Cent. Eur. J. Math.

DOI: 10.2478/s11533-013-0381-y

VERS ITA

Central European Journal of Mathematics

On the nonlocal Cauchy problem for semilinear fractional order evolution equations

Research Article

JinRong Wang1'2*, Yong Zhou3î, Michal FeCkan4'5*

1 School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, China

2 Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

3 Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China

4 Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynska dolina, Bratislava 842 48, Slovakia

5 Mathematical Institute, Slovak Academy of Sciences, Stefanikova 49, Bratislava 814 73, Slovakia

Received 29 May SOI 3; accepted 1 September SOI 3

Abstract: In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507-516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465-4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O'Regan fixed point theorem.

MSG: 26A33, 34A12, 47D06, 34G20

Keywords: Fractional order evolution equations • Nonlocal Cauchy problem • Mild solution • Existence © Versita Sp. z o.o.

* E-mail: sci.jrwang@gzu.edu.cn î E-mail: yzhou@xtu.edu.cn

* E-mail: Michal.Feckan@fmph.uniba.sk

Springer

1. Introduction

The nonlocal condition has a better effect on the solution and Is more precise for physical measurements than the classical condition alone. For the contribution to the nonlocal Cauchy problem for nonlinear evolution equations we refer the reader to Byszewski [5, 6], Jackson [14], Deng [8], Liang et al. [17], Ntouyas and Tsamatos [25] and other papers (see for instance [4, 7, 11-13, 18, 30, 35] and references therein).

Boucherif and Precup [2] explored a new approach and conditions to study existence of solutions to the following initial value problem for first order differential equations with nonlocal conditions:

x'(t) = F(t,x(t)), for a.e. t e J = [0,1],

x(0) + ^ akx(tk) = 0, k = 1,2.....m,

where F: J x r — r is a given function and ak are real numbers with ^m=1 ak = —1 and tk, k = 1, 2,..., m, are given points satisfying 0 < t1 < t2 < ... < tm < 1. The idea was to put less restrictive conditions on F by splitting the growth condition on F into two parts, one for t e [0, tm] and the other for t e [tm, 1].

In [3] Boucherif and Precup adopted the idea of [2] via fixed point methods and presented existence results for mild solutions to the following nonlocal Cauchy problem for first order evolution equations:

x'(t)+ Ax(t) = f(t,x(t)), t e J,

x(0) + ^ akx(tk) = 0, k = 1,2.....m,

where A: D(A) C X — X is the generator of a C0-semigroup {T(t) : t > 0} on a Banach space X and f: J x X — X is a given function.

In [23, 24] Nica and Precup developed further the approach and techniques of [2] and applied them in order to study the nonlocal Cauchy problem for first order nonlinear differential systems.

Recently, fractional order differential equations found application in studies related with viscoelasticity, electrical circuits, nonlinear oscillation of earthquake and etc. There appeared a number monographs which provide with the main theoretical tools for the qualitative analysis of fractional order differential equations, and at the same time show the interconnection as well as the contrast between integer order differential models and fractional order differential models [1, 9, 15, 16, 19, 20, 27, 29].

A pioneering work on existence of solutions to the following initial value problem for fractional order differential equations with nonlocal conditions:

CDa0tx(t) = f(t, x(t)), a e (0,1), t e J, x(0) + G(u) = x0, x0 e X,

where the symbol CDat denotes the Caputo fractional derivative of order a with the lower limit zero, f: J x X — X and the nonlocal term G: C(J,X) — X, is due to N'Guerekata [21]. In [22] N'Guerekata noted that the results from [21] hold only in finite dimensional spaces. Dong et al. [10] revisited the above problem and presented some new existence results under certain suitable conditions, extending the results of [21] to infinite dimensional spaces.

Zhou and Jiao [36] studied the following nonlocal Cauchy problem for fractional order evolution equations:

cD0) tx(t) = Ax(t) + f(t, x(t)), a e (0,1), t e J, x(0) + G(x) = x0, x0 e X.

They gave a suitable definition of a mild solution associated with characteristic solution operators of this problem and established existence results in the case when f and G satisfy Lipschitz continuous and growth conditions on J via Banach and Krasnoselskii fixed point theorems.

Motivated by [2, 3, 33, 34, 36] we investigate existence of mild solutions to the following Cauchy problem for fractional order evolution equations with nonlocal conditions:

CDa0tx(t) = Ax(t) + f(t, x(t)), a e (0,1), t e J,

m (1) x(0) = 2_ akx(tk), k = 1,2.....m.

We develop the approach and techniques from the above papers and establish two new existence results under general and weak assumptions on f by utilizing fractional calculus and Schaefer and O'Regan fixed point theorems. We give a suitable definition of a mild solution to equation (1) by introducing a bounded operator B = [/ — Yjm=-\ akT(tk)] . Our first existence result relies on a growth condition on J and the second one relies on a growth condition involving two parts, one for [0, tm], and the other for [tm, 1]. Our assumptions on f are more general and less restrictive than those imposed in [34, 36].

2. Preliminaries

Let C(J,X) be the Banach space of all X-valued continuous functions from J into X endowed with the norm ||x||c(j,X) = supteJ ||x(t)||. For brevity, we denote ||x||c = ||x||c(j,X).

Definition 2.1 ([15]).

The fractional integral of order y with the lower limit a e r for a function f: [a, to) ^ r is

"»-rirt lv-hds' Y> 0

provided that the righthand side is point-wise defined on [a, to), where r( •) is the gamma function. The Riemann-Liouville derivative of order y with the lower limit zero for a function f: [0, to) —> r is

LDy0tf (•)-=-—, f , f ds, •> 0. n - 1 <y<n. "■• ( » r(n - y» dtn J0 (t - s»Y+1-n

The Caputo derivative of order y for a function f : [0, to» ^ r is

D/ (t»-LD0Jf (•» - Yl*- f V)(0»\ , t> 0, n - 1 < y < n. \ -=0 ' '

Remark 2.2.

If f is an abstract function with values in X, then the integrals in the definition are understood in Bochner's sense. Suppose M — sup || T(t)|| and define

T(t»- f Za(9»T(ta9» dQ, S(t) — af 6ta(9»T(ta9» d6, t > 0,

^a(9» — 1 9-1-1/aOa(9-1/a» > 0, a

Oa(9» — 1 Y(-1»n-19-na-1 r(na +1» sin(nna», 9 e (0^», n '— n!

n —1

where Za is a probability density function defined on (0, oo), that is

Za(9) > 0, 0 e (0, TO), / Za(9) de = 1.

In a recent paper, Zhou and Jiao [37] gave some basic properties of t and s which will play an important role in the sequel.

Lemma 2.3 ([37, Lemmas 3.2-3.4]).

(i) For any fixed t > 0 and any x e X, ||t(i)x| < M\\x\\ and ||§(i)x| < M\\x\\/r(a).

(ii) {T(t) : t > 0} and {s(t) : t > 0} are strongly continuous.

(iii) For each t > 0, t(t) and S(t) are compact operators if T(t) is compact.

Further properties of t and s were explored by Wang and Zhou [31, 32]. Suppose that there exists the bounded operator B: X — X given by

I - ^ ak7(tk

Applying [33, Theorem 3.3, Remark 3.4] we can give two sufficient conditions for the existence and boundedness of the operator B.

Lemma 2.4.

The operator B defined in (2) exists and is bounded if one of the following two conditions holds: (Q) there are real numbers ak such that

m ^ KI < 1;

(C2) T(t) is compact for each t > 0 and the homogeneous linear nonlocal problem

CDa0tx(t)= Ax(t), a e (0,1), t e J, x(0) = ^ akx(tk),

has no non-trivial mild solutions.

Proof. Under assumption (C1), from Lemma 2.3 (i) and (3) we have

^ akT(tk

< M ¿\ak| < 1.

Thus by the Neumann theorem, B exists and it is bounded. Under assumption (C2), it is obvious that mild solutions to (4) have the form x(t) = T(t)x(0), hence

x(0) = ¿ akx(tk) = ¿ akT(tk)x(0).

By Lemma 2.3 (iii), t(tk) is compact for each tk > 0, k = 1, 2,..., m. Thus ^™=1 ak7(tk) is also compact. Since problem (4) has no non-trivial mild solutions, one obtains the desired result applying the Fredholm alternative theorem. □

Similarly to [36], one can Introduce the following definition of mild solutions to (1).

Definition 2.5.

A function x G C(J,X) is called a mild solution to (1) if it satisfies the following equation:

x(t) = T(t) ^ akB(g(tk)) + g(t), t G J, (5)

g(tk ) = f\tk - s)a-1S(tk - s)f (s,x(s)) ds, (6)

g(t) = f (t - s)a-1S(t - s)f(s,x(s)) ds, t G J. (7)

Remark 2.6.

Due to [36] a mild solution to fractional evolution equation (1) with the initial condition is x(t) = T(t)x(0) + g(t), so taking into account our nonlocal condition, we get

x(0) = ^ akT(tk)x(0) + ^ akg(tk).

k=1 k=1

So x(0) = Z_mk=i akB(g(tk)) and hence x(t) = T(t) ^mk=i akB(g(tk)) + g(t), it is exactly (5).

3. First existence result

Our first existence result Is based on the well-known Schaefer fixed point theorem [28].

Theorem 3.1.

Let F: X —> X be a continuous mapping of X into X which is compact on each bounded subset of X. Then either

(i) the equation x = AFx has a solution for A = 1, or

(ii) the set {x e X : x = AFx for some A e (0,1)} is unbounded.

In this section, we will study our problem under the following assumptions: (Ht) f : J x X ^ X satisfies the Carathéodory conditions.

(H2) There is a function h such that Igth(t) exists for all t G J and Igrh(-) G C((0,1], r+) with Hm(^0+ Iojh{t) = 0 and a nondecreasing continuous function Q: r+ ^ r+ such that

\\f(t,*)|| < h(t)Q(\\x\\)

for all x G X and for almost every t G J.

Remark 3.2.

In our previous works [34, 36], we assumed that there exists a function h e L1/ai(J, r+), a e [0, a), where LP(J, r+) denotes the Banach space of all Lebesgue measurable functions h: J ^ r+ with the norm of h given by

\\h\kp(j,ii+)

\h(t)\vPdt

1 < p < <X),

inf sup \h(t)\, p = œ,

M=° tGj-J

where ^(J) is the Lebesgue measure on J. However, it is not difficult to verify that the old (strong) condition h e Lvai(j, r+), a1 e [0, a), implies a new (weak) condition /0ah( ■) e C((0,1], r+) with Hm /0ath(t) = 0.

(H3) The inequality Umsup p(M2BQ(p) £\ak\/a0tkh(tk)+ MQ(p)sup /0ath(t)| > 1 holds.

\ k=i , teJ , I

(H4) T(t) is compact for each t > 0. We consider the following problem:

CD0>(t)= Ax(t)+ Xf(t,x(t)), a e (0,1], X, t e J, x(0) = £ akx(tk). (8)

Define an operator F: C(J,X) ^ C(J,X) as follows:

(Fx)(t) = (Fix)(t) + (F2x)(t), t e J, where FL: C(J,X) —> C(J,X), i = 1,2, are given by the formulas

(F1x)(t) = T(t) £ akB(g(tk)), Fx)(t) = g(t),

where B is the operator defined in (2), g(tk) is defined in (6) and g(t) is defined in (7). Obviously, a mild solution to equation (8) is a solution to the operator equation

x = XFx (9)

and conversely. Thus, we can apply the Schaefer fixed point theorem to derive the existence of solutions to equation (1).

Lemma 3.3.

There exists a constant R* > 0 independent of the parameter X e J such that ||x||C < R* for every solution x to equation (9).

Proof. Denote R0 = ||x||C. Taking into account our conditions and Lemma 2.4 (C1), (C2), it follows from (5) that

||x(t)|| < ||(F1x)(t)|| + ||(F2x)(t)|| < M £\ak\||B||||g(tk)|| + ||g(t)||, t e J. (10)

Note that

ftk M itk

||g(tk)||</ (tk - s)a-1HS(tk - S)||f(s,x(s))| ds < — (tk - s)a-1h(s)Q(|x|C) ds

J0 r(a) J0

< ^^ fk(tk - s)a-1h(s) ds = MQ(R0)/a0tkh(tk), k = 1, 2.....m,

l( a) J0

f (t - s)a-ih(s) ds = MQ(R0)sup /0th(t), t e J. (11)

l( a) J0 teJ

From (10)-(11), one has

R0= Hc < M2||B||Q(R0) £\ak\/g,tl!h(tk)+ MQ(R0)sup /a^^h(t), t e J,

which Implies

rJm2^B^Q(R0) ¿\ak\Ia0tkh(tk) + MQ(Ro)sup Ia0th(t)\ < 1. (12)

\ k=i tGJ I

However, according to (H3), there exists R* > 0 such that for all R > R* we have

R |m2||B||Q(R) Y_\ak\Ig.tkh(tk) + Mfi(R)sup ^J > 1. (13)

Now, comparing (12) and (13), we deduce that R0 < R*. As a result, we find that ||x||C < R*. This completes the proof. □

Let BR* = {x G C(J,X) : ||x||C < R*}. Then BR* is a bounded closed and convex subset in C(J,X). By Lemma 3.3, we can derive the following result.

Lemma 3.4.

The operator F maps BR* into itself.

Lemma 3.5.

The operator F : BR* ^ BR* is completely continuous.

Proof. For our purpose, we only need to check that Ft : BR* ^ BR*, i = 1, 2, is completely continuous. Firstly, by repeating the procedure of our previous work (see Step III in the proof of [36, Theorem 3.1]), one can obtain that F2 : BR* —> BR* is completely continuous. We only emphasize that the main difference is that the condition h G L1 ai(J, r+), a G [0, a), is replaced by the new condition I0a h( • ) G C((0,1], r+) with Um I0ath(t) = 0.

Secondly, one can check that F1 : BR* —> BR* is continuous (by (Hi), (H2) and Lemma 2.3 (i)) and F1 : BR* ^ BR* is compact since t(t) is compact for each t > 0 (by (H4) and Lemma 2.3 (iii)). □

Now, we can state the main result of this section.

Theorem 3.6.

Assume that (H1)-(H4) hold and condition (C1 ) (or (C2)) is satisfied. Then equation (1) has at least one solution u G C(J,X) and the set of solutions to equation (1) is bounded in C(J,X).

Proof. Obviously, the set {x G C(J,X) : x = XFx, 0 < A < 1} is bounded due to Lemma 3.4. Now we can apply Theorem 3.1 to derive that F has a fixed point in BR* which is just the mild solution to equation (1). □

4. Second existence result

Our second existence result is based on the O'Regan fixed point theorem [26].

Theorem 4.1.

Let U be an open set in a closed, convex set c of X. Assume 0 e U, T(U) is bounded and T: U ^ c is given by T = T| + T2 where T1: U ^ X is completely continuous, and T2: U —> X is a nonlinear contraction. Then either

(i) T has a fixed point in U, or

(ii) there is a point x e dU and X e (0,1) with x = XT(x).

In addition to (H^, (H4) and (C|) (or (C2)), motivated by Boucherlf and Precup [2, 3], we Introduce the following two assumptions:

(H5) There exists a function h such that Igth(t) exists for every t G [0,tm] and Igh( ■) G C((0,tm], r+) with Um(^o+ I§th(t) = 0 and a nondecreasing continuous function Q: r+ ^ r+ such that ||f(t,x)|| < h(t)Q(\\x||) for all x G X and for a.e. t G [0, tm], and for every t G [tm, 1] there exists a function l such that If tl(t) exists and I".l( ■) G C([tm, 1], r+) such that ""

t ¥(t,x)||< l(t), (14)

for all x G X and for a.e. t G [tm, 1]. Moreover, Q has the property

r>MQ(r)l ¿ K WB! + 1 | sup 0th(t) (15)

\ k=1 I tG[°,,tm]

for all r> R* > 0.

(H6) There exists a function q such that I"itq(t) exists for every t G [tm, 1] and I".q(.) G C([tm, 1], R+) with M sup Igtq(t) < 1 and a nondecreasing continuous function ^: r+ ^ r+ with ^(r) < r for r > 0 such that

tG\tw,1] '

Hf(t,x) - f(t,y)H < q(t)Y(||x - y||)

for a.e. t G [tm, 1] and for all x, y G X. Consider equation (8) again and the equivalent equation

x = ATx, (16)

where T: C(J,X) ^ C(J,X) is defined by (Tx)(t) = (T1 x)(t) + (T2x)(t), t G J, T: C(J,X) ^ C(J,X), i = 1, 2, are given

(T x)(t) =

(T2x)(t) =

T(t) i akB(g(tk)) + g(t), t G [0,tm),

" r t"

T(t)5~ akB(g(tk))+ (t - s)a-1S(t - S)f(s,x(s)) ds, t G [" 1],

k=1 Jo

0, t G [0, t"),

J (t - s)a-1S(t - s) f(s,x(s)) ds, t G [t", 1].

We first prove that solutions to equation (16) are a priori bounded.

Lemma 4.2.

There exist R* > 0, i = 1, 2, independent of the parameter A, such that ||x|C([0,tt],X) < R* and ||x|C([tm,i],X) < R*, that is ||x||c < R* = max {R*, R*} for every solution x of the equation (16).

Proof. Case 1. We prove that there exists R* > 0 such that ||x|C([0,tt],X) < R*. For t G [0, tm] and A G J, denote R[0t"] = ||x||C(\o,t"],x), we have

||x(t)|| < AUhxm + H(T2x)(t)|| < M £\ak|||B||||g(tk)H + ||g(t)||

" M i tk M it

< MY_\ak\||B|| ^ (tk - s)a-1h(s)Q{R{otm]) ds + (t - s)a-1 h(s)Q{R0.tm]) ds

k=1 l( a) J0 l( a) J0

< MQ{R[0,tm]H il\ak\||B|| +1) sup I^h(t),

k=1 G0"

which Implies

R^ < MQ(R[0A]H ¿1 K\||B|| + 1 sup a,h(t).

\ k=1 I t^[0'tm]

From (15) we find that there exists R* > R[o,tm] > 0 such that ||x||c([0,tm],x) < R*-

Case 2. We prove that there exists R* > 0 such that ||x||c([tm,i],x) < R*- For t e [tm, 1] and A e J, keeping in mind our assumptions, we find that

\\x(t)\\ < M £\ak \\\e\\ ra V (tK - sp^OR*) ds k=1 ( )

+ M y (t - s)a-1h(s)Q(R*) ds + M f (t - s)a-1h(s) ds r( «Mo r( a) Jtr„

< MQ(R*) \ak\\\B\\ + 1 sup /«tA(t)+ M sup /^/(t),

\ k=1 j te[0,tm] te[tm,1]

which Implies ||x\\c([tm,i],x) < R*, where

R2 = M

Q(R*) £\ak\\\B\\ + 1 sup 0Mt)+ sup /U(t)

\ k=1 j te[0,tm] te[tm,1]

Let R* = max {R*,R2*}. Then all solutions of the equation (16) satisfy \\x\\C < R*, where R* is independent of the parameter A. □

Denote d = {x G C(J,X) : \\x\\C < R* + 1}. We can proceed as in the proof of Lemma 4.2 to derive the following result.

Lemma 4.3.

T(d) is bounded.

One can proceed as in the proof of Lemma 3.5 to obtain the following result.

Lemma 4.4.

The operator T1 : d ^ C(J, X) is completely continuous.

Lemma 4.5.

The operator T2 : d ^ C(J, X) is a nonlinear contraction.

Proof. From the definition of T2 we only need to show that T2: d ^ C([tm, 1],X) is a nonlinear contraction. In fact, for any x, y G d and t G [tm, 1], we have

\\(T2X)(t) - (T2y)(t)\\ <ft(t - s) a-1 ||s(t - s)[f(s, x(s)) - f(s, y(s))]|| ds < Mïit(t - s)a-1q(s)^(\x(s) - y(s)\\) ds

< M^(l'x( - y\\C ) f (t - s) a-1q(s) ds < (m sup /ltq(t))n\\x - y\\C ), r(a) Jtm \ te[W] I

which implies \\T2x - T2y\\C < 1J(\\x - y\\C). □

Now, we are ready to present the main result of this section.

Theorem 4.6.

Assume that (Ht), (N4), (H5) and (H6) hold and condition (C|) (or (C)) is satisfied. Then equation (1) has at least one solution u G C(J, X).

Proof. By Lemma 4.2 we see that (ii) in Theorem 4.1 does not hold for U = d. Therefore, from Theorem 4.1, T has a fixed point in d which is just the mild solution to the equation (1). This completes the proof. □

Finally, we try to change the conditions (H5) and (H6) to the following parallel conditions: (H5') Condition (H5) is assumed without (14).

(He') Denoting 5 = Urn \nf^V(r)jr < 1, condition (H6) is assumed in addition with

M5 sup Igtq(t) < 1.

tG[tm,1] ,

Corollary 4.7.

The existence result in Theorem 4.6 also holds even if (H5) and (He) are replaced by the conditions (H5') and (He') respectively.

Proof. Indeed, we can modify Case 2 in the proof of Lemma 4.2 as follows:

T( a) I

' r( a) I

< MQ(R*) I jr\0k|||e|| + 1 I sup Ia0,th(t) +

m KA rtfc KA ftm

*(t)||< MY_K \||B|| ^ (tk - s)a-1h(s)Q(R^) ds + — (t - s)a-1 h(s)Q(R1) ds

r( a) J0 r( a) Jo

M r (t - s)a-1 If (s, 0)|| ds + M ft (t - s) a-1q(s) Y(||x(s)||) ds

M supte|W]||f(t, 0)| (1 - tm)a

k=i 1 tG\0^

M f (t - s)a-1q(s)(5||x(s)|| + 5i) ds,

r( a + 1)

for some > 0. Then we have

1 - M5suptGitm,i]Io,tq(t)

Mfi(RÎ) £\ak\||6|| +1\ sup OMt)

tG[0,tm]

sup Hf(t, 0) | (1 - tm)a+ M5isup 0tq(t)

i( a+\) tG[tm,t] tG[tm,i]

The rest proof is standard. So we omit it here.

Acknowledgements

The authors thank the referee for valuable comments and suggestions which improved their paper.

The first author acknowledges the support by National Natural Science Foundation of China (11201091) and Doctor Project of Guizhou Normal College (13BS010). The second author acknowledges the support by National Natural Science Foundation of China (11271309), Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001) and Key Projects of Hunan Provincial Natural Science Foundation of China (12JJ2001). The third author acknowledges the support by Grants VEGA-MS 1/0071/14, VEGA-SAV 2/0029/13 and APVV-0134-10.

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