Semilinear Volterra Integrodifferential Problems with Fractional Derivatives in the Nonlinearities

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:𝑢^(𝑡)=𝐴𝑢(𝑡)+𝑓𝑢(𝑡()0+)=^𝑡0𝑢𝑔₀𝑡,𝑠,𝑢(𝑠),𝐷^𝛽1𝑢(𝑠),…,𝐷^𝛽𝑛𝑢(𝑠)𝑑𝑠,𝑡>0,∈𝑋,𝑢^(0)=𝑢₁∈𝑋,(1.1)

where 0<𝛽_𝑖≤1, 𝑖=1,…,𝑛. Here the prime denotes time differentiation and 𝐷^𝛽𝑖, 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), 𝑡≥0 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=⋯=𝛽_𝑛=0 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 𝛽₁=𝛽₂=⋯=𝛽_𝑛=1, the underlying space is the space of continuously differentiable functions.

In this work, when 0<𝛽_𝑖<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(1−𝛽)^𝑡0𝑢^(𝑠)𝑑𝑠(𝑡−𝑠)^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 ^𝐶𝐷^𝛽𝑢(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 𝐸_𝛽 and 𝐹𝑆_𝛽 (see (4.1)) instead of the classical ones 𝐸 and 𝐶¹ (see [1–7]).

To simplify our task we will treat the following simpler problem𝑢^(𝑡)=𝐴𝑢(𝑢𝑡)(0+)𝑓=(𝑢𝑡₀)+^𝑡0𝑔𝑡,𝑠,𝑢(𝑠),𝐷^𝛽𝑢(𝑠)𝑑𝑠,𝑡>0,∈𝑋,𝑢^(0)=𝑢₁∈𝑋,(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 (𝐼^𝛼ℎ)(𝑥)=1Γ(𝛼)^𝑥𝑎ℎ(𝑡)𝑑𝑡(𝑥−𝑡)^1−𝛼,𝑥>𝑎(2.1) is called the Riemann-Liouville fractional integral of ℎ of order 𝛼>0 when the right side exists.

Here Γ is the usual Gamma functionΓ(𝑧)∶=^∞0𝑒^−𝑠𝑠^𝑧−1𝑑𝑠,𝑧>0.(2.2)

Definition 2.2. The (left hand) Riemann-Liouville fractional derivative of order 0<𝛼<1 is defined by 𝐷^𝛼𝑎ℎ(𝑥)=Γ(11−𝛼)𝑑𝑑𝑥^𝑥𝑎ℎ(𝑡)𝑑𝑡(𝑥−𝑡)^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 ^𝐶𝐷^𝛼𝑎ℎ(𝑥)=Γ(11−𝛼)^𝑥𝑎ℎ^(𝑡)𝑑𝑡(𝑥−𝑡)^𝛼,𝑥>𝑎.(2.4)

Remark 2.4. The fractional integral of order 𝛼 is well defined on 𝐿^𝑝, 𝑝≥1 (see [10]). Further, from Definition 2.2, it is clear that the Riemann-Liouville fractional derivative is defined for any function ℎ∈𝐿^𝑝, 𝑝≥1 for which 𝑘_1−𝛼∗ℎ is differentiable (where 𝑘_1−𝛼(𝑡)∶=𝑡^−𝛼/Γ(1−𝛼) and ∗ is the incomplete convolution). In fact, as domain of 𝐷^𝛼0=𝐷^𝛼 we can take 𝐷(𝐷^𝛼)=ℎ∈𝐿^𝑝(0,𝑇)∶𝑘_1−𝛼∗ℎ∈𝑊^1,𝑝(0,𝑇),(2.5) where 𝑊^1,𝑝(0,𝑇)∶=𝑢∶∃𝜑∈𝐿^𝑝(0,𝑇)∶𝑢(𝑡)=𝐶+^𝑡0𝜑(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: 𝐷^𝛼𝑎ℎ(𝑥)=Γ(11−𝛼)ℎ(𝑎)(𝑡−𝑎)^𝛼+^𝑥𝑎ℎ^(𝑡)𝑑𝑡=1(𝑥−𝑡)^𝛼Γ(1−𝛼)(𝑡ℎ−(𝑎𝑎))^𝛼+^𝐶𝐷^𝛼𝑎ℎ(𝑥),𝑥>𝑎.(2.7) See [10–15] for more on fractional derivatives.

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

The associated sine family 𝑆(𝑡), 𝑡∈𝐑 is defined by𝑆(𝑡)𝑥∶=^𝑡0𝐶(𝑠)𝑥𝑑𝑠,𝑡∈𝐑,𝑥∈𝑋.(2.8)

It is known (see [9, 16]) that there exist constants 𝑀≥1 and 𝜔≥0 such that||𝐶(𝑡)||≤𝑀𝑒^𝜔|𝑡|,𝑡∈𝐑,||𝑆(𝑡)−𝑆𝑡₀||≤𝑀||||^𝑡𝑡₀𝑒^𝜔|𝑠|𝑑𝑠||||,𝑡,𝑡₀∈𝐑.(2.9)

If we define 𝐸∶={𝑥∈𝑋∶𝐶(𝑡)𝑥isoncecontinuouslydifferentiableon𝐑}(2.10) then we have the following.

Lemma 2.5 (see [9, 16]). Assume that (H1) is satisfied. Then (i)𝑆(𝑡)𝑋⊂𝐸, 𝑡∈𝐑,(ii)𝑆(𝑡)𝐸⊂𝐷(𝐴), 𝑡∈𝐑,(iii)(𝑑/𝑑𝑡)𝐶(𝑡)𝑥=𝐴𝑆(𝑡)𝑥, 𝑥∈𝐸,   𝑡∈𝐑,(iv)(𝑑²/𝑑𝑡²)𝐶(𝑡)𝑥=𝐴𝐶(𝑡)𝑥=𝐶(𝑡)𝐴𝑥, 𝑥∈𝐷(𝐴), 𝑡∈𝐑.

Lemma 2.6 (see [9, 16]). Suppose that (H1) holds, 𝑣∶𝐑→𝑋 a continuously differentiable function and 𝑞(𝑡)=∫^𝑡0𝑆(𝑡−𝑠)𝑣(𝑠)𝑑𝑠. Then, 𝑞(𝑡)∈𝐷(𝐴), 𝑞^(𝑡)=∫^𝑡0𝐶(𝑡−𝑠)𝑣(𝑠)𝑑𝑠 and 𝑞^(𝑡)=∫^𝑡0𝐶(𝑡−𝑠)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 𝐷^𝛽𝑢(𝑡) be continuous.

Definition 2.8. A continuously differentiable solution of the integrodifferential equation 𝑢(𝑡)=+𝐶(𝑡)𝑢₀+𝑆(𝑡)𝑢₁+^𝑡0𝑆(𝑡−𝑠)𝑓(𝑠)𝑑𝑠_𝑡0𝑆(𝑡−𝑠)^𝑠0𝑔𝑠,𝜏,𝑢(𝜏),^𝐶𝐷^𝛽𝑢(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., 𝐷^𝛽𝑢(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 [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 𝐶¹([0,𝑇]) (in theory, absolute continuity is enough).

Let 𝑋_𝐴=𝐷(𝐴) endowed with the graph norm ‖𝑥‖_𝐴=‖𝑥‖+‖𝐴𝑥‖. We need the following assumptions on 𝑓 and 𝑔:(H2)𝑓∶𝐑⁺→𝑋 is continuously differentiable,(H3)𝑔∶𝐑⁺×𝐑⁺×𝑋_𝐴×𝑋→𝑋 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 is‖‖‖‖𝑔𝑔₁𝑡,𝑠,𝑥₁,𝑦₁−𝑔𝑡,𝑠,𝑥₂,𝑦₂‖‖≤𝐴_𝑔‖‖𝑥₁−𝑥₂‖‖_𝐴+‖‖𝑦₁−𝑦₂‖‖,𝑡,𝑠,𝑥₁,𝑦₁−𝑔₁𝑡,𝑠,𝑥₂,𝑦₂‖‖≤𝐴_𝑔1‖‖𝑥₁−𝑥₂‖‖_𝐴+‖‖𝑦₁−𝑦₂‖‖,(3.1) for some positive constants 𝐴_𝑔 and 𝐴_𝑔1.

Theorem 3.1. Assume that (H1)–(H4) hold. If 𝑢₀∈𝐷(𝐴) and 𝑢₁∈𝐸 then there exists 𝑇>0 and a unique function 𝑢∶[0,𝑇]→𝑋, 𝑢∈𝐶([0,𝑇];𝑋_𝐴)∩𝐶²([0,𝑇];𝑋) which satisfies (1.3) with Caputo fractional derivative ^𝐶𝐷^𝛽𝑢.

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+^𝑡0𝑆(𝑡−𝑠)𝑓(𝑠)𝑑𝑠_𝑡0𝑆(𝑡−𝑠)^𝑠0𝑔𝑠,𝜏,𝑢(𝜏),^𝐶𝐷^𝛽𝑢(𝜏)𝑑𝜏𝑑𝑠.(3.2) Notice that 𝐶(𝑡)𝑢₀∈𝐷(𝐴) because 𝑢₀∈𝐷(𝐴) and we have 𝐴𝐶(𝑡)𝑢₀=𝐶(𝑡)𝐴𝑢₀. Also from the facts that 𝑢₁∈𝐸 and 𝑆(𝑡)𝐸⊂𝐷(𝐴) (see (ii) of Lemma 2.5) it is clear that 𝑆(𝑡)𝑢₁∈𝐷(𝐴). Moreover, it follows from Lemma 2.6, (H2) and (H3) that both integral terms in (3.2) are in 𝐷(𝐴). Therefore, 𝐾𝑢∈𝐶([0,𝑇];𝐷(𝐴)). In addition to that we have from Lemma 2.6, (𝐴𝐾𝑢)(𝑡)=+𝐶(𝑡)𝐴𝑢₀+𝐴𝑆(𝑡)𝑢₁+^𝑡0𝐶(𝑡−𝑠)𝑓^(𝑠)𝑑𝑠+𝐶(𝑡)𝑓(0)−𝑓(𝑡)_𝑡0−𝐶(𝑡−𝑠)𝑔𝑠,𝑠,𝑢(𝑠),^𝐶𝐷^𝛽𝑢(𝑠)+^𝑠0𝑔₁𝑠,𝜏,𝑢(𝜏),^𝐶𝐷^𝛽𝑢(𝜏)𝑑𝜏𝑑𝑠_𝑡0𝑔𝑡,𝜏,𝑢(𝜏),^𝐶𝐷^𝛽𝑢(𝜏)𝑑𝜏,𝑡∈[0,𝑇].(3.3) Next, a differentiation of (3.2) yields (𝐾𝑢)^(𝑡)=+𝑆(𝑡)𝐴𝑢₀+𝐶(𝑡)𝑢₁+^𝑡0𝐶(𝑡−𝑠)𝑓(𝑠)𝑑𝑠_𝑡0𝐶(𝑡−𝑠)^𝑠0𝑔𝑠,𝜏,𝑢(𝜏),^𝐶𝐷^𝛽𝑢(𝜏)𝑑𝜏𝑑𝑠,𝑡∈[0,𝑇].(3.4) Therefore, 𝐾𝑢∈𝐶¹([0,𝑇];𝑋) (remember that 𝑢∈𝐶¹([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)∶=sup_{0≤𝑡≤𝑇}‖𝑢(𝑡)−𝑣(𝑡)‖+‖𝐴(𝑢(𝑡)−𝑣(𝑡))‖+‖‖𝑢^(𝑡)−𝑣^(𝑡)‖‖.(3.5)
For 𝑢, 𝑣 in 𝐶¹, we can write ‖(𝐾≤𝑢)(𝑡)−(𝐾𝑣)(𝑡)‖_𝑡0^0𝑡−𝑠𝑀𝑒^𝜔𝜏𝑑𝜏𝐴_𝑔^𝑠0‖𝑢(𝜏)−𝑣(𝜏)‖_𝐴+‖‖^𝐶𝐷^𝛽𝑢(𝜏)−^𝐶𝐷^𝛽𝑣(𝜏)‖‖𝑑𝜏𝑑𝑠,(3.6) and since ‖‖^𝐶𝐷^𝛽𝑢(𝜏)−^𝐶𝐷^𝛽𝑣(𝜏)‖‖≤1Γ(1−𝛽)^𝜏0≤𝜏(𝜏−𝜎)^−𝛽‖‖𝑢^(𝜎)−𝑣^(𝜎)‖‖𝑑𝜎_1−𝛽Γ(2−𝛽)sup_{0≤𝑡≤𝑇}‖‖𝑢^(𝑡)−𝑣^(𝑡)‖‖,(3.7) it appears that ‖(𝐾𝑢)(𝑡)−(𝐾𝑣)(𝑡)‖≤𝑀𝐴^𝑔𝑇²2max1,Γ(𝑇2−^1−𝛽𝛽)^𝑇0𝑒^𝜔𝜏𝑑𝜏𝜌(𝑢,𝑣).(3.8) Moreover, ‖(𝐴≤𝐾𝑢)(𝑡)−(𝐴𝐾𝑣)(𝑡)‖_𝑡0+𝑀𝑒^{𝜔(𝑡−𝑠)}𝐴_𝑔‖𝑢(𝑠)−𝑣(𝑠)‖_𝐴+‖‖^𝐶𝐷^𝛽𝑢(𝑠)−^𝐶𝐷^𝛽𝑣(𝑠)‖‖𝑑𝑠_𝑡0𝑀𝑒^{𝜔(𝑡−𝑠)}𝐴_𝑔1^𝑠0‖𝑢(𝜏)−𝑣(𝜏)‖_𝐴++‖‖^𝐶𝐷^𝛽𝑢(𝜏)−^𝐶𝐷^𝛽𝑣(𝜏)‖‖𝑑𝜏𝑑𝑠_𝑡0𝐴_𝑔‖𝑢(𝑠)−𝑣(𝑠)‖_𝐴+‖‖^𝐶𝐷^𝛽𝑢(𝑠)−^𝐶𝐷^𝛽𝑣(𝑠)‖‖𝑑𝑠(3.9) implies that ‖(𝐴≤𝐾𝑢m)(a𝑡x)−(1𝐴,𝐾𝑣𝑇)(𝑡_1−𝛽)‖Γ(2−𝛽)𝐴^𝑔𝑇+𝑀𝐴_𝑔+𝐴_𝑔1𝑇^𝑇0𝑒^{𝜔(𝑇−𝑠)}𝑑s𝜌(𝑢,𝑣).(3.10)
In addition to that, we see that‖‖(𝐾𝑢≤)^(𝑡)−(𝐾𝑣)^(𝑡)_𝑡0‖‖𝑀𝑒^{𝜔(𝑡−𝑠)}𝐴_𝑔^𝑠0‖𝑢(𝜏)−𝑣(𝜏)‖_𝐴≤𝑀𝐴_𝑔+‖‖^𝐶𝐷^𝛽𝑢(𝜏)−^𝐶𝐷^𝛽𝑣(𝜏)‖‖𝑑𝜏𝑑𝑠^𝑡0𝑒^{𝜔(𝑡−𝑠)}^𝑠0‖𝑢(𝜏)−𝑣(𝜏)‖^𝐴+𝜏^1−𝛽Γ(2−𝛽)sup_{0≤𝜎≤𝜏}≤max1,𝑇_1−𝛽‖‖𝑢^(𝜎)−𝑣^(𝜎)‖‖𝑑𝜏𝑑𝑠Γ(2−𝛽)𝑀𝐴^𝑔𝑇^𝑇0𝑒^{𝜔(𝑇−𝑠)}𝑑𝑠𝜌(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 𝑢∈𝐶¹. Furthermore, it is clear (from (3.4), Lemmas 1, and 2) that 𝑢∈𝐶²([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 𝑡^𝛾). 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 consider𝐸_𝛽𝐹∶𝑆=_𝛽𝑥∈𝑋∶𝐷^𝛽𝐶(𝑡)𝑥iscontinuouson𝐑⁺∶=𝑣∈𝐶([0,𝑇])∶𝐷^𝛽𝑣∈𝐶([0,𝑇])(4.1) equipped with the norm ‖𝑣‖_𝛽∶=‖𝑣‖_𝐶+‖𝐷^𝛽𝑣‖_𝐶 where ‖⋅‖_𝐶 is the uniform norm in 𝐶([0,𝑇]).

We will use the following assumptions:

(H5) 𝑓∶𝐑⁺→𝑋 is continuous,

(H6) 𝑔∶𝐑⁺×𝐑⁺×𝑋×𝑋→𝑋 is continuous and Lipschitzian, that is‖‖𝑔𝑡,𝑠,𝑥₁,𝑦₁−𝑔𝑡,𝑠,𝑥₂,𝑦₂‖‖≤𝐴_𝑔‖‖𝑥₁−𝑥₂‖‖+‖‖𝑦₁−𝑦₂‖‖,(4.2) for some positive constant 𝐴_𝑔.

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 𝐼^1−𝛼𝑅(𝑡)𝑥∈𝐶¹([0,𝑇]),  𝑇>0, then one has 𝐷^𝛼^𝑡0𝑅(𝑡−𝑠)𝑥𝑑𝑠=^𝑡0𝐷^𝛼𝑅(𝑡−𝑠)𝑥𝑑𝑠+lim_𝑡→0+𝐼^1−𝛼𝑅(𝑡)𝑥,𝑥∈𝑋,𝑡∈[0,𝑇].(4.3)

Proof. By Definition 2.2 and Fubini's theorem we have 𝐷^𝛼^𝑡0𝑅(𝑡−𝑠)𝑥𝑑𝑠=1Γ(1−𝛼)𝑑𝑑t^𝑡0𝑑𝜏(𝑡−𝜏)^𝛼^𝜏0=1𝑅(𝜏−𝑠)𝑥𝑑𝑠Γ(1−𝛼)𝑑𝑑𝑡^𝑡0𝑑𝑠^𝑡𝑠𝑅(𝜏−𝑠)𝑥(𝑡−𝜏)^𝛼=1𝑑𝜏Γ(1−𝛼)^𝑡0𝑑𝑠𝜕𝜕𝑡^𝑡𝑠𝑅(𝜏−𝑠)𝑥(𝑡−𝜏)^𝛼𝑑𝜏+1Γ(1−𝛼)lim^{𝑠→𝑡−}^𝑡𝑠𝑅(𝜏−𝑠)𝑥(𝑡−𝜏)^𝛼𝑑𝜏.(4.4) These steps are justified by the assumption 𝐼^1−𝛼𝑅(𝑡)𝑥∈𝐶¹([0,𝑇]). Moreover, a change of variable 𝜎=𝜏−𝑠 leads to 𝐷^𝛼^𝑡0𝑅(𝑡−𝑠)𝑥𝑑𝑠=1Γ(1−𝛼)^𝑡0𝑑𝑠𝜕𝜕𝑡^0𝑡−𝑠𝑅(𝜎)𝑥(𝑡−𝑠−𝜎)^𝛼+1𝑑𝜎Γ(1−𝛼)lim_𝑡→0⁺^𝑡0𝑅(𝜎)𝑥(𝑡−𝜎)^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 𝑥∈𝑋 and 𝑡∈[0,𝑇]𝐷^𝛼^𝑡0𝑆(𝑡−𝑠)𝑥𝑑𝑠=^𝑡0𝐷^𝛼𝑆(𝑡−𝑠)𝑥𝑑𝑠=^𝑡0𝐼^1−𝛼𝐶(𝑡−𝑠)𝑥𝑑𝑠.(4.6)

Proof. First, from (2.7), we have 𝑑𝑑𝑡𝐼^1−𝛼𝑆(𝑡)𝑥=𝐷^𝛼𝑆(𝑡)𝑥=1Γ(1−𝛼)𝑆(𝑡0^𝛼)𝑥+^𝑡0(𝑡−𝑠)^−𝛼𝑑𝑆(𝑠)=1𝑑𝑠𝑥𝑑𝑠Γ(1−𝛼)^𝑡0(𝑡−𝑠)^−𝛼𝐶(𝑠)𝑥𝑑𝑠=𝐼^1−𝛼𝐶(𝑡)𝑥.(4.7) Notice that this means that (𝑑/𝑑𝑡)𝐼^1−𝛼𝑆(𝑡)𝑥=𝐼^1−𝛼𝐶(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 𝐼^1−𝛼𝐶(𝑡)𝑥 is continuous on [0,𝑇] (actually, the operator 𝐼^𝛼 has several smoothing properties, see [11]) and therefore 𝐼^1−𝛼𝑆(𝑡)𝑥∈𝐶¹([0,𝑇]). We can therefore apply Lemma 4.1 to obtain 𝐷^𝛼^𝑡0𝑆(𝑡−𝑠)𝑥𝑑𝑠=^𝑡0𝐷^𝛼𝑆(𝑡−𝑠)𝑥𝑑𝑠+lim_𝑡→0+𝐼^1−𝛼𝑆(𝑡)𝑥,𝑥∈𝑋,𝑡∈[0,𝑇].(4.8) Next, we claim that lim_𝑡→0⁺𝐼^1−𝛼𝑆(𝑡)𝑥=0. This follows easily from the definition of 𝑆(𝑡) and 𝐼^1−𝛼. Indeed, we have ||𝐼^1−𝛼𝑆(𝑡)𝑥||≤1Γ(1−𝛼)^𝑡0(𝑡−𝑠)^−𝛼||𝑆(𝑠)𝑥||𝑑𝑠≤𝑡^1−𝛼Γ(2−𝛼)sup_{0≤𝑡≤𝑇}||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)∈𝐸_𝛽×𝑋, then there exists 𝑇>0 and a unique mild solution 𝑢∈𝐹𝑆_𝛽 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+^𝑡0𝑆(𝑡−𝑠)𝑓(𝑠)𝑑𝑠_𝑡0𝑆(𝑡−𝑠)^𝑠0𝑔𝑠,𝜏,𝑢(𝜏),𝐷^𝛽𝑢(𝜏)𝑑𝜏𝑑𝑠.(4.10) It is clear that 𝐾𝑢∈𝐶([0,𝑇];𝑋) when 𝑢∈𝐹𝑆_𝛽. From Corollary 4.2, we see that 𝐷^𝛽(𝐾𝑢)(𝑡)=+𝐷^𝛽𝐶(𝑡)𝑢₀+𝐷^𝛽𝑆(𝑡)𝑢₁+^𝑡0𝐼^1−𝛽𝐶(𝑡−𝑠)𝑓(𝑠)𝑑𝑠_𝑡0𝐼^1−𝛽𝐶(𝑡−𝑠)^𝑠0𝑔𝑠,𝜏,𝑢(𝜏),𝐷^𝛽𝑢(𝜏)𝑑𝜏𝑑𝑠.(4.11) Therefore 𝐾𝑢∈F𝑆_𝛽 and maps 𝐹𝑆_𝛽 to 𝐹𝑆_𝛽 because 𝑢₀∈𝐸_𝛽, 𝐷^𝛽𝑆(𝑡)𝑢₁=𝑑𝑑𝑡𝐼^1−𝛽𝑆(𝑡)𝑢₁=^𝐶𝐷^𝛽𝑆(𝑡)𝑢₁=𝐼^1−𝛽𝐶(𝑡)𝑢₁,(4.12) and the integral terms are obviously continuous. For 𝑢,𝑣∈𝐹𝑆_𝛽, we find ‖(𝐾≤𝑢)(𝑡)−(𝐾𝑣)(𝑡)‖_𝑡0^0𝑡−𝑠𝑀𝑒^𝜔𝜏𝑑𝜏𝐴_𝑔^𝑠0‖𝑢(𝜏)−𝑣(𝜏)‖+‖‖𝐷^𝛽≤𝑀𝐴_𝑔𝑢(𝜏)−𝐷^𝛽𝑣(𝜏)‖‖𝑑𝜏𝑑𝑠𝑇²2^𝑇0≤𝑀𝐴_𝑔𝑒^𝜔𝜏𝑑𝜏sup_{0≤𝑡≤𝑇}‖𝑢(𝑡)−𝑣(𝑡)‖+sup_{0≤𝑡≤𝑇}‖‖𝐷^𝛽𝑢(𝑡)−𝐷^𝛽𝑣(𝑡)‖‖𝑇²2^𝑇0𝑒^𝜔𝜏𝑑𝜏‖𝑢(𝑡)−𝑣(𝑡)‖_𝛽.(4.13) Further, ‖‖𝐷≤^𝛽𝐾𝑢(𝑡)−𝐷^𝛽𝐾𝑣(𝑡)_𝑡0‖‖‖‖‖𝐼_1−𝛽≤𝑀𝐴_𝑔𝐶(𝑡−𝑠)^𝑠0𝑔𝑠,𝜏,𝑢(𝜏),𝐷^𝛽𝑢(𝜏)−𝑔𝑠,𝜏,𝑣(𝜏),𝐷^𝛽𝑣(𝜏)𝑑𝜏𝑑𝑠‖‖‖^𝑡0(𝑡−𝑠)^1−𝛽𝑒^{𝜔(𝑡−𝑠)}Γ(2−𝛽)𝑠𝑑𝑠sup_{0≤𝑡≤𝑇}‖𝑢(𝑡)−𝑣(𝑡)‖+‖‖𝐷^𝛽≤𝑀𝐴_𝑔𝑢(𝑡)−𝐷^𝛽𝑣(𝑡)‖‖𝑇^2−𝛽Γ(2−𝛽)^𝑇0𝑒^{𝜔(𝑇−𝑠)}𝑑𝑠‖𝑢(𝑡)−𝑣(𝑡)‖_𝛽.(4.14) Thus, for 𝑇 sufficiently small, 𝐾 is a contraction on the complete metric space 𝐹𝑆_𝛽 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.