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Chelyabinsk Physical and Mathematical Journal. 2022. Vol. 7, iss 1. P. 80-96.
DOI: 10.47475/2500-0101-2022-17106
p-ALMOST PERIODIC TYPE FUNCTIONS IN Rn
M. Kostic
University of Novi Sad, Novi Sad, Serbia marco.s @verat.net
We investigate various classes of multi-dimensional (S, D, B)-asymptotically (w, p)-periodic type functions, multi-dimensional quasi-asymptotically p-almost periodic type functions and multi-dimensional p-slowly oscillating type functions of the form F : I x X ^ Y, where n G N, 0 = I C Rn, w G Rn \ {0}, X and Y are complex Banach spaces and p is a binary relation on Y. The main structural properties of these classes of almost periodic type functions are deduced. We also provide certain applications of our results to the abstract Volterra integro-differential equations.
Keywords: (S, D, B)-asymptotically (w, p)-periodic type functions, quasi-asymptotically p-almost periodic type functions, remotely p-almost periodic type functions, p-slowly oscillating type functions, abstract Volterra integro-differential equations.
1. Introduction and preliminaries
The class of almost periodic functions was introduced by the Danish mathematician H. Bohr around 1925 and later generalized by many others. For the basic source of information about almost periodic functions and their applications, we refer the reader to [1-8].
Suppose that X is a complex Banach space, and I = R or I = [0, to). In [9], E. Alvarez, A. Gomez and M. Pinto [9] have considered the following notion: A continuous function f : I ^ X is said to be (w, c)-periodic (w > 0, c G C \ {0}) if and only if f (t + w) = cf (t) for all t G I; see also [10; 11] for some applications made. Following M. Feckan, K. Liu and J. Wang [12, Definition 2.2], we say that a continuous function f : I ^ X is (w, T)-periodic if and only if f (t + w) = Tf (t) for all t > 0; here, T denotes a linear isomorphism on X. In [12], the authors have investigated the existence and uniqueness of (w, T)-periodic solutions for various classes of impulsive evolution equations, linear and semilinear Cauchy problems.
Further on, we would like to emphasize the following:
(i) The class of S-asymptotically w-periodic functions, where w > 0, was introduced by H. R. Henriquez, M. Pierri and P. Taboas in [13]; for some applications of S-asymptotically w-periodic functions, see [14; 15] and the list of references quoted in [16].
(ii) In [17; 16], and [18] (a joint work with M. T. Khalladi, M. Pinto, A. Rahmani and D. Velinov), the author of this paper has investigated the class of S-asymptotically (w, c)-periodic functions and the class of quasi-asymptotically c-almost periodic functions, where c G C \ {0}. Any S-asymptotically (w, c)-periodic function f : I ^ X is quasi-asymptotically c-almost periodic if |c| < 1, while the converse
The work is partially supported by grant 451-03-68/2020/14/200156 of Ministry of Science and Technological Development, Republic of Serbia.
statement is not true in general. The Stepanov classes of multi-dimensional quasi-asymptotically c-almost periodic functions have been analyzed in the above-mentioned research articles, as well.
(iii) The main purpose of research articles [19; 20], written in a collaboration with A. Chavez, K. Khalil and M. Pinto, have been to investigate various classes of multi-dimensional (Stepanov) almost periodic functions.
(iv) The notion of c-almost periodicity, introduced in our joint research article [21] with M. T. Khalladi, A. Rahmani, M. Pinto and D. Velinov, was extended to multidimensional setting in [22]. It should be also emphasized that various classes of multi-dimensional (w, c)-almost periodic type functions have been analyzed in [23].
(v) Stepanov and Weyl classes of multi-dimensional p-almost periodic type functions have recently been introduced and analyzed in [24].
(vi) The class of remotely almost periodic functions has been introduced by D. Sarason in [25] and later reconsidered by the same author in [26] (see also the list of references quoted in our joint paper [27] with V. Kumar). As observed in [27, Example 2.4], there is a bounded, uniformly continuous, slowly oscillating function f : [0, to) ^ co whose range is not relatively compact in co, the Banach space of all numerical sequences tending to zero equipped with the sup-norm.
The main aim of this paper is to continue the research studies raised in the above-mentioned papers by introducing and investigating the following classes of multidimensional almost periodic type functions:
(i) multi-dimensional (S, D, B)-asymptotically (w, p)-periodic type functions;
(ii) multi-dimensional quasi-asymptotically p-almost periodic functions;
(iii) multi-dimensional remotely p-almost periodic functions;
(iv) multi-dimensional p-slowly oscillating type functions,
as well as to provide certain applications of the introduced notion to the abstract Volterra integro-differential equations and the partial differential equations.
The organization of paper can be briefly described as follows. Section 2 investigates the main features of (S, D, B)-asymptotically (w, p)-periodic type functions and (S, B)-asymptotically (wj,pj, DjjNn-periodic type functions. Section 3 is devoted to the study of multi-dimensional quasi-asymptotically p-almost periodic type functions; in Subsection 3.1., we analyze multi-dimensional remotely p-almost periodic type functions. The main aim of Section 4 is to provide the main definitions and results about multidimensional p-slowly oscillating type functions. The final section of paper is reserved for some applications to the abstract Volterra integro-differential equations. Notation and terminology. Suppose that X, Y, Z and T are given non-empty sets. Let us recall that a binary relation between X into Y is any subset p C X x Y. If p C X x Y and a C Z x T with Y n Z = 0, then we define p-1 C Y x X and a • p = a o p C X x T by p-1 := {(y, x) G Y x X : (x, y) G p} and
a o p := {(x, t) G X x T : 3y G Y n Z such that (x, y) G p and (y, t) G a},
respectively. As is well known, the domain and range of p are defined by D(p) := {x G X : 3y G Y such that (x, y) G X x Y} and R(p) := {y G Y : 3x G X such that (x, y) G X x Y}, respectively; p(x) := {y G Y : (x,y) G p} (x G X), x p y ^ (x,y) G p. If p is a binary relation on X and n G N, then we define pn inductively; p-n := (pn)-1. Set p(X') := {y : y G p(x) for some x G X'} (X' C X) and Nn := {1, • • -,n} (n G N).
Unless stated otherwise, in the remainder of this paper we will always assume that (X, || ■ ||) and (Y, || ■ ||Y) are two complex Banach spaces. By L(X, Y) we denote the Banach space of all bounded linear operators from X into Y with L(X, X) being denoted L(X); the symbol I stands for the identity operator on Y. If t0 G Rn and e > 0, then we define B(t0,e) := {t G Rn : |t — t0| < e}, where | ■ | denotes the Euclidean norm in Rn; by (ei, e2, ...,en) we denote the standard basis of Rn. If D C Rn and M > 0, then we set DM := {t G D : |t| > M}. We will always assume henceforth that B is a collection of non-empty subsets of X such that, for every x G X, there exists B G B with x G B.
2. (S, D, B)-Asymptotically (w, p)-periodic type functions and (S, B)-asymptotically (wj, pj, Dj)jGNn-periodic type functions
The notion of (S, D, B)-asymptotical (w, c)-periodicity and the notion of (S, B)-asymptotical (wj, cj, Dj)jeNn-periodicity, introduced recently in [16, Definition 2.1, Definition 2.2], can be further generalized as follows:
Definition 1. Let w G Rn \ {0}, p be a binary relation on Y, w + I C I, D C I C Rn and the set D be unbounded. A continuous function F : I x X ^ Y is said to be (S, D, B)-asymptotically (w, p)-periodic if and only if for each B G B, t G D and x G B there exists an element yt;x G p(F(t; x)) such that
lim ||F(t + w; x) — yt-JL = 0, uniformly in x G B.
Definition 2. Let wj G R \ {0}, pj be a binary relation on Y, wjej +1 C I, Dj C I C Rn and the set Dj be unbounded (1 < j < n). A continuous function F : I x X ^ Y is said to be (S, B)-asymptotically (wj,pj, Dj)jeNn -periodic if and only if for each j G Nn, B G B, t G Dj and x G B there exists an element yj;t;x G pj(F(t; x)) such that
lim 1F (t + wj ej; x) — yj;t;x Hv = 0, uniformly in x G B.
The class of (D, B)-asymptotically (w, p)-periodic functions and the class of (D, B)-asymptotically (wj, pj)jeNn-periodic functions can be also introduced. An analogue of [16, Proposition 2.4] can be proved in this context; details can be left to the interested readers.
We will provide the proof of the first part of the following simple result for the sake of completeness:
Proposition 1. 1. Let w G Rn\{0}, w +1 C I, D C I C Rn and the set D be unbounded. If for each B G B there exists eB > 0 such that the sequence (Fk(■; ■)) of (S, D, B)-asymptotically (w, p)-periodic functions converges uniformly to a function F(■; ■) on the set B° ^UB(x,eB), then F(■; ■) is (S, D, B)-asymptotically (w,p)-periodic, provided that D(p) is a closed subset of Y and p is continuous in the following sense:
(Cp) For each e > 0 there exists 5 > 0 such that, for every y1, y2 G Y with ||y1 — y2||Y < 8, we have ||z1 — z2||Y < e/3 for every z1 G p(y1) and z2 G p(y2).
2. Let w^- G R \ {0}, w^-e^- + I C I, D^- C I C Rn and the set D^- be unbounded (1 < j < n). If for each B G B there exists eB > 0 such that the sequence (Fk(■; ■)) of (S, B)-asymptotically (wj,pj, Dj)jeNn-periodic functions converges uniformly to a function F(■; ■) on the set B° ^UB(x,eB), then the function F(■; ■) is (S, B)-asymptotically (wj ,pj, Dj)jeNn -periodic, provided that for each j G Nn we have that D(pj) is a closed subset of Y and condition (Cp) holds with p = pj therein.
Proof. We will prove only the first assertion. By the proofs of [19, Proposition 2.7, Proposition 2.8], it follows that the function F(•; •) is continuous. Since D(p) is closed, we have F(t; x) G D(p) for all t G I and x G X. Let e > 0 and B G B be fixed. Further on, let a number 8 > 0 be chosen in accordance with the continuity of relation p. Due to our assumptions, for any number e0 = min(e/3, 8), we can find a positive integer k G N such that ||Fk(t; x) — F(t; x) ||Y < e0 for all x G B° and t G I. Fix now a point t G I and an element x G B. Then there exist an element G p(Fk(t; x)) and a sufficiently large real number M > 0, independent of x G B, such that, for every t G D with |t| > M, we have ||Fk(t + w; x) — ykx||Y < e0/3. Let yt ;x G p(F(t; x)) be arbitrarily chosen. Then the final conclusion follows from the next computation:
||F(t + w; x) — yt;x||Y < ||F(t + w; x) — Ffc(t; x)||y + ||Ffc(t + w; x) — yt^^
+ 11 yk;x — yt;x| y < 3 • (e/3) = e.
□
In the following important result, we examine the convolution invariance of function spaces introduced in Definition 1 and Definition 2:
Theorem 2. Suppose that h G L1(Rn) and F : Rn x X m Y is a continuous function satisfying that for each B G B there exists a finite real number eB > 0 such that ||F(t;x)||y < where B• = B° U UxedBB(x,eB). Suppose, further, that p = A is a closed linear operator on Y satisfying that:
(D) For each t G Rn and x G B, the function s m AF(t — s; x), s G Rn is Bochner integrable; for each B G B, the function s m supxeB ||AF(s; x)||Y, s G Rn is bounded.
Then the function
(h * F)(t; x) := í h(a)F(t — a; x) da, t G Rn, x G X
J Rn
is well defined and for each B G B we have supteRn,xeB- ||(h * F)(t; x)|Y < Furthermore, the following holds:
(i) Suppose that D = Rn. If the function F(•; •) is (S, Rn, B)-asymptotically (w,A)-periodic, then the function (h* F)(•; •) is (S, Rn, B)-asymptotically (w, A)-periodic.
(ii) Suppose that Dj = Rn for all j G Nn. If for each j G Nn condition (D) holds with the closed linear operator A replaced therein with the closed linear operator Aj, and the function F(•; •) is (S, B)-asymptotically (wj, Aj, Rra)jeNn-periodic, then the function (h * F)(•; •) is likewise (S, B)-asymptotically (wj, Aj, Rra)jeNn-periodic.
Proof. The proof is very similar to the proof of [16, Theorem 2.6], which contains small typographical errors; because of that, we will provide all relevant details of proof. We will consider the issue (i), only. It can be easily shown that the function (h*F)(•; •) is well defined as well as that supteRn xeB- ||(h * F)(t; x)||Y < for all B G B. The continuity of function (h * F)(•; •) at the fixed point (t0; x0) G Rn x X follows from the existence of a set B G B such that x0 G B, the assumption supteRn,xeB- ||F(t; x)||Y < and the dominated convergence theorem. Fix now a real number e > 0 and a set B G B. Then there exists a sufficiently large real number M1 > 0 such that ||F(t + w; x) — AF(t; x)||Y < e, provided |t| > M1 and x G B. Since A is closed and condition (D) holds, for every t G Rn and x G B, we have that the element zt,x := A((h * F)(t; x)) =
JR„ h(s)A(F(t — s; x)) ds is well defined. Therefore, due to the second part of condition (D), we have that there exists a finite constant cB > 1 such that
(h * F )(t + w; x) — zt,x < / |h(a)|-||F(t + w — a; x) — AF(t — a; x)||Y da =
Y it"
= |h(t — a)| ■ ||F(a + w; x) — AF(a; x)||Y da+
+ / |h(t — a)| ■ ||F(a + w; x) — AF(a; x)||Y da <
Jlal>M 1
< e|h|Li(Rn) + / |h(t — a)| ■ ||F (a + w; x) — AF (a; x)||Y da <
J M<M1
< e|h|Li(Rn) + cW |h(t — a)| da.
./|o-|<M1
The final conclusion simply follows from the above computation and the existence of a finite real number M2 > 0 such that J|ct|>M2 |h(a)| da < e; observe only that, if |t| > Mi + M2, then for each a G Rn with |a| < Mi we have |t — a| > M2. □
The proof of following result is very similar to the proof of [16, Proposition 2.7]:
Proposition 2. Let wj G R \ {0}, Tj G L(X), wjej +1 C 1, D^ C I C Rn and the set D.,-be unbounded (1 < j < n). If F : 1 x X ^ Y is (S, B)-asymptotically (wj,Tj, Dj)jeNn-periodic and the set D consisting of all tuples t G Dn such that t + w^e for
all j G Nn-i is unbounded in Rn, then the function F(■; ■) is (S, D, B)-asymptotically (w, T)-periodic, with w := ^™=i wjej and T := Hn=1 Tj.
The proof of following proposition can be omitted, as well:
Proposition 3. Let w, a G Rn\{0}, w +1 C 1 and a +1 C 1. Suppose that the functions F : 1 x X ^ Y and G : 1 x X ^ Y are (S, D, B)-asymptotically (w, p)-periodic ((S, B)-asymptotically (wj,pj,Dj)jeNn-periodic). Then the following holds:
(i) The function F(-; ■) is (S, —D, B)-asymptotically (—w, p)-periodic ((S, B)-asymptotically (—wj,pj, —Dj)jeNn-periodic), where F(t; x) := F(—t; x), t G —1, x G X.
(ii) Set a := {(|yi|Y, ||y2||Y) | 3t G 1 3x G X : yi = F(t; x) and y2 G p(yi)} and aj by replacing p by pj above (1 < j < n). Then the function ||F(■; -)||Y is (S, D, B)-asymptotically (w, a)-periodic ((S, B)-asymptotically (wj, aj, Dj)jeNn-periodic).
(iii) Define D(pi) := R(F + G) and pi(F(t; x) + G(t; x)) := {y + z : y G p(F(t; x)); z G pi(G(t; x))} for all t G 1 and x G B. Then the function [F + G](-; ■) is (S, D, B)-asymptotically (w, pi)-periodic.
(iv) Suppose that A G C \ {0}. Set pA := {A(yi, y2)| 3t G 1 3x G X : yi = F(t; x) and y2 G p(yi)}. Then the function AF(■; ■) is (S, D, B)-asymptotically (w, pA) -periodic.
(v) If a + D C D (a + D^- C D^- for all j G Nn) and y G X, then the function Fa,y : 1 x X ^ Y defined by Fa,y(t; x) := F(t + a; x + y), t G 1, x G X is (S, D, By)-asymptotically (w, p)-periodic ((S, By)-asymptotically (wj,pj, Dj)jeNn-periodic), where By := {—y + B : B G B}.
(vi) If w G Rn \ {0}, w +1 C 1, the function E : 1 x X ^ C is (S, D, B)-asymptotically (w, pi)-periodic and the function H : 1 xX ^ Y is (S, D, B)-asymptotically (w, p2)-periodic, where pi is a binary relation on C and p2 is a binary relation on Y, then
the function F(■) := E(-)H(■) is (S, D, B)-asymptotically (o, p)-periodic, where D(p) := R(EH) and p(E(t; x)F(t; x)) := {eh : e G pi(E(t; x)), h G p2(H(t; x))} for all t G I and x G B, provided that for each set B G B we have sup№€B |E(t; x)| +supteJ;x€B suPhep2(H(t;x)) ||h||Y < ro or suptej||H(t; x)||Y + suPte/;x€B SUPeepi(E(t;x)) |e| <
(vii) Let oj G R \ {0} and ojej + I C I (1 < j < n). Suppose that the function E : I x X ^ C is (S, B)-asymptotically (oj, Pj,1, Dj)jeNn -periodic and the function H : I ^ X is (S, B)-asymptotically (oj,Pj,2, Dj)jeNn -periodic, where pj,1 is a binary relation on C and pj,2 is a binary relation on Y for all j G Nn. Set D(pj) := R(EH) and pj(E(t; x)H(t; x)) := {eh : e G pj,1 (E(t; x)), h G Pj,2(H(t; x))} for all j G Nn, t G I and x G B. Then the function F(■) := E(-)H(■) is (S, B)-asymptotically (oj ,pj, Dj)jeNn -periodic, provided that for each set B G B and for each integer j G Nn we have
SUPte/;x€B |E(t; x)| + suPte/;xeB SUPhepj,2(H(t;x)) ||h|Y < ro or SUPteJ||H(t; x)|Y + SUPteJ;x€B SUPeepJ.1(E(t;x)) |e| <
3. Multi-dimensional quasi-asymptotically p-almost periodic type functions
In this section, we investigate various classes of multi-dimensional quasi-asymptotically p-almost periodic functions. The following notion generalizes the notion of D-quasi-asymptotical (B, I', c)-almost periodicity and the notion of D-quasi-asymptotical (B, I', c)-uniform recurrence (see [16, Definition 3.1]):
Definition 3. Suppose that D C I C Rn, 0 = I' C Rn, 0 = I C Rn, the sets D and I' are unbounded, F : I x X ^ Y is a continuous function and I + I' C I. Then we say that:
(i) F(■; ■) is D-quasi-asymptotically (B, I', p)-almost periodic if and only if for every B G B and e > 0 there exists l > 0 such that for each t0 G I' there exists t G B(t0, l) fi I' such that there exists a finite real number M(e, t) > 0 such that, for every t G DM(e,r) such that t + t G Dm(e,r) and for every x G B, there exists an element yt;x G p(F(t; x)) such that ||F(t + t; x) — yt;x||Y < e.
(ii) F(■; ■) is D-quasi-asymptotically (B, I', p)-uniformly recurrent if and only if for every B G B there exist a sequence (rfc) in I' and a sequence (Mk) in (0, ro) such that |rk| = Mk = +ro and, for every t G D and x G B, there exists an element yt;x G p(F(t; x)) such that
lim sup 1F (t + rfc; x) — ytJL = 0.
t,t+rfceDMfc ;xeB
If I' = I, then we also say that F(■; ■) is D-quasi-asymptotically (B, p)-almost periodic (D-quasi-asymptotically (B, p)-uniformly recurrent); furthermore, if X G B, then it is also said that F(■; ■) is D-quasi-asymptotically (I', p)-almost periodic (D-quasi-asymptotically (I', p)-uniformly recurrent). If I' = I and X G B, then we also say that F(■; ■) is D-quasi-asymptotically p-almost periodic (D-quasi-asymptotically p-uniformly recurrent). We remove the prefix "D-" in the case that D = I, remove the prefix " (B,)" in the case that X G B and remove the prefix "p-" if p = I.
As easily approved, the notion of D-asymptotical Bohr (B, I', p)-almost periodicity (uniform recurrence) (of type 1), introduced recently in [16, Subsection 2.2], is a
special case of the notion of D-quasi-asymptotical (B, I', p)-almost periodicity (uniform recurrence).
The interested reader may try to generalize [16, Proposition 3.2] in our new framework. The following result can be also formulated for (S, B)-asymptotical (wj ,pj, Dj )jeNn -periodicity; the proof is very similar to the proof of [16, Proposition 3.3 (i)] and therefore omitted:
Proposition 4. Let w G I \ {0}, p = T G L(X), ||T|| < 1, w + I Ç I,
w + D Ç D and D Ç I Ç Rn. Set I' := w ■ N. If a continuons function F : I x X ^ Y is (S, D, B)-asymptotically (w,T)-periodic, then the function F(■; ■) is D-quasi-asymptotically (B,I',T)-almost periodic.
The spaces introduced in Definition 3 do not have a linear vector structure. Many structural properties like the translation invariance, the invariance under homotheties, the invariance under reflections at zero with respect to the first variable, can be straightforwardly formulated for D-quasi-asymptotically (B, I', p)-almost periodic functions and D-quasi-asymptotically (B, I', p)-uniformly recurrent functions. The formulation of [16, Proposition 3.5 (i)] is slightly incorrect and we only want to note here that this simple result can be formulated for D-quasi-asymptotically (B,I', p)-almost periodic functions and D-quasi-asymptotically (B,/', p)-uniformly recurrent functions, provided that p = T G L(Y) is a linear isomorphism. Concerning [16, Proposition 3.5 (ii)], we can simply prove the following result (cf. also the proof of Proposition 1):
Proposition 5. Suppose that (Fk (■; ■)) is a sequence of D-quasi-asymptotically (B,I',p)-almost periodic functions, resp. D-quasi-asymptotically (B, I', p)-uniformly recurrent functions. Suppose, further, that for each B G B there exists a finite real number eB >0 such that limfc^+œ Fk(t; x) = F(t; x) for all t G R, uniformly in x G B^ = B°UUB(x,eB). Then the function F(■; ■) is D-quasi-asymptotically (B, I', p)-almost periodic, resp. D-quasi-asymptotically (B, I', p)-uniformly recurrent, provided that D(p) is a closed subset of Y and p satisfies condition (Cp).
Using the argumentation contained in the proof of Theorem 2, we may deduce a result concerning the convolution invariance of D-quasi-asymptotical (B, I', p)-almost periodicity and D-quasi-asymptotical (B, I', p)-uniform recurrence, provided that p = A is a closed linear operator on Y satisfying condition (D); this can be also done for remotely p-almost periodic type functions considered in the subsequent subsection.
3.1. Remotely p-almost periodic type functions
In [27, Section 3], we have recently analyzed relations between quasi-asymptotical c-almost periodicity and remote c-almost periodicity. The following notion generalizes the notion introduced in [27, Definition 3.2]:
Definition 4. Suppose that F : I x X ^ Y is a continuous function and p is a binary relation on Y.
(i) It is said that F(■; ■) is D-remotely (B, I', p)-almost periodic if and only if F(■; ■) is D-quasi-asymptotically (B, I', p)-almost periodic and for each B G B the function
F(■; ■) is uniformly continuous on I x B; that is
(VB e B) (Ve > 0) (35 > 0) (Vt', t'' e I) (Vx', x'' e B)
t' - t''| + ||x - x'11 <5 ^
(ii) It is said that F(■; ■) is D-remotely (B, I', p)-uniformly recurrent if and only if F(■; ■) is D-quasi-asymptotically (B, I', p)-uniformly recurrent and for each B e B the function F(■; ■) is uniformly continuous on I x B.
(iii) It is said that F(■; ■) is D-remotely (B, I', p)-almost periodic of type 1 if and only if F(■; ■) is D-quasi-asymptotically (B, I', p)-almost periodic and
(VB e B) (Ve > 0) (35 > 0) (Vt', t'' e I) (Vx e B) t' - t''1 <5 ^
(iv) It is said that F(■; ■) is D-remotely (B, I', p)-uniformly recurrent of type 1 if and only if F(■; ■) is D-quasi-asymptotically Bohr (B, I', p)-uniformly recurrent and (1) holds.
We will provide all details of the proof of the following result for completeness (cf. also the proof of [16, Proposition 2.2]):
Proposition 6. Suppose that D C I C Rn, 0 = I' C Rn, 0 = I C Rn, the sets D and I' are unbounded, I + I' C I, and the function F : I x X ^ Y is D-quasi-asymptotically (-remotely) (B, I', p)-almost periodic ((B, I', p)-uniformly recurrent), where p is a binary relation on Y satisfying R(F) C D(p) and p(y) is a singleton for any y e R(F). Then the function F(■; ■) is D'-quasi-asymptotically (-remotely) (B, I' — I', I)-almost periodic ((B, I' — I', I)-uniformly recurrent), provided that D C I, the set D' is unbounded and D' - I' C D.
Proof. We will consider the class of D-quasi-asymptotically (B,I',p)-almost periodic functions, only. Let e > 0 and B e B be given in advance. For a given point t0 e I' — I', we can always find two points t,1,, t^ e I' such that t0 = t,1, —10. Then there exists a real number l > 0, two points Ti e B^^l) H I' and t2 e B(t0,l) H I' as well as two finite real numbers M(e,Ti) > 0 and M(e,r2) > 0 such that, for every t e Dmax(M(e,Tl),M(e,r2)) and x e B, we have
||F(t + ti; x) - p(F(t; x))||Y < e/2 and ||F(t + t2; x) - p(F(t; x))|Y < e/2. This implies ^F(t + ti; x) - F(t + T2; x)11Y < e, t e Dmax(M(c,ti),m(e,T2)), x e B, i.e.,
^F (v + [t2 - tJ ; x) - F (v; x) 11Y < e, v e Ti + Dmax(M (e,Ti),M (e,T2)), x e B. (2)
It is clear that t2 - t1 e B(t2 - t0, 2l) H (I' - I'). Suppose now that t e D' and |t| > |t1| +max(M(e,T1),M(e,T2)). Since D' -1' C D, it is clear that t - t1 e D' -1' C D and |t - t^1| > max(M(e, t^1), M(e, t2)). This implies the required conclusion by definition and (2). □
In connection with Proposition 7, it should be also noted that the conclusions from [16, Proposition 2.5, Example 2.8, Example 2.10, Theorem 2.16, Proposition 2.24] can be reformulated for multi-dimensional quasi-asymptotically (remotely) almost periodic functions. We will not consider here the extension type theorems for multi-dimensional quasi-asymptotically (remotely) almost periodic functions; see [16, Theorem 2.27, Theorem 2.28] for more details about this subject.
F(t'; x') - F(t"; x")
< e
Y
F (t'; x) - F (t''; x)
< e
Y
4. p-Slowly oscillating type functions in Rn
We start this section by introducing the following notion (cf. also [27, Definition 2.1, Definition 2.5]; unless stated otherwise, we will always assume here that p is a binary relation on Y and p, is a binary relation on Y (1 < j < n)):
Definition 5. Let 0 = 1 С Rn, D С 1 С Rn and the set D be unbounded. Define A/ := {ш G Rn\{0} : ш+1 С /}. Then we say that a continuous function F : 1x! ^ Y is (D, B, p)-slowly oscillating if and only if, for every ш G A/, B G B, t G D and x G B, there exists an element yt;x G p(F(t; x)) such that
lim IIF(t + ш; x) — yt-JL = 0, uniformly in x G B. (3)
Definition 6. Let D, С 1 С Rn and the set D, be unbounded (1 < j < n). Define
В/ := {(шь...,ш„) G (R \ {0})n : ш,б, + 1 С 1 for all j G N„}.
Then we say that a continuous function F : 1 x X ^ Y is (B, (D,, p,)jeNn)-slowly oscillating if and only if, for every (ш1, ...,wn) G B/, j G , В G B, t G D and x G B, there exists an element yj;t;x G p,(F(t; x)) such that
lim ||F(t + ш,б,; x) — y,;t;x||v = 0, uniformly in x G B.
In other words, a continuous function F : 1 x X ^ Y is (D, B,p)-slowly oscillating ((B, (D,, p,),eNn)-slowly oscillating) if and only if the function F(■; ■) is (S, D, B)-asymptotically (ш, p)-periodic ((S, B)-asymptotically (ш,, p,, D,),eNn-periodic) for all ш G A/ ((ш1,..., шп) G B/). This enables one to simply transfer all structural results from the second section of paper to the multi-dimensional slowly oscillating type functions introduced in Definition 5, Definition 6.
As before, if D = 1, then we omit the term "D" from the notation; if X = {0}, then we omit the term "B" from the notation. The usual notion of a slowly oscillating function is obtained by plugging that p = I.
In [27, Proposition 2.2], we have explained that it is not so logical to study the class of (D, B,c)-slowly oscillating functions by replacing the term ||F(t + ш; x) — F(t; x)||y in (3) by the term ||F(t + ш; x) — cF(t; x) ||Y, where c G C \ {0,1}; the reason is quite simple because then the space of all (D, B, c)-slowly oscillating functions coincides, under certain very reasonable assumptions, with the space C0,D,B(1 x X : Y). On the other hand, it is very meaningful to further explore the use of general binary relations, even linear isomorphisms here:
Example 1. Suppose that
0, 1,
where a, c G C; then T is a linear isomorphism on C2 provided that a = 0. If the function f1 : R ^ C satisfies lim|t|^+TO f1(t) = 0 and the function f2 : R ^ C is slowly oscillating, then the vectorial function /(t) := (f1(t), f2(t)), t G R is T-slowly oscillating. Therefore, the function /(•) does not vanish as |t| ^
Using [16, Proposition 2.20] (see also [27, Proposition 2.2] for the last conclusion), we can simply deduce the following result in the finite-dimensional spaces:
T :=
a c
Proposition 7. Suppose that k e N, T = A = [a^] is a non-zero complex matrix of format k x k, I = R or I = [0, to), D C I, D is unbounded, u e R \ {0}, u + I C I, and the function F : I m Ck is D-remotely (I', A)-almost periodic (D-remotely (I', A)-uniformly recurrent; D-quasi-asymptotically (I', A)-almost periodic; D-quasi-asymptotically (I', A)-uniformly recurrent; (D, A)-slowly oscillating; (S, D)-asymptotically (u, A)-periodic). If F = (F1, ■ ■ -,Fk), then there exists a non-trivial linear combination u(-) of functions F1, ■ ■ -,Ffc and a number A e ap(A) \ {0} such that the function u(-) is D-remotely (I', A)-almost periodic (D-remotely (I', A)-uniformly recurrent; D-quasi-asymptotically (I', A)-almost periodic; D-quasi-asymptotically (I', A)-uniformly recurrent; (D, A)-slowly oscillating; (S, D)-asymptotically (u, A)-periodic); furthermore, if F(■) is (D, A)-slowly oscillating and D = I, then u(-) is slowly oscillating.
On the other hand, we can use [27, Proposition 2.2] in order to see that there exists a large class of slowly oscillating functions which are not T-slowly oscillating, where T is a linear isomorphism on Y; for example, the function t m exp(ita), t > 0 is slowly oscillating but not (-I)-slowly oscillating (0 < a < 1).
We will only mention in passing that the statement of [27, Proposition 3.7] admits a reformulation for multi-dimensional slowly oscillating type functions introduced in Definition 5, Definition 6. Suppose now that F : I x X m Y and G : I x Y m Z are given continuous functions. Then we define the multi-dimensional Nemytskii operator W : I x X m Z by
W(t; x) := G(t; F(t; x)), t e I, x e X. (4)
Concerning composition principles for multi-dimensional p-slowly oscillating type functions, we will clarify the following result which can be also reformulated for the class of (B, (Dj,pj)jeNn)-slowly oscillating type functions:
Theorem 5. Suppose that D C I C Rn, the set D is unbounded, F : I x X m Y is (D, B, p)-slowly oscillating and G : I x Y m Z is (D, BG, pG)-slowly oscillating, where p(R(F)) e BG and pG is a binary relation on Z. Let a be another binary relation on Z. Suppose, further, that there exists a finite real constant L > 0 such that
||G(t; y) - G(t; y')||Z < L||y - y'||y, t e I, y, y' e R(F) U p(R(F)), (5)
as well as that, for every B e B, t e I, x e B, as well as for every yt;x e p(F(t; x)) and for every zt,yt e pG(G(t; yt;x)), ther^e exists an element ut;x e a(W(t; x)) such that
lim yztyt.x - ut;JL = 0, uniformly in x e B. (6)
Then the function W(■; ■), given by (4), is (D, B, a)-slowly oscillating.
Proof. Let u e A/,B e B, t e D and x e B be given. Then there exists an element yt;x e p(F(t; x)) such that (3) holds. Since G(-; ■) is (D, BG,pG)-slowly oscillating and p(R(F)) e BG, we know that there exist an element zt,yt e pG(G(t; yt;x)) such that
lim HG(t + u; yt;^) — ztyt = 0, uniformly in x G B. (7)
Due to our assumption, there exists an element ut;x e a(W(t; x)) such that (6) holds. Keeping in mind (3), (5) and (7), the final conclusion simply follows from the next
decomposition:
|G(t + w; F(t + w; x)) - <
< ||G(t + w; F(t + w; x)) - G(t + w; yt;x)^Z + ||G(t + w; yt;x) - ||Z+ + |Zt;yt;x - < L^F (t + w; X) - yt;x^y + ^G(t + w; yt;x) - Zt;yt;x +
+ |zt;yt;x - •
□
Suppose that (5) holds. For our applications, it would be worth mentioning that Theorem 5 holds in the following particular cases:
(i) p = I and R(F) e Then the requirements of Theorem 5 holds provided that a = pG because the choice zt;yt = ut;x can be made (t e I; x e B).
(ii) p = I, R(F) e and a = pG = IZ, the identity operator on Z.
5. Applications to the abstract Volterra integro-differential equations
This section is devoted to applications of our abstract theoretical results to the abstract Volterra integro-differential equations and the partial differential equations.
1. Our results about the convolution invariance of introduced function spaces can be simply incorporated in the qualitative analysis of solutions for the inhomogeneous heat equation in Rn. We will only briefly describe here that certain applications can be made even if the binary relation p is not a linear function (see Theorem 2); suppose, for example, that p(z) := zk, z e C, where k e N, and Y = BUG(Rn), the space of all bounded, uniformly continuous functions F : Rn ^ C equipped with the sup-norm. We know that the Gaussian semigroup
(G(t)F)(x) := (4nt)
-(n/2)
I y I
F(x - y)e-^ dy, t > 0, f G Y, x G Rn
can be extended to a bounded analytic G0-semigroup of angle n/2, generated by the Laplacian AY acting with its maximal distributional domain in Y. Let t0 > 0 be a fixed real number, D = Rn and the non-zero function F : Rn ^ C be (S, Rn)-asymptotically (w, p)-periodic for some w e Rn \ {0}. Let us define a new binary relation a on C by
n
a := I i (G(to)F)(x), (4nta)-(n/2^ [F(x - y)]ke-^ dy ) : x G Rn
Then the function Rn 9 x ^ u(x,t0) = (G(t0)F)(x) G C is (S,Rn)-asymptotically (w, a)-periodic. Towards this end, fix a real number e > 0. Then there exists a sufficiently large real number M > 0 such that |F(x + w) - [F(x)]k| < (4nt0)n/2e/2 for all x e Rn with |x| > M. Let Mi > 0 be any real number such that
i e-'y'2/4i° dy < (4nto)n/2e/[2(||F+ ||F|U)].
JRn \B(0,Mi)
n
Then, for every x E Rn with |x| > M + Mi, we have:
u (x + t, to) - a (u(x,to) ) < (4n"o) (n/2W F(x - y + t) - [F(x - y)]
V / J»n
e 4to
dy <
< (4nio)-(n/2)(4nio)n/2e/2+
+ (4ntc
-(n/2)
'y€B(x,M ) \ - (n/2) ( il ri ilk
F (x - y + t ) - [F (x - y)]
e 4to dy <
< e/2 + (4nto)-(n/2^||F+ ||F|U) / e-W dy <
./yeB(x,M )
< e/2 + (4nto)
-(n/2)
^\B(o,Mi)
e-|y|2/4t0 dy < e/2 + e/2 = e.
This simply implies the required conclusion. Observe, finally, that the assumption on function F(■) does not imply that F(x) = 0 in general. For example, consider
any continuous function F0 : [0,1] M [0,1] such that Fo(0) = F0(1) = 0 and F0(1/2) = 1. It is clear that there exists a bounded, uniformly continuous function F : R M [0,1] such that F(t) = F0(t) for all t E [0,1] and F(t +1) = [F(t)]2 for all t E R. The function F(■) does not vanish at plus infinity since F(n + (1/2)) = 1 for all n E Z.
It is also worth noting that the nonlinear functions p : C M C can be used in the analysis of remarkable examples [4, Examples 4, 5, 7, 8; pp. 32-34] given in the research monograph of S. Zaidman.
2. In this issue (see also [28]), we consider the following wave equation in R3:
utt(t,x) = d2Axu(t,x), x E R3, t > 0; u(0,x) = g(x), ut(0,x) = h(x), (8)
where d > 0, g E C3(R3 : R) and h E C2(R3 : R). By the Kirchhoff formula (see e.g., [29, Theorem 5.4, pp. 277-278]; we will use the same notation), the function
, \ d u(t,x) := di
1
4nd2t
'dBdt(x)
g(a) da
+
1
4nd21 JdBdt(
x)
h(a) da
-1 I g(x + dtw) + 7" I
4n </dBi(o) 4n ./dBi(o)
Vg(x + dtw) ■ wdw + —
4n
dBi (o)
h(x + dtw) dw,
t > 0, x E R3, is a unique solution of problem (8) which belongs to the class C2([0, to) x R3). Let a number to > 0 be fixed, let w' E R3 \ {0}, and let c E C \ {0}. Then it can be simply shown that the function x M u(t0,x), x E R3 will be (S,D)-asymptotically (w', cI)-periodic, provided that the functions #(•), Vg(-) and h(-) are (S, D)-asymptotically (w', cI)-periodic and D + dto№(0) C D. We can similarly consider the following wave equation in R2 :
utt(t,x) = d2Axu(t,x), x E R2, t> 0; u(0,x) = g(x), ut(0,x) = h(x), (9)
where d > 0, g E C3(R2 : R) and h E C2(R2 : R). Let us only recall that, due to the Poisson formula (see e.g., [29, Theorem 5.5, pp. 280-281]), the function
/ x d
u(t,x) := —
g(a)
2nd JdBdt(x) d2t2 - |x - y|2
da
+
h(a)
2nd J dB dt(x) ^ d2t2 - |x - y|2
da =
=d
g(x + dta) 2 f Vg(x + dta) ■ a i ^ f h(x + dta)
Bi(o) ^ 1 - |a|2
da + d t
Bi(o) V1 - |a|2
da + dt
'bi(o) ^1 - |a|2
da,
2
y
k
2
k
2
1
1
t > 0, x e R2, is a unique solution of problem (9) which belongs to G2([0, ro) x R3).
3. Concerning applications to semilinear Cauchy problems, we will continue here our former analyses of the semilinear Hammerstein integral equation of convolution type on Rn; see e.g., [19] and [16]. Consider the following integral equation
y(t) = f k(t - s)G(s; y(s)) ds, t e Rn, (10)
J Rn
where G : Rn x X M X is (Rn, B, I)-slowly oscillating with B being the collection of all bounded subsets of X. Then the equation (10) has a unique bounded, slowly oscillating solution, provided that supteRn;xeB ||G(t; x)|| < ro for every bounded subset B of X, and there exists a finite real constant L > 0 such that
||G(t; y) - G(t; y')||X < L||y - y'||X, t e Rn, y e X, y' e X.
Keeping in mind our assumption on the function G(; •), we can apply Theorem 3 in order to see that for any bounded, slowly oscillating function y : Rn M X the function t M G(t; y(t)), t e Rn is bounded and slowly oscillating, as well. The space consisting of all bounded, slowly oscillating functions in Rn is a Banach space equipped the sup-norm, which is also convolution invariant due to Theorem 2. Denote this space by BSO(Rn; X); by the foregoing, we have that the mapping
BSO(Rn; X) 9 y M i k(- s)G(s; y(s)) ds e BSO(Rn; X)
J Rn
is well defined. If we assume that L JRn |k(t)| dt < 1, then we can apply the Banach contraction principle to obtain the required.
In the one-dimensional setting, Theorem 3 can be applied in the qualitative analysis of solutions for various classes of semilinear (fractional) integro-differential Cauchy problems; see also [30, Lemma 3.5] and [8].
6. Conclusions and final remarks
In this paper, we have analyzed multi-dimensional (S, D, B)-asymptotically (w,p)-periodic type functions, multi-dimensional quasi-asymptotically (remotely) p-almost periodic type functions, and multi-dimensional p-slowly oscillating functions, where p is a general binary relation acting on the pivot Banach space Y. We have clarified many structural properties for introduced classes of functions and furnished certain applications of our results to the abstract Volterra integro-differential equations and the partial differential equations.
It is worth noting that we can also analyze the Stepanov classes of quasi-asymptotically p-almost periodic type functions under the following conditions (for further information concerning the Lebesgue spaces with variable exponents Lp(x), the reader may consult the monograph [31] by L. Diening, P. Harjulehto, P. Hastuso, M. Ruzicka and the list of references quoted in our joint research articles [32; 33] with T. Diagana); here we use the notation from [34]): Q is a fixed compact subset of Rn with positive Lebesgue measure, 0 = A C Rn satisfies A + Q C A, D C A C Rn, 0 = A' C A C Rn, the sets D and A' are unbounded, A + A' C A, and
(MD - B)s : 0 : [0, ro) M [0, ro), p e P(Q), F : A x (0, ro) x A'm (0, ro), F : A x N M (0, ro), F : A M (0, ro), p is a binary relation on Y and pj is a binary relation on Y (1 < j < n).
For example, we can consider the following generalization of the notion introduced in [16, Definition 4.1] (cf. also [16, Definition 4.2, Definition 4.3]):
Definition 7. Let (MD - B)S hold.
1. A function F : A x X M Y is called Stepanov-[Q, B, A', D,p,0, F,p]-quasi-asymptotically almost periodic, resp. Stepanov-[Q, B, A', D,p, 0, F, p]-quasi-asymptotically uniformly recurrent, if and only if for every B e B and e > 0 there exists l > 0 such that for each t0 e A' there exists r e B(t0,l) H A' such that there exists a finite real number M(e,r) > 0 such that, for every t e DM(e,T) such that t + t e DM(e,r), for every x e B and for almost every u e Q, there exists an element yt+u;x e p(F(t + u; x)) such that
sup F(t,e,r)0(||F(t + u + t; x) - yt+u;x|^lp(u)(n) < ^
teDM (E,T) :t+T GDm (e,T );x€B
resp. there exist a strictly increasing sequence (rk) in A' whose norms tending to plus infinity and a sequence (Mk) of positive real numbers tending to plus infinity such that, for every t e D, x e B and for almost every u e Q, there exists an element yt+u;x e p(F(t + u; x)) such that
lim sup F(t,k)0( ^F (t + u + rfc; x) - yt+uJy)lp(u)(n) = 0.
teroMfc :t+Tfc eroMfc
2. Let u e Rn\{0}, u + A C A, D C A C Rn and the set D be unbounded. A function F : A x X M Y is said to be Stepanov [S, Q, B, D,p, 0, F]-asymptotically (u, p)-periodic if and only if, for every B e B, t e D, x e B and for almost every u e Q, there exists an element yt+u;x e p(F(t + u; x)) such that
lim F(t)0(1F(t + u + u; x) - yt+uJL) lp(u)(q) = 0, uniformly in x e B.
3. Let uj e R \ {0}, ujej + A C A, Dj C A C Rn and the set Dj be unbounded (1 < j < n). A function F : A x X M Y is said to be [S, Q, B, D,p, 0, F]-asymptotically (uj,pj, Dj)jeNn-periodic if and only if, for every j e Nn, B e B, t e D, x e B and for almost every u e Q, there exists an element yj;t+u;x e pj (F(t + u; x)) such that
lim F(t)0( ^F (t + uj ej + u; x) - yj;t+u;x|^lp(u) (n) = 0, uniformly in x e B.
In the most important case, when p(x) = p e [1, ro), 0(x) = x, Q = [0,1]n, and the functions F, F, F are identically equal to one, we can simply transfer a great number of our results from the ordinary classes to Stepanov classes by using the multi-dimensional Bochner transform Fa : A x X M Yn, defined by
FA(t; x) (u) := F(t + u; x), t e A, u e Q, x e B;
here, Yn denotes the collection of all functions f : Q M Y. We close the paper with the observation that the statement of [16, Theorem 4.6], which investigates the convolution invariance of Stepanov-[Q, B, A', D,p, 0, F, c]-quasi-asymptotical almost periodicity (c e C \ {0}), can be generalized by assuming that p = T is a linear continuous function therein. This can be applied to the inhomogeneous heat equation in Rn, for example.
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Corrections received 03.03.2022.
Челябинский физико-математический журнал. 2022. Т. 7, вып. 1. С. 80-96.
УДК 517.518.6 DOI: 10.47475/2500-0101-2022-17106
ФУНКЦИИ р-ПОЧТИ ПЕРИОДИЧЕСКОГО ТИПА В R
М. Костич
Университет Нови-Сада, Нови-Сад, Сербия marco.s @verat.net
Исследованы различные классы многомерных функций (5, В, В)-асимптотически (ш, р)-периодического типа, многомерные функциии квазиасимптотически р-почти периодического типа и многомерные функции р-медленно осциллирующего типа вида ^ : I х X ^ У, где п € М, 0 = I С М", ш € М" \ {0}, X и У — комплексные банаховы пространства и р — бинарное отношение на У. Получены главные структурные свойства этих классов функций почти периодического типа. Получены некоторые приложения данных результатов к абстрактным интегро-дифференциальным уравнениям Вольтерры.
Ключевые слова: функции (Б, В, В)-асимптотически (ш, р)-периодического типа, функции квазиасимптотически р-почти периодического типа, функции удалённо р-почти периодического типа, функции р-медленно осциллирующего типа, абстрактные интегро-дифференциальные уравнения Вольтерры.
Поступила в 'редакцию 14-10.2021. После переработки 03.03.2022.
Сведения об авторе
Костич Марко, профессор, факультет технических наук, Университет Нови-Сада, Нови-Сад, Сербия; e-mail: [email protected].
Работа частично поддержана грантом 451-03-68/2020/14/200156 Министерства науки и технологического развития Сербии.
