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22Математические заметки СВФУ Апрель—июнь, 2019. Том 26, №2
UDC 517.98
ERRATUM AND ADDENDUM TO THE PAPER
"WEYL-ALMOST PERIODIC AND ASYMPTOTICALLY WEYL-ALMOST PERIODIC PROPERTIES OF SOLUTIONS TO LINEAR AND SEMILINEAR ABSTRACT VOLTERRA
INTEGRO-DIFFERENTIAL EQUATIONS", MAT. ZAM. SVFU, 25, N 2, 65—84 (2018) M. Kostic
Abstract. The main aim of this paper is to perceive some inconsistencies made in the paper "Weyl-almost periodic and asymptotically Weyl-almost periodic properties of solutions to linear and semilinear abstract Volterra integro-differential equations", recently published in Mathematical Notes of SVFU, as well as to propose alternative solutions for problems considered.
DOI: 10.25587/SVFU.2019.102.31520 Keywords: Weyl almost periodicity, erratum and addendum.
1. Introduction and mistakes observed
The proof of Proposition 2.1 in [1] is not correct in the case that p > 1. Proposition 2.1 in [1] is true for the class of equi-Weyl-p-almost periodic functions with exponent p =1. In the proof of this statement, we have claimed that any Weyl-p-almost periodic function is automatically Weyl p-bounded (equivalently, Stepanov p-bounded, Sp-bounded for short), which is not known in the existing literature (p > 1). If we consider the situation of Proposition 2.1 in [1] with p =1 and add the condition on Sx-boundednes of function g(^) therein, we will obtain a correct result for the class of Weyl-1-almost periodic functions, as well.
The main mistake made in the proof of Proposition 2.1 in [1] can be simply described as follows. We have obtained a correct estimate
œ / t-k \ 1/P
||G(t + T) - G(t)||< £ ||R(-)||L,[k,k+i] ( f ||g(s + T) - g(s)||p ds I , t G R
k=° \t-(k+1)
(1.1)
The author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.
© 2019 M. Kostic
in the proof, where || • || denotes the norm of the pivot space X. In the fifth line after the equation [1, (2.3)], the author has hustily interchanged (1.1) with the estimate
||G(t + t) -G(t)|| <
£||R(-)IU'[*,*+1] / ||g(s + t) - g(s)||p dsl , t G R,
\k=° t-(k+1)
which is not true. Unfortunately, the remaining part of proof, starting from that point, is correct.
The misleading proof of Proposition 2.1 in [1] also implies the obvious incorrectness in the formulation and proof of Proposition 2.5 in [1], where we have analyzed finite convolution product. In this paper, we will explain how one can state a correct result closely connected with the assertion of Proposition 2.5 in [1]. We also observe a small incorrectness in the formulation of Theorem 3.5 (Theorem 3.6).
2. Addendum
We will use the same notion and notation as in [1]. In contrast to [1], in this paper we analyze the quasi asymptotical almost periodicity of function G(-) defined through infinite convolution product with a given (equi-)Weyl-p-almost periodic forcing term g(-). The notion of quasi asymptotical almost periodicity is introduced in the following definition, which seems to be unknown in the existing literature:
Definition 2.1. Suppose that I = [0, to) or I = R. Then we say that a bounded continuous function f : I ^ X is quasi asymptotically almost periodic iff for each e > 0 there exists a finite number L(e) > 0 such that any interval I' C I of length L(e) contains at least one number t G I' satisfying that there exists a finite number M(e, t) > 0 such that
If (t + t) - f (t)|| < e, t G I, |t| > M(e, t).
In the case that I = [0, to), the usually considered class of asymptotically almost periodic functions is contained in the class of quasi asymptotically almost periodic functions (the number M depends only on e and not on t for asymptotically almost periodic functions). Dependence of M on t is crucial in the following example:
Example 2.2. Let I = R and let f (•) be any bounded scalar-valued continuous function such that f (t) = 1 for all t > 0 and f (t) = 0 for all t < — 1. Then it can be simply proved that f (•) is quasi asymptotically almost periodic and that f (•) is not asymptotically almost periodic in the sense that for each e > 0 there exist two finite numbers L(e) > 0 and M(e) > 0 such that any interval I' C I of length L(e) contains at least one number t G I' satisfying that ||f (t + t) — f (t)|| < e for all t G R with |t| > M(e).
Further analysis of quasi asymptotically almost periodic functions is without scope of this paper.
For our purposes, we also need to introduce the following definition:
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M. Kostic
Definition 2.3. Let 1 < p < to, A : (0, to) x I ^ [1, to) be a given mapping, and let f £ Lfoc(I : X). It is said that the function f (■) is A-Weyl-p-almost periodic, f £ A; W|p(I : X) for short, iff for each e > 0 we can find a real number L > 0 such that any interval I' C I of length L contains a point t £ I' such that
DP, [f (■ + t),f (•)] < e for all I > A(e,T).
It is clear that any Weyl-p-almost periodic function is A-Weyl-p-almost periodic for a certain mapping A : (0, to) x I ^ [1, to).
We continue by stating the following result:
Proposition 2.4. Suppose that 1 < p < to, 1/p + 1/q = 1, (R(t))t>0 C L(X, Y) is a strongly continuous operator family, the mapping t ^ ||R(t) ||, t £ (0,1] is an element of the space Lq[0,1] and there exists a finite constant M > 0 such that
^Z1/p||R(0||Lq[ki,(k+1)i] < M for all I > 1.
(2.1)
k=1
If g : R ^ X is equi-Weyl-p-almost periodic and the condition
Ve > 0 V? > 0 Vr £ R 3M(e,1,r) > 0:^ ||g(s + r) - g(s)||p ds < e, |t| > M(e,?,r)
(2.2)
t-i
holds true, then the function G : R ^ Y, given by
t
G(t) := J R(t - s)g(s) ds, t £ R,
(2.3)
is well-defined and quasi asymptotically almost periodic.
Proof. Without loss of generality, we may assume that X = Y Since any equi-Weyl-p-almost periodic function g : R ^ X is Sp-bounded, arguing as in the proof of [1, Proposition 2.1], we can see that G(-) is bounded and continuous on the real line. Therefore, it suffices to show that G(-) satisfies the condition from Definition 2.1. So, let a number e > 0 be given. Then there exist two numbers l = l(e) and L = L(e) > 0 such that any subinterval I of R of length L contains a number t £ I such that
sup
x£R
x+l
J ||g(t + r) - g(t)||P dt
1/p
e.
(2.4)
Arguing as in the proof of [2, Proposition 2.11], where the case ? = 1 has been examined, we get
/ t \ 1/p
t-kl
Wg(s + r) - g(s)||p ds
\t-(k+i)i
l|G(t + r) - G(t)W <£||R(-)||Lq[ki,(k+i)i]
k=1
+ ||R(^)|Lq[o,i] ( f ||g(s + r) - g(s)||p ds| , t £ R. (2.5)
1/p
Kt-l
Due to the estimate (2.4), we get ||G(t + t) - G(t)||< Me + ||R(0|U,[o,i] ^ - ||g(s + t) - g(s)||p ds^ , t G R.
If ||R(^)|Lq[0,i] = 0, then ||G(t + t) - G(t)|| < Me for all t G R; otherwise, using (2.2) with this l, t, and e replaced therein by e/||R(-)||L<j[0j1], we get the existence of a finite number M(e, t) = M(e,l(e),T) > 0 such that ||G(t + t) - G(t)|| < (M + 1)e for |t| > M(e, t). This completes the proof. □
Remark 2.5. Suppose that t ^ ||R(t)||, t G (0,1] is an element of the space Lq[0,1]. Then the condition (2.1) holds in the case that there exist two finite numbers M > 0 and Z < -1 such that ||R(t)|| < Mtz, t > 1. To see this, it suffices to observe that, in case p > 1, there is an absolute constant c > 0 such that for each l > 1 one has:
^ A Jf ^ ^
k=1 |Zq + 1 k=1 k=1
The analysis is similar in case p = 1.
Remark 2.6. If lim ||R(t)|| = 0 and g G LP (R : X) has compact support, t—
then the function G(-) always belongs to the space C0(R : Y), consisting of all continuous functions f : R ^ Y satisfying lim ||f (t)| = 0.
|t| —+TO
Remark 2.7. The validity of requirements of Proposition 2.1 in [1] for the class of equi-Weyl-p-almost periodic functions implies the validity of requirements of Proposition 2.4.
We can similarly prove the following proposition concerning the class of A-Weyl-p-almost periodic functions (then the estimates (2.4), (2.5) hold for any l > A(e, t), so that the final statement follows by applying (2.6)):
Proposition 2.8. Suppose that 1 < p < to, 1/p + 1/q = 1, (R(t))t>0 C L(X, Y) is a strongly continuous operator family, the mapping t ^ ||R(t) ||, t G (0,1] is an element of the space Lq[0,1] and there exists a finite constant M > 0 such that (2.1) holds. Let A : (0, to) x R ^ [1, to) be a given mapping. Assume that the following condition holds:
Ve > 0 Vt G R 3l(e,T) > A(e, t) 3M(e,T) > 0 :
t
||RO)||Lq[0,i(£,T)] I ||g(s + T) - g(s)|p < e, |t| > M(e,T). (2.6)
t-l(e,T)
If g : R ^ X is A-Weyl-p-almost periodic and Sp-bounded, then the function G : R ^ Y, given by (2.3), is well-defined and quasi asymptotically almost periodic.
The question when the resulting function G(-) will be asymptotically almost periodic is interesting but we will not consider this item here.
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M. Kostic
Concerning Proposition 2.5 in [1], the following should be said:
(i) Let the assumptions of this proposition hold good for the class of equi-Weyl-p-almost periodic functions. Suppose additionally that there exists a finite constant M > 0 such that (2.1) holds as well as that the function g(-) satisfies the condition (2.2). Then the resulting function H(•) belongs to the same class as in [1], with term W,Pap([0, to) : Y) replaced therein with the space consisting of all restrictions of quasi asymptotically almost periodic Y-valued functions to the non-negative real axis.
(ii) Let the assumptions of this proposition hold good for the class of Weyl-p-almost periodic functions. Suppose that A : (0, to) x R ^ [1, to) is a given mapping. Suppose additionally that there exists a finite constant M > 0 such that (2.1) holds as well as that the function g(-) is A-Weyl-p-almost periodic, Sp-bounded and satisfies the condition (2.6). Then the resulting function H(•) belongs to the same class as in [1], with term Wpp([0, to) : Y) replaced therein with the space consisting of all restrictions of quasi asymptotically almost periodic Y-valued functions to the non-negative real axis.
Finally, we would like to note the following. In the formulation of Theorem 3.5 in [1] (Theorem 3.6 in [1]), we have not imposed any condition on (equi-)Weyl-almost periodicity of function f (•, •), so that the mapping t ^ f (t, 0), t G R need not be Sp-bounded, as incorrectly stated in the proof of Theorem 3.5 in [1]. Both results, Theorem 3.5 and Theorem 3.6 in [1], are true if, in the initial formulations of these theorems, we add the condition that the mapping t ^ f (t, 0), t G R is Sp-bounded.
For more details about (equi-)Weyl-almost periodic functions and their applications, we refer the reader to [3].
REFERENCES
1. M. Kostic, "Weyl-almost periodic and asymptotically Weyl-almost periodic properties of solutions to linear and semilinear abstract Volterra integro-differential equations," Mat. Zamet. SVFU, 25, No. 2, 65-84 (2018).
2. M. Kostic, "Existence of generalized almost periodic and asymptotic almost periodic solutions to abstract Volterra integro-differential equations," Electron. J. Differ. Equ., 2017, No. 239, 1-30 (2017).
3. M. Kostic, "Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations," Banach J. Math. Anal. 13, 64-90 (2019).
Submitted May 23, 2019 Revised May 23, 2019 Accepted June 3, 2019
Marko Kostic
Faculty of Technical Sciences, University of Novi Sad,
6 Trg D. Obradovica, 21125 Novi Sad, Serbia [email protected]
