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)+∫0tg(t,s,u(s),Dβ1u(s),…,Dβnu(s))ds,t>0,u(0)=u0∈X,u′(0)=u1∈X,

where 0<βi≤1, 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), 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=⋯=βn=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=⋯=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(1-β)∫0tu′(s)ds(t-s)β,0<β<1,

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 [1–7]).

To simplify our task we will treat the following simpler problemu′′(t)=Au(t)+f(t)+∫0tg(t,s,u(s),Dβu(s))ds,t>0,u(0)=u0∈X,u′(0)=u1∈X,
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
(Iαh)(x)=1Γ(α)∫axh(t)dt(x-t)1-α,x>a
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):=∫0∞e-ssz-1ds,z>0.

Definition 2.2.

The (left hand) Riemann-Liouville fractional derivative of order 0<α<1 is defined by
(Daαh)(x)=1Γ(1-α)ddx∫axh(t)dt(x-t)α,x>a,
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
(CDaαh)(x)=1Γ(1-α)∫axh′(t)dt(x-t)α,x>a.

Remark 2.4.

The fractional integral of order α is well defined on Lp, p≥1 (see [10]). Further, from Definition 2.2, it is clear that the Riemann-Liouville fractional derivative is defined for any function h∈Lp, p≥1 for which k1-α*h is differentiable (where k1-α(t):=t-α/Γ(1-α) and * is the incomplete convolution). In fact, as domain of D0α=Dα we can take
D(Dα)={h∈Lp(0,T):k1-α*h∈W1,p(0,T)},
where
W1,p(0,T):={u:∃φ∈Lp(0,T):u(t)=C+∫0tφ(s)ds}.
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:
(Daαh)(x)=1Γ(1-α)[h(a)(t-a)α+∫axh′(t)dt(x-t)α]=1Γ(1-α)h(a)(t-a)α+(CDaαh)(x),x>a.
See [10–15] for more on fractional derivatives.

We will assume the following.

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:=∫0tC(s)xds,t∈R,x∈X.

It is known (see [9, 16]) that there exist constants M≥1 and ω≥0 such that|C(t)|≤Meω|t|,t∈R,|S(t)-S(t0)|≤M|∫t0teω|s|ds|,t,t0∈R.

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

Lemma 2.5 (see [<xref ref-type="bibr" rid="B15">9</xref>, <xref ref-type="bibr" rid="B16">16</xref>]).

Assume that (H1) is satisfied. Then

S(t)X⊂E, t∈R,

S(t)E⊂D(A), t∈R,

(d/dt)C(t)x=AS(t)x, x∈E, t∈R,

(d2/dt2)C(t)x=AC(t)x=C(t)Ax, x∈D(A), t∈R.

Lemma 2.6 (see [<xref ref-type="bibr" rid="B15">9</xref>, <xref ref-type="bibr" rid="B16">16</xref>]).

Suppose that (H1) holds, v:R→X a continuously differentiable function and q(t)=∫0tS(t-s)v(s)ds. Then, q(t)∈D(A), q′(t)=∫0tC(t-s)v(s)ds and q′′(t)=∫0tC(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 Dβu(t) be continuous.

Definition 2.8.

A continuously differentiable solution of the integrodifferential equation
u(t)=C(t)u0+S(t)u1+∫0tS(t-s)f(s)ds+∫0tS(t-s)∫0sg(s,τ,u(τ),CDβu(τ))dτds
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).

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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 XA=D(A) endowed with the graph norm ∥x∥A=∥x∥+∥Ax∥. We need the following assumptions on f and g:

f:R+→X is continuously differentiable,

g:R+×R+×XA×X→X is continuous and continuously differentiable with respect to its first variable,

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

‖g(t,s,x1,y1)-g(t,s,x2,y2)‖≤Ag(‖x1-x2‖A+‖y1-y2‖),‖g1(t,s,x1,y1)-g1(t,s,x2,y2)‖≤Ag1(‖x1-x2‖A+‖y1-y2‖),
for some positive constants Ag and Ag1.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];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+∫0tS(t-s)f(s)ds+∫0tS(t-s)∫0sg(s,τ,u(τ),CDβu(τ))dτds.
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+∫0tC(t-s)f′(s)ds+C(t)f(0)-f(t)+∫0tC(t-s)[g(s,s,u(s),CDβu(s))+∫0sg1(s,τ,u(τ),CDβu(τ))dτ]ds-∫0tg(t,τ,u(τ),CDβu(τ))dτ,t∈[0,T].
Next, a differentiation of (3.2) yields
(Ku)′(t)=S(t)Au0+C(t)u1+∫0tC(t-s)f(s)ds+∫0tC(t-s)∫0sg(s,τ,u(τ),CDβu(τ))dτds,t∈[0,T].
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):=sup0≤t≤T(‖u(t)-v(t)‖+‖A(u(t)-v(t))‖+‖u′(t)-v′(t)‖).

For u, v in C1, we can write
‖(Ku)(t)-(Kv)(t)‖≤∫0t(∫0t-sMeωτdτ)Ag∫0s(‖u(τ)-v(τ)‖A+‖CDβu(τ)-CDβv(τ)‖)dτds,
and since
‖CDβu(τ)-CDβv(τ)‖≤1Γ(1-β)∫0τ(τ-σ)-β‖u′(σ)-v′(σ)‖dσ≤τ1-βΓ(2-β)sup0≤t≤T‖u′(t)-v′(t)‖,
it appears that
‖(Ku)(t)-(Kv)(t)‖≤MAgT22max(1,T1-βΓ(2-β))(∫0Teωτdτ)ρ(u,v).
Moreover,
‖(AKu)(t)-(AKv)(t)‖≤∫0tMeω(t-s)Ag(‖u(s)-v(s)‖A+‖CDβu(s)-CDβv(s)‖)ds+∫0tMeω(t-s)Ag1∫0s(‖u(τ)-v(τ)‖A+‖CDβu(τ)-CDβv(τ)‖)dτds+∫0tAg(‖u(s)-v(s)‖A+‖CDβu(s)-CDβv(s)‖)ds
implies that
‖(AKu)(t)-(AKv)(t)‖≤max(1,T1-βΓ(2-β))[AgT+M(Ag+Ag1T)](∫0Teω(T-s)ds)ρ(u,v).

In addition to that, we see that‖(Ku)′(t)-(Kv)′(t)‖≤∫0tMeω(t-s)Ag∫0s(‖u(τ)-v(τ)‖A+‖CDβu(τ)-CDβv(τ)‖)dτds≤MAg∫0teω(t-s)∫0s(‖u(τ)-v(τ)‖A+τ1-βΓ(2-β)sup0≤σ≤τ‖u′(σ)-v′(σ)‖)dτds≤max(1,T1-βΓ(2-β))MAgT(∫0Teω(T-s)ds)ρ(u,v).
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 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:={x∈X:DβC(t)xiscontinuousonR+}FSβ:={v∈C([0,T]):Dβv∈C([0,T])}
equipped with the norm ∥v∥β:=∥v∥C+∥Dβv∥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)‖≤Ag(‖x1-x2‖+‖y1-y2‖),
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 I1-αR(t)x∈C1([0,T]), T>0, then one has
Dα∫0tR(t-s)xds=∫0tDαR(t-s)xds+limt→0+I1-αR(t)x,x∈X,t∈[0,T].

Proof.

By Definition 2.2 and Fubini's theorem we have
Dα∫0tR(t-s)xds=1Γ(1-α)ddt∫0tdτ(t-τ)α∫0τR(τ-s)xds=1Γ(1-α)ddt∫0tds∫stR(τ-s)x(t-τ)αdτ=1Γ(1-α)∫0tds∂∂t∫stR(τ-s)x(t-τ)αdτ+1Γ(1-α)lims→t-∫stR(τ-s)x(t-τ)αdτ.
These steps are justified by the assumption I1-αR(t)x∈C1([0,T]). Moreover, a change of variable σ=τ-s leads to
Dα∫0tR(t-s)xds=1Γ(1-α)∫0tds∂∂t∫0t-sR(σ)x(t-s-σ)αdσ+1Γ(1-α)limt→0+∫0tR(σ)x(t-σ)αdσ.
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]Dα∫0tS(t-s)xds=∫0tDαS(t-s)xds=∫0tI1-αC(t-s)xds.

Proof.

First, from (2.7), we have
ddtI1-αS(t)x=DαS(t)x=1Γ(1-α)[S(0)xtα+∫0t(t-s)-αdS(s)dsxds]=1Γ(1-α)∫0t(t-s)-αC(s)xds=I1-αC(t)x.
Notice that this means that (d/dt)I1-αS(t)x=I1-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 I1-αC(t)x is continuous on [0,T] (actually, the operator Iα has several smoothing properties, see [11]) and therefore I1-αS(t)x∈C1([0,T]). We can therefore apply Lemma 4.1 to obtain
Dα∫0tS(t-s)xds=∫0tDαS(t-s)xds+limt→0+I1-αS(t)x,x∈X,t∈[0,T].
Next, we claim that limt→0+I1-αS(t)x=0. This follows easily from the definition of S(t) and I1-α. Indeed, we have
|I1-αS(t)x|≤1Γ(1-α)∫0t(t-s)-α|S(s)x|ds≤t1-αΓ(2-α)sup0≤t≤T|S(t)x|.

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 u∈FSβ 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+∫0tS(t-s)f(s)ds+∫0tS(t-s)∫0sg(s,τ,u(τ),Dβu(τ))dτds.
It is clear that Ku∈C([0,T];X) when u∈FSβ. From Corollary 4.2, we see that
Dβ(Ku)(t)=DβC(t)u0+DβS(t)u1+∫0tI1-βC(t-s)f(s)ds+∫0tI1-βC(t-s)∫0sg(s,τ,u(τ),Dβu(τ))dτds.
Therefore Ku∈FSβ and maps FSβ to FSβ because u0∈Eβ,
DβS(t)u1=ddtI1-βS(t)u1=CDβS(t)u1=I1-βC(t)u1,
and the integral terms are obviously continuous. For u,v∈FSβ, we find
‖(Ku)(t)-(Kv)(t)‖≤∫0t(∫0t-sMeωτdτ)Ag∫0s(‖u(τ)-v(τ)‖+‖Dβu(τ)-Dβv(τ)‖)dτds≤MAgT22(∫0Teωτdτ)(sup0≤t≤T‖u(t)-v(t)‖+sup0≤t≤T‖Dβu(t)-Dβv(t)‖)≤MAgT22(∫0Teωτdτ)‖u(t)-v(t)‖β.
Further,
‖(DβKu)(t)-(DβKv)(t)‖≤∫0t‖I1-βC(t-s)∫0s[g(s,τ,u(τ),Dβu(τ))-g(s,τ,v(τ),Dβv(τ))]dτds‖≤MAg∫0t(t-s)1-βeω(t-s)Γ(2-β)sdssup0≤t≤T(‖u(t)-v(t)‖+‖Dβu(t)-Dβv(t)‖)≤MAgT2-βΓ(2-β)(∫0Teω(T-s)ds)‖u(t)-v(t)‖β.
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).

Acknowledgment

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

BahajM.Remarks on the existence results for second-order differential inclusions with nonlocal conditionsBenchohraM.NtouyasS. K.Existence of mild solutions of second order initial value problems for delay integrodifferential inclusions with nonlocal conditionsBenchohraM.NtouyasS. K.Existence results on the semiinfinite interval for first and second order integrodifferential equations in Banach spaces with nonlocal conditionsBenchohraM.NtouyasS. K.Existence results for multivalued semilinear functional-differential equationsByszewskiL.LakshmikanthamV.Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach spaceHernandezM. E.Existence of solutions to a second order partial differential equation with nonlocal conditionsHernandezM. E.HenríquezH. R.McKibbenM. A.Existence of solutions for second order partial neutral functional differential equationsTravisC. C.WebbG. F.Compactness, regularity, and uniform continuity properties of strongly continuous cosine familiesTravisC. C.WebbG. F.Cosine families and abstract nonlinear second order differential equationsSamkoS. G.KilbasA. A.MarichevO. I.GorenfloR.VessellaS.KilbasA. A.SrivastavaH. M.TrujilloJ. J.MillerK. S.RossB.OldhamK. B.SpanierJ.PodlubnyI.TravisC. C.WebbG. F.An abstract second-order semilinear Volterra integro-differential equation