# Sensitivity of a Fractional Integrodifferential Cauchy Problem of Volterra Type

Authors: Dariusz Idczak; Andrzej Skowron; Stanisław Walczak
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We prove a theorem on the existence and uniqueness of a solution as well as on a sensitivity (i.e., differentiable dependence of a solution on a functional parameter) of a fractional integrodifferential Cauchy problem of Volterra type. The proof of this result is based on a theorem on diffeomorphism between Banach and Hilbert spaces. The main assumption is the Palais-Smale condition.


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## Body

## 1. Introduction

In the paper, we consider the following fractional Integrodifferential Cauchy problem of Volterra type of orderα∈(0,1):
(1)Da+αx(t)=∫atΦ(t,s,x(s))ds+g(t),t∈[a,b]a.e.,Ia+1-αx(a)=0,
where Φ:PΔ×ℝn→ℝn, PΔ={(t,s)∈[0,1]×[0,1];t≥s}, and g∈L2=L2([a,b],ℝn). We consider the existence and uniqueness of a solution in the space Ia+α(L2) (the range of the right-sided integral Riemann-Liouville operator Ia+α:L2→L2) as well as sensitivity, that is, differentiable dependence of a solution on a functional parameter g∈L2.Fractional functional systems, including Integrodifferential ones, have recently been studied by several authors. The reasons for this interest are numerous applications of fractional differential calculus in physics, chemistry, biology, economics, signal processing, image processing, aerodynamics, and so forth. Integrodifferential systems are investigated in finite and infinite dimensional spaces, with Riemann-Liouville and Caputo derivatives as well as with different types of initial and boundary conditions, local, nonlocal, involving values of solutions or their fractional integrals, delay [1–7]. Applied methods also are of different type. They are based on Banach, Brouwer, Schauder, Schaefer, Krasnoselskii fixed point theorems, nonlinear alternative Leray-Schauder type, strongly continuous operator semigroups, the reproducing kernel Hilbert space method, and so forth.We propose a new method for the study problems of type (1), namely, a theorem on diffeomorphism between Banach and Hilbert spaces obtained by the authors in paper [8]. This theorem is based on the Palais-Smale condition. In the mentioned work, an application of this result to study problem of type (1) with α=1 is given. In the paper, we use the line of the proof presented therein. The main difference between cases of α∈(0,1) and α=1 is that, in the first case, the elements of the solution space Ia+α(L2) are not, in general, continuous functions on [a,b] as it is when α=1 (cf. Remark 10).The paper is organized as follows. In the second section, we recall some facts from the fractional calculus and formulate a theorem on diffeomorphism between Banach and Hilbert spaces. Third section is devoted to the existence and uniqueness of a solution as well as sensitivity of problem (1) (Theorem 9). Let us point that Lemma 7 in itself is a general result on the existence and uniqueness of a solution to problem (1) under a Lipschitz condition with respect to the state variable, imposed on the integrand. Strengthening the smoothness assumptions about the integrand and Palais-Smale condition allows us to prove sensitivity of (1).To our best knowledge, sensitivity of fractional systems of type (1) has not been studied by other authors so far.


## 2. Preliminaries

### 2.1. Fractional Calculus

Letα>0, h∈L1=L1([a,b],ℝn). By the left-sided Riemann-Liouville fractional integral of h on the interval [a,b], we mean (cf. [9]) a function Ia+αh given by
(2)(Ia+αh)(t)=1Γ(α)∫ath(τ)(t-τ)1-αdτ,t∈[a,b]a.e.,
where Γ is the Euler function.One can show that the above integral exists and is finite a.e. on[a,b]. Moreover, if h∈Lp=Lp([a,b],ℝn), 1≤p<∞, then Ia+αh∈Lp and
(3)∥Ia+αh∥Lp≤(b-a)αΓ(α+1)∥h∥Lp.In [10], the following useful theorem is proved.Theorem 1.
The operatorIa+α:Lp→Lp, 1≤p<∞, is compact, that is, it maps bounded sets onto relatively compact ones.Now, letα∈(0,1). One says that [9] x∈L1 possesses the left-sided Riemann-Liouville derivative Da+αx of order α∈(0,1) on the interval [a,b], if the integral Ia+1-αx is absolutely continuous on [a,b] (more precisely, if Ia+1-αx has an absolutely continuous representant a.e. on [a,b]). By this derivative one means the classical derivative D1(Ia+1-αx), that is,
(4)(Da+αx)(t)=1Γ(1-α)D1(∫atx(τ)(t-τ)αdτ),t∈[a,b]a.e.One has ([9], Theorem 2.4).Theorem 2.
Ifh∈L1, then Da+αIa+αh=h a.e. on [a,b]. If x∈Ia+α(L1), then Ia+αDa+αx=x a.e. on [a,b].From the second part of the above theorem and (3), it follows that if x∈Ia+α(Lp), then
(5)∥x∥Lp≤(b-a)αΓ(α+1)∥Da+αx∥Lp.Of course,Ia+α(Lp) with the norm
(6)∥x∥Ia+α(Lp)=∥Da+αx∥Lp,x∈Ia+α(Lp)is complete. Whenp=2, the above norm is generated by the following scalar product:
(7)〈x,y〉Ia+α(L2)=∫abDa+αx(t)Da+αy(t)dt,x,y∈Ia+α(L2).Next, one will use the following.Lemma 3.
If a sequence(xn) is weakly convergent in Ia+α(Lp) to some x0, then it is convergent to x0 with respect to the norm in Lp, and the sequence (Da+αxn) is weakly convergent in Lp to Da+αx0.Proof.
To prove the second part of the theorem, it is sufficient to observe that the linear operatorDa+α:Ia+α(Lp)→Lp is continuous. Consequently, it preserves weak convergence. To prove the first part, let us observe that, from Theorem 1, it follows that Ia+α:Lp→Lp maps weakly convergent sequences onto strongly convergent (with respect to the norm) ones. Thus, the sequence (xn)=(Ia+αDa+αxn) is convergent to x0=Ia+αDa+αx0 with respect to the norm in Lp.


### 2.2. A Theorem on a Diffeomorphism

In [8], we proved the following theorem.Theorem 4.
LetX be a real Banach space, let H be a real Hilbert space. If f:X→H is a C1-mapping (i.e., differentiable in Frechet sense on X with the differential f′ continuous on X) such that(α)
for anyy∈H, the functional
(8)φ:X∋x⟼(12)∥f(x)-y∥2∈ℝ
satisfies Palais-Smale condition,(β)
for anyx∈X, f′(x):X→H is “one-one” and “onto”,then f is a diffeomorphism (i.e., it is “one-one”, “onto” and the inverse f-1:H→X is differentiable in Frechet sense).Let us recall that aC1-functional φ:X→ℝ satisfies Palais-Smale condition if any sequence (xk) such that
(9)|φ(xk)|≤M∀k∈ℕ,andsomeM>0,φ′(xk)⟶0,admits a convergent subsequence (here,φ′(xk) is the Frechet differential of φ at xk).


## 2.1. Fractional Calculus

Letα>0, h∈L1=L1([a,b],ℝn). By the left-sided Riemann-Liouville fractional integral of h on the interval [a,b], we mean (cf. [9]) a function Ia+αh given by
(2)(Ia+αh)(t)=1Γ(α)∫ath(τ)(t-τ)1-αdτ,t∈[a,b]a.e.,
where Γ is the Euler function.One can show that the above integral exists and is finite a.e. on[a,b]. Moreover, if h∈Lp=Lp([a,b],ℝn), 1≤p<∞, then Ia+αh∈Lp and
(3)∥Ia+αh∥Lp≤(b-a)αΓ(α+1)∥h∥Lp.In [10], the following useful theorem is proved.Theorem 1.
The operatorIa+α:Lp→Lp, 1≤p<∞, is compact, that is, it maps bounded sets onto relatively compact ones.Now, letα∈(0,1). One says that [9] x∈L1 possesses the left-sided Riemann-Liouville derivative Da+αx of order α∈(0,1) on the interval [a,b], if the integral Ia+1-αx is absolutely continuous on [a,b] (more precisely, if Ia+1-αx has an absolutely continuous representant a.e. on [a,b]). By this derivative one means the classical derivative D1(Ia+1-αx), that is,
(4)(Da+αx)(t)=1Γ(1-α)D1(∫atx(τ)(t-τ)αdτ),t∈[a,b]a.e.One has ([9], Theorem 2.4).Theorem 2.
Ifh∈L1, then Da+αIa+αh=h a.e. on [a,b]. If x∈Ia+α(L1), then Ia+αDa+αx=x a.e. on [a,b].From the second part of the above theorem and (3), it follows that if x∈Ia+α(Lp), then
(5)∥x∥Lp≤(b-a)αΓ(α+1)∥Da+αx∥Lp.Of course,Ia+α(Lp) with the norm
(6)∥x∥Ia+α(Lp)=∥Da+αx∥Lp,x∈Ia+α(Lp)is complete. Whenp=2, the above norm is generated by the following scalar product:
(7)〈x,y〉Ia+α(L2)=∫abDa+αx(t)Da+αy(t)dt,x,y∈Ia+α(L2).Next, one will use the following.Lemma 3.
If a sequence(xn) is weakly convergent in Ia+α(Lp) to some x0, then it is convergent to x0 with respect to the norm in Lp, and the sequence (Da+αxn) is weakly convergent in Lp to Da+αx0.Proof.
To prove the second part of the theorem, it is sufficient to observe that the linear operatorDa+α:Ia+α(Lp)→Lp is continuous. Consequently, it preserves weak convergence. To prove the first part, let us observe that, from Theorem 1, it follows that Ia+α:Lp→Lp maps weakly convergent sequences onto strongly convergent (with respect to the norm) ones. Thus, the sequence (xn)=(Ia+αDa+αxn) is convergent to x0=Ia+αDa+αx0 with respect to the norm in Lp.


## 2.2. A Theorem on a Diffeomorphism

In [8], we proved the following theorem.Theorem 4.
LetX be a real Banach space, let H be a real Hilbert space. If f:X→H is a C1-mapping (i.e., differentiable in Frechet sense on X with the differential f′ continuous on X) such that(α)
for anyy∈H, the functional
(8)φ:X∋x⟼(12)∥f(x)-y∥2∈ℝ
satisfies Palais-Smale condition,(β)
for anyx∈X, f′(x):X→H is “one-one” and “onto”,then f is a diffeomorphism (i.e., it is “one-one”, “onto” and the inverse f-1:H→X is differentiable in Frechet sense).Let us recall that aC1-functional φ:X→ℝ satisfies Palais-Smale condition if any sequence (xk) such that
(9)|φ(xk)|≤M∀k∈ℕ,andsomeM>0,φ′(xk)⟶0,admits a convergent subsequence (here,φ′(xk) is the Frechet differential of φ at xk).


## 3. Main Result

Let us consider problem (1) with g∈L2. We assume that function Φ=(Φ1,…,Φn):PΔ×ℝn→ℝn satisfies the following conditions:(A1)
Φ is measurable in (t,s)∈PΔ and continuously differentiable in Frechet sense in x∈ℝn,(A2)
there exist functionsa, b∈L2(PΔ)=L2(PΔ,ℝ) such that
(10)|Φ(t,s,x)|≤a(t,s)|x|+b(t,s),
for (t,s)∈PΔ a.e., x∈ℝn,
(11)∥a∥L2(PΔ)<Γ(α+1)2(b-a)α,(A3)
there exists a functionc∈L2(PΔ) such that
(12)|Φx(t,s,x)|≤c(t,s),
for (t,s)∈PΔ a.e., x∈ℝn,
(13)∫atc2(t,s)ds≤C,
for t∈[a,b] a.e. and some C>0.We shall show that the operator(14)f:Ia+α(L2)∋x⟼Da+αx(·)-∫a·Φ(·,s,x(s))ds∈L2satisfies assumptions of Theorem4 with the spaces X=Ia+α(L2), Y=L2. Namely, we have the following.Lemma 5.
The operatorf is well-defined C1-mapping with the differential f′(x) at any x∈Ia+α(L2) given by
(15)f′(x):Ia+α(L2)∋h⟼Da+αh(·)-∫a·Φx(·,s,x(s))h(s)ds∈L2,
where Φx is the Jacobi matrix of Φ with respect to x.Proof.
Well-definiteness of  f. Since Φ is the Caratheodory function with respect to (t,s)∈PΔ and x∈ℝn, the function
(16)PΔ∋(t,s)⟼Φ(t,s,x(s))∈ℝn
is measurable. From (A2), it follows that it belongs to L1. The Fubini theorem implies integrability of the function
(17)[a,b]∋t⟼∫atΦ(t,s,x(s))ds∈ℝn.
Moreover,(18)|∫atΦ(t,s,x(s))ds|2≤(∫ata(t,s)|x(s)|ds+∫atb(t,s)ds)2≤((∫atb2(t,s)ds)1/2)(∫ata2(t,s)ds)1/2(∫at|x(s)|2ds)1/2+(∫at12ds)1/2(∫atb2(t,s)ds)1/2)2≤2(∫ata2(t,s)ds)(∫ab|x(s)|2ds)+2(b-a)(∫atb2(t,s)ds).
The right-hand side is integrable on[a,b]. So, function (17) belongs to L2.
Differentiability of  f. Continuous differentiability of the first term of f follows from the linearity and continuity of the operator Da+α:Ia+α(L2)→L2.
So, let one consider the second term, that is, the operator(19)g:Ia+α(L2)∋x⟼∫a·Φ(·,s,x(s))ds∈L2.
One will check that the operator(20)g′(x):Ia+α(L2)∋h⟼∫a·Φx(·,s,x(s))h(s)ds∈L2
is the Frechet differential ofg at x∈Ia+α(L2).
First, let one observe thatg′(x) is well defined. Of course, the function
(21)PΔ∋(t,s)⟼Φx(t,s,x(s))h(s)∈ℝn,
where h∈Ia+α(L2), is measurable. By (A3) it belongs to L1. From the Fubini theorem it follows that the function
(22)[a,b]∋t⟼∫atΦx(t,s,x(s))h(s)ds
is integrable. Moreover, similarly as in the case off,
(23)|∫atΦx(t,s,x(s))h(s)ds|2≤(∫atc(t,s)|h(s)|ds)2≤(∫atc2(t,s)ds)(∫at|h(s)|2ds)≤C(∫ab|h(s)|2ds).
So, function (22) belongs to L2.
Linearity ofg′(x) is obvious. Its continuity follows from the following estimations (cf. (5)):
(24)∥g′(x)h∥L22=∫ab|∫atΦx(t,s,x(s))h(s)ds|2dt≤C(b-a)∥h∥L22≤C(b-a)2α+1Γ(α+1)2∥h∥Ia+α(L2)2.
Now, one will check thatg′(x) is the Gateaux differential of g at x, that is,
(25)limλ⟶0g(x+λh)-g(x)λ=g′(x)h
inL2, for any x, h∈Ia+α(L2). Indeed, let (λk) be a sequence of real numbers converging to 0 and consider the limit
(26)limk⟶∞∫ab|∫at(Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk-Φx(t,s,x(s))h(s)(Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk)ds|∫at(Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk|2dt.
From the differentiability ofΦ with respect to x, it follows that, for t∈[a,b] a.e., the sequence of functions
(27)[a,t]∋s⟼Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk-Φx(t,s,x(s))h(s)∈ℝn
converges pointwise a.e. on[a,t] to the zero function. From the mean value theorem applied to any coordinate function
(28)[0,1]∋τ⟼Φj(t,s,x(s)+τλkh(s))∈ℝ
(j=1,…,n), it follows that functions (27) indexed by k∈ℕ are commonly pointwise (a.e. on [a,t]) bounded by an integrable function (cf. (A3)). So,
(29)∫at(Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk-Φx(t,s,x(s))h(s)(Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk)ds⟶0,
fort∈[a,b] a.e. Moreover, using once again the mean value theorem, one obtains
(30)|∫at‍(Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk-Φx(t,s,x(s))h(s)(Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk)ds|2≤(∫at|Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk-Φx(t,s,x(s))h(s)|Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk|ds(∫at|Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk)2≤(n+1)2(∫at‍c(t,s)|h(s)|ds)2≤(n+1)2∫atc2(t,s)ds∫ab|h(s)|2ds≤(n+1)2C∥h∥L2.
Consequently,(31)limk⟶∞∫ab|∫at(Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk-Φx(t,s,x(s))h(s)Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk)ds|∫at(Φ(t,s,x(s)+λkh(s))-Φ(t,s,x(s))λk|2dt=0,
that is, (25) holds true.
Continuity of
g
′. Let (xk) be a sequence converging in Ia+α(L2) to some x0. Similarly, as mentioned above, one obtains
(32)∥(g′(xk)-g′(x0))h∥L22=∫ab|∫at(-Φx(t,s,x0(s)))Φx(t,s,xk(s))-Φx(t,s,x0(s)))h(s)ds|∫at(Φx(t,s,xk(s))|2dt≤∫ab(∫at|Φx(t,s,xk(s))-Φx(t,s,x0(s))|Φx(t,s,xk(s))||h(s)|ds(∫at|Φx(t,s,xk(s)))2dt≤∫ab∫at|Φx(t,s,xk(s))-Φx(t,s,x0(s))|2dsdt(∫ab|h(s)|2ds)≤((b-a)αΓ(α+1))2×∫ab∫at|Φx(t,s,xk(s))-Φx(t,s,x0(s))|2dsdt×∥h∥Ia+α(L2)2
for anyh∈L2. Convergence
(33)∫ab∫at|Φx(t,s,xk(s))-Φx(t,s,x0(s))|2dsdt⟶0
follows from (A3), the Krasnoselskii theorem on the continuity of the Nemytskii operator and from (5).
So,g being continuously differentiable in Gateaux sense is continuously differentiable in Frechet sense.Now, one will prove.Lemma 6.
For any fixedy∈L2, the functional
(34)φ:Ia+α(L2)∋x⟼12∥-∫a·‍Φ(·,s,x(s))ds-y(·)∥L22Da+αx(·)-∫a·‍Φ(·,s,x(s))ds-y(·)∥L22∈ℝ
satisfies Palais-Smale condition.Proof.
It is easy to see that(35)φ(x)≥12∥x∥Ia+α(L2)2+12∥y∥L22-φ1(x)+φ2(x)-φ3(x),
forx∈Ia+α(L2), where
(36)φ1(x)=∫ab(Da+αx(t))y(t)dt,φ2(x)=∫ab(Da+αx(t))∫atΦ(t,s,x(s))dsdt,φ3(x)=∫aby(t)∫atΦ(t,s,x(s))dsdt.
Of course,(37)|φ1(x)|≤∥y∥L2∥x∥Ia+α(L2).
Moreover,(38)|φ2(x)|≤∫ab|Da+αx(t)|∫at(a(t,s)|x(s)|+b(t,s))dsdt≤∫ab|Da+αx(t)|((∫ata2(t,s)ds)1/2×(∫at|x(s)|2ds)1/2+(b-a)1/2×(∫atb2(t,s)ds)1/2)dt≤∥x∥L2∥x∥Ia+α(L2)∥a∥L2(PΔ)+(b-a)1/2∥x∥Ia+α(L2)∥b∥L2(PΔ)≤(b-a)αΓ(α+1)∥a∥L2(PΔ)∥x∥Ia+α(L2)2+(b-a)1/2∥x∥Ia+α(L2)∥b∥L2(PΔ),(39)|φ3(x)|≤∫ab|y(t)|∫at(a(t,s)|x(s)|+b(t,s))dsdt≤(b-a)αΓ(α+1)∥x∥Ia+α(L2)∥y∥L2∥a∥L2(PΔ)+(b-a)1/2∥y∥L2∥b∥L2(PΔ).
So,(40)φ(x)≥12∥x∥Ia+α(L2)2+12∥y∥L22-∥y∥L2∥x∥Ia+α(L2)-(b-a)αΓ(α+1)∥a∥L2(PΔ)∥x∥Ia+α(L2)2-(b-a)1/2∥x∥Ia+α(L2)∥b∥L2(PΔ)-(b-a)αΓ(α+1)∥x∥Ia+α(L2)∥y∥L2∥a∥L2(PΔ)-(b-a)1/2∥y∥L2∥b∥L2(PΔ)=d0∥x∥Ia+α(L2)2+d1∥x∥Ia+α(L2)+d2,
forx∈Ia+α(L2), where
(41)d0=(12-(b-a)αΓ(α+1)∥a∥L2(PΔ)),d1=-∥y∥L2-(b-a)1/2∥b∥L2(PΔ)-(b-a)αΓ(α+1)∥y∥L2∥a∥L2(PΔ),d2=12∥y∥L22-(b-a)1/2∥y∥L2∥b∥L2(PΔ).
Since,d0>0 (by (A1)), therefore φ is coercive, that is, φ(x)→∞ as ∥x∥Ia+α(L2)→∞.
Let us fix a sequence(xk)⊂Ia+α(L2) such that
(42)|φ(xk)|≤M∀k∈ℕandsomeM>0,φ′(xk)⟶0.
The first condition and coercivity of φ imply that the sequence (xk) is bounded. So, without loss of the generality, we may assume that it weakly converges in Ia+α(L2) to some x0. Lemma 5 implies that φ is of class C1 and
(43)φ′(x)h=∫ab‍Da+αx(t)Da+αh(t)dt-∫ab‍y(t)Da+αh(t)dt-∫ab‍Da+αh(t)∫at‍Φ(t,s,x(s))dsdt-∫ab‍Da+αx(t)∫at‍Φx(t,s,x(s))h(s)dsdt+∫ab‍y(t)∫at‍Φx(t,s,x(s))h(s)dsdt+∫ab‍(∫at‍Φ(t,s,x(s))ds)×(∫at‍Φx(t,s,x(s))h(s)ds)dtو
forx, h∈Ia+α(L2). Consequently,
(44)(φ′(xk)-φ′(x0))(xk-x0)=∥xk-x0∥Ia+α(L2)2+∑i=16‍ψi(xk),
where
(45)ψ1(xk)=-∫ab(Da+αxk(t)-Da+αx0(t))×(∫at(-Φ(t,s,x0(s)))Φ(t,s,xk(s))-Φ(t,s,x0(s)))ds(∫at(Φ(t,s,xk(s)))dt,(46)ψ2(xk)=-∫abDa+αxk(t)×(∫atΦx(t,s,xk(s))(xk(s)-x0(s))ds)dt,(47)ψ3(xk)=∫abDa+αx0(t)×(∫atΦx(t,s,x0(s))(xk(s)-x0(s))ds)dt,(48)ψ4(xk)=∫aby(t)×(∫ab(Φx(t,s,xk(s))-Φx(t,s,x0(s)))×(xk(s)-x0(s))ds(∫aby(t))dt,(49)ψ5(xk)=∫ab(∫at‍Φ(t,s,xk(s))ds)×(∫atΦx(t,s,xk(s))(xk(s)-x0(s))ds)dt,(50)ψ6(xk)=-∫ab(∫atΦ(t,s,x0(s))ds)×(∫atΦx(t,s,x0(s))(xk(s)-x0(s))ds)dt.
The left-hand side converges to 0 because(51)(φ′(xk)-φ′(x0))(xk-x0)=φ′(xk)(xk-x0)-φ′(x0)(xk-x0)
andφ′(xk)→0 as well as xk⇀x0 weakly in Ia+α(L2). Terms ψi(xk), i=1,…,6, also converge to 0. This follows from the strong convergence of the sequence (xk) to x0 in L2 and weak convergence of the sequence (Da+αxk(t)) to Da+αx0(t) in L2 (cf. Lemma 3) as well as from the Krasnoselskii theorem on the continuity of the Nemytskii operator.
Indeed, from the Krasnoselskii theorem, it follows that the sequence(52)(∫a·(Φ(·,s,xk(s))-Φ(·,s,x0(s)))ds)
converges pointwise a.e. on[a,b] to the zero function. Moreover, in the same way as in the proof of Lemma 5, one can check that the sequence
(53)(|∫a·‍(Φ(·,s,xk(s))-Φ(·,s,x0(s)))ds|2)
is bounded on[a,b] by an integrable function. This means that
(54)∫ab|∫at(Φ(t,s,xk(s))-Φ(t,s,x0(s)))ds|2dt⟶0,
that is, the sequence(∫a·‍(Φ(·,s,xk(s))-Φ(·,s,x0(s)))ds) converges in L2 to the zero function.
Similarly, ifχk(t,s), k∈ℕ, are functions belonging to L2(PΔ), commonly bounded on PΔ by a function χ∈L2(PΔ), then the sequence
(55)(∫a·χk(·,s)(xk(s)-x0(s))ds)
converges pointwise a.e. on [a,b] to the zero function. Moreover,
(56)|∫at‍χ(t,s)(xk(s)-x0(s))ds|2≤(∫at|χk(t,s)||(xk(s)-x0(s))|ds)2≤(∫at|χ(t,s)|2ds)(∫ab|xk(s)-x0(s)|2ds)≤const(∫at|χ(t,s)|2ds),
where const is a constant which bounds the sequence (xk) in L2. So, sequence (55) converges in L2 to the zero function. Applying these facts to the functions
(57)χk(t,s)=Φx(t,s,xk(s)),χk(t,s)=Φx(t,s,x0(s)),χk(t,s)=Φx(t,s,xk(s))-Φx(t,s,x0(s)),
and using (54) one asserts that ψi(xk)→0, for i=1,…,6.
Consequently,∥xk-x0∥Ia+α(L2)2→0, that is, φ satisfies Palais-Smale condition.Now, one will show thatf satisfies assumption (β) from Theorem 4. More precisely, one will prove a more general result, namely.Lemma 7.
IfΨ=Ψ(t,s,h):PΔ×ℝn→ℝnis(B1)
measurable in(t,s)∈PΔ,(B2)
there exists a functiond∈L2(PΔ) such that
(58)|Ψ(t,s,h1)-Ψ(t,s,h2)|≤d(t,s)|h1-h2|
for (t,s)∈PΔ a.e., h1, h2∈ℝn,
(59)∫atd2(t,s)ds≤D
for t∈[a,b] a.e. and some D>0,(B3)
Ψ(·,·,0)∈L2(PΔ),
then the operator(60)Λ:Ia+α(L2)∋h⟼Da+αh(·)-∫a·Ψ(·,s,h(s))ds∈L2
is well defined, “one-one” and “onto”.Proof.
First, let us observe thatΛ is well defined. Indeed, for any h∈L2 (in particular, for h∈Ia+α(L2)), one has
(61)|∫atΨ(t,s,h(s))ds|≤∫at|Ψ(t,s,h(s))-Ψ(t,s,0)|ds+∫at|Ψ(t,s,0)|ds≤∫atd(t,s)|h(s)|ds+∫at|Ψ(t,s,0)|ds≤(∫atd2(t,s)ds)1/2(∫ab|h(s)|2ds)1/2+∫at|Ψ(t,s,0)|ds,
for t∈[a,b] a.e., and the right-hand side belongs to L2.
Now, let one consider some auxiliary problem(62)Da+αh(t)=v(t),t∈[a,b]a.e.,Ia+1-αh(a)=0.
with a fixedv∈L2. Of course, problem (62) has a unique solution hv(t)=Ia+αv(t) in the space Ia+α(L2) (cf. [11]).
To end the proof, it is sufficient to show that the operator(63)K:L2∋v⟼∫a·Ψ(·,s,hv(s))ds+g(·)∈L2
with any fixedg∈L2 possesses a unique fixed point.
One will show that there exist constantsκ∈(0,1), l∈ℕ such that
(64)∥Kv1-Kv2∥l≤κ∥v1-v2∥,
for anyv1, v2∈L2, where ∥·∥l is the Bielecki norm in L2 given by
(65)∥v∥l2=∫abe-lt|v(t)|2dt,v∈L2.
Indeed, one has(66)∥Kv1-Kv2∥l2=∫abe-lt|Kv1(t)-Kv2(t)|2dt≤∫abe-lt(∫at|Ψ(t,s,hv1(s))-Ψ(t,s,hv2(s))|ds)2dt≤∫abe-lt(∫atd(t,s)|hv1(s)-hv2(s)|ds)2dt=∫abe-lt(∫atd(t,s)|Ia+α(v1-v2)(s)|ds)2dt≤∫abe-lt∫atd2(t,s)ds∫at|Ia+α(v1-v2)(s)|2dsdt≤D((b-a)αΓ(α+1))2∫abe-lt∫at|(v1-v2)(s)|2dsdt≤D((b-a)αΓ(α+1))21l∫abe-lt|(v1-v2)(t)|2dt=Dl((b-a)αΓ(α+1))2∥v1-v2∥l2,
forv1, v2∈L2. It is sufficient to choose l∈ℕ, such that D/k((b-a)α/Γ(α+1))<1.Applying Lemma7 to the function
(67)Ψ(t,s,h)=Φx(t,s,x(s))hwith a fixedx∈Ia+α(L2), one obtains the following.Lemma 8.
Operatorf satisfies (β).Theorem4 and Lemmas 5, 6, and 8 imply the following.Theorem 9.
Problem (1) possesses a unique solution xg∈Ia+α(L2), for any g∈L2, and the operator
(68)L2∋g⟼xg∈Ia+α(L2)
is differentiable in Frechet sense.Remark 10.
Whenα∈(1/2,1), all elements of Ia+α(L2) are continuous (cf. [9], Theorem 3.6). Consequently, growth condition (A3) can be slightly weakened just like in [8].


## 4. Conclusions

In the paper, sensitivity of a fractional Integrodifferential Cauchy problem of Volterra type has been investigated. Namely, it has been proved that problem (1) possesses (under the appropriate assumptions) a unique solution xg∈Ia+α(L2) for any fixed functional parameter g∈L2 and the dependence (68) is differentiable in Frechet sense. In the next paper, sensitivity of such a problem with an integral term of Fredholm type as well as of a problem containing the both terms will be considered.



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*Source: 129478-2013-10-21.xml*