Abstract and Applied Analysis

Volume 2011, Article ID 510314, 11 pages

http://dx.doi.org/10.1155/2011/510314

## Semilinear Volterra Integrodifferential Problems with Fractional Derivatives in the Nonlinearities

^{1}Université de la Rochelle, Avenue Michel Crépeau, 17042 La Rochelle Cedex 1, France^{2}Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Comenius University, Mlynska Doliná, 84248 Bratislava, Slovakia^{3}Department 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.

#### Abstract

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:

where , . Here the prime denotes time differentiation and , denotes fractional time differentiation (in the sense of Riemann-Liouville or Caputo). The operator is the infinitesimal generator of a strongly continuous cosine family , of bounded linear operators in the Banach space , and are nonlinear functions from to and to , respectively, and are given initial data in . The problem with or 1 has been investigated by several authors (see [1–7] 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 , the underlying space is the space of continuously differentiable functions.

In this work, when , , 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 . 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 form

then one is attracted by 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 (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 *-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 and (see (4.1)) instead of the classical ones and (see [1–7]).

To simplify our task we will treat the following simpler problem with . 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
is called the Riemann-Liouville fractional integral of of order when the right side exists.

Here is the usual Gamma function

*Definition 2.2. *The (left hand) Riemann-Liouville fractional derivative of order is defined by
whenever the right side is pointwise defined.

*Definition 2.3. *The fractional derivative of order in the sense of Caputo is given by

*Remark 2.4. *The fractional integral of order is well defined on , (see [10]). Further, from Definition 2.2, it is clear that the Riemann-Liouville fractional derivative is defined for any function , for which is differentiable (where and is the incomplete convolution). In fact, as domain of we can take
where
In particular, we know that the absolutely continuous functions ( 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:
See [10–15] for more on fractional derivatives.

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

The associated sine family , is defined by

It is known (see [9, 16]) that there exist constants and such that

If we define then we have the following.

Lemma 2.5 (see [9, 16]). *Assume that (H1) is satisfied. Then *(i)*, ,*(ii)*, ,*(iii)*, , ,*(iv)*, , .*

Lemma 2.6 (see [9, 16]). *Suppose that (H1) holds, a continuously differentiable function and . Then, , and .*

*Definition 2.7. *A function is called a classical solution of (1.3) if , satisfies the equation in (1.3) and the initial conditions are verified.

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

*Definition 2.8. *A continuously differentiable solution of the integrodifferential equation
is called mild solution of problem (1.3).

In case of Riemann-Liouville fractional derivative the (continuous) solution is merely *-differentiable* (i.e., 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

For the sake of comparison with the results in the next section we prove here existence and uniqueness of solutions in the space . This is the space where we usually look for mild solutions in case the first-order derivative of appears in the nonlinearity (see [1–7]). 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 (in theory, absolute continuity is enough).

Let endowed with the graph norm . We need the following assumptions on and :(H2) is continuously differentiable,(H3) is continuous and continuously differentiable with respect to its first variable,(H4) and (the derivative of with respect to its first variable) are Lipschitz continuous with respect to the last two variables, that is for some positive constants and .

Theorem 3.1. *Assume that (H1)–(H4) hold. If and then there exists and a unique function , which satisfies (1.3) with Caputo fractional derivative .*

*Proof. *We start by proving existence and uniqueness of mild solutions in the space of continuously differentiable functions . To this end we consider for
Notice that because and we have . Also from the facts that and (see (ii) of Lemma 2.5) it is clear that . Moreover, it follows from Lemma 2.6, (H2) and (H3) that both integral terms in (3.2) are in . Therefore, . In addition to that we have from Lemma 2.6,
Next, a differentiation of (3.2) yields
Therefore, (remember that ) and maps into .

Now we want to prove that is a contraction on endowed with the metric

For , in , we can write
and since
it appears that
Moreover,
implies that

In addition to that, we see that
These three relations (3.8), (3.10), and (3.11) show that, for small enough, is indeed a contraction on , and hence there exists a unique mild solution . Furthermore, it is clear (from (3.4), Lemmas 1, and 2) that 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 . From the proof of Theorem 3.1 it can be seen that existence and uniqueness of mild solutions hold when . In case of Riemann-Liouville fractional derivative one can still prove well-posedness in 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 ). This also will require . 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 . Namely, for , we consider equipped with the norm where is the uniform norm in .

We will use the following assumptions:

(H5) is continuous,

(H6) is continuous and Lipschitzian, that is for some positive constant .

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 , , then one has
*

*Proof. *By Definition 2.2 and Fubini's theorem we have
These steps are justified by the assumption . Moreover, a change of variable leads to
This is exactly the formula stated in the lemma.

Corollary 4.2. *For the sine family associated with the cosine family one has, for and *

*Proof. *First, from (2.7), we have
Notice that this means that 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 it is clear that is continuous on (actually, the operator has several smoothing properties, see [11]) and therefore . We can therefore apply Lemma 4.1 to obtain
Next, we claim that . This follows easily from the definition of and . Indeed, we have

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 , then there exists and a unique mild solution of problem (1.3) with Riemann-Liouville fractional derivative.*

*Proof. *For , consider the operator
It is clear that when . From Corollary 4.2, we see that
Therefore and maps to because ,
and the integral terms are obviously continuous. For , we find
Further,
Thus, for sufficiently small, is a contraction on the complete metric space and hence there exists a unique mild solution to (1.3).

#### Acknowledgment

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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