Ulam-Hyers stability analysis to a three-point boundary value problem for fractional differential equations
UDC 517.9 We study the problem of existence and uniqueness of solution of a three-point boundary-value problem for a differential equation of fractional order. Further, we investigate various kinds of the Ulam stability, such as the Ulam–Hyers stability, the generalized Ulam–Hyers stability, the Ula...
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2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507014920142848 |
|---|---|
| author | Shah, K. Ali, A. Shah, K. Ali, A. |
| author_facet | Shah, K. Ali, A. Shah, K. Ali, A. |
| author_sort | Shah, K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-04-07T10:38:23Z |
| description | UDC 517.9
We study the problem of existence and uniqueness of solution of a three-point boundary-value problem for a differential equation of fractional order. Further, we investigate various kinds of the Ulam stability, such as the Ulam–Hyers stability, the generalized Ulam–Hyers stability, the Ulam–Hyers–Rassias stability, and the generalized Ulam–Hyers–Rassias stability for the analyzed problem.We also provide examples to explain our results. |
| first_indexed | 2026-03-24T02:02:35Z |
| format | Article |
| fulltext |
UDC 517.9
A. Ali, K. Shah (Univ. Malakand, Chakdara Dir (Lower), Khyber Pakhtunkhawa, Pakistan)
ULAM – HYERS STABILITY ANALYSIS
OF A THREE-POINT BOUNDARY-VALUE PROBLEM
FOR FRACTIONAL DIFFERENTIAL EQUATIONS
АНАЛIЗ СТАБIЛЬНОСТI ЗА УЛАМОМ ТА ХАЙЄРСОМ
ТРИТОЧКОВОЇ ГРАНИЧНОЇ ЗАДАЧI
ДЛЯ ДРОБОВИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
We study the problem of existence and uniqueness of solution of a three-point boundary-value problem for a differential
equation of fractional order. Further, we investigate various kinds of the Ulam stability, such as the Ulam – Hyers stability,
the generalized Ulam – Hyers stability, the Ulam – Hyers – Rassias stability, and the generalized Ulam – Hyers – Rassias
stability for the analyzed problem. We also provide examples to explain our results.
Вивчається проблема iснування та єдиностi розв’язку триточкової граничної задачi для дробового диференцiального
рiвняння. Крiм того, дослiджено рiзнi типи стабiльностi даної проблеми за Уламом, що включають стабiльнiсть
за Уламом та Хайєрсом, узагальнену стабiльнiсть за Уламом та Хайєрсом, стабiльнiсть за Уламом, Хайєрсом та
Рассiасом, а також узагальнену стабiльнiсть за Уламом, Хайєрсом та Рассiасом. Наведено приклади, що пояснюють
отриманi результати.
1. Introduction. Classical calculus has been generalized from integer order to arbitrary order. At
the end of sixteenth century (1695), in a letter to Leibnitz, L. Hospital asked about the derivative of z
with respect to t of order \alpha = 1/2. This was a question which moved minds towards generalization
of integer order derivatives to fractional order. Lacroix was the first person who introduced fractional
order derivative for first time [18]. Later on a great contribution in this field was made by researchers
like Abel, Fourier, Riemann, Liouville, Grunwald, Letnikov and others, for detail see [11, 15, 20].
Now a days fractional calculus is the most developing and interesting area of research. There has
been a lot of development in this field. This course has got great attention and importance for its
many applications in various fields of science, engineering and technology like physics, chemistry,
dynamics, control system, optimization theory, computer networking systems, mathematical biology,
bioengineering, aerodynamics, electrodynamics, signal and image processing, mathematical model-
ing, etc. (see, for instance, [8, 9, 16, 19]). One of the most well-known area of research in fractional
differential equations is concerning to the existence theory. For the last one hundred years this area
was very well explored by many authors, for detail see [2, 4, 23, 28, 31]. Benchohra et al. [5],
studied existence and uniqueness of solutions to the following antiperiodic boundary-value problem
(BVP) provided by
cD\alpha z(t) = \Theta (t, z(t),cD\alpha - 1z(t)), 0 \leq t \leq 1, 1 \leq \alpha < 2,
z(0) = - z(1), z\prime (0) = - z\prime (1).
c\bigcirc A. ALI, K. SHAH, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2 147
148 A. ALI, K. SHAH
In same line Shah et al. [27], studied the following BVP for multipoints:
- cD\alpha z(t) = \Theta (t, z(t),cD\alpha - 1z(t)), 0 < t < 1, 1 < \alpha \leq 2,
z(0) = 0, z(1) =
m - 2\sum
i=1
\delta iz(\vargamma i), where \delta i, \vargamma i \in (0, 1) with
m - 2\sum
i=1
\delta i(\vargamma i) < 1.
To receive the existence and uniqueness results, the researchers used the classical fixed point theory
of cone type. Besides from the aforesaid theory, they also applied pre estimate method known
as topological degree method, Schauder’s degree method and Brouwer’s degree method, etc., for
instance, we refer to [1, 3, 10, 26].
Another important area of research which has attracted more attention from researchers is devoted
to the stability analysis of differential equations of both classical and fractional order. Historically,
S. M. Ulam [29], did a fundamental question about the stability of functional equations which was
answered in 1941 by Hyers [12] in Banach spaces. Obloza was the first to report Hyers – Ulam
stability for linear differential equations. Later on this result was generalized and extended by Rassias,
Jung and others, for instance, we refer to [13, 14, 25]. Recently Benchohra and his co-author [7],
established Ulam – Hyers stability, generalized Ulam – Hyers stability, Ulam – Hyers – Rassias stability
and generalized Ulam – Hyers – Rassias stability for the following initial value problem of implicit
fractional order differential equation:
cD\alpha z(t) = \Theta (t, z(t),cD\alpha z(t)), 0 \leq t \leq 1, 0 < \alpha \leq 1,
z(0) = z0,
where cD\alpha is the Caputo fractional derivative and \Theta : J\times \Re \times \Re \rightarrow \Re is a given continuous function,
z0 \in \Re , J = [0, T ], T > 0 and \Re denotes the set of real numbers.
The aim of this paper is to investigate the existence and uniqueness results of solution and then to
establish the above four types of Ulam stabilities for the following boundary-value implicit fractional
order differential equation:
cD\alpha z(t) = \Theta (t, z(t), D\alpha z(t)), 0 \leq t \leq 1, 1 \leq \alpha < 2,
z(0) = 0, z(1) = \delta z(\vargamma ), \delta , \vargamma \in (0, 1),
(1)
where cD\alpha is the Caputo fractional derivative and \Theta : J \times \Re \times \Re \rightarrow \Re is a given continuous
function. Here we remark that over all in the subject of fractional calculus huge research is in
progress in recent times which addresses existence theory, numerical analysis and stability theory, we
present some recent work as [33 – 44].
2. Preliminaries. Now to receive the aforementioned goals, we remind some basic definitions
and lemmas which will be used in our results.
Definition 1 [22]. The arbitrary order integral of a function h \in L1([0, T ],\Re +) of order \alpha \in
\in (0,\infty ) is defined by
I\alpha h(t) =
1
\Gamma (\alpha )
t\int
0
(t - s)\alpha - 1h(s)ds,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
ULAM – HYERS STABILITY ANALYSIS OF A THREE-POINT BOUNDARY-VALUE PROBLEM . . . 149
provided that integral on the right is pointwise defined on (0,\infty ), where \Gamma is the Euler Gamma
function defined as \Gamma (\alpha ) =
\int \infty
0
t\alpha - 1e - tdt, \alpha > 0.
Definition 2 [15]. The Caputo fractional arbitrary order derivative of order \alpha of function h is
defined by
cD\alpha h(t) =
1
\Gamma (n - \alpha )
t\int
0
(t - s)n - \alpha - 1hnh(s)ds,
provided that integral on the right is pointwise defined on (0,\infty ), where n=[\alpha ]+1 and [\alpha ] denotes
the integer part of the real number \alpha .
Lemma 1 [17]. For a fractional derivative and integral of order \alpha ,we have the following result:
I\alpha cD\alpha h(t) = h(t) + b0 + b1t+ b2t
2 + . . .+ bn - 1t
n - 1,
where bi \in \Re , i = 0, 1, 2, 3, . . . , n - 1.
Lemma 2 [5]. The space \~C defined by
\~C(J,\Re ) = \{ z \in C(J,\Re ) : cD\alpha z \in C2(J,\Re )\}
with the norm
| | z| | \infty = \mathrm{S}\mathrm{u}\mathrm{p} \{ | z(t)| : t \in [0, 1]\}
is a Banach space under the defined norm.
Definition 3 [24]. The equation (1) is said to be Ulam – Hyers stable if there exists a positive
real number \aleph such that for every \varepsilon > 0 and for each solution w \in C1(J,\Re ) of the inequality
| cD\alpha w(t) - \Theta (t, w(t),cD\alpha w(t))| \leq \varepsilon , t \in J, (2)
there exists a solution z \in C1(J,\Re ) of the equation (1) such that | w(t) - z(t)| \leq \aleph \varepsilon , t \in J.
Definition 4 [24]. The equation (1) is said to be generalized Ulam – Hyers stable if there exists
\mu \in C(\Re +,\Re +), \mu (0) = 0, such that for each solution z \in C1(J,\Re +) of the inequality (2), there
exists a solution w \in C1(J,\Re +) of the equation (1) such that
| w(t) - z(t)| \leq \mu \varepsilon , t \in J.
Definition 5 [24]. The equation (1) is said to be Ulam – Hyers – Rassias stable with respect to
\Psi \in C(J,\Re +) if there exists a nonzero positive real number \aleph such that for each \varepsilon > 0 and for
each solution w \in C1(J,\Re ) of the inequality
| cD\alpha w(t) - \Theta (t, w(t),cD\alpha w(t))| \leq \varepsilon \Psi (t), t \in J, (3)
there exists a solution z \in C1(J,\Re ) of the equation (1) such that
| w(t) - z(t)| \leq \aleph \varepsilon \Psi (t), t \in J.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
150 A. ALI, K. SHAH
Definition 6 [24]. The equation (1) is said to be generalized Ulam – Hyers – Rassias stable with
respect to \Psi \in C(J,\Re ), if there exists a real number \aleph \Psi > 0 such that for each solution w \in
\in C1(J,\Re ) of the inequality
| cD\alpha w(t) - \Theta (t, w(t),cD\alpha w(t))| \leq \Psi (t), t \in J, (4)
there exists a solution z \in C1(J,\Re ) of the equation (1) such that | w(t) - z(t)| \leq \aleph \Psi \Psi (t), t \in J.
Remark 1. A function w \in C1(J,\Re ) is a solution of the inequality (2) if there exists a function
h \in C(J,\Re ) (dependent on z) such that
(I) | h(t)| \leq \varepsilon for all t \in J ;
(II) cD\alpha w(t) = \Theta (t, w(t),cD\alpha w(t)) + h(t), t \in J.
Definition 7. A function x \in C1(J) is said to be a solution of the problem (1) if x satisfies (1)
and the boundary conditions on J.
3. Existence and stability analysis. The concerned section is devoted to establish conditions
for the existence of at least one solution to BVP (1) and also to discuss the four different kinds of
stability for the afore said problem.
Theorem 1. Let h \in C(J,\Re ), then the equivalent Fredholm integral equation of the given
BVP (1) is z(t) =
\int 1
0
\scrH (t, s)h(s)ds, where \scrH (t, s) is the Green’s function given by
\scrH (t, s) =
1
\Gamma (\alpha )
\left\{
\delta t
\Delta
(\vargamma - s)\alpha - 1 - t
\Delta
(1 - s)\alpha - 1, 0 \leq t \leq s \leq \vargamma \leq 1,
\delta t
\Delta
(\vargamma - s)\alpha - 1 - t
\Delta
(1 - s)\alpha - 1 + (t - s)\alpha - 1, 0 \leq s \leq t \leq \vargamma \leq 1,
- t
\Delta
(1 - s)\alpha - 1, 0 \leq \vargamma \leq t \leq s \leq 1,
- t
\Delta
(1 - s)\alpha - 1 + (t - s)\alpha - 1, 0 \leq \vargamma \leq s \leq t \leq 1,
where \Delta = 1 - \delta \vargamma .
Proof. Let us consider a linear BVP given by
cD\alpha z(t) = h(t), 1 \leq \alpha < 2, t \in [0, 1]. (5)
Applying Lemma 1, we have
z(t) = b0 + b1t+ I\alpha h(t). (6)
By using initial and boundary conditions z(0) = 0 and z(1) = \delta z(\eta ), we get b0 = 0 and
b1 =
1
\Delta
[\delta I\alpha h(\vargamma ) - I\alpha h(1)].
Inserting these values of b0 and b1 in equation (6), we have
z(t) =
t
\Delta
[\delta I\alpha h(\vargamma ) - I\alpha h(1)] + I\alpha h(t) =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
ULAM – HYERS STABILITY ANALYSIS OF A THREE-POINT BOUNDARY-VALUE PROBLEM . . . 151
=
\delta t
\Gamma \alpha
\vargamma \int
0
(\vargamma - s)\alpha - 1h(s)ds - t
\Delta \Gamma \alpha
1\int
0
(1 - s)\alpha - 1h(s)ds+
+
1
\Gamma \alpha
t\int
0
(t - s)\alpha - 1h(s)ds
which implies that
z(t) =
1\int
0
\scrH (t, s)h(s)ds,
where \scrH (t, s) is the Green’s function.
Therefore, in view of above theorem, our considered problem becomes
z(t) =
1\int
0
\scrH (t, s)\Theta (s, z(s),cD\alpha z(s))ds, t \in [0, 1]. (7)
Theorem 1 is proved.
The given assumptions are useful in the proof of the following theorems. Assume that there exist
\varpi (t) \in C(J,\Re +) and a continuous nondecreasing function \varphi : [0,\infty ) \rightarrow (0,\infty ) such that
(A1) \Theta : J \times \Re \times \Re \rightarrow \Re is continuous;
(A2) | \Theta (t, z, w)| \leq \varpi (t)\varphi (w) for z, w \in \Re ;
(A3) \varphi (\varpi )\varpi \ast \scrH \ast \leq \varpi , where \varpi \ast = \mathrm{s}\mathrm{u}\mathrm{p}\{ \varpi (s) : s \in J\} and \scrH \ast = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
\int 1
0
| \scrH (t, s)| ds;
(A4) there exists a constant \lambda > 0 such that for each t \in J and for all z, w, \=z, \=w \in \Re , we have
| \Theta (t, z, w) - \Theta (t, \=z, \=w)| \leq \lambda (| z - \=z| + | w - \=w| ).
Theorem 2. Under the assumptions (A1) - (A3), there exists at least one solution of the con-
cerned BVP (1).
Proof. To prove the required result, we use Schauder fixed point theorem. Let zn be a sequence
such that zn \rightarrow z, where z \in (J,\Re ). Let \sigma > 0 such that \| zn\| \leq \sigma for each t \in J . Then considered
a bounded set
D =
\bigl\{
z \in C(J,\Re ) : \| z\| \leq \varpi
\bigr\}
\subset C(J \times \Re ,\Re ),
and defined an operator
\digamma : D \rightarrow D by \digamma z(t) = z(t), t \in J.
We have to show that the operator has at least one fixed point. To prove this, consider
| \digamma zn(t) - \digamma z(t)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
\scrH (t, s)[\Theta (s, zn(s),
cD\alpha zn(s)) - \Theta (s, z(s),cD\alpha z(s))]ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
| \scrH (t, s)| | \Theta (s, zn(s),
cD\alpha zn(s)) - \Theta (s, z(s),cD\alpha z(s))| ds.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
152 A. ALI, K. SHAH
By using Lebesgue dominated convergent theorem, we have | \digamma zn(t) - \digamma z(t)| n\rightarrow \infty \rightarrow 0, which
implies that \digamma is continuous.
Next, we show that \digamma is bounded. For this we will show that \digamma (D) \subseteq D. Let z \in D \in and
consider
| \digamma z(t)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
\scrH (t, s)\Theta (s, z(s),cD\alpha z(s)ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
| \scrH (t, s)| | \Theta (s, z(s),cD\alpha z(s)| ds \leq
\leq \varpi (t)\varphi | w| \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
1\int
0
| \scrH (t, s)| ds \leq
\leq \varpi (t)\varphi | w| \scrH \ast \leq
\leq \varpi \ast \varphi (\varpi )\scrH \ast \leq \varpi .
Thus, | \digamma z(t)| \leq \varpi .
This shows that \digamma is bounded and hence \digamma (D) \subseteq D. For showing that \digamma is equicontinuous, let
t1, t2 \in J with t1 < t2, consider
| \digamma z(t2) - \digamma z(t1)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
\scrH (t2, s) - \scrH (t1, s)\Theta (s, z(s),cD\alpha z(s)ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
| \scrH (t2, s) - \scrH (t1, s)| | \Theta (s, z(s),cD\alpha \alpha z(s)| ds \leq
\leq \varpi (t)\varphi (w)
1\int
0
| \scrH (t2, s) - \scrH (t1, s)| ds.
Now, if t2 \rightarrow t1, then \varpi (t)\varphi (w)
\int 1
0
| \scrH (t2, s) - \scrH (t1, s)| ds \rightarrow 0, consequently, | \digamma z(t2) - \digamma z(t1)| \rightarrow
\rightarrow 0, which implies that \digamma is equicontinuous. So by Arzelá – Ascoli theorem, \digamma has at least one
fixed point and hence the corresponding BVP(1) has at least one solution.
Theorem 2 is proved.
Theorem 3. Under the assumptions (A1) and (A4) with the additional condition 2\scrH \ast \lambda < 1,
the BVP (1) has a unique solution.
Proof. To prove the required result, we use Banach contraction principle. Define a mapping
\digamma : (J \times \Re ,\Re ) \rightarrow C(J \times \Re ,\Re ) by
\digamma z(t) = z(t) =
1\int
0
\scrH (t, s)\Theta (s, z(s))ds.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
ULAM – HYERS STABILITY ANALYSIS OF A THREE-POINT BOUNDARY-VALUE PROBLEM . . . 153
Obviously, \digamma z(t) is continuous, because \scrH (t, s) and \Theta are continuous. Let z, \=z \in C(J,\Re ) and
t \in J, consider
| \digamma z(t) - \digamma \=z(t)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
\scrH (t, s)[\Theta (s, z(s),cD\alpha z(s)) - \Theta (s, \=z(s),cD\alpha \=z(s))]ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
| \scrH (t, s)| | \Theta (s, z(s),cD\alpha z(s)) - \Theta (s, \=z(s),cD\alpha \=z(s))| ds \leq
\leq \scrH \ast \lambda (\| z - \=z\| \infty + \| cD\alpha z - c D\alpha \=z\| \infty ) \Rightarrow
\Rightarrow | \digamma z(t) - \digamma \=z(t)| \leq 2\scrH \ast \lambda \| z - \=z\| \infty .
Here,
\scrH \ast = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
1\int
0
| \scrH (t, s)| ds.
Since 2\scrH \ast \lambda < 1, so by Banach contraction theorem \digamma is contraction and so has a unique fixed poind
and hence the corresponding BVP (1) has a unique solution.
Theorem 3 is proved.
Theorem 4. If the assumptions (A1), (A4) along with the conditions \scrH \ast \lambda \not = 1 - \lambda and \lambda \not = 1
hold, then the BVP(1) is Ulam – Hyers stable.
Proof. Let (A1), (A4) and the conditions \scrH \ast \lambda \not = 1 - \lambda and \lambda \not = 1 hold. Let w \in C(J,\Re ) be
a solution of the inequality (1) and z \in (J,\Re ) be a unique solution of the Cauchy problem
cD\alpha z(t) = \Theta (t, z(t),cD\alpha z(t)) for all t \in J, 1 \leq \alpha < 2.
By Theorem 1, we have
z(t) =
1\int
0
\scrH (t, s)h(s)ds,
where h \in C(J,\Re ) satisfies the functional equation
y(t) = \Theta
\left( t,
1\int
0
\scrH (t, s)h(s)ds, h(t)
\right) .
Hence, we take \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| w(t) -
1\int
0
\scrH (t, s)hw(s)ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \varepsilon . (8)
On the other hand, we get, for t \in J,
| w(t) - z(t)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| w(t) -
1\int
0
\scrH (t, s)hz(s)ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
154 A. ALI, K. SHAH
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| w(t) -
1\int
0
\scrH (t, s)hw(s)ds+
1\int
0
\scrH (t, s)hw(s)ds -
1\int
0
\scrH (t, s)hz(s)ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| w(t) -
1\int
0
\scrH (t, s)hw(s)ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
\scrH (t, s)[hw(s) - hz(s)]ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \varepsilon +
1\int
0
| \scrH (t, s)| | hw(s) - hz(s)| ds (via using the inequality (8)), (9)
where hw(t) = \Theta (t, w(t), hw(t)) and hz(t) = \Theta (t, z(t), hz(t)). We have, for all t \in J,
| hw(t) - hz(t)| = | \Theta (t, w(t), hw(t)) - \Theta (t, z(t), hz(t))| \leq
\leq \lambda | w(t) - z(t)| + \lambda | hw(t) - yz(t)| (by using (A4)) \leq
\leq \lambda
1 - \lambda
| w(t) - z(t)| .
Hence from above inequality (9), we obtain
| w(t) - z(t)| \leq \varepsilon +
\scrH \ast \lambda
1 - \lambda
| w(t) - z(t)| \Rightarrow
\Rightarrow | w(t) - z(t)| \leq \varepsilon
1 - \scrH \ast \lambda
1 - \lambda
, \scrH \ast \lambda \not = 1 - \lambda and \lambda \not = 1 \Rightarrow
\Rightarrow | w(t) - z(t)| \leq C\varepsilon ,
where C =
1
1 - \scrH \ast \lambda
1 - \lambda
with \scrH \ast \lambda \not = 1 - \lambda and \lambda \not = 1.
So, equation (1) is Ulam – Hyers stable. By putting \Psi (\varepsilon ) = C\varepsilon , \Psi (0) = 0, in this case the
equation (1) is generalized Ulam – Hyers stable.
Theorem 4 is proved.
Theorem 5. Assume that (A1), (A4) hold, then the equation (1) is Ulam – Hyers – Rassias stable
if \scrH \ast \lambda \not = 1 - \lambda and \lambda \not = 1.
Proof. Let w \in J be any solution of the inequality
| cD\alpha w(t) - \Theta (t, w(t),cD\alpha w(t))| \leq \varepsilon \Phi (t), t \in J, (10)
and z \in J be the unique solution of the considered Cauchy problem (1). Then, for \varepsilon > 0, we get
| w(t) - z(t)| \leq \varepsilon \Phi (t). (11)
In view of Theorem 1, we get
z(t) =
1\int
0
\scrH (t, s)h(s)ds,
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ULAM – HYERS STABILITY ANALYSIS OF A THREE-POINT BOUNDARY-VALUE PROBLEM . . . 155
where y \in C(J,\BbbR ) satisfies the functional equation
h(t) = \Theta
\left( t,
1\int
0
\scrH (t, s)ds, h(t)
\right) .
Hence, we obtain, from inequality (11),\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| w(t) -
1\int
0
\scrH (t, s)hw(s)ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \varepsilon \Phi (t). (12)
Also, we have, for t \in J,
| w(t) - z(t)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| w(t) -
1\int
0
\scrH (t, s)hz(s)ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \varepsilon \Phi (t) =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| w(t) -
1\int
0
\scrH (t, s)hw(s)ds+
1\int
0
\scrH (t, s)hw(s)ds -
1\int
0
\scrH (t, s)hz(s)ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| w(t) -
1\int
0
\scrH (t, s)hw(s)ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
\scrH (t, s)[hw(s) - hz(s)]ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| .
By using inequality (12), we get
| w(t) - z(t)| \leq \varepsilon \Phi (t) +
1\int
0
| \scrH (t, s)| | hw(s) - hz(s)| ds, (13)
where hw(t) = \Theta (t, w(t), hw(t)) and hz(t) = \Theta (t, z(t), hz(t)). So, we have, for all t \in J,
| hw(t) - hz(t)| = | \Theta (t, w(t), hw(t)) - \Theta (t, z(t), hz(t))| \leq
\leq \lambda | w(t) - z(t)| + \lambda | hw(t) - hz(t)| (by using (A4)) \Rightarrow | hw(t) - hz(t)| \leq
\leq \lambda
1 - \lambda
| w(t) - z(t)| .
So, inequality (13) becomes
| w(t) - z(t)| \leq \varepsilon \Phi (t) +
1\int
0
| \scrH (t, s)| \lambda
1 - \lambda
| w(t) - z(t)| \leq
\leq \varepsilon \Phi (t) +\scrH \ast \lambda
1 - \lambda
| w(t) - z(t)| \Rightarrow
\Rightarrow | w(t) - z(t)| \leq \varepsilon \Phi (t)
1 - \scrH \ast \lambda
1 - \lambda
\biggl(
where
\scrH \ast \lambda
1 - \lambda
\not = 1
\biggr)
\Rightarrow
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
156 A. ALI, K. SHAH
\Rightarrow | w(t) - z(t)| \leq \aleph \varepsilon \Phi (t)
\left( where \aleph =
1
1 - \scrH \ast \lambda
1 - \lambda
\right)
with \scrH \ast \lambda \not = 1 - \lambda and \lambda \not = 1. So, equation (1) is Ulam – Hyers – Rassias stable. By taking \Psi (\varepsilon ) =
= \aleph \varepsilon \Phi (t), we have \Psi (0) = 0. This shows that equation (1) is generalized Ulam – Hyers – Rassias
stable.
Theorem 5 is proved.
4. Examples. To demonstrate the established results in previous section, we provide the follo-
wing examples.
Example 1. We consider
cD
3
2 z(t) =
1
80
\biggl(
t \mathrm{s}\mathrm{i}\mathrm{n} z(t) - z(t) \mathrm{c}\mathrm{o}\mathrm{s} t
\biggr)
+
| cD
3
2 z(t)|
40 + | cD
3
2 z(t)|
, t \in [0, 1],
z(0) = 0, z(1) =
1
4
z
\biggl(
1
3
\biggr)
.
(14)
From the BVP (1), we see that \alpha =
3
2
, \delta =
1
4
, \vargamma =
1
3
and the nonlinear function
\Theta (t, z, w) =
1
80
\biggl(
t \mathrm{s}\mathrm{i}\mathrm{n} z(t) - z(t) \mathrm{c}\mathrm{o}\mathrm{s} t
\biggr)
+
| cD
3
2 z(t)|
40 + | cD
3
2 z(t)|
is clearly continuous and the Green’s function is
\scrH (t, s) =
1
\Gamma
\biggl(
3
2
\biggr)
\left\{
3t
11
\biggl(
1
3
- s
\biggr) 1
2
- 12t
11
(1 - s)
1
2 , 0 \leq t \leq s \leq 1
3
< 1,
3t
11
\biggl(
1
3
- s
\biggr) 1
2
- 12t
11
(1 - s)
1
2 + (t - s)
1
2 , 0 \leq s \leq t \leq 1
3
< 1,
- 12t
11
(1 - s)
1
2 , 0 <
1
3
\leq t \leq s \leq 1,
- 12t
11
(1 - s)
1
2 + (t - s)
1
2 , 0 <
1
3
\leq s \leq t \leq 1.
Now for any z, \=z, w, \=w \in \Re and t \in [0, 1], we get
| \Theta (t, z, w) - \Theta (t, \=z, \=w)| \leq
\leq 1
80
| t| | \mathrm{s}\mathrm{i}\mathrm{n} z - \mathrm{s}\mathrm{i}\mathrm{n} \=z| + 1
80
| \mathrm{c}\mathrm{o}\mathrm{s} t| | z - \=z| +
\bigm| \bigm| \bigm| \bigm| | w|
40 + | w|
- | \=w|
40 + | \=w|
\bigm| \bigm| \bigm| \bigm| \leq
\leq 1
40
| z - \=z| + 1
40
| w - \=w| .
Therefore, we have \lambda =
1
40
and computing
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
ULAM – HYERS STABILITY ANALYSIS OF A THREE-POINT BOUNDARY-VALUE PROBLEM . . . 157
\scrH \ast = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
1\int
0
| \scrH (t, s)| ds \leq 2
11\Gamma
\biggl(
3
2
\biggr) 1\int
0
(1 - s)
1
2ds =
8
33
\surd
\pi
.
Now using Theorem 3, we see that 2\scrH \ast \lambda =
2
165
\surd
\pi
< 1. Hence, the BVP (1) has a unique solution.
Further, as \scrH \ast \lambda \not = 1 - \lambda and \lambda \not = 1 are also satisfied, hence, by Theorem 4, the given BVP (1) is
Ulam – Hyers stable and, hence, generalized Ulam – Hyers stable. Also it can be easily derived that
the given BVP is Ulam – Hyers – Rassias stable and, hence, generalized Ulam – Hyers – Rassias stable
by applying Theorem 5, because it is obvious that \aleph =
1
1 - \scrH \ast \lambda
1 - \lambda
\not = 0.
Example 2. We consider
cD
3
2 z(t) =
2 + | z(t)| + | cD
3
2 z(t)|
120 e2t
\bigl(
1 + | z(t)| + | cD
3
2 z(t)|
\bigr) , t \in [0, 1],
z(0) = 0, (15)
z(1) =
1
3
z
\biggl(
1
2
\biggr)
.
From the BVP (2), we see that \alpha =
3
2
, \delta =
1
3
, \vargamma =
1
2
and the nonlinear function
\Theta (t, z, w) =
2 + | z| + | w|
120 e2t(1 + | z| + | w| )
is clearly continuous and the Green’s function is
\scrH (t, s) =
1
\Gamma
\biggl(
3
2
\biggr)
\left\{
2t
5
\biggl(
1
2
- s
\biggr) 1
2
- 6t
5
(1 - s)
1
2 , 0 \leq t \leq s \leq 1
2
< 1,
2t
5
\biggl(
1
3
- s
\biggr) 1
2
- 6t
5
(1 - s)
1
2 + (t - s)
1
2 , 0 \leq s \leq t \leq 1
2
< 1,
- 6t
5
(1 - s)
1
2 , 0 \leq 1
2
\leq t \leq s \leq 1,
- 6t
5
(1 - s)
1
2 + (t - s)
1
2 , 0 <
1
2
\leq s \leq t \leq 1.
Now, for any z, \=z, w, \=w \in \Re and t \in [0, 1], we get
| \Theta (t, z, w) - \Theta (t, \=z, \=w)| = 1
120e2t
\bigm| \bigm| \bigm| \bigm| 2 + | z| + | w|
1 + | z| + | w|
- 2 + | \=z + | \=w|
1 + | \=z| + | \=w|
\bigm| \bigm| \bigm| \bigm| \leq
\leq 1
120
\bigm| \bigm| \bigm| \bigm| 2 + | z| + | w|
1 + | z| + | w|
- 2 + | \=z| + | \=w|
1 + | \=z| + | \=w|
\bigm| \bigm| \bigm| \bigm| =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
158 A. ALI, K. SHAH
=
1
120
\bigm| \bigm| \bigm| \bigm| | z| - | \=z| + | w| - | \=w|
(1 + | z| + | w| )(1 + | \=z| + | \=w| )
\bigm| \bigm| \bigm| \bigm| \leq
\leq 1
120
\bigm| \bigm| \bigm| \bigm| | z - \=z| + | w - \=w|
(1 + | z| + | w| )(1 + | \=z| + | \=w| )
\bigm| \bigm| \bigm| \bigm| \leq
\leq 1
120
\bigm| \bigm| \bigm| \bigm| | z - \=z| + | w - \=w|
\bigm| \bigm| \bigm| \bigm| \leq
\leq 1
120
| z - \=z| + 1
120
| w - \=w| .
Therefore, we have \lambda =
1
120
and computing
\scrH \ast = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
1\int
0
| \scrH (t, s)| ds \leq
1\int
0
\scrH (1, s)ds \leq 1
5\Gamma (3/2)
1\int
0
(1 - s)
1
2ds =
4
15
\surd
\pi
.
Now using Theorem 3, we see that 2\scrH \ast \lambda \leq 1
225
\surd
\pi
< 1, hence, the BVP (2) has a unique solution.
Further, as \scrH \ast \lambda \not = 1 - \lambda and \lambda \not = 1 are also satisfied, hence, by Theorem 4, the given BVP (2) is
Ulam – Hyers stable and, hence, generalized Ulam – Hyers stable. Also it can be easily derived that
the given BVP is Ulam – Hyers – Rassias stable and, hence, generalized Ulam – Hyers – Rassias stable
by applying Theorem 5, because it is obvious that \aleph =
1
1 - \scrH \ast \lambda
1 - \lambda
\not = 0.
5. Conclusion. By use of Arzelá – Ascoli theorem, Lebesgue’s dominated convergent theorem
and Banach contraction principle, we have deduced the sufficient conditions for existence and unique-
ness of solution for our considered problem (1). Also under certain assumptions and conditions, we
have deduced the Ulam – Hyers stability results for the solution of the said problem by adopting the
definitions from [24].
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|
| id | umjimathkievua-article-371 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:35Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6c/cc9c29ddfd5172e301a6ac162367486c.pdf |
| spelling | umjimathkievua-article-3712020-04-07T10:38:23Z Ulam-Hyers stability analysis to a three-point boundary value problem for fractional differential equations Аналіз стабільності за Уламом та Хайєрсом триточкової граничної задачі для дробових диференціальних рівнянь Shah, K. Ali, A. Shah, K. Ali, A. дробові диференціальні рівняня гранична задача стабільності за Уламом та Хайєрсом three-points boundary value problem Ulam-Hyers stability differential equations UDC 517.9 We study the problem of existence and uniqueness of solution of a three-point boundary-value problem for a differential equation of fractional order. Further, we investigate various kinds of the Ulam stability, such as the Ulam–Hyers stability, the generalized Ulam–Hyers stability, the Ulam–Hyers–Rassias stability, and the generalized Ulam–Hyers–Rassias stability for the analyzed problem.We also provide examples to explain our results. УДК 517.9 Вивчається проблема існування та єдиності розв'язку триточкової граничної задачі для дробового диференціального рівняння.&nbsp;Крім того, досліджено різні типи стабільності даної&nbsp;проблеми за Уламом, що включають стабільність за Уламом та Хайєрсом, узагальнену стабільність за Уламом та Хайєрсом, стабільність за Уламом, Хайєрсом та Рассіасом, а також узагальнену стабільність за Уламом, Хайєрсом та Рассіасом. Наведено приклади, що пояснюють отримані результати. Institute of Mathematics, NAS of Ukraine 2020-02-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/371 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 2 (2020); 147-160 Український математичний журнал; Том 72 № 2 (2020); 147-160 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/371/8686 |
| spellingShingle | Shah, K. Ali, A. Shah, K. Ali, A. Ulam-Hyers stability analysis to a three-point boundary value problem for fractional differential equations |
| title | Ulam-Hyers stability analysis to a three-point boundary value problem for fractional differential equations |
| title_alt | Аналіз стабільності за Уламом та Хайєрсом триточкової граничної задачі для дробових диференціальних рівнянь |
| title_full | Ulam-Hyers stability analysis to a three-point boundary value problem for fractional differential equations |
| title_fullStr | Ulam-Hyers stability analysis to a three-point boundary value problem for fractional differential equations |
| title_full_unstemmed | Ulam-Hyers stability analysis to a three-point boundary value problem for fractional differential equations |
| title_short | Ulam-Hyers stability analysis to a three-point boundary value problem for fractional differential equations |
| title_sort | ulam-hyers stability analysis to a three-point boundary value problem for fractional differential equations |
| topic_facet | дробові диференціальні рівняня гранична задача стабільності за Уламом та Хайєрсом three-points boundary value problem Ulam-Hyers stability differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/371 |
| work_keys_str_mv | AT shahk ulamhyersstabilityanalysistoathreepointboundaryvalueproblemforfractionaldifferentialequations AT alia ulamhyersstabilityanalysistoathreepointboundaryvalueproblemforfractionaldifferentialequations AT shahk ulamhyersstabilityanalysistoathreepointboundaryvalueproblemforfractionaldifferentialequations AT alia ulamhyersstabilityanalysistoathreepointboundaryvalueproblemforfractionaldifferentialequations AT shahk analízstabílʹnostízaulamomtahajêrsomtritočkovoígraničnoízadačídlâdrobovihdiferencíalʹnihrívnânʹ AT alia analízstabílʹnostízaulamomtahajêrsomtritočkovoígraničnoízadačídlâdrobovihdiferencíalʹnihrívnânʹ AT shahk analízstabílʹnostízaulamomtahajêrsomtritočkovoígraničnoízadačídlâdrobovihdiferencíalʹnihrívnânʹ AT alia analízstabílʹnostízaulamomtahajêrsomtritočkovoígraničnoízadačídlâdrobovihdiferencíalʹnihrívnânʹ |