Sensitivity of a Fractional Integrodifferential Cauchy Problem of Volterra Type

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.

1. Introduction

In the paper, we consider the following fractional Integrodifferential Cauchy problem of Volterra type of order 𝛼∈(0,1): 𝐷^𝛼_𝑎+𝑥(𝑡)=^𝑡_𝑎Φ((𝑠(𝑡),[]𝐼𝑡,𝑠,𝑥))𝑑𝑠+𝑔𝑡∈𝑎,𝑏a.e.,_{1−𝛼𝑎+}𝑥(𝑎)=0, (1) where Φ∶𝑃_Δ×ℝ^𝑛→ℝ^𝑛, 𝑃_Δ={(𝑡,𝑠)∈[0,1]×[0,1];𝑡≥𝑠}, and 𝑔∈𝐿²=𝐿²([𝑎,𝑏],ℝ^𝑛). We consider the existence and uniqueness of a solution in the space 𝐼^𝛼_𝑎+(𝐿²) (the range of the right-sided integral Riemann-Liouville operator 𝐼^𝛼_𝑎+∶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 𝐼^𝛼_𝑎+(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

Let 𝛼>0, ℎ∈𝐿¹=𝐿¹([𝑎,𝑏],ℝ^𝑛). By the left-sided Riemann-Liouville fractional integral of ℎ on the interval [𝑎,𝑏], we mean (cf. [9]) a function 𝐼^𝛼_𝑎+ℎ given by 𝐼^𝛼_𝑎+ℎ𝑡)=1(Γ(𝛼)^𝑡_𝑎ℎ(𝜏)(𝑡−𝜏)^1−𝛼𝑑𝜏,𝑡∈[𝑎,𝑏]a.e., (2) 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∈𝐿^𝑝=𝐿^𝑝([𝑎,𝑏],ℝ^𝑛), 1≤𝑝<∞, then 𝐼^𝛼_𝑎+ℎ∈𝐿^𝑝 and 𝐼^𝛼_𝑎+ℎ_𝐿𝑝≤(𝑏−𝑎)^𝛼Γ()‖ℎ‖𝛼+1^𝐿𝑝. (3)

In [10], the following useful theorem is proved.

Theorem 1. The operator 𝐼^𝛼_𝑎+∶𝐿^𝑝→𝐿^𝑝, 1≤𝑝<∞, is compact, that is, it maps bounded sets onto relatively compact ones.

Now, let 𝛼∈(0,1). One says that [9] 𝑥∈𝐿¹ possesses the left-sided Riemann-Liouville derivative 𝐷^𝛼_𝑎+𝑥 of order 𝛼∈(0,1) on the interval [𝑎,𝑏], if the integral 𝐼^1−𝛼_𝑎+𝑥 is absolutely continuous on [𝑎,𝑏] (more precisely, if 𝐼^1−𝛼_𝑎+𝑥 has an absolutely continuous representant a.e. on [𝑎,𝑏]). By this derivative one means the classical derivative 𝐷¹(𝐼^1−𝛼_𝑎+𝑥), that is, 𝐷^𝛼_𝑎+𝑥𝑡)=1(Γ()𝐷1−𝛼^1𝑡_𝑎𝑥(𝜏)(𝑡−𝜏)^𝛼,[]𝑑𝜏𝑡∈𝑎,𝑏a.e. (4)

One has ([9], Theorem 2.4).

Theorem 2. If ℎ∈𝐿¹, then 𝐷^𝛼_𝑎+𝐼^𝛼_𝑎+ℎ=ℎ a.e. on [𝑎,𝑏]. If 𝑥∈𝐼^𝛼_𝑎+(𝐿¹), then 𝐼^𝛼_𝑎+𝐷^𝛼_𝑎+𝑥=𝑥 a.e. on [𝑎,𝑏].

From the second part of the above theorem and (3), it follows that if 𝑥∈𝐼^𝛼_𝑎+(𝐿^𝑝), then ‖𝑥‖_𝐿^𝑝≤(𝑏−𝑎)^𝛼Γ()𝐷𝛼+1^𝛼𝑎+𝑥_𝐿𝑝. (5)

Of course, 𝐼^𝛼_𝑎+(𝐿^𝑝) with the norm ‖𝑥‖_𝐼^𝛼_𝑎+(𝐿^𝑝₎=𝐷^𝛼_𝑎+𝑥_𝐿𝑝,𝑥∈𝐼^𝛼_𝑎+𝐿^𝑝 (6)

is complete. When 𝑝=2, the above norm is generated by the following scalar product: 𝑥,𝑦_{𝐼𝛼𝑎+(𝐿2)}=^𝑏_𝑎𝐷^𝛼_𝑎+𝑥(𝑡)𝐷^𝛼_𝑎+𝑦(𝑡)𝑑𝑡,𝑥,𝑦∈𝐼^𝛼_𝑎+𝐿². (7)

Next, one will use the following.

Lemma 3. If a sequence (𝑥_𝑛) is weakly convergent in 𝐼^𝛼_𝑎+(𝐿^𝑝) to some 𝑥₀, then it is convergent to 𝑥₀ with respect to the norm in 𝐿^𝑝, and the sequence (𝐷^𝛼_𝑎+𝑥_𝑛) is weakly convergent in 𝐿^𝑝 to 𝐷^𝛼_𝑎+𝑥₀.

Proof. To prove the second part of the theorem, it is sufficient to observe that the linear operator 𝐷^𝛼_𝑎+∶𝐼^𝛼_𝑎+(𝐿^𝑝)→𝐿^𝑝 is continuous. Consequently, it preserves weak convergence. To prove the first part, let us observe that, from Theorem 1, it follows that 𝐼^𝛼_𝑎+∶𝐿^𝑝→𝐿^𝑝 maps weakly convergent sequences onto strongly convergent (with respect to the norm) ones. Thus, the sequence (𝑥_𝑛)=(𝐼^𝛼_𝑎+𝐷^𝛼_𝑎+𝑥_𝑛) is convergent to 𝑥₀=𝐼^𝛼_𝑎+𝐷^𝛼_𝑎+𝑥₀ with respect to the norm in 𝐿^𝑝.

2.2. A Theorem on a Diffeomorphism

In [8], we proved the following theorem.

Theorem 4. Let 𝑋 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 any 𝑦∈𝐻, the functional 1𝜑∶𝑋∋𝑥⟼2𝑓(𝑥)−𝑦²∈ℝ (8) satisfies Palais-Smale condition,(𝛽) for any 𝑥∈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 a 𝐶¹-functional 𝜑∶𝑋→ℝ satisfies Palais-Smale condition if any sequence (𝑥_𝑘) such that 𝜑𝑥_𝑘𝜑≤𝑀∀𝑘∈ℕ,andsome𝑀>0,_′𝑥_𝑘⟶0, (9)

admits a convergent subsequence (here, 𝜑^′(𝑥_𝑘) is the Frechet differential of 𝜑 at 𝑥_𝑘).

3. Main Result

Let us consider problem (1) with 𝑔∈𝐿². We assume that function Φ=(Φ₁,…,Φ_𝑛)∶𝑃_Δ×ℝ^𝑛→ℝ^𝑛 satisfies the following conditions:(A₁)Φ is measurable in (𝑡,𝑠)∈𝑃_Δ and continuously differentiable in Frechet sense in 𝑥∈ℝ^𝑛,(A₂) there exist functions 𝑎, 𝑏∈𝐿²(𝑃_Δ)=𝐿²(𝑃_Δ,ℝ) such that |Φ(𝑡,𝑠,𝑥)|≤𝑎(𝑡,𝑠)|𝑥|+𝑏(𝑡,𝑠), (10) for (𝑡,𝑠)∈𝑃_Δ a.e., 𝑥∈ℝ^𝑛, ‖𝑎‖_𝐿²_(𝑃Δ)<Γ(𝛼+1)2(𝑏−𝑎)^𝛼, (11)(A₃) there exists a function 𝑐∈𝐿²(𝑃_Δ) such that Φ_𝑥(𝑡,𝑠,𝑥)≤𝑐(𝑡,𝑠), (12) for (𝑡,𝑠)∈𝑃_Δ a.e., 𝑥∈ℝ^𝑛, ^𝑡_𝑎𝑐²(𝑡,𝑠)𝑑𝑠≤𝐶, (13) for 𝑡∈[𝑎,𝑏] a.e. and some 𝐶>0.

We shall show that the operator 𝑓∶𝐼^𝛼_𝑎+𝐿²∋𝑥⟼𝐷^𝛼_𝑎+𝑥(⋅)−^⋅_𝑎Φ(⋅,𝑠,𝑥(𝑠))𝑑𝑠∈𝐿² (14)

satisfies assumptions of Theorem 4 with the spaces 𝑋=𝐼^𝛼_𝑎+(𝐿²), 𝑌=𝐿². Namely, we have the following.

Lemma 5. The operator 𝑓 is well-defined 𝐶¹-mapping with the differential 𝑓^′(𝑥) at any 𝑥∈𝐼^𝛼_𝑎+(𝐿²) given by 𝑓^′(𝑥)∶𝐼^𝛼_𝑎+𝐿²∋ℎ⟼𝐷^𝛼_𝑎+ℎ(⋅)−^⋅_𝑎Φ_𝑥(⋅,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠∈𝐿², (15) where Φ_𝑥 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)∈𝑃_Δ and 𝑥∈ℝ^𝑛, the function 𝑃_Δ∋(𝑡,𝑠)⟼Φ(𝑡,𝑠,𝑥(𝑠))∈ℝ^𝑛 (16)
is measurable. From (A₂), it follows that it belongs to 𝐿¹. The Fubini theorem implies integrability of the function [𝑎,𝑏]∋𝑡⟼^𝑡_𝑎Φ(𝑡,𝑠,𝑥(𝑠))𝑑𝑠∈ℝ^𝑛. (17)
Moreover, ^𝑡_𝑎Φ(𝑡,𝑠,𝑥(𝑠))𝑑𝑠²≤^𝑡_𝑎𝑎(𝑡,𝑠)|𝑥(𝑠)|𝑑𝑠+^𝑡_𝑎𝑏(𝑡,𝑠)𝑑𝑠²≤^𝑡_𝑎𝑎²(𝑡,𝑠)𝑑𝑠^1/2𝑡_𝑎|𝑥(𝑠)|²𝑑𝑠^1/2+^𝑡_𝑎1²𝑑𝑠^1/2𝑡_𝑎𝑏²(𝑡,𝑠)𝑑𝑠^1/22≤2^𝑡_𝑎𝑎²(𝑡,𝑠)𝑑𝑠^𝑏_𝑎|𝑥(𝑠)|²()𝑑𝑠+2𝑏−𝑎^𝑡_𝑎𝑏²(𝑡,𝑠)𝑑𝑠. (18)
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 𝐷^𝛼_𝑎+∶𝐼^𝛼_𝑎+(𝐿²)→𝐿².
So, let one consider the second term, that is, the operator 𝑔∶𝐼^𝛼_𝑎+𝐿²∋𝑥⟼^⋅_𝑎Φ(⋅,𝑠,𝑥(𝑠))𝑑𝑠∈𝐿². (19)
One will check that the operator 𝑔^′(𝑥)∶𝐼^𝛼_𝑎+𝐿²∋ℎ⟼^⋅_𝑎Φ_𝑥(⋅,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠∈𝐿² (20)
is the Frechet differential of 𝑔 at 𝑥∈𝐼^𝛼_𝑎+(𝐿²).
First, let one observe that 𝑔^′(𝑥) is well defined. Of course, the function 𝑃_Δ∋(𝑡,𝑠)⟼Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)∈ℝ^𝑛, (21) where ℎ∈𝐼^𝛼_𝑎+(𝐿²), is measurable. By (A₃) it belongs to 𝐿¹. From the Fubini theorem it follows that the function [𝑎,𝑏]∋𝑡⟼^𝑡_𝑎Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠 (22)
is integrable. Moreover, similarly as in the case of 𝑓, ^𝑡_𝑎Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠²≤^𝑡_𝑎𝑐(𝑡,𝑠)|ℎ(𝑠)|𝑑𝑠²≤^𝑡_𝑎𝑐²(𝑡,𝑠)𝑑𝑠^𝑡_𝑎|ℎ(𝑠)|²𝑑𝑠≤𝐶^𝑏_𝑎|ℎ(𝑠)|²𝑑𝑠. (23)
So, function (22) belongs to 𝐿².
Linearity of 𝑔^′(𝑥) is obvious. Its continuity follows from the following estimations (cf. (5)): 𝑔^′(𝑥)ℎ²_𝐿2=^𝑏_𝑎^𝑡_𝑎Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠²(ℎ‖𝑑𝑡≤𝐶𝑏−𝑎)‖²_𝐿2()≤𝐶𝑏−𝑎^2𝛼+1Γ(𝛼+1)²‖ℎ‖²_{𝐼𝛼𝑎+𝐿2}. (24)
Now, one will check that 𝑔^′(𝑥) is the Gateaux differential of 𝑔 at 𝑥, that is, lim_𝜆⟶0𝑔(𝑥+𝜆ℎ)−𝑔(𝑥)𝜆=𝑔^′(𝑥)ℎ (25)
in 𝐿², for any 𝑥, ℎ∈𝐼^𝛼_𝑎+(𝐿²). Indeed, let (𝜆_𝑘) be a sequence of real numbers converging to 0 and consider the limit lim_𝑘⟶∞^𝑏_𝑎^𝑡_𝑎Φ𝑡,𝑠,𝑥(𝑠)+𝜆_𝑘ℎ(𝑠)−Φ(𝑡,𝑠,𝑥(𝑠))𝜆_𝑘−Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠²𝑑𝑡. (26)
From the differentiability of Φ with respect to 𝑥, it follows that, for 𝑡∈[𝑎,𝑏] a.e., the sequence of functions []Φ(𝑠)𝑎,𝑡∋𝑠⟼𝑡,𝑠,𝑥+𝜆_𝑘ℎ(𝑠)−Φ(𝑡,𝑠,𝑥(𝑠))𝜆_𝑘−Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)∈ℝ^𝑛 (27)

converges pointwise a.e. on [𝑎,𝑡] to the zero function. From the mean value theorem applied to any coordinate function [0,1]∋𝜏⟼Φ_𝑗𝑡,𝑠,𝑥(𝑠)+𝜏𝜆_𝑘ℎ(𝑠)∈ℝ (28)
(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, ^𝑡_𝑎Φ𝑡,𝑠,𝑥(𝑠)+𝜆_𝑘ℎ(𝑠)−Φ(𝑡,𝑠,𝑥(𝑠))𝜆_𝑘−Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠⟶0, (29)
for 𝑡∈[𝑎,𝑏] a.e. Moreover, using once again the mean value theorem, one obtains ^𝑡_𝑎Φ𝑡,𝑠,𝑥(𝑠)+𝜆_𝑘ℎ(𝑠)−Φ(𝑡,𝑠,𝑥(𝑠))𝜆_𝑘−Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠²≤^𝑡_𝑎Φ𝑡,𝑠,𝑥(𝑠)+𝜆_𝑘ℎ(𝑠)−Φ(𝑡,𝑠,𝑥(𝑠))𝜆_𝑘−Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠²≤(𝑛+1)^2𝑡_𝑎𝑐(𝑡,𝑠)|ℎ(𝑠)|𝑑𝑠²≤(𝑛+1)^2𝑡_𝑎𝑐²(𝑡,𝑠)𝑑𝑠^𝑏_𝑎|ℎ(𝑠)|²≤()𝑑𝑠𝑛+1²𝐶‖ℎ‖_𝐿². (30)
Consequently, lim_𝑘⟶∞^𝑏_𝑎^𝑡_𝑎Φ𝑡,𝑠,𝑥(𝑠)+𝜆_𝑘ℎ(𝑠)−Φ(𝑡,𝑠,𝑥(𝑠))𝜆_𝑘−Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠²𝑑𝑡=0, (31)
that is, (25) holds true.
Continuity of 𝑔^′. Let (𝑥_𝑘) be a sequence converging in 𝐼^𝛼_𝑎+(𝐿²) to some 𝑥₀. Similarly, as mentioned above, one obtains 𝑔^′𝑥_𝑘−𝑔^′𝑥₀ℎ²_𝐿2=^𝑏_𝑎^𝑡_𝑎Φ_𝑥𝑡,𝑠,𝑥_𝑘(𝑠)−Φ_𝑥𝑡,𝑠,𝑥₀(𝑠)ℎ(𝑠)𝑑𝑠²𝑑𝑡≤^𝑏_𝑎^𝑡_𝑎Φ_𝑥𝑡,𝑠,𝑥_𝑘(𝑠)−Φ_𝑥𝑡,𝑠,𝑥₀(𝑠)|ℎ(𝑠)|𝑑𝑠²𝑑𝑡≤^𝑏_𝑎^𝑡_𝑎Φ_𝑥𝑡,𝑠,𝑥_𝑘(𝑠)−Φ_𝑥𝑡,𝑠,𝑥₀(𝑠)²𝑑𝑠𝑑𝑡^𝑏_𝑎|ℎ(𝑠)|²≤()𝑑𝑠𝑏−𝑎^𝛼Γ(𝛼+1)²×^𝑏_𝑎^𝑡_𝑎Φ_𝑥𝑡,𝑠,𝑥_𝑘(𝑠)−Φ_𝑥𝑡,𝑠,𝑥₀(𝑠)²×‖ℎ‖𝑑𝑠𝑑𝑡_{2𝐼𝛼𝑎+𝐿2} (32)
for any ℎ∈𝐿². Convergence ^𝑏_𝑎^𝑡_𝑎Φ_𝑥𝑡,𝑠,𝑥_𝑘(𝑠)−Φ_𝑥𝑡,𝑠,𝑥₀(𝑠)²𝑑𝑠𝑑𝑡⟶0 (33)
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 fixed 𝑦∈𝐿², the functional 𝜑∶𝐼^𝛼_𝑎+𝐿²1∋𝑥⟼2𝐷^𝛼_𝑎+𝑥(⋅)−^⋅_𝑎Φ(⋅,𝑠,𝑥(𝑠))𝑑𝑠−𝑦(⋅)²_𝐿2∈ℝ (34) satisfies Palais-Smale condition.

Proof. It is easy to see that 𝜑(𝑥)≥12‖𝑥‖²_{𝐼𝛼𝑎+𝐿2}+12𝑦²_𝐿2−𝜑₁(𝑥)+𝜑₂(𝑥)−𝜑₃(𝑥), (35)
for 𝑥∈𝐼^𝛼_𝑎+(𝐿²), where 𝜑₁(𝑥)=^𝑏_𝑎𝐷^𝛼_𝑎+𝑥(𝑡)𝑦(𝑡)𝜑𝑑𝑡,₂(𝑥)=^𝑏_𝑎𝐷^𝛼_𝑎+𝑥(𝑡)^𝑡_𝑎Φ((𝑠𝜑𝑡,𝑠,𝑥))𝑑𝑠𝑑𝑡,₃(𝑥)=^𝑏_𝑎𝑦(𝑡)^𝑡_𝑎Φ(𝑡,𝑠,𝑥(𝑠))𝑑𝑠𝑑𝑡. (36)
Of course, 𝜑₁(𝑥)≤𝑦_𝐿2‖𝑥‖_𝐼^𝛼_𝑎+(𝐿²₎. (37)
Moreover, 𝜑₂(𝑥)≤^𝑏_𝑎𝐷^𝛼_𝑎+𝑥(𝑡)^𝑡_𝑎(𝑎(𝑥(𝑠(≤𝑡,𝑠)|)|+𝑏𝑡,𝑠))𝑑𝑠𝑑𝑡_𝑏𝑎𝐷^𝛼_𝑎+𝑥(𝑡)^𝑡_𝑎𝑎²(𝑡,𝑠)𝑑𝑠^1/2×^𝑡_𝑎|𝑥(𝑠)|²𝑑𝑠^1/2+(𝑏−𝑎)^1/2×^𝑡_𝑎𝑏²(𝑡,𝑠)𝑑𝑠^1/2≤‖𝑥‖𝑑𝑡_𝐿2‖𝑥‖_𝐼^𝛼_𝑎+(𝐿²₎‖𝑎‖_𝐿²_(𝑃Δ)+(𝑏−𝑎)^1/2‖𝑥‖_𝐼^𝛼_𝑎+(𝐿²₎‖𝑏‖_𝐿²_(𝑃Δ)≤(𝑏−𝑎)^𝛼Γ()‖𝑎‖𝛼+1^{𝐿2(𝑃Δ)}‖𝑥‖²_{𝐼𝛼𝑎+𝐿2}+(𝑏−𝑎)^1/2‖𝑥‖_𝐼^𝛼_𝑎+(𝐿²₎‖𝑏‖_𝐿²_(𝑃Δ),𝜑₃(𝑥≤)_𝑏𝑎𝑦(𝑡)^𝑡_𝑎(𝑎(𝑥(𝑠(≤()𝑡,𝑠)|)|+𝑏𝑡,𝑠))𝑑𝑠𝑑𝑡𝑏−𝑎^𝛼Γ()‖𝑥‖𝛼+1^{𝐼𝛼𝑎+(𝐿2)}𝑦_𝐿2‖𝑎‖_𝐿²_(𝑃Δ)+(𝑏−𝑎)^1/2𝑦_𝐿2‖𝑏‖_𝐿²_(𝑃Δ). (38)(39)
So, 𝜑(𝑥)≥12‖𝑥‖²_{𝐼𝛼𝑎+𝐿2}+12𝑦²_𝐿2−𝑦_𝐿2‖𝑥‖_𝐼^𝛼_𝑎+(𝐿²₎−(𝑏−𝑎)^𝛼Γ()‖𝑎‖𝛼+1^{𝐿2(𝑃Δ)}‖𝑥‖²_{𝐼𝛼𝑎+𝐿2}−(𝑏−𝑎)^1/2‖𝑥‖_𝐼^𝛼_𝑎+(𝐿²₎‖𝑏‖_𝐿²_(𝑃Δ)−(𝑏−𝑎)^𝛼Γ()‖𝑥‖𝛼+1^{𝐼𝛼𝑎+(𝐿2)}𝑦_𝐿2‖𝑎‖_𝐿²_(𝑃Δ)−(𝑏−𝑎)^1/2𝑦_𝐿2‖𝑏‖_𝐿²_(𝑃Δ)=𝑑₀‖𝑥‖²_{𝐼𝛼𝑎+𝐿2}+𝑑₁‖𝑥‖_𝐼^𝛼_𝑎+(𝐿²₎+𝑑₂, (40)
for 𝑥∈𝐼^𝛼_𝑎+(𝐿²), where 𝑑₀=12−(𝑏−𝑎)^𝛼Γ()‖𝑎‖𝛼+1^{𝐿2(𝑃Δ)},𝑑₁=−𝑦_𝐿2−(𝑏−𝑎)^1/2‖𝑏‖_𝐿²_(𝑃Δ)−(𝑏−𝑎)^𝛼Γ()𝑦𝛼+1^𝐿2‖𝑎‖_𝐿²_(𝑃Δ),𝑑₂=12𝑦²_𝐿2−(𝑏−𝑎)^1/2𝑦_𝐿2‖𝑏‖_𝐿²_(𝑃Δ). (41)
Since, 𝑑₀>0 (by (A₁)), therefore 𝜑 is coercive, that is, 𝜑(𝑥)→∞ as ‖𝑥‖_𝐼^𝛼_𝑎+(𝐿²₎→∞.
Let us fix a sequence (𝑥_𝑘)⊂𝐼^𝛼_𝑎+(𝐿²) such that 𝜑𝑥_𝑘𝜑≤𝑀∀𝑘∈ℕandsome𝑀>0,_′𝑥_𝑘⟶0. (42) The first condition and coercivity of 𝜑 imply that the sequence (𝑥_𝑘) is bounded. So, without loss of the generality, we may assume that it weakly converges in 𝐼^𝛼_𝑎+(𝐿²) to some 𝑥₀. Lemma 5 implies that 𝜑 is of class 𝐶¹ and 𝜑^′(𝑥)ℎ=^𝑏_𝑎𝐷^𝛼_𝑎+𝑥(𝑡)𝐷^𝛼_𝑎+ℎ(𝑡)𝑑𝑡−^𝑏_𝑎𝑦(𝑡)𝐷^𝛼_𝑎+ℎ(𝑡)−𝑑𝑡_𝑏𝑎𝐷^𝛼_𝑎+ℎ(𝑡)^𝑡_𝑎Φ(𝑡,𝑠,𝑥(𝑠))𝑑𝑠𝑑𝑡−^𝑏_𝑎𝐷^𝛼_𝑎+𝑥(𝑡)^𝑡_𝑎Φ_𝑥((𝑠ℎ(𝑠)+𝑡,𝑠,𝑥))𝑑𝑠𝑑𝑡_𝑏𝑎𝑦(𝑡)^𝑡_𝑎Φ_𝑥((𝑠ℎ(𝑠)+𝑡,𝑠,𝑥))𝑑𝑠𝑑𝑡_{𝑏𝑎𝑡𝑎}Φ((𝑠×𝑡,𝑠,𝑥))𝑑𝑠_𝑡𝑎Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ(𝑠)𝑑𝑠𝑑𝑡𝑤 (43)
for 𝑥, ℎ∈𝐼^𝛼_𝑎+(𝐿²). Consequently, 𝜑^′𝑥_𝑘−𝜑^′𝑥₀𝑥_𝑘−𝑥₀=𝑥_𝑘−𝑥₀²_{𝐼𝛼𝑎+𝐿2}+⁶_𝑖=1𝜓_𝑖𝑥_𝑘, (44) where 𝜓₁𝑥_𝑘=−^𝑏_𝑎𝐷^𝛼_𝑎+𝑥_𝑘(𝑡)−𝐷^𝛼_𝑎+𝑥₀(𝑡)×^𝑡_𝑎Φ𝑡,𝑠,𝑥_𝑘(𝑠)−Φ𝑡,𝑠,𝑥₀(𝑠)𝜓𝑑𝑠𝑑𝑡,₂𝑥_𝑘=−^𝑏_𝑎𝐷^𝛼_𝑎+𝑥_𝑘(𝑡)×^𝑡_𝑎Φ_𝑥𝑡,𝑠,𝑥_𝑘(𝑠)𝑥_𝑘(𝑠)−𝑥₀(𝑠)𝜓𝑑𝑠𝑑𝑡,₃𝑥_𝑘=^𝑏_𝑎𝐷^𝛼_𝑎+𝑥₀(𝑡)×^𝑡_𝑎Φ_𝑥𝑡,𝑠,𝑥₀(𝑠)𝑥_𝑘(𝑠)−𝑥₀(𝑠)𝜓𝑑𝑠𝑑𝑡,₄𝑥_𝑘=^𝑏_𝑎𝑦(𝑡)×^𝑏_𝑎Φ_𝑥𝑡,𝑠,𝑥_𝑘(𝑠)−Φ_𝑥𝑡,𝑠,𝑥₀(𝑠)×𝑥_𝑘(𝑠)−𝑥₀(𝑠)𝜓𝑑𝑠𝑑𝑡,₅𝑥_𝑘=^𝑏_𝑎^𝑡_𝑎Φ𝑡,𝑠,𝑥_𝑘(𝑠)×𝑑𝑠_𝑡𝑎Φ_𝑥𝑡,𝑠,𝑥_𝑘(𝑠)𝑥_𝑘(𝑠)−𝑥₀(𝑠)𝜓𝑑𝑠𝑑𝑡,₆𝑥_𝑘=−^𝑏_𝑎^𝑡_𝑎Φ𝑡,𝑠,𝑥₀(𝑠)×𝑑𝑠_𝑡𝑎Φ_𝑥𝑡,𝑠,𝑥₀(𝑠)𝑥_𝑘(𝑠)−𝑥₀(𝑠)𝑑𝑠𝑑𝑡. (45)(46)(47)(48)(49)(50)
The left-hand side converges to 0 because 𝜑^′𝑥_𝑘−𝜑^′𝑥₀𝑥_𝑘−𝑥₀=𝜑^′𝑥_𝑘𝑥_𝑘−𝑥₀−𝜑^′𝑥₀𝑥_𝑘−𝑥₀ (51)
and 𝜑^′(𝑥_𝑘)→0 as well as 𝑥_𝑘⇀𝑥₀ weakly in 𝐼^𝛼_𝑎+(𝐿²). Terms 𝜓_𝑖(𝑥_𝑘), 𝑖=1,…,6, also converge to 0. This follows from the strong convergence of the sequence (𝑥_𝑘) to 𝑥₀ in 𝐿² and weak convergence of the sequence (𝐷^𝛼_𝑎+𝑥_𝑘(𝑡)) to 𝐷^𝛼_𝑎+𝑥₀(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 ^⋅_𝑎Φ⋅,𝑠,𝑥_𝑘(𝑠)−Φ⋅,𝑠,𝑥₀(𝑠)𝑑𝑠 (52)
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 ^⋅_𝑎Φ⋅,𝑠,𝑥_𝑘(𝑠)−Φ⋅,𝑠,𝑥₀(𝑠)𝑑𝑠² (53)
is bounded on [𝑎,𝑏] by an integrable function. This means that ^𝑏_𝑎^𝑡_𝑎Φ𝑡,𝑠,𝑥_𝑘(𝑠)−Φ𝑡,𝑠,𝑥₀(𝑠)𝑑𝑠²𝑑𝑡⟶0, (54)
that is, the sequence (∫^⋅_𝑎(Φ(⋅,𝑠,𝑥_𝑘(𝑠))−Φ(⋅,𝑠,𝑥₀(𝑠)))𝑑𝑠) converges in 𝐿² to the zero function.
Similarly, if 𝜒_𝑘(𝑡,𝑠), 𝑘∈ℕ, are functions belonging to 𝐿²(𝑃_Δ), commonly bounded on 𝑃_Δ by a function 𝜒∈𝐿²(𝑃_Δ), then the sequence ^⋅_𝑎𝜒_𝑘(⋅,𝑠)𝑥_𝑘(𝑠)−𝑥₀(𝑠)𝑑𝑠 (55) converges pointwise a.e. on [𝑎,𝑏] to the zero function. Moreover, 𝜒(𝑡,𝑠)𝑥^𝑡_𝑎𝑘(𝑠)−𝑥₀(𝑠)𝑑𝑠²≤^𝑡_𝑎𝜒_𝑘(𝑡,𝑠)𝑥_𝑘(𝑠)−𝑥₀(𝑠)𝑑𝑠²≤^𝑡_𝑎𝜒(𝑡,𝑠)²𝑑𝑠^𝑏_𝑎𝑥_𝑘(𝑠)−𝑥₀(𝑠)²𝑑𝑠≤const^𝑡_𝑎𝜒(𝑡,𝑠)²𝑑𝑠, (56) where const is a constant which bounds the sequence (𝑥_𝑘) in 𝐿². So, sequence (55) converges in 𝐿² to the zero function. Applying these facts to the functions 𝜒_𝑘(𝑡,𝑠)=Φ_𝑥𝑡,𝑠,𝑥_𝑘(𝑠),𝜒_𝑘(𝑡,𝑠)=Φ_𝑥𝑡,𝑠,𝑥₀(𝑠),𝜒_𝑘(𝑡,𝑠)=Φ_𝑥𝑡,𝑠,𝑥_𝑘(𝑠)−Φ_𝑥𝑡,𝑠,𝑥₀(𝑠), (57) and using (54) one asserts that 𝜓_𝑖(𝑥_𝑘)→0, for 𝑖=1,…,6.
Consequently, ‖𝑥_𝑘−𝑥₀‖²_{𝐼𝛼𝑎+(𝐿2)}→0, that is, 𝜑 satisfies Palais-Smale condition.

Now, one will show that 𝑓 satisfies assumption (𝛽) from Theorem 4. More precisely, one will prove a more general result, namely.

Lemma 7. If Ψ=Ψ(𝑡,𝑠,ℎ)∶𝑃_Δ×ℝ^𝑛→ℝ^𝑛is(𝐵₁) measurable in (𝑡,𝑠)∈𝑃_Δ,(𝐵₂) there exists a function 𝑑∈𝐿²(𝑃_Δ) such that Ψ𝑡,𝑠,ℎ₁−Ψ𝑡,𝑠,ℎ₂≤𝑑(𝑡,𝑠)ℎ₁−ℎ₂ (58) for (𝑡,𝑠)∈𝑃_Δ a.e., ℎ₁, ℎ₂∈ℝ^𝑛, ^𝑡_𝑎𝑑²(𝑡,𝑠)𝑑𝑠≤𝐷 (59) for 𝑡∈[𝑎,𝑏] a.e. and some 𝐷>0,(𝐵₃)Ψ(⋅,⋅,0)∈𝐿²(𝑃_Δ),
then the operator Λ∶𝐼^𝛼_a+𝐿²∋ℎ⟼𝐷^𝛼_𝑎+ℎ(⋅)−^⋅_𝑎Ψ(⋅,𝑠,ℎ(𝑠))𝑑𝑠∈𝐿² (60) is well defined, “one-one” and “onto”.

Proof. First, let us observe that Λ is well defined. Indeed, for any ℎ∈𝐿² (in particular, for ℎ∈𝐼^𝛼_𝑎+(𝐿²)), one has ^𝑡_𝑎Ψ(𝑡,𝑠,ℎ(𝑠))𝑑𝑠≤^𝑡_𝑎|Ψ((𝑠(+𝑡,𝑠,ℎ))−Ψ𝑡,𝑠,0)|𝑑𝑠_𝑡𝑎|Ψ(≤𝑡,𝑠,0)|𝑑𝑠_𝑡𝑎𝑑(𝑡,𝑠)|ℎ(𝑠)|𝑑𝑠+^𝑡_𝑎|Ψ(≤𝑡,𝑠,0)|𝑑𝑠_𝑡𝑎𝑑²(𝑡,𝑠)𝑑𝑠^1/2𝑏_𝑎|ℎ(𝑠)|²𝑑𝑠^1/2+^𝑡_𝑎|Ψ(𝑡,𝑠,0)|𝑑𝑠, (61) for 𝑡∈[𝑎,𝑏] a.e., and the right-hand side belongs to 𝐿².
Now, let one consider some auxiliary problem 𝐷^𝛼_𝑎+ℎ(𝑡)(𝑡)[]𝐼=𝑣,𝑡∈𝑎,𝑏a.e.,_{1−𝛼𝑎+}ℎ(𝑎)=0. (62)
with a fixed 𝑣∈𝐿². Of course, problem (62) has a unique solution ℎ_𝑣(𝑡)=𝐼^𝛼_𝑎+𝑣(𝑡) in the space 𝐼^𝛼_𝑎+(𝐿²) (cf. [11]).
To end the proof, it is sufficient to show that the operator 𝐾∶𝐿²∋𝑣⟼^⋅_𝑎Ψ⋅,𝑠,ℎ_𝑣(𝑠)𝑑𝑠+𝑔(⋅)∈𝐿² (63)
with any fixed 𝑔∈𝐿² possesses a unique fixed point.
One will show that there exist constants 𝜅∈(0,1), 𝑙∈ℕ such that 𝐾𝑣₁−𝐾𝑣_2𝑙≤𝜅𝑣₁−𝑣₂, (64)
for any 𝑣₁, 𝑣₂∈𝐿², where ‖⋅‖_𝑙 is the Bielecki norm in 𝐿² given by ‖𝑣‖²_𝑙=^𝑏_𝑎𝑒^−𝑙𝑡|𝑣(𝑡)|²𝑑𝑡,𝑣∈𝐿². (65)
Indeed, one has 𝐾𝑣₁−𝐾𝑣₂²_𝑙=^𝑏_𝑎𝑒^−𝑙𝑡𝐾𝑣₁(𝑡)−𝐾𝑣₂(𝑡)²≤𝑑𝑡_𝑏𝑎𝑒^{−𝑙𝑡𝑡}_𝑎Ψ𝑡,𝑠,ℎ_𝑣1(𝑠)−Ψ𝑡,𝑠,ℎ_𝑣2(𝑠)𝑑𝑠²≤𝑑𝑡_𝑏𝑎𝑒^{−𝑙𝑡𝑡}_𝑎𝑑(𝑡,𝑠)ℎ_𝑣1(𝑠)−ℎ_𝑣2(𝑠)𝑑𝑠²=𝑑𝑡_𝑏𝑎𝑒^{−𝑙𝑡𝑡}_𝑎𝑑(𝑡,𝑠)𝐼^𝛼_𝑎+𝑣₁−𝑣₂(𝑠)𝑑𝑠²≤𝑑𝑡_𝑏𝑎𝑒^{−𝑙𝑡𝑡}_𝑎𝑑²(𝑡,𝑠)𝑑𝑠^𝑡_𝑎𝐼^𝛼_𝑎+𝑣₁−𝑣₂(𝑠)²()𝑑𝑠𝑑𝑡≤𝐷𝑏−𝑎^𝛼Γ(𝛼+1)^2𝑏_𝑎𝑒^{−𝑙𝑡𝑡}_𝑎𝑣₁−𝑣₂(𝑠)²()𝑑𝑠𝑑𝑡≤𝐷𝑏−𝑎^𝛼Γ(𝛼+1)²1𝑙^𝑏_𝑎𝑒^−𝑙𝑡𝑣₁−𝑣₂(𝑡)²=𝐷𝑑𝑡𝑙(𝑏−𝑎)^𝛼Γ(𝛼+1)²𝑣₁−𝑣₂²_𝑙, (66)
for 𝑣₁, 𝑣₂∈𝐿². It is sufficient to choose 𝑙∈ℕ, such that 𝐷/𝑘((𝑏−𝑎)^𝛼/Γ(𝛼+1))<1.

Applying Lemma 7 to the function Ψ(𝑡,𝑠,ℎ)=Φ_𝑥(𝑡,𝑠,𝑥(𝑠))ℎ (67)

with a fixed 𝑥∈𝐼^𝛼_𝑎+(𝐿²), one obtains the following.

Lemma 8. Operator 𝑓 satisfies (𝛽).

Theorem 4 and Lemmas 5, 6, and 8 imply the following.

Theorem 9. Problem (1) possesses a unique solution 𝑥_𝑔∈𝐼^𝛼_𝑎+(𝐿²), for any 𝑔∈𝐿², and the operator 𝐿²∋𝑔⟼𝑥_𝑔∈𝐼^𝛼_𝑎+𝐿² (68) is differentiable in Frechet sense.

Remark 10. When 𝛼∈(1/2,1), all elements of 𝐼^𝛼_𝑎+(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 𝑥_𝑔∈𝐼^𝛼_𝑎+(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.

Acknowledgment

The project was financed with funds of the National Science Centre, granted on the basis of decision DEC-2011/01/B/ST7/03426.