On a solution to fractional order integrodifferential equations with analytic semigroups

Nonlinear Analysis 71 (2009) 3690–3698

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

On a solution to fractional order integrodifferential equations with analytic semigroups Dwijendra N. Pandey ∗ , Amit Ujlayan, D. Bahuguna Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India

article

info

Article history: Received 29 November 2008 Accepted 4 February 2009 Keywords: Fractional order differential equation Parabolic equation Analytic semigroup Mild solution Local and global existence and uniqueness

abstract In this paper we study a class of fractional order integrodifferential equations considered in an arbitrary Banach space. Using the theory of analytic semigroups we establish the existence, uniqueness and regularity of a mild solution to these fractional order integrodifferential equations. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Although the tools of the fractional differential calculus have been available and applicable to modelling several physical phenomena, the investigation of the theory of fractional differential equations has been started during the past three decades or so. The nonlinear oscillations of earthquake [1] is one of such important models which uses the fractional derivatives. The fluid-dynamic traffic model with fractional derivatives [2] can eliminate the deficiency arising from the assumptions of continuum traffic flow. Also based on experimental data fractional partial differential equations for seepage flow in porous media are suggested in [3]. We refer [4,5] for more details about the theory of fractional differential equations. Jaradat et al. [6] have considered the following fractional order differential equation in a Banach space E, dη u(t ) dt η

+ Au(t ) = f (t , u(t ), Gu(t ), Su(t )),

t > t0 , η ∈ (0, 1],

u(t0 ) = u0 ,

(1.1)

where A generates a strongly continuous semigroup. The authors in [6] have used the theory of semigroup and standard fixed point theorems to prove the existence and uniqueness of a solution. In this paper we are concerned with the following fractional order integrodifferential equation in a Banach space X : dη u(t )

+ Au(t ) = f (t , u(t )) + K (u)(t ), dt η u(t0 ) = u0 ,

t > t0 ,

(1.2)

where K (u)(t ) =

Z

t

a(t − s)g (s, u(s))ds. t0

∗

Corresponding author. E-mail addresses: [email protected] (D.N. Pandey), [email protected] (A. Ujlayan), [email protected] (D. Bahuguna).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.018

(1.3)

D.N. Pandey et al. / Nonlinear Analysis 71 (2009) 3690–3698

3691

In (1.2), we assume that −A generates an analytic semigroup {S (t ), t ≥ 0}, on a Banach space X , the function a is real valued and locally integrable on [t0 , ∞), the nonlinear maps f and g are defined on [t0 , ∞) × X into X . Our analysis is motivated by the work of Muslim [7]. The author in [7] has discussed the existence and uniqueness of a solution of a semilinear version of the problem considered here. Authors in [8,9] have discussed the existence and uniqueness of a solution of the following integrodifferential equation under the above said assumptions on the generator A of an analytic semigroup and the nonlinear maps f , g and the kernel a du(t ) dt

+ Au(t ) = f (t , u(t )) +

t

Z

a(t − s)g (s, u(s)),

t ∈ (0, T ],

0

u(0) = u0 .

(1.4)

We first establish that under Assumption F, stated below, there exists a unique local mild solution to (1.2). We also study the regularity of the mild solution to (1.2). Finally, at the end we give an example of a class of parabolic integrodifferential equations as an application of the results obtained for the abstract integrodifferential equation of fractional order (1.2). Eq. (1.2) represents an abstract formulation of certain classes of parabolic integrodifferential equations. 2. Preliminaries Let X denote the Banach space and let J denote the closure of the interval [t0 , T ), t0 < T ≤ ∞. Let −A be the infinitesimal generator of an analytic semigroup {S (t ), t ≥ 0} in X . We note that if −A is the infinitesimal generator of an analytic semigroup then −(A + α I ) is invertible and generates a bounded analytic semigroup for α > 0 large enough. This allows us to reduce the general case in which −A is the infinitesimal generator of an analytic semigroup to the case in which the semigroup is bounded and the generator is invertible. Hence for convenience, we suppose that

kS (t )k ≤ M for t ≥ 0 and 0 ∈ ρ(−A), where ρ(−A) is the resolvent set of −A. It follows that for 0 ≤ α ≤ 1, Aα can be defined as a closed linear invertible operator with its domain D(Aα ) being dense in X . We denote by Xα the Banach space D(Aα ) equipped with norm

kxkα = kAα xk which is equivalent to the graph norm of Aα . We have Xβ ,→ Xα

for 0 < α < β

and the imbedding is continuous. By a mild solution to (1.2) on J we mean a continuous function u defined from J into X satisfying the following integral equation u(t ) = S (t − t0 )u0 +

1

Γ (η)

Z

t

(t − s)η−1 S (t − s)[f (s, u(s)) + K (u)(s)]ds,

t ∈ J.

(2.1)

t0

We say that (1.2) has a local mild solution if there exists a T0 , t0 < T0 ≤ T and a continuous function u defined from J0 = [t0 , T0 ] into X such that u is a mild solution to (1.2) on J0 . To establish the existence of a unique mild solution to (1.2) in later sections, we shall require the following assumption on the maps f and g. Assumption F. Let U be an open subset of [0, ∞) × Xα . A function f is said to satisfy Assumption F if for every (t , x) ∈ U there exist a neighborhood V ⊂ U of (t , x) and constants L0 > 0, such that

kf (s, u) − f (s, v)k ≤ L0 ku − vkα

(2.2)

for all (s, u) and (s, v) in V . To begin with the analysis we need some basic definitions and properties from the fractional calculus theory. Definition 2.1. A real function f (x), x > 0 is said to be in the space Cµ , µ ∈ R if there exist a real number p (>µ), such that f (x) = xp f1 (x), where f1 ∈ C [0, ∞) and it is said to be in the space Cµm iff f (m) ∈ Cµ , m ∈ N.

3692

D.N. Pandey et al. / Nonlinear Analysis 71 (2009) 3690–3698

Definition 2.2. The Riemann–Liouville fractional integral operator of order 0 ≤ α , of a function f ∈ Cµ , µ ≥ −1 is defined as I α f (x) =

x

Z

1

Γ (α)

(x − t )α−1 f (t )dt , α > 0, x > 0,

0

I 0 f (x) = f (x).

(2.3)

While modeling some real world phenomena with fractional differential equation the Riemann–Liouville derivative has certain disadvantage. Therefore, we shall introduce a modified fractional differential operator Dα∗ proposed by M. Caputo. Definition 2.3. The fractional derivative of f (x) in the Caputo sense is defined as Dαx f (x) = I m−α Dm f (x)

=

x

Z

1

Γ (m − α)

(x − t )m−α−1 Dm f (t )dt ,

m for m − 1 ≤ α < m, m ∈ N , x > 0, f ∈ C− 1.

(2.4)

0

3. Local existence of mild solutions As pointed out earlier, we may suppose without loss of generality that the analytic semigroup generated by −A is bounded and that −A is invertible. Furthermore, we assume that 0 < T < ∞ to establish local existence. We shall continue to use the notations considered in the previous section. Lemma 3.1. The evolution problem (1.2) is equivalent to the following integral equation u(t ) = u0 +

1

Γ (η)

t

Z

( t − s)

η−1

1

(−Au(s))ds +

Z

Γ (η)   Z s × f (s, u(s)) + a(s − τ )g (τ , u(τ ))dτ ds, t0

t

(t − s)η−1

t0

t ≥ t0 .

(3.1)

t0

Proof. For the details, we refer to Lemma 1.1 in [10].



The existence of a solution to (1.2) is associated with the following equivalent integral equation: u(t ) = S (t − t0 )u0 +

1

Γ (η)

t

Z

  Z s (t − s)η−1 S (t − s) f (s, u(s)) + a(s − τ )g (τ , u(τ ))dτ ds,

t0

t ≥ t0 .

(3.2)

t0

With these simplifications we have the following theorem. Theorem 3.2. Suppose that the operator −A generates the analytic semigroup S (t ) with kS (t )k ≤ M, t ≥ 0 and that 0 ∈ ρ(−A). If the maps f and g satisfy Assumption F and the real valued map a is integrable on J, then (1.2) has a unique local mild solution for every u0 ∈ Xα . Furthermore if the kernel a satisfies the following condition (H) There exist constants C0 > 0 and 0 < β ≤ 1 such that

|a(t ) − a(s)| ≤ C0 |t − s|β for all t , s ∈ J. Then the obtained mild solution is also Hölder continuous. Proof. We shall use the notions and notations introduced in the preceding section. We fix a point (t0 , u0 ) in the open subset U of [0, ∞) × Xα and choose t10 > t0 and r > 0 such that (2.2), with some constant L0 > 0 holds for the functions f and g on the set V = {(t , x) ∈ U : t0 ≤ t ≤ t10 , kx − u0 kα ≤ r }. Let B1 = sup kf (t , u0 )k t0 ≤t ≤t10

and B2 = sup kg (t , u0 )k. t0 ≤t ≤t10

(3.3)

D.N. Pandey et al. / Nonlinear Analysis 71 (2009) 3690–3698

3693

Choose t1 > t0 such that 1

kS (t − t0 ) − I kkAα u0 k ≤

2

r,

for t0 ≤ t ≤ t1

(3.4)

and

( t1 − t0 < min t1 − t0 , 0

hr

Cα

2

−1

(η − α)Γ (η)(L(r ))

1

−1 η−α

i

) (3.5)

where L(r ) = (L0 r + B1 ) + aT (L0 r + B2 ) and Cα is a positive constant depending on α satisfying

kAα S (t )k ≤ Cα t −α ,

for t > t0 ,

(3.6)

and

Z

T

|a(s)|ds.

aT =

(3.7)

t0

Let Y = C ([t0 , t1 ]; X ) be endowed with the supremum norm

kykY = sup ky(t )k. t0 ≤t ≤t1

Then Y is a Banach space. We define a map on Y by Fy where Fy is given by t

Z

1

Fy(t ) = S (t − t0 )Aα u0 +

Γ (η)

(t − s)η−1 Aα S (t − s)[f (s, A−α y(s)) +

t0

Z

s

a(s − τ )g (τ , A−α y(τ ))dτ ]ds.

(3.8)

t0

It follows from Assumption F on the functions f and g, (3.6) and (3.7) that F : Y −→ Y and for every y ∈ Y , Fy(t0 ) = Aα u0 . Let S be the nonempty closed and bounded set given by

S = {y ∈ Y : y(t0 ) = Aα u0 , ky(t ) − Aα u0 k ≤ r } .

(3.9)

Then for y ∈ S we have α

α

kFy(t ) − A u0 k ≤ kS (t − t0 ) − I kkA u0 k + × kf (s, A

−α

1

Z

Γ (η)

y(s)) − f (s, u0 )kds +

t

(t − s)η−1 kAα S (t − s)k

t0 t

Z

kAα S (t − s)k

t0 s

Z

|a(s − τ )|kg (τ , A

×

−α



y(τ )) − g (τ , u0 )kdτ ds

t0

+ + ≤

1

2 ≤r

t

Z

1

Γ (η)

t

Z

1

Γ (η)

r + Cα

(t − s)η−1 kAα S (t − s)kkf (s, u0 )kds

t0

(t − s) t0

η−1

α

kA S (t − s)k

Z

s



|a(s − τ )|kg (τ , u0 )kdτ ds t0

1

Γ (η)

(η − α)−1 L(r )(t1 − t0 )η−α (3.10)

where the last two inequalities follow from (3.4) and (3.5). Thus, we have that F : S −→ S . Now we show that F is a strict contraction on S which will ensure the existence of a unique continuous function satisfying Eq. (2.1). For y and z in S , then we have

kFy(t ) − Fz (t )k ≤

Z

1

Γ (η) 1

t

(t − s)η−1 kAα S (t − s)kkf (s, A−α y(s)) − f (s, A−α z (s))kds

t0

Z

t

(t − s)η−1 kAα S (t − s)k Γ (η) t0 Z s  −α −α × |a(s − τ )|kg (τ , A y(τ )) − g (τ , A z (τ ))kdτ ds. +

t0

(3.11)

3694

D.N. Pandey et al. / Nonlinear Analysis 71 (2009) 3690–3698

Using Assumption F on f and g and (3.6), (3.7), we get

  Z t 1 (t − s)η−1 kAα S (t − s)kd ky − z kY kFy(t ) − Fz (t )k ≤ L0 (1 + aT ) Γ (η) t0 ≤ L0 (1 + aT )Cα ≤ ≤ ≤

1 r 1 r 1 2

1

Γ (η)

L0 r (1 + aT )Cα L(r )Cα

1

Γ (η)

(η − α)−1 (t1 − t0 )η−α ky − z kY

1

Γ (η)

(η − α)−1 (t1 − t0 )η−α ky − z kY

(η − α)−1 (t1 − t0 )η−α ky − z kY

ky − z kY ,

(3.12)

using (3.5) in the last inequality. Thus F is a strict contraction map from S into S and therefore by the Banach contraction principle there exists a unique fixed point y of F in S , i.e., there is a unique y ∈ S such that Fy = y. −α

Let u = A

(3.13)

y. Then for t ∈ [t0 , t1 ], we have

u(t ) = A−α y(t )

= S ( t − t 0 ) u0 +

t

Z

1

Γ (η)

(t − s)η−1 S (t − s)[f (s, u(s)) + K (u)(s)]ds.

t0

Thus u is a unique local mild solution to (1.2). Next we prove the Hölder continuity of the solution, for this we assume that the kernel a satisfies the condition (H ). Since it follows that there exist T0 , t0 < T0 < T and a function u such that u is a unique mild solution to (1.2) on J0 = (t0 , T0 ) given by u(t ) = S (t − t0 )u0 +

t

Z

1

Γ (η)

(t − s)η−1 S (t − s)[f (s, u(s)) + K (u)(s)]ds,

t ∈ J0 ,

(3.14)

t0

where K (u)(t ) =

Z

t

a(t − s)g (s, u(s))ds. t0

Let

v(t ) = Aα u(t ).

(3.15)

Then α

v(t ) = S (t − t0 )A u0 +

1

Z

Γ (η)

t

( t − s)

η−1 α



A S (t − s) f (s, A

−α

v(s)) +

t0

Z

s

a(s − τ )g (τ , A

−α



v(τ ))dτ ds.

(3.16)

t0

For simplification, set f˜ (t ) = f (t , A−α v(t )),

(3.17)

g˜ (t ) = g (t , A−α v(t )). Then (3.16) can be rewritten as

v(t ) = S (t − t0 )Aα u0 +

1

Γ (η)

Z

t t0

  Z s (t − s)η−1 Aα S (t − s) f˜ (s) + a(s − τ )˜g (τ )dτ ds.

(3.18)

t0

Since u(t ) is continuous on J0 and the maps f and g satisfy Assumption F, it follows that f˜ and g˜ are continuous, and therefore bounded on J0 . Let N1 = sup kf˜ (t )k t ∈J0

N2 = sup k˜g (t )k. t ∈J0

(3.19)

D.N. Pandey et al. / Nonlinear Analysis 71 (2009) 3690–3698

3695

Next, we have Aα [v(t2 ) − v(t1 )] = [(S (t2 − t0 ) − S (t1 − t0 ))]Aα ut0   Z Z t2 1 + (t2 − s)η−1 [(S (t2 − s))Aα ] f˜ +

Γ (η)

Zt1t1

s



a(s − τ )˜g dτ ds



t0

   Z s a(s − τ )˜g dτ ds ]A (S (t2 − s)) f˜ + Γ (η) 0     Z s t0 Z t1 1 η−1 α ˜ + a(s − τ )˜g dτ ds . (t1 − s) A (S (t2 − s) − S (t1 − s)) f + Γ (η) 0 t0 

−

1

[(t1 − s)

η−1

η−1

− (t2 − s)

α

kAα [v(t2 ) − v(t1 )]k ≤ kI1 k + kI2 k + kI3 k + kI4 k.

(3.20) (3.21)

Since part (d) of Theorem 6.13 in Pazy [11] on page 74 states that for 0 < β ≤ 1 and x ∈ D(Aβ ),

k(S (t ) − I )xk ≤ Cβ t β kAβ xk, hence if 0 < β < 1 is such that 0 < α + β < 1 and y ∈ D(Aα ) then Aα y ∈ D(Aβ ). Therefore for t , s ∈ (0, T ], we have

k(S (t ) − I )Aα S (s)xk ≤ Cβ t β kAα+β S (s)xk ≤ Cβ Cα+β t β s−(α+β) kxk.

(3.22)

Where we have

kI1 k ≤ k[(S (t2 − t0 ) − S (t1 − t0 ))]Aα u0 k ≤ Cβ Cα+β (t2 − t1 )β (t1 − t0 )−(α+β) ku0 k;

(3.23)

for I2 , we have

k I2 k ≤ ≤

Z

1

Γ (η)

t2

  Z s k(t2 − s)η−1 k (S (t2 − s))Aα kf˜ k + |a(s − τ )|k˜g kdτ ds

t1

t0

1

(η − α)Γ (η)

Cα [N1 + aT0 N2 T0 ](t2 − t1 )η−α ;

(3.24)

and for I3 , we have

k I3 k ≤

t1

Z

1

Γ (η)

Cα [N1 + aT0 N2 T0 ]

(t1 − s)λ−1 [(t1 − s)−λµ − (t2 − s)−λµ ]ds.

(3.25)

0

With λ = 1 − α, µ = 1−α and α 6= 1. Here we can use the calculation presented in Theorem 3.2 of [10] to find the upper bound of integral and thus we get 1−η

k I3 k ≤

1

µ−1

Γ (η)

Cα [N1 + aT0 N2 T0 ](µδ1

(1 − c )λ(µ−1)−1 )(t2 − t1 )λ(1−µ)

(3.26)

1

µ

where c = (1 − λ ) λµ and 0 < r1 ≤ 1. Along the same lines we may calculate the bound of I4 as

k I4 k ≤ ≤

Γ (η) 1

Cα [N1 + aT0 N2 T0 ]

(t1 − s)η−1 [(t1 − s)−α − (t2 − s)−α ]ds

0

µ−1

Γ (η) µ

t1

Z

1

Cα [N1 + aT0 N2 T0 ](µδ2

(1 − c )λ(µ−1)−1 )(t2 − t1 )λ(1−µ) ,

(3.27)

1

with c = (1 − λ ) λµ , λ = η, µ = αη , and 0 < r2 ≤ 1. Hence from (3.23)–(3.27), Hölder continuity of v(t ) follows.



4. Global existence In order to establish the global existence of a mild solution to (1.2), we need the following lemma. Lemma 4.1. Let u0 (t , s) ≥ 0 be continuous on 0 ≤ s ≤ t ≤ T < ∞. If there are positive constants A, B and α such that u0 (t , s) := A + B

Z s

t

(t − σ )α−1 u0 (σ , s)dσ ,

(4.1)

3696

D.N. Pandey et al. / Nonlinear Analysis 71 (2009) 3690–3698

for 0 ≤ s < t ≤ T , then there is a constant C such that u0 ( t , s ) ≤ C . Proof. For 0 ≤ s < t ≤ T , we have t

Z

(t − τ )α−1 (τ − s)dτ = (t − s)α+β−1

s

Γ (α)Γ (β) , Γ (α + β)

(4.2)

which holds for every α, β > 0. Integrating (4.1) n − 1 times using (4.2) and replacing t − s by T , we get u0 ( t , s ) ≤ A

j n−1  X BT α α

j =0

+

(BΓ (α))n Γ (nα)

t

Z

(t − σ )nα−1 u0 (σ , s)dσ .

(4.3)

s

Let n be large enough so that nα > 1. We majorize (t − σ )nα−1 by T nα−1 to obtain u0 (t , s) ≤ c1 + c2

t

Z

u0 (σ , s)dσ .

(4.4)

s

Application of Gronwall’s inequality leads to u0 (t , s) ≤ c1 ec2 (t −s) ≤ c1 ec2 T ≤ C .

(4.5)

This completes the proof of the lemma.



The following theorem establishes the global existence of a mild solution to (1.2). Theorem 4.2. Let 0 ∈ D(A) and let −A be the infinitesimal generator of an analytic semigroup S (t ) satisfying

kS (t )k ≤ M for t ≥ t0 . Let f , g : [t0 , ∞) × Xα −→ X satisfy Assumption F. If there exist continuous nondecreasing functions k1 and k2 from [t0 , ∞) into [0, ∞) such that

kf (t , x)k ≤ k1 (t )(1 + kxkα ) for t ≥ t0 , x ∈ Xα , kg (t , x)k ≤ k2 (t )(1 + kxkα ) for t ≥ t0 , x ∈ Xα ,

(4.6)

then the initial value problem (1.2) has a unique mild solution u on [t0 , ∞) for every u0 ∈ Xα . Proof. From Theorem 3.2 it follows that there exist a T0 , t0 < T0 and a unique solution u on J0 = [t0 , T0 ]. If

ku(t )kα ≤ C

(4.7)

for t ∈ J0 for some positive constant C , then the solution u(t ) may be continued further on the right of T0 . Therefore it suffices to prove that if a mild solution u to (1.2) exists on [t0 , T ], t0 < T < ∞ then ku(t )kα is bounded as t ↑ T . Since u(t ) is a mild solution. Therefore we have u(t ) = S (t − t0 )u0 +

t

Z

1

Γ (η)

η−1

( t − s)



S (t − s) f (s, u(s)) +

t0

Z

s



a(s − τ )g (τ , u(τ ))dτ ds.

(4.8)

t0

Making use of the fact that S (t ) commutes with A and that

kS (t )k ≤ M , kAα S (t )k ≤ Cα t −α for t ≥ t0 in (4.8), after applying Aα and taking norms on both sides, we get

ku(t )kα ≤ M kAα u0 k + Cα

1

Γ (η)

Z

t t0

Z s

ds. (t − s)η−α−1 f ( s , u ( s )) + a ( s − τ ) g (τ , u (τ )) d τ

(4.9)

t0

For the last term in (4.9), we have the estimate

Z

s

|a(s − τ )|kg (τ , u(τ ))dτ k ≤ aT k2 (T ) t0

Z

s

[1 + ku(τ )kα ]dτ . t0

(4.10)

D.N. Pandey et al. / Nonlinear Analysis 71 (2009) 3690–3698

3697

Incorporating the estimate of (4.10) in (4.9), we get α

ku(t )kα ≤ M kA u0 k + Cα [k1 (T ) + aT k2 (T )]

1

Γ (η)

t

Z

(t − s)η−α−1

t0

  Z s (1 + ku(τ )kα )dτ ds. × 1 + ku(s)kα +

(4.11)

t0

After a slight modification in (4.11), we get 1

ku(t )kα ≤ M kAα u0 k + Cα t

Z +

(1 + T )[k1 (T ) + aT k2 (T )]

T η−α

Γ (η) η−α   Z s ku(τ )kα dτ ds. (t − s)η−α ku(s)kα +

(4.12)

t0

t0

The estimate in (4.12) is of the type

ku(t )kα ≤ C1 + C2

Z

t

(t − s)

η−α

  Z s ku(τ )kα dτ ds, ku(s)kα +

(4.13)

t0

t0

for some positive constants C1 and C2 depending on η, α and T only. Integrating (4.13) over (t0 , t ), we get

Z

t

ku(ξ )kα dξ ≤ C1 T + C2

ξ

Z tZ

t0

t0

  Z s (ξ − s)η−α ku(s)kα + ku(τ )kα dτ dsdξ .

t0

(4.14)

t0

Changing the order of integration in (4.14), we obtain

Z

t

ku(ξ )kα dξ ≤ C1 T + C2

Z tZ

t0

t0

t

  Z s (ξ − s)η−α ku(s)kα + ku(τ )kα dτ dξ ds.

s

(4.15)

t0

We rewrite (4.15) as

Z

t

ku(ξ )kα dξ ≤ C1 T + C2

Z t Z

t0

t0

≤ C1 T +

ξ

(ξ − s)η−α dξ

  Z s ku(s)kα + ku(τ )kα dτ ds

t0

C2 T

(4.16)

t0

Z

η−α+1

t

(t − s)

η−α

  Z s ku(s)kα + ku(τ )kα dτ ds.

t0

(4.17)

t0

The estimate (4.17) is of the form

Z

t

ku(ξ )kα dξ ≤ C3 + C4 t0

Z

t

(t − s)

η−α

  Z s ku(s)kα + ku(τ )kα dτ ds,

t0

(4.18)

t0

for some positive constants C3 and C4 , depending on α and T only. Adding (4.13) and (4.18), we have

ku(t )kα +

Z

t

ku(ξ )kα dξ ≤ C5 + C6 t0

Z

t

( t − s)

η−α

t0



ku(s)kα +

Z

s



ku(τ )kα dτ ds,

(4.19)

t0

for some positive constants C5 and C6 , depending on η, α and T only. Applying Lemma 4.1 to (4.19), we conclude that

ku(t )k ≤ C on [t0 , T ]. This completes the proof of the theorem.



5. Applications Let Ω ⊂ Rn be a bounded domain with smooth boundary ∂ Ω . Consider the linear partial differential operator A(x, D) =

X

aα (x)Dα .

(5.1)

|α|≤2m

¯ for each multi-index α . We assume that A(x, D) is strongly where aα (x) is a real or complex valued function defined on Ω elliptic, i.e., there exists a constant c > 0 such that X |α|=2m

aα (x)ξ α ≥ c |ξ |2m

(5.2)

3698

D.N. Pandey et al. / Nonlinear Analysis 71 (2009) 3690–3698

¯ and ξ ∈ Rn . Consider the parabolic integrodifferential equation for all x ∈ Ω ∂ η u(x, t ) + A(x, D)u(x, t ) = f (x, t , u(x, t ), Du(x, t ), . . . , D2m−1 u(x, t )) ∂tη Z t a(t − s)g (x, s, u(x, s), Du(x, s), . . . , D2m−1 u(x, s)) ds, +

x ∈ Ω , t > t0 ,

(5.3)

t0

u(x, t0 ) = u0 (x) x ∈ Ω u(x, t ) = 0,

x ∈ Ω , t ∈ [t0 , T ), t0 < T ≤ ∞,

j

where D stands for any jth order derivative. We assume that f and g are continuously differentiable functions of all their variables, except possibly in x. The parabolic integrodifferential equation (5.3) can be reformulated as the following abstract integrodifferential equation in X = Lp (Ω ): dη u(t ) dt η

+ Ap u(t ) = F (t , u(t )) +

Z

t

a(t − s)G(s, u(s)) ds,

t > t0 ,

(5.4)

t0

u(t0 ) = u0 , where Ap : D(Ap ) ⊂ X −→ X given by 2m,p

D(Ap ) = W 2m,p (Ω ) ∩ W0

(Ω ),

Ap u = A(x, D)u + λu

for u ∈ D(Ap ), λ > 0

and F , G : [t0 , T ) × D(Ap ) −→ X are the Nemyckii operators given by F (t , u)(x) = f (x, t , u(x, t ), Du(x, t ), . . . , D2m−1 u(x, t ))

(5.5)

G(t , u)(x) = g (x, t , u(x, t ), Du(x, t ), . . . , D

(5.6)

2m−1

u(x, t ))

where we assume the usual sufficient Caratheodory and growth conditions on the functions f and g for the Nemyckii operators in (5.5) and (5.6) to be well defined. Here we assume that λ is large enough so that Ap is invertible. It follows that −Ap is the infinitesimal generator of an analytic semigroup on X . Also, from imbedding theorems it follows that Xα is ¯ ) for 1 − 1/2m < α < 1 and p large enough. It can be verified that Assumption F is continuously imbedded in C 2m−1 (Ω satisfied by F and G. Under suitable assumptions on the kernel a, Theorem 4.2 ensures the existence of a unique global mild solution to (5.4) for p large enough which in turn guarantees the existence of a unique global mild solution to (5.3). Acknowledgments The authors would like to thank the referees for the valuable suggestions. The work of first author is partially supported by the CSIR India (CSIR File no. 9/92(386)/2005-EMR-I). References [1] D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996) 609–625. [2] J.H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. 15 (2) (1999) 86–90. [3] J.H. He, Approximate analytical solutions for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg. 167 (1998) 57–68. [4] F. Mainardi, Fractional calculus some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York, 1997, p. 291348. [5] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, 1993. [6] O.K. Jaradat, A. Al-Omrai, S. Momani, Existence of the mild solutions for fractional semilinear initial value problems, Nonlinear Anal. 69 (9) (2008) 3153–3159. [7] M. Muslim, Existence and approximation of solutions to fractional differential equations, Math. Comput. Modelling (2008) (in press). [8] D. Bahuguna, Existence, uniqueness and regularity of solutions to semilinear nonlocal functional differential equations, Nonlinear Anal. 57 (2004) 1021–1028. [9] D. Bahuguna, S.K. Srivastava, S. Singh, Approximation of solutions to semilinear integrodifferential equations, Numer. Funct. Anal. Optim. 22 (2001) 487–504. [10] M.M. El-Borai, Some probability densities and fundamental solutions of fractional differential equations, Chaos Solitons Fractals 14 (3) (2002) 433–440. [11] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.