Sensitivity of a Fractional Integrodifferential Cauchy Problem of Volterra Type

and Applied Analysis 3 (A 3 ) 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,

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][2][3][4][5][6][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  = 1 is given.In the paper, we use the line of the proof presented therein.The main difference between cases of  ∈ (0, 1) and  = 1 is that, in the first case, the elements of the solution space   + ( 2 ) are not, in general, continuous functions on [, ] as it is when  = 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.

Fractional Calculus
By the left-sided Riemann-Liouville fractional integral of ℎ on the interval [, ], we mean (cf.[9]) a function   + ℎ given by (2) where Γ is the Euler function.
One can show that the above integral exists and is finite a.e. on In [10], the following useful theorem is proved.
Let us recall that a  1 -functional  :  → R satisfies Palais-Smale condition if any sequence (  ) such that      (  )     ≤  ∀ ∈ N, and some  > 0, admits a convergent subsequence (here,   (  ) is the Frechet differential of  at   ).
We shall show that the operator satisfies assumptions of Theorem 4 with the spaces  =   + ( 2 ),  =  2 .Namely, we have the following.
Lemma 5.The operator  is well-defined  1 -mapping with the differential   () at any  ∈   + ( 2 ) given by where Φ  is the Jacobi matrix of Φ with respect to .
Proof.Well-definiteness of .Since Φ is the Caratheodory function with respect to (, ) ∈  Δ and  ∈ R  , the function is measurable.From (A 2 ), it follows that it belongs to  1 .The Fubini theorem implies integrability of the function Moreover, The right-hand side is integrable on One will check that the operator is the Frechet differential of  at  ∈   + ( 2 ).First, let one observe that   () is well defined.Of course, the function where ℎ ∈   + ( 2 ), is measurable.By (A 3 ) it belongs to  1 .From the Fubini theorem it follows that the function for  ∈   + ( 2 ), where Since,  0 > 0 (by The first condition and coercivity of  imply that the sequence (  ) is bounded.So, without loss of the generality, we may assume that it weakly converges in   + ( 2 ) to some  0 .Lemma 5 implies that  is of class for , ℎ ∈   + ( 2 ).Consequently, where The left-hand side converges to 0 because and   (  ) → 0 as well as   ⇀  0 weakly in   + ( 2 ).Terms   (  ),  = 1, . . ., 6, also converge to 0. This follows from the strong convergence of the sequence (  ) to  0 in  2 and weak convergence of the sequence (  +   ()) to   +  0 () in  2 (cf.Lemma 3) as well as from the Krasnoselskii theorem on continuity of the Nemytskii operator.for  ∈ [, ] a.e. and some  > 0, then the operator is well defined, "one-one" and "onto".
Proof.First, let us observe that Λ is well defined.Indeed, for any ℎ ∈  2 (in particular, for ℎ ∈   + ( 2 )), one has         ∫ for  ∈ [, ] a.e., and the right-hand side belongs to  2 .Now, let one consider some auxiliary problem with a fixed V ∈  2 .Of course, problem (62) has a unique solution ℎ V () =   + V() in the space   + ( 2 ) (cf. [11]).To end the proof, it is sufficient to show that the operator with any fixed  ∈  2 possesses a unique fixed point.

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   ∈   + ( 2 ) for any fixed functional parameter  ∈  2 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.