# On a Differential Equation Involving Hilfer-Hadamard Fractional Derivative

Authors: M. D. Qassim; K. M. Furati; N.-E. Tatar
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.


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## Body

## 1.  Introduction

Fractional derivatives have proved to be very efficient and adequate to describe many phenomena with memory and hereditary processes. These phenomena are abundant in science, engineering (viscoelasticity, control, porous media, mechanics, electrical engineering, electromagnetism, etc.) as well as in geology, rheology, finance, and biology. Unlike the classical derivatives, fractional derivatives have the ability to characterize adequately processes involving a past history. We are witnessing a huge development of fractional calculus and methods in the theory of differential equations. Indeed, after the appearance of the papers by Bagley and Torvik [1–3], researchers started to deal directly with differential equations containing fractional derivatives instead of ignoring them as it used to be the case. For analytical treatments, we may refer the reader to [4–36], and for some applications, one can consult [1–3, 8, 25, 26, 26, 27, 27–31, 33, 34, 37–49] to cite but a few.We will consider the problem:(1.1)(Da+α,βu)(t)=f[t,u(t)],0<α<1,0≤β≤1,t>a>0,(Da+(β-1)(1-α)u)(a)=u0≥0,
where 𝒟a+α,βu is a new type of fractional derivative we will define below and u0 is a given constant. This new fractional derivative interpolates the Hadamard fractional derivative and its Caputo counterpart [26, 34], in the same way the Hilfer fractional derivative interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative. That is why we are naming it after Hilfer and Hadamard.A nonexistence result for global solutions of the problem (1.1) will be proved when f[t,u(t)]≥(log(t/a))μ|u(t)|m for some m>1 and μ∈R. That is we consider the Cauchy problem:(1.2)(Da+α,βu)(t)≥(log⁡ta)μ|u(t)|m,t>a>0,m>1,μ∈R,(Da+γ-1u)(a)=u0≥0,
where γ=α+β-αβ and show that no solutions can exist for all time for certain values of μ and m. Clearly, sufficient conditions for nonexistence provide necessary conditions for existence of solutions. In addition, we construct an example for which there exist solutions for some powers m and in some appropriate space.The existence and uniqueness of solutions for problem (1.1) has been discussed in [50] in the space Cδ;1-γ,μα,β[a,b] defined by(1.3)Cδ;1-γ,μα,β[a,b]={y∈C1-γ,log⁡[a,b],Da+α,βy∈Cμ,log⁡[a,b]},
where(1.4)Cγ,log⁡[a,b]={g:(a,b]⟶R:(log⁡xa)γg(x)∈C[a,b]}
for 0≤μ<1 and C0,log[a,b]=C[a,b].We also point out here that the case where𝒟a+α,β is the usual Riemann-Liouville fractional derivative has been studied in [26] (see also references therein). There are very few papers [26, 29] dealing with the pure Hadamard case, that is, when β=0.The rest of the paper is divided into three sections. In Section2, we present some definitions, notations, and lemmas which will be needed later in our proof. Section 3 is devoted to the nonexistence result and Section 4 contains an example of existence of solutions.


## 2.  Preliminaries

In this section, we present some background material for the forthcoming analysis. For more details, see [25, 26, 33, 42, 51, 52].Definition 2.1.
The spaceXcp(a,b)(c∈R,1≤p≤∞) consists of those real-valued Lebesgue measurable functions g on (a,b) for which ∥g∥Xcp<∞, where
(2.1)‖g‖Xcp=(∫ab|tcg(t)|pdtt)1/p,1≤p<∞,c∈R,‖g‖Xc∞=esssup⁡a≤x≤b|xcg(x)|,c∈R.
In particular, when c=1/p, we see that X1/pp(a,b)=Lp(a,b).Definition 2.2.
LetΩ=[a,b](0<a<b<∞) be a finite interval and 0≤γ<1, we introduce the weighted space Cγ,log[a,b] of continuous functions g on (a,b]:
(2.2)Cγ,log⁡[a,b]={g∈C(a,b]:(log⁡xa)γg(x)∈C[a,b]}.
In the space Cγ,log[a,b], we define the norm:
(2.3)‖g‖Cγ,log⁡=‖(log⁡xa)γg(x)‖C,‖g‖C0,log⁡=‖g‖∞.Definition 2.3.
Letδ=x(d/dx) be the δ-derivative, for n∈N, we denote by Cδ,γn[a,b] (0≤γ<1) the Banach space of functions g which have continuous δ-derivatives on [a,b] up to order n-1 and have the derivative δng of order n on (a,b] such that δng∈Cγ,log[a,b]:
(2.4)Cδ,γn[a,b]={g:(a,b]⟶R:δkg∈C[a,b],k=0,…,n-1,δng∈Cγ,log⁡[a,b]}
with the norm:
(2.5)‖g‖Cδ,γn=∑k=0n-1‖δkg‖C+‖δng‖Cγ,log⁡.
When n=0, we set
(2.6)Cδ,γ0[a,b]=Cγ,log⁡[a,b].Definition 2.4.
Let(a,b)(0≤a<b≤∞) be a finite or infinite interval of the half-axis R+ and let α>0. The Hadamard left-sided fractional integral 𝒥a+αf of order α>0 is defined by
(2.7)(Ja+αf)(x)∶=1Γ(α)∫ax(log⁡xt)α-1f(t)dtt,a<x<b
provided that the integral exists. When α=0, we set
(2.8)Ja+0f=f.Definition 2.5.
Let(a,b)(0≤a<b≤∞) be a finite or infinite interval of the half-axis R+ and let α>0.The Hadamard right-sided fractional integral 𝒥b-αf of order α>0 is defined by
(2.9)(Jb-αf)(x)∶=1Γ(α)∫xb(log⁡tx)α-1f(t)dtt,a<x<b,
provided that the integral exists. When α=0,we set
(2.10)Jb-0f=f.Definition 2.6.
The left-sided Hadamard fractional derivative of order0≤α<1 on (a,b) is defined by
(2.11)(Da+αf)(x)∶=δ(Ja+1-αf)(x),
that is,
(2.12)(Da+αf)(x)=(xddx)1Γ(1-α)∫ax(log⁡xt)-αf(t)dtt,a<x<b.
When α=0, we set
(2.13)Da+0f=f.Definition 2.7.
The right-sided Hadamard fractional derivative of orderα(0≤α<1) on (a,b) is defined by
(2.14)(Db-αf)(x)∶=-δ(Jb-1-αf)(x),
that is,
(2.15)(Db-αf)(x)=-(xddx)1Γ(1-α)∫xb(log⁡tx)-αf(t)dtt,a<x<b.
When α=0, we set
(2.16)Db-0f=f.Lemma 2.8.
Ifα>0,β>0 and  0<a<b<∞, then
(2.17)(Ja+α(log⁡ta)β-1)(x)=Γ(β)Γ(α+β)(log⁡xa)β+α-1,(Da+α(log⁡ta)β-1)(x)=Γ(β)Γ(β-α)(log⁡xa)β-α-1.
In particular, if  β=1, then the Hadamard fractional derivative of a constant is not equal to zero:
(2.18)(Da+α1)(x)=1Γ(1-α)(log⁡xa)-α,
when  0<α<1.Lemma 2.9.
Let0<a<b<∞,α>0, and  0≤μ<1.(a)
Ifμ>α>0, then 𝒥a+α is bounded from Cμ,log[a,b] into Cμ-α,log[a,b]. In particular, 𝒥a+α is bounded in Cμ,log[a,b].(b)
Ifμ≤α, then 𝒥a+α is bounded from Cμ,log[a,b] into C[a,b]. In particular, 𝒥a+α is bounded in Cμ,log[a,b].This lemma justifies the following oneLemma 2.10 (the semigroup property of the fractional integration operator𝒥a+α).
Letα>0,β>0, and  0≤μ<1. If  0<a<b<∞, then, for  f∈Cμ,log[a,b],
(2.19)Ja+αJa+βf=Ja+α+βf
holds at any point  x∈(a,b]. When  f∈C[a,b], this relation is valid at any point  x∈[a,b].Lemma 2.11.
Let0≤α<1 and  0≤γ<1. If  f∈Cγ,log1[a,b], then the fractional derivatives  𝒟a+α and  𝒟b-α exist on  (a,b] and  [a,b), respectively,  (a>0) and can be represented in the forms:
(2.20)(Da+αf)(x)=f(a)Γ(1-α)(log⁡xa)-α+1Γ(1-α)∫ax(log⁡xt)-αf′(t)dt,(Db-αf)(x)=f(b)Γ(1-α)(log⁡bx)-α-1Γ(1-α)∫xb(log⁡tx)-αf′(t)dt,
respectively.Lemma 2.12 (fractional integration by Parts).
Letα>0 and  1≤p≤∞. If  φ∈Lp(R+) and  ψ∈X-1/pq, then
(2.21)∫0∞φ(x)(J+αψ)(x)dxx=∫0∞ψ(x)(J-αφ)(x)dxx,
where  1/p+1/q=1.Definition 2.13.
The fractional derivativec𝒟a+αf of order α(0<α<1) on (a,b) defined by
(2.22)cDa+αf=Ja+1-αδf,
where δ=x(d/dx), is called the Hadamard-Caputo fractional derivative of order α.Now, motivated by the Hilfer fractional derivative introduced in [41, 42], we introduce the new fractional derivative which we call Hilfer-Hadamard fractional derivative of order 0<α<1 and type 0≤β≤1:(2.23)(Da+α,βu)(t)=Ja+β(1-α)(tddt)(Ja+(1-β)(1-α)u)(t).
The Hilfer fractional derivative interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative. This new one interpolates the Hadamard fractional derivative and its caputo counterpart. Indeed, for β=0, we find the Hadamard fractional derivative as defined in Definition 2.6 and, for β=1, we find its Caputo type counterpart (Definition 2.13).Theorem 2.14 (Young’s inequality).
Ifa and  b are nonnegative real numbers and  p and  q are positive real numbers such that  1/p+1/q=1, then one has
(2.24)ab≤app+bqq.
Equality holds if and only if  ap=bq.


## 3. Nonexistence Result

Before we state and prove our main result, we will start with the following lemma.Lemma 3.1.
Ifα>0 and  f∈C[a,b], then
(3.1)(Ja+αf)(a)=lim⁡t→a(Ja+αf)(t)=0,(Jb-αf)(b)=lim⁡t→b(Jb-αf)(t)=0.Proof.
Sincef∈C[a,b], then on [a,b] we have |f(t)|<M for some positive constant M. Therefore,
(3.2)|(Ja+αf)(t)|≤1Γ(α)∫at(log⁡ts)α-1|f(s)|dss≤MΓ(α)∫at(log⁡ts)α-1dss≤MΓ(α+1)(log⁡ta)α.
As α>0, we see that
(3.3)(Ja+αf)(a)=lim⁡t→a(Ja+αf)(t)=0.
In a similar manner, we prove the second part of the lemma.The proof of the next result is based on the test function method developed by Mitidieri and Pokhozhaev in [52].Theorem 3.2.
Assume thatμ∈R and  m<(1+μ)/(1-γ). Then, Problem (1.2) does not admit global nontrivial solutions in  C1-γ,logγ[a,b], where
(3.4)C1-γ,log⁡γ[a,b]={y∈C1-γ,log⁡[a,b]:Da+γy∈C1-γ,log⁡[a,b]}
when  u0≥0.Proof.
Assume that a nontrivial solution exists for all timet>a. Let φ(t)∈C1([a,∞)) be a test function satisfying φ(t)≥0,φ(t) is non-increasing and such that
(3.5)φ(t):={1,a≤t≤θT,0,t≥T,
for some T>a and some θ (θ<1) such that a<θT<T. Multiplying the inequality in (1.2) by φ(t)/t and integrating over [a,T], we get
(3.6)∫aTφ(t)(Da+α,βu)(t)dtt≥∫aT(log⁡ta)μ|u(t)|mφ(t)dtt.
Observe that the integral in left-hand side exists and the one in the right-hand side exists for m<(1+μ)/(1-γ) when u∈C1-γ,logγ[a,b]. Moreover, from the definition of (𝒟a+α,βu)(t), we can rewrite (3.6) as
(3.7)∫aTφ(t)(Ja+β(1-α)tddtJa+1-γu)(t)dtt≥∫aT(log⁡ta)μ|u(t)|mφ(t)dtt.
By virtue of Lemma 2.12 (after extending by zero outside [a,T]), we may deduce from (3.7) that
(3.8)∫aTddt(Ja+1-γu)(t)(JT-β(1-α)φ(t))(t)dt≥∫aT(log⁡ta)μ|u(t)|mφ(t)dtt.
Notice that Lemma 2.12 is valid in our case since ((log(t/a))(1-γ)(𝒟a+γu)∈C[a,T] implies that |(log(t/a))(1-γ)(𝒟a+γu)(t)|≤M on [a,T] for some positive constant M)
(3.9)∫aT|t-1/p(Da+γu)(t)|p′dtt≤M∫aTt1-p′(log⁡ta)-p′(1-γ)dtt≤M∫a∞t1-p′(log⁡ta)-p′(1-γ)dtt.
Let s=(p′-1)(log(t/a)), then by the definition of the Gamma function,
(3.10)∫aT|t-1/p(Da+γu)(t)|p′dtt≤Ma1-p′(p′-1)1-p′(1-γ)∫0∞s-p′(1-γ)e-sds≤Ma1-p′(p′-1)1-p′(1-γ)Γ(1-p′(1-γ))<∞.
Hence, t(d/dt)(𝒥a+1-γu)t=(𝒟a+γu)(t)∈X-1/pp′ (and φ∈Lp) for some p>1/γ.
An integration by parts in (3.8) yields
(3.11)[(Ja+1-γu)(t)(JT-β(1-α)φ)(t)]t=aT-∫aT(Ja+1-γu)(t)ddt(JT-β(1-α)φ)(t)dt≥∫aT(log⁡ta)μ|u(t)|mφ(t)dtt,
or
(3.12)-u0(JT-β(1-α)φ)(a+)-∫aT(Ja+1-γu)(t)ddt(JT-β(1-α)φ)(t)dt≥∫aT(log⁡ta)μ|u(t)|mφ(t)dtt
because (𝒥T-β(1-α)φ)(T)=0 (see Lemma 3.1) and
(3.13)(Ja+1-γu)(a+)=(Da+γ-1u)(a+)=u0.
Multiplying by t/t inside the integral in the left hand side of (3.12), we see that
(3.14)L∶=∫aT(Ja+1-γu)(t)(-tddt)(JT-β(1-α)φ)(t)dtt≥∫aT(log⁡ta)μ|u(t)|mφ(t)dtt.
It appears from Definition 2.7 that
(3.15)L=∫aT(Ja+1-γu)(t)(DT-1-β(1-α)φ)(t)dtt,
and from Lemma 2.11, we see that
(3.16)L=∫aT(Ja+1-γu)(t)[φ(T)Γ(β(1-α))(log⁡Tt)β(1-α)-1-1Γ(β(1-α))∫tT(log⁡st)β(1-α)-1φ′(s)ds]dtt.
Since φ(T)=0 and
(3.17)1Γ(β(1-α))∫tT(log⁡st)β(1-α)-1φ′(s)ds=(JT-β(1-α)δφ)(t),
the last equality becomes
(3.18)L=-∫aT(Ja+1-γu)(t)(JT-β(1-α)δφ)(t)dtt≥∫aT(log⁡ta)μ|u(t)|mφ(t)dtt.
Note that δφ∈Lp and by the same argument as the one used at the beginning of the proof we may show that 𝒥a+1-γu∈X-1/pp′ since 𝒥a+1-γu∈C1-γ,log[a,T].
Therefore, Lemma2.12 again allows us to write
(3.19)L=-∫aTδφ(t)(Ja+β(1-α)Ja+1-γu)(t)dtt,
and by the semigroup property Lemma 2.10(3.20)L=-∫aTδφ(t)(Ja+1-αu)(t)dtt.
On the other hand,
(3.21)∫aTδφ(t)(Ja+1-αu)(t)dtt=1Γ(1-α)∫aTδφ(t)∫at(log⁡ts)-αu(s)sdsdtt≤1Γ(1-α)∫aT|δφ(t)|∫at(log⁡ts)-α|u(s)|sdsdtt.
As φ is nonincreasing, we have φ(s)≥φ(t) for all t≥s and 1/φ1/m(s)≤1/φ1/m(t), m>1. Also, it is clear that
(3.22)φ′(t)=0,t∈[a,θT].
Therefore,
(3.23)L≤1Γ(1-α)∫aT|δφ(t)|∫at(log⁡ts)-α|u(s)|φ1/m(s)sφ1/m(s)dsdtt≤1Γ(1-α)∫θTT|δφ(t)|φ1/m(t)∫at(log⁡ts)-α|u(s)|φ1/m(s)sdsdtt.
Definition 2.4 allows us to write
(3.24)L≤∫θTT|δφ(t)|φ1/m(t)(Ja+1-α|u|φ1/m)(t)dtt.
By the same argument as the one used at the beginning of the proof, we may show that |u(t)|φ1/m(t)∈X-1/pp′(|u(t)|φ1/m(t)≤|u(t)|). Moreover, it is easy to see that δφ(t)/φ1/m(t)∈Lp (for, otherwise, we consider φλ(t) with some sufficiently large λ). Thus, we can apply Lemma 2.12 to get
(3.25)L≤∫θTT|u(t)|φ1/m(t)(JT-1-α|δφ|φ1/m)(t)dtt.
Next, we multiply by (log(t/a))μ/m.(log(t/a))-μ/m inside the integral in the right-hand side of (3.25):
(3.26)L≤∫θTT(JT-1-α|δφ|φ1/m)(t)|u(t)|φ1/m(t)(log⁡(t/a))μ/m(log⁡(t/a))μ/mdtt.
For μ≥0, we have (log(t/a))-μ/m≤(log(θT/a))-μ/m (because -μ/m<0 and t>θT). It follows that
(3.27)L≤(log⁡θTa)-μ/m∫θTT(JT-1-α|δφ|φ1/m)(t)(log⁡ta)μ/m|u(t)|φ1/m(t)dtt.
By using the Young inequality (see Theorem 2.14), with m and m′ such that 1/m+1/m′=1, in the right-hand side of (3.27), we find
(3.28)L≤1m∫θTT(log⁡ta)μφ(t)|u(t)|mdtt+(log⁡(θT/a))-μm′/mm′∫θTT(JT-1-α|δφ|φ1/m)m′(t)dtt≤1m∫aT(log⁡ta)μφ(t)|u(t)|mdtt+(log⁡(θT/a))-μm′/mm′∫θTT(JT-1-α|δφ|φ1/m)m′(t)dtt.
Clearly, from (3.14) and (3.28), we see that
(3.29)(log⁡(θT/a))-μm′/mm′∫θTT(JT-1-α|δφ|φ1/m)m′(t)dtt≥(1-1m)∫aT(log⁡ta)μφ(t)|u(t)|mdtt,
or
(3.30)1m′∫aT(log⁡ta)μφ(t)|u(t)|mdtt≤(log⁡(θT/a))-μm′/mm′∫θTT(JT-1-α|δφ|φ1/m)m′(t)dtt.
Therefore, by Definition 2.5, we have
(3.31)∫aT(log⁡ta)μφ(t)|u(t)|mdtt≤(log⁡(θT/a))-μm′/mΓm′(1-α)∫θTT(∫tT(log⁡st)-α|δφ(s)|φ1/m(s)dss)m′dtt.
The change of variable σT=t yields
(3.32)∫aT(log⁡ta)μφ(t)|u(t)|mdtt≤(log⁡(θT/a))-μm′/mΓm′(1-α)∫θ1(∫σTT(log⁡sσT)-α|φ′(s)|φ(s)1/mds)m′dσσ.
Another change of variable r=s/T gives
(3.33)∫aT(log⁡ta)μφ(t)|u(t)|mdtt≤(log⁡(θT/a))-μm′/mΓm′(1-α)∫θ1(∫σ1(log⁡rσ)-α|φ′(r)|φ(r)1/mdr)m′dσσ.
We may assume that the integral term in the right-hand side of (3.33) is convergent, that is,
(3.34)1Γm′(1-α)∫θ1(∫σ1(ln⁡rσ)-α|φ′(r)|φ(r)1/mdr)m′dσ≤C,
for some positive constant C, for otherwise we consider φλ(r) with some sufficiently large λ. Therefore
(3.35)∫aT(log⁡ta)μφ(t)|u(t)|mdtt≤C(log⁡θTa)-μm′/m.
If μ>0, then
(3.36)(log⁡θTa)-μm′/m⟶0,
as T→∞. Finally, from (3.35), we obtain
(3.37)lim⁡T→∞∫aT(log⁡ta)μφ(t)|u(t)|mdtt=0.
We reach a contradiction since the solution is not supposed to be trivial.
In the caseμ=0 we have -μm′/m=0 and the relation (3.35) ensures that
(3.38)lim⁡T→∞∫aT(log⁡ta)μφ(t)|u(t)|mdtt≤C.
Moreover, it is clear that
(3.39)(log⁡θTa)-μ/m∫θTT(JT-1-α|δφ|φ1/m)(t)(log⁡ta)μ/m|u(t)|φ1/m(t)dtt≤(log⁡θTa)-μ/m[∫θTT(JT-1-α|δφ|φ1/m)m′(t)dtt]1/m′[∫θTT(log⁡ta)μ|u(t)|mφ(t)dtt]1/m.
This relation, together with (3.27) (relations (3.28) and (3.31) also are used without θ), implies that
(3.40)∫aT(log⁡ta)μφ(t)|u(t)|mdtt≤K[∫θTt(log⁡ta)μ|u(t)|mφ(t)dtt]1/m
for some positive constant K, with
(3.41)lim⁡T→∞∫θTT(log⁡ta)μ|u(t)|mφ(t)dtt=0
due to the convergence of the integral in (3.38). This is again a contradiction.
Ifμ<0, we have (log(t/a))-μ/m≤(log(T/a))-μ/m (because -μ/m>0 and t<T). Then, the change of variables t=(T/a)σ and s=(T/a)r in (3.27) yields
(3.42)∫aT(log⁡ta)μφ(t)|u(t)|mdtt≤(log⁡(T/a))1-μm′/mΓm′(1-α)∫ln⁡θT/ln⁡(T/a)ln⁡T/ln⁡(T/a)(∫σln⁡T/ln⁡(T/a)(ln⁡(T/a)r(T/a)σ)-α|φ′(r)|φ1/m(r)dr)m′dσ,
or
(3.43)∫aT(log⁡ta)μφ(t)|u(t)|mdtt≤(log⁡(T/a))1-αm′-μm′/mΓm′(1-α)∫ln⁡θT/ln⁡(T/a)ln⁡T/ln⁡(T/a)(∫σln⁡T/ln⁡(T/a)(r-σ)-α|φ′(r)|φ1/m(r)dr)m′dσ.
The expression |φ′(r)|/φ1/m(r) may be assumed bounded (or else we use φλ(r) with a large value of λ). Hence,
(3.44)∫aT(log⁡ta)μφ(t)|u(t)|mdtt≤C(log⁡Ta)-m′-μm′/m
for some positive constant C.Although we are concerend here about nonexistence of solutions, using standard techniques, one may show the existence of local solutions of Problem (1.1) with 1<m<(1+μ)/(1-γ). However, according to Theorem 3.2, such a solution cannot be continued for all time in C1-γ,logγ[a,b]. This is a phenomenon which occurs often in parabolic and hyperbolic problems with sources of polynomial type. In the absence of strong dissipations, these sources are the cause of blowup in finite time (of local solutions). For this reason, they are called blowup terms.


## 4.  Example

For our example, we need the following lemma.Lemma 4.1.
The following result holds for the fractional derivative operator𝒟a+α,β:
(4.1)(Da+α,β[(log⁡sa)γ-1])(t)=Γ(γ)Γ(γ-α)(log⁡ta)γ-α-1,t>a;γ>0,
where 0<α<1 and  0≤β≤1.Proof.
We observe from Lemma2.8 that
(4.2)(Ja+(1-α)(1-β)(log⁡sa)γ-1)(t)=Γ(γ)Γ((1-α)(1-β)+γ)(log⁡ta)γ+(1-α)(1-β)-1.
Therefore,
(4.3)(tddt)(Ja+(1-α)(1-β)(log⁡sa)γ-1)(t)=[γ+(1-α)(1-β)-1]Γ(γ)Γ((1-α)(1-β)+γ)(log⁡ta)γ+(1-α)(1-β)-2,
which, in light of the definition of 𝒟a+α,β, yields
(4.4)(Da+α,β[(log⁡sa)γ-1])(t)=Γ(γ)Γ((1-α)(1-β)+γ-1)(Ja+β(1-α)(log⁡sa)γ+(1-α)(1-β)-2)(t).
From Lemma 2.8 again, we have
(4.5)(Da+α,β[(log⁡sa)γ-1])(t)=Γ(γ)Γ(β(1-α)+γ+(1-α)(1-β)-1)(log⁡ta)β(1-α)+γ+(1-α)(1-β)-2=Γ(γ)Γ(γ-α)(log⁡ta)γ-α-1.
The proof is complete.Example 4.2.
Consider the following differential equation of Hilfer-Hadamard-type fractional derivative of order0<α<1 and type 0≤β≤1:
(4.6)(Da+α,βy)(t)=λ(log⁡ta)μ[y(t)]m(t>a>0;m>1)
with real λ, μ∈R+ (λ≠0). Suppose that the solution has the following form:
(4.7)y(t)=c(log⁡ta)ν.
Our aim next is to find the values of c and ν. By using Lemma 4.1 we have
(4.8)(Da+α,β[c(log⁡sa)ν])(t)=cΓ(ν+1)Γ(ν-α+1)(log⁡ta)ν-α.
Therefore,
(4.9)cΓ(ν+1)Γ(ν-α+1)(log⁡ta)ν-α=λ(log⁡ta)μ[c(log⁡ta)ν]m.
It can be directly shown that ν=(α+μ)/(1-m) and c=[Γ((α+μ)/(1-m)+1)/λΓ((mα+μ)/(1-m)+1)]1/(m-1). If (mα+μ)/(1-m)>-1, that is, m>(1+μ)/(1-α), then (4.6) has the exact solution:
(4.10)y(t)=[Γ((α+μ)/(1-m)+1)λΓ((mα+μ)/(1-m)+1)]1/(m-1)(log⁡ta)(α+μ)/(1-m).
This solution satisfies the initial condition when (α+μ)/(1-m)≥γ-1>-1. Note that there is an overlap of the interval of existence in this example and the interval of nonexistence in the previous theorem. This may be explained by the fact that this solution is in C1-γ,log[a,b] but not in C1-γ,logγ[a,b].



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*Source: 391062-2012-06-03.xml*