Variational Methods and Critical Point Theory 2013View this Special Issue
Research Article | Open Access
Sensitivity of a Fractional Integrodifferential Cauchy Problem of Volterra Type
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.
In the paper, we consider the following fractional Integrodifferential Cauchy problem of Volterra type of order : where , , and . We consider the existence and uniqueness of a solution in the space (the range of the right-sided integral Riemann-Liouville operator ) as well as sensitivity, that is, differentiable dependence of a solution on a functional parameter .
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 . 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 is given. In the paper, we use the line of the proof presented therein. The main difference between cases of and is that, in the first case, the elements of the solution space are not, in general, continuous functions on as it is when (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.
2.1. Fractional Calculus
Let , . By the left-sided Riemann-Liouville fractional integral of on the interval , we mean (cf. ) a function given by where is the Euler function.
One can show that the above integral exists and is finite a.e. on . Moreover, if , , then and
In , the following useful theorem is proved.
Theorem 1. The operator , , is compact, that is, it maps bounded sets onto relatively compact ones.
Now, let . One says that  possesses the left-sided Riemann-Liouville derivative of order on the interval , if the integral is absolutely continuous on (more precisely, if has an absolutely continuous representant a.e. on ). By this derivative one means the classical derivative , that is,
One has (, Theorem 2.4).
Theorem 2. If , then a.e. on . If , then a.e. on .
From the second part of the above theorem and (3), it follows that if , then
Of course, with the norm
is complete. When , the above norm is generated by the following scalar product:
Next, one will use the following.
Lemma 3. If a sequence is weakly convergent in to some , then it is convergent to with respect to the norm in , and the sequence is weakly convergent in to .
Proof. To prove the second part of the theorem, it is sufficient to observe that the linear operator is continuous. Consequently, it preserves weak convergence. To prove the first part, let us observe that, from Theorem 1, it follows that maps weakly convergent sequences onto strongly convergent (with respect to the norm) ones. Thus, the sequence is convergent to with respect to the norm in .
2.2. A Theorem on a Diffeomorphism
In , we proved the following theorem.
Theorem 4. Let be a real Banach space, let be a real Hilbert space. If is a -mapping (i.e., differentiable in Frechet sense on with the differential continuous on ) such that) for any , the functional satisfies Palais-Smale condition,() for any , is “one-one” and “onto”,then is a diffeomorphism (i.e., it is “one-one”, “onto” and the inverse is differentiable in Frechet sense).
Let us recall that a -functional satisfies Palais-Smale condition if any sequence such that
admits a convergent subsequence (here, is the Frechet differential of at ).
3. Main Result
Let us consider problem (1) with . We assume that function satisfies the following conditions:() is measurable in and continuously differentiable in Frechet sense in ,() there exist functions , such that for a.e., , () there exists a function such that for a.e., , for a.e. and some .
We shall show that the operator
satisfies assumptions of Theorem 4 with the spaces , . Namely, we have the following.
Lemma 5. The operator is well-defined -mapping with the differential at any given by where is the Jacobi matrix of with respect to .
Proof. Well-definiteness of . Since is the Caratheodory function with respect to and , the function
is measurable. From (), it follows that it belongs to . The Fubini theorem implies integrability of the function
The right-hand side is integrable on . So, function (17) belongs to .
Differentiability of . Continuous differentiability of the first term of follows from the linearity and continuity of the operator .
So, let one consider the second term, that is, the operator
One will check that the operator
is the Frechet differential of at .
First, let one observe that is well defined. Of course, the function where , is measurable. By () it belongs to . From the Fubini theorem it follows that the function
is integrable. Moreover, similarly as in the case of ,
So, function (22) belongs to .
Linearity of is obvious. Its continuity follows from the following estimations (cf. (5)):
Now, one will check that is the Gateaux differential of at , that is,
in , for any , . Indeed, let be a sequence of real numbers converging to and consider the limit
From the differentiability of with respect to , it follows that, for a.e., the sequence of functions
converges pointwise a.e. on to the zero function. From the mean value theorem applied to any coordinate function
(), it follows that functions (27) indexed by are commonly pointwise (a.e. on ) bounded by an integrable function (cf. ()). So,
for a.e. Moreover, using once again the mean value theorem, one obtains
that is, (25) holds true.
Continuity of . Let be a sequence converging in to some . Similarly, as mentioned above, one obtains
for any . Convergence
follows from (), the Krasnoselskii theorem on the continuity of the Nemytskii operator and from (5).
So, being continuously differentiable in Gateaux sense is continuously differentiable in Frechet sense.
Now, one will prove.
Lemma 6. For any fixed , the functional satisfies Palais-Smale condition.
Proof. It is easy to see that
for , where
for , where
Since, (by ()), therefore is coercive, that is, as .
Let us fix a sequence such that 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 to some . Lemma 5 implies that is of class and
for , . Consequently, where
The left-hand side converges to 0 because
and as well as weakly in . Terms , , also converge to . This follows from the strong convergence of the sequence to in and weak convergence of the sequence to in (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
converges pointwise a.e. on to the zero function. Moreover, in the same way as in the proof of Lemma 5, one can check that the sequence
is bounded on by an integrable function. This means that
that is, the sequence converges in to the zero function.
Similarly, if , , are functions belonging to , commonly bounded on by a function , then the sequence converges pointwise a.e. on to the zero function. Moreover, where is a constant which bounds the sequence in . So, sequence (55) converges in to the zero function. Applying these facts to the functions and using (54) one asserts that , for .
Consequently, , that is, satisfies Palais-Smale condition.
Now, one will show that satisfies assumption () from Theorem 4. More precisely, one will prove a more general result, namely.
Lemma 7. If is() measurable in ,() there exists a function such that
for a.e., , ,
for a.e. and some ,(),
then the operator is well defined, “one-one” and “onto”.
Proof. First, let us observe that is well defined. Indeed, for any (in particular, for ), one has
for a.e., and the right-hand side belongs to .
Now, let one consider some auxiliary problem
with a fixed . Of course, problem (62) has a unique solution in the space (cf. ).
To end the proof, it is sufficient to show that the operator
with any fixed possesses a unique fixed point.
One will show that there exist constants , such that
for any , , where is the Bielecki norm in given by
Indeed, one has
for , . It is sufficient to choose , such that .
Applying Lemma 7 to the function
with a fixed , one obtains the following.
Lemma 8. Operator satisfies ().
Theorem 9. Problem (1) possesses a unique solution , for any , and the operator is differentiable in Frechet sense.
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 for any fixed functional parameter 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.
The project was financed with funds of the National Science Centre, granted on the basis of decision DEC-2011/01/B/ST7/03426.
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