CC BY
34
UDC 517.95 Вестник СПбГУ. Математика. Механика. Астрономия. 2021. Т. 8 (66). Вып. 3
MSC 34D05, 34K37, 47H08, 47H10
Asymptotically almost periodic solutions of fractional evolution equations
Nguyen Duc Huy, Vu Trong Luong
VNU University of Education, Vietnam National University, 144, Xuanthuy, Caugiay, Hanoi, Vietnam
For citation: Nguyen Duc Huy, Vu Trong Luong. Asymptotically almost periodic solutions of fractional evolution equations. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2021, vol. 8(66), issue 3, pp. 475-483. https://doi.org/10.21638/spbu01.2021.309
We study the asymptotic behavior of solutions of nonlinear fractional evolution equations in Banach spaces. Asymptotically almost periodic solutions on half line are obtained by establishing a sharp estimate on the resolvent operator family of evolution equations. In particular, when the semigroup generated by A is exponentially stable then the conditions of the existence asymptotically almost periodic solutions is satisfied. An application to a fractional partial differential equation with initial boundary condition is also considered. Keywords: fractional evolution equations, almost periodic solutions, resolvent operator family.
1. Introduction. We consider nonlinear fractional evolution equations of the form
D^u(t) = Au(t) + f (t,u(t)), t > 0, u(0) = x, 0 < a < 1, (1.1)
where the (unbounded) linear operator A generates a strongly continuous semigroup on a Banach space X, f : R+ x X ^ X, Dgu(t) stands for the derivative of the function u in the sense of Caputo. Unlike ordinary differential equations, fractional differential equations generally do not have periodic solutions (see [1]). For the above reason, some authors have studied the problem of the existence of almost periodic solutions of fractional differential equations (see [2, 3]).
The existence of asymptotic almost periodic solutions to equations on the half line is an interesting topic in the qualitative theory of differential equations. We refer the reader to the monographs [4] and the papers [5, 6] for a more complete account of the methods as well as results concerning this topic in the last decades. The reader is referred to [7-10] and the references therein for more information. Almost periodic solutions, asymptotically periodic solutions and other types of behavior of solutions to fractional differential equations have been studied in many papers (see, e. g., [11-14] and references therein). In papers [11-13], the authors have investigated Equation (1.1) with 1 < a < 2, so, the operator of mild solutions is simpler than that in the case 0 < a < 1. In fact, in Laplace transform of equation in this case, the factor Aa—1 is analytic in A, 1 < a. In [14] the author considers the fractional integro-differential equations with the fractional derivative is understood in Weyl's sense in the following form
Dau(t) = Au(t) + f a(t - s)Au(s)ds + f (t, u(t)), t e R, 0 < a < 2. (1.2)
J—w
© St. Petersburg State University, 2021
He proved that if f is almost periodic and satisfies some Lipschitz type conditions, then there exists a unique almost periodic mild solution of (1.2) given by
u(t) = / Sa(t — s)f (s,u(s))ds,
•J —w
where Sa (t) is the a-resolvent family generated by A. In [15], the authors consider the semilinear fractional differential equation
Dg+1u(t) + ^Dg u(t) = Au(t) + Dg f (t,u(t)), t > 0, 0 < a < £ < 1, ^ > 0,
where A is an w-sectorial operator of angle ^n/2 with w < 0. Under some conditions on f they obtained the asymptotic behavior of bounded solutions. However, when the input data is not fractional derivative of function f, the problem has not been studied yet. Hence, motivated by this open question we are going to study the asymptotic behavior of bounded solutions of problem (1.1). To this end, we will need to prove some estimates on a-resolvent family generated by A in Section 3. The main results of this paper are Theorems 3.6, 4.3 that appear to be new to our best knowledge.
2. Preliminaries. 2.1. Almost periodic functions. A subset E c R is said to be relatively dense if there exists a number l > 0 (inclusion length) such that every interval [a, a + l] contains at least one point of E. Let f be a continuous function on R taking values in a complex Banach space X. f is said to be almost periodic in the sense of Bohr if to every e > 0 there corresponds a relatively dense set T(e, f) (of e-periods) such that
sup ||f (t + T) — f (t)|| < e, Vt G T(e, f).
te r
If f is almost periodic function, then (Approximation Theorem [16, Chap. 2]) it can be approximated uniformly on R by a sequence of trigonometric polynomials, i. e., a sequence of functions in t G R of the form
N (n)
Pn(t):=X) a„ifce!A"-kt, n = 1, 2,...; A„jfc G R, a„jfc G X, t G R.
k=1
Of course, every function which can be approximated by a sequence of trigonometric polynomials is almost periodic. Specifically, the exponents of the trigonometric polynomials (i. e., the reals An,k in (2.1) below) can be chosen from the set of all reals A (Fourier exponents) such that the following integrals (Fourier coefficients)
1 rT
a(X,f):= lim — / f(t)e-iXtdt
T —yw 2T J — t
are different from 0. As is known, there are at most countably such reals A, the set of which will be denoted by <rb(f) and called Bohr spectrum of f.
We denote by
AP(X) := {f G BUC(R, X) : f is almost periodic}
end define
AP(R+, X) := {f |r+ ,f G AP(X)}.
From the definition of almost periodicity it is easy to prove that
sup ||f(t)|| = sup ||f(t)|| for all f G AP(X).
te R te R+
Therefore, the operator of restriction of a function from AP(X) to the half line R+ is actually an invertible isometry from AP(X) onto AP(R+, X). Later on we will sometimes identify the function f G AP(X) with its restriction to the half line R+. Now, let us denote by
Co(X):= {/ G BUC(R+,X): lim ||f(t)|| = 0},
t^+w
and define the space of asymptotically almost periodic functions
AAP(R+ ,X) := AP(R+ ,X) © C0(X). 2.2. Fractional differentiation in Caputo's sense. Below we denote
ta-1
(i) = tttT' t > °> a > r(a)
Let 0 < a,t > a, and a is a fixed number. Then, the fractional operator
Jau(t):=(ga * u)(t)= / ga(t - t)f (t)dr (2.1)
a
is called fractional Riemann — Liouville integral of order a.
The Caputo fractional derivative of u of order a is defined by
{1 it u(n)(T)
Jn~au^(t) = —.-- ---dr, n-l<a<nGN,
W r(n — a) Ja (t - t)«+1-™
u(n)(t), a = n G N.
When 0 < a < 1, we have JaD gu(t) = u(t) — u(a), and
DgU (A) = Aa — Aa-1 u(a),
where Dgu(A) is Laplace transform of Dgu.
3. Cauchy Problem for linear case. Consider the problem
D gu(t) = Au(t) + f (t), u(0) = x, 0 < a < 1, (3.1)
where f G BC(R+, X), A : D(A) c X ^ X is a closed linear operator that generates a C0-semigroup in X. By using the Laplace transform, we have
u(A) = Sa(A)x + ((.)^Pa)(A) • f(A). (3.2)
where (.)°-1Pa(A) = (Aa — A)-1,Sa(A) = Aa-1(Aa — A)-1. The inversion of the Laplace transform shows that u(t) has the following form
t(t) = Sa(t)x + / (t - s)a-1Pa(t - s)f (s)ds. Jo
Ю
Hence, mild solutions of (3.1) are defined as functions u : R+ ^ X satisfy
u(t) = Sa(t)x + [ (t - s)a-1Pa(t - s)f (s)ds, t > 0. (3.3)
J 0
By the subordination principle (see [17]), Sa and Pa, a G (0,1], exist if A generates a C0-semigroup {T(t)}t>0. The explicit formulas of Sa and Pa were given in [18, 19]:
pœ
Sa(t) = $a(s)T (tas)ds,
o
fœ
a
a(
o
/• œ
Pa(t) = a / вФ^в^(tas)ds,
o
where is a probability density function defined on (0, to), that is, $a(t) > 0 and Jo" $a(t)dt = 1. Moreover, has the expression
a
V>a(0) = - ^(-l)™-1^-«™-1^^-^!! sin(n7ra), e G (0, oo). n ^^ n!
n= 1
We recall the definition and properties of Mittag —Leffler functions. The Mittag —Leffler function Ea, ß is defined as follows
EaAz) = ^0ia>ß>0> ZGC-
Lemma 3.1. These Mittag — Leffler functions have following properties for a G (0, 1), t G R:
(i) Ea,i(t),E
a, a
(t) > 0;
(ii) (EaA)'(t) = ¿£7a,a(i);
(iii) lim Ea1(t)= lim Ea a(t) = 0.
t — — " ' t —)— ^o '
Proof. The assertions (i) and (iii) are implied from [20, Lemmas 2.1 and 2.2]. It follows from the definition of Mittag —Leffler function that
nt™-1 1 tn 1
(Ea¡1)'(t) = ]Г --— = - ———— = -Ea,a(t), t G IR.
n=iГ(ап + 1) а П=о Г(а(п + 1)) а
Therefore the assertion (ii) hold.
Definition 3.2 [21]. A strongly measurable family of operators
{T(t)}t>о С L(X)
is called uniformly integrable (or strongly integrable) if
/ ||T(t)||dt< то.
о
In this paper, for any uniformly integrable family of such operators
{T(t)}t>o C L(X),
we will use the notation ||T|| := /J" ||T(t)||dt. In order to study the asymptotic almost periodic solutions to problem (1.1), we need the following assumption.
(A) The operator A generates a Co-semigroup {T(t)}t>o which is uniformly integrable and lim ||Sa(t)|| =0.
Remark 3.3. Assume that the semigroup {T(t)}t>0 generated by A is exponentially stable, i.e., there are positive numbers a, M such that
||T(t)|| < Me-"1, t > 0.
By the fact that (see [22])
TO
/ Фа(%—dtf = Ea,i(-z), Jo
/• w
/ а0Фа(%—d<9 = Ea,a(-z),
o
we have
/• w w
пади / Фа(0)||Т(flta)||d0 </ Фа(%—^= Ea,i(-ata),
oo
and
ww
||Pa(t)|| < а/ 0Фа(0)||Т (tfta)||dtf < mJ 0Фа(0)е—d<9 = MEa,a(-ata).
oo
From the estimates above, we obtain
||Sa(t)||, ||P«(t)|H 0, t
and
/ (t - s)a-1|Pa(t - s)||ds < m/ (t - s)a-1Ea,a(-a(t - s)a)ds <
J—w J—w
M
< — £7ail(-a(t-s)a)
a
M a
Lemma 3.4. Leí f G AP(X), and set
w(t) = / (t - s)a—1Pa(t - s)f (s)ds.
—w
Then w belongs to AP(X).
Proof. If f e AP(X), then for each e > 0 there exists a relatively dense set T(e, f) such that
sup ||f (t + т) - f (t)||< e, Vr e T(e, f).
t£lR
w
For each т e T(e, f), we have
SUp ||w(t + T) - w(t)|| =SUp || / (t - s)a-1Pa(t - s)[f (s + T) - f (s)]ds||<
te R te R J-то
/t Z't /'TO
(t - s)a-1||Pa(t - s)||ds < e / (t - s)a-1 a / 0Фа(0)|Т((t - s)a0)||d0ds <
-то J —oo J 0
/»^o /»то
< e Фа(0)^0 / ||T(s)||ds = e||T||,
00
and therefore, w is almost periodic.
The essential tool in the establishing of our results is given in the following lemma. Lemma 3.5. Let f e AAP(R+, X), and Ff is defined by
(Ff )(t) = f\t - s)a-1Pa (t - s)f (s)ds.
0
Then Ff belongs to AAP(R+, X).
Proof. If f e AAP(R+, X), f = g + h, where g e AP(X) and h e C0(X), then we have that (Ff )(t) = Gf (t) + Hf (t), where
(Gf )(t)= Г (t - s)a-1Pa(t - s)g(s)ds,
J -то
(Hf )(t) = I (t - s)a-1Pa(t - s)f (s)ds - I (t - s)a-1Pa(t - s)g(s)ds.
0 -то
By Lemma 3.4 Gf e AP(X). Next, we will prove Hf e C0(X). From h e C0(X), for each e > 0 there exists a number T > 0 such that ||h(s)|| < e, Vs > T. Then for all t > 2T, we have
ft/2 ft ||(Hf)(t)||<||h|| / (t - sr-1||Pa(t - s)||ds + e (t - s)a-1 ||Pa(t - s)||ds+ 0 t/2
/0 /»то
(t - s)a-1||Pa(t - s)||ds < (||g|| + ||h||) / Sa-1|Pa (s)||ds + e||T ||.
то t/2
Note that
то
/ Sa-1|Pa(s)|ds < ||T|| < TO,
0
So (Hf )(t) — 0, as t —у to. Hence, Ff e AAP(R+, X).
As an immediate consequence of Lemma 3.5, we obtain the following theorem.
Theorem 3.6. Let (A) hold. Then, for each f e AAP(R+,X) all mild solutions u of (3.1) are in AAP(R+ ,X).
4. Asymptotic behavior of mild solution for semilinear problem. In what follows we study existence asymptotically almost periodic mild solutions of problem (3.1). We denote by M(R+ x X,X) the set of all functions f : R+ x X — X such that f (., x) e AAP(R+, X) uniformly for each x e K, where K is any bounded subset of X.
Let f € M(R+ x X, X), the Nemyskii superposition operator is defined as follows
N(у)(.) := f(.,у(.))
for у € AAP(R+,X).
Theorem 4.1 [15]. Given f € M(R+ x X, X), suppose that there exists a constant L > 0 such that
||f (t,x) - f (t,y)|| < L||x - y||,
for all t € R+,x,y € X, and let у € AAP(R+ ,X). Then N(y) € AAP(R+,X).
Motivated by formula (3.3), we have the following definition of mild solutions for semilinear problem.
Definition 4.2. A function u : R+ ^ X is said to be a mild solution of problem (1.1) if it satisfies the integral equation
u(t) = Sa(t)x + i (t - s)a-1 Pa(t - s)f (s,u(s))ds, t > 0, x € X. (4.1) Jo
Theorem 4.3. Let Assumption (A) be made, and let f : R+ x X ^ X be a function in M(R+ x X, X). Assume further that there exists a bounded integrable function L : R+ ^ R+ satisfying
|f (t,x) - f (t, y)| < L(t)||x - y||,
for all t € R+, x, y € X. Then all mild solution u of (1.1) are asymptotic almost periodic whenever L := supt>0 ||L(t)|| < ||T||-1.
Proof. We define the operator F on the space AAP(R+, X) by
F(u)(t) = Sa(t)x + i (t - s)a-1 Pa(t - s)f (s,u(s))ds := Sa(t)x + Fa(t).
o
Note that, by the assumption (A), we have lim ||Sa(t)x|| = 0. It implies that
Sa(t)x € C0(X). It follows from Theorem 4.1 that the functions s ^ f(s,u(s)) is in AAP(R+, X); then according to Lemma 3.5, F € AAP(R+,X). Hence, F is well defined on AAP(R+,X). Then u is an asymptotically almost periodic mild solution of problem (1.1) iff it is a fixed point of operator F. Let u, v be in AAP(R+, X). We have following estimate
||F(u)(t) - F(v)(t)|| < ||u - v|| /\t - s)a-1|Pa(t - s)||L(s)ds < L||T||||u - v||.
Jo
This proves that F is a contraction, so by Banach fixed point theorem there exists a unique u € AAP(R+, X). The proof is completed.
Let Q be a bounded domain in Rn with sufficiently smooth boundary dQ. We consider the following problem:
Dau(t, x) = Au(t, x) + f (t, u(t, x)), u(0, x) = y(x), x € Q. (4.2)
Let X = L2(Q), A = Д with D(A) = H2(Q)ПЩ1(Q). Then problems (4.2) is in the form of the abstract model (1.1).
It is known that A generates a compact (and hence equicontinuous) semigroup T(■), which is exponentially stable, i. e. ||T(t)|| < e-Ali, with Ai > 0 being the first eigenvalue of A (see, e. g., [23]). Therefore, the assumption (A) is satisfied. As a result of Theorem 4.3, we obtain the following theorem.
Theorem 4.4. Let f : R+ x X ^ X be a function on M(R+ x X, X) and assume that there exists a bounded integrable function L : R+ ^ R+ satisfying
|f (t,x) - f (t,y)|| < L(t)||x - y||,
for all t € R+, x, y € X. Then all mild solution u of (4.2) are asymptotic almost periodic whenever L := supi>0 ||L(t)|| < A1.
The authors wish to thank VNU University of Education for support. References
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Received: September 13, 2020 Revised: March 14, 2021 Accepted: March 19, 2021
Authors' information:
Nguyen Duc Huy — PhD; [email protected]
Vu Trong Luong — PhD; [email protected], [email protected]
Асимптотически почти периодические решения дробных эволюционных уравнений
Д.Х.Нгуен, Т.Л.Ву
Вьетнамский национальный университет, Вьетнам, Ханой, Хуантхуы, Цаугиаы, 144
Для цитирования: Nguyen Duc Huy, Vu Trong Luong. Asymptotically almost periodic solutions of fractional evolution equations // Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия. 2021. Т. 8(66). Вып. 3. С. 475-483. https://doi.org/10.21638/spbu01.2021.309
Изучается асимптотическое поведение решений нелинейных дробных эволюционных уравнений в банаховых пространствах. Асимптотически почти периодические решения на полулинии получаются путем установления строгой оценки семейства резольвентных операторов эволюционных уравнений. В частности, когда полугруппа, порожденная A, экспоненциально устойчива, выполняются условия существования асимптотически почти периодических решений. Рассматривается также применение полученных результатов при решении дробных дифференциальных уравнений в частных производных с начальным граничным условием.
Ключевые слова: дробные эволюционные уравнения, почти периодические решения, семейство резольвентных операторов.
Статья поступила в редакцию 13 сентября 2020 г.;
после доработки 14 марта 2021 г.; рекомендована в печать 19 марта 2021 г.
Контактная информация: Нгуен Дук Хай — [email protected]
Ву Тронг Луонг — [email protected], [email protected]
