CC BY
15
Chelyabinsk Physical and Mathematical Journal. 2020. Vol. 5, iss. 4, part 1. P. 4-15-4-27.
DOI: 10.47475/2500-0101-2020-15403
ASYMPTOTICALLY (w,c)-ALMOST PERIODIC TYPE SOLUTIONS OF ABSTRACT DEGENERATE NON-SCALAR VOLTERRA EQUATIONS
M. KostiC1'", V.E. Fedorov2'36
1Novi Sad University, Novi Sad, Serbia 2 Chelyabinsk State University, Chelyabinsk, Russia "marco.s@verat.net, bkar@csu.ru
We analyze asymptotically (w, c)-almost periodic type solutions of abstract degenerate non-scalar Volterra equations in Banach spaces. In order to do that, we primarily consider the exponentially decaying rate of (A, k, B)-regularized C-pseudoresolvent families. We illustrate the obtained results with some examples and possible applications.
Keywords: asymptotically (w,c)-almost periodic function, asymptotically (w,c)-almost automorphic function, abstract degenerate non-scalar Volterra equation.
1. Introduction and preliminaries
There are by now only a few relevant references concerning abstract non-scalar Volterra equations, degenerate or non-degenerate in time variable. Concerning non-degenerate abstract Volterra equations of non-scalar type, mention should be made of the research monograph [1] by J.Priiss, the article [2] by M.Jung and the article [3] by M. Kostic. In [4], we have explained how the methods proposed in [1] and [3] can be helpful in the analysis of abstract degenerate Volterra equations of non-scalar type.
The theory of almost periodic functions, almost automorphic functions and their applications is a very popular and growing field of research. The main aim of this paper is to initate the study of the existence and uniqueness of asymptotically almost periodic type solutions of the abstract degenerate non-scalar Volterra equations. In actual fact, we investigate asymptotically (u, c)-almost periodic type solutions of the abstract degenerate non-scalar Volterra equations in Banach spaces.
The organization and main ideas of paper can be briefly described as follows. After collecting the preliminary definitions about asymptotically (u, c)-almost periodic functions and asymptotically (u,c)-almost automorphic functions in Subsection 1.1, we remind ourselves of the various notions of (A, k, B)-regularized C-pseudoresolvent families introduced in [4] (Section 2). Theorem 1, Theorem 2 and Proposition 1 are new contributions of ours given in this section. In Section 3, we analyze the existence and uniqueness of asymptotically (u, c)-almost periodic type solutions of the abstract degenerate Cauchy problem
t
Bu(t) = f (t) + j A(t - s)u(s) ds, t E [0, t); (1)
0
see Section 2 for the notion and more details.
We use the standard notation throughout the paper. Let (X, || ■ ||X) and (Y, || ■ ||Y) be two non-trivial complex Banach spaces such that Y is continuously embedded in X, and let L(X, Y) denote the space consisting of all linear continuous operators from X into Y; L(X) = L(X, X). By || ■ ||L(X,Y) we denote the norm in L(X, Y). The following condition on a scalar valued function k(t) will be used:
(P1) k(t) is Laplace transformable, i.e., it is locally integrable on [0, to) and there exists p e R such that fc(A) := L(k)(A) := /0e-Aifc(t) dt := e-Aifc(t) dt exists
for all A e C with Re A > p. Put abs(k) :=inf{Re A : fc(A) exists}, and denote by L-1 the inverse Laplace transform.
Suppose now that c e C\{0} and w > 0. The class of (w, c)-periodic functions has recently been introduced and analyzed by E. Alvarez, A. Gomez and M. Pinto [5] and E.Alvarez, S.Castillo, M.Pinto [6]. A continuous function f : I ^ X is said to be (w, c)-periodic if and only if f (t + w) = cf (t) for all t e I. The space of all (w, c)-periodic functions f : I ^ X will be denoted with Pw,c(I : X). For c =1 and c = — 1, we obtain the spaces of w-periodic functions f : I ^ X and w-antiperiodic functions f : I ^ X; furthermore, for c = eifcw, we obtain the space of Bloch (w, k)-periodic functions f : I ^ X. Due to [5, Proposition 2.2], we know that the function f : I ^ X is (w, c)-periodic if and only if the function
fw,c(t):= c-(i/w)f(t), t e I, (2)
is periodic of period w.
The (w, c)-periodic functions play an important role in the qualitative analysis of solutions to the Mathieu linear differential equation
y"(t) + [a — 2q cos 2t]y(t) = 0,
arising in modeling of seasonally forced population dynamics. The (w, c)-periodic functions arise also as solutions of the linear delayed equations (see e.g., [5, Example 2.5]) and the Lasota — Wazewska equation with (w,c)-pseudo periodic coefficients
y'(t) = -<fy(t) + h(t)e-a(i)y(i-r), t > 0.
This equation describes the survival of red blood cells in the blood of an animal. The authors of [5] have also analyzed the existence and uniqueness of mild (w, c)-periodic solutions to the abstract semilinear integro-differential equation
Dau(t) = Au(t) + i a(t - s)Au(s) ds + f (t,u(t)), t e R,
J-xi
where Dau(t) denotes the Weyl — Liouville fractional derivative of order a > 0, a e L11oc([0, to)) is a scalar-valued kernel, the function f (■, ■) enjoys some properties and A generates an a-resolvent operator family on X.
1.1. Asymptotically (w,c)-almost periodic type functions
The notion of almost periodicity was introduced by a Danish mathematician Harald Bohr (1924-1926), the younger brother of the Nobel Prize-winning physicist Niels Bohr [7]. Suppose that f : I ^ X is a continuous function. Given e > 0, we call t > 0 an e-period for f (■) if
||f (t + t) - f (t)||< e, t e I.
The set consisting of all e-periods for f(•) is denoted by id(f,e). We say that f(•) is almost periodic if for each e > 0 the set $(f, e) is relatively dense in [0, ro), which means that there exists l > 0 such that any subinterval of I of length l meets id(f,e). Further on, we say that a function f : I ^ X is asymptotically almost periodic if and only if there exist an almost periodic function h : R ^ X and a function 0 E C0(I : X) such that f (t) = h(t) + 0(t) for all t E I. This is equivalent to saying that, for every e > 0, we can find numbers l > 0 and M > 0 such that every subinterval of I of length l contains, at least, one number t such that ||f (t + t) — f (t)|| < e provided |t|, |t + t| > M.
The class of almost automorphic functions, which extends the class of almost periodic functions, was introduced by an American mathematician Salomon Bochner in 1962 [8]. It is said a continuous function f (•) is almost automorphic if and only if for every real sequence (bn) there exist a subsequence (an) of (bn) and a map g : R ^ X such that
lim f(t + an) = g(t) and lim g(t — an) = f (t), (3)
n—n—>■<<
pointwise for t E R. If the convergence of limits appearing in (3) is uniform on compact subsets of R, then we say that f (•) is compactly almost automorphic. It is well known that an almost automorphic function f (•) is compactly almost automorphic if and only if it is uniformly continuous as well as that an almost automorphic function is always bounded. The function f : I ^ X is said to be asymptotically (compactly) almost automorphic if and only if there exist a (compactly) almost automorphic function h : R ^ X and a function 0 E C0(I : X) such that f (t) = h(t) + 0(t) for all t E I.
The following definitions has been introduced in a recent research paper by M.T. Khalladi, M. Kostic, A. Rahmani and D.Velinov:
Definition 1. Let c E C\{0} and u > 0. Then it is said that a continuous function f : I ^ X is (u,c)-almost periodic ((u,c)-almost automorphic/compactly (u,c)-almost automorphic) if and only if the function fu,c(^), defined by (2), is almost periodic (almost automorphic/compactly almost automorphic).
Definition 2. Let c E C, |c| > 1 and u > 0. Then it is said that a continuous function f : I ^ X is asymptotically (u,c)-almost periodic, resp. asymptotically (compactly) (u,c)-almost automorphic, if and only if the function fu,c(^), defined by (2), is asymptotically almost periodic, resp. asymptotically (compactly) almost automorphic.
It is clear that any (u, c)-periodic function is (u, c)-almost periodic as well as that the converse statement does not hold in general. We will be primarily focused to the notion introduced in Definition 1 and Definition 2, although one can similarly analyze the notion of (u, c)-asymptotical periodicity introduced recently by E. Alvarez, S. Castillo and M. Pinto. The Stepanov, Weyl and Besicovitch generalizations of classes introduced in Definition 1 and Definition 2 will not be considered here, as well (see [9; 10] and references cited therein for more details about this subject).
For further information concerning almost periodic functions and their applications, we refer the reader to the research monographs [11] by A.S. Besicovitch, [12] by T. Diagana, [13] by A.M. Fink, [14] by G.M. N'Guerekata, [10] by M. Kostic and [15] by S. Zaidman.
2. Solution operator families for abstract degenerate non-scalar Volterra equations
Concerning the abstract degenerate Volterra integro-differential equations, we can recommend for the reader the research monographs [16] by A. Favini, A.Yagi, [17] by
M. Kostic and [18] by G.A. Sviridyuk, V.E. Fedorov.
Let the operator C e L(X) be injective, and let t e (0, to]. We use the symbol B to denote a closed linear operator with domain and range contained in X; by
|| ■ ||[d(B)] := || ■ + ||B ■ hx
we denote the corresponding graph norm and by [D(B)] = (D(B), || ■ ||[D(B)]) we denote
the corresponding Banach space. If Z is a general topological space and Z0 C Z, then by _z
Z0 we denote the adherence of Z0 in Z. We will basically follow the notation employed in the monograph of J. Priiss [1] and our paper [4].
We start by recalling the following notion introduced in [4] (see also [17, Section2.9]):
Definition 3. Let k e C([0,t)) and k = 0, let t e (0, to], f e C([0,t) : X), and let A e L1oc([0,T) : L(Y, X)). Then we say that a function u e C([0,t) : [D(B)]) is:
(i) a strong solution of (1) if and only if u e L^c([0,t) : Y) and (1) holds on [0,t),
(ii) a mild solution of (1) if and only if there exist a sequence (fn) in C([0,t) : X) and a sequence (un) in C([0,t) : [D(B)]) such that un(t) is a strong solution of (1) with f (t) replaced by fn(t) and that limn^x fn(t) = f (t) as well as limn^x un(t) = u(t), uniformly on compact subsets of [0,t).
The following definition will be invaluably important in our further work ( [4]):
Definition 4. Let t e (0, to], k e C([0,t)), k = 0 and A e Loc([0,T) : L(Y, X)). A family (S(t))te[0,r) in L(X, [D(B)]) is called an (A, k, B)-regularized C-pseudoresolvent family if and only if the following holds:
(51) The mappings t M S(t)x, t e [0,t) and t M BS(t)x, t e [0,t) are continuous in X for every fixed x e X, BS(0) = k(0)C and S(t)C = CS(t), t e [0, t).
(52) Put U(t)x := /0 S(s)xds, x e X, t e [0,t). Then (S2) means U(t)Y C Y, U(t)r e L(Y), t e [0,t) and (U(t)|Y)te[0,r) is locally Lipschitz continuous in L(Y).
(53) The resolvent equations
BS(t)y = k(t)Cy +/ A(t - s) dU(s)y, t e [0,t), y e Y, (4)
0
BS(t)y = k(t)Cy +/ S(t - s)A(s)yds, t e [0,t), y e Y, (5)
0
hold; (4), resp. (5), is called the first resolvent equation, resp. the second resolvent equation.
An (A, k, B)-regularized C-pseudoresolvent family (S(t))te[0,r) is said to be an (A, k, B)-regularized C-resolvent family if additionally:
(54) For every y e Y, we have S(-)y e L£c([0, t) : Y).
An operator family (S(t))te[0,r) in L(X, [D(B)]) is called a weak (A, k, B)-regularized C-pseudoresolvent family if and only if (S1) and (5) hold. Finally, a weak (A, k,B)-regularized C-pseudoresolvent family (S(t))te[0,r) is said to be a-regular (a e L1oc([0, t)))
_x
if and only if a * S(-)x e C([0, t) : Y), x e YX.
As is well known, condition (S3) can be rewritten in the following equivalent form: (S3)'
BU(t)y = Q(t)Cy + f A(t - s)U(s)yds, t e [0,t), y e Y,
J 0
BU(t)y = Q(t)Cy + f U(t - s)A(s)yds,t e [0,r), y e Y.
0
We also need the following definition from [4]:
Definition 5. Let k e C([0, œ)), k = 0, A e L^oc([0, œ) : L(Y,X)), a e (0,n], and let (S(t))t>0 Ç L(X, [D(B)]) be a (weak) (A,k,B)-regularized C-(pseudo)resolvent family. Then it is said that (S(t))t>0 is an analytic (weak) (A,k,B)-regularized C-(pseudo)resolvent family of angle a, if there exists an analytic function S : ^ L(X, [D(B)]) satisfying S(t) = S(t), t > 0, S(z)x = S(0)x and
BS(z)x = BS(0)x for all 7 e (0,a) and x e X. We say that (S(t))t>0 is an exponentially bounded, analytic (weak) (A, k, B)-regularized C-(pseudo)resolvent family, resp. bounded analytic (weak) (A, k, B)-regularized C-(pseudo)resolvent family, of angle a, if (S(t))t>0 is an analytic (weak) (A, k, B)-regularized C-(pseudo)resolvent family of angle a and for each 7 e (0, a) there exist MY > 0 and > 0, resp. = 0, such that ||S(z)||L(X) + ||BS(z)||L(X) < MYeM'<lzl, z e . Since no confusion seems likely, we shall identify S(■) and S(-) in the sequel.
In [4], we have also introduced the notion of an (A,k,B)-regularized C-uniqueness family with a view to analyze the uniqueness of solutions of the abstract Cauchy problem (1):
Definition 6. Let t e (0, œ], k e C([0,t)), k = 0 and A e L^oc([0,r) : L(Y, X)). A strongly continuous operator family (V(t))te[0;T) Ç L(X) is said to be an (A,k,B)-regularized C-uniqueness family if and only if
V(t)By = k(t)Cy + Î V(t - s)A(s)yds, t e [0,r), y e Y n D(B).
0
We will use the following statements proved in [4, Proposition 2]:
[P]: Assume that (V(t))te[0,r) is an (A,k,B)-regularized C-uniqueness family, f E C([0,t) : X) and u(t) is a mild solution of (1). Then we have (kC*u)(t) = (V*f )(t), t E [0,t ).
[Q]: Assume that (S(t))te[0'T) is an (A, 1, B)-regularized C-pseudoresolvent family, C-1f E C([0,t) : X) and f (0) = 0. Then we know that the following statements hold:
(a) Let C-1f E AC\oc([0,T) : Y) and (C-1f)' E L\oc([0,t) : Y). Then the function t M- u(t), t E [0, t) given by
t t u(t) = J S(t — s)(C-1f )'(s) ds = j dU(s)(C-1f )'(t — s) 00
is a strong solution of (1). Moreover, u E C([0,t) : Y).
_x
(b) Let (C-1f)' G L11oc([0, T) : X) and Y = X. Then the function
u(t) = / S(t - s)(C-1f)'(s) ds, t G [0, t) Jo
is a wild solution of (1).
(c) Let C-1g G WlO'c1([0,T) : YX), a G LfOc([0,T)), f (t) = (a * g)(t), t G [0,t) and let (S(t))te[o,r) be a-regular. Then the function
u(t) = / S(t - s)(a * (C-1g)')(s) ds, t G [0,t) o
is a strong solution of (1).
The uniqueness of solutions in (a), (b) or (c) holds provided that for each y G Y n D(B) we have S(t)By = BS(t)y, t G [0,t).
Even in case that B = C = I and k(t) = 1, there exist examples of global not exponentially bounded (A, k, B)-regularized C-pseudoresolvent families (see e.g., [1, Example 6.2, pp. 165-166]). For our purposes, it will be crucial to examine whether the operator family (S(t))t>0 is exponentially decaying as the time variable goes to plus infinity. The existence of a number e0 > 0 such that
j e-£i||A(t)||L(Y>X) dt < TO, e > eo, (6)
o
which has been used in [1] and [3; 4], is not sufficient to ensure the exponential decaying of (S(t))t>0 as t ^ +to. Therefore, we must impose some extra conditions ensuring this property of (S(t))t>0, which will be extremely important for us.
Now we will state two simple results concerning this problematic. The both of them are basically deduced in [4]:
Theorem 1. Assume e0 > 0, k(t) satisfies (P1), u > max(abs(k), e0), (6) holds, a G (0,n/2], there exists an analytic mapping H : u + £n +Q, ^ L(X, [D(B)]) such that
(i) BH(A)y - H(A)A(A)y = fc(A)Cy, y G Y, Re A > u, fc(A) = 0; H(A)C = CH(A), Re A > u,
(ii) SUPA€w+S n
2
|(A - u)H(A)|L(x) + ||(A - u)BH(A)|L(x)
< to for all y G (0, a),
(iii) there exists an operator F G L(X, [D(B)]) such that BFx = k(0)Cx, x G X and
limA^+x;fc(A)=0 AH(A)x = Fx, x G X, and
x
(iv) limA^+X fc(A)=0 ABH(A)x = k(0)Cx, x G X, provided that Y = X.
If there exists a real number u0 < 0 such that the mapping H : u + £n +a ^ L(X, [D(B)]) can be analytically extended to the sector u0 + £n +a and condition (ii) holds with the number u replaced by the number u0 therein, then there exists a weak analytic (A,k,B)-regularized C-pseudoresolvent family (S(t))t>0 of angle a such that
sup
zes7
le-W0Z S (z )|l(x ) + H^ BS (z )|L(x)
< TO for all y G (0, a). (7)
Proof. By [4, Theorem 3], we know that there exists a weak analytic (A, k, B)-regularized C-pseudoresolvent family (S(t))t>0 of angle a, satisfying that the estimate (7) holds with the number w0 replaced by the number w. The final statement follows easily from this fact, [19, Theorem 2.6.1], the uniqueness theorem for the Laplace transform and the assumption we have made after the formulation of conditions (i)-(iv). □
We can similarly deduce the validity of the following result which corresponds to [4, Theorem 4]:
Theorem 2. Assume a G (0,n/2], e0 > 0, k(t) satisfies (P1) and (6) holds. Let u > max(abs(k), e0), and let there exist an analytic mapping H : u + £n +a M L(X, [D(B)]) such that H|Y : u + £n M L(Y) is an analytic mapping, as well as that:
(i) one has for all y G (0, a)
sup
Ae^+s2+7
|(A - w)H(A)||lw + ||(A - w)BH(A)^ + ||(A - w)H(A)||L(y}
<00 ;
(ii) BH(A)y - H(A)A(A)y = k(A)Cy, y G Y, Re A > w, k(A) = 0; BH(A)y -A(A)H(A)y = k(A)Cy, y G Y, Re A > w, k(A) = 0; H(A)C = CH(A), Re A > wo;
(iii) there exists an operator F G L(X, [D(B)]) such that BFx limA^+œ,fc(A)=o AH(A)x = FX x G X;
k(0)Cx, x G X,
(iv) ^(A)=0 ABH(A)x = k(0)Cx, x G X, provided that Y = X.
If there exists a real number u0 < 0 such that the mapping H : u + £| M L(X, [D(B)]) can be analytically extended to the sector u0 + £n +a, the mapping H|Y : u+£n M L(Y) can be analytically extended to the sector u0 + £n +a, and condition (i) holds with the number u replaced by the number u0 therein, then there exists an analytic (A,k,B)-regularized C-resolvent family (S(t))t>0 of angle a such that
sup
zes.
= -WQZ
S (z )|
L(X )
+ lie
-WQZ
BS(z)L(„) + sup He-^S(z) L(X ) zes.
lL(Y )
< oo
and the mapping t M U(t) G L(Y), t > 0 can be analytically extended to the sector £a.
Remark 1. Concerning Theorem 1, it should be noted that we can also impose condition that there exist a negative real number u < 0, a real number P G (0,1] and a number a0 G (0,n/2) such that H(■) is analytic on the region Q = u0 + £(n/2)+a, continuous on Q and satisfies the estimate
sup a en
(1 + |A|)-^ H (A)||l(x ) + ||(1 + |A|)-^ BH (A)|
L(X )
< TO.
Then the integral computation carried out in the proof of [19, Theorem 2.6.1] shows that there exists a weak analytic (A,k,B)-regularized C-pseudoresolvent family (S(t))t>0 of angle a such that
sup
zes7
1z ris (z)|l(x ) +
|e-WQZ |z|^-1BS (z)
lL(X )
< to for all y G (0, a).
e
A similar comment can be given in the case of consideration of Theorem 2.
Clearly, it is not trivial to practically verify the requirements of Theorem 1 and Theorem 2 as well as that these theorems are not suitable for applications to the abstract fractional differential equations of non-scalar type. But, in many concrete situations, the requirements of these theorems can be very simply verified:
Example 1. Suppose that X = Y, B = C = I, k(t) = 1, u0 < 0, 0 < a < n/2 and D is a closed linear operator in X such that for each number 7 e (0, a) there exists a finite real number MY > 0 such that
sup
AewQ+s(n/2)+7
A(A - D)
-1
x
(A - uo)(X - D)
-1
< 00.
Define A(^) through A(A) := (2D)/(X) — (D2)/(A2), A = 0. Then the assumptions of Theorem 2 hold true because for each 7 e (0,n/2) we have
sup
AewQ+S(n/2)+7
A — ^o
A
x
sup
AewQ+S(n/2)+7
sup
AewQ+S(n/2)+7
I - A(A) A -
A
-1
x
2D D2
1 - T + D
1
A(A - D)-1 x (A - uo){A - D)-1
<.
Further possibilities to apply Theorem 1 and Theorem 2 will be considered somewhere else. In [3, Theorem3] and [4, Theorem 2], we have considered the hyperbolic perturbation results for the abstract non-scalar Volterra equations. Before proceeding further, we want also to observe that it is very difficult to say whether the perturbed resolvent solution family will be exponentially decaying if the initial resolvent solution family is exponentially decaying as time marches to plus infinity.
Concerning the exponential decaying rate at infinity of an (A, k, B)-regularized C-pseudoresolvent family (S(t))t>0, we would like to stress that, in [10, Remark 2.6.15], we have presented a simple idea which can be also applied in the qualitative analysis of asymptotically almost periodic type solutions of the abstract degenerate non-scalar Volterra integral equations. This will be the starting point for our investigations carried out in the remainder of paper. First of all, we will clarify the following result which can be also formulated for analytic (A, k, B)-regularized C-pseudoresolvent families:
Proposition 1. Suppose that z E C, a E Llc([0,T)), k = 0, A E Llc([0,T) : L(Y,X)) and (S(t))te[0,T) is an (A,k,B)-regularized C-pseudoresolvent family (weak (A,k,B)-regularized C-pseudoresolvent family). Define
kz(t) := e-ztk(t), Az(t) := e-ztA(t), and Sz(t) := e-ztS(t), t E [0,t).
Then (Sz(t))te[0,T) is an (Az,kz,B)-regularized C-pseudoresolvent family (weak (Az,kz, B)-regularized C-pseudoresolvent family). Furthermore, (S(t))te[0,T) is a-regular if and only if (Sz(t))ie[0,T) is az-regular, where az(t) := e-zta(t), t E [0,t), and (Sz(t))te[0,T) is an (Az, kz, B)-regularized C-resolvent family if (S(t))te[0,T) is an (A, k, B)-regularized C-resolvent family and Re z < 0.
Proof. We will provide the main details of proof for (A, k, B)-regularized C-pseudoresolvent families, only. It is clear that condition (S1) holds true. In order to
show (S2), define Uz(t) := /0 Sz(s)xds, x G X, t G [0,t) and observe that the partial integration implies
Uz(t)x = e-ztU(t)x + zi e-zsU(s)xds, x G X, t G [0,t). (8)
Jo
This simply yields that Uz(t)Y C Y, Uz(t)|y G L(Y), t G [0,t) and (U(t)|y)te[o>r) is locally Lipschitz continuous in L(Y). We will prove only the first resolvent equation in (S3)' because the second resolvent equation in (S3)' [or (S3)] can be deduced almost trivially. So, let y G Y and t G [0, t) be fixed. Applying (8) twice and using the first resolvent equation in (S3)' for (S(t))te[0;T), we get
BUz (t)y = e
-zt
ft nt
e-zsk(s)Cyds + / A(t - s)U(s)yds
+
t
+ z / e-Jo
-zt
e zrk(r)Cydr + / A(s — r)U(r)ydr
ds
e-zsk(s)Cyds + ze / e
i't i's
zt I —zs I
e zrk(r)Cydrds
+
+ / A(t — s)e U(s)yds + z Jo
e-zA(0 * 1 * e-zU0)y
(t).
The use of partial integration yields that
zt
i e-zsk(s)Cyds + zezt / e-zs oo
e zrk(r)Cydrds
= e-zsk(s)Cy ds o
and the required statement simply follows because the above equality yields
BUz (t)y
e-zsk(s)Cy ds + / e-z(t-s)A(t — s)
e-zsU(s)y + z / e-zrU(r)ydr o
ds.
In order to see that (S(i))te[0,r) is a-regular if and only if (Sz(i))te[0,r) is az-regular, it suffices to observe that
(«z * Sz(-)x) (t) = e-zi(a * S(-)x)(t), t G [0,t), x G Y The remainder of proof for (A,k,B)-regularized C-resolvent families is trivial. □
o
o
s
s
o
o
t
o
o
o
t
t
s
o
s
o
o
3. Asymptotically (w,c)-almost periodic type solutions of (1)
In this section, we will analyze the existence and uniqueness of asymptotically (w, c)-almost periodic type solutions of the abstract Cauchy problem (1). First of all, we will state the following lemma whose proof is very simple and therefore omitted:
Lemma 1. Let k G C([0, t)) and k = 0, let t G (0, to], z G C, f G C([0, t) : X), and let A G L11oc([0,T) : L(Y, X)). Suppose that (V(t))te[0;T) C L(X) is an (A, k, B)-regularized C-uniqueness family. Define fz(t) := e-ztf (t), V^(t) := e-ztV(t) and Az(t) := e-ztA(t) for all t G [0, t). Then we have:
(i) if u(-) is a strong (mild) solution of problem (1), then uz(■) = e-z>u(-) is a strong (mild) solution of problem obtained by replacing respectively f (■) and A(-) in (1) by fz(■) and Az(■);
(ii) (Vz(t))t>o C L(X) is an (Az,kz, B)-regularized C-uniqueness family.
Now we will prove the following proposition:
Proposition 2. Let k E C([0, <)), k = 0, u0 > 0, u > 0, 1 > uu0, A E L11oc([0, <) : L(Y,X)) and (V(t))t>0 C L(X) is an (A, k, B)-regularized C-uniqueness family such that \\V(t)\\ < Me^0t, t > 0. If u(^) is a mild solution of (1) and f (•) is asymptotically (u,e)-almost periodic (asymptotically (u,e)-almost automorphic, asymptotically compactly (u,e)-almost automorphic), then the function (kC * u)(^) is likewise asymptotically (u,e)-almost periodic (asymptotically (u,e)-almost automorphic, asymptotically compactly (u,e)-almost automorphic).
Proof. Let z = 1/u. Due to our assumptions, we have that the operator family (Vz(t) = e-ztV(t))t>0 is exponentially decaying. By Lemma 1(i), uz(•) is a strong (mild) solution of problem obtained by replacing respectively f (•) and A(-) in (1) by fz(•) and Az(•). Due to Lemma 1 (ii), we have that (Vz(t))t>0 C L(X) is an (Az,kz,B)-regularized C-uniqueness family. Applying now [P], we get that
(kz C * Uz )(t)= (Vz * fz )(t), t > 0, i.e., e-z\kC * u)(t)= (Vz * fz )(t), t > 0.
We have that fz(•) is asymptotically almost periodic (asymptotically almost automorphic, asymptotically compactly almost automorphic), so that the function
t M (Vz * fz)(t), t > 0 has the same property [10]. This implies the required statement. □
It is clear that, if (S(t))te[0,T) C L(X, [D(B)]) is a weak (A,k,B)-regularized C-pseudoresolvent family and BS(t)y = S(t)By, t E [0,t), y E Y fl D(B), then (S(t))te[o,r) C L(X) is an (A, k, B)-regularized C-uniqueness family. Using this observation, [P], [Q] and Proposition 2, we may deduce the following:
Proposition 3. Suppose that (S(t))t>0 C L(X, [D(B)]) is an (A, 1, B)-regularized C-pseudoresolvent family, BS(t)y = S(t)By, t E [0,t), y E Y f D(B), u0 > 0, u > 0, 1 > uu0, \\S(t)\\ < Me^0t, t > 0 and f(•) is asymptotically (u,e)-almost periodic (asymptotically (u,e)-almost automorphic, asymptotically compactly (u,e)-almost automorphic). Then we have the following:
(i) Let C-1f E ACioc([0, <) : Y), (C-1f)' E LjOc([0, <) : Y) and f (0) = 0. Then there exists a unique strong solution u(^) of (1); moreover, u E C([0,t) : Y) and the mapping t M- C fQ u(s) ds, t > 0 is asymptotically (u,e)-almost periodic (asymptotically (u,e)-almost automorphic, asymptotically compactly (u,e)-almost automorphic).
_X
(ii) Let (C-1f)' E LjOc([0, <) : X) and Y = X. Then there exists a unique mild solution u(^) of (1); moreover, the mapping t M C J* u(s) ds, t > 0 is asymptotically (u,e)-almost periodic (asymptotically (u,e)-almost automorphic, asymptotically compactly (u,e)-almost automorphic).
(iii) Let C-1g E W¿¿([0, <) : YX), a E LlM <)), f (t) = (a * g)(t), t > 0 and let (S(t))t>0 be a-regular. Then there exists a unique strong solution u(^) of (1); moreover, the mapping t M C /0 u(s) ds, t > 0 is asymptotically (u,e)-almost periodic (asymptotically (u,e)-almost automorphic, asymptotically compactly (u,e)-almost automorphic).
It is worth noting that Proposition 3 can be deduced directly, as well as that some sufficient conditions ensuring the above features of mapping t M u(t), t > 0 can be also achieved. We will explain this only in the case of consideration of part (i). So, let us assume that (S(t))t>0 C L(X, [D(B)]) is an (A, 1, B)-regularized C-pseudoresolvent family as well as that C-1/ G ACioc([0, to) : Y), (C-1/)' G LOc([0, to) : Y) and / (0) = 0. Then the function t M u(t), t > 0 given by u(t) = /0 S(t - s)(C-1/)'(s) ds is a strong solution of (1). Let u0 > 0, u > 0, 1 > uu0, let ||0(t)|| < Mew°*, t > 0, and let the mapping (C-1/)'(•) be asymptotically (u, e)-almost periodic (asymptotically (u,e)-almost automorphic, asymptotically compactly (u,e)-almost automorphic). Then we have
e-t/w u(t) = e
= e-t/"
S(t - s)(C-1f )'(s) ds
ft _
e-(t-s)/wS(t - s) e-s/w(C-1 f )'(s)
ds, t > 0
t
o
o
Since the operator family (e i/wS(t))t>0 is exponentially decaying, it follows that the function t M e-i/wu(t), t > 0 is asymptotically almost periodic (asymptotically almost automorphic, asymptotically compactly almost automorphic). Hence, the mapping t M u(t), t > 0 is asymptotically (u,e)-almost periodic (asymptotically (u,e)-almost automorphic, asymptotically compactly (u,e)-almost automorphic).
Concerning the abstract non-degenerate Volterra equations of non-scalar type, it is clear that the above results can be applied to numerous problems in linear (thermo-)viscoelasticity and electrodynamics with memory (cf. [1, Chapter 9, Chapter 13] for more details); for example, in the analysis of viscoelastic Timoshenko beam in case of non-synchronous materials. In both cases, degenerate and non-degenerate, we can make many applications of our results with the regularizing operator C = I; see e.g., [3, Corollary 1, Example 1, Example 2] and the paragraph following [4, Theorem 2].
References
1. PriissJ. Evolutionary Integral Equations and Applications, Basel, Boston, Berlin, Birkhauser, 1993.
2. JungM. Duality theory for solutions to Volterra integral equation. Journal of Mathematical Analysis and Applications, 1999, vol. 230, pp. 112-134.
3. Kostic M. Generalized well-posedness of hyperbolic Volterra equations of non-scalar type. Ann. Acad. Rom. Sci. Ser. Math. Appl., 2014, vol. 6, pp. 19-45.
4. KostiC M. Abstract degenerate non-scalar Volterra equations. Chelyabinsk Physical and Mathematical Journal, 2016, vol. 1, pp. 104-112.
5. Alvarez E., Gomez A., Pinto M. (w, c)-Periodic functions and mild solution to abstract fractional integro-differential equations. Electronic Journal of Qualitative Theory of Differential Equations, 2018., vol. 16, pp. 1-8.
6. Alvarez E., Castillo S., Pinto M. (w,c)-Pseudo periodic functioins, first order Cauchy problem and Lasota — Wazewska model with ergodic and unbounded oscillating production of red cells. Boundary Value Problems, 2019, vol. 106, pp. 1-20.
7. BohrH. Zur theorie der fastperiodischen Funktionen I; II; III. Acta Mathematica, 1924, vol. 45, pp. 29-127; 1925, vol. H6, pp. 101-214; 1926, vol. HT, pp. 237-281.
8. Bochner S. A new approach to almost periodicity. Proceedings of the National Academy of Sciences, 1962, vol. 48, pp. 2039-2043.
9. Fedorov V., Kostic M. A note on (asymptotically) Weyl-almost periodic properties of convolution products. Chelyabinsk Physical and Mathematical Journal, 2019, vol. 4, pp. 195-206.
10. KostiC M. Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations. Berlin, W. de Gruyter, 2019.
11. Besicovitch A.S. Almost Periodic Functions. New York, Dover Publications Inc., 1954.
12. DiaganaT. Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces. New York, Springer-Verlag, 2013.
13. Fink A.M. Almost Periodic Differential Equations. Berlin, Springer-Verlag, 1974.
14. N'Guerekata G.M. Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Dordrecht, Kluwer Acad. Publ., 2001.
15. ZaidmanS. Almost-Periodic Functions in Abstract Spaces. Boston, Pitman, 1985.
16. FaviniA., YagiA. Degenerate Differential Equations in Banach Spaces. New York, Chapman and Hall/CRC, 1998.
17. KostiC M. Abstract Degenerate Volterra Integro-Differential Equations. Belgrade, Mathematical Institute SANU, 2020.
18. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, VSP, 2003.
19. Arendt W., Batty C.J.K., HieberM., NeubranderF. Vector-valued Laplace Transforms and Cauchy Problems. Basel, Birkhauser/Springer Basel AG, 2001.
Article received 05.07.2020. Corrections received 20.09.2020.
Челябинский физико-математический журнал. 2020. Т. 5, вып. 4, ч. 1. С. 4-15-4-27.
УДК 517.95+517.98 DOI: 10.47475/2500-0101-2020-15403
РЕШЕНИЯ АБСТРАКТНЫХ ВЫРОЖДЕННЫХ НЕСКАЛЯРНЫХ УРАВНЕНИЙ ВОЛЬТЕРРА АСИМПТОТИЧЕСКИ (ш, с)-ПОЧТИ ПЕРИОДИЧЕСКОГО ТИПА
М. Костич1'", В. Е. Федоров2'3'6
1 Университет Нови-Сада, Нови-Сад, Сербия
2 Челябинский государственный университет, Челябинск, Россия "marco.s@verat.net, bkar@csu.ru
Анализируются решения абстрактных вырожденных нескалярных уравнений Воль-терра асимптотически (w, с)-почти периодического типа. Для этого сначала рассматривается показатель экспоненциального убывания (A, к, В)-регуляризованных семейств C-псевдорезольвент. Полученные результаты иллюстрируются некоторыми примерами и возможными приложениями.
Keywords: асимптотически (w, c)-почти периодическая функция, асимптотически (w, c)-почти автоморфная функция, абстрактное вырожденное нескалярное уравнение Вольтерра.
Поступила в редакцию 05.07.2020. После переработки 20.09.2020.
Сведения об авторах
Костич Марко, профессор, факультет технических наук, Университет Нови-Сада, Нови-Сад, Сербия; e-mail: marco.s@verat.net.
Федоров Владимир Евгеньевич, доктор физико-математических наук, профессор, профессор кафедры математического анализа, Челябинский государственный университет, Челябинск, Россия; e-mail: kar@csu.ru.
