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Abstract
We investigate a new class of boundary value problems of a nonlinear coupled system of sequential fractional differential equations and inclusions involving Caputo fractional derivatives and boundary conditions. We use standard fixed-point theory tools to deduce sufficient criteria for the existence and uniqueness of solutions to the problems at hand. Examples are discussed to illustrate the validity of the proposed results.
1. Introduction
Fractional calculus has gotten a lot of attention over the last two decades. Consequently, there has been a burgeoning interest in the theory and applications of fractional differential equations (FDEs) under various types of initial and boundary conditions (BCs); see, for example [Citation1–11] and the references cited therein. The feature of fractional differentiation and integration has significantly improved the consideration of mathematical modellings of many real-life problems within fractional settings. As a result, it has been realized that this subject has applications in a wide range of technical and physical sciences, including electrodynamics of complex media, control theory ecology, viscoelasticity, biomathematics, electrical circuits, electroanalytical chemistry, aerodynamics and blood flow phenomena. For additional information, the reader can consult the papers [Citation12–24].
In the remarkable monograph [Citation25], the concept of two fractional order operators (i.e. Sequential Fractional Derivative (SFD)) was discussed. It has been recognized that SFDs and non-SFDs are inextricably linked and thus there have appeared some recent work on sequential fractional differential equations (SFDEs) [Citation26–30]. Apart from the study of fractional-order boundary value problems (BVPs) for equations and inclusions [Citation31–33], the study of systems of coupled FDEs has accelerated and attracted interested researchers. Examples on applications of coupled systems include disease models, Lorenz system, ecological models, Duffing system, synchronization of chaotic systems, etc. [Citation34–38]. For the theoretical development of coupled systems of FDEs, we refer the readers to [Citation39–41] and the references cited therein. In [Citation42, Citation43], the authors investigated SFDEs with different types of BCs. Sufficient conditions are utilized to demonstrate the consequence of existence and uniqueness to the analysed equation. Numerous mathematicians and applied researchers have attempted to use fractional calculus to model real-world processes. It has been deduced in biology that the membranes of biological organism cells have fractional-order electrical conductance [Citation44] and thus are classified in groups of non-integer-order models. Fractional derivatives are the most successful in the field of rheology because they embody essential features of cell rheological behaviour [Citation45]. In most biological systems, such as HIV infection, hepatitis C virus (HCV) infection, and cancer spread, fractional-order ordinary differential equations are naturally related to systems with long-time memory. Additionally, they are related to fractals, which occur frequently in biological systems. Wang and Li [Citation46] analysed the global dynamics of HIV infection of CD4+ cells. Arafal et al. [Citation47] studied fractional modelling dynamics of HIV and CD4+ T cells during primary infection. As a result, fractional-order differential equations are thought to be a better tool than integer-order differential equations for describing hereditary properties of various materials and processes. Fractional-order models have become more realistic and practical than their classical integer-order counterparts as a result of this advantage, and their dynamics behaviour is also as stable as their integer-order counterparts. Due to the fact that theoretical results can aid in the development of a more complete understanding of the dynamic behaviour of biological processes, the study of abstract fractional dynamic models is becoming increasingly relevant and important in the modern era. On the other hand, the BCs in (Equation1(1)
(1) ) are referred to as coupled BCs; they are encountered in the study of reaction–diffusion equations, Sturm–Liouville problems and mathematical biology, among other fields [Citation48–50]. Recently in [Citation51], Ahmad et al. discussed the existence and uniqueness of coupled system of FDEs with a novel class of coupled boundary conditions specified by
(1)
(1) where
is the Caputo fractional derivatives (CFD) of order
,
,
are continuous functions and A is nonnegative constant. The main results are established by converting the system (Equation1
(1)
(1) ) to a fixed point equivalent problem and solving it using standard fixed point theorems. As far as we know, the single-valued and multi-valued maps for the solutions of nonlinear coupled SFDEs with coupled boundary conditions have been rarely investigated. Motivated by the HIV infection model and its application background, we investigate the consequences of existence for a nonlinear coupled system of Caputo-type SFDEs subject to coupled boundary conditions of the form:
(2)
(2)
where
,
,
,
are continuous functions,
is the collection of non-empty subsets of
, and ϕ is positive constant. Further existence investigation is carried out for the following nonlinear coupled differential inclusion
(3)
(3)
under the same assumptions. Ahmad et al. [Citation51] reported the research to study the existence of positive solutions for nonlinear coupled system of fractional differential equations complemented with boundary conditions. We should point out that the term “sequential” is used in this context in the sense that the operator
can be written as the composition of the operators
. In this article, authors have extend the boundary value problem of Ahmad et al. [Citation51] to nonlinear coupled system of sequential Caputo fractional differential equations and inclusions having the value of unknown functions v and w at the interval endpoints
being zero, whereas the impact of the sum of the unknown functions on an arbitrary domain
of the given interval
remaining constant. Furthermore, the authors have emphasized the major results on existence, uniqueness of solutions for sequential fractional differential equations and inclusions compared against [Citation51]. Unlike the paper [Citation51], the main results of this paper are entirely different in the sense that we consider the main problems in frame of sequential fractional derivative, use different techniques based on Schaefer's, Banach's, Covitz-Nadler's, and nonlinear alternative for Kakutani fixed point theorems and investigate the nonlinear coupled differential inclusion (Equation3
(3)
(3) ) which was not considered in [Citation51]. Additionally, to our knowledge, there are no published outcomes relating system (Equation3
(3)
(3) ). Section 2 provides essential preliminaries along with an auxiliary lemma that is critical for deriving the solution to the given problem. Section 3 is devoted to the main results in which we study the existence and uniqueness of solutions for systems (Equation2
(2)
(2) ) and (Equation3
(3)
(3) ), separately. In Section 4, particular examples consistent with the studied systems and the main theorems are provided.
2. Preliminaries
This section explores several definitions of multi-valued maps and lemmas that are necessary for proving the primary results [Citation13, Citation16, Citation52, Citation53].
Let be a normed space and that
,
.
A multi-valued map is
convex valued if
is convex
;
upper semi-continuous (u.s.c.) on
if, for each
; the set
is a non-empty closed subset of
and if, for each open set
of
containing
, there exists an open neighbourhood
of
such that
;
lower semi-continuous (l.s.c.) if the set
is open for any open set
in
;
completely continuous (c.c) if
is relatively compact (r.c) for every
.
A map of multi-valued is said to be measurable if, for every
, the function
is measurable.
A multi-valued map is said to be Caratheodory if
is measurable for each
;
is u.s.c for almost all
.
Further a Caratheodory function is called
-Caratheodory if
for each
, ∃
∋
with
and for a.e.
.
Definition 2.1
The Riemann–Liouville fractional integral of order ϱ with the lower limit zero for a function is defined as
provided the RHS is point-wise defined on
, where
is the gamma function, which is defined by
. Note that the above integral exists on
when
.
Definition 2.2
The Riemann–Liouville fractional derivative of order ,
,
for a function
is defined as
Notice that the Riemann–Liouville fractional derivative of order
exists almost everywhere on
if
.
Definition 2.3
The Caputo derivative of order for a function
can be written as
Note that the Caputo fractional derivative of order
exists almost everywhere on
if
.
Lemma 2.1
[Citation54]
Let a closed convex subset of a Banach space
and
be an open subset of
with
. In addition,
is an u.s.c compact map. Then either
has fixed point in
or
∃
and
such that
.
Lemma 2.2
[Citation55]
Let be a complete metric space. If
is a contraction, then Fix
.
Lemma 2.3
[Citation56]
Let be a completely continuous operator in Banach Space
and the set
is bounded. Then
has a fixed point in
.
Lemma 2.4
Let and
. Then the integral solution for the linear system of SFDEs:
(4)
(4) is given by
(5)
(5)
(6)
(6)
Proof.
As argued in [Citation43], the general solution of the system (Equation4(4)
(4) ) can be written as
(7)
(7)
where
are arbitrary constants. Using BCs (Equation4
(4)
(4) ) in (Equation7
(7)
(7) ), we obtain
(8)
(8) and
(9)
(9) The solutions (Equation5
(5)
(5) ) and (Equation6
(6)
(6) ) are obtained by substituting the values of
and
in (Equation7
(7)
(7) ) respectively.
3. Main results
The main results are stated and proved in this section. The results are carried out separately for systems (Equation2(2)
(2) ) and (Equation3
(3)
(3) ).
3.1. Existence results for system (2)
Define as the Banach space endowed with norm
, for
. Using Lemma 2.4, we convert system (Equation2
(2)
(2) ) into a fixed point problem as
, the following operator
is defined by
(10)
(10) where
(11)
(11) and
(12)
(12) Following that, we initiate the hypotheses that will be used to demonstrate the paper's primary research results.
Let be continuous functions.
(Q1) | There exist continuous positive functions | ||||
(Q2) | There exist non-negative constants |
Theorem 3.1
Suppose that holds. Furthermore, the assumption is that
(15)
(15) where
are defined by (Equation13
(13)
(13) ) and (Equation14
(14)
(14) ). Then the problem (Equation2
(2)
(2) ) has at least one solution on
.
Proof.
We begin by demonstrating that the operator
is c.c. Note that Ψ is continuous as the functions
and
are continuous. Now let
be bounded. Then ∃ positive constants
and
such that
for all
, we have
Thus
Thus the operator Ψ is uniformly bounded as a result of the preceding inequality. Let Ψ prove that it determines bounded sets into equicontinuous sets of
, let
,
and
). Then
Analogously, we can obtain
Take note that in the limit
, the RHS of the preceding inequalities tends to zero independently of
. Thus the operator
is equicontinuous, and hence by, Arzela–Ascoli theorem,
is c.c. Next, it will be verified that the set
is bounded. For any
, we have
Using
defined by (Equation13
(13)
(13) )–(Equation14
(14)
(14) ), we get
In consequence, we get
Then, using Equation (Equation15
(15)
(15) ), we conclude that
This demonstrates that the set Θ is bounded. Hence, by Schaefer's fixed point theorem, there exists a solution of (Equation2
(2)
(2) ).
Next, we express our second result, which is based on Banach's fixed point theorem and concerns the existence of a unique solution to (Equation2(2)
(2) ).
Theorem 3.2
Suppose that hold and that
(16)
(16) where
,
and
are defined by (Equation13
(13)
(13) )–(Equation14
(14)
(14) ). Then the problem (Equation2
(2)
(2) ) has a unique solution.
Proof.
Let us choose , and
and fix
Then we show that
where
, we have
which leads to
when the norm for
. Equivalently, for
, one can obtain
Therefore, for any
, we have
which demonstrates that Ψ maps
into itself. To demonstrate that the operator Ψ is a contraction, let
. Then, in view of
, we obtain
Clearly, the preceding inequalities imply that
As a result, in light of the assumption (Equation16
(16)
(16) ), the operator \Psi is a contraction. As a result of Banach's contraction mapping theorem, Ψ has an unique fixed point. This demonstrates that system (Equation2
(2)
(2) ) has a unique solution on
.
If and
are valid, then Theorem (3.1) has the special case form shown below.
Remark 3.1
There exist positive functions and
which are continuous functions such that
Then system (Equation2
(2)
(2) ) has at least one solution on
.
Remark 3.2
According to the assumptions of Theorem 3.1, if
non-negative constants, and the criteria of the functions
have the following form:
there are real constants
so
and (Equation15
(15)
(15) ) becomes
3.2. Existence results for system (3)
Definition 3.1
A function satisfying the BCs
and for which ∃ functions
∋
, a.e on
and
(17)
(17) and
(18)
(18) is referred to as a coupled solution for system (Equation3
(3)
(3) ). Let
and
define the sets of
selections for each
. Using Lemma 2.4, the following operators
are defined by:
(19)
(19) and
(20)
(20) where
(21)
(21) and
(22)
(22)
Following that, we define the operator by
where
and
are defined in (Equation19
(19)
(19) ) and (Equation20
(20)
(20) ), respectively.
In this portion, we prove the existence of solutions for the BVP (Equation3(3)
(3) ) via nonlinear alternative of Leray–Schauder [Citation55]. Following that, we initiate the assumptions that will be used to demonstrate the paper's primary research results.
(Q3) |
| ||||
(Q4) | ∃ continuous increasing functions | ||||
(Q5) | ∃ a constant | ||||
(Q6) |
| ||||
(Q7) |
|
Theorem 3.3
Assume that ,
and
holds. Then system (Equation3
(3)
(3) ) has at least one solution on
.
Proof.
Consider the operators which is given by (Equation19
(19)
(19) ) and (Equation20
(20)
(20) ) respectively. Using the assumption
, the sets
and
are non-empty for each
. Then, for
,
for
, we have
(23)
(23) and
(24)
(24) where
,
, and so
. The operator Λ will be shown to satisfy the Leray–Schauder nonlinear alternative hypotheses in several steps. To begin, we demonstrate that
has a convex value. Let
, i = 1, 2. Then ∃
,
, i = 1, 2, ∋ for each
, we have
(25)
(25) and
(26)
(26) Let
. Then, for each
, we have
(27)
(27) and
(28)
(28) We may deduce that
and
have convex values since
and
have convex values. Clearly,
,
, and thus
. For a non-negative number
, let
be a bounded set in
. Then ∃
,
such that
(29)
(29) and
(30)
(30) Then we have
and
Thus we get
Then we prove that Λ is equicontinuous. Let
with
. Then ∃
,
such that
and
Analogously, we can obtain
As a consequence, the operator
is equicontinuous, hence the operator
is c.c according to the Arzela–Ascoli theorem. We know from [Citation53, Proposition 1.2] that a c.c operator has a closed graph if it is upper semicontinuous. As a result, we must demonstrate that Λ has a closed graph. Let
,
and
, then we must demonstrate
. Remember that
implies that ∃
,
such that
and
Consider the
continuous linear operators provided by
(31)
(31) and
(32)
(32) We can deduce from [Citation57] that
is a closed graph operator. In addition, we have
for all n. Since
,
, it follows that
,
such that
and
(i.e.)
. Let
. Then ∃
,
such that
(33)
(33) and
(34)
(34) For each
, we obtain
which implies that
According to
,
exists such that
. Let us fix
It should be noted that operator
is c.c and u.v.c. There is no
∋
for some
by
selection. As a result, we can deduce from the Leray–Schauder nonlinear alternative [Citation55] that Λ has a fixed point
, which is a solution of system (Equation3
(3)
(3) ).
Let denote a metric space generated from the normed space
, and let
be defined by
, where
and
. Then
is a metric space and
is a generalized metric space (see [Citation57]).
Definition 3.2
A multi-valued operator is called
δ-Lipschitz iff ∃
∋
for each
; and
a contraction iff it is δ-Lipschitz with
.
The following result makes use of Covitz and Nadler's theorem for multi-valued maps [Citation54].
Theorem 3.4
Assume that and
holds. Then system (Equation3
(3)
(3) ) has at least one solution on
provided that
(35)
(35)
Proof.
Assuming that the sets
and
are non-empty for each
,
and
have measurable selections (see Theorem III.6 in [Citation58]). Next we demonstrate that the operator Λ fulfils the theorem of Covitz and Nadler's [Citation54]. Next we demonstrate that
for each
. Let
such that
in
. Then
and ∃
and
such that
and
Due to the fact that
and
have compact values, we pass onto subsequences (referred to as sequences) to ensure that
and
converge to
and
in
. Hence
and
for each
and that
and
As a result,
, implying that
is closed. Following that, we demonstrate that ∃ (defined by (Equation35
(35)
(35) )) such that
Let
and
. Then ∃
and
∋, for each
, we have
and
Using
, we have
and
So, ∃
and
such that
and
Define
by
and
There are functions
,
that are an observable selection for
because the multi-valued operators
and
are measurable (Proposition III.4 in [Citation25]). And
,
such that ∀
, we have
and
Let
and
Hence,
Thus
Similarly, we can define that
Therefore
Similarly, by swapping the positions of
and
, we can obtain
In light of the assumption, Λ is a contraction (Equation35
(35)
(35) ). As a result of fixed point theorem of Covitz and Nadler's [Citation54], Λ has a fixed point
that is a solution to system (Equation3
(3)
(3) ).
4. Examples
In consistence with systems (Equation2(2)
(2) ) and (Equation3
(3)
(3) ) and the main theorems, we provide some examples in this section.
Example 4.1
Consider the following system:
(36)
(36) where
. Utilizing the above data, we get
, where
and
are respectively given by Equation13
(13)
(13) and Equation14
(14)
(14) . To illustrate Theorem 3.1, we will use
Next,
and
are continuous and fulfil the hypothesis
with
Also
Thus, by Theorem 3.1, ∃ a solution to system (Equation36
(36)
(36) ) on
.
For the application of Theorem 3.2, we choose
(37)
(37) where
and
are continuous and fulfil the hypothesis (
) with
and
and
. Further, we get
Example 4.2
Consider system (Equation36(36)
(36) ) again. It is clear that all the assumptions of Theorem 3.2 are fulfilled. Consequently, ∃ a unique solution to system (Equation36
(36)
(36) ).
Example 4.3
Consider the following system:
(38)
(38) where
and
, and on the other hand,
Utilizing the above data, we get
and
All the assumptions of Theorem 3.4 are fulfilled. As a result, ∃ a solution to system (Equation38
(38)
(38) ).
5. Concluding remarks
In this research work, we studied a new BVP involving a coupled system of nonlinear SFDEs and inclusions of the Caputo type and supplemented with coupled integral boundary conditions. We have studied a new type of coupled boundary condition that determines the sum of unknown functions at the boundary points and along any arbitrary segment of the domain. Under these conditions, we solved a nonlinear Caputo SFDEs and inclusion system. The consequences of existence and uniqueness were examined via single-valued and multi-valued maps. It is also possible to broaden the scope of this investigation to include fractional differential and integral operators of the Riemann–Liouville and Hadamard types. Our results aren't only novel in the context of the problem, but they also lead to some novel cases involving specific parameter choices. The results of this paper are limited to a few intriguing instances with adequate values for the systems parameters. For instance, our results correspond to those for new coupled Stieltjes boundary conditions if we set in
in (Equation2
(2)
(2) ) and (Equation3
(3)
(3) ), and also we can obtained new existence results through special cases as stated in Remark 3.1 and Remark 3.2. We believe that the results discussed in this paper are of great significance for the scientific audience. Future research could focus on different concepts of stability and existence concerning a neutral time-delay system/inclusion and a time-delay system/inclusion with finite delay.
Declarations
Availability of data and material
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Authors' contributions
All authors contributed equally and significantly to this paper. All authors have read and approved the final version of the manuscript.
Acknowledgements
J. Alzabut would like to thank Prince Sultan University and OSTİM Technical University for supporting this research.
Disclosure statement
No potential conflict of interest was reported by the author(s).
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