A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions
UDC 517.9 We develop the existence theory for a more general class of nonlocal integro-multipoint boundary value problems of Caputo  type  fractional integro-differential inclusions. Our results include the convex and  non-convex cases for the given problem and...
Збережено в:
| Дата: | 2021 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2021
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/388 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507015938310144 |
|---|---|
| author | Ahmad, B. Ntouyas, S. K. Alsaedi, A. Ahmad, B. Ntouyas, S. K. Alsaedi, A. Ahmad, B. Ntouyas, S. K. Alsaedi, A. |
| author_facet | Ahmad, B. Ntouyas, S. K. Alsaedi, A. Ahmad, B. Ntouyas, S. K. Alsaedi, A. Ahmad, B. Ntouyas, S. K. Alsaedi, A. |
| author_sort | Ahmad, B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:03:02Z |
| description | UDC 517.9
We develop the existence theory for a more general class of nonlocal integro-multipoint boundary value problems of Caputo  type  fractional integro-differential inclusions. Our results include the convex and  non-convex cases for the given problem and rely on standard fixed point theorems for multivalued maps.  The obtained results are illustrated with the aid of examples. |
| doi_str_mv | 10.37863/umzh.v73i6.388 |
| first_indexed | 2026-03-24T02:02:36Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i6.388
UDC 517.9
B. Ahmad (King Abdulaziz Univ., Saudi Arabia),
S. K. Ntouyas (Univ. Ioannina, Greece and King Abdulaziz Univ., Saudi Arabia),
A. Alsaedi (King Abdulaziz Univ., Saudi Arabia)
A STUDY OF A MORE GENERAL CLASS OF NONLOCAL
INTEGRO-MULTIPOINT BOUNDARY-VALUE PROBLEMS
OF FRACTIONAL INTEGRO-DIFFERENTIAL INCLUSIONS*
ВИВЧЕННЯ ВIДНОСНО ЗАГАЛЬНОГО КЛАСУ НЕЛОКАЛЬНИХ
IНТЕГРАЛЬНИХ БАГАТОТОЧКОВИХ КРАЙОВИХ ЗАДАЧ
ДЛЯ ДРОБОВИХ IНТЕГРО-ДИФЕРЕНЦIАЛЬНИХ ВКЛЮЧЕНЬ
We develop the existence theory for a more general class of nonlocal integro-multipoint boundary value problems of
Caputo type fractional integro-differential inclusions. Our results include the convex and non-convex cases for the given
problem and rely on standard fixed point theorems for multivalued maps. The obtained results are illustrated with the aid
of examples.
Запропоновано теорiю iснування для вiдносно загального класу нелокальних iнтегральних багатоточкових крайових
задач для дробових iнтегро-диференцiальних включень типу Капуто. Нашi результати охоплюють опуклi та неопуклi
випадки даної проблеми i базуються на стандартних теоремах про нерухому точку для багатозначних вiдображень.
Отриманi результати проiлюстровано вiдповiдними прикладами.
1. Introduction. The tools of fractional calculus revolutionized the mathematical modeling of
various phenomena occurring in sciences and engineering by producing fractional-order models for
them, which are found to be more informative and realistic than their integer-order counterparts.
The interest in this branch of mathematical analysis owes to the nonlocal nature of fractional order
operators which are capable to trace the history of processes and materials involved in the phe-
nomenon at hand. Examples include continuum and statistical mechanics [1], protein dynamics [2],
chaos and fractional dynamics [3], bio-engineering [4], chaos synchronization [5], viscoelasticity [6],
ecology [7], infectious diseases [8, 9], financial economics [10], etc.
Widespread applications of fractional calculus motivated many researchers to develop the theory
of initial and boundary-value problems arising in the fractional order models associated with nume-
rous real world phenomena. In particular, boundary-value problems of nonlinear fractional differen-
tial equations and inclusions have been extensively studied by several researchers during the past few
decades, for instance, see [11 – 18]. Recently, in [19], the authors proved some existence results for
fractional differential equations with non-separated type nonlocal multipoint and multistrip boundary
conditions.
In this paper, we consider the inclusions (multivalued) case of the problem addressed in [19]
and investigate the existence of solutions for the problem at hand. In precise terms we study the
following multivalued problem:
cDqx(t) \in F (t, x(t), cD\beta x(t), I\gamma x(t)), 1 < q \leq 2, 0 < \beta < 1, \gamma > 0, t \in [0, 1],
* This paper was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Saudi Arabia
under grant no. (KEP-PhD-40-130-41).
c\bigcirc B. AHMAD, S. K. NTOUYAS, A. ALSAEDI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6 763
764 B. AHMAD, S. K. NTOUYAS, A. ALSAEDI
ax(0) + bx(1) =
m - 2\sum
i=1
\alpha ix(\sigma i) +
p - 2\sum
j=1
rj
\eta i\int
\xi j
x(s)ds,
(1.1)
cx\prime (0) + dx\prime (1) =
m - 2\sum
i=1
\delta ix
\prime (\sigma i) +
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
x\prime (s)ds,
0 < \sigma 1 < \sigma 2 < . . . < \sigma m - 2 < . . . < \xi 1 < \eta 1 < \xi 2 < \eta 2 < . . . < \xi p - 2 < \eta p - 2 < 1,
where cD(.) denotes the Caputo fractional derivative of order (.), I(.) denotes the Riemann –
Liouville integral of fractional order (.), F : [0, 1]\times \BbbR 3 \rightarrow \scrP (\BbbR ) is a multivalued map, \scrP (\BbbR ) is the
family of all nonempty subsets of \BbbR , and a, b, c, d are real constants and \alpha i, \delta i, i = 1, 2, . . . ,m - 2,
rj , \gamma j , j = 1, 2, . . . , p - 2, are real constants.
Differential inclusions play a key role in studying dynamical systems and stochastic processes.
An important application of differential inclusions can be found in the area of sweeping processes,
which are modeled by evolution differential inclusions. For a detailed account of this subject and
its applications, we refer the reader to the texts [20, 21]. Differential inclusions help to study
sweeping process [22], granular systems [23, 24], nonlinear dynamics of wheeled vehicles [25],
control problems [26, 27], synchronization process [28], etc.
This paper is organized as follows. In Section 2, we recall some useful preliminaries from multi-
valued analysis and fractional calculus. Section 3 contains the main results. The first existence result
dealing with convex valued maps involved in (1.1) is proved by applying the nonlinear alternative of
Leray – Schauder type, while the second result for the problem (1.1) is concerned with the non-convex
valued maps and relies on a fixed point theorem for contractive multivalued maps due to Covitz and
Nadler. The methods used in our analysis are known, however their exposition in the framework of
problem (1.1) facilitates the enhancement of the topic. Examples illustrating the main results are also
constructed. Finally, we discuss some special cases of the work presented in this article.
2. Background material. Let X =
\bigl\{
x : x \in C([0, 1],\BbbR ) and cD\beta x \in C([0, 1], R)
\bigr\}
denotes
the space equipped with the norm \| x\| X = \| x\| + \| cD\beta x\| = \mathrm{s}\mathrm{u}\mathrm{p}t\in [0,1] | x(t)| + \mathrm{s}\mathrm{u}\mathrm{p}t\in [0,1] | cD\beta x(t)| .
Observe that (X, \| \cdot \| X) is a Banach space.
In the forthcoming analysis, we need the following spaces: \scrP cl(X) = \{ Y \in \scrP (X) : Y is closed\} ,
\scrP b(X) = \{ Y \in \scrP (X) : Y is bounded\} , \scrP cp(X) = \{ Y \in \scrP (X) : Y is compact\} , and \scrP cp,c(X) =
= \{ Y \in \scrP (X) : Y is compact and convex\} .
Next we state some known results related to our proposed work.
Lemma 2.1 ([29], Proposition 1.2). If G : X \rightarrow \scrP cl(X) is u.s.c., then \itG \itr (G) is a closed subset
of X \times Y , i.e., for every sequence \{ xn\} n\in \BbbN \subset X and \{ yn\} n\in \BbbN \subset Y, if when n \rightarrow \infty , xn \rightarrow x\ast ,
yn \rightarrow y\ast and yn \in G(xn), then y\ast \in G(x\ast ). Conversely, if G is completely continuous and has a
closed graph, then it is upper semicontinuous.
Lemma 2.2 [30]. Let X be a Banach space. Let F : [0, 1] \times X3 \rightarrow \scrP cp,c(X) be an L1-
Carathéodory multivalued map and let \Theta be a linear continuous mapping from L1
\bigl(
[0, 1], X
\bigr)
to
C
\bigl(
[0, 1], X
\bigr)
. Then the operator
\Theta \circ SF,x : C
\bigl(
[0, 1], X
\bigr)
\rightarrow \scrP cp,c
\bigl(
C([0, 1], X)
\bigr)
, x \mapsto \rightarrow (\Theta \circ SF,x)(x) = \Theta (SF,x)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
A STUDY OF A MORE GENERAL CLASS OF NONLOCAL INTEGRO-MULTIPOINT . . . 765
is a closed graph operator in C
\bigl(
[0, 1], X
\bigr)
\times C
\bigl(
[0, 1], X
\bigr)
.
Lemma 2.3 (Nonlinear alternative for Kakutani maps [31]). Let E be a Banach space, C a
closed convex subset of E, U an open subset of C and 0 \in U. Suppose that F : U \rightarrow \scrP cp,c(C) is a
upper semicontinuous compact map. Then or (i) F has a fixed point in U, or (ii) there is a u \in \partial U
and \lambda \in (0, 1) with u \in \lambda F (u).
3. Existence results.
Definition 3.1. A function x \in C2([0, 1],\BbbR ) is said to be a solution of the boundary-value
problem (1.1) if ax(0) + bx(1) =
\sum m - 2
i=1
\alpha ix(\sigma i) +
\sum p - 2
j=1
rj
\int \eta i
\xi j
x(s)ds, cx\prime (0) + dx\prime (1) =
=
\sum m - 2
i=1
\delta ix
\prime (\sigma i) +
\sum p - 2
j=1
\gamma j
\int \eta j
\xi j
x\prime (s)ds, and there exists function v \in L1
\bigl(
[0, 1],\BbbR
\bigr)
such that
v(t) \in F
\bigl(
t, x(t), cD\beta x(t), I\gamma x(t)
\bigr)
a.e. on [0, 1] and
x(t) =
t\int
0
(t - s)q - 1
\Gamma (q)
v(s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
v(s)ds +
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
v(u)du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
v(s)ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
v(s)ds +
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
v(u)du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
v(s)ds
\right] ,
where
\Delta 1 = c+ d - \mu 1 \not = 0, \Delta 2 = a+ b - \mu 2 \not = 0, \Delta 3 = b - \mu 3,
\mu 1 =
m - 2\sum
i=1
\delta i -
p - 2\sum
j=1
\gamma j(\eta j - \xi j), \mu 2 =
m - 2\sum
i=1
\alpha i -
p - 2\sum
j=1
rj(\eta j - \xi j), (3.1)
\mu 3 =
m - 2\sum
i=1
\alpha i\sigma i -
p - 2\sum
j=1
rj
\eta 2j - \xi 2j
2
.
In the above definition, we have used Lemma 2.5 derived in [19]. For the sake of convenience,
we set
\Lambda =
1
\Gamma (q + 1)
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
| \Delta 2t - \Delta 3|
| \Delta 1\Delta 2|
\Biggl[
m - 2\sum
i=1
| \delta i|
\sigma q - 1
i
\Gamma (q)
+
p - 2\sum
j=1
| \gamma j |
\Gamma (q + 1)
| \eta qj - \xi qj | +
| d|
\Gamma (q)
\Biggr]
+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
766 B. AHMAD, S. K. NTOUYAS, A. ALSAEDI
+
1
| \Delta 2|
\left[ m - 2\sum
i=1
| \alpha i|
\sigma q
i
\Gamma (q + 1)
+
p - 2\sum
j=1
| rj |
\Gamma (q + 2)
| \eta q+1
j - \xi q+1
j | + | b|
\Gamma (q + 1)
\right] , (3.2)
\Lambda 1 =
1
\Gamma (q)
+
1
| \Delta 1|
\left[ m - 2\sum
i=1
| \delta i|
\sigma q - 1
i
\Gamma (q)
+
p - 2\sum
j=1
| \gamma j |
\Gamma (q + 1)
| \eta qj - \xi qj | +
| d|
\Gamma (q)
\right] , (3.3)
L1 = 1 +
1
\Gamma (\gamma + 1)
. (3.4)
Our first existence result is concerned with the case that the multivalued map F has convex
values (upper semicontinuous case) and its proof is based on Lemma 2.3.
Theorem 3.1. Assume that:
(H1) F : [0, 1]\times \BbbR 3 \rightarrow \scrP (\BbbR ) is L1-Carathéodory and has nonempty compact and convex values;
(H2) there exist a function \phi \in C([0, 1],\BbbR +) and a nondecreasing, subhomogeneous (that is,
\Omega (\mu x) \leq \mu \Omega (x) for all \mu \geq 1 and x \in \BbbR +) function \Omega : \BbbR + \rightarrow \BbbR + such that \| F (t, x)\| \scrP :=
:= \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
| w| : w \in F (t, x, y, z)
\bigr\}
\leq \phi (t)\Omega
\bigl(
\| x\| + \| y\| + \| z\|
\bigr)
for each (t, x, y, z) \in [0, 1]\times \BbbR 3;
(H3) there exists a constant M > 0 such that
M\biggl(
\Lambda +
\Lambda 1
\Gamma (2 - \beta )
\biggr)
\| \phi \| L1\Omega (M)
> 1,
where \Lambda , \Lambda 1 and L1 are defined by (3.2) – (3.4).
Then the boundary-value problem (1.1) has at least one solution on [0, 1].
Proof. Define an operator \Omega F : C
\bigl(
[0, 1],\BbbR
\bigr)
\rightarrow \scrP
\bigl(
C
\bigl(
[0, 1],\BbbR
\bigr) \bigr)
by
\Omega F (x) =
\left\{
h \in C
\bigl(
[0, 1],\BbbR
\bigr)
:
h(t) =
\left\{
\int t
0
(t - s)q - 1
\Gamma (q)
v(s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\Biggl[ \sum m - 2
i=1
\delta i
\int \sigma i
0
(\sigma i - s)q - 2
\Gamma (q - 1)
v(s)ds+
+
\sum p - 2
j=1
\gamma j
\int \eta j
\xi j
\biggl( \int s
0
(s - u)q - 2
\Gamma (q - 1)
v(u)du
\biggr)
ds - d
\int 1
0
(1 - s)q - 2
\Gamma (q - 1)
v(s)ds
\Biggr]
+
+
1
\Delta 2
\Biggl[ \sum m - 2
i=1
\alpha i
\int \sigma i
0
(\sigma i - s)q - 1
\Gamma (q)
v(s)ds+
+
\sum p - 2
j=1
rj
\int \eta j
\xi j
\biggl( \int s
0
(s - u)q - 1
\Gamma (q)
v(u)du
\biggr)
ds - b
\int 1
0
(1 - s)q - 1
\Gamma (q)
v(s)ds
\Biggr]
\right\}
for v \in SF,x, where SF,x :=
\bigl\{
v \in L1([0, 1],\BbbR ) : v(t) \in F (t, x(t), cD\beta x(t), I\gamma x(t)) for a.e. t \in
\in [0, 1]
\bigr\}
denotes the set of selections of F. We split the proof into several steps to show that the
operator \Omega F satisfies the assumptions of Lemma 2.3. As a first step, we show that \Omega F is convex
for each x \in C
\bigl(
[0, 1],\BbbR
\bigr)
. This step is obvious since SF,x is convex (F has convex values), and
therefore we omit the proof.
In the second step, we show that \Omega F maps bounded sets (balls) into bounded sets in C
\bigl(
[0, 1],\BbbR
\bigr)
.
For a positive number \rho , let B\rho =
\bigl\{
x \in C
\bigl(
[0, 1],\BbbR
\bigr)
: \| x\| \leq \rho
\bigr\}
be a bounded ball in C
\bigl(
[0, 1],\BbbR
\bigr)
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
A STUDY OF A MORE GENERAL CLASS OF NONLOCAL INTEGRO-MULTIPOINT . . . 767
Then, for each h \in \Omega F (x), x \in B\rho , there exists v \in SF,x such that
h(t) =
t\int
0
(t - s)q - 1
\Gamma (q)
v(s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
v(s)ds+
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
v(u)du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
v(s)ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
v(s)ds+
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
v(u)du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
v(s)ds
\right] .
Then, for t \in [0, 1], we have
| h(t)| \leq
t\int
0
(t - s)q - 1
\Gamma (q)
| v(s)| ds+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
| \Delta 2t - \Delta 3|
| \Delta 1\Delta 2|
\left[ m - 2\sum
i=1
| \delta i|
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
| v(s)| ds+
+
p - 2\sum
j=1
| \gamma j |
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
| v(u)| du
\right) ds+ | d|
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
| v(s)| ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
| \alpha i|
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
| v(s)| ds+
+
p - 2\sum
j=1
| rj |
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
| v(u)| du
\right) ds+ | b|
1\int
0
(1 - s)q - 1
\Gamma (q)
| v(s)| ds
\right] \leq
\leq
\Biggl(
1
\Gamma (q + 1)
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
| \Delta 2t - \Delta 3|
| \Delta 1\Delta 2|
\Biggl[
m - 2\sum
i=1
| \delta i|
\sigma q - 1
i
\Gamma (q)
+
p - 2\sum
j=1
| \gamma j |
\Gamma (q + 1)
| \eta qj - \xi qj | +
| d|
\Gamma (q)
\Biggr]
+
+
1
| \Delta 2|
\Biggl[
m - 2\sum
i=1
| \alpha i|
\sigma q
i
\Gamma (q + 1)
+
p - 2\sum
j=1
| rj |
\Gamma (q + 2)
| \eta q+1
j - \xi q+1
j | + | b|
\Gamma (q + 1)
\Biggr] \Biggr)
\| \phi \| \Omega (L1\| x\| X)\leq
\leq \Lambda \| \phi \| L1\Omega (\| x\| X),
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
768 B. AHMAD, S. K. NTOUYAS, A. ALSAEDI
which, on taking the norm for t \in [0, 1] yields \| h\| \leq \Lambda \| \phi \| L1\Omega (\| x\| X). Also, we get
| h\prime (t)| \leq
t\int
0
(t - s)q - 2
\Gamma (q - 1)
| v(s)| ds+ 1
| \Delta 1|
\left[ m - 2\sum
i=1
| \delta i|
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
| v(s)| ds+
+
p - 2\sum
j=1
| \gamma j |
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
| v(u)| du
\right) ds+ | d|
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
| v(s)| ds
\right] \leq
\leq
\Biggl\{
1
\Gamma (q)
+
1
| \Delta 1|
\Biggl[
m - 2\sum
i=1
| \delta i|
\sigma q - 1
i
\Gamma (q)
+
p - 2\sum
j=1
| \gamma j |
\Gamma (q + 1)
| \eta qj - \xi qj | +
| d|
\Gamma (q)
\Biggr] \Biggr\}
\| \phi \| \Omega (L1\| x\| X) \leq
\leq \Lambda 1\| \phi \| L1\Omega (\| x\| X),
which, in view of the definition of Caputo fractional derivative with 0 < \beta < 1, implies that
\bigm| \bigm| cD\beta h(t)
\bigm| \bigm| \leq t\int
0
(t - s) - \beta
\Gamma (1 - \beta )
\bigm| \bigm| h\prime (s)\bigm| \bigm| ds \leq \Lambda 1\| \phi \| L1\Omega
\bigl(
\| x\| X
\bigr) t\int
0
(t - s) - \beta
\Gamma (1 - \beta )
ds \leq
\leq 1
\Gamma (2 - \beta )
\Lambda 1\| \phi \| L1\Omega
\bigl(
\| x\| X
\bigr)
.
Hence,
\| h\| X = \| h\| + \| cD\beta h\| \leq
\biggl(
\Lambda +
\Lambda 1
\Gamma (2 - \beta )
\biggr)
\| \phi \| L1\Omega (r). (3.5)
Now we show that \Omega F maps bounded sets into equicontinuous sets of C
\bigl(
[0, 1],\BbbR
\bigr)
. Let t1, t2 \in
\in [0, 1] with t1 < t2 and x \in B\rho . For each h \in \Omega F (x), we obtain\bigm| \bigm| h(t2) - h(t1)
\bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
\Gamma (\alpha )
t1\int
0
[(t2 - s)\alpha - 1 - (t1 - s)\alpha - 1]| v(s)| ds+ 1
\Gamma (\alpha )
t2\int
t1
(t2 - s)\alpha - 1| v(s)| ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
+
| t2 - t1|
| \Delta 1|
\left[ m - 2\sum
i=1
| \delta i|
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
| v(s)| ds +
+
p - 2\sum
j=1
| \gamma j |
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
| v(u)| du
\right) ds+ | d|
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
| v(s)| ds
\right] \leq
\leq \| \phi \| L1\Omega (r)
\Gamma (\alpha + 1)
| 2(t2 - t1)
\alpha + t\alpha 1 - t\alpha 2 | +
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
A STUDY OF A MORE GENERAL CLASS OF NONLOCAL INTEGRO-MULTIPOINT . . . 769
+
\| \phi \| L1\Omega (r)| t2 - t1|
| \Delta 1|
\left[ m - 2\sum
i=1
| \delta i|
\sigma q - 1
i
\Gamma (q)
+
p - 2\sum
j=1
| \gamma j |
\Gamma (q + 1)
| \eta qj - \xi qj | +
| d|
\Gamma (q)
\right]
and \bigm| \bigm| cD\beta h(t2) - cD\beta h(t1)
\bigm| \bigm| \leq
\leq 1
\Gamma (1 - \beta )
\left\{
t1\int
0
\bigm| \bigm| (t1 - s)\beta - (t2 - s)\beta
\bigm| \bigm|
(t1 - s)\beta (t2 - s)\beta
| h\prime (x)(s)| ds+
t2\int
t1
| (t2 - s) - \beta | | h\prime (x)(s)| ds
\right\} \leq
\leq \Lambda 1
\Gamma (1 - \beta )
\left\{
t1\int
0
\bigm| \bigm| (t1 - s)\beta - (t2 - s)\beta
\bigm| \bigm|
(t1 - s)\beta (t2 - s)\beta
ds+
t2\int
t1
\bigm| \bigm| (t2 - s) - \beta
\bigm| \bigm| ds
\right\} \| \phi \| L1\Omega (r).
Obviously, the right-hand side of each of the above two inequalities tends to zero independently
of x \in \scrB \rho as t2 - t1 \rightarrow 0. Since F satisfies the above assumptions, therefore it follows by the
Arzelá – Ascoli theorem that F : C
\bigl(
[0, 1],\BbbR
\bigr)
\rightarrow \scrP
\bigl(
C([0, 1],\BbbR )
\bigr)
is completely continuous.
In next step, we show that \Omega F is upper semicontinuous. To this end it is sufficient to show that
\Omega F has a closed graph, by Lemma 2.1. Let xn \rightarrow x\ast , hn \in \Omega F (xn) and hn \rightarrow h\ast . Then we need to
show that h\ast \in \Omega F (x\ast ). Associated with hn \in \Omega F (xn), there exists vn \in SF,xn such that, for each
t \in [0, 1], we get
hn(t) =
t\int
0
(t - s)q - 1
\Gamma (q)
vn(s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
vn(s)ds+
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
vn(u)du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
vn(s)ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
vn(s)ds+
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
vn(u)du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
vn(s)ds
\right] .
Thus it suffices to show that there exists v\ast \in SF,x\ast such that, for each t \in [0, 1],
h\ast (t) =
t\int
0
(t - s)q - 1
\Gamma (q)
v\ast (s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
v\ast (s)ds+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
770 B. AHMAD, S. K. NTOUYAS, A. ALSAEDI
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
v\ast (u)du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
v\ast (s)ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
v\ast (s)ds+
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
v\ast (u)du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
v\ast (s)ds
\right] .
Let us consider the linear operator \Theta : L1
\bigl(
[0, 1],\BbbR
\bigr)
\rightarrow C
\bigl(
[0, 1],\BbbR
\bigr)
given by
v \mapsto \rightarrow \Theta (v)(t) =
t\int
0
(t - s)q - 1
\Gamma (q)
v(s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
v(s)ds+
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
v(u)du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
v(s)ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
v(s)ds+
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
v(u)du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
v(s)ds
\right] .
Since
\| hn(t) - h\ast (t)\| =
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
(t - s)q - 1
\Gamma (q)
(vn(s) - v\ast (s))ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
\bigl(
vn(s) - v\ast (s)
\bigr)
ds+
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
\bigl(
vn(u) - v\ast (u)
\bigr)
du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
\bigl(
vn(s) - v\ast (s)
\bigr)
ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
\bigl(
vn(s) - v\ast (s)
\bigr)
ds+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
A STUDY OF A MORE GENERAL CLASS OF NONLOCAL INTEGRO-MULTIPOINT . . . 771
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
\bigl(
vn(u) - v\ast (u)
\bigr)
du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
(vn(s) - v\ast (s))ds
\right]
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \rightarrow 0
as n \rightarrow \infty , therefore, it follows by Lemma 2.2 that \Theta \circ SF is a closed graph operator. Further, we
have hn(t) \in \Theta (SF,xn). As xn \rightarrow x\ast , we obtain
h\ast (t) =
t\int
0
(t - s)q - 1
\Gamma (q)
v\ast (s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
v\ast (s)ds+
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
v\ast (u)du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
v\ast (s)ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
v\ast (s)ds+
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
v\ast (u)du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
v\ast (s)ds
\right]
for some v\ast \in SF,x\ast .
Finally, we show there exists an open set U \subseteq C([0, 1],\BbbR ) with x /\in \Omega F (x) for any \theta \in (0, 1)
and all x \in \partial U. Let \theta \in (0, 1) and x \in \theta \Omega F (x). Then there exists v \in L1([0, 1],\BbbR ) with v \in SF,x
such that, for t \in [0, 1], we get
| x(t)| \leq
t\int
0
(t - s)q - 1
\Gamma (q)
| v(s)| ds+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
| \Delta 2t - \Delta 3|
| \Delta 1\Delta 2|
\left[ m - 2\sum
i=1
| \delta i|
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
| v(s)| ds+
+
p - 2\sum
j=1
| \gamma j |
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
| v(u)| du
\right) ds+ | d|
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
| v(s)| ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
| \alpha i|
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
| v(s)| ds +
+
p - 2\sum
j=1
| rj |
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
| v(u)| du
\right) ds+ | b|
1\int
0
(1 - s)q - 1
\Gamma (q)
| v(s)| ds
\right] \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
772 B. AHMAD, S. K. NTOUYAS, A. ALSAEDI
\leq
\left( 1
\Gamma (q + 1)
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
| \Delta 2t - \Delta 3|
| \Delta 1\Delta 2|
\left[ m - 2\sum
i=1
| \delta i|
\sigma q - 1
i
\Gamma (q)
+
p - 2\sum
j=1
| \gamma j |
\Gamma (q + 1)
| \eta qj - \xi qj | +
| d|
\Gamma (q)
\right] +
+
1
| \Delta 2|
\left[ m - 2\sum
i=1
| \alpha i|
\sigma q
i
\Gamma (q + 1)
+
p - 2\sum
j=1
| rj |
\Gamma (q + 2)
| \eta q+1
j - \xi q+1
j | + | b|
\Gamma (q + 1)
\right] \right) \| \phi \| L1\Omega
\bigl(
\| x\| X
\bigr)
\leq
\leq \Lambda \| \phi \| L1\Omega
\bigl(
\| x\| X
\bigr)
,
which on taking the norm for t \in [0, 1] yields
\| x\| \leq \Lambda \| \phi \| L1\Omega
\bigl(
\| x\| X
\bigr)
.
In a similar manner, one can obtain that | x\prime (t)| \leq \Lambda 1\| \phi \| L1\Omega (\| x\| X). In consequence, we have
\| cD\beta x(t)| \leq
t\int
0
(t - s) - \beta
\Gamma (1 - \beta )
\| x\prime (s)| ds \leq \Lambda 1
\Gamma (2 - \beta )
\| \phi \| L1\Omega (\| x\| X).
Hence,
\| x\| X = \| x\| + \| cD\beta x\| \leq
\Biggl(
\Lambda +
\Lambda 1
\Gamma (2 - \beta )
\Biggr)
\| \phi \| L1\Omega (\| x\| X), (3.6)
which implies that
\| x\| X\Biggl(
\Lambda +
\Lambda 1
\Gamma (2 - \beta )
\Biggr)
\| \phi \| L1\Omega (\| x\| X)
\leq 1.
In view of (H3), there exists M such that \| x\| \not = M. Let us set
U =
\bigl\{
x \in C
\bigl(
[0, 1],\BbbR
\bigr)
: \| x\| < M
\bigr\}
.
Note that the operator \Omega F : U \rightarrow \scrP
\bigl(
C([0, 1],\BbbR )
\bigr)
is upper semicontinuous and completely conti-
nuous. From the choice of U, there is no x \in \partial U such that x \in \theta \Omega F (x) for some \theta \in (0, 1).
Consequently, by Lemma 2.3, we deduce that \Omega F has a fixed point x \in U which is a solution of the
problem (1.1).
Theorem 3.1 is proved.
Now we prove the existence of solutions for the problem (1.1) with a nonconvex valued right-
hand side of the inclusion (Lipschitz case) by applying a fixed point theorem for multivalued maps
due to Covitz and Nadler [32]. Let us briefly recall some preliminary concepts needed to establish
the desired result.
Let Hd : \scrP (X)\times \scrP (X) \rightarrow \BbbR \cup \{ \infty \} be a mapping defined by
Hd(A,B) = \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\mathrm{s}\mathrm{u}\mathrm{p}
a\in A
d(a,B), \mathrm{s}\mathrm{u}\mathrm{p}
b\in B
d(A, b)
\Bigr\}
,
where d(A, b) = \mathrm{i}\mathrm{n}\mathrm{f}a\in A d(a; b) and d(a,B) = \mathrm{i}\mathrm{n}\mathrm{f}b\in B d(a; b). Then (\scrP b,cl(X), Hd) is a metric space
and (\scrP cl(X), Hd) is a generalized metric space (see [21]).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
A STUDY OF A MORE GENERAL CLASS OF NONLOCAL INTEGRO-MULTIPOINT . . . 773
Definition 3.2. A multivalued operator N : X \rightarrow \scrP cl(X) is called \gamma -Lipschitz if and only if
there exists \gamma > 0 such that Hd(N(x), N(y)) \leq \gamma d(x, y) for each x, y \in X, and a contraction if
and only if it is \gamma -Lipschitz with \gamma < 1.
Lemma 3.1 [32]. Let (X, d) be a complete metric space. If N : X \rightarrow \scrP cl(X) is a contraction,
then \mathrm{F}\mathrm{i}\mathrm{x}N \not = \varnothing .
Theorem 3.2. Suppose that the following assumptions hold:
(A1) F : [0, 1] \times \BbbR 3 \rightarrow \scrP cp(\BbbR ) is such that F (\cdot , x(t), cD\beta x(t), I\gamma x(t)) : [0, 1] \rightarrow \scrP cp(\BbbR ) is
measurable for each x \in \BbbR ;
(A2) Hd(F (t, x, y, z), F (t, \=x, \=y, \=z)) \leq \varrho (t)
\bigl[
| x - \=x| + | y - \=y| + | z - \=z|
\bigr]
for almost all t \in [0, 1]
and x, y, z, \=x, \=y, \=z \in \BbbR with \varrho \in C
\bigl(
[0, 1],\BbbR +
\bigr)
and d(0, F (t, 0, 0, 0)) \leq \varrho (t) for almost all t \in [0, 1].
Then the boundary-value problem (1.1) has at least one solution on [0, 1] if
\| \varrho \| L1
\biggl(
\Lambda +
\Lambda 1
\Gamma (2 - \beta )
\biggr)
< 1,
where \Lambda , \Lambda 1 and L1 are defined by (3.2) – (3.4).
Proof. Notice that the set SF,x is nonempty for each x \in C([0, 1],\BbbR ) by the assumption (A1),
so F has a measurable selection (see Theorem III.6 [33]). Now we show that the operator \Omega F :
C
\bigl(
[0, 1],\BbbR
\bigr)
\rightarrow \scrP
\bigl(
C([0, 1],\BbbR )
\bigr)
(defined in the beginning of the proof of Theorem 3.1) satisfies the
assumptions of Lemma 3.1. To show that \Omega F (x) \in \scrP cl
\bigl(
(C[0, 1],\BbbR )
\bigr)
for each x \in C
\bigl(
[0, 1],\BbbR
\bigr)
, let
\{ un\} n\geq 0 \in \Omega F (x) be such that un \rightarrow u (n \rightarrow \infty ) in C([0, 1],\BbbR ). Then u \in C
\bigl(
[0, 1],\BbbR
\bigr)
and there
exists vn \in SF,x such that, for each t \in [0, 1],
un(t) =
t\int
0
(t - s)q - 1
\Gamma (q)
vn(s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
vn(s)ds +
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
vn(u)du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
vn(s)ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
vn(s)ds +
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
vn(u)du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
vn(s)ds
\right] .
As F has compact values, we pass onto a subsequence (if necessary) to obtain that vn converges to
v in L1([0, 1],\BbbR ). Thus, v \in SF,x and, for each t \in [0, 1], we have
un(t) \rightarrow u(t) =
t\int
0
(t - s)q - 1
\Gamma (q)
v(s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
v(s)ds+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
774 B. AHMAD, S. K. NTOUYAS, A. ALSAEDI
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
v(u)du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
v(s)ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
v(s)ds +
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
v(u)du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
v(s)ds
\right] .
Hence, u \in \Omega F (x).
Next we show that there exists \delta < 1 such that
Hd(\Omega F (x),\Omega F (\=x)) \leq \delta \| x - \=x\| X for each x, \=x \in C2
\bigl(
[0, 1],\BbbR
\bigr)
.
Let x, \=x \in C2
\bigl(
[0, 1],\BbbR
\bigr)
and h1 \in \Omega F (x). Then there exists v1(t) \in F (t, x(t), cD\beta x(t), I\gamma x(t))
such that, for each t \in [0, 1],
h1(t) =
t\int
0
(t - s)q - 1
\Gamma (q)
v1(s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
v1(s)ds +
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
v1(u)du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
v1(s)ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
v1(s)ds+
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
v1(u)du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
v1(s)ds
\right] .
By (A2), we have
Hd
\bigl(
F (t, x, y, z), F (t, \=x, \=y, \=z)
\bigr)
\leq \varrho (t)
\bigl[
| x - \=x| + | y - \=y| + | z - \=z|
\bigr]
.
So, there exists w \in F (t, \=x, \=y, \=z) such that
| v1(t) - w| \leq \varrho (t)
\bigl[
| x(t) - \=x(t)| + | y(t) - \=y(t)| + | z(t) - \=z(t)|
\bigr]
, t \in [0, 1].
Define \scrV : [0, 1] \rightarrow \scrP (\BbbR ) by
\scrV (t) =
\Bigl\{
w \in \BbbR : | v1(t) - w| \leq \varrho (t)
\bigl[
| x(t) - \=x(t)| + | y(t) - \=y(t)| + | z(t) - \=z(t)|
\bigr] \Bigr\}
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
A STUDY OF A MORE GENERAL CLASS OF NONLOCAL INTEGRO-MULTIPOINT . . . 775
Since the multivalued operator \scrV (t)\cap F (t, \=x, \=y, \=z) is measurable (Proposition III.4 [33]), there exists
a function v2(t) which is a measurable selection for \scrV (t) \cap F (t, \=x, \=y, \=z). So, v2(t) \in F (t, \=x, \=y, \=z)
and, for each t \in [0, 1], we get | v1(t) - v2(t)| \leq \varrho (t)
\bigl[
| x(t) - \=x(t)| + | y(t) - \=y(t)| + | z(t) - \=z(t)|
\bigr]
.
For each t \in [0, 1], let us define
h2(t) =
t\int
0
(t - s)q - 1
\Gamma (q)
v2(s)ds+
\Delta 2t - \Delta 3
\Delta 1\Delta 2
\left[ m - 2\sum
i=1
\delta i
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
v2(s)ds +
+
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
v2(u)du
\right) ds - d
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
v2(s)ds
\right] +
+
1
\Delta 2
\left[ m - 2\sum
i=1
\alpha i
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
v2(s)ds +
+
p - 2\sum
j=1
rj
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
v2(u)du
\right) ds - b
1\int
0
(1 - s)q - 1
\Gamma (q)
v2(s)ds
\right] .
Thus,
\bigm| \bigm| h1(t) - h2(t)
\bigm| \bigm| = t\int
0
(t - s)q - 1
\Gamma (q)
| v1(s) - v2(s)| ds+
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
| \Delta 2t - \Delta 3|
| \Delta 1\Delta 2|
\left[ m - 2\sum
i=1
| \delta i|
\sigma i\int
0
(\sigma i - s)q - 2
\Gamma (q - 1)
| v1(s) - v2(s)| ds+
+
p - 2\sum
j=1
| \gamma j |
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 2
\Gamma (q - 1)
| v1(u) - v2(u)| du
\right) ds+
+ | d|
1\int
0
(1 - s)q - 2
\Gamma (q - 1)
| v1(s) - v2(s)| ds
\right] +
1
| \Delta 2|
\left[ m - 2\sum
i=1
| \alpha i|
\sigma i\int
0
(\sigma i - s)q - 1
\Gamma (q)
| v1(s) - v2(s)| ds +
+
p - 2\sum
j=1
| rj |
\eta j\int
\xi j
\left( s\int
0
(s - u)q - 1
\Gamma (q)
| v1(u) - v2(u)| du
\right) ds+ | b|
1\int
0
(1 - s)q - 1
\Gamma (q)
| v1(s) - v2(s)| (s)ds
\right] \leq
\leq \| \varrho \|
\left( 1
\Gamma (q + 1)
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
| \Delta 2t - \Delta 3|
| \Delta 1\Delta 2|
\Biggl[
m - 2\sum
i=1
| \delta i|
\sigma q - 1
i
\Gamma (q)
+
p - 2\sum
j=1
| \gamma j |
\Gamma (q + 1)
| \eta qj - \xi qj | +
| d|
\Gamma (q)
\Biggr]
+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
776 B. AHMAD, S. K. NTOUYAS, A. ALSAEDI
+
1
| \Delta 2|
\left[ m - 2\sum
i=1
| \alpha i|
\sigma q
i
\Gamma (q + 1)
+
p - 2\sum
j=1
| rj |
\Gamma (q + 2)
| \eta q+1
j - \xi q+1
j | ++
| b|
\Gamma (q + 1)
\right] \right) \Bigl[ \| x - \=x\| X +
+
1
\Gamma (\gamma + 1)
\| x - \=x\| X
\Bigr]
\leq \| \varrho \| \Lambda L1\| x - \=x\| X .
Hence, \| h1 - h2\| \leq \| \varrho \| \Lambda L1\| x - \=x\| X . In a similar manner, we obtain
| h\prime 1(t) - h\prime 2(t)| \leq \| \varrho \|
\left\{ 1
\Gamma (q)
+
1
| \Delta 1|
\left[ m - 2\sum
i=1
| \delta i|
\sigma q - 1
i
\Gamma (q)
+
p - 2\sum
j=1
| \gamma j |
\Gamma (q + 1)
| \eta qj - \xi qj | +
| d|
\Gamma (q)
\right] \right\} \times
\times
\biggl(
1 +
1
\Gamma (\gamma + 1)
\biggr)
\| x - \=x\| X = \| \varrho \| \Lambda 1L1\| x - \=x\| X
and
\bigm| \bigm| cD\beta h1(t) - cD\beta h2(t)
\bigm| \bigm| \leq t\int
0
(t - s) - \beta
\Gamma (1 - \beta )
| h\prime 1(t) - h\prime 2(t)| ds \leq
1
\Gamma (2 - \beta )
\| \varrho \| \Lambda 1L1\| x - \=x\| X .
Thus,
\| h1 - h2\| X \leq \| \varrho \| L1
\Biggl(
\Lambda +
\Lambda 1
\Gamma (2 - \beta )
\Biggr)
\| x - \=x\| X .
Analogously, interchanging the roles of x and x, we obtain
Hd(\Omega F (x),\Omega F (\=x)) \leq \| \varrho \| L1
\Biggl(
\Lambda +
\Lambda 1
\Gamma (2 - \beta )
\Biggr)
\| x - \=x\| X .
Since \Omega F is a contraction, it follows by Lemma 3.1 that \Omega F has a fixed point x which is a solution
of (1.1).
Theorem 3.2 is proved.
Examples. Consider the nonlocal integro-multipoint boundary multivalued (inclusion) problem
cD8/5x(t) \in F (t, x(t), cD3/4x(t), I1/2x(t)), t \in [0, 1],
ax(0) + bx(1) =
3\sum
i=1
\alpha ix(\sigma i) +
4\sum
j=1
rj
\eta i\int
\xi j
x(s)ds, (3.7)
cx\prime (0) + dx\prime (1) =
3\sum
i=1
\delta ix
\prime (\sigma i) +
4\sum
j=1
\gamma j
\eta j\int
\xi j
x\prime (s)ds.
Here, q = 8/5, \beta = 3/4, \gamma = 1/2, m = 5, p = 6, \sigma 1 = 1/15, \sigma 2 = 2/15, \sigma 3 = 3/15, \xi 1 = 1/4,
\eta 1 = 5/16, \xi 2 = 6/16, \eta 2 = 7/16, \xi 3 = 8/16, \eta 3 = 9/16, \xi 4 = 10/16, \eta 4 = 11/16, \alpha 1 = 2,
\alpha 2 = 1, \alpha 3 = 1, \delta 1 = 2, \delta 2 = 3, \delta 3 = - 1, r1 = 1, r2 = 1, r3 = 3, r4 = - 2, \gamma 1 = - 3,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
A STUDY OF A MORE GENERAL CLASS OF NONLOCAL INTEGRO-MULTIPOINT . . . 777
\gamma 2 = - 1, \gamma 3 = 1, \gamma 4 = 2, a = 1, b = 1, c = - 2, d = 4. Using the given data in (3.1) –
(3.4), we find that \Delta 1 = - 2.0625, \Delta 2 = - 1.8125, \Delta 3 = 0.5939, \mu 1 = 4.0625, \mu 2 = 3.8125,
\mu 3 = 0.4061,\Lambda = 6.6286,\Lambda 1 = 4.3363, L1 = 2.1284.
In order to illustrate Theorem 3.1, we take
F (t, x(t), cD3/4x(t), I1/2x(t)) =
=
\Biggl[
1\surd
t2 + 144
\Biggl(
1
3
\mathrm{s}\mathrm{i}\mathrm{n}(x(t)) +
1
2
| cD3/4x(t)| \bigl(
1 + | cD3/4x(t)|
\bigr) + 1
\pi
\mathrm{t}\mathrm{a}\mathrm{n} - 1
\Bigl(
I1/2x(t)
\Bigr)
+
1
2
\Biggr)
,
1
t+ 15
\Biggl(
1
16
e - x4(t) +
1
5
\mathrm{s}\mathrm{i}\mathrm{n}(cD1/2x(t)) +
1
10
| I1/4x(t)| \bigl(
1 + | I1/4x(t)|
\bigr) + 1
2
\Biggr) \Biggr]
. (3.8)
It is easy to find that \phi (t) =
1\surd
t2 + 144
with \| \phi \| = 1/12,\Omega (M) = 11/6. By the condition (H3),
we find that M > 3.7111. Thus all the assumptions of Theorem 3.1 hold and consequently the
problem (3.7) with F given by (3.8) has a solution on [0, 1].
Next we illustrate Theorem 3.2 by taking the map
F (t, x(t), cD3/4x(t), I1/2x(t)) =
=
\biggl[
1
t+ 30
x(t) +
1
t2 + 45
\mathrm{c}\mathrm{o}\mathrm{s}(D3/4x(t)) +
1
t2 + 49
I1/2x(t),
1
t2 + 64
\mathrm{t}\mathrm{a}\mathrm{n} - 1(x(t)) +
1
t2 + 36
cD3/4x(t) +
1
t2 + 49
\mathrm{t}\mathrm{a}\mathrm{n} - 1(I1/2x(t)) +
1
32
\biggr]
. (3.9)
By the condition (A2), we get \varrho (t) = 1/(t+30) with \| \varrho \| = 1/30. Then \| \varrho \| L1[\Lambda +\Lambda 1/\Gamma (5/4)] \approx
\approx 0.8097 < 1. Clearly the hypothesis of Theorem 3.2 is satisfied. Therefore, there exists at least one
solution for the problem (3.7) with F given by (3.9) on [0, 1].
4. Conclusions. We have addressed a more general problem of fractional order differential inclu-
sions involving a multivalued map depending on the unknown function together with its lower-order
fractional derivative and Riemann – Liouville integral, supplemented with non-separated boundary
conditions containing finite many nonlocal points and strips on the given interval [0, 1]. The exis-
tence results obtained for the problem at hand are not only new but also yield several new results as
special cases by fixing the parameters involved in the problem. Some of the these results are listed
below.
We obtain the results for the inclusion problem with periodic/antiperiodic type boundary con-
ditions of the form x(0) = - (b/a)x(1), x\prime (0) = - (d/c)x\prime (1) by taking rj = \gamma j = \alpha j = \delta j = 0,
j = 1, . . . , p in (1.1). Further, the results for antiperiodic boundary conditions follow with (b/a) =
= 1 = (d/c).
Our results correspond to non-separated nonlocal multipoint and multistrip conditions, respec-
tively, by taking rj = 0 = \gamma j , j = 1, . . . , p, and \alpha j = 0 = \delta j , j = 1, . . . , p, in (1.1).
For a = c = 0, b = d = 1, we get the results for the inclusion problem with terminal nonlocal
multipoint and multistrip conditions:
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
778 B. AHMAD, S. K. NTOUYAS, A. ALSAEDI
x(1) =
m - 2\sum
i=1
\alpha ix(\sigma i) +
p - 2\sum
j=1
rj
\eta i\int
\xi j
x(s)ds, x\prime (1) =
m - 2\sum
i=1
\delta ix
\prime (\sigma i) +
p - 2\sum
j=1
\gamma j
\eta j\int
\xi j
x\prime (s)ds.
Existence theorems for the inclusion problem with purely nonlocal multipoint and multistrip
conditions follow by choosing a = c = b = d = 0 in the results of this paper.
In the scenario of generality of fractional order differential inclusions and boundary conditions,
the present work is quite versatile in nature and significantly contributes to the existing literature on
fractional order multivalued boundary-value problems. Moreover, a variety of new results follow
from the ones obtained in this paper by specializing the parameters involved in the problem, which
enhances the utility/scope of the work.
References
1. F. Mainardi, Some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.),
Fractals and Fractional Calculus in Continuum Mechanics, Springer, Berlin (1997), p. 291 – 348.
2. W. Glockle, T. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68, 46 – 53
(1995).
3. G. M. Zaslavsky, Hamiltonian chaos and fractional dynamics, Oxford Univ. Press, Oxford (2005).
4. R. L. Magin, Fractional calculus in bioengineering, Begell House Publ. (2006).
5. Z. M. Ge, C. Y. Ou, Chaos synchronization of fractional order modified Duffing systems with parameters excited by
a chaotic signal, Chaos, Solitons and Fractals, 35, 705 – 717 (2008).
6. F. Mainardi, Fractional calculus and waves in linear viscoelasticy, World Sci., Singapore (2010).
7. M. Javidi, B. Ahmad, Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton-zooplankton
system, Ecol. Modelling, 318, 8 – 18 (2015).
8. N. Nyamoradi, M. Javidi, B. Ahmad, Dynamics of SVEIS epidemic model with distinct incidence, Int. J. Biomath., 8,
№ 6, Article 1550076 (2015), 19 p.
9. A. Carvalho, C. M. A. Pinto, A delay fractional order model for the co-infection of malaria and HIV/AIDS, Int. J.
Dyn. and Control, 5, 168 – 186 (2017).
10. H. A. Fallahgoul, S. M. Focardi, F. J. Fabozzi, Fractional calculus and fractional processes with applications to
financial economics. Theory and application, Elsevier/Acad. Press, London (2017).
11. Z. B. Bai, W. Sun, Existence and multiplicity of positive solutions for singular fractional boundary-value problems,
Comput. Math. Appl., 63, 1369 – 1381 (2012).
12. B. Ahmad, S. K. Ntouyas, A. Alsaedi, A study of nonlinear fractional differential equations of arbitrary order with
Riemann – Liouville type multistrip boundary conditions, Math. Probl. Eng., 2013, Article ID 320415 (2013), 9 p.
13. L. Zhang, G. Wang, B. Ahmad, R. P. Agarwal, Nonlinear fractional integro-differential equations on unbounded
domains in a Banach space, J. Comput. and Appl. Math., 249, 51 – 56 (2013).
14. B. Ahmad, S. K. Ntouyas, Existence results for higher order fractional differential inclusions with multi-strip fractional
integral boundary conditions, Electron. J. Qual. Theory Different. Equat., 2013, № 20 (2013), 19 p.
15. J. R. Graef, L. Kong, M. Wang, Existence and uniqueness of solutions for a fractional boundary-value problem on a
graph, Fract. Calc. and Appl. Anal., 17, 499 – 510 (2014).
16. X. Liu, Z. Liu, Existence results for fractional semilinear differential inclusions in Banach spaces, J. Appl. Math. and
Comput., 42, 171 – 182 (2013).
17. H. Ergoren, Impulsive functional differential equations of fractional order with variable moments, Ukr. Math. J., 68,
№ 9, 1340 – 1352 (2017).
18. B. Ahmad, R. Luca, Existence of solutions for sequential fractional integro-differential equations and inclusions with
nonlocal boundary conditions, Appl. Math. and Comput., 339, 516 – 534 (2018).
19. B. Ahmad, S. K. Ntouyas, A. Alsaedi, W. Shammakh, R. P. Agarwal, Existence theory for fractional differential
equations with non-separated type nonlocal multi-point and multistrip boundary conditions, Adv. Difference Equat.,
№ 89 (2018), 20 p.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
A STUDY OF A MORE GENERAL CLASS OF NONLOCAL INTEGRO-MULTIPOINT . . . 779
20. M. Monteiro, D. P. Manuel, Differential inclusions in nonsmooth mechanical problems. Shocks and dry friction,
Progress in Nonlinear Differential Equations and their Applications, vol. 9, Birkhäuser, Basel (1993).
21. M. Kisielewicz, Stochastic differential inclusions and applications, Springer Optim. and Appl., vol. 80, Springer, New
York (2013).
22. S. Adly, T. Haddad, L. Thibault, Convex sweeping process in the framework of measure differential inclusions and
evolution variational inequalities, Math. Program. Ser. B, 148, 5 – 47 (2014).
23. P. Richard, M. Nicodemi, R. Delannay, P. Ribiere, D. Bideau, Slow relaxation and compaction of granular system,
Nature Mater., 4, 121 – 128 (2005).
24. J. C. Quezada Guajardo, L. Sagnol, C. Chazallon, Shear test on viscoelastic granular material using contact dynamics
simulations, EPJ Web Conf., 140, Article 08009 (2017).
25. J. Bastien, Study of a driven and braked wheel using maximal monotone differential inclusions: applications to the
nonlinear dynamics of wheeled vehicles, Arch. Appl. Mech., 84, 851 – 880 (2014).
26. F. L. Pereira, J. Borges de Sousa, A. Coimbra de Matos, An algorithm for optimal control problems based on
differential inclusions, Proc. 34th Conf. Decision and Control, New Orleans, LA - December 1995.
27. M. Korda, D. Henrion, C. N. Jones, Convex computation of the maximum controlled invariant set for polynomial
control systems, SIAM J. Control and Optim., 52, 2944 – 2969 (2014).
28. M.-F. Danca, Synchronization of piecewise continuous systems of fractional order, Nonlinear Dynam., 78, 2065 – 2084
(2014).
29. K. Deimling, Multivalued differential equations, Walter De Gruyter, Berlin, New York (1992).
30. A. Lasota, Z. Opial, An application of the Kakutani – Ky Fan theorem in the theory of ordinary differential equations,
Bull. Acad. Polon. Sci. Ser. Sci. Math., Astronom., Phys., 13, 781 – 786 (1965).
31. A. Granas, J. Dugundji, Fixed point theory, Springer-Verlag, New York (2005).
M. Kisielewicz, Differential inclusions and optimal control, Kluwer, Dordrecht, The Netherlands, 1991.
32. H. Covitz, S. B. Nadler (Jr.), Multivalued contraction mappings in generalized metric spaces), Israel J. Math., 8, 5 – 11
(1970).
33. C. Castaing, M. Valadier, Convex analysis and measurable multifunctions, Lect. Notes Math., 580, Springer-Verlag,
Berlin ect. (1977).
Received 05.10.18,
after revision — 08.01.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
|
| id | umjimathkievua-article-388 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:36Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ff/6f55e72e6fb6d52a939bdf0d5bffbdff.pdf |
| spelling | umjimathkievua-article-3882022-03-26T11:03:02Z A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions A study of a more general class of nonlocal integro-multipoint boundary value problems of fractional integro-differential inclusions A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions Ahmad, B. Ntouyas, S. K. Alsaedi, A. Ahmad, B. Ntouyas, S. K. Alsaedi, A. Ahmad, B. Ntouyas, S. K. Alsaedi, A. Fractional dierential inclusion multipoin integral boundary condition fixed poin Fractional dierential inclusion multipoin integral boundary condition fixed poin UDC 517.9 We develop the existence theory for a more general class of nonlocal integro-multipoint boundary value problems of Caputo&nbsp; type&nbsp; fractional integro-differential inclusions. Our results include the convex and&nbsp; non-convex cases for the given problem and rely on standard fixed point theorems for multivalued maps.&nbsp; The obtained results are illustrated with the aid of examples. УДК 517.9 Вивчення вiдносно загального класу нелокальних iнтегральних багатоточкових крайових задач для дробових iнтегро-диференцiйних включень&nbsp; Запропоновано теорію існування для відносно загального класу нелокальних інтегральних багатоточкових крайових задач для дробових інтегро-диференціальних включень типу Капуто.&nbsp;Наші результати охоплюють опуклі та неопуклі випадки даної проблеми і базуються на стандартних теоремах про нерухому точку для багатозначних відображень.&nbsp;Отримані результати проілюстровано відповідними прикладами.&nbsp; Institute of Mathematics, NAS of Ukraine 2021-06-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/388 10.37863/umzh.v73i6.388 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 6 (2021); 763 - 799 Український математичний журнал; Том 73 № 6 (2021); 763 - 799 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/388/9025 Copyright (c) 2021 B. Ahmad, S. K. Ntouyas, A. Alsaedi |
| spellingShingle | Ahmad, B. Ntouyas, S. K. Alsaedi, A. Ahmad, B. Ntouyas, S. K. Alsaedi, A. Ahmad, B. Ntouyas, S. K. Alsaedi, A. A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions |
| title | A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions |
| title_alt | A study of a more general class of nonlocal integro-multipoint boundary value problems of fractional integro-differential inclusions A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions |
| title_full | A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions |
| title_fullStr | A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions |
| title_full_unstemmed | A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions |
| title_short | A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions |
| title_sort | study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions |
| topic_facet | Fractional dierential inclusion multipoin integral boundary condition fixed poin Fractional dierential inclusion multipoin integral boundary condition fixed poin |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/388 |
| work_keys_str_mv | AT ahmadb astudyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ntouyassk astudyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT alsaedia astudyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ahmadb astudyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ntouyassk astudyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT alsaedia astudyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ahmadb astudyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ntouyassk astudyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT alsaedia astudyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ahmadb studyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ntouyassk studyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT alsaedia studyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ahmadb studyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ntouyassk studyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT alsaedia studyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ahmadb studyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT ntouyassk studyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions AT alsaedia studyofamoregeneralclassofnonlocalintegromultipointboundaryvalueproblemsoffractionalintegrodifferentialinclusions |