A short note on the solvability of impulsive fractional differential equations with Caputo derivatives

Accepted Manuscript A short note on the solvability of impulsive fractional differential equations with Caputo derivatives Zhenbin Fan PII: DOI: Reference:

S0893-9659(14)00204-3 http://dx.doi.org/10.1016/j.aml.2014.06.015 AML 4577

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Applied Mathematics Letters

Received date: 13 April 2014 Revised date: 20 June 2014 Accepted date: 20 June 2014 Please cite this article as: Z. Fan, A short note on the solvability of impulsive fractional differential equations with Caputo derivatives, Appl. Math. Lett. (2014), http://dx.doi.org/10.1016/j.aml.2014.06.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A short note on the solvability of impulsive fractional differential equations with Caputo derivatives ∗ Zhenbin Fan† a

Department of Mathematics, Changshu Institute of Technology, Suzhou, Jiangsu 215500, PR China

Abstract: In this paper, we prove the existence and non-existence of solutions to two impulsive fractional differential equations with strong or weak Caputo derivatives in Euclidean space, respectively. Keywords: Caputo derivatives, impulsive differential equations, existence and uniqueness. MSC(2010): 34A08; 34A12.

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Introduction and preliminaries

The theory of fractional differential equations has received much attention over the past twenty years, since they are important in describing the natural models such as diffusion processes, stochastic processes, finance and hydrology. We refer for instance to the books [6, 7], the recent papers [1–5] and the references therein. However, there are still some basic problems on the existence of solutions to impulsive fractional differential equations in Euclidean space R. In this paper, we give the appropriate definitions of solutions to impulsive fractional differential equations with Caputo derivatives and then derive the existence and non-existence of solutions to the following impulsive fractional differential equations   D α u(t) = h(t), t ∈ (0, b]\{t1 , t2 , · · · , tm },   s (1.1) u(0) = u0 ,    + u(ti ) = u(t− i = 1, 2, · · · , m, i ) + yi ,   D α u(t) = h(t), t ∈ (0, b]\{t1 , t2 , · · · , tm },   w u(0) = u0 ,    + u(ti ) = u(t− i = 1, 2, · · · , m, i ) + yi ,

(1.2)

where 0 < α < 1, 0 = t0 < t1 < t2 < · · · < tm < tm+1 = b, u0 ∈ R, yi ∈ R for i = 1, 2, · · · , m and h ∈ C([0, b], R). The main results in this paper show that it is not suitable to consider the impulsive conditions for the fractional differential equation (1.2) with weak Caputo derivatives. ∗

The work was supported by the NSF of China (11001034,11171210) and the Qing Lan Project of Jiangsu Province of China. † Corresponding author. [email protected] (Z. Fan).

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Let b > 0 be fixed, N, R and R+ be the set of positive integers, real numbers and nonnegative real numbers, respectively. We denote by C([0, b], R) the space of all real valued continuous functions on [0, b] with the norm kuk = sup{|u(t)|, t ∈ [0, b]}, L1 ([0, b], R) the space of real valued Lebesgue Rb integrable functions on [0, b] with the norm kf kL1 = 0 |f (t)| dt. Let C n ([0, b], R) = {u : u(k) ∈ C([0, b], R), k = 0, 1, · · · , n} for n ∈ N. We recall some definitions about the fractional differentiation and integration (see [6, 7]). Definition 1.1. The fractional order integral of the function f ∈ L1 ([0, b], R) of order α ∈ R+ is defined by Z t 1 α Jt f (t) = (t − s)α−1 f (s) ds, Γ(α) 0

where Γ is the Gamma function.

Definition 1.2. The Riemann-Liouville fractional order derivative of order α of a function f ∈ L1 ([0, b], R) given on the interval [0, b] is defined by (if it exists) DLα f (t)

1 dn = Γ(n − α) dtn

Z

t

0

(t − s)n−α−1 f (s) ds,

where α ∈ (n − 1, n], n ∈ N. Definition 1.3. The (strong) Caputo fractional order derivative of order α of a function f ∈ L1 ([0, b], R) given on the interval [0, b] is defined by (if it exists) Dsα f (t)

1 = Γ(n − α)

Z

t 0

(t − s)n−α−1 f (n) (s) ds,

where α ∈ (n − 1, n], n ∈ N. Definition 1.4. The (weak) Caputo fractional order derivative of order α of a function f ∈ L1 ([0, b], R) given on the interval [0, b] is defined by (if it exists) α Dw f (t)

=

DLα [f (t) −

n−1 X tk k=0

k!

f (k)(0)],

where α ∈ (n − 1, n], n ∈ N. Remark 1.5. Assume f ∈ C n ([0, b], R), α ∈ (n − 1, n], then α Dw f (t)

1 = Γ(n − α)

Z

t 0

(t − s)n−α−1 f (n) (s) ds = Jtn−α f (n) (t) = Dsα f (t),

t ∈ [0, b].

Note that for strong Caputo derivatives in Definition 1.3, f ∈ C n ([0, b], R) is not necessarily required. In fact, f (n) ∈ L1 ([0, b], R), for example, f (n−1) be of bounded variation, can guarantee the existence of Dsα f (t) on [0, b].

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2

Main results

In this section, we first give the appropriate definition of solutions to equations (1.1) and (1.2). Then we prove the existence and non-existence of solutions to equations (1.1) and (1.2), respectively. We always assume that 0 < α < 1 in this section. Let I = [0, b]. We introduce the Banach space P C(I, R) = {u : I → R : u is continuous at t 6= ti , i = 1, 2, · · · , m

− + and there exist u(t− i ) and u(ti ), i = 1, · · · , m with u(ti ) = u(ti )}

with the norm kukP C := sup{|u(t)| : t ∈ I}. In this section, we always work in P C(I, R) since we mainly consider the impulsive equations. Definition 2.1. A function u ∈ P C(I, R) is called a solution to impulsive fractional differential equation (1.1) if it satisfies (1.1). Definition 2.2. A function u ∈ P C(I, R) is called a solution to impulsive fractional differential equation (1.2) if it satisfies (1.2). Consider the following Cauchy problems ( α Ds u(t) = h(t), u(0) = u0 ,

(

α Dw u(t) = h(t),

u(0) = u0 .

t ∈ (0, b],

(2.1)

t ∈ (0, b],

(2.2)

The solutions to equations (2.1) and (2.2) are also understood in the sense of Definition 2.1 and 2.2, respectively. That is, u ∈ P C(I, R) satisfies the equations (2.1) and (2.2), respectively. Then, we have the following results. Lemma 2.3. Suppose that h ∈ C 1 (I, R). Then the following function u is a solution of the equation (2.1) in P C(I, R) Z t 1 (t − s)α−1 h(s) ds, t ∈ [0, b]. u(t) = u0 + Γ(α) 0 Proof. Suppose that

u(t) = u0 +

1 Γ(α)

Z

t 0

(t − s)α−1 h(s) ds,

t ∈ [0, b].

Then, it is easy to see that u(0) = u0 and u ∈ C(I, R) ⊆ P C(I, R). Moreover, for fixed 0 < t ≤ b (we only consider the left limit if t = b), we have

u(t + r) − u(t) r→0 r Z t Z t+r 1 α−1 = lim h(t − s)sα−1 ds] h(t + r − s)s ds − [ r→0 rΓ(α) 0 0 R t+r Z t h(t + r − s)sα−1 ds 1 h(t + r − s) − h(t − s) α−1 = lim [ s ds + t ]. r→0 Γ(α) 0 r r lim

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Note that for fixed t, s and any r such that 0 ≤ t − s ≤ b and 0 ≤ t + r − s ≤ b, there exists θ ∈ (0, 1) such that h(t + r − s) − h(t − s) = h′ (t − s + rθ). r Then h(t + r − s) − h(t − s) | − h′ (t − s)| r can be smaller than any given positive number ε uniformly for 0 ≤ s ≤ t when |r| is sufficiently small since h ∈ C 1 ([0, b], R). So we obtain that u(t + r) − u(t) r Z t 1 [ h′ (t − s)sα−1 ds + tα−1 h(0)] = Γ(α) 0 Z t 1 = [ (t − s)α−1 h′ (s) ds + tα−1 h(0)], Γ(α) 0 lim

r→0

which implies that u′ (t) exists for 0 < t ≤ b and 1 u (t) = Γ(α) ′

Z

t 0

(t − s)α−1 h′ (s) ds +

tα−1 h(0) tα−1 h(0) = Jtα h′ (t) + , Γ(α) Γ(α)

Thus, Dsα u(t) = Jt1−α u′ (t) = Jt1−α Jtα h′ (t) + Jt1−α

0 < t ≤ b.

tα−1 h(0) = Jt1 h′ (t) + h(0) = h(t). Γ(α)

Therefore, u satisfies the equation (2.1). That is, u is a solution of the equation (2.1). Lemma 2.4. Suppose that h ∈ C(I, R). Then there is a unique solution u ∈ P C(I, R) of the equation (2.2) given by Z t 1 (t − s)α−1 h(s) ds, t ∈ [0, b]. u(t) = u0 + Γ(α) 0 Proof. Suppose that

1 u(t) = u0 + Γ(α)

Z

t 0

(t − s)α−1 h(s) ds,

t ∈ [0, b].

Then, it is easy to see that u(0) = u0 and u ∈ C(I, R) ⊆ P C(I, R). Moreover, for 0 < t ≤ b, in view of h ∈ C(I, R), α Dw u(t) =

d 1−α d [Jt (u(t) − u(0))] = [Jt1−α Jtα h(t)] = h(t), dt dt

which implies that u satisfies the equation (2.2). That is, u is a solution of the equation (2.2). Now, we show the uniqueness of solutions to the equation (2.2). Suppose that u and v are two solutions to equation (2.2) in P C(I, R). Let w = u − v. Then w ∈ P C(I, R) is the solution of the following equation ( α Dw w(t) = 0, t ∈ (0, b], w(0) = 0.

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It follows that for every t ∈ (0, b], we obtain α Dw w(t) =

d 1−α [J (w(t) − w(0))] = 0, dt t

Jt1−α w(t) = Jt1 0 = 0, Jt1 w(t) = Jtα Jt1−α w(t) = 0.

Rt That is, 0 w(s) ds = 0 for any t ∈ [0, b]. So, w(t) = 0 for almost all t ∈ [0, b] since w ∈ P C(I, R) ⊆ L1 (I, R), which implies that w|(ti ,tt+1 ) = 0 almost everywhere on (ti , tt+1 ) for i = 0, 1, · · · , m. Thus, w|(ti ,tt+1 ) ≡ 0 on (ti , tt+1 ) by the continuity of w|(ti ,tt+1 ) for i = 0, 1, · · · , m. On the other hand, w ∈ P C(I, R) implies that w ≡ 0 on [0, b]. That is, u ≡ v on [0, b]. Now, let us turn to the impulsive fractional differential equations (1.1) and (1.2). Theorem 2.5. Suppose that h ∈ C 1 ([0, b], R). Then there is a solution u ∈ P C(I, R) of impulsive equation (1.1) given by  Z t 1    u + (t − s)α−1 h(s) ds, 0 ≤ t ≤ t1 , 0   Γ(α)  0  Z t    u + y + 1  (t − s)α−1 h(s) ds, t1 < t ≤ t2 ,  0 1 Γ(α) 0 (2.3) u(t) = ..    .     Z t m  X  1   y + (t − s)α−1 h(s) ds, tm < t ≤ b. u +  0 i  Γ(α) 0 i=1

Proof. Suppose that u be defined by (2.3). It’s easy to verify that u ∈ P C(I, R), u(0) = u0 and − u(t+ i ) = u(ti ) + yi for i = 1, 2 · · · , m. For 0 < t ≤ t1 , u clearly satisfies the equation (1.1) by Lemma 2.3. Now, let tj < t ≤ tj+1 for some index j ∈ {1, 2 · · · , m}. Then Z s  1   (s − τ )α−1 h(τ ) dτ, 0 ≤ s ≤ t1 , u0 +   Γ(α)  0    .. . u(s) =   Z s j  X  1    yi + (s − τ )α−1 h(τ ) dτ, tj < s ≤ t.  u0 + Γ(α) i=1

0

Thus, by the proof of Lemma 2.3, u(s) is differentiable almost everywhere on [0, t] and Z t 1 (t − s)−α u′ (s) ds Dsα u(t) = Γ(1 − α) 0 Z t1 Z t 1 1 −α ′ = (t − s) u (s) ds + · · · + (t − s)−α u′ (s) ds Γ(1 − α) 0 Γ(1 − α) tj Z t Z s 1 1 −α d [u0 + (t − s) (s − τ )α−1 h(τ ) dτ ] ds = Γ(1 − α) 0 ds Γ(α) 0 =h(t).

Therefore u given by (2.3) is a solution of equation (1.1). 5

Theorem 2.6. Suppose that h ∈ C([0, b], R) and yi 6= 0 for i = 1, 2 · · · , m. Then there is no solutions to impulsive equation (1.2) in P C(I, R). Proof. First, we show that if equation (1.2) has a solution, then the solution is unique. Suppose that u and v are two solutions to equation (1.2) in P C(I, R). Let w = u − v. Then w ∈ P C(I, R) is the solution of the following equation   D α w(t) = 0, t ∈ (0, b]\{t1 , t2 , · · · , tm },   w (2.4) w(0) = 0,    + − w(ti ) = w(ti ), i = 1, 2, · · · , m.

α w(t), we obtain that [J 1−α (w(t) − w(0))]′ = 0 for t ∈ (t , t By the definition of Dw i i+1 ), i = 0, 1 · · · , m. t 1−α It follows that Jt (w(t) − w(0)) = ci for t ∈ (ti , ti+1 ), where ci are constants, i = 0, 1 · · · , m. Thus, Jt1−α (w(t) − w(0)) = 0 for t ∈ [0, b] because of the continuity of integration, which implies Rt that 0 w(s) ds = 0 for all t ∈ [0, b]. So, w(t) = 0 for almost all t ∈ [0, b] since w ∈ P C(I, R) ⊆ L1 (I, R), which implies that w|(ti ,tt+1 ) = 0 almost everywhere on (ti , tt+1 ) for i = 0, 1, · · · , m. Thus, w|(ti ,tt+1 ) ≡ 0 on (ti , tt+1 ) by the continuity of w|(ti ,tt+1 ) for i = 0, 1, · · · , m. On the other hand, − conditions w(t+ i ) = w(ti ), i = 1, 2 · · · , m imply that w ≡ 0 on [0, b]. Therefore, u ≡ v on [0, b]. Now, we assume that u ∈ P C(I, R) is the unique solution of equation (1.2). Next we will look for a contradiction. Also, by Lemma 2.4, we assume that v is the unique solution of equation (2.2) in P C(I, R). Let w = u − v. Then w ∈ P C(I, R) is the unique solution of the following equation   D α w(t) = 0, t ∈ (0, b]\{t1 , t2 , · · · , tm },   w (2.5) w(0) = 0,    − w(t+ i = 1, 2, · · · , m. i ) = w(ti ) + yi ,

However, for the equation

(

α Dw w(t) = 0,

w(0) = 0,

t ∈ (0, b]\{t1 , t2 , · · · , tm },

(2.6)

we have proven in equation (2.4) if w ∈ P C(I, R) is a solution of it then w|(ti ,tt+1 ) ≡ 0 for i = − 0, 1, · · · , m. It follows that, for the equation (2.6) the impulsive conditions w(t+ i ) = w(ti ) + yi for i = 1, 2, · · · , m, can not be satisfied. That is, the equation (2.5) has no solutions in P C(I, R). This is a contradiction. Consequently, the impulsive equation (1.2) has no solutions in P C(I, R). Remark 2.7. According to Theorem 2.6, it is not suitable to consider the impulsive conditions for the fractional differential equation (1.2) with weak Caputo derivatives. On the other hand, it seems to be difficult to prove the uniqueness of solutions in Lemma 2.3 and Theorem 2.5 because the absolute continuity of f is not required in the definition of strong Caputo derivatives. At last, we consider the following two simple examples to illustrate the main results in this paper:  1  4  t ∈ (0, 2]\{1},   Ds u(t) = t, u(0) = 0,     u(1+ ) = u(1− ) + 2, 6

(2.7)

 1  4    Dw u(t) = t, One can verify that u given by

t ∈ (0, 2]\{1},

u(0) = 0,     u(1+ ) = u(1− ) + 2.    

16 5 t4 , 5Γ( 41 ) u(t) = 16 5   t4 , 2 + 5Γ( 41 )

(2.8)

0 ≤ t ≤ 1, 1
is a solution of equation (2.7). However, it does not satisfy the equation (2.8). In fact, for 1 < t ≤ 2, we have 1 d 3 16 5 2 −1 d 3 t4 ] = t 4 + t. Dw4 u(t) = Jt4 (u(t) − u(0)) = Jt4 [2 + dt dt 5Γ( 14 ) Γ( 43 ) On the other hand, we know, by Theorem 2.6, the equation (2.8) has no solutions in P C([0, 2], R).

References [1] Z. Fan, Approximate controllability of fractional differential equations via resolvent operators, Advances in Difference Equations, 2014:54 (2014). [2] Z. Fan, Characterization of compactness for resolvents and its applications, Appl. Math. Comput. 232 (2014) 60-67. [3] S. Kumar, N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Diff. Equs. 252 (2012) 6163-6174. [4] N. I. Mahmudov, Approximate controllability of fractional neutral evolution equations in Banach spaces, Abstract and Applied Analysis, Vol 2013, Article ID 531894, 11 pages. [5] G. Mophou, G. M. N’Gu´er´ekata, V. Valmorin, Asymptotic behavior of mild solutions of some fractional functional integro-differential equations, Afr. Diaspora J. Math. (N.S.) 16(1) (2013) 1-89. [6] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [7] J. Pr¨ uss, Evolutionary Integral Equations and Applications, Birkh¨ auser, Basel, Berlin, 1993.

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