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Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the nonlocal conditions
Advances in Difference Equations volumeÂ 2011, ArticleÂ number:Â 18 (2011)
Abstract
A secondorder abstract problem of neutral type with derivatives of noninteger order in the nonlinearity as well as in the nonlocal conditions is investigated. This model covers many of the existing models in the literature. It extends the integer order case to the fractional case in the sense of Caputo. A fixed point theorem is used to prove existence of mild solutions.
AMS Subject Classification
26A33, 34K40, 35L90, 35L70, 35L15, 35L07
1 Introduction
In this paper, we investigate the following neutral secondorder abstract differential problem
with 0 â‰¤ Î±, Î², Î³ â‰¤ 1. Here, the prime denotes time differentiation and ^{C} D ^{Îº}, Îº = Î±, Î², Î³ denotes fractional time differentiation (in the sense of Caputo). The operator A is the infinitesimal generator of a strongly continuous cosine family C(t), t â‰¥ 0 of bounded linear operators in the Banach space X and f, g are nonlinear functions from R ^{+} Ã— X Ã— X to X, u ^{0} and u ^{1} are given initial data in X. The functions p : [C(I; X)]^{2} â†’ X, q : [C(I; X)]^{2} â†’ X are given continuous functions (see the example at the end of the paper).
This problem has been studied in case Î±, Î², Î³ are 0 or 1 (see [1â€“8]). Wellposedness has been established using different fixed point theorems and the theory of strongly continuous cosine families in Banach spaces. We refer the reader to [7, 9, 10] for a good account on the theory of cosine families.
Fractional nonlocal conditions are the natural generalization of the integer order nonlocal conditions as studied by Hernandez [5] and others. They include the discrete case where the solution is prescribed at some finite number of times.
Time delay is a natural phenomena which occurs in many problems (see [11, 12]). It is caused for instance by the finite switching speed of amplifiers in electronic networks or finite speed for signal propagation in biological networks. We can trace problems with delays back to Volterra who introduced past states in population dynamics. It has been also introduced by Boltzmann in viscoelasticity in the form of a convolution. When there is a dependence on all past states we usually call such a delay a distributed delay. There are in fact several types of delays. The importance of delays has been pointed out by many researchers and we are now witnessing a growing interest in such problems. An important class of delayed differential equations (or functional differential equations) is the class of neutral differential equations. In this type of problems the delayed argument occurs in the derivative of the state variable. This is the case, for instance, when a growing population consumes more (or less) food than matured one or when this term appears in the constitutive relationship between the stress and the strain. In fact, neutral differential equations arise naturally in biology, ecology, electronics, economics, epidemiology, control theory and mechanics [11â€“18]. More precisely, they appear in the study of oscillatory systems, electrical networks containing lossless transmission line (highspeed computers, distributed nonlumped transmission line, lossless transmission line terminated by a tunnel diode and lumped parallel capacitor) [11, 13, 15, 18], vibrating masses attached to an elastic bar [11, 12], automatic control, neuromechanical systems and some variational problems (Euler equations) [14, 16, 17]. For the sake of simplicity and since the case where time delay exists in the function "g" has been already studied before (at least for some types of delays) we shall focus on the distributed delay present in the nonlinearity "f ".
We consider the case (g â‰¢ 0) and prove existence of mild solutions under different conditions on the different data. In particular, this work may be viewed as an extension of the work in [6] to the fractional order case. Indeed, the work in [6] is concerned with the firstorder derivatives whereas here we treat the fractional order case where some difficulties arise because of the nonlocal nature of the fractional derivatives. In addition to that, to the best of the author's knowledge, fractional derivatives are introduced here for such problems for the first time.
The next section of this paper contains some notation and preliminary results needed in our proofs. Section 3 treats the existence of a mild solution in the space of continuously differentiable functions. An example is provided to illustrate our finding.
2 Preliminaries
In this section, we present some notation, assumptions and preliminary results needed in our proofs later.
Definition 1. The integral
is called the RiemannLiouville fractional integral of h of order Î± > 0 when the right side exists.
Here, Î“ is the usual Gamma function
Definition 2. The fractional derivative of h of order Î± > 0 in the sense of Caputo is given by
In particular
See [19â€“22] for more on fractional derivatives and fractional integrals.
We will assume that (H1) A is the infinitesimal generator of a strongly continuous cosine family C(t), t âˆˆ R, of bounded linear operators in the Banach space X.
The associated sine family S(t), t âˆˆ R is defined by
It is known (see [7, 8, 10]) that there exist constants M â‰¥ 1 and Ï‰ â‰¥ 0 such that
For simplicity, we will designate by and bounds for C(t) and S(t) on I = [0, T], respectively.
If we define
then we have
Assume that (H 1) is satisfied . Then

(i)
S(t)X âŠ‚ E, t âˆˆ R,

(ii)
S(t)E âŠ‚ D(A), t âˆˆ R,

(iii)
,
 (iv)
Suppose that (H1) holds, v : R â†’ X a continuously differentiable function and . Then, q(t) âˆˆ D(A), and
Definition 3. A continuously differentiable function u satisfying the integrodifferential equation
is called a mild solution of problem (1).
This definition follows directly from the definition of the cosine family and (1), see [6, 7].
3 Existence of mild solutions
In this section, we prove existence of a mild solution in the space C ^{1}(I; X). Before we proceed with the assumptions on the different data we recall that E is a Banach space when endowed with the norm x_{ E } = x + sup_{0â‰¤tâ‰¤1}AS(t)x, x âˆˆ E (see [23]). It is also wellknown that AS(t) : E â†’ X is a bounded linear operator. By B _{ r }(x, X) we will denote the closed ball in X centered at x and of radius r.
The assumptions on f, g, p and q are (H2)

(i)
f(t,.,.) : X Ã— X â†’ X is continuous for a.e. t âˆˆ I.

(ii)
For every (x, y) âˆˆ X Ã— X, the function f(.,x, y) : I â†’ X is strongly measurable.

(iii)
There exist a nonnegative continuous function K _{ f } (t) and a continuous nondecreasing positive function Î©_{ f } such that
for (t, x, y) âˆˆ I Ã— X Ã— X.

(iv)
For each r > 0, the set f(I Ã— B _{ r } (0, X ^{2})) is relatively compact in X.
(H3)

(i)
The function g takes its values in E and g : I Ã— X Ã— X â†’ X is continuous.

(ii)
There exist a nonnegative continuous function K _{ g }(t), a continuous nondecreasing positive function Î©_{ g } and two positive constants C _{1}, C _{2} such that
and
for (t, x, y) âˆˆ I Ã— X Ã— X.

(iii)
The family of functions {t â†’ g(t, u, v); u, v âˆˆ B _{ r }(0, C(I; X))} is equicontinuous on I.

(iv)
For each r > 0, the set g(I Ã— B _{ r }(0, X ^{2})) is relatively compact in E.
(H4) u ^{0}+p : [C(I; X)]^{2} â†’ E (takes its values in E) and q : [C(I; X)]^{2} â†’ X are completely continuous.
The positive constants N _{ p } and N _{ q } will denote bounds for u ^{0} + p(u, v)_{ E } and q(u, v), respectively. To lighten the statement of our result we denote by
and
We are now ready to state and prove our result.
Theorem 1. Assume that (H1)(H4) hold. If l > 0 and
then problem (1) admits a mild solution u âˆˆ C ^{1}([0, T]).
Proof. Note that by our assumptions and for u, v âˆˆ C([0, T]); the maps
and
are well defined, and map [C([0, T])]^{2} into C([0, T]). These maps are nothing but the right hand side of (2) and its derivative. We would like to apply the LeraySchauder alternative [which states that either the set of solutions of (6) (below) is unbounded or we have a fixed point in D (containing zero) a convex subset of X provided that the mappings Î¦ and Î¨ are completely continuous]. To this end, we first prove that the set of solutions (u _{ Î» } , v _{ Î» }) of
is bounded. Then, we prove that this map is completely continuous. Therefore, there remains the alternative which is the existence of a fixed point. We have from (4)
and from (5)
Then
and
where
and
Taking the sup in the relation (7) and max sup in the relation (8) and adding the resulting expressions we end up with
where Î›(z) is equal to the expression
and
or simply
With
and
provided that
If we designate by Ï†_{ Î» }(t) the right hand side of (9), then
Î˜_{Î»} (t) â‰¤ Ï†_{ Î» }(t), t âˆˆ I and
We infer that
This (with (3)) shows that Î˜_{Î»}(t) and thereafter the set of solutions of (6) is bounded in [C(I; X)]^{2} :
It remains to show that the maps Î¦ and Î¨ are completely continuous. From our hypotheses it is immediate that
is completely continuous. To apply AscoliArzela theorem we need to check that
is equicontinuous on I. Let us observe that
for t âˆˆ I and h such that t + h âˆˆ I. In virtue of (H1) and (H3), for t âˆˆ I and Îµ > 0 given, there exists Î´ > 0 such that
for s âˆˆ [0, t] and , when h < Î´. This together with (H2), (H3) and the fact that S(t) is Lipschitzian imply that
for some positive constant N _{ l } : The equicontinuity is therefore established.
On the other hand, for t âˆˆ I, as (s,Î¾) â†’ C(t  s)Î¾ is continuous from to X and is relatively compact,
is relatively compact as well in X. As for Î¦_{3} := Î¦  Î¦_{1} + Î¦_{2} we decompose it as follows
and select the partition of [0, t] in such a manner that, for a given Îµ > 0
for , when s, s' âˆˆ [s _{ i }, s _{ i+1}] for some i = 1,..., k  1: This is possible in as much as
is bounded (by (H2)(iii)) and the operator S is uniformly Lipschitz on I. This leads to
where
and co(U(t, s _{ i }, r)) designates its convex hull. Therefore, is relatively compact in X. By AscoliArzela Theorem, is relatively compact in C(I; X) and consequently Î¦_{3} is completely continuous. Similarly, we may prove that Î¨ is completely continuous.
We conclude that (Î¦, Î¨) admits a fixed point in [C([0, T])]^{2} .
Remark 1. In the same way we may treat the more general case
where 0 â‰¤ Î± _{ i }, Î² _{ j }, Î³ _{ k } â‰¤ 1, i = 1,..., n, j = 1,..., m, k = 1,...,r.
Remark 2. If g does not depend on u'(t), that is for g(t, u(t)), we may avoid the condition that g must be an Evalued function. We require instead that g be continuously differentiable and apply Lemma 2 to
to obtain
instead of
in (5).
Example As an example we may consider the following problem
in the space X = L ^{2}([0, Ï€]). This problem can be reformulated in the abstract setting (1). To this end, we define the operator Ay = y" with domain
The operator A has a discrete spectrum with n ^{2}, n = 1, 2,... as eigenvalues and , n = 1, 2,... as their corresponding normalized eigenvectors. So we may write
Since A is positive and selfadjoint in L ^{2}([0, Ï€]), the operator A is the infinitesimal generator of a strongly continuous cosine family C(t), t âˆˆ R which has the form
The associated sine family is found to be
One can also consider more general nonlocal conditions by allowing the Lebesgue measure ds to be of the form dÎ¼(s) and dÎ·(s) (LebesgueStieltjes measures) for nondecreasing functions Î¼ and Î· (or even more general: Î¼ and Î· of bounded variation), that is
These (continuous) nonlocal conditions cover, of course, the discrete cases
which have been extensively studied by several authors in the integer order case.
For u, v âˆˆ C([0, T]; X) and x âˆˆ [0, Ï€], defining the operators
allows us to write (10) abstractly as
Under appropriate conditions on F, G, P and Q which make (H2)(H4) hold for the corresponding f, g, p and q, Theorem 1 ensures the existence of a mild solution to problem (10).
Some special cases of this problem may be found in [24â€“28]. They model some phenomena with hereditary properties. See also [29â€“33] for some problems with fractional boundary conditions.
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The author is very grateful for the financial support provided by King Fahd University of Petroleum and Minerals through the project No. IN 100007.
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Tatar, Ne. Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the nonlocal conditions. Adv Differ Equ 2011, 18 (2011). https://doi.org/10.1186/16871847201118
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DOI: https://doi.org/10.1186/16871847201118