Solvability of boundary-value problems for nonlinear fractional differential equations
We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations $$D^{α}u(t)+λ[f(t,u(t))+q(t)]=0,\; 0 < t < 1, \; u(0) = 0,\; u(1) = βu(η),$$ where $λ > 0$ is a parameter, $1 < α ≤ 2,\; η ∈ (0, 1),\; β ∈ \mat...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508953901793280 |
|---|---|
| author | Guo, Y. Го, Ю. |
| author_facet | Guo, Y. Го, Ю. |
| author_sort | Guo, Y. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:41:02Z |
| description | We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations
$$D^{α}u(t)+λ[f(t,u(t))+q(t)]=0,\; 0 < t < 1, \; u(0) = 0,\; u(1) = βu(η),$$
where $λ > 0$ is a parameter, $1 < α ≤ 2,\; η ∈ (0, 1),\; β ∈ \mathbb{R} = (−∞,+∞),\; βη^{α−1} ≠ 1,\; D^{α}$ is a Riemann–Liouville differential operator of order $α$, $f: (0,1)×\mathbb{R}→\mathbb{R}$ is continuous, $f$ may be singular for $t = 0$ and/or $t = 1$, and $q(t) : [0, 1] → [0, +∞)$. We give some sufficient conditions for the existence of nontrivial solutions to the formulated boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. In particular, we do not use the assumption of nonnegativity and monotonicity of $f$ essential for the technique used in almost all available literature. |
| first_indexed | 2026-03-24T02:33:24Z |
| format | Article |
| fulltext |
УДК 512.662.5
Y. Guo (Qufu Normal University, China)
SOLVABILITY OF BOUNDARY-VALUE
PROBLEMS FOR NONLINEAR FRACTIONAL
DIFFERENTIAL EQUATIONS*
РОЗВ’ЯЗУВАНIСТЬ КРАЙОВИХ ЗАДАЧ ДЛЯ НЕЛIНIЙНИХ
ДРОБОВИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
We consider the existence of nontrivial solutions of boundary-value problem for the nonlinear fractional
differential equation
Dαu(t) + λ[f(t, u(t)) + q(t)] = 0, 0 < t < 1,
u(0) = 0, u(1) = βu(η),
where λ > 0 is a parameter, 1 < α ≤ 2, η ∈ (0, 1), β ∈ R = (−∞,+∞), βηα−1 6= 1, Dα is the
Riemann – Liouville differential operator of order α, and f : (0, 1)×R→ R is continuous, f may be singular
at t = 0 and/or t = 1, q(t) : [0, 1] → [0,+∞) is continuous. We give some sufficient conditions for the
existence of nontrivial solutions to the above boundary-value problems. Our approach is based on Leray –
Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity of f
which was essential for the technique used in almost all existed literature.
Розглянуто iснування нетривiальних розв’язкiв крайової задачi для нелiнiйних дробових диференцiаль-
них рiвнянь
Dαu(t) + λ[f(t, u(t)) + q(t)] = 0, 0 < t < 1,
u(0) = 0, u(1) = βu(η),
де λ > 0 — параметр, 1 < α ≤ 2, η ∈ (0, 1), β ∈ R = (−∞,+∞), βηα−1 6= 1,Dα — диференцiальний
оператор Рiманна – Лiувiлля порядку α, функцiя f : (0, 1)× R→ R неперервна, причому f може бути
сингулярною при t = 0 та (або) t = 1, q(t) : [0, 1] → [0,+∞) неперервна. Наведено деякi достатнi
умови для iснування нетривiальних розв’язкiв вказаних крайових задач. Застосований у дослiдженнях
пiдхiд базується на нелiнiйнiй альтернативi Лереа – Шаудера. Зокрема, не використовується припущення
про невiд’ємнiсть, а також монотоннiсть функцiї f, що було iстотним для методики, застосованої майже
у всiх описаних у лiтературi дослiдженнях.
1. Introduction. Fractional calculus has played a significant role in engineering, science,
economy, and other fields. Many papers and books on fractional calculus, fractional
differential equations have appeared recently, (see [1, 6 – 9]). It should be noted that
most of papers and books on fractional calculus are devoted to the solvability of linear
initial fractional differential equations in terms of special functions [5]. Recently, there are
some papers deal with the existence and multiplicity of solutions (or positive solutions)
of nonlinear initial fractional differential equations by the use of techniques of nonlinear
analysis (fixed-point theorems, Leray – Shauder theory, etc.), see [7 – 10]. However, there
are few papers consider the three-point problem for linear ordinary differential equations
of fractional order, see [11, 12]. No contributions exist, as far as we know, concerning
the existence and multiplicity of positive solutions of the following problem:
Dαu(t) + λ[f(t, u(t)) + q(t)] = 0, 0 < t < 1,
u(0) = 0, u(1) = βu(η),
(1.1)
*The authors were supported financially by the NNSF of China (10801088).
c© Y. GUO, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9 1211
1212 Y. GUO
where λ > 0 is a parameter, 1 < α ≤ 2, η ∈ (0, 1), β ∈ R = (−∞,+∞) are real
numbers, βη 6= 1, and Dα
0+ is the Riemann – Liouville differential operator of order
α, and f : (0, 1) × R → R is continuous, f may be singular at t = 0 and/or t = 1,
q(t) : [0, 1] → [0,+∞) is continuous. As far as we known, there has no paper which
deal with the boundary-value problem for nonlinear fractional differential equation (1.1).
In [7], the authors consider the existence and multiplicity of positive solutions of
nonlinear fractional differential equation boundary-value problem
Dα
0+u(t) + f(t, u(t)) = 0, 0 < t < 1,
u(0) = u(1) = 0,
(1.2)
where 1 < α ≤ 2 is a real number. Dα
0+ is the standard Riemann – Liouville fractional
derivative, and f : [0, 1]× [0,+∞)→ [0,+∞) is continuous.
In [10], the authors consider the existence and multiplicity of positive solutions of
nonlinear fractional differential equation boundary-value problem
Dαu(t) + a(t)f(u(t)) = 0, 0 < t < 1,
u(0) = u′(1) = 0,
(1.3)
where 1 < α ≤ 2 is a real number. Dα is the Riemann – Liouville differential operator of
order α, and f : [0, 1]×[0,+∞)→ [0,+∞) is continuous, a is a positive and continuous
function on [0, 1].
Motivated by the work mentioned above, in this paper, we establish serval sufficient
conditions of the existence of nontrivial solutions for the above nonlinear fractional
differential equations (1.1). Here, by a nontrivial solution of (1.1) we understand a
function u(t) 6≡ 0 which satisfies (1.1). Our results are new. Particularly, we do not
use the nonnegative assumption and monotonicity which was essential for the technique
used in almost all existed literature on f.
2. Preliminaries. For completeness, in this section, we will demonstrate and study
the definitions and some fundamental facts of fractional order.
Definition 2.1 ([6], Definition 2.1). For a positive function f(x) given in the interval
[0,∞), the integral
Isf(x) =
1
Γ(s)
x∫
0
f(t)
(x− t)1−s
dt, x > 0,
where s > 0, is called Riemann – Liouville fractional integral of order s.
Definition 2.2 [6, p. 36 – 37]. For a positive function f(x) given in the interval
[0,∞), the expression
Dsf(x) =
1
Γ(n− s)
(
d
dx
)n x∫
0
f(t)
(x− t)s−n+1
dt,
where n = [s] + 1, [s] denotes the integer part of number s, is called the Riemann –
Liouville fractional derivative of order s.
Remark. If f ∈ C[0, 1], then DsIsf(x) = f(x).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
SOLVABILITY OF BOUNDARY-VALUE PROBLEMS FOR NONLINEAR FRACTIONAL . . . 1213
In order to rewrite (1.1), (1.2) as an integral equation, we need to know the action of
the fractional integral operator Is on Dsf for a given function f. To this end, we first
note that if λ > −1, then
Dstλ =
Γ(λ+ 1)
Γ(λ− s+ 1)
tλ−s,
Dsts−k = 0, k = 1, 2, . . . , n,
where n = [s].
The following two lemmas, found in [7], are crucial in finding an integral representati-
on of the boundary-value problem (1.1).
Lemma 2.1. Let α > 0, u ∈ C[0, 1], then the differential equation
Dαu(t) = 0
has solutions u(t) = c1t
α−1 + c2t
α−2 + . . . + cnt
α−n, ci ∈ R, i = 0, 1, . . . , n, n =
= [α] + 1.
From the lemma above, we deduce the following statement.
Lemma 2.2. Let α > 0, u ∈ C[0, 1], then
IαDαu(t) = u(t) + c1t
α−1 + c2t
α−2 + . . .+ cnt
α−n
for some ci ∈ R, i = 0, 1, . . . , n, n = [α] + 1.
The following theorems will play major role in our next analysis.
Lemma 2.3 [3, 4]. Let X be a real Banach space, Ω be a bounded open subset of
X, 0 ∈ Ω, T : Ω → X is a completely continuous operator. Then, either there exists
x ∈ ∂Ω, µ > 1 such that T (x) = µx, or there exists a fixed point x∗ ∈ ∂Ω.
3. Main results. In this section, we give our main results. First, we have the
following lemma.
Lemma 3.1. If 1 < α ≤ 2, βηα−1 6= 1, u ∈ C[0, 1]. Let h(t) ∈ C[0, 1] be a
given function, then the boundary-value problem
Dαu(t) + h(t) = 0, 0 < t < 1,
u(0) = 0, u(1) = βu(η),
(3.1)
has a unique solution
u(t) = − 1
Γ(α)
t∫
0
(t− s)α−1h(s)ds+
tα−1
1− βηα−1
1
Γ(α)
1∫
0
(1− s)α−1h(s) ds−
− βtα−1
1− βηα−1
1
Γ(α)
η∫
0
(η − s)α−1h(s) ds.
Proof. By the Lemma 2.2, we can reduce the equation of problem (3.1) to an
equivalent integral equation
u(t) = − 1
Γ(α)
t∫
0
(t− s)α−1h(s)ds+ c1t
α−1 + c2t
α−2
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1214 Y. GUO
for someconstants c1, c2 ∈ R. As boundary conditions for problem (3.1), we have c2 = 0
and
c1 =
1
1− βηα−1
1
Γ(α)
1∫
0
(1− s)α−1h(s)ds− β
η∫
0
(η − s)α−1h(s)ds
.
Therefore, the unique solution of (3.1) is
u(t) = − 1
Γ(α)
t∫
0
(t− s)α−1h(s)ds+
tα−1
1− βηα−1
1
Γ(α)
1∫
0
(1− s)α−1h(s) ds−
− βtα−1
1− βηα−1
1
Γ(α)
η∫
0
(η − s)α−1h(s)ds
which completes the proof.
The lemma is proved.
Let E = C[0, 1] be endowed with the maximum norm ‖u‖ = max
0≤t≤1
|u(t)|. Clearly,
it follows that (E, ‖ · ‖) is a Banach space.
Theorem 3.1. Suppose that f(t, 0) 6≡ 0, t ∈ [0, 1], βηα−1 6= 1, and there exist
nonnegative functions r ∈ C[0, 1], p ∈ C(0, 1) (p may be singular at t = 0 and/or
t = 1) such that
(H1)
∫ 1
0
(1− s)α−1p(s)ds < +∞;
(H2) the function f satisfies
|f(t, u)| ≤ p(t)|u|+ r(t), a.e. (t, u) ∈ (0, 1)× R,
and there exists t0 ∈ [0, 1] such that p(t0) 6= 0.
Then there exists a constant λ∗ > 0, such that for any 0 < λ ≤ λ∗, the boundary-
value problem (1.1) has at least one nontrivial solution u∗ ∈ C[0, 1].
Proof. Let
A =
(
1 +
∣∣∣∣ 1
1− βηα−1
∣∣∣∣)
1∫
0
(1− s)α−1p(s)ds+
∣∣∣∣ β
1− βηα−1
∣∣∣∣
η∫
0
(η − s)α−1p(s)ds,
B =
(
1 +
∣∣∣∣ 1
1− βηα−1
∣∣∣∣)
1∫
0
(1− s)α−1k(s)ds+
∣∣∣∣ β
1− βηα−1
∣∣∣∣
η∫
0
(η − s)α−1k(s)ds,
where k(s) = r(s) + q(s). By Lemma 3.1, problem (1.1) has a solution u = u(t) if and
only if u solves the operator equation
(Tu)(t) = − λ
Γ(α)
t∫
0
(t− s)α−1
[
f(s, u(s)) + q(s)
]
ds+
+
tα−1
1− βηα−1
λ
Γ(α)
1∫
0
(1− s)α−1
[
f(s, u(s)) + q(s)
]
ds−
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
SOLVABILITY OF BOUNDARY-VALUE PROBLEMS FOR NONLINEAR FRACTIONAL . . . 1215
− βtα−1
1− βηα−1
λ
Γ(α)
η∫
0
(η − s)α−1
[
f(s, u(s)) + q(s)
]
ds
in E. So we only need to seek a fixed point of T in E. In view of nonnegativeness and
continuity of (t−s)α−1, tα−1
1− βηα−1
(1−s)α−1, βtα−1
1− βηα−1
(η−s)α−1 and continuity of
[f(t, u) + q(t)] and (H1), by Ascoli – Arzela Theorem, it is well known that this operator
T : E → E is a completely continuous operator.
Since |f(t, 0)| ≤ r(t), a.e., t ∈ [0, 1], we know
∫ 1
0
[r(t) + q(t)]dt > 0. From
p(t0) 6= 0, we easily obtain
∫ 1
0
p(s)ds > 0. Let
m =
B
A
, Ω =
{
u ∈ C[0, 1] : ‖u‖ < m
}
Suppose u ∈ ∂Ω, µ > 1 such that Tu = µu. Then
µm = µ‖u‖ = ‖Tu‖ = max
0≤t≤1
|(Tu)(t)| ≤
≤ max
0≤t≤1
λ
Γ(α)
t∫
0
(t− s)α−1|f(s, u(s)) + q(s)|ds+
+ max
0≤t≤1
tα−1
|1− βηα−1|
λ
Γ(α)
1∫
0
(1− s)α−1|f(s, u(s)) + q(s)|ds+
+ max
0≤t≤1
|β|tα−1
|1− βηα−1|
λ
Γ(α)
η∫
0
(η − s)α−1|f(s, u(s)) + q(s)|ds ≤
≤ λ
Γ(α)
1∫
0
(1− s)α−1(|f(s, u(s))|+ q(s))ds+
+
1
|1− βηα−1|
λ
Γ(α)
1∫
0
(1− s)α−1(|f(s, u(s))|+ q(s))ds+
+
∣∣∣∣ β
1− βηα−1
∣∣∣∣ λ
Γ(α)
η∫
0
(η − s)α−1(|f(s, u(s))|+ q(s))ds ≤
≤
(
1 +
∣∣∣∣ 1
1− βηα−1
∣∣∣∣) λ
Γ(α)
1∫
0
(1− s)α−1[p(s)|u(s)|+ r(s) + q(s)]ds+
+
∣∣∣∣ β
1− βηα−1
∣∣∣∣ λ
Γ(α)
η∫
0
(η − s)α−1[p(s)|u(s)|+ r(s) + q(s)]ds ≤
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1216 Y. GUO
≤ λ
Γ(α)
[(
1 +
∣∣∣∣ 1
1− βηα−1
∣∣∣∣)
1∫
0
(1− s)α−1p(s)ds+
+
∣∣∣∣ β
1− βηα−1
∣∣∣∣
η∫
0
(η − s)α−1p(s)ds
]
‖u‖+
+
λ
Γ(α)
[(
1 +
∣∣∣∣ 1
1− βηα−1
∣∣∣∣)
1∫
0
(1− s)α−1[r(s) + q(s)]ds+
+
∣∣∣∣ β
1− βηα−1
∣∣∣∣
η∫
0
(η − s)α−1[r(s) + q(s)]ds
]
.
Choose λ∗ =
Γ(α)
2A
. Then when 0 < λ ≤ λ∗, we have
µ‖u‖ ≤ 1
2
‖u‖+
B
2A
.
Consequently,
µ ≤ 1
2
+
B
2mA
= 1.
This contradicts µ > 1, by Lemma 2.3, T has a fixed point u∗ ∈ Ω, since f(t, 0) 6≡ 0,
then when 0 < λ ≤ λ∗, the boundary-value problem (1.1) has at least one nontrivial
solution u∗ ∈ C[0, 1].
Theprem 3.1 is proved.
Remark. Though the paper [13] devoted to a much more general case of multipoint
problems for equations of arbitrary order α > 1, our condition on f is obvious more
general than [13]. For example, for Example 4.1, our results are not covered by Salem’s
results.
Theorem 3.2. Suppose that f(t, 0) 6≡ 0, t ∈ [0, 1], βηα−1 6= 1, and there exist
nonnegative functions p ∈ C(0, 1) (p may be singular at t = 0 and/or t = 1) such that
(H1)
∫ 1
0
(1− s)α−1p(s)ds < +∞;
(H2) the function f satisfies∣∣f(t, u1)− f(t, u2)
∣∣ ≤ p(t)|u1 − u2|, a.e. (t, ui) ∈ (0, 1)× R, i = 1, 2,
and there exists t0 ∈ [0, 1] such that p(t0) 6= 0.
Then there exists a constant λ∗ > 0, such that for any 0 < λ ≤ λ∗, the boundary-
value problem (1.1) has an unique nontrivial solution u∗ ∈ C[0, 1].
Proof. In fact, if u2 = 0, then we have |f(t, u1)| ≤ p(t)|u1| + |f(t, 0)|, a.e.
(t, u1) ∈ [0, 1]×R. From Theorem 3.1, we know the boundary-value problem (1.1) has
a nontrivial solution u∗ ∈ C[0, 1].
But in this case, we prefer to concentrate on the uniqueness of nontrivial solutions
for the boundary-value problem (1.1). Let T be given in Theorem 3.1, we shall show
that T is a contraction. In fact,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
SOLVABILITY OF BOUNDARY-VALUE PROBLEMS FOR NONLINEAR FRACTIONAL . . . 1217
‖Tu1 − Tu2‖ = max
0≤t≤1
∣∣(Tu1)(t)− (Tu2)(t)
∣∣ ≤
≤ max
0≤t≤1
λ
Γ(α)
t∫
0
(t− s)α−1
∣∣f(s, u1(s))− f(s, u2(s))
∣∣ds+
+ max
0≤t≤1
λt
|1− βηα−1|
1
Γ(α)
1∫
0
(1− s)α−1
∣∣f(s, u1(s))− f(s, u2(s))
∣∣ds+
+ max
0≤t≤1
λ|β|t
|1− βηα−1|
1
Γ(α)
η∫
0
(η − s)α−1
∣∣f(s, u1(s))− f(s, u2(s))
∣∣ds ≤
≤ max
0≤t≤1
λ
Γ(α)
t∫
0
(t− s)α−1p(s)|u1 − u2|ds+
+ max
0≤t≤1
λt
|1− βηα−1|
1
Γ(α)
1∫
0
(1− s)α−1p(s)|u1 − u2|ds+
+ max
0≤t≤1
λ|β|t
|1− βηα−1|
1
Γ(α)
η∫
0
(η − s)α−1p(s)|u1 − u2|ds ≤
≤ λ
Γ(α)
1∫
0
(1− s)α−1p(s)|u1 − u2|ds+
+
λ
|1− βηα−1|
1
Γ(α)
1∫
0
(1− s)α−1p(s)|u1 − u2|ds+
+
λ|β|
|1− βηα−1|
1
Γ(α)
η∫
0
(η − s)α−1p(s)|u1 − u2|ds ≤
≤ λ
Γ(α)
1∫
0
(1− s)α−1p(s)ds+
1
|1− βηα−1|
1∫
0
(1− s)α−1p(s)ds+
+
|β|
|1− βηα−1|
η∫
0
(η − s)α−1p(s)ds
‖u1 − u2‖.
If we choose λ∗ =
Γ(α)
2A
, where A as in the Theorem 3.1. Then when 0 < λ ≤ λ∗, we
have
‖Tu1 − Tu2‖ ≤
1
2
‖u1 − u2‖.
So T is indeed a contraction. Finally we use the Banach fixed point theorem to deduce
the existence of an unique solution to the boundary-value problem (1.1).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1218 Y. GUO
Corollary 3.1. Suppose that f(t, 0) 6≡ 0, and
0 ≤M = lim sup
|u|→+∞
max
0≤t≤1
|f(t, u)|
|u|
< +∞,
M + 1− ε
αΓ(α)
[
1 +
1
|1− βηα−1|
+
|β|ηα
|1− βηα−1|
]
≤ 1,
where ε > 0 such that M +1−ε > 0. Then there exists a constant λ∗ > 0, such that for
any 0 < λ ≤ λ∗, the boundary-value problem (1.1) has at least one nontrivial solution
u∗ ∈ C[0, 1].
Proof. Let ε > 0 such that M + 1− ε > 0. By (3.2), there exists H > 0 such that
|f(t, u)| ≤ (M + 1− ε)|u|, |u| ≥ H, 0 ≤ t ≤ 1.
Let N = maxt∈[0,1],|u|≤H |f(t, u)|. Then for any (t, u) ∈ [0, 1]× R, we have
|f(t, u)| ≤ (M + 1− ε)|u|+N.
From Theorem 3.2 we know the boundary-value problem (1.1) has at least one nontrivial
solution u∗ ∈ C[0, 1].
4. Examples.
Example 4.1. Consider the following third-order three-point problem:
D
3/2
0+ y(t) = λ
(
y
3t2 sin t
4
√
1− t
+ t3
)
+ λ cos t, 0 < t < 1,
y(0) = 0, y(1) = 2
√
2y
(
1
2
)
,
(4.1)
where f(t, y) = y
3t2 sin t
4
√
1− t
+ t3, q(t) = cos t. We choose p(t) =
1√
1− t
, r(t) = t3,
then
A =
(
1 +
√
2√
2
) 1∫
0
√
1− s 1√
1− s
ds+
4√
2
1/2∫
0
√
1/2− s 1√
1− s
ds =
= 2 + 2
√
2
(
1
3
− ln 3
4
)
=
12 + 4
√
2− 3
√
2 ln 3
6
,
and
Γ(3/2)
2A
=
3
√
π
24 + 8
√
2− 6
√
2 ln 3
≈ 0.204568.
Choose λ∗ =
3
√
π
24 + 8
√
2− 6
√
2 ln 3
≈ 0.204568, then by Theorem 3.1, (4.1)
has a nontrivial solution y∗ ∈ C[0, 1] for any λ ∈
(
0,
3
√
π
24 + 8
√
2− 6
√
2 ln 3
]
≈
≈ (0, 0.204568].
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
SOLVABILITY OF BOUNDARY-VALUE PROBLEMS FOR NONLINEAR FRACTIONAL . . . 1219
Example 4.2. Consider the following second-order boundary-value problem:
−D1.5
0+y(t) =
1√
1− t
(y − cos y) + λte2t−1, 0 < t < 1,
y(0) = y(1) = 0.
(4.2)
In this example f(t, y(t)) =
1√
1− t
(y − cos y), then
∣∣f(t, y1(t))− f(t, y2(t))
∣∣ ≤ p(t)|y1 − y2|,
where p(t) =
1√
1− t
, by Computation, we get λ∗ =
3
√
π
24 + 8
√
2− 6
√
2 ln 3
≈ 0.204568.
Choose λ∗ =
3
√
π
24 + 8
√
2− 6
√
2 ln 3
≈ 0.204568, then by Theorem 3.2, (4.2) has a
nontrivial solution y∗ ∈ C[0, 1] for any λ ∈
(
0,
3
√
π
24 + 8
√
2− 6
√
2 ln 3
]
≈ (0, 0.204568].
Acknowledgment. The authors express their deep gratitude for the referee’s important
advices.
1. Agrawal O. P. Formulation of Euler – Larange equations for fractional variational problems // J. Math.
Anal. Appl. – 2002. – 272. – P. 368 – 379.
2. Delbosco D., Rodino L. Existence and uniqueness for a nonlinear fractional differential equation // J.
Math. Appl. – 1996. – 204. – P. 609 – 625.
3. Deimling K. Nonlinear functional analysis. – Berlin: Springer, 1985.
4. Liu B. Positive solutions of a nonlinear three-point boundary-value problem // Comput. Math. Appl. –
2002. – 44. – P. 201 – 211.
5. Podlubny I. Fractional differential equations // Math. Sci. and Eng. – 1999. – 198.
6. Samko S. G., Kilbas A. A., Marichev O. I. Fractional integral and derivatives (theorey and applications).
– Switzerland: Gordon and Breach, 1993.
7. Bai Zhanbing, Lü Haishen. Positive solutions for boundary-value problem of nonlinear fractional di-
fferential equation // J. Math. Anal. and Appl. – 2005. – 311. – P. 495 – 505.
8. Zhang Shu-qin. The existence of a positive solution for a nonlinear fractional differential equation //
Ibid. – 2000. – 252. – P. 804 – 812.
9. Zhang Shu-qin. Existence of positive solution for some class of nonlinear fractional differential equations
// Ibid. – 2003. – 278, № 1. – P. 136 – 148.
10. Kaufmann Eric R., Mboumi E. Positive solutions of a boundary-value problem for a nonlinear fractional
differential equations // Electron. J. Qual. Theory Different. Equat. – 2008. – № 3. – P. 1 – 11.
11. Kilbas A. A., Trujillo J. J. Differential equations of fractional order: methods, results and problems. II //
Appl. Anal. – 2002. – 81. – P. 435 – 493.
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fractional derivatives in the lower terms // Dokl. Akad. Nauk SSSR. – 1977. – 234. – P. 308 – 311.
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Received 08.01.09,
after revision — 13.06.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
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| id | umjimathkievua-article-2949 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:33:24Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/34/7b3ebf0b9ba4abb9a3ca853448105934.pdf |
| spelling | umjimathkievua-article-29492020-03-18T19:41:02Z Solvability of boundary-value problems for nonlinear fractional differential equations Розв'язність крайових задач для нелінійних дробових диференціальних рівнянь Guo, Y. Го, Ю. We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations $$D^{α}u(t)+λ[f(t,u(t))+q(t)]=0,\; 0 < t < 1, \; u(0) = 0,\; u(1) = βu(η),$$ where $λ > 0$ is a parameter, $1 < α ≤ 2,\; η ∈ (0, 1),\; β ∈ \mathbb{R} = (−∞,+∞),\; βη^{α−1} ≠ 1,\; D^{α}$ is a Riemann–Liouville differential operator of order $α$, $f: (0,1)×\mathbb{R}→\mathbb{R}$ is continuous, $f$ may be singular for $t = 0$ and/or $t = 1$, and $q(t) : [0, 1] → [0, +∞)$. We give some sufficient conditions for the existence of nontrivial solutions to the formulated boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. In particular, we do not use the assumption of nonnegativity and monotonicity of $f$ essential for the technique used in almost all available literature. Розглянуто існування нетривіальних розв'язків крайової задачі для нелінійних дробових диференціальних рівнянь $$D^{α}u(t)+λ[f(t,u(t))+q(t)]=0,\; 0 < t < 1, \; u(0) = 0,\; u(1) = βu(η),$$ де $λ > 0$ — параметр, $1 < α ≤ 2,\; η ∈ (0, 1),\; β ∈ \mathbb{R} = (−∞,+∞),\; βη^{α−1} ≠ 1,\; D^{α}$ —диференціальний оператор Рімана-Ліувілля порядку $α$, функція $f: (0,1)×\mathbb{R}→\mathbb{R}$ неперервна, до того ж $f$ може бути сингулярною при $t = 0$ та (або) $q(t) : [0, 1] → [0, +∞)$ неперервна. Наведено деякі достатні умови для існування нетривіальних розв'язків вказаних крайових задач. Застосований у дослідженнях підхід базується на нелінійній альтернативі Лерея - Шаудера. Зокрема, не використовується припущення про невід'ємність, а також монотонність функції $f$ , що було істотним для методики, застосованої майже в усіх описаних у літературі дослідженнях. Institute of Mathematics, NAS of Ukraine 2010-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2949 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 9 (2010); 1211–1219 Український математичний журнал; Том 62 № 9 (2010); 1211–1219 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2949/2648 https://umj.imath.kiev.ua/index.php/umj/article/view/2949/2649 Copyright (c) 2010 Guo Y. |
| spellingShingle | Guo, Y. Го, Ю. Solvability of boundary-value problems for nonlinear fractional differential equations |
| title | Solvability of boundary-value problems for nonlinear fractional differential equations |
| title_alt | Розв'язність крайових задач для нелінійних дробових диференціальних рівнянь |
| title_full | Solvability of boundary-value problems for nonlinear fractional differential equations |
| title_fullStr | Solvability of boundary-value problems for nonlinear fractional differential equations |
| title_full_unstemmed | Solvability of boundary-value problems for nonlinear fractional differential equations |
| title_short | Solvability of boundary-value problems for nonlinear fractional differential equations |
| title_sort | solvability of boundary-value problems for nonlinear fractional differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2949 |
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