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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 510314, 11 pages
Research Article

Semilinear Volterra Integrodifferential Problems with Fractional Derivatives in the Nonlinearities

1Université de la Rochelle, Avenue Michel Crépeau, 17042 La Rochelle Cedex 1, France
2Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Comenius University, Mlynska Doliná, 84248 Bratislava, Slovakia
3Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received 29 January 2011; Accepted 7 April 2011

Academic Editor: Allan C. Peterson

Copyright © 2011 Mokhtar Kirane 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.


A second-order semilinear Volterra integrodifferential equation involving fractional time derivatives is considered. We prove existence and uniqueness of mild solutions and classical solutions in appropriate spaces.

1. Introduction

In this work we discuss the following problem:𝑢(𝑡)=𝐴𝑢(𝑡)+𝑓(𝑡)+𝑡0𝑔𝑡,𝑠,𝑢(𝑠),𝐷𝛽1𝑢(𝑠),,𝐷𝛽𝑛𝑢(𝑠)𝑑𝑠,𝑡>0,𝑢(0)=𝑢0𝑋,𝑢(0)=𝑢1𝑋,(1.1)

where 0<𝛽𝑖1, 𝑖=1,,𝑛. Here the prime denotes time differentiation and 𝐷𝛽𝑖, 𝑖=1,,𝑛 denotes fractional time differentiation (in the sense of Riemann-Liouville or Caputo). The operator 𝐴 is the infinitesimal generator of a strongly continuous cosine family 𝐶(𝑡), 𝑡0 of bounded linear operators in the Banach space 𝑋, 𝑓 and 𝑔 are nonlinear functions from 𝐑+ to 𝑋 and 𝐑+×𝐑+×𝑋××𝑋 to 𝑋, respectively, 𝑢0 and 𝑢1 are given initial data in 𝑋. The problem with 𝛽1==𝛽𝑛=0 or 1 has been investigated by several authors (see [17] and references therein, to cite a few). Well-posedness has been proved using fixed point theorems and the theory of strongly continuous cosine families in Banach spaces developed in [8, 9]. This theory allows us to treat a more general integral or integrodifferential equation, the solutions of which are called “mild” solutions. In case of regularity (of the initial data and the nonlinearities), the mild solutions are shown to be classical. In case 𝛽1=𝛽2==𝛽𝑛=1, the underlying space is the space of continuously differentiable functions.

In this work, when 0<𝛽𝑖<1, 𝑖=1,,𝑛, we will see that mild solutions need not be that regular (especially when dealing with Riemann-Liouville fractional derivatives). It is the objective of this paper to find the appropriate space and norm where the problem is solvable. We first consider the problem with a fractional derivative in the sense of Caputo and look for a mild solution in 𝐶1. Under certain conditions on the data it is shown that this mild solution is classical. Then we consider the case of fractional derivatives in the sense of Riemann-Liouville. We prove existence and uniqueness of mild solution under much weaker regularity conditions than the expected ones. Indeed, when the nonlinearity involves a term of the form1Γ(1𝛽)𝑡0𝑢(𝑠)𝑑𝑠(𝑡𝑠)𝛽,0<𝛽<1,(1.2)

then one is attracted by 𝑢(𝑠) in the integral and therefore it is natural to seek mild solutions in the space of continuously differentiable functions. This is somewhat surprising if instead of this expression one is given 𝐶𝐷𝛽𝑢(𝑡) (the latter is exactly the definition of the former). However, this is not the case when we deal with the Riemann-Liouville fractional derivative. Solutions are only 𝛽-differentiable and not necessarily once continuously differentiable. It will be therefore wise to look for solutions in an appropriate “fractional” space. We will consider the new spaces 𝐸𝛽 and 𝐹𝑆𝛽 (see (4.1)) instead of the classical ones 𝐸 and 𝐶1 (see [17]).

To simplify our task we will treat the following simpler problem𝑢(𝑡)=𝐴𝑢(𝑡)+𝑓(𝑡)+𝑡0𝑔𝑡,𝑠,𝑢(𝑠),𝐷𝛽𝑢(𝑠)𝑑𝑠,𝑡>0,𝑢(0)=𝑢0𝑋,𝑢(0)=𝑢1𝑋,(1.3) with 0<𝛽<1. The general case can be derived easily.

The rest of the paper is divided into three sections. In the second section we prepare some material consisting of notation and preliminary results needed in our proofs. The next section treats well-posedness when the fractional derivative is taken in the sense of Caputo. Section 4 is devoted to the Riemann-Liouville fractional derivative case.

2. Preliminaries

In this section we present some assumptions and results needed in our proofs later. This will fix also the notation used in this paper.

Definition 2.1. The integral (𝐼𝛼1)(𝑥)=Γ(𝛼)𝑥𝑎(𝑡)𝑑𝑡(𝑥𝑡)1𝛼,𝑥>𝑎(2.1) is called the Riemann-Liouville fractional integral of of order 𝛼>0 when the right side exists.

Here Γ is the usual Gamma functionΓ(𝑧)=0𝑒𝑠𝑠𝑧1𝑑𝑠,𝑧>0.(2.2)

Definition 2.2. The (left hand) Riemann-Liouville fractional derivative of order 0<𝛼<1 is defined by 𝐷𝛼𝑎1(𝑥)=𝑑Γ(1𝛼)𝑑𝑥𝑥𝑎(𝑡)𝑑𝑡(𝑥𝑡)𝛼,𝑥>𝑎,(2.3) whenever the right side is pointwise defined.

Definition 2.3. The fractional derivative of order 0<𝛼<1 in the sense of Caputo is given by 𝐶𝐷𝛼𝑎1(𝑥)=Γ(1𝛼)𝑥𝑎(𝑡)𝑑𝑡(𝑥𝑡)𝛼,𝑥>𝑎.(2.4)

Remark 2.4. The fractional integral of order 𝛼 is well defined on 𝐿𝑝, 𝑝1 (see [10]). Further, from Definition 2.2, it is clear that the Riemann-Liouville fractional derivative is defined for any function 𝐿𝑝, 𝑝1 for which 𝑘1𝛼 is differentiable (where 𝑘1𝛼(𝑡)=𝑡𝛼/Γ(1𝛼) and is the incomplete convolution). In fact, as domain of 𝐷𝛼0=𝐷𝛼 we can take 𝐷(𝐷𝛼)=𝐿𝑝(0,𝑇)𝑘1𝛼𝑊1,𝑝,(0,𝑇)(2.5) where 𝑊1,𝑝(0,𝑇)=𝑢𝜑𝐿𝑝(0,𝑇)𝑢(𝑡)=𝐶+𝑡0.𝜑(𝑠)𝑑𝑠(2.6) In particular, we know that the absolutely continuous functions (𝑝=1) are differentiable almost everywhere and therefore the Riemann-Liouville fractional derivative exists a.e. In this case (for an absolutely continuous function) the derivative is summable [10, Lemma 2.2] and the fractional derivative in the sense of Caputo exists. Moreover, we have the following relationship between the two types of fractional derivatives: 𝐷𝛼𝑎1(𝑥)=Γ(1𝛼)(𝑎)(𝑡𝑎)𝛼+𝑥𝑎(𝑡)𝑑𝑡(𝑥𝑡)𝛼=1Γ(1𝛼)(𝑎)(𝑡𝑎)𝛼+𝐶𝐷𝛼𝑎(𝑥),𝑥>𝑎.(2.7) See [1015] for more on fractional derivatives.

We will assume the following.(H1)𝐴 is the infinitesimal generator of a strongly continuous cosine family 𝐶(𝑡), 𝑡𝐑, of bounded linear operators in the Banach space 𝑋.

The associated sine family 𝑆(𝑡), 𝑡𝐑 is defined by𝑆(𝑡)𝑥=𝑡0𝐶(𝑠)𝑥𝑑𝑠,𝑡𝐑,𝑥𝑋.(2.8)

It is known (see [9, 16]) that there exist constants 𝑀1 and 𝜔0 such that||𝐶||(𝑡)𝑀𝑒𝜔|𝑡|||𝑆𝑡,𝑡𝐑,(𝑡)𝑆0||||||𝑀𝑡𝑡0𝑒𝜔|𝑠|||||𝑑𝑠,𝑡,𝑡0𝐑.(2.9)

If we define 𝐸={𝑥𝑋𝐶(𝑡)𝑥isoncecontinuouslydierentiableon𝐑}(2.10) then we have the following.

Lemma 2.5 (see [9, 16]). Assume that (H1) is satisfied. Then (i)𝑆(𝑡)𝑋𝐸, 𝑡𝐑,(ii)𝑆(𝑡)𝐸𝐷(𝐴), 𝑡𝐑,(iii)(𝑑/𝑑𝑡)𝐶(𝑡)𝑥=𝐴𝑆(𝑡)𝑥, 𝑥𝐸,   𝑡𝐑,(iv)(𝑑2/𝑑𝑡2)𝐶(𝑡)𝑥=𝐴𝐶(𝑡)𝑥=𝐶(𝑡)𝐴𝑥, 𝑥𝐷(𝐴), 𝑡𝐑.

Lemma 2.6 (see [9, 16]). Suppose that (H1) holds, 𝑣𝐑𝑋 a continuously differentiable function and 𝑞(𝑡)=𝑡0𝑆(𝑡𝑠)𝑣(𝑠)𝑑𝑠. Then, 𝑞(𝑡)𝐷(𝐴), 𝑞(𝑡)=𝑡0𝐶(𝑡𝑠)𝑣(𝑠)𝑑𝑠 and 𝑞(𝑡)=𝑡0𝐶(𝑡𝑠)𝑣(𝑠)𝑑𝑠+𝐶(𝑡)𝑣(0)=𝐴𝑞(𝑡)+𝑣(𝑡).

Definition 2.7. A function 𝑢()𝐶2(𝐼,𝑋) is called a classical solution of (1.3) if 𝑢(𝑡)𝐷(𝐴), satisfies the equation in (1.3) and the initial conditions are verified.

In case of Riemann-Liouville fractional derivative then we require additionally that 𝐷𝛽𝑢(𝑡) be continuous.

Definition 2.8. A continuously differentiable solution of the integrodifferential equation 𝑢(𝑡)=𝐶(𝑡)𝑢0+𝑆(𝑡)𝑢1+𝑡0+𝑆(𝑡𝑠)𝑓(𝑠)𝑑𝑠𝑡0𝑆(𝑡𝑠)𝑠0𝑔𝑠,𝜏,𝑢(𝜏),𝐶𝐷𝛽𝑢(𝜏)𝑑𝜏𝑑𝑠(2.11) is called mild solution of problem (1.3).

In case of Riemann-Liouville fractional derivative the (continuous) solution is merely 𝛽-differentiable (i.e., 𝐷𝛽𝑢(𝑡) exists and is continuous).

It follows from [8] that, in case of continuity of the nonlinearities, solutions of (1.3) are solutions of the more general problem (2.11).

3. Well-Posedness in 𝐶1([0,𝑇])

For the sake of comparison with the results in the next section we prove here existence and uniqueness of solutions in the space 𝐶1([0,𝑇]). This is the space where we usually look for mild solutions in case the first-order derivative of 𝑢 appears in the nonlinearity (see [17]). We consider fractional derivatives in the sense of Caputo. In case of Riemann-Liouville fractional derivatives we can pass to Caputo fractional derivatives through the formula (2.7) provided that solutions are in 𝐶1([0,𝑇]) (in theory, absolute continuity is enough).

Let 𝑋𝐴=𝐷(𝐴) endowed with the graph norm 𝑥𝐴=𝑥+𝐴𝑥. We need the following assumptions on 𝑓 and 𝑔:(H2)𝑓𝐑+𝑋 is continuously differentiable,(H3)𝑔𝐑+×𝐑+×𝑋𝐴×𝑋𝑋 is continuous and continuously differentiable with respect to its first variable,(H4)𝑔 and 𝑔1 (the derivative of 𝑔 with respect to its first variable) are Lipschitz continuous with respect to the last two variables, that is𝑔𝑡,𝑠,𝑥1,𝑦1𝑔𝑡,𝑠,𝑥2,𝑦2𝐴𝑔𝑥1𝑥2𝐴+𝑦1𝑦2,𝑔1𝑡,𝑠,𝑥1,𝑦1𝑔1𝑡,𝑠,𝑥2,𝑦2𝐴𝑔1𝑥1𝑥2𝐴+𝑦1𝑦2,(3.1) for some positive constants 𝐴𝑔 and 𝐴𝑔1.

Theorem 3.1. Assume that (H1)–(H4) hold. If 𝑢0𝐷(𝐴) and 𝑢1𝐸 then there exists 𝑇>0 and a unique function 𝑢[0,𝑇]𝑋, 𝑢𝐶([0,𝑇];𝑋𝐴)𝐶2([0,𝑇];𝑋) which satisfies (1.3) with Caputo fractional derivative 𝐶𝐷𝛽𝑢.

Proof. We start by proving existence and uniqueness of mild solutions in the space of continuously differentiable functions 𝐶1([0,𝑇]). To this end we consider for 𝑡[0,𝑇](𝐾𝑢)(𝑡)=𝐶(𝑡)𝑢0+𝑆(𝑡)𝑢1+𝑡0+𝑆(𝑡𝑠)𝑓(𝑠)𝑑𝑠𝑡0𝑆(𝑡𝑠)𝑠0𝑔𝑠,𝜏,𝑢(𝜏),𝐶𝐷𝛽𝑢(𝜏)𝑑𝜏𝑑𝑠.(3.2) Notice that 𝐶(𝑡)𝑢0𝐷(𝐴) because 𝑢0𝐷(𝐴) and we have 𝐴𝐶(𝑡)𝑢0=𝐶(𝑡)𝐴𝑢0. Also from the facts that 𝑢1𝐸 and 𝑆(𝑡)𝐸𝐷(𝐴) (see (ii) of Lemma 2.5) it is clear that 𝑆(𝑡)𝑢1𝐷(𝐴). Moreover, it follows from Lemma 2.6, (H2) and (H3) that both integral terms in (3.2) are in 𝐷(𝐴). Therefore, 𝐾𝑢𝐶([0,𝑇];𝐷(𝐴)). In addition to that we have from Lemma 2.6, (𝐴𝐾𝑢)(𝑡)=𝐶(𝑡)𝐴𝑢0+𝐴𝑆(𝑡)𝑢1+𝑡0𝐶(𝑡𝑠)𝑓(+𝑠)𝑑𝑠+𝐶(𝑡)𝑓(0)𝑓(𝑡)𝑡0𝑔𝐶(𝑡𝑠)𝑠,𝑠,𝑢(𝑠),𝐶𝐷𝛽+𝑢(𝑠)𝑠0𝑔1𝑠,𝜏,𝑢(𝜏),𝐶𝐷𝛽𝑢(𝜏)𝑑𝜏𝑑𝑠𝑡0𝑔𝑡,𝜏,𝑢(𝜏),𝐶𝐷𝛽[].𝑢(𝜏)𝑑𝜏,𝑡0,𝑇(3.3) Next, a differentiation of (3.2) yields (𝐾𝑢)(𝑡)=𝑆(𝑡)𝐴𝑢0+𝐶(𝑡)𝑢1+𝑡0+𝐶(𝑡𝑠)𝑓(𝑠)𝑑𝑠𝑡0𝐶(𝑡𝑠)𝑠0𝑔𝑠,𝜏,𝑢(𝜏),𝐶𝐷𝛽[].𝑢(𝜏)𝑑𝜏𝑑𝑠,𝑡0,𝑇(3.4) Therefore, 𝐾𝑢𝐶1([0,𝑇];𝑋) (remember that 𝑢𝐶1([0,𝑇];𝑋)) and 𝐾 maps 𝐶1 into 𝐶1.
Now we want to prove that 𝐾 is a contraction on 𝐶1 endowed with the metric 𝜌(𝑢,𝑣)=sup0𝑡𝑇𝑢𝑢(𝑡)𝑣(𝑡)+𝐴(𝑢(𝑡)𝑣(𝑡))+(𝑡)𝑣.(𝑡)(3.5)
For 𝑢, 𝑣 in 𝐶1, we can write (𝐾𝑢)(𝑡)(𝐾𝑣)(𝑡)𝑡00𝑡𝑠𝑀𝑒𝜔𝜏𝐴𝑑𝜏𝑔𝑠0(𝑢𝜏)𝑣(𝜏)𝐴+𝐶𝐷𝛽𝑢(𝜏)𝐶𝐷𝛽𝑣(𝜏)𝑑𝜏𝑑𝑠,(3.6) and since 𝐶𝐷𝛽𝑢(𝜏)𝐶𝐷𝛽1𝑣(𝜏)Γ(1𝛽)𝜏0(𝜏𝜎)𝛽𝑢(𝜎)𝑣𝜏(𝜎)𝑑𝜎1𝛽Γ(2𝛽)sup0𝑡𝑇𝑢(𝑡)𝑣,(𝑡)(3.7) it appears that (𝐾𝑢)(𝑡)(𝐾𝑣)(𝑡)𝑀𝐴𝑔𝑇22𝑇max1,1𝛽Γ(2𝛽)𝑇0𝑒𝜔𝜏𝑑𝜏𝜌(𝑢,𝑣).(3.8) Moreover, (𝐴𝐾𝑢)(𝑡)(𝐴𝐾𝑣)(𝑡)𝑡0𝑀𝑒𝜔(𝑡𝑠)𝐴𝑔(𝑢𝑠)𝑣(𝑠)𝐴+𝐶𝐷𝛽𝑢(𝑠)𝐶𝐷𝛽+𝑣(𝑠)𝑑𝑠𝑡0𝑀𝑒𝜔(𝑡𝑠)𝐴𝑔1𝑠0𝑢(𝜏)𝑣(𝜏)𝐴+𝐶𝐷𝛽𝑢(𝜏)𝐶𝐷𝛽+𝑣(𝜏)𝑑𝜏𝑑𝑠𝑡0𝐴𝑔𝑢(𝑠)𝑣(𝑠)𝐴+𝐶𝐷𝛽𝑢(𝑠)𝐶𝐷𝛽𝑣(𝑠)𝑑𝑠(3.9) implies that 𝑇(𝐴𝐾𝑢)(𝑡)(𝐴𝐾𝑣)(𝑡)max1,1𝛽Γ𝐴(2𝛽)𝑔𝐴𝑇+𝑀𝑔+𝐴𝑔1𝑇𝑇0𝑒𝜔(𝑇𝑠)𝑑s𝜌(𝑢,𝑣).(3.10)
In addition to that, we see that(𝐾𝑢)(𝑡)(𝐾𝑣)(𝑡)𝑡0𝑀𝑒𝜔(𝑡𝑠)𝐴𝑔𝑠0𝑢(𝜏)𝑣(𝜏)𝐴+𝐶𝐷𝛽𝑢(𝜏)𝐶𝐷𝛽𝑣(𝜏)𝑑𝜏𝑑𝑠𝑀𝐴𝑔𝑡0𝑒𝜔(𝑡𝑠)𝑠0𝑢(𝜏)𝑣(𝜏)𝐴+𝜏1𝛽Γ(2𝛽)sup0𝜎𝜏𝑢(𝜎)𝑣𝑇(𝜎)𝑑𝜏𝑑𝑠max1,1𝛽Γ(2𝛽)𝑀𝐴𝑔𝑇𝑇0𝑒𝜔(𝑇𝑠)𝑑𝑠𝜌(𝑢,𝑣).(3.11) These three relations (3.8), (3.10), and (3.11) show that, for 𝑇 small enough, 𝐾 is indeed a contraction on 𝐶1, and hence there exists a unique mild solution 𝑢𝐶1. Furthermore, it is clear (from (3.4), Lemmas 1, and 2) that 𝑢𝐶2([0,𝑇];𝑋) and satisfies the problem (1.3).

4. Existence of Mild Solutions in Case of R-L Derivative

In the previous section we proved existence and uniqueness of classical solutions provided that (𝑢0,𝑢1)𝐷(𝐴)×𝐸. From the proof of Theorem 3.1 it can be seen that existence and uniqueness of mild solutions hold when (𝑢0,𝑢1)𝐸×𝑋. In case of Riemann-Liouville fractional derivative one can still prove well-posedness in 𝐶1 by passing to the Caputo fractional derivative with the help of (2.7) (with a problem of singularity at zero which may be solved through a multiplication by an appropriate term of the form 𝑡𝛾). This also will require (𝑢0,𝑢1)𝐸×𝑋. Moreover, from the integrofractional-differential equation (2.11) it is clear that the mild solutions do not have to be continuously differentiable. In this section we will prove existence and uniqueness of mild solutions for the case of Riemann-Liouville fractional derivative for a less regular space than 𝐸×𝑋. Namely, for 0<𝛽<1, we consider𝐸𝛽=𝑥𝑋𝐷𝛽𝐶(𝑡)𝑥iscontinuouson𝐑+𝐹𝑆𝛽[]=𝑣𝐶(0,𝑇)𝐷𝛽[])𝑣𝐶(0,𝑇(4.1) equipped with the norm 𝑣𝛽=𝑣𝐶+𝐷𝛽𝑣𝐶 where 𝐶 is the uniform norm in 𝐶([0,𝑇]).

We will use the following assumptions:

(H5) 𝑓𝐑+𝑋 is continuous,

(H6) 𝑔𝐑+×𝐑+×𝑋×𝑋𝑋 is continuous and Lipschitzian, that is𝑔𝑡,𝑠,𝑥1,𝑦1𝑔𝑡,𝑠,𝑥2,𝑦2𝐴𝑔𝑥1𝑥2+𝑦1𝑦2,(4.2) for some positive constant 𝐴𝑔.

The result below is mentioned in [15, Lemma 2.10] (see also [15]) for functions. Here we state it and prove it for Bochner integral.

Lemma 4.1. If 𝐼1𝛼𝑅(𝑡)𝑥𝐶1([0,𝑇]),  𝑇>0, then one has 𝐷𝛼𝑡0𝑅(𝑡𝑠)𝑥𝑑𝑠=𝑡0𝐷𝛼𝑅(𝑡𝑠)𝑥𝑑𝑠+lim𝑡0+𝐼1𝛼[]𝑅(𝑡)𝑥,𝑥𝑋,𝑡0,𝑇.(4.3)

Proof. By Definition 2.2 and Fubini's theorem we have 𝐷𝛼𝑡01𝑅(𝑡𝑠)𝑥𝑑𝑠=𝑑Γ(1𝛼)𝑑t𝑡0𝑑𝜏(𝑡𝜏)𝛼𝜏0=1𝑅(𝜏𝑠)𝑥𝑑𝑠𝑑Γ(1𝛼)𝑑𝑡𝑡0𝑑𝑠𝑡𝑠𝑅(𝜏𝑠)𝑥(𝑡𝜏)𝛼=1𝑑𝜏Γ(1𝛼)𝑡0𝜕𝑑𝑠𝜕𝑡𝑡𝑠𝑅(𝜏𝑠)𝑥(𝑡𝜏)𝛼1𝑑𝜏+Γ(1𝛼)lim𝑠𝑡𝑡𝑠𝑅(𝜏𝑠)𝑥(𝑡𝜏)𝛼𝑑𝜏.(4.4) These steps are justified by the assumption 𝐼1𝛼𝑅(𝑡)𝑥𝐶1([0,𝑇]). Moreover, a change of variable 𝜎=𝜏𝑠 leads to 𝐷𝛼𝑡01𝑅(𝑡𝑠)𝑥𝑑𝑠=Γ(1𝛼)𝑡0𝜕𝑑𝑠𝜕𝑡0𝑡𝑠𝑅(𝜎)𝑥(𝑡𝑠𝜎)𝛼+1𝑑𝜎Γ(1𝛼)lim𝑡0+𝑡0𝑅(𝜎)𝑥(𝑡𝜎)𝛼𝑑𝜎.(4.5) This is exactly the formula stated in the lemma.

Corollary 4.2. For the sine family 𝑆(𝑡) associated with the cosine family 𝐶(𝑡) one has, for 𝑥𝑋 and 𝑡[0,𝑇]𝐷𝛼𝑡0𝑆(𝑡𝑠)𝑥𝑑𝑠=𝑡0𝐷𝛼𝑆(𝑡𝑠)𝑥𝑑𝑠=𝑡0𝐼1𝛼𝐶(𝑡𝑠)𝑥𝑑𝑠.(4.6)

Proof. First, from (2.7), we have 𝑑𝐼𝑑𝑡1𝛼𝑆(𝑡)𝑥=𝐷𝛼1𝑆(𝑡)𝑥=Γ(1𝛼)𝑆(0)𝑥𝑡𝛼+𝑡0(𝑡𝑠)𝛼𝑑𝑆(𝑠)=1𝑑𝑠𝑥𝑑𝑠Γ(1𝛼)𝑡0(𝑡𝑠)𝛼𝐶(𝑠)𝑥𝑑𝑠=𝐼1𝛼𝐶(𝑡)𝑥.(4.7) Notice that this means that (𝑑/𝑑𝑡)𝐼1𝛼𝑆(𝑡)𝑥=𝐼1𝛼𝐶(𝑡)𝑥 which is in accordance with a general permutation property valid when the function is 0 at 0 (see [10, 15]). It also shows that in this case the Riemann-Liouville derivative and the Caputo derivative are equal. Now from the continuity of 𝐶(𝑡) it is clear that 𝐼1𝛼𝐶(𝑡)𝑥 is continuous on [0,𝑇] (actually, the operator 𝐼𝛼 has several smoothing properties, see [11]) and therefore 𝐼1𝛼𝑆(𝑡)𝑥𝐶1([0,𝑇]). We can therefore apply Lemma 4.1 to obtain 𝐷𝛼𝑡0𝑆(𝑡𝑠)𝑥𝑑𝑠=𝑡0𝐷𝛼𝑆(𝑡𝑠)𝑥𝑑𝑠+lim𝑡0+𝐼1𝛼[].𝑆(𝑡)𝑥,𝑥𝑋,𝑡0,𝑇(4.8) Next, we claim that lim𝑡0+𝐼1𝛼𝑆(𝑡)𝑥=0. This follows easily from the definition of 𝑆(𝑡) and 𝐼1𝛼. Indeed, we have ||𝐼1𝛼||1𝑆(𝑡)𝑥Γ(1𝛼)𝑡0(𝑡𝑠)𝛼||||𝑡𝑆(𝑠)𝑥𝑑𝑠1𝛼Γ(2𝛼)sup0𝑡𝑇||||.𝑆(𝑡)𝑥(4.9)

We are now ready to state and prove our main result of this section.

Theorem 4.3. Assume that (H1), (H5), and (H6) hold. If (𝑢0,𝑢1)𝐸𝛽×𝑋, then there exists 𝑇>0 and a unique mild solution 𝑢𝐹𝑆𝛽 of problem (1.3) with Riemann-Liouville fractional derivative.

Proof. For 𝑡[0,𝑇], consider the operator (𝐾𝑢)(𝑡)=𝐶(𝑡)𝑢0+𝑆(𝑡)𝑢1+𝑡0+𝑆(𝑡𝑠)𝑓(𝑠)𝑑𝑠𝑡0𝑆(𝑡𝑠)𝑠0𝑔𝑠,𝜏,𝑢(𝜏),𝐷𝛽𝑢(𝜏)𝑑𝜏𝑑𝑠.(4.10) It is clear that 𝐾𝑢𝐶([0,𝑇];𝑋) when 𝑢𝐹𝑆𝛽. From Corollary 4.2, we see that 𝐷𝛽(𝐾𝑢)(𝑡)=𝐷𝛽𝐶(𝑡)𝑢0+𝐷𝛽𝑆(𝑡)𝑢1+𝑡0𝐼1𝛽+𝐶(𝑡𝑠)𝑓(𝑠)𝑑𝑠𝑡0𝐼1𝛽𝐶(𝑡𝑠)𝑠0𝑔𝑠,𝜏,𝑢(𝜏),𝐷𝛽𝑢(𝜏)𝑑𝜏𝑑𝑠.(4.11) Therefore 𝐾𝑢F𝑆𝛽 and maps 𝐹𝑆𝛽 to 𝐹𝑆𝛽 because 𝑢0𝐸𝛽, 𝐷𝛽𝑆(𝑡)𝑢1=𝑑𝐼𝑑𝑡1𝛽𝑆(𝑡)𝑢1=𝐶𝐷𝛽𝑆(𝑡)𝑢1=𝐼1𝛽𝐶(𝑡)𝑢1,(4.12) and the integral terms are obviously continuous. For 𝑢,𝑣𝐹𝑆𝛽, we find (𝐾𝑢)(𝑡)(𝐾𝑣)(𝑡)𝑡00𝑡𝑠𝑀𝑒𝜔𝜏𝐴𝑑𝜏𝑔𝑠0(𝐷𝑢𝜏)𝑣(𝜏)+𝛽𝑢(𝜏)𝐷𝛽𝑣(𝜏)𝑑𝜏𝑑𝑠𝑀𝐴𝑔𝑇22𝑇0𝑒𝜔𝜏𝑑𝜏sup0𝑡𝑇𝑢(𝑡)𝑣(𝑡)+sup0𝑡𝑇𝐷𝛽𝑢(𝑡)𝐷𝛽𝑣(𝑡)𝑀𝐴𝑔𝑇22𝑇0𝑒𝜔𝜏𝑑𝜏𝑢(𝑡)𝑣(𝑡)𝛽.(4.13) Further, 𝐷𝛽𝐷𝐾𝑢(𝑡)𝛽𝐾𝑣(𝑡)𝑡0𝐼1𝛽𝐶(𝑡𝑠)𝑠0𝑔𝑠,𝜏,𝑢(𝜏),𝐷𝛽𝑢(𝜏)𝑔𝑠,𝜏,𝑣(𝜏),𝐷𝛽𝑣(𝜏)𝑑𝜏𝑑𝑠𝑀𝐴𝑔𝑡0(𝑡𝑠)1𝛽𝑒𝜔(𝑡𝑠)Γ(2𝛽)𝑠𝑑𝑠sup0𝑡𝑇𝐷𝑢(𝑡)𝑣(𝑡)+𝛽𝑢(𝑡)𝐷𝛽𝑣(𝑡)𝑀𝐴𝑔𝑇2𝛽Γ(2𝛽)𝑇0𝑒𝜔(𝑇𝑠)(𝑑𝑠𝑢𝑡)𝑣(𝑡)𝛽.(4.14) Thus, for 𝑇 sufficiently small, 𝐾 is a contraction on the complete metric space 𝐹𝑆𝛽 and hence there exists a unique mild solution to (1.3).


The third author is very grateful for the financial support provided by King Fahd University of Petroleum and Minerals through Project no. IN100007.


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