ψ(t)−ψ(u)))g(h(ψ(u)))ψ0(u)du
= g−1 Z t
0
g
(f◦ψ)(ψ−1(ψ(t)−ψ(u)))
g[(h◦ψ)(u)]ψ0(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⊗gand f⊗ψg, then the results of Theorem 41can be written, like Eq.(2) forψ-fractional derivatives and integrals, as a conjugation of operators
⊗ψ=Qψ◦ ⊗ ◦ Q−1ψ ,Q−1ψ namely
⊗ψ(f,g) =Qψ ⊗ Q−1ψ f,Q−1ψ 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 ofψ-exponential order c>0. Then,
Lψ⊕
f⊗ψh =Lψ⊕{f}Lψ⊕{h}.
Proof. In fact, note that by Theorem41and Theorem33, we have (Lψ⊕)◦(⊗ψ) = L⊕◦Q−1ψ
◦ Qψ◦ ⊗ ◦ Q−1ψ ,Q−1ψ
= (L⊕◦ ⊗)◦ Q−1ψ ,Q−1ψ therefore
(Lψ⊕)(f,h) = (L⊕◦ ⊗)◦(Q−1ψ f,Q−1ψ h)
= L⊕(Q−1ψ f)L⊕(Q−1ψ h)
= Lψ⊕{f}Lψ⊕{h}
which conclude the proof.
The proof of Theorem43follows directly from Theorem41.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Theorem 43. Let a generator g:J→[0,∞]of the pseudo-addition⊕and the pseudo multiplicationbe an increasing function. Letψ, h and p be ofψ-exponential order, piecewise continuous function over each finite interval[0,T], and let a and b constants. Then,
(1) f⊗ψh=h⊗ψ f .
(2) (f⊗ψh)⊗ψp= f⊗ψ h⊗ψp . (3) f⊗ψ(ah⊕bp) =a f⊗ψh⊕b f⊗ψp.
5. Existence and uniqueness
In this section, we will discuss the existence and uniqueness of the solution of the nonlinear pseudo- fractional differential equation in terms of the Mittag-Leffler function of one and two parameters.
Lemma 44. For0<α≤1, the solution ofEq.(4)is x(t) = x0g−1
(ψ(t)−ψ(t0))γ−1
Z ⊕
[t0,t]g−1 ψ0(s)
Eα,γ g(A) (ψ(t)−ψ(s))α ds
⊕ Z ⊕
[t0,t]
g−1
ψ0(s) (ψ(t)−ψ(s))α−1
Eα,α g(A) (ψ(t)−ψ(s))α
f(s,x(s))ds.
Proof. Applying the operatorI1−γ⊕,,t;ψ0+(·)on both sides of the Eq.(4) and using the Theorem15, one has x(t)
h
g−1(C1)g−1(ψ(t)−ψ(a))γ−k i
=I1−γ;ψ⊕,,t0+(Ax(t)⊕f(t,x(t))).
This implies (34)
x(t) =g−1(C1)g−1(ψ(t)−ψ(t0))γ−k⊕ Z ⊕
[t0,t]
g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
(Ax(s)⊕f(s,x(s)))ds
whereC1= It1−γ;ψ0+ g(x(t)) Γ(γ) .
Note that, the Eq.(34) can be rewritten in the form (35)
x(t) =I1−γ;ψ⊕,,t0+x(t0)
Γ(γ) g−1(ψ(t)−ψ(t0))γ−k⊕ Z ⊕
[t0,t]
g−1 ψ0(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
(36) x0(t) = x0
Γ(γ)g−1(ψ(t)−ψ(t0))γ−1 and
xm(t) = x0
Γ(γ)g−1(ψ(t)−ψ(t0))γ−1 (37)
⊕ Z ⊕
[t0,t]
g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
(Axm−1(s)⊕f(s,x(s)))ds.
Using the Eq.(36) and Eq.(37), we find x1(t) = x0(t)⊕
Z ⊕
[t0,t]g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
(Ax0(s)⊕f(s,x(s)))ds
= x0
Γ(γ)g−1(ψ(t)−ψ(t0))γ−1⊕ Z ⊕
[t0,t]
g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
A x0
Γ(γ)g−1(ψ(s)−ψ(t0))γ−1⊕f(s,x(s))
ds
= x0
Γ(γ)g−1(ψ(t)−ψ(t0))γ−1⊕ A x0
Γ(γ) Z ⊕
[t0,t]
g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
g−1(ψ(s)−ψ(t0))γ−1ds
⊕ Z ⊕
[t0,t]
g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
f(s,x(s))ds
= x0
Γ(γ)g−1(ψ(t)−ψ(t0))γ−1⊕A x0
Γ(γ)Iα;ψ⊕,,t0+
g−1(ψ(t)−ψ(t0))γ−1
⊕Iα;ψ⊕,,t0+f(t,x(t)).
On the other hand, we have
x2(t) = x0
Γ(γ)g−1(ψ(t)−ψ(t0))γ−1⊕ Z ⊕
[t0,t]
g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
(Ax1(s)⊕f(s,x(s)))ds
= x0
Γ(γ)g−1(ψ(t)−ψ(t0))γ−1⊕ Z ⊕
[t0,t]
g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
A x0
Γ(γ)g−1(ψ(s)−ψ(t0))γ−1ds
⊕ Z ⊕
[t0,t]
g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
A2 x0
Γ(γ)Iα;ψ⊕,,t0+g−1
(ψ(t)−ψ(s))γ−1
ds
⊕ Z ⊕
[t0,t]
g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
AIα;ψ⊕,,t0+f(s,x(s))ds
⊕ Z ⊕
[t0,t]
g−1 ψ0(s) (ψ(t)−ψ(s))α−1 Γ(α)
!
f(s)ds
= x0 Z ⊕
[t0,t]
2 k=0
∑
Akg−1 ψ0(s) (ψ(t)−ψ(s))αk−1 Γ(αk)
!g−1
(ψ(s)−ψ(t0))γ−1
Γ(γ) ds
⊕ Z ⊕
[t0,t]
2
∑
k=1
Akg−1 ψ0(s) (ψ(t)−ψ(s))αk−1 Γ(αk)
!
f(s,x(s))ds.
(38)
Continuing this process, we derive the following relation
xm(t) = x0 Z ⊕
[t0,t]
m
∑
k=0
Akg−1 ψ0(s) (ψ(t)−ψ(s))αk−1 Γ(αk)
!g−1
(ψ(s)−ψ(t0))γ−1
Γ(γ) ds
⊕ Z ⊕
[t0,t]
m k=0
∑
Akg−1 ψ0(s) (ψ(t)−ψ(s))αk+α−1 Γ(αk+α)
!
f(s,x(s))ds.
(39)
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