Studia Scientiarum Mathematicarum Hungarica 51 (2), 141–154 (2014)DOI: 10.1556/SScMath.51.2014.2.1271

ON THE EXISTENCE OF MILD SOLUTIONS FOR A CLASS OFFRACTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL

CONDITIONS IN THE α-NORM

MOHAMED I. ABBAS

Department of Mathematics, Faculty of Science, Alexandria University, Moharrem Bey 21511,Alexandria, Egypt

e-mail: m i abbas77@yahoo.com

Communicated by P. Vertesi

(Received October 15, 2011; accepted February 25, 2014)

Abstract

This paper concerns the existence of mild solutions for some fractional Cauchy prob-lem with nonlocal conditions in the α-norm. The linear part of the equations is assumedto generate an analytic compact bounded semigroup, and the nonlinear part satisfies someLipschitz conditions with respect to the fractional power norm of the linear part. By usinga fixed point theorem of Sadovskii, we establish some existence results which generalizeones in the case of fractional order derivative.

1. Introduction and Perliminaries

Recently fractional differential equations emerged as a new branch ofapplied mathematics which have been used for many mathematical modelsin science and engineering (see for instance [10], [11], [13] and [15]).

The purpose of this paper is to prove the existence of mild solutions ofthe following fractional Cauchy problem with nonlocal conditions

(1)

Dq[x(t) + F

(t, x(t)

)] = −Ax(t) +G

(t, x(t)

), t ∈ [0, T ],

x(0) + g(x) = x0 ∈ X,

2010 Mathematics Subject Classification. Primary 34K05, 34A12, 34A40.Key words and phrases. Fractional differential equation, analytic compact semigroup,

nonlocal condition, α-norm.

0081–6906/$ 20.00 c⃝ 2014 Akademiai Kiado, Budapest

142 M. I. ABBAS

where T > 0, 0 < q 5 1, −A generates an analytic compact semigroup(Q(t)

)t=0

of uniformly bounded linear operators on a Banach space X. Thederivative Dq is understood here in the Riemann–Liouville sense.

The nonlocal Cauchy problem was considered by Byszewski [1] and theimportance of nonlocal conditions in different fields has been discussed in[1], [2] and the references therein. For example, in [2] the author describedthe diffusion phenomenon of a small amount of gas in a transparent tube byusing the formula

g(x) =

p∑i=1

cix(ti)

where ci, i = 0, 1, . . . , p are given constants and 0 < t0 < t1 < · · · < tp < T .In this case the above equation allows the additional measurement at ti,i = 0, 1, . . . , p. Subsequently several authors have investigated the problemfor different types of nonlinear differential equations and integrodifferentialequations including functional differential equations in Banach spaces, e.g.,[3], [4]. El-Borai [5], [6] studied fundamental solution of fractional evolutionequations in a Banach space.

Very recently N’Guerekata [7] discussed the existence of solutions of frac-tional abstract differential equations with a nonlocal initial condition. Moti-vated by this work we study in this paper the existence of solutions of frac-tional semilinear evolution equations in Banach spaces by using fractionalcalculus and fixed point theorems.

In our investigation of the existence of mild solutions of Eq. (1), we as-sume that the nonlinear part satisfies some Lipschitz conditions with respectto the fractional power norm of −A, and use a fixed point of Sadovskii todiscuss existence results of solutions of Eq. (1).

For the rest of this article, we set J = [0, T ]. We denote by X a Ba-nach space with norm ∥.∥ and −A : D(A) → X is the infinitesimal genera-tor of a compact analytic semigroup of uniformly bounded linear operators(Q(t)

)t=0

. This means that there exists M > 1 such that∥∥Q(t)∥∥ 5 M.

Without loss of generality, we assume that 0 ∈ ρ(A). Then it is possible todefine the fractional power Aα for 0 < α < 1, as a closed linear operator onits domain D(Aα) with inverse A−α (see [12]). The following are the basicproperties of Aα.

Theorem 1.1 ([12], pp. 69–75).

(i) Xα = D(Aα) is a Banach space with the norm ∥x∥α := ∥Aαx∥ for x ∈D(Aα).

(ii) Q(t) : X → Xα for each t > 0.

ON THE EXISTENCE OF MILD SOLUTIONS 143

(iii) AαQ(t)x = Q(t)Aαx for each x ∈ D(Aα) and t = 0.

(iv) For every t > 0, AαQ(t) is bounded on X and there exist Mα > 0 andδ > 0 such that ∥∥AαQ(t)

∥∥ 5 Mα

tαe−δt 5 Mα

tα.

(v) A−α is a bounded linear operator in X with D(Aα) = Im(A−α).

(vi) If 0 < α 5 β, then D(Aβ) → D(Aα).

Let Xα be the Banach space D(Aα) endowed with the norm |.|α. Fromnow on, we denote by ∥.∥α the operator norm in Xα and by Cα the operator

norm of A−α, i.e., Cα := ∥A−α∥. We use, as usual, ∥f∥p to denote the Lp-

norm of f whenever f ∈ Lp(0, T ) for some p with 1 5 p < ∞. Let E be the

Banach space C([0, T ],Xα

)endowed with the supnorm given by

|x|∞ := sup05t5T

∣∣x(t)∣∣α

for x ∈ Xα.

To begin with the analysis we need some basic definitions and propertiesfrom the fractional calculus theory (see [10], [11] and [13]).

Definition 1.1. A real function f(x); x > 0 is said to be in the spaceCµ, µ ∈ R if there exist a real number p(> µ), such that f(x) = xpf1(x),

where f1 ∈ C[0,∞) and it is said to be in the space Cmµ iff f (m) ∈ Cµ, m ∈ N.

Definition 1.2. The Riemann–Liouville fractional integral operator oforder q = 0, of a function f ∈ Cµ, µ = −1 is defined as

Iqf(x) =1

Γ(q)

x∫0

(x− t)q−1f(t) dt, q > 0, x > 0,

I0f(x) = f(x).

While modeling some real world phenomena with fractional differentialequation the Riemann–Liouville derivative has certain disadvantage. There-fore, we shall introduce a modified fractional differential operator Dq

∗ pro-posed by M. Caputo.

Definition 1.3. The fractional derivative of f(x) in the Caputo senseis defined as

Dqxf(x) = Im−qDmf(x)

144 M. I. ABBAS

=1

Γ(m− q)

x∫0

(x− t)m−q−1Dmf(t) dt,

for m− 1 5 q < m, m ∈ N, x > 0, f ∈ Cm−1.

The following are the basic assumptions of this paper.

(H1) The function F : [0, T ]×Xα → X1 is a continuous function and thereexists L1 > 0 such that the function AF satisfies the Lipschitz condi-tion: ∥∥AF (t1, x1)−AF (t2, x2)

∥∥ 5 L1

(|t1 − t2|+ |x1 − x2|α

),

for all (t1, x1), (t2, x2) ∈ [0, T ]× Xα.

(H2) There is a β with α 5 β 5 1 such that the function G : [0, T ]×Xα →Xβ satisfies the two conditions:

(i) For each t ∈ [0, T ], the function G(t, .) : Xα → Xβ is continuousand for each x ∈ Xα, the function G(., x) : [0, T ] → Xβ is stronglymeasurable.

(ii) There exists a positive function µ ∈ L1([0, T ],R+

)such that:∥∥AβG(t, x)

∥∥ 5 µ(t),

for all (t, x) ∈ [0, T ]× Xα.

(H3) g ∈ C(E,Xα) is completely continuous. There exist L2, L3 > 0 suchthat: ∣∣g(x)∣∣

α5 L2|x|∞ + L3.

2. Existence of mild solutions

Definition 2.1 (see [9]). A continuous solution x(t) of the integralequation:

x(t) = Q(t)[x0 + F(0, x(0)

)− g(x)]− F

(t, x(t)

)+

1

Γ(q)

t∫0

(t− s)q−1AQ(t− s)F(s, x(s)

)ds

ON THE EXISTENCE OF MILD SOLUTIONS 145

+1

Γ(q)

t∫0

(t− s)q−1Q(t− s)G(s, x(s)

)ds,

will be called a mild solution of the nonlocal Cauchy problem (1), for eacht ∈ [0, T ].

Remark 2.1. Observe as in [8] that by Theorem 1.1 (ii) and (iii), therestriction Qα(t) of Q(t) to Xα is exactly the part of Q(t) in Xα. Let x ∈ Xα.Since∥∥Q(t)x

∥∥α=∥∥AαQ(t)x

∥∥ =∥∥Q(t)Aαx

∥∥ 5∥∥Q(t)

∥∥∥Aαx∥ =∥∥Q(t)

∥∥∥x∥α,and as t decreases to 0∥∥Q(t)x− x

∥∥α=∥∥AαQ(t)x−Aαx

∥∥ =∥∥Q(t)Aαx−Aαx

∥∥ → 0,

for all x ∈ Xα, it follows that Qα(t) is a strongly continuous semigroup on Xα

and∥∥Qα(t)

∥∥α5∥∥Q(t)

∥∥ for all t = 0.

We have the following result from [8].

Lemma 2.1.(Qα(t)

)t=0

is an immediately compact semigroup in Xα,

and hence it is immediately norm-continuous.

Theorem 2.1. Suppose that the hypotheses (H1), (H2) and (H3) hold.If x0 ∈ Xα and the following inequality holds

(2) ML2 +

((M + 1)C1−α +

MαT−α+q

(−α+ q)Γ(q)

)L1 < 1,

then Eq. (1) has a mild solution on [0, T ].

Proof. We define the function P on E by

(Px)(t) := Q(t)[x0 + F(0, x(0)

)− g(x)]− F

(t, x(t)

)+

1

Γ(q)

t∫0

(t− s)q−1AQ(t− s)F(s, x(s)

)ds

+1

Γ(q)

t∫0

(t− s)q−1Q(t− s)G(s, x(s)

)ds, 0 5 t 5 T.

146 M. I. ABBAS

For each positive integer k, let Bk =x ∈ E : |x|∞ 5 k

.

It is sufficient to show that P has a fixed point. We first show that thereis a positive number k such that PBk ⊂ Bk. If this were not the case,then foreach k > 0, there exist xk ∈ Bk and tk ∈ [0, T ] such that

∣∣(Pxk)(tk)∣∣α> k.

We see by (H1), (H2) and (H3) that

k <∣∣(Pxk)(tk)

∣∣α

5 |Q(tk)[x0 − g(xk) + F(0, xk(0)

)]|α+ |F

(tk, xk(tk)

)|α

+1

Γ(q)

tk∫0

(tk − s)q−1|AQ(tk − s)F(s, xk(s)

)|αds

+1

Γ(q)

tk∫0

(tk − s)q−1|Q(tk − s)G(s, xk(s)

)|αds

5∥∥Q(tk)

∥∥ .(|x0|α +∣∣g(xk)∣∣α + ∥Aα−1AF

(0, xk(0)

)∥)

+ ∥Aα−1AF(tk, xk(tk)

)∥

+1

Γ(q)

tk∫0

(tk − s)q−1∥AαQ(tk − s)AF(s, xk(s)

)∥ ds

+1

Γ(q)

tk∫0

(tk − s)q−1∥Aα−βQ(tk − s)AβG(s, xk(s)

)∥ ds

5∥∥Q(tk)

∥∥ .(|x0|α +∣∣g(xk)∣∣α +

∥∥Aα−1∥∥ .(∥AF

(0, xk(0)

)−AF (0, 0)∥

+∥∥AF (0, 0)

∥∥))+∥∥Aα−1

∥∥ .(∥AF ( tk, xk(tk)) −AF (0, 0)∥+∥∥AF (0, 0)

∥∥)+

1

Γ(q)

tk∫0

(tk − s)−α+q−1(∥AF(s, xk(s)−AF (0, 0)∥+

∥∥AF (0, 0)∥∥) ds

+1

Γ(q)

tk∫0

(tk − s)q−1∥∥Aα−β

∥∥∥∥Q(tk − s)∥∥∥AβG

(s, xk(s)

)∥ ds

ON THE EXISTENCE OF MILD SOLUTIONS 147

5 M(|x0|α + L2|xk|∞ + L3

)+

((M + 1)

∥∥Aα−1∥∥ +

MαT−α+q

(−α+ q)Γ(q)

)(L1|xk|∞ +

∥∥AF (0, 0)∥∥)

+ L1T

(∥∥Aα−1∥∥ +

T−α+q

(−α+ q)Γ(q)

)+M

∥∥Aα−β∥∥ T q

Γ(q + 1)∥µ∥L1 .

Dividing on both sides by k and taking the lower limit as k → +∞, we have

ML2 +

((M + 1)C1−α +

MαT−α+q

(−α+ q)Γ(q)

)L1 = 1,

which contradicts (2). Hence PBk ⊂ Bk for some k ∈ N.To show that the operator P has a fixed point on Bk, we first decompose

P as P = P1 + P2, where P1 : Bk → Bk is defined by

(P1x)(t) := Q(t)F(0, x(0)

)− F

(t, x(t)

)+

1

Γ(q)

t∫0

(t− s)q−1AQ(t− s)F(s, x(s)

)ds, ∀t ∈ [0, T ],

and P2 : Bk → Bk is defined by

(P2x)(t) := Q(t)[x0 − g(x)

]+

1

Γ(q)

t∫0

(t− s)q−1Q(t− s)G(s, x(s)

)ds, ∀t ∈ [0, T ].

Then we show that P1 is a contraction and P2 is a compact operator.We first show that P1 is a contraction. To see this, observe that for

t ∈ [0, T ] and x1, x2 ∈ Bk, we have by assumption (H1)∣∣(P1x1)(t)− (P1x2)(t)∣∣α

5 |Q(t)[F(0, x1(0)

)− F

(0, x2(0)

)]|α+ |F

(t, x1(t)

)− F

(t, x2(t)

)|α

+1

Γ(q)

t∫0

(t− s)q−1|AQ(t− s)[F(s, x1(s)

)− F

(s, x2(s)

)]|αds

148 M. I. ABBAS

5 MC1−αL1

∣∣x1(0)− x2(0)∣∣α+ C1−αL1

∣∣x1(t)− x2(t)∣∣α

+L1

Γ(q)

t∫0

Mα(t− s)−α+q−1 ds sup05s5T

∣∣x1(s)− x2(s)∣∣α

5 L1

((1 +M)C1−α +

MαT−α+q

(−α+ q)Γ(q)

)sup

05s5T

∣∣x1(s)− x2(s)∣∣α

and hence

|P1x1 − P1x2|∞ 5 L0|x1 − x2|∞,

where L0 := L1((1 +M)C1−α + MαT−α+q

(−α+q)Γ(q)). From (2), it is obvious that

L0 < 1 and this implies that P1 is a contraction.Now, we show that P2 is compact. Let xn ⊂ Bk with xn → x in Bk,

then by (H2)(i), we have

G(s, xn(s)

)→ G

(s, x(s)

), n → ∞,

for each s ∈ [0, T ]. Also, since

∥G(s, xn(s)

)−G

(s, x(s)

)∥ 5 2Cβµ(s), ∀s ∈ [0, T ],

and since the function θα(t− .) is integrable on [0, t], then (H3) and thedominated convergence theorem ensure that:

|P2xn − P2x|∞

= sup05t5T

∣∣∣∣∣Q(t)[g(x)− g(xn)

]

+1

Γ(q)

t∫0

(t− s)q−1Q(t− s)[G(s, xn(s)

)−G

(s, x(s)

)] ds

∣∣∣∣∣α

5 sup05t5T

(|Q(t)

[g(x)− g(xn)

]|α

+1

Γ(q)

t∫0

(t− s)q−1|Q(t− s)[G(s, xn(s)

)−G

(s, x(s)

)]|αds

)

ON THE EXISTENCE OF MILD SOLUTIONS 149

5 M∣∣g(x)− g(xn)

∣∣α

+Mα

Γ(q)sup

05t5T

( t∫0

(t− s)−α+q−1∥G(s, xn(s)

)−G

(s, x(s)

)∥ ds

)

→ 0 as n → ∞,

i.e. P2 is continuous.We shall show that

(P2x)(t) : x ∈ Bk

is relatively compact in Xα, for

all t ∈ [0, T ]. Obviously,(P2x)(0) : x ∈ Bk

is relatively compact in Xα

by (H3). Fix t ∈ (0, T ] and for each h ∈ (0, T ) and x ∈ Bk, we define theoperator P2,h by

(P2,h)(t) = Qα(t)[x0 − g(x)

]+

1

Γ(q)

t−h∫0

(t− s)q−1Qα(t− s)G(s, x(s)

)ds

= Qα(t)[x0 − g(x)

]+

Qα(h)

Γ(q)

t−h∫0

(t− s)q−1Qα(t− h− s)G(s, x(s)

)ds.

Lemma (2.1) ensures that the sets(P2,hx)(t) : x ∈ Bk

are relatively com-

pact in Xα, since∣∣(P2x)(t)− (P2,hx)(t)∣∣α

5 1

Γ(q)

t∫t−h

(t− s)q−1|Qα(t− s)G(s, x(s)

)|αds

5 1

Γ(q)

t∫t−h

(t− s)q−1∥AαQ(t− s)G(s, x(s)

)∥ ds

5 1

Γ(q)

t∫t−h

(t− s)q−1∥Aα−βQ(t− s)AβG(s, x(s)

)∥ ds

5 MCβ−α∥µ∥L1

Γ(q + 1)hq,

150 M. I. ABBAS

for every x ∈ Bk. Therefore, we deduce that P2x : x ∈ Bk is relativelycompact in Xα for all t ∈ (0, T ] and since it is compact at t = 0 we have therelatively compactness in Xα for all t ∈ [0, T ].

It remains to show that P2x : x ∈ Bk is a family of equicontinuousfunctions. To do this, we fix t1 > 0 and let t2 > t1. Then∣∣(P2x)(t2)− (P2x)(t1)

∣∣α

5∥∥Qα(t2)−Qα(t1)

∥∥α

∣∣x0 − g(x)∣∣α

+1

Γ(q)

t1∫0

(t2 − s)q−1|[Q(t2 − s)−Q(t1 − s)

]G(s, x(s)

)|αds

+1

Γ(q)

t1∫0

[(t2 − s)q−1 − (t1 − s)q−1]|Q(t1 − s)G

(s, x(s)

)|αds

+1

Γ(q)

t2∫t1

(t2 − s)q−1|Q(t2 − s)G(s, x(s)

)|αds

= A1 +A2 +A3 +A4,

where

A1 =∥∥Qα(t2)−Qα(t1)

∥∥α

∣∣x0 − g(x)∣∣α,

A2 =1

Γ(q)

t1∫0

|(t2 − s)q−1[Q(t2 − s)−Q(t1 − s)]G(s, x(s)

)|αds,

A3 =1

Γ(q)

t1∫0

|[(t2 − s)q−1 − (t1 − s)q−1]|Q(t1 − s)G

(s, x(s)

)|αds

A4 =1

Γ(q)

t2∫t1

|(t2 − s)q−1Q(t2 − s)G(s, x(s)

)|αds.

Actually, A1, A2, A3 and A4 tend to 0 independently of x as t2 − t1 → 0.Indeed, since Qα(.) is immediately norm-continuous in Xα by Lemma (2.1),it follows by the relative compactness of g(Bk) that A1 tends to 0 indepen-dently of x as t2 − t1 → 0.

ON THE EXISTENCE OF MILD SOLUTIONS 151

Moreover, we have

A2 51

Γ(q)

t1∫0

(t2 − s)q−1∥Aα[Q(t2 − s)−Q(t1 − s)

]G(s, x(s)

)∥ ds

5 1

Γ(q)

t1∫0

(t2 − s)q−1∥Aα−β[Q(t2 − s)−Q(t1 − s)

]AβG

(s, x(s)

)∥ ds

5 Cβ−α∥µ∥L1

Γ(q)

t1∫0

(t2 − s)q−1∥[Q(t2 − s)−Q(t1 − s)

]∥ ds.

Therefore, the continuity of Q(.) allows us to conclude that A2 tends to 0independently of x as t2 − t1 → 0.

Also, we have

A3 51

Γ(q)

t1∫0

∣∣(t2 − s)q−1 − (t1 − s)q−1∣∣∥AαQ(t1 − s)G

(s, x(s)

)∥ ds

5 1

Γ(q)

t1∫0

∣∣(t2 − s)q−1 − (t1 − s)q−1∣∣∥Aα−βQ(t1 − s)AβG

(s, x(s)

)∥ ds

5 MCβ−α∥µ∥L1

Γ(q)

t1∫0

∣∣(t2 − s)q−1 − (t1 − s)q−1∣∣ ds

5 MCβ−α∥µ∥L1

Γ(q + 1)|t2 − t1|q.

Hence, A3 tends to 0 independently of x as t2 − t1 → 0.Finally, since

A4 51

Γ(q)

t2∫t1

|(t2 − s)q−1∥AαQ(t2 − s)G(s, x(s)

)∥ ds

5 1

Γ(q)

t2∫t1

|(t2 − s)q−1∥Aα−βQ(t2 − s)AβG(s, x(s)

)∥ ds

152 M. I. ABBAS

5 MCβ−α∥µ∥L1

Γ(q)

t2∫t1

∣∣(t2 − s)q−1∣∣ ds

5 MCβ−α∥µ∥L1

Γ(q + 1)|t2 − t1|q.

then A4 tends to 0 independently of x as t2− t1 → 0. Consequently, P2 mapsBk into a family of equicontinuous functions.

Thus, the Ascoli–Arzela theorem guarantees that P2 is a compact op-erator, and hence P = P1 + P2 is a condensing map on Bk. By the fixedpoint theorem of Sadovskii [14], P has a fixed point x(.) in Bk, and there-fore, the nonlocal Cauchy problem (1) has a mild solution. This completesthe proof.

3. Application

Consider the partial differential equation of fractional order

(3)∂α

∂tα[u(x, t) + F

(t, u(x, t)

)] = −

∑|q|52m

aq(x)Dqxu(x, t) + G

(t, u(x, t)

),

where t ∈ [0, T ], x = (x1, . . . , xn) ∈ Ω, Ω is abounded domain in the n-dimensional Euclidean space Rn with smooth boundary ∂Ω, Dq

x = Dq1x1 . . .

Dqnxn , Dxi =

∂∂xi

, q = (q1, . . . , qn) is an n-dimensional multi-index, and |q| =q1 + · · ·+ qn.

We assume that the nonlocal condition holds

(4) u(x, 0) +

p∑i=1

ciu(x, ti) = u0(x).

Let X = L2(Ω) be the set of all square integrable functions on Ω. Wedenote by Cm(Ω) the set of all continuous real-valued functions defined onΩ which have continuous partial derivatives of order less than or equal to m.

By Cm0 (Ω) we denote the set of all functions f ∈ Cm(Ω) with compact

supports. Let Hm0 (Ω) be the completion of Cm

0 (Ω) with respect to the norm

∥f∥m =

[ ∑|q|5m

∫Ω

[Dq

xf(x)]2

dx

]1/2.

ON THE EXISTENCE OF MILD SOLUTIONS 153

Define the operator A : D(A) j X → X by

D(A) = H2m(Ω) ∩Hm0 (Ω),

A =∑

|q|52m

aq(x)Dqxu(x, t).

Let us suppose the following conditions:

(i) The operator A =∑

|q|=2m aq(x)Dqx is strongly elliptic in Ω. In other

words, there exists a constant c > 0 such that

(−1)m∑

|q|=2m

aq(x)ξq = c|ξ|2m

for all x ∈ Ω, 0 < t 5 T and for all ξ = (ξ1, . . . , ξn) = (0, . . . , 0), and ξ,(ξq = ξq11 . . . ξqnn , |ξ|2 = ξ21 + · · ·+ ξ2n

).

(ii) All the coefficients aq are continuous and bounded on Ω, for all |q| =2m.

Then the operator A generates an equicontinuous C0 semigroup on X(see [12]).

We assume that1. F,G : [0, T ]× X → X are defined by

F (t, z)(x) = F(t, z(x)

), x ∈ Ω

G(t, z)(x) = G(t, z(x)

), x ∈ Ω

2. g : C([0, T ],X

)→ X is given by

g(z)(x) =

p∑i=1

ciz(ti)(x), x ∈ Ω.

Now, let us observe that the problem (3), (4) may be reformulated asthe abstract problem (1) under the above conditions.

By using Theorem 2.1, the problem (3), (4) has at least one mild solution

u ∈ C([0, T ], L2(Ω)

)provided that the hypotheses (H1)–(H3) hold.

154 M. I. ABBAS: ON THE EXISTENCE OF MILD SOLUTIONS

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