Existence and attractivity results for Hilfer fractional differential equations
We present some results of the existence of attracting solutions of some fractional differential equations of Hilfer type. The results of the existence of solutions are applied to the Schauder fixed point theorem. We prove that all solutions are uniformly locally attracting.
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nasplib_isofts_kiev_ua-123456789-1773292025-02-23T18:34:21Z Existence and attractivity results for Hilfer fractional differential equations Існування притягувальних розв’язків дробових диференціальних рівнянь Хілфера Существование притягивающих решений дробных дифференциальных уравнений Хилфера Abbas, S. Benchohra, M. Henderson, J. We present some results of the existence of attracting solutions of some fractional differential equations of Hilfer type. The results of the existence of solutions are applied to the Schauder fixed point theorem. We prove that all solutions are uniformly locally attracting. Наведено результати iснування притягувальних розв’язкiв деяких дробових диференцiальних рiвнянь хiлферовського типу. Результати iснування розв’язкiв застосовано для теореми Шаудера про нерухому точку. Доведено, що всi розв’язки однорiдно локально притягувальнi. 2018 Article Existence and attractivity results for Hilfer fractional differential equations / S. Abbas, M. Benchohra, J. Henderson // Нелінійні коливання. — 2018. — Т. 21, № 3. — С. 295-304 — Бібліогр.: 32 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/177329 517.9 en Нелінійні коливання application/pdf Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
We present some results of the existence of attracting solutions of some fractional differential equations of Hilfer type. The results of the existence of solutions are applied to the Schauder fixed point theorem. We prove that all solutions are uniformly locally attracting. |
| format |
Article |
| author |
Abbas, S. Benchohra, M. Henderson, J. |
| spellingShingle |
Abbas, S. Benchohra, M. Henderson, J. Existence and attractivity results for Hilfer fractional differential equations Нелінійні коливання |
| author_facet |
Abbas, S. Benchohra, M. Henderson, J. |
| author_sort |
Abbas, S. |
| title |
Existence and attractivity results for Hilfer fractional differential equations |
| title_short |
Existence and attractivity results for Hilfer fractional differential equations |
| title_full |
Existence and attractivity results for Hilfer fractional differential equations |
| title_fullStr |
Existence and attractivity results for Hilfer fractional differential equations |
| title_full_unstemmed |
Existence and attractivity results for Hilfer fractional differential equations |
| title_sort |
existence and attractivity results for hilfer fractional differential equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2018 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/177329 |
| citation_txt |
Existence and attractivity results for Hilfer fractional differential equations / S. Abbas, M. Benchohra, J. Henderson // Нелінійні коливання. — 2018. — Т. 21, № 3. — С. 295-304 — Бібліогр.: 32 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
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2025-11-24T11:26:40Z |
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2025-11-24T11:26:40Z |
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| fulltext |
UDC 517.9
EXISTENCE AND ATTRACTIVITY RESULTS
FOR HILFER FRACTIONAL DIFFERENTIAL EQUATIONS
IСНУВАННЯ ПРИТЯГУВАЛЬНИХ РОЗВ’ЯЗКIВ ДРОБОВИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ ХIЛФЕРА
S. Abbas
Tahar Moulay Univ. Saı̈da
P.O. Box 138, EN-Nasr, Saı̈da, 20000, Algeria
e-mail: said.abbas@univ-saida.dz,
abbasmsaid@yahoo.fr
M. Benchohra
Djillali Liabes Univ. Sidi Bel-Abbès
P.O. Box 89, Sidi Bel-Abbès, 22000, Algeria
e-mail: benchohra@univ-sba.dz
J. Henderson
Baylor Univ.
Waco, Texas, 76798-7328, USA
e-mail: JohnnyHenderson@baylor.edu
We present some results of the existence of attracting solutions of some fractional differential equations
of Hilfer type. The results of the existence of solutions are applied to the Schauder fixed point theorem.
We prove that all solutions are uniformly locally attracting.
Наведено результати iснування притягувальних розв’язкiв деяких дробових диференцiальних рiв-
нянь хiлферовського типу. Результати iснування розв’язкiв застосовано для теореми Шаудера про
нерухому точку. Доведено, що всi розв’язки однорiдно локально притягувальнi.
1. Introduction. Fractional differential equations have recently been applied in various areas
of engineering, mathematics, physics and bio-engineering, and other applied sciences [1, 2]. For
some fundamental results in the theory of fractional calculus and fractional differential equations
we refer the reader to the monographs of Abbas et al. [3 – 5], Samko et al. [6], Kilbas et al. [7] and
Zhou [8, 9], the papers by Abbas et al. [10 – 13], Benchohra et al. [14 – 16], Lakshmikantham et
al. [17 – 19], and the references therein.
Recently, considerable attention has been given to the existence of solutions of initial and
boundary value problems for fractional differential equations with Hilfer fractional derivative;
see [1, 20 – 25]. In [4, 10 – 13, 26 – 30], Abbas et al. presented some results on the local and
global attractivity of solutions for some classes of fractional differential equations involving
both the Riemann –Liouville and the Caputo fractional derivatives by employing some fixed
point theorems. Motivated by the above papers, in this article we discuss the existence and the
attractivity of solutions for the following problem of Hilfer fractional differential equations of the
form
© S. Abbas, M. Benchohra, J. Henderson, 2018
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3 295
296 S. ABBAS, M. BENCHOHRA, J. HENDERSON
(
Dα,β
0 u
)
(t) = f(t, u(t)), t ∈ R+ := [0,+∞),(
I1−γ0 u
)
(t)
∣∣∣
t=0
= φ,
(1)
where α ∈ (0, 1), β ∈ [0, 1], γ = α+ β − αβ, φ ∈ R, f : R+ ×R→ R is a given function, I1−γ0
is the left-sided mixed Riemann –Liouville integral of order 1− γ, and Dα,β
0 is the generalized
Riemann –Liouville derivative operator of order α and type β, introduced by Hilfer in [1]. This
paper initiates the concept of local attractivity of solutions of problem (1).
2. Preliminaries. Let C be the Banach space of all continuous functions v from I := [0, T ],
T > 0, into R with the supremum (uniform) norm
‖v‖∞ := sup
t∈I
|v(t)|.
As usual, AC(I) denotes the space of absolutely continuous functions from I into R. We denote
by AC1(I) the space defined by
AC1(I) :=
{
w : I → R :
d
dt
w(t) ∈ AC(I)
}
.
By L1(I), we denote the space of Lebesgue-integrable functions v : I → R with the norm
‖v‖1 =
T∫
0
|v(t)|dt.
By Cγ(I) and C1
γ(I), we denote the weighted spaces of continuous functions defined by
Cγ(I) =
{
w : (0, T ]→ R : t1−γw(t) ∈ C
}
,
with the norm
‖w‖Cγ := sup
t∈I
∣∣t1−γw(t)
∣∣ ,
and
C1
γ(I) =
{
w ∈ C :
dw
dt
∈ Cγ
}
,
with the norm
‖w‖C1
γ
:= ‖w‖∞ + ‖w′‖Cγ .
Let BC := BC(R+) be the Banach space of all bounded and continuous functions from R+ into
R. By BCγ := BCγ(R+) we denote the weighted space of all bounded and continuous functions
defined by
BCγ =
{
w : (0,+∞)→ R : t1−γw(t) ∈ BC
}
,
with the norm
‖w‖BCγ := sup
t∈R+
∣∣t1−γw(t)
∣∣ .
In the following we denote ‖w‖BCγ by ‖w‖BC .
Now, we give some results and properties of fractional calculus.
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
EXISTENCE AND ATTRACTIVITY RESULTS FOR HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 297
Definition 2.1 [4, 6, 7]. The left-sided mixed Riemann – Liouville integral of order r > 0 of
a function w ∈ L1(I) is defined by
(Irθw) (t) =
1
Γ(r)
t∫
0
(t− s)r−1w(s)ds for a.e. t ∈ I,
where Γ(·) is the (Euler’s) Gamma function defined by
Γ(ξ) =
∞∫
0
tξ−1e−tdt, ξ > 0.
Notice that for all r, r1, r2 > 0 and each w ∈ C, we have Ir0w ∈ C, and
(Ir10 I
r2
0 w) (t) = (Ir1+r20 w)(t) for a.e. t ∈ I.
Definition 2.2 [4, 6, 7]. The Riemann – Liouville fractional derivative of order r ∈ (0, 1] of
a function w ∈ L1(I) is defined by
(Dr
0w) (t) =
(
d
dt
I1−r0 w
)
(t) =
=
1
Γ(1− r)
d
dt
t∫
0
(t− s)−rw(s)ds for a.e. t ∈ I.
Let r ∈ (0, 1], γ ∈ [0, 1) and w ∈ C1−γ(I). Then the following expression leads to the left
inverse operator as follows.
(Dr
0I
r
0w) (t) = w(t) for all t ∈ (0, T ].
Moreover, if I1−r0 w ∈ C1
1−γ(I), then the following composition is proved in [6]:
(Ir0D
r
0w) (t) = w(t)− (I1−r0 w)(0+)
Γ(r)
tr−1 for all t ∈ (0, T ].
Definition 2.3 [4, 6, 7]. The Caputo fractional derivative of order r ∈ (0, 1] of a function
w ∈ L1(I) is defined by
(cDr
0w) (t) =
(
I1−r0
d
dt
w
)
(t) =
=
1
Γ(1− r)
t∫
0
(t− s)−r d
ds
w(s)ds for all t ∈ I.
In [1], R. Hilfer studied applications of a generalized fractional operator having the
Riemann –Liouville and the Caputo derivatives as specific cases (see also [22 – 24].
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
298 S. ABBAS, M. BENCHOHRA, J. HENDERSON
Definition 2.4 (Hilfer derivative). Let α ∈ (0, 1), β ∈ [0, 1], w ∈ L1(I), I
(1−α)(1−β)
0 w ∈
∈ AC1(I). The Hilfer fractional derivative of order α and type β of w is defined as(
Dα,β
0 w
)
(t) =
(
I
β(1−α)
0
d
dt
I
(1−α)(1−β)
0 w
)
(t) for all t ∈ I. (2)
Properties. Let α ∈ (0, 1), β ∈ [0, 1], γ = α+ β − αβ, and w ∈ L1(I).
1. The operator
(
Dα,β
0 w
)
(t) can be written as
(
Dα,β
0 w
)
(t) =
(
I
β(1−α)
0
d
dt
I1−γ0 w
)
(t) =
(
I
β(1−α)
0 Dγ
0w
)
(t) for all t ∈ I.
Moreover, the parameter γ satisfies
γ ∈ (0, 1], γ ≥ α, γ > β, 1− γ < 1− β(1− α).
2. The generalization (2) for β = 0, coincides with the Riemann-Liouville derivative and for
β = 1 with the Caputo derivative,
Dα,0
0 = Dα
0 and Dα,1
0 = cDα
0 .
3. If Dβ(1−α)
0 w exists and is in L1(I), then(
Dα,β
0 Iα0 w
)
(t) =
(
I
β(1−α)
0 D
β(1−α)
0 w
)
(t) for a.e. t ∈ I.
Furthermore, if w ∈ Cγ(I) and I1−β(1−α)0 w ∈ C1
γ(I), then(
Dα,β
0 Iα0 w
)
(t) = w(t) for a.e. t ∈ I.
4. If Dγ
0w exists and is in L1(I), then
(
Iα0D
α,β
0 w
)
(t) = (Iγ0D
γ
0w) (t) = w(t)− I1−γ0 (0+)
Γ(γ)
tγ−1 for a.e. t ∈ I.
Corollary 2.1. Let h ∈ Cγ(I). Then the linear problem
(
Dα,β
0 u
)
(t) = h(t), t ∈ I := [0, T ],(
I1−γ0 u
)
(t)
∣∣∣
t=0
= φ,
has a unique solution given by
u(t) =
φ
Γ(γ)
tγ−1 + (Iα0 h) (t).
From the above corollary, we have the following lemma.
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
EXISTENCE AND ATTRACTIVITY RESULTS FOR HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 299
Lemma 2.1. Let f : I × R → R be such that f(·, u(·)) ∈ BCγ for any u ∈ BCγ . Then
problem (1) is equivalent to the problem of the solutions of the integral equation
u(t) =
φ
Γ(γ)
tγ−1 + (Iα0 f(·, u(·))) (t).
Let ∅ 6= Ω ⊂ BC, and let G : Ω→ Ω, and consider the solutions of the equation
(Gu)(t) = u(t). (3)
We introduce the following concept of attractivity of solutions for equation (3).
Definition 2.5. Solutions of equation (3) are locally attractive if there exists a ball B(u0, η)
in the space BC such that, for arbitrary solutions v = v(t) and w = w(t) of equations (3)
belonging to B(u0, η) ∩ Ω, we have
lim
t→∞
(v(t)− w(t)) = 0. (4)
When the limit (4) is uniform with respect to B(u0, η) ∩ Ω, solutions of equation (3) are said to
be uniformly locally attractive (or equivalently that solutions of (3) are locally asymptotically
stable).
Lemma 2.2 [31, p. 62]. Let D ⊂ BC. Then D is relatively compact in BC if the following
conditions hold:
(a) D is uniformly bounded in BC;
(b) the functions belonging to D are almost equicontinuous on R+, i.e., equicontinuous on
every compact of R+;
(c) the functions from D are equiconvergent, that is, given ε > 0 there exists T (ε) > 0 such
that |u(t)− limt→∞ u(t)| < ε for any t ≥ T (ε) and u ∈ D.
In the sequel we will make use of the following fixed point theorems.
Theorem 2.1 (Schauder fixed point theorem, [32]). Let E be a Banach space and Q be
a nonempty bounded convex and closed subset of E, and let N : Q → Q be a compact and
continuous map. Then N has at least one fixed point in Q.
3. Existence and attractivity results. Let us start by defining what we mean by a solution
of the problem (1).
Definition 3.1. By a solution of the problem (1) we mean a measurable function u ∈ BCγ
that satisfies the condition (I1−γ0 u)(0+) = φ, and the equation
(
Dα,β
0 u
)
(t) = f(t, u(t)) on R+.
The following hypotheses will be used in the sequel.
(H1) The function t 7→ f(t, u) is measurable on R+ for each u ∈ BCγ , and the function
u 7→ f(t, u) is continuous on BCγ for a.e. t ∈ R+,
(H2) There exists a continuous function p : R+ → R+ such that
|f(t, u)| ≤ p(t)
1 + |u|
for a.e. t ∈ R+ and each u ∈ R,
and
lim
t→∞
t1−γ(Iα0 p)(t) = 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
300 S. ABBAS, M. BENCHOHRA, J. HENDERSON
Set
p∗ = sup
t∈R+
t1−γ (Iα0 p) (t).
Now, we present a theorem concerning the existence and the attractivity of solutions of our
problem (1).
Theorem 3.1. Assume that the hypotheses (H1) and (H2) hold. Then the problem (1) has
at least one solution defined on R+. Moreover, solutions of problem (1) are uniformly locally
attractive.
Proof. Consider the operator N such that, for any u ∈ BCγ ,
(Nu)(t) =
φ
Γ(γ)
tγ−1 +
t∫
0
(t− s)α−1 f(s, u(s))
Γ(α)
ds.
The operator N maps BCγ into BCγ . Indeed the map N(u) is continuous on R+ for any
u ∈ BCγ , and for each t ∈ R+, we have
∣∣t1−γ(Nu)(t)
∣∣ ≤ |φ|
Γ(γ)
+
t1−γ
Γ(α)
t∫
0
(t− s)α−1|f(s, u(s))|ds ≤
≤ |φ|
Γ(γ)
+
t1−γ
Γ(α)
t∫
0
(t− s)α−1p(s) ds ≤ |φ|
Γ(γ)
+ p∗.
Thus
‖N(u)‖BC ≤
|φ|
Γ(γ)
+ p∗ := R. (5)
Hence, N(u) ∈ BCγ . This proves that the operator N maps BCγ into itself.
By Lemma 2.1, the problem of finding solutions of the problem (1) is reduced to finding the
solutions of the operator equation N(u) = u. Equation (5) implies that N transforms the ball
BR := B(0, R) = {w ∈ BCγ : ‖w‖BC ≤ R} into itself.
We shall show that the operator N satisfies all the assumptions of Theorem 2.1. The proof
will be given in several steps.
Step 1. N is continuous.
Let {un}n∈N be a sequence such that un → u in BR. Then, for each t ∈ R+, we have
∣∣t1−γ(Nun)(t)− t1−γ(Nu)(t)
∣∣ ≤ t1−γ
Γ(α)
t∫
0
(t− s)α−1|f(s, un(s))− f(s, u(s))| ds. (6)
Case 1. If t ∈ [0, T ], T > 0, then, since un → u as n → ∞ and f is continuous, by the
Lebesgue dominated convergence theorem, equation (6) implies
‖N(un)−N(u)‖BC → 0 as n→∞.
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
EXISTENCE AND ATTRACTIVITY RESULTS FOR HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 301
Case 2. If t ∈ (T,∞), T > 0, then from the hypotheses and (6), we get
∣∣t1−γ(Nun)(t)− t1−γ(Nu)(t)
∣∣ ≤ 2
t1−γ
Γ(α)
t∫
0
(t− s)α−1p(s) ds. (7)
Since un → u as n→∞ and t1−γ(Iα0 p)(t)→ 0 as t→∞, then (7) gives
‖N(un)−N(u)‖BC → 0 as n→∞.
Step 2. N(BR) is uniformly bounded.
This is clear since N(BR) ⊂ BR and BR is bounded.
Step 3. N(BR) is equicontinuous on every compact subset [0, T ] of R+, T > 0.
Let t1, t2 ∈ [0, T ], t1 < t2, and let u ∈ BR. Then we have∣∣∣t1−γ2 (Nu)(t2)− t1−γ1 (Nu)(t1)
∣∣∣ ≤
≤
∣∣∣∣∣∣t1−γ2
t2∫
0
(t2 − s)α−1
f(s, u(s))
Γ(α)
ds− t1−γ1
t1∫
0
(t1 − s)α−1
f(s, u(s))
Γ(α)
ds
∣∣∣∣∣∣ ≤
≤ t1−γ2
t2∫
t1
(t2 − s)α−1
|f(s, u(s))|
Γ(α)
ds+
+
t1∫
0
∣∣∣t1−γ2 (t2 − s)α−1 − t1−γ1 (t1 − s)α−1
∣∣∣ |f(s, u(s))|
Γ(α)
ds ≤
≤ t1−γ2
t2∫
t1
(t2 − s)α−1
p(s)
Γ(α)
ds+
t1∫
0
∣∣∣t1−γ2 (t2 − s)α−1 − t1−γ1 (t1 − s)α−1
∣∣∣ p(s)
Γ(α)
ds.
Thus, from the continuity of the function p and by setting p∗ = supt∈[0,T ] p(t), we get∣∣∣t1−γ2 (Nu)(t2)− t1−γ1 (Nu)(t1)
∣∣∣ ≤ p∗T
1−γ+α
Γ(1 + α)
(t2 − t1)α+
+
p∗
Γ(α)
t1∫
0
∣∣∣t1−γ2 (t2 − s)α−1 − t1−γ1 (t1 − s)α−1
∣∣∣ ds.
As t1 → t2, the right-hand side of the above inequality tends to zero.
Step 4. N(BR) is equiconvergent.
Let t ∈ R+ and u ∈ BR. Then we have
∣∣t1−γ(Nu)(t)
∣∣ ≤ |φ|
Γ(γ)
+
t1−γ
Γ(α)
t∫
0
(t− s)α−1|f(s, u(s))| ds ≤
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 3
302 S. ABBAS, M. BENCHOHRA, J. HENDERSON
≤ |φ|
Γ(γ)
+
t1−γ
Γ(α)
t∫
0
(t− s)α−1p(s) ds ≤ |φ|
Γ(γ)
+ t1−γ (Iα0 p) (t).
Since t1−γ(Iα0 p)(t)→ 0, as t→ +∞, we get
|(Nu)(t)| ≤ |φ|
t1−γΓ(γ)
+
t1−γ(Iα0 p)(t)
t1−γ
→ 0 as t→ +∞.
Hence,
|(Nu)(t)− (Nu)(+∞)| → 0 as t→ +∞.
As a consequence of Steps 1 to 4, together with the Lemma 2.2, we can conclude that N : BR →
→ BR is continuous and compact. From an application of Schauder’s theorem (Theorem 2.1),
we deduce that N has a fixed point u which is a solution of the problem (1) on R+.
Step 5. The uniform local attractivity of solutions.
Let us assume that u0 is a solution of problem (1) with the conditions of this theorem. Taking
u ∈ B(u0, 2p
∗), we have∣∣t1−γ(Nu)(t)− t1−γu0(t)
∣∣ =
∣∣t1−γ(Nu)(t)− t1−γ(Nu0)(t)
∣∣ ≤
≤ t1−γ
Γ(α)
t∫
0
(t− s)α−1|f(s, u(s))− f(s, u0(s))|ds ≤
≤ 2t1−γ
Γ(α)
t∫
0
(t− s)α−1p(s)ds ≤ 2p∗.
Thus, we get
‖N(u)− u0‖BC ≤ 2p∗.
Hence, we obtain that N is a continuous function such that
N (B (u0, 2p
∗)) ⊂ B (u0, 2p
∗) .
Moreover, if u is a solution of problem (1), then
|u(t)− u0(t)| = |(Nu)(t)− (Nu0)(t)| ≤
≤ 1
Γ(α)
t∫
0
(t− s)α−1|f(s, u(s))− f(s, u0(s))| ds ≤ 2 (Iα0 p) (t).
Thus
|u(t)− u0(t)| ≤
2t1−γ (Iα0 p) (t)
t1−γ
. (8)
By using (8) and the fact that limt→∞ t
1−γ (Iα0 p) (t) = 0, we deduce that
lim
t→∞
|u(t)− u0(t)| = 0.
Consequently, all solutions of problem (1) are uniformly locally attractive.
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EXISTENCE AND ATTRACTIVITY RESULTS FOR HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 303
4. An example. As an application of our results we consider the following problem of Hilfer
fractional differential equation of the form
(
D
1/2,1/2
0 u
)
(t) = f(t, u(t)), t ∈ R+,(
I
1/4
0 u
)
(t)|t=0 = 1,
(9)
where f(t, u) =
ct−1/4 sin t
64(1 +
√
t)(1 + |u|)
, t ∈ (0,∞), u ∈ R,
f(0, u) = 0, u ∈ R,
and c =
9
√
π
16
. Clearly, the function f is continuous.
The hypothesis (H2) is satisfied withp(t) =
ct−1/4| sin t|
64(1 +
√
t)
, t ∈ (0,∞),
p(0) = 0.
Also, we have
t1−γI
1/2
0 p(t) =
t1/4
Γ
(
1
2
) t∫
0
(t− τ)−1/2p(τ)dτ ≤ 1
4
t−1/4 → 0 as t→∞.
All conditions of Theorem 3.1 are satisfied. Hence, the problem (9) has at least one solution
defined on R+, and solutions of this problem are uniformly locally attractive.
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Received 16.11.16
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