Scholarly article on topic 'Fractional neutral evolution equations with nonlocal conditions'

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Academic research paper on topic "Fractional neutral evolution equations with nonlocal conditions"

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Fractional neutral evolution equations with nonlocal conditions

Hamdy M Ahmed*

Correspondence: Hamdy_17eg@yahoo.com Higher Institute of Engineering, El-Shorouk Academy, P.O. 3, El-ShoroukCity, Cairo, Egypt

Abstract

In the present paper, we deal with the fractional neutral differential equations involving nonlocal initial conditions. The existence of mild solutions are established. The results are obtained by using the fractional power of operators and the Sadovskii's fixed point theorem. An application to a fractional partial differential equation with nonlocal initial condition is also considered. MSC: 26A33; 34K30; 34K37; 34K40

Keywords: fractional calculus; semilinear neutral differential equations; semigroups; nonlocal conditions; mild solutions; Sadovskii fixed-point theorem

1 Introduction

The nonlocal condition, which is a generalization of the classical condition, was motivated by physical problems. The pioneering work on nonlocal conditions is due to Byszewski (see [1-3]). Existence results for semilinear evolution equations with nonlocal conditions were investigated in [4-6]. Neutral differential equations arises in many areas of applied mathematics and such equations have received much attention in recent years. A good guide to the literature for neutral functional differential equations is the Hale book [7].

Fractional differential equations describe many practical dynamical phenomena arising in engineering, physics, economy and science. In particular, we can find numerous applications in viscoelasticity, electrochemistry, control, electromagnetic, seepage flow in porous media and in fluid dynamic traffic models (see [8-10]). The result obtained is a generalization and a continuation of some results reported in [11-15].

The main purpose of this paper is to study the existence of mild solutions of semilinear neutral fractional differential equations with nonlocal conditions in the following form

cDa[x(t)+F(t, x(t), x(b1(t)),...,x(bm(t)))] + Ax(t) = G(t,x(t),x(fl1(f)),...,x(fl„(f))), t e J = [0,b], x(0)+ g (x)=xo, (1.1)

ft Spri

ringer

where -A is the infinitesimal generator of an analytic semigroup and the functions F, G and g are given functions to be defined later. The fractional derivative cDa, 0 < a < 1 is understood in the Caputo sense.

© 2013 Ahmed; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Preliminaries

Throughout this paper, X will be a Banach space with the norm || • || and -A : D(A) ^ X is the infinitesimal generator of an analytic compact semigroup of uniformly bounded linear operators {S(t), t > 0}. This means that there exists a M > 1 such that || S(t) ||< M. We assume without loss of generality that 0 e p (A). This allows us to define the fractional power AY, for 0 < y < 1, as a closed linear operator on its domain D(AY) with inverse A-y. We will introduce the following basic properties of Ay.

Theorem 2.1 (see [16])

(1) Xy = D(Ay) is a Banach space with the norm || x ||y = || Ayx ||, x e Xy .

(2) S(t): X ^ Xy for each t >0 and AyS(t)x = S(t)AYx for each x e Xy and t > 0.

(3) For every t >0, AyS(t) is bounded on X and there exists a positive constant Cy such that

A S(t)|| < ^. (.1)

(4) If 0 < ¡3 < y < 1, then D(Ay) D(A3) and the embedding is compact whenever the resolvent operator ofA is compact.

Let us recall the following known definitions.

Definition 2.1 (see [17-19]) The fractional integral of order a >0 with the lower limit zero for a function f can be defined as

1 f * 'af (t) = -pr-r

r(a) Jo

f(s) . f _

as, t > 0

(t - s)

provided the right-hand side is pointwise defined on [0,to), where r(-) is the Gamma function.

Definition 2.2 (see [17-19]) The Caputo derivative of order a with the lower limit zero for a function f can be written as

cDaf (t) = , * f"l(s\ ds = In-af(n)(t), t > 0,0 < n - 1 < a < n.

J ' r(n - a) J0 (t - s)a+1-n J '

If f is an abstract function with values in X, then the integrals appearing in the above definitions are taken in Bochner's sense.

We list the following basic assumptions of this paper.

(H1) F: J x Xm+1 ^ X is a continuous function, and there exists a constant ¡3 e (0,1) and Mi,M2 > 0 such that the function APF satisfies the Lipschitz condition:

AF(s1,x0,x1,...,xm)-APF(S2,y0,y1,...,ym)\ <MA |s1 -S21 + , max ||x; -,

V i=0,1,...,m /

for 0 < s1, s2 < b, xi, yi e X, i = 0,1,..., m and the inequality

\APF(t,x0,x1,...,xm)\< M^ max ||xi| +1, (2.2)

holds for (t,xq,x1,...,xm) e J x Xm+1.

(H2) The function G: J x Xn+1 ^ X satisfies the following conditions:

(i) for each t e J, the function G(t, ■): Xn+1 ^ X is continuous and for each

(xo,x1,...,xn) e Xn+1 the function G(-,xo,x1,...,xn): J ^ X is strongly measurable;

(ii) for each positive number q e N, there is a positive function hq(-): [0, b] ^ R+ such that

sup ||G(t,xo,x1,...,xn)|| < hq(t),

\\xo\\,-,\\xn\\<q

the function s ^ (t - s)1-ahq(s) e i1([0, t], R+) and there exists a A >0 such that

0 (t - s)1-a hq(s) ds lim inf—---= A < to, t e [0, b],

q^TO q

(H3) ai,bj e C(J,J), i = 1,2,...,n, j = 1,2,...,m. g e C(E,X), here and hereafter E = C(J,X), and g satisfies that:

(i) There exist positive constants M3 and M4 such that \\g(x) \\ < M3 \\x\\ + M4 for all x e E;

(ii) g is a completely continuous map.

At the end of this section, we recall the fixed-point theorem of Sadoviskii [20], which is used to establish the existence of the mild solution of the nonlocal Cauchy problem (1.1).

Theorem 2.2 (Sadovskii's fixed-point theorem) Let $ be a condensing operator on a Ba-nach space X, that is, $ is continuous and takes bounded sets into bounded sets, and fi($(B)) < fi(B) for every bounded set B of X with 11(B) > 0. If $(Y) c Y for a convex, closed and bounded set Y ofX, then $ has a fixed point in X (where ¡x(-) denotes Kura-towski's measure ofnoncompactness).

3 Main result

In this section, we study the existence of mild solutions for the neutral fractional differential equations with nonlocal conditions (1.1), so we introduce the concept of a mild solution.

Definition 3.1 (see [21, 22]) A continuous function x(-): J ^ X is said to be a mild solution of the nonlocal Cauchyproblem (1.1) if the function (t - s)a-1ATa (t - s)F(s, x(s), x(b1(s)), ...,x(bm(s))), s e [0, b) is integrable on [0, b) and the following integral equation is verified:

x(t)=Sa(t)[x0 + F(0,x(0),x(bi(0)),...,x(bm(0))) -g(x)]

- F(t, x(t), x(b1(t)),...,x(bm(t)))

- i (t-s)a-1ATa(t-s)F(s,x(s),x(b1(s)),...,x(bm(s)))ds

+ i (t-s)a-1Ta(t-s)G(s,x(s),x(a1(s)),...,x(an(s))) ds, 0 < t < b, (3.1) 0

/> TO /> TO

Sa (t)x = na (0 )S(tae) xdO, Ta (t)x = a 0qa (0 )S(tae)xde 00

with na is a probability density function defined on (0, to), that is na (0) > 0, 0 e (0, to)

and /Qœ na (0 ) d0 = 1.

Remark 0n (0 ) d0 =

r(1+a) '

Lemma 3.1 (see [22]) The operators Sa (t) and Ta (t) have the following properties:

(I) for any fixed x e X, ||S„ (t)x|| <M\\x\\, T (t)x\\ < ^ M;

(II) {Sa(t), t > 0} and {Ta(t), t > 0} are strongly continuous;

(III) for every t >0, Sa (t) and Ta(t) are also compact operators;

(IV) for any x e X, £ e (0,1) and S e (0,1), we have AT a (t)x = A1-e Ta (t)Ap x and

\ASTa(t)\< aSa-S)), t e (0,V]-

Theorem 3.1 If the assumptions (H1)-(H3) are satisfied and x0 e X, then the nonlocal Cauchy problem (1.1) has a mild solution provided that

Lo = Mi

Ci«r(1 + P )baP

(M + 1)Mo + 1 P (-

y ' pr(1 + aP )

< 1 (3.2)

MoM2 + M3 +

C1-pr(1 + P )baP M2 , .

+ MoM2 + 1 P \ x-2 < 1, (3.3)

0 2 Pr(1 + aP ) v '

r(a + 1)_ where M0 = \\A~P

Proof For the sake of brevity, we rewrite that

(t, x(t), x(bi(t)),...,x(bm(t))) = (t, v(t)) and (t, x(t), ^a1(t)),...,^a„(t)y) = (t, u(t)).

Define the operator $ on E by

($x)(t) = Sa(t)[x0 + F(0, v(0)) -g(x)] - F(t, v(t)) - f (t - s)a-1ATa(t - s)F(s, v(s))<

+ i (t - s)a-1Ta(t - s)G(s, u(s)) ds, t e J. J0

For each positive integer q, let Bq = {x e E: \x(t) \ < q,0 < t < b}. Then for each q, Bq is clearly a bounded closed convex set in E. From Lemma 3.1 and (2.2) yield

i (t - s)a-1ATa (t - s)F(s, v(s)) ds Jo

f || (t - s)a-1A1-P Ta (t - s)APF(s, v(s)) I ds

i (t - s)aP-11APF(s, v(s))| ds

aC1-Pr(1 + P ) r(1 + aP )

^ Ci_¿r(i + p )bap " pr(l + ap )

^ Ci-pr(l + p)bap " pr(l + ap )

M2( max +l)

\i=l,2,...,m /

M2(q + l) (3.4)

it follows that (t - s)a ATa (t - s)F(s, v(s)) is integrable on J, by Bochner's theorem [23] so $ is well defined on Bq. Similarly, from (H2)(ii), we obtain

í (t - s)a-1Ta(t - s)G(s, «(s)) ds < Í || (t - s)a-1Ta(t - s)G(s, «(s))| ds Jo Jo

< aM Cn ^a-l

Í (t - s)a-l|G(s,4s)) || ds Jo

r(a +1) aM

-n/ (t - s)a-%(s) ds. (3-5)

r(a + 1) Jo

We claim that there exists a positive number q such that $Bq c Bq. If it is not true, then for each positive number q, there is a function xq(-) e Bq, but $xq £ Bq, but ||$xq(t)|| > q for some t(q) e J, where t(q) denotes that t is dependent of q. However, from equations (2.2), (3.4) and (3.5) and (H3)(i), we have

q < ||($*q)(t)|| <M[ ||xo || + MoM2(q + 1) + (M3q + M4)] + MoM2(q + 1)

+ Cle'rT Z' M2<q+»+fr+m ft - ^ ds. (3.)

+ afi) r(a + 1),/o

Dividing both sides of (3.6) by q and taking the lower limit as q ^ we get

MoM2 + M3 +

r(a + l)_

Cl-pr(l + P )bap M2 + M0M2 + l p ( p) —2 > l. 0 2 pr(l + ap ) >

This contradicts (3.3). Hence, for positive q, $Bq c Bq.

Next, we will show that the operator $ has a fixed point on Bq, which implies that equation (1.1) has a mild solution. We decompose $ as $ = $1 + $2, where the operators $1 and $2 are defined on Bq, respectively, by

($lx)(t) = 5a (t)F(0, v(0)) - F(t, v(t)) - Í (t - s)a-lATa (t - s)F(s, v(s)) ds

($2*)(t)=Sa(t)[*0-g(*)] + Í (t - s)a-lTa(t - s)G(s, u(s)) ds,

for t e J. We will show that $1 verifies a contraction condition while $2 is a compact operator.

To prove that satisfies a contraction condition, we take xi,x2 e Bq, then for each t e J and by condition (Hi) and (3.2), we have

|($ixi)(t)-($ix2)(t)|| < ||5a (t)[F (0, Vi(0)) - F (0, V2(0))]| + ||F(t, vi(t)) - F (t, V2

+ f (t - s)a-iATa (t - s)[F(s, n(s)) - F (s, V2(s))] ds Jo

C\ r(i + p )M\baP

< (M + i)MoMi sup |Xi(s)-X2(s)1 + i-PoW,-—i- sup |Xi(s)-X2(s)1.

o<s<b ¡¡i (i + ap) o<s<b

Hence,

|($iXi)(t)-($iX2)(t)| < Mi

(M + i)Mo +

Ci_pr(i + p )bap pr(i + ap )

sup ||Xi(s) - X2(s) |

= L0 sup |xi(s)-x2(s)|.

|($ixi)(i)-($ix2)(i)| <L0 sup ||xi(s)-x2(s)||,

and by assumption 0 < L0 < 1, we see that is a contraction. To prove that $2 is compact, firstly we prove that $2 is continuous on Bq. Let {xn} c Bq with Xn ^ x in Bq,thenforeach s e J, Un (s) ^ u(s), and by (H2)(i), we have G(s,un(s)) ^ G(s,u(s)), as n ^to. By the dominated convergence theorem, we have

||$2*n - $2X||

Sa (t) [£(*)-£(*„)]

i (t-s)a-iTa(t-s)[G(s,un(s)) - G(s,u(s))] ds Jo

as n ^ to, that is continuous.

Next, we prove that the family {$2x : x e Bq} is a family equicontinuous functions. To do this, let e >0 small, 0 < i1 < t2, then

||($2X)(i2)-($2X)(il)||

< ||5a (t2)-5a (ti)| |X0- g(x)||

r ti-e

+ / |(t2- s)a-1Ta (t2- s)-(ti- s)a-1Ta (ti- s)||||G(s, u(s)) | ds

+ /ti |(t2- s)a-iTa (t2- s) - (ti - s)a-1Ta (ti- s)||G(s, u(s))| ds

+ /t2 |(t2 - s)a-1Ta (t2 - s)| |G(s, u(s))|| ds.

We see that ||($2x)(t2)-($2x)(ti)| tends to zero independently of x e Bq as t2 ^ ti,withe sufficiently small since the compactness of Sa (t) for t >0 (see [i6]) implies the continuity of Sa (t) for t > 0 in t in the uniform operator topology. Similarly, using the compactness of the set g(Bq) we can prove that the function $2x, x e Bq are equicontinuous at t = 0. Hence, $2 maps Bq into a family of equicontinuous functions.

It remains to prove that V(t) = {($2x)(t): x e Bq} is relatively compact in X. Obviously, by condition (H3), V(0) is relatively compact in X.

Let 0 < t < b be fixed, 0 < e < t, arbitrary S > 0, for x e Bq, we define

($2,Sx) (t)= J na(0)S(t?0) [x0-g(x)] d0

pt-e pto

+ a / 0(t - s)a-ina(0)S((t - s)a0)G(s, u(s)) d0 ds J0 Js

= S(eaS) J na(0)S(ta0 - eaS)[x0 -g(x)] d0

p t-e p to

xS(eaS) / 0(t - s)a-ina (0)S((t - s)a0 - eaS) G(s, u(s)) d0 ds. J0 JS

Since S(eaS), eaS > 0 is a compact operator, then the set Ve,S(t) = {($2,Sx)(t): x e Bq} is relatively compact in X for every e, 0 < e < t and for all S >0. Moreover, for every x e Bq, we have

|| ($2*)(t) - x) (t) | < / na (0 )S(f0) [X0 - g(x)} d0

n0(t - s)a-ina(0)S((t - s)a0)G(s, u(s)) d0 ds

0(t - s)a-ina(0(t - s)a0)G(s, u(s)) d0 ds

r- t-e /> to

- / 0(t - s)a-ina(0)S((t - s)a^G(s, u(s)) d0 ds J0 Js

i na(0)S(ta0)[x0-g(x)] d0 J0

n0 (t - s)a-ina (0 )S((t - s)a0 )G(s, u(s)) d0 ds

p t p TO

/ / 0(t - s)a-ina(0)S((t - s)a0)Gs, u(s)) d0 ds

J t-e J S

<M[yx01| + M3IXH + Mi]j na(0)d0

+ aM^j^ (t - s)a-ihq(s) d^ 0na(0) d0

+ aMl (t - s)a-ihq(s) ds

0na (0 ) d0

< M[IIX0I + M3|x| + M4] / na(0) d0

+ aM^(t - s)a-1hq(s)jo (0) d0

+ f (t - sr-ihq(s) ds.

r(l + a) Jt-e

Therefore, there are relative compact sets arbitrary close to the set V(t), t > 0. Hence, the set V(t), t > 0 is also relatively compact in X.

Thus, by Arzela-Ascoli theorem $2 is a compact operator. Those arguments enable us to conclude that $ = $l + $2 is a condensing map on Bq, and by the fixed-point theorem of Sadovskii there exists a fixed point x(-) for $ on Bq. Therefore, the nonlocal Cauchy problem (l.l) has a mild solution, and the proof is completed. □

4 Example

Let X = L2([0, n ], R), we consider the following fractional neutral partial differential equations

u(t,z)+ I a(z,y)u(t,y) dy = 32u(t,z) +3zh(t, u(t,z)), 0 < t < b,0 < z < n,

u(t,0) = u(t, n ) = 0, 0 < t < b,

u(0,z) + y"] I k(z,y)u(ti,y) dy = uo(z), 0 < z < n, (4.l)

i=i j0

where cda is a Caputo fractional partial derivative of order 0< a < l, b >0, z e [0, n ], p is a positive integer, 0 < t0 < tl < ••• < tp < b.

U0(z) e X = L2([0,n],R), k(z,y) e L2([0,n] x [0,n],R).

We define an operator A by Af = —f" with the domain

D(A) = {f(■) e X :/,/absolutely continuous,f" e X,f(0) = f (n) = 0-

Then -A generates a strongly continuous semigroup (5(t))t>0 which is compact, analytic, and self-adjoint. Furthermore, -A has a discrete spectrum, the eigenvalues are -k2, n e N, with the corresponding normalized eigenvectors un(z) = (2/n)1/2 sin(nz). We also use the following properties:

(a) Iff e D(A), then Af = n2 (/, un>un.

(b) For each f e X, A-1/2f = K f, un>un. In particular, yA-1/21| = 1.

(c) The operator A1/2 is given by

A1/2f ^ nf,

un > un

on the space D(A1/2) = (f (■) e X, £ nf, un>un e X}.

The system (4.1) can be reformulated as the following nonlocal Cauchy problem in X:

cDa [x(t)+F(t, x(t), x(bi(t)),...,x(bm(t)))] + Ax(t) = G(t,x(t),x(fli(i)),...,x(fl„(i))), t e J = [0,b], x(0) + g(x) =xo,

where x(t) = u(t, ■) that is (x(t))(z) = u(t, z), t e [0, b], z e [0, n]. The function F :[0, b] x X ^ X is given by

(F(t,p))(z)= / a(z,y)p(y)dy

holds for (p, t) e [0, b] x X ^ X and z e [0, n]. The function G :[0, b] x X ^ X is given by

(G(t, p))(z) = dzh(t, u(t, z))

holds for (p, t) e [0, b] x X ^ X and z e [0, n ], and the function g : E ^ X is given by

g(x) = J2 Kgx(ti),

where Kg(u)(z) = /0 k(z,y)u(y) dy, for z e [0, n].

We can take a = 1 and G(t, x) = ^173 sinx, then (H2) is satisfied. Furthermore, assume that M3 = M4 = (p + 1) [/0n fn k2 (z, y) dy dz]1/2. Then (H3) is satisfied (noting that Kg : X ^ X is completely continuous). Moreover, we assume the following conditions hold: (i) The function a(z,y), z,y e [0, n] is measurable and

I I a2(z,y) dydz < to.

(ii) The function dza(z,y) is measurable, a(0,y) = a(n,y) = 0, and let

fn fn 2 1

/ / (3za(z,y) dydz

Therefore, the conditions (H1)-(H3) are all satisfied. Hence, according to Theorem 3.1, system (4.1) has a mild solution provided that (3.2) and (3.3) hold.

Competing interests

The author declares that he has no competing interests. Acknowledgements

I would like to thank the referees and Professor Ravi Agarwalfor their valuable comments and suggestions. Received: 27 November 2012 Accepted: 10 April 2013 Published: 24 April 2013

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doi:10.1186/1687-1847-2013-117

Cite this article as: Ahmed: Fractional neutral evolution equations with nonlocal conditions. Advances in Difference Equations 2013 2013:117.

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