AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation39106210.1155/2012/391062391062Research ArticleOn a Differential Equation Involving Hilfer-Hadamard Fractional DerivativeQassimM. D.FuratiK. M.TatarN.-E.AhmadBashir1Department of Mathematics and Statistics, King Fahd University of Petroleum and MineralsDhahran 31261Saudi Arabiakfupm.edu.sa201236201220122712201110042012140420122012Copyright © 2012 M. D. Qassim et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 , 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 , and for some applications, one can consult [13, 8, 25, 26, 26, 27, 2731, 33, 34, 3749] to cite but a few.

We will consider the problem:(Da+α,βu)(t)=f[t,u(t)],  0<α<1,  0β1,  t>a>0,(Da+(β-1)(1-α)u)(a)=u00, 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 μR. That is we consider the Cauchy problem:(Da+α,βu)(t)(logta)μ|u(t)|m,t>a>0,  m>1,  μR,(Da+γ-1u)(a)=u00, 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  in the space Cδ;1-γ,μα,β[a,b] defined byCδ;1-γ,μα,β[a,b]={yC1-γ,log[a,b],Da+α,βyCμ,log[a,b]}, whereCγ,log[a,b]={g:(a,b]R:(logxa)γg(x)C[a,b]} for 0μ<1 and C0,log[a,b]=C[a,b].

We also point out here that the case where 𝒟a+α,β is the usual Riemann-Liouville fractional derivative has been studied in  (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 Section 2, 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 space Xcp(a,b)  (cR,1p) consists of those real-valued Lebesgue measurable functions g on (a,b) for which gXcp<, where gXcp=(ab|tcg(t)|pdtt)1/p,  1p<,  cR,gXc=esssupaxb|xcg(x)|,cR. In particular, when c=1/p, we see that X1/pp(a,b)=Lp(a,b).

Definition 2.2.

Let Ω=[a,b]  (0<a<b<) be a finite interval and 0γ<1, we introduce the weighted space Cγ,log[a,b] of continuous functions g on (a,b]: Cγ,log[a,b]={gC(a,b]:(logxa)γg(x)C[a,b]}. In the space Cγ,log[a,b], we define the norm: gCγ,log=(logxa)γg(x)C,gC0,log=g.

Definition 2.3.

Let δ=x(d/dx) be the δ-derivative, for nN, we denote by Cδ,γn[a,b] (0γ<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 δngCγ,log[a,b]: Cδ,γn[a,b]={g:(a,b]R:δkgC[a,b],k=0,,n-1,δngCγ,log[a,b]} with the norm: gCδ,γn=k=0n-1δkgC+δngCγ,log. When n=0, we set Cδ,γ0[a,b]=Cγ,log[a,b].

Definition 2.4.

Let (a,b)(0a<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 (Ja+αf)(x)=1Γ(α)ax(logxt)α-1f(t)dtt,  a<x<b provided that the integral exists. When α=0, we set Ja+0f=f.

Definition 2.5.

Let (a,b)(0a<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 (Jb-αf)(x)=1Γ(α)xb(logtx)α-1f(t)dtt,  a<x<b, provided that the integral exists. When α=0, we set Jb-0f=f.

Definition 2.6.

The left-sided Hadamard fractional derivative of order 0α<1 on (a,b) is defined by (Da+αf)(x)=δ(Ja+1-αf)(x), that is, (Da+αf)(x)=(xddx)1Γ(1-α)ax(logxt)-αf(t)dtt,  a<x<b. When α=0, we set Da+0f=f.

Definition 2.7.

The right-sided Hadamard fractional derivative of order α  (0α<1) on (a,b) is defined by (Db-αf)(x)=-δ(Jb-1-αf)(x), that is, (Db-αf)(x)=-(xddx)1Γ(1-α)xb(logtx)-αf(t)dtt,  a<x<b. When α=0, we set Db-0f=f.

Lemma 2.8.

If α>0,  β>0 and 0<a<b<, then (Ja+α(logta)β-1)(x)=Γ(β)Γ(α+β)(logxa)β+α-1,(Da+α(logta)β-1)(x)=Γ(β)Γ(β-α)(logxa)β-α-1. In particular, if β=1, then the Hadamard fractional derivative of a constant is not equal to zero: (Da+α1)(x)=1Γ(1-α)(logxa)-α, when 0<α<1.

Lemma 2.9.

Let 0<a<b<,  α>0, and 0μ<1.

If μ>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].

If μα, 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 one

Lemma 2.10 (the semigroup property of the fractional integration operator <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M101"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>𝒥</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mtext>a</mml:mtext></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>).

Let α>0,  β>0, and 0μ<1. If 0<a<b<, then, for fCμ,log[a,b], Ja+αJa+βf=Ja+α+βf holds at any point x(a,b]. When fC[a,b], this relation is valid at any point x[a,b].

Lemma 2.11.

Let 0α<1 and 0γ<1. If fCγ,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: (Da+αf)(x)=f(a)Γ(1-α)(logxa)-α+1Γ(1-α)ax(logxt)-αf(t)dt,(Db-αf)(x)=f(b)Γ(1-α)(logbx)-α-1Γ(1-α)xb(logtx)-αf(t)dt, respectively.

Lemma 2.12 (fractional integration by Parts).

Let α>0 and 1p. If φLp(R+) and ψX-1/pq, then 0φ(x)(J+αψ)(x)dxx=0ψ(x)(J-αφ)(x)dxx, where 1/p+1/q=1.

Definition 2.13.

The fractional derivative   c𝒟a+αf of order α  (0<α<1) on (a,b) defined by   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 0β1:(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).

If a and b are nonnegative real numbers and p and q are positive real numbers such that 1/p+1/q=1, then one has abapp+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.

If α>0 and fC[a,b], then (Ja+αf)(a)=limta(Ja+αf)(t)=0,(Jb-αf)(b)=limtb(Jb-αf)(t)=0.

Proof.

Since fC[a,b], then on [a,b] we have |f(t)|<M for some positive constant M. Therefore, |(Ja+αf)(t)|1Γ(α)at(logts)α-1|f(s)|dssMΓ(α)at(logts)α-1dssMΓ(α+1)(logta)α. As α>0, we see that (Ja+αf)(a)=limta(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 .

Theorem 3.2.

Assume that μR and m<(1+μ)/(1-γ). Then, Problem (1.2) does not admit global nontrivial solutions in C1-γ,logγ[a,b], where C1-γ,logγ[a,b]={yC1-γ,log[a,b]:Da+γyC1-γ,log[a,b]} when u00.

Proof.

Assume that a nontrivial solution exists for all time t>a. Let φ(t)C1([a,)) be a test function satisfying φ(t)0,  φ(t) is non-increasing and such that φ(t):={1,  atθT,0,  tT, 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 aTφ(t)(Da+α,βu)(t)dttaT(logta)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 uC1-γ,logγ[a,b]. Moreover, from the definition of (𝒟a+α,βu)(t), we can rewrite (3.6) as aTφ(t)(Ja+β(1-α)tddtJa+1-γu)(t)dttaT(logta)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 aTddt(Ja+1-γu)(t)(JT-β(1-α)φ(t))(t)dtaT(logta)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) aT|t-1/p(Da+γu)(t)|pdttMaTt1-p(logta)-p(1-γ)dttMat1-p(logta)-p(1-γ)dtt. Let s=(p-1)(log(t/a)), then by the definition of the Gamma function, aT|t-1/p(Da+γu)(t)|pdttMa1-p(p-1)1-p(1-γ)0s-p(1-γ)e-sdsMa1-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 φLp) for some p>1/γ.

An integration by parts in (3.8) yields [(Ja+1-γu)(t)(JT-β(1-α)φ)(t)]t=aT-aT(Ja+1-γu)(t)ddt(JT-β(1-α)φ)(t)dtaT(logta)μ|u(t)|mφ(t)dtt, or -u0(JT-β(1-α)φ)(a+)-aT(Ja+1-γu)(t)ddt(JT-β(1-α)φ)(t)dtaT(logta)μ|u(t)|mφ(t)dtt because (𝒥T-β(1-α)φ)(T)=0 (see Lemma 3.1) and (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 L=aT(Ja+1-γu)(t)(-tddt)(JT-β(1-α)φ)(t)dttaT(logta)μ|u(t)|mφ(t)dtt. It appears from Definition 2.7 that L=aT(Ja+1-γu)(t)(DT-1-β(1-α)φ)(t)dtt, and from Lemma 2.11, we see that L=aT(Ja+1-γu)(t)[φ(T)Γ(β(1-α))(logTt)β(1-α)-1-1Γ(β(1-α))tT(logst)β(1-α)-1φ(s)ds]dtt. Since φ(T)=0 and 1Γ(β(1-α))tT(logst)β(1-α)-1φ(s)ds=(JT-β(1-α)δφ)(t), the last equality becomes L=-aT(Ja+1-γu)(t)(JT-β(1-α)δφ)(t)dttaT(logta)μ|u(t)|mφ(t)dtt. Note that δφLp and by the same argument as the one used at the beginning of the proof we may show that 𝒥a+1-γuX-1/pp since 𝒥a+1-γuC1-γ,log[a,T].

Therefore, Lemma 2.12 again allows us to write L=-aTδφ(t)(Ja+β(1-α)Ja+1-γu)(t)dtt, and by the semigroup property Lemma 2.10L=-aTδφ(t)(Ja+1-αu)(t)dtt. On the other hand, aTδφ(t)(Ja+1-αu)(t)dtt=1Γ(1-α)aTδφ(t)at(logts)-αu(s)sdsdtt1Γ(1-α)aT|δφ(t)|at(logts)-α|u(s)|sdsdtt. As φ is nonincreasing, we have φ(s)φ(t) for all ts and 1/φ1/m(s)1/φ1/m(t), m>1. Also, it is clear that φ(t)=0,t[a,θT]. Therefore, L1Γ(1-α)aT|δφ(t)|at(logts)-α|u(s)|φ1/m(s)sφ1/m(s)dsdtt1Γ(1-α)θTT|δφ(t)|φ1/m(t)at(logts)-α|u(s)|φ1/m(s)sdsdtt. Definition 2.4 allows us to write LθTT|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 LθTT|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): LθTT(JT-1-α|δφ|φ1/m)(t)|u(t)|φ1/m(t)(log(t/a))μ/m(log(t/a))μ/mdtt. For μ0, we have (log(t/a))-μ/m(log(θT/a))-μ/m (because -μ/m<0 and t>θT). It follows that L(logθTa)-μ/mθTT(JT-1-α|δφ|φ1/m)(t)(logta)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 L1mθTT(logta)μφ(t)|u(t)|mdtt+(log(θT/a))-μm/mmθTT(JT-1-α|δφ|φ1/m)m(t)dtt1maT(logta)μφ(t)|u(t)|mdtt+(log(θT/a))-μm/mmθTT(JT-1-α|δφ|φ1/m)m(t)dtt. Clearly, from (3.14) and (3.28), we see that (log(θT/a))-μm/mmθTT(JT-1-α|δφ|φ1/m)m(t)dtt(1-1m)aT(logta)μφ(t)|u(t)|mdtt, or 1maT(logta)μφ(t)|u(t)|mdtt(log(θT/a))-μm/mmθTT(JT-1-α|δφ|φ1/m)m(t)dtt. Therefore, by Definition 2.5, we have aT(logta)μφ(t)|u(t)|mdtt(log(θT/a))-μm/mΓm(1-α)θTT(tT(logst)-α|δφ(s)|φ1/m(s)dss)mdtt. The change of variable σT=t yields aT(logta)μφ(t)|u(t)|mdtt(log(θT/a))-μm/mΓm(1-α)θ1(σTT(logsσT)-α|φ(s)|φ(s)1/mds)mdσσ. Another change of variable r=s/T gives aT(logta)μφ(t)|u(t)|mdtt(log(θT/a))-μm/mΓm(1-α)θ1(σ1(logrσ)-α|φ(r)|φ(r)1/mdr)mdσσ. We may assume that the integral term in the right-hand side of (3.33) is convergent, that is, 1Γm(1-α)θ1(σ1(lnrσ)-α|φ(r)|φ(r)1/mdr)mdσC, for some positive constant C, for otherwise we consider φλ(r) with some sufficiently large λ. Therefore aT(logta)μφ(t)|u(t)|mdttC(logθTa)-μm/m. If μ>0, then (logθTa)-μm/m0, as T. Finally, from (3.35), we obtain limTaT(logta)μφ(t)|u(t)|mdtt=0. We reach a contradiction since the solution is not supposed to be trivial.

In the case μ=0 we have -μm/m=0 and the relation (3.35) ensures that limTaT(logta)μφ(t)|u(t)|mdttC. Moreover, it is clear that (logθTa)-μ/mθTT(JT-1-α|δφ|φ1/m)(t)(logta)μ/m|u(t)|φ1/m(t)dtt(logθTa)-μ/m[θTT(JT-1-α|δφ|φ1/m)m(t)dtt]1/m[θTT(logta)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 aT(logta)μφ(t)|u(t)|mdttK[θTt(logta)μ|u(t)|mφ(t)dtt]1/m for some positive constant K, with limTθTT(logta)μ|u(t)|mφ(t)dtt=0 due to the convergence of the integral in (3.38). This is again a contradiction.

If μ<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 aT(logta)μφ(t)|u(t)|mdtt(log(T/a))1-μm/mΓm(1-α)lnθT/ln(T/a)lnT/ln(T/a)(σlnT/ln(T/a)(ln(T/a)r(T/a)σ)-α|φ(r)|φ1/m(r)dr)mdσ, or aT(logta)μφ(t)|u(t)|mdtt(log(T/a))1-αm-μm/mΓm(1-α)lnθT/ln(T/a)lnT/ln(T/a)(σlnT/ln(T/a)(r-σ)-α|φ(r)|φ1/m(r)dr)mdσ. The expression |φ(r)|/φ1/m(r) may be assumed bounded (or else we use φλ(r) with a large value of λ). Hence, aT(logta)μφ(t)|u(t)|mdttC(logTa)-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 operator 𝒟a+α,β: (Da+α,β[(logsa)γ-1])(t)=Γ(γ)Γ(γ-α)(logta)γ-α-1,  t>a;  γ>0, where 0<α<1 and 0β1.

Proof.

We observe from Lemma 2.8 that (Ja+(1-α)(1-β)(logsa)γ-1)(t)=Γ(γ)Γ((1-α)(1-β)+γ)(logta)γ+(1-α)(1-β)-1. Therefore, (tddt)(Ja+(1-α)(1-β)(logsa)γ-1)(t)=[γ+(1-α)(1-β)-1]Γ(γ)Γ((1-α)(1-β)+γ)(logta)γ+(1-α)(1-β)-2, which, in light of the definition of 𝒟a+α,β, yields (Da+α,β[(logsa)γ-1])(t)=Γ(γ)Γ((1-α)(1-β)+γ-1)(Ja+β(1-α)(logsa)γ+(1-α)(1-β)-2)(t). From Lemma 2.8 again, we have (Da+α,β[(logsa)γ-1])(t)=Γ(γ)Γ(β(1-α)+γ+(1-α)(1-β)-1)(logta)β(1-α)+γ+(1-α)(1-β)-2=Γ(γ)Γ(γ-α)(logta)γ-α-1. The proof is complete.

Example 4.2.

Consider the following differential equation of Hilfer-Hadamard-type fractional derivative of order 0<α<1 and type 0β1: (Da+α,βy)(t)=λ(logta)μ[y(t)]m(t>a>0;m>1) with real λ, μR+ (λ0). Suppose that the solution has the following form: y(t)=c(logta)ν. Our aim next is to find the values of c and ν. By using Lemma 4.1 we have (Da+α,β[c(logsa)ν])(t)=cΓ(ν+1)Γ(ν-α+1)(logta)ν-α. Therefore, cΓ(ν+1)Γ(ν-α+1)(logta)ν-α=λ(logta)μ[c(logta)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: y(t)=[Γ((α+μ)/(1-m)+1)λΓ((mα+μ)/(1-m)+1)]1/(m-1)(logta)(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].

Acknowledgments

The authors wish to express their thanks to the referees for their suggestions. The authors are also very grateful for the financial support and the facilities provided by King Fahd University of Petroleum and Minerals through the Project no. In101003.