Existence of solutions for a fractional-order boundary value problem
UDC 517.9 We investigate the existence of solutions for a fractional-order boundary-value problem by using some fixed point theorems.As applications, examples are given to illustrate the main results.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512233359933440 |
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| author | Karaca , I. Y. Oz , D. Karaca , I. Y. Oz , D. D. |
| author_facet | Karaca , I. Y. Oz , D. Karaca , I. Y. Oz , D. D. |
| author_sort | Karaca , I. Y. |
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| datestamp_date | 2025-03-31T08:49:28Z |
| description | UDC 517.9
We investigate the existence of solutions for a fractional-order boundary-value problem by using some fixed point theorems.As applications, examples are given to illustrate the main results. |
| doi_str_mv | 10.37863/umzh.v72i12.6033 |
| first_indexed | 2026-03-24T03:25:32Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i12.6033
UDC 517.9
I. Y. Karaca, D. Oz (Ege Univ., Izmir, Turkey)
EXISTENCE OF SOLUTIONS FOR A FRACTIONAL-ORDER
BOUNDARY-VALUE PROBLEM
IСНУВАННЯ РОЗВ’ЯЗКIВ КРАЙОВОЇ ЗАДАЧI ДРОБОВОГО ПОРЯДКУ
We investigate the existence of solutions for a fractional-order boundary-value problem by using some fixed point theorems.
As applications, examples are given to illustrate the main results.
За допомогою теорем про нерухому точку вивчено проблему iснування розв’язкiв крайової задачi дробового порядку.
Як застосування наведено приклади, що iлюструють отриманi результати.
1. Introduction. The history of the theory of fractional calculus goes back to seventeenth century,
when in 1695 the derivative of order \alpha =
1
2
was defined by Leibnitz in his “Letter to L’Hospital”.
From that time, the theory be attractive to mathematics as well as physics, biology, engineering
and economy. The first application of fractional calculus was due to Abel in his solution to the
Tautochrone problem [8]. We refer to the books by Agarwal et al. [9], Kilbas et al. [1] and
Podlubny [2].
The existence of positive solutions for fractional-order nonlinear boundary-value problems has
been studied by many authors using the fixed point theorem in cones. To identify a few, we refer the
reader to [3 – 7, 10 – 12] and references therein.
Some studies in the literature are as follows:
X. Su [11] studied the multipoint boundary-value problem
D\alpha u(t) = f(t, v(t), D\mu v(t)), 0 < t < 1,
D\beta v(t) = g(t, u(t), D\nu u(t)), 0 < t < 1,
u(0) = u(1) = v(0) = v(1) = 0,
where 1 < \alpha , \beta < 2, \mu , \nu > 0, \alpha - \nu \geq 1, \beta - \mu \geq 1, f, g : [0, 1]\times \BbbR \times \BbbR \rightarrow \BbbR are given functions
and D is the standard Riemann – Liouville differentiation. X. Su obtained the existence of solutions a
boundary-value problem for a coupled differential system of fractional order by using Schauder fixed
point theorem.
Rehman and Khan [7] studied the multipoint boundary-value problem
D\alpha
t y(t) = f(t, y(t), D\beta
t y(t)), t \in (0, 1),
y(0) = 0, D\beta
t y(1) -
m - 2\sum
i=1
\eta iD
\beta
t y(\xi i) = y0,
where 1 < \alpha \leq 2, 0 < \beta < 1, 0 < \xi i < 1, i = 1, 2, . . . ,m - 2, \eta i \leq 0 with
\sum m - 2
i=1
\eta i\xi
\alpha - \beta - 1
i < 1
and D\alpha
t represents the standard Riemann – Liouville fractional derivative. They obtained the unique-
ness existence of solutions by means of the Banach fixed point theorem.
c\bigcirc I. Y. KARACA, D. OZ, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12 1651
1652 I. Y. KARACA, D. OZ
J. Graef, L. Kong, Q. Kong and M. Wang [4] study of the nonlinear fractional boundary-value
problem
- D\alpha
0+u = F (t, u), 0 < t < 1,
u(0) = 0, u(1) - aI\alpha 0+u(1) = b,
where 1 < \alpha \leq 2, 0 \leq a < \Gamma (\alpha + 1), b \in \BbbR + with \BbbR + = [0,\infty ), b = 0 if a = 0, and \Gamma is
the Gamma function. The authors study a type of nonlinear fractional boundary-value problem with
nonhomogeneous integral boundary conditions and the existence and uniqueness of positive solutions
are discussed.
Keyu Zhang and Jiafa Xu [5] consider the unique positive solution for the fractional boundary-
value problem
D\alpha
0+u(t) = - f(t, u(t)), t \in [0, 1],
u(0) = u\prime (0) = u\prime (1) = 0,
where \alpha \in (2, 3] is a real number, D\alpha
0+ is the standard Riemann – Liouville fractional derivative of
order \alpha . By using the method of upper and lower solutions and monotone iterative technique, they
also obtain that there exists a sequence of iterations uniformly converges to the unique solution.
Motivated by the above works, we study the fractional-order nonlinear boundary-value problem
D\alpha u(t) = f
\Bigl(
t, u(t), D\beta u(t)
\Bigr)
, t \in [0, 1],
u(0) = u\prime (0) = 0, (1.1)
D\alpha - 2u\prime (1) -
m - 2\sum
i=1
\eta iI
\delta u\prime (\xi i) = A.
Here, 2 < \alpha \leq 3, \beta \leq \alpha - 1, 0 < \xi 1 < \xi 2 < . . . < \xi m - 2 < 1, i = 1, . . . ,m - 2, \eta i > 0 and \delta > 0
with \Gamma (\alpha + \delta - 1) -
\sum m - 2
i=1
\eta i\xi
\alpha +\delta - 2
i \not = 0, A \in \BbbR , D\alpha standard Riemann – Liouville fractional
derivative and I\alpha standard Riemann – Liouville fractional integral.
Throughout this paper we assume that following conditions hold:
(H1) f : I \times \BbbR \times \BbbR \rightarrow \BbbR is a continuous function;
(H2) there exists a constant k > 0 such that
| f(t, u, v) - f (t, \=u, \=v)| \leq k (| u - \=u| + | v - \=v| )
for each t \in I and all u, \=u, v, \=v \in \BbbR .
By using Schauder fixed point theorem and the Banach contraction principle, we get the existence
of at least one solution.
2. Preliminaries. For the convenience of the reader, we present here some necessary definitions
from fractional calculus theory.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
EXISTENCE OF SOLUTIONS FOR A FRACTIONAL-ORDER BOUNDARY-VALUE PROBLEM 1653
Definition 2.1. The Riemann – Liouville fractional integral of order \alpha > 0 of a function f :
(0,\infty ) \rightarrow \BbbR is given by
I\alpha f(t) =
1
\Gamma (\alpha )
t\int
0
(t - s)\alpha - 1f(s)ds
provided that the right-hand side is pointwise defined on (0,\infty ).
Definition 2.2. The Riemann – Liouville fractional derivative of order \alpha > 0 of a continuous
function f : (0,\infty ) \rightarrow \BbbR is given by
D\alpha f(t) =
1
\Gamma (n - \alpha )
\biggl(
d
dt
\biggr) t\int
0
f(s)
(t - s)\alpha - n+1
ds,
where n - 1 \leq \alpha < n, provided that the right-hand side is pointwise defined on (0,\infty ).
Lemma 2.1. Let \alpha > 0, then I\alpha t D
\alpha
t u(t) = u(t) + c1t
\alpha - 1 + c2t
\alpha - 2 + c3t
\alpha - 3 + . . . + cnt
\alpha - n,
where ci \in \BbbR , i = 1, 2, . . . , n, n = [\alpha ] + 1. Here, I\alpha t stands for the standard Riemann – Liouville
fractional integral of order \alpha > 0 and D\alpha
t denotes the Riemann – Liouville fractional derivative.
Lemma 2.2. Assume that the conditions (H1) and (H2) are satisfied. Let
\Delta = \Gamma (\alpha )
\Biggl[
\Gamma (\alpha + \delta - 1) -
m - 2\sum
i=1
\eta i\xi
\alpha +\delta - 2
i
\Biggr]
.
If h \in C[0, 1], the fractional boundary-value problem
D\alpha u(t) = h(t), t \in [0, 1], 2 < \alpha \leq 3, \beta \leq \alpha - 1, \delta > 0,
u(0) = u\prime (0) = 0, (2.1)
D\alpha - 2u\prime (1) -
m - 2\sum
i=1
\eta iI
\delta u\prime (\xi i) = A,
has a unique solution
u(t) =
1\int
0
G(t, s)h(s)ds+
\Gamma (\alpha + \delta - 1)A
\Delta
t\alpha - 1, t \in [0, 1], (2.2)
where G(t, s) is the Green function and is given by
G(t, s) =
1
\Gamma (\alpha )
\left\{
(t - s)\alpha - 1
\Gamma (\alpha )
+
t\alpha - 1
\Delta
\Bigl( \sum m - 2
j=i
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\Bigr)
,
0 \leq s \leq t, \xi i - 1 \leq s \leq \xi i, i = 1, 2, . . . ,m - 1,
t\alpha - 1
\Delta
\Bigl( \sum m - 2
j=i
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\Bigr)
,
0 \leq t \leq s, \xi i - 1 \leq s \leq \xi i, i = 1, 2, . . . ,m - 1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1654 I. Y. KARACA, D. OZ
Proof. In view of Lemma 2.1 and (2.1), we have
u(t) = I\alpha h(t) + c1t
\alpha - 1 + c2t
\alpha - 2 + c3t
\alpha - 3 for c1, c2, c3 \in \BbbR . (2.3)
The boundary conditions u(0) = 0 and u\prime (0) = 0 satisfied that c2 = c3 = 0. We can be written
u(t) = I\alpha h(t) + c1t
\alpha - 1 for c1 \in \BbbR .
From here
u\prime (t) = I\alpha - 1h(t) + (\alpha - 1)c1t
\alpha - 2 for c1 \in \BbbR .
Using the relations D\alpha t\beta =
\Gamma (\beta + 1)
\Gamma (\beta - \alpha + 1)
t\beta - \alpha , I\alpha t\beta =
\Gamma (\beta + 1)
\Gamma (\alpha + \beta + 1)
t\alpha +\beta , (2.3) reduces to
D\alpha - 2u\prime (t) =
t\int
0
h(s)ds+ c1\Gamma (\alpha ). (2.4)
By using the boundary condition D\alpha - 2u\prime (1) -
\sum m - 2
i=1
\eta iI
\delta u\prime (\xi i) = A and (2.4), we obtain
c1 =
\sum m - 2
i=1
\eta i
\int \xi i
0
(\xi i - s)\alpha +\delta - 2h(s)ds - \Gamma (\alpha + \delta - 1)
\int 1
0
h(s)ds+ \Gamma (\alpha + \delta - 1)A
\Delta
.
Therefore the unique solution of problem (2.1) is given by
u(t) =
1
\Gamma (\alpha )
t\int
0
(t - s)\alpha - 1h(s)ds+
t\alpha - 1
\Delta
m - 2\sum
i=1
\eta i
\xi i\int
0
(\xi i - s)\alpha +\delta - 2h(s)ds -
- t\alpha - 1
\Delta
\Gamma (\alpha + \delta - 1)
1\int
0
h(s)ds+
t\alpha - 1
\Delta
\Gamma (\alpha + \delta - 1)A. (2.5)
For 0 \leq t \leq \xi 1, (2.5) can be expressed as follows:
u(t) =
t\int
0
\left[ (t - s)\alpha - 1
\Gamma (\alpha )
+
t\alpha - 1
\Delta
\left( m - 2\sum
j=1
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\right) \right] h(s)ds+
+
\xi 1\int
t
t\alpha - 1
\Delta
\left( m - 2\sum
j=1
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\right) h(s)ds+
+
m - 2\sum
i=2
\xi i\int
\xi i - 1
t\alpha - 1
\Delta
\left( m - 2\sum
j=i
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\right) h(s)ds -
-
1\int
\xi m - 2
t\alpha - 1\Gamma (\alpha + \delta - 1)
\Delta
h(s)ds+
t\alpha - 1\Gamma (\alpha + \delta - 1)
\Delta
A.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
EXISTENCE OF SOLUTIONS FOR A FRACTIONAL-ORDER BOUNDARY-VALUE PROBLEM 1655
For \xi k - 1 \leq t \leq \xi k, 2 \leq k \leq m - 2, (2.5) can be expressed as follows:
u(t) =
\xi 1\int
0
\left[ (t - s)\alpha - 1
\Gamma (\alpha )
+
t\alpha - 1
\Delta
\left( m - 2\sum
j=k
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\right) \right] h(s)ds+
+
k - 2\sum
i=2
\xi i\int
\xi i - 1
\left[ (t - s)\alpha - 1
\Gamma (\alpha )
+
t\alpha - 1
\Delta
\left( m - 2\sum
j=i
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\right) \right] h(s)ds+
+
t\int
\xi k - 1
\left[ (t - s)\alpha - 1
\Gamma (\alpha )
+
t\alpha - 1
\Delta
\left( m - 2\sum
j=k
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\right) \right] h(s)ds+
+
\xi k\int
t
t\alpha - 1
\Delta
\left( m - 2\sum
j=k
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\right) h(s)ds+
+
m - 2\sum
i=k+1
\xi i\int
\xi i - 1
t\alpha - 1
\Delta
\left( m - 2\sum
j=i
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\right) h(s)ds -
-
1\int
\xi m - 2
t\alpha - 1\Gamma (\alpha + \delta - 1)
\Delta
h(s)ds+
t\alpha - 1\Gamma (\alpha + \delta - 1)
\Delta
A.
For \xi m - 2 \leq t \leq 1, (2.5) can be expressed as follows:
u(t) =
m - 2\sum
i=1
\xi i\int
\xi i - 1
\left[ (t - s)\alpha - 1
\Gamma (\alpha )
+
t\alpha - 1
\Delta
\left( m - 2\sum
j=i
\eta j(\xi j - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\right) \right] h(s)ds+
+
t\int
\xi m - 2
\biggl(
(t - s)\alpha - 1
\Gamma (\alpha )
- t\alpha - 1\Gamma (\alpha + \delta - 1)
\Delta
\biggr)
h(s)ds -
-
1\int
t
t\alpha - 1\Gamma (\alpha + \delta - 1)
\Delta
h(s)ds+
t\alpha - 1\Gamma (\alpha + \delta - 1)
\Delta
A.
Hence, the unique solution of boundary-value problem (1.1) is given by
u(t) =
1\int
0
G(t, s)h(s)ds+
\Gamma (\alpha + \delta - 1)A
\Delta
t\alpha - 1.
Lemma 2.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1656 I. Y. KARACA, D. OZ
3. Main results. Let C[0, 1] be space of continuous functions defined on [0,1]. The space
E =
\Bigl\{
u : u \in C[0, 1], D\beta u \in C[0, 1]
\Bigr\}
equipped with the norm
\| u\| E = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
\biggl\{
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
| u(t)| , \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
\bigm| \bigm| \bigm| D\beta u(t)
\bigm| \bigm| \bigm| \biggr\}
is a Banach space.
For convenience, we define the following constants: N = \mathrm{m}\mathrm{a}\mathrm{x}t\in [0,1]
\bigm| \bigm| f \bigl( t, u(t), D\beta u(t)
\bigr) \bigm| \bigm| + 1
and
M = N
\left[ 1
\Gamma (\alpha - \beta + 1)
+
\Gamma (\alpha )
\Bigl[ \sum m - 2
i=1
\eta i\xi
\alpha +\delta - 1
i + \Gamma (\alpha + \delta )
\Bigr]
| \Delta | \Gamma (\alpha - \beta )(\alpha + \delta - 1)
\right] +
\Gamma (\alpha + \delta - 1)\Gamma (\alpha )| A|
| \Delta | \Gamma (\alpha - \beta )
.
Lemma 3.1. Assume that (H1) and (H2) hold. Then the operator T : \Omega \rightarrow \Omega is completely
continuous.
Proof. Define an operator T : E \rightarrow E by
Tu(t) =
1\int
0
G(t, s)h(s)ds+
\Gamma (\alpha + \delta - 1)A
\Delta
t\alpha - 1.
Fixed points of the operator T are solutions of the boundary-value problem (1.1). In view of the
continuity of f and G, the operator T is continuous.
Firstly, we define
\Omega = \{ u \in E : \| u\| E \leq M, t \in [0, 1],M > 0\}
and prove that T : \Omega \rightarrow \Omega . Let N = \mathrm{m}\mathrm{a}\mathrm{x}t\in [0,1] | f(t, u(t), D\beta u(t)| + 1. For every u \in \Omega and for
every u \in \Omega , we have
| (Tu)(t)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
G(t, s)f
\Bigl(
s, u(s), D\beta u(s)
\Bigr)
ds+
t\alpha - 1\Gamma (\alpha + \delta - 1)A
\Delta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
G(t, s)f
\Bigl(
s, u(s), D\beta u(s)
\Bigr)
ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
\bigm| \bigm| \bigm| \bigm| t\alpha - 1\Gamma (\alpha + \delta - 1)A
\Delta
\bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
| G(t, s)|
\bigm| \bigm| \bigm| f(s, u(s), D\beta u(s))
\bigm| \bigm| \bigm| ds+ \bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + \delta - 1)A
\Delta
\bigm| \bigm| \bigm| \bigm| \leq
\leq N
1\int
0
| G(t, s)| ds+ \Gamma (\alpha + \delta - 1)| A|
| \Delta |
\leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
EXISTENCE OF SOLUTIONS FOR A FRACTIONAL-ORDER BOUNDARY-VALUE PROBLEM 1657
\leq N
\left[ 1
\Gamma (\alpha + 1)
+
\sum m - 2
i=1
\eta i\xi
\alpha +\delta - 1
i + \Gamma (\alpha + \delta )
| \Delta | (\alpha + \delta - 1)
\right] +
\Gamma (\alpha + \delta - 1)| A|
| \Delta |
\leq M < \infty .
In view of relation D\beta I\alpha = I\beta - \alpha and (2.5), we have the following estimates:\bigm| \bigm| \bigm| D\beta (Tu)(t)
\bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \bigm| I\alpha - \beta f
\Bigl(
t, u(t), D\beta u(t)
\Bigr)
+
+
\Biggl[
1
\Delta
m - 2\sum
i=1
\eta i
\xi i\int
0
(\xi i - s)\alpha +\delta - 2f(s, u(s), D\beta u(s))ds -
- \Gamma (\alpha + \delta - 1)
\Delta
1\int
0
f
\Bigl(
s, u(s), D\beta u(s)
\Bigr)
ds+
\Gamma (\alpha + \delta - 1)A
\Delta
\Biggr]
D\beta t\alpha - 1
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
(t - s)\alpha - \beta - 1
\Gamma (\alpha - \beta )
\bigm| \bigm| \bigm| f \Bigl( s, u(s), D\beta (s)
\Bigr) \bigm| \bigm| \bigm| ds+
+
\Biggl[
1
| \Delta |
m - 2\sum
i=1
\eta i
\xi i\int
0
(\xi i - s)\alpha +\delta - 2
\bigm| \bigm| \bigm| f \Bigl( s, u(s), D\beta (s)
\Bigr) \bigm| \bigm| \bigm| ds+
+
\Gamma (\alpha + \delta - 1)
| \Delta |
1\int
0
\bigm| \bigm| \bigm| f \Bigl( s, u(s), D\beta (s)
\Bigr) \bigm| \bigm| \bigm| ds+ \Gamma (\alpha + \delta - 1)| A|
| \Delta |
\Biggr]
\Gamma (\alpha )
\Gamma (\alpha - \beta )
t\alpha - \beta - 1 \leq
\leq N
\Biggl[ 1\int
0
(1 - s)\alpha - \beta - 1
\Gamma (\alpha - \beta )
ds+
\Gamma (\alpha )
| \Delta | \Gamma (\alpha - \beta )
m - 2\sum
i=1
\eta i
\xi i\int
0
(\xi i - s)\alpha +\delta - 2ds+
+
\Gamma (\alpha + \delta - 1)\Gamma (\alpha )
| \Delta | \Gamma (\alpha - \beta )
\Biggr]
+
\Gamma (\alpha + \delta - 1)\Gamma (\alpha )
| \Delta | \Gamma (\alpha - \beta )
=
= N
\left[ 1
\Gamma (\alpha - \beta + 1)
+
\Gamma (\alpha )
\Bigl[ \sum m - 2
i=1
\eta i\xi
\alpha +\delta - 1
i + \Gamma (\alpha + \delta )
\Bigr]
| \Delta | \Gamma (\alpha - \beta )(\alpha + \delta - 1)
\right] +
+
\Gamma (\alpha + \delta - 1)\Gamma (\alpha )| A|
| \Delta | \Gamma (\alpha - \beta )
= M < \infty .
Therefore, \| (Tu)(t)\| E \leq M. Thus we have T : \Omega \rightarrow \Omega . Now we show that T is completely
continuous. For this, let
N = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
\bigm| \bigm| \bigm| f \Bigl( t, u(t), D\beta u(t)
\Bigr) \bigm| \bigm| \bigm| + 1
for u \in \Omega and t1, t2 \in [0, 1] be such that t1 < t2. Then we have
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1658 I. Y. KARACA, D. OZ\bigm| \bigm| \bigm| \bigm| (Tu)(t2) - (Tu)(t1)
\bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
(G(t2, s) - G(t1, s))f(s, u(s), D
\beta u(s))ds+
\Gamma (\alpha + \delta - 1)A
\Delta
\bigl(
t\alpha - 1
2 - t\alpha - 1
1
\bigr) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
| G(t2, s) - G(t1, s)|
\bigm| \bigm| \bigm| f \Bigl( s, u(s), D\beta u(s)
\Bigr) \bigm| \bigm| \bigm| ds+ \Gamma (\alpha + \delta - 1)| A|
| \Delta |
\bigm| \bigm| t\alpha - 1
2 - t\alpha - 1
1
\bigm| \bigm| \leq
\leq N
1\int
0
\bigm| \bigm| \bigm| \bigm| (t2 - s)\alpha - 1 - (t1 - s)\alpha - 1
\Gamma (\alpha )
+
t\alpha - 1
2 - t\alpha - 1
1
\Delta
\times
\times
\Biggl(
m - 2\sum
i=1
\eta i(\xi i - s)\alpha +\delta - 2 - \Gamma (\alpha + \delta - 1)
\Biggr) \bigm| \bigm| \bigm| \bigm| ds+ \Gamma (\alpha + \delta - 1)| A|
| \Delta |
\bigl(
t\alpha - 1
2 - t\alpha - 1
1
\bigr)
\leq
\leq N (t\alpha 2 - t\alpha 1 )
\Gamma (\alpha + 1)
+
N
\bigl(
t\alpha - 1
2 - t\alpha - 1
1
\bigr)
| \Delta |
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum m - 2
i=1
\eta i
\Bigl(
\xi \alpha +\delta - 1
i
\Bigr)
- \Gamma (\alpha + \delta )
\alpha + \delta - 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
+
\Gamma (\alpha + \delta - 1)| A|
| \Delta |
\bigl(
t\alpha - 1
2 - t\alpha - 1
1
\bigr)
,
\bigm| \bigm| \bigm| \bigm| D\beta (Tu)(t2) - D\beta (Tu)(t1)
\bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| I\alpha - \beta
\Bigl(
f
\Bigl(
t2, u(t2), D
\beta (t2)
\Bigr)
- f
\Bigl(
t1, u(t1), D
\beta (t1)
\Bigr) \Bigr)
+
+
\Biggl(
1
\Delta
m - 2\sum
i=1
\eta i
\xi i\int
0
(\xi i - s)\alpha +\delta - 2f
\Bigl(
s, u(s), D\beta u(s)
\Bigr)
ds -
- \Gamma (\alpha + \delta - 1)
\Delta
1\int
0
f
\Bigl(
s, u(s), D\beta u(s)
\Bigr)
ds+
\Gamma (\alpha + \delta - 1)A
\Delta
\Biggr)
D\beta
\bigl(
t\alpha - 1
2 - t\alpha - 1
1
\bigr) \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| N
\left( t2\int
0
(t2 - s)\alpha - \beta - 1
\Gamma (\alpha - \beta )
ds -
t1\int
0
(t1 - s)\alpha - \beta - 1
\Gamma (\alpha - \beta )
ds
\right) +
\Gamma (\alpha )
\Delta \Gamma (\alpha - \beta )
\Bigl(
t\alpha - \beta - 1
2 - t\alpha - \beta - 1
1
\Bigr)
\times
\times
\left( N
m - 2\sum
i=1
\eta i
\xi i\int
0
(\xi i - s)\alpha +\delta - 2ds - N\Gamma (\alpha + \delta - 1) + \Gamma (\alpha + \delta - 1)A
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
N
\Gamma (\alpha - \beta + 1)
\bigm| \bigm| \bigm| t\alpha - \beta
2 - t\alpha - \beta
1
\bigm| \bigm| \bigm| + N\Gamma (\alpha )
| \Delta | \Gamma (\alpha - \beta )
\bigm| \bigm| \bigm| t\alpha - \beta - 1
2 - t\alpha - \beta - 1
1
\bigm| \bigm| \bigm| m - 2\sum
i=1
\eta i
\xi \alpha +\delta - 1
i
\alpha + \delta - 1
+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
EXISTENCE OF SOLUTIONS FOR A FRACTIONAL-ORDER BOUNDARY-VALUE PROBLEM 1659
+
\Gamma (\alpha + \delta - 1)\Gamma (\alpha )
| \Delta | \Gamma (\alpha - \beta )
\bigm| \bigm| \bigm| t\alpha - \beta - 1
2 - t\alpha - \beta - 1
1
\bigm| \bigm| \bigm| | A - N | .
Now using the fact that the functions t\alpha - \beta - 1
2 - t\alpha - \beta - 1
1 , t\alpha - 1
2 - t\alpha - 1
1 and t\alpha 2 - t\alpha 1 are uniformly
continuous on [0,1], we conclude that T\Omega is equicontinuous. Also T\Omega is a uniformly bounded set.
We have T\Omega \subset \Omega . By the Arzela – Ascoli theorem, T : \Omega \rightarrow \Omega is completely continuous.
Lemma 3.1 is proved.
Theorem 3.1 [9]. (Schauder fixed point theorem). Let C be a convex subset of a Banach space,
U be an open subset of C with 0 \in U. Then every completely continuous map N : U \rightarrow C has at
least one of the following two properties:
(A1) N has a fixed point in U
or
(A2) there is an x \in \partial U and \lambda \in (0, 1) with x = \lambda Nx.
Theorem 3.2 [10]. Let (X, d) be a complete metric space. T : X \rightarrow X is a contraction map if
there exist a constant 0 \leq k < 1 such that d(Tx, Ty) \leq kd(x, y). The set of fixed points of T is
given by F (T ) = \{ x \in X : Tx = x\} . Then each contraction map T : X \rightarrow X has an unique fixed
point.
Theorem 3.3. Assume that (H1) and (H2) hold. If
\rho
M
\geq 1 for \rho > 0,
where M is as previously defined, then the boundary-value problem (1.1) has a solution u = u(t)
such that
0 \leq u(t) \leq \rho , t \in [0, 1].
Proof. Let U = \{ u \in E| \| u\| < \rho \} . Our aim is to show that u \not = \lambda Tu with \lambda \in (0, 1) and
u \in \partial U. For this, let u = \lambda Tu for \lambda \in (0, 1). Then, for t \in [0, 1], we obtain
| \lambda (Tu)(t)| < | (Tu)(t)| \leq N
\left[ 1
\Gamma (\alpha + 1)
+
\sum m - 2
i=1
\eta i\xi
\alpha +\delta - 1
i + \Gamma (\alpha + \delta )
| \Delta | (\alpha + \delta - 1)
\right] +
+
\Gamma (\alpha + \delta - 1)| A|
| \Delta |
\leq M,
| \lambda D\beta (Tu)(t)| < | D\beta (Tu)(t)| \leq
\leq N
\left[ 1
\Gamma (\alpha - \beta + 1)
+
\Gamma (\alpha )
\Bigl[ \sum m - 2
i=1
\eta i\xi
\alpha +\delta - 1
i + \Gamma (\alpha + \delta )
\Bigr]
| \Delta | \Gamma (\alpha - \beta )(\alpha + \delta - 1)
\right] +
+
\Gamma (\alpha + \delta - 1)\Gamma (\alpha )| A|
| \Delta | \Gamma (\alpha - \beta )
= M.
Hence, we have \lambda (Tu)(t) < M and \lambda D\beta (Tu)(t) < M. Since u = \lambda Tu, we get \rho = \| u\| =
= \| Tu\| < M. From here
\rho
M
< 1. This is a contradiction with our hypothesis. Then the boundary-
value problem (1.1) has a solution u = u(t) such that
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1660 I. Y. KARACA, D. OZ
0 \leq u(t) \leq \rho , t \in [0, 1].
Theorem 3.3 is proved.
Theorem 3.4. Assume that (H1) and (H2) hold. If
k <
\Biggl[
2
\Gamma (\alpha - \beta + 1)
+
2\Gamma (\alpha )
\Delta \Gamma (\alpha - \beta )(\alpha + \delta - 1)
\Biggl(
m - 2\sum
i=1
\eta i
\Bigl(
\xi \alpha +\delta - 1
i + 1
\Bigr)
+ \Gamma (\alpha + \delta )
\Biggr) \Biggr] - 1
,
then the boundary-value problem (1.1) has an unique solution.
Proof. By assumption (H2), we have following estimates:
\bigm| \bigm| \bigm| \bigm| (Tu)(t) - (Tv)(t)
\bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
G(t, s)f
\Bigl(
s, u(s), D\beta u(s)
\Bigr)
ds+
\Gamma (\alpha + \delta - 1)A
\Delta
t\alpha - 1 -
-
1\int
0
G(t, s)f
\Bigl(
s, v(s), D\beta v(s)
\Bigr)
ds - \Gamma (\alpha + \delta - 1)A
\Delta
t\alpha - 1
\bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
G(t, s)
\Bigl(
f(s, u(s), D\beta u(s)) - f(s, v(s), D\beta v(s))
\Bigr)
ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
1\int
0
| G(t, s)|
\bigm| \bigm| \bigm| f \Bigl( s, u(s), D\beta u(s)
\Bigr)
- f
\Bigl(
s, v(s), D\beta v(s)
\Bigr) \bigm| \bigm| \bigm| \leq
\leq k
1\int
0
\Bigl(
| u(s) - v(s)| +
\bigm| \bigm| \bigm| D\beta u(s) - D\beta v(s)
\bigm| \bigm| \bigm| \Bigr) | G(t, s)| ds \leq
\leq k
1\int
0
\Bigl(
\mathrm{m}\mathrm{a}\mathrm{x} | u(s) - v(s)| +\mathrm{m}\mathrm{a}\mathrm{x}
\bigm| \bigm| \bigm| D\beta u(s) - D\beta v(s)
\bigm| \bigm| \bigm| \Bigr) | G(t, s)| ds \leq
\leq 2k\| u(s) - v(s)\|
1\int
0
| G(t, s)| ds \leq
\leq 2k\| u(s) - v(s)
\left[ 1\int
0
(t - s)\alpha - 1
\Gamma (\alpha )
ds+
1\int
0
t\alpha - 1
| \Delta |
m - 2\sum
j=1
\eta j(\xi j - s)\alpha +\delta - 2ds
\right] \leq
\leq 2k\| u(s) - v(s)\|
\left[ 1\int
0
(1 - s)\alpha - 1
\Gamma (\alpha )
ds+
1\int
0
\sum m - 2
j=1
\eta j(\xi j - s)\alpha +\delta - 2
| \Delta |
ds
\right] <
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
EXISTENCE OF SOLUTIONS FOR A FRACTIONAL-ORDER BOUNDARY-VALUE PROBLEM 1661
< 2k\| u(s) - v(s)\|
\left[ 1
\Gamma (\alpha - \beta + 1)
+
\Gamma (\alpha )
\sum m - 2
j=1
\eta j
\Bigl(
\xi \alpha +\delta - 1
j + 1
\Bigr)
+ \Gamma (\alpha + \delta )
| \Delta | (\alpha + \delta - 1)\Gamma (\alpha - \beta )
\right] =
= \eta \| u(s) - v(s)\| ,\bigm| \bigm| \bigm| D\beta (Tu)(t) - D\beta (Tv)(t)
\bigm| \bigm| \bigm| \leq k
\bigl(
| u(s) - v(s)| +
\bigm| \bigm| D\beta u(s) - D\beta v(s)
\bigm| \bigm| \bigr)
\Gamma (\alpha - \beta )
\times
\times
\left[ 1\int
0
(1 - s)\alpha - \beta - 1ds+
\Gamma (\alpha )
| \Delta |
m - 2\sum
i=1
\eta i
\xi i\int
0
(\xi i - s)\alpha +\delta - 2ds - \Gamma (\alpha + \delta - 1)\Gamma (\alpha )
| \Delta |
1\int
0
ds
\right] <
< 2k\| u(s) - v(s)\|
\left[ 1
\Gamma (\alpha - \beta + 1)
+
\Gamma (\alpha )
\sum m - 2
i=1
\eta i
\Bigl(
\xi \alpha +\delta - 1
i + 1
\Bigr)
+ \Gamma (\alpha + \delta )
| \Delta | (\alpha + \delta - 1)\Gamma (\alpha - \beta )
\right] =
= \eta \| u(s) - v(s)\| .
Hence, it follows that \| (Tu)(t) - (Tv)(t)\| < \eta \| u(t) - v(t)\| , where
\eta = 2k
\Biggl[
1
\Gamma (\alpha - \beta + 1)
+
\Gamma (\alpha )
| \Delta | \Gamma (\alpha - \beta )(\alpha + \delta - 1)
\Biggl\{
m - 2\sum
i=1
\eta i
\Bigl(
\xi \alpha +\delta - 1
i + 1
\Bigr)
+ \Gamma (\alpha + \delta )
\Biggr\} \Biggr]
< 1.
Therefore, by the contraction mapping principle the boundary-value problem (1.1) has an unique
solution.
Theorem 3.4 is proved.
4. Examples.
Example 4.1. Let m = 4, \eta 1 =
1
4
, \eta 2 =
1
8
, \alpha =
5
2
, \delta =
1
2
, \beta =
1
2
, \xi 1 =
1
2
, \xi 2 = 1, A =
1
32
,
f(t, u, v) =
u+ v
8 (1 + t2)
. We consider the fractional boundary-value problem
D
5
2u(t) = f
\Bigl(
t, u(t), D
1
2u(t)
\Bigr)
, t \in [0, 1],
u(0) = u\prime (0) = 0, (4.1)
D
1
2u(1) -
2\sum
i=1
\eta iI
1
2u\prime (\xi i) = A.
It is clearly that (H1) and (H2) are provided. Also,
\rho
M
\geq 1 for \rho \geq 3, 9183673469.
Then all conditions of Theorem 3.3 hold. Hence with Theorem 3.3, the boundary-value problem
(4.1) has a solution u = u(t) such that
0 \leq u(t) \leq \rho , t \in [0, 1].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1662 I. Y. KARACA, D. OZ
Example 4.2. Let m = 4, \eta 1 =
25
4
, \eta 2 =
9
4
, \alpha =
5
2
, \beta =
1
2
, \delta =
3
2
, \xi 1 =
1
5
, \xi 2 =
1
3
, A \in \BbbR ,
f(t, u, v) =
et
1013
(u+ v). We consider the fractional boundary-value problem
D
5
2u(t) = f
\Bigl(
t, u(t), D
1
2u(t)
\Bigr)
, t \in [0, 1],
u(0) = u\prime (0) = 0, (4.2)
D
1
2u(1) -
2\sum
i=1
\eta iI
3
2u\prime (\xi i) = A.
It is clearly that (H1) and (H2) are provided. For u, \=u, v, \=v, we have
| f(t, u, v) - f(t, \=u, \=v)| \leq k (| u - \=u| + | v - \=v| ) for k <
135
1013
.
Then all conditions of Theorem 3.4 hold. Hence with Theorem 3.4, the boundary-value problem
(4.2) has an unique solution.
References
1. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-
Holland Math. Stud., 204 (2006).
2. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equati-
ons, to methods of their solution and some of their applications, Acad. Press, 198 (1998).
3. I. Yaslan, M. Gunendi, Positive solutions of high-order nonlinear multipoint fractional equations with integral
boundary conditions, Fract. Calc. and Appl. Anal., 19, № 4, 989 – 1009 (2016).
4. J. Graef, L. Kong, Q. Kong, M. Wang, Uniqueness of positive solutions of fractional boundary value problems with
non-homogeneous integral boundary conditions, Fract. Calc. and Appl. Anal., 15, № 3, 509 – 528 (2012).
5. K. Zhang, J. Xu, Unique positive solution for a fractional boundary value problem, Fract. Calc. and Appl. Anal., 16,
№ 4, 937 – 948 (2013).
6. M. Dalir, M. Bashour, Applications of fractional calculus, Appl. Math. Sci., 4, № 21, 1021 – 1032 (2010).
7. M. ur Rehman, R. A. Khan, Existence and uniqueness of solutions for multi-point boundary-value problems for
fractional differential equations, Appl. Math. Lett., 23, № 9, 1038 – 1044 (2010).
8. N. Abel, Solutions de quelques problèmes à laide dintégrales définies, Euvres complètes de Niels Henrik Abel, 1,
11 – 18 (1823).
9. R. P. Agarwal, M. Meehan, D. O’ Regan, Fixed point theory and applications, Vol. 141, Cambridge Univ. Press
(2001).
10. S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math.,
3, № 1, 133 – 181 (1922).
11. X. Su, Boundary-value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett.,
22, № 1, 64 – 69 (2009).
12. Y. Guo, Solvability of boundary-value problems for nonlinear fractional differential equations, Ukr. Math. J., 62,
№ 9, 1409 – 1419 (2011).
Received 21.11.17,
after revision — 21.05.18
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
|
| id | umjimathkievua-article-6033 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:32Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c2/8ecc66af788d3918e8c91a1c041355c2.pdf |
| spelling | umjimathkievua-article-60332025-03-31T08:49:28Z Existence of solutions for a fractional-order boundary value problem Existence of solutions for a fractional-order boundary value problem Karaca , I. Y. Oz , D. Karaca , I. Y. Oz , D. D. fractional calculus boundary value problem fixed point theorems fractional calculus boundary value problem fixed point theorems UDC 517.9 We investigate the existence of solutions for a fractional-order boundary-value problem by using some fixed point theorems.As applications, examples are given to illustrate the main results. UDC 517.9 Існування розв’язкiв крайової задачi дробового порядку За допомогою теорем про нерухому точку вивчено проблему існування розв'язків крайової задачі дробового порядку.Як застосування наведено приклади, що ілюструють отримані результати. Institute of Mathematics, NAS of Ukraine 2020-12-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6033 10.37863/umzh.v72i12.6033 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 12 (2020); 1651-1662 Український математичний журнал; Том 72 № 12 (2020); 1651-1662 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6033/8873 |
| spellingShingle | Karaca , I. Y. Oz , D. Karaca , I. Y. Oz , D. D. Existence of solutions for a fractional-order boundary value problem |
| title | Existence of solutions for a fractional-order boundary value problem |
| title_alt | Existence of solutions for a fractional-order boundary value problem |
| title_full | Existence of solutions for a fractional-order boundary value problem |
| title_fullStr | Existence of solutions for a fractional-order boundary value problem |
| title_full_unstemmed | Existence of solutions for a fractional-order boundary value problem |
| title_short | Existence of solutions for a fractional-order boundary value problem |
| title_sort | existence of solutions for a fractional-order boundary value problem |
| topic_facet | fractional calculus boundary value problem fixed point theorems fractional calculus boundary value problem fixed point theorems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6033 |
| work_keys_str_mv | AT karacaiy existenceofsolutionsforafractionalorderboundaryvalueproblem AT ozd existenceofsolutionsforafractionalorderboundaryvalueproblem AT karacaiy existenceofsolutionsforafractionalorderboundaryvalueproblem AT ozd existenceofsolutionsforafractionalorderboundaryvalueproblem AT d existenceofsolutionsforafractionalorderboundaryvalueproblem |